Application of Multivariate Statistical Approaches to

Application of Multivariate Statistical Approaches to Complex Robotic System Analyses

by

Javad Sovizi June 2016

A dissertation submitted to the Faculty of the Graduate School of the State University of New York at Buffalo in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mechanical and Aerospace Engineering

ProQuest Number: 10127645

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ProQuest 10127645 Published by ProQuest LLC (2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346

Copyright by Javad Sovizi 2016

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The dissertation of Javad Sovizi was reviewed by the following:

Dr. Venkat Krovi Professor, Mechanical and Aerospace Engineering University at Buffalo, The State University of New York Academic Advisor, Committee Chair Dr. Rahul Rai Assistant Professor, Mechanical and Aerospace Engineering University at Buffalo, The State University of New York Committee Member Dr. M. Amin Karami Assistant Professor, Mechanical and Aerospace Engineering University at Buffalo, The State University of New York Committee Member Dr. Manoranjan Majji Assistant Professor, Mechanical and Aerospace Engineering University at Buffalo, The State University of New York Committee Member Dr. Karthik Dantu Assistant Professor, Computer Science and Engineering University at Buffalo, The State University of New York Outside Reader

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Acknowledgements All my achievements, including this dissertation, are dedicated to my loving parents whom I have not seen for four years.

Submitting my dissertation is the end of an almost five-years-long journey, away from home. At the end of this journey, I would like to deeply appreciate the generous help and support of many people who made it not only possible but worthwhile. First and foremost, I would like to express my sincere and profound gratitude to my advisor, Prof. Venkat Krovi, for his excellent research guidance, for providing lots of freedom but still holding me on the right track, and for being truly supportive and having trust in me. I am greatly thankful to Prof. Gary Dargush for being supportive, patient and caring, on numerous occasions during my stay at University at Buffalo. I would like to thank Dr. Sonjoy Das for teaching me the probability and random matrix theory, and for reviewing my research papers. I also thank the members of my dissertation committee, Dr. Rahul Rai, Dr. Amin Karami, Dr. Manoranjan Majji and Dr. Karthik Dantu for their comments and suggestions. Special thanks to Dr. Rai, for reviewing my papers and his machine learning course. I am thankful to all my friends in Buffalo, especially Reza Madankan who has always been an amazing companion, and a source of support and encouragement in any situation over the past few years. I would like to thank Suren Kumar for being a genuine friend and a wonderful lab-mate/colleague in the last two years of my stay at Buffalo. Also, thanks to my friend and colleague, Aliakbar Alamdari, for our great friendship and research collaborations. I would also like to thank Mark Tomaszewski, Nagavenkat Adurthi and Kumar Vishwajeet for their friendship and support. Last but not least, I would love to thank my parents and two brothers, Jalal, and Amirhossein, for their love, encouragement and sacrifices to ensure my success in pursuing higher education. Javad Sovizi Spring 2016

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Contents Acknowledgements

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List of Tables

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List of Figures

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Abstract

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1 Introduction 1.1 Our Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3

2 Mathematical Preliminaries 2.1 Matrix-Variate Distributions . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multivariate Moments and Cumulants . . . . . . . . . . . . . . . . . 2.3 von Mises-Fisher Distribution and Maximum Likelihood Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 MaxEnt Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 6 8

3 Uncertainty Characterization in Manipulator Systems: Random Matrix Analysis 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Motion Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Forward and Inverse Kinematics . . . . . . . . . . . . . . . . . 3.2.2 Random Jacobian Matrix . . . . . . . . . . . . . . . . . . . . 3.2.2.1 Maximum Entropy Formulation of the Jacobian Perturbation Matrix . . . . . . . . . . . . . . . . . . . . 3.2.2.2 Gaussian Jacobian Matrix Formulation . . . . . . . . 3.2.3 Physical Justifications . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Numerical Illustration: 2R Planar Serial Manipulator . . . . . 3.2.4.1 Simulation Based on the Wishart Perturbation Model 3.2.4.2 Simulation Based on Gaussian Jacobian Matrix . . . 3.2.5 Consistency Study . . . . . . . . . . . . . . . . . . . . . . . . v

10 12

13 13 17 17 19 20 24 29 31 32 35 39

. . . . . . . . . . . . .

40 43 44 48 48 48 56 61 64 65 67 69 72

4 Wrench Uncertainty Quantification: Random Vector Analysis 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Motivational Examples . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Wrench Uncertainty Quantification . . . . . . . . . . . . . . . . . . . 4.2.1 Planar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1.1 Output Force Vector Statistics . . . . . . . . . . . . 4.2.1.2 Output Moment Statistics . . . . . . . . . . . . . . . 4.2.1.3 Base Configuration and Cables Tension Optimization 4.2.1.4 Numerical Study: Reconfiguration Planning . . . . . 4.2.1.5 Experimental Study . . . . . . . . . . . . . . . . . . 4.2.2 Spatial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2.1 Joint Moment Generating Function: . . . . . . . . . 4.2.2.2 Kullback-Leibler Divergence Minimization: . . . . . . 4.2.2.3 Output Force Statistics . . . . . . . . . . . . . . . . 4.2.2.4 Numerical Study: Reconfiguration Planning . . . . . 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 74 76 78 80 82 82 82 85 86 89 98 103 104 105 107 108 111

3.3

3.4

3.2.5.1 Kinematically Redundant Parallel Manipulator 3.2.5.2 RMT Application to EOM . . . . . . . . . . . . 3.2.5.3 Numerical Simulations . . . . . . . . . . . . . . 3.2.6 Experimental Study . . . . . . . . . . . . . . . . . . . . 3.2.6.1 Experimental Setup: KUKA youBot . . . . . . 3.2.6.2 Motion Uncertainty Characterization . . . . . . 3.2.6.3 RMT-based Particle Filtering . . . . . . . . . . Wrench Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 RMT-based Formulation . . . . . . . . . . . . . . . . . . 3.3.2 Parametric Formulation . . . . . . . . . . . . . . . . . . 3.3.3 Numerical Study . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Experimental Study: Cooperative iRobots . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Workspace and Wrench Capacity Quantification: Analysis 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Workspace Density . . . . . . . . . . . . . . 5.1.2 Force Capacity . . . . . . . . . . . . . . . . 5.2 Manipulator Workspace Characterization . . . . . . 5.2.1 Probabilistic Forward Kinematics . . . . . . 5.2.2 Density Characterization from Characteristic 5.2.3 Moment-Based Density Estimation . . . . .

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. . . . . . . . . . . . .

Random Vector 114 . . . . . . . . . . 114 . . . . . . . . . . 114 . . . . . . . . . . 118 . . . . . . . . . . 119 . . . . . . . . . . 120 Function . . . . 121 . . . . . . . . . . 123

5.3

5.4

5.2.3.1 Closed-form Expressions of the Moments . . . . . . . 5.2.3.2 MaxEnt Density Estimation . . . . . . . . . . . . . . 5.2.4 Numerical Study: MaxEnt-Based Worskspace Characterization 5.2.4.1 Serial Arm with Revolute Joints . . . . . . . . . . . 5.2.4.2 Serial Arm with Revolute and Prismatic Joints . . . Force Capacity in Parallel Architectures . . . . . . . . . . . . . . . . 5.3.1 Force Capacity Quantification . . . . . . . . . . . . . . . . . . 5.3.2 Simultaneous Force Capacity and Uncertainty Quantification . 5.3.3 Numerical Analysis: Configuration Optimization . . . . . . . . 5.3.3.1 von Mises Kernel Approximation . . . . . . . . . . . 5.3.3.2 Moments Validation . . . . . . . . . . . . . . . . . . 5.3.3.3 Configuration Optimization: Wrench Capacity . . . . 5.3.3.4 Simultaneous Characterization . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 124 125 126 127 131 132 135 137 137 138 140 142 144

6 Conclusion

147

References

149

Appendices

158

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List of Tables 3.1 4.1 4.2 5.1 5.2

3-(P)RRR PM specifications; Nominal values and STD of (zero-mean) Gaussian perturbations. . . . . . . . . . . . . . . . . . . . . . . . . .

45

Mean and standard deviations of the resultant force vector and moment: closed-form solution versus Monte Carlo simulation. . . . . . . 90 Estimated parameters of the iRobots used in the experimental study. 102 Mean and standard deviations of the resultant force vector (without uncertainty): Monte Carlo simulation and closed-form solution. . . . 139 Mean and standard deviations of the resultant force vector (with uncertainty; σ ˜i = 50 ∀i): Monte Carlo simulation and closed-form solution.139

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List of Figures 1.1

Sketch of this dissertation. . . . . . . . . . . . . . . . . . . . . . . . .

2R serial chain; (a) kinematic configuration (b) desired trajectory and Yoshikawa manipulability (hot colors represent higher manipulability). 3.2 Standard deviation of ωθ1 and ωθ2 (Wishart perturbation matrix approach) with ordinates on the left axis. The Jacobian determinant, end-effector and joint velocities of the nominal (deterministic) manipulator over the time of the simulation with ordinates on the right axis. 3.3 KS joint density estimate of ωθ1 and ωθ2 at t = 1, 5, 7.5 and 9 s Wishart perturbation matrix approach. . . . . . . . . . . . . . . . . . 3.4 Ellipsoids of covariance matrix of ω at different time points (Wishart perturbation matrix approach). Note that the center of ellipsoids are located at the end-effector position in the corresponding time point. The contour plot represents Yoshikawa measure of the manipulability. 3.5 Wishart perturbation matrix approach: (a) States uncertainty propagation (hot color represents higher densities); (b) standard deviation of the system states versus the simulation time. . . . . . . . . . . . . 3.6 Standard deviation of ωθ1 and ωθ2 (Gaussian Jacobian matrix approach) with ordinates on the left axis.The Jacobian determinant, endeffector and joints velocities of the nominal (deterministic) manipulator over the time of the simulation with ordinates on the right axis. . . . 3.7 Ellipsoids of calculated Σ at different time points (Gaussian Jacobian matrix approach). Note that the center of ellipsoids are located at the end-effector position in the corresponding time point. The contour plot represents Jacobian Frobenius norm measure of the manipulability. . 3.8 KS joint density estimate of ωθ1 and ωθ2 at t = 1, 5, 7.5 and 9 s Gaussian Jacobianm matrix approach. . . . . . . . . . . . . . . . . . 3.9 Gaussian Jacobian matrix approach: (a) States uncertainty propagation (hot color represents higher densities); (b) standard deviation of the system states versus the simulation time. . . . . . . . . . . . . . . 3.10 Redundant planar parallel manipulator . . . . . . . . . . . . . . . . . 3.11 Response of the nominal (deterministic) system. . . . . . . . . . . . . 3.12 Time evolution of the dispersion parameters. . . . . . . . . . . . . . .

2

3.1

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33 34

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38 41 45 46

3.13 Mean and STD of the end-effector position and orientation errors corresponding to RV-based and RMT-based approaches. . . . . . . . . . 3.14 Experimental setup for motion uncertainty analysis: KUKA youBot with reflective markers on its three links; Kinect sensor to observe robot configuration; and OptiTrack cameras for motion capturing (ground truth data). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Left ordinate: realizations (solid red lines) of the KUKA youBot joint angles in a trajectory tracking task performed in T = 3.2 seconds (for each realization). Dashed black lines are the mean of each ensemble overlaid on top of the realizations. Right ordinate: standard deviation of the process noise (shown by dashdot lines) obtained from experimental data and using Eq. (3.51). . . . . . . . . . . . . . . . . . . . . 3.16 Joint state error when the mean of model-generated realizations is compared with the mean of real experimental observations. All three models can appropriately capture the mean in this example where the maximum error is less than 0.02 rad. . . . . . . . . . . . . . . . . . . 3.17 Standard deviation of the joint angles obtained from experimental (accurate) observations, and three stochastic models simulations. Red solid line shows the experimental results. The results obtained from additive Gaussian noise model is shown by cyan dotted lines. Blue dashed lines and black dot-dash lines correspond to RMT-based models with Wishart perturbation and Gaussian noise matrix, respectively. 3.18 Estimation of the joint angles of the KUKA youBot using Kinect measurement. (a) is the initial and final configuration and (b) shows the most extended configuration while tracking the desired trajectory. In order to improve the estimation of the joint angles, we allowed the length of the segments to change in a certain range. . . . . . . . . . . 3.19 Noise characteristics of the Kinect sensor. (a) shows the mean and (b) shows the standard deviation. It can be seen that Kinect noise is biased and its characteristics are time-varying. . . . . . . . . . . . . . 3.20 State estimation errors in particle filtering. First row shows the results corresponding to the additive Gaussian noise model. Second row shows the results corresponding to the RMT-based model with Wishart perturbation matrix. Third row corresponds to the RMT-based stochastic model based on Gaussian noise matrix. Red lines are the 1 − σ error bounds and blue lines show the actual error. . . . . . . . . . . . . . . 3.21 Representative examples of the cooperative systems: (a) aerial towing/suspension using flying robots; (b) airplane hangar where bases can be re-adjusted along the rails. . . . . . . . . . . . . . . . . . . . . 3.22 Schematic of a planar cable robot. Solid lines show the nominal (mean) state of the system (tension and orientation of the cables) and dashed lines show the perturbed state. . . . . . . . . . . . . . . . . . . . . . . x

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60

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3.23 Error histograms. For 2000 samples of system state and parameters, the Euclidean distances (error) between the output wrench variances obtained by RMT-based model and those given by parametric model are calculated. 5 histograms with different colors correspond to 5 systems with different number of cables. . . . . . . . . . . . . . . . . . . 3.24 Experimental evaluation of RMT-based models in capturing the wrench (only force vector here) uncertainty. Three different configurations are shown in three rows. First column shows the configuration of the iRobots. The KS density of the (accurately) measured force is shown in second column. The KS density of the data points obtained from RMT-based model given by Eqs. (3.61) and (3.66) is shown in the third column. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1

4.2 4.3

4.4

4.5

4.6

4.7

Cooperative aerial towing example: gray ellipsoids are the uncertainty ellipsoids in 2D plane. (c) shows a safer configuration that minimizes the probability of collision for this specific task compared to non-optimal configuration in (b). . . . . . . . . . . . . . . . . . . . . Cable-based suspension and measurement system: airplane hangar where bases can be re-adjusted along the rails. . . . . . . . . . . . . . Schematic of a planar cable robot. Solid lines show the nominal (mean) state of the system (tension and orientation of the cables) and dashed lines show the perturbed state. . . . . . . . . . . . . . . . . . . . . . Three planar systems with m = 4, 5 and 6 cables and their corresponding statistical information used for Monte Carlo validation of the closed-form solutions given by Eqs. (4.11)-(4.14) and (4.17)-(4.18). . . Optimization of the tension and orientation of the cables to minimize (a) the tension factor (b) Var(Fx ) with 2 cables (c) Var(Fx ) with 6 cables (d) Var(Fy ) with 2 cables (e) Var(Fy ) with 6 cables. The prescribed task is to provide the static wrench W d = [45 N, 60 N, 0 N.m]T at the end effector. Each base can move 2π rad as shown by dashed lines (constraint (4.23)). The minimum and maximum tension of the cables are set to be 0 N and 100 N for all cables (constraint (4.24)). Additionally, σθi = 100 and σTi = 2 N ∀i = 1, . . . , m. . . . . . . . . . Optimization of the tension and orientation of the cables to (a)-(b) maximize the tension factor and (c)-(d) minimize Var(M ). The desired static wrench is W d = [45 N, 60 N, 120 N.m]T . Each base can move along the arcs shown by dashed lines whose lengths are π rad. The minimum and maximum tension of the cables are set to be and 0 N and 40 N, respectively. Also, σθi = 100 and σTi = 2 N ∀i = 1, . . . , m. . 4-CDDR: the gray boxes show the domains over which the bases can be relocated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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76 78

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94 95

4.8 4.9

4.10 4.11

4.12

4.13

4.14

4.15 4.16

4.17

5.1

Uncertainty level in force and moment over the simulation time: the location of the bases is optimized to reduce the force uncertainty level. Uncertainty level in force and moment over the simulation time: the location of the bases is optimized to reduce the moment uncertainty level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the uncertainty level in (a) force and (b) moment versus number of cables for homogeneous and non-homogeneous agents. . . . Experimental setup. Three iRobots are tied to the force transducer located at the origin of the global coordinate system shown by blue lines. The task is to provide a pre-defined output force at the end effector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The traction force provided by each iRobot corresponding to different motion commands. Three motion commands are sent consecutively with a pause after first and second command to separate the data points. The average traction values are shown above the corresponding data points for each motion command. The data acquisition frequency is 50 Hz and we collect the force data for 50 s, i.e., 2500 data points for each motion command. . . . . . . . . . . . . . . . . . . . . . . . . Orientational uncertainty of the force vectors. The data points corresponding to three motion commands are shown by blue, black and red dots and their mean direction is shown by a solid line drawn from the center of the circles. The points shown by green are the rotated data points in order to use MLE approach for estimating σθi ’s (refer to the text for more details). . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental application of the optimization scenarios. First row corresponds to T F maximization. Second and third rows show the configuration and cable tensions that minimize Var(Fx ) and Var(Fy ), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial system: random force vectors can be modeled using spherical statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Configuration optimization for spatial system (maximum 2% error in mean output): diamonds/lines show the configuration and ellipses represent the covariance matrix of output force uncertainty. Red color shows the non-optimal and blue is the optimal configuration. . . . . Configuration optimization for spatial system (maximum 10% error in mean output): diamonds/lines show the configuration and ellipses represent the covariance matrix of output force uncertainty. Red color shows the non-optimal and blue is the optimal configuration. . . . .

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101 104

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Workspace density approximation: a 4-link serial arm with revolute joints. Link lengths are l1 = 0.3 m, l2 = 0.25 m, l3 = 0.2 m, l4 = 0.15 m. Manipulator can reach its base. . . . . . . . . . . . . . . . . . . . . . 128

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5.2

5.3

5.4 5.5

5.6 5.7

5.8 5.9

Workspace density approximation: a 2-link serial arm with revolute and prismatic joints. Link length limits are l1l = 0.15 m, l1u = 0.5 m, l2l = 0.05 m, l2u = 0.5 m. . . . . . . . . . . . . . . . . . . . . . . . . . 129 Workspace density approximation: a 2-link serial arm with revolute and prismatic joints. Link length limits are l1l = 0.24 m, l1u = 0.4 m, l2l = 0.08 m, l2u = 0.16 m. Link length limits are chosen such that manipulator can not reach its base. . . . . . . . . . . . . . . . . . . . 130 A schematic of the force diagram of a cooperative system with m agents.133 von Mises kernel approximation to bounded uniform distributions on the circle: (a) φu − φl = 60◦ and (b) φu − φl = 120◦ . Solid blue line is the true density, solid black line is the approximated density and dashed lines are 8 von Mises kernels. . . . . . . . . . . . . . . . . . . 138 Optimal base configurations in order to maximize the range of output load in specific directions. . . . . . . . . . . . . . . . . . . . . . . . . 140 Coefficient of variation as a measure of wrench capacity: change versus increasing number of agents when range of agents orientation is (a) 60◦ and (b) 120◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Variation of the uncertainty level and force capacity for the systems with different number of agents. . . . . . . . . . . . . . . . . . . . . . 143 Variation of the uncertainty versus the system lay out in a 3-agent system. First and second rows correspond to the wrench uncertainty in x and y directions, respectively. Third row shows the variation of the uncertainty in both x and y directions. . . . . . . . . . . . . . . . 145

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Abstract Ever-increasing development of robotic systems to different areas of application often leads to a complex architecture of the system. Complexity, on one hand, increases the difficulty in modeling and escalates the computational burden due to the curse of dimensionality, and hence, exacerbates the modeling uncertainty. However, on the other hand, more complex structures often provide more flexibility in design of such systems. These flexibilities can be exploited in order to improve a variety of the performance indices. In this dissertation, we are seeking two main objectives: (i) systematic approaches to better encounter the stochastic behavior of the complex robotic systems when only partial/limited information is available (which is a result of complexity), and (ii) probabilistic frameworks to systematically exploit the flexibilities offered by the complexity in order to optimize both the deterministic and probabilistic performance indices. Along our first objective, we adopt and develop ideas based on random matrix theory (RMT) in order to model the behavior of the robotic manipulators when only limited information is available to construct the stochastic model. We perform our analyses in both kinematic and static arrangements that provide frameworks for motion and wrench uncertainty characterizations, respectively. Our second objective is followed by the analyses of random vectors relying on the theories from directional statistics. We develop probabilistic formulations that utilize the redundancy (and in some sense the complexity) in the multi-agent systems to (i) reduce (or manipulate) the uncertainty in the system response, and (ii) quantify and improve the capacity of the system output in both kinematic (system workspace) and static (wrench capacity) frameworks. We back up our developments with different numerical simulations to investigate the critical aspects of different scenarios and validate our approaches using real-system experimental analyses.

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Chapter 1 Introduction Robotic systems are increasingly developing toward taking part in applications as complex as surgery, micro- and nano-manipulations, cooperative transportation, to name a few. For performing complicated tasks, one may expect the architecture of the system to also grow in complexity. On one hand, this induces further modeling and computational efforts in order to characterize stochastic behavior of such systems. First, the detailed formulation of such systems is not feasible due to the limited available information as well as curse of dimensionality. Second, approximating models applicable to the simpler systems are unable to appropriately capture the behavior of such systems. A second perspective is to take advantages of the flexibilities that are naturally offered by a complex system. One illustrative example is cooperative and distributed systems where a complex task is shared among several monolithic robotic agents. Such systems often have (kinematic) redundancy (a technical replacement for flexibility) and can perform an identical task in several (often infinite) different configurations. One can use the redundancy in such systems to improve several performance indices. Hence, dealing with the complexity, we first need to have systematic frameworks 1

that are (i) less information-demanding, and (ii) yet capable of appropriately capturing the system stochastic behavior. Second, we need frameworks that systematically and efficiently quantify both the deterministic and probabilistic behaviors of the (hyper-)redundant systems. This allows us to optimize the design parameters or system lay out (off-line or real-time), in order to improve the performance indices depending on the particular application. Figure 1.1 shows the sketch of the study carried out in this dissertation.

Figure 1.1: Sketch of this dissertation.

1.1

Our Contribution

This dissertation is configured around the objectives mentioned above. Our main interest is to develop and apply different methodologies from multivariate statistics, particularly random matrix theories and directional statistics, to the complex problems induced by the complexity/redundancy of the robotic systems. For articulated manipulator systems, we propose RMT-based models that capture the uncertainty by introducing the perturbations in system level (macro-level) rather 2

than state level (micro-level). We show that how such approaches provide models that require less information to be constructed and yet appropriately capture the stochastic behavior of the system. Relying on fundamental theories in directional statistics, we develop frameworks to address the output wrench uncertainty problem in (cable-based) multi-agent systems with parallel architectures. Furthermore, we extend our development to address well-known deterministic problems in the robotic systems: workspace density characterization in a kinematic arrangement, and load capacity quantification in a static arrangement. We show that how such formulations can be exploited along with the redundancy of the system in order to optimize system design parameters (system lay out) to improve different performance indices that primarily concern the uncertainty level in system output. We back up our developments in each part by numerical analyses as well as realsystem experiments in order to properly validate the effectiveness of the proposed approaches.

1.2

Outline

Chapter 2 briefly summarizes a basic set of mathematical preliminaries from RMT and directional statistics that are used in the rest of the dissertation. Chapter 3, addresses our first goal dealing with complex systems. RMT-based formulations for uncertainty characterizations in articulated manipulator systems are developed in this chapter. We start with kinematic analysis in order to address motion uncertainty and continue to static analysis for RMT-based formulation of the output load uncertainty. In chapter 4, we are solely focused on formulation of the output wrench uncertainty in multi-agent systems. We quantify the uncertainty by formulating the problem as sum of random vectors and providing closed-form expressions of the statistical moments. Through this, we set the ground for our optimization procedures to optimize system 3

lay out for reducing (manipulating) the uncertainty level. Chapter 5 is mainly our probabilistic formulation of the deterministic problems. We couple random vector analyses with maximum entropy (MaxEnt) method to approximate the workspace density of the redundant serial arms. This chapter continues with the formulation of the deterministic load capacity and simultaneous formulation of the load capacity and uncertainty in the multi-agent systems. We conclude our work and discuss the directions of the future work in Chapter 6.

4

Chapter 2 Mathematical Preliminaries In this chapter, we briefly review some special concepts in probability theory and multivariate statistics that are used in the subsequent chapters. We directly point to the concepts used in this dissertation and refer the reader to the literature for further details. Matrix-variate distributions and useful theorems are introduced in Sec. 2.1. In chapter 3, we use such families of distributions and related theorems to study the motion and wrench uncertainty in complex manipulator systems. Then, in Sec. 2.2, we review the multivariate moment and cumulant analyses that are used in chapters 4 and 5. In addition, distributions from directional statistics are presented in Sec. 2.3. In chapter 4 and 5, we use such distributions to model both the probabilistic and deterministic behaviors of the robotic agents. We finalize this chapter by the brief review of the MaxEnt method that is used in chapter 5 in order to formulate the workspace density of the serial arms.

5

2.1

Matrix-Variate Distributions

Relying on the existing literature [1], we present the theorems and definitions that are used in chapter 3 to develop the proposed RMT-based uncertainty characterization models. In this dissertation, bold letters are used to show the scalar and multidimensional random variables. Definition 1: The p-dimensional random vector X is said to have a multivariate normal distribution with mean vector M and covariance matrix Σ, denoted by Np (M, Σ), if its probability density function (pdf) is given by − 21 p

pX (X) = (2π)

− 21

|Σ|



1 etr − Σ−1 (X − X)(X − X)T 2

 (2.1)

where etr(.) = exp{tr(.)}, M ∈ Rp and Σ > 0. Note that A > 0 if A is positive definite matrix that is also denoted by A ∈ M+ p (A is a p × p matrix). For scalar case, the pdf is simplified to   1 (x − x)2 px (x) = √ exp − 2σx2 σx 2π

(2.2)

Definition 2: The p × q random matrix X is said to have a matrix-variate Gaussian distribution with mean matrix M ∈ Rp×q and covariance matrix Σ ⊗ Ψ + T where Σ ∈ M+ p and Ψ ∈ Mq , denoted as X ∼ Np,q (M, Σ ⊗ Ψ), if vec(X ) ∼

Npq (vec(M T ), Σ ⊗ Ψ). In Definition 2,   a1  . . vec(Ap×q ) =  .   aq

(2.3)

where ai , i = 1, . . . , q is the ith column of A. In addition, ⊗ represents Kronecker 6

products defined by 

 a11 B a12 B . . .   a21 B a22 B . . .  A⊗B = .  ..   am1 B am2 B . . .



a1n B   a2n B       amn B

(2.4)

The density function of X is then given by n 1 o 1 1 1 (2.5) pX (X) = (2π)− 2 qp |Σ|− 2 q |Ψ|− 2 p etr − Σ−1 (X − M )Ψ−1 (X − M )T 2 Definition 3: Assume that yi ’s (i = 1, . . . , k) are independent standard normal random variables and let z=

k X

yi2

i=1

Then, z has chi-squared distribution with k degrees of freedom, denoted as z ∼ χ2k . Now, a generalization to higher dimension is performed by constructing Y = (Y1 , . . . , Yn ), Yi ∈ Rp . If n ≥ p and Yi ∼ Np (0, Σ) then S = Y Y T has a Wishart distribution with parameters n and Σ ∈ M+ p , denoted by S ∼ Wp (n, Σ), whose density function is  pS (S) =

2

1 np 2

Γp

1 n 2




−1
1 |Σ| ) |S| 2 (n−p−1) etr − 12 Σ−1 S 1 n 2

where S ∈ M+ p

(2.6)

In the following, we present the theorems from RMT literature that are mainly used in this dissertation. Refer to [1] and references therein for more details and proofs of these theorems. Theorem 1: Let X ∼ Np,q (M, Σ ⊗ Ψ), then X T ∼ Nq,p (M T , Ψ ⊗ Σ). 7

Theorem 2: Let X ∼ Np,q (0, Σ ⊗ Iq ), q ≥ p, then XX T > 0 with probability one. Theorem 3: Let X ∼ Np,q (0, Σ⊗Iq ) and define S = XX T , q ≥ p, then S ∼ Wp (q, Σ). Theorem 4: Let S ∼ Wp (d, Σ), then E [S] = dΣ. Theorem 5: Let X ∼ Np,q (M, Σ ⊗ Ψ) and A be a q × q constant matrix, then E[XAX T ] = tr{AT Ψ}Σ + M AM T .

2.2

Multivariate Moments and Cumulants

For a (scalar) random variable x, rth (statistical) moment is defined as

r

Z

xr px (x)dx

E[x ] =

(2.7)

Dx

where px (x) is the pdf and Dx is the support of x. The first moment is the mean (i.e., E[x] = x) and the variance is the second central moment defined by σx2 = E[(x−x)2 ]. Now, let X = [x1 , . . . , xn ]T be an n-dimensional random vector. The elements of rth moment of the random vector X, denoted by ms1 ,s2 ,...,sr , can be written as

ms1 ,s2 ,...,sr = E[xs1 xs2 ... xsr ]

(2.8)

where sk ∈ {1, . . . , n} (∀k = 1, . . . , r). From the general representation in (2.8), the elements of the mean (first moment) vector and second moment matrix can be ˜ s1 ,s2 = E[xs1 xs2 ], respectively. Similarly, the elements obtained as µs1 = E[xs1 ] and Σ of the covariance matrix are Σs1 ,s2 = E[(xs1 − µs1 )(xs2 − µs2 )]. Moment generating function (mgf), if exists, is an alternative to the pdf for representation of a distribution. For a scalar random variable x, mgf is defined as Mx (η) = E[exp{ηx}]. The multivariate counterpart, and for random vector X, is the joint moment generating function (jmgf) defined as MX (v) = E[exp{v T X}]. Mo-

8

ments of the distribution can be obtained by differentiating jmgf. However, analytical analyses are sometimes simpler when using joint cumulant generating function (jcgf). Given the jmgf, the jcgf is KX (v) = log MX (v). The coefficients of the Taylor expansion of the jcgf around the origin are the cumulants. From the Taylor expansion of the multivariate functions, it can be seen that the rth cumulant, denoted by cs1 ,s2 ,...,sr , is cs1 ,s2 ,...,sr =

∂ r KX (v) |(0,0) ∂vs1 . . . ∂vsr

(2.9)

Cumulants can be written in terms of the moments. For the first and second cumulants we have

cs1 = ms1

(2.10)

cs1 ,s2 = ms1 ,s2 − ms1 ms2

(2.11)

In fact, from Eqs. (2.10) and (2.11), the first cumulant is simply the first moment (mean) and second cumulant is the second central moment (covariance). For special case of n = 2, we have X = [x1 , x2 ]T and v = [η, ζ]T . The mean vector and covariance matrix can be written in terms of the cumulants as

E[X] = [m1 , m2 ]T = [c1 , c2 ]T   c1,1 c1,2  Cov(X) =   c1,2 c2,2 in which ci =

∂KX (η,ζ) |(0,0) ∂xi

and ci,j =

∂ 2 KX (η,ζ) |(0,0) ∂xi ∂xj

(2.12) (2.13)

where KX (η, ζ) = log MX (η, ζ) =

log E[exp{ηx1 + ζx2 }]. We refer the reader to [2] for a quick review and to [3, 4] for detail discussions on the multivariate moments and cumulants.

9

2.3

von Mises-Fisher Distribution and Maximum Likelihood Parameter Estimation

When dealing with orientational data, typically represented by unit vectors on sphere (in 3-dimensional space) or circle (in plane), von Mises-Fisher distribution is one of the most useful distributions to represent data points scattered around a mean orientation. von Mises-Fisher distribution is, in fact, the counterpart of the Gaussian distribution on the unit hypersphere with a symmetric and unimodal density function that peaks at the mean orientation. An n-dimensional unit vector u, defined on (n − 1)-dimensional hypersphere (sphere when n = 3 and circle when n = 2), has von Mises-Fisher distribution with mean vector u and dispersion parameter σu , denoted by u ∼ vMF(u, σu ), if its pdf is given by [5] σun/2−1 pu (u) = exp{σu uT u} (2π)n/2 In/2−1 (σu ) where Iν (z) =

( 12 z)ν 1 π 2 Γ(ν+ 21 )

Rπ 0

(2.14)

e±zcos(θ) sin(θ)2ν dθ is the νth order (Re(ν) > −1/2) modi-

fied Bessel function of the first kind. An interesting properties of the von Mises-Fisher distributions is expressed in terms of the following theorem, and will be used in our experimental study in chapter 4. Theorem 6: Let H ∈ SO(n), if u ∼ vMF(u, σu ) then Hu ∼ vMF(Hu, σu ). Theorem 6 states that the dispersion parameter (uncertainty level) in von MisesFisher distribution is invariant under rotation. For planar case when n = 2, the von Mises-Fisher distribution reduces to the von Mises distribution whose density function is given by

pθ (θ) =

1 exp{σθ cos(θ − θ)}, θ ∈ [0, 2π] 2πI0 (σθ ) 10

(2.15)

where θ is the angle of random (unit) vector u in 2D coordinate system and θ is the mean orientation. Different studies have addressed the maximum likelihood estimation (MLE) of the parameters of the von Mises-Fisher distribution, i.e., u and σu [5, 6, 7]. We briefly review the MLE of the parameters of the von Mises distribution for circular data. Let θ1 , . . . , θN be N observations of the random variable θ (angles of the unit vectors u1 , . . . , uN ) that has von Mises distribution with mean θ and dispersion parameter σθ , denoted as θ ∼ M(θ, σθ ). It can be shown that the MLE of θ is simply the direction of the sum of the sample unit vectors i.e., ˆθ = angle(u1 + · · · + uN ). Furthermore, it can be shown that the likelihood function is maximized by

¯ σ ˆθ = A−1 (R)

(2.16)

√ (σ) ¯ = C¯ 2 + S¯2 in which C¯ = 1 PN cos(θi ) and S¯ = and R where A(σ) = II01 (σ) N i=1 P N 1 ¯ N i=1 sin(θi ). The approximate solutions to Eq. (2.16) for different values of R are provided in [5] as    ¯+R ¯3 + 5 R ¯5,  2R  6    ¯ + 0.43 , σ ˆθ ' −0.4 + 1.39R ¯  1−R      1  3 ¯ − 4R ¯ 2 + 3R ¯, R

¯ < 0.53 R ¯ ≤ 0.85 0.53 ≤ R

(2.17)

¯ 0.85 < R

Refer to [5, 6] for additional details on von Mises-Fisher distribution and estimation of its parameters.

11

2.4

MaxEnt Method

MaxEnt method is a well-known density estimation approach that results in exponential approximation to the underlying pdf of a random variable when partial information is available in terms of its statistical moments. Entropy, a quantitative measure of the distribution uncertainty, is maximized subject to satisfying the available information that are typically moment constraints. Entropy measure was initially introduced by Shannon [8] for discrete random variables. By extending the idea of Shannon entropy to continuous random variables, the differential entropy of a density function is given by Z S(p) = −

px (x) ln{px (x)}dx

(2.18)

Dx

where in MaxEnt approach, S(p) is maximized subject to Z px (x)dx = 1

(2.19)

Dx

Z gi (x)px (x)dx = µgi

i = 1, . . . , m

(2.20)

Dx

that results in

px (x) = Cexp{−

m X

λi gi (x)}

(2.21)

i=1

in which C and λi ’s are obtained using the normalization (Eq. (2.19)) and moment (Eq. (2.20)) constraints. The generalization of this formulation to higher dimensions can be used to estimate the density function of the random vectors and matrices as in chapters 3 and 5 in this dissertation. See ref. [9] for details on the MaxEnt method.

12

Chapter 3 Uncertainty Characterization in Manipulator Systems: Random Matrix Analysis 3.1

Introduction

Application of robotic manipulators is continuously advancing toward increasingly complex tasks that require high levels of reliability and safety. Some representative examples of such applications are minimally invasive surgery, micro- and nanomanipulation and cooperative payload transportation, among many others. The complexity of the task often leads to the complexity in kinematic structure of the robotic system. As a result, the modeling uncertainty (due to all ignored/unknown factors that contribute to deviation of the actual response from deterministic predictions) increases leading to a poor fidelity of the stochastic models of the system. Research in motion and kinematic uncertainties in the robotic systems dates as far back as work by Smith and Cheeseman [10] in 1986. They developed a first order

13

model for the propagation of the uncertainty under a series of the frame transformations, given the statistics of the uncertainty associated with a single transformation. Following their work, several researchers performed different studies that are mainly focused on the simultaneous localization and map building (SLAM) [11, 12, 13, 14] in the mobile robotic systems. The problem of uncertainty propagation focusing on the manipulator systems has been addressed by Wang and Chirikjian [15, 16]. In their work, they addressed the problem of error propagation on the Euclidean motion groups in which the distal frame pdf is derived using the convolution of the densities corresponding to each link along the serial chain. In another effort [16], they extended their method to the second order approximation of larger error propagation using the theory of Lie algebras and Lie groups. An improved stochastic model that is specialized to complex robotic manipulators is a key factor in better estimation of the system states and subsequently improved control performance. Additionally, complex and reconfigurable architecture of several manipulator systems allow the designer to optimize the level of the uncertainty in the system response. This further motivates appropriate characterizations that result in stochastic models with higher fidelity by which the uncertain behavior of the actual system can be properly captured. To achieve this goal, it is necessary to have: (i) a descriptive mathematical model of the physical system that embraces all the effective (random and deterministic) variables; (ii) statistical information on the random variation of all the variables; and (iii) a reliable and efficient technique to propagate the uncertainty. While the last element can be addressed by one of the many uncertainty quantification techniques that pass the uncertainty through the nonlinear model of the system, the first two elements of the characterization platform are often difficult or intractable to achieve. Despite the body of literature on the propagation of the uncertainty in the robotic systems, methods for the characterization of the uncertainty 14

have not received much attention. Hence, techniques for uncertainty characterization and subsequently design of systems with lower level of the uncertainty (or, in the other words, less sensitive to the uncertainties) still need to be further developed in systematic frameworks. There is a plethora of the methods reported in the literature where detailed formulation of the governing equations is proposed in order to capture the effect of different factors ignored in the simpler models. However, there are always unidentified factors that may affect the system response. In addition, incorporating all the effects that are known to the designer typically results in a highly nonlinear and complex model to which the solution is computationally demanding. Moreover, characterization (finding the statistical information) of each single variable requires several experimental analyses that can not be pursued due to the cost and time needed for such detailed investigations. To this end, in this chapter we propose and examine system-level approaches based on the random matrix theory (RMT) that provide several advantages over the parametric uncertainty characterization models. The idea behind the system-level RMT-based models is in fact the consistent modeling of the uncertainty at the macro-level rather than modeling the system state level (micro-level) parameters. This requires only limited information for formulating a probabilistic model of the system. It will be shown in the subsequent sections that a scalar parameter (that represents the level of uncertainty) is the only information used to construct the matrix-variate probability models. In addition, system level formulations are aware of the system state and configuration that play important role in response variation especially in articulated manipulators. For example, formulating the Jacobian matrix as a random matrix (addressed in this chapter) provides information to the probabilistic models of the state and configuration (such as close-to-singularity configurations). Finally, macro-level formulations provide compu15

tational efficiency in Monte Carlo based approaches by directly sampling the system matrices and eliminating the need for propagating the uncertainty jointly induced by several random variables at the micro-level. In addition to the physics and statistics, RMT has recently been applied to many engineering applications. In electrical engineering and wireless communications, RMT plays an important role in analyzing multi-input multi-output (MIMO) channels performance [17, 18, 19, 20, 21]. Furthermore, RMT-based models have been extensively developed in several recent works by Soize [22, 23, 24, 25, 26, 27], Soize et. al [24], Adhikari [28, 29], Adhikari et al. [30, 31, 32, 33] and Das & Ghanem [34, 35] for uncertainty characterization in the multi-scale mechanics and vibration problems. We organize our RMT-based analyses into two main parts: kinematic (motion uncertainty) analysis and static (wrench uncertainty) analysis. For motion uncertainty analysis, we formulate the Jacobian matrix as a random matrix. Jacobian matrix includes information on the dexterity and configuration of the manipulator such as singularity (or close-to-singularity) regions in which the system shows higher sensitivity to the perturbations. We construct the formulation based on two approaches. First, we adopt the method developed in [22, 23] and [28, 29] and extend its application to the robotic manipulators. Jacobian matrix in this model adopts a random perturbation matrix whose mean and certain normalized standard deviation of its norm are known. The density function of the perturbation matrix is derived using MaxEnt method that yields a Wishart distribution. In our second approach, we perturb the Jacobian matrix with a Gaussian noise matrix whose uncertainty level is adaptively changing depending on the manipulator dexterity. We finalize this part by a series of experimental studies on KUKA youBot and validating the proposed RMT-based models. In the second part, we provide RMT-based model in order to quantify the uncer16

tainty in cooperative (parallel) architectures, mainly cable robot systems. We show that how these models can be used to quantify the wrench uncertainty when the information about individual agents are partially known by only upper and lower bounds. Our model is in fact the probabilistic formulation of the static wrench relations where the (static) Jacobian and tension vectors are modeled by matrix-variate (and multivariate) Gaussian distributions. In fact, the problem turns into a Gaussian product problem to which different analytical treatments can be performed. We perform both numerical and experimental analyses to investigate the applicability of the proposed model dealing with the complex multi-agent systems.

3.2 3.2.1

Motion Uncertainty Forward and Inverse Kinematics

A routine problem in manipulators kinematic analysis is to express the end-effector position and orientation as a function of the joint space variables (forward kinematics) and vice versa (inverse kinematics). Forward kinematics relation may be stated as

x(t) = g(q(t))

(3.1)

where x is the vector of position and orientation of the end-effector, q is the vector of joint variables, t is time and g is a vector function that is, in general, nonlinear and maps from joint space to operational/task space. Given the trajectory of the end-effector, the inverse of function g, if exists, can be used to obtain the joint variables as a function of time. However, in many cases g −1 does not exist or is not explicitly available and requires high level of algebraic and geometric intuitions. Thus, numerical methods, that are applicable to all kinematic structures, are exploited for which the following differential kinematics relation is 17

used: x(t) ˙ = J(q(t))q(t) ˙

(3.2)

where J(q(t)) is Jacobian matrix of the manipulator. Note that for notational simplicity, the time variable t is suppressed from the subsequent equations and J implies J(q(t)). By assuming that J is non-singular, from Eq. (3.2), one can write

q˙ = J −1 x˙ d

(3.3)

that defines inverse differential kinematics relation in which x˙ d is the desired endeffector velocity vector. Then, joint variables vector, q, can be numerically obtained in a discrete time domain as

qk+1 = qk + q˙k ∆t and k = 0, . . . , m

(3.4)

where qk implies q(tk ) and tk+1 = tk + ∆t. Substituting Eq. (3.3) in Eq. (3.4) gives qk+1 = qk + Jk−1 x˙ dk ∆t

(3.5)

Thus, given the end-effector trajectory, xd , and the Jacobian of the manipulator, J, the joint variables can be determined in the discrete time points, tk ’s. Following the discussion in the preceding section, the Jacobian matrix in Eq. (3.5) (Jk ) can be considered as a random matrix. Next subsection establishes probabilistic formulations of the Jacobian matrix using random matrix theory.

18

3.2.2

Random Jacobian Matrix

Dealing with a random system, the inverse differential kinematics equation (3.3) can be considered as a general stochastic differential equation given by dq = f (q, ω, t) dt

(3.6)

where q is the vector of random joint states, f is a vector-function describing the kinematic structure of the manipulator and ω is the process noise. In the discrete time domain, Eq. (3.6) can be treated as a stochastic difference equation. In special case of the additive independent noise, one can write

qk+1 = f (qk ) + ωk+1 = qk + J(qk )−1 x˙ dk ∆t + ωk+1

(3.7)

where J(.) is the manipulator Jacobian matrix, x˙ dk is the desired end-effector velocity at time tk and ∆t = tk+1 − tk is the constant time step. However, due to the facts discussed above, we consider the stochastic difference equation as

qk+1 = qk + Jk−1 x˙ dk ∆t

(3.8)

where Jk is the random Jacobian matrix by which the uncertainty is introduced to the system at time tk+1 . In fact, uncertainty of the state in subsequent time instant, i.e., qk+1 is due to (i) propagation of the uncertainty of qk , and (ii) the uncertainty introduced by the process noise at time tk+1 . In contrast with the model in Eq. (3.7) that perturbs the joint states by an additive noise, we perturb the kinematic structure of the system by formulating the manipulator Jacobian as random matrix. This ultimately adds a perturbation to the states at tk+1 , however, it also includes system

19

level effects when we consistently perturb the Jacobian rather than the joint states. For instance, in low-dexterity configurations (e.g., close-to-singularity), perturbation of the Jacobian results in introducing higher level of uncertainty compared with highdexterity regions where manipulator provides improved precision. Moreover, endeffector desired velocity (and hence the required joint velocities) is a significant factor that affects the uncertainty level and is taken into account by perturbing the Jacobian matrix. On top of such advantages that typically result in improved fidelity of the model, the fact that RMT-based model is less-information demanding motivates development and application of such models to several complex systems. We discuss the benefits of the model given by Eq. (3.8) in further details in experimental validation section (Sec. 3.2.6). In this section, we are interested in characterizing p(Jk |qk = qk ) that facilitates statistical formulation of the subsequent configuration of the robot (joint states) when its current configuration is given. In construction of this probability model, we use the conditional mean of the Jacobian matrix given the current state of the system (simply called mean of the Jacobian matrix) denoted by J k , i.e., J k = E[Jk |qk = qk ] and also assume that J k = J(qk ). 3.2.2.1

Maximum Entropy Formulation of the Jacobian Perturbation Matrix

In our first approach we adopted the method proposed by Soize [22, 23] to construct the random symmetric positive definite matrices of the structural systems. The n × n random symmetric and positive definite matrix A (A ∈ M+ n ) is written as A = LTA GA LA

20

(3.9)

where LA is an upper triangular matrix corresponding to Cholesky decomposition of A which is the mean of random matrix A, and GA is a random symmetric positive definite matrix (GA ∈ M+ n ) with identity mean. Although Jacobian matrix is, in general, rectangular and can also be singular, in several common cases the number of joint space variables is identical to the number of end-effector degrees of freedom that results in a square Jacobian matrix. In addition, prescribed task for the manipulator is commonly defined such that it does not reach the singularity regions, however, it may cross the close-to-singularity regions. Then, Jacobian matrix is non-singular during the operation time. Thus, let us assume that the Jacobian matrix is a non-singular square matrix with dimensions n. In a similar fashion, we decompose the mean of the Jacobian matrix as

J k = LJ k UJ k

(3.10)

Equation (3.10) is LU decomposition of Jacobian mean matrix in which UJ k is an upper triangular matrix and LJ k is such that LJ k UJ k = J k . Now, we can write

Jk = LJ k BUJ k

(3.11)

where B ∈ M+ n . In Eq. (3.11) the random perturbing matrix B is a symmetric and positive definite matrix although the Jacobian matrix is not, in general. From Eq. (3.11) the expected value of the random matrix B can be obtained as follows:

E[Jk ] = E[LJ k BUJ k ] = LJ k E[B]UJ k

21

(3.12)

Equating the right hand side of Eq. (3.10) and Eq. (3.12) gives

E[B] = In

(3.13)

Normalization constraint (necessary for every proper pdf) along with the mean constraint given by Eq. (3.13) construct the constraints as follows: 1. Z B∈M+ n

pB (B)dB = 1

(3.14)

BpB (B)dB = B = In

(3.15)

2. Z E [B] = B∈M+ n

A well-known approach for approximation of the pdf when constraints in the form (3.14) and (3.15) (moment constraints) are available, is the maximum entropy (MaxEnt) approach. This information-theoretic method proposed by Jaynes [36] relies on the entropy of a density function, initially introduced by Shannon [8] for discrete random variables. Based on MaxEnt principle, among all admissible density functions (that satisfy the constraints) the one with the maximum entropy is the proper density function. MaxEnt method [9] has been frequently utilized to obtain the density function of the random variates using the known information. Soize [22, 23, 24], Adhikari [28] and Ghanem & Das [37] used the MaxEnt method to derive the probability model of the structural system matrices (mass, damping, stiffness and frequency response function matrices) when the mean system is known. Das and Ghanem [34, 35] further addressed this problem incorporating additional upper and lower bound constraints on the system matrices. This allows for appropriate constructions of less uncertain probability models when bounding information are available. 22

In the following, the procedure for the MaxEnt derivation of the pdf of B is reviewed. For further detail on this procedure we refer the reader to refs. [22, 23, 28]. Entropy measure for density function pB (B) is defined as [9, 36] Z S(p) = − B∈M+ n

pB (B) ln{pB (B)}dB

(3.16)

Method of Lagrangian multipliers is typically used to address the MaxEnt problem for maximizing S(p) subject to the constraints (that are described by Eq. (3.15) and (3.14) in our case of study). Hence, Lagrangian functional is constructed as Z

Z

pB (B) ln{pB (B)}dB − (λ0 − 1) B∈M+ n   Z  pB (B)dB − 1 − tr Λ1 BpB (B)dB − B

L(pB (B)) = −

B∈M+ n

(3.17)

B∈M+ n

where (λ0 − 1) ∈ R and Λ1 ∈ M+ n are Lagrange multipliers corresponding to the normalization and mean constraints, respectively. Using the variational and matrix calculus, it can be shown that (see ref. [28])

pB (B) =

rnr |B|−r 1 etr{−rB −1 B} and r = (n + 1) Γn (r) 2

(3.18)

Comparing Eq. (3.18) with Eq. (2.6) shows that B has a Wishart distribution B with parameters d = n + 1 and Σ = n + 1 . The distribution of B, derived as above, is maximally uncertain [28] as the minimum number of constraints are used to obtain the density function. An additional constraint that can be applied to the random matrix B is the existence of its inverse ˙ This constraint is written moments that implies the existence of the moments of q. as E[kB −1 kγF ] < ∞ 23

(3.19)

where k.kF is Frobenius norm and γ is the order of inverse moment. It is shown [22, 28] that in this case random matrix B has a Wishart distribution with parameters B d = θ + n + 1 and Σ = where θ = 2γ. θ+n+1 From the formulation of the pdf of B only θ remains undetermined. The normalized standard deviation of B is defined as [22, 28]

2 = σB

E[kB − E[B]k2F ] kE[B]k2F

(3.20)

Here, θ can be derived from Eq. (3.20) as [28]

θ=

1 2 σB

2

1+

{tr (B)}
tr B 2

! − (n + 1)

(3.21)

in which σB is called dispersion parameter of random matrix B which is a measure of uncertainty that was initially introduced by Soize [22]. Calculating this parameter from experimental data is further discussed in Sec. 3.2.6. In the Monte Carlo simulation, having Wishart distribution parameters, samples of random matrix B can be generated using existing Wishart sampling algorithms (see, for example, [37]). Here we use MATLAB function wishrnd to draw samples of B. Matlab automatically converts d parameter value to the nearest integer for non-integer values of θ obtained from Eq. (3.21). Then, the samples of the random Jacobian matrix can be obtained by substituting the samples of B in Eq. (3.11). Algorithm 1 summarizes the procedure for Monte Carlo simulation of Jk and subsequently qk+1 . 3.2.2.2

Gaussian Jacobian Matrix Formulation

MaxEnt formulation of the Jacobian matrix can be effective in characterizing the motion uncertainty (we will discuss this shortly in the Sec. 3.2.6). However, it requires 24

Algorithm 1 Monte Carlo simulation of Jk . Inputs: qk , σB and n; Output: Sample of Jk and sample of qk+1 1: 2: 3: 4: 5: 6: 7:

Substitute qk in the J(qk ) to obtain Jk Calculate θ from Eq. (3.21) In×n Set d = θ + n + 1 and Σ = θ+n+1 Generate a sample B of B (MATLAB’s command wishrnd may be useful here) Calculate LJ k and UJ k from LU decomposition of Jk Substitute LJ k , B and UJ k in Eq. (3.11) to obtain a sample Jk of Jk Substitute Jk in Eq. (3.8) to obtain a sample of qk+1

a proper decomposition and ultimately provides the pdf of the perturbation matrix rather than Jacobian itself. More specifically, the perturbation matrix represents the underlying fluctuation/perturbation that equivalently perturbs any system regardless of its kinematic structure and once used in the model described by Eq. (3.8), the system-specific uncertainty is resulted. While the distribution of the perturbation matrix facilitates sampling q, further analytical manipulations to find the closed-form expressions that quantify qk uncertainty level are difficult or intractable. Particularly for motion planning algorithms, it becomes important to efficiently quantify the uncertainty associated with several admissible trajectories. A sketch of such algorithm is described in Sec. 3.2.6 (Eq. (3.71)). Hence, in this subsection, we propose an alternative probabilistic model for the formulation of the random Jacobian matrix. Here, norm of the Jacobian matrix (as a measure of the manipulability) is also incorporated in the probabilistic formulation. In addition, it directly provides the density of Jacobian matrix. Here, we assume that the mean Jacobian matrix (at every time point) is perturbed with an additive Gaussian noise matrix. Covariance matrix of the noise is then obtained in an optimization problem that adaptively varies the level of the uncertainty depending on the manipulator dexterity level captured by the Jacobian matrix. Objective function of this optimization problem is defined to be differential entropy of the pdf of a function of

25

the noise matrix. Along with the positive semi-definiteness of the covariance matrix, we also define a specific Jacobian norm inequality constraint to be satisfied in the optimization problem. We assume that Jk is perturbed with an additive Gaussian noise matrix Jν k ∼ Nn,n (0, I ⊗ Σk ). So, random Jacobian matrix Jk is

Jk = Jk + Jν k

(3.22)

⇒ Jk ∼ Nn,n (Jk , I ⊗ Σk )

where I ⊗ Σk is the covariance matrix of Jk (or Jν k ) and implies that the rows of the perturbation matrix are independent random vectors [1]. Now, let us define Yk = Jν Tk Jν k . Using Theorems 1 and 3 in section 2.1, Yk ∼ Wn (n, Σk ). The objective is to find Σk such that the entropy of pY (Y ) is maximized. Differential entropy of the Wishart density function pS (S) (S ∈ M+ p ) with parameters d and Σ is given by [38]

S(pS (S)) = − ln B(Σ, d) −

dp d−p−1 E [ln S] + 2 2

(3.23)

where  !−1 p Y d+1−i B(Σ, d) = |Σ| 2 π Γ 2 i=1   p X d+1−i E [ln S] = ψ + p ln 2 + ln |Σ| 2 i=1 − d2

dp 2

p(p−1) 4

(3.24) (3.25)

in which Γ(.) is the univariate gamma function and ψ(a) = d ln Γ(a) is the digamma da function. Substituting Eq. (3.24) and (3.25) in Eq. (3.23) yields

S(pS (S)) = c1 ln |Σ| + c2 26

(3.26)

where p+1 2    p X 1 d 1−i 1 c2 = p(p + 1) ln 2 + p(p − 1) ln π + ln Γ + 2 4 2 2 i=1   p X 1 dp 1 ψ (d + 1 − i) + − (d − p − 1) 2 2 2 i=1 c1 =

Here, we have Yk ∼ Wn (n, Σk ). So, in Eq. (3.26) p = n = d. Before proceeding to the optimization problem, we review some features of the manipulator Jacobian matrix. Special functional form of the elements of the Jacobian matrix allows us to consider upper bound (u) on the Frobenius norm of the Jacobian matrix (as known information). So, we have

kJk kF ≤ u ⇒ kJk k2F ≤ u2

(3.27)

Equation (3.27) also implies h i E kJk k2F ≤ u2

From Eq. (3.22) and given that kAkF =

p

(3.28)

tr{AAT }, we can write kJk k2F as

n o kJk k2F = kJk + Jν k k2F = tr [Jk + Jν k ][Jk + Jν k ]T

(3.29)

Using the linearity of the tr{.} operator and given tr{A} = tr{AT } and tr{AB} = tr{BA}, we can then write Eq. (3.29) as

kJk k2F = tr{Jν Tk Jν k } + 2tr{Jν k Jk T } + tr{Jk J k T }

27

(3.30)

Now, from inequality (3.28) together with Eq. (3.30) and using the linearity of the mean and tr{.} operator we have n h io n h io T T tr E Jν k Jν k + 2tr E Jν k Jk ≤ u2 − kJk k2F

(3.31)

h i h i In relation (3.31), substituting Yk = Jν Tk Jν k and E Jν k Jk T = E Jν k Jk T = 0 gives tr{E[Yk ]} ≤ u2 − kJk k2F

(3.32)

Given that Yk ∼ Wn (n, Σk ) (Theorem 3), then using Theorem 4, E[Yk ] = nΣk . So, we finally get tr{nΣk } ≤ u2 − kJk k2F

(3.33)

In Eq. (3.26), c1 and c2 can be ignored in the cost function as they are independent of the Σ. So, the cost function for maximizing the entropy S(pYk (Yk )) is f = ln |Σk | (given c1 > 0). In addition, the optimization constraints are the Jacobian norm inequality described in (3.33) along with the positive semi-definiteness of Σk . However, applying only norm inequality constraint described by inequality (3.33) may yield a maximally uncertain distribution. To avoid unrealistic uncertainty in the resulting distribution, we multiply the right hand side of the inequality (3.33) by factor α2 where α is a positive constant parameter. Finally, the matrix optimization problem is given by

Maximize f = ln |Σk |

(3.34)

Subject to tr{nΣk } ≤ α2 (u2 − kJk k2F )

(3.35)

Σk ≥ 0

(3.36) 28

Here, α can be determined based on the experiment or prior knowledge of the system. For example, once the values of Σk ’s are determined from a set of experimental measurements, the best estimate of the parameter α, satisfying inequality (3.35), may be readily obtained. We postpone further discussions on estimating model parameters (u and α here) to section 3.2.6. Algorithm 2 summarizes the Monte Carlo simulation based on the Gaussian Jacobian matrix model. Algorithm 2 Monte Carlo simulation based on Gaussian Jacobian matrix model. Inputs: qk , n, u and α; Output: Σk , Jk and qk+1 Substitute qk in the J(qk ) to obtain Jk Find kJk kF Obtain Σk by solving (3.34)-(3.36) Given Jk and Σk , generate Jk from matrix-variate normal distribution (MATLAB’s command mvnrnd may be useful here) 5: Substitute Jk in Eq. (3.8) to obtain a sample of qk+1

1: 2: 3: 4:

In the experimental study, we show that modeling the inverse of the Jacobian matrix as random matrix (and then following the same procedure described in this subsection) provides smoother uncertainty bounds. Adopting this model in q˙k = Jk−1 x˙ dk facilitates integrating new uncertainty-quantifying terms (closed-form expressions are tractable due to the Gaussianity) into the motion planning algorithms that are commonly used to generate optimal trajectories for the robotic manipulators. Although motion planning is not the focus of this paper, we briefly describe the procedure and provide a sketch of the optimization setup in Sec. 3.2.6.

3.2.3

Physical Justifications

Robotic manipulators are more sensitive to the disturbances and show more fluctuations while working in the regions close to the singularity or when their kinematic manipulability degrades. Kinematic manipulability of the robotic manipulator refers to its ability in moving or re-orienting the end-effector in different directions. This 29

changes with the joint configuration of the robot during the task performance. To quantify the manipulability, different measures and indices have been proposed based on the Jacobian matrix of the manipulator. Yoshikawa manipulability measure, defined in Eq. (3.37), is one of the most commonly used measures in the robotic studies.

W =

p

det(JJ T )

(3.37)

Another manipulability measure is the weighted (squared) Frobenius norm of the Jacobian matrix, given by [39] Φ=

1 tr(JJ T ) n

(3.38)

where n is the workspace dimension. From Eq. (3.37) and (3.38), it can be seen that Yoshikawa measure has information about the singularity regions, however, the weighted Frobenius norm does not provide this information. Referring to Eq. (3.11), when the manipulator configuration is close to the singular configurations or when, based on Yoshikawa measure, it loses its kinematic manipulability, the variation in Jk−1 (that determines the variation in the response qk+1 ) increases. This also holds in the Gaussian Jacobian matrix formulation, as we have modeled Jk instead of Jk−1 . One may consider Jk−1 as random matrix in similar formulation and assumes J−1 k as known information. However, the resulting model does not take into account the singularity regions and the change of the (Yoshikawa) manipulability measure. In addition to this feature of both models, there is a noteworthy fact associated with the model based on the Gaussian Jacobian matrix formulation. In fact the covariance adapting rule described in inequality (3.35) uses the Frobenius norm of the Jacobian matrix that is, based on Eq. (3.38), a measure of the kinematic manipulability. This constraint increases the upper bound on the trace of

30

Σk when kJ k k (kinematic manipulability) decreases, resembling the instability and higher variations in these working regions.

3.2.4

Numerical Illustration: 2R Planar Serial Manipulator

In this section, we implement both developed methods (Wishart perturbation matrix and Gaussian Jacobian matrix) for uncertainty characterization in a 2R planar serial chain, shown in Fig. 3.1 (a), with two identical links (a = 1.5 m). A trajectory, described in Eq. (3.39) and (3.40), is first defined to be followed by the manipulator end-effector such that the robot experiences both high-manipulability (start and end of the path) and close to singularity configurations (middle of the path). The total time of the simulation is 10 seconds and ∆t = 0.005 s.

xdee (t) = 0.3t − 1.5

(3.39)

d yee (t) = 0.04t2 − 0.4t + 1.5

(3.40)

In Fig. 3.1 (b), the contour plot illustrates Yoshikawa manipulability corresponding to each end-effector position. There are two close to singularity regions, one near the base of the manipulator and one near the workspace boundary. The difference is that Yoshikawa measure reduces in both these regions, however, the Jacobian Frobenius norm measure reduces only around the base of the manipulator (because it does not provide information about the singularity regions). As depicted in Fig. 3.1 (b), the trajectory is defined such that the manipulator end-effector passes the regions near the robot base (close to the singularity), in which both the manipulability measures decreases.

31

(a)

(b)

Figure 3.1: 2R serial chain; (a) kinematic configuration (b) desired trajectory and Yoshikawa manipulability (hot colors represent higher manipulability).

3.2.4.1

Simulation Based on the Wishart Perturbation Model

Using Algorithm 1, assuming σB = 0.25 (for all time points) and q0 ∼ N (q0 , 1 × 10−3 × I2×2 ), we performed a Monte Carlo analysis with 1000 simulations of the defined trajectory tracking task. Fig. 3.2 shows the standard deviation of the ωθ1 and ωθ2 versus the time of the simulation. Joint density function of ωθ1 and ωθ2 at every time point (tk ) can be estimated using 2D kernel smoothing (KS) function estimation. Fig. 3.3 shows these densities corresponding to 4 different time points (t = 1, 5, 7.5 and 9 s). To illustrate the evolution of the covariance matrix of ω over the time, the sample covariance matrices are calculated at several time points and their corresponding ellipsoids are plotted in Fig. 3.4. For a bi-variate Gaussian distribution with mean µ and covariance matrix Σ, the equal probability contours are defined by

(X − µ)T Σ−1 (X − µ) = c2 32

(3.41)

−3

Standard deviation (rad)

2

x 10

4

ωθ1 std ωθ2 std | det(J )| Vx m/s Vy m/s θ˙1 rad/s θ˙2 rad/s

3

2

1 1

0

0 0

2

4

6

8

−1 10

t (s)

Figure 3.2: Standard deviation of ωθ1 and ωθ2 (Wishart perturbation matrix approach) with ordinates on the left axis. The Jacobian determinant, end-effector and joint velocities of the nominal (deterministic) manipulator over the time of the simulation with ordinates on the right axis. Eq. (3.41) represents an (in general) ellipsoid centered at µ. Constant c determines the fraction of the probability mass enclosed by the ellipsoid. Here c is chosen such that 90% of the probability mass is enclosed. Note that, in Fig. 3.4, for illustration purpose, the ellipsoids are not centered at mean, but at the corresponding end-effector position. In addition their sizes are magnified with the same factor to be visualized in the manipulator workspace (x-y plane). The results shown in Fig. 3.4 verifies those in Fig. 3.2 and additionally illustrate the correlation between ωθ1 and ωθ2 at different time points. To show the propagation of the uncertainty, the evolution of the kernel density estimates of θ1 and θ2 over the simulation time are shown in Fig. 3.5 (a). In addition, Fig. 3.5 (b) shows the variation of standard deviation of θ1 and θ2 over the time.

33

Figure 3.3: KS joint density estimate of ωθ1 and ωθ2 at t = 1, 5, 7.5 and 9 s - Wishart perturbation matrix approach.

Figure 3.4: Ellipsoids of covariance matrix of ω at different time points (Wishart perturbation matrix approach). Note that the center of ellipsoids are located at the end-effector position in the corresponding time point. The contour plot represents Yoshikawa measure of the manipulability.

34

0.18

θ1 std θ2 std

Standard deviation (rad)

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

2

4

6

8

10

t (s)

(a)

(b)

Figure 3.5: Wishart perturbation matrix approach: (a) States uncertainty propagation (hot color represents higher densities); (b) standard deviation of the system states versus the simulation time.

3.2.4.2

Simulation Based on Gaussian Jacobian Matrix

Similar to the simulation in section 3.2.4.1, for an identical manipulator, task and initial condition, a Monte Carlo analysis with 1000 simulations is performed based on the Gaussian Jacobian matrix approach using Algorithm 2. The parameter α is selected to be equal to 0.06 and u = 3.5. A drawback of this method is that it requires the solution of a nonlinear optimization problem to obtain the noise covariance. One effective treatment to this problem is to update the covariance matrix in every N time steps (instead of every time step), as the norm of the Jacobian matrix may be not rapidly changing in several tasks/systems. However, for systems with higher speed or tasks close to the singularity regions, the value of N may be adaptively selected in different speeds/regions. Similarly, the standard deviation of the ωθ1 and ωθ2 versus the time are plotted in Fig. 3.6. Comparing to the results of Wishart perturbation matrix approach, depicted in Fig. 3.2, this model shows similar trend of the noise standard deviations versus time. However, in this approach, the change (increase) of the standard deviations in the areas close to the base of the manipulator (close to the singularity regions) with

35

respect to the high-manipulability areas, is more significant than those in Fig. 3.2 (Wishart perturbation matrix approach). As discussed before, the reason behind this fact is that Gaussian Jacobian matrix formulation incorporates both manipulability measures (Yoshikawa and Jacobian norm) that are decreasing in the areas near the base. However, Wishart perturbation matrix approach takes into account only the effect of Yoshikawa measure. Fig. 3.7 shows the ellipsoids corresponding to Σ calculated at different time points (tk ’s) along with the Jacobian norm contour in the manipulator workspace. The ellipsoids are centered at the end-effector position corresponding to the time point in which Σ is calculated. As the Jacobian, in this case, is a 2 × 2 matrix, to maximize the entropy measure that is ln |Σk |, the off diagonal term of Σk (2 × 2) has to be zero. This can be seen in Fig. 3.7 where the (generally) ellipsoids of the Σk ’s are converted to the circles. The increase in the circles sizes when the end-effector approaches the robot base illustrates the increase of the adapting noise standard deviations governed by inequality (3.35). Fig. 3.8 illustrates the KS estimation of the joint density function of the ωθ1 and ωθ2 (at t = 1, 5, 7.5 and 9 s) obtained based on the Gaussian Jacobian matrix approach. Similar to section 3.2.4.1, the evolutions of the kernel density estimates of θ1 and θ2 over the time and the variation of θ1 and θ2 standard deviations versus the time are shown in Fig. 3.9 (a) and Fig. 3.9 (b), respectively. Indeed, the resulting propagation of the uncertainty in the joint state variables (here are θ1 and θ2 ) depends on the uncertainty characterization models as well as the pre-defined task (trajectory) that acts as an input to the system. Here, we examined the proposed uncertainty characterization methods and investigated the propagation of the uncertainty (based on these methods) for a trajectory tracking 36

−3

Standard deviation (rad)

2

x 10

4

ωθ1 std ωθ2 std | det(J )| Vx m/s Vy m/s θ˙1 rad/s θ˙2 rad/s

3

2

1 1

0

0 0

2

4

6

8

−1 10

t (s)

Figure 3.6: Standard deviation of ωθ1 and ωθ2 (Gaussian Jacobian matrix approach) with ordinates on the left axis.The Jacobian determinant, end-effector and joints velocities of the nominal (deterministic) manipulator over the time of the simulation with ordinates on the right axis.

Figure 3.7: Ellipsoids of calculated Σ at different time points (Gaussian Jacobian matrix approach). Note that the center of ellipsoids are located at the end-effector position in the corresponding time point. The contour plot represents Jacobian Frobenius norm measure of the manipulability.

37

Figure 3.8: KS joint density estimate of ωθ1 and ωθ2 at t = 1, 5, 7.5 and 9 s - Gaussian Jacobianm matrix approach.

0.16

θ1 std θ2 std

Standard deviation (rad)

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

2

4

6

8

10

t (s)

(a)

(b)

Figure 3.9: Gaussian Jacobian matrix approach: (a) States uncertainty propagation (hot color represents higher densities); (b) standard deviation of the system states versus the simulation time.

38

task. Characterizing the uncertainty and subsequently investigating its propagation in a given system, in fact, provide a systematic framework facilitating the design of the systems that are more robust to the uncertainties.

3.2.5

Consistency Study

In complex dynamic systems, several parameters may be uncertain and vary over the operation time, representing stochastic processes. For example, in robotic manipulators, mass of the different parts, effective length of the links, position of the bases, etc. can be assumed as random variables at every time point, that induce disturbing (uncertain) forces (external disturbing forces are treated separately). Time varying masses, (e.g., moving platform carrying a time varying mass), deflection and bending of the links at different configurations, flexibilities at the joints and bases of the manipulators, clearances, imperfections, etc. can cause these fluctuations. However, information about the variation of all random parameters and their correlations are not known for several complex systems. Here, we perform Monte Carlo simulations based on both conventional random variable scheme and the developed random matrix method. While in random variable based approach, several random parameters are independently sampled and substituted in nonlinear relations at every time point, the RMT-based simulation will only use a scalar dispersion parameter to generate the sample of the dynamic system matrices. As discussed above, this is beneficial when dealing with complex robotic systems in which limited information is available on the detailed fluctuation of the system parameters. However, there exist a question of consistency of RMT-based method. We are aiming to investigate the consistency of the RMT-based approach when compared with its parametric random variable counterpart. It is expected that results obtained from these two approaches differ to some extend. This is because 39

of the loss of information in construction RMT-based model and more generally due to different schemes in modeling the randomness in the system. However, when a detail parametric approach is feasible (detail statistical information are available), the results from RMT-based models should have an acceptable agreement (follow the trend) with those obtained from the random variable approach. We examine this in this part by applying both models to a kinematically redundant parallel manipulator. In the following, we introduce the parallel architecture and review the kinematics and dynamics of our case study in order to set the ground for the uncertainty analysis (consistency investigation). 3.2.5.1

Kinematically Redundant Parallel Manipulator

Parallel manipulators provide several advantages compared to their serial counterparts. These advantages include higher stiffness, higher accuracy, higher stability, higher payload to weight ratio, etc. However, smaller workspace, several singularity regions within their workspace and complexity of the kinematic/dynamic models are some major drawbacks associated with them. Recently, researchers have studied the kinematics and dynamics of the redundant planar parallel manipulators whose (nonredundant) basis topology is 3-RRR (underline denotes the actuated joint) parallel manipulator. For example, Ebrahimi et al. [40] introduced a 3-(PR)RR redundant manipulator in which a prismatic actuator is added to all three limbs of the base manipulator. They investigated the kinematics, workspace and the singularities of the proposed 3-(PR)RR manipulator. In addition Kotlarski et al. [41] proposed a 3-(P)RRR manipulator, in which the prismatic actuator is added to only one limb for singularity elimination and enlarging the workspace, and its dynamic identification was performed in another effort by Thanh et al. [42]. The kinematic and dynamic formulation of the 3-(P)RRR redundant planar parallel manipulator [42], shown in

40

Fig. 3.10, is reviewed next. The loop-closure constraints for each serial limb of the mechanism at the platform pivots are given by     xGi + li1 cos(θi ) + li2 cos(θi + ψi ) − xE − ri cos(φE + βi ) 0 fi =  =  yGi + li1 sin(θi ) + li2 sin(θi + ψi ) − yE − ri sin(φE + βi ) 0

(3.42)

The general EOM of the whole system shown in Fig. (3.10) can be written as

M (q)¨ q + h(q, q) ˙ + ΦTq λ = τ

(3.43)

where q = [δ, θ1 , ψ1 , θ2 , ψ2 , θ3 , ψ3 , xE , yE , φE ]T and M (q) = diag(M1 , M2 , M3 , M4 ) is the inertial mass matrix of the system, h(q, q) ˙ = [hT1 , hT2 , hT3 , hT4 ]T is the Coriolis and centripetal force vector, and τ = [τ1T , τ2T , τ3T , τ4T ]T is the input vector, in which Mi , hi and τi (i = 1, . . . , 4) are the mass matrix, Coriolis and centripetal force vector and external force/moment vector of the three limbs and moving platform, respectively. Differentiating f = [f1T , f2T , f3T ]T (fi , (i = 1, . . . , 3) are given in Eq. (3.42))

Figure 3.10: Redundant planar parallel manipulator [42].

41

with respect to the time yields 

 ∂f Φq q˙ = q˙ = 0 ∂q

(3.44)

To eliminate the vector of Lagrangian multipliers, λ, Eq. (3.43) is pre-multiplied by RT where R is the null space of the Φq , i.e., Φq R = 0. Then, RT M (q)¨ q + RT h(q, q) ˙ = RT τ

(3.45)

The state vector q can be decomposed into dependent (qD ) and independent (z) co T T T , z where z = [xE , yE , φE , δ]T , qD = [θ1 , ψ1 , θ2 , ψ2 , θ3 , ψ3 ]T ordinates as q = K qD h i  T T T ∂q and K = ∂q∂qD ∂z , z˙ . After is the constant permutation matrix. Then, q˙ = K q˙D some algebra, it can be shown that q˙ = Rz˙ where   −1 −ΦqD Φz  R=K  I4×4

in which I4×4 is the identity matrix and ΦqD =

h

∂f ∂qD

i

and Φz =

 ∂f  . In Eq. (3.45), ∂z

writing h(q, q) ˙ = C(q, q) ˙ q˙ and substituting q˙ and q¨ = R¨ z + R˙ z˙ yield the unconstrained EOM of the system in minimal (independent) coordinates as:

˜ z¨ + C˜ z˙ = Eτ ˜ a M

(3.46)

˜ = RT M (q)R, C˜ = RT M (q)R˙ + RT C(q, q)R, where M ˙ τa = [τθ1 , τθ2 , τθ3 , Fδ ]T is the vector of the actuator forces/torques and E˜ is the corresponding transformation matrix. For more details on kinematic and dynamic formulation of 3-(P)RRR manipulator, the reader is referred to Refs. [41, 42], and to Refs. [43, 44] for details on

42

the well-known methods in inverse dynamics of the constrained multibody systems. 3.2.5.2

RMT Application to EOM

Here, in addition to the Jacobian matrix, the inertial mass, Coriolis and centripetal matrices are treated as random matrices. The EOM of the deterministic mean dynamic system can be expressed as

M(q)¨ q + C(q, q) ˙ q˙ + N(q) = E(q)τ

(3.47)

in which under-bar represents the mean value that will be defined shortly, q = [q1 q2 . . . qn ]T is the system state vector, M(q) ∈ M+ n is the inertial mass matrix, C(q, q) ˙ ∈ Rn×n is the Coriolis and centripetal matrix, N(q) ∈ Rn×1 is the gravitational force vector, E(q) ∈ Rn×n is the input transformation matrix and τ ∈ Rn×1 is the input to the system. Now, the stochastic counterpart of the deterministic system described in Eq. (3.47) can be written as

M q¨ + C q˙ + N = Eτ

(3.48)

In order to utilize the constructed probability model in the simulation of the random dynamic system described in Eq. (3.48), the following general matrix factorizations are considered:

M = M1 M2 ,

C = C1 C2 ,

E = E1 E2

(3.49)

and the system random matrices are modeled as

M = M1 AM M2 ,

C = C1 AC C2 ,

43

E = E1 AE E2

(3.50)

An appropriate matrix factorization (e.g., Cholesky, QR, LU, etc.) must be selected depending on the properties of the corresponding random matrices (M , C and E) as symmetry, positive definiteness, etc. Moreover, from Eq. (3.50) and using Eq. (3.49), we can simply conclude that E[AM ] = E[AC ] = E[AE ] = In where In is n × n identity matrix. 3.2.5.3

Numerical Simulations

The initial state of the 3-(P)RRR manipulator, shown in Fig. 3.10, is set to be z0 = [xE 0 , yE 0 , φE 0 , δ0 ]T = [0.3 0.3 0 0.05]T . The desired task is to drive the state to zend = [0.4 0.4 0 0.05]T where the end-effector velocity and acceleration are zero at both start and end points (rest to rest maneuver). A simple PD control scheme ˆ + (Kd e˙ + Kp e) where (.)+ is is used to perform this task. The control law is τa = (E) the Moore–Penrose pseudo-inverse operator that is the conventional matrix inverse in cases of square matrices, e = zdes − z is the error vector that implies e˙ = z˙des − z, ˙ and Kd and Kp are differential and proportional gain matrices, respectively. Kd and Kp are chosen to give a stable feedback control system. Here, we choose Kd = 5I4×4 and Kp = 15I4×4 . Response of the system described by Eq. (3.46), when using this control law and nominal parameter values, is shown in Fig. 3.11. Nominal system specifications are also tabulated in Tab. 3.1. Monte Carlo Simulations: At every time point, several system parameters as the mass of moving platform (mp ), length of the links (Li1 and Li2 ), orientation of the prismatic actuator (α) and position of the bases (XGi , YGi , XGp and YGp ) are considered as random variables and their samples are generated by adding a zeromean Gaussian perturbation to their nominal values. The nominal values of these random parameters along with the standard deviation (STD) of the corresponding Gaussian perturbations are summarized in Tab. 3.1.

44

0.5

0.45

0.45 yE (m)

xE (m)

0.5

0.4 0.35

0.4 0.35

0.3 0

2

4

0.3 0

6

2

Time (s)

4

6

4

6

Time (s)

−4

x 10

0.055

0

δ (m)

φE (rad)

5

−5 −10 0

0.05 0.045

2

4

0.04 0

6

2

Time (s)

Time (s)

Figure 3.11: Response of the nominal (deterministic) system. Table 3.1: 3-(P)RRR PM specifications; Nominal values and STD of (zero-mean) Gaussian perturbations. Li1 Li2 mi1 mi2 XCi1 XCi2 mp ri m m kg kg m m kg m

βi mδ rad kg Unit 90 i=1 Nominal 0.5 0.6 0.1 0.1 0.2 0.3 1.5 0.107 210 i=2 1 330 i=3 value 0.01 0.01 – – 0.01 0.01 0.05 – – – STD

α XG i rad m 0

YG i m

0 i=2 0 i=2 0.7 i=3 0 i=3

0.1 0.01

0.01

XGp YGp m m 0.35 0.7 0.01 0.01

Following this procedure, 500 simulations are performed for t ∈ [0, 6] s (with time step size ∆t = 0.005 s) and the realizations of the system response (xE , yE and φE ) are obtained for statistical analysis. In addition, to perform a similar analysis with nearly the same level of uncertainty in the RMT-based approach, we obtained the dispersion parameters of the system matrices (using their samples constructed by substituting the samples of the system random parameters) at different time points using Eq. (3.20). Figure 3.12 shows the time evolution of the dispersion parameters ˜,C ˜ and E. ˜ Using the dispersion parameters (the only information corresponding to M needed for RMT-based method), we are able to perform RMT-based Monte Carlo 45

0.45

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0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

1

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Figure 3.12: Time evolution of the dispersion parameters. analysis (with approximately the same level of uncertainty) as follows. At each time point, the mean matrices are calculated by substituting the nominal values of the parameters and accurately observed state. Subsequently, obtained mean matrices are decomposed using a proper matrix factorization (Eq. (3.49)). Here, we use Cholesky decomposition for mass matrix, and LU decomposition for Coriolis and transformation matrices. Then, using the constructed probability model, samples of the perturbation matrices are drawn from Wishart distribution. Substituting the samples in Eq. (3.50) ˜,C ˜ and E ˜ at the corresponding time point and the system generates a sample of the M state at the subsequent time step is obtained. Using this procedure, we performed 500 Monte Carlo simulations and, similar to the random variable based (RV-based) analysis, the realizations of xE , yE and φE are obtained for the statistical analysis. Figure 3.13 shows the mean and STD of the end-effector position and orientation error (denoted by eX and eφ , respectively) versus the time for both RV-based and RMT-based analyses. From the realizations of the xE (t), yE (t) and φE (t), eX and p eφ are obtained as eX (t) = (xE (t) − xdE )2 + (yE (t) − yEd )2 and eφ (t) = |φE (t) − φdE | where xdE = 0.4 m, yEd = 0.4 m and φdE = 0 rad are desired end-effector x-position, y-position and orientation, respectively. From Fig. 3.13, it can be seen that RMT-based results are in good agreement with 46

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Figure 3.13: Mean and STD of the end-effector position and orientation errors corresponding to RV-based and RMT-based approaches. those obtained from RV-based analysis. The mean position and orientation errors corresponding to RMT-based analysis (dotted red lines) are closely following the mean error trajectories obtained from RV-based method (solid black lines). Fig. 3.13 also shows that in both position and orientation error cases, the RMT-based STD trajectory has similar trend (over the simulation time) to that of the RV-based approach. The STD of end-effector position error obtained from RMT-based analysis is only slightly higher than the STD corresponding to RV-based method. This difference is relatively more for orientation error. As noted above, there are some unknown (and unmodeled) uncertainties that are taken into account by RMT-based approach. For example, perturbing the random parameters does not perturb the zero components in the system matrices, i.e., does not include any variation due to unmodeled coupling in the system, while perturbation of the random matrices incorporates such coupling effects. This can result in higher level of variations in the system response.

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3.2.6

Experimental Study

Random Jacobian matrix approach, developed in Sec. 3.2.2, is investigated here using a KUKA youBot arm shown in Fig. 3.14. The goal of our experiment on youBot is twofold: (i) to investigate the performance of the RMT-based models and if they provide higher fidelities and outperform the conventional approaches, and (ii) to provide more accurate bounds on the joint state estimation error when RMT-based kinematic motion model is used coupled with the low-cost Kinect sensor measurements. 3.2.6.1

Experimental Setup: KUKA youBot

We use marker-based motion capturing system in order to provide (approximate) ground truth values of the joint angles at each time instant. Three reflective markers are attached to each link of the youBot arm and OptiTrack cameras are used to capture the trajectory of the markers. Note that youBot arm has 5 joints (excluding the end-effector gripper) from which two of them are set to be fixed during the experiment: first joint that is located at the base and rotates the entire arm; and the joint located at the wrist. This makes the system a three-links serial chain. We use low-cost Kinect sensor as our sensing modality. The experimental setup is shown in Fig. 3.14. 3.2.6.2

Motion Uncertainty Characterization

A desired end-effector trajectory is first defined. Then, we repeat the experiment (trajectory tracking) for 100 times while the desired trajectory remains unchanged. This provides us with 100 realizations of the joint angles of the robot arm captured through relatively accurate motion capturing system. The data set is used as ground truth for our further statistical analysis. Realizations of the joint angles along with their corresponding mean trajectories are shown in Fig. 3.15.

48

OptiTrack camera

KUKA youBot arm

Microsoft Kinect

Reflective markers

Figure 3.14: Experimental setup for motion uncertainty analysis: KUKA youBot with reflective markers on its three links; Kinect sensor to observe robot configuration; and OptiTrack cameras for motion capturing (ground truth data). Right ordinate in Fig. 3.15 shows the sample estimate of the standard deviation of the process noise ωk+1 in Eq. (3.15), calculated from experimental data as follows. Let us denote the true (measured) state of the system at kth time instant in ith realization by q˜ki . Now, the ith realization of the process noise can be obtained as i i i i ωk+1 = qk+1 − q˜k+1 = q˜ki + J(˜ qki )−1 x˙ dk ∆t − q˜k+1

(3.51)

A visual inspection of Fig. 3.15 shows the dynamic nature of the uncertainty level in the process noise ωk+1 in Eq. (3.7). Now, we examine three stochastic models in capturing the uncertainty in the system state depicted in Fig. 3.15: (i) an inverse differential kinematics model perturbed by an additive Gaussian noise; (ii) an RMT-based model with Wishart perturbation matrix (using Algorithm 1); and (iii)

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Figure 3.15: Left ordinate: realizations (solid red lines) of the KUKA youBot joint angles in a trajectory tracking task performed in T = 3.2 seconds (for each realization). Dashed black lines are the mean of each ensemble overlaid on top of the realizations. Right ordinate: standard deviation of the process noise (shown by dashdot lines) obtained from experimental data and using Eq. (3.51). an RMT-based model with Gaussian noise matrix (using Algorithm 2). Karydis et al. [45] developed a general methodology to evaluate the fidelity of stochastic models corresponding to different robotic platforms. Adopting their terminology and notation, a model M to describe the stochastic behavior of a system can be parameterized by p-dimensional vector ξ ∈ Rp . So, once the structure of the model is proposed, the 50

problem turns into finding the best ξ such that M(ξ) captures the system response most accurately. Here, for additive Gaussian noise model, the parameter vector is ω ω ω ] that is in fact the elements of the noise covariance matrix. , . . . , σ3,3 , σ1,2 ξ = [σ1,1

For RMT-based model with Wishart perturbation matrix, we have ξ = σB that is the scalar dispersion parameter and with Gaussian noise matrix the model parameter vector is ξ = [α, u]. Once a set of experimental measurements are available, different strategies can be used to estimate the parameter vector ξ. For example in [45], a lease-squares method is proposed to minimize the distance between model output and average of the experimentally-obtained samples at each time instant. Here, we first calculate the samples of process noise at each time point using Eq. (3.51). Then, for additive Gaussian noise model, the covariance matrix of the noise is considered to be timeinvariant and equal to the mean of the sample estimates of covariance matrices at ˆ ˆ ω = (1/M ) PM Σ each time instant, i.e., Σ k=1 k . Capturing for 3.2 seconds with 100 frames per second (fps) sampling frequency results in M = 320. For RMT-based model with Wishart perturbation matrix, we calculate σB as follows. First, from the model proposed in Eq. (3.8), the process noise added to the state at tk+1 is ωk+1 = (Jk−1 − J −1 ˙ dk ∆t k )x

(3.52)

i Hence, given q˜ki and ωk+1 from Eq. (3.51) and substituting Jk from Eq. (3.10), we

have i ((J i1k Bk+1 J i2k )−1 − J ik

−1

i )x˙ dk ∆t = ωk+1

(3.53)

where J ik = J(˜ qki ). Then, ith sample of the perturbation matrix B at time tk+1 (i.e.,

51

i Bk+1 ) can be obtained by solving the following optimization problem.

Minimize F = −1 i Bk+1

i α1 kωk+1

  −1 −1 i −1 i −1 i i −1 i − J 2 k Bk+1 J 1 k − J k x˙ dk ∆tk+α2 kBk+1 − Ik (3.54)

s.t. i Bk+1

−1

>0

(3.55)

i i where α1 + α2 = 1. Assuming the inverse of Bk+1 rather than Bk+1 as optimization

variable facilitates using the convex optimization packages, and here, we use CVX [46, 47] for solving the problem described by (3.54)-(3.55). Last term in the objective i to be close to its mean, i.e., In . Using the samples of Bk function enforces Bk+1

(at tk ), σBk can be calculated using Eq. (3.20). Finally, we calculate the dispersion P ˆBk . In RMT-based model with Gaussian noise parameter as σ ˆB = (1/M ) M k=1 σ matrix, we set uˆ = max {kJ −1 ˜ k k} + u k=1,...,M

(3.56)

where u˜ is a positive constant that extends the deterministic upper bound on Jacobian norm due to the uncertainty. Note that for this model, we formulate the inverse of the Jacobian matrix as a random matrix because it provides smoother bounds on the uncertainty level. Here, we choose u˜ and α such that (in 100 realizations) the mean and variance of the model outputs take the minimum distance from those of actual experiments, i.e.,

Minimize F = u ˜,α

M  X

 qk )} β1 kq k − q˜k k+β2 tr{Cov(qk ) − Cov(˜

(3.57)

k=1

where β1 +β2 = 1. The results of our parameter estimations for three different models

52

are 

Add Gauss:

 0.0330 −0.0406 0.0592     ˆ ω = −0.0406 0.0547 −0.0839 × 10−3 Σ     0.0592 −0.0839 0.2457

RMT- Wish:

σ ˆB = 0.25

RMT- Gauss: u ˆ = 18, α ˆ = 0.1

Given the models parameters, we generated 100 realizations of the joint states for each stochastic model. Joint state at each time instant are perturbed with a sample of Gaussian noise (in MATLAB using mvrnd function) when additive Gaussian noise model is adopted. When using RMT-based model with Wishart perturbation matrix, a sample of Jacobian matrix at each time instant is used to propagate the state that can be drawn using Algorithm 1. Finally, when RMT-based model with Gaussian noise matrix is used, at each time step we sample the Jacobian matrix using the procedure described in Algorithm 2. Figure 3.16 shows the error corresponding to each model. Error is computed as the Euclidean distance between the mean of joint states generated through model simulations and that of real experimental data at each time instant. The error corresponding to Wishart-based model shows higher variations than the other two models in an average sense. However, the maximum error for all models is less than 0.02 rad. This implies the fact that all approaches are capable of capturing the system response up to the first statistical moment with an acceptable error. However, the focus is more on the uncertainty level that is characterized with higher level moments. Standard deviations of the joint state corresponding to the experimental data and three stochastic models are shown in Figure 3.17. Red solid lines in Fig. 3.17 are standard deviations corresponding to actual experiment, that are in fact identical to those plotted in Fig. 3.15 by black dot-dash lines. Standard deviations corresponding to the generated realizations of the joint states using additive

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0.016 Additive Gaussian RMT−Wishart RMT−Gaussian

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Figure 3.16: Joint state error when the mean of model-generated realizations is compared with the mean of real experimental observations. All three models can appropriately capture the mean in this example where the maximum error is less than 0.02 rad. Gaussian noise model, Wishart-based model and the model based on Gaussian noise matrix, are shown with cyan dotted lines, blue dashed lines and black dot-dash lines, respectively. From Fig. 3.17, it is clear that the additive Gaussian noise model is unable to properly capture the dynamics of the uncertainty level that is inherent to the actual platform. This is in fact the main drawback of such models that motivates RMT-based approaches. While the results from additive Gaussian noise model remain almost time-invariant, it can be seen that the variation in uncertainty level is properly captured by both RMT-based models. There exists an excessive uncertainty in the third joint angle (q3 ) that can not be captured as well as other two joints through RMT-based models. We recognized an unusual clearance in the third joint of our KUKA youBot arm and the excessive raise of the uncertainty is believed to be due to this abnormality. The fact that RMT-based models provide higher fidelities compared to other conventional approaches (as depicted in Fig. 3.17) motivates application of these models to practical estimation and control problems when sequential Monte Carlo filtering approaches

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Figure 3.17: Standard deviation of the joint angles obtained from experimental (accurate) observations, and three stochastic models simulations. Red solid line shows the experimental results. The results obtained from additive Gaussian noise model is shown by cyan dotted lines. Blue dashed lines and black dot-dash lines correspond to RMT-based models with Wishart perturbation and Gaussian noise matrix, respectively. are used. More accurate bounds on the state estimation errors are expected when the conventional models are replaced by RMT-based models. We now examine this through an experiment where RMT-based models are used coupled with Kinect measurements to estimate the state and the bounds on the estimation errors.

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3.2.6.3

RMT-based Particle Filtering

Sequential Monte Carlo sampling techniques rely on the samples of the random variables rather their joint pdf in order to approximate the moments of the posterior. This enables these approaches to be used for state estimation in nonlinear systems (such as the case of our study here) where conventional Kalman filters can not be employed. Here, our purpose is to integrate the RMT-based sampling approaches with a sequential importance sampling filter that estimates the joint angles of the KUKA youBot arm when low-cost Kinect measurements are available (see Fig. 3.14). For this purpose, we run the experiment described in Sec. 3.2.6.1 for 50 more trials and simultaneously collect the joint angles data using OptiTrack motion capturing (as our ground truth) and Kinect sensor (as our sensing modality). For our filtering purpose, we use the same model parameters obtained from our earlier experiment with 100 realizations. In some sense, the set of data obtained from the first experiment with 100 realizations is our training data set and 50 new realizations construct our testing data set. Kinect sensor can stream depth data with a sampling rate of 30 fps. In each frame, a 2.5-dimensional data structure is provided that contains the information on the depth of each pixel in the Kinect field of view. This information can be used to track moving objects in the scene. In case of an articulated system such as KUKA youBot arm, one sensor can be used to simultaneously track all the links. This feature along with their low cost makes such sensors desirable for many applications. However, on the other hand, the low quality measurement limits their applications when high-accuracy perceptions of the system states are required. There are commercial software tools (such as nuiCapture) that, in addition to Kinect data acquisition, provide skeleton (articulation) detection and tracking. However, these tracking platforms are mainly built for human motions. For a custom articulated system, i.e., KUKA youBot arm, a customized algorithm is required to detect and track the articulated arm from the point cloud. In computer vision literature, there exist several such algorithms that can solve similar problems. However, here we provide a simple algorithm

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based on the least-squares method that proves to be adequate for our specific system and application. We use our prior knowledge of the articulation in KUKA youBot arm in order to detect and track the robot links. In each frame, finding the points corresponding to joint 1 (shown by O1 in Fig. 3.18) and the end-effector (shown by EE) is straightforward. This is because O1 position does not change during the robot motion and relative position of the point end-effector respect to other points is known (from prior information about the end-effector trajectory). Now, in order to fit three segments (corresponding to three links) to the point cloud, we need to find the optimal position of joint 2 and 3, i.e., points O2 and O3 in Fig. 3.18, respectively. Hence, assuming the point cloud representing the youBot arm is described by P = {p1 , . . . , pI }, we solve the following optimization problem to complete the articulation detection.   X  I 3 X s s Minimize F = γ1 (1/I) min{dsp1i , dsp2i , dsp3i } + γ2 |daj − duj | x1 ,y1 ,x2 ,y2

i=1

(3.58)

j=1

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motion of the robot links is in a plane when the base joint is fixed. In Eq. (3.58), dpji is the EE EE O3 O2 O3

O2

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Figure 3.18: Estimation of the joint angles of the KUKA youBot using Kinect measurement. (a) is the initial and final configuration and (b) shows the most extended configuration while tracking the desired trajectory. In order to improve the estimation of the joint angles, we allowed the length of the segments to change in a certain range.

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Figure 3.19: Noise characteristics of the Kinect sensor. (a) shows the mean and (b) shows the standard deviation. It can be seen that Kinect noise is biased and its characteristics are time-varying. shortest distance between the ith point in the point cloud to the jth segment. Moreover, s

s

daj and duj are the maximum distance of the points above and under the jth segment. First term in (3.58) fits the segments by first finding the point that belong to that segment (link) and minimizing the distance from the points. However, the second term is also necessary to keep the segment passing through the middle of the point cloud corresponding to each link. This is because the density of the points captured by Kinect is different at different locations on the robot link and solely minimizing the distance (first term in the objective function) shifts the segment toward the high density areas, and hence, less symmetry and lower accuracy. The mean and standard deviation of the Kinect noise over the experiment time are shown in Fig. 3.19. Now with the Kinect based observations and by using the OptiTrack measurements as the ground truth data, one can examine the performance of three stochastic motion models in joint state estimation. Let us consider the joint state sequence as a Markov process denoted by {q1 , . . . , qM } (M = 320) and qk ∈ Dq where Dq is the domain defined by the

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robot joint limits. Also, let us denote the Kinect observation at time tk by yk and yk ’s are conditionally independent given the state at tk . At each time instant tk , we seek to estimate the expectations of the posterior defined by Z fk (qk )p(qk |yk )dqk

E(fk (qk )) =

(3.59)

qk ∈Dq

We used sequential importance sampling [48] to estimate the mean (fk (qk ) = qk ) and covariance (fk (qk ) = [qk − q k ][qk − q k ]T ) of the random state qk for 50 test trials. The results corresponding to three different models are shown in Fig. 3.20. Blue lines in Fig. 3.20 show the actual estimation error that is the distance between the estimated mean and the ground truth value of the state (captured by the OptiTrack cameras). Red lines show the ±1-σ error bounds on the state estimates. First row of Fig. 3.20 corresponds to the additive Gaussian noise model, second row corresponds to the RMT-based model with Wishart perturbation and third row shows the results corresponding to the RMT-based model with Gaussian noise matrix. Comparing the actual error (shown by blue lines) for different models, the results are supporting those that are shown in Fig. 3.16 and imply that all three models are capable of capturing the mean state with approximately the same level of accuracy. However, comparing the estimates of the error bounds (shown by red lines) shows a significant difference and proves the superiority of the RMT-based models. When additive Gaussian noise model is used (first row of Fig. 3.20), error bounds can not be reliably estimated in the commonly used particle filtering technique. The estimates on the error bounds are approximately constant over the time while the actual error (shown by the blue lines) is highly varying. The estimates of the error bounds are significantly improved and properly capture the variation of the actual error when RMT-based models are used (second and third rows). This shows that both RMT-based stochastic models outperform the commonly used additive Gaussian noise model in estimating the uncertainty level. The bounds corresponding to Wishart-based model are more conservative compared to those

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Figure 3.20: State estimation errors in particle filtering. First row shows the results corresponding to the additive Gaussian noise model. Second row shows the results corresponding to the RMT-based model with Wishart perturbation matrix. Third row corresponds to the RMT-based stochastic model based on Gaussian noise matrix. Red lines are the 1 − σ error bounds and blue lines show the actual error. from Gaussian noise matrix model. Modeling the inverse of Jacobian matrix in Gaussian RMT-based model results in less conservative results. Despite the small difference, the performance of both the models are relatively close and one may use them interchangeably depending on the particular problem and available information.

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Additionally, the Gaussian RMT-based model can be adopted to augment the motion uncertainty in the motion planning algorithms. Motion planning is not addressed in this work, however, we briefly discuss the procedure an provide an sketch of the optimization problem as follows. Assume q as discrete stochastic process denoted by {q0 , q1 , . . . , qM } and qk ∈ Dq (∀k = 0, . . . , M ) where Dq is the domain on which the joint angles are defined (feasible range of motion of the joints). In order to choose the least uncertain joint trajectory for performing the desired task described by {xd0 ; x˙ d0 , . . . , x˙ dM −1 } one may solve the following optimization problem: Minimize

q 0 ,q 1 ,...,q M ∈Dq

F =

M X

Cov(q˙k )

(3.60)

k=1

s.t. q k+1 − q k = Jk−1 x˙ dk

(∀k = 0, . . . , M − 1)

If the Gaussian RMT-based model is adopted, one can consider q˙k = Jk−1 x˙ dk , and hence, the covariance in (3.60) has a closed-form expression given in Sec. 3.3.1. Note that this is a rough sketch of the proposed approach and practical motion planning algorithms include many factors in the formulation such as collision avoidance, minimization of time and energy/fuel, etc., that are not the focus of this work.

3.3

Wrench Uncertainty

A systematic treatment to the wrench uncertainty in multi-agent cooperative systems is presented in this section. One may study the wrench uncertainty from two perspectives: (structural) failure and reliability analysis where the distributions and bounds on the output load are required; and motion variability that results from non-deterministic forces/moments. The former mainly falls into structural system analysis topics and the latter is of more interest in the robotic applications especially in cooperative tasks. The main idea behind cooperative robotic systems design is to distribute a task, unachievable for a single agent, among several agents that may or may not be identical. In order to im-

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prove the flexibility and fault tolerance of such systems the number of cooperating agents are typically more than required, hence, kinematic redundancy. Redundancy brings complexity to the design and control of such architectures. However, it gives a freedom to designers to optimize different performance indices while maintaining a prescribed system response. Special lay out of such cooperative systems makes them vulnerable to (i) configuration/calibration uncertainty and (ii) actuation/sensing uncertainty that ultimately result in an uncertain output wrench. Hence, neglecting the effect of these variations in such loosely interconnected systems may result in lack of stability or failure in several applications. Two representative examples of these systems are shown in Fig. 3.21.

(a)

(b)

Figure 3.21: Representative examples of the cooperative systems: (a) aerial towing/suspension using flying robots; (b) airplane hangar where bases can be re-adjusted along the rails. For instance, in aerial towing problem (Fig. 3.21 (a)), flying robots (such as commonly used quadcopters) are inherently uncertain in motion (especially when loaded) that leads to a highly uncertain configuration. A comprehensive study of such systems can be found in the literature [49, 50, 51, 52, 53, 54]. Uncertain position of the flying robots has been reported to be responsible for control errors (for example, see [50]). Similarly in measurement systems such as aircraft hangar [55, 56] (shown in Fig. 3.21 (b)), configuration of the agents is uncertain due to the calibration errors, flexibility and displacement of the bases, etc. Furthermore, actuation and sensing uncertainties impact the performance of the sys-

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tems shown in Fig. 3.21 (a) and (b), respectively. In a wrench-delivery (forward) problem such as aerial towing, there exist an intrinsic imperfection of the actuators in providing the exact (commanded) force. Likewise, in a wrench-sensing (inverse) problem such as platform wrench measurement shown in Fig. 3.21 (b), uncertainty in sensing the force/tension corresponding to each individual agent (cable tension) is inevitable. We incorporate these uncertainties in a static formulation of the output wrench. In a wide range of applications, the operation condition is such that a quasi-static analysis can significantly characterize the system response (for example, see [52]). Hence, our formulation is applicable to several physical systems working in such conditions. Characterization of the uncertainty introduced by each agent and formulating the resultant effect are practically and theoretically difficult or, in some cases, intractable (where individual agents parameters are not known). The detailed parametric formulation of the output wrench uncertainty is the subject of next chapter. Here, we show that random matrix theories can be used to systematically model the wrench uncertainty with significantly less characterization efforts. Since limited available information is used in formulating the RMT-based model, it provides lower fidelity compared to the parametric approaches in which all random entities are carefully modeled. We will show that such (RMT-based) approaches can still represent the real system behavior with sufficient accuracy and become particularly more beneficial when complexity of the system increases (e.g., cooperative systems with large number of agents). In an RMT-based framework and based on some Gaussianity assumptions, a model is constructed that provides the closed-form expression of the resultant wrench uncertainty level (output wrench covariance). Such closed-form expressions become very useful in the case when reducing output uncertainty is among design and motion planning criteria. For instance, new generation of the cable-based architectures have become highly reconfigurable by adding mobility to the bases (agents) [57, 58, 59, 60, 61, 62]. Reconfiguration planning algorithms can benefit from such formulations by augmenting the objective function with

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covariance terms that can be computed efficiently. It also facilitates analytical investigation and systematic trade-offs in multi-agent systems design to study the sensitivity of the response uncertainty respect to different design parameters. We show that elaborate and (roughly speaking) information-demanding parametric models (developed in the next chapter) can be replaced by less demanding RMT-based models at the expense of reducing the accuracy. Under many circumstances and especially in multi-agent systems with large number of cooperative robots, the loss of accuracy due to utilizing RMT-based models can be sufficiently low such that the fidelity of the model is yet acceptable.

3.3.1

RMT-based Formulation

Wrench vector in a system with m agents and n degrees of freedom at the end-effector can be written as W = ST

(3.61)

where W ∈ Rn is random wrench vector, S ∈ Mn×m is random (static) Jacobian matrix and T ∈ Rm is random force vector. Randomness of the Jacobian matrix and force vector are due to the configuration/calibration and actuation/sensing uncertainties, respectively. Let us assume S ∼ Nn,m (S, ΣS ⊗ ΨS ) and T ∼ Nm (T , ΣT ). Further, let us assume S and T are independent. As Gaussian models can properly represent the behavior of many physical systems, one appropriate assumption is to consider random actuation vector T to have a multivariate normal distribution. The Gaussianity assumption on S matrix is made by neglecting the deviation of the actual distribution of this matrix from the matrix-variate normal distribution. This facilitates subsequent analytical treatments and the error induced by this approximation can remain bounded an sufficiently low in several cases depending on the system complexity. Moreover, assuming that in a physical system actuators performance is independent of their arrangement, the S and T independence assumption can be justified.

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Now, covariance of the random wrench vector W can be derived as follows:

Cov(W ) = E[W W T ] − E[W ]E[W ]T = E[ST T T S T ] − E[S]E[T ]E[T ]T E[S]T

(3.62)

The first term on the right hand side of Eq. (3.62) can be written as

T

T

Z

Z

Z

Z

T

ST S pS,T (S, T )dT dS = E[ST T S ] = DS DT   T = E SE[T ]S = tr{E[T ]T ΨS }ΣS + SE[T ]S T

S DS



T pT (T )dT S T pS (S)dS

DT

(3.63)

where T = T T T . The last equality in Eq. (3.63) results from Theorem 5. Furthermore, using Theorem 5 we can also write

E[T ] = ΣT + T T T

(3.64)

Substituting Eq. (3.64) into (3.63), we get    T E[ST T S ] = tr ΣT + T T ΨS ΣS + S[ΣT + T T T ]S T T

T

(3.65)

Finally, substituting Eq. (3.65) into Eq. (3.62) gives    T Cov(W ) = tr ΣT + T T ΨS ΣS + SΣT S T

3.3.2

(3.66)

Parametric Formulation

More elaborate formulation of the wrench covariance can be developed when the behavior of all agents are statistically identifiable. This implies that, sources of uncertainty are known in some reasonable details and information on the distributions of the random variables are available. Configuration and actuation uncertainties are two major sources of the uncertainty that

65

impact the performance of the multi-agent cooperative systems. Hence, the random total wrench at the platform can be written as the sum of individual wrenches as (a planar system is considered here) 

     F T cos(θ ) f i i ix  x  m  m    X     X   T sin(θ )  f  = W = i i  Fy  =    iy    i=1   i=1   M mi rix fi y − riy fi x

(3.67)

where θi = θi +θiν and Ti = T i +Tiν are the random orientation and tension of the ith agent force vector in which θiν and Tiν are the perturbation terms. In Eq. (4.3), ri = rix eˆ1 + riy eˆ2 is the vector in global frame x − y pointing from the origin of the local frame u − v to the attachment point of the ith cable, as shown in Fig. 4.3.

Figure 3.22: Schematic of a planar cable robot. Solid lines show the nominal (mean) state of the system (tension and orientation of the cables) and dashed lines show the perturbed state. We assume that the magnitudes of the force vectors (Ti ’s) have Gaussian distributions with mean T i and standard deviation σTi , i.e., Ti ∼ N (T i , σT2i ). Moreover, we use von Mises distribution, in order to model random orientation of the force vectors. So, we have θi ∼ vMF(θi , σθi ) ∀i = 1, . . . , m. We also incorporate the following independence assumptions: (i) θi and θj are independent ∀i, j = 1, . . . , m where i 6= j; (ii) Ti and Tj are independent ∀i, j = 1, . . . , m where i 6= j; and (iii) Ti and θj are independent

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∀i, j = 1, . . . , m. In the next chapter, we show that the variances of the wrench vector elements can be obtained in closed-form expressions that are given here (for convenience) as follows:

Var(Fx ) = Var(Fy ) =

m X i=1 m X

a ¯i cos2 (θi ) − ¯bi cos(2θi )

(3.68)

a ¯i sin2 (θi ) + ¯bi cos(2θi )

(3.69)

i=1



I1 (σθi ) a ¯i = + − T I0 (σθi ) i
2 2 ¯bi = I1 (σθi ) T i + σTi I0 (σθi )σθi T 2i

1 where In (z) = π

Z

σT2i

2

π

exp{z cos(x)} cos(nx)dx is the nth order (n is integer) modified Bessel 0

function of the first kind. We use the closed-form expressions of the force variance given by Eqs. (3.68) and (3.69) to estimate the parameters of the model given by Eq. 3.66 when only some bounds on the system parameters are known. The parameter estimation scheme is described in Sec. 3.3.3 in the context of a numerical example.

3.3.3

Numerical Study

We are aiming to optimally find the parameters of the RMT-based model described in Eq. (3.66) such that the model captures the behavior of the systems whose parameters fall into the specific known bounds. Following information are known in our numerical simulations: (i) upper and lower bounds on σθi ’s i.e., σθl ≤ min{σθ1 , . . . , σθm } and σθu ≥ max{σθ1 , . . . , σθm }; (ii) σTi ’s, the lower bound σTl ≤ min{σT1 , . . . , σTm } and the upper bound σTu ≥ max{σT1 , . . . , σTm }; and (iii) the domain on which the system mean state is defined, i.e., T li ≤ T i ≤ T ui and θli ≤ θi ≤ θui (∀i = 1, . . . , m). Now, let us consider the output force in a planar cable-based system in which all cables are attached to one point on the platform (zero moment). The static Jacobian matrix S is a 2 × m matrix whose first and second rows are the cosine’s and sine’s of the force vector

67

orientations, i.e.,   cos(θ1 ), . . . , cos(θm ) S=  sin(θ1 ), . . . , sin(θm )

(3.70)

Relying on the independence of θi ’s, it is a valid assumption to consider the columns of S as independent random vectors, hence, in Eq. (3.66) ΨS = Im [1]. So, ΣS is the only parameter that needs to be optimally chosen given the lower and upper bounds on the system state and parameters. In 1000 Monte Carlo simulations, a system sample is first obtained by drawing a sample of σθi , σTi , T i and θi from uniform distributions with corresponding (given) bounds. In our numerical simulation we set σθl = 100, σθu = 800, σTl = 0.5 N, σTu = 1 N and the bounds on the mean tension and orientation of the cables are T li = 3 N, T ui = 5 N, θli = π/4 rad and θui = π rad (∀i = 1, . . . , m). Then, the following optimization problem is solved to obtain the optimal covariance matrix ΣS : Minimize F = kdiag(Cov(W )) − [Var(Fx ), Var(Fy )]T k ΣS ≥0

(3.71)

in which Cov(W ) is given by Eq. (3.66) and Var(Fx ) and Var(Fy ) are given by Eqs. (3.68) and (3.69), respectively. Finally, we set the optimal value of ΣS to be the average of 1000 ΣS ’s obtained in Monte Carlo runs. Figure 3.23 shows the error histograms when the RMT-based model is tested for 2000 random system realizations. Note that the system samples are different from those (1000 samples) used for parameter estimation. As shown in Fig. 3.23, for a simple system with only m = 3 agents and in 2000 trials, the errors remain less than 25 % and the average error is 4 % which is significantly lower than the maximum error. The notable fact is that by increasing the number of agents, error histograms are continuously shifting toward lower values. For example, maximum error for a system with m = 20 agents is 7 % and the error average is 2 %. This indicates a significant improvement of the RMT-based model performance when compared to the results corresponding to a system with only 3 agents.

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Figure 3.23: Error histograms. For 2000 samples of system state and parameters, the Euclidean distances (error) between the output wrench variances obtained by RMTbased model and those given by parametric model are calculated. 5 histograms with different colors correspond to 5 systems with different number of cables. These results encourage replacing the detailed parametric models by RMT-based models especially when the number of uncertain agents grows in complex multi-agent systems. However, a more general conclusion can be drawn after careful theoretical analysis that can be the subject of the future work. In Sec. 3.3.4, we examine the performance of the RMTbased model developed in this section, in capturing the output wrench of an experimental setup consisting of three mobile robots (iRobot).

3.3.4

Experimental Study: Cooperative iRobots

A system consisting of three iRobots is used to evaluate the RMT-based wrench uncertainty model (Eqs. (3.61) and (3.66)). iRobots are tied to an ATI force transducer using cables, as shown in Fig. 3.24. A desired output wrench W d = [3 N, 5 N, 0 N.m]T is aimed to be generated at the platform (that is the force transducer here) as a result of the individual tractions provided by each iRobot. Three different configurations of the iRobots, shown in the first column of Fig. 3.24, are considered to provide this wrench. Let us refer

69

to the configuration in the first, second and third rows as C1, C2 and C3, respectively. While all configurations provide (approximately) the same mean wrench at the platform, the uncertainty profile is significantly different. The idea is to investigate the effectiveness of the Gaussian product model in capturing the real system output in a cooperative system. After identifying the characteristics of each iRobot, the bounds on the system parameters are set to be σθl = 169.35, σθu = 2.416 × 103 , σTl = 0.48, and σTu = 2.27. The mean state bounds for three different configurations are (∀i = 1, . . . , 3)

C1:

T li = 1.82 N, T ui = 13.89 N

θli = 0 rad, θui = 3.8 rad

C2:

T li = 1.82 N, T ui = 13.89 N

θli = π/4 rad, θui = 2π rad

C3:

T li = 1.82 N, T ui = 13.89 N

θli = 0 rad, θui = 4.45 rad

ˆ c1 = (1e−4)diag([4.771, 6.716]), Following the procedure described in Sec. 3.3.3, we obtained Σ S ˆ c2 = (1e − 4)diag([6.019, 5.557]) and Σ ˆ c3 = (1e − 4)diag([5.366, 5.861]). Σ S S Experimental observations as well as the results obtained using RMT-based model are illustrated in Fig. 3.24. Density plots in the second and third columns are obtained by applying kernel smoothing (KS) density estimation technique to the experimental observations (second column) and to the RMT-based wrench samples (third column). Hot colors show higher and cold colors represent lower densities. The results demonstrate the effectiveness of the RMT-based model in capturing the uncertainty. In all three configurations, the mean and variances of the output force obtained using the RMT-based model is adequately close to those obtained from (relatively accurate) experimental measurements. Note that the optimization problem described in (3.71) targets only the diagonal terms of ΣS . This is because only the distance between variances of the force vector are enforced to be minimized and the covariance (Cov(FX , FY )) terms are not included in the objective function. The optimal values of the off-diagonal terms can be found by slight modification of (3.71) where the objective function is augmented with the distance between the covariance

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Robots configuration

Experimental observations

RMT-based model

Figure 3.24: Experimental evaluation of RMT-based models in capturing the wrench (only force vector here) uncertainty. Three different configurations are shown in three rows. First column shows the configuration of the iRobots. The KS density of the (accurately) measured force is shown in second column. The KS density of the data points obtained from RMT-based model given by Eqs. (3.61) and (3.66) is shown in the third column.

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terms corresponding to the parametric and RMT-based models. This will further improve the accuracy of the RMT-based model results.

3.4

Conclusion

We employed random matrix theories to formulate the Jacobian matrix of a robotic manipulator as random matrix. Two approaches were considered to model the Jacobian matrix. In the first approach, based on a model proposed by Soize [22, 23, 24], Jacobian matrix was perturbed by a positive definite perturbation matrix whose density function was obtained through direct implementation of the MaxEnt method. In the second approach, we assumed that the mean Jacobian matrix was perturbed by an additive and zero mean Gaussian noise matrix. An optimization problem was constructed to obtain the covariance of the noise matrix (or covariance of the random Jacobian matrix) in which the differential entropy of the pdf of a function of the noise matrix is maximized. The constraint of the optimization problem was derived based on the boundedness of the Jacobian norm, that allowed adapting the noise covariance matrix based on the working configuration of the manipulator system. Followed by numerical studies on a 2R serial link (for simplicity in visualization), we reported the results of our real experiment on KUKA youBot system. We showed that how the proposed models require less information and are superior to conventional parametric models in capturing motion uncertainty especially when used in a particle filtering framework for estimating the error bounds. Based on some Gaussianity assumptions, we also proposed an RMT-based model that captures the output wrench uncertainty for which the closed-form expression of the covariance matrix was derived. This was aimed to be an alternative model to a parametric formulation (that we develop in the next chapter) when detail information on the system parameters is not available in complex multi-agent platforms. We validated the proposed model using an experimental setup consisting of three iRobots that provided a static wrench at the end-effector. It was shown that, if only lower and upper bounds are known (in con-

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trast with the parametric formulation that requires the exact values of the parameters), the RMT-based model can still provide acceptable results in capturing the output wrench uncertainty. RMT-based models, despite all advantages discussed in this chapter, provide less flexibility compared to the parametric approaches where each variable is individually modeled. For instance, special structure of the system matrices (for example in decoupled dynamical systems) may be difficult to be enforced in such approaches. Hence, unknown uncertainties can be incorporated into stochastic model of the system that needs to be carefully considered depending on the system under study.

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Chapter 4 Wrench Uncertainty Quantification: Random Vector Analysis 4.1

Introduction

Cable based parallel architectures are of significant interest due to the advantages they offer compared to their rigid-link counterparts. Replacing articulated arms by light-weight cable elements results in simpler construction, lower cost, simpler transportation, larger workspace, ease of reconfiguration and lower interaction of the robot structure with the environment. However, one consequence of using cable elements is the increase of system complexity due to the unilateral force constraint of the cables. Hence, in order to keep the robot fully restrained m = n + 1 cables are at least required for a cable robot with n degrees-offreedom (n-DOF). Nonetheless, in order to improve the wrench-closure-workspace (WCW) [63], many of the cable-based systems are redundantly restrained (m > n + 1), hence, cooperation of several actuation units (e.g., pulley and cables, gantry cranes, etc.). Uncertainty in cable robot architectures arises mainly due to their loosely inter-connected,

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light and flexible structure as well as cooperation of several agents that individually introduce the uncertainty to the system. On the other hand, their special structure and actuation method lead to more challenging strategies of the sensing and control compared with the articulated robotic systems. This is even more critical when dealing with new class of the cable robots with mobile bases. Base mobility in these systems, however, motivates developing reconfiguration schemes in order to lower the effect of uncertainties on the system output. In several applications of such cooperative systems, it is critical to reduce the system response uncertainty in a given direction. We show that, using the redundancy, we can achieve this by reconfiguring the system while the desired mean output is maintained for any new configuration. Hence, a systematic and efficient quantification of the system response uncertainty, given its configuration, is necessary. Efficiency in uncertainty quantification is crucial because, as we will discuss in Sec. 4.2.1.3, measures of the output uncertainty are constructing the objective function and constraints of the configuration optimization problem. The significance of the computational efficiency is further increased when the reconfiguration is aimed to be performed in real-time. To this end, in this chapter, we provide a probabilistic framework where the forces applied to the platform by each individual agent are treated as random vectors. The statistics of the output wrench are characterized through solving the problem of sum of random vectors. Using the closed-form expressions of the mean and variances of the output wrench, we cast the reconfiguration problem into a polynomial optimization that can be efficiently solved in real-time applications. Using both numerical and real-system experiments we show the effectiveness of the proposed approach in realizing the configurations that minimize the uncertainty in certain directions. It is noteworthy that, in addition to reconfiguration planning, our quantification of the output wrench uncertainty provides useful statistical information for higher-level control and facilitates a systematic design trade-off between agents with different levels of precision. The rest of this section is devoted to the motivational examples, statement of the prob-

75

Multi-agent aerial towing configuration

Non-optimal configuration

B

A

(a)

(b)

Optimal

B

A

(c)

Figure 4.1: Cooperative aerial towing example: gray ellipsoids are the uncertainty ellipsoids in 2D plane. (c) shows a safer configuration that minimizes the probability of collision for this specific task compared to non-optimal configuration in (b). lem and sketch of our approach as well as its connection to the real system applications. Furthermore, related literature is reviewed and the contribution of our study to the existing literature is discussed.

4.1.1

Motivational Examples

A motivational example is discussed here to support the necessity and effectiveness of the uncertainty quantification and reconfiguration planning in cable-based cooperative systems. We mainly consider cooperative payload transportation using flying robots (aerial towing), shown in Fig. 4.1, as an illustrative example. Cooperative systems consist of aerial robots have received a significant amount of attention in the last decade. The low-cost off-the-shelf flying robots such as quadrotors, are easily accessible and multiple number of them can be employed for different cooperative tasks. There exist several studies that particularly focus on different aspects of the design and control of such systems. For example see [49, 50]. In many of such applications, the system is mainly aimed to compensate the gravitational force and move the payload in an almost quasi-static condition. Low-cost flying robots are inherently uncertain, hence, neither the load provided by individual agents nor

76

their spatial position can be identified deterministically. Consequently, the resultant wrench delivered at the platform is random due to the uncertainty in tension and orientation of the suspending cables. The main idea is that, although the uncertainty introduced by each individual agent can not be reduced, however, the overall effect of these uncertainties can be minimized by reconfiguration of these flying units. Clearly, re-distribution of the tension on the cables is simultaneously considered in order to maintain a prescribed mean output wrench. Fig. 4.1 illustrates this concept where the objective is to transport the payload from point A to B without collision with the walls. In Fig. 4.1 (b), the output uncertainty, corresponding to the configuration of the agents, is shown by a gray ellipsoid. The major axis of the ellipse is crossing the walls, hence, an unsafe configuration for this particular task. The reconfiguration can be performed such that the probability of collision is minimized. The schematic uncertainty ellipse in Fig. 4.1 (c) illustrates this matter where the major axis is aligned with the straight path from A to B. Note that, in addition to re-orienting the ellipse, the uncertainty can also be reduced, i.e., the length of the minor axis can be minimized. In addition to transporting/manipulating systems, the proposed approach here is applicable to other classes of the cable robots. One main example is cable-based suspension and measurement systems such as aircraft model hangars [55, 56] (Fig. 4.2) that are commonly used in a static setup. In contrast to towing or manipulating systems that deliver the wrench at the platform, cable based measurement systems are used to characterize the applied wrench on the platform. Given the configuration of the system, this is feasible through sensing the tension of the cables. However, sensing uncertainty and uncertainty in characterizing the exact configuration of the robot are inevitable. Using the developed approach in this chapter the density of the platform wrench can be characterized and optimal sensing configurations can be identified to improve the estimations of the true wrench applied to the prototype under the experiment.

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Figure 4.2: Cable-based suspension and measurement system: airplane hangar where bases can be re-adjusted along the rails.

4.1.2

Problem Statement

Here, we provide a mathematical presentation of the problem that is tackled in this chapter. We present the problem for a planar system, however, both the planar (Sec. 4.2.1) and spatial (Sec. 4.2.2) systems are investigated. Figure 4.3 shows a planar cable-based archib ] tecture with m cables. Let us describe the configuration of the bases by Q = [X1b , . . . , Xm

where Xib is the location of ith base in the global frame x−y. Moreover, assume that the attachment point of each cable to the platform is known and fixed in the local coordinate u−v. Now, let us denote the prescribed task by T (t) = (X d (t), W d (t)) where X d (t) and W d (t) are the desired platform pose and output wrench at time t (in global coordinates), respectively. Given T (t), a feasible domain Q can be defined such that if Q ∈ Q then geometrical and wrench-feasible constraints will be satisfied during the task. Furthermore, let us define the mean or nominal state of the system (solid lines in Fig. 4.3) by S(t) = (T (t), Θ(t)) where T = [T 1 (t), . . . , T m (t)]T and Θ(t) = [θ1 (t), . . . , θm (t)]T are the vectors of mean tension and orientation of the cables, respectively. When the base locations are given, the system state trajectory can be identified by the desired task, i.e., S(t) = S(T (t)|Q). Note that Q is not a function of time as for current study we assume that base locations are fixed during the task, although they can be optimally relocated for every new task. In the real setup, the state of the system is uncertain. In fact for an uncertain system

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we have S(t) = (T (t), Θ(t)) = (T (t) + T ν (t), Θ(t) + Θν (t)) where T ν (t) and Θν (t) are the random perturbations corresponding to the tension and orientation vectors, respectively. Schematic of a sample of random state is depicted in Fig. 4.3 by dashed lines. Randomness in the tension and orientation of the cables simply results in a random wrench vector on the platform, denoted by W (t) = [F (t), M (t)]T . For notational simplicity, we drop the time variable t in the rest of the chapter. Now, the problem can be described as follows:

Figure 4.3: Schematic of a planar cable robot. Solid lines show the nominal (mean) state of the system (tension and orientation of the cables) and dashed lines show the perturbed state.

argmin UW (Q|T )

(4.1)

Q

s.t. Q ∈ Q

(4.2)

where UW (Q|T ) is a scalar-valued function that quantifies the uncertainty of the output wrench corresponding to the base configuration Q for a given task T . In order to enable solving the optimization problem described in (4.1) and (4.2), the objective function UW needs to be efficiently calculated. To construct UW one needs the mean of the output wrench vector to minimize its distance from the desired output wrench, W d . Moreover, in order to lower the uncertainty level, a function of the variances of the output force and moment is simultaneously minimized. To this end, we seek to develop the closed-form expressions of the mean and variances of the resultant force and moment given the mean state of the

79

system, i.e., E[F |S(Q; T )], E[M |S(Q; T )], E[(Fx − F x )2 |S(Q; T )], E[(Fy − F y )2 |S(Q; T )] and E[(M − M )2 |S(Q; T )]. Monte Carlo based approaches to calculate these identities are computationally prohibitive to be used in such optimization problems as the Monte Carlo simulation is required at each iteration during the search for the optimal values. The computation cost and convergence problem is even more critical when dealing with redundant systems with multiple agents (i.e., large m’s) especially in real-time applications.

4.1.3

Literature Review

Representation and statistical manipulation of the uncertainty have been the subject of several studies in robotic community. Early contributions in this area dates back to the work by Smith and Cheeseman [64] on the spatial uncertainty defined on the coordinate frames that was applied to the navigation problem of the mobile robots. Followed by their work and in order to adopt the convenient representation of the motion, quantification of the uncertainty in the form of homogeneous transforms and on the motion groups received special interest. Su and Lee [65] used the differential homogeneous transform to represent the uncertainty. They addressed the kinematic error propagation in a robotic manipulator used in an assembly task. Error propagation on the Euclidean motion group (with a focus on robotic manipulator applications) have been comprehensively studied by Wang and Chirikjian [66, 67, 68, 69]. In their work, the propagation of uncertainty is first studied by convolution on SE(3) and, using the theory of Lie algebras and Lie groups, first- [66, 67] and second-order [68, 69] approximations to the mean and covariance were provided. This enabled the recursive approximation of the mean and covariance of the whole manipulator. The robustness of the second-order approximation as well as its superiority compared to the first-order method were verified in different numerical examples. Cable-based cooperative systems, that will mostly benefit from the approach developed in this chapter, have been extensively considered in many recent studies. In towing applications, the cooperative agents are commonly mobile robots such as quadrotors for aerial

80

towing and wheeled robots for ground towing. Cheng et al. [70] proposed a planar system in which a payload was towed/carried with multiple mobile robots (Scarab) attached to the platform through cables. A comprehensive treatment to the aerial towing systems consist of multiple flying robots has been addressed by [49, 50, 51, 52, 53, 54]. It can be seen that the performance of the cooperative system has been degraded by inevitable uncertainty in the flying robot position (and provided force). For example in [50], fluctuations in controlled position of the individual robots as well as random oscilliations of the platform around equilibrium configurations has been reported. Base mobility and reconfiguration problem also applies to the new class of the cablebased manipulators. Although conventional cable robots use fixed bases with varying cable lengths [71, 72, 73], recent studies [57, 58, 59, 60, 61, 62] focus on reconfigurable systems where the cable lay out can be changed through mobility of the bases. The configuration optimization has been considered to improve some performance indices such as dexterity [58], tension factor [60], [74], power consumption & stiffness [62], etc. However, in this work our main focus is to reduce the uncertainty of the generated wrench at the platform that directly impacts the platform position uncertainty. An overview of the literature reveals that: (i) new class of reconfigurable cable-based systems are increasingly evolving and extending their application to a broad range of physical problems; (ii) these systems are significantly impacted by the uncertainty especially with recent growing attitudes toward using low-cost and rapidly-prototyped robotic agents in many distributed systems; (iii) systematic treatment to the uncertainty in these systems and approaches toward reducing the system output fluctuations have not been addressed in the literature. To this end, we propose a probabilistic approach based on the sum of random vectors to characterize the uncertainty of the output wrench in cable-based cooperative systems. Relying on our analytical treatment to the output wrench uncertainty, we propose a reconfiguration scheme in order to reduce the uncertainty level in a given direction.

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The problem of sum of random vectors has found several applications in statistics and engineering. Specifically, in electrical and communications engineering, investigation of the sum of random phasors [75, 76] is critical to the characterization of the random signals (that are basically the sum of sinusoidal signals). Application examples include multi-path propagation in communication channels [77], communication cochannel interference [78], light scattering from random phase screens [79], among many others. Despite of extensive application of this problem in electrical engineering, there are not many studies on the sum of random vectors and its applications to robotic design problems.

4.2 4.2.1

Wrench Uncertainty Quantification Planar Systems

In this section, we develop the probabilistic formulation of the resultant wrench in planar systems. Using vector-analysis, the total wrench at the payload can be written as the sum of individual wrenches as 

     F f T cos(θ ) x i i i x    m  m    X   X    f  =  T sin(θ )  W = i i  Fy  =  iy      i=1   i=1   M rix fi y − riy fi x mi

(4.3)

where θi = θi + θiν and Ti = T i + Tiν are the random orientation and tension of the ith cable in which θiν and Tiν are the perturbation terms. in Eq. (4.3) ri = rix eˆ1 + riy eˆ2 is the vector in global frame x − y pointing from the origin of the local frame u − v to the attachment point of the ith cable, as shown in Fig. 4.3.

4.2.1.1

Output Force Vector Statistics

As discussed in Sec. 4.1, closed-form expressions of the mean and variances of the output wrench facilitate efficient reconfiguration of the mobile bases in order to reduce the uncertainty level on the output wrench. In this subsection, our focus is only on the

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force vector statistics and the output moment is statistically formulated in the following subsection. Referring to Sec. 2.2, the first and second order derivatives of the jcgf at the origin are the mean and variances, respectively. To derive the jcgf one needs to first construct the jmgf that for the random force vector F = [Fx , Fx ]T can be written as

MFx ,Fy (η, ζ) = E[exp{ηFx + ζFy }]

(4.4)

and substituting Fx and Fy from Eq. (4.3) results in

MFx ,Fy (η, ζ) = ET1 ,...,Tm ,θ1 ,...,θm

Y m

 exp{ηTi cos(θi ) + ζTi sin(θi )}

(4.5)

i=1

Some assumptions can be made on random variables θi ’s and Ti ’s with fair validity in many practical applications. Especially in loosely-interconnected cable-based architectures the performance of an agent is not significantly affected by other agents. One may consider the fixed-base cable robots as the most interconnected and cable-based ground and aerial towing systems as least interconnected systems. This leaves Ti (almost) independent from Tj and similarly θi (almost) independent from θj (∀i, j = 1, . . . , m where i 6= j). Additionally, we are assuming that for each agent the cable tension and cable orientation are independent random variables, i.e., Ti is independent from θi (∀i = 1, . . . , m). Care must be taken when making this assumption as the fluctuation of a single agent (orientation uncertainty) may increase when demanding higher traction. With these assumptions the jmgf in Eq. (4.5) can be written as

MFx ,Fy (η, ζ) =

m Y

Mfi x ,fi y (η, ζ)

(4.6)

i=1

where Mfi x ,fi y (η, ζ) is Z Mfx i ,fy i (η, ζ) =

DTi

Z Dθi

exp{ηTi cos(θi ) + ζTi sin(θi )}pθi (θi )pTi (Ti )dθi dTi (4.7)

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and DTi and and Dθi are the domains where Ti and θi are defined, respectively. As Gaussian models reliably represent the uncertain quantities in several physical systems, we assume the tension in each cable is normally distributed with mean T i and standard deviation σTi , i.e., Ti ∼ N (T i , σT2i ). Furthermore, the cable orientation is assumed to have von Mises distribution with mean θi and dispersion parameter σθi , i.e., θi ∼ M (θi , σθi ). The validity of these assumptions are further investigated in Sec. 4.2.1.5 using experimental data. Note that Ti ’s are defined on [−∞, +∞], however, for relatively large (and positive) values of the mean and small values of the variance, negative tensions are at the tail of the distribution. This is necessary in order to respect the unilateral force constraint in cablebased systems. Substituting densities given in (2.2) and (2.15) into (4.7) and simplifying the exponential term results in 1 Mfx i ,fy i (η, ζ) = √ 2πσTi I0 (σθi ) in which z(Ti ) =

Z

+∞

−∞

  (Ti − T i )2 exp − I0 (z(Ti ))dTi 2σT2i

(4.8)

q a2i + b2i where ai = ηTi + σθi cos(θi ) and bi = ζTi + σθi sin(θi ). Refer to

Appendix A for detailed derivations. Now, jcgf for the random force vector can be written as

KFx ,Fy (η, ζ) = log MFx ,Fy (η, ζ) = log

m Y i=1

=

m X

Mfx i ,fy i (η, ζ) =

m X

log Mfx i ,fy i (η, ζ)

i=1

Kfx i ,fy i (η, ζ)

(4.9)

i=1 F On the other hand, the first and second cumulants (cF s1 and cs1 ,s2 ) are defined as

cF s1 =

∂KFx ,Fy (η, ζ) ∂ 2 KFx ,Fy (η, ζ) |(0,0) , cF = |(0,0) s1 ,s2 ∂xs1 ∂xs1 ∂xs2

84

(4.10)

where

   xj 0 = η

if j 0 = 1

  xj 0 = ζ

if j 0 = 2

Now, substituting (4.8) in (4.9) and using the relations given in (4.10), and after some algebra (see the details in Appendix A), the mean and variances of the force vector can be obtained as

E[Fx ] =

cF 1

m X I1 (σθi ) = T cos(θi ) I0 (σθi ) i

(4.11)

I1 (σθi ) T sin(θi ) I0 (σθi ) i

(4.12)

E[Fy ] = cF 2 =

i=1 m X i=1

Var(Fx ) =

cF 1,1

=

Var(Fy ) = cF 2,2 =

m X i=1 m X

a ¯i cos2 (θi ) − ¯bi cos(2θi )

(4.13)

a ¯i sin2 (θi ) + ¯bi cos(2θi )

(4.14)

i=1



I1 (σθi ) + − a ¯i = T I0 (σθi ) i
2 2 ¯bi = I1 (σθi ) T i + σTi I0 (σθi )σθi T 2i

σT2i

2

Note that the first equalities in Eqs. (4.11) and (4.12) are due to Eq. (2.12) and first equalities in Eqs. (4.13) and (4.14) are due to Eq. (2.13).

4.2.1.2

Output Moment Statistics

The moment generating function (mgf) of the resultant moment on the platform can be similarly written as (using the same independency assumptions)

MM (η) = EM [exp{ηM }] = ET1 ,...,Tm ,θ1 ,...,θm

Y m

 exp{η (ri Ti cos(θi ) + vi Ti sin(θi ))}

i=1

=

m Y

Mmi (η)

(4.15)

i=1

85

where ri = −riy and vi = rix . Using densities given in (2.15) and (2.2) Mmi (η) can be obtained as

Mmi (η) = √

1 2πσTi I0 (σθi )

Z

+∞

exp{− −∞

(Ti − T i )2 }I0 (z(Ti ))dTi 2σT2i

(4.16)

q where z(Ti ) = a2i + b2i in which ai = ηri Ti + σθi cos(θi ) and bi = ηvi Ti + σθi sin(θi ). Following similar procedure as discussed in section 4.2.1.1 we obtain the mean and variance of the resultant moment as m X I1 (σθi ) T (ri cos(θi ) + vi sin(θi )) I (σ ) i i=1 0 θi " m X
I0 (σθi ) I1 (σθi ) 2 M 2 Var(M ) = c1,1 = T + σTi (ri cos(θi ) I0 (σθi ) i I1 (σθi ) i=1 # m 2 2 X (v − r ) cos(2θ ) − 2v r sin(2θ ) i i i i i i 2 − + vi sin(θi )) + σθi i=1  2 I1 (σθi ) T (ri cos(θi ) + vi sin(θi )) I0 (σθi ) i

E[M ] = cM 1 =

4.2.1.3

(4.17)

(4.18)

Base Configuration and Cables Tension Optimization

Closed-form expressions of the mean and variances given by Eqs. (4.11)-(4.14) and (4.17)-(4.18) verify the dependency of wrench uncertainty level on the cable tensions (T i ’s) as well as orientation of the cables (θi ’s). We use these closed-form expressions in an optimization framework to find optimal configuration of the bases and tension of the cables for a prescribed output wrench such that the uncertainty in resultant force and moment are

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minimized. The optimization problem can be formulated as follows:

argmin

J(Var(Fx ), Var(Fy ), Var(M ))

(4.19)

θ1 ,...,θm ,T 1 ,...,T m

s.t. |E[Fx ] − Fxd | − eFx ≤ 0

(4.20)

|E[Fy ] − Fyd | − eFy ≤ 0

(4.21)

|E[M ] − M d | − eM ≤ 0

(4.22)

ΘL < Θ < ΘU

(4.23)

TL < T < TU

(4.24)

where J is the objective function that quantifies the output wrench uncertainty. Constraints described by (4.20)-(4.22) ensure the mean of resultant wrench is sufficiently close to the prescribed wrench. Superscript d indicates the desired (prescribed) values and eFx , eFy and eM are the maximum acceptable distances between mean and the desired values of the wrench vector elements. Inequality constraint (4.23) ensures the kinematic feasibility of the cable orientations (typically to avoid the collision) in which ΘL and ΘU are respectively the vectors of lower and upper bounds on the mean orientation of the agents. Moreover, inequality (4.24) guarantees that optimal tensions are consistent with the force capacity of the agents where T L and T U are respectively the vectors of lower and upper bounds on the mean tension of the cables. Depending on application scenario, different variant of optimization problems can be formulated by selecting various combinations of objective functions. One such formulation based on the assumption of linear combination of force and moment variances can be written as follows:

J = γ1 Var(Fx ) + γ2 Var(Fy ) + γ3 Var(M )

87

(4.25)

where

P3

i=1 γi

= 1. The general optimization problem described by (4.19)-(4.24) with

objective function given by (4.25) can be converted into polynomial optimization problem assuming xi = cos(θi ), yi = sin(θi ) and zi = T i as follows:

argmin

J=

m X

x1 ,...,xm ,y1 ,...,ym ,z1 ,...,zm

ai x2i + bi yi2 + ci x2i zi2 + di yi2 zi2 + ei xi yi + fi xi yi zi2

(4.26)

i=1

s.t. m X d pi xi zi − Fx − eFx ≤ 0 i=1 m X

|

i=1 m X

(4.27)

pi yi zi − Fyd − eFy ≤ 0

(4.28)

qi xi zi + wi yi zi − M d | − eM ≤ 0

(4.29)

i=1

xli ≤ xi ≤ xui

(4.30)

yil ≤ yi ≤ yiu

(4.31)

zil ≤ zi ≤ ziu

(4.32)

x2i + yi2 = 1

(4.33)

The coefficients in (4.26)-(4.33) can be written in terms of the distributions parameters as 

pi pi γ1 + γ3 ri2 + σT2i γ2 + γ3 vi2 ai = 1− σθi σθi  

pi pi bi = σT2i 1 − γ2 + γ3 vi2 + σT2i γ1 + γ3 ri2 σθi σθi  

pi pi ci = 1 − p2i − γ1 + γ3 ri2 + γ2 + γ3 vi2 σθi σθi  

pi pi di = 1 − p2i − γ2 + γ3 vi2 + γ1 + γ3 ri2 σθ σ θi  i    pi pi ei = 2γ3 ri vi σT2i 1 − 2 , fi = 2γ3 ri vi 1 − p2i − 2 σθi σθi I1 (σθi ) pi = , qi = pi ri , wi = pi vi I0 (σθi ) σT2i



88

There are many available packages that can be used to solve such polynomial optimization problems such as GloptiPoly [80] (used in this work), MATLAB optimization toolbox (e.g., fmincon function along with sqp algorithm), etc. It is noteworthy that in special case of minimizing the trace of the force covariance matrix by choosing γ1 = 1/2, γ2 = 1/2 and γ3 = 0, using Eqs. (4.13) and (4.14) we get

J = Var(Fx ) + Var(Fy ) =

m X

c˜i T 2i + d˜i

(4.34)

i=1

1 c˜i = 2



I1 (σθi ) 1− I0 (σθi )

2 > 0,

1 d˜i = σT2i 2

that shows the trace of the force covariance matrix is invariant with respect to the orientation of the cables. Moreover, since c˜i > 0, ∀i = 1, . . . , m then the uncertainty optimization problem turns into minimization of the cable tensions.

4.2.1.4

Numerical Study: Reconfiguration Planning

We first validate the closed-form expressions of the statistical moments given by Eqs. (4.11)(4.14) and (4.17)-(4.18) by a series of the Monte Carlo simulations for different number of agents (cables) in the system. Then, an example of a planar cable robot is considered to validate the effectiveness of the proposed approach in quantifying the wrench uncertainty and efficient optimization of the system state.

Monte Carlo Validation: Three planar systems with m = 4, m = 5 and m = 6 cables are considered in order to validate the closed-form expressions of the statistical moments. The geometrical and statistical information corresponding to each system is shown in Fig. 4.4. Table 4.1 compares the mean and variances of the resultant force and moment given by Eqs. (4.11)-(4.14) and (4.17)-(4.18) with the corresponding values obtained through 10000 Monte Carlo simulations. The results tabulated in Tab. 4.1 verify the validity of the developed analytical solutions to the statistical moments of the output wrench. Given the

89

Figure 4.4: Three planar systems with m = 4, 5 and 6 cables and their corresponding statistical information used for Monte Carlo validation of the closed-form solutions given by Eqs. (4.11)-(4.14) and (4.17)-(4.18). Table 4.1: Mean and standard deviations of the resultant force vector and moment: closed-form solution versus Monte Carlo simulation. m=4 m=5 m=6 32.05 39.46 41.17 Eq. (4.11) κ1F = E[Fx ] 32.09 39.55 41.11 MC -7.54 5.30 -4.39 Eq. (4.12) κ2F = E[Fy ] -7.51 5.23 -4.18 MC -0.1 -0.99 -0.78 Eq. (4.17) κ1M = E[M ] -0.11 -0.99 -0.77 MC 74.47 78.19 81.14 Eq. (4.13) 2 κ1,1 F = E[(Fx − E[Fx ]) ] 76.9 77.71 80.72 MC 110.65 112.09 112.75 Eq. (4.14) 2 κ2,2 F = E[(Fy − E[Fy ]) ] 112.56 113.35 112.57 MC 0.46 0.5 0.55 Eq. (4.18) 2 κ1,1 M = E[(M − E[M ]) ] 0.46 0.51 0.54 MC

fact that Eqs. (4.11)-(4.14) and (4.17)-(4.18) are the exact solutions, the small differences between the analytical and Monte Carlo based results are expected to be further reduced by increasing the number of Monte Carlo realizations.

Reconfiguration Planning ((Quasi-)Static System): Here we inspect the proposed reconfiguration strategy in a static cable-based architecture that provides a static wrench at the platform (end-effector). The bases are kept fixed during the task, however,

90

they can adaptively reconfigure to improve the performance index (that is uncertainty level here) for each specific task. The real-system experimental proof of the concept and validation of the simulation results reported in this section are presented in Sec. 4.2.1.5. In the first set of our examples, the desired output moment is set to be zero. So, all cables are attached to one point on the platform. In optimization of the tension and orientation of the cables we examine three objective functions. First is the tension factor given by T F = min(T )/max(T ). Maximizing the T F provides an improved tension balance among P the cables and it has been shown [74] that it is equivalent to minimizing JT F = m i=1 T i . So, Pm in (4.26) we have J = JT F = i=1 zi . Second and third objective functions are J = Var(Fx ) (by choosing γ1 = 1, γ2 = 0 and γ3 = 0) and J = Var(Fy ) (by choosing γ1 = 0, γ2 = 1 and γ3 = 0), respectively. For all optimization problems we set W d = [45 N, 60 N, 0 N.m]T , σθi = 100 and σTi = 2 N ∀i = 1, . . . , m. The lower and upper bounds on the mean tension of the cables are assumed to be T Li = 0 N and T U i = 100 N ∀i = 1, . . . , m. Assuming a minimum tension of 0 N allows the algorithm to remove a cable if the global optimum can be achieved by fewer number of cables. Moreover, θi ’s can be chosen anywhere on the circle, i.e., θi ∈ [0, 2π]. This may be not feasible in real systems (e.g., iRobot cooperative system described in the next section) where different agents must maintain a certain distance to avoid collision. However, for simulation purpose and to identify minimum possible uncertainty level, a bound is not prescribed for the cable angles in this example. For the next example where the output moment uncertainty is concerned and for the real system experiment, described in Sec. 4.2.1.5, θi ’s are restricted to certain bounds. For the optimal T i ’s and θi ’s corresponding to each objective function, we generate 10000 samples of the output force vector by sampling Ti ’s from Gaussian distribution and θi ’s from von Mises distribution, and substituting the samples into Eq. (4.3). A pdf is approximated from the generated samples using (bi-variate) KS density approximation technique. Figure 4.5 shows the optimal configuration and cable tensions as well as the resulting KS joint pdf of the output force corresponding to each optimization problem. In the configuration

91

m=2

TF maximization:

(a) m=2

m=6

(b)

(c)

(d)

(e)

γ1 = 1 γ2 = 0 γ3 = 0

γ1 = 0 γ2 = 1 γ3 = 0

Figure 4.5: Optimization of the tension and orientation of the cables to minimize (a) the tension factor (b) Var(Fx ) with 2 cables (c) Var(Fx ) with 6 cables (d) Var(Fy ) with 2 cables (e) Var(Fy ) with 6 cables. The prescribed task is to provide the static wrench W d = [45 N, 60 N, 0 N.m]T at the end effector. Each base can move 2π rad as shown by dashed lines (constraint (4.23)). The minimum and maximum tension of the cables are set to be 0 N and 100 N for all cables (constraint (4.24)). Additionally, σθi = 100 and σTi = 2 N ∀i = 1, . . . , m. plots, the acceptable range of the θi ’s are shown by dashed circular arcs and the (optimal) location of the bases are shown by diamond markers. In KS joint density plots, hot colors

92

show higher densities and lower densities are shown by cold colors. Figure 4.5 (a) corresponds to a system with m = 2 cables and the objective is to maximize T F . Second row (Figs. 4.5 (b) and (c)) shows the results when the objective is solely minimizing Var(Fx ). Comparing Fig. 4.5 (b) with Fig. 4.5 (a) indicates the uncertainty in Fx can not be notably reduced when m = 2. However, increasing the number of cables facilitates a significant reduction in the Var(Fx ) where it reduces from 20.68 N2 in Fig. 4.5 (a) to 9.95 N2 in Fig. 4.5 (c). Third row of Fig. 4.5 illustrates the results when the objective is to reduce Var(Fy ). Similarly, reduction in uncertainty level of Fy is not significant when only 2 cables are used, however, a major reduction from 15.16 N2 (Fig. 4.5 (a)) to 7.6 N2 (Fig. 4.5 (e)) can be achieved in a system with 6 cables. Roughly speaking, and referring to Figs. 4.5 (c) and (e), by optimizing the tension and orientation of the cables we are in fact rotating the uncertainty ellipsoid to align its minor axis with a direction along which the maximum precision is required. Furthermore, the results show that, in addition to reorienting the ellipsoid, the length of minor axis can also be reduced. The uncertainty in output moment can be similarly treated as the output force vector. Figure 4.6 shows the results corresponding to T F maximization (first row) and minimization of the Var(M ) (second row). The platform (gray circles) is chosen to be circular for simplicity in the presentation of the results, however, it can be of any general geometrical shape. There are 10 possible attachment points on a circular arc with a radius 2 m whose center is matching with the center of the circular platform. We examine two systems
with m = 3 and m = 4 cables and the optimization problem is solved for 10 m choices of the attachment points to find the one that provides the least uncertainty level in output moment. For the results shown in Fig. 4.6 we have W d = [45 N, 60 N, 120 N.m]T , σθi = 100 and σTi = 2 N ∀i = 1, . . . , m. Additionally, the mean tension of the cables are restricted to T Li = 0 N and T U i = 40 N ∀i = 1, . . . , m. The domain from which θi ’s can be chosen are shown by dashed arcs whose lengths are π rad. As shown in Fig. 4.6, the variance of the

93

m=4

(a)

(b)

(c)

(d)

γ1 = 0, γ2 = 0, γ3 = 1

TF maximization:

m=3

Figure 4.6: Optimization of the tension and orientation of the cables to (a)-(b) maximize the tension factor and (c)-(d) minimize Var(M ). The desired static wrench is W d = [45 N, 60 N, 120 N.m]T . Each base can move along the arcs shown by dashed lines whose lengths are π rad. The minimum and maximum tension of the cables are set to be and 0 N and 40 N, respectively. Also, σθi = 100 and σTi = 2 N ∀i = 1, . . . , m. output moment can be reduced from 56.04 N.m2 (when T F is maximized) to 39.02 N.m2 (when Var(M ) is minimized) in a system with m = 3 cables. Similarly, when m = 4, Var(M ) is reduced from 61.91 (N.m)2 to 37.09 (N.m)2 . It is also noteworthy that the attachment points can be chosen depending on the objective/application of the system. For example, referring to Figs. 4.6 (b) and (d), the algorithm chooses a different set of attachment points when the objective is changed to minimize the output moment variance.

Reconfiguration Planning: 4-CDDR with Moving Platform In order to examine our proposed approach for a cable robot with moving platform, we consider an example of a cable-direct-driven robot (CDDR) with 4 cables, shown in Fig. 4.7. The objective is to relocate the bases of the robot for a given task (trajectory following

94

Figure 4.7: 4-CDDR: the gray boxes show the domains over which the bases can be relocated. while providing a constant wrench) in order to reduce the uncertainty of the resultant force or moment during the process. In doing so, the objective function value is obtained as follows. For a given location of the bases and a desired task, at each time instant, the cables orientation is obtained using forward kinematics relations and cables tension is given by T = J + W d + (I4 − J + Jz)

(4.35)

where J is the static Jacobian matrix, I4 is 4 × 4 identity matrix and z is an arbitrary 4-dimensional vector. Now, given the mean state trajectory, the variances of the force or moment can be calculated at every time point using (4.13)-(4.14) and (4.18). The objective function is assumed to be the sum of the variances over the simulation time points. Depending on the application, one may also assume the objective function to be the maximum value of the variance over the time, corresponding to the given base locations. In Fig. 4.7, the desired task is to move the platform without change of its orientation from A1 to A2 while providing a constant wrench W d = [15 N, 2 N, 5 N.m]T at the platform. The base locations are constrained to be in the gray boxes with given dimensions.

95

These constraints can be defined such that the positive cable tension is ensured during the process. Other geometrical factors such as cables interference can also be incorporated in posing these constraints. For this analysis, we assume σθi = 19, σTi = 1 N, ∀i = 1, . . . , 4. √ √ Figure 4.8 shows the variation of eF = Var(Fx )+Var(Fy ) and σM = Var(M ) corresponding to the base locations that reduce the force uncertainty. Similar results when reducing the moment uncertainty is sought are shown in Fig. 4.9. 75

7.5 Non−optimal base locations Optimal base locations

Non−optimal base locations Optimal base locations 7

70

6.5

σM

eF

6 65

5.5 5

60

4.5 55 0

1

2

3

4

4 0

5

1

2

3

t (s)

t (s)

(a)

(b)

4

5

Figure 4.8: Uncertainty level in force and moment over the simulation time: the location of the bases is optimized to reduce the force uncertainty level. As shown in Figs. 4.8 and 4.9, there are base configurations that have reduced the force or moment uncertainty in all time points of the process. However, depending on the problem, there may be not an specific base configuration that results in lower uncertainty in both force and moment over all process time.

Effect of Redundancy: One important question that may arise in designing redundant cable architectures is the effect of increasing the number of cables on the system performance that, from the perspective of the current work, is the wrench uncertainty level. Intuitively, it is expected that distributing the task among several agents should result in averaging the error and reduction of the uncertainty (a main advantage of the parallel architectures compared to their serial counterparts). However, this depends on the characteristics of the

96

70

6.5 Non−optimal base locations Optimal base locations

Non−optimal base locations Optimal base locations

6

69.5 5.5 69

5

σM

eF

4.5 68.5

4 68

3.5 3

67.5 2.5 67 0

1

2

3

4

2 0

5

1

2

3

t (s)

t (s)

(a)

(b)

4

5

Figure 4.9: Uncertainty level in force and moment over the simulation time: the location of the bases is optimized to reduce the moment uncertainty level. individual agents and needs to be carefully quantified for systematic design trade-off. Figure. 4.10 shows the variation of the uncertainty level versus the number of cables used in the architecture for systems with 4 to 15 cables. The quantities depicted in Fig. 4.10 (a) and (b) are

Cv F

and CvM , respectively, that are defined as q q Var(Fx ) + Var(Fy )/ E[Fx ]2 + E[Fy ]2 p = Var(M )/|E[Mx ]|

CvF =

(4.36)

CvM

(4.37)

The prescribed wrench is W d = [60 N, 25 N, 5 N.m]T . The first 4 agents are considered to be homogeneous, i.e., they have identical uncertainty characteristics that are σθi = 10, σTi = 2 N, i = 1, . . . , 4. Two approaches are considered in increasing the number of agents. In the first approach, all agents that are added to the base 4-cable system are identical to 4 primary agents. In the second approach, every new agent has improved uncertainty characteristics compared with the last one added to the system. In fact, σθi increases (then uncertainty level decreases) from 10 for 4th agent to 573 for 15th agent. Similarly, the standard deviation of the cable tension reduces from 2 N for 4th agent to 0.1 N corresponding to 15th agent. The results corresponding to the first approach (homogeneous

97

agents) are shown by black diamonds and those corresponding to the second approach are shown by blue squares in Fig. 4.10. 0.3

3.5 Homogeneous agents Non−homogeneous agents

0.28

Homogeneous agents Non−homogeneous agents 3

0.26 0.24 Cv M

Cv F

2.5 0.22 0.2

2 0.18 0.16

1.5

0.14 0.12 4

6

8

10 12 Number of cables

14

1 4

16

(a)

6

8

10 12 Number of cables

14

16

(b)

Figure 4.10: Variation of the uncertainty level in (a) force and (b) moment versus number of cables for homogeneous and non-homogeneous agents. The results depicted in Fig. 4.10 verify and also quantify the improvement of the uncertainty level on the platform by adding either identical agents or agents with better uncertainty characteristics. However, comparing the results corresponding to these approaches shows that improvement of the precision is limited through adding identical agents. The saturation effects can be seen in CvF and CvM after adding 13th agent. However, these values corresponding to the second approach are constantly decreasing that represents the compensation of the effect of perturbations induced by highly uncertain agents through adding new agents with higher precision level.

4.2.1.5

Experimental Study

We apply our method to a cooperative system shown in Fig. 4.11. There are three mobile robots (iRobot Create) that are attached to a force transducer through cables. In our experiment the desired output moment is considered to be 0 and, therefore, all cables are attached to one point on the force transducer.

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Figure 4.11: Experimental setup. Three iRobots are tied to the force transducer located at the origin of the global coordinate system shown by blue lines. The task is to provide a pre-defined output force at the end effector. Agents Force Capacity and Uncertainty Characterization: We choose (nominally) identical robots, however, their characteristics are different and hence need to be identified individually. The quantities of interest to construct the optimization problem (described by (4.26)-(4.33)) are σTi , σθi , T iL and T iU for i = 1, . . . , 3. To find an estimate of these quantities, we send a 50-seconds long motion command to the iRobot that is tied to the force transducer by a cable. We measure the force in both x and y directions. This process is repeated for two more motion commands that provide more tractions. There is a pause between the motion commands in order to facilitate segmenting the data points. Figure 4.12 shows the magnitude of the force vector versus time for three iRobots used in our experiment. The mean of the traction samples for each motion command is shown above the corresponding data points. We use these mean values to estimate T iL and T iU . Moreover, we center the entire data points (traction values) around zero traction by subtracting the corresponding mean from each set of the data points. Then, a sample estimate of σTi is obtained to be used in our optimization problem. The force vector orientation can be represented by the random unit vector u with von Mises-Fisher distribution given by Eq. (2.14) defined on the unit sphere Sn−1 = {u ∈ Rn : kuk= 1} where in planar system n = 2 and S1 is a unit circle. Let U j = {uj1 , . . . , ujN } be

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Figure 4.12: The traction force provided by each iRobot corresponding to different motion commands. Three motion commands are sent consecutively with a pause after first and second command to separate the data points. The average traction values are shown above the corresponding data points for each motion command. The data acquisition frequency is 50 Hz and we collect the force data for 50 s, i.e., 2500 data points for each motion command. the set of N samples of u corresponding to the jth motion command. It can be seen that the data sets U j (j = 1, . . . , 3) have different sample mean vectors, as shown in Fig. 4.13. However, in order to use the MLE approach presented in Sec. 2.3 for estimating σθi , all data points must share the same mean and dispersion parameter. This problem can be addressed using Theorem 6 (Sec. 2.3) as follows [7]. An arbitrary mean vector is first chosen. Here we choose the north pole, i.e., unp = [0, 1]T . Using Rodrigues’ formula [81] we calculate a

Figure 4.13: Orientational uncertainty of the force vectors. The data points corresponding to three motion commands are shown by blue, black and red dots and their mean direction is shown by a solid line drawn from the center of the circles. The points shown by green are the rotated data points in order to use MLE approach for estimating σθi ’s (refer to the text for more details). 100

Experimental results

(a)

(b)

(c)

(d)

(e)

(f)

γ1 = 0, γ2 = 1, γ3 = 0

γ1 = 1, γ2 = 0, γ3 = 0

TF maximization:

Numerical simulation

Figure 4.14: Experimental application of the optimization scenarios. First row corresponds to T F maximization. Second and third rows show the configuration and cable tensions that minimize Var(Fx ) and Var(Fy ), respectively.

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rotation matrix Rj that rotates the sample mean vector uj (corresponding to data set U j ) to unp , i.e., unp = Rj uj . Then, the data set U is constructed as follows:

U = {Rj U j , j = 1, . . . , 3}

(4.38)

The samples in U have common mean vector unp and dispersion parameter σu . Now, we can use Eq. (2.17) to estimate σθi . Table 4.2 summarizes the estimated parameters corresponding to three iRobots used in this study.

Table 4.2: Estimated parameters of the iRobots used in the experimental study. iRobot 1 TˆiL (N) 3 TˆiU (N) 5 σ ˆTi (N) 0.95 σ ˆ θi 1.208e3

iRobot 2 3 6 1.13 338.7

iRobot 3 4 6 1.13 778.7

Minimizing the uncertainty level: We implement different optimization scenarios simulated in Sec. 4.2.1.4 on the cooperative system shown in Fig. 4.11. The desired task is determined to provide an output wrench W d = [3 N, 5 N, 0 N.m]T . However, as the system output is uncertain, we can only achieve the desired wrench in an average sense. We set the acceptable distance from the mean wrench vector to be 1% of the desired wrench. Figure 4.14 shows both the simulation and real system experimental results corresponding to three optimization scenarios: T F maximization (Figs. 4.14 (a) and (b)), minimization of Var(Fx ) (Figs. 4.14 (c) and (d)) and minimization of the Var(Fy ) (Figs. 4.14 (e) and (f)). Referring to Fig. 4.14, the results obtained through experimental studies are in a good agreement with simulation results. The first row, shows the result of our attempt to maximize the tension factor and simultaneously maintaining a mean output wrench close to the desired wrench. The simulation results predict Var(Fx ) and Var(Fy ) to be 2.36 N2 and 1.16 N2 , respectively. The sample estimate of these variances, obtained from the experimental measurements, are 2.68 N2 and 1.43 N2 , respectively. The KS joint density plot

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for experimental data is also closely resembling the one obtained through the numerical simulation. In the second row of Fig. 4.14, we are aiming to (roughly speaking) rotate the uncertainty ellipsoid such that its minor axis is aligned with the x direction, i.e., the variance of Fx is minimized. Figure 4.14 (d) verifies that reconfiguration of the iRobots according to the optimal solution can successfully minimize the variance of Fx . It can be seen that Var(Fx ) reduces from 2.68 N2 when we maximize T F (Figs. 4.14 (b)) to 0.35 N2 when the objective is minimizing Var(Fx ) ( (Figs. 4.14 (d))). Similarly, when the objective is to minimize Var(Fy ), the simulation predicts the variance to be 0.49 N2 and Var(Fy ) is 0.43 N2 in the actual system that is a significant reduction compared with Var(Fy ) = 1.43 N2 when T F is maximized. Different factors are responsible for the small difference between simulation and experimental results. In our simulation we ignored the dependency of the σTi and σθi on T i . Moreover, there is an unavoidable error in adjusting the orientation of the cables (configuring the iRobots) that is not taken into account in our formulation. However, neglecting these factors enables the analytical treatment to the problem as developed in the preceding sections, and as presented in Fig. 4.14, the proposed model can still provide high fidelity, hence, justifying these assumptions.

4.2.2

Spatial Systems

By transitioning to spherical statistics (from circular statistics in the previous section) we can extend our formulation to 3-D applications. Aerial towing with flying robots (quadrotors) are representative examples of such systems. Let us consider the output force in the output wrench vector for our analysis in this section. One can extend the analysis to the output moment using the force statistics formulation and similar procedure described in previous section (discussed for 2D systems). The output force is similarly modeled by random vector sum as F = [Fx , Fy , Fz ]T = Pm ~ i where Ti is the random magnitude of force vector f~i (corresponding to i’th i=1 Ti u ~ i is the random unit vector that represents the (random) orientation of f~i . agent) and u

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𝑧 𝒇3

𝒇2

𝝓1

𝑦

𝒇𝑚

𝒇1 𝜽1 𝑥

Figure 4.15: Spatial system: random force vectors can be modeled using spherical statistics.

4.2.2.1

Joint Moment Generating Function:

Using the same argument as previous section, f~i ’s are independent random vectors, and hence, one can write (likewise Eq. (4.9))

KFx ,Fy ,Fz (η, ζ, γ) =

m X

log Mf~i (η, ζ, γ) =

i=1

m X

Kf~i (η, ζ, γ)

(4.39)

i=1

and Mf~i (η, ζ, γ) is Mf~i (η, ζ, γ) = Mfi x ,fi y ,fi z (η, ζ, γ) = E[exp{ηfi x + ζfi y + γfi z }] Z Z = exp{ηfi x + ζfi y + γfi z }p(f~i )df~i = exp{ηfi x + ζfi y + γfi z } p(Ti , ~ui ) dTi d~ui (4.40) | {z } p(Ti )p(~ ui )

In the spherical coordinates (see Fig. 4.15) one can write

f~i = Ti [cos(θi ), sin(θi ) cos(φi ), sin(θi ) sin(φi )]T

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Hence, Eq. (4.40) can be re-written as Z Mf~i (η, ζ, γ) =

exp{Ti [η cos(θi ) + ζ sin(θi ) cos(φi ) + γ sin(θi ) sin(φi )]}

p(Ti )p(θi , φi )dTi dθi dφi

(4.41)

In Eq. (4.41) we use the relation p(~ui )d~ui = p(θi , φi )dθi dφi because one can write ~ i = f (θi , φi ) = [cos(θi ), sin(θi ) cos(φi ), sin(θi ) sin(φi )]T . In Eq. (4.41), p(θi , φi ) repreu sents orientational uncertainty that can be appropriately captured using von Mised-Fisher distribution, given by Eq. (2.14), that is a well-know distribution to model a Gaussianlike random orientation on (hyper-)sphere. In spherical polar coordinates, the pdf for von Mises-Fisher distribution is [5]

pθ,φ (θ, φ) =

κ exp{κ[cos(θ) cos(α) + sin(θ) sin(α) cos(φ − β)]} sin(θ) 4π sinh(κ)

(4.42)

where mean direction µ ~ is [cos(α), sin(α) cos(β), sin(α) sin(β)]T . While von Mises-Fisher distribution is primarily the appropriate distribution for modeling the orientational uncertainty of the agents, using its density function into jmgf’s introduces difficulties in analytical evaluation of the integrals. We, instead, independently perturb θi and φi with von Mises-distributed noises. Hence, p(θi , φi ) ≈ p˜(θi , φi ) = p˜(θi )˜ p(φi ) = 1 exp{σθi (2π)2 I0 (σθi )I0 (σφi )

4.2.2.2

cos(θi − θi ) + σφi cos(φi − φi )}.

Kullback-Leibler Divergence Minimization:

In order to closely characterize the physical system behavior, primarily described by the von Mises-Fisher distribution, we use the Kullback-Liebler (KL) divergence measure to minimize the distance between the two densities, p(θi , φi ) and p˜(θi , φi ) that is defined as p˜ D(˜ pkp) = Ep˜[log ] = p

Z log θ,φ∈[0,2π]

p˜(θ, φ) p˜(θ, φ)dθdφ p(θ, φ)

(4.43)

Note that while the support for the von Mises-Fisher density function is defined by

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θ ∈ [0, π], φ ∈ [0, 2π], the domain for the approximate density p˜(θ, φ) extends to θ, φ ∈ [0, 2π] as θ and φ are independently perturbed by von Mises distribution that is defined over the entire circle. This can be safely considered for our spherical analysis by assuming c0 < θi < π − cπ such that P (π < θ < 2π) is minimal. In some sense, this is similar to the Gaussianity assumption on cable tensions (Ti ’s) where we assumed that the mean tensions are sufficiently greater than zero and the uncertainty is relatively small such that negative tensions are at the tail of the distribution. Now, by substituting p˜(θ, φ) and p(θ, φ) in Eq. (4.43) and after some algebra one can show that D(˜ pkp) simplifies to I1 (σφ i ) sinh(κ) I1 (σθi ) I1 (σθi ) + σθi + σφ i − κ cos(αi ) cos(θi ) πκI0 (σθi )I0 (σφ i ) I0 (σθi ) I0 (σφ i ) I0 (σθi ) I1 (σθi ) I1 (σφ i ) (4.44) − κ sin(αi ) sin(θi ) cos(βi − φi ) − Eθi [log sin θi ] I0 (σθi ) I0 (σφ i )

Di (˜ pkp) = log

In order to find the optimal concentration parameters of p˜i corresponding to ith agent, i.e., σ ˆθi and σ ˆφ i , we solve the following optimization problem: argmin Di (˜ pkp)

(4.45)

σθ i ,σφ i

σθi and σφ i ≥ 0

In summary, the von Mises-Fisher distribution is the appropriate distribution to capture the orientational uncertainty of the agents in an actual setup. One may use the MLE technique to estimate the concentration parameter of the von Mises-Fisher density function for each agent and then find the optimal corresponding values for the concentration parameters of p˜(θi , φi ) by minimization of the KL divergence measure through (4.45). In our numerical study and configuration optimization we follow the same procedure that will be discussed in further details in Sec. 4.2.2.4.

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4.2.2.3

Output Force Statistics

Now with Ti ∼ N (T i , σT2i ) and p˜θi ,φi (θi , φi ), Eq. (4.41) can be re-arranged as 1 Mf~i (η, ζ, γ) = 5/2 (2π) σTi I0 (σθi )I0 (σφi )

Z exp{f (Ti , θi , φi ; η, ζ, γ)}exp{−

(Ti − T i )2 }dTi dθi dφi 2σT2i (4.46)

where

f (Ti , θi , φi ; η, ζ, γ) = Ti [η cos(θi ) + ζ sin(θi ) cos(φi ) + γ sin(θi ) sin(φi )] + σθi cos(θi − θi ) + σφi cos(φi − φi ) Using Mf~i (η, ζ, γ) in Eq. (4.39) and after some algebra, we derive the following closedform expressions for the mean and variances (first and second cumulants, respectively) of the 3-dimensional output force vector as

E[Fx ] = cF 1 = E[Fy ] = cF 2 = E[Fz ] = cF 3 =

m X i=1 m X i=1 m X

rθi T i cos(θi )

(4.47)

rθi rφi T i sin(θi ) cos(φi )

(4.48)

rθi rφi T i sin(θi ) sin(φi )

(4.49)

i=1

 m  X
rθ cos(θi )2 − i cos(2θi ) σT2i + T 2i − rθ2i T 2i cos(θi )2 σθi i=1   m  X rφ rθ cos(φi )2 − i cos(2φi ) = 1 − cos(θi )2 + i cos(2θi ) σθi σ φi

Var(Fx ) = cF 1,1 = Var(Fy ) = cF 2,2

(4.50)

i=1


2

σT2i + T i − rθ2i rφ2 i T 2i sin(θi )2 cos(φi )2 (4.51)     m X rφ rθ Var(Fz ) = cF = 1 − cos(θi )2 + i cos(2θi ) 1 − cos(φi )2 + i cos(2φi ) 3,3 σ θi σ φi i=1


2

σT2i + T i − rθ2i rφ2 i T 2i sin(θi )2 sin(φi )2

(4.52)

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where

rθi =

I1 (σθi ) I0 (σθi )

and

rφi =

I1 (σφi ) I0 (σφi )

Similar to planar case, cumulants are derived by calculating the partial derivatives of KFx ,Fy ,Fz (η, ζ, γ) at the origin and simplifying the integrals.

4.2.2.4

Numerical Study: Reconfiguration Planning

Likewise Sec. 4.2.1.4, in this section we construct an optimization framework to optimally locate the agents in order to minimize the output force variance in particular directions. The optimization problem for planar systems, described by (4.19)-(4.24), is modified as follows for spatial system shown in Fig. 4.15:

argmin

J(Var(Fx ), Var(Fy ), Var(Fz ))

(4.53)

θ1 ,...,θm ,φ1 ,...,φm ,T 1 ,...,T m

s.t. |E[Fx ] − Fxd | − eFx ≤ 0

(4.54)

|E[Fy ] − Fyd | − eFy ≤ 0

(4.55)

|E[Fz ] − Fzd | − eFz ≤ 0

(4.56)

ΘL < Θ < ΘU

(4.57)

ΦL < Φ < ΦU

(4.58)

TL < T < TU

(4.59)

The prescribed task for the flying agents is to compensate an external load that requires Fxd = 10 N, Fyd = 50 N and Fzd = 300 N to be generated at the platform. Depending on the prescribed task and the architecture of the system, infinite solutions might exist for the configuration of the agents and distribution of the tension among the cables. Here, we use 5 agents for this task and find two different solutions: one is solely obtained by solving the feasibility problem, and another is obtained by solving the optimization problem

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(4.53)-(4.59) in order to minimize uncertainty level in a particular direction. A solution is feasible if the resultant force compensates the external load within an acceptable error, i.e., constraints (4.54)-(4.56), collision between agents is avoided, i.e., constraints (4.57) and (4.58), and avoids actuator saturation as well as slackness of the cables, i.e., constraint (4.59). Here, we restrict the orientation of the agents such that angle(~ui , ~uj ) > 10◦ (∀i 6= j). U Tension of the cables are also restricted through T L i = 5 N and T i = 100 N (∀i = 1, . . . , m).

Fig. 4.16 shows the results corresponding to eFx = 0.02Fxd , eFy = 0.02Fyd and eFz = 0.02Fzd , in three different rows. First row corresponds to the minimization of the output load variance in x direction and the results corresponding to y and z directions are shown in the second and third rows, respectively. Second column in each row is the 2D visualization of the 3D plot in the first column (for better illustration of the differences). Diamonds represent the location of the agents and ellipses represent the covariance matrix of the random output load corresponding to each configuration. Red color is used to show the results for solely feasibility problem analysis (non-optimal configuration) and blue shows the optimal solution. Similar to the numerical analyses performed for the planar case, we first optimize the configuration and tension distribution using the closed-form expressions of the moments and once optimal configuration is found, we characterize the uncertainty (covariance ellipses) corresponding to the optimal configuration through exhaustive Monte Carlo simulation. It can be seen that in all three cases, the variance of the output load has significantly decreased by re-configuration of the agents to the optimal configuration. The optimization procedure reduces the variance of Fx from 743.82 N2 to 458.02 N2 . Along y direction it reduces from 1620.18 N2 to 542.4 N2 and along z direction the variance reduces from 120.69 N2 to 53.9 N2 . Note that the results shown in Fig. 4.16 are generally local optima’s that provide better uncertainty results and the improvement is dependent on different factors such as the initial condition as well as the constraints structure (e.g., eF. , upper and lower bounds, etc.). For more tolerance on the mean output error, i.e., eFx = 0.1Fxd , eFy = 0.1Fyd and

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Minimization of Var(Fx )

(b)

(c)

(d)

(e)

(f)

Minimization of Var(Fz )

Minimization of Var(Fy )

(a)

Figure 4.16: Configuration optimization for spatial system (maximum 2% error in mean output): diamonds/lines show the configuration and ellipses represent the covariance matrix of output force uncertainty. Red color shows the non-optimal and blue is the optimal configuration. 110

eFz = 0.1Fzd , further reduction on uncertainty level is achieved, as illustrated in Fig. 4.17. Generally, once the optimization framework is established, one can use different techniques such as converting into polynomial optimization, as described in Sec. 4.2.1.3, in order to seek the global optimum.

4.3

Conclusion

Quantification of wrench uncertainty in cable robot systems was considered in this chapter. By taking advantage of the special lay out of the parallel (cable-based) architectures, we were able to formulate the resultant wrench in an uncertain system through sum of the random vectors. The uncertainty induced by each agent was considered due to the variation in the tension and orientation of the cables that were modeled by Gaussian and von Mises distributions, respectively. The closed-form expressions of the mean and variances of the resultant force and moment were provided. This enabled us to efficiently and accurately quantify the wrench uncertainty for a given state of the system. This, in turn, facilitates off-line or on-line optimization procedures that adaptively optimize the system state (configuration and cable tensions) in order to improve precision indices (functions of force and moment variances), i.e., reducing the uncertainty level. We provided examples of the static cable-based system to examine the effectiveness of the proposed approach. Moreover, in an experimental study we implemented the developed method on a cooperative system (consists of three iRobots) where the analysis of the system output verified the applicability and acceptable accuracy of the proposed approach in reducing the uncertainty level. We first established our development by analyzing planar systems and, next, extended our analyses to spatial systems by using spherical statistics. Our objective in this chapter was mainly configured around uncertainty aspect of reconfiguration in multi-agent systems with mobile bases. However, for real system applications, several other factors need to be simultaneously taken into account. Dynamic analysis of the platform, slackness of cables or maximum number of cables that can be in tension, ca-

111

Minimization of Var(Fx )

(b)

(c)

(d)

(e)

(f)

Minimization of Var(Fz )

Minimization of Var(Fy )

(a)

Figure 4.17: Configuration optimization for spatial system (maximum 10% error in mean output): diamonds/lines show the configuration and ellipses represent the covariance matrix of output force uncertainty. Red color shows the non-optimal and blue is the optimal configuration. 112

bles interference that may introduce specific constraints into the optimization problem, and several other factors are needed to be considered in real applications. We, in this chapter, aimed to add one important element, i.e., configuration/state dependent uncertainty level, to these factors.

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Chapter 5 Workspace and Wrench Capacity Quantification: Random Vector Analysis 5.1

Introduction

In this chapter, we extend our probabilistic framework based upon sum of random vectors to address the deterministic problems in both kinematic and static arrangements. In a kinematic setup, we characterize the workspace density of the redundant planar serial arms with both revolute and prismatic joints. Moreover, in a static formulation, we model the output load capacity of a multi-agent system with focus on cable-based parallel architectures.

5.1.1

Workspace Density

Workspace characterization is an important step in the design of almost every articulated manipulator system. More particularly, in kinematically redundant systems, workspace can be characterized through a probability density function whose value at each pose/point in

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the workspace represents the number of robot configurations that locate the end-effector in that pose/point. Overall performance and accuracy of the manipulator is evidently higher when it is working in the regions for which the workspace density is higher. Hence, workspace density can be directly used for inverse kinematics [82] and motion planning in redundant manipulators. Moreover, once the workspace density is given, one can find the boundaries of the workspace (that is of interest in many design procedures) by identifying the regions with near-zero density values. Here, we extend our analysis presented in the previous chapter and couple that with MaxEnt method to construct the workspace density function of the redundant serial arms with both revolute and prismatic joints. The main idea is to model the deterministic motion of the links, i.e., angular motion of the revolute and linear motion of the prismatic joints, using uniform distributions (defined on the circle and line for revolute and prismatic joints, respectively). Several studies in the literature [83, 84, 85] address different varients of the manipulator workspace problem such as workspace boundaries detection, singularity-free workspace characterization, etc. Majority of proposed approaches are relying on the exhaustive Monte Carlo simulations where several possible end-effector poses are generated by sampling the joint variables and passing the samples through the forward kinematics relations. However, these brute-force approaches are not computationally efficient and, particularly in complex redundant systems with several joints, one may not be able to pursue such schemes. Following the same argument made in [82], if one discretizes the motion of a revolute joint into, say, M discrete values, for an N −link arm, M N points need to be computed. The total number of points grows rapidly with the number of links. The computational difficulty will be further increased with adding prismatic joints to each link that results in M 2N points to be calculated. Hence, a systematic treatment to the problem based upon probabilistic formulation is necessary for the complex redundant system analysis. Using uniform distributions for

115

modeling the motion of manipulator links in a deterministic system has been previously considered by Ebert-Uphoff and Chirikjian [86] and Dong et al. [82]. In [82], a closed-form solution of the workspace density function was provided by solving the convolution integrals of the frame densities. Dealing with the convolutions, two main approaches were employed. One based on the Gaussian distribution and central limit theorem and another based on the motion-group Fourier transform. Furthermore, the application of workspace density in solving the inverse kinematics problem was discussed. In another work by Wang and Chirikjian [87], the elements of SE(N ) were assumed as random variates and a diffusion equation was constructed for the manipulator density function that resulted in an explicit solution. In the following, we briefly review the method proposed in [82] based on motiongroup Fourier transform. In Sec. 5.2.4, we compare our MaxEnt based workspace densities with those obtained through motion-group Fourier transform technique as well as those obtained from Monte Carlo simulations. Using forward kinematics, the end-effector pose in a 2-link planar serial arm is

hEE = h1 h2

(5.1)

where hi ∈ SE(2) is the homogeneous transformation from frame (link) i to i − 1. Now, given ph1 (h1 ) and ph2 (h2 ), the pdf of hEE can be written as Z phEE (hEE ) = (ph1 ? ph2 )(hEE ) = Z = h1 ∈SE(2)

Z p(hEE , h1 )dh1 =

h1 ∈SE(2)

h1 ∈SE(2)

ph2 (h−1 1 hEE )ph1 (h1 )dh1

p(hEE |h1 )ph1 (h1 )dh1 (5.2)

Similarly, for an N -link manipulator one can show that

phEE (hEE ) = (ph1 ? ph2 ? . . . ? phN )(hEE )

(5.3)

Given that the Fourier transform of convolution of densities is the product of Fourier

116

transform of individual pdf’s, one can write

F (phEE ) =

N Y

F (phi )

(5.4)

i=1

where F (p(h))(s) is the Fourier transform of p(h), given by Z

p(h)U (h−1 , s)d(h)

F (p(h))(s) = pˆ(s) =

(5.5)

h∈SE(2)

in which U (h−1 , s) is the unitary representation of the SE(2) and d(h) is the natural biinvariant integration measure for SE(2) [88]. Additionally, the inverse Fourier transform is

F

−1

1 (ˆ p(s)) = p(h) = (2π)2

Z

∞

tr{ˆ p(s)U (h, s)}sd(s)

(5.6)

0

It is shown that [82]

F (phEE ) =

N Y

J0 (sLi )F (ph1 )

(5.7)

i=2

where Li is the length of ith link. Hence, the pdf of end-effector pose is obtained by taking inverse Fourier transform of RHS of Eq. (5.7) as

phEE (hEE ) =

1 (2π)2

Z 0

N ∞Y

J0 (sLi )tr{ˆ ph1 (s)U (hEE , s)}sd(s)

(5.8)

i=2

Refer to [86, 82, 87, 88] for further details on workspace density characterization using Fourier analysis on motion groups. Here, we formulate the workspace density function of a serial arm with both revolute and prismatic joints by modeling the joint states as random variables with uniform distributions in the forward kinematics equations. Randomness of the joint angles represent the revolute joints motion and random link lengths are due to the prismatic joints motion. It can be

117

seen that forward kinematics equations can be treated as sum of random vectors and hence the closed-form expressions of its moments are available. Then, we utilized the maximum entropy principle to construct the workspace density using its moments.

5.1.2

Force Capacity

A variety of the parallel architecture configurations are formed when multiple robotic agents are attached to the same platform and cooperate to manipulate the payload. Examples include cable robot systems with fixed/mobile bases and fixed/varying cable lengths [89], cooperative mobile robots with articulated arms attached to the same platform [90], etc. Physical cooperation between robotic agents is critical to accomplishing cooperative payload manipulation of large payloads that exceed the capacity of the individual agents. Such multi-agent systems can potentially realize many benefits: achieve greater flexibility, accomplish better performance and provide greater reliability and fault-tolerance than a single monolithic agent [91]. However, the consequence of replacing a monolithic agent by a distributed cooperating team of loosely-interconnected agents is the increase of system complexity. Depending on the architecture of the system and allocation of the tasks, the range of possible resultant forces generated on the platform (called force capacity here) can vary in magnitude and direction. In addition, the uncertainty of each individual agent affects the overall force uncertainty that also depends on the system architecture and task allocation, as discussed in previous chapter. Hence, it is critical to have a framework that facilitates optimizing the system layout and task allocations to optimize both (i) the resultant force capacity, and (ii) resultant force uncertainty within the system. Two main differences in solving this problem compared with its kinematic counterpart (workspace density characterization) are independency of the agents and the range of angular motions. In the sequel, we will see that, in contrast with the workspace problem, the contribution of agents in the output load of a parallel architecture can be safely considered

118

as independent random vectors. However, while in workspace analysis we consider revolute joints to have no mechanical locks (i.e., θi ∈ [0, 2π]), when analyzing the output force, range of agents motion is restricted to a limited arc on the circle. These constraints are imposed for motion feasibility and avoiding the collision between the agents. Further discussion on how to treat different problems are presented in the following sections. Although we are interested in workspace density function, for the load capacity analysis we are only interested in quantifying the first two moments of the (deterministic) output load in order to optimize the configuration for maximizing the range of possible loads in a particular (given) direction. The rest of this chapter is organized as follows. In Sec. 5.2, we formulate the workspace density using sum of random vectors and maximum entropy principle. It is also followed by our numerical experiments where we compare our results with those obtained from Fourier analysis on motion groups developed in [82] and exhaustive Monte Carlo simulations. In Sec. 5.3, we derive the closed-from expressions for the mean and variances of the resultant force vector in a (deterministic) multi-agent system. In addition, we extend the analysis to the simultaneous characterization of the load capacity and (unwanted) load uncertainty. Similarly, these analyses are followed by numerical experiments where the system lay out is optimized to improve the output load range in certain directions and avoid configurations that are vulnerable to uncertainty. We conclude the chapter by a brief discussion in Sec. 5.4.

5.2

Manipulator Workspace Characterization

In this section, we characterize the workspace of a serial planar arm by formulating the forward kinematics relations in a probabilistic framework. The characterization can be performed in two arrangements: (i) direct derivation of jpdf of the end-effector position using the characteristic function, and (ii) estimation of the jpdf from the statistical moments. First approach is mainly relying on the special shape of the characteristic function of the distribution. We argue that the closed-form expression of the workspace density, given

119

as integral of a function of the manipulator parameters, is intractable in many cases of the complex systems with several links. In the second approach, we estimate the density function using the closed-form expressions of the higher statistical moments in the MaxEnt scheme. We compare the performance of our formulations with the model proposed in [82] and the results obtained from Monte Carlo simulations.

5.2.1

Probabilistic Forward Kinematics

Forward kinematics relations of an m-links serial chain can be written as

x = l1 cos(θ1 ) + y = l1 sin(θ1 ) +

m X i=2 m X

li cos(θ1 + · · · + θi )

(5.9)

li sin(θ1 + · · · + θi )

(5.10)

i=2

Θ=

m X

θi

(5.11)

i=1

where θi is the relative angle of ith link, x and y are the position and Θ is the orientation of the end-effector in Cartesian coordinate system. Now, assuming a general arm with m revolute joints and m prismatic joints, state of each joint can be modeled by a uniform distribution that represents the range of motion of the joint. Here, we assume that the prismatic joint corresponding to ith link enables the length of the link to change from lil to liu (li ∈ [lil , liu ]). Additionally, we assume that there is no mechanical lock at each revolute joint that results in θi ∈ [0, 2π] (∀i = 1, . . . , m). Referring to Eqs. (5.9)-(5.11), we first need to solve for sum of the random joint angles, i.e., αi = θ1 + · · · + θi . Because uniform distributions are used to model the (linear and angular) motion of the links, random variable θi (∀i = 1, . . . , m) has the following density function defined on the circle:

pθi (θi ) =

120

1 2π

(5.12)

denoted as θi ∼ Uc (0, 2π). It can be shown that [5] if θi ∼ Uc (0, 2π) then αi ’s are independent and have uniform distributions on the circle, i.e., αi ∼ Uc (0, 2π) (∀i = 1, . . . , m). Now, one may define the problem as finding the closed-form solution for pX (X) where X is given by     Pm x  i=1 li cos(αi ) X =   = P  m y l sin(α ) i i=1 i

(5.13)

in which li ∼ U (lil , liu ) and αi ∼ Uc (0, 2π), ∀i = 1, . . . , m. We know that prismatic joint states (li ’s) are independent random variables. Because revolute joints can swing a full circle (no mechanical lock), αi ’s can also be considered as independent random variables. Finally, from the mechanics of the robot motion, it is clear that li is independent from αj (∀i, j = 1, . . . , m).

5.2.2

Density Characterization from Characteristic Function

Here, we first construct the characteristic function for random vector X as

ΨX (η, ζ) = Ex,y [exp{j(ηx + ζy)}]

where j =

√

(5.14)

−1. Substituting x and y from Eq. (5.13) in (5.14) results in

m m h i Y h i X ΨX (η, ζ) = El1 ,...,lm ,α1 ,...,αm exp{ jli (ηci + ζsi )} = Eli ,αi exp{jli (ηci + ζsi )} i=1

=

m Y

i=1

Ψli ,αi (η, ζ)

(5.15)

i=1

where ci = cos(αi ) and si = sin(αi ). Second equality in Eq. (5.15) is due to the independency of the random variables. Given that li ’s and αi ’s are distributed uniformly (on the

121

line and circle, respectively), the characteristic function can be written as

ΨX (η, ζ) =

m Z Y i=1

liu

2π

hZ

lil

0

|

i exp{ jli (ηci + ζsi ) }pαi (αi )dαi pli (li )dli |√ {z } =jli η 2 +ζ 2 sin(αi +βi ) {z } √ =J0 (li

=C

m Z lu Y i lil

i=1

where C =

Qm

u i=1 1/(li

J0 (li

η 2 +ζ 2 )

p η 2 + ζ 2 )dli

(5.16)

− lil ) and Jν (.) is the νth order Bessel function of the first kind.

Given the shape of the ΨX (η, ζ), random vector X is spherically symmetric. Definition 5.1 [92, 93] describes this systematically. Definition 5.1: Random vector X ∈ Rn is spherically symmetric with weight matrix
A if and only if its joint characteristic function is of the form ΨX (u) = h (uAuT )1/2 where h is a function on [0, ∞] and A is a n × n positive-definite matrix. For a spherically symmetric random vector with weight matrix A, if the function h is absolutely integrable, then the jpdf of X is [94]   pX (X) = |A|−1/2 gn (XA−1 X T )1/2

(5.17)

where

−n/2 1−n/2

gn (s) = (2π)

Z

∞

s

h(λ)λn/2 J(n−2)/2 (λs)dλ

0

From Eq. (5.16), we have h(λ) = C

Qm R liu i=1 ll i

J0 (li λ)dli . Substituting h(λ) in Eq. (5.17),

and given that n = 2 for planar manipulator and A = In , the jpdf of X (end-effector position) is px,y (x, y) = (2π)−1 C

Z

l1u

Z

u lm

... l1l

l lm

hZ

∞

i p J0 (l1 λ)J0 (l2 λ) . . . J0 (lm λ)J0 (λ x2 + y 2 )λdλ dl1 . . . dlm

0

(5.18)

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Although Eq. (5.18) can be considered as a closed-form expression for the workspance density function, however, solving the integral inside the bracket is difficult/intractable depending on the number of links and the values of the prismatic joint states. This motivates moment-based density estimation approaches to the workspace density function characterization.

5.2.3

Moment-Based Density Estimation

Workspace density function can be properly approximated by finite number of its statistical moments. The optimal number of moments for density construction is dependent on the distribution as well as the accuracy required for the particular application. Here, we first analytically study the closed-form expressions of the moments, and next, formulate the workspace density using MaxEnt method.

5.2.3.1

Closed-form Expressions of the Moments

In one approach, the moments of the end-effector position in 2D plane are directly calculated and, in an alternative approach, the statistical moments of the exponential function of the end-effector position are obtained. The latter is proposed as it provides relaxation in analytical formulation of the higher moments and the workspace density can be ultimately obtained by log-transformation of the MaxEnt jpdf of transformed workspace. There are advantages and drawbacks associated with these schemes that will be discussed in simulations and results section (Sec. 5.2.4). In the first approach, we consider the end-effector position as the random variable and directly calculate the moments of X in Eq. (5.13). Because we know (from characteristic function) that the wrokspace distribution is spherically symmetric, all odd moments of the distribution are zero when the manipulator base is located at the origin. So, for r = (p + q)th moment we have  Z m m m Y X X 1  p q  µp,q = E[xp y q ] = 1 { l c } { li si } dli dαi if p + q is even (5.19) i i m (2π) i=1 liu − lil l1 ,...,lm i=1 α1 ,...,αm i=1   µp,q = 0 if p + q is odd (5.20) Instead of the expression given in Eq. (5.19), one can calculate the moments through calculating the cumulants which is the sum of cumulants corresponding to each agent (as discussed in the previous chapter). However, calculation of the higher order cumulants may be equally cumbersome. In fact, direct computation of the moments given by Eqs. (5.19) and (5.20) is practical because of the following reasons. First, for MaxEnt density approximation, finite number of moments are

123

typically adequate in order to construct the density functions with acceptable accuracy. Second, unlike configuration optimization problem in which the moments need to be calculated at each iteration of the search for optimal configuration (when solving the optimization problem), here we need to calculate the moments once for a given manipulator (or given manipulator parameters). Third, for calculating upto rth moment (p + q ≤ r), the integral in Eq. (5.19) needs to be computed only when p + q is an even number and otherwise the moment is zero (once the base of manipulator is transferred to the origin of the coordinate system) that results in a significant reduction of the computational need. An alternative approach to direct moment computation is to formulate the pdf of exponential of the end-effector position variables. This facilitates the closed-form solution to the moments of arbitrary order. Once the density is constructed through MaxEnt formulation, one can use the inverse mapping technique to obtain the actual density of the end-effector position. Here, we transform the ˜ = [x, ˜ y] ˜ T = [ex , ey ]T . From pX˜ (˜ ˜ y˜ = pX (x, y)dxdy, one end-effector position vector as X x, y˜)dxd can simply show that pX (x, y) = pX˜ (exp{x}, exp{y})exp{x + y} (inverse mapping). Now, µ ˜p,q is m h i X ˜ p y˜q ] = El1 ,...,lm ,α1 ,...,αm exp{ µ ˜p,q = E[x li (pci + qsi )}

(5.21)

i=1

Substituting the uniform density functions (for li ’s and αi ’s) in Eq. (5.21) results in

µ ˜p,q = C

m Z Y i=1

liu

I0 (li ρ)dli

(5.22)

lil

Qm where similar to Eq. (5.16) C = i=1 1/(liu − lil ), Iν (.) is the νth order modified Bessel funcp R tion and ρ = p2 + q 2 . One can show that [95] I0 (x)dx = Λ(x) = [x + (πx/2) L1 (x)]I0 (x) − ν R 2π 2( 21 z ) sinh(z cos(θ)) sin2ν (θ)dθ is the modified Struve [(πx/2) L0 (x)]I1 (x) where Lν (z) = √πΓ(ν+ 1 ) 0 2

function. Hence, µ ˜p,q is

µ ˜p,q =

5.2.3.2

m i C Yh u l Λ(l ρ) − Λ(l ρ) i i ρm i=1

(5.23)

MaxEnt Density Estimation

MaxEnt relations introduced in Sec. 2.4 can be extended to bi-variate case (planar serial arm) in order to construct workspace density function pX (x, y). With first R moments (r = 1, . . . , R),

124

MaxEnt problem can be formulated as Z

ymax

Z

xmax

−

r=1 p=0

ymax

Z

xmax

xp y r−p pX dxdy − µp,r−p

λr,p ymin

ymin

xmin

ymin

Z

ymax

Z

xmax

pX ln{pX }dxdy − λ0

Maximize L (pX ) = − R X r X

Z

 pX dxdy − 1

xmin

 (5.24)

xmin

where λ0 and λr,p ’s are Lagrangian multipliers corresponding to the normalization and moment constraints, respectively. Integral bounds are in fact the maximum reachable distances from the manipulator base that can be determined based on the prior information on the kinematic structure and its given parameters as xmin = ymin = exp{dmin } and xmax = ymax = exp{dmax } where Pm Pm u dmin = − i=1 liu and dmax = i=1 li . In Eq. (5.24), first term is the differential entropy of pX (x, y). Second integral corresponds to the normalizing constraint that ensures the density function integrates to one over the entire domain of the random variate. Last term in Eq. (5.24) corresponds to the moment constraints. Using the calculus of variations, it can be shown that the MaxEnt jpdf (that maximizes L (pX )) is

pX (x, y) = exp{−λ0 −

r R X X

λr,p xp y r−p }

(5.25)

r=1 p=0

where λ0 can be written in terms of λr,p ’s using the normalization constraint as Z

xmax

Z

ymax

exp{λ0 } =

exp{− xmin

ymin

r R X X

λr,p xp y r−p }dxdy

(5.26)

r=1 p=0

and λr,p ’s are determined using the moment constraints. Refer to [9] for more details on moment problems.

5.2.4

Numerical Study: MaxEnt-Based Worskspace Characterization

We categorize our numerical experiments into two parts: (i) serial arms with only revolute joints, and (ii) serial arms with both revolute and prismatic joints. For manipulators with only revolute joints, we re-visit the equation of the moments of the workspace density in which li ’s are no longer random. Hence, in Eq. (5.19) length parameters are constant (and given) values and integration

125

is performed only on the angular random variables. Moreover, for these manipulators we compare our results with those obtained from Fourier analysis on motion groups proposed in [82] as well as exhaustive Monte Carlo simulations.

5.2.4.1

Serial Arm with Revolute Joints

Choosing an appropriate numerical technique to solve the moment problem, i.e., calculating λr,p in Eq. (5.25) depends on the number of moment constraints as well as the integral dimensions. There are a plethora of studies in the literature that propose efficient computational techniques for solving the moment problem. The most well-known approach is the standard Newton method [96, 97]. Here, we use the iterative method for our bi-variate case as follows. If we have moment constraints upto order R, for a 2-dimensional random vatiate, it results in K =

(2+R)! 2! R!

constraints (moment

equations to be satisfied). Let us denote the set of moment indices by I = {(p, q)| p + q ≤ R} where Ii = (pi , qi ), i = 1, . . . , K. The iterative method to update Λ = [λ1 , . . . , λK ]T is given by −1 n

Λn+1 − Λn = (Gn )

b

(5.27)

n where Gn = gi,j , i, j = 1, . . . , K, and

n gi,j

Z

xmax

Z

ymax

=

x xmin

pi +pj qi +qj

y

exp{−

ymin

K X

λnk xpk y qk }dxdy

(5.28)

k=1

n n − µpK ,qK ]T . Also, in Eq. (5.27), bn = [g1,1 − µp1 ,q1 , . . . , gK,1

˜ is typically more complex in shape When exponential transformation is used, the jpdf of X compared with pX (X). Hence, one may use higher moments in MaxEnt formulation to capture the jpdf with adequate accuracy. Dealing with higher moment constraints in MaxEnt problem, the convergence in optimization problem is facilitated by pre-conditioning [98] the moments through linear transformations of the end-effector position vector such that the new random vector has zero mean and identity covariance matrix. To obtain zero-mean random vector we have ˜s = X ˜ −X ˜ X

(5.29)

˜ and to get the identity covariance matrix, one can write ˜ is the mean (first moment) of X, where X

126

˜ sr = AX ˜s X

(5.30)

in which A = V D1/2 where V and D are the matrix of eigenvectors and eigenvalues of the covariance ˜ s. matrix of X Figure 5.1 shows the results corresponding to a 4-Link serial arm with revolute joints. Length of the links are assumed to be l1 = 0.3, l2 = 0.25, l3 = 0.2 and l4 = 0.15. First column in Fig. 5.1 shows the results corresponding to the transformed workspace (exponential transformation), i.e., ˜ The densities of log-transformation of X, ˜ i.e., X, are shown in the second column. First X. row is the MaxEnt results, second row shows the Monte Carlo simulation using 625e4 points, and third row shows the contour plots of both the MaxEnt and Monte Carlo based densities for better comparison. Effectiveness of the MaxEnt approach can be verified by the results illustrated in Fig. 5.1. Besides the good agreement with Monte Carlo based results, curves shown in Fig. 5.1 (e) and (f) (blue lines) provide improved smoothness compared to the Monte Carlo based densities that are more representative of the actual workspace density. In Fig. 5.1 (f), black lines show the contours corresponding to the workspace density obtained through motion-group Fourier transform approach proposed in [82] (briefly reviewed in Sec. 5.1.1). The results are close to those obtained by the exhaustive Monte Carlo method and provide improved smoothness compared to the estimated density from the data points. Compared with Monte Carlo results, in some regions the Fourier transform technique provides better accuracy than MaxEnt approach. MaxEnt density estimation can be further improved if higher moments are employed (current results are obtained with upto 8th moment constraints) or by eliminating the exponential transformation step and calculating the moments directly from Eq. (5.19) and (5.20). It is also noteworthy that, sampling approach as well as non-parametric density estimation techniques (here we use bi-variate kernel smoothing method) can influence the accuracy and smoothness of the Monte Carlo based densities.

5.2.4.2

Serial Arm with Revolute and Prismatic Joints

Given the moments of the workspace density, the MaxEnt approach can be applied to the serial arms with both revolute and prismatic joints without further difficulties. Figure 5.2 shows the contours of the workspace density of a 2-link serial arm with revolute and prismatic joints.

127

X

MaxEnt solution

˜ X

(b)

(c)

(d)

(e)

(f)

Contour comparison

MC solution

(a)

Figure 5.1: Workspace density approximation: a 4-link serial arm with revolute joints. Link lengths are l1 = 0.3 m, l2 = 0.25 m, l3 = 0.2 m, l4 = 0.15 m. Manipulator can reach its base. 128

˜ X

(b)

(c)

(d)

(e)

(f)

X

(a)

Figure 5.2: Workspace density approximation: a 2-link serial arm with revolute and prismatic joints. Link length limits are l1l = 0.15 m, l1u = 0.5 m, l2l = 0.05 m, l2u = 0.5 m. First row of Fig. 5.2 ((a), (b) and (c)) shows the density of exponential transformation of the ˜ The workspace density after inverse mapping is shown in the second row workspace, i.e., pX˜ (X). of Fig. 5.2 ((d), (e) and (f)). First column shows the results when only upto 3rd moments (10 constraints) are used in the MaxEnt formulation. Second ((b) and (e)) and third ((c) and (f)) columns show the results corresponding to upto 5th (21 constraints) and 6th moments (28 constraints) in the MaxEnt formulation. One can skip the transformation step and directly approximate the workspace density with lower moments (e.g., upto 4th moments) at the cost of computing integrals in Eqs. (5.19) and (5.20). Above examples are targeting manipulators whose link lengths are such that end-effector can reach manipulator base. This is commonly the case when multiple links are used in a (hyper)redundant arm. This results in a unimodal workspace density whose exponential transformation is not significantly twisted. However, depending on the design parameters, e.g., link lengths in revolute

129

X

MaxEnt solution

˜ X

(b)

(c)

(d)

(e)

(f)

Contour comparison

MC solution

(a)

Figure 5.3: Workspace density approximation: a 2-link serial arm with revolute and prismatic joints. Link length limits are l1l = 0.24 m, l1u = 0.4 m, l2l = 0.08 m, l2u = 0.16 m. Link length limits are chosen such that manipulator can not reach its base. 130

joint arms or link length limits in revolute-prismatic joint arms, manipulator may not be able to reach its base. This results in zero-density regions around the base and further complexity will be introduced when exponential transformation is performed for such manipulators. Figure 5.3 shows the results corresponding to an example of a 2-link arm with both revolute and prismatic joints. Link length limits are chosen as l1l = 0.24 m, l1u = 0.4 m, l2l = 0.08 m, l2u = 0.16 m. Referring to Figs. 5.3 (c) and (d), it can be seen that workspace density takes a relatively complex shape after exponential transformation that leads to further computation effort in MaxEnt approximation. Figures 5.3 (e) and (f) illustrate the contour comparison of the Monte Carlo and MaxEnt approximation results. While accuracy of the approximation is relatively decreased compared to the results corresponding to above examples, shape of the workspace density is significantly retrieved by inverse mapping as shown in the Figs. 5.3 (a) and (b). Similarly, one can obtain improved approximation accuracy by direct computation of the moment integrals, however, the computational cost increases with both increasing the number of links (which is the case for (hyper-)redundant systems) and calculating higher moments.

5.3

Force Capacity in Parallel Architectures

Output wrench capacity of a multi-agent system (that is, roughly speaking, the range of loads that can be provided by the entire system) is a crucial design criteria in order to avoid changing system architecture to compensate new external loads. Each agent has its individual load capacity that is determined by the minimum and maximum force (magnitude) it can provide as well as its range of motion that determines the range of orientations it can apply the forces. Although agents individual load capacity is independent of the configuration of agents, however, the wrench capacity of the entire system is highly dependent on the kinematic lay out of the system or, for example in a payload transport system, the distribution of agents around the payload. With new class of mobile-base cable robots, such quantification of system wrench capacity allows re-configuration of the system in order to compensate a wider range of external loads depending on the specific task. In section 5.2, we analyzed the workspace of a serial chain through probabilistic formulation of the forward kinematics relations. In this section, we follow similar approach for the static framework that is applicable to several parallel architectures especially loosely-interconnected cable-based systems. We are aiming to characterize the output wrench capacity of entire system when the infor-

131

mation on individual agents are available. In contrast with the workspace analysis, we are primarily interested in quantifying the statistical moments of the output wrench rather than the density function. These moments are used in Sec. 5.3.3 for base configuration optimization in order to improve the output wrench capacity of the entire system for a particular demand (desired task).

5.3.1

Force Capacity Quantification

Figure 5.4 shows a sketch of the static force diagram for a multi-agent cooperative system with m agents. The resultant force applied to the platform is the vector sum of the forces applied by each individual agent. So, Fp =

m X

fi

(5.31)

i=1

in which Fp = [Fx , Fy ]T , fi = [fxi , fy i ]T , fxi = Ti cos(φi ) and fy i = Ti sin(φi ) where Ti is the magnitude of the force applied by ith agent along the unit vector ~u = cos(φi )ˆ ex + sin(φi )ˆ ey . Following the idea of previous section and assuming that each agent can provide a range of magnitudes in a range of directions, their individual force capacity can be modeled through probability density functions. Subsequently, the pdf of the resultant force on the platform, i.e., pFp (Fp ) = pFx ,Fy (Fx , Fy ), provides useful information on the force capacity of the multi-agent system. Following our physical perception from the multi-agent system, one almost valid assumption is to consider that each agent performs independently from the others that, in turn, implies the independence of the random vectors fi ’s. In addition, the direction of the applied force by each agent may be assumed independent of the magnitude of the force applied to the platform. This will lead to the problem of sum of independent random vectors that are not, in general, identically distributed. The main difference of the static problem with the kinematic formulation is that each agent can only apply the force in a limited range of directions as shown in Fig. 5.4. As an example, φl1 and φu1 are the lower and upper bounds on the feasible working orientation of the agent 1. These limitations are inevitable in physical systems to avoid the collision and cables interference. To model the force capacity of the ith agent, we consider Ti ∼ U (Til , Tiu ) with pdf pTi (Ti ) =
1 Ti ∈ [Til , Tiu ] where Til and Tiu are the minimum and maximum possible forces that ith T u −T i i

i

agent can provide. Since, we are mainly interested in cable-based systems with unilateral force constraints, it is assumed that Til > 0 (∀i = 1, . . . , m). Similarly, we consider φi ∼ Uc (φli , φui ). Due to imposing the limits on the angular random variables (φi ’s), further analytical developments of

132

Figure 5.4: A schematic of the force diagram of a cooperative system with m agents. the output force moments is made feasible through approximating the pφi (φi ) = 1/(φui − φli ) by the von Mises kernels as

pφi (φi ) ≈ p˜φi (φi ) =

M X

ci,k g(φi ; µi,k , σi,k )

(5.32)

k=1

where φi ∈ [0, 2π] and

g(φi ; µi,k , σi,k ) =

1 exp{σi,k cos(φi − µi,k )} 2πI0 (σi,k )

In Eq. (5.32), ci,k , µi,k and σi,k are fitting parameters that depend on φli and φui . In addition, PM ci,k ’s satisfy k=1 ci,k = 1 (ci,k ≥ 0). von Mises Kernel estimation to the orientational density functions has been studied in the past [5, 99, 100, 101]. However, the systematic approaches proposed in the literature mainly tackle the unimodal density functions on the circle that represent observational data. Here, while number of terms in Eq. (5.32), i.e., M , is considered to be given, we estimate Ξi = [ci,1 , . . . , ci,M , µi,1 , . . . , µi,M , σi,1 , . . . , σi,M ]T through ˆ i = argmin Ξ Ξi

Z

φu i

φli

h p˜φi (φi ) −

φui

i2 1 dφi l − φi

(5.33)

ˆ i is performed only one time for each agent and the results It is noteworthy that calculating Ξ can be used in the optimization algorithm for real-time applications. Hence, the computational cost involved with Eq. (5.33) does not affect the efficiency of the real-time configuration optimization.

133

Similarly, due to the independency of the random variables, jmgf can be written as

MFx ,Fy (η, ζ) = EFx ,Fy [exp{(ηFx + ζFy )}] =

m Y

MTi ,φi (η, ζ)

(5.34)

i=1

Substituting p˜φi (φi ), we get

MTi ,φi (η, ζ) =

Z Tiu m X ci,k 1 I0 (zi,k (Ti ))dT Tiu − Til k=1 I0 (σi,k ) Til

(5.35)

where

2 zi,k (Ti ) = Ti2 (η 2 + ζ 2 ) + σi,k + 2Ti σi,k (η cos(µi,k ) + ζ sin(µi,k ))


1/2

Similar to Eq. (4.9) and due to the independency of agents, we have

KFx ,Fy (η, ζ) =

m X

log MTi ,φi (η, ζ) =

i=1

m X

KTi ,φi (η, ζ)

(5.36)

i=1

From jcgf in Eq. (5.36), we obtain the first cumulants of the output force (mean of output force) as

κ1 =

m X M X Ci,k I1 (σi,k )(T u + T l ) i

i=1 k=1

κ2 =

i

2I0 (σi,k )

m X M X Ci,k I1 (σi,k )(T u + T l ) i

i=1 k=1

2I0 (σi,k )

134

i

cos(µi,k )

(5.37)

sin(µi,k )

(5.38)

and second cumulants (variances) as #
" m X M X Ci,k (Tiu )3 − (Til )3 sin2 (µi,k ) 2
g(σi,k ) cos (µi,k ) + I1 (σi,k ) − κ = σi,k 3I0 (σi,k ) Tiu − Til i=1 k=1 2 m X M X Ci,k I1 (σi,k )(Tiu + Til ) cos(µi,k ) 2I0 (σi,k ) i=1 k=1 #
" m X M u 3 l 3 2 X C (T ) − (T ) cos (µ ) i,k i,k i i
g(σi,k ) sin2 (µi,k ) + I1 (σi,k ) κ2,2 = − u l σi,k 3I (σ ) T − T 0 i,k i i i=1 k=1 2 m X M X Ci,k I1 (σi,k )(Tiu + Til ) sin(µi,k ) 2I0 (σi,k ) i=1 1,1

(5.39)

(5.40)

k=1

where

5.3.2

g(p) = I0 (p) −

I1 (p) p

Simultaneous Force Capacity and Uncertainty Quantification

In previous section we addressed a deterministic problem (wrench capacity quantification) through a probabilistic formulation. This sets the ground to incorporate the effect of uncertainties to the formulation. In multi-agent systems, each agent introduces uncertainty to the system individually and the accumulation of these uncertainties can be realized at the platform. In previous chapter, we addressed this problem in detail when the system configuration is given. When the wrench capacity and uncertainty are simultaneously considered in the design procedure, one may seek a measure of system overall uncertainty in the following sense: Z U =k

˜ |W )dW k Cov(W

(5.41)

DW

˜ |W ) captures the level of wrench uncertainty when the output wrench is W . In where Cov(W fact, a multi-agent system with a given lay out has a range of output wrench described by DW . ˜ |W . Then, corresponding to each output wrench, there exists a fluctuation that is described by W Information about the total uncertainty corresponding to a specific lay out (quantified in the sense of Eq. (5.41)) can be used to investigate the precision in delivering the required wrench. Here, we incorporate the effect of orientational uncertainty. This is performed by adding a ˜i with von Mises distribution to the uniformly perturbing zero-mean random variable (angle) φ

135

distributed φi ’s. Note that uncertainty of the force/traction magnitude can also be added without much difficulties. We perturb the force vector relation (5.31) as     m ˜i ) X F˜x Ti cos(φi + φ  F˜p =   = f˜i =  ˜i ) ˜ Fy Ti sin(φi + φ i=1

(5.42)

˜i ’s are the angular perturbation terms. Due to the independence of φ ˜i ’s, we can similarly where φ write ˜ y }] MF˜x ,F˜y (η, ζ) = EF˜x ,F˜y [exp{η F˜x + ζ F Y m n o ˜ ˜ = ET1 ,...,Tm ,φ1 ,...,φm ,φ˜1 ,...,φ˜m exp ηTi cos(φi + φi ) + ζTi sin(φi + φi ) i=1

=

m Y

   m n o Y ETi Eφi Eφ˜i exp ηTi cos(φi + φ˜i ) + ζTi sin(φi + φ˜i ) = Mf˜xi ,f˜yi (η, ζ)

i=1

(5.43)

i=1

Using p˜φi (φi )’s given by Eq. (5.32), and pTi (Ti ) =

1 Tiu −Til


Ti ∈ [Til , Tiu ] along with the density

of the perturbing angle given by pφ˜i (φ˜i ) =

1 ˜i ∈ [0, 2π] exp{˜ σi cos(φ˜i )}, φ 2πI0 (˜ σi )

in jmgf equation and simplifying the exponential terms, we get

Mf˜xi ,f˜yi (η, ζ) =

m X k=1

ci,k 2πI0 (σi,k )I0 (˜ σi )

Z

Tiu

Til

Z 0

2π

q exp{˜ σi cos(φ˜i )}I0 ( a ˜2i,k + ˜b2i,k )dφ˜i dTi

(5.44)

where    a ˜

i,k

= ηTi cos φ˜i + ζTi sin φ˜i + σi,k sin(µi,k )

  ˜bi,k = ζTi cos φ˜i − ηTi sin φ˜i + σi,k cos(µi,k ) Following similar procedure as in chapter 4 and after some algebra, the first and second cumulants

136

can be obtained as

κ ˜1 =

m X M X Ci,k I1 (σi,k )I1 (˜ σi )(T u + T l ) i

i=1 k=1

i

2I0 (σi,k )I0 (˜ σi )

cos(µi,k )

(5.45)

m X M X Ci,k I1 (σi,k )I1 (˜ σi )(T u + T l )

i i sin(µi,k ) (5.46) 2I (σ )I (˜ σ ) 0 i,k 0 i i=1 k=1
" m X M  X Ci,k (Tiu )3 − (Til )3 I1 (σi,k )  1,1
κ ˜ = 2I (˜ σ ) − g(˜ σ ) − h(µ , σ ˜ ) g(σ )h (µ , σ ˜ ) + 0 i i i,k i i,k 1 i,k i σi,k 3I0 (σi,k )I0 (˜ σi ) Tiu − Til i=1 k=1 2 m X M X Ci,k I1 (σi,k )I1 (˜ σi )(Tiu + Til ) − cos(µi,k ) (5.47) 2I0 (σi,k )I0 (˜ σi ) i=1 k=1
" m X M  X Ci,k (Tiu )3 − (Til )3 I1 (σi,k )  2,2
g(σ )h (µ , σ ˜ ) + κ ˜ = I (˜ σ ) + g(˜ σ ) − h (µ , σ ˜ ) 0 i i 2 i,k i i,k 2 i,k i σi,k 3I0 (σi,k )I0 (˜ σi ) Tiu − Til i=1 k=1   M m 2 X X Ci,k I1 (σi,k )I1 (˜ σi )(Tiu + Til ) − sin(µi,k ) (5.48) 2I0 (σi,k )I0 (˜ σi ) i=1 k=1   where h1 (p, q) = g(q) cos2 (p) + I0 (q) − g(q) sin2 (p)   h2 (p, q) = g(q) sin2 (p) + I0 (q) − g(q) cos2 (p)

κ ˜2 =

5.3.3

Numerical Analysis: Configuration Optimization

In this section, we present numerical experiments to study the proposed approach for the wrench capacity analysis. First, we show the result of von Mises kernel approximation to the bounded uniform distributions on the circle. Second, we validate the closed-form expressions of the moments given in Sec. 5.3.1 and 5.3.2 using Monte Carlo sampling approach. Third, we present and numerically investigate the optimization framework to increase the range of possible output loads in a certain direction. Finally, we propose and investigate a measure on simultaneous characterization of the load capacity and uncertainty by which we study the effect of system redundancy on these factors.

5.3.3.1

von Mises Kernel Approximation

In our numerical experiments in the following subsections, we consider the range of angular motion to be either 60◦ or 120◦ . Solution to (5.33) depends only on the number of kernels and range ˆ of angular motion rather than the absolute lower and upper bounds on the motion. We calculate Ξ

137

with M = 8 kernels for both 60◦ and 120◦ ranges of motion. The resultant approximating densities along with the true densities are plotted in Fig. 5.5.

(a)

(b)

Figure 5.5: von Mises kernel approximation to bounded uniform distributions on the circle: (a) φu − φl = 60◦ and (b) φu − φl = 120◦ . Solid blue line is the true density, solid black line is the approximated density and dashed lines are 8 von Mises kernels.

5.3.3.2

Moments Validation

Table (5.1) compares the mean and variances given by Eqs. (5.37)-(5.40) with the corresponding values given by exhaustive Monte Carlo simulations for systems with m = 1, m = 2 and m = 3 agents. It can be seen that for all three systems, the results provided by developed closed-form expressions are closely following the values obtained by Monte Carlo simulations. The agent(s) force capacity and configuration of these three systems are described in (5.49). The units of the force vector magnitudes and directions are N and rad, respectively. Similarly, in Tab. (5.2), mean and variances of the resultant force vector, given by Eqs. (5.45)(5.48), that capture both resultant force capacity and uncertainty, are validated by Monte Carlo simulations. The working configuration of the robots and force capacity of individual agents are identical to those described in (5.49). Acceptable accuracy of the closed-form expressions can be

138

concluded from the results summarized in Tab. 5.2.

(m = 1) :

    ◦ φl1 = 15◦ , φu 1 = 75    T1l = 20, T1u = 25

(m = 2) :

   ◦  φl1 = 15◦ , φu  1 = 75        l  ◦ φ2 = 150◦ , φu 2 = 210    l u   T1 = 20, T1 = 25       T l = 20, T u = 25 2 2

(m = 3) :

    ◦  φl1 = 93◦ , φu  1 = 123        l ◦ u ◦   φ2 = 165 , φ2 = 195        ◦ φl3 = 225◦ , φu 3 = 255

(5.49)

   l u   T1 = 20, T1 = 22        T2l = 20, T2u = 22          T3l = 20, T3u = 22

Table 5.1: Mean and standard deviations of the resultant force vector (without uncertainty): Monte Carlo simulation and closed-form solution. m=1 m=2 m=3 15.164 -6.279 -37.512 Eq. (5.37) κ1 = E[Fx ] 15.192 -6.289 -37.535 MC 15.161 15.163 1.775 Eq. (5.38) κ2 = E[Fy ] 15.192 15.183 1.764 MC √ p 4.925 5.203 4.315 Eq. (5.39) κ1,1 = E[(Fx − E[Fx ])2 ] 4.834 5.141 4.223 MC √ p 4.926 8.365 3.919 Eq. (5.40) κ2,2 = E[(Fy − E[Fy ])2 ] 4.834 8.258 3.837 MC

Table 5.2: Mean and standard deviations of the resultant force vector (with uncertainty; σ ˜i = 50 ∀i): Monte Carlo simulation and closed-form solution. m=1 m=2 m=3 15.012 -6.216 -37.135 Eq. (5.45) κ ˜ 1 = E[F˜x ] 15.042 -6.222 -37.144 MC 15.001 15.011 1.757 Eq. (5.46) κ ˜ 2 = E[F˜y ] 15.031 15.021 1.746 MC q √ 5.378 5.721 5.731 Eq. (5.47) κ ˜ 1,1 = E[(F˜x − E[F˜x ])2 ] 5.293 5.683 5.702 MC q √ 5.65 9.431 5.598 Eq. (5.48) κ ˜ 2,2 = E[(F˜y − E[F˜y ])2 ] 5.304 9.052 5.181 MC

139

5.3.3.3

Configuration Optimization: Wrench Capacity

Max range of Fy

Agents range of motion: 120◦

Agents range of motion: 60◦

Max range of Fx

Max range of Fx and Fy (simultaneous maximization)

Figure 5.6: Optimal base configurations in order to maximize the range of output load in specific directions. Using Eqs. (5.37)-(5.40) one can efficiently find the optimal working configuration of a multiagent system that maximizes the resultant force capacity and provides the mean resultant force close

140

to a predefined (preferred) force vector. We have

M aximize J = w1 Var(Fx ) + w2 Var(Fy ) α1 ,...,αm

= w1 κ1,1 + w2 κ2,2

(5.50)

s.t. kE[F ] − F d k= 0

where F d is the desired force vector and w1 and w2 (w1 + w2 = 1) are positive scalar weights. A physical interpretation to optimization problem (5.50) is that the multi-agent system is primarily required to compensate an external load −F d , however, by optimizing the configuration, a larger variation of the external loads (around −F d ) can be compensated by the system. It is noteworthy that number of optimization variables in (5.50) is equal to the number of agents. By shifting the initial mean angles µi,k ∀k by variable αi (i.e., µnew + αi ∀k), we find the optimal αi ’s that i,k = µi,k

minimizes the objective function given in Eq. (5.50). This, along with availability of the closed-form expressions of the moments, leads to an efficient optimization procedure that can be implemented in real-time applications. Optimal working configurations corresponding to F d = 15 × [cos(π/4), sin(π/4)]T are shown in Fig. 5.6 for a system with m = 3 agents. Agents are homogeneous, i.e., have the same range of force magnitudes and directions. We have Til = 10 and Tiu = 15 ∀i for all cases shown in Fig. (5.6). In the first row of Fig. (5.6), range of orientation is set to φui − φli = π/3 ∀i and in the second row φui − φli = 2π/3 ∀i. Small arrows represent the mean direction (φi ) of each agent force vector. Then, ith agent covers φi ± (φui − φli )/2. First and second columns correspond to maximization of the range of output force along x (w1 = 1 and w2 = 0) and y (w1 = 0 and w2 = 1) directions, respectively. Third column compromises between both x and y directions (w1 = 0.5 and w2 = 0.5). In addition to configuration optimization, our development allows the designer to quantitatively analyze the effect of increasing the number of cooperating agents on the performance of the system, that is the output wrench capacity here. Figure 5.7 shows the change in coefficients of variations √ √ (Cv x = κ1,1 /κ1 and Cv y = κ2,2 /κ2 ) respect to the number of agents used in the multi-agent system (m = 1, . . . , 50). As shown in this figure, both Cv x and Cv y generally increase by adding more agents to the system. Moreover, Cv x + Cv y is almost monotonically increasing by m. The rate of improvement is faster for lower number of the agents and decreases for higher m’s. Also,

141

comparing the results in Fig. 5.7 (a) and (b), it can be seen that a faster rate of improvement is realized when agents range of motion is increased from 60◦ to 120◦ . Note that these coefficients of variations are corresponding to the configurations obtained through solving optimization problem given by Eq. (5.50), some of which are illustrated in Fig. 5.6.

(a)

(b)

Figure 5.7: Coefficient of variation as a measure of wrench capacity: change versus increasing number of agents when range of agents orientation is (a) 60◦ and (b) 120◦ .

5.3.3.4

Simultaneous Characterization

Using the simultaneous quantification of the resultant force capacity and uncertainty given by Eqs. (5.45)-(5.48), one can quantify the overall uncertainty corresponding to a specific lay out of the system as κerr



1,1 2,2 T

κ κ ˜ ] − [κ1,1 κ2,2 ]T

[˜

=

1,1 2,2 T

[κ κ ]

(5.51)

Considering U as a rigorous quantification of the uncertainty associated with a specific system lay out, κerr in Eq. (5.51) is a coarse but efficient estimation to uncertainty level that can be calculated using expressions given in Eqs. (5.45)-(5.48). The argument for introducing κerr as in Eq. (5.51) is as following. One can represent the pdf of wrench capacity through a set of its statistical moments up to the order, say r. This (partially) describes the fully deterministic system. Now, if we add the uncertainty to the wrench capacity formulation, the random output wrench takes a different distribution that can also be represented with a set of its statistical moments (up to rth

142

Figure 5.8: Variation of the uncertainty level and force capacity for the systems with different number of agents. moment). We argue that if one specific architecture minimizes the effect of uncertainty, it results in a less different wrench probability distribution once the uncertainty is added to wrench capacity formulation. That difference can be quantified by the distance of two statistical moment vectors. Eq. (5.51) measures the normalized distance for the second central moments (variances). Our first numerical experiment using Eq. (5.51) involves studying the effect of redundancy on the uncertainty level in multi-agent systems. We increase the number of individual robots and re-apply the optimization problem described by (5.50) to find the lay out by which the wrench capacity is maximized and the prescribed mean constraint is satisfied. Then, we calculate κerr given the uncertainty level for each individual agent, i.e., σ ˜i ’s. We assume two approaches in deciding on σ ˜i ’s: first we increase number of agents and σ ˜i = σ ˜j ∀i, j = 1, . . . , m; second, we assume σ ˜i < σ ˜j when i < j. In fact, in the second approach the new agent added to the system is more accurate than all other agents already in use. Note than when using von Mises distribution, higher σ ˜i implies less uncertain distribution. As shown in Fig. 5.7, adding agents to the system improves the load capacity of the entire system. Intuitively, by constantly adding more accurate agents to the system we also expect to improve the entire system performance in terms of output wrench uncertainty. However, when the agents provide similar accuracy, one may not achieve lesser uncertainty in the output wrench. The results shown in Fig. 5.8 verifies these facts.

143

In Fig. 5.8, the output load capacity (Cv x + Cv y ) (x axis) and κerr (z axis) are plotted versus number of agents in the system (y axis). It can be seen that κerr does not show much changes against increasing the number of agents when all agents have identical uncertainty parameter (black line in Fig. 5.8). However, the output uncertainty level (almost) constantly decreases by adding agents with higher precision (blue line in Fig. 5.8). This type of quantification facilitates systematic trade-offs on number and quality of the agents in multi-agent system designs. Our further numerical investigation involves the sensitivity of κerr to the working lay out of the system. In order to visualize the sensitivity of the uncertainty level, we consider an example of three-agent system. Figure 5.9 illustrates variation of different uncertainty measures versus the system lay out defined by the mean orientation of agents, i.e., φ1 , φ2 and φ3 . We use three axes for φ1 , φ2 and φ3 and show the uncertainty level using the colorbar. Using Fig. 5.9 one may characterize the configurations near the maximum-capacity configuration that result in lower level of uncertainty in the desired direction. First row of Fig. 5.9 depicts κ1,1 − κ1,1 | which quantifies the uncertainty in x direction. It can be seen that maximum κxerr = |˜ (unwanted) uncertainty level along x axis is when all three agents are aligned along y axis. The 2-dimensional plot of such cases is shown in Fig. 5.9 (b) for better visualization. However, with two agents aligned along y axis, one can reduce the uncertainty level by re-configuring the third agent as shown in Fig. 5.9 (a) when φ3 is π and 2π. Second row of Fig. 5.9 shows the results corresponding to κ2,2 − κ2,2 |. Similarly, maximum uncertainty along y axis happens when all three agents are κyerr = |˜ aligned along x axis, and it can be reduced by changing at least one agent’s orientation. Third row shows the results for κxy κ1,1 κ ˜ 2,2 ] − [κ1,1 κ1,1 ]k. There are multiple peaks of the κxy err = k[˜ err in the configuration space that one may avoid by slightly shifting the (load capacity-)optimal configurations to the nearest point where the uncertainty level is adequately low.

5.4

Conclusion

Probabilistic frameworks relying on the theories from directional statistics were developed by which both the deterministic and probabilistic design problems were tackled. In kinematic arrangement, we addressed the workspace density characterization in redundant serial arms with both revolute and prismatic joints. In probabilistic forward kinematics relations, we formulated the workspace problem as sum of random vectors. The closed-form expressions of the moments were

144

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.9: Variation of the uncertainty versus the system lay out in a 3-agent system. First and second rows correspond to the wrench uncertainty in x and y directions, respectively. Third row shows the variation of the uncertainty in both x and y directions. 145

used in a maximum entropy formulation in order to approximate the density of the end-effector position in Cartesian coordinates. Applying this approach to the static problem, we were able to quantify the overall load capacity of a multi-agent system, with a focus on the cable-based parallel architectures. It was shown that one can efficiently optimize the working architecture (lay out) of the system in order to maximize the possible range of output loads in a particular direction. We also developed an approach by which both the load capacity and precision of the system in delivering those loads can be simultaneously studied.

146

Chapter 6 Conclusion Categorized by chapters, we summarize the study that was performed in this dissertation and point to the directions of future work.

Chapter 3: Using random matrix theory, we developed and applied uncertainty characterization approaches that are applicable to complex robotic manipulators. Physical intuition to the performance of the robotic manipulators was exploited in development of these models that enabled us to properly capture the stochastic behavior of such complex systems. We showed that such RMT-based models require less information compared with their parametric counterparts and yet capture the uncertainty appropriately. Furthermore, due to the system level modeling based on random matrix theories, (in several circumstances) they become superior in capturing the real system behavior. We focused our developments on kinematic and static arrangements, however, further extensions to the dynamic formulation can be the subject of future research. Further experimental investigations can lead to draw more general conclusions on the probability distribution of the system matrices, particularly inertial mass and Jacobian matrices. Ultimately, by having a prior knowledge of the matrix-variate distributions, one can use such RMT-based models in designing robust control schemes for different robotic manipulator systems.

Chapter 4: Relying on the theories in directional statistics, we formulated the wrench uncertainty problem in a multi-agent system as a problem of sum of random vectors. By taking advantage of the redundancy in the system and using our developed probabilistic framework, we proposed op-

147

timization procedures (to be performed off-line or in real-time) in order to reduce, or in some sense manipulate, the uncertainty in the output wrench of such systems. We developed the theory for both planar and spatial systems, and performed experimental study for the static planar systems. Further investigation of such methodology on spatial system through extensive experimental analyses is a promising line of research. This will benefit several emerging cooperative systems which consist of multiple low-cost agents that introduce uncertainty to the overall response of such systems.

Chapter 5: We addressed deterministic problems (in kinematic and static arrangements) by probabilistic formulations. In our kinematic analysis, we coupled the random vector analyses with maximum entropy method to approximate the workspace density of redundant serial arms with both revolute and prismatic joints. In the static setting, we developed a framework to quantify the load capacity of the (cable-based) parallel architectures as a function of the system lay out. Additionally, we studied the simultaneous characterization of the output load (unwanted) uncertainty and capacity. These developments enabled systematic optimization and design trade-offs in order to find system lay outs that provide maximum range of output loads in particular directions with acceptable precision. As a future study, the workspace density analyses can be extended to spatial manipulators and particularly re-configurable systems. Efficient characterization of the workspace density is important in real-time reconfiguration planning of many evolving (highly) reconfigurable systems. Similarly, the output wrench capacity analyses, carried out for planar systems in this dissertation, can be extended to spatial cable-based architectures.

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Appendices

158

Appendix A Derivation of Eq. (4.8) Substituting densities given in (2.2) and (2.15) into (4.7) gives

Mfx i ,fy i (η, ζ) =

Z

1

√

 exp ηTi cos(θi ) + ζTi sin(θi ) + σθi cos(θi − θi )

Z

2π 2πI0 (σθ )σTi DTi  (Ti − T i )2 − dθi dTi 2σT2 i

Dθi

(1)

By writing cos(θi − θi ) = cos(θi ) cos(θi ) + sin(θi ) sin(θi ), and given Ti ∈ [−∞, +∞] and θi ∈ [0, 2π] ∀i = 1, . . . , m, we re-write Eq. (1) as

Mfx i ,fy i (η, ζ) =

1

√ 2π 2πI0 (σθ )σTi

∞

 Z 2π   (Ti − T i )2 exp − exp{ai cos(θi ) + bi sin(θi )}dθi dTi 2σT2 i −∞ 0

Z

(2) where ai = [ηTi + σθi cos(θi )] and bi = [ζTi + σθi sin(θi )]. Second integral in Eq. (2) can be written Z 2π Z 2π q q as exp{ai cos(θi ) + bi sin(θi )}dθi = exp{ a2i + a2i sin(θi + βi )}dθi = 2πI0 ( a2i + a2i ) = 0 0 p p 2πI0 (z(Ti )) where sin(βi ) = ai / a2i + b2i and cos(βi ) = bi / a2i + b2i . Substituting this in Eq. (2) gives the expression in Eq. (4.8).

Derivation of Eq. (4.11)-(4.14) For notational simplicity we denote Mfx i ,fy i simply by Mi and Kfx i ,fy i by Ki . Using Eq. (4.9) and (4.10), and given Mi |(0,0,) = 1, we can write m

cF 1 =

m

X ∂Ki (η, ζ) X ∂Mi (η, ζ) ∂KFx ,Fy (η, ζ) |(0,0) = |(0,0) = |(0,0) ∂η ∂η ∂η i=1 i=1

(3)

Now, from Eq. (4.8), we get

cF 1

=

m X ∂Mi (η, ζ) i=1

∂η

|(0,0) =

We know zi = z(Ti ) =

m X i=1

p

√

1 2πσTi I0 (σθi )

Z

+∞

−∞

  (Ti − T i )2 ∂I0 (zi ) ∂zi exp − |(0,0) dTi 2σT2 i ∂zi ∂η

(4)

a2i + b2i where ai = ηTi + σθi cos(θi ) and bi = ζTi + σθi sin(θi ). Then,

159

first derivative of zi respect to η is ∂zi 1 = (η 2 Ti2 + 2ηTi σθi cos(θi ) + σθ2i + ζ 2 Ti2 + 2ζTi σθi sin(θi ))−1/2 (2Ti2 η + 2Ti σθi cos(θi )) ∂η 2 and its value at the origin (η, ζ) = (0, 0) is second variable ζ, we get

∂zi ∂ζ |(0,0) =

∂zi ∂η |(0,0) =

Ti cos(θi ). Following a similar procedure for

Ti sin(θi ). In Eq. (4), the derivative of the Bessel function is

again a Bessel function with higher order, i.e.,

∂I0 (zi ) ∂zi

= I1 (zi ). Substituting into Eq. (4) gives

" #   Z +∞ m m 2 X X (T − T ) 1 ) I (σ ) cos(θ I1 (σθi ) i 1 θ i i i F √ Ti exp − dTi = c1 = T cos(θi ) 2 I (σ ) 2σ I (σ ) i 2πσTi −∞ 0 θi Ti i=1 i=1 0 θi

(5)

That is the expression given in Eq. (4.11). Note that, in Eq. (5), the term inside the bracket is simply the mean of random variable Ti . Replacing η by ζ and following the same procedure, the expression given by Eq. (4.12) will be derived. In order to derive the variance terms given in Eqs. (4.13) and (4.14), one needs to obtain second derivatives of the jmgf as

Var(Fx ) =

"  2 # m X ∂ 2 KFx ,Fy (η, ζ) ∂Mi ∂ 2 Mi = |(0,0) = − ∂η ∂η 2 ∂ 2 η2 i=1

cF 1,1

(6) (0,0)

Last equality in Eq. (6) is resulted by taking into account the fact that jmgf is 1 at the origin. The first term in the bracket is m

X 1 ∂ 2 Mi √ 2 2 = ∂ η 2πσTi I0 (σθi ) i=1

Z

+∞

−∞

(Ti − T i )2 exp − 2σT2 i 

"

∂ 2 I0 (zi ) ∂zi2



∂zi ∂η

2

∂I0 (zi ) ∂ 2 zi + ∂zi ∂η 2

# dTi (0,0)

(7)

Given Iν (z) =

( 12 z)ν

1 π2

Γ(ν+ 12 )

I (z ) I0 (zi ) − 1zi i . Hence,

Rπ 0

ezcos(θ) sin(θ)2ν dθ, then we can derive

∂ 2 I0 (zi ) ∂zi2

= I0 (zi )−

√ 2 πΓ(3/2)I1 (zi ) = zi π

I (σ ) I0 (σθi ) − 1σθ θi . i p 2 Given the closed-form expression of zi = ai + b2i , one can simply verify that ∂ 2 I0 (zi ) |(0,0) = ∂zi2

∂ 2 zi sin2 (θi ) |(0,0) = Ti2 2 ∂η σθi

160

Now, substituting all the expressions into Eq. (7), it can be simplified as   Z +∞ m X ∂ 2 Mi (Ti − T i )2 1 √ = exp − 2σT2 i ∂ 2 η2 2πσTi I0 (σθi ) −∞ i=1 " #  2 I1 (σθi ) 2 sin (θ i ) 2 2 I0 (σθi ) − Ti cos (θi ) + I1 (σθi )Ti dTi σθ i σθ i

(8)

that results in   Z m X (Ti − T i )2 I0 (σθi ) cos2 (θi ) − (I1 (σθi )/σθi ) cos(2θi ) +∞ 2 ∂ 2 Mi √ dTi = T exp − i 2σT2 i ∂ 2 η2 2πσTi I0 (σθi ) −∞ i=1 m   X
I1 (σθi ) = cos2 (θi ) − cos(2θi ) T 2i + σT2 i I (σ )σ 0 θi θi i=1

(9)

Finally, "  2 # m X ∂ 2 Mi ∂Mi Var(Fx ) = − ∂η ∂ 2 η2 i=1 −

I12 (σθi ) 2 Ti I02 (σθi )

2

cos (θi ) =

=

(0,0) m X

T 2i

i=1

m  X

cos2 (θi ) −

i=1

+

σT2 i





I1 (σθi ) cos(2θi ) T 2i + σT2 i I0 (σθi )σθi


I1 (σθi ) T 2i + σT2 i I12 (σθi ) 2  2 T i cos (θi ) − − 2 cos(2θi ) I0 (σθi )σθi I0 (σθi ) (10)

Right hand side of Eq. (10) is the expression given by Eq. (4.13). Eq. (4.14) can be similarly derived when η is replaced by ζ.

161