Received: 20 March 2016
Revised: 25 July 2016
Accepted: 16 August 2016
DOI 10.1002/rcs.1774
ORIGINAL ARTICLE
Data‐driven methods towards learning the highly nonlinear inverse kinematics of tendon‐driven surgical manipulators Wenjun Xu1
|
Jie Chen2
Henry Y.K. Lau2
|
Hongliang Ren1*
|
1
Department of Biomedical Engineering, National University of Singapore, Singapore
Abstract
2
Background
Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Hong Kong SAR
Accurate motion control of flexible surgical manipulators is crucial in tissue
manipulation tasks. The tendon‐driven serpentine manipulator (TSM) is one of the most widely
Correspondence Hongliang Ren, Department of Biomedical Engineering, National University of Singapore, Singapore. Email:
[email protected]
adopted flexible mechanisms in minimally invasive surgery because of its enhanced maneuverability in torturous environments. TSM, however, exhibits high nonlinearities and conventional analytical kinematics model is insufficient to achieve high accuracy.
Methods
To account for the system nonlinearities, we applied a data driven approach to
encode the system inverse kinematics. Three regression methods: extreme learning machine (ELM), Gaussian mixture regression (GMR) and K‐nearest neighbors regression (KNNR) were implemented to learn a nonlinear mapping from the robot 3D position states to the control inputs.
Results
The performance of the three algorithms was evaluated both in simulation and phys-
ical trajectory tracking experiments. KNNR performed the best in the tracking experiments, with the lowest RMSE of 2.1275 mm.
Conclusions
The proposed inverse kinematics learning methods provide an alternative and
efficient way to accurately model the tendon driven flexible manipulator. K E Y W O RD S
data‐driven methods, inverse kinematics, surgical robotics, tendon‐driven serpentine manipulator
1
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I N T RO D U CT I O N
conventional modeling approach based on the assumption of constant curvature is insufficient.
With the advancement of robotic assisted minimally invasive surger-
High motion control accuracy is required in surgical applications
ies including single port surgeries, we have witnessed the emergence
such as targeting a diseased location or dissecting diseased tissue.
of various kinds of flexible and continuum robots for surgical pur-
To enhance motion control accuracy for flexible surgical manipula-
poses. Examples include, but are not limited to, the tendon‐driven
tors, researchers have designed feedforward controllers to compen-
serpentine manipulator,1 tendon‐driven continuum manipulator,2 con-
sate for part of the nonlinear effects by modeling friction and
centric tube robot (CTR)3 and pneumatically driven soft tentacles.4
backlash.5,6 With miniaturized position sensors and vision sensors
Precise motion control for these robots, however, is extremely chal-
playing more important roles in surgical robotics, feedback controls
lenging, primarily because an accurate kinematics model is usually
were investigated during the last decade. The PID controller was
complicated and analytically inaccessible. These flexible manipulators
one of the earliest attempts,7,8 which greatly enhanced the control
usually incorporate elastic beam elements to enable continuum bend-
accuracy compared with open loop control. The disadvantages are
ing, and introduce nonlinearities to the system. More specifically, for
twofold: the control loop performs poorly when the manipulator
tendon driven manipulators, the focus of this paper, cable tensioning
has to change the bending direction; actuation commands and con-
and slack cause friction and backlash in the system, making the sys-
trol action decomposition depends on an inaccurate kinematics
tem even more complicated than most continuum manipulators.
model. Intelligent control schemes such as fuzzy logic were later
Therefore,
proposed,9,10 which rely on a reliable inverse dynamic model.
these
systems
exhibit
high
Int J Med Robotics Comput Assist Surg 2016; 1–11
nonlinearities
and
a
wileyonlinelibrary.com/journal/rcs
Copyright © 2016 John Wiley & Sons, Ltd.
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2
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However, the design of multiple membership functions can be
2
ET AL.
MATERIALS AND METHODS
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troublesome. As can be learned from previous studies, before a reliable control-
2.1
|
The TSM system
ler can be applied, the core requirement for control of a flexible manipulator is to obtain an accurate, nonlinearity compensated inverse
An in‐house TSM system (shown in Figure 1) was built to facilitate
kinematics model. As analytical solution is usually unattainable and
minimally invasive robotic trans‐oral surgery.16–18 The system com-
numerical solutions require exhaustive iterative processes; a data‐
prises a one section tendon‐driven serpentine section having 21 iden-
driven approach has proven to be a more efficient alternative. Machine
tical spherical joints with an outer diameter of 7.5 mm and an inner
learning algorithms are efficient at approximating nonlinear functions
diameter of 5 mm were connected in series. Two antagonistic tendon
when provided with proper dataset. A feedforward neural network
pairs, controlled by two DC servos are inserted through tendon guides
(FNN) has been applied to learn the nonlinear inverse kinematic model
in each individual joint and terminated at the distal end. A design sche-
Another regres-
matic of an example one section TSM is shown in Figure 2. By varying
sion method based on FNN called ELM was applied to train the inverse
the tension in the tendon pairs, the TSM is able to bend and rotate. An
equilibrium model of a soft arm in a faster manner based on prior
elastic silicone tube is introduced in the central cavity of the joints to
11
12,13
of a CTR
and soft tendon driven manipulators.
Moreover, a statistics based learning approach such as
enhance compliance. A movable rigid constraint tube is inserted inside
Gaussian process regression (GPR) has been proposed to learn an
of the elastic tube to change the length of the bending portion and
accurate kinematics controller for a cable driven robot.15.
thus, enlarge the workspace. The flexible manipulator was placed on
14
knowledge.
The aforementioned approach was developed for either contin-
a motion stage, so that insertion movement is enabled. The working
uum manipulators or cable driven rigid robots. For TSM, however,
mechanism of TSM involves three spaces: actuation space defined by
nonlinear effects come not only from cable tensioning and elastic ele-
motor movements: ξm, configuration space parameterized by its bend-
ments, but also from the interaction of two adjacent joints in the ser-
ing angle θ and bending direction angle ϕ, as well as working space
pentine structure. Hence, TSM is an even more challenging platform.
defined by robot tip position in 3D space ξe.
Theoretically, any regression methods that are capable of approximat-
An accurate inverse kinematics model is crucial to position the
ing nonlinear functions are applicable in this context. In this paper, we
TSM to the targeted diseased site or manipulate along the desired
aim to evaluate three popular regression methods, namely Gaussian
trajectory in a reliable and fast manner. The TSM system features
mixture regression (GMR), extreme learning machine (ELM) and K‐
high nonlinearities for the following reasons1: cable elongation is
nearest neighbors regression (KNNR) for learning the highly nonlinear
introduced when high tension is applied2; tendon slack is likely to
inverse kinematics of the challenging TSM robot. Each method pre-
happen after the system has been run for a certain time3; unmodeled
sents its own distinct features: GMR is probability based, KNNR
friction introduced by tendon and joint surface is unavoidable4; hys-
requires no training process, and ELM features a neural network struc-
teresis and time delay in the system could be brought about by elas-
ture. Their performances were evaluated on a robot‐specific dataset
tic components. The kinematics analysis and Jacobian matrix of TSM
and by real world tracking experiments. The reasons we choose the
are detailed in our previous papers,16,19 and show that the analytical
above‐mentioned three regression methods are three‐fold: first, GMR, ELM, and KNNR represent three distinct categories of regression methods, e.g. density based, artificial neural network based, sample based, respectively. Second, GMR, ELM, and KNNR are all efficient and effective. GMR has the tight structure of a parametric model, yet still retains the flexibility of a nonparametric method. ELM is a single‐ hidden layer feedforward neural network (SLFN) which randomly chooses hidden nodes and analytically determines the output weights. In theory, ELM tends to provide good generalization performance at extremely fast learning speed. KNNR requires no training process, and with increase of the number of samples, the performance of KNNR will improve correspondingly. And finally, previous literature has demonstrated the effectiveness of the proposed regression methods on approximating the nonlinear inverse kinematics models of other flexible manipulators. The rest of the paper is structured as follows: the next section first introduces the tendon‐driven flexible robotic system and defines the problem. Then three regression algorithms that are able to approximate nonlinear functions are detailed. The third section describes the process of model learning and compares the performance of the proposed three learning algorithms. Evaluation results for the performance of the learned models by a trajectory tracking experiment are also presented. The final section provides a discussion and concludes the paper.
FIGURE 1 Experimental setup of the tendon‐driven serpentine manipulator (TSM) used in this work. Two DC servo motors are used to drive the flexible bending segment, and a step motor is used for translation. An EM tracker is attached to the tip of the manipulator to measure the 3D position. The system is controlled by C++ and MATLAB program communicating through RS232 serial port
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AL.
To encode the observed data, GMM20 is expressed by the following probability density function: K pðξ i Þ ¼ ∑k¼1 pðkÞN ξ i jμk ; ∑k 9 8 < ðξ −μ ÞT ∑−1 ðξ −μ Þ= 1 i i k k k N ξ i jμk ; ∑k ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi exp:− ; 2 2 ð2πÞ3 ∑k
(1)
∑k¼1 pðkÞ ¼ 1 K
where K is the number of Gaussian components in GMM, p(k) is the prior probability of the kth component and N(ξi| μk, ∑k) is a Gaussian distribution, μk is the mean vector, and FIGURE 2
Planar tendon‐driven serpentine manipulator. As shown in the figure, the backbone has two parts. One is the rigid vertebras and the other is the elastic tube. The elastic tube serves as the torsion spring on each joint. It constrains the joint rotations. The tube cross section area is uniform. As a result, the constraint on each joint is the same. On the other hand, the elastic tube deformation is also confined by the vertebras rotations. The tube has two parts, one is attached to the vertebras and the other is in the joints. During bending, only the second part deforms. When the robotic arm bends, the tube axis is coincident with the robotic arm neutral axis, i.e. the tube length is constant, or the tube deformation is pure bending. With this design, the robotic arm has high axial rigidity and can bend significantly
solution to the inverse Jacobian is difficult to obtain. The highly nonlinear inverse model can be denoted by the mapping from the robot end effector state to the motor input: ξm(t) = f(ξe(t)), where ξmis
∑k
is the covariance matrix.
Given the training data and formulas of GMM, the Expectation maximization (EM) algorithm21 is applied to estimate the parameters of GMM, and k‐means22 is applied to initialize the EM algorithm. Gaussian mixture regression (GMR)23 is used to extract a generalized motor trajectory from the GMM. The output motor commands ξm can be computed from the joint probability distribution of GMM and take the expected values. Robot state ξe is used to query for the estimation of corresponding motor commands. Since ξi = (ξe, ξm)i, μk and
∑k can also be written by: μk ¼ ðμek ; μmk Þ; ∑k ¼
∑eek ∑emk ∑mek ∑mmk
! (2)
given each component k and robot state ξe, the conditional probability distribution of ξm can be obtained: f pðξ m jξ e ; kÞ ¼ N ξ m jeξ mk ; ∑ mm
the controller input from the DC servos and ξeis the 3D position vector of the end effector. The linear stage input is neglected here
(3)
−1
eξ mk ¼ μmk þ ∑ ∑ ðξ e −μek Þ eek mek
since it doesn't account for the nonlinearities in the system and is separately controlled for insertion purpose only. Machine learning
f ∑ mmk ¼ ∑mmk −∑mek ∑eek ∑emk
algorithms that are capable of approximating nonlinear functions
−1
are exploited.
the conditional probability of ξm with respect to ξe can be defined by: K
3 | HEURISTIC METHODS FOR NONLINEAR MODEL APPROXIMATION
pðξ m jξ e Þ ¼
∑ βk N
k¼1
f ξ m jeξ mk ; ∑ mmk
(4)
pðkÞN ξ e jμek ; ∑eek pðkÞpðξ e jkÞ βk ¼ K ¼ K ∑k¼1 pðkÞpðξ e jkÞ ∑k¼1 πk N ξ e jμek ; ∑eek
Three regression methods: GMM, ELM and KNNR are investigated in this section.
according to the equations above, ξm at state ξe can be estimated by: eξ m ¼ ∑ βk ~ξ mk k¼1 K
3.1 | Gaussian mixture models and Gaussian mixture regression
(5)
f 2f ∑ mm ¼ ∑k¼1 ðβk Þ ∑ mmk K
Gaussian mixture regression is a multivariate nonlinear regression approach which can approximate a continuous function from a multidimensional input ξe to a multidimensional output ξm. The training dataset ξ ¼ fξ i gNi¼1 consists of N pairs of observations of the motor movements (left and right DC servo motors) and end‐effector states. Within each data pair, ξm ∈ R2 denotes the movements of the two servo
therefore, at different end‐effector states ξe, a smooth generalized form of the motor movements eξ m can be generated with this method.
3.2
|
K‐nearest neighbors regression
motors, ξe ∈ R3 represents the state of the end‐effector and ξican be
In the field of pattern recognition and machine learning, K‐nearest
rewritten as ξi = (ξe, ξm)i ∈ R5.
neighbors algorithms (KNN)24 is among the simplest ones, while it is
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b H ¼ HH† ξ m b ξ m ¼ f β;
also one of the most efficient methods. Many techniques have been developed to enhance the performance of KNN during recent decades, 25
such as weighted KNN,
26
condensed KNN,
27
tunable KNN,
ET AL.
(11)
model
based KNN28 and so on. The goal of KNN in a regression problem is to learn a function fKNNR(ξe) : ξe → ξm from N observed input–output
4
RESULTS
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samples:
4.1
h i ξ e 1 ; ξ m 1 ; ξ e 2 ; ξ m 2 ; …; ξ e N ; ξ m N
(6)
∑k¼1 kξ e −ξ ek k2 ef KNNR ðξ ek Þ K
f KNNR ðξ e Þ ¼
|
Model training and testing
In this section, the aforementioned algorithms are implemented to approximate the highly nonlinear inverse kinematics model of TSM. The performance of each algorithm was evaluated by a self‐collected
∑k¼1 kξ e −ξ ek k2 K
TSM dataset, which contains observations of robot states and motor inputs. The training and testing processes were conducted on a
In fact, KNNR is a locally weighted regression method, like locally 29,30
weighted projection regression (LWPR)
2.50 GHz Intel (R) Core™ i5‐2520 M PC with 8 GB RAM.
and locally Gaussian pro-
cess regression (LGPR),31,32 and it also suffers from the same problems
4.2
LWPR and LGPR have. They all need to store all the training data for
A dataset comprising sample pairs of (ξe, ξm) was collected for learning
future prediction, which costs a lot of memory space.
|
The TSM dataset
and evaluation purposes. A data collection process was carried out on the TSM setup illustrated in Figure 1. The robot state ξe was measured
3.3
|
Extreme learning machine regression
by an electromagnetic (EM) sensor (trakSTAR 2, Ascension Technology
The third learning algorithm we proposed was ELM, which features a
Corp, VT, USA) mounted at the tip of the TSM, and the corresponding
single‐hidden layer feedforward neural network. ELM was first intro-
DC servo motor movements ξm were measured by integrated
duced in 2005 by Huang et al. to expedite the learning process of
encoders. Although the inverse model directly captures the relation
the conventional single layer FNN.33,34 The most prominent feature
between end effector position and actuator commands, data were col-
of ELM is that the input weights and the hidden layer biases are
lected based on interpolation in the configuration space defined by
randomly assigned, so the output weights can be analytically
(θ, φ). The configuration space was explored by articulating within
determined. ELM exhibits extremely fast learning speed without
the bending angle range θ ∈ [−π/2 , π/in 13 equidistant steps, and
scarifying generalization accuracy, which is important in potential
meanwhile rotating within the range φ ∈ [0, 2π]at the speed of
applications of real‐time control and adaptation for TSM.
0.628 rad/s. The procedure was repeated five times at five different
To learn the nonlinear inverse kinematics of TSM, consider N N observations of input–output pairs ξ je ; ξ jm , where ξe ∈ R3 and
translational distances. For each observation, estimation of the corre-
e nodes in hidξm ∈ R . The standard single layer neural networks with N
resulting end effector position (ξe, ξm) were observed by sensors intro-
den layer and nonlinear activation function ef ELM ðξ e Þ can approximate
duced earlier. As such, the feasible workspace is fully explored, which
the mapping with zero error mean:
resulted in a total of 20 000 sample pairs. The collected data were
j¼1
2
~ N
∑
j ~ i¼1 βi f ELM ωi ⋅ξ e
þ bi ¼
¼ 1; …N
(7)
is the hidden layer biases. ωi and bi are initialized randomly and remain unchanged in the training process.12 can be rewritten in matrix form: Hβ ¼ ξ m according to,
introduced in.35 However, the actual actuator commands and the
preprocessed by de‐noising and filtering, followed by a normalization ξ im ; j
where βi is the output weight vector, ωi is the input weight vector, and bi
33
sponding actuator commands eξ m are given by a partial‐inverse relation
(8)
process into [−1,1]. The dataset is visualized in Figure 3.
4.3 | Inverse kinematics learning with Gaussian mixture regression Given the training data and formulas of GMM, the Expectation maximization (EM) algorithm is applied to estimate the parameters of GMM, and k‐means algorithm initializes the EM algorithm.
training the ELM network is equivalent to solving the
following least square minimization problem: b Hβ−ξ m ¼ minβ kH; β−ξ m k
ξ m ¼ ~f GMR ðξ e Þ ¼
K
∑ Hk ðξ e Þ
Ak ξ e þ Bk
(12)
k¼1
(9) where K is the number of Gaussian distributions in GMM and p(k) is the prior probability of the kth component; other symbols were defined
the analytical solution is given by:
earlier. Figure 4 (left) illustrates the results of the Gaussian Mixture βb ¼ H† ξ m
(10)
where H† is the pseudo‐inverse of the hidden layer output matrix H. Given a learned model f ~ β; H , the control input can be generalized by:
Model, where K = 9 (based on Bayesian Information Criterion) is applied and Figure 4 (right) demonstrates the output of Gaussian mixture regression to generalize motor movements with respect to a given end‐effector reference trajectory (a circle with radius 30 mm).
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AL.
The final prediction accuracy (See Figure 5) for left and right motor
the training set can be found in Figure 8. The optimal value of k was
movements are 87.39% and 95.04%, respectively. The errors between
determined to be 2 based on Bayesian Information Criterion as illus-
original data and predicted data are plotted in Figure 6. Prediction
trated by Figure 11. Figure 9 illustrates results of 2NNR on the testing
accuracy of GMR with respect to the number of Gaussian components
set for left motor movement as well as right motor movement. The
K were investigated and plotted in Figure 7.
prediction accuracy for the left and the right motor movements are 90.4957% and 95.9142%, respectively. Figure 10 plots the errors between the original data and the predection.
4.4 Inverse kinematics learning with K‐nearest neighbors regression |
K‐nearest neighbors regression (KNNR) is implemented in this section to make predictions for the testing set. KNNR is an instance‐based learning algorithm for regression, which does not need a pre‐training process. KNNR calculates the distance between the testing sample and every sample in the training set based on their features, then it finds the K training samples nearest to the testing sample. With these K training samples, KNNR can define the predicted output for the testing sample. The KNNR model can be expressed by: ~ ~f KNNR ðξ e Þ ¼ ∑k¼1 ξ e −ξ ek f KNNR ðξ ek Þ K ∑k¼1 ξ e −ξ ek K
(13)
generally speaking, KNNR is able to approximate any function, however, when the dimensions of the data increase, it can be easily fooled by the ‘curse of dimensionality’, and dimensionality reduction techniques are often used to solve this problem. For more details, please refer to.25,26 A toy example showing the results of 1NNR on
FIGURE 5 Prediction accuracy of GMR with respect to the number of Gaussian components K. Upper: Prediction accuracy for the left DC servo motor movements. Lower: Prediction accuracy for the right DC servo motor movements
Visualization of the TSM dataset. Left: Observed robot tip position in Cartesian space ξe = [x, y, z]; Right: Executed actuator commands ξm = [Rm, Lm, Zb]
FIGURE 3
FIGURE 4
Results of the Gaussian Mixture Model (GMM) and Gaussian Mixture Regression (GMR) to encode DC servo motor commands and robot end‐effector states into a joint probabilistic model. Left: Visualization of the training dataset, the green lines represent the end‐effector states, while the blue dots are the corresponding motor commands. Middle: Encode servo motor commands and robot end‐effector states into a single joint probabilistic model with GMM, the ellipses denote the independent Gaussian components (covariance matrices), and the black crosses represent the corresponding centers (means). Right: The green line denotes a reference circle, and the dark blue line is the corresponding servo motor commands generalized from GMR, the blue shadow area represents the variance
6
FIGURE 6
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ET AL.
Absolute error between prediction data by GMR and original data
FIGURE 7
Prediction accuracy of GMR with respect to the number of Gaussian components K. Left: Prediction accuracy for the left DC servo motor movements. Right: Prediction accuracy for the right DC servo motor movements. From the figures we can see that the best number for K is 9
4.5 | Inverse kinematics learning with extreme learning machine
optimized value after several rounds of trial and error. The activation function ef ELM ðhi Þ was chosen to be the sigmoid function:
ELM was applied to the collected dataset to learn the inverse
~f ELM ðhi Þ ¼
kinematics in this section. 90% of the samples in the dataset form
1 ; i ¼ 1; 2; …; N 1 þ e−hi
(14)
the training set, and 10% of the dataset are extracted to form the testing set randomly. In order to select the optimized number of hidden
where hi = ωiξe + bi.
neurons in the ELM model, training and testing accuracy under differ-
The learning process includes implementation of the following
ent numbers of hidden neurons were investigated and plotted in
steps1: randomly assign input weights ωi and bias vector bi2; compute
Figure 12. The number of hidden nodes is set to be 45 as this is the
hidden layer output matrix H3; compute output layer weight matrix β
FIGURE 8 1‐NN algorithm applied to a dataset consisting of 2D robot end‐effector states. The number of K (in this toy example K = 1) will be determined based on Bayesian Information Criterion (BIC). The circles represent centers of different clusters and different clusters are represented by different colors
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AL.
FIGURE 9
Prediction accuracy of KNNR with respect to the number of nearest neighbors K. Upper: Prediction accuracy for the left DC servo motor movements. Lower: Prediction accuracy for the right DC servo motor movements
FIGURE 10
FIGURE 11
FIGURE 12 Training and testing accuracies versus the number of hidden neurons in ELM (the number of hidden neurons were selected to be 45)
Absolute error between prediction data by KNNR and original data
Prediction accuracy of kNNR with respect to the number of nearest neighbors K. Left: Prediction accuracy for the left DC servo motor movements. Right: Prediction accuracy for the right DC servo motor movements. From the figures we can see that the best number for K is 2
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using Equation 104; apply the learned model to the testing set to
as ξd, the estimated motor movements can be generated by:
make predictions. The above process was carried out 100 times.
~ξ m ¼ f ðξ d Þ
The prediction results for the two actuator movements are visualized in Figure 13. Figure 14 plots the errors between original data and the prediction. The performance of the three regression methods using the TSM dataset is shown in Table 1. ELM outperformed the other two algorithms with the highest prediction accuracy of 98.2%, which is very inspiring, considering the limited size of the dataset. The average training time is 0.358 s, which is acceptable.
ET AL.
(15)
where f is the inverse kinematics model learned in the previous section. This forms a feedforward controller, which is expected to compensate for the system internal nonlinearities and enhance the tracking accuracy. It should be noted that, the control input of the insertion axis x was generated separately by a proportional controller. With f learned by the three regression algorithms, three sets of tracking experiments were conducted. The output trajectory and the reference trajectory are plotted in Figure 15. The tracking error is rep-
4.6
|
Real world experiments
resented by the RMSE:
To evaluate further the performance of the learned inverse kinematics ε¼
model in real world scenarios, a trajectory tracking experiment was designed. A circular trajectory with a diameter of 60 mm in yoz plane with x = 45 mm was predefined for the TSM end effector to follow. The set up was shown in Figure 1. Denote the reference trajectory
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u i i 2 ∑ t i¼1 ξ e −ξ d2 N
(16)
The average tracking accuracy for each set of experiments was computed and is shown in Table 1. The results demonstrated that KNNR approximated the system internal model more accurately and reliably than ELM and GMR, which is reflected by the RMSE of 2.1275 mm. To illustrate the superiority of proposed heuristic methods, we also conducted experiments with a constant curvature based IK model derived in,36 and the RMSE of IK is 8.3359 mm. The three regression methods enhanced the tracking accuracy by over 75% compared with traditional IK.
5
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DISCUSSION AND CONCLUSION
Three regression methods, namely GMR, KNNR and ELM were used to learn the highly nonlinear inverse kinematics model of a tendon‐ driven serpentine surgical manipulator in this work. The prediction accuracy of the three methods on our own dataset is encouraging, with ELM demonstrating the highest prediction accuracy of 98.2% and KNNR providing the shortest training time. ELM, however, tends FIGURE 13
Generalization performance of ELM on a testing set consisting of 100 samples
FIGURE 14
to suffer from overfitting when the hidden nodes and activation function are not chosen properly. Statistical based GMR provides
Absolute error between prediction data by ELM and original data
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TABLE 1
Performance of the proposed three regression methods on testing set and real world experiments Prediction accuracy for the Left DC Servo Motor / %
Prediction Accuracy for the Right DC Servo Motor / %
Training Time / s
Root mean square error (RMSE) / mm
GMR with 9 Gaussians
87.3900%
95.0400%
0.9157
2.5556
2‐NNR
90.4957%
95.9142%
0.0288
2.1527
ELM Inverse Kinematics
FIGURE 15
98.2000%
98.2000%
0.3580
2.3277
Not Applicable
Not Applicable
Not Applicable
8.3359
Experimental results of the TSM robot tracking a circle. Upper: GMR, KNNR; Lower: ELM and IK
the worst prediction accuracy. Moreover, it requires the longest
sensor, which is 1.4 mm according to its technical specifications, this
training time as it uses the iterative EM based parameter tuning
error performance is quite impressive for this challenging platform.
method to find the optimal solution. Conversely, the training and
Among the three regression methods, KNNR produces the highest
testing in KNNR are conducted concurrently and thus, training time
accuracy,
is minimized. However, the generalization accuracy of KNNR radically
Although ELM has the highest prediction accuracy, the RMSE of
reduced if the samples were not within the training set; in other
ELM is higher than that of KNNR, and there exist three possible
words, a large and dense dataset is required. ELM assigns the
reasons. First, ELM is much more sensitive to noise which leads to
weights to the input layer and hidden layer randomly and defines
overfitting, this phenomenon becomes even more prominent when
the output weights analytically, which also greatly reduces the train-
the RBF kernel is used as the activation function. Secondly, as KNNR
ing period. Based on the performance on the dataset, one can con-
is a sample based regression method, its generalization capability is
clude that ELM and KNNR are expected to enhance control
more stable and reliable when the training dataset is large enough.
accuracies in real world experiments and can be applied in real time
And finally, due to the high nonlinearities inherent in the TSM as well
learning and control problems.
as the measurement error of EMT sensor, the repeatability of the
In real world trajectory tracking experiments, all three regression
demonstrating
its
superior
generalization
capability.
system is relatively low.
methods outperforms traditional IK and reduced the RMSE error to
To conclude, we proposed three data‐driven approaches to learn
around 2 mm. Considering the measurement error of the EMT
an accurate inverse kinematics model for a flexible surgical
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manipulator. Although we evaluated the algorithms on our specific TSM robot, our approach could serve as a general paradigm to solve the highly‐nonlinear inverse kinematics problem for other flexible or continuum robots such as concentric tube robots or the emerging soft robots, given that they all incorporate elastic elements and suffer from friction and hysteresis. From our evaluation and experiments results, KNNR was proven to be the most reliable regression method for our specific system. In the future, we will focus on two potential applications. On the one hand, the learned inverse kinematics model can serve as a feedforward controller to compensate for the internal nonlinearity and predict a feasible control input. Furthermore, the feedforward controller is to be integrated into the feedback loop to achieve more robust position control. On the other hand, higher level problems such as trajectory planning, task level learning and adaptation can be enabled by utilizing dynamic planning optimization schemes. ACKNOWLEDGMENTS This work was supported by the Singapore Academic Research Fund under the Grant R397000227112, Office of Naval Research Global, ONRG‐NICOP‐N62909‐15‐1‐2029,
and
Singapore
Millennium
Foundation under the Grant R‐397‐000‐201‐592 awarded to Dr Hongliang Ren. RE FE R ENC E S
ET AL.
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How to cite this article: Xu, W., Chen, J., Lau, H. Y. K., and Ren, H. (2016), Data‐driven methods towards learning the highly nonlinear inverse kinematics of tendon‐driven surgical manipulators, Int J Med Robotics Comput Assist Surg, doi: 10.1002/rcs.1774
