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Forward dynamics is used and the reaction forces are determined in different simulating conditions. ... ergonomics, medicine, sports or biomedical engineering. ..... Computational Dynamics: Theory and Applications of Multibody Systems.
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ScienceDirect Procedia - Social and Behavioral Sciences 137 (2014) 57 – 63

SEC-IASR 2013

MBS dynamics for assessing the reaction forces and torques in elbow joint Elena Mereutaa***, Daniel Ganeaa, Claudiu Mereutaa , and Silviu Epurea a

”Dunarea de Jos” University of Galati, 47 Domneasca Street, Galati 800008, Romania

Abstract The paper presents the multi-body system method (MBS) as a proper tool for assessing the magnitude of reaction forces and torques in human upper limb joints. Forward dynamics is used and the reaction forces are determined in different simulating conditions. The steps to be followed in MBS modeling are presented together with the principles of modeling the bones, the muscles and the joints of the human upper limb. The simulation model provides results for the reaction forces and torques in the elbow joint during the flexion of the forearm. © Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license © 2014 2014 The The Authors. Selection and peer-review under responsibility of the Sports, Education, Culture-Interdisciplinary Approaches in Scientific (http://creativecommons.org/licenses/by-nc-nd/3.0/). Research Selection Conference. and peer-review under responsibility of the Sports, Education, Culture-Interdisciplinary Approaches in Scientific Research Conference. Keywords: upper limb, dynamic model, reaction forces, MBS

1. Introduction The most representative studies on modelling the human body are those concerning the assessment of internal forces and the reactions, due to the fact that there are no methods for measuring them. Multi-body dynamic modelling (MBS) of the human body provides these parameters which are of great importance in areas such as ergonomics, medicine, sports or biomedical engineering. MBS considers that the system is made up of a number of solid or flexible bodies, connected at the joints and that can interact with each other and/or with the environment. Constraints are described using mathematical equations. In addition, the forces that may act on the multi body system are internal and external like gravity, friction, etc. (Ganea, D., Mereuta, C., Tudoran, M.S., Mereuta, E., 2011). Dynamics of multi-body systems enable solving problems as diverse as real-time simulation (forward dynamics link), inverse dynamics, synthesis, optimization, contact and impact (Schiehlen W., 2006, 2007). Forward dynamics

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Elena Mereuta, Tel.: +040-745-590-600 E-mail address: [email protected]

1877-0428 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Sports, Education, Culture-Interdisciplinary Approaches in Scientific Research Conference. doi:10.1016/j.sbspro.2014.05.252

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is used more and more to investigate the effects of the restoration of muscles, muscle pathologies and design of assisting motion devices (Ackermann, 2007). In forward dynamics the muscle forces and joint torques can be used as inputs, in which case the time variation of joint angles will be part of the movement results. If the time variations of joint angle are used as input, the motion of the system is described by the motion of the centre of mass of the entire system and the time variation of the entire guidance system (Mereuta, E., Ciubucciu-Ionete, L.G., Tudoran, M.S., 2009). This is known as the simulation model (Mereuta, E., Tudoran, M.S., Mereuta, C., Ciubucciu-Ionete, L.G, 2011). a.

MBS Dynamic modelling steps

The main steps in developing a dynamic simulation using MBS are (fig.1): x featuring the system as a multibody system (defining the bodies and the constrained degrees of mobility); x establishing the reference systems attached to bodies (local systems for mobile bodies, and the global system attached to the fixed body); x ascertaining the geometry of bodies with respect to local reference systems; x defining the bodies orientation (matrix of transition from local system to the global reference system); x expressing the global coordinates of the most important points; x expressing analytical equations defining the geometric and kinematic constraints of the model; x formulating the differential equations of motion, using different computational formalisms (Newton - Euler and Lagrange); x expressing mechanical characteristics of bodies; x computing the system of differential algebraic equations.

Fig. 1. MBS principles

The bones of the upper limb are considered to be the kinematic elements of the dynamic model, the joints are modelled through rotational joints and muscles of the arm are modelled as a system of two semi couplings which act like linear actuators (Y. Bar-Cohen, 2003 M. Hackel, 2007; Incerti G., 2011; Alaydi JY, 2012). Considering time optimization the bone-muscle link was modelled using a spherical joint. It was necessary to block one of its rotations making it behave like a universal joint.

Elena Mereuta et al. / Procedia - Social and Behavioral Sciences 137 (2014) 57 – 63

2. Dynamic model for human upper link The dynamic model (fig. 2) is composed as follows: 4 bones (scapula, humerus, ulna and radius), 12 semi couplings, corresponding to six muscle fibres, 6 cylindrical joints, 12 spherical joints, 14 applied forces, 2 rotational joints, 2 rigid joints. Bones are modelled as rigid bodies and the bone material is considered to be isotropic with 1300 Kg/m3 density (Chandler RF, 1975 XQ Dai, 2006). The 12 semi couplings are mobile elements and form the anterior muscles of the arm (biceps and brachialis), and the posterior one (triceps). The biceps is modelled by the long and the short fiber, and the triceps is modelled through three of its fibers (long, lateral and medial fiber). The mechanical structure that simulates the motion of the upper limb is actuated by a single driving force; the remaining 13 are applied forces for locking certain movement. The applied moment is acting on the shoulder and is designed to maintain the humerus in the same position during the movement. Thus, only one rotational joint is active, allowing the model to simulate flexion-extension of the forearm. The spatial kinematic chain of the upper limb has one degree of freedom.

Fig. 2. Upper limb model

Forward dynamics aims at determining the reaction forces and moments of elbow and shoulder joint. The virtual biomechanical model is simulated in two load cases: first case in which on the human upper limb act only the muscle forces as applied forces, and the second case in which there are also external forces. Forearm flexion is simulated under the following conditions: 1. The driving force acting on the long end of the biceps; 2. The driving force acting on the short end of the biceps; 3. The driving force acts on both ends of the biceps. The first two loading conditions meet some possible dysfunctions of the human upper limb. 3. Results and discussions When the driving force acts on the long end of the biceps, the time variation of the elbow joint reaction force decreases rapidly at the beginning of the simulation and then remains approximately constant, in both cases I and II (fig.3). The shock generated by the linear law of motion causes sharp drop, which lasts only 0.05 seconds and has a value of 13.4 N for case I, and respectively 3220 N for case II. The time variation of reaction torques occurring in the elbow (fig. 4a) sharp decreases by 63.1 Nmm, followed by a smooth decrease from 73.5 Nmm to 63.1 Nmm, in the first case. For the second case the amplitude of the shock

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load is 114741 Nmm, lasting 0.05 seconds, followed by a smooth decrease from 15502.4 Nmm to 24860.5 Nmm recorded at the end of the simulation (fig. 4b). Elbow joint reaction force-case I (no external forces) 16 14

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When the driving force acts on the short end of the biceps (the second load case), the elbow reaction force shows the same time variation as the case shown above. The difference consists in the value of the shock values which is smaller, only 3.1 N. After that decrease, it remains approximately constant at a value of 2.4 N in case I (fig.5a). For the second case the amplitude of the shock load is 280 N, lasting for 0.05second, followed by a decrease from 838.3 N to 586.1 N recorded at the end of the simulation (fig. 5b). Elbow joint reaction force-case II

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The time variation of the elbow reaction torques decreases by 121.9 Nmm, then smoothly increases from 58.1 Nmm to 74.7 Nmm, for the first loading case (fig.6a). For the second case the amplitude of the shock load is 17637.2 Nmm, then it decreases from 21849.3 Nmm to 18217.5 Nmm recorded at the end of the simulation (fig.6b). Elbow joint reaction torque-case II

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b) Fig. 6. Elbow joint reaction torques – case II

In the third loading case, due to the fact that there are two driving forces, two special cases are emphasized: a). The elbow joint is modelled using a cylindrical joint For the first loading case variation the elbow reaction forces sharply increase up to 2.5 N, then stabilize for 1.8 seconds, and then slowly decrease down to 2.3 N (fig.7a). For the second loading case the time variation curve has only two parts: an increasing one up to 835.8 N followed by a slowly decreasing one down to 575.8N (fig.7b). Elbow joint reaction force-case III (cylindrical joint)

Elbow joint reaction force-case III (cylindrical joint)

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b) Fig. 7 Elbow joint reaction forces- case III

The variation curves of the elbow reaction torque occurring are approximately shaped the same as before, reaching a maximum load of 114.7 Nmm in the first loading case, respectively 33356.8 Nmm in the second one (fig.8). b). The shoulder joint is modelled using a cylindrical joint For the first loading case the elbow reaction force sharply increases up to 13.9 N, then slowly decreases from 2.6 N to 2.2 N (fig.9a). In the second loading case the curve rises sharply to 3777.1 N and then slowly decreases from 873.5 N to 541.4 N (fig.9b).

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Elbow joint reaction torque-case III (cylindrical joint)

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The time variation of the elbow reaction torques presents a sharp drop of 459.3 Nmm followed by a smooth decrease from 71.3 Nmm to 65.7 Nmm for the first loading case (fig.10a). For the second loading case the shock magnitude is 132611 Nmm, lasting 0.05 seconds, followed by a smooth decrease from 26084.4 Nmm to 15591.5 Nmm registered at the end of the simulation (fig.10b). Elbow joint reaction torque-case III (shoulder joint - cylindrical joint)

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b) Fig. 10 Elbow joint reaction torques

4. Conclusions Using methods and techniques for multi-body modelling with optimization techniques provides the possibility of evaluating the approximate values of reaction forces and torques in human upper limb joints. It is possible to analyze complex models, thus facilitating both the simulation and the interpretation of the results.

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The same MBS could provide information on the reaction forces and torques in shoulder joint, as well as in hand joint. Modifying the magnitude of the loads, we can observe the differences in joints reaction forces and torques from one loading case to another. It is important to be aware of the stresses to which the joints are subjected in order to plan training programs for improving the performances or to design devices that might assist a disabled person. The experimental results have revealed that there is an important stress of the elbow joint at the beginning of each motion, considered to be a shock compared to the following stages of the motion. References Ackermann, M. (2007). Dynamics and energetics of walking with prostheses, Institute of Engineering and Computational Mechanics University of Stuttgart. Alaydi, J.Y. (2012). Modeling and Simulation of High-Performance Symmetrical Linear Actuator, International Journal of Scientific & Engineering Research, 3(8). Bar-Cohen, Y. (2003). Biologically Inspired Intelligent Robotics. Proceedings of the SPIE Smart Structures Conference, San Diego, CA., Mar 26. Chandler, R.F., Clauser, C.E., McConville, J.T., Reynolds, H.M., & Young, J.W. (1975). Investigation of the Inertial Properties of the Human Body. (DOT HS-801-430), Wright-Patterson Air Force Base, Ohio. Dai, X.Q., Li, Y., Zhang, M., & Cheung, J.T. (2006). Effect of sock on biomechanical response of foot during walking. Clinical Biomechanics. Ganea, D., Mereuta, C., Tudoran, M.S., & Mereuta, E. (2011), Experimental method for determining the ground reaction and the orthostatic position, Annals of “Dunarea de Jos” University of Galati, Fascicle XV, 27-32. Incerti, G., (2011). A technique to reduce swing of suspended loads transported by linear actuators, World Academy of Science, Engineering and Technology, 59. Mereuta, E, Ciubucciu-Ionete, L.G., & Tudoran, M.S. (2009). Principles of MBS Analysis in Biomechanics. The Annals of “Dunarea de Jos” University of Galati, Fascicle XV, 123-127. Mereuta, E., Tudoran, M.S., Mereuta, C., & Ciubucciu-Ionete, L.G. (2011). Biomechanical models for the kinematics of upper limb. Annals of “Dunarea de Jos” University of Galati, Fascicle II, 280-285. Schiehlen, W. (2006). Computational Dynamics: Theory and Applications of Multibody Systems. European Journal of Mechanics A/Solids, 25, 566-594. Schiehlen, W. (2007). Computational Dynamics: Research Trends In Multibody System Dynamics. Multibody System Dynamics, 18, 3-13.

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