The Preliminary Study of Optimal Planning for Front Chin-ups

Transcription

1 Journal of Medical and Biological Engineering, 25(3): he Preliminary Study of Optimal Planning for Front Chin-ups Ching-Hua Chiu * Office of Physical Education and Sport, National Chung Hsing University, aichung, aiwan, 42, ROC Received 2 Aug 25; Accepted 2 Sep 25 Abstract he primary purpose of this study was to design a dynamic system for front chin-ups. Applying the principle of minimum energy expenditure control, this study used this system to simulate the optimal trajectory for angular positions of front chin-ups. Additionally, this study investigated the ways to avoid possible injuries caused by front chin-ups. he research method of this study involved the following steps: () Based on Lagrange-Euler (L-E) equations of motion, this study defined the primary body segments as a dynamic system with 4 degrees of freedom, (2) to achieve the goal of minimum energy control, the Open-loop Linear Quardatic Controller was used. hen under the established boundary conditions, the optimal trajectory for body joints was simulated, (3) the length, mass, center of mass position, and inertia for segments of three subjects were measured, and C ++ computer language was used to edit a computer program. he algorithm of the computer program was used to simulate hand suspension motions of three subjects performing front chin-ups. here were two principal findings from the simulation results. First, it was found that under the established boundary conditions and the control of minimum energy expenditure, the dynamic system can simulate the optimal trajectory for front chin-ups. Second, it was suggested that the three subjects perform a repetition of front chin-up with the time between sec and.5sec. If the time spent in doing a repetition was longer than.5sec, the energy expended would be smaller. However, the time spent in supporting shoulder and elbow joints would become longer, which could easily lead to sore and pain of arm muscles. Keywords: Segment, Energy, Center of mass Introduction Basically, in moving a joint from one position to another, it is the most efficient way when the minimum energy is expended. he efficiency can be achieved by controlling the path of the motion. o find the optimal trajectory for the path, trajectory planning for robots can be used. As for human motions, only a quarter of the energy produced by muscles can really work, and the remaining three-fourths vaporizes as heat[]. If a job involves repetitive motions for a long time, it is necessary to reduce energy expenditure. Following the optimal trajectory can reduce mechanical energy expenditure, thus extending working or exercising time and avoiding muscle fatigue. Consequently, the optimal trajectory control is an issue worth researching. In the past, some researchers proposed various studies concerning the reduction of energy expenditure. Beckett and Chang used Lagrange s equations to design a dynamic system for human legs[2]. Applying the principle of minimum energy expenditure, they analyzed the thigh and shank motions in the swing phase of walking. Lin and Dillman used Lagrange s * Corresponding author: Ching-Hua Chiu el: ; Fax: chungoodman@yahoo.com.tw equations to design a dynamic system for lower limbs[3]. Applying the principle of minimum energy expenditure, they analyzed the optimal stride length for each of their seven subjects. Researchers also proposed various studies on human dynamic systems. Ramey and Yang developed a -segment dynamic system for human body [4]. his system was applied to sports and dance. It was also applied to simulate hitch-kick, somersault, and long jump. Aleshinsky and Zatsiorsky applied a 5-segment model to design a 3D dynamic system for human body [5]. hey used this system to simulate dynamics of walking and running. Huston and Passerello applied Kane s equations and the 5-segment body model propose by Hanavan to design a dynamic system with 34 degrees of freedom [6]. hey also simulated simple jumping with a computer. Many other researchers also designed a dynamic system for certain segments of human body [7-]. he design of previous human dynamic systems tends to be complicated and poorly-organized. As a consequence, it is difficult to edit a computer program and transform equations into the state-space of a control system. Nowadays, L-E equations are applied to homogeneous transformation matrices to design dynamic systems. A complete dynamic system comprises an inertial acceleration-related symmetric matrix, a nonlinear Coriolis and centrifugal force vector, and a gravity

2 3 J. Med. Biol. Eng., Vol. 25. No.3 25 loading force vector. he above three can be written in a matrix form. hus, the L-E equations become formatted, systematic, and generalized. hat makes the edition of computer programs and the transformation into the state-space become easy [,2,3]. Generally, human motions come from the movement of body joints. here are various trajectories for moving a joint from its initial position to another. he purpose of trajectory planning is to find the optimal trajectory for the path of a moving joint. o achieve the goal of finding the optimal trajectory, it is necessary to conduct path control over the angular positions of each joint axis. In the human multibody dynamic system, optimizing a motion requires the establishment of cost function. aking front chin-ups as an example, this study attempted to simulate the optimal trajectory for suspending hands. First, L-E equations were used to establish a dynamic system. his system defined four types of homogeneous transformation matrices. Second, this study adopted the human body model proposed by Zatsiorsky and Seluyanov [4]. In this study, the human body model was defined to have 4 degrees of freedom. he generalized coordinate and generalized velocity for each degree of freedom acted as the input variables of this dynamic system. hen, the Open-loop Linear Quardatic Controller was used to transform the L-E equations into the state-space of a control system. With the established boundary conditions and the principle of minimum energy expenditure, the optimal trajectory for the angular position of each joint axis was simulated. Additionally, this study explored the ways to avoid possible injuries caused by front chin-ups. Methods Human body model Figure presented the human body segment lengths. he symbol B i ( i=,..,2) represented body segments. For example, the following representations were used: B, integration of right and left hands; B 2, integration of right and left forearms; B 3, integration of left and right upper arms; B 4, head; B 5, upper trunk; B 6, lower trunk; B 7, left thigh; B 8, left shank; B 9, left foot; B, right thigh; B, right shank; B 2, right foot. he symbol O represented the axis center of hands grasping a horizontal bar. he remaining O i ( i=,,) represented joints. In figure (b), l i represented body segment lengths. For example, the following representations were used: l, length from the axis center of hands grasping a horizontal bar to wrist joint; l 2, forearm length; l 3, upper arm length; l 4, head length; l 5, upper trunk length; l 6, lower trunk length; l 7, left thigh length; l 8, left shank length; l 9, left foot length; l, right thigh length; l, right shank length; l 2, right foot length. In Figure (b), p i = (x i,y i, zi), and pi represented the translation from the origin of the ith link coordinate frame relative to the i-th link coordinate frame. he center of mass vector for a certain segment was represented with r i. Lagrange-Euler equations of motion Figure (a) and (b) showed a coordinate system based on the right-hand rule. his coordinate X Y Z was established (a) (b) (c) (d) Figure. Human body segment lengths in the axis center of hands grasping a horizontal bar with (x, y, z ) as the origin. Due to the structure of human body joints, this system was composed of 4 degrees of freedom. hey acted as the variables of this system, and each degree of freedom involved a coordinate frame. he generalized forceτ i, which was defined in accordance with L-E equations of motion, could be written as followed []: n j τ i = race ( U j= i k = jk J j U ji )& q& k n j j + race ( U j= i k = i m= n - ( m jgu jir j) j= i i=,2,3,.,4 jkm J j U ji ) q& kq& m q i =generalized coordinate; q& i =generalized velocity; q&i & =generalized acceleration; i- A i =homogeneous transformation matrix of the ith coordinate frame relative to the i-th coordinate frame A i =homogeneous transformation matrix from coordinate frame to the ith coordinate frame; A i = A A 2 2 A 3 i- A i Uij = Ai q j ( i, j =,2,3,..., 4) ()

3 Optimal Planning for Front Chin-ups 3 Uijk = Uij qk ( i, j, k =,2,3,..., 4) ; J i =pseudo-inertiam matrix (Appendix A); g =[,,- g,]; g=9.862 ms -2 ; m j =the mass of the ith segment; r i =the center of mass for the ith segment; r i = ( x i, y i, z i,) Equation () could be writer in a matrix form as followed[2,5]: τ =M( q ) q& +V( q, q& )+G( q ) (2) In equation (2), the symbolτ represented a 4 matrix, and τ =[,,.., ] ; q represented a 4 matrix, τ τ 2 q q 2 τ 4 4 ; and q = [,,.., q ] q& represented a 4 matrix, and q& =[ q&, q& 2,.., q& 4 ] ; q& & represented a 4 matrix, and q& & =[ q& &, q& & 2,.., q& & 4 ] ; M( q ) represented a 4 4 inertial acceleration-related symmetric matrix; V( q, q& ) represented a 4 nonlinear Coriolisand and centrifugal force vector; G( q& ) represented a 4 gravity loading force vector. he power P i was the product of generalized force multiplied by generalized velocity[6]; therefore, Pi could be written as Pi = τ iq& i (i=,2,3,,4). he parameters for body segments in this dynamic system comprised the length, mass and inertia tensor for each segment. Zatsiorsky and Seluyanov used gamma ray to conduct a research on parameters for human body segments [4]. he data they collected are more complete and precise than those of other researchers. Consequently, the dynamic system of this study based the parameters for body segments on their research result. C ++ computer language was used to design the algorithm of the dynamic system. Four types of homogeneous transformation matrices Equations (3) to (8) showed the four homogeneous transformation matrices used in this study. In the basic homogeneous rotation matrices, q i represented generalized coordinates, and it was assumed that cqi = cos q i, sq i = sin q i. he first type was defined as i A = ; the second type was defined as i was defined as i i Ai = Si P i = ( xi, y i, zi ) i A i = z, q A i = t x, q i y, q ; the third type ; the fourth type was defined as (i referred to positive integers), which represented a 4 4 identity matrix. In the matrix,, which represented the translation from the origin of the ith link coordinate frame relative to the i-th link coordinate frame[3]. cqi y, q = sqi sqi cqi qi = t x, q xi yi zi (3) (4) Variables forms of i- A i q A = t-x,q able. Variables qi and forms of i Description q 2 ranslation of hand B A 2 = relative to the based t-z,q coordinate system q 3 2 A 3 = y,q Rotation of hand B relative to the based coordinate system q 4 3 A 4 = y,q Rotation of forearm B 2 relative to handb q 5 4 A 5 = y,q Rotation of upper arm B 3 relative to forearm B 2 q 6 5 A 6 = y,q Rotation of head B 4 relative to upper arm B 3 q 7 6 A 7 = y,q Rotation of upper trunk B 5 relative to upper arm B 3 q 8 7 A 8 = y,q Rotation of lower trunk B 6 relative to upper trunk B 5 q 9 8 A 9 = y,q Rotation of left thigh B 7 relative to lower trunk B 6 q 9 A = y,q Rotation of left shank B 8 relative to left thigh B 7 q A = y,q Rotation of left foot B 9 relative to left shank B 8 q2 A 2 = y,q Rotation of right thigh B relative to lower trunk q3 2 A 3 = y,q B 6 Rotation of right shank B relative to right thigh B q4 3 A 4 = y,q Rotation of right foot B 2 relative to right shank B t z, q = qi S = (6) i i m m m + n 2 n Sn = Sm + Sm + 2 Sn Sn (m and n are positive integers, and n>m) Variables of the dynamic system Figure (a) and (b) presented the homogeneous transformation of the dynamic system from the base coordinate frame to each segment. here were 4 degrees of freedom in this system, and they were defined in able. Homogeneous transformation able 2 presented position vector p i and the homogeneous transformation of each segment relative to the base coordinate frame OX Y Z. o o o Minimum energy control he L-E dynamic system established in this study could be transformed into the linear state-space of a control system [2]. In the state-space, the state variable was represented with x : x=[ q, q& ] (8) In equation (8), q was defined as the position vector and q& as the velocity vector. he linear state-space of the control system (xεr 28 ) could be written as followed: I x& = x+ u (9) I A i (5) (7)

4 32 J. Med. Biol. Eng., Vol. 25. No able 2. Homogeneous transformation of A i p i i x i y i z i A i ransformation A = ransformation of hand B A relative to the base coordinate frame A 2= A 2 3 -l A 3= A 3 4 -l 2 A 4= A 4 ransformation of forearm B 2 relative to the base coordinate frame 5 -l 3 A 5= A 5 ransformation of upper arm B 3 relative to the base coordinate frame 6 l 4 A 6= A 6 ransformation of head B 4 relative to the base coordinate frame 7 -l 5 A 7= A 5 5 S 6 6 A 7 ransformation of upper trunk B 5 relative to the base coordinate frame 8 -l 6 A 8= A 5 5 S 6 6 A 8 ransformation of lower trunk B 6 relative to the base coordinate frame 9 -l 7 A 9= A 5 5 S 6 6 A 9 ransformation of left thigh B 7 relative to the base coordinate frame -l 8 A = A 5 5 S 6 6 A ransformation of left shank B 8 relative to the base coordinate frame l 9 A = A 5 5 S 6 6 A ransformation of left foot B 9 relative to the base coordinate frame 2 -l A 2= A 5 5 S 6 6 A 8 8 S A 2 ransformation of right thigh B relative to the base coordinate frame 3 -l A 3= A 5 5 S 6 6 A 8 8 S A 3 ransformation of right shank B relative to the base coordinate frame 4 l 2 A 4= A 5 5 S 6 6 A 8 8 S A 4 ransformation of right foot B 2 relative to the base coordinate frame In equation (9), I represented a 4 4 identity matrix, and u= [ q& &, q& & 2,.., q& & 4 ] represented an acceleration input: u= -M - (q)[v(q, q& )+G(q)]+M - (q)τ () his system adopted Open-loop Linear Quardatic Controller as the control method. he whole controlling process was divided into two time points. he first time point was defined as t with x(t ) as the initial state. he second time point was defined as t 2 with x(t 2 ) as the final state. he state variables x(t ) and x(t 2 ) were known to be boundary conditions. he cost function of the minimum energy expenditure J during the interval of [t, t 2 ]could be written as followed [5]: 2 t J= u Ru dt () 2 t R was the symmetric matrix (R=[ I ]). A video camera was used to shoot the process of performing front chin-ups. Based on the shot images, a model of front chin-ups was founded. o test the practicability of the algorithm proposed in this study, the boundary conditions for positions and velocities of each subject s joints were established. Among the 4 parameters for the position of each joint, five of them were defined as major parameters. he angular position for the axis center of hands grasping a horizontal bar on the Y-axis was represented with q3 [,6 ] ; the angular position for elbow joints was represented with q5 [,7 ] ; the angular position for shoulder joints was represented with q7 [, 55 ] ; the angular position for left hip joint was represented with q9 [, 2 ] ; the angular position for right hip joint was represented with q2 [, 2 ] ; the angular position for left ankle joint was represented with q [,6 ] ; the able 3. Height and weight of each subject Normal subjects A B C Height(m) Weight(kg) angular position for right ankle joint was represented with q4 [,6 ]. Other parameters for the position of each joint were assumed to be. Conversely, the boundary conditions for velocity were assumed to be. he reason was that subjects were motionless at both the initial state and final state of performing a front chin-up. Results hree subjects of different shapes participated in this study. able 3 presented their height and weight. he length for each segment of the three subjects ( l i ) was measured and presented in Figure (a) and (b). Applying the segmental parameters proposed by Zatsiorsky and Seluyanov(able 3.) [4], this study calculated the mass, inertia, and center of mass vector for the ith segment. he translation from the origin of the ith link coordinate frame relative to the i-th link coordinate frame was represented with p i, which was also calculated. According to the algorithm of optimal control, the boundary conditions for controlled joint axes must be established before the calculation of optimal trajectory, moment, and power for joint axes. his study took simulation of the front chin-ups done by subject C as an example. Boundary conditions such as position vector and velocity vector for each joint were established. Figure 2 (a) presented the initial state of simulated front chin-up. In the initial state

5 Optimal Planning for Front Chin-ups 33 (a) (b) (c) (d) (e) (f) Figure 2. Simulation of the optimal trajectory for subject C was divided into six phases. he interval between two continuous phases was.4sec Phase (a) represented the initial state. Phase (f) represented the final state. he whole process took 2. sec. Angle(degrees) hand elbow shoulder R-L-hip R-L-ankle Force(N) X-axis Z-axis time (s) time(s) (a) (b) Power(J/s) time(s) hand elbow shoulder R-L-hip R-L-ankle Moment(Nm) time(s) hand elbow shoulder R-L-hip R-L-ankle (c) Figure 3. (a)he optimal trajectory for subject C. he angular position for subject C's axis center of hands grasping a horizontal bar on the Y-axis was represented with q 3; the angular position for elbow joints was represented with q 5; the angular position for shoulder joints was represented with q 7. R-L-hip represented hip joints, and the angular position for left hip joint was represented with q 9; the angular position for right hip joint was represented with q 2; R-L-ankle represented ankle joints, and the angular position for left ankle joint was represented with q ; the angular position for right ankle joint was represented with q 4. Figure 3(b) presented the force acted upon the axis center of hands grasping a horizontal bar. Figure 3(c) presented the loaded moment for each controlled joint axis. Figure 3(d) showed the output power for each controlled joint axis. (d)

6 34 J. Med. Biol. Eng., Vol. 25. No Minimum of Energy(J) Figure 4. he minimum energy expended by subject A, B, and C performing a repetition of front chin-up with different periods of time. t = s, subject C was simulated suspending both hands on a horizontal bar. Figure 2(f) presented the final state of simulated front chin-up. In the final state t 2 = 2s, both legs and the left hand stayed in the same position. he angular position for subject C's axis center of hands grasping a horizontal bar on the Y-axis was represented with q 3, and q3 [, 6 ] ; the angular position for elbow joints was represented with q 5, and q5 [,7 ] ; the angular position for shoulder joints was represented with q 7, and q 7 [, 55 ] ; the angular position for neck joint was represented with q 6, and q 6 [, 55 ] ; the angular position for left hip joint was represented with q9, and q 9 [, 2 ] ; the angular position for right hip joint was represented with q 2, and q 2 [, 2 ] ; the angular position for left ankle joint was represented with q, and q [ 35, 6 ] ; the angular position for right ankle joint was represented with q 4, and q 4 [ 35,6 ]. he other boundary conditions such as position vectors and velocity vectors were assumed to be. Under the above-mentioned boundary conditions for front chin-ups, the trajectory for subject C doing a repetition of front chin-up with 2 seconds was simulated. Figure 3(a) presented the graph of the optimal trajectory calculated with control equations. Figure 3(b) presented the force acted upon the axis center of hands grasping a horizontal bar. Figure 3(c) presented the moment for each controlled joint axis. Figure 3(d) presented the output power for each controlled joint axis. Subject A, B, and C were simulated spending different periods of time performing a repetition of front chin-up with the minimum energy expenditure. Figure 4 presented the simulation results. Discussion ime(s) A set of algorithm was designed in this study. o test the practicability of the algorithm, this study applied segmental parameters of three subjects to simulate the subjects doing A B C front chin-ups [4]. Figure 3(a) presented the graph for the optimal trajectory for the angular position of each controlled joint. here are four principal findings from the simulation results. First, it was found that the five curves showing the optimal trajectory for five joints are smooth with no peak value. his proves that the algorithm met the requirements of optimal trajectory. So the algorithm of this system is reliable in simulating the optimal trajectory of hand suspension motions such as flying rings, horizontal bar gymnastics, and rock climbing. Second, figure 3(b) presented the curves for the vertical force (Z-axis) and the horizontal force (X-axis) acted upon the axis center of hands grasping a horizontal bar. he vertical force increased initially and then decreased; this is due to the reacting force in the initial stage of front chin-up. he horizontal force was negative in value at first, then increased, and finally decreased. hat's because swing happened in the process of front chin-up. hird, figure 3(c) presented the loaded moment for each joint. It was found that elbow joints required the largest moment, and shoulder joints required the second largest moment. Fourth, figure 3(d) presented the output power of each joint. he output power of a certain joint was determined by the product of the joint's angular position multiplied by its moment obtained during the movement time. It was found that the elbow joints released the largest power, and the shoulder joints released the second largest power. It means that most of the power required in front chin-up derives from muscles of elbow and shoulder joints. But it also implies that the muscles of the two parts are most likely to get injured. he algorithm of this system is proved to be reliable in simulating the optimal trajectory of hand suspension motions such as flying rings, horizontal bar gymnastics, and rock climbing. o get a clear picture of energy expenditure, this study simulated front chin-ups performed with different periods of time. his study adopted Open-loop Linear Quardatic Controller as the control method. he whole controlling process was defined to have the initial state and the final state. he variables of the two states acted as the established boundary conditions [2]. he initial state for each subject had the same boundary conditions as their final state. Figure 4 presented the amount of energy expenditure for front chin-ups performed with different periods of time. he time spent on a repetition of front chin-up was set to be from.5sec to sec. It was found that the amount of energy expenditure for front chin-ups varied with different periods of time. he shorter the time spent on a repetition of front chin-up, the larger energy expenditure it required. he mechanical energy expenditure reached the maximum at.5sec and then dropped abruptly. he change of the curve became less dramatic when more than s was spent. At.5sec, the curve remained steady. It could be concluded that the three subjects of different shapes had better perform a repetition of front chin-up with the time between s and.5s. If they spend less than sec performing a repetition of front chin-up, they need to expend larger energy and the loaded moment for joints will become larger, too. his may lead to injury. If they spend more than.5sec performing a repetition of front chin-up, they expend smaller energy. But the

7 Optimal Planning for Front Chin-ups 35 time spent in supporting shoulder and elbow joints will become longer, and the muscles of other parts have to exert more energy to help complete the motion. his will easily cause the sore and pain of muscles. In reality, people move continuously in performing a motion. o cover the issue, this study suggests the use of sectional optimal control. he way for sectional optimal control is to set more time points in simulating a motion[7]. hen boundary conditions for these time points are established. Linking the sectional movements can form a complete trajectory. Setting appropriate values or boundary conditions is necessary, too. With boundary conditions set appropriately, the simulated optimal trajectory can better represent real human motions. References [] S. K. Powers, and E.. Howley, Exercise Physiology: heory and application to fitness and performance. 4th ed. New York: McGraw-Hill, 2,ch6: 98-. [2] R. Beckett, and K. Chang, An Evaluation of the Kinemetics of Gait by Minimum Energy, J. Biomechanics, :47-59, 968. [3] D. C. Lin, and C. J. Dillman, Optimal Stride Length in Running, Biomechanics in Sports., In J.erauds (Eds.), Research Center for Sports, , 983. [4] M. R. Ramey, and A.. Yang, A Simulation Procedure for Human Motion Studies. J. Biomechanics, 4:23-23, 98. [5] S. Y. Aleshinsky, and S. Y. Zatsiorsky, Human Locomotion in Space Analyzed Biomechanically hrough a Multi-Link Chain Model, J. Biomechanics. :-8, 978. [6] R. L. Huston, and C.E. Passerelle, he Mechanical of human body motion. In: Ghista, D.N. (Ed.), Human Body Dynamics: Impact, occupational, and athletic aspects. Oxford University, ,982. [7] R. Raikova,. A general approach for modelling and mathematical investigation of the human upper limb. Journal of Biomechanics., 25: , 992. [8] B.Burkett, J. Smeathers, and. Barker, A computer model to simulate the Swing phase of transfemoral prosthesis, Journal of Applied Biomechanics., 2: 25-37, 24. [9] C. S. Gregersen, and M.L. Hull, Non-driving intersegmental knee moments in cycling computed using a model that includes three-dimensional kinematics of shank/foot and the effect of simplifying assumptions, Journal of Biomechanics., 36: 83-83, 23. [] P. Pan,M.A. Peshkin,J.E. Colgate,and K.M. Lynch, Static single-arm force generation with kinematic constraints, J.Neurophysiol.,93: ,24. [] K. S. Fu, R. C. Gonzalez, and C. S. G. Lee, Robotics: control, sensing, vision, and intelligence. McGraw-Hill, New York,987,ch.3: [2] F. L. Lewis, C.. Abdallah, and D.M. Dawson, Control of Robot Manipulators. Macmillan, New York, 993,ch2: [3] C.H. Chiu, he study of the dynamic modeling for the support phase in running, PhD.thesis, National aiwan Normal University Press, 999. [4] V. Zatsiorsky, and V. Seluyanov, he mass and inertia characteristics of the main segment of the human body. Biomechanics Ⅷ-B., In: Matsui, H. Kobayashi, K.(Eds), Human Kinetics. Champaign, IL, 983, [5] F. L. Lewis, and V. L. Syrmos, Optimal Control. Wiley and Sons, New York, 995,ch3: [6] S. Y. Aleshinsky, An energy sources and fractions approach to the mechanical energy expenditure problem-Ⅰ. basic concepts, description of the model. analysis of a one-link system movement, J. Biomechanics., 9(4): , 986. [7] S. E. Engelbrecht, Minimum principles in motor control, Journal of Mathematical Psychology., 45: ,2. Appendix A Matrix Ji he J i matrix of L-E equations could be written as followed[]: J i = ( Ixx + Iyy + Izz) / 2 Ixy Ixz mixi Ixy ( Ixx Iyy + Izz) / 2 Iyz miyi Ixz Iyz ( Ixx + Iyy Izz) / 2 mizi he parameters for body segments in mixi miyi (2) mizi mi matrix comprised the mass mi, center of mass position r i = ( xi,y i,zi, ), moment of inertia I I I ), and products of inertia (Ixy, I xz, I yz) for the ith segment. J i ( xx, yy, zz

8 36 J. Med. Biol. Eng., Vol. 25. No.3 25