Computer Algorithm of Optimal Planning for the Support of a Single Leg

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1 International Journal of Sport and Exercise Science, (): -8 Computer Algorithm of Optimal Planning for the Support of a Single Leg Ching-Hua Chiu * Graduate Institute of Sports & Health Management, National Chung Hsing University, Taichung, Taiwan, 42, ROC Received 29 Oct 28; Accepted 28 Dec 28 Abstract Walking, umping, and running (WJR) are inborn abilities of human beings. Since these three motions are indispensable to our daily lives, it is worthy of our exploration to enhance their exercise efficiency. In terms of biomechanics, performing these three motions involves the support from a single leg and maintaining the coordination and balance of body movements. Consequently, this study attempted to provide a set of algorithm for the support from a single leg, and thus establish mechanical minimum-energy expenditure (MMEE) control system for WJR. To formulate the algorithm, this study adopted seven types of homogeneous transformation matrices (CH-7T), Lagrange-Euler equations of motion (LEEM), and minimum-energy control (MEC). The simulation results showed that under the boundary conditions for the initial and the final states of the support phase for a single leg, the set of algorithm could compute the optimized traectory, generalized force, and the MMEE for each controlled segment. The results proved that the adoption of CH-7T, LEEM, and MEC was successful in establishing an MMEE control system for the support phase of a single leg. Consequently, the designed algorithm and the system are capable of developing an optimal control model for the support phase of a single leg in RJW. Keywords: Running, Boundary conditions, Generalized force, Optimal control Introduction Since the stored energy of the human body is limited, the energy expenditure must be reduced to the minimum to ensure an extended time of activity. For example, marathon runners, marching soldiers, and astronauts who stay in the outer space for a long period of time need to save as much energy as possible. These situations suggest that it is necessary to design a control system of mechanical minimum-energy expenditure (MMEE). Among the everyday activities, walking, umping, and running (WJR) are the most common. For the three activities, the support from one single leg can provide energy to move the body. Therefore, the researcher in this study attempted to design a control system of mechanical minimum-energy expenditure (MMEE) for one single leg in the support phase. In the past, researchers adopted various ways to design a dynamic system for human body, such as Newton-Euler equations of motion [], Generalized d Alembert equations of motion, and Lagrange-Euler equations of motion (LEEM) [2]. Equations of human motions comprise equations of the whole human body [3-7], equations of arm motions [8], equations of leg motions [9-], and the dynamic system of a single oint[]. * Corresponding author: Ching-Hua Chiu Tel: Fax: chungoodman@yahoo.com.tw Due to the large number of human oints, the dynamic system for one single leg in the support phase is very complicated. In designing the dynamic system, this study adopted the human body model proposed by Zatsiorsky and Seluyanov [2]. Based on the 5 segments in this model, this dynamic system was defined to have 38 degrees of freedom (DoF). The generalized coordinate, generalized velocity, and generalized acceleration for each DoF acted as input variables in establishing the equations of the dynamic system. In the past, the LEEM which were used to establish a dynamic system comprised Denavit-Hartenberg (D-H) matrix [2]. The D-H matrix has its advantages. For one thing, this matrix can present equations in a simple way. For another, with this matrix, LEEM can be transformed into a control system. However, a dynamic system which adopts the D-H matrix can only be applied to a single kinematic chain of robot arm motion. Conversely, the human body, characterized by its branching segments, forms a multi-kinetic chains system. Consequently, D-H matrix is not suitable to establish equations for human motions [4]. Recently, the researcher of this study has successfully established an MMEE control system for the lower and upper limbs of the human body by adopting CH-7T [4, 3-4], LEEM, and MEC. The results of that study showed that moving the arm in the optimized traectory simulated through computer could reduce 7% of mechanical energy expenditure, and that 9% of energy expenditure for the leg motion could be reduced. Inspired by this previous study, the researcher

2 2 International Journal of Sport and Exercise Science, Vol.. No. 29 attempted to an optimal control system for WJR in this study, which can reduce the mechanical energy expenditure for WJR and enhance the exercise efficiency. The researcher adopted CH-7T, LEEM, and MEC to design a set of algorithm for the support from a single leg, and thus form the future foundation of establishing an optimal control system for WJR. Nomenclature Materials and Methods τ i =generalized force q i =generalized coordinate q& i =generalized velocity q& & i =generalized acceleration i- A i =the homogeneous transformation matrix of the ith relative to the i-th coordinate frame(appendix A) A =the coordinate transformation matrix from coordinate frame to the th, and the equation could be written as A = A A 2 2 A 3 - A g =[,,- g,], g=9.862 m/s 2 m = the mass of the th link r =( x, y, z,) T, position of the center of mass J = pseudo-inertia matrix (Appendix B) τ(t) =[τ(t), τ2(t),..,τ38(t)] T, an 38 generalized force vector q (t) =[ q ( t), q 2 ( t),, 38( t ), an 38 vector of the oint variable q& (t) =[ q& ( t), q& 2( t),, q& 38( t ) ]T, an 38 vector of the oint velocity u(t) =[ ( t ) 2 ( 38 (, an 38 vector of the acceleration M( q (t)) =an38 38 inertial acceleration-related symmetric matrix V( q (t), q& (t)) =an 38 nonlinear Coriolis and centrifugal force vector G( q (t)) =an 38 gravity loading force vector Fig. human body model. Subect The subect was a male student in the department of physical education. The subect was in good shape. He was aged 2, 8cm in height, and 75kg in weight. The segment lengths, CoM(r ), the mass of segment(m ), and the inertia of the subect were measured []. Human body model In this study, the human body was assumed to be a right body with no friction between segments. Based on a 5-segment body model, this study designed a three dimensional MMEE control system with 38 DoF (Fig. -2[4,3,4]).The right toe tip was represented by O, and O7 represented the left toe tip. The remaining O i (i=,,6, 8,,5) represented oints. The CoM position for each segment was represented with G i (i=,2,,5). In Fig., l i represented body segment lengths. Lengths between oints were represented with Fig 2. The translation from the origin of the ith link relative to the i-th link p i and the CoM for th link, r.

3 International Journal of Sport and Exercise Science, (): -8 3 the symbol s as followed: s, length from right hip oint to the central point of two hip oints; s 2, length from left hip oint to the central point of two hip oints; s 3, length from the central point of two shoulder oints (CPTSJ) to right shoulder oint; s 4, length from the CPTSJ to left shoulder oint; s 5, length from the CPTSJ to waist oint O 8 [3]. Dynamics System q i q q 2 q 3 i- A i A =T t-x,q A 2 =T t-y,q 2 A 3 =T t-z,q Table. The homogeneous transformation matrix i- A i and A A Transformation A = A Transformation of right hand B relative to the base A 2= A 2 A 3= A 3 q 4 3 A 4 =T y,q A 4= A 4 q 5 4 A 5 =T z,q A5= A5 q 6 5 A 6 =T x,q A6= A6 q 7 q 8 q 9 q q q 2 6 A 7 =T y,q A 7= A 7 Transformation of right forearm B 2 relative to the base 7 A 8 =T z,q A 8= A 8 8 A 9 =T y,q A 9= A 9 Transformation of right upper arm B 3 relative to the base coordinate system 9 A =T z,q A = A A =T x,q A= A Transformation of upper trunk B 5 relative to the base A 2 =T y,q A2= A2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 2 q 2 q 22 q 23 q 24 q 25 q 26 q 27 q 28 q 29 q 3 q 3 q 32 q 33 q 34 q 35 q 36 q 37 q 38 2 A 3 =T y,q 3 A 4 =T x,q 4 A 5 =T z,q 5 A 6 =T y,q 6 A 7 =T y,q 7 A 8 =T z,q 8 A 9 =T y,q 9 A 2 =T x,q 2 A 2 =T z,q 2 A 22 =T y,q 22 A 23 =T x,q 23 A 24 =T z,q 24 A 25 =T y,q 25 A 26 =T x,q 26 A 27 =T z,q 27 A 28 =T y,q 28 A 29 =T z,q 29 A 3 =T y,q 3 A 3 =T x,q 3 A 32 =T y,q 32 A 33 =T x,q 33 A 34 =T z,q 34 A 35 =T y,q 35 A 36 =T z,q 36 A 37 =T y,q 37 A 38 =T x,q A 3= A 3 A4= A4 Transformation of lower trunk B 6 relative to the base A5= A5 A 6= A 6 A 7= A 7 Transformation of right thigh B relative to the base A 8= A 8 A 9= A 2 2 S 8 8 A 9 A 2= A 2 2 S 8 8 A 2 Transformation of right shank B relative to the base A 2= A 2 2 S 8 8 A 2 Transformation of right foot B 2 relative to the base A 22= A 2 2 S 8 8 A 22 A 23= A 2 2 S 8 8 A 23 Transformation of left thigh B 3 relative to the base A 24= A 2 2 S 8 8 A 24 A 25= A 2 2 S 8 8 A 2 2 S A 25 A 26= A 2 2 S 8 8 A 2 2 S A 26 Transformation of left shank B 4 relative to the base A 27= A 2 2 S 8 8 A 2 2 S A 27 Transformation of left foot B 5 relative to the base A 28= A 2 2 S 8 8 A 2 2 S A 28 A 29= A 2 2 S 8 8 A 2 2 S A 29 Transformation of left upper arm B 7 relative to the base A 3= A 2 2 S 8 8 A 2 2 S A 3 A 3= A 2 2 S 8 8 A 2 2 S A 3 A 32= A 2 2 S 8 8 A 2 2 S 3 3 A 32 Transformation of left forearm B 8 relative to the base A 33= A 2 2 S 8 8 A 2 2 S 3 3 A 33 A 34= A 2 2 S 8 8 A 2 2 S 3 3 A 34 Transformation of left hand B 9 relative to the base A 35= A 2 2 S 8 8 A 2 2 S 3 3 A 35 A 36= A 2 2 S 8 8 A 2 2 S 3 3 A 36 Transformation of the head B 4 relative to the base A 37= A 2 2 S 8 8 A 2 2 S 3 3 A 37 A 38= A 2 2 S 8 8 A 2 2 S 3 3 A 38 In A i, wh e n i =,.., 8, v ec tor p 2 = [, s +s 2, ] links t he B 4 and B 5 oi n t. In A i, w h e n i=9,, 3 8, v ec t or p 2 = [, s, l 4] li n k s the B 4 an d B 8. In A i, w h en i=,, 24, v e ct o r p 2 = [,, l 8] lin ks t h e B 8 a n d B 9 oint. In A i, w h e n i= 25,, 3, ve c tor p 2 = [, -s 3, s 5] lin ks th e B 8 a n d B oin t. In A i, wh e n i=32,,38, ve c tor p 2 = [, s 4, s 5 ] links t h e B 8 a nd B 3 oint.

4 4 International Journal of Sport and Exercise Science, Vol.. No. 29 I x& (t) = x ( t ) + u ( t ) I (3) In Eq. (3), the state variable was represented with x(t)=[ q(t) T q& (t) ] T, I represented a identity matrix. u(t) represented a control input: u(t)=-m - ( q (t) )[V( q (t), q& (t) ))+G( q (t) ))]+M - ( q (t) )τ(t) (4) T Fig 3. OLKCs for the support phase of a single leg. Starting from the origin of the base (X,Y, Z ), four OLKCs were established with the right toe tip as the support point.the first OLKC consisted of B, B 2, B 3, B 4, B 5, B 6, and B 7. The second OLKC consisted of B, B 2, B 3, B 4, B 8,and B 9.The third OLKC consisted of B, B 2, B 3, B 4, B 8, B, B, and B 2 The fourth OLKC consisted of B, B 2, B 3, B 4, B 8, B 3, B 4, and B 5. A coordinate system was presented in this study, and the origin of this coordinate system was represented with (x, y, z )(Fig. -2). Due to the structure of human body oints, the dynamic system in this study was defined to comprise 38 DoF. They acted as variables of this dynamic system. Each DoF involves a. Starting from the origin of the XYZ, four Open-Loop kinematic chains (OLKCs) were established with the right toe tip as the support point (Fig. 2-3).By applying the four OLKCs, the researcher once used CH-7T (Table [4,3]), Appendix A) and LEEM to establish the dynamic system for the support from a single leg [2, 4, 5]: n τ i = = i n = i n = i k = i k = T Trace( W J W ) q& & k + m= k Trace( W km J i W T i ) q& k q& m ( m gw i r ) i=,2,3,.,38 () In Eq. (), W k was defined as A / q k (i=,2,3,,38). W i was defined as A / q i (i,=,2,3,,38). W km was defined as W k / q m (i,k,m=,2,3,,38). The definition and the usage of the coordinate transformation matrix A could be found in Table (Appendix A). Minimum-Energy Control The equation of motion () could be written in a matrix form as followed [5-6]): τ(t)=m( q (t)) u +V( q (t)), q& (t) )+G( q (t)) (2) The equation of dynamics for the support phase was then transformed into the state-space of the control system (Lewis et al., 993[5]): By adopting the MEC, this study divided the process of simulating the motion into two time points. The first time point was defined as t with x(t ) being the initial state. The second time point was defined as tf with x(t f ) being the final state. The state variables x(t ) and x(t f )were defined as established boundary conditions. The cost function of the minimum-energy control during the interval of t [t, t f ] could be written as followed [6]: t J min = 2 f t u(t) T R u(t) dt (5) R was a symmetric control weighting matrix (R=[I]). Following the standard procedures of MEC in solving problems, this study utilized Eq. (3)-(5) under established boundary conditions, x(t ) and x(t f ), to obtain optimal control input u*(t) and optimal traectories x*(t)= [q*(t) T q& *(t) T ]. Next, the resulting u(t) and x(t) were planted in Equation (2) to obtain generalized force τ(t). With the computed optimal traectories x*(t) and generalized force τ*(t), the MMEE for the ith link during the period from t to t f was defined as E i, which could be written as followed []: t f Ei = t τ i *(t) q& i *(t) dt i=,2,3,,38 (6) Example The subect was simulated standing in a rising elevator with his right foot on the elevator floor, his left knee bent, and his upper trunk leaning forward. Then he straightened his left leg and waist, and both arms were lifted from the sides of the body and remained elevated. Since boundary conditions are needed in the simulation, this study referred to the real support motion of a single leg to find out the boundary conditions for the six main oints involved in the motion (q 3, q 3, q 6, q 9, q 26, q 33 ). Then the simulation was conducted under these boundary conditions (Appendix B). Data Analysis The computer program language Borland C ++ (Builder 5. version) was used to edit the designed algorithm into applied software. The reason why Borland C ++ was adopted was that it is an obect-oriented programming, easy for users to develop other applied software. The input (Table 3 [6]) and output parameters involved in the calculation were as

5 International Journal of Sport and Exercise Science, (): -8 5 followed: Input data: () the relative position vector between two ends of a certain body segment (p i ). (2) the mass of the th link (m ). (3) the CoM for th link (r ). (4) inertia (I i ). (5) initial state x(t ); t. (6) final state x(t f ); t f. Output data: () optimal traectories x*(t). (2) optimal control u*(t). (3) generalized force τ*(t). (4) MMEE for ith link E i. Results Fi g 4. Th e s i mu la ted m ot ion i n a ri s ing e le va t o r. A c om p u ter w as u sed to s im u lat e th e MM E E o f t h is m ot ion. Th e s i mulation t o o k. 8 s an d wa s d ivided in to n ine p h as es. Th e i n t erva l b e twee n tw o cont in u o u s p h as es wa s.s. (a ) t h e ini ti al s tate t = s (i) th e fi n a l s tate t f =.8s. In this study, a subect was simulated standing on one leg in a rising elevator. Fig. 4(a)[4] presented the initial state of the simulated motion. In the initial state t =s, the subect stood in the elevator, and his right toe was at the position of.m on the Z-axis of the OX Y Z coordinate system. Fig. 4(i) presented the final state of the simulated motion. In the final state t f =.8s, the right toe was at the position of.8m on the Z-axis of the OX Y Z coordinate system [4].

6 6 International Journal of Sport and Exercise Science, Vol.. No. 29 Ti me (s ) Ta b le 2. Sim u lat i o n o f M M E E L Hi-Y L Kn -Y Wa -Y L Sh -X E 3(J ) E 6 (J ) E 9(J ) E 3 3(J ) RS h -X E 2 6( J).6 (t =, t f=.6) (t =, t f=.8) (t =, t f=) L Hi -Y: le ft h i p oi n t ; L K n -Y: left k n e e oin t; Wa -Y: w ai s t oi n t ; R Sh -X : ri gh t sh o u lder oi n t (X - ax is ); L Sh - X : le ft s h ou ld er o in t[4]. Fig 5. (a) The motion traectory for the right toe on the Z-axis. (b) The loaded force for the right toe on the Z-axis[4]. Fig. 5(a) presented the motion traectory of the right toe. Fig. 5(b) presented the loaded vertical force of the right toe τ 3 = [362N, 4N]. Fig. 6(a) presented the angular positions for left hip oint, left knee oint, right shoulder oint, and left shoulder oint. Fig 6(b) presented the moments for left hip oint, left knee oint, right shoulder oint, and left shoulder oint. Fig. 6(c) presented the angular position for the waist. Fig. 6(d) presented the moment for the waist[4]. Fig 6. (a) Graphs for the changing angular position of each oint: L Hi-Y, the graph for left hip oint, and the angular position q 3 ; L Kn-Y, the graph for left knee oint, and the angular position q 6 ; R Sh-X, the graph for the right shoulder oint, and the angular position q 26; L Sh-X, the graph for left shoulder oint, and the angular position q 33. (b) The moment for left hip oint, left knee oint, right shoulder oint, and left shoulder oint. (c) The angular position for the waist q 9 ( Wa-Y). (d) The moment for the waist (Wa-Y) [4]. The MMEE of moving in a rising elevator for.8s was calculated under the above-mentioned established boundary conditions for five human body oints(table 2)[4]. To obtain the minimum energy expended for the same motion performed within a shorter or longer period of time, this study also simulated the same motion for.6s and.s. Table 2 presented the MMEE for each of the five oints[4]. Discussion In recent years, many researchers have attempted to develop a dynamic system. Their accomplishments can be attributed to the advance of computer hardware and numerical methods. Following the trend, this study was also conducted with computer hardware and numerical methods. The primary reason why this study adopted LEEM to establish the equations for human motions is that LEEM has an intact structure, making it easy to design the state-space of a control system [2, 6, 7, 8]. Applying this dynamic system, this study simulated the subect moving in a rising elevator for.8s (Fig. 5(a)). The vertical force imposed on the right toe τ 3 was larger in the beginning, amounting to 362 N. The force decreased gradually to 4 N (Fig. 5(b)). In the dynamic system, generalized forces τ, τ 2, and τ 3 were external forces from the elevator imposed on the human body. It meant that this amount of energy derived from the elevator, not from the human body. Because of this, energy expenditure E, E 2, and E 3 couldn't be counted in the mechanical energy expenditure of human body. The energy produced by the human body oints were represented with the symbols from E 4 to E 38 []. Fig. 6 presented the graphs of the changing angular position for each of the five oints. The angular position for left hip oint was defined as q 3 [-2, ]. The loaded moment of left hip oint was less than 8Nm (Table 2), and its MMEE was.9 J. The angular position for left knee oint was defined as q 6 [9, ]. Its moment was larger than that of left hip oint, and its MMEE was 9. J. The angular position for the right shoulder oint was defined as q 26 [, -35 ]. The angular position for the left shoulder oint was defined as q 33 [, 35 ]. The right shoulder oint obtained as much moment and MMEE as the left shoulder oint. The MMEE a shoulder oint expended was 3.25 J. The angular position for the waist oint was defined as q 9 [6, ]. The MMEE the waist required was 69.59J. Table 2 indicated that when the subect moved in a rising elevator for.8s, the waist oint required the largest energy

7 International Journal of Sport and Exercise Science, (): -8 7 expenditure, the shoulder oint required the second largest energy expenditure, and the left hip oint required the smallest energy expenditure. Two other simulations were done to make a comparison among three different periods of rising time:.6s,.8s, and.s. The result from the algorithm showed that the rising time of.6ses required the largest energy expenditure to accomplish the same motion. The rising time of.s required the smallest energy expenditure. In conclusion, with longer time spent in performing the motion came the less energy expenditure. More energy was expended when shorter time was spent in performing the motion. The researcher once applied the same methodology to a previous study on front chin-ups of three subects [3]. The above-mentioned conclusion was similar to the results obtained from this previous study on front chin-ups. The above simulation results showed that under the boundary conditions for the initial and final states of the support phase for a single leg, this set of algorithm could locate the optimal traectory and compute the MMEE for each controlled segment. The results proved that the adoption of CH-7T, LEEM, and MEC was successful in establishing an MMEE control system for the support phase of a single leg. Consequently, the designed algorithm and the dynamic system are capable of developing an optimal control model for the support phase of one leg in WJR. Reference [] Lee, S.H., Kim,., Park, F.C., Kim, M., and Bobrow, J.E. (25). Newton-Type Algorithms for Dynamics-Based Robot Movement Optimization. IEEE Transactions on Robotics, 2( 4), [2] Fu, K.S., Gonzalez, R.C., and Lee, C. S. G. (987). Robotics: control, sensing, vision, and intelligence. McGraw-Hill, New York. [3] Huston, R.L. Passerelle, C.E. (982). The mechanics of human body motion. In: Ghista, D.N. (Eds.), Human Body Dynamics: Impact, occupational, and athletic aspects. Oxford University, pp [4] Chiu, C.H. (999). The study of the dynamic modeling for the support phase in running. Ph.D. Dissertation, National Taiwan Normal University. [5] Hatze, H. (2). The fundamental problem of myoskeletal inverse dynamics and its implications. Journal of Biomechanics, 35, 9-5. [6] Chiu, C.H. (25a). The preliminary study of optimal planning for front chin-ups. Journal of Medical and Biological Engineering,25(3), [7] Park, W., Chaffin, D.B., Martin, B.J., and Faraway, J.J. (25). A computer algorithm for representing spatial-temporal structure of human motion and a motion generalization method. Journal of Biomechanics, 38, pp [8] Raikova, R. (992). A general approach for modeling and mathematical investigation of the human upper limb. Journal of Biomechanics, 25, [9] Burkett, B.S., Meathers, J., and Barker, T. (24). A computer model to simulate the Swing phase of transfemoral prosthesis. Journal of Applied Biomechanics, 2, [] Gregersen, C.S., and Hull, M.L. (23). Non-driving inter segmental knee moments in cycling computed using a model that includes three-dimensional kinetics of shank/foot and the effect of simplifying assumptions. Journal of Biomechanics. 36, [] Aleshinsky, S.Y. (986). 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), [2] Zatsiorsky, V., and Seluyanov,V. (983). The mass and inertia characteristics of the main segment of the human body. In: Matsui, H., Kobayashi, K.(Eds), Biomechanics Ⅷ-B, Champaign: Human Kinetics Publisher, pp [3] Chiu, C.H., (25b). Simulation of positional center of gravity for different human motions, Journal of Medical and Biological Engineering, 25(3), [4] Chiu, C.H. (26). Optimization of Human Motions. Taiwan: Yee Lee Book Publisher, ch.7:-26. [5] Lewis, F.L., Abdallah, C.T., and Dawson, D.M., 993, Control of Robot Manipulators. Macmillan, New York, pp [6] Lewis, F. L., and Syrmos, V. L. (995). Optimal Control. Wiley and Sons. New York, pp [7] Engelbrecht, S.E. (2). Minimum principles in motor control. Journal of Mathematical Psychology, 45, [8] Chiu, C.H., Sun, T.C., Kao, M.F., Lee, S.S., Shu, K.H., and Wong, T.L. (27). A Numerical Estimation of Energy Expenditure for Wheelchair User, Journal of Medical and Biological Engineering, 27(2), Appendix A The homogeneous coordinate transformation matrices applied in LEEM came from CH-7T proposed by Chiu [4, 3, 4]. q i is a oint variable associated with the ith link, and it was assumed that cq i =cosq i, sq i =sinq i. The translation from the origin of the ith link relative to the i-th link was represented with pi and, p i = ( x i, yi, zi ) T (Fig. 2). CH-7T was defined as followed: T x,q = T y,q = T z,q = cqi sqi cqi sqi cqi sqi sqi cqi sqi cqi sqi cqi q i T t-x,q = T t-y,q = T t-z,q = q i q i x i y i z i xi yi zi x i y i z i (A) (A2) (A3) (A4) (A5) (A6)

8 8 International Journal of Sport and Exercise Science, Vol.. No. 29 i- S i = (A7) i- S i is a 4 4 identity matrix, m S n = m S m+ m+ S m+2 n-2 S n- n- S n (m and n are positive integers, n>m) Appendix B Matrix Ji could be written as followed[2]: ( Ixx+ Iyy+ Izz)/2 Ixy Ixz mx i i J i = Ixy ( Ixx Iyy+ Izz)/2 Iyz my i i Ixz Iyz ( Ixx+ Iyy Izz)/2 mz i mx i i my i i mz i i mi (B) This Ji matrix requires parameters for human body segments. In Table 3 [4], the moments of inertia for the link i were defined as (I xx, I yy, I zz ), and cross-products of inertia were defined as (I xy, I xz, I yz ). i x i Table 3. The subect's segment parameters, the initial state x (t ), a n d th e fi n a l s tate x (t f). p i(m) r i(m) Ii(kgm 2 ) x(t ) x(t f) y i z i x i y i z i m i(kg) I I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 q i q& i q i q& i m m m m 2 m m m m 3 m m.5m m x y 2 z x y 2 z In A i, when i=,..,8, p 2=[,.268, ] T. In A i when i=9,,38, p 2=[,.34,.7] T. In A i when i=,,24, p 2=[,,.436] T. In A i when i=25,,3, p 2=[, -.93,.46] T. In Ai when i=32,,38, p 2=[,.93,.46] T. I = (-I xx + I yy + I zz )/2, I 2 = I xy, I 3 = I xz, I 4 =I xy, I 5 = (I xx-i yy + I zz )/2, I 6 = I yz, I 7 = I xz, I 8 = I yz, I 9 = (I xx + I yy - I zz )/2.