Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 75

where

T !

X

n 0 T j ð q Þ

@

0 T j ð q Þ
@

Tr @ I j @

M ik ð q Þ 5 i ; k 1 ; 2 ; ...; n (4.24)

5

q k q i

j 5 max ði;kÞ

! T !

X X X

2 0 T j ð q Þ
@

n n n

Tr @ I j @

0 T j ð q Þ
@

; q Þ 5 q k _

_

V i ð q q m i ; k ; m 5 1 ; 2 ; ...; n

q k @ q m q i

k 5 1 m 5 1 j 5 max ði;k;mÞ

(4.25)

0 T i
@

J i 5 @ r i (4.26)

q i

0 T s
@

J s 5 @ r s (4.27)

q s

where J i is the Jacobian matrix for link i and r i is the position of the center
of mass of link i with respect to the i th coordinate system; I j is the aug-
mented inert i a matrix for link j ; J s is the Jacobian matrix for the external
load f s and r s is the position where external load is applied in the s th

coordi-

nate system.

4.4.1 Sensitivity analysis
Sensitivity analysis means calculation of derivatives of various quantities with
respect to the state variables. We note from Equations (4.6 4.10) that the regular
Lagrangian equations are coupled, nonlinear, and second-order differential equa-
tions. M ð q Þ is an n 3 n matrix and V ð q ; q Þ is an n 3 1 vector. Each term involves
summation and state variables. Direct sensitivity analysis gives the n 3 n sensitiv-
ity matrix as

q 1 @ P i J i m i g

1 @ J s f s
@

@τ
@ q 5 @ M ð q Þ q q 1 @

V ð q ; q Þ
@

(4.28)

@ @ q q

4.5 Recursive Lagrangian equations
We begin with a general formulation for forward recursive kinematics. Recursive
dynamics methods have been shown to allow for efficient simulation of dynamic
systems with large DOFs regardless of whether they are open, closed, or branched
loops. Recursive dynamics averts the need for the reformulation of the dynamic
equations for human systems. Furthermore, recursive dynamics provide for an
increased stability in numerical performance.