Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 70

in more convenient calculation of joint torques, especially for a skeletal model,
compared with the Newton Euler formulation, which is based on the Cartesian
coordinates. Recursive dynamics will be introduced and sensitivity analysis will
be derived.
In this chapter, the recursive Lagrangian dynamics and sensitivity formulation
for the system are presented. Implementation aspects of sensitivity calculations
for the complex articulated human mechanism are addressed. The developed for-
mulation can systematically treat open-loop, closed-loop and branched mechanical
systems. In addition, the sensitivity analysis needed for the optimization process
is easier to implement. The formulation is based on DH transformation matrices
and external forces, and moments at any point of the mechanism are included in
the recursive formulation.
In order to demonstrate this formulation, an optimal time trajectory plan-
ning problem for a two-link human arm model is solved. Initial and final con-
ditions are given. Total

travel time is minimized subject to the joint torque

limits.
We begin by developing the equations of motion. We first formulate a general
static torque equation in vector form. This chapter is an adaptation of our work as
it has appeared in Xiang et al. (2009a,b, 2010a,b,c) .

4.2 General static torque
To obtain a relationship for the torque in terms of linear and angular velocity vec-
tors, we use the chain rule in joint space as follows. For the vector of angular
joint variables q q n ] T ; the Jacobian (J(q)) provides a direct relationship

between the velocity in joint and Cartesian spaces.
v
ω

5 [q 1 q 2 ...

5 J ð q Þ q (4.1)

where, v is the translational velocity of the end-effector and is the angular

velocity of the end-effector frame. J ð q Þ is the augmented Jacobian matrix
( Sciavicco and Siciliano, 2000 ) of the kinematic structure defined by

ω

J x
J ω

J ð q Þ 5

(4.2)

This also indicates that the virtual displacements have the similar relationship:
δ

5

x
δθ

J ð q Þδ ' δ q q (4.3)

δ δθ δ

where q are the virtual displacement vectors of Cartesian linear,

Cartesian angular, and joint variables, respectively.
We begin with the calculation of the torque at each joint. To account for all of
the elements that enter into calculating the torque at a given joint, we apply the

x, , and