Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 81

equation of motion. Thus, the derivatives of a torque with respect to the control

points and knot points can be computed using the chain rule as:

τ

@τ
@ p i 5 @τ q @ p i 1 @τ

q

q @ _ p i 1 @τ

q

q @ €

q

(4.58)

@ _ @ € @ @ @ @ p i

@τ
@ t i 5 @τ q @ q t i 1 @τ q @ q t i 1 @τ q @ q

(4.59)

@ @ @ _ @ @ € @ t i

4.6 Examples using a 2-DOF arm
We first derive the recursive Lagrangian sensitivity equations for the 2-DOF
system. We then use these derivations in numerical examples to illustrate the
use of the Lagrangian recursive formulations in solving general-purpose

problems.
The first problem is the time-optimal trajectory-planning design
without gravity and external forces. The reason for using this example is to
validate the numerical results, as these solutions have been well studied and
are readily available in the literature ( Dissanayake et al., 1991 ; Wang et al.,
2005 ).
The second problem is to optimize a lifting motion of the arm with a mixed
performance criterion. Both gravity and external force are considered. Sensitivity
results with the recursive algorithm and the closed-form formulation are numeri-
cally compared.
For the purpose of demonstrating the formulation for dynamics, consider a
2-DOF arm constrained to move in the vertical plane—shown in Figure 4.2 (verti-
cal motion only to reduce the complexity for this example while taking gravity
into consideration). We shall first derive the recursive Lagrangian sensitivity
equations for the arm. This system consists of two links whose lengths are L 1 and
L 2 , and moments of inertia are I 1 and I 2 as shown in Figure 4.2 . The relative joint
angles are denoted by q 1 and q 2 , respectively, and are controlled by the joint actu-
ating torques τ 2 . The two segmental links of the arm are considered rigid,

and the relative joint angles are selected as independent generalized coordinates.
The system is assumed to lie in the vertical plane restricting the motion to 2
DOFs. Actuator torques (muscle actions) drive the arm from the initial position
( q 1 ð 0 Þ;

τ 1 and

q 2 ðTÞ ) in the time interval T . In addition,
the arm is at rest at the initial and final points. The data for the arm are given in
Table 4.1 .
This arm can be modeled as a planar kinematic chain as shown in Figure 4.3 ,
where l 1 and l 2 are the distances from the local coordinate system to the center of
mass location.

q 2 ð 0 Þ ) to the final position ( q 1 ðTÞ;