Predictive Dynamics - Human Motion Simulation: Predictive Dynamics - page 111

J 2 ðq; τ; tÞ 5 max

t A ½ 0 ;T τ

(5.36)

J 3 ðq ; τ; tÞ 5 T (5.37)

J 4 ðq ; τ; tÞ 5 c (5.38)

The first performance measure is to minimize the integral of squares of the
joint torque for the entire time domain, which is a form of mechanical energy.
The second is to minimize the maximum torque over the entire time domain, and
the third is to minimize the total travel time, T , subjected to the same boundary
conditions. The final performance measure is to solve for only a feasible solution
where c is a constant.
The optimization problem is discretized into a nonlinear programming prob-
lem and then solved by SNOPT with various performance measures as defined
in Equations (5.35 5.38) . The optimal solution yields the joint angle, velocity,
and external torque history as depicted in Figures 5.11, 5.12, and 5.13 ,

respectively.
Figures 5.12 and 5.13 show that the performance measures torque-square
and min max torque successfully predict joint angle and velocity response.
However, only torque-square predicts joint torque correctly. Minimizing the
total time or a constant fails to predict the response of the dynamic system. This
is explained by the fact that the natural motion always obeys an energy-saving
rule so the energy-related performance measure is more appropriate for predict-
ing the dynamic motion.

0.0

-0.5

-1.0

-1.5

Torque square
Constant
Min-max
Minimum time
Forward dynamics

-2.0

-2.5

0.0 0.1 0.2 0.3 0.4

Time (s)

FIGURE 5.11
Joint angle prediction of the single pendulum, Case 1.