Predictive Dynamics - Human Motion Simulation: Predictive Dynamics - page 108

τ h can be expressed in terms of interpolating
functions and discrete nodal DOF (control points) P q and P τ .
q h 5

The discretized state q h and force

q ð P q ; hÞ; τ h 5 τð P τ ; hÞ (5.29)

Thus, the discretized predictive dynamics problem is formulated as:

min
q; τ

Jð q h ; τ h ; t h Þ

s : t : :

τ h 2 f ð q h ; q h ; q h ; t h Þ 5 0
g ðϒ h Þ # 0
q

(5.30)

L U

# q h # q

L U

τ #τ h #τ

In general, all the unknowns and the equations of motion should be scaled to
improve the numerical performance of the nonlinear optimization solver.
Appropriate scale factors are chosen so as to obtain quantities that have the same
magnitude order, Oð 1 Þ . It is noted that scaling of a constraint does not change the
constraint boundary, so it has no effect on the optimum solution.

Jðs q s τ τ h ; s t

min
q; τ

q h ; t h Þ

: s τ τ h 2 f ðs q s q s q s t

q h ; q h ;

s : t : q h ; t h Þ 5 0

(5.31)

g ðϒ h Þ # 0
s q

L

s q s q

U

q q h # q

#

s τ τ

L

s τ τ h # s τ τ

U

#

5.9 Numerical example: single pendulum
Before applying predictive dynamics to a large problem, such as a human with
many DOF, we will examine the method's implementation in a simpler and more
basic problem to illustrate its power. The example below considers a simple
pendulum.

5.9.1 Description of the problem
The natural swinging motion of a single pendulum subjected to external torque is
considered. The swinging motion is first treated as a forward dynamics problem
with the known external force and solved by the multi-body dynamics solver
ADAMS. The solution is assumed to be the true response of the system used to
evaluate the results obtained with the predictive dynamics formulation. Predictive
dynamics is implemented based on the available information about the system.
Four cases are examined with predictive dynamics as listed in Table 5.2 .