Predictive Dynamics - Human Motion Simulation: Predictive Dynamics - page 103

Time Boundary

Boundary Set

Constraints

Geometrical Boundary

Response Set

FIGURE 5.5
Constraints for a biosystem.

angle at mid-swing time point, etc. The overall set of constraints is depicted in
Figure 5.5 .

5.6.1 Feasible set
The feasible set of solutions for the problem is an important issue for predictive
dynamics. An infeasible set will result in a null solution space for the system.
This situation should always be avoided while formulating a predictive dynamics
problem. For a biosystem, feasibility of all the constraints can be tested by solv-
ing the predictive dynamics problem with a constant objective function as
follows:

min
q; τ

Jð q ; τ ; t Þc

s : t : :

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

(5.19)

L U

# q # q

L U

τ #τ#τ

where c is a constant.
The solution of Equation (5.19) implies that the output set ðq f

f

Þ satisfies all lin-
ear and nonlinear constraints, but does not optimize any performance measure for the
biosystem. This is a feasible solution of the predictive dynamics problem. There are
two purposes of obtaining a feasible solution for the system: one is to test the feasi-
bility of all the constraints, and the other is to get a solution that might be used as a
good initial guess for the predictive dynamics with a physical performance measure.

; τ

5.6.2 Minimal set of constraints
It is obvious that the more information about the biosystem that is available, the
more accurate the predictive dynamics solution is. As an extreme case, all the dis-
placement and force histories can be available in the time domain, .

However, in most cases only minimal information about the biosystem is avail-
able so that predictive dynamics seeks the minimal constraint set g ðϒ minimal Þ and

Ω , Γ