Predictive Dynamics - Human Motion Simulation: Predictive Dynamics - page 107

Wrist

d

r 1

r 2

Hip

FIGURE 5.8
Self-avoidance constraint between the wrist and hip.

where subscripts L and R represent the DOFs of the leg, arm and shoulder joints which
satisfy the symmetry conditions with the contra-lateral leg, arm and shoulder joints;
the subscript S represents the DOFs of spine, neck and global joints which satisfy the
symmetry their conditions at the initial and final times; x , y , z are the global axes.

5.7.2.2 Ground clearance
To avoid foot drag motion, ground clearance constraint is imposed during the
walking motion. Instead of controlling the maximum height of the swing leg, the
maximum knee flexion at mid-swing is used to formulate ground clearance con-
straint. Biomechanical experiments have shown that the maximum knee flexion
of normal gait is around 60 degrees regardless of the subject's age and gender.
This constraint is expressed as

2 ε # q knee 2 60 # ε; t 5 t midswing (5.27)

where ε is a small range of motion, i.e., ε 5 5 degrees.

5.8 Discretization and scaling
The predictive dynamics problem in Equation (5.2) is actually an optimal control
problem with boundary conditions and some state constraints. The classical method
to solve the optimal control problem is to derive the optimality condition for the
continuous variable optimization problem. However, beyond boundary conditions
the continuous method has difficulty dealing with discrete state constraints. The
most efficient way to solve a complex optimal control problem is to use nonlinear
optimization techniques. The basic idea is to discretize the governing equations of
motion using a suitable numerical method and define finite dimensional approxi-
mation for the state and control variables. This process transforms the system dif-
ferential equations into algebraic equations with parametric representation of the
state and control variables. The performance measures and the constraints are also
evaluated in terms of discrete state and control values. Therefore, the original opti-
mal control problem is transformed into a nonlinear programming (NLP) problem.
The time domain is first discretized into n intervals with step size h i , as follows:

h i 0 t 0 # t 1 # ... t n 2 1 # t n 5 T and t i 1 1 2 t i (5.28)

5 5