Predictive Dynamics: Lifting - Human Motion Simulation: Predictive Dynamics - page 192

the knot points but also between the adjacent knots, so that a smooth motion can
be generated.
Since q ,

P Þ is an explicit
function of the knot vector and control points. Thus, the derivatives of a torque
with respect to the control points can be computed using the chain rule as
@τ
@

q , and _ q are functions of t and P, torque € τ 5 τð t ;

P i 5 @τ q @ q P i 1 @τ q @ q P i 1 @τ q @ q

(8.18)

@ @ @ _ @ @ € @ P i

Equations (8.10 8.12) and (8.18) are used to calculate accurate gradients for
the optimization formulation to improve the computational efficiency.

8.5.2 Objective functions
For the dynamic lifting motion prediction problem, two performance measures are
investigated. The first one is to minimize the dynamic effort, which is defined as
the time integral of the squares of all joint torques; the second one is to maximize
the stability, which can be transformed to minimize the time integral of the dis-
tance squares between ZMP and the foot support boundaries. The weighted sum of
the two objective functions is used as the performance measure as follows:

ð T
t 5 0 τ ð P

w 2 N ð T
t 5 0

X dt

nb

T 2

i

Fð P Þ 5 w 1 N ; tÞ

τ ð P

; tÞ dt sðtÞ (8.19)

1

i 1

5

where s i is the distance between ZMP and the i th foot support boundary at time- t ;
nb is the number of foot support boundaries; w 1 and w 2 are weighting coefficients
for the two objective functions ranging from 0 to 1, respectively; and

w 1 1 1. The symbol N ðÞ is a normalization operator. A general function-

transformation method ( Marler and Arora, 2005 ) is used to determine the normali-
zation operator in Equation (8.19) .

w 2 5

8.5.3 Constraints
We shall consider two types of constraints for the predictive dynamics task of lift-
ing. One type is the time-dependent constraints, which include joint limits, torque
limits, ground penetration, dynamic balance, foot locations, vision, hand orientation,
and collision avoidance. These constraints are imposed throughout the time interval.
The second type is time-independent constraints, which comprise the initial and
final box locations and the initial and final static conditions; these constraints are
considered only at the starting and ending time points for the lifting motion.
For the lifting task, joint angle limits, torque limits, ground penetration, foot
locations, and dynamic balance constraints are detailed by Xiang et al. (2009a,b) ,
and a symmetric walking motion is simulated using a one-step formulation. The
vision, hand orientation, collision avoidance, initial and final box locations, and