Posture Prediction and Optimization - Human Motion Simulation: Predictive Dynamics - page 62

in Figure 3.10 and corresponds to the z i 2 1 axis. Furthermore, the positions of
the target and end-effector must be given in terms of a common coordinate
system. For this purpose, a global coordinate system is chosen to correspond to
the zeroth coordinate frame of the model. Hence, the global position (0,0,0) is
located at the center of the torso. The end-effector for this model is a point on
the thumb given by the local position (L 8 , L 9 , 0) relative to the last coordi-

2

nate frame.

3.12 Optimization algorithm
To solve this problem we have used a number of commercial optimization soft-
ware programs. However, these programs have consistently resulted in a violated
distance constraint. In an effort to compensate for this, the distance constraint is
combined with the objective and given a large weighting factor. This forms the
following unconstrained objective:

0 X

nTIME

x

ð t j Þ

end - eff

2

path

f ð q Þ 5

: ð q ð t j ÞÞ 2 x

f discomfort 1 1000 (3.24)

j 5 1

Furthermore, additional terms to control the shape of the resulting joint profile
are added to the objective function. The first is a term to control the inconsistency
of the curve—that is, to prevent joint curves that produce back and forth move-
ments. This inconsistency is mathematically defined by:

!

X X

nTIME nDOF

f inconsistancy ð q Þ 5 ðj sign ð q i ð t j ÞÞ 2 trend i j 1 1 Þj q i ð t j Þj (3.25)

j 5 1 i 5 1

where

1if q i ð t Þ $ _ 0

sign ð _ q i ð t ÞÞ 5 (3.26)

1f q i ð t Þ , _ 0

2

and

1if ð q i 2 q i Þ $

0

trend i 5 (3.27)

1f ð q i 2

q i Þ ,

0

2

The second is a term to ensure smooth joint movement by minimizing the sec-
ond derivative of the joint profile. The non-smoothness added to the objective is
given by:

!

X X

nTIME nDOF

2

f nonsmoothness ð € q Þ 5 ð € q i ð t j ÞÞ (3.28)

j 5 1 i 5 1