Predictive Dynamics - Human Motion Simulation: Predictive Dynamics - page 101

5. Stability

ð T

f Sdt (5.11)

5

0

where S represents the stability quantity, which can be defined in different
ways. One definition is the deviation of ZMP position from the center of the
support polygon or a prescribed ZMP trajectory ( Huang et al., 2001; Xiang
et al., 2010b ). The second definition is the deviation of the trunk from vertical
position ( Gubina et al., 1974; Kim et al., 2008 ).
6. Maximum absolute value of joint torque

f max

i

fjτ i jg (5.12)

5

A common approach for treating cost function in Equation (5.12) is to
introduce an additional unknown parameter λ ( Rasmussen et al., 2001; Xiang

et al., 2009a ):

Min : λ

s : t : τ i # λ;

2 τ i # λ;

(5.13)

i ; ; ...; n

5 1 2

7. Dynamic effort as a cost function
A performance measure that is well studied in the biomechanics literature is
the minimizing of the squares of all actuating torques or minimizing the maxi-
mum torque for all joints.
The dynamic effort , which is represented as time integral of the squares of all joint
torques, is used as a performance measure to be minimized for the walking motion.
This is also sometimes called the total torque effort. The predicted motion depends
strongly on the adopted objective function F . As an example, dynamic effort some-
time used as the performance criterion for the walking problem can be written as:

ð T T dt

τð q tÞ

jτj max

; τð q tÞ

jτj max

;

Fð q Þ 5 (5.14)

U

t 5 0

where jτj max is the maximum absolute value of joint torque limit.

5.5 Inner optimization
An alternative way to define the performance measure is to use the inner optimi-
zation (nested optimization) method. The basic idea is to construct a local search
space of cost functions S J with a specific functional form based on some insight
into the physical processes governing the biosystem.
S J ð q

t j p ÞJð q

; τ; ; τ; t Þ (5.15)

where p is the parameter vector that needs to be determined.