Biomedical Engineering Reference
In-Depth Information
where
A
d
zmp
is the position vector from point A to ZMP, and
B
d
zmp
is the position
vector from point B to ZMP.
7.9
Optimization formulation
The objective of this section is to formulate the optimization problem for walking
in terms of three major components: design variables, objective function, and con-
straints. At a given walking velocity (
V
) and a step length (
L
), the time duration for
the step is calculated as
T
L
=
V.
The double support time duration is taken as
5
T
DS
5
α
2
αÞT
.
Single support is detailed into rear foot support (mid-stance) and forefoot support
(terminal stance). Their time durations are set to
T
; as a result, the single support time duration is given as
T
SS
5
ð
1
β
T
SS
and
ð
1
2
βÞT
SS
, respectively.
The parameters
are obtained from the literature (
Ayyappa, 1997
).
The walking task is formulated as a nonlinear optimization problem. A general
mathematical form is defined as: Find the optimal joint trajectories q(t) and joint tor-
ques
α
and
β
τ
(t) to minimize a human performance measure subject to physical constraints:
Find : q
; τ
To :
min
Fð
q
; τÞ
(7.25)
Sub
:
h
i
5
0
;
i
1
; ...;
m
5
g
j
#
0
;
j
5
1
; ...;
k
where h
i
are the equality constraints and g
j
are the inequality constraints.
7.9.1
Design variables
In the current formulation
,
the design variables are the joint angle profiles q
ðtÞ
.
The joint torques
t
Þ
are calculated using the governing differential equations.
This is called the inverse dynamics procedure where the differential equations are
not integrated. This has also been called the differential inclusion formulation.
τð
q
;
7.9.2
Objective function
The predicted motion depends strongly on the adopted objective function
F
.In
this work, the dynamic effort, the time integral of squares of all the joint torques,
is used as the performance criterion for the walking problem:
ð
T
T
dt
τð
q
tÞ
jτj
max
;
τð
q
tÞ
jτj
max
;
Fð
q
Þ
5
(7.26)
U
t
5
0
where
jτj
max
is the maximum absolute value of joint torque limit.
Figure 7.11
illustrates the optimization problem containing the three compo-
nents of a formulation. Note that we have used the simple form of the minimiza-
tion function as the integration of the torque square versus the normalized form in
Equation (7.26)
.
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