Biomedical Engineering Reference
In-Depth Information
q
i
(t)
(0,0,0,) 0
T/2
T (T,T,T)
FIGURE 8.5
B-spline discretization of a joint profile.
lifting motion is specified. The initial and final postures are optimized along with
the lifting motion instead of specifying them from the experiments.
8.5.1
Design variables and time discretization
Because we must calculate the joint profile across time, we will discretize the time
domain using a B-spline. The time domain is discretized by using cubic B-spline
functions. Thus a joint profile
qðtÞ
is parameterized as follows:
X
m
q
j
ð
t
;
P
Þ
5
N
i
ð
t
Þp
i
0
t
T
(8.16)
#
#
i
1
5
where
N
i
ð
t
Þ
represents the basis functions; t
5
ft
0
; :::;
t
s
g
is the knot vector; and
P
j
5
fp
1
; :::;
p
m
g
is the control points vector. With this representation, the control
points become the optimization design variables. In this study, the knot vector is
specified and fixed in the optimization process.
We formulate the lifting task as a general nonlinear programming (NLP)
problem. To find the optimal control points P for the lifting motion, a human per-
formance measure,
Fð
P
Þ
, is minimized subject to physical constraints as follows:
Find: P
To:
min Fð
P
Þ
(8.17)
Sub
:
h
i
5
0
;
i
1
; ...;
m
5
g
j
#
0
;
j
1
; ...;
k
5
where
h
i
are the equality constraints and
g
j
are the inequality constraints.
We represent the joint angle profile for each DOF by five control points. As a
result, there are 275 design variables (55 DOF
5 control points). In addition, the
total time duration is discretized into four evenly distributed segments, and five
time grid points are used for the entire motion as shown in
Figure 8.5
, where the
horizontal scale shows the knot vector. Multiplicity at the ends is used in the knot
vector. For B-splines, the multiplicity property guarantees that the initial and final
joint angle values of a DOF are exactly those corresponding to the initial and
final control point values. The time-dependent constraints are imposed not only at
3
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