Biomedical Engineering Reference
In-Depth Information
affects only a small portion of the curve. This is desirable when using these coeffi-
cients as design variables in an optimization in order to shape the joint profile.
The pth-degree B-spline curve C(u) is a linear combination of pth-degree basis
functions N
k,p
(u), defined as follows:
X
n
k
5
0
N
k
;
p
ð
u
Þ
P
k
C
ð
u
Þ
5
(3.17)
where
u
k
u
k
1
p
2
u
u
k
1
p
1
1
2
u
2
N
k
;
p
ð
u
Þ
5
u
k
N
k
;
p
2
1
ð
u
Þ
1
u
k
1
1
N
k
1
1
;
p
2
1
ð
u
Þ;
(3.18)
u
k
1
p
1
1
2
and
1fu
k
#
u
u
k
1
1
#
N
k
;
0
ð
u
Þ
5
(3.19)
0
otherwise
...
,
u
m
} is a non-decreasing sequence of real numbers called the knot vector; each u
i
is a knot. The number of knots m
1
Here, the curve has n
1
1 coefficients P
k
(0
#
k
#
n). The vector U
5
{u
0
,
1, the number of coefficients n
1
1, and the
degree p are related by
m
5
n
1
p
1
1
(3.20)
For calculation of joint acceleration, the joint profiles must be at least twice
differentiable; thus, it is necessary to use at least a third-degree B-spline curve.
Furthermore, using a knot vector that begins and ends with multiplicity p
1
1 will
ensure that
the joint
trajectories interpolate the initial and final
joint values.
Hence, the joint profiles are defined by
X
n
k
5
0
N
k
;
3
ð
u
Þ
P
ð
i
Þ
q
i
ð
u
Þ
5
(3.21)
k
f
P
ð
i
Þ
0
P
ð
i
Þ
1
P
ð
i
Þ
n
where q
i
is the ith joint profile,
;
; ...;
g
is the unknown coefficient
vector for q
i
, and 1
i
nDOF (where nDOF is the number of DOF). Hence,
#
#
1)
(nDOF) coefficients to determine.
For example, for a 15-DOF kinematic model, the first and last coefficients are
equal to the initial and final values; therefore, there are only 13
there are (n
1
1
1
11
2
2
5
coefficients per curve. Thus, there will be (11)
(nDOF)
11
15
165 design
5
5
variables for optimization.
Lastly, note that using B-spline curves gives a result with normalized time
between zero and one second. The resulting motion can be scaled to longer
durations.
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