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In-Depth Information
to form the twist representation is given by
R
)
ρ
=
(
I
3
×
3
−
t
(7.41)
2
(
1
−
cos
ψ)
a) By equating the rigidmotion
R
X
+
t
with the equivalent screw transfor-
mation
R
(
X
−
ρ)
+
dv
+
ρ
(where
d
is the linear displacement along the
screw axis), show that
t
=
(
I
3
×
3
−
R
)ρ
+
dv
.
R
)
R
)ρ
b)
.
c) Use the Rodrigues formula and the fact that
Show that
(
I
3
×
3
−
t
=
(
2
I
3
×
3
−
R
−
ρ
v
=
0 to show that
R
)ρ
=
(
2
I
3
×
3
−
R
−
2
(
1
−
cos
ψ)ρ
, proving Equation (
7.41
).
7.6 The exponential map of a 3
×
3 matrix is defined by a matrix Taylor series:
1
2
M
2
1
6
M
3
exp
(
M
)
=
I
3
×
3
+
M
+
+
+···
(7.42)
∞
1
k
M
k
=
!
k
=
0
In Equation (
7.12
) we have the special case of exp
[
v
]
×
, where
[
v
]
×
is the
skew-symmetric matrix defined in Equation (
5.39
).
a)
2
vv
−
2
I
3
×
3
and
3
2
]
×
.
b) By considering the first 6 terms of the exponential Taylor series,
conclude that
Show that
[
v
]
×
=
v
[
v
]
×
=−
v
[
v
−
1
cos
v
vv
exp
[
v
]
×
=
cos
v
I
3
×
3
+
sinc
v
[
v
]
×
+
(7.43)
2
v
thus proving the Rodrigues formula.
7.7
Show how to modify Equation (
7.26
) when the angle between the vectors
X
j
(
t
)
−
X
i
(
t
)
and
f
ij
(
θ
(
t
))
should be as close as possible to a given angle
φ
.
7.8
Show how the constraint that a specific joint angle remain in the limits
θ
l
≤
θ
i
(
)
≤
θ
u
can be expressed in the formof Equation (
7.27
) (assuming the
joint is directly parameterized with angles, not quaternions or twists).
7.9 Show how to formulate limits on (a) joint velocity and (b) joint acceleration
as soft constraints in an inverse kinematics cost function. Be sure to specify
whether your limits are in 3D space or joint angle space.
7.10 Give specific examples of (a) a linear low-pass and (b) a nonlinear noise-
removing filter that can be applied independently to each joint parameter
channel
t
θ
i
(
)
.
7.11 Explain why simply linearly interpolating quaternions is incorrect.
7.12 Using spherical linear interpolation, compute the rotation matrix that is
one-third of the way from
t
−
0.4129
0.8744
0.2547
R
1
=
−
0.6783
−
0.1086
−
0.7267
to
−
0.6077
−
0.4729
0.6380
