CSE4421/5324: Assignment 1
Burton Ma
Posted: Jan 21, 2012
Due: Feb 03, 2012
1. Find the
4
×
4
homogeneous transformation matrix (showing the numeric values for all 16 elements)
T
0
1
where:
(a)
{
1
}
has the same orientation as
{
0
}
and the origin of
{
1
}
is translated relative to the origin of
{
0
}
by
d
0
1
=
±
5

5 10
²
T
.
(b) The origin of
{
1
}
is coincident with the origin of
{
0
}
, and
ˆ
x
0
1
=

ˆ
z
0
0
,
ˆ
y
0
1
=

ˆ
x
0
0
, and
ˆ
z
0
1
= ˆ
y
0
0
.
(c) The origin of
{
1
}
is translated relative to the origin of
{
0
}
by
d
0
1
=
±
0 0

10
²
T
, and the
orientation of
{
1
}
relative to
{
0
}
is the same as in part (b).
(d) The origin of
{
0
}
is translated relative to the origin of
{
1
}
by
d
1
0
=
±
0 0

10
²
T
, and the
orientation of
{
1
}
relative to
{
0
}
is
ˆ
x
0
1
=
±
0
.
9971

0
.
0292

0
.
0705
²
T
,
ˆ
y
0
1
=
±

0
.
0292 0
.
7083

0
.
7053
²
T
, and
ˆ
z
0
1
=
±
0
.
0705 0
.
7053 0
.
7053
²
T
.
Explain how you derived the solution for part (d).
2. Find the missing elements of the following rotation matrices. Show your work, or explain your rea
soning. It may be the case that there is no unique solution, in which case you should ﬁnd all possible
solutions.
(a)
·
1
0
·
0
0
·
0

1
(b)
·
√
3
/
2 0
·
0
1
√
3
/
2
·
0
(c)
0 0 1
· ·
0
· ·
0
1
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View Full Document3. Consider the following
4
×
4
homogeneous transformation matrices:
R
x,a
:
rotation about
x
by an angle
a
R
y,a
:
rotation about
y
by an angle
a
R
z,a
:
rotation about
z
by an angle
a
D
x,a
:
translation along
x
by a distance
a
D
y,a
:
translation along
y
by a distance
a
D
z,a
:
translation along
z
by a distance
a
Write the matrix product giving the overall transformation for the following sequences (do not perform
the actual matrix multiplications):
(a) The following rotations all occur in the moving frame.
i. Rotate about the current
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 Winter '11
 BURTON
 Rotation, Frame, Rotation matrix, angle rotation matrix

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