All 10 Chapters Covered
SOLUTION MANUAL
, 2
CHAPTER ONE
Problem 1.1
Draw the approximate workspace for the following robot. Assume the dimensions of the base and
other parts of the structure of the robot are as shown.
Estimated student time to complete: 15-25 minutes
Prerequisite knowledge required: Text Section(s) 1.14
Solution:
The workspace shown is approximate.
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 3
Problem 1.2
Draw the approximate workspace for the following robot. Assume the dimensions of the base and
other parts of the structure of the robot are as shown.
Estimated student time to complete: 20-30 minutes
Prerequisite knowledge required: Text Section(s) 1.14
Solution:
The workspace shown is approximate.
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 4
Problem 1.3
Draw the approximate workspace for the following robot. Assume the dimensions of the base and
other parts of the structure of the robot are as shown.
Estimated student time to complete: 10-15 minutes
Prerequisite knowledge required: Text Section(s) 1.14
Solution:
The workspace shown is approximate.
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 5
CHAPTER TWO
Problem 2.1
Write a unit vector in matrix form that describes the direction of the cross product of
p 5i 3k and q 3i 4j 5k .
Estimated student time to complete: 5-10 minutes
Prerequisite knowledge required: Text Section(s) 2.4
Solution:
⎡i k⎤j
r p q ⎢5 0 3 ⎥ i 0 12 j 25 9 k 20 0 12i 16j 20k
⎢⎣ 3 4 5 ⎥⎦
2
r r r2
2
28.28
144 256 400
⎡ x12 ⎤y z
⎢ 28.28 ⎥ ⎡ 0.424⎤
⎢ ⎥
r ⎢ 16 ⎥ ⎢ 0.566⎥
⎢ 28.28 ⎥ ⎢ ⎥
⎢ 20 ⎥ ⎢⎣ 0.707 ⎥⎦
⎢ ⎥
⎢⎣ 28.28 ⎥⎦
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 6
Problem 2.2
A vector p is 8 units long and is perpendicular to vectors q and r described below. Express
the vector in matrix form.
⎡0.3⎤ ⎡ rx ⎤
⎢
0.5⎥
qunit runit ⎢⎢0.4⎥
⎢ ⎥
⎥
⎢q y
⎣0⎦
⎥
⎢0.4
⎥
⎢
0
⎥
⎣
⎦
Estimated student time to complete: 15-20 minutes
Prerequisite knowledge required: Text Section(s) 2.4
Solution:
The two vectors given are unit vectors. Therefore, each missing component can be found as:
qy 10.09 0.16 0.866
rx 0.768
10.25 0.16
Since p is perpendicular to the other two vectors, it is in the direction of the cross product of the
two. Therefore:
⎡⎢ i j k ⎤
⎥
p 0.3 0.866 0.4⎥ i 0.346 0.2 j 0.12 0.307 k 0.15 0.665
⎢
⎢⎣0.768 0.5 0.4⎥⎦
i 0.146 j 0.187 k 0.515
Since q and r are not perpendicular to each other, the resulting p is not a unit vector. Vector p can
be found as:
p i 0.146 j 0.187 k 0.515
0.567
8
0.146 0.187 0.515
2 2 2
p
w
0.567 14.1
p w i 0.146 j 0.187 k 0.515
p i 2.06 j 2.64 k 7.27
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 7
Problem 2.3
Will the three vectors p, q, and r in Problem 2.2 form a traditional frame? If not, find the necessary
unit vector s to form a frame between p, q, and s.
Estimated student time to complete: 15-20 minutes
Prerequisite knowledge required: Text Section(s) 2.4
Solution:
As we saw in Problem 2.2, since q r is not a unit vector, it means that q and r and not
perpendicular to each other, and therefore, they cannot form a frame. However, p and q are
perpendicular to each other, and we can select s to be perpendicular to those two. Of course, p is
not a unit length, therefore we use the unit vector representing it.
p 0.146 0.187 0.515
2 2 2 0.567
1
w
1.764
0.567
p w i 0.146 j 0.187 k 0.515
p i 0.257 j 0.33 k 0.908
p i 0.257 j 0.33 k 0.908
⎡ i j k ⎤
s ⎢0.257 0.33 0.908⎥ i 0.918 j 0.375 k 0.124
⎢ ⎥
⎣⎢ 0.3 0.866 0.4 ⎥⎦
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 8
Problem 2.4
Suppose that instead of a frame, a point T
P 3, 5,7 in space was translated a distance
T
of d 2, 3, 4 . Find the new location of the point relative to the reference frame.
Estimated student time to complete: 5 minutes
Prerequisite knowledge required: Text Section(s) 2.6
Solution:
As for a frame,
⎡1 0 0 2⎤ ⎡ 3⎤ ⎡ 5 ⎤
⎢0 ⎥ ⎢ 5⎥ ⎢ 8 ⎥
1 0 3
Pnew
⎢0 0 1 4⎥⎢7⎥ ⎢11⎥
⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣0 0 0 1 ⎦ ⎣1 ⎦ ⎣1 ⎦
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 9
Problem 2.5
The following frame B was moved a distance of the T
frame relative to the reference frame. d 5, 2, 6 . Find the new location of
⎡0 1 0 2⎤
⎢1 0 0 4⎥
B ⎢ ⎥
⎢0 0 1 6⎥
⎢ ⎥
0 0 0 1
Estimated
student time to complete: 5-10 minutes
Prerequisite knowledge required: Text Section(s) 2.6
Solution:
The transformation matrix representing the translation is used to find the new location as:
⎡ 1 0 0 5⎤ ⎡ 0 1 0 2⎤ ⎡0 1 0 7⎤
⎢ 0 1 0 2⎥ ⎢ 1 0 0 4⎥ ⎢1 0 0 6⎥
Bnew 1 12⎥
⎢0 0 1 6⎥ ⎢0 0 1 6⎥ ⎢0
0
⎢ ⎥⎢ ⎥ ⎢ ⎥
0 0 0 10 0 0 1 0 0 0 1
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SOLUTION MANUAL
, 2
CHAPTER ONE
Problem 1.1
Draw the approximate workspace for the following robot. Assume the dimensions of the base and
other parts of the structure of the robot are as shown.
Estimated student time to complete: 15-25 minutes
Prerequisite knowledge required: Text Section(s) 1.14
Solution:
The workspace shown is approximate.
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 3
Problem 1.2
Draw the approximate workspace for the following robot. Assume the dimensions of the base and
other parts of the structure of the robot are as shown.
Estimated student time to complete: 20-30 minutes
Prerequisite knowledge required: Text Section(s) 1.14
Solution:
The workspace shown is approximate.
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 4
Problem 1.3
Draw the approximate workspace for the following robot. Assume the dimensions of the base and
other parts of the structure of the robot are as shown.
Estimated student time to complete: 10-15 minutes
Prerequisite knowledge required: Text Section(s) 1.14
Solution:
The workspace shown is approximate.
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 5
CHAPTER TWO
Problem 2.1
Write a unit vector in matrix form that describes the direction of the cross product of
p 5i 3k and q 3i 4j 5k .
Estimated student time to complete: 5-10 minutes
Prerequisite knowledge required: Text Section(s) 2.4
Solution:
⎡i k⎤j
r p q ⎢5 0 3 ⎥ i 0 12 j 25 9 k 20 0 12i 16j 20k
⎢⎣ 3 4 5 ⎥⎦
2
r r r2
2
28.28
144 256 400
⎡ x12 ⎤y z
⎢ 28.28 ⎥ ⎡ 0.424⎤
⎢ ⎥
r ⎢ 16 ⎥ ⎢ 0.566⎥
⎢ 28.28 ⎥ ⎢ ⎥
⎢ 20 ⎥ ⎢⎣ 0.707 ⎥⎦
⎢ ⎥
⎢⎣ 28.28 ⎥⎦
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 6
Problem 2.2
A vector p is 8 units long and is perpendicular to vectors q and r described below. Express
the vector in matrix form.
⎡0.3⎤ ⎡ rx ⎤
⎢
0.5⎥
qunit runit ⎢⎢0.4⎥
⎢ ⎥
⎥
⎢q y
⎣0⎦
⎥
⎢0.4
⎥
⎢
0
⎥
⎣
⎦
Estimated student time to complete: 15-20 minutes
Prerequisite knowledge required: Text Section(s) 2.4
Solution:
The two vectors given are unit vectors. Therefore, each missing component can be found as:
qy 10.09 0.16 0.866
rx 0.768
10.25 0.16
Since p is perpendicular to the other two vectors, it is in the direction of the cross product of the
two. Therefore:
⎡⎢ i j k ⎤
⎥
p 0.3 0.866 0.4⎥ i 0.346 0.2 j 0.12 0.307 k 0.15 0.665
⎢
⎢⎣0.768 0.5 0.4⎥⎦
i 0.146 j 0.187 k 0.515
Since q and r are not perpendicular to each other, the resulting p is not a unit vector. Vector p can
be found as:
p i 0.146 j 0.187 k 0.515
0.567
8
0.146 0.187 0.515
2 2 2
p
w
0.567 14.1
p w i 0.146 j 0.187 k 0.515
p i 2.06 j 2.64 k 7.27
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 7
Problem 2.3
Will the three vectors p, q, and r in Problem 2.2 form a traditional frame? If not, find the necessary
unit vector s to form a frame between p, q, and s.
Estimated student time to complete: 15-20 minutes
Prerequisite knowledge required: Text Section(s) 2.4
Solution:
As we saw in Problem 2.2, since q r is not a unit vector, it means that q and r and not
perpendicular to each other, and therefore, they cannot form a frame. However, p and q are
perpendicular to each other, and we can select s to be perpendicular to those two. Of course, p is
not a unit length, therefore we use the unit vector representing it.
p 0.146 0.187 0.515
2 2 2 0.567
1
w
1.764
0.567
p w i 0.146 j 0.187 k 0.515
p i 0.257 j 0.33 k 0.908
p i 0.257 j 0.33 k 0.908
⎡ i j k ⎤
s ⎢0.257 0.33 0.908⎥ i 0.918 j 0.375 k 0.124
⎢ ⎥
⎣⎢ 0.3 0.866 0.4 ⎥⎦
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 8
Problem 2.4
Suppose that instead of a frame, a point T
P 3, 5,7 in space was translated a distance
T
of d 2, 3, 4 . Find the new location of the point relative to the reference frame.
Estimated student time to complete: 5 minutes
Prerequisite knowledge required: Text Section(s) 2.6
Solution:
As for a frame,
⎡1 0 0 2⎤ ⎡ 3⎤ ⎡ 5 ⎤
⎢0 ⎥ ⎢ 5⎥ ⎢ 8 ⎥
1 0 3
Pnew
⎢0 0 1 4⎥⎢7⎥ ⎢11⎥
⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣0 0 0 1 ⎦ ⎣1 ⎦ ⎣1 ⎦
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 9
Problem 2.5
The following frame B was moved a distance of the T
frame relative to the reference frame. d 5, 2, 6 . Find the new location of
⎡0 1 0 2⎤
⎢1 0 0 4⎥
B ⎢ ⎥
⎢0 0 1 6⎥
⎢ ⎥
0 0 0 1
Estimated
student time to complete: 5-10 minutes
Prerequisite knowledge required: Text Section(s) 2.6
Solution:
The transformation matrix representing the translation is used to find the new location as:
⎡ 1 0 0 5⎤ ⎡ 0 1 0 2⎤ ⎡0 1 0 7⎤
⎢ 0 1 0 2⎥ ⎢ 1 0 0 4⎥ ⎢1 0 0 6⎥
Bnew 1 12⎥
⎢0 0 1 6⎥ ⎢0 0 1 6⎥ ⎢0
0
⎢ ⎥⎢ ⎥ ⎢ ⎥
0 0 0 10 0 0 1 0 0 0 1
© Copyrighted 2010.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other
electronic media system or the Internet without prior expressed consent of the copyright owner.
, 1
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CLICK ON THE L.I.N.K
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