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 + 4 j + 5k .
Estimated student time to complete: 5-10 minutes
Prerequisite knowledge required: Text Section(s) 2.4
Solution:
⎡i j k⎤
r = p × q = ⎢⎢5 0 3 ⎥⎥ = i ( 0 − 12 ) − j ( 25 − 9 ) + k ( 20 − 0 ) = −12i − 16 j + 20k
⎢⎣3 4 5 ⎥⎦
λ = rx2 + ry2 + rz2 = 144 + 256 + 400 = 28.28
⎡ −12 ⎤
⎢ 28.28 ⎥
⎢ ⎥ ⎡ −0.424 ⎤
−16 ⎥ ⎢
r= ⎢ = −0.566 ⎥⎥
⎢ 28.28 ⎥ ⎢
⎢ ⎥ ⎢⎣ 0.707 ⎥⎦
⎢ 20 ⎥
⎣⎢ 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 ⎤
⎢q ⎥ ⎢ 0.5⎥
qunit = ⎢ y ⎥ runit = ⎢ ⎥
⎢ 0.4 ⎥ ⎢0.4 ⎥
⎢ ⎥ ⎢ ⎥
⎣ 0 ⎦ ⎣ 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:
q y = 1 − 0.09 − 0.16 = 0.866
rx = 1 − 0.25 − 0.16 = 0.768
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 ⎤
λ 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:
λ p = i ( 0.146 ) + j ( 0.187 ) − k ( 0.515 )
λp = ( 0.146 ) + ( 0.187 ) + ( 0.515) = 0.567
2 2 2
8
w= = 14.1
0.567
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.
λp = ( 0.146 ) + ( 0.187 ) + ( 0.515) = 0.567
2 2 2
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 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.
T
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 1 0 3 ⎥⎥ ⎢⎢ 5 ⎥⎥ ⎢⎢ 8 ⎥⎥
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 d = ( 5, 2, 6 ) . Find the new location of
T
the frame relative to the reference frame.
⎡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 =⎢ =
⎢0 0 1 6 ⎥ ⎢0 0 −1 6 ⎥ ⎢0 0 −1 12⎥
⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣0 0 0 1 ⎦ ⎣0 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.
, 10
Problem 2.6
For frame F, find the values of the missing elements and complete the matrix
representation of the frame.
⎡ ? 0 −1 5 ⎤
⎢ ? 0 0 3⎥
F =⎢ ⎥
⎢ ? −1 0 2 ⎥
⎢ ⎥
⎣0 0 0 1 ⎦
Estimated student time to complete: 10 minutes
Prerequisite knowledge required: Text Section(s) 2.4
Solution:
⎡ nx 0 −1 5⎤
⎢n 0 0 3 ⎥⎥
F =⎢ y
⎢ nz −1 0 2⎥
⎢ ⎥
⎣0 0 0 1⎦
⎡i j k⎤
⎢ ⎥
From n × o = a ⎢ nx n y nz ⎥ = − i
⎢⎣ 0 0 −1⎥⎦
Or: i ( −n y ) − j ( −nx ) + k ( 0 ) = −i , and therefore: n y = 1, nx = 0, nz = 0
⎡0 0 −1 5⎤
⎢1 0 0 3 ⎥⎥
F =⎢
⎢0 −1 0 2⎥
⎢ ⎥
⎣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.
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 + 4 j + 5k .
Estimated student time to complete: 5-10 minutes
Prerequisite knowledge required: Text Section(s) 2.4
Solution:
⎡i j k⎤
r = p × q = ⎢⎢5 0 3 ⎥⎥ = i ( 0 − 12 ) − j ( 25 − 9 ) + k ( 20 − 0 ) = −12i − 16 j + 20k
⎢⎣3 4 5 ⎥⎦
λ = rx2 + ry2 + rz2 = 144 + 256 + 400 = 28.28
⎡ −12 ⎤
⎢ 28.28 ⎥
⎢ ⎥ ⎡ −0.424 ⎤
−16 ⎥ ⎢
r= ⎢ = −0.566 ⎥⎥
⎢ 28.28 ⎥ ⎢
⎢ ⎥ ⎢⎣ 0.707 ⎥⎦
⎢ 20 ⎥
⎣⎢ 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 ⎤
⎢q ⎥ ⎢ 0.5⎥
qunit = ⎢ y ⎥ runit = ⎢ ⎥
⎢ 0.4 ⎥ ⎢0.4 ⎥
⎢ ⎥ ⎢ ⎥
⎣ 0 ⎦ ⎣ 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:
q y = 1 − 0.09 − 0.16 = 0.866
rx = 1 − 0.25 − 0.16 = 0.768
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 ⎤
λ 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:
λ p = i ( 0.146 ) + j ( 0.187 ) − k ( 0.515 )
λp = ( 0.146 ) + ( 0.187 ) + ( 0.515) = 0.567
2 2 2
8
w= = 14.1
0.567
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.
λp = ( 0.146 ) + ( 0.187 ) + ( 0.515) = 0.567
2 2 2
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 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.
T
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 1 0 3 ⎥⎥ ⎢⎢ 5 ⎥⎥ ⎢⎢ 8 ⎥⎥
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 d = ( 5, 2, 6 ) . Find the new location of
T
the frame relative to the reference frame.
⎡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 =⎢ =
⎢0 0 1 6 ⎥ ⎢0 0 −1 6 ⎥ ⎢0 0 −1 12⎥
⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣0 0 0 1 ⎦ ⎣0 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.
, 10
Problem 2.6
For frame F, find the values of the missing elements and complete the matrix
representation of the frame.
⎡ ? 0 −1 5 ⎤
⎢ ? 0 0 3⎥
F =⎢ ⎥
⎢ ? −1 0 2 ⎥
⎢ ⎥
⎣0 0 0 1 ⎦
Estimated student time to complete: 10 minutes
Prerequisite knowledge required: Text Section(s) 2.4
Solution:
⎡ nx 0 −1 5⎤
⎢n 0 0 3 ⎥⎥
F =⎢ y
⎢ nz −1 0 2⎥
⎢ ⎥
⎣0 0 0 1⎦
⎡i j k⎤
⎢ ⎥
From n × o = a ⎢ nx n y nz ⎥ = − i
⎢⎣ 0 0 −1⎥⎦
Or: i ( −n y ) − j ( −nx ) + k ( 0 ) = −i , and therefore: n y = 1, nx = 0, nz = 0
⎡0 0 −1 5⎤
⎢1 0 0 3 ⎥⎥
F =⎢
⎢0 −1 0 2⎥
⎢ ⎥
⎣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.
