This set of Multiple Choice Questions & Answers (MCQs) focuses on “Robotics – Set 6”.

1. For the axis/angle representation of rotation matrix R, which of the following represent the equivalent angle θ, where Tr() represents trace.

a) θ = sin((Tr(R)-1)/2)

b) θ = cos^{-1}((Tr(R)-1)/2)

c) θ = sin^{-1}((Tr(R)-1)/2)

d) θ = cos((Tr(R)-1)/2)

View Answer

Explanation: To derive R

_{k,θ},rotate the vector k into one of the coordinate axes, say z

_{0}, then rotate about z

_{0}by θ and ﬁnally rotate k back to its original position. Then rotate k into z

_{0}by ﬁrst rotating about z

_{0}by −α, then rotating about y

_{0}by −β. Since all rotations are performed relative to the ﬁxed frame o

_{0}x

_{0}y

_{0}z

_{0}the matrix R

_{k,θ}is obtained as R

_{k,θ}= R

_{z,α}R

_{y,β}R

_{z,θ}R

_{y,−β}R

_{z,−α}. R = R

_{k,θ}, k is a unit vector deﬁning the axis of rotation, and θ is the angle of rotation about k. Given an arbitrary rotation matrix R with components (r

_{ij}), the equivalent angle θ and equivalent axis k are given by the expressions θ = cos

^{-1}((Tr(R)-1)/2).

2. For the axis/angle representation of rotation matrix Rij, which of the following represent the equivalent axis k?

a) \(k = \frac{1}{2 cotθ} \begin{bmatrix}

r32-r23\\

r13-r31\\

r21-r12

\end{bmatrix}\)

b) \(k = \frac{1}{2 sinθ} \begin{bmatrix}

r32-r23\\

r13-r31\\

r21-r12

\end{bmatrix}\)

c) \(k = \frac{1}{2 tanθ} \begin{bmatrix}

r32-r23\\

r13-r31\\

r21-r12

\end{bmatrix}\)

d) \(k = \frac{1}{2 cosθ} \begin{bmatrix}

r32-r23\\

r13-r31\\

r21-r12

\end{bmatrix}\)

View Answer

Explanation: According to the definition for the axis/angle representation of rotation matrix R, the equivalent axis k is derived as \(k = \frac{1}{2 sinθ} \begin{bmatrix}

r32-r23\\

r13-r31\\

r21-r12

\end{bmatrix}\). This can be obtained from R

_{k,θ}= R

_{z,α}R

_{y,β}R

_{z,θ}R

_{y,−β}R

_{z,−α}.

3. For the axis/angle representation of rotation matrix R, if R is the identity matrix, then what is the value θ ,k where θ is the equivalent angle and k is the equivalent axis ?

a) 0^{0}, undefined

b) 90^{0}, 0

c) 45^{0}, 1

d) 60^{0}, 1

View Answer

Explanation: According to the axis/angle representation of rotation matrix R, equivalent angle

θ = cos

^{-1}((Tr(R)-1)/2), where Tr(R) is 1 for identity matrix, θ = 0

^{0}, and equivalent axis k is \(k = \frac{1}{2 sinθ} \begin{bmatrix}

r32-r23\\

r13-r31\\

r21-r12

\end{bmatrix}\), so k is undefined since θ = 0

^{0}.

4. Consider R to be generated by a rotation of 90◦ about z_{0} followed by a rotation of 30◦ about y0 followed by a rotation of 60◦ about x_{0}. Then which of the following denotes R?

a) R = R_{x,60} R_{y,30} R_{z,90 }

b) R = R_{x,30} R_{y,60} R_{z,0 }

c) R = R_{x,30} R_{y,60} R_{z,90 }

d) R = R_{x,60} R_{y,30} R_{z,0}

View Answer

Explanation: A rotation matrix R can also be described as a product of successive rotations about the principal coordinate axes x

_{0}, y

_{0}, and z

_{0}taken in a speciﬁc order. Since R is generated by a rotation of 90◦ about z

_{0}followed by a rotation of 30◦ about y

_{0}followed by a rotation of 60◦ about x

_{0}. So, R can be denoted as R = R

_{x,60}R

_{y,30}R

_{z,90}.

5. For rotation matrix R generated by a rotation of 90◦ about z_{0} followed by a rotation of 30◦ about y0 followed by a rotation of 60◦ about x_{0}. According to axis/angle representation find the value of equivalent angle?

a) 120^{0}

b) 60^{0}

c) 90^{0}

d) 30^{0}

View Answer

Explanation: Rotation matrix R can be denoted as R = R

_{x,60}R

_{y,30}R

_{z,90}.

Equivalent angle θ = cos

^{-1}((Tr(R)-1)/2 )

R = \(\begin{bmatrix}

0 & -√3/2&1/2\\

1/2&-√3/4&-3/4\\

√3/2&1/4&-√3/4

\end{bmatrix}\)

Trace of R = Tr(R) = 0

θ = cos

^{-1}(-1/2)

θ = 120

^{0}

6. Which of the following is termed as the orientation of the frame {2}, which is rotated about of the three principles axes of frame {1}?

a) Principal Axes Representation

b) Fixed Angle Representation

c) Euler Angle Representation

d) Equivalent Angle Axis Representation

View Answer

Explanation: Principal Axes Representation is represented as the rotation of one frame with respect to another frame by some angle. Principal Axes Representation is also known as “fundamental rotation matrix”.

7. Which of the following representation uses the denotation of rotation of matrix as

R_{xyz} (θ_{3} θ_{2} θ_{1}) = R_{z}(θ_{3}) R_{y}(θ_{2}) R_{x}(θ_{1})?

a) Principal Axes Representation

b) Fixed Angle Representation

c) Euler Angle Representation

d) Equivalent Angle Axis Representation

View Answer

Explanation: According to Fixed Angle Representation:

Consider fixed frame {1} and moving frame {2} to be initially coincident. Consider sequence of rotations, first moving frame {2} is rotated by angle θ1 about x axis to frame {2’}. This is denoted by R

_{x}(θ

_{1}). Next frame {2’} is rotated by angle θ

_{2}about y axis to give frame {2’’}. This is denoted by R

_{y}(θ

_{2}). Finally it is rotated by an angle θ

_{3}about z axis to frame {2}. This is denoted by R

_{z}(θ

_{3}). So, R

_{xyz}(θ

_{3}θ

_{2}θ

_{1}) = R

_{z}(θ

_{3}) R

_{y}(θ

_{2}) R

_{x}(θ

_{1}) is Fixed Angle Representation.

8. Which of the following representation uses the denotation of rotation of matrix as

R_{wvu }(θ_{3} θ_{2} θ_{1}) = R_{w}(θ_{1}) R_{v’}(θ_{2}) R_{u’’}(θ_{3}) or R_{xyz} (θ_{3} θ_{2} θ_{1}) = R_{zyx} (θ_{1} θ_{2} θ_{3})?

a) Principal Axes Representation

b) Fixed Angle Representation

c) Euler Angle Representation

d) Equivalent Angle Axis Representation

View Answer

Explanation: According to Euler Angle Representation:

The moving frame instead of rotating about the principal axes of the fixed frame, can rotate about its own axes. Consider rotations of frame {2} with respect to frame {1}, starting from the position where the two frames are initially coincident. To begin with frame {2} is rotated by an angle θ

_{1}about its w axis coincident with z axis of frame {1}. The rotated frame is now {2’} and is denoted as R

_{w}(θ

_{1}).Next, moving frame {2’} is rotated by an angle θ

_{2}about its v’ axis. The rotated v’ axis to frame {2’’} and is denoted as R

_{v’}(θ

_{2}). Finally, frame {2’’} is rotated by an angle θ

_{3}about its u’’ axis, the rotated u axis to give frame {2} and is denoted as R

_{u’’}(θ

_{3}). Therefore it is denoted as R

_{wvu}(θ

_{3}θ

_{2}θ

_{1}) = R

_{w}(θ

_{1}) R

_{v’}(θ

_{2}) R

_{u’’}(θ

_{3}) or R

_{xyz}(θ

_{3}θ

_{2}θ

_{1}) = R

_{zyx}(θ

_{1}θ

_{2}θ

_{3}).

9. For a rotation matrix R which is rotated by φ degrees about the current y-axis followed by a rotation of θ degrees about the current z-axis. Then, what is the matrix R?

a) R = R_{y,φ} R_{z,θ}

b) R = R_{z,φ} R_{y,θ}

c) R = R_{y,90-φ} R_{z,90-θ}

d) R = R_{z,90-φ} R_{y,90-θ}

View Answer

Explanation: It is important to remember that the order in which a sequence of rotations are carried out, and consequently the order in which the rotation matrices are multiplied together.

Considering c

_{θ}= cosθ, s

_{θ}= sinθ for trigonometric functions.

R = R

_{y, φ}R

_{z, θ}

=\(\begin{bmatrix}

cφ & 0 & sφ\\

0 & 1 & 0\\

-sφ & 0 & cφ

\end{bmatrix}\) \(\begin{bmatrix}

cθ & -sθ & 0\\

sθ & cθ & 0\\

0 & 0 & 1

\end{bmatrix}\) =\(\begin{bmatrix}

cφcθ & -cφsθ & sφ\\

sθ & cθ & 0\\

-sφcθ & sφsθ & cφ

\end{bmatrix}\)

10. Rotation of a frame by φ degrees about the current y-axis followed by a rotation of θ degrees about the current z-axis is same as rotation of a frame by θ degrees about the current z-axis followed rotation by φ degrees about the current y-axis. True or False?

a) True

b) False

View Answer

Explanation: Rotation of a frame by φ degrees about the current y-axis followed by a rotation of θ degrees about the current z-axis is

R = R

_{y, φ}R

_{z, θ}

=\(\begin{bmatrix}

cφ & 0 & sφ\\

0 & 1 & 0\\

-sφ & 0 & cφ

\end{bmatrix}\) \(\begin{bmatrix}

cθ & -sθ & 0\\

sθ & cθ & 0\\

0 & 0 & 1

\end{bmatrix}\)

=\(\begin{bmatrix}

cφcθ & -cφsθ & sφ\\

sθ & cθ & 0\\

-sφcθ & sφsθ & cφ

\end{bmatrix}\)

Rotation of a frame by θ degrees about the current z-axis followed rotation by φ degrees about the current y-axis is

R’ = R_{z, θ} R _{y, φ}

= \(\begin{bmatrix}

cθ & -sθ & 0\\

sθ & cθ & 0\\

0 & 0 & 1

\end{bmatrix}\) \(\begin{bmatrix}

cφ & 0 & sφ\\

0 & 1 & 0\\

-sφ & 0 & cφ

\end{bmatrix}\)

=\(\begin{bmatrix}

cθcφ & -sθ & cθsφ\\

sθcφ & cθ & sθsφ\\

-sφ & 0 & cφ

\end{bmatrix}\)
Therefore, it can be concluded by R is not same as R’.

**Sanfoundry Global Education & Learning Series – Robotics.**

To practice all areas of Robotics, __ here is complete set of Multiple Choice Questions and Answers__.