2. Some parts of this question refer to the attached research paper: Mili Shah (2013), Solving the Robot-World/Hand-Eye Calibration Problem Using the Kronecker Product, Journal of Mechanisms and Robotics, Volume 5, Issue 3 (2013). You do not need to add a bibliography to reference this paper, and you may refer to it in your answer as: Shah (2013). (a) A rotation matrix is a 3 × 3 matrix R such that det(R) R and S be rotation matrices, and let t, u e R³. = 1 and R-1 = RT. Let (i) Prove that RS is a rotation matrix. r11 r12 ri13 [R t (ii) Given R = r21 r22 r23 and t = E R°, we write for the r31 T32 r33 4 × 4 matrix r11 r12 r13 r21 r22 r23 r31 r32 r33 1 From page 1 of the attached paper, the product [R tS u |0 1 R' v is a matrix of the form where R' is a rotation matrix and v E R³. 1 Write down expressions for R' and v in terms of R, S, t and u. For this question only, you do not need to give any reasons for your answer. (b) Suppose a robot is posed n times, leading to the equations RA, RX = Ry RB, where RA;, Rx, Ry and RB, are rotation matrices, for j = 1, 2,..., n. Write a few sentences to summarise the results of the attached paper by Shah on: (i) the number of poses needed to obtain unique matrices Rx and Ry which satisfy these equations; and (ii) how the position errors for the method presented in the attached paper compare to the position errors for the method of Li et al, on simulated data and real-world data.
2. Some parts of this question refer to the attached research paper: Mili Shah (2013), Solving the Robot-World/Hand-Eye Calibration Problem Using the Kronecker Product, Journal of Mechanisms and Robotics, Volume 5, Issue 3 (2013). You do not need to add a bibliography to reference this paper, and you may refer to it in your answer as: Shah (2013). (a) A rotation matrix is a 3 × 3 matrix R such that det(R) R and S be rotation matrices, and let t, u e R³. = 1 and R-1 = RT. Let (i) Prove that RS is a rotation matrix. r11 r12 ri13 [R t (ii) Given R = r21 r22 r23 and t = E R°, we write for the r31 T32 r33 4 × 4 matrix r11 r12 r13 r21 r22 r23 r31 r32 r33 1 From page 1 of the attached paper, the product [R tS u |0 1 R' v is a matrix of the form where R' is a rotation matrix and v E R³. 1 Write down expressions for R' and v in terms of R, S, t and u. For this question only, you do not need to give any reasons for your answer. (b) Suppose a robot is posed n times, leading to the equations RA, RX = Ry RB, where RA;, Rx, Ry and RB, are rotation matrices, for j = 1, 2,..., n. Write a few sentences to summarise the results of the attached paper by Shah on: (i) the number of poses needed to obtain unique matrices Rx and Ry which satisfy these equations; and (ii) how the position errors for the method presented in the attached paper compare to the position errors for the method of Li et al, on simulated data and real-world data.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 41EQ
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Transcribed Image Text:2. Some parts of this question refer to the attached research paper:
Mili Shah (2013), Solving the Robot-World/Hand-Eye Calibration Problem Using the
Kronecker Product, Journal of Mechanisms and Robotics, Volume 5, Issue 3 (2013).
You do not need to add a bibliography to reference this paper, and you may refer to it in
your answer as: Shah (2013).
RT. Let
(a) A rotation matrix is a 3 x 3 matrix R such that det(R) = 1 and R-1
R and S be rotation matrices, and let t, u E R³.
(i) Prove that RS is a rotation matrix.
r11 r12
r13
R t
(ii) Given R
r21
r22
r23
and t =
E R°, we write
for the
0 1
r31
r32
r33
4 x 4 matrix
r11
r12
r13
r21
r22
r23 Y
r31
r32 r33
1
From page 1 of the attached paper, the product
[R tS u
0 1
R' v
is a matrix of the form
where R' is a rotation matrix and v E R³.
Write down expressions for R' and v in terms of R, S, t and u. For this
question only, you do not need to give any reasons for your answer.
(b) Suppose a robot is posed n times, leading to the equations
RA, RX = Ry RB,
where RA;, Rx, Ry and RB, are rotation matrices, for j = 1, 2,..., n. Write a few
sentences to summarise the results of the attached paper by Shah on:
(i) the number of poses needed to obtain unique matrices Rx and Ry which
satisfy these equations; and
(ii) how the position errors for the method presented in the attached paper
compare to the position errors for the method of Li et al, on simulated data
and real-world data.
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