ECE 417/598: Homework 2

ECE 417/598: Homework 2
Max marks: 90 marks. ETA: 90 min

You can also use the following template to fill
in your answers: hw2.cpp
1 Jan 26 Lecture
Problem 1 In class we proved the Rodrigues
formula that converts from axis-angle representation (θ, kˆ), where θ is the angle of rotation
and kˆ is the axis of rotation (kkˆk = 1). Let
K = [kˆ]× be the cross product matrix of kˆ. The
cross product matrix of kˆ = [kx, ky, kz]
> (such
that k
2
x + k
2
y + k
2
z = 1) is defined as,
K = [kˆ]× =


0 −kz ky
kz 0 −kx
−ky kx 0

 (1)
The corresponding rotation matrix is given by,
R(θ, kˆ) = I + sin θK + (1 − cosθ)K2
. (2)
An exponential of a square matrix M is defined
as
exp(M) = X∞
n=0
1
n!
Mk = I +
1
1!M +
1
2!M2 + . . .
(3)
Recall the series expansion of sin θ, and cos θ,
sin θ = θ −
θ
3
3! +
θ
5
5! − . . . (4)
cos θ = 1 −
θ
2
2! +
θ
4
4! − . . . (5)
1. First prove that K3 = −K. (15 marks, 15
minutes)
2. As a result note that K4 = −K2
, K5 = K,
and so on. In general, K2n+1 = (−1)nK
and K2n+2 = (−1)nK2
. Using the expansion of sin θ and cos θ, prove that R(θ, kˆ) =
exp(θK). (30 marks, 30 minutes)
Problem 2 Write a pair of functions in C++
that converts rotation matrix from axis-angle
representation and vice versa. Recall that
R(θ, kˆ) = I + sin θK + (1 − cosθ)K2
. (6)
and to get axis-angle back from a given rotation
matrix
R =


r11 r12 r13
r21 r22 r23
r31 r32 r33

 , (7)
we have
θ = cos−1

tr(R) − 1
2

(8)
kˆ =
1
2 sin θ


r32 − r23
r13 − r31
r21 − r12

 if θ 6= 0 or π. (9)
If θ = 0 or π, then
kˆ = ±


p
p
(r11 + 1)/2
p
(r22 + 1)/2
(r33 + 1)/2

 (10)
(30 marks. Estimated time: 30 min)
2 Jan 31 Lecture
Problem 3 Recall the definition of DenavitHartenberg parameters from the video. Recall
that transformation between two joints for the
defined parameters d, θ, r, α is given by,
T = Tz(θ, d)Tx(α, r), (11)
1
where
Tx(α, r) =




1 0 0 r
0 cos(α) − sin(α) 0
0 sin(α) cos(α) 0
0 0 0 1




(12)
Tz(θ, d) =




cos(θ) − sin(θ) 0 0
sin(θ) cos(θ) 0 0
0 0 1 d
0 0 0 1




(13)
For the robot given below find transformation
matrix from joint 4 to joint 1 assuming the joint
angles to be θ1, θ2, θ3 respectively. Write the expression for 3T4(θ3),
2T3(θ2),
1T2(θ1) and then
1T4(θ1, θ2, θ3) in terms of the first three transformations. You do not need to expand the expression of 1T4(θ1, θ2, θ3).
1
2
X-axis
Z-axis
Y-axis
3
4
Indicates out of the page towards viewer
X Indicates into the page away from viewer
X
X-axis
Z-axis
Y-axis
1
2
3 4
d =
r =
r =
(15 marks. 15 min)
3 ECE 598 only
Write a short review of the following paper On
continuity of rotation representations in Neural
networks. We have not covered all the concepts
covered in this paper; you can skip the parts that
you do not understand. In the review answer the
following questions evaluating the paper,
1. Problem: What problem is the paper trying
to solve?
2. Approach: What is the proposed approach
to solve the problem?
3. Contribution: What is the paper’s novel
contribution?
4. Evidence: Do they any experiments or proof
that their approach/contributions work?
5. Results: Are the results of the paper justified by evidence and a direct result of the
contibutions?
(Ungraded. 3-5 hrs)
2