EE106b - Spring 2017
Assignment 1
Assignment 1: Review Rigid Transformations,
Kinematics, and Dynamics
Due: Thursday Feb. 2, 2017 at 9:30am
This assignment covers material from MLS Chapters 2-4. Your homework will be submitted
electronically through GradeScope. No hardcopies will be accepted.
We don’t mind if you work with other students on your homework. However, each student
must write up and turn in their own assignment (i.e. no copy & paste). If you worked with
other students, please acknowledge who you worked with at the top of your homework.
Question 1. (10 points)
Properties of rotation matrices. MLS Problem 2.3
Question 2. (10 points)
Rigid point set registration. Robots often need to understand the location of known objects
in 3D space relative to their own body reference frame. The process of determining the best
estimate of the rigid transformation between observed points from a sensor and a set of
known points is called rigid point set registration. Applications of this method include
autonomous driving and 3D scanning.
(a) Let x1 , ..., xN ∈ R3 be a set of observed points in the world and y1 , ..., yN ∈ R3 be a
set of target points, where xi corresponds to point yi .
Derive a closed-form solution to:
∗
∗
R ,t =
argmin
N
X
R∈SO(3),t∈R3 i=1
kyi − (Rxi + t)k22
Hint: You can use the known solution to the orthogonal Procrustes problem,
https://en.wikipedia.org/wiki/Orthogonal Procrustes problem.
(b) In real settings we often do not have perfect point matches. Let x1 , ..., xN ∈ R3 be a
set of observed points in the world and y1 , ..., yN ∈ R3 be a set of target points with
unknown correspondences. One idea is to choose correspondences based on the closest
point, given the current transformation.
Propose an algorithm to solve:
R∗ , t∗ =
argmin
N
X
min kyj − (Rxi + t)k22
R∈SO(3),t∈R3 i=1 j=1,...,N
Justify your answer.
1
EE106b - Spring 2017
Assignment 1
Question 3. (10 points)
Kinematics and Jacobians. MLS Problem 3.3
Question 4. (10 points)
Equations of Motion. MLS Problem 4.1
2