Cs223a Homework Answers

Introduction to Robotics (CS223A) Homework # 1

(Winter 2007/2008) Due: Wednesday, January 23

Some tips for doing CS223A problem sets:

• Use abbreviations for trigonometric functions (e.g. cθ for cos(θ), s1 or sθ1 for sin(θ1)) in situations where it would be tedious to repeatedly write sin, cos, etc.

• Unless instructed otherwise, leave square roots in symbolic form rather than writing out their decimal values.

• If you give a vector as an answer, make sure that you specify what frame it is given in (if it is not clear from context). The same rule applies to rotation and transformation matrices.

1. A frame {B} and a frame {A} are initially coincident. Frame {B} is rotated about ŶB by an angle θ, and then rotated about the new ẐB by an angle φ. Determine the 3× 3 rotation matrix, A

B R, which will transform the coordinates of a position vector from BP, its value in

frame {B}, into AP, its value in frame {A}.

2. We are given a single frame {A} and a position vector AP described in this frame. We then transform AP by first rotating it about ẐA by an angle φ, then rotating about ŶA by an angle θ. Determine the 3×3 rotation matrix operator, R(φ, θ), which describes this transformation.

3. (a) Given a transformation matrix:


AT =

1 0 0 1 0 cos(θ) − sin(θ) 2 0 sin(θ) cos(θ) 3 0 0 0 1

Find A B T

(b) Given θ = 45◦ and BP = [

4 5 6 ]T

, compute AP .

4. Given the following 3× 3 matrix:

R =

1√ 2

0 1√ 2

−1 2

1√ 2



−1 2

− 1√ 2



(a) Show that it is a rotation matrix.

(b) Determine a unit vector that defines the axis of rotation and the angle (in degrees) of rotation.

(c) What are the Euler parameters ε1, ε2, ε3, ε4 of R?

Academic Integrity

[The following is owed to Stuart Kurtz]

The University of Chicago is a scholarly academic community. You need to both understand and internalize the ethics of our community. A good place to start is with the Cadet's Honor Code of the US Military Academy: "A Cadet will not lie, cheat, or steal, or tolerate those who do." It is important to understand that the notion of property that matters most to academics is ideas, and that to pass someone else's ideas off as your own is to lie, cheat, and steal.

The University has a formal policy on Academic Honesty, which is somewhat more verbose than West Point's. Even so, you should read and understand it.

We believe that student interactions are an important and useful means to mastery of the material. We recommend that you discuss the material in this class with other students, and that includes the homework assignments. So what is the boundary between acceptable collaboration and academic misconduct? First, while it is acceptable to discuss homework, it is not acceptable to turn in someone else's work as your own. When the time comes to write down your answer, you should write it down yourself from your own memory. Moreover, you should cite any material discussions, or written sources (e.g. ).

The University's policy, for its relative length, says less than it should regarding the culpability of those who know of misconduct by others, but do not report it. An all too common case has been where one student has decided to "help" another student by giving them a copy of their assignment, only to have that other student copy it and turn it in. In such cases, we view both students as culpable and pursue disciplinary sanctions against both.

For the student collaborations, it can be a slippery slope that leads from sanctioned collaboration to outright misconduct. But for all the slipperyness, there is a clear line: present only your ideas as yours and attribute all others.

If you have any questions about what is or is not proper academic conduct, please ask your instructors.

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