robotics material

E. Ledoux 1 Spring 2019

ET 4860

Homework 2: Rigid Body Transformations

Purpose: to map one coordinate frame to another in 3D space

Problem 1

Given the coordinate frames shown, where the tie fighter is located at p0 = [3, 4, 0]T and θ = 30 deg, find R01 and p1

Problem 2

Given the coordinate frames shown, where 𝑜𝑜0 0 = �

0 0 0 � , 𝑜𝑜1

0 = � 6 1 0 � , 𝑝𝑝1 = �

−5 5 0 � , and θ = 60 deg, find

𝑅𝑅1 0 , 𝑇𝑇1

0, 𝑝𝑝0, 𝑎𝑎𝑎𝑎𝑎𝑎 𝑇𝑇0 1

E. Ledoux 2 Spring 2019

Problem 3

Given the following rotation sequence, find the resultant rotation matrix R (write rotation order only).

1) Rotate around fixed X axis by angle θ 2) Rotate around fixed Z axis by angle 𝜙𝜙 3) Rotate around current Z axis by angle α 4) Rotate around fixed Y axis by angle β 5) Rotate around current X axis by angle δ

Problem 4

Given a yaw of ψ = 10 deg, pitch of θ = 20 deg, and roll of φ = 30 deg,

(a) find rotation matrix R30 (b) find rotation matrix R03 (c) if p0 = [1, 2, 0]T find p3 (d) Is p in the new xy plane?

E. Ledoux 3 Spring 2019

Problem 5

Given the following matrix 𝑅𝑅 = � 0.75 −0.433 0.5 0.65 0.625 −0.433

−0.125 0.65 0.75 �, is it a rotation matrix?

Prove using two properties.

Problem 6

Derive separately the rotation matrices for rotating

(a) About the X axis by angle φ (b) About the Y axis by angle θ

Problem 7

Answer the following questions about lab A1:

(a) What is pulse width modulation? (b) Why should you always connect an LED in series with a

resistive load? (c) Is the circuit shown acceptable? If not, draw a better one.

E. Ledoux 4 Spring 2019

Problem 8

Answer the following questions about lab A2:

(a) What is the difference between an active buzzer and a passive buzzer? (b) How does a tilt switch work? Support your answer with illustrations.

Problem 9

Using the guidelines discussed in class and posted on D2L (Robot Selection Guide, see “Basic Selection” section), spec a robot to move a 20 kg object from “pick” point (x, y, z) = (2, 2, 0) m to “place” point (2, - 2, 1) m using a 10 kg EOAT. Justify your choice by providing:

(a) make and model: ____________________________________

(b) payload: ____________ (c) reach: _____________