ECE 470 Introduction to Robotics Homework 3 solved

Description

Question 1.
A manipulator arm is designed as illustrated by the following figure. It can be
assumed that the mass distributions of the links are insignificant and can be treated as
lumped equivalent masses m1 and m2.
a) Write down the position of masses m1 and m2 in terms of 𝜃𝜃1 and 𝜃𝜃2 referenced
from the given frame. (2 marks)
b) Obtain the velocities v1 and v2 of the mass m1 and m2, respectively. (4 marks)
c) Show that the total kinetic energy of the system, K can be written as
𝐾𝐾 = 1
2 �𝑚𝑚1𝑙𝑙1
2 + 𝑚𝑚2𝑙𝑙1
2
�𝜃𝜃̇
1
2
+ 𝑚𝑚2𝑙𝑙1𝑙𝑙2 cos(𝜃𝜃2 − 𝜃𝜃1) 𝜃𝜃̇
1𝜃𝜃̇
2 + 1
2
𝑚𝑚2𝑙𝑙2
2
𝜃𝜃̇
2
2
(4 marks)
d) Obtain the total potential energy of the system. (3 marks)
e) Write down the Lagrangian L. (2 marks)
f) Obtain the equation of motion. (5 marks)