Homework 7 (Assigned on 10/20/2023, due at 5:00pm on 10/27/2023)
1. A 2-DOF planar robot manipulator is shown in the figure below. At a configuration of the manipulator,
𝑓𝑓𝑥𝑥
when a force 𝒇𝒇 =
is applied to the tip of the manipulator, torques (i.e., 𝜏𝜏1 and 𝜏𝜏2 ) will be generated in
𝑓𝑓𝑦𝑦
the joints. Find the configuration(s) at which the two joint torques are the same, namely 𝜏𝜏1 = 𝜏𝜏2 . (100 points)
Hint: 1) You need to calculate the Jacobian matrix of the manipulator first.
2) To find configurations means to find sets of joint angles (𝜃𝜃1 , 𝜃𝜃2 ) 𝑌𝑌
𝑃𝑃
expressed in terms of 𝑓𝑓𝑥𝑥 , 𝑓𝑓𝑦𝑦 , 𝑙𝑙1 , and 𝑙𝑙2 .
𝑙𝑙2
𝑓𝑓𝑥𝑥
𝒇𝒇
=
θ2
𝑓𝑓𝑦𝑦
𝜏𝜏1
𝑂𝑂
𝑙𝑙1
θ1
𝜏𝜏2
𝑋𝑋
1