ECE 470: Introduction to Robotics Homework 3
Figure 1
Question 1.
After realizing that the two-link arm in Homework 2 may not have sufficient DOF, one
of our friends decided to add an additional link connected by a joint at Point P such that
the final design is as shown in Figure 1. The forward kinematics of the manipulator
can be described by the homogenous matrices
𝟏𝟏𝐓𝐓𝟎𝟎 = �
𝑐𝑐1 −𝑠𝑠1 0
𝑠𝑠1 𝑐𝑐1 0
0 0 1
0
0
0
0 0 0 1
� ; 𝟐𝟐𝐓𝐓𝟏𝟏 = �
𝑐𝑐2 −𝑠𝑠2 0
0 0 −1
𝑠𝑠2 𝑐𝑐2 0
L1
0
0
0 0 0 1
� ; 𝟑𝟑𝐓𝐓𝟐𝟐 = �
𝑐𝑐3 −𝑠𝑠3 0
𝑠𝑠3 𝑐𝑐3 0
0 0 1
L2
0
0
0 0 0 1
�
a) Given
3𝑇𝑇0 = �
C1C23 −C1S23 𝑖𝑖
S1C23 −S1S23 𝑗𝑗
S23 C23 𝑘𝑘
𝑥𝑥
𝑦𝑦
𝑧𝑧
0 0 0 1
�,
solve i, j, k, x, y, and z in terms of q1, q2 and q3 (You may use some results from
Homework 1 directly) (4 Points)
b) Determine the 3-by-3 Jacobian when multiplied with the joint velocity produce the
linear velocity of Point P. Express the Jacobian in Frame {0}. (3 Points)
c) Write down an expression that relates (𝐹𝐹𝑥𝑥 𝐹𝐹𝑦𝑦 𝐹𝐹𝑧𝑧) 3
0 T, the force vector acting on
Frame {3} in Frame {0} with the joint torques, (𝜏𝜏1 𝜏𝜏2 𝜏𝜏3)T. (1 Points)
d) Referring to the Jacobian, comment on the contribution of Joint 3 towards the motion
and force of the point P (i.e. origin of Frame {3}). (2 Points)
Question 2
In Figure 1, a new frame, Frame {4} is assigned to the wrist of the robot such that
4𝐓𝐓3 = �
1 0 0
0 1 0
0 0 1
L3
0
0
0 0 0 1
�
a) Find the absolute velocity of the origin of Frame {4}, i.e. 4
V4 (7 Points)
b) Determine the Jacobian in terms of Frame {4}. (3 Points)
