Z x y q 2 0 q 3 q 1 z x y q 2 q 3 0 q 1 4 exercise

z
x
y
q
2
=0
q
3
q
1
z
x
y
q
2
q
3
=0
q
1
4 Exercise Robot 1
i θ d a α
1 0 q
1
0 0
2 q
2
0 0 - 90
◦
3 0 q
3
0 0
4.1 Sketch
In a right-handed XYZ coordinate frame, draw a sketch of this robot showing the spatial relation-
ships between the coordinate frames associated with each link.
x
0
z
2
z
0
z
x
y

2
x
1,2
x
3
z
3
z
1
d
1
d
3

2
d
1
d
3
12
MECEE 4602: Intro to Robotics, Fall 2018

4.2 Forward Kinematics
Compute the transform matrix
b
T
ee
from the base of the robot to its end-effector.
4.3 Inverse Kinematics
Assume we require the end-effector to be at position [ a, b, c ]
T
, and we do not care about end-effector
orientation. Show how to compute the values for all the robot joints such that the end-effector
achieves the desired position. Be sure to list all possible solutions. If the number of solutions
depends on certain characteristics of a, b and c , explain how and why.
13
MECEE 4602: Intro to Robotics, Fall 2018

4.4 Differential Kinematics
Compute the manipulator Jacobian with respect to end-effector position (and ignoring end-effector
orientation). Find all the joint configurations where the Jacobian becomes singular; for each of
them, briefly explain why the robot is in a singular configuration (for example, which direction of
end-effector motion is impossible or which robot joint has no effect on end-effector position).
5.1 Sketch
In a right-handed XYZ coordinate frame, draw a sketch of this robot showing the spatial relation-
ships between the coordinate frames associated with each link.
x
0 z
1
z
0
x
3
z
3
z
x
y

1

2
d
3
x
1,2
z
2
l
1
14
MECEE 4602: Intro to Robotics, Fall 2018

5.2 Forward Kinematics
Compute the translation part of the transform matrix
b
T
ee
from the base of the robot to its
end-effector.