Assume that the pose of a moving robot with respect to an inertial (world) frame is given as a function of time t by T w r (t) = Rw r (t) d w r (t) =   1 0 0 t 0 cos πt 4 sin πt 4 2t 0 − sin πt 4 cos πt 4 0 0 0 0 1  

(a) Compute the linear velocity wv
w
r
(t) and angular velocity wω
w
r
(t) of the robot with respect to the
world frame represented in the world frame as a function of t. (6 points)
A note on the representation: ω
w
r
is the angular velocity of the robot with respect to the world,
and the superscript on the left indicates which frame this quantity is represented in. The same
applies to other quantities in this question.
(b) Compute the angular velocity rω
w
r
(t) of the robot with respect to the world frame represented in
the robot frame as a function of t. (6 points)
(c) Now suppose p is a fixed point on the rigid body of the robot, with parameter y in the robot
frame: p
r
(y) = (0, y, 0). What is the velocity ˙p
w
(y)
(t) of this point as a function of t in the world
frame? What is the function of velocity for y = 1, y = 0.1 and y = 0 respectively? Verify that
when y = 0, you have the same linear velocity as you obtained in question (a