Introduction to Robotics Homework 5

ECE 470: Introduction to Robotics Homework 5
Question 1.
Consider the single-link manipulator arm in Figure 1(a) as shown also in Figure 10.4
(Craig, Introduction to Robotics 3rd Ed.).
a) Given that the revolute joint moves the link over 2 cubic segments in 6s from
an initial angle 𝜽𝟎 = 𝟏𝟓 𝐝𝐞𝐠 to rest at a final position 𝜽𝒇 = 𝟗𝟎 𝐝𝐞𝐠 through
a via point 𝜽𝒗 = 𝟑𝟎 𝐝𝐞𝐠 at 𝒕𝒗 = 𝟑𝒔 with a velocity of 𝜽̇
𝒗 = 𝟏𝟓 𝐝𝐞𝐠/𝐬,
obtain the 8 parameters of the 2-segment cubic polynomial. (10 Points)
Figure 1
b) Imagine you decided to create a prismatic-revolute (PR) 2-dof robotic
manipulator and mounted the single-link on a horizontally moving cart as
shown in Figure 1(b),
I. Illustrate the configuration space of this PR manipulator with a sketch
given that the joint limits are 𝑥 ⊂ [−𝑑, 𝑑] and 𝜃 ⊂ [−𝜋, 𝜋].
(2 Points)
II. Describe a possible path in the configuration space if a vertical straight
path is desired from point (0, −𝒍) to (0,𝒍) in the workspace
(Hint: circular motion projects to orthogonal axes as sinusoidal motion)
(3 Points)
III. Assuming the motor at 𝒒𝟐 rotates at a constant speed of 𝝎, suggest a
trajectory for 𝒒𝟏 (2 Points)
IV. Suggest a control scheme if the manipulator is tasked to performance
ultrasound imaging over a region by sliding the probe along the 𝒙
direction at a vertically downward controlled contact force with the
surface. You may assume an additional joint 𝒒𝟑 to orientate the
ultrasound transducer as shown.in Figure 2 (3 Points)
Figure 2
Question 2 (Optional Bonus Question)
Find the new equation of motion relating 𝒇 and 𝝉 to 𝒙̈, 𝒙̇, 𝒙,𝜽̈, 𝜽̇ and 𝜽 if the singlelink manipulator is mounted on a horizontally moving cart as shown in Figure 1(b).
(10 Points)