E28 mobile robotics - fall 2015 - homework 5 define the

Question

E28: Mobile Robotics - Fall 2015 - HOMEWORK 5

1. Inverse kinematics

Consider the three-link planar arm depicted below:

The planar arm has one prismatic (sliding) joint which places the shoulder a distance d above the base, and two revolute (hinge) joints at the shoulder and elbow with corresponding joint angles θ₁ and θ₂, respectively. We will derive the inverse kinematics equations for placing the end effector (hand) at an arbitrary position and orientation.

a. Define the transformation from the hand coordinate frame to the base frame as

Explain why specifying a particular pose (x_H, y_H, θ_H) of the hand also specifies the position (x_E, y_E) of the elbow in the base frame, and solve for the elbow position in terms of the end effector pose.

b. Solve for the values of d and θ₁ which yield a particular position (x_E, y_E) of the elbow.

c. Finally, solve for θ₂ in terms of your answers for (a) and (b) above.

2. Differential kinematics

Continuing from in the example above, let (x_S, y_S), (x_E, y_E), and (x_H, y_H) all represent the base-frame coordinates of the shoulder, elbow, and hand, respectively. Finally, let θ_H represent the angle between the hand frame and base frame, as above.

a. Write out the 3 x 3 geometric Jacobian matrix given by

You should compose each of the elements of J using only the coordinates above, as well as the numbers 0 and 1.

b. Let l₁ = l₂ = 0.25 m. Show that J is not invertible when d = 1.0 m and θ₁ = θ₂ = 0 rad. To do so, you may compute the determinant of J or show that some of its rows or columns are linearly dependent. Geometrically, what does this linear dependence mean?