Unless otherwise indicated, all the problems are from the course text, Craig, Introduction to Robotics. Unless otherwise indicated, each problem is worth 10 points.

1. 5.18
   Note: the problem is only concerned with the linear part of the rigid body velocity.
2. 5.19
3. For the three link planar robot illustrated in in the Figure,
   1. determine the forward kinematics from the base frame, S, to the tool frame T including the orientation of the tool frame (in class, I only did (x,y) for the two link manipulator);
   2. compute the Jacobian;
   3. determine at least two singular configurations; and,
   4. (optional - 10 points extra ) modify the animation code presented in class to drive the mechanism from the starting configuration given in the code along a desired trajectory that takes it near one of the workspace interior singularities. Verify that large joint velocities occur near the singularity.
   
4. In Homework 2, you determined the forward kinematics for the mechanism in problem 3.18, Figure 3.38. Assume that we are interested in the (x,y,z) location of the end effector (note that you will have to add a frame at the end effector -- before your forward kinematic computations only went to the frame attached to the last link at the axis of rotation). Assume that each joint angle is 30^o.
   1. Compute the Jacobian for this system; and
   2. determine at least one singular configuration.
   3. What is the direction of the maximum mechanical advantage?
   4. What is the direction of the maximum velocity amplitude?
5. In Homework 2, you determined the forward kinematics for the mechanism in problem 3.20, Figure 3.40. Assume that we are interested in the (x,y,z) location of the end effector. Assume that d₁ = 0 and that the two revolute joint angles are 30^o
   1. compute the Jacobian for this system; and,
   2. determine at least one singular configuration.
   3. What is the direction of the worst velocity amplitude?
   4. What is the direction of the worst mechanical advantage?
6. In Homework 3, you determined the forward kinematics for the mechanism in problem 3.21, Figure 3.41. Assume that we are interested in the (x,y,z) location of the end effector. Assume that d₁ = 2, d₂ = 1 and d₃ = 4.
   1. What is the direction of the maximum force fidelity?
   2. What is the direction of the maximum velocity fidelity?