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. Each project group must turn in a summary of the work accomplished to date on Project 2. Attach this to the homework for only one group member -- everyone will get the same credit.
2. In class, singularities were presented as a generally bad thing since very large joint velocities were required near them. However, they are good in the sense that they maximize mechanical advantage. Show this for the mechanism from the previous homework illustrated below by finding a large force (and/or torque) near one of the singular configurations that requires very small joint torques to maintain.
3. In Homework 2, you determined the forward kinematics for the mechanism in problem 3.18, Figure 3.38 and in the last homework you computed the Jacobian and one singular configuration. Again assuming that we are interested in the (x,y,z) location of the end effector if each joint angle is 30^o, and a force, F = (25,25,25) is applied at the origin of the end effector frame, what are the joint torques required to maintain the force?
4. In Homework 2, you determined the forward kinematics for the mechanism in problem 3.20, Figure 3.40 and in the last homework you computed its Jacobian and one singular configuration. Again assuming that we are interested in the (x,y,z) location of the end effector if each joint angle is 30^o, and a force, F = (25,25,25) is applied at the origin of the end effector frame, what are the joint torques required to maintain the force?
5. 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. What are the singular configurations for this mechanism?
6. Consider the four dimensional SCARA type robot considered in class last week. Assume that each of the three rotational joints is at 30^o, and the prismatic joint is at the reference configuration, i.e., theta_4 = 0.
   1. What is the direction of the maximum mechanical advantage?
   2. What is the direction of the maximum velocity amplitude?