Exercise 25.4

Suppose that you are working with the robot in Exercise AB-manipulator-ex and you are given the problem of finding a path from the starting configuration of figure figRobot2 to the ending configuration. Consider a potential function $D(A, {Goal})^2 + D(B, {Goal})^2 + \frac{1}{D(A, B)^2}$ where $D(A,B)$ is the distance between the closest points of A and B.

1. Show that hill climbing in this potential field will get stuck in a local minimum.

2. Describe a potential field where hill climbing will solve this particular problem. You need not work out the exact numerical coefficients needed, just the general form of the solution. (Hint: Add a term that “rewards” the hill climber for moving A out of B’s way, even in a case like this where this does not reduce the distance from A to B in the above sense.)