Recovering Plans from a Sequence of Preimages

As the algorithm proceeds, the interval of orientations corresponding to each preimage is shifted to fit within the following preimage. Shifting the set of part orientations is equivalent to rotating the gripper. Thus, the relative position of adjacent preimages defines a gripper orientation.

The sequence of intervals defines a -step plan for orienting the part up to symmetry. To recover the plan we work backward from the final interval, (a point). Let be the leftmost point in interval : . Let the relative orientation of with respect to be . An action applied at angle will cause all orientations in to be aligned with the gripper.

Next we relate the relative orientation of to that of . This is . That is, by rotating the gripper by radians and then squeezing, we convert all orientations in interval to orientations in interval . We proceed backwards until we reach . Then by reversing and negating the sequence: we have a set of gripper angles that defines an open-loop plan for converting all orientations in to the single orientation .

Note that at each step we align the intervals precisely against their left edge. We can allow some error margin in gripper angle by noting the relative difference in size between neighboring intervals in the sequence. For example is smaller than so we can adjust the gripper angle by half the difference in size. Let .

The -step squeezing plan is given by