The Planning Algorithm

We define the parts-feeding problem as follows:

• Given a list of vertices describing a planar part,

• Find a list of gripper angles corresponding to a plan for orienting the part up to symmetry in its transfer function.

The algorithm is given below:

Figures 5, 6, and 7 illustrate how the algorithm proceeds for a rectangular part. All orientations in map into the single orientation in when the frictionless gripper is closed. So is the image of , and is the preimage of . Step 3 searches for the widest interval whose image is smaller . This interval becomes . We can implement step 3 geometrically using a square box of dimension . We position the box over the step function such that the range of output angles contained in the box is smaller than the range of input angles. That is, the function must enter on the box's left-hand edge and exit on the box's right-hand edge as illustrated in Figure 7. Continuing, wider and wider intervals are found until an interval of width equal to a period of symmetry in the transfer function is found.

Figure 6: In step 2, the widest single step in the transfer function is identified. All the orientations in (horizontal bar at bottom), map into the single final orientation: (dot at PI).

Figure 7: In step 3, we identify the largest interval whose image is smaller than h1 = |THETA1|. This can be visualized by left-aligning a box of dimension h1 with each step. If the squeeze function emerges from the right edge of the box, then the corresponding image is smaller than h1. The largest such interval in this case is THETA2 (shown at bottom).