B Davenport-schinzel Sequences the Appendix a Mechanical Constraints on Part Pose

(a) q C q C q C Figure 10: (a): A two-axis gripper poised over a trapezoidal part P. There are two pairs of jaws, one formed by H 1 ; L 1 and the other H 2 ; L 2. Both pairs of jaws close in towards a central point, C. The pair H 1 ; L 1 closes in as far as possibly (without deforming the part) before H 2 ; L 2 does. (b): Possible nal poses for P after a single complete grasp. A sequence of integers U = (u 1 ; : : :; u m) is called an (n; s) Davenport-Schinzel sequence 8] if 1. 1 u i n, 2. For every i < m we have u i 6 = u i+1 , and 3. There do not exist s + 2 indices 1 i 1 < i 2 < : : : < i s+2 m such that u i1 = u i3 = u i5 = = a, u i2 = u i4 = u i6 = = b, and a 6 = b. By s (n) is understood the maximum length, m, of such an (n; s) Davenport-Schinzel sequence. The problem of estimating s (n) has been studied repeatedly beginning with the discoverers and Szemeredi 23] and ending 4 with Agarwal, Sharir, and Shor 2]. Szemeredi showed the generic bound s (n) = O(n log n) for any s 3. It is fairly easily shown that 1 (n) = n and 2 (n) = 2n ? 1. These are the results used in this paper. Better bounds for s (n) may be found in 2]. 20 If noise sensitivity is known in advance, we can deene a \separation threshold" for transformed registration marks. That is, we can compute the lower envelope over the entire plane and use the threshold to slice the envelope at a particular height, d s , given by the sensitivity. This will deene corresponding regions of the plane where a registration mark can be safely placed. Such regions could be useful if we later want to move the mark due to functional requirements. Another alternative is the brute-force solution that searches over a large number of candidate points and explicitly nding the minimum distance between each point. For m points and k transforms, this would require O(mk 2) time and does not guarantee an optimal placement …