Mechanical Analysis

We assume that:

1. The gripper has two linear jaws arranged in parallel.
2. The squeezing motion is orthogonal to the jaws.
3. The part is a rigid planar polygon of known shape.
4. The part's initial position is unconstrained provided it lies somewhere between the two jaws. The part remains between the jaws throughout grasping.
5. All motion occurs in the plane and is slow enough that inertial forces are negligible. The scope of this quasi-static model is discussed in [13] and [16].
6. Both jaws make contact simultaneously (pure squeezing).
7. Once contact is made between a jaw and the part, the two surfaces remain in contact throughout the grasp. A grasp continues until further motion would deform the part.
8. There is zero friction between part and the jaws.
9. We can design a binary filter that accepts a particular orientation of the part and rejects all others.

These assumptions are similar to those made by Brost [2], and Taylor, Mason and Goldberg [21]. Assumptions 2, 6, and 8 simplify the analysis and improve the combinatorics of the search. By restricting gripper motion to be orthogonal to the jaws (Assumption 2), we obtain a one-dimensional action space. The frictionless gripper (Assumption8) insures that the state space is the finite set of stable part orientations. Assumtion 6 simplifies the mechanical analysis. In a later section (Discussion), we discuss how this assumption can be relaxed by considering the related class of push-grasp actions.

Figure 5: The diameter function (top) and squeeze function (bottom) for the rectangular part.

When a part is grasped with the frictionless gripper, it assumes one of a finite number of stable orientations corresponding to local minima in a diameter function. Let a two-dimensional part be described with a continuous curve, , in the plane. The distance between two parallel tangent lines varies with the orientation of the lines. Let be the set of planar orientations. The diameter function, , is the distance between parallel tangents at angle . For polygonal parts, the diameter function is piecewise sinusoidal (Figure 5; top).

When a part is grasped between the jaws, the distance between the jaws corresponds to the diameter. Closing the jaws changes the diameter and thus the relative orientation of the part. The jaws continue closing until the diameter is at a local minimum that also defines a stable orientation of the part. The diameter function can be viewed as a potential energy function for a conservative system [6].

During a squeeze action, part motion is determined by the diameter function. That is, given an initial orientation of the part with respect to the gripper, the part's final orientation can be determined from the diameter function. A transfer function, relating initial orientations to final orientations, can be represented with a piecewise constant squeeze function , .

We define the squeeze function such that if is the initial orientation of the part with respect to the gripper, is the final orientation of the part with respect to the gripper. The squeeze function can be derived from the diameter function as follows. All orientations that lie between a pair of adjacent local maxima in the diameter function will map into the same final orientation. The squeeze function is constant over this interval of orientations. Each local maximum in the diameter function corresponds to a discontinuity in the squeeze function. In order for the squeeze function to be single-valued, we assume that all steps are closed on the left (Figure 5; bottom).

Note that the squeeze function has period due to rotational symmetry in the gripper. Rotational symmetry in the part also introduces periodicity into the squeeze function. In general the squeeze function has period such that

Periodicity in the squeeze function gives rise to aliasing, where the part in orientation behaves identically to the part in . Any sequence of actions that maps to will map to . This implies that no sequence of squeeze-grasp actions can map orientations and into a single final orientation. Thus a part can only be oriented up to symmetry in its squeeze function.

For a given part, let be the smallest period in its transfer function. We say that a plan orients a part up to symmetry if the set of possible final states includes exactly states that are equally spaced on .

The algorithm requires that the transfer function is a monotonically nondecreasing step function on the space of planar orientations. The squeeze function satisfies these conditions as does the push-grasp function defined in the Discussion section.