Mark H. Overmars

Computing Immobilizing Grasps of Polygonal Parts

We present the first algorithms for computing all placements of (frictionless) point fingers that put a polygonal part in form closure and all placements of point fingers that achieve second-order immobility of a polygonal part. Our algorithms run in O(n2+∈ +K) and O(n2 log2 n + K) time in the case of form closure and second-order immobility, respectively, where n is the number of vertices of the polygon, K is the description size of the resulting set of finger placements, and ∈ is an arbitrarily small constant. The basis of our algorithm is a translation of the problem into geometric searching problems, which are solved using efficient data structures. Our results can be extended to the problem of computing all placements of a line and two points that put a polygonal part in form closure. The resulting algorithm runs in O(n2 log2 n + K) time, where K is again the description size of the output.

Trap Design for Vibratory Bowl Feeders

The vibratory bowl feeder is the oldest and still most common approach to the automated feeding (orienting) of industrial parts. In this paper, the authors consider a class of vibratory bowl filters that can be described by removing polygonal sections from the track; this class of filters is referred to as traps. For an n-sided polygonal part and an m-sided polygonal trap, an O(n2m log n) algorithm is given to decide whether the part in a specific orientation will safely move across the trap or will fall through the trap and thus be filtered out. For an n-sided convex polygonal part and m-sided convex polygonal trap, this bound is improved to O((n+m) log n). Furthermore, the authors show how to design various trap shapes, ranging from simple traps to general polygons, which will filter out all but one of the different stable orientations of a given part. Although the runtimes of the design algorithms are exponential in the number of trap parameters, many industrial part feeders use few-parameter traps (balconies, canyons, slots); in these cases, the running times of the algorithms range from linear to low-degree polynomial.

Computing fence designs for orienting parts

A common task in automated manufacturing processes is to orient parts prior to assembly. We consider sensorless orientation of a polygonal part by a sequence of fences. We show that any polygonal part can be oriented by a sequence of fences placed along a conveyor belt, thereby settling a conjecture by Wiegley et al. (1997), and present the first polynomial-time algorithm to compute the shortest such sequence. The algorithm is easy to implement and runs in time O(n3 logn), where n is the number of vertices of the part.

Geometric Eccentricity and the Complexity of Manipulation Plans

Abstract. Complexity bounds for algorithms for robotic motion and manipulation can be misleading when they are constructed with pathological ``worst-case scenarios that rarely appear in practice. Complexity can in some cases be reduced by characterizing nonpathological objects in terms of intuitive geometric properties. In this paper we consider the number of push and push—squeeze gripper actions needed to orient a planar part without sensors and improve on the upper bound of O(n) for polygonal parts given by Chen and Ierardi in [1]. We define the geometric eccentricity of a planar part based on the length-to-width ratio of a distinguished type of bounding box. We show that any part with a given eccentricity can be oriented with a plan whose maximum length depends only on the eccentricity and not on the description complexity of the part. The analysis also applies to curved parts, providing the first complexity bound for nonpolygonal parts. Our results also yield new bounds on part feeders that use fences and conveyor belts.

Orienting polyhedral parts by pushing

A common task in automated manufacturing processes is to orient parts prior to assembly. We consider sensorless orientation of an asymmetric polyhedral part by a sequence of push actions, and show that is it possible to move any such part from an unknown initial orientation into a known final orientation if these actions are performed by a jaw consisting of two orthogonal planes. We also show how to compute an orienting sequence of push actions.We propose a three-dimensional generalization of conveyor belts with fences consisting of a sequence of tilted plates with curved tips; each of the plates contains a sequence of fences. We show that it is possible to compute a set-up of plates and fences for any given asymmetric polyhedral part such that the part gets oriented on its descent along plates and fences.

Geometry and Part Feeding

Many automated manufacturing processes require parts to be oriented prior to assembly. A part feeder takes in a stream of identical parts in arbitrary orientations and outputs them in uniform orientation. We consider part feeders that do not use sensing information to accomplish the task of orienting a part; these feeders include vibratory bowls, parallel jaw grippers, and conveyor belts and tilted plates with so-called fences. The input of the problem of sensorless manipulation is a description of the part shape and the output is a sequence of actions that moves the part from its unknown initial pose into a unique final pose. For each part feeder we consider, we determine classes of orientable parts, give algorithms for synthesizing sequences of actions, and derive upper bounds on the length of these sequences.

Fixturing hinged polygons

We study the problem of fixturing a chain of hinged objects in a given placement with frictionless point contacts. We define the notions of immobility and robust immobility - which are comparable to the second and first order immobility for a single object - to capture the intuitive requirement for the fixture of a chain of hinged objects. Robust immobility differs from immobility in that it additionally requires insensitivity to small perturbations of contacts. We show that (p+2) frictionless point contacts can immobilize any chain of p/spl ne/3 polygons without parallel edges; six contacts can immobilize any chain of three such polygons. Any chain of p arbitrary polygons can be immobilized with at most (p+4) contacts. We also show that /spl lceil/(6/5)(p+2)/spl rceil/ contacts suffice to robustly immobilize p polygons without parallel edges, and that /spl lceil/(5/4)(p+2)/spl rceil/ contacts can robustly immobilize p/spl ne/3 arbitrary polygons, and eight contacts can robustly immobilize three polygons.

Trap design for vibratory bowl feeders

The vibratory bowl feeder is the oldest and still most common approach to the automated feeding (orienting) of industrial parts. We consider a class of vibratory bowl filters that can be described by removing polygonal sections from the track; we refer to this class of filters as traps. For an n-sided convex polygonal part and m-sided convex polygonal trap, we give an O((n+m)log(n+m)) algorithm to decide if the part will be rejected by the trap, and an O((nm(n+m))/sup 1+/spl epsiv//) algorithm which deals with non-convex parts and traps. We then consider the problem of designing traps for a given part, and consider two rectilinear subclasses, balconies and gaps. We give linear and O(n/sup 2/) algorithms for designing feeders and have tested the results with physical experiments using a commercial inline vibratory feeder.

Orienting parts by inside-out pulling

A common task in automated manufacturing processes is that of orienting (or feeding) parts prior to assembly. We propose a new type of feeder. We consider sensorless orientation of polygonal parts with elevated edges by pull actions with an overhead finger. We show that any asymmetric convex polygonal part can be oriented by a sequence of pull operations. We give an O(n/sup 3/) algorithm to compute the shortest sequence of pull operations to orient a convex polygonal part with n vertices, if such a sequence exists. We also show that there exist non-convex parts that cannot be fed by a sequence of pull operations.