In this paper, we include Coulomb friction at contacts between bodies and give a characterization of the entire set of stable orientations of an assembly under uniform gravity. Our characterization is based on the concept of potential stability, which describes a necessary but not sufficient condition for the stability of an assembly. Orientations that are computed as being unstable, however, are guaranteed to fall apart. Our characterization reveals that the set of stable orientations maps out a convex region on the unit-sphere of directions and corresponds to a spherical analog of a planar polygon-the region is bounded by a sequence of vertices joined by great arcs. Linear programming techniques are used to automatically find this set of vertices, yielding a description of the range of stable orientations for any assembly. For frictionless assemblies, our characterization of stable orientations is exact. For assemblies with friction, some conservative approximations associated with the use of a linearized Coulomb law are made.

The stability of an assemblage of contacting rigid bodies without friction is investigated. A method is presented for finding an orientation of the assembly so that the assembly remains motionless under gravity. If no stable orientation exists for an assembly, the method finds the "least" unstable orientation. The metric used to measure stability is based on the second time-rate of change of the gravitational potential energy, and the desired orientation for an assembly is expressed in terms of an optimization problem involving changes in potential energy. The problem of finding stable or maximally-stable orientations is formulated as a constrained maximin problem. The maximin problem is shown to be a variant of standard zero-sum matrix games, and can be solved using linear programming. The method is the first general method for automatically determining stable orientations. Example assemblies are presented.>

Previous work by Mattikalli et al. (1993) considered the stability of assemblies of frictionless contacting bodies with uniform gravity. A linear programming-based technique was described that would automatically determine a single stable orientation for an assembly (if such an orientation existed). In this paper, we give an exact characterization of the entire set of stable orientations of any assembly under uniform gravity. Our characterization reveals that the set of stable orientations maps out a convex region on the unit-sphere of directions. The region is bounded by a sequence of vertices adjoined with great arcs. Linear programming techniques are used to automatically find this set of vertices, yielding a precise description of the range of stable orientations for any frictionless assembly.>

High level assembly plans prescribe the sequence in which parts come together along with their motions. At each stage of assembly, the stability of subassemblies is an important concern. The authors address the problem of the gravitational stability of assemblies of frictionless rigid bodies. Solution methods to the problems of determining if an assembly is stable, and finding a stable orientation for a given for a given assembly are proposed. The solution methods for the two problems compute constrained motions of individual parts that decrease the gravitational potential energy, in order to determine the stability and search for stable orientations. Linear programming is used to obtain numerical solutions. The problem of finding a stable orientation is formulated as a maximin problem. The solution to this problem is the first general method for automatically determining stable orientations. The program to solve the two problems takes as input a geometric model of the assembly and information related to grounding constraints. Preliminary results are presented.

A method to determine constraints on translational and rotational motion of planar and 3-D objects from their contact geometry is presented. Translations are represented by spatial vectors and rotations by axes in space. For each of these, a geometric realization (M/sub a/) of the space of motion parameters is created. Subspaces in M/sub a/ that represent the range of values of motion parameters that are disallowed due to the contact are identified. The geometric realization makes it easier to visualize results, provides a good measure of the extent of restraints between objects, reduces computations by eliminating redundant constraints, and simplifies computation of new constraints. The proposed representation can be used effectively to automate the evaluation of motion constraints.>