Raju S. Mattikalli

Motion constraints from contact geometry: representation and analysis

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.<<ETX>>

Subassembly identification and motion generation for assembly: a geometric approach

A method for automated generation of an assembly procedure for a given assembly is presented. The procedure is generated from the parts geometry and topology model, and thus explicit specification of mating relations between parts is not required. The approach also prevents wrong mating relations from being presented and makes checking for validity unnecessary. The geometric considerations in the generation of the assembly sequence make it possible to evaluate assembly performance directly and to easily generate detailed assembly plans, such as grasping, transferring, and manipulation, based on the given geometry of the assembly. Algorithms are presented for finding a collision-free path for disassembly motions. The efficiency and feasibility of the algorithms is demonstrated through a case study

Finding all stable orientations of assemblies with friction

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.

Finding all gravitationally stable orientations of assemblies

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.<<ETX>>

Generation of partial medial axis for disassembly motion planning

An efficient approach is presented for determining disassembly motion plans of a subassembly in the free space within its parent subassembly. A two-step approach is presented to generating motion plans. First, all possible paths within any free space of the parent subassembly are generated using a partial medial axis. This is followed by a graph search for an optimal global path. Secondly, a collision-free motion is planned using the global path and the geometry of the moving subassembly. The authors focus on the first problem, i.e. generation of partial medial axis for disassembly motion planning. A method is given to determine a partial medial axis when the parent subassembly is a 2D polygon. A set of critical points within the given polygon is identified, and these critical points are then connected based on geometry constraints.<<ETX>>

Stability of assemblies

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.

Two-disk motion planning strategy

The problem of planning the motion of a polygonal object through a set of planar obstacles is addressed. A two-disk motion planning strategy is proposed to navigate the object within the free space between the obstacles from an initial location to a final location. This method makes use of the medial axis transform of the free space. Two minimal overlapping disks are determined that fully enclose the moving object, and then the centers of the two disks are constrained to move continuously along a path on the medial axis. Efforts are also directed to the problem of finding the two enclosing disks for a moving object which is considered as a polygon. The problem has been considered as being optimally cutting a polygon into two smaller polygons such that each of smaller polygons can be covered by a minimal disk. It is proved that if the cut is optimal, the resultant minimal disks for two smaller polygons have equal diameters.<<ETX>>

Determining two minimal circumscribing discs for a polygon

Abstract The paper proposes a 2-disc motion planning strategy to navigate an object within free space between obstacles from an initial location to a final location. The method makes use of the medial axis transform of the free space. Two minimal overlapping discs that fully enclose the moving object are determined, and then the centres of the two discs are constrained to move continuously along a path on the medial axis. The paper focuses on the problem of finding the two enclosing discs for an arbitrary moving polygon object. The problem is reduced to one of finding an optimal straight line cut through the polygon such that each of the subpolygons can be covered by a minimal disc. The paper discusses how an arbitrary polygon is cut, and how the two minimal overlapping discs that cover the given polygon are determined. Simulations are presented for a variety of polygons. The method has been used for planning part disassembly motion from inside to outside assembly.

Developing compatible grasp and fixture plans

Introduces a method for producing geometrically compatible grasp and fixture plans which minimize the closing force required of a gripper loading an object into a fixture and the closing force required of the fixture to keep the object stable during subsequent assembly operations. The authors formulate the problem for parallel-jaw tools by considering equilibrium equations with the gripper and fixture positions as unknown parameters. Algebraic constraints on the positions of the gripper and fixture ensure geometric compatibility. The resulting maximin problem is solved for a two and a three-dimensional example. Extensions of this formulation to other gripper geometries can easily be made.

Motion Planning Using Medial Axis

Abstract This paper addresses the problem of planning the motion of a polygonal object through a set of planar obstacles. We propose a two-disk motion planning strategy to navigate the object within the free space between the obstacles from an initial location to a final location. This method makes use of the Medial Axis Transform (MAT) of the free space which can be generated efficiently using the method developed in [11]. We determine two minima] overlapping disks that fully enclose the moving object, and then constrain the centers of the two disks to move continuously aJong a path on the medial axis. In this paper we direct our efforts to the problem of finding the two enclosing disks for a moving object which is considered as a polygon. The problem has been considered as being optimally cutting a polygon into two smaller polygons such that each of smaller polygons can be covered by a minimal disk. We have proved that if the cut is optimal. the resultant minimal disks for two smaller polygons have equal diameters. Based on the geometry of a convex polygon, we formulated the problem as an optimization problem to determine a local optimum for a given edge pair. For a concave polygon, we propose a method to create a hypothetical convex polygon for approximating a given concave polygon, and then perform optimization using the method described above. We have proved that the maximal radii of the disks covering the hypothetical convex polygon generated from a concave polygon is 1.25 times of the radii of the disks that covers the original concave polygon. This shows that the approximation of a concave polygon using a minimal convex polygon does not yield too conservative solution. Simulations are presented for a variety of polygons. The method is being used for disassembly motion planning of a subassembly within its parent subassembly.