Analysis of Restraints to Translational and Rotational Motion from the Geometry of Contact

The automatic generation of high-level assembly plans from geometric models makes extensive use of restraints imposed by components on one another. The geometry of contact can be used to find the nature of these restraints. In this paper, we present a method to determine restraints to translational and rotational motion of planar as well as 3-D objects. A set of parameters that represent any possible translational/rotational motion is identified. A geometric representation of the space of motion parameters (Ma) is created. Restraints due to individual mating surface elements are computed by identifying subspaces inMa which represent motion parameters that are ‘disallowed’ due to the contact. These are then superposed to generate the required solution. Restraints that are redundant are easily identified and are not computed. Information about restraints is used extensively in assembly planning to determine degrees-of-freedom of components and subassemblies, to analyze the stability of subassemblies, to plan proper grasps, etc. We believe that the proposed representation can be used effectively in making these evaluations, taking us closer to generating correct assembly plans. This research was supported in part by the Engineering Design Research Center an NSF Engineering Research Center at Carnegie Mellon University Notation dof degree of freedom <2 two dimensional Euclidean space <3 three dimensional Euclidean space Ma space of all motion parameters Mp space of permissible motion parameters Md space of disallowed motion parameters M+ (for planar rotations) space of motion parameters parallel to +Z ` line in M that marks the boundary betweenMd and Mp ~r, ~d direction vector of the axis of rotation ~p position vector of a point on the axis of rotation P an arbitrary point on a plane tangent to ~d CA center of curvature at point A on a curve ( ; ; x; y) algebraic representation of an axis of rotation Tp plane tangent to the direction sphere at point p Sf constraint sphere generated by contact face f