Optimal Placement of Fixture Clamps: Minimizing the Maximum Clamping Forces

This paper addresses the problem of synthesizing robust optimal clamping schemes on three-dimensional parts with planar faces, with and without friction. Given a work part with a pre-defined deterministic 3-2-1 location scheme, a set of polygonal convex regions on its faces are defined as areas of admissible clamp positions. A known set of external disturbing wrenches is also given. The frictionless case is considered first, and a new linear program is formulated to provide clamp locations that minimize the maximum clamping force. A transformation of the solution is presented that permits the extraction of both, the optimal positions of the clamps and the magnitude of the corresponding maximum clamping force. Friction is introduced next, and a linear program is presented that minimizes the maximum normal clamping force. We also extend the earlier formulations to support the case of frictionless clamping contacts on cylindrical faces. The result is a nonlinear (but convex) optimization problem that can be easily solved. Finally, we discuss a linear algebraic technique to find the directions, and associated relative motion rates, along which the clamps can be moved while maintaining the clamping forces constant. These lines of constant clamping force, as we name them, identify an equivalence set of clamping schemes (such that the maximum clamping force, given a set of disturbing wrenches, stays invariant).