Optimal Placement of Fixture Clamps: Maintaining Form Closure and Independent Regions of Form Closure

This paper addresses the problem of computing frictionless optimal clamping schemes with form closure on three-dimensional parts with planar and cylindrical faces. Given a work part with a pre-defined 3-2-1 locator scheme, a set of polygonal convex regions on the clamping faces are defined as the admissible positions of the clamps. The work part-fixture contact is assumed to be of the point-surface type. Using extended screw theory, we present a linear algebraic method that computes the sub-regions of the clamping faces such that clamps located within them are guaranteed to achieve form closure. These are termed dependent regions of form closure, since the clamps must be placed according to a precise relationship. We develop methods to compute these regions on work parts with planar and cylindrical faces. This result is incorporated into a new linear programming formulation to compute frictionless optimal clamping schemes. Clamping schemes with form closure are robust when uncertainty in knowledge of the external loads acting on the work part is present. Next, we extend the method to compute maximal independent regions of form closure. These are sub-regions of the dependent regions of form closure where the clamps can be placed completely independent of each other while maintaining form closure. When the clamps are placed within the independent regions of form closure, the clamping scheme is made robust against errors in their positions.