Algebraic Structure and Geometric Interpretation of Rigid Complex Fixture Systems

The rigid complex fixture is presented as a mathematical model for the analysis and design of a fixture that contains more fingers than needed for immobilizing an object alone. But the fixture with more fingers may act in violation of the approachability and accessibility conditions. So we propose the design principle for a complex fixture: the finger set must be accessible; the distance from a workpiece to the finger set should be selected such that the workpiece can be in contact with all locators simultaneously; the clamping force should enable the workpiece to be in contact with all of the locators (approachable). Furthermore, we develop a new efficient procedure of analysis and verification in screw space based on the linear programming formulation and the geometric interpretation of the rigid fixture model. In addition, we introduce a new quantitative test for force closure, captured by a function that measures how far a fixture is from achieving force closure. Locating the table problem, as a complex fixture prototype, is investigated in detail to gain a great insight into the mathematical model and the analysis procedure for the rigid complex fixture. Note to Practitioners-This paper is concerned with the parameter design and verification of a complex fixture that consists of more locators, supports, and clamps than needed for force closure. There are two essential issues related to the design problems: 1) the test on whether the workpiece can be in contact with all locators simultaneously (approachable) and 2) the test on whether the workpiece can reach the desired location smoothly (accessible). A procedure for system analysis and performance verification of the rigid complex fixture is outlined in this paper. In the procedure, a linear programming with its dual formulation is applied to establish the sufficient condition for the stable state and to recognize the locators and supports. A performance index is provided to verify if a fixture is accessible, a- - nd a quantitative test is used to check whether a fixture has force closure. In the future, the linear and nonlinear compliance models of complex fixtures will be discussed and incorporated with this rigid model for the implementation of a complete analysis and design tool of the complex fixture