A geometric approach to automated fixture layout design

Fixtures are used in many manufacturing processes to hold objects. Fixture layout design is to arrange fixturing elements (fixels) on the object surface such that the object can be held in form-closure and totally immobilized. It is well known that 4/7 fixels are sufficient for immobilizing a 2D/3D object without rotational symmetry and their locations satisfy form-closure if and only if the convex hull of their primitive wrenches forms a 3D/6D simplex in the wrench space containing the origin as an interior point. This paper presents a method for finding form-closure locations of 4/7 fixels with enhanced immobilization capability. First, the Gilbert-Johnson-Keerthi distance algorithm and the Gram-Schmidt process are used to yield the fixel locations such that the simplex with vertices at their primitive wrenches is 3D/6D and contains the origin. Then, an interchange algorithm is developed for altering the fixel locations to meet form-closure and increase an immobilization capability index of fixture layouts. The meanings of this index in fixture localization accuracy and force balance capability are elucidated as well. Its value is proved to be equal to the minimum distance from the wrench origin to the facets of the wrench simplex in terms of a unit-invariant norm for wrench vectors. Without using any general optimization techniques, this method determines an optimal fixture layout very efficiently, so that it can be tried with various initial conditions to attain a result approaching the global optimum or with other good performance qualities.

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