Tolerance Analysis by Polytopes: Taking into account degrees of freedom with cap half-spaces
Résumé
To determine the relative position of any two surfaces in a system, one approach is to use
operations (Minkowski sum and intersection) on sets of constraints. These constraints are
made compliant with half-spaces of R^n where each set of half-spaces defines an operand
polyhedron. These operands are generally unbounded due to the inclusion of degrees of
invariance for surfaces and degrees of freedom for joints defining theoretically unlimited
displacements. To solve operations on operands, Minkowski sums in particular, "cap" halfspaces
are added to each polyhedron to make it compliant with a polytope which is by
definition a bounded polyhedron. The difficulty of this method lies in controlling the influence
of these additional half-spaces on the topology of polytopes calculated by sum or intersection.
This is necessary to validate the geometric tolerances that ensure the compliance of a
mechanical system in terms of functional requirements.
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