Tolerance analysis by polytopes: Taking into account degrees of freedom with cap half-spaces

• In tolerance analysis, sets of constraints can be compliant with operand polyhedra.
• These operands are generally unbounded due to the inclusion of degrees of freedom.
• Cap half-spaces are added to each polyhedron to make it compliant with a polytope.
• The influence of the cap half-spaces on the topology of polytopes must be controlled.
• It is necessary to ensure the compliance of a mechanism in terms of requirements.

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 RnRn 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” half-spaces 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.