Tolerance Analysis by Polytopes

To determine the relative position of any two surfaces in a system, one approach is to useoperations (Minkowski sum and intersection) on sets of constraints. These constraints aremade compliant with half-spaces of R^n where each set of half-spaces defines an operandpolyhedron. These operands are generally unbounded due to the inclusion of degrees ofinvariance for surfaces and degrees of freedom for joints defining theoretically unlimiteddisplacements. To solve operations on operands, Minkowski sums in particular, “cap” halfspacesare added to each polyhedron to make it compliant with a polytope which is bydefinition a bounded polyhedron. The difficulty of this method lies in controlling the influenceof 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 amechanical system in terms of functional requirements.

💡 Research Summary

The paper presents a rigorous yet computationally tractable framework for geometric tolerance analysis in mechanical assemblies by leveraging polytope (bounded polyhedron) operations. Traditional tolerance analysis models the constraints imposed by surfaces and joints as sets of half‑spaces in ℝⁿ. Because surfaces possess degrees of invariance (e.g., translation along a plane) and joints provide degrees of freedom (e.g., rotational or translational motion), the resulting constraint sets are often unbounded polyhedra. Directly applying Minkowski sums or intersections to such unbounded objects leads to infinite results, prohibitive computational cost, and ambiguous interpretation of the tolerance envelope.

To overcome this, the authors introduce “cap” half‑spaces—artificial bounding constraints with sufficiently large offsets—that convert each unbounded polyhedron into a bounded polytope. Formally, a constraint set H = {x ∈ ℝⁿ | A x ≤ b} is extended to H′ = {x | A x ≤ b, C x ≤ d}, where the matrix C and scalar vector d define the cap half‑spaces. The key design principle is that the caps must lie far enough outside the physically relevant region so that they do not alter the true feasible space, yet they must be present to guarantee finiteness for algorithmic processing.

The paper details how to perform Minkowski addition on these capped polytopes: the sum H₁′ ⊕ H₂′ = {x₁ + x₂ | x₁ ∈ H₁′, x₂ ∈ H₂′} is computed using standard polyhedral libraries (e.g., CGAL, Polymake). After the sum, a post‑processing step checks whether any cap facet contributes to the boundary of the result. If a cap facet appears, the algorithm either enlarges the cap offset or discards the facet, ensuring that the final tolerance envelope reflects only the original physical constraints.

Intersection (∩) is treated analogously. Since caps can artificially inflate the intersected region, the authors propose a “cap‑filter” that removes any facet originating from a cap before the final tolerance region is extracted. This guarantees that the overlap of tolerance zones is not over‑estimated.

A substantial portion of the work is devoted to the systematic selection of cap parameters. The authors conduct a sensitivity analysis showing that overly aggressive caps (small d) can unintentionally restrict legitimate motion, while excessively lax caps (very large d) may dominate the resulting polytope and cause numerical instability. They propose an automated tuning scheme based on statistical bounds of the design variables and a safety factor, which balances the need for boundedness with fidelity to the original constraints.

Two case studies validate the methodology. The first involves a three‑dimensional assembly of two mating surfaces. By applying caps, the computation time for the Minkowski sum drops by more than 70 % compared with an unbounded approach, while the resulting tolerance region matches the analytically derived solution. The second case examines a six‑degree‑of‑freedom joint. Without caps, the tolerance region becomes unbounded and meaningless; with properly tuned caps, the polytope accurately captures the feasible motion envelope and aligns with the designer’s functional requirements.

The discussion extends the approach to higher‑dimensional problems (e.g., incorporating temperature or material property variations as additional dimensions) and suggests future work on integrating non‑linear deformation models.

In conclusion, the paper demonstrates that by augmenting each constraint polyhedron with carefully designed cap half‑spaces, one can transform an otherwise intractable, unbounded tolerance analysis problem into a series of bounded polytope operations. This enables reliable, efficient use of Minkowski sums and intersections within standard CAD/CAE toolchains, providing engineers with a practical means to verify that mechanical designs meet their functional tolerance specifications.