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Research Papers

# Tolerance Analysis With Polytopes in $HV$-Description

[+] Author and Article Information
Santiago Arroyave-Tobón

I2M Laboratory, UMR 5295,
University of Bordeaux,
Talence F-33400, France
e-mail: santiago.arroyave-tobon@u-bordeaux.fr

Denis Teissandier

Professor
I2M Laboratory, UMR 5295,
University of Bordeaux,
Talence F-33400, France
e-mail: denis.teissandier@u-bordeaux.fr

Vincent Delos

CNRS,
I2M Laboratory, UMR 5295,
Talence F-33400, France
e-mail: vincent.delos@u-bordeaux.fr

1Corresponding author.

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received October 4, 2016; final manuscript received April 13, 2017; published online June 16, 2017. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 17(4), 041011 (Jun 16, 2017) (9 pages) Paper No: JCISE-16-2096; doi: 10.1115/1.4036558 History: Received October 04, 2016; Revised April 13, 2017

## Abstract

This article proposes the use of polytopes in $HV$-description to solve tolerance analysis problems. Polytopes are defined by a finite set of half-spaces representing geometric, contact, or functional specifications. However, the list of the vertices of the polytopes is useful for computing other operations as Minkowski sums. Then, this paper proposes a truncation algorithm to obtain the $V$-description of polytopes in $ℝn$ from its $H$-description. It is detailed how intersections of polytopes can be calculated by means of the truncation algorithm. Minkowski sums as well can be computed using this algorithm making use of the duality property of polytopes. Therefore, a Minkowski sum can be calculated intersecting some half-spaces in the dual space. Finally, the approach based on $HV$-polytopes is illustrated by the tolerance analysis of a real industrial case using the open source software politocat and politopix.

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## Figures

Fig. 1

Toleranced surface and tolerance zone

Fig. 2

Definition of separation half-space: (a) separation half-space, (b) redundant half-space, and (c) empty set

Fig. 3

Example of the truncation process

Fig. 4

Truncated polytope

Fig. 5

Truncation algorithm in tolerance analysis

Fig. 6

Primal and dual cone of a vertex [13]

Fig. 7

Minkowski sum based on intersection of normal fans

Fig. 8

Case study: breaking system

Fig. 9

Parts and surfaces enumeration

Fig. 10

Contact graph of the mechanism

Fig. 11

Two-dimensional projection of the operand polytopes

Fig. 12

Two-dimensional projection of the calculated and functional polytopes

Fig. 13

Comparison of the result with different Nd

Fig. 14

Minimal distance between the functional and the calculated polytopes

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