Kinematic decomposition of geometric constraints 

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Geometric model

Designers usually manipulate only the nominal geometry of products through CAD systems. However, the need to consider geometric deviations has led to alternative geometric represen-tation models.
Skin model shapes are models based on non-ideal geometries representing given instances of manufactured parts, including form defects (see Figure 1.4). Different strategies to generate skin model shapes have been proposed in (Favreliere, 2009; Zhang et al., 2011; Schleich and Wartzack, 2014; Yan and Ballu, 2017; Homri et al., 2017; Zhu et al., 2017). The advantage of this kind of representation is that it can simulate geometric deviations that are expected, predicted or already observed in real manufacturing processes (Anwer et al., 2013).
However, the level of detail of the models, and consequently of the simulations, restricts their use to simple assemblies, as shown in Figure 1.5. Furthermore, designers tend to neglect form defects, assuming that contact clearances are of a higher-order (Adragna et al., 2010; Dumas, 2014); this cannot be assumed at the nano-scale (Samper, 2007).
Tolerance simulations are often based on features of perfect form, called substitute features. These are associated to the real feature following a given criterion (minimization of the sum of the squared distances, minimization of the maximal distance) to minimize the form defects. Figure 1.6 illustrates an example of this association process.
In the most approximate case, a real surface is represented as a point (1D tolerancing). In 2D tolerancing, real surfaces are represented by line segments. In 3D tolerancing, perfect form surfaces are used to simulate position and orientation deviations. Seven surface classes are used: spherical, planar, cylindrical, helical, rotational, prismatic and complex. Each of these classes is derived from the displacement subgroup that leaves it globally invariant (Hervé, 1994).

Estimation of limits

To compute the worst-case deviation propagation scenario in an assembly, the extreme values for each tolerance and contact are assumed. By doing this, it is ensured that 100% of the products are going to be assembled and/or work as expected, which becomes interesting for prototypes or small production series. However, it implies tighter tolerances, and consequently, higher manufacturing costs due to additional machining operations or more expensive machining equipment (Jeang, 1994).
The statistical approach considers that it is more economical to reject a small percentage of production than to use a more accurate manufacturing process (Fleming, 1987). Using this idea, tolerance values are relaxed to some extent, which is interesting for high production rates. The challenge is then to find the right compromise between the cost of the rejected production and the cost of the increase in the tolerance values.
The objective of a statistical simulation is to estimate the probability distribution of Y considering the probability distributions of the contributors. When real manufacturing data are available, more accurate tolerance propagation simulations can be performed (Dantan and Qureshi, 2009). This can be done either by analytic or stochastic methods. When analytic methods are used, it is assumed that the contributors X1, X2, …Xn are all normally distributed. Their nominal values are then set at the mean and from the standard deviation of the input variables the probability distribution of the output can be estimated (Morse, 2004; Shen et al., 2005; Ghie et al., 2010): s σY = ∂X1 2 σX2 1 + ∂X2 2 σX2 + … ∂Xn 2 σX2 (1.2).
When f is not available in analytic form, the partial derivatives cannot be computed. Stochastic methods, such as the Monte Carlo simulation, are then required to generate a popu-lation of input parameters and estimate σY (Chase et al., 1995; Yang et al., 2013; Qureshi et al., 2012). A representative number of feature instances are generated, varying their position and orientation according to a given probability distribution (see for example the sample illustrated in Figure 1.11). In this way, intersections and Minkowski sums are avoided. The propagation is computed by means of linear optimization methods combined with reliability calculation algorithms (Dantan and Qureshi, 2009; Beaucaire et al., 2013; Mansuy et al., 2013b). One of the advantages of this strategy is that sensitivity indices can be estimated in relation to the tolerance values, which produces very interesting results for tolerance decision-making (Ziegler and Wartzack, 2015).

Combinatorial geometry

The combinations and arrangements of geometric objects is known as combinatorial geometry. It is concerned not only with theoretical aspects but also applied ones (Weibel, 2007). The applied branch of the combinatorial geometry is called computational geometry. It deals with algorithms computing geometrical objects and solving geometrical problems. Since many general problems in sciences can be solved by geometrical models, the field of applications is very large. In biology, Pey and Planes (2014) uses polyhedral cones to represent metabolic networks (see Figure 2.1a). In robotics, Firmani et al. (2008) and Dai (2016) represent sets of reaction forces by means of polyhedral cones (see Figure 2.1b). In CAD, Peternell and Steiner (2007) employ polytopes combinatorics (particularly Minkowski sums) in solid modelling (see Figure 2.1c). In manufacturing, Inui and Ohta (2007) use Minkowski sums for computing tool paths (see Figure 2.1d). Grandguillaume et al. (2017) represent kinematic limits of machine-tools with polytopes for choosing tooling orientation.

Modelling sets of constraints with 6D polyhedra

In mechanical design, a tolerance zone represents the limits of the manufacturing defects for a given feature. When the feature is considered as a discrete set of points Ni, this restriction is transferred to each of them. These geometric constraints can be modelled as algebraic constraints:
S1 ⊆ T Z ⇔ ∀Ni ∈ S0 : dsup ≥ tNi · ni ≥ dinf (2.1).
where S1 is the substitute surface related to the nominal feature S0, T Z is the tolerance zone defined offsetting S0 from dinf to dsup and tNi is the translation displacement of S1 in relation to S0 at point Ni (see Figure 2.3).

Modelling stack-up of deviations by operations with polytopes

When considering rigid parts, the defects propagation in a mechanical system depends on how the constituting parts mated. The cumulative stack-up of deviations between any couple of surfaces of an assembly can be simulated operating with geometric and contact constraints. To do this, all the constraints must be expressed under the same reference system and at the same point. The set of required operations can be determined according to the topological structure of the assembly (serial or parallel).
As presented in Eq. 2.5, the native input data used for defining polyhedra in geometric tolerancing is the H-representation (set of closed half-spaces). As we expose next, the V-representation (set of vertices) is also required for computing some operations. So we handle polytopes in their HV-representation.

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Modelling parallel architectures – intersections

The interaction of geometric deviations when parts are mated in parallel, i.e. sharing mul-tiple contacts, can be modelled as the intersection of the derived sets of constraints. In the typical example of a clamp, the misalignment between the connected parts can be calculated intersecting the polyhedra derived from the pin-hole joints, as depicted in Figure 2.7.
Figure 2.7: Modelling stack-up of deviations as the intersection of polyhedra (Gouyou et al., 2016). Algorithmically, the computation of the intersection of polyhedra is not complicated. It demands joining together the constraints of both operands and removing the redundant ones. Then, an intersection of two polyhedra 1 and 2 can be computed with the H-representation of the operands (Teissandier, 2012; Arroyave-Tobón et al., 2017c): kmax1 kmax2 \1 \2 k =1 k =11 ∩ 2 = H¯k+1 ∩ H¯k+2 (2.6) .

Modelling serial architectures – Minkowski sums

Fleming (1988) and Srinivasan (1993) established the correlation between cumulative defect limits on parts in contact and the Minkowski sum of sets of constraints. In other words, if several parts are mated in a serial configuration, the stack-up of their geometric deviations can be calculated summing the geometric and contact polyhedra involved in the toleranced chain (see Figure 2.8). Definition 2.3.1 (Minkowski sum) Let P1 and P2 be two polytopes. Their Minkowski sum is defined as: n o P1 ⊕ P2 = a + b, a ∈ P1, b ∈ P2.
Because their unbounded nature, Minkowski sum of polyhedra is challenging in computa-tional geometry. This is why few works has been published in this subject. A graphical example of the sum of polyhedra is presented in Figure 2.9.
Fukuda (2004) presented an algorithm to compute Minkowski sums of polytopes, mentioning the possibility of applying the same procedure for the case of polyhedra with at least one vertex (pointed polyhedra) by treating infinite rays as points at ‘infinity’. However, due to the degrees of freedom (or invariance), the polyhedra manipulated in tolerancing usually do not have vertices. In fact, each degree of freedom (or invariance) implies a sweeping operation of a polytope along a straight line, placing the vertices at infinity.
Figures 2.10a and 2.10b present examples of Minkowski sums of 2D and 3D polytopes respectively. Probably the most common technique for computing sums of polytopes consists in adding all the vertices of the operands (Wu et al., 2003; Peternell and Steiner, 2007). It implies, afterwards, the computation of the convex hull of the calculated cloud of points, which can be expensive in an affine space of dimension 6. In addition, this method has to deal with the identification of the points which are not vertices but which are located on the boundary of the calculated polytope.
Although some improvements in the calculation of the sum of the vertices of the operands are presented in (Weibel, 2007; Mansuy et al., 2011; Delos and Teissandier, 2015c), these algorithms do not provide the H-description of the calculated polytope (required in tolerance analysis to compute subsequent intersections). Other methods have been proposed in the literature (Fogel and Halperin, 2007; Lien, 2010; Li and McMains, 2014) but they are only applicable in a 3-dimensional space and can not be generalized to higher dimensions.
Within the context of geometric tolerancing, Mansuy et al. (2011) propose a method for calculating separately the sum of the most disadvantageous vertices with respect to a functional polytope. Even if this method avoids the computation of Minkowski sums, the set of computed vertices is only representative of a given functional condition. In addition, the authors only consider the case of serial tolerance chains made up of features of the same invariance class and in a particular relative position (i.e. a set of parallel planes).
Teissandier and Delos (2011) proposed a method for summing HV-polytopes in a 3D space. The generalization in Rn is formalized in (Delos and Teissandier, 2015b). With this algorithm it is possible to compute Minkowski sums taking advantage of the duality property of polytopes. This property, proved by Ziegler (1995), states that the normal fan of a Minkowski sum of two polytopes P1 ⊕ P2 is the common refinement of the normal fans of its summands: N(P1 ⊕ P2) = N(P1) ∧ N(P2) (2.7).

Table of contents :

1 Review of geometric tolerancing approaches 
1.1 Tolerancing model
1.1.1 Physical model
1.1.2 Geometric model
1.1.3 Variation model
1.1.4 Assembly behaviour model
1.2 Computation strategy
1.2.1 Estimation of limits
1.2.2 Solution direction
1.3 Positioning this thesis in the geometric tolerancing map
1.4 Conclusions
2 Geometric tolerancing with 6D polytopes 
2.1 Combinatorial geometry
2.2 Modelling sets of constraints with 6D polyhedra
2.3 Modelling stack-up of deviations by operations with polytopes
2.3.1 Modelling parallel architectures – intersections
2.3.2 Modelling serial architectures – Minkowski sums
2.3.3 Checking requirements satisfaction: inclusion test
2.4 Truncation algorithm
2.4.1 Computing intersections and sums
2.4.2 PolitoCAT and politopix software tools
2.5 Polytopes and cap half-spaces
2.6 Case study: solution by caps-based method
2.6.1 Operands and operations definition
2.6.2 Simulation run
2.6.3 Analysis of results
2.7 Conclusions
3 Controlling the effects of DOF propagation 
3.1 Cap half-space definition
3.2 Tracing caps: Minkowski sums
3.2.1 Decomposition theorem
3.2.2 Caps propagation theorem
3.2.3 Algorithm: tracing caps in sums
3.2.4 Caps removal
3.2.5 Operands commutativity in sums with caps removal
3.2.6 Algorithm: sum with caps removal
3.3 Tracing caps: intersections
3.4 Tracing caps: inclusion test
3.5 Case study: solution by caps removal method
3.5.1 Caps removal
3.5.2 Influence of summation order
3.6 Conclusions
4 Kinematic decomposition of geometric constraints 
4.1 Kinematic and tolerance analysis
4.1.1 Theory of displacement subgroups
4.1.2 Theory of screws
4.2 Prismatic polyhedra
4.2.1 Definition and properties
4.2.2 Polyhedra decomposition in geometric tolerancing
4.3 Sum of prismatic polyhedra
4.3.1 Summing the underlying polytopes of two prismatic polyhedra
4.3.2 Sum of projections of decomposed polyhedra
4.3.3 Algorithm: projection-based sum
4.4 Simulation feasibility test
4.5 Prismatic polyhedra and ISO standards compatibility
4.5.1 Not fully constrained tolerance zones
4.5.2 Datum features
4.6 Case study: solution by projection-based method
4.6.1 Kinematic analysis and feasibility test
4.6.2 Simulation run
4.6.3 Analysis of results
4.7 Conclusions


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