Tolerance envelopes of parametric planar part models Yaron Ostrovsky-Berman , Leo Joskowicz , Fabian Schwarzer
School of Computer Science and Engineering The Hebrew University of Jerusalem, Jerusalem 91904, Israel Robotics Laboratory, Computer Science Department Stanford University, Stanford, CA 94305 E-mail:
[email protected]
Abstract: We present a framework for modeling parametric variation in planar parts and for efficiently computing approximations of their tolerance envelopes. Part features are specified by algebraic equations defining their position and shape as a function of parameters whose nominal values vary along tolerance intervals. Their tolerance envelopes model perfect form Least and Most Material Conditions (LMC/MMC). We derive geometric properties of the tolerance envelopes and describe efficient algorithms for computing first-order linear approximations with successive accuracy. We show that the tolerance envelope of a parametric arc-line polygonal part with features has segments and can be computed in time, where is the maximum number of non-zero partial feature functions derivatives evaluated at nominal parameter values. Our implementation shows that the algorithms are practical on part models with tens of parameters.
Keywords: parametric models, tolerance envelopes, LMC and MMC boundaries
1. INTRODUCTION A key problem in tolerance analysis is computing the tolerance enveloppe of a part from its tolerance specifications. Tolerance specifications define a family of parts consisting of all valid instances of the part. The tolerance zone of a part is the difference between the smallest volume containing all part instances and the largest volume contained in all part instances. Its boundaries, called the part tolerance enveloppe, define the worst-case variability of the part features, and thus model perfect form Most and Least Material Conditions (MMC/LMC). Part tolerance zones are useful in mechanism design, assembly sequencing and validation, robotics, and metrology, among many others. Recent research in Computer-Aided Tolerancing (CAT) describes methods for defining and computing tolerance zones for individual features from their tolerance specifications [ASME 14.5M, 94], [Clem´ent et al., 97], [Whitney et al., 94]. However, many issues regarding tolerance zones for entire parts remain open: what is their geometric complexity, what are good approximations, and how to efficiently compute them. Previous works are limited by the descriptive power of their variational models, by the quality of the approximations they produce, and by their computational efficacy.
We have identified the need to develop a computational framework for systematically studying part tolerance zones defined by parametric tolerance specifications. The framework should reflect current tolerancing practice, incorporate common tolerancing assumptions, and expose the computational trade-offs of approximations. Of the two commonly used tolerance specification methods [Voelcker, 93], geometric and parametric, we chose the latter because it has a simple semantics, can describe most geometric specifications, and is best suited for functional tolerancing. We present a framework for modeling parametric variation in planar parts and for efficiently computing approximations of their worst-case tolerance envelopes. Part features are specified by algebraic equations defining their position and shape as a function of parameters whose nominal values vary along tolerance intervals. We derive geometric properties of the tolerance envelopes and describe efficient algorithms for computing successive first-order linear approximations, which are the most frequently used. Specifically, we describe three successive approximations of the tolerance envelope of a parametric segments and arc-line polygonal part with segments. The most refined one has can be computed in time, where is the maximum number of non-zero partial feature functions derivatives evaluated at nominal parameter values. The algorithms are practical on part models with tens of parameters.
2. PREVIOUS WORK Several models have been proposed to model parametric part variations. These include simple-shaped regions around boundary points [Goldberg et. al, 98], [Yap and Chang, 97] and fixed-distance boundary offsets [Latombe et al., 97], which are computationally efficient but are often inaccurate since they ignore parameter dependencies. [Pino et al., 01] describe a kinematic model to simulate the “motion” of the features tolerance zone but do not describe how to compute the entire part tolerance zone. [Kethara and Wilheim, 01] model tolerance envelopes with B-splines and describe how to verify if a part is in tolerance without explicitly constructing its tolerance envelope. [Sacks and Joskowicz, 98] developed a kinematic tolerance analysis method that computes contact tolerance zones of planar parametric parts in configuration space. They use the same parametric part model; their contact zones are complementary to our part tolerance zones. Several CAT packages provide tools for computing part tolerance zones [Solomons et al., 97]. Some compute tolerance zones from many randomly generated part shape instances, which is expensive and incomplete, as parts typically have hundreds of features defined by tens of parameters. ADAPT [ADAPT, 2000], developed by Ford for internal use, computes the tolerance envelopes of parametric planar parts with procedural definitions [Hinze, 94]. The procedures indicate how to compute the tolerance zone of each feature and how to compose them to get the part envelope. It has one procedure for each of the many feature definition cases and incorporates ad-hoc simplifying assumptions that preclude quantifying the approximation error. These drawbacks motivate our work.
3. PARAMETRIC PART MODEL be a planar part model whose boundary consists of line and arc segments, and let be a vector of parameters defining them. Each parameter has a nominal value and a tolerance interval . A part instance, , is the part defined by a specific parameter vector ; is the nominal part. We choose the for line and arc seg following vertex-based parametric representation ments. Let and be two consecutive boundary vertices. The parametric representation of a line segment connecting them is: "!# for %$ '& (1) Let
+ -, . for %$ /& (2) where #3 547698 ;:9 ! 3 47698 ;:9 . @ 47698 :9 is the height of the triangle connecting the segment to the arc point, , is the C E B D ! . vector perpendicular to is the arc angle: , and is the length of , , and A J K λ) G)I L9M λ,αN (1−λ) F GH λF The parametric representation of a circular arc connecting them is:
)( * "!# + !0 21
O
α
and = P P5Q P = SRUTWV .YXYZ[ ]=\ ^`_ P5Q SRTWV .YXaZb[ =\ ^?_ for %$ '& and P5e , are the The inner and outer tolerance envelopes of a part model, Pdc boundaries of the intersection and the union of all the instances, respectively: P5c fRUTV .gXaZ[ =h ^`_ Pie fRUTWV .YXYZ[ =\ ^`_ The tolerance envelopes of a point and of a segment of the part model are the boundaries of the union of all their instances, respectively:
In the following, we assume that the parametric part model describes geometrically valid part instances (same topology, no boundary discontinuities or self-intersections), and that tolerance parameter intervals are much smaller (an order of magnitude or more) than their nominal values.
4. TOLERANCE ENVELOPES: DEFINITION AND PROPERTIES
9
We first define the linear approximation of our vertex, line, and arc segment parameterizations. The linear approximation of vertex is defined by the first order terms of the Taylor series expansion:
=< => = < => = =
= < =>
(3)
is the partial derivative value of parameter for the nomiwhere nal parameter value and is the tolerance interval offset. Since tolerance intervals are usually small, higher-order terms are comparatively negligible. Similarly, the linear approximations of a line and arc segments are, respectively:
!# =< => = < =>
( ( * "!# = < => = < => %& $)< => (
(4)
(5)
The key property of these approximations is that they depend only on the parameters that define the vertex coordinates, which are those with non-zero partial derivatives. The number of such parameters, , is usually much smaller than the total number of part model parameters . In the following, we refer to as the maximum number of dependent vertex part parameters. Consider now the tolerance envelope of a point on the part boundary. According to Eq. 3, the maximal displacement of a vertex in direction occurs at extremal parameter offset values . The sign of each such parameter offset depends on the direction and on the partial derivatives :
a
$ .,
V
< =?> . ,- 00 VV .2.2141 f3 && -/ T#6 +87 Z:9
