Relative position computation for assembly planning with planar toleranced parts∗ Yaron Ostrovsky-Berman and Leo Joskowicz School of Engineering and Computer Science The Hebrew University of Jerusalem, Israel Email:
[email protected]
Abstract This paper presents a new framework for worst-case toleranced assembly planning of planar mechanical systems. Unlike most assembly planners, which produce plans for nominal parts, our framework incorporates the inherent imprecision of the manufacturing process, which introduces uncertainties in the shape and size of the assembly parts. It accounts for the uncertainty of the relative part placements in the assembled state and allows planning for all assembly instances. Our framework is more general than existing approaches in terms of the part model, the tolerance specification model, and the type of motions used in the assembly sequences. We describe efficient algorithms for computing the sensitivity of part positioning to variations in part geometries, and for incorporating these computations into the geometric core of existing assembly planners for nominal parts. We show that the cost of accounting for toleranced parts in planning is a multiplicative factor which is a polynomial of low degree in the number of tolerance parameters. Our implementation and experiments on five assembly models show that tolerancing significantly reduces the volume of the space of valid motions for assembly sequence planning, and that for translational motions this reduction depends linearly on the size of the tolerance intervals. We conclude that geometric computation for assembly planning with tolerance parts is time efficient and practical. Keywords: motion planning, assembly planning, geometric constraint solving, tolerance analysis, tolerance envelopes.
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Introduction
Assembly sequence planning of mechanical systems has received significant attention due to its practical importance in manufacturing. Nearly all assembly planners produce plans for nominal parts, assuming that the part geometry, position, and orientation in the assembled state are fixed and known. However, the inherent imprecision of the manufacturing process introduces uncertainties in the shape and size of the assembly parts, which results in uncertainty in their relative placement in the assembled state. Thus, the nominal assembly plan may not be feasible for certain instances of the parts, and a valid plan for one instance may not be suitable for another. Manufacturing variability is controlled through tolerances, which allow engineers to specify limits on part and assembly variability. A manufactured part instance passes inspection if its ∗ An earlier version of this paper appears in the proceedings of IEEE International Conference on Robotics and Automation 2005 and was selected as one of the three finalists of the Kayamori Best Automation Paper Award.
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dimensions (lengths, radii, and angles) and geometric characteristics (form, profile, orientation, location, and runout) meet the design specifications. Since it is impossible to know in advance which instances of the parts will have to be assembled and to plan an appropriate assembly sequence, an assembly sequence for toleranced parts must be feasible for the parts that meet the design specifications. To achieve this goal, the assembly planner must perform worst case tolerance analysis based on the geometry of the parts and their relative positioning relations. The geometry of the parts is affected by tolerancing on feature dimensions. A chain of related features accumulates tolerances such that the last feature in the chain can tolerate larger variations. This is called a stack up analysis of tolerances in parts. Similarly, the relative positioning of parts is affected by tolerances on the precision of the assembly operations. The last part in a chain of related parts accumulates tolerances due to imperfect placement of parts relative to each other and imperfect geometry of parts in the chain. The tolerance stack up in this case is on the translations and rotations that the parts have to undergo to satisfy the relative positioning relations. From the perspective of mechanical engineering, the goal is to perform geometric worst case analysis of tolerance chains of two-dimensional elements (parts) with changing contacts and closed loops, for functionality analysis and assembly sequence planning. Chains of one-dimensional elements are well understood, and assemblies with open and closed loops of fixed contacts (joints) have been studied extensively, but assembly planning requires a twodimensional tolerance stack up analysis of parts with varying contacts. In this paper we present a framework for geometric worst case analysis of assemblies with toleranced parts. We introduce a constraint-based assembly specification which determines the relative positioning of parts with imperfect geometries. We analyze the sensitivity of part positions to variation in part dimensions, and perform geometric computations for assembly planning of toleranced parts that replace their counterparts for nominal geometries in many existing planners. The framework is based on our previously developed parametric tolerancing model for planar parts whose boundaries consist of line and circular arc segments, and whose vertices are described by standard elementary functions of part dimensions, which vary within tolerance intervals. This model is general in its semantics and complies with a large subset of the international tolerancing standard [3]. It allows dependencies between neighboring vertices, and variations in lengths, radii, and angles, and therefore subsumes simpler tolerancing models used for assembly planning [8, 21]. We model the relation between parts in the assembly with geometric constraints based on distances between points, lines, and arcs. These are sufficient to model all types of clearance, contact, and angularity relations between part features. The relations constrain the degrees of freedom of parts such that for every instance of part geometry, there exists a unique rigid transformation for relative positioning. To model the relative position constraints in the entire assembly, we introduce the assembly graph, a generalization of Latombe’s relation graph [21] that includes closed chains, parts with three degrees of freedom, and conditional constraints, which are deliberate over-constrained specifications. From the tolerance specification of the individual parts, and from the relative positioning constraints, we obtain the sensitivity of individual part positioning to the variation in assembly parameters (part dimensions and distance relations) in the form of sensitivity matrices. For each assembly part vertex, we compute the sensitivity matrix, which accounts for all possible variations of the vertex coordinates. Using the sensitivity matrices, we obtain the relative position envelopes and the configuration space envelopes of parts, which are the main geometric 2
tools for functionality analysis, and assembly planning algorithms, respectively. We demonstrate our framework on algorithms based on the motion space approach for assembly planning [15]. We show how to replace the nominal motion space objects with their toleranced counterparts in probabilistic roadmap methods and in three useful assembly sequence types: single step infinite translations, infinitesimal rigid motions, and multiple step translations. Our analysis shows that the added complexity of accounting for tolerancing in assembly planning is a multiplicative factor, which is a polynomial of low degree in the number of tolerance parameters. In a series of experiments on five assembly models, we quantify the effect of tolerancing on the assembly by measuring the increase in blocked translational motions as a function of the tolerance interval size, and determine that the reduction in the volume of valid motions is linear in the size of the tolerance intervals. We conclude that geometric computation for assembly planning with tolerance parts is time efficient and practical. This paper is organized as follows. In Section 2 we review work related to tolerancing, relative position computation, and assembly planning. In Section 3 we describe the relative positioning constraints and the assembly graph. In Section 4 we demonstrate the framework on the motion space algorithms for assembly planning. In Section 5 we discuss the implementation and experiments. We conclude in Section 6 with extensions and future work.
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Related work
The current dimensioning and tolerancing standard [3] defines geometric tolerance zones for toleranced part features. The features must be contained in this zone, which is typically two parallel planes or two concentric cylinders. Since the feature can vary in an arbitrary manner within the tolerance zone, there are many ways to model the geometry of a valid part instance. The standard does not specify how the part should be represented for assembly tolerance analysis. There are numerous models and representations of toleranced parts and features. Juster [18] surveys models developed prior to 1992. Models based on simple bounding volumes for vertex variation [1, 7, 9, 17, 21] have the advantage of allowing exact geometrical analysis of tolerance accumulation but cannot model parameter dependency induced by actual tolerance specifications and assembly relations. Requicha [26] and Rossignac [27] present a tolerancing model based on offsetting operations on solid model boundaries. Their models have a strong mathematical basis but depart from practices defined by current standards. Models of toleranced features based on rigid transformations of small displacements and rotations were recently developed to model variation within the geometric tolerance zone defined by the standard [6, 10, 28, 39]. These models were used to analyze chains of parts in the assembly. However, the acceptable domain for the transformation variables (called deviation space or tolerance map) is a high dimensional convex polytope, which necessitates exponential assembly instance samples for worst case analysis. The tolerancing model we use in this paper [24] is equivalent to the parametric tolerance specifications [2, 35] and complies with a large subset of geometric specifications [3] while allowing polynomial time worst case geometric analysis. The relative position of imperfect planar parts was studied by Turner [38], who reduces the problem to solving a non-linear system of constraints for a given cost function. Sodhi and Turner [33] later extended this work for 3D parts. Li and Roy [22] show how to find the relative position of polyhedral parts with mating planes constraints. These methods compute the placement of a single instance of the assembly, and cannot be extended to analyze the entire variational class of the assembly. Inui et al. [17] propose a method for bounding the volume of the configuration space representing position uncertainties between two parts. However, their 3
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