[10] J. U. Turner, Tolerances in Computer-Aided Geometric Design, Ph.D. Thesis, Rensselaer Polytechnic Institute, 1987.

Transcription

1 Automation, pp , New Mexico, USA, [8] A. A. G. Requicha, Representation of Tolerances in Solid Modeling: Issues and Alternative Approaches, in Solid Modeling by Computers: from Theory to Applications, J. W. Boyse and M. S. Pickett, Eds. New York: Plenum, pp. 3-22, [9] A. A. G. Requicha and S. C. Chan, Representation of Geometric Features, Tolerances, and Attributes in Solid Modelers Based on Constructive Geometry, IEEE of Robotics and Automation, Vol RA-2, No. 3, pp , [10] J. U. Turner, Tolerances in Computer-Aided Geometric Design, Ph.D. Thesis, Rensselaer Polytechnic Institute, [11] J. Guilford and J. Turner, Advanced Analysis & Synthesis for Geometric Tolerances, Manufacturing Review, Vol. 6, No. 4, pp , [12] W. Hsu and C. S. G. Lee and S. F. Su, Feedback Evaluation of Assembly Plans, Proceedings of the 1990 IEEE International Conference on Robotics and Automation, pp , [13] M. Inui and M. Miura and F. Kimura, Analysis of Position Uncertainties of Parts in an Assembly using Configuration Space in Octree Representation, ACM Solid Modeling, pp , [14] S. Lee and C. Yi, Statistical Representation and Computation of Tolerance and Clearance for Assembly Evaluation, Robotica, Vol 16, pp , [15] J. Stark and J. W. Woods, Probability, Random Processes, and Estimation Theory for Engineers, Prentice Hall, 1986

2 gant. The result of the computation can be used to identify one or more problem parallel chains. These parallel chains would be the ones that have very low assemblability. This identification can be used by the designer to re-design the tolerances on those parts involved in these parallel chains. Finally, the approach brings the tolerance technology one step closer to the real-time computation of a product assemblability. This will be a very important tool for the designer. There are a number issues which have not been addressed in this paper. The accuracy of the proposed method need to be studied in detail. The margin of error for the proposed approach need to be studied further. The proposed approach only analyses the given tolerances for the assemblability. However, the result could be used in automatic tolerance design. Such issue has not been addressed in this paper. Lastly, the issues of integrating the approach to existing CAD or solid modeling systems have not be addressed. One of the challenging problem is the representation of tolerance. Existing tolerance representation is in 2D [6] such as geometric tolerances, or 3D such as tolerance zone [1] that must be converted into more computational representation. References [1] A. A. G. Requicha, Toward a Theory of Geometric Tolerancing, The International of Robotics Research, Vol. 2, No. 4, pp , [2] D. E. Whitney and O. L. Gilbert, Representation of Geometric Variations Using Matrix Transforms for Statistical Tolerance Analysis in Assemblies, 1993 IEEE International Conference on Robotics and Automation, pp , [3] O. Bjorke, Computer-Aided Tolerancing, ASME PRESS New York, [4] G. Allen, Tolerances and Assemblies in CAD/CAM Systems, Manufacturing Review, Vol. 6, No. 4, pp , [5] D. Shalon and D. Gossard and K. Ulrich and D. Fitzpatrick, Representing Geometric Variations in Complex Structural Assemblies on CAD Systems, DE-Vol. 44-2, Advances in Design Automation, ASME, Vol. 2, pp , [6] Dimensioning and Tolerancing ANSI Y14.5M, American Society of Mechanical Engineers, USA, [7] S. Lee and C. Yi, Tolerance Analysis for Multi-Chain Assemblies with Sequence and Functionality Constraints, Proceedings of the 1997 IEEE International Conference on Robotics and

3 The number of Tolerance Sweep Operations (TSOs) performed by the algorithm is n(p-1) + 2M, and the number of Adjustable Displacement Sweep Operations (ADSOs) is n(p-1) + M, where M is the number of mating nodes, denoted by double circles. The sweep operations can be pre-computed for each segment (or chain) between every pair of part nodes. Since a TSO is performed for all mating features, it needs 2M TSOs to solve all segments. Note that a mating node is composed of two mating features. Now, in the worst case, there are P-1 future nodes to do an TSO for each backpropagation.since there are n P-chains, n(p-1) TSOs are performed. Therefore, in the worst case, n(p-1) + 2M TSOs are performed. Similarly, since an ADSO is performed for each mating nodes, there are M ADSOs to perform in order to pre-compute all segments. n(p-1) ADSOs are performed during the back-propagation as TSOs.Therefore, in the worst case, n(p-1) + M ADSOs are performed. 6 Conclusion We presented an analytic approach for evaluating the assemblability of a product. The assemblability of a product was evaluated based on both tolerances and clearances, where clearances were used as adjustable space to compensate for possible deviations caused by tolerances. With an analytic method to solve a parallel chain from its two serial chains, an algorithm to solve for the whole assembly was proposed. The proposed method has a big advantage over the simulation based methods. It is computationally more efficient. This is an important feature since this allows for a possibility of real-time computing. But, its drawback is the accuracy of the results. Since the results are the approximations, from the initial representations to the computation of the assemblability, the final result of the product assemblability is subject to some margin of error. However, even the simulation based methods are prone to some margin of error in reality. There are several contributions of this work. First, we developed an analytic methods to compute the propagations of clearances on a serial chain, based on the ellipsoidal boundary of nominal clearances. The computation is fast with very good approximations. Second, we developed an augmented space method to compute the assemblability. This method is very convenient and ele-

4 Table 1: Algorithm execution steps of the multi-chain. P-chain Sequence Pool of Nodes under Consideration Pool of Nodes for Future P-chains 1 {P 1, P 2, P 3 } = S 1 {P 1, P 2, P 3 } 2 {P 4, P 5 } S 1 = S 2 {P 1, P 2, P 3, P 4, P 5 } 3 S 2 = S 3 {P 1, P 2, P 4, P 5 } 4 {P 6 } S 3 = S 4 {P 1, P 2, P 4, P 5, P 6 } 5 S 4 = S 5 {P 1, P 2, P 4, P 6 } 6 S 5 = S 6 {P 1, P 2, P 6 } 7 S 6 = S 7 1. The CH p is solved at P 3, which is underlined and bold faced in the table. After the computation of tolerance and adjustable displacement of P 3, the result back-propagates to P 1 and P 2. When the P-chain 2 is considered, the part nodes attached to it, e.g., P 4 and P 5, are added to the pool of nodes to consider. There are P 1, P 2, P 3, P 4, and P 5 which are attached to the future P-chains, i.e., 3, 4, 5, 6, and 7. P 4 is selected as a node for which P-chain 2 is solved. When the solution of P 4 is calculated, its solution back-propagates to P 1, P 2, P 3, and P 5. This process continues, and can be traced as shown in Table 1. The computational complexity of the algorithm is provided in terms of the number of Sweep Operations (SOs) and Intersection Operations (IOs), computations involved during serial chain and CH p computations, respectively. In the worst case, the number of IOs is np where P is the number of part nodes that join two or more CH p s. n is the number of P-chains in a multi-chain. During the back-propagation process as in step 6 of the algorithm, there are at most P - 1 P- nodes which need IO. Since the iteration ends when n P-chains are computed, there are at most n(p - 1) IOs to compute. However, for each P-chain computation as in step 5, one IO is performed. Therefore, there are n IOs to perform for forward-propagations. Finally, there are np IOs to perform.

5 5 4 P P 5 P 6 P P 3 P 1 Figure 10: A multi-chain of seven parallel chains 7. Let i = i + 1. If i = n, compute the solution of S n at any node of S n, and stop. Otherwise, add all the part node of S i to L c. Go to step 4. Let us explain the algorithm using the multi-chain shown in Figure 10 as an example.the pool of nodes under consideration, when the P-chain 1 is selected, are P 1, P 2, and P 3 as shown in Table

6 affect the previously computed features. Then, the next CH p is solved from the two serial chains as described before, each starting from the solved features of solved CH p s. The algorithm ends when the last CH p of the multi-chain is solved. The algorithm is described next. Multi-Chain Algorithm: 1. Let S i, for i=1,..., n, be a sequence of n CH p s, P-chains, of a multi-chain. 2. Let i=1. Then, collect the every part nodes in S i that are also in other P-chains. For example, P 2 in Figure 10 is part of P-chains 1, 3, 6, and 7. Let L c be a list of this part nodes under consideration. 3. Let L f be a list of nodes for future P-chains.These are the nodes in L c that are attached to P- chains that are not processed, yet. 4. Select one node, n j, in L f that is also in the current S i. 5. Compute the solution of S i at n j. S i is solved using the CH p algorithm described before. However, it must be noted that the propagations to the two mating features (M 1 and M 2 ) of n j must start from two nodes, n 1 and n 2, of L c. n 1 and n 2 are found by tracing back from M 1 and M 2 through their serial chains until the serial chains end. 6. Back-propagate the effect of S i to all other nodes in L c. The back-propagation to the nodes of L c can be computed by finding the shortest paths to the nodes from n j. Note that every node of L c has its current solution. When back-propagation is computed, there are two types of node to consider: (1) a node in a serial chain and (2) a node in L c. In the case of (1), back-propagation is done like a serial chain computation. In the case of (2), back-propagation is done like a CH p computation

7 In the augmented space the assemblability is defined as whether or not the clearance hypersphere with its center point in the tolerance hyper-ellipsoid can intersect the hyper-plane. If so, we can define the center of this intersection as a new tolerance, and the intersection as clearance. The semantics of the intersection is that this is the space where two serial chains can be assembled, or the amount of space the mated features can adjust the position. Therefore, the center of the intersection can be used as a new tolerance, whereas the intersection can be used as a new clearance. The intersection of the constraint hyper-plane and the tolerance hyper-ellipsoid can be used as a new tolerance. Statistically the most of the centers of the intersections of clearance hyper-ellipsoids and the hyper-plane will be inside this intersection. The new nominal clearance can be obtained from the intersection of the nominal clearance hyper-sphere and the hyper-plane. Also, the new variable clearance can be obtained from the intersection of the variable hyper-ellipsoid and the hyper-plane. 5 Multiple Parallel Chains In general, most assemblies have more than one parallel chain, CH p. Therefore, the assemblability of a product must be solved in terms of multiple CH p s, called multi-chain. A multi-chain is defined as a set of minimal loops (CH p s). This is illustrated in Figure 10. In this figure, there are 7 CH p s forming a multi-chain. Each node in the graph represents either a pair of mating features or a reference feature, such as a datum, of a part. In a multi-chain, each CH p affects other CH p s. That is, one CH p imposes a constraint on other CH p s. For example, when a CH p is completed, the tolerances and clearances of each mating features in the chain is more constrained than when they were on a serial chain. Therefore, other CH p s now must use the more constrained tolerances and clearances. Briefly, each CH p is solved sequentially. That is, when a solution of a CH p is computed at some mating feature, its effect is propagated back (back-propagation) to other features of already solved CH p s. Back-propagation is necessary because the completion (or assembly) of a CH p may

8 X 2 variable clearance probability distribution projected on projectio hyper-plane X 1 projection hyper-plane Figure 9: A variable clearance and its distribution on the projection line. 4.4 Tolerance and Clearance of a Parallel Chain A parallel chain, CH p, has tolerance and clearance. That is, statistically a mating feature of a CH p can deviate from its nominal position when it is assembled within the CH p. Similarly, it can have clearance. In this section, we describe an approach to compute the tolerance and clearance of a CH p in the augmented space.

9 X 2 d ti t i X 1 clearance hyper-ellipsoid Figure 8: Tolerance t i at distance d ti and clearance associated to t i. the variable clearance hyper-ellipsoid projected on the projection space. The projection space is a space orthogonal to the constraint hyper-plane. Figure 9 illustrates the projection space and the assemblability.the assemblability is shown by the hatched area of the probability distribution. The bound is located at d ti - 1 from the origin, shown by a dotted line, which is determined from the unit hyper-sphere of nominal clearance, and the distance of t i from the constraint hyper-plane.

10 X 2 tolerance hyper-ellipsoid δd X 1 X 1 = X 2 P t (d) Figure 7: A tolerance hyper-ellipsoid and slices where d max is equal to one plus the maximum distance of the center point of the variable hyperellipsoid from the constraint hyper-plane. The value one comes from the unit radius of the nominal clearance hyper-sphere. P c (d) can be computed as follows. Any tolerance instance, t i, is at distance d ti from the constraint hyper-plane. t i may be assemblable if there is large clearance associated with it, as shown in Figure 8. However, the probability of assembly of t i depends on the statistical variability associated to the nominal clearance. This probability is computed from the probability distribution of

11 4.3 The Assemblability Computation Based on Augmented Space Whether an instance of a CH p can be assembled successfully can be determined by the existence of intersection between the clearances of two serial chains at the deviated positions. The existence of an intersection signifies that the both ends of two serial chains can be brought to this intersection for assembling. This section describes the computation of the assemblability of a CH p based on augmented space. We define a hyper-ellipsoid, X T Σ X = 1, in the augmented space, X = (X 1, X 2 ), to represent the accumulated tolerances from the two serial chains altogether. By the same token, we define a hyper-ellipsoid in the augmented space to represent the accumulated clearances from the two serial chains. Whether or not the two clearances have an intersection at a chosen pair of tolerance instances can be identified by the following equivalence: The nominal clearance hyper-ellipsoid defined in the augmented space and located at the point representing the chosen pair of tolerances has an intersection with the hyper-plane representing the constraint, X 1 = X 2. For the convenience in computation, we can apply the Whitening transformation so as to make the clearance hyperellipsoid a unit hyper-sphere. The tolerance hyper-ellipsoid is partitioned into thin slices such that all the tolerances, t i, in the same slice have equal probability of assembly. The slices can be formed by displacing the constraint hyper-plane by small distance, δ d from the origin along the direction of constraint space normal, as shown in Figure 7. Now, we define P t (d) as the probability of a slice between two hyper-planes located at d and d + δ d in tolerance hyper-ellipsoid, and P c (d) as the assemblability associated with all the t i s on the hyper-plane of distance d from the constraint plane. Then, the assemblability of the whole tolerance can be computed from the equation (2). d max 2 P t ( d ) P c ( d ) (2) d= 0

12 ellipse 1 ellipse 2 accumulated ellipse Figure 6: The accumulation of two nominal clearance ellipses. The variable range of a clearance can be calculated firstly by approximating the range by an ellipsoid. This can be done by using the one third of ranges as standard deviations. Then, the covariance matrix of a range can be calculated by V i Σ r V -1 i, where Σ r is the diagonal matrix composed of the range variances, and V i are Eigen vectors of the nominal clearance ellipsoid matrix, E i. Now ranges can be treated as tolerances, and their propagations and accumulations can be calculated using the same approach as tolerance described in the previous section.

13 The second ellipsoid can be transformed into Y space by letting X = W -1 Y in X T E n c2 X = 1 as ( W 1 Y) T E n c2( W 1 Y) = 1. Then, we can obtain the ellipsoid matrix, y E n c2, in Y space from Y T (W -1T E n c2 W -1T ) Y = 1, as W 1T E n c2w 1 T = E n y c2 Now, two ellipsoids can be added. The axis lengths of the second ellipsoid can be computed from its Eigen value matrix as y E n c2 = V 2 Λ 2 V T 2, where Λ 2 is a diagonal matrix consisting of inverse squares of the axis lengths of the second ellipsoid. All axis lengths of Λ 2 are added by one. The added matrix can be converted back to an ellipsoid matrix form E n c12, as y E n c12 = V 2 E n c12 V T 2 which is the accumulate clearance ellipsoid matrix in Y space. Finally, this result is transformed back to X space as X T ( W T E n )X = 1 y c12 W which is the accumulated ellipsoid of the first and the second ellipsoids. Figure 6 shows the solution of the accumulation of two ellipses. The figure shows that a flat horizontal ellipse 1 is swept around the flat vertical ellipse 2. This swept area is the actual solution of the accumulation of the two ellipses. However, this area is approximated by an accumulated ellipse computed from the approach described before. The result will be conservative since the accumulated ellipse is always going to be smaller than or equal to the actual accumulation.

14 where V 1 and Λ 1 are the Eigen vector matrix and Eigen value diagonal matrix of E n c1, respectively. The transformation W is obtained from the following derivations: λ 1 0 Λ 1 = 0 λ 2 = Λ Λ 2 Λ Λ 2 = Λ Λ 2 I Λ Λ 2 where λ 1 and λ 2 correspond to inverse squares of the axis lengths of the first ellipsoid, and I is an identity matrix. Letting Ψ = λ λ 2 we have T ( ΨV 1 X1 ) T I( ΨV T 1X 1 ) = 1 Replacing W = Ψ V T 1, we have (WX) T I (WX) = 1. The first ellipsoid is now a unit circle in Y variable where Y = WX. This unit sphere can be easily added to the second ellipsoid in the same Y space.

15 where (x,y,z) T as the position vector of the next feature relative to the previous feature [12]. The propagated variance, Σ 1, is added or accumulated to the covariance matrix, Σ 2, of the next coordinate frame as Σ 3 = Σ 1 + Σ 2 to obtain the total tolerance covariance, Σ 3. [3,12] 4.2 Clearance Propagation Clearance is defined by a nominal ellipsoid and a variable ellipsoid. A nominal ellipsoid represents the clearance from nominal dimensions of mating features. A variable ellipsoid represents the variation of the nominal ellipsoid due to the tolerances associated with the individual features. This section will describe in detail the approach to compute the propagation and accumulation of clearances in a serial chain in terms of clearance ellipsoids. A nominal ellipsoid of a small deviation random variable, X, is defined as X T E n c X = 1, where E n c is the weight matrix of the ellipsoid. The propagated nominal ellipsoid is defined as X T E n cp X = 1, where E n cp = J T E n c J, with Jacobian J. The accumulation of a propagated ellipsoid and the next ellipsoid can be approximated using a Whitening transformation. The method transforms the first clearance ellipsoid into a unit sphere using the Whitening transformation, W. The second clearance ellipsoid is transformed according to W. Since the first ellipsoid is a unit sphere, it can be easily added to the second ellipsoid.that is, the axis lengths of the second ellipsoid is increase by one. The added ellipsoid is transformed back using W to obtain the accumulated ellipsoid. More formally, let X T E n c1 X = 1 and X T E n c2 X = 1 define the first and the second ellipsoids. Replacing E n c1 = V 1 Λ 1 V T 1 into the first ellipsoid equation, we get T ( V 1 X1 ) T Λ 1 ( V T 1X 1 ) = 1

16 4 Computation of a Parallel Chain Assemblability A parallel chain, CH p,is formed by the interconnection of mating features forming a closed loop. If any one of CH p s of a product fails to fit, the assembling of the product will fail as well. Therefore, the assemblability of the product can be computed in terms of CH p s. The assemblability of a CH p can be computed by breaking the chain into two serial chains, and measuring the assemblability of these two serial chains. Each serial chain has its ending feature which will have to fit to each other. Therefore, we first compute the possible tolerance and clearance at the ending feature of a serial chain. Then, the assemblability of the CH p s is computed in terms of the tolerances and clearances at the ending features. The tolerances and clearances propagate along the chain from the starting feature to the ending feature. We assume here that parts in a serial chain will always assemble. However, the tolerances and clearances will propagate and accumulate such that the ending feature will have the sum of the tolerances and clearances of all the features in the serial chain. Next, we will describe the analytic approach to compute the tolerance and clearance propagations. 4.1 Tolerance Propagation The propagation of a tolerance ellipsoid can be computed from its covariance matrix, Σ, as Σ 1 = J Σ J T [12, 15], where J is a Jacobian defined as J = R T R T M T, 0 R T with R as the rotation matrix, 0 as the zero matrix, and M as M = 0 z y z 0 x y x 0,

17 Gaussian-Sigmoid (analytic solution) Gaussian Simulated solution Sigmoid Figure 5: Optimal approximation of clearance distribution using Gaussian-Sigmoid filtering nominal and variation clearance ellipsoids are computed using Gaussian-Sigmoid filtering method. The distribution of clearance samples, simulated solution, has a flat-top bell shape, as shown in Fig. 5. The Gaussian-Sigmoid distribution is shown also in the figure and the equation is given in equation (1)., FX ( ) X T Σ 1 X e ( 2π) n 2 Σ 1 2 = (1) Two parameters, Σ and T, in Gaussian-Sigmoid function control the shape of the analytic solution. These parameters can be optimized to best approximate the simulated solution. For the details of the optimization, refer to [14].

18 0.10 θ 1.80 ± 0.01 y P x P1 x 1.70 ± 0.01 (a) (b) (c) Figure 4: The clearance between two parts, P1 and P2, and the clearance zone. Figure 4 describes the clearance. There are two 2D parts P1 and P2 mated through the peg and hole mating features, as shown in Figure 4(a). The peg feature had the width of 1.70 and the hole feature has the width of Each has tolerance of ±0.01. These tolerances mean that the width of each feature can vary by ±0.01. That is, only the forms of the features are allowed to vary. Note that the tolerances are small so that there is not problem to fit P1 and P2. Figure 4(b) shows the clearance zone similar to that of tolerance zone, but with one difference. The clearance zone can vary statistically, but bounded by the given tolerances, as shown by the gray areas. When transformed into the variation space, this clearance zone becomes a diamond shape boundary, like in the pose tolerance example, and the statistical variation is maintained around the nominal shape, as shown in Figure 4(c). Clearance is represented by two ellipsoids: nominal ellipsoid and variation ellipsoid.the nominal clearance ellipsoid, E n c, approximates the nominal clearance, and the variation clearance ellipsoid, E v c, approximates the statistical variation on nominal clearance due to tolerances. The clearance ellipsoids can be optimally approximated using Monte Carlo Method [14]. Briefly, many samples of possible clearances are randomly generated and transformed into the deviation space. These samples are generated statistically within the given tolerances.the samples in deviation space are bounded volumes, are statistically distributed. From this sample distribution, the

19 hole axis tolerance zone 1 1 θ A B x B 3 (a) A (b) Figure 3: (a) A hole feature and the position tolerance of 0.2. (b) Tolerance modeled in the deviation space of x and θ, where ellipse boundary approximates the real boundary, diamond. 3 Clearance Representation The clearance between two mating features allows a small play between the features after the mating. This small plays between the features can be used to adjust the poses of the parts in a partial assembly in order to compensate for the pose deviations caused by the tolerances. If the clearances are sufficiently large, more parts may be assembled successfully to the partial assembly, called a sub-assembly. In contrast, without clearances it may be very difficult, if not impossible, to assemble the parts. If mating features are of perfect shape, or nominal dimension, the corresponding clearances between the features are also nominal. That is, this nominal clearances do not have any variations. However, individual features with tolerances, such as form tolerances, can create variability to the forms of the individual features. This variation is also statistical in nature as in pose tolerances, and is related to the clearances.

20 datum, or datums, in location, orientation, runout, and profile. Where as, individual features are features which relate to perfect geometric counterparts of themselves as the desired forms. No datum is used. The types of tolerances associated with the related features can create pose errors of features. They make the assembling more difficult. That is, when mating features deviate from their nominal mating pose, the assembly parts may fail to fit. These types of tolerances are called pose tolerance. We assume normal distribution for tolerances because specific manufacturing processes are not known at a design stage, many manufacturing processes have Gaussian distribution, and central limit theorem can be applied. With this assumption, we represent pose tolerances by ellipsoids in a deviation space. The deviation space is defined by kinematic parameters of translation and rotation. The ellipsoid of a pose tolerance optimally represents the statistical distribution of the pose deviation of a feature [2]. In brief, the meaning of tolerance of a feature can be captured by a tolerance zone, which defines the maximum boundary of allowed deviation. This tolerance zone is transformed into the deviation space. That is, all pose variations of the feature permitted by the pose tolerance is captured by the boundary. Then, the probability distribution of the pose variations in the boundary is optimally approximated by an ellipsoid. Figure 3 describes the representation of a position tolerance of a 2D hole. Figure 3(a) shows the position tolerance of 0.2 specified to the hole, and the tolerance zone. The meaning of the tolerance zone is that the axis of the hole can vary within this zone. This allowed variation is transformed into the deviation frame of x and θ, where x is the translation variation along the x axis and θ is the rotation variation. The maximum rotation variation can be computed using a small angle approximation. Figure 3(b) shows the boundary of the position tolerance in the deviation frame. The diamond shape results from the linear approximation of the dependency between the two parameters, x and θ.using Gaussian distribution assumption with 3σ limit, the diamond boundary is approximated by the ellipse.

21 Bjorke [3] proposed statistical approaches to a tolerance analysis based on functionality. The goal was to derive a tolerance chain equation for one or more functional dimensions. Then, these sum dimensions were checked for functionality criteria. The analysis works on one dimensional space along the sum dimension. Turner [10,11] showed that a tolerance specification can be expressed as an in-tolerance region of a normed vector space, where points outside this region represents an assembly with one or more out of tolerance parts. Several possible tolerance analysis methods such as linear programming, Monte Carlo, and least square fitting were proposed. Hsu and Lee [12] modeled tolerances using differential transforms, and characterized them using means and covariances. They proposed an analytical method to compute the pose uncertainty of an object during an assembly task. Based on a configuration space, Inui, Miura, and Kimura [13] proposed an algorithm for analyzing the position uncertainty of two parts due to tolerances.the problem was formulated as a problem of computing a variation bound of the configuration of two parts. Then the bound was checked if it satisfied the position constraint in an assembly. Allen [4] suggested both worst case and statistical schemes for checking the assemblability of a product design based on CAD model. The fit condition was checked by using an intersection operation between part models. The worst case was simulated by allowing part models to have maximum materials. A statistical scheme was simulated by using sample variants of parts. Lee and Yi have proposed a statistical approach to an assemblability analysis, where tolerances were modeled by ellipsoids and the propagations were computed based on Monte Carlo Method[7,14]. 2 Tolerance Representation Tolerance is defined as the total amount by which a specific dimension is permitted to vary. Geometric tolerance is the general term applied to the category of tolerances used to control form, profile, orientation, location, and runout [6]. There are two kinds of features to which a geometric characteristic is applicable: related and individual. Related features are features which relate to a

22 In this work, we make the following assumption. (1) Parts are rigid such that they do not deform nor have subparts that translate or rotate. The evaluation becomes much complex when dealing with such parts since they introduce more uncertainties to the evaluation, although they contribute to increasing the assemblability. (2) There exists some clearance between two mating features, like peg and hole mating. When mating is of different type, such as two planar faces, the clearance can be obtained based on the functionality constraint. For example, the functionality may require that they must be within some limit either along the plane of the faces, if they are planar faces, or off along the normal of the faces, or both. (3) The nominal parts never fail to assemble. Therefore we assume that the part geometries and assembly are designed correctly. (4) Tolerances are assumed to have a Gaussian distribution. Although some parts may have different probability distributions, many manufacturing processes, as well as the sum of the parts in the assembly, will produce normal distribution based on Limit Theorem. 1.2 Related Work ANSI Y14.5M [6] described geometric dimensioning and tolerancing for designing and manufacturing components of mechanical products. The main intended use was for manufacturing and inspection. If specified correctly by the designers, tolerances can also guarantee the assemblability between two parts with multiple mating features. Requicha [1,8] proposed tolerance zones, which are regions constructed by offsetting nominal boundaries of features. The objective was to formalize and generalize the tolerance practice and to establish a suitable bases for incorporating tolerances to solid modeling systems [9]. Whitney and Gilbert [2] proposed transform matrices to represent tolerances defined in ANSI standard [6]. Using Chi-Square error scheme, they computed the value of an optimizing factor to obtain optimal parameter values. Tolerances analysis methods at the assembly level can be classified into two categories: functionality and assemblability. The functionality analysis identifies and analyzes critical dimensions in an assembly. Such dimensions may be critical to the proper function of an assembly. The assemblability analysis deals with calculating a possible part pose in an assembly or with analyzing the probability of successful assembly.

23 P n P 12 P 22 P 11 P 21 P b Figure 2: An assembly of six parts with clearances. help assembling the parts. In this figure, there are six parts mated through peg and hole features. Clearances shown between the parts with some exaggeration. However, they show that the parts will fit because of clearances allow the parts be adjusted in position. Therefore, clearances play an important role in the evaluation of assemblability. They provide the adjustability to part positions during assembling. The assemblability problem, therefore, is to determine the probability of successfully assembling the parts given the dimensions and tolerances of the parts, mating constraints, and an assembly sequence. Mating constraints include positions of mating features in the assembly, clearances between mating features, and functionality of the parts. An assembly sequence is required because different assembly sequences, of the same assembly, can have different assemblability [7]

24 loop Figure 1: A simple two parts assembly with one parallel chain. when the mating features deviate from their nominal positions such that the loop cannot be closed. In this paper, we consider only the case of global matings. Whether or not a CH p can be assembled depends not only on tolerances but also on clearances. Tolerances allow parts to vary by small amount. Depending on the type of tolerances used, part features can vary either in position or dimension, or both. (See [6] for the details of geometric tolerancing.) The tolerances that allow position variations can cause more problems for assembly. Because each tolerance cause a part to deviate by small displacement, the total effect could accumulate to be large enough for the parts in CH p fail to assemble. However, clearances between two mating features can help compensate the position deviations. A clearance is an allowed infinitesimal space between two mating features, usually to fit two mating features like a peg and a hole. They not only help the assembly operation between two features or two parts, but they also provide adjustability to CH p.that is they can accumulate in a chain such that these clearances can help adjust the position of the whole chain. Figure 2 shows an example of how clearances can

25 1 Introduction It is well known that the manufacturing processes are inherently imprecise, producing parts that vary [1,2,3,4]. These varying parts not only affect the functionality and manufacturability, but also affect the assemblability of the product. Therefore, one of the objectives of concurrent engineering should be to design the optimal tolerances of the parts, such that they not only satisfy the functionality but also lower the cost of manufacturing and assembling. In general, larger tolerances decrease both the manufacturing cost and the quality of the product, but on the other hand, they increase the assembling cost. The manufacturing cost can be decreased with large tolerances because low end machines can be used to produce the parts and because parts can be manufactured in shorter manufacturing time. However, the quality of the parts, and therefore the product, may be decreased because assembled parts may not function well. The assembling cost can increase with large tolerances because the parts can fail more frequently to assemble, increasing the scrap and rework cost [5]. This paper is about the effect of the tolerance design of the parts on the product assemblability. 1.1 Problem Parts fail to assemble if they form a loop in the assembly. A simple loop can be formed by a pair of parts each having two mating features. For example, one part has two cylindrical pegs and the other part has two cylindrical holes. The pegs and holes are the mating features and they must fit. Figure 1 illustrates a loop formed by the mating features. That is, two parts must be joined by peghole and peg-hole loop. They will fail to assemble (fit) when the both pegs cannot be fit into the their mating holes. A loop formed by two or more parts is called a parallel chain (CH p ). CH p may fail to assemble mainly because of two reasons: local mating and global mating. A local mating refers to the mating between two features only. For instance, if the diameter of a cylindrical peg is larger than the diameter of a cylindrical hole, these two features will never fit. A global mating refers to the mating of CH p. That is, although all local mating are successful, the assembly of CH p may fail

26 Abstract Since manufacturing processes are inherently imprecise producing parts that vary, the assembly parts sometimes can fail to fit together. These failures directly affect the product cost due to rework, rejection tags, or engineering changes. Consequently, the tolerance design should not only consider the machining cost of the parts but also the product assemblability. In this paper, we propose an approach to evaluate the assemblability of a product design based on the analytic computation of propagations of tolerances and clearances, and the analytic computation of the assemblability. A particular attention is paid to the computation of a parallel chain. Then, a complete algorithm is given for solving the whole assembly based on the parallel chain computation. The proposed approach has a significant computational advantage over the simulation based approaches.

27 Analytic Evaluation of Assemblability for Tolerance Design Sukhan Lee 1 Chunsik Yi 2 Samsung Advanced Institute of Technology 1 P. O. Box 111 Suwon, Korea Unigraphics Solutions Inc. 2 Computer-Aided Manufacturing Hope Street Cypress, CA 90630