Generic Fixture Design Algorithms for Minimal Modular Fixture Toolkits Aaron S. Wallack (
[email protected]) Computer Science Division, University of California at Berkeley Abstract
Fixturing, i.e. the process of immobilizing a workpiece for manufacturing or assembly operations, is a fundamental task in manufacturing. Fixtures can either be fabricated from scratch or assembled from a toolkit of modular components; the latter approach is termed modular xturing. Recently, many researchers have proposed generate and test xture design algorithms for minimal modular xture toolkits where minimal xture toolkits incorporate a single degree of freedom and constrain xture elements to a nite number of locations. Generate and test strategies have succeeded because minimal xture toolkits can only xture generic workpieces in a nite number of ways. In this paper, we present alternative minimal toolkits and two generic xture design algorithms: a complete enumeration algorithm based upon the generate and test paradigm, and an ecacious heuristic hill climbing algorithm. Our generate and test design algorithm is based upon a duality we observed between xture design and object recognition from sparse probe data. In particular, we reduce the task of completely enumerating xtures to the task of constructing complete indexing tables, a previously solved problem [18, 19].
1 Introduction
Fixturing encompasses the design and assembly of devices for locating and immobilizing a workpiece during a manufacturing operation such as machining or assembly. Fixtures can be fabricated from scratch, or assembled from a toolkit of precisely manufactured, mass produced xture elements; the latter approach is termed modular xturing. Fixturing accounts for 10-20% of total manufacturing costs [7], and as such, its automation is greatly desired by industry. Hazen and Wright presented a thorough review of xturing and automated xture design in the mechanical engineering, manufacturing, and robotics literature [9]. Hazen and Wright's review implied that Supported by Fannie and John Hertz Fellowship and NSF FD93-19412, currently employed at Cognex Corp., Natick Mass.
most of the research in automated xturing systems involved expert systems. Unfortunately, without geometric analysis, expert systems are incapable of computing xture designs, they can only describe \types" of xtures. The robotics community has also made signi cant contributions in terms of grasping, eciently enumerating useful grasps, and grasp quality metrics; the reader is referred to [14, 17] for an overview of these results. Mishra [15] analyzed the task of designing xtures for objects using a toe clamp toolkit; his research sparked renewed interest in the algorithmic community. Recently, many robotics researchers have proposed modular xture toolkits replete with complete xture design algorithms (which enumerate all valid xture con gurations). If we enumerate all candidate xture designs, then we can nd the optimal xture (with respect to expected forces and torques), and/or a single xture capable of xturing a series of workpieces (thereby reducing changeover time). Fixture toolkits must incorporate at least one degree of freedom to handle workpiece variations. Assuming a frictionless point contact model, each xture element contact provides one wrench; k + 1 wrenches are necessary to positively span a k dimensional wrench space, i.e. achieve force closure (the ability to resist arbitrary forces and torques). Force closure therefore requires k + 1 xture element contacts. Since k + 1 degrees of freedom are sucient to generically satisfy k + 1 constraints, xture toolkits must provide at least one degree of freedom in addition to the k degrees of freedom inherent in the workpiece's pose. When the xture only provides a single degree of freedom, the number of degrees of freedom is exactly equal to the number of constraints; therefore, only a nite number of xture con gurations can simultaneously contact a generic workpiece, which is a prerequisite for xturing. We use the term minimal to characterize xture toolkits which incorporate a single degree of freedom and where xture elements are placed in a nite number of locations. Wallack and Canny [18, 21], Brost and Goldberg [2], and Overmars et al. [16] developed
xture design algorithms for minimal modular xture toolkits: the xture vise (Figure 1), the translating clamp, and the translating clamp and wall toolkits respectively. In this report, we generalize these complete xture design algorithms [18, 21, 2, 16], i.e., we present a generic complete xture design algorithm for minimal xture toolkits.
Figure 1: Wallack and Canny's xture vise toolkit. 1.1 Object Recognition from Sparse Probe Data Object recognition is a classical problem in machine vision and robotics. Although object recognition is commonly associated with CCD cameras, simple sensors, such as linear distance sensors, are also capable of recognizing and localizing objects. Furthermore, simple sensors can be more accurate than complex sensors. Wallack and Canny demonstrated that crossbeam sensors and scanning beam sensors can quickly (0.1 seconds) and accurately (0:025mm) recognize and localize objects [18, 20, 22]. 1.2 Duality Between Modular Fixture Design and Indexing for Sparse Sensing It turns out that modular xture design and object recognition are dual problems. Goldberg and Mason [8] alluded to the duality between xturing and object recognition by using a parallel jaw gripper to identify objects. Consider the subtasks of localizing a workpiece from sparse probe data and verifying a candidate xture con guration.Both of these subtasks involve determining the object's pose achieving simultaneous contact with the probe elements; the main di erence being that xture veri cation also involves verifying force closure. The task of automatically designing xtures involves enumerating all force closure xture element and workpiece con gurations. In this paper, we present a duality between xture design and indexing.
1.2.1 Indexing
Indexing is a well understood constant-time object recognition technique which involves predicting all conceivable observations [4, 6, 10, 11] by constructing a lookup table containing all predictable observations; indexing coordinates are extracted from the
sensed data and they are used to index the table entry containing the object's identity. For correctness (i.e., in order for indexing techniques to always recognize recognizable objects), indexing tables should include table entries characterizing every predictable observation. Recall that automated xture design also involves enumerating every possible feature combination. It turns out that every xture con guration corresponds to at least one indexing table entry. Indexing has mainly been used with image data, obviating the need to construct complete indexing tables because image data usually includes multiple cues for each object; if the object's image provides 10 indexing coordinates; if the indexing table only contains entries for 90% of the valid entries, objects will be recognized with probability 0:9999999999. In contrast, indexing table completeness is critical for sparse sensing techniques, where each observation only provides a handful of data points. While developing a method for recognizing objects from sparse sensor data, i.e. applying indexing to sparse sensing applications, Wallack and Canny [18, 19] needed to construct complete indexing tables and subsequently developed a generic strategy for constructing complete lookup tables by reducing the problem to enumerating cells in an arrangement in con guration space. It turns out that Wallack and Canny's [18, 19] complete indexing table construction strategy does not even characterize the cells in the arrangement; rather it enumerates at least one witnesses (curve intersections) for each cell. Interestingly, these witnesses characterize con gurations where the probe position is an exact multiple of the indexing table discretization. These multiples exactly correspond to the evenly spaced lattice positions for the xels. Therefore, since each xture con guration corresponds to at least one indexing table entry, we can enumerate all valid xture con gurations by constructing a complete indexing table and then testing for force closure. In addition, it turns out that reducing the problem to enumerating cells in an arrangement in con guration space also simpli es the task of verifying force closure. Wallack and Canny [21] characterized force closure algebraically in terms of intersections of k + 1 halfspaces bounded by algebraic curves. These halfspaces can be used to eciently prune away nonforce closure con gurations; therefore, we can incorporate force closure constraint into the con guration space arrangement cell enumeration algorithm. The duality between xture design and object recognition also implies that we can always parameterize minimal xture toolkits in terms of k degrees of freedom. Dense parameterizations, i.e., parame-
terizations involving the least number of degrees of freedom, are important because irrelevant and redundant degrees of freedom signi cantly adversely a ect performance. Although a minimal xture toolkit may seem to have k+1 degrees of freedom, k for the object pose and 1 for the toolkits actuator, we can always parameterize a minimal toolkit in terms of only k degrees of freedom because we are only ultimately concerned with con gurations where the xture elements simultaneously contact the workpiece. 1.3 Outline This paper continues as follows: in section two, we present alternative minimal modular xturing toolkits. In section three, we present two pairs of dual systems. In section four, we describe a generic complete xture design algorithm for minimal xture toolkits. In section ve, we present a heuristical xture design algorithm since enumerating a single xture design is more appropriate for some applications. We conclude by highlighting the main contributions.
2 Alternative Modular Fixture Toolkits
2.1 Scissor Jaw Fixture Toolkit Figure 2 depicts a planar scissor jaw xturing toolkit. Pegs are inserted into the lattice holes inlaid in xture plates which can rotationally pivot. Workpieces are xtured by torquing the two xture plates together.
Top View Instantiation #1
Isometric View Instantiation #2
Figure 2: Two jaw scissor xture vise 2.2 Three-Jaw Fixture Chuck Figure 3 depicts a planar three jaw chuck modular xturing toolkit. Pegs are inserted into the lattice holes inlaid in xture plates mounted on the three jaws of a chuck. Workpieces are xtured by closing the chuck. 2.3 Four-Jaw Fixture Chuck Figure 4 depicts a planar four jaw chuck modular xturing toolkit. Pegs are inserted into the lattice holes inlaid in xture plates mounted on the four jaws of a chuck. Workpieces are xtured by closing the chuck.
Top View Instantiation #1
Isometric View Instantiation #2
Figure 3: Three jaw xture chuck
Top View Instantiation #1
Isometric View Instantiation #2
Figure 4: Four jaw xture chuck 2.4 Three Dimensional Tetrahedral Chuck Figure 5 depicts a tetrahedral four jaw chuck modular xturing toolkit. Pegs are inserted into the lattice holes inlaid in xture plates mounted on the four jaws of a tetrahedral chuck. Workpieces are xtured by closing the chuck.
Figure 5: Tetrahedral four jaw xture chuck
3 Dual Fixture Design and Object Recognition Systems
In this section, we detail two examples of the duality between modular xturing and sparse object recognition: the xture vise toolkit and horizontal probing, and the four jaw xture chuck and orthogonal probing. 3.1 Fixture Vise and Horizontal Point Probing Horizontal probing involves recognizing and localizing objects using parallel distance sensors (Figure 6). Consider the xture vise toolkit shown in Figure 1. The four xture element contacts could alternatively
Probed Object
Figure 6: The task is to determine the object's identity and pose from the probe data correspond to four horizontal probe contact points (Figure 7). Under certain criteria, tuples of horizontal probe contacts exactly correspond to xture vise con gurations. 1. The lattice xel constraints dictate that the horizontal probe lines must be separated by integral multiples of the row spacing 2. The lattice xel constraints also dictate that two pairs of probe points (which may share one point) where the points in the pair are separated in x by integral multiples of the column spacing 3. The xture vise non-overlapping constraint dictates that contact points must be compatible with non-overlapping jaws, so when the two pairs of points are disjoint, one pair must lie entirely to the left of another, and when the two pairs of points share a point, then the remaining point must lie completely to the left or right of the other three.
2D View of Modular Fixture Configuration Achieving Simultaneous Contact
Horizontal Probe Measurements from an Object in Some Pose
Figure 7: Fixture vise con gurations can alternatively be characterized in terms of horizontal probe data where two pairs of probe measurements di er by integral multiples of the lattice spacing.
3.1.1 Fixture Con gurations Correspond to Indexing Table Entries
Assume that indexing vectors contain all the pairwise relative di erences between probe measurements. Since the indexing table contains entires for every
sensing con guration and since every modular xture con guration corresponds to a specialized sensing con guration, every modular xture con guration has a corresponding entry in the indexing table. Furthermore, candidate xture vise con gurations correspond to indexing table entries where two of these indexing coordinates (relative positional measurements) achieve integral values. Actually, these two systems are not completely dual because the xture vise can xture workpieces using internal holes, whereas a horizontal probe sensor cannot measure internal holes. 3.2 Four Jaw Fixture Chuck and Orthogonal Point Probing In this section, we detail the duality between the four jaw xture chuck (section 2.3) and orthogonal point probing (Figure 8). Again, we can de ne criteria under which orthogonal point probe con gurations correspond to xture con gurations: 1. The lattice xel constraints dictate that the distances between the orthogonal probe lines and a reference frame must be separated by integral multiples of the lattice spacing 2. The xture chuck's separation corresponds to four probe point measurements which exactly di er by integral multiples of the lattice spacing 3. The contact points must be compatible with nonoverlapping jaws
Lattice Rows and Columns
Orthogonal Probe Measurements
Figure 8: Fixture chuck con gurations can alternatively be characterized in terms of orthogonal probe data where all four probe measurements di er by integral multiples of the lattice spacing.
4 Generic Fixture Design Algorithm
4.1 Constructing Complete Indexing Tables Indexing (section 1.2.1) is a constant time object recognition technique for interpreting sensed features in terms of model features. Indexing techniques involve extracting indexing coordinates from the sensed
data, discretizing these coordinates, and using the discretize coordinates to index a table entry containing the correspondence information (Figure 9).
Correspondence boundary curves
Y 5.632 => 0 6.073 => 0.441 7.290 => 1.558 14.661 => 9.029 16.149 => 9.517 13.173 => 7.541
16.149 14.661 13.173
6.073 5.632 7.290
1. Probe object and measure contact positions
0 0 0 0
2 1 1 2
0.441 => 0 1.558 => 2 9.029 => 9 9.517 => 10 7.541 => 8
0 2 9 10 8
3. Discretize indexing coordinates and make indexing vector
2. Normalize indexing coordinates
Indexing Table 9 10 9 Id Features 9 10 9 1 A,B,A,E,E,E 9 10 8 1 A,B,A,E,E,E 9 10 8 1 A,B,A,E,E,E
C
D
B E A F E
B A A
E E
−1 1 9 10 7 1 B,B,A,E,E,E
4. Look up indexing entry in table
5. Return correspondence interpretation
Figure 9: Indexing techniques involve discretizing the sensed data to index the entry containing valid interpretations In section 3.1.1 we explained why every xture con guration corresponds to at least one indexing table entry. Therefore, an algorithm for constructing complete indexing tables for generic sparse sensing systems can also be used to enumerate xture designs. Wallack and Canny's complete indexing table construction strategy [18, 19] involves partitioning con guration space according to the discretized indexing coordinates and the correspondence interpretations (Figure 10). Thereby, every valid indexing table entry corresponds to a cell in that con guration space arrangement. Wallack and Canny's approach has the advantage that it only involves two types of partitioning curves: discretization boundary curves, curves which characterize con gurations separating con gurations whose indexing coordinates quantize to di erent values and correspondence boundary curves, separating regions of con gurations involving di erent interpretations (Figure 10). We also showed that the intersections of discretization boundary curves correspond to simultaneous contact con gurations in the dual xturing context because integral relative probe measurements exactly characterize the lattice constraint. Consequently, a reasonable xture design strategy would involve constructing complete indexing tables, and verifying the xture con guration corresponding to each entry for force closure; this approach can be
Discretization boundary curves
θ
Figure 10: We can project the boundaries down onto con guration space where the cells de ned by the arrangement of discretization boundary curves and correspondence boundary curves correspond to all the indexing table entries. further improved by incorporating the force closure constraint into the enumeration scheme. 4.2 Characterizing Force Closure In the robotics, mechanical, and manufacturing literature, there are many techniques for verifying force closure, i.e., determining whether a xture can resist all external forces and torques (wrenches). For each combination of xels and features, Wallack and Canny [18, 21] characterized force closure in terms of the k k minors of a k (k + 1) wrench matrix having the same sign. Geometrically, each of these minors characterizes the signed volume of the polytope spanned by the k wrenches multiplied by 1 depending upon whether the removed column index is odd or even; consider the minors Mi and Mj and the vector vi;j normal to the other wrenches: fw1; w2; : : :; wi?1; wi+1; : : :; wj ?1; wj +1; : : :; wk+1g. Mi characterizes the signed volume of all but the ith wrench multiplied by 1 depending upon whether i is even or odd. The signed volumes of fw1; w2; : : :; wi?1; wi+1; : : :; wk+1g and fw1; w2; : : :; wj ?1; wi; wj +1; : : :; wi?1; wi+1; : : :; wk+1g have the same sign if and only if wi and wj point in the same directions with respect to normal v. If wi and wj both pointed in the same direction with respect to v, then no positive combination of fwg could counter a force in the opposite direction (v or ?v). Correspondingly, if for all i; j, wi and wj point in opposite directions with respect to the normal vi;j , then a positive combination of fwg can counter any wrench. We now show that if wi and wj point in the same direction with respect to v, then their minors Mi ,Mj will have di erent signs. We show that the e ect of multiplying the determinant by 1 for even/odd columns and the e ect of multiplying the determinant by ?1
for swapping column vectors yield a ?1 multiplicative e ect irrespective of i; j. Consider adjacent i; j, the remaining column will occupy the same place in the submatrix, but they will have di erent even/odd polarity; thereby the subminors should have opposite signs if wi v = wj v. Next, we can the case of adjacent i; j to prove the hypothesis for the case of non-adjacent i; j 0 . If we were to move column j to the right an even number of columns to j 0 , then we would not a ect the odd/even polarity, but the signed volume would be a ected in that all of the columns of the minor from j + 1 to j 0 would shift left by 1 column. This involves performing an even number of column swaps, each one having a ?1 multiplicative effect on the signed volume. The case of shifting j an odd number of columns follows in the same manner. 2 Consider parameterizing each element of the wrench matrix by the object's pose so that we can express all of the k k minors as functions of the object's pose parameters. Since we are only concerned with con gurations which bound the force closure con gurations, we only need to look for con gurations where selected minors are exactly zero (since every continuous value must pass through zero to change sign). Still, each force closure con guration is only concerned with k + 1 of the minors, whereas we must keep track of (mk ) minors if there are m contacts. We can simultaneously monitor all the situations by extending con guration space into a Riemann surface which characterizes a separate sheet of con guration space for each xel contact con guration. Thereby, we can incorporate force closure constraints into our geometric enumeration scheme, although such an implementation would be relatively complicated. 4.3 Brute Force Fixture Design Algorithm In this section, we present a brute force generate and test xture design algorithm for arbitrary workpieces and arbitrary minimal toolkits. The generate phase includes three steps: enumerating all tuples of k+1 features, enumerating all xel con gurations possibly achieving simultaneous contact with each feature tuple, and enumerating all simultaneous contact con gurations for each combination of features and xels. The test phase involves checking each simultaneous contact con guration for force closure. The rst step involves enumerating tuples of k + 1 features capable of providing a force closure contact con guration [1, 2, 3, 5, 12, 13]. The second step, enumerating xel con gurations possibly achieving simultaneous contact with the features is slightly more involved; forsaking performance, we can always use the most general geometric con-
Force Closure Boundary Curves
Y
θ
Figure 11: The indexing algorithm can be extended by including con guration space curves characterizing when any of the k k minors equals zero for each combination of xels and features. These force closure curves bound the regions (shown in grey) achieving force closure for each particular combination of xels and features. straint that the contact points ( xels) can never be separated by distances which are less than or greater than distances between points on the two features. For example if two points on two features can only be separated by at least 5 and at most 10 units, then we only need to consider xel con gurations where those xels are separated by between 5 and 10 units. Since features are nite, this provides a nite bound on the number of xel con gurations we need to consider. The third step, computing simultaneous contact poses, can always be solved algebraically. Each contact constraint can be formulated algebraically (in terms either k or k + 1 degrees of freedom) yielding a multivariate system; then, this multivariate system can be solved via a combination of algebraic and numerical techniques. 4.4 Equivalence Classes It is important to identify xture con guration equivalence classes, because highlighting redundant or irrelevant degrees of freedom, can reduce the dimensionality of the con guration space. One example of equivalence classes occur in the xture vise toolkit, xture con gurations can be shifted up or down lattice rows or right or left lattice columns. In Brost and Goldberg's translating clamp toolkit design algorithm [2], they assumed that the rst xel was always placed at the origin, and that the part's orientation was between 0 and 2 .
5 Heuristic Fixture Design
Since our complete enumeration strategies may involve processing k + 1 tuples k 2 f3; 6g of model fea-
tures, enumerating xtures for models with hundreds of features may take hours if not days. For time critical applications, where a single valid xture suces, a directed search approach may be more appropriate. We focus on the task of eciently identifying candidate xtures because there are already many di erent metrics for comparing grasps [5, 14]. Our directed search strategy identi es promising candidate feature tuples and then, for each candidate. searches for a xel con guration achieving simultaneous contact and force closure. 5.1 Feature Tuple Quality Metric Although there are many algorithms in the literature which can check whether a feature tuple can admit a force closure grasp [1, 2, 3, 5, 12, 13, 21], none of these algorithms are specialized for the lattice constraints associated with minimal xture toolkits without enumerating candidate xel con gurations [2, 21]. Many of these algorithms are based on the wrench polytope (the convex hull of the wrenches corresponding to the extremal points on the features). Assuming contact is allowed anywhere on the features, every wrench within the convex hull can be achieved by a positive combination of the wrenches corresponding to the extremal contacts. Therefore, contacts on the features can resist arbitrary wrenches if and only if the origin lies in the interior of the wrench space convex hull. For minimal xture toolkits, we are also concerned with the probability that a set of constrained xture elements can achieve simultaneous contact. We propose an alternative metric for ranking the feature tuples; instead of focusing solely on the entire convex hull, we consider the k + 1 polytopes spanned by combinations of k features. It turns out that the signed volumes of these features correspond to the minors of the k k + 1 wrench matrix (section 4.2). We score a wrench tuple as the larger of the smallest positive and additive inverse of the least negative minors multiplied by 1 depending upon whether all the minors have the same sign. Furthermore, these minors can be weighted by either the areas of the features or the moments of inertia of the features along the nonprincipal axes to approximate the probability that a constrained set of contacts can achieve simultaneous contact. 5.2 Discrete Hill Climbing Heuristic In this section, we describe a heuristic search strategy for nding a xel con guration for a particular feature tuple. The hill climbing search strategy modi es the position of one xel at a time until arriving at a xel con guration achieving simultaneous contact and providing force closure with the feature tuple. We
determine which xel to move and where to move it by computing the metric for all neighboring xel con gurations. Consequently, the entire workpiece shifts slightly with respect to all of the xels. Our proposed metric incorporates discrepancies with respect to achieving simultaneous contact as well as discrepancies with respect to providing force closure. Consider two separate error estimates Esc, Efc : Esc , an error estimate of the discrepancy from achieving simultaneous contact, and Efc , an error estimate of the deviance from achieving force closure. Esc(x; y; ) is de ned as the sum of the squared minimum distances between the xels rotated and translated by (x; y; ) and the corresponding features. Efc (x; y; ) is de ned in terms of the k + 1 minors of the wrench matrix (section 4.2); Efc (x; y; ) is de ned to be the larger of the smallest positive and additive inverse of the least negative minors multiplied by 1 depending upon whether all the minors have the same sign. In this way, Efc (x; y; ) is continuous and piecewise di erentiable. These error estimates can be computed exactly in constant time because for each feature/ xel pair, the closest point on the feature either corresponds to an interior feature point or boundary feature (boundary vertices for two dimensional models, boundary edges and vertices for three dimensional systems). We can heuristically iteratively determine the closest contact points (boundary/non-boundary). In the worst case, we only need to consider a nite number of possibilities: 34 for two dimensional systems, and 77 for triangulated three dimensional polyhedra. We can always characterize Esc(x; y; ) as the sum of up to k squared dot products between the point and a feature since we can always algebraically express distances between points and planar features. If the closest boundary feature is a point, we can de ne k orthogonal feature planes through that point, and compute the distance between the closest feature point and the other point by summing the squared dot product distance between the point and the k features. We compute Efc (x; y; ; ) by considering the contact points at the closest points on the objects. Using algebraic methods, we can compute the minimum of the map Esc(x; y; ; ) + Efc (x; y; ; ) over the entire (x; y; ) con guration space in constant time (i.e., as in simulated annealing, we hope to avoid local extrema by varying ). We can solve for the global minimum in constant time using a combination of algebraic and numerical techniques since the map can be characterized algebraically, the partial derivatives r (Esc+Efc ) can also be characterized algebraically, providing a multivariate system where the roots corre-
spond to local extrema, which must include the global minimum. Unfortunately, since Efc and Esc are de ned in terms of maximums and minimums, computing the global minima is non-trivial.
5.2.1 Computing Global Maximum of MinMax Functions
Although it is easy to compute global extrema of algebraic functions, it is much more dicult to compute global extrema of continuous piecewise di erentiable algebraic functions, because we can no longer exploit the fact that extrema correspond to the roots of r f. Thereby, computing the global maximum of an error function de ned as the of the minimum of a set of functions, i.e., minfjM1j; jM2j; : : :; jMk+1jg is rather hairy. This is because the global maximum may not correspond to a local maximum of any of the individual functions: fjM1j; jM2j; : : :; jMk+1jg, but rather, it may correspond to a critical point where Mi (~x) = Mj (~x). Therefore, we not only have to check local maxima, but also maxima along critical curves.
6 Conclusion
In this report, we presented a complete xture design algorithm for minimal xture toolkits, where minimal xture toolkits involve xels inseted into a lattice of locations and a single degree of articulation. Minimal xture toolkits exhibit the property that only a nite number of con gurations can immobilize a generic object assuming a frictionless point contact model. Of course, this nite characteristic is not without its disadvantages. Without additional machinery for rotating these xtures, minimal xture toolkits may be dicult to integrate with vertical assembly robots because three dimensional tetrahedral chuck xtures can only immobilize xtures in a nite number of poses. In this paper, we presented two general algorithms for generating xture designs: a general complete algorithm based upon a duality we observed between modular xturing and sparse sensing, and a heuristic algorithm designed for eciency.
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