Shape Tolerance for Robot Gripper Jaws 1 Tao Zhang, Lawrence Cheung and Ken Goldberg 2 ALPHA Lab, IEOR and EECS Dept., UC Berkeley

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1 Shape Tolerance for Robot Gripper Jaws Tao Zhang, Lawrence Cheung an Ken Golberg ALPHA Lab, IEOR an EECS Dept., UC Bereley ABSTRACT In [9] we specifie robot grippers that can orient an grasp parts with an arrangement of trapeoial aw moules. Since aw moules may be imprecisely machine, we efine a parametric tolerance class such that part alignment is guarantee for all aw geometry in the class. This tolerance class is erive base on analysis of toppling, motion traectory, an form-closure. Given maximal aw geometry from the previous algorithm, we escribe an O(n 3 ) algorithm to compute the parametric tolerance class base on maximal an minimal aw specifications. We have implemente the algorithm an report results from physical experiments. for a given part: both align the part to the esire orientation an achieve a form-closure grasp on the part. The tolerance class is boune by J max an J min. J max J min. INTRODUCTION Although grippers are wiely use for automate manufacturing, assembly, an pacing, the esign of gripper aws is often a-hoc an suboptimal. In inustry, 4 DOF robots, such as SCARA arms, an DOF parallel aw grippers are common ue to their low cost an high reliability. The combination of these two evices is inematically limite to orienting parts in the horiontal plane. Zhang an Golberg [9] gave an algorithm to esign aws base on trapeoial moules that will align parts in the vertical plane an grasp them in form closure. The algorithm fins aws that achieve maximal contact at the final grasp configuration to maximie resistance to applie forces. For many inustrial applications, it may be preferable to use aws with smaller contact area, for example to minimie gripper weight for high-velocity transfer. Furthermore, machine aws may not precisely comply with the specifie maximal contact geometry. In this paper we consier variations in aw shape an efine a tolerance class for aws base on maximal an minimal contact areas. Let J max enote the aw specification from [9] that achieves maximal linear contacts with the part at its esire final orientation. Let J enote an instance of aw geometry that is efine by an arrangement of trapeoial aw moules. We say J is amissible if it will rotate the part to the esire orientation an achieve form-closure. Let J min enote the amissible aw geometry with minimal contacts at the esire orientation of the part. Figure shows J max an J min Figure Blac trapeois illustrate the maximum an minimum bounaries of the aw tolerance class for the part shown in gray. As illustrate in Figure, the gripper with trapeoial aw moules rotates the part from its initial resting orientation (a) to the esire final orientation (b) for assembly. Our gripper esign buils on recent results in toppling manipulation [8]. Zhang et al. [7] propose the toppling graph that can be use to ientify the location of contacts permitting toppling. Zhang et al. [8] apply toppling to grasping an fin four frictionless point contacts that will align a given part in the vertical plane. [9] gives an O(n 5 ) algorithm to compute the maximal aw esign with linear contacts that has the following properties: () It is able to align the part from the initial orientation to the esire final orientation; () It has maximal (linear) contact with the part at the esire orientation of the part; an (3) It achieves a form-closure grasp on the part at its esire orientation. In this paper, we evelop a tolerance class specifie as a range of trapeoial aws. Each aw moule is etermine by the locations of two vertices that mae contact with the part in its final grasp configuration. The line segment between these two vertices represents an accessible segment on an ege of the part at its esire orientation. The accessible segment correspons to an ege of the aw moule that is neither horiontal nor vertical. This wor was supporte in part by the National Science Founation uner CDA an Presiential Faculty Fellow Awar IRI Research funing was also provie by Aept Technology, For Motors, an California State MICRO Grant For more information contact: golberg@ieor.bereley.eu.

2 Figure Gripper with trapeoial aw moules rotates the part in the gravitational plane to facilitate assembly. There are three types of vertices in the set of accessible segments: one pushing vertex, one toppling vertex, an other vertices name tips. As illustrate in Figure 3, we efine a single variational parameter λ along the ege at each tip. As is common in tolerance analysis, we assume perfect form: all aw mo - ules in the tolerance class have perfect linear eges. We efine the tolerance class by fixing the pushing an toppling vertices an computing how far the tips can be expane or contracte along the accessible segments. Note that we efine a single common tolerance parameter λ for all tips. λ λ [9], [], [], an [5] survey the status quo an mathematical approaches to tolerancing. Neumann [0] escribes a new stanar, Y4.5M, which provies a mathematical basis for imensioning an tolerancing. A funamental problem in geometric tolerancing is classification: given a part, is it within tolerance? Yap an Chang [6] give an example using a -imensional probe moel. Configuration space can provie a theoretical basis for tolerance analysis. Donal [5] stuies part manipulation with geometric uncertainty. He consiers shape variations as an aitional imension in a generalie configuration space an escribes multi-step error etection an recovery strategies. Josowic et al. [6] present inematic tolerance in term of configuration space an evelop a worst-case tolerance analysis algorithm for -DOF planar pairs. Sac an Josowic [3] exten the analysis to multi-pair planar mechanisms with statistical geometric variation. They also moel general planar part pairs using 3- imensional configuration-space to capture both quantitative an qualitative inematic variation [4]. Latombe et al. [7] consiers assembly sequence planning problem with tolerance parts. They give a polynomial time algorithm to ecie if an assembly sequence exists given the specifie tolerances. Their tolerance moel is similar to ours in that both approaches fix the relative orientation of eges. Aella an Mason [] evelop a planner to generate orienting plans for tolerance polygonal parts. Their tolerance moel is efine by circular uncertainty ones aroun the nominal positions of the COM an the vertices. Chen et al. [4] propose parameteric tolerance classes for sensorless part orienating an fixturing. Each are efine by a uncertainty one at part vertices. They evelop algorithms to compute the bounaries of the tolerance class. Brost an Peters [3] give an algorithm to esign 3D moular fixtures for tolerance parts that are specifie by an uncertainty polygon at each vertex. Bohringer et al. [] show that tolerance parts can be oriente using an elliptic force fiel. 3. PROBLEM DEFINITION Figure 3 Variational parameters along an accessible segment of a aw moule. We present an O(n 3 ) algorithm for testing if a aw specification J is amissible. Given J max, we then present an O(n 3 ) algorithm to compute the tolerance class.. RELATED WORK Let I enote the input to the maximal aw esign algorithm in [9]: the n-sie convex proection of an extrue polygonal part, its COM, its initial an esire orientations, vertex clearance raius ε, µ t an µ s : friction coefficients of gripper-part an surfacepart, respectively. We first consier problem (), testing if a given J is amissible (will rotate the part an hol it in form-closure). The input to problem () is <I, J>. The output is binary: yes if J is amissible; no if not. We then consier problem (), fining the lower bounary of the tolerance class. The input of problem () is <I, J max >. The output is J min. Figure 4 Notation.

3 As shown in Figure 4, the part sits on a worsurface at an initial resting pose. We efine the Worl frame, W, to be a Cartesian coorinate system originating at pivot point P with X-axis on the surface pointing right, Z-axis vertical to the surface pointing up. The pushing contact, A, is a istance A from the surface; the toppling contact, A, is a istance A from the surface. Starting from the pivot, we consier each ege of the part in counter-clocwise orer, namely e, e,, e n. The ege e i, with vertices v i at (x i, i ) an v (i+) at (x (i+), (i+) ), is in irection ψ i from the X- axis. Let enote the rotation angle of the part from the +X irection; initially =0 an at the final orientation =. We say an ege e is visible if it can be seen from +X irection; invisible, otherwise. We assume the part can be treate as a rigi extrusion of a polygon; both the part an the aws are rigi; part geometry an location of the COM are nown; part motion is sufficiently slow to apply quasi-static analysis. 4. TOLERANCE ANALYSIS The tolerance analysis is a combination of toppling, motion traectory, an form-closure stuy. 4. Toppling graph Our analysis involves the graphical construction of a set of shape functions that represent the mechanics of grasping. All of these functions are piecewise sinusoial an epenent on. They map from part orientation to height: S +, where S is the set of planar orientations. The shape functions inclue vertex functions V (), toppling functions H (), an amming functions J (). J3 H3 H J Z V5 V Figure 5 Toppling Graph. The toppling graph, which consists of these shape functions, helps us to ientify the range of the contact permits toppling. Each function represents a particular property of the part, an the graph escribes properties of the part uring grasping. h h V3 V q For toppling to be successful, there must exist a horiontal line at height h that has the following characteristics:. h > H i (), if V i () < h < V i+ (); #. h > J i (), if V i () < h < V i+ (); # 3. h < max (V i ()), where 0 < <. # 3 i where i is the inex of visible eges. The first three criteria can be escribe as: the toppling contact A must be above the toppling function, the amming function, an the liftoff function. When the horiontal line crosses a vertex function, there is a contact ege switch. Therefore, A must satisfie criteria an for the new contact ege. The thir criterion requires that A must mae contact with the part. For example, we want to rotate a sample part 35º for assembly. Figure 5 illustrates the toppling graph of the part given A = 0.5cm. Note that H an J equal 0. We can see that A at A = h is unable to topple the part to the esire orientation because the line goes uner H 3 after rotating to ; A at A = h is capable to perform the tas. Notice that A switches contact ege from e to e at. Given A, the toppling graph allows us to fin the feasible range of A such that the corresponing A an A can rotate the part from the resting orientation to the esire orientation. 4. Traectory analysis To ensure no portion of the aw blocs the part rotation, we efine quasi-vertex functions to represent the motion traectory of vertices. The part performs both rotation an linear translation uring toppling. We ecompose the part motion into pure rotation an pure translation. The part first rotates about pivot point P to semi-position, an then translates to actual-position. Let ( x, ) an ( x, ) enote the actual-position an the semiposition of vertex v after the part is topple by, respectively. Let ( x, ) an ( x, ) enote the actual-position an the semi-position of vertex v after the part is topple to its esire orientation, respectively. Let x t an x t enote the istance between the actual-position an the semi-position of any point after the part is topple by an, respectively. To obtain a quasi-vertex function, we efine a frame of reference F at the esire orientation of the part originating at v. The Z-axis of F is the interior normal of ege e (-), an the X-axis is on ege e (-) obeying the right-han rule. Given A, the quasi-vertex function Q (, A ) inicates the location of v in F as the part rotates, which can be shown to be: Q () = xq Q = (, A) (, ) A

4 ( + xt xt)cos( ψ + ) + ( )sin( ψ ( + xt xt)sin( ψ + ) + ( )cos( ψ # 4 where ( x t = x A - x cos 0 = x sin A A < m+, an x t = x A - A < l+. m ' m+ sin 0 + cos )( m, ) + ) + ) m + m - x m if m < ' ( A l )( ) ' ' l+ l+ l - x l if l < l equivalent to solve a system of equations an can be one in O(n 3 ). Secon, we nee to test if J is able to rotate the part to the esire final orientation. Since the part is rolle by the pushing contact an the toppling contact, we nee to ientify these two points. These two contacts are only vertices that eep touch with the part uring the toppling phase. This can be one easily in time O(n). Known the height of the pushing contact A, we construct the corresponing toppling graph. If h = A satisfies inequality # ~ #3, the pair of A an A can rotate the part to the final orientation. Since the time to obtain a toppling graph is O(n), this step taes O(n). Finally, we nee to consier if any portion of J will bloc the part s traectory. x We represent the motion traectory of the eges of the part base upon the quasi-vertex functions, an then we erive the accessible segments of the aws. To guarantee no obstacle blocs the part rotation, the aws shoul stay away from the motion traectory of the eges. The quasi-vertex function escribes the motion traectory of the part. Note that the quasi-vertex function is the proection of configuration-space (x,, ) onto the plane of (x, ); the shape function is the proect of configuration-space (x,, ) onto the plane of (, ). Therefore, the shape function an the quasivertex function are both the ecomposition of the configuration-space. The reason we apply C-space ecomposition is that, in orer to reuce complexity, we only nee to eep a portion of the configurationspace information that is necessary for certain analysis. W Frame F Frame Figure 6 Motion traectory of the part. From equation #4, we have: O I x 4.3 Problem (): Checing if J is amissible To solve problem () checing if a given J is amissible, we first test if J will achieve form-closure on the part. Starting from the one closest to P, we orer the vertices of J in counter-clocwise (a, b ), (a, b ),, (a m, b m ). Let V enote the unit normal vector pointing inwar at (a, b ), V a (V b ) enote the X (Z)- axis proection of this unit vector, an T enote the torque of V respective to P. V a Va V3a... Vma Let M enote V b V b V b... V, let w 3 mb T T T3... Tm enote [ ω ω ω ω ] T 3... m. It is well nown that J generates a form-closure grasp on the part if an only if w >0, s.t. Mw = 0. Therefore, to chec the form-closure grasps is x Q + Q = ( + x x ) + ( ). t t Therefore, the intersection between Q () an the X- axis of F is at: x = where Q, ) = ( ' ) ( ~ ' ~ ) + ~ ' ' x + xt x + xt ~ satisfies: ( ~ A ( ~ + ~ xt xt)sin( ψ + ) + ( ~ )cos( ψ + = 0. Figure 6 illustrates the motion of the part. O is the origin of F at v, an I is the intersection between )

5 the quasi-vertex function an the X-axis of frames F. Therefore, x is the length of OI. We proect OI to the X-axis of W, an the length of the resulting segment is + x ) ( + x ) ; we proect OI ( t ~ ~ t to the Z-axis of W, an the length of the resulting segment is ~ '. If x of J is smaller than that of J max (where an correspon to all the visible eges), no portion of the aws will become an obstacle in the traectory. We compute x of J an that of J max, an compare these two values for all an. Thus the algorithm to solve Problem () runs in time O(n 3 ). 4.4 Problem (): Computing the tolerance class [J min, J max ] efines a tolerance class: the uncountable set of grippers with aws having eges parallel to J min an J max an volume within these lower an upper bounaries. We first prove the convexity of the tolerance class: if J [J min, J max ], J must be amissible. Then we escribe an algorithm to fin J min by searching the upper bon of λ. Lemma. If J [J min, J max ], J must be amissible. Proof: Note that both J min an J max are amissible by efinition. () Can J topple the part to esire orientation? First, we consier the toppling conitions. Since the pushing contact an the toppling contact are the same for all J in the class, the toppling conition of J is the same as that of J max. Secon, we consier the part s motion traectory conitions. The smaller J, the less liely it will bloc the motion traectory of the part. Since J max is the geometry shape that guarantees no collision in the part s motion traectory an J is smaller than J max, J satisfies the motion traectory conitions. Therefore, J is able to topple the part to esire orientation. () Can J achieve a form-closure grasp on the part at its esire orientation? J is larger than J min. Since J min achieve a formclosure grasp on the part at its esire orientation, J must have the same property. In summary, J can topple the part to esire orientation an achieve a form-closure grasp on the part at its esire orientation. Therefore, J is amissible. g Lemma. λ must be nonnegative for J. Proof: Assume that there exists a λ that is negative for certain aw geometry J. Then, the total length of the contact eges of J are longer than that of J max because λ < 0. Since J max has maximal contacts with the part at the esire orientation, some portion of J will become an obstacle in the part rotation traectory. Therefore, λ can only be nonnegative. g Numerical Algorithm: We use binary search to fin the maximum variational parameter λ. By Lemma, λ must be nonnegative. We choose a small positive number δ. Starting with λ = δ, we use the algorithm for Problem () to chec if the corresponing J is amissible. If so, we try λ = δ, an so on, until λ is sufficiently large that J is not amissible. We then interpolate to a esire level of accuracy an the corresponing J is J min. 5. IMPLEMENTATION RESULTS We verify our shape tolerance algorithms by the following example. The part is initially at the stable orientation efine by the vertices at (0,0), (5., 0), (64., 57.), (37.5, 96.), (-3., 44.6), an COM at (.9, 4.3). We nee to rotate the part 0º to final orientation for assembly. a 46.4 b.64 a b a b a b a b a b a b a b Table Optimal aw esign: vertex location. We fin the optimal gripper aw esign as shown in Figure 7. Table inicates the location of the aw moule vertices. The toppling contact is at (a, b ) an the pushing contact is at (a 6, b 6 ). We apply our algorithm to fin the upper bon of λ equals as illustrate in Figure 8. Figure 7 J max rotates the part an grasps it securely at the esire orientation. We conucte physical experiments to verify our results on the example part. Two sets of aws were machine from aluminum. The friction coefficients are µ t = an µ s = The first set is J max as shown in Figure 7. The secon set is J min as shown in Figure 8. We installe these two sets of the aws onto an AeptOne inustrial robot. We teste each J max an J min 50 times to align the part an observe ero failures. Figure 7 an 8 illustrate, in sequence,

6 both aw sets successfully rotate the part to the esire orientation an grasp it securely. (a) (a) (a3) Figure 8 Physical experiments for J min. 6. DISCUSSION AND FUTURE WORK It is very ifficult to characterie the grasping proprieties of the aws with uncountable shape uncertainty. Therefore, we inten to quicly chec the orientability of aws uring interactive esign cycle. Algorithms with low complexity, such as those escribe in this paper, can provie rapi feebac to esigners. We propose a rigorous parametric tolerance class to aress the shape uncertainty of the gripper aws. We stuy shape tolerance of the aws in terms of toppling graph, part s motion traectory, an formclosure. We present a fast checing algorithm, an use it to compute the tolerance class. We implement the algorithms an illustrate with physical examples. In the future, we will stuy sensitivity to changes in friction coefficient an consier alternative materials for gripper aws. We will also consier sensitivity in aw shape normal to the contacting surfaces, which may ustify use of eformable materials such as rubber for the contacting surfaces. The iea is to esign aws that are also robust to variations in part shape. REFERENCES [] S. Aella an M. Mason. Orienting tolerance polygonal parts, Int. J. Robot. Res., vol. 9, no., pp , 000. [] K. Bohringer, B. Donal, L. Kavrai, an F. Lamiraux. Part orientation with one or two stable equilibria using programmable force fiels, IEEE Trans. Robot. Automat., vol. 6, no., pp , 000. [3] R. Brost an R. Peters. Automatic esign of 3-D fixtures an assembly pallets, Int. J. Robot. Res., vol. 7, no., pp. 43-8, 998. [4] J. Chen, K. Golberg, M. Overmas, D. Halperin, K. Bohringer, an Y. Zhuang. Shape tolerance in feeing an fixturing, in Roin Robotics: The Algorithmic Perspective, e. P. Agarwal, L. Kavrai, an M. Mason. A. K. Peters, 999. [5] B. Donal. Planning multi-step error etection an recovery strategies, Int. J. Robot. Res., vol. 9, no., pp. 3-60, 990. [6] L. Josowic, E. Sacs, an V. Srinivasan. Kinematic tolerance analysis, Computer- Aie Design, Vol. 9, No., pp , 996. [7] J.-C. Latombe, R. Wilson, an F. Caals. Assembly sequencing with tolerance parts, Computer-Aie Design, Vol. 9, No., pp , 997. [8] K. Lynch. Toppling manipulation, in IEEE Int. Con. Robot. Automat., Detroit, 999, pp [9] J. Meaows. Geometric Dimensioning an Tolerancing, Marcel Deer, Inc., 995. [0] A. Neumann. The new Y4.5M stanar on imensioning an tolerancing, Manufacturing Review, Vol. 7, No., pp. 9-5, 994. [] A. Requicha. Mathematical efinition of tolerance specifications, Manufacturing Review, Vol. 6, No. 4, pp , 993. [] U. Roy, C. Liu, an T. Woo. Review of imensioning an tolerancing: representation an processing, Computer-Aie Design, Vol. 3, No. 7, pp , 99. [3] E. Sacs an L. Josowic. Parametric inematic tolerance analysis of planar mechanisms, Computer-Aie Design, Vol. 9, No. 5, pp , 997. [4] E. Sacs an L. Josowic. Parametric inematic tolerance analysis of general planar systems, Computer-Aie Design, Vol. 30, No. 9, pp , 998. [5] H. Voelcer. A current perspective on tolerancing an metrology, Manufacturing Review, Vol. 6, No. 4, pp , 993. [6] C. Yap an E. Chang. Issues in the metrology of geometric tolerancing, in J.-P. Laumon an M. Overmas, eitors, Algorithms for Robotic Motion an Manipulation, pp , A. K. Peters, 997. [7] T. Zhang, G. Smith, R. Berretty, M. Overmars, an K. Golberg. The toppling graph: esigning pin sequences for part feeing, in Proc. IEEE Int. Conf. Robot. Automat., San Francisco, 000, pp [8] T. Zhang, G. Smith an K. Golberg. Compensatory grasping with the parallelaw gripper, in 4 th Int. Worshop Algorithmic Founations Robot., Hanover, NH, 000. [9] T. Zhang an K. Golberg. Design of gripper aws base on trapeoial moules, in Proc. IEEE Int. Conf. Robot. Automat., Seoul, Korea, 00.