C.3

SANDIA REPORTSAND97-0063 • UC–705Unlimited ReleasePrinted January 1997

A 3-d Modular Gripper Design Tool

Russell G. Brown, Randy C. Brost

Prepared bySandia National LaboratoriesAlbuquerque, New Mexico 87185 and Livermore, California 94550for the United States Department of Energyunder Contract DE-AC04-94AL85000

Approved for public release; distribution is unlimited.

Issued by Sandia National Laboratories, operated for the United StatesDepartment of Energy by Sandia Corporation.

NOTICE: This report was prepared as an account of work sponsored by anagency of the United States Government. Neither the United States Govern-ment nor any agency thereof, nor any of their employees, nor any of theircontractors, subcontractors, or their employees, makes any warranty,express or implied, or assumes any legal liability or responsibility for theaccuracy, completeness, or usefulness of any information, apparatus, prod-uct, or process disclosed, or represents that its use would not infringe pri-vately owned rights. Reference herein to any specific commercial product,process, or service by trade name, trademark, manufacturer, or otherwise,does not necessarily constitute or imply its endorsement, recommendation,or favoring by the United States Government, any agency thereof or any oftheir contractors or subcontractors. The views and opinions expressedherein do not necessarily state or reflect those of the United States Govern-ment, any agency thereof or any of their contractors.

Printed in the United States of America. This report has been reproduceddirectly from the best available copy.

Available to DOE and DOE contractors fromOffice of Scientific and Technical InformationPO BOX62Oak Ridge, TN 37831

Prices available from (615) 576-8401, FTS 626-8401

Available to the public fromNational Technical Information ServiceUS Department of Commerce5285 Port Royal RdSpringfield, VA 22161

NTIS price codesPrinted copy A03Microfiche copy AO1

SAND-97-0063Unlimited Release

Printed January 1997

DistributionCategory UC-705

.

A 3-d Modular Gripper Design Tool

Russell G. BrownRandy C. Brost

Intelligent Systems and Robotics CenterSandia National Laboratories

Albuquerque, NM 87185-5800

Abstract

Modular fixturing kits are precisely machined sets of components used for flexible, short-turnaround construction of fixtures for a variety of manufacturing purposes. A modular-vise is a parallel-jaw vise, where each jaw is a modular fixture plate with a regular grid ofprecisely positioned holes. A modular vise can be used to locate and hold parts for machining,assembly, and inspection tasks. To fixture a part, one places pins in some of the holes so thatwhen the vise is closed, the part is reliably located and completely constrained. The modularvise concept can be adapted easily to the design of modular parallel-jaw grippers for robots.By attaching a grid plate to each jaw of a parallel-jaw gripper, we gain the ability to easilyconstruct high-quality grasps for a wide variety of parts from a standard set of hardware.

Wallack and Canny [WC94, WC96] developed a previous algorithm for planning planargrasp configurations for the modular vise. In this paper, we expand this work to produce a3-d fixture/gripper design tool. We describe several analyses we have added to the planaralgorithm to improve its utility, including a three-dimensional grasp quality metric based ongeometric and force information, three-dimensional geometric loading analysis, and inter-gripper interference analysis to determine the compatibility of multiple grasps for handingthe part from one gripper to another. Finally, we describe two applications which combinethe utility of modular vise-style grasping with inter-gripper interference: The first is thedesign of a flexible part-handling subsystem for a part cleaning workcell under developmentat Sandia National Laboratories; the second is the automatic design of grippers that supportthe assembly of multiple products on a single assembly line.

ii

.

Contents

1 Introduction

2 Related Research

3 Design and Analysis Algorithms

3.1 Problem Statement . . . . . . . . . . . .

3.2 Algorithm Synopsis . . . . . . . . . . . .

3.3 3-d Fixture Synthesis . . . . . . . . . . .

3.4 Loading Analysis . . . . . . . . . . . . .

3.5” Force Analysis . . . . . . . . . . . . . .

3.5.1 A Force-Based Quality Metric . .

3.5.2 Squeeze Force Calculation . . . .

3.6 Inter-Gripper Interference Analysis . . .

4 Applications

4.1 Part Handling for the Automated Com-ponent Cleaning Workcell . . . . . . . .

4.2 Fixture Loading and Unloading . . . . .

5 Conclusion

6 Acknowledgements

1

3

5

5

5

6

!’

7

8

8

10

10

1015

17

17

...111

iv

.

1 Introduction

Part holdlng is one of the fundamental problems of au-tomated manufacturing. Fixturing is a requirement formany manufacturing processes, including machining,assembly, and inspection. For machining purposes, fix-turing is necessary to immobilize the part against theforces exerted on it by the machine tool. For assem-bly, the part must be immobilized sufficiently to resistinsertion, fastening, and pallet transfer forces. For in-spection tasks, it is necessary to locate the part accu-rately. Part grasping is also required by all of these pro-cesses, but is particularly important for assemb~y. Forexample, automated assembly equipment must graspparts which are to be inserted into an assembly; thesegrasps must reliably locate the part and hold the partin the desired position and orientation while the inser-tion takes place. The part must not slip due to reactionforces caused by the insertion. Thus part grasping is apervasive problem in automated assembly.

At the same time, it is becoming increasingly impor-tant to develop part grasping solutions quickly. Manycommercial firms face increasing pressure to bringproducts to maxket quickly, which motivates the de-velopment of methods that speed the implementationof production processes. To manufacture a product,design engineers must carefully design and implementeach aspect of the manufacturing process. In this pa-per, we address the part grasping portion of this prob-lem, presenting a method for speeding both the designand construction of robot grippers.

In the last few years, a number of researchers havepublished methods for automatic design of modular f3x-tures. Most relevant to our work is the planar mod-ular fixture design algorithm by Wallack and Canny[WC94, WC96]. This algorithm designs fixtures for amodular vise, which is a parallel-jaw vise, where eachjaw is a fixture plate with a regular grid of preciselypositioned holes. To fixture a part, one places pins inselected grid holes such that as the vise closes, it com-pletely constrains the motion of the part. In this paper,we apply this concept to the design of robot grippers.By placing a grid-plate on each jaw of a parallel-jawgripper, we develop a modular gripper system whichcan grasp a wide variety of parts.

The output of the Wallack and Canny algorithm isthe set of all two-dimensional fixtures which providetwodmensional form closure for a twodmensionalpart. In other words, the fixturing elements are cir-cles, and the part features are line segments or circulararcs in the zy-plane.

Figure 1: Two views of the valve housing test part

In thk paper, we present several extensions we havemade to the modular vise algorithm. We have addedthe ability to design fixtures for three-dimensionalparts. This is done by extracting legal contact featuresfrom 3-d solid CAD models and by analyzing the CADmodels to determine heights for the fixture elements.In addition, we have added an analysis that determineswhether a given fixture is easy to load. Thk determineswhether it is possible to insert the part into the openvise with nonzero clearance, and then close the vise toobtain the desired grasp. We have also added the abil-ity to analyze the behavior of a fixture under a given setof applied disturbance forces, which in turn gives us aquality metric based on how well the fixture will resistthe expected disturbance forces. Thk necessarily re-quires an analysis of contact friction. Finally, we havealso implemented an inter-gripper interference analysisto determine the compatibility of multiple grasps forhanding off the part from one gripper to another.

We demonstrate these results using two examples:One is the automatic design of grippers for a partcleaning workcell under development at Sandia Na-tional Laboratories. The second is the design of a singlegripper capable of loading and unloadlng two dissim-ilar products from an assembly pallet. Thk could beused to allow zero-time product changeover in an agileassembly line.

An Example

Figure 1 shows two views of a sample part that needsto be fixtured. This valve housing is roughly two incheswide, long, and high. We selected a 7/16” (llmm) gridspacing for the modular vise, based on the scale of thepart, and a finger radius of 3/32” (2.4mm), based onstrength and rigidity requirements for the grasp. Theplanar portion of the modular vise algorithm produced125 fixtures for thk part. Of these, 42 were acceptableunder 3-d stability considerations. Figure 2 shows a

1

.

.

.

.

. . .;: .,:

...

Figure 2: An example fixture that was automatically gen-erated. When rendering grasps, our program only drawsvalid contact edges. Edges visible from the top axe drawnbold; edges visible from below are drawn with thinner, greylines. The vise jaws are shown &shed.

top view of the valve housing in one of the fixturesproduced by the algorithm. Figure 3 shows an obliqueview of the same fixture. This example is explained indetail in Section 4.

Overview

The next section provides a review of related researchin the areas of fixturing, grasping, and mechanics anal-ysis. Section 3 provides a detailed description of the Figure 3: An oblique view of the fixture shown in Figure 2.algorithms and analyses that we have added to Wal- The heights of the fixture pins are governed by contactedlack and Canny’s work, including generation of three-dimensional fixtures, loadability analysis, and quality

surfaces and by overhanging surfaces, as demonstrated bythe short pin at lower left.

analysis. Section 4 describes two applications of themodular-vise algorithms to real problems. Finally, Sec-tion 5 discusses some of the limitations to thk workwhich we have discovered.

2 Related Research

This work builds upon several results in the fields ofgrasping, fixture design, and mechanics analysis.

As explained above, our fixture synthesis algorithmis a direct extension of Wallack and Canny’s planar al-gorithm [WC94, WC96]. A very similar algorithm wasreported by Brost and Goldberg for standard modularfixture kits [BG96], and later extended to 3-d by Brostand Peters [BP96, BP97]. These algorithms share anumber of common geometric analysis procedures withthe results presented here; however, all of these algo-rithms address frictionless form closure, while thk pa-per necessarily includes an analysis of force closure withcontact friction.

In the same vein, a number of prior papers ap-proached the topic of grasping and fixture design fromthe perspective of form- and force-closure. Form clo-sure is a kinematic condition where a set of contactsprevents motion of an object without requiring contactfriction. Similarly, force closure is a condition wherea set of contacts prevents motion of the object, butonly when sufficient friction is present. These formu-lations of the grasping problem have led to a numberof landmark results: Reuleaux showed that four fric-tionless contacts are necessary to produce form clo-sure in the plane [Reu76], and Somoff and Lakshmi-narayana later showed that seven contacts are nec-essary in 3-d [SomOO, Lak78]. Mishra, et al showedthat twelve points are sufficient for 3-d bodies that arenot surfaces of revolution [MSS87]; Markenscoff, et allater tightened this bound to seven points [MNP90].Nguyen presented algorithms for the construction offorce closure grasps in both 2-d and 3-d [Ngu88]. Un-fortunately, none of these results were directly appli-cable to our problem, because they dld not considerthe gripper’s limited actuation force capacity, whichis an important constraint in many practical graspingproblems. Nonetheless, these results provided numer-ous insights that were quite helpful while developingthe current result.

Our work is also inspired by a number of priorpractically-oriented fixture design results. For exam-ple, several Ph.D. theses addressed the topic of fixturedesign, including analysis methods both with and with-out friction [Eng87, Hay90, Kim93]. All of these thesesdrew a clear connection between the intended task andthe required analysis, which is a key idea underpinningthe quality analysis we present in Section 3.5.1. Wealso drew inspiration from the seminal work of Asadaand By, who developed a reconfigurable flxturing sys-tem and associated analysis methods [AB85].

Our quaWy analysis models contacts in terms ofwrenches, which provide a unified representation of theforce and moment exerted by a contact. Our wrenchformulation of contact force and the aggregate wrenchmatrix W are drawn directly from Salisbury’s graspanalysis for dextrous hands [Sa182]. This work waslater extended by Kerr and Roth [KR86] and oth-ers. Of particular relevance is the work of Nguyen[Ngu88, Ngu89], who developed the notion of forec-losure under friction, which we employ to eliminatecertain problematic ambiguous cases. See Grupen, etal [GHM89] and Pertin-Tkoccaz [PT89] for surveys ofthe grasping literature.

There have been a number of prior results that de-veloped quantitative measures of grasp or fixture qual-ity that are closely related to the quahty analysis wepresent here. For example, Tkinkle reported a quanti-tative measure of the form closure condition that is abased on a linear program formulation that is very sim-ilar to ours [Tri92]. De Meter also employs a similarlinear program formulation when analyzing assemblyfixtures [Met93]. Our formulation dtiers from thesein that it focuses particularly on the case of analyzingthe constraint provided by a parallel-jaw gripper withfriction. Bicchi reports a force-closure grasp qualitymetric in [Bic95]. Bicchl’s approach uses a Lyapunovformulation that is very different from ours, but is sim-ilar in that it directly relates contact forces to gripperactuator limits. Bicchl accomplishes this through a Ja-cobian transformation, which can be viewed as a gen-eralization of our calculation of the squeezing force F’qdescribed in Section 3.5.1. In order to fully capturethe semantics of our squeeze force calculation, Bicchi’sformulation would need to be extended to include ac-tuator coupling between the fingers. Other approachesto fixture evaluation include finite element methods,such as those used by Menassa and DeVries [MD89].

Ponce reported an algorithm that designs grasp con-figurations for a unique 3-d modular vise [Pon96].Ponce’s vise differs from ours in that the fixture platesare on the inner faces of the jaws, while in our vise,the fixture plates are on top of the jaws. The configu-ration of Ponce’s vise allows full 3-d form closure withfrictionless contacts, while our system requires frictionto prevent motion in some directions. However, thkadditional constraint capability comes at the expenseof more restricted access to the held part.

Ci

4

.

3 Design and Analysis Algorithms

3.1 Problem Statement

Assumptions

The primary assumption we make is that the work-piece and all iixture elements are rigid bodies — ouralgorithms provide no deformation analysis.

Our quality analysis assumes Coulomb tilction, andrequires a value for the static coefficient of friction. Wedo recognize that rigid body mechanics has inherent in-consistencies and ambiguities when Coulomb friction ispresent, and take steps to avoid some of the difficultiesthk causes. Thk is discussed in Section 3.5.

We also &sume that the user specifies the desiredvertical orientation of the part. We justify this assump-tion with the observation that, for fixturing purposes,“up” is defined by the task. For example, holes aretypically drilled vertically from above, so the vertica~orientation of the part may be determined by the axisalong which a hole is to be drilled.

Finally, we assume that pins can be cut to any de-sired length.

Input

●

●

●

●

Part Specification: An ACIS@ solid-model of thepart to fixture, including a specification of ‘(up”and part material.

Fixture Kit Specification: The grid spacing, loca-tor radius, and minimum and maximum jaw sep-arations of the modular vise or parallel jaw grip-per. The maximum squeezing force which can beapplied by the vise or gripper jaws.

Friction Data: The static ~lction coefficient p foreach pair of part/pin materials.

Disturbance Forces: A list of the physical distur-bance wrenches which will be applied to the partwhile it is held in the fixture.

output

Given this input, the design tool outputs a list of fixturedesigns. A fixture is described by:

. The position and height of each locating pin.

● The part’s position relative to the fixture.

● The quality score for the fixture, based on its abil-ity to resist the expected disturbance forces.

Each output design is geometrically feasible and obeysthe jaw travel limits. Further, each fixture is easy toload — the gripper fingers can close from the open posi-tion to the grasp position without interference. Finally,all returned fixtures can resist the expected distur-bance forces, without exceeding the available squeez-ing force or violating friction constraints. Fixtures aregiven quality scores according to the excess squeezingforce that is available, and output in order of decreas-ing quality. The user can select the optimal grasp, oruse subjective evaluation criteria to select among thetop-ranked several grasps.

3.2 Algorithm Synopsis

The 3-d fixture generation algorithm is composed of anumber of well-defined phases:

1.

2.

3.

4.

5.

6.

Convert from 3-d to 2-d. For each valid contactsurface in the 3-d CAD model, construct a corre-sponding 2-d edge projected on the zg-plane.

Generate planar jixtures. Invoke the planar fixturedesign algorithm on the projected edges obtainedin phase 1.

Convert planar jixtures into 3-d jixtures. This isaccomplished by probing the CAD representationto determine the maximum possible and maximumuseful heights for each pin and taking the smallerof the two.

Loading analysis. Ensure that the modular visecan obtain the desired grasp without interferencewhen it closes on the paxt. Discard fixtures whichfail thk test.

Qualitg analysis. Rank the remaining iixtures by aquality measure that characterizes the robustnessof those fixtures to external disturbance forces.Discard unstable iixtures.

Inter-gripper interference analysis. If the applica-tion requires a gripper which can load or unloadanother fixture, or handoff between two grippers,determine which grippers are compatible with oneanother. Discard pairings where gripper fingers orlocator pins collide.

The following sections will explain these steps in detail.

3.3 3-d Fixture Synthesis

Our code interfaces to 3-d CAD data using theACE@ solid modelling system. Using the ACIS datastructures, we extract line segments and circular arcsfrom the part, keeping only those features which are ac-cessible from below. These 3-d features are projectedonto the zy-plane to produce planar edges and arcs.

We extract these 2-d features as follows: We firstsearch the ACIS model for all linear edges and thoseelliptical arcs whose projections onto the xy-plane areapproximately circular. We project all of these gener-alized edges onto the my-plane. We then find all pair-wise intersections between any two of these generalizededges and divide up the edges into sections (edgelets)which are not intersected by any of the other edges. Wethen determine whether each edgelet is visible from be-low, using the ACIS ray-shooting facility. A ray is shotstraight up from below the part at the (z, y) locationof the edgelet’s midpoint. If that ray first intersectsthe part at the feature which generated the edgelet,then the edgelet is visible from below. If the ray in-tersects the part at a lower height, then the edgelet isoccluded, so we discard it. Once we have filtered theedgelets, we recombine visible, adjacent edgelets fromthe same original feature. The combked edgelets arethen passed to the planar fixture generation algorithm.Since our implementation of the planar algorithm onlyaccepts linear edges, at thk time we convert all circulararcs to piecewise linear approximations.

The limited set of geometric feature types used inthe initial projection affects the completeness of thealgorithm, but not its correctness, as any locator pinwhich intersects a part feature of an unhandled typewill be detected later in the algorithm. If the objecthas only planar faces, then the algorithm is completeand generates all feasible fixture designs. If the objectcontains other surface types such as NURBS patches,then some valid fixtures may be overlooked, but noincorrect ilxtures will be returned.

After we generate the planar fixtures, we converteach planar fixture to a 3-d fixture by computing ap-propriate heights for the pins. We first transform thepart into its fixtured pose, and then for each pin invokethe following procedure:

1.

2.

3.

4.

5.

Create two rays which point straight up. If thepin is located at position (z+, yP), then both raysare anchored at (ZP, VP,z– ), where z– is a largenegative value. If the pin has diameter d, then rayup-large has diameter d+, where d+ = d+ c. Ray

up-small has the same anchor and direction, buta diameter of d_, where d_ = d – e.

Shoot rays up-large and up-small at the part(Figure 4). The z coordinate at which up-largehits the part becomes pin-cent act -rein, the min-imum height at which the pin touches the part.

If up-small hits the part, the z coordinate of thathit becomes pin-height -max, the maximum pos-sible height for the pin. In this case, we concludethat the pin contacts a non-silhouette edge of thepart.

If pin-height-max is defined, create a raydown-large of diameter d+, pointing down from

(z,, y,, pin-height-max– c). Shoot this ray atthe part (Figure 4, left side). The z coordl-nate where this ray first hits the part becomespin-cent act -max. The actual pin will have heightpin-contact -max+ d, where 6 is a small distance.That is, we make the pin as tall as is useful, pro-viding for a small protrusion of the pin above thehighest contact point. Thk height is reduced ifnecessary to avoid exceeding pin-height -max.

Otherwise, create a ray down-large of diameterd+, pointing down from (x=, yP, z+), where z+ is alarge positive value. Shoot down-large at the part(Figure 4, right side), and set pin-contact-maxto the z coordinate of that hit. In this case, theactual pin height is set to pin-contact –max+ 6.

This procedure determines the height of each pin,and also the extremal contact heights for each pin,pin-contact-rein and pin-contact-max. These val-ues are used later in the force analysis.

. .......~...-.i

pin-Contact -

Min I

1

..

Di.tl-Contact

w

I

‘d+c

Figure 4: Determining pin heights. On the left, the casewhere the pin touches a non-silhouette edge of the part,visible only from below. On the right, the silhouette edgecase.

3.4 Loading Analysis

Wallack and Canny’s algorithm returns all grasps thatprovide planar form closure. Unfortunately, somegrasps returned by the algorithm cannot be easilyachieved, because they require a complex rotating mo-tion to acquire the part (Figure 5a), or a tight-toleranceinsertion to reach the grasp configuration (Figure 5b).We prefer grasps whkh may be attained by simplyopening the gripper to full extent, placing it aroundthe part, and then closing the gripper until the squeez-ing force is resisted by the part being grasped. We referto such grasps as loadable and use a filter to select onlythose grasps:

First we verify that edge normals point to the leftfor edges touching left-jaw pins and to the right foredges touching right-jaw pins. For each such grasp, weverify that the pins can move from their fully-openedposition to the grasp position without intersecting thepart. We do this by determining whether a set of axis-

(a)

Figure 5: The basic planarmodulm vise algorithm returnssome fixtures which cannot be loaded by simply closing thejaws from their open position.

—— ——(a) (b)

Figure 6: The loadability test: (a) loadable. (b) not.

aligned rectangular prisms intersects the part, as shownin Figure 6. The heights of these prisms are chosen tomatch the corresponding pin heights. Our implemen-tation also allows the user to select a faster 2-d testthat assumes that the pins are infinitely tall.

3.5 Force Analysis

In addition to these purely geometric grasp analysismethods, we’ve added a grasp quality analysis thatconsiders grasp stability. All grasps returned by the 2-dfixture generation algorithm provide form closure, sofriction is not required to resist in-plane disturbances.However, in three dimensions we cannot use form clo-sure as an indication of grasp adequacy, for two mainreasons.

One reason is that we clearly cannot provide 3-dform closure with parallel, vertical, frictionless cylin-ders. No matter what grasp configuration is chosen,vertical translation is always possible. Vertical con-straint could be obtained by adding supports and topclamps to achieve full 3-d form closure (Figure 7a), orby adding hooks to the ends of the pins (Figure 7b),but for a gripper which may need to lift the part off alevel surface or insert it into a hole, these approachesare generally infeasible. Therefore, we must use frictionto hold the part.

7

(a) (b)

Figure 7: (a) One means of obtaining 3-d form closurewith modular elements. In addition to side locators, sup-port pads and top clamps are used to constrain the part.(b) This approach isn’t feasible for parallel-jaw grippers—how do we grasp a part resting on a flat surface?

The second reason for looklng beyond form closure todetermine grasp quality is this: form closure assumesrigidity and the ability to apply forces of arbitrary mag-nitude — for any given disturbance force, there existssufficient reaction force from the contacts to resist thatdisturbance. Unfortunately, for robot grippers theseare not realistic assumptions. Most parallel-jaw grip-pers are pneumatically actuated, and therefore have astrictly limited amount of force available to resist dis-turbance forces. We clearly need to consider the mag-nitude of the forces required to restrain the part.

3.5.1 A Force-Based Quality Metric

We want to maximize the margin for error provided bya gripper at its rated maximum squeezing force. This isequivalent to minimizing the squeezing force requiredto hold the part while executing the intended task. Fora given application, we measure grasp quality from theset of disturbance wrenches which are likely to be en-countered by the grasp in that application. These canbe determined by examining the operations that takeplace in the application. For example, if a pin is to beinserted in a given position on the held part, then theinsertion forces caused by the pin, applied at the inser-tion point, are appropriate disturbance wrenches forthat operation. For each candidate grasp, we computethe minimum squeezing force F,q required to resist eachdisturbance wrench. The maximum F,q over all of thedisturbance wrenches becomes the qutilty measure forthe grasp. The best grasps, then, are the ones whichrequire the smallest squeezing force to be able to resistall disturbance wrenches.

We have used a rigid-body mechanics approach todetermining the minimum gripper force required to op-pose the expected disturbance wrenches. The metricaccepts a candidate grasp, a set of expected distur-bance wrenches W, and a force limit Flim~t describingthe maximum squeezing force that may be exerted bythe gripper. Given this input, the metric calculates thesqueezing force Fmaz required to counter the worst-case

wrench in W. If this value is greater than the maximumavailable squeezing force Fl~~at, then the grasp is infea-sible and we discard it. Otherwise, Fmaz becomes thebasis for comparing alternative fixture designs; designswith the smallest F~~z are preferred because these pro-vide the greatest margin of safety relative to the max-imum available squeezing force FIMt.

3.5.2 Squeeze Force Calculation

The key calculation required to compute this metric isthe calculation of the squeezing force F~q required toresist a given 6-dof disturbance force FD, expressed inthe form [~m~Y ~. ~~ ~y 7Z]*. This calculation is donefor each F~ ~ W. The maximum F,q is taken to beFmaz. We now consider how F,q may be calculated.This calculation must include friction, since motion inthe *Z direction cannot be resisted without friction.

We would like to establish the minimum Fsq thatwould guarantee that a given grasp will resist a givendisturbance F~. Unfortunately, because of the ambi-guities that are present in Newtonian rigid-body dy-namics with Coulomb friction, this problem is knownto be NP-complete in 2-d and remains open in 3-d[Bar90, Erd94, TPSL95]. Consequently, we insteadcompute the minimum gripper force F~q which can op-pose the disturbance F~ in static equilibrium. Thisleaves open the possibility of object motion in ambigu-ous cases.

We partialy address this concern by disallowing sit-uations where contacts with the fingers on a single jawcan produce force closure with no participation by con-tacts on the second jaw. Such grasps suffer from wedg-ing, which has a number of disadvantages. For exam-ple, the grasp shown in Figure 8 may resist arbitrarydisturbances using the contacts on the left jaw alone,depending on the internal strain created by these con-tacts. (Consider the effect of wedging the part tightlybetween the fingers, like a door stop.) Since our rigid-body model does not represent this internal strain, thiscase may lead to ambiguities. We discard such fixtures,thereby decreasing the likelihood of encountering anambiguity. An additional reason to discard such graspsis that when the jaws open to release the part, the partmay remain stuck between the fingers.

Modeling Contact and Squeezing Force

Each gripper pin contacts the part at a single point,a vertical line segment, or an arbitrary series of pointsand line segments. For any of these cases, we can modelthe pin/part contact using one or two contact points.If the pin contacts a single edge at a point, then thereis only a single point contact. If the pin contacts two

8

,

.Figure 8: A grasp which provides frictional force closurewith a single jaw. Because the dashed line connecting thecontacts on the left lies Within the friction cones at eachcontact, these contacts alone provide force closure [Ngu88].

or more edges or contacts one or more surfaces along aline, then we can treat thk as two point contacts — thehighest and lowest contacts on the pin. Thus we havebetween four and eight point contacts with friction,which we treat as hard finger contacts: the finger canimpart no torque about the contact normaJ [MS85].

Figure 9 shows a modular vise holdlng a part. Inthis case, all pins have a single line contact with thepart, giving us eight point contacts to consider. Thefigure also shows the contact points and the approxi-mate directions of the contad, reaction forces inducedby these contacts, along with a disturbance force FD.If the part is in equilibrium under these nine force vec-tors, then it must be the case that the vector sum ofthe nine forces, ~i Fi + FD, is zero.

The total squeezing force exerted in such an equi-librium is the sum of the forces exerted by the fingersparallel to the actuation axis of the vise, in the ac-tuated direction for each finger. Thus for the left twopins, the contribution to the total squeezing force is thez-component of F1 through FA. For the right two pins,the contribution is the negative of the x-component ofF5 through F8. In general, if the gripper closes par-allel to the z-axis, then the contribution of F~ towardthe squeezing force Fsq is the z-component of Fi timeso(i), where we define c(i) to be 1 for contacts on theleft jaw and –1 for contacts on the right jaw, so thatF,q = ~i o(i) [1 OO]- F,.

To determine whether the static equilibrium obeysfriction constraints, we must break the contact reactionforces into their normal and tangentizd components andapply the rules of Coulomb friction. For each contacti, F; is the total force at that contact point. Let FT,~be the tangential force due to friction at the contadpoint, while FN,t is the normal component of the totalforce. Coulomb’s Law requires lFz-,i[ ~ p[FN,iIfor eachcontact i, where p is the coeilicient of friction.

w +SqueezingDhection

Y

)-x

z

Figure 9: At left, a three-dimensional object held in apamdlel-jaw gripper with cylindrical pins. At right, theforces acting upon the object. F1 through F8 are the equiv-alent point contact forces caused by the line contacts be-tween the object and the pins. FD is an external dMmr-bance force acting on the object.

We also define

[1100

FT.. ,;= 010 .FT,i000

and

[1000

FT=,i = 000 . FT,i001

These are the components of each FT,; which are par-allel to the zy-plane and orthogonal to it, respectively.We now describe a formulation of force closure whichprovides a way to calculate the minimum squeezingforce consistent with static equilibrium under Coulombfriction.

Linear Program Formulation

Chapter 5 of [MS85] describes a method that uses thewrench representation to determine whether a givengrasp resists a given dkturbance wrench. For a set ofn frictional hard-finger contact points on a part, we ap-ply thk method starting with a set of 3n unit wrencheswhich decompose each contact force into its normal andtangential components. These 3n wrench vectors arearranged as a 6 by 3n matrix TV, with the normal-force wrenches FN,i forming the first n columns, thehorizontal tangential wrenches FT=V,i in the second n—.columns, and the vertical tangential wrenches, FTZ,~inthe thwd n columns. A grasp can resist a disturbance

9

wrench FD if there exists a 3n-element vector of con-tact wrench magnitudes c, such that:

e

●

e

Wc+FD=O,

The first n elements of c are positive (a contactcan push but not pull), and

The magnitudes of the tangential-force wrenchesat each contact do not exceed A times the magni-tude of the normal-force wrench.

We can then formulate the minimum squeezing forceproblem as an optimization problem, where we wantto minimize Fsq given the above conditions. Unfortu-nately, the third condition is nonlinear, since it requirescomputing the magnitude of FT,i. Since we don’t havea general method for solving nonlinear optimizationproblems, we linearize the problem using the methodproposed by Thkle, et al [TPSL95]: Instead of re-stricting the magnitude of the total tangential force tono more than p times the normal force, we restrict eachof FTZ9,; and FTZ,i to be no more than * times the@normal force. Thus, instead of restricting the contactforce to lie in a circular cone, we restrict the force to liein a square pyramid inscribed inside the circulax cone.

This approximation allows us to write the followinglinear program:

minimize

F.q = ~ o(z)([l O O]. (FN,i + FT,i))i

subject to

l: WC+ FD=O

3: vi, lc2il< p’ci

4: vi, [c3~l< ,LJ’c~

where p’ = %, and i c [1, n], where n is the num-

ber of contact points. Condition 1 ensures that staticequilibrium is possible. Condition 2 ensures that eachcontact normal force is non-negative, and conditions3-4 ensure that the total contact force lies within thefriction pyramid. Solving this linear program identifiesthe minimum squeezing force for which the system canmaintain static equilibrium while obeying our approx-imated friction constraints.

We repeat this for all expected disturbance forcesand take the maximum to obtain the desired worst-case required squeezing force Fmaz.

3.6 Inter-Gripper Interference Analysis

Another type of analysis which our tool provides isinter-gripper interference analysis. This is useful insituations where there is a gripper mounted on a robotarm which is placing a part in a stationary modularvise fixture, or removing a part from such a fixture.Such a situation might arise in manufacturing applica-tions when it is necessary to automatically refixture apart so that new features can be exposed for machin-ing, assembly, or cleaning operations. This analysisis performed on existing grasp designs, and returns allpairs of grasps with which the part can be handed fromone grasp to the other without interference between thefingers of the two grippers.

The analysis is performed after grasp designs aregenerated, filtered, and sorted for quality. We assumethat the two sets of grasps are generated from oppo-site sides of the part. Thk means that the featuresagainst which pins are placed in the two sets of graspsmay be different, but more importantly means thatthere is a reflection across the y-axis in the informa-tion passed to the 2-d fixture generation algorithm. Todetermine whether a pair of grasps is compatible, wecheck for (i) interference between the pins when bothgrippers are closed, (ii) interference between the pinswhen the first gripper opens or closes, and (iii) interfer-ence between the pins when the second gripper opensor closes. If no interference is found, then the part maybe handed off from one grasp to the other. An exampleof this analysis is shown in the next section.

4 Applications

4.1 Part Handling for the AutomatedComponent Cleaning Workcell

The Automated Component Cleaning project is aninitiative within Sandia’s Intelligent Systems andRobotics Center to design a workcell capable of per-forming flexible, high-quality cleaning of machinedparts using an environmentally benign alcohol spray.The workcell uses a six degree-of-freedom robot arm

equipped with a focused spray-nozzle to clean theparts, using an automatically formulated spraying mo- -.tion to clean machine oil and similar impurities out ofholes and concavities on the part.

10

.

Figure 10: The ACC workcell:

.

>

1. the airlock. 2. the pallet. 3. the arm’s linear track. 4. the arm’s rotational actuator.

5. the arm gripper. 6. the pedestal gripper. 7. the 6-dof sprayer arm. 8. the spray nozzle.

The

a.

b.

c.

d.

e.

process is the following:

The part enters the workcell.

The exposed features on the part are cleaned witha high pressure spray.

The pert is flipped over so that previously oc-cluded features become exposed.

The fllp side of the part is cleaned.

The part exits the workcell.

Figure 10 shows a model of the workcell and its part-handling hardware. Because the cleaning spray is aflammable alcohol solution, the workcell is kept under apositive pressure nitrogen atmosphere. For this reason,the workcell is equipped with an airlock, through which

apart enters on a pallet. Once the part is in the airlock,a two degree of freedom arm (a linear actuator andone degree of rotational actuation) with a parallel-jawgripper extracts the part from the pallet and carries itinto the workcell. Before spraying begins, a pedestal-mounted gripper closes onto the part and the 2-dofarm releases the part. After releasing the part, the 2-dof arm rotates 180 degrees to avoid interfering withthe cleaning process. After the first cleaning operationhas been accomplished, the 2-dof arm rotates back intoplace and reacquires the part. The pedestal gripperreleases the part, and then the 2-dof arm rotates 180degrees again, so that the part is sitting above the armgripper. At this time, the second half of the cleaningoperation takes place. After cleaning is completed, the2-dof arm rotates back to its original position, carriesthe part into the airlock, and places it back on thepallet. The arm then retracts, and the part can then be

11

(b)

—(c)

Figure 11: The ACC pedestal and arm grippers. (a) The

pedestal gripper holding the pmt. (b) The handoff. (c) Thewm rotated 180 degrees, holdlng the part.

removed from the airlock. Figure 11 shows a detailedview of the arm fpedestal handoff.

To allow the workcell to handle a variety of parts,the arm and pedestal grippers are each configured asa modular vise. The pallet is configured as one-half ofa modular vise; since the part rests on the flat surfaceof the pallet, the pallet must locate the part but doesnot need to grip it. Note that in order to allow thesedevices to grasp and hand off arbitrary parts, theirrelative positions must be adjustable. Thus the thearm gripper can be adjusted in 0, while the pedestalgripper can be adjusted in z and y. Similarly, the palletis adjustable in z, y, and 0.

We developed a design program which allows theuser to configure the workcell grippers and pallet tohandle an arbitrary part, subject to the limitations of

the workcell hardware. The program begins by gen-erating candidate grasps for the arm gripper, pedestal

-“”’’’”’’””’“tF

. . . . {: .,, ,,.;:.:..+.:.-j

(a).

. .

. . . .::.

(b).,

.

(c)

Figure 12: Top views of the ACC grippers. The “race-track” shapes show the paths followed by fingers on theother grippers during a handoff. (a) The arm gripper, withthe pallet (dashed) and pedestal finger paths shown. (b)The pedestal gripper, with the arm finger paths shown. (c)The pallet, also with the arm finger paths shown. Thisinterference-tiee grasp triple was automatically generatedby the design tool.

12

I

(a)

(b)

I--J(c)

(c)

Figure 14: A closeup of the physical ACC grippers.

(a) The =m gripp=- (b) The pedestal gripper. andFigure 13: The ACC grippers holding the valve housing. (c) Both grippers simultaneously holding the part. The(a) The arm gripper. (b) The pedestal gripper. (c) Both pin lengths were extended to allow spray clearance fromgrippers simultaneously holding the part. low angles.

13

gripper, and pallet. Since the part “up” orientation isthe same for the pallet and pedestal gripper, the al-gorithm produces pallet designs by simply taking all

pedestal gripper designs that have three fingers on onejaw, and ignoring the single finger on the other jaw.

The program then generates triples of grasps whichwill successfully handle the part. We use the inter-gripper interference analysis described in Section 3.6 toensure that the grasps for the arm and the pallet arecompatible and also that the grasps for the arm andthe pedestal are compatible. This produces a seriesof compatible (m-m, pallet) and (a,rm, pedestal) grasppairs. A join operation is then used to generate alltriples where the arm is compatible with both the pal-let and the pedestal. “Join” is a set operator from re-lational databases; in thk case, it produces all 3-tuples(arm, pedestal, pallet) for which (arm, pedesta~) and(arm, pallet) are pairs of noninterfering grasps.

Figures 12–14 show an example grasp triple foundby our code. A study of Figure 12 confirms that thereis no interference between the gripper fingers duringhandoffs. Physical tests with the hardware verified thatthese grippers successfully hold the part during sprayoperations, and also that the handoff operation worksproperly.

To assist the user in selecting the best triple, ourprogram ranks the triples by minimizing the maximumsqueeze force required by the arm or pedestal graspsto resist the expected disturbances. The next sectionexplains how these disturbance forces were estimated.

Disturbance Force Calculations

Two types of disturbances act on a part in the ACCworkcell: contact forces exerted during spray cleaning,and inertial forces exerted during arm motion.

When the cleaning spray hits the part, it exerts aforce on the part at the point of contact. We can esti-mate the magnitude of this force as follows: Based onthe manufacturer’s specifications of the spray nozzle,the nozzle aperture is 1.2mm dhrneter, and produces aflow of 0.07- at 500psi. A flow of 0.07- through a1.2mm diameter aperture gives a flow velocity of 62%(flow velocity = volume per unit time/ area). Further,0.07~ at a density of l.0~ gives a mass flow rate

of 0.07& (mass flow = volume flow x density). Sincethe imparted force = flow velocity x mass flow [HR81],the force of the spray exiting the nozzle is 4.2N, justunder one pound. A conservative model of the force ex-perienced by the part is to neglect aerodynamic dragand assume that the part reverses this flow, doublingthe force. Thus, we use a value of 8.4N for the forceexerted on the part by the spray.

Figure 15: The disturbance forces used for the grippersqueeze force analysis.

This is the magnitude of the force exerted on thepart at a particular instant in the cleaning process,when the cleaning spray hits a specific point on thepart surface. To characterize the set of forces exertedon the part over the entire cleaning process, we gen-erate a large number of random 8.4N forces, takingcare to get a decent sampling of force directions andpositions. We then determine where that force first in-tersects the part, and erect an 8.4N force at that inter-section point, normal to the surface. A more accurateforce characterization could be obtained by simulatingthe cleaning operation in detail, but we chose not touse this method because of its computational expense.

There are two types of arm transfer motions: A lin-ear motion when the part is moved in and out of theairlock, and a rotational motion when the part is swunginto the inverted position. The linear motion is veryslow, so its accelerations are negligible. For the ro-tational motion, the largest accelerations occur whenthe arm hits its motion stops. Accelerometer mea-surements indicated that the peak acceleration is 6g(59=). Since the mass of the example part is 0.18kg,the maximum dkturbance force is llN, acting on thecentroid of the part in the *Z direction.

Figure 15 shows the set of forces used for the grip-per squeeze force analysis. Given a measured fric-tion coefficient of 0.2, the worst-case squeezing forcerequired to resist these disturbances is 77N for thearm grasp shown in Figures 13–14, and 39N for thepedestal grasp, which does not incur the llN acceler-ation. The maximum squeeze force capacity of eachgripper is 140N.

.

14

.

4.2 Fixture Loading and Unloading

In this section, we describe the use of modular grippersfor flexible loading and unloading of assembly fixtures.Consider the beginning and the end of an automatedassembly line: At the beginning of the line, a stream ofsome base part (e.g, a chassis or housing) is presentedto the line for assembly, perhaps on a series of trays.The bwe parts need to be removed fkom the trays andplaced into assembly fixtures, so that they can be fedthrough the line, with some assembly operation oc-curring at each station. Similarly, at the end of theline, the assembled or semi-assembled product must betransferred from the fixtures into trays or boxes. Thegripper used to load and unload these fixtures mustavoid interference with the fixture. At the end of theassembly process, additional constraints on the graspresult from the presence of attached parts which maynot be rigid enough to permit grasping of the assemblyby those parts.

We have implemented code which takes a descrip-tion of a fixture, along with the part description andthe description of any attached parts which create con-straints, and produces a list of configurations for amodular gripper whkh can be used to load or unloadthe ikture. By treating the fixture and attached partsas constraint regions, thk code can design a gripperwhich can be used to load the part into the fixture atthe start of the assembly, and also to remove it fromthe fixture at the end.

Further, this code can identify gripper designs whichcan be used to load and unload a variety of differentparts, with no hardware change. This can be very use-ful for flexible assembly lines which have frequent prod-uct changeover, perhaps among several related prod-ucts. In the rest of this section, we explain how thiscode was used to design a single modular gripper whichcan load or unload either of two products from their as-sembly fixture.

The example products are a chassis from a personalcassette player and a plastic hot-melt glue gun (Fig-ure 16). We desire a gripper that can grasp either thechassis or glue gun with no attached parts and alsoafter assembly operations have been performed. Thkfits well with the zero-time product changeover philos-ophy used in the design of the assembly ilxture, whkhwas designed to support the assembly of either product(Figure 17). See [BP96, BP97] for a discussion of thedesign of the fixture.

We ran our code on this example, using a 3/8” gridspacing and 1/4” diameter gripper pins. Figure 18shows the set of graspable features extracted for eachpart. For the glue gun, we obtained 24,367 grasps

Pp(a) (b)

(c) (d)

Figure 16: The two pats for the fixture loading/unloadlng

example. (a) and (c) show the base parts that are loadedinto the fixture, while (b) and (d) show the assemblies thatare removed from the iixtures.

Figure 17: The assembly fixture.

which could successfully hold the empty housing. Af-ter avoiding interference with the added parts or theassembly fixture, 5,563 grasps remained which couldsuccessfully insert or remove the glue gun housing fromthe fixture in either state. For the cassette chassis, thecode produced 6,753 grasps for the empty chassis. Be-cause a large percentage of the chassis perimeter is oc-cupied by gears, buttons, ad the side-locators of thefixture, only 48 of these grasps allowed the chassis to beremoved from the fixture after assembly. The commongrasp code identified three (gun-grasp, cassette-grasp)

pairs, one of which is shown in Figures 19 and 20. Therobustness of each grasp was verified by loading andunloading the fixture several times using a robot ma-nipulator.

15

—

(a) (b)

(c) (d)

Figure 18: The extracted features for the glue gun andcassette chassis examples: (a) and (c) show the features ex-tracted from the base parts. (b) and (d) show the features

extracted from the populated assemblies sitting in their aa-sembly fixtures. The small circles at the part boundariesrepresent the side-locating pins on the fixtures; similar cir-

cles within the part boundzwies are the vertical supportpads.

(a) (b)

Figure 19: The common gripper for the gluegun and thecassette chassis. The views shown are in the gripper frameof reference, as though seen through transparent plates.The small open circles are the pins of the aasembly fixture.

(b)

Figure 20: The physical gripper for the glue gun and thecassette chassis, shown loadlng the sssembly pallet. Thegripper is attached to an Adept One manipulator, whosebase is visible in the background.

16

.

5 Conclusion

The algorithms described in this paper extend Wal-lack and Canny’s planar modular vise algorithm to 3-d.Our code accepts 3-d CAD models as input and out-puts 3-d gripper designs. Loadability analysis ensuresthat the returned grasps can be easily achieved. Wehave added a force analysis that takes a gripper designand a set of expected dkiturbance forces and returnsthe minimum squeeze force that can oppose all of theexpected disturbances. To quickly find the best grasps,we rank them using this minimum squeeze force. Wealso implemented inter-gripper interference algorithmsto analyze grasp handoffs.

We have applied our code to two examples. In thefirst example, we used the code to design triples ofnon-interfering grasps for the Sandia Automated Com-ponent Cleaning workcell. In the second example, weused the code to design a gripper for loading and un-loading assembly fixtures on a mixed-product assemblyline. We performed physical tests to verify the successof the resulting designs in both cases.

In the course of this research, we’ve spent substantialtime considering the possibilities offered by modularparallel-jaw grippers, amdalso talking with automationengineers from industry. The biggest reaction we’vehad from our industrial contacts is concern about thecylindrical gripper fingers. The problem is primarilyone of synthesizing grippers that do not interfere withthe surrounding assembly during insertion operations.In many cases, it is impossible to select a diameter ofcylindrical finger which will provide sufficient rigiditywhile avoiding interference during the insertion. Pos-sible solutions to this problem might involve the use ofan expanded set of finger primitives, such as familiesof flat, circular arc, or angled corner pieces.

Another limitation on both the analysis we have per-formed and the possibilities offered by the algorithm isthat we assume that a part being grasped contacts onlythe fingers of the gripper, and only has contact withvertical surfaces on the fingers. While we have alreadyobserved that it is not feasible to use horizontal con-tacts to completely constrain the part, it is reasonableto use them to add stability. For example, we couldequip the fingers with shoulders that can oppose distur-bance forces with a significant component directed to-ward the palm of the gripper. Another approach wouldbe to allow the part to contact the palm of the gripper.Extending the synthesis and force analysis algorithmsto handle these additionzd contacts would be relativelystraightforward.

Unfortunately, including these additional contacts

significantly complicates the problem of robustly ac-quiring the part. Given real, imperfect manipulators,it is not possible to exactly match the z heights of thegripper and part during the grasping motion, and theresulting misalignment could cause problems when thefingers close. Solving this problem requires some formof compliant grasping strategy. One possible strat-egy would be to (i) Position the gripper over the part,above the final z height, (ii) Apply a small squeezingforce whkh closes the jaws on the part without apply-ing significant holding force, (iii) Apply a downwardforce, so that the horizontal shoulders of the pins orgripper palm slide into contact with the part, and fi-nally (iv) Tighten the jaws completely, ensuring a se-cure grasp. It may be possible to implement this orother strategies with a passive mechanism attached tothe gripper.

6 Acknowledgements

We thank Aaron Wallack for providing his planar fix-ture design code, and Ralph Peters and Randy Wil-son for their comments on a draft of this paper. Wethank Ron Simon for fabricating the gripper hardware,and Gilbert Benavides and Lilita Meirans for their helpdeveloping the ACC example. ACIS@ is a registeredtrademark of Spatial Technologies.

References

[AB85]

[Bar90]

[BG96]

[Bic95]

[BP96]

[BP97]

H. Asada and A. B. By. Kinematic analysis ofworkpart fixturing for flexible assembly with au-tomatically reconfigurable fixtures. IEEE Jew-nal of Robotics and Automation, RA-1(2) :86–94,June 1985.

D. Baraff. Determining frictional inconsistencyfor rigid bodies is rip-complete. Technical ReportTR 90-1112, Cornell University Department ofComputer Science, April 1990.

R. C. Brost and K. Y. Goldberg. A com-plete algorithm for designing planar ilxtures us-ing modular components. IEEE Transactions onRobotics and Automation, 12(1):31-46, February1996.

A. Bicchi. On the closure properties of roboticgrasping. International Journal of Robotics Re-search, 14(4):319-334, August 1995.

R. C. Brost and R. R. Peters. Automatic de-sign of 3-d tixtures and assembly pallets. InProceedings, IEEE International Conference onRobotics and Automation, pages 495–502, April1996.

R. C. Brost and R. R. Peters. Automatic designof 3-d fixtures and assembly pallets. Technical

17

Report SAND 95-2411, Sandia National Labora-tories, January 1997.

[Eng87] P. J. Englert. Principles for Part Setup andWorkholding in Automated Manufacturing. PhDthesis, Carnegie Mellon University Departmentof Mechanical Engineering, December 1987.

[Erd94]

[GHM89]

[Hay90]

[HR81J

[Kim93]

[KR86]

[Lak78]

[MD89]

[Met93]

[MNP90]

[MS85]

[MSS87]

[Ngu88]

[Ngu89]

M. Erdmann. On a representation of frictionin configuration space. International Journal ofRobotics Research, 13(3):240-271, June 1994.

R. A. Grupen, T. C. Henderson, and I. D. Mc-Cammon. A survey of general purpose manipula-tion. International Journal of Robotics Research,8(1) :38-62, February 1989.

C. C. Hayes. Machining Planning: A Model ofan Expert Level Planning Process. PhD thesis,Carnegie Mellon University Robotics Institute,December 1990.

D. Halliday and R. Resnick. Fundamentals ofPhysics, ,%’ndEd. John Wiley and Sons, NewYork, 1981. Page 144.

K. H. Kim. A System foT Automated FixturePlanning with Modular Fixtures. PhD thesis,Carnegie Mellon University Robotics Institute,May 1993.

J. Kerr and B. Roth. Analysis of multifingeredhands. International Journal of Robotics Re-search, 4(4):3–17, Winter 1986.

K. Lakshminarayana. The mechanics of formclosure. Technical Report 78-DET-32, ASME,1978.

R. J. Menassa and W. R. DeVries. Locatingpoint synthesis in fixture design. Annals of theCIRP, 38(1):165-169, 1989.

E. C. De Meter. Restraint analysis of assemblywork carriers. Robotics and Computer IntegratedManufacturing, 10(4):257-265, 1993.

X. Markenscoff, L. Ni, and C. H. Papadimitriou.The geometry of grasping. International Journalof Robotics Research, 9(1) :61–74, February 1990.

Matthew T. Mason and J. Kenneth Salisbury.Robot Hands and the Mechanics of Manipula-tion. MIT Press, Cambridge, Massachusetts,1985.

B. Mishra, J. T. Schwartz, and M. Sharir. Onthe existence and synthesis of MultiFinder posi-tive grips. Algorithmic, 2(4):641–558, 1987.

V. Nguyen. Constructing force-closure grasps.International Journal of Robotics Research,7(3):3-16, June 1988.

V. Nguyen. Constructing stable grasps. Interna-tional Journal of Robotics Research, 8(1):26–37,February 1989.

[Pon96]

[PT89]

[Reu76]

[Sa182]

[SomOO]

[TPSL95]

[Tri92]

pVc94]

~C96]

J. Ponce. On planning immobilizing fixtures forthree dimensional parts. In Proceedings, IEEEInternational Conference on Robotics and Au-tomation, pages 509–514, April 1996.

J. Pertin-Troccaz. Grasping: A state of theart. In O. Khatib, J. J. Craig, and T. Lozano-P&ez, editors, The Robotics Review I, pages 71-98. MIT Press, Cambridge, Massachusetts, 1989.

F. Reuleaux. The Kinematics of Machinery.Macmillan, 1876. Republished by Dover, NewYork, 1963.

J. K. Salkbury. Kinematic and Force Analysisof Articulated Hands. PhD thesis, Stanford Uni-versity Depart ment of Mechanical Engineering,May 1982. Reprinted in Robot Hands and theMechanics of Manipulation, MIT Press, Cam-bridge, Massachusetts, 1985.

P. Somoff. Uber gebiete von schraubengeschwin-dlgkeiten eines festen korpers bei verschiedenerzal von stutzfiachen. Zeitschrift fur Mathematikund Physik, 45:245–306, 1900.

J. C. Trinkle, J-S. Pang, S. Sudarsky, and G. Lo.On dynamic multi-rigid-body contact problemswith coulomb friction. Technical Report 95-003,Texas A&M University Department of Com-puter Science, January 1995.

J. C. Winkle. A quantitative test for form closuregrasps. In Proceedings, IEEE/RSJ InternationalConference on Intelligent Robots and Systems,pages 1670-1677, July 1992.

A. S. Wallack and J. F. Canny. Planning formodular and hybrid iixtures. In Proceedings,IEEE International Conference on Robotics andAutomation, pages 520-527, May 1994.

A. S. Wallack and J. F. Canny. Modular tixturedesign for generalized polyhedra. In Proceedings,IEEE International Conference on Robotics andAutomation, pages 830-837, April 1996.

18

Distribution:

MS 9018 Central Technical Files, 8940-2

0899 Technical Library, 44140619 Review & Approval Desk, 12630

For DOE/OSTI