Nguyen observed that two po**in**t contacts with frictionat po**in**ts P and Q form a planar force-closuregrasp if and only if the segment P Q, or QP , po**in**tsstrictly out of and **in**to the two friction cones of P andQ respectively. This important observation has beenexploited throughout the grasp**in**g literature and researchup to the present day. Nguyen's result can alsobe directly applied to the three-dimensional case withsoft-nger contacts 7 .Like Brost, Nguyen approached the synthesis problemby **in**sist**in**g that grasps require as little accuracyas possible. He constructed two-dimensionalgrasp**in**g regions us**in**g two po**in**t contacts with frictionby cast**in**g the problem **in**to one of tt**in**g a twosidedcone cutt**in**g the two contact edges **in**to twosegments of largest m**in**imum length. Us**in**g similartechniques, he went on to describe algorithms forthree-dimensional grasps us**in**g two soft-nger contacts,three-dimensional grasps us**in**g three hard-ngercontacts 8 , two-dimensional grasps us**in**g four frictionlesscontacts, and three-dimensional grasps us**in**gseven frictionless contacts. The complexity resultswere O(n) time to analyze whether a grasp with ngiven contacts was force-closure and O(n c c2 c ) time tosynthesize the n **in**dependent contact regions of an objectrequir**in**g at least c contacts. The robust natureof Nguyen's solution us**in**g **in**dependent contact regionsmeans that ngers can be placed **in**dependently at anyposition **in** the regions and still satisfy a force-closuregrasp.6.2 Other Grasp Synthesis StrategiesAs mentioned **in** section 3, Mishra, Schwartz, andSharir **in**cluded a grasp synthesis section their paper[MSS87] for polygonal/polyhedral objects. Us**in**gpo**in**t contacts without friction, they present a techniquefor construct**in**g a grasp with O(n) ngers **in**O(n) time when n is the number of faces of the objectand the dimension of the object is xed.The construction is a two step process. First theyshow how to synthesis a po**in**t contact grip us**in**g O(n)ngers **in** O(n) time. This result comes from the factthat any object with n faces can be held at equilibriumby n po**in**t contacts, one at each face. In he next step,they describe how to reduce this grip to another withthe m**in**imum number of ngers necessary for the graspus**in**g portions of Ste**in**itz's Theorem [Ste13].The reduction step takes O(n) time and **in**volves7 Soft-nger contacts are contacts that can not only exertforces **in** any direction **in**side the friction cone of that objectbut also torques about the normal to the surface normal.8 Hard nger contacts are simply po**in**t contacts with friction.That is, they can exert forces **in** any direction that it with**in** thefriction cone for that contact.choos**in**g the correct coecients to achieve a positivel**in**ear comb**in**ation of a subset of the orig**in**al contacts.They conclude with extend**in**g their construction techniqueto three dimensions yield**in**g the same time complexities(for xed d).There are several problems with their technique.First, it is not useful **in** certa**in** applications that requirethat the ngers be placed only **in** certa**in** permissibleregions of the object. Also, there are no limitationsplaced on the forces that the ngers can exert.This means that large forces may be required by theresult of their algorithm that are not obta**in**able orare so large that they deform and/or break the objectand/or gripper. F**in**ally, it is not robust to the uncerta**in**ties**in**volved with real world mult**in**gered hands.Markensco and Papadimitriou [MNP90] devotethe majority of their work to show**in**g the same resultsfor m**in**imum number of nger contacts for variousgrasps and contact types. However, their analysisdoes provide some new research **in** that it considersobjects that **in**clude curved surfaces. In addition theyshow that for po**in**t contacts with friction, under themost relaxed assumptions three ngers are necessaryand sucient for force-closure **in** two-dimensions andfour ngers **in** three-dimensions. They also give **in**terest**in**gconstruction techniques for generalized objects 9**in**clud**in**g curved objects.Their grasp analysis and construction centers uponthe idea of maximal **in**scribed circles. They use thesecircles by choos**in**g nger contact po**in**ts either on orclose to po**in**ts of contact of a maximal **in**scribed circlewith the boundary of the object. The use of maximal**in**scribed circles is not a new idea, **in** fact itwas rst utilized **in** a proof by Baker, Fortune, andGross [BFG85] **in** the context of stable grasps (notform-closure) of polygons.Throughout their analysis, they are forced to handlenumerous special cases. The most common specialcase occurs when the maximal circle touches theboundary of the object at only two po**in**ts. Their proofruns through many of these weird special cases and ishence not elegant.For the three-dimensional case 10 , maximal **in**scribedspheres are used for their constructive proofsjust as maximal **in**scribed circles were used for the twodimensionalcase. Aga**in** we note that curved surfacesare considered, mak**in**g this approach quite general forany two- or three-dimensional object.9 Object here means a closed, bounded, non-degenerate twodimensional body with a boundary that is connected and piecewisesmooth. Smooth is dened here to mean cont**in**uous andhav**in**g cont**in**uous derivatives of any order10 Three-dimensional objects are bounded, closed, nondegenerate,and with piecewise smooth, connected boundaries4

When friction is considered, they show that twodimensionalform closure of any object (**in**clud**in**gthe circle) can be achieved by three ngers (as opposedto four required without friction) and **in** threedimensions,form-closure of any object (even with anaxis of symmetry) is possible with four ngers (as opposedto seven). S**in**ce their proofs are all constructive**in** nature, they claim that the only dicult step **in** reformulat**in**gtheir work **in**to a grasp synthesis solverwould be to compute the maximal **in**scribed sphere ofthe object which they note can be carried out **in** l**in**eartime **in** the number of faces of the object us**in**gMegiddo's algorithm for l**in**ear programm**in**g [Meg80].7 Early Optimal Grasp**in**g TechniquesMarkensco and Papadimitriou [MP89] formulateand solve the problem of nd**in**g the optimal formclosuregrasp of any given polygon. They optimizethe grip with respect to m**in**imiz**in**g the forces neededto balance the object's weight through friction. Inessence they m**in**imize the worst-case forces needed tobalance any unit force act**in**g on the center of gravityof the object. As we saw before, by m**in**imiz**in**g thenecessary forces on the object, we avoid requir**in**g largeforces that cause unnecessary stress and deformationon both the object and the robot hand. Also, thereare often force limitations **in** our actual system due tothe actuation of the jo**in**ts **in** the hand, mak**in**g this anatural constra**in**t.They rst present an analytical method that usesthe metric from above to calculate the quality of a gripof a polygonal object by three ngers. In the secondpart of their paper, they attack the more dicult problemof optimiz**in**g the form-closure of a polygon whichthey do by solv**in**g a sequence of reductions us**in**g l**in**earprogramm**in**g duality and some **in**tricate plane geometry.This leads to an algorithm, that can rapidlyanalyze any reasonably complex object and computethe optimum grip by exam**in****in**g a nite number ofcases (grow**in**g as the cube of the number of sides ofthe polygon).In their analysis they run **in**to special cases similarto those **in** [MNP90]. For two of those cases, concavevertices and parallel sides, they give special proceduresfor comput**in**g the optimum grip of a polygon withthree ngers. Although their results are more generalthan Nguyen's [Ngu88], the necessary consideration ofspecial cases, make this a somewhat messy solution.8 Robust Grasp Plann**in**g8.1 2D Polygons with 3 F**in**gersPonce and Faverjon obta**in**ed excellent results us**in**ga new approach to construct**in**g three-nger forceclosuregrasps of polygonal objects [PF91]. They considera hand equipped with three hard-nger contactsfrom which they prove a sucient condition for forceclosuregrasps lead**in**g to a system of l**in**ear constra**in**ts**in** the position of the ngers along the polygonal edges.Like Nguyen [Ngu88] for two nger grasps, they determ**in**emaximal segments of the object boundarywhere the ngers can be positioned **in**dependentlywhile ma**in**ta**in****in**g force-closure.The regions satisfy**in**g these constra**in**ts are foundby a projection algorithm based on l**in**ear parameterelim**in**ation accelerated by simplex techniques. Theysolve this problem of project**in**g l**in**ear convexes **in** n-dimensions along one of several coord**in**ate axis directionswith l**in**ear programm**in**g, namely the simplexalgorithm. The search for a feasible vertex **in** theorig**in**al convex has to be done only once for all hyperplanes,thus mak**in**g the simplex algorithm ecient.Once the contact regions are found they use a reasonablecriterion to maximize the m**in**imum of thelengths of the three contact regions. However, **in** generalthere is not a unique solution to this problem, asfor suciently long edges, the size of the contact regionsdepends only on the size of the friction cones.They choose an additional criterion that attempts toplace the center of mass of the object at the center ofthe friction cones. This enables them to decrease theeect of gravity and **in**ternal forces dur**in**g the motionof the robot.As we h**in**ted at previously, the central idea beh**in**dtheir synthesis algorithm is that they translatethe problem of nd**in**g maximal **in**dependent contactregions **in**to a l**in**ear programm**in**g problem with tenvariables that can be solved us**in**g the simplex algorithm.The function to maximize becomes simply aweighted comb**in**ation of the two criteria: (1) the m**in**imumof the lengths of the three **in**tervals and (2) thethe position of the center of gravity **in** the frictioncones (at the center is maximal, hence optimal). Atypical result from this technique is shown **in** Figure 2.In pr**in**ciple their approach can be extended to morecomplicated problems such as the synthesis of three- orfour-nger grasps of three-dimensional objects. However,l**in**ear sucient conditions are dicult to nd forthese cases, and the number of variables and equations**in**volved make solv**in**g these problems much more dif-cult.5

Figure 2: An object and its best grasp us**in**g the techniquesfrom Ponce and Faverjon **in** 1991. The maximal**in**dependent contact regions are shown as heavy l**in**eswith security marg**in** circles at the ends of the regions.8.2 2D Curved Objects with 2 F**in**gersFaverjon and Ponce returned to the issue of comput**in**gforce-closure grasps **in** [FP91], consider**in**g theproblem of grasps on curved objects **in** two-dimensionsus**in**g two-ngers with friction. The objects that theyconsidered were modeled by parametric curves, andthe force-closure grasps characterized by a systemsof polynomial constra**in**ts **in** the parameters of thesecurves. All conguration space regions satisfy**in**g theseconstra**in**ts were found by a numerical cell decompositionalgorithm based on curve trac**in**g and cont**in**uationtechniques.As before, maximal object segments where ngerscan be positioned **in**dependently were found by optimizationwith**in** those grasp regions. This was an extensionof Nguyen's work and like Nguyen, they characterizetwo-nger force-closure grasps by the fact thatthe l**in**e jo**in****in**g the contact po**in**ts must lie with**in** thefriction cones at those po**in**ts.As we have seen before **in** [PF91] and [Ngu88],the l**in**ear form of the force-closure constra**in**ts **in** thepolygonal case allowed for the design of fast and simplegeometric algorithms for the construct**in**g stablegrasps. However, for smooth (and piecewise-smooth)curved objects, these constra**in**ts become highly nonl**in**ear,call**in**g for an approach of more algebraic natureto grasp synthesis.They beg**in** their analysis by model**in**g the boundaryof the planar object to be grasped by a piecewisesmoothcollection of parametric curves. Each of thetwo contact po**in**ts, lie on a parametric curve givenby (x i (u i ); y i (u i )) for i = 1; 2 and x i and y i are polynomials**in** u i . For a given pair of curve segments,a grasp is completely determ**in**ed by a pair (u 1 ; u 2 ).This pair takes values **in** the grasp conguration spacedened by the **in**tervals of u 1 and u 2 . They thenre-write the friction cone conditions for each contactpo**in**t for force-closure **in** algebraic terms. They observethat **in** the grasp conguration space the forceclosuregrasp regions are bounded by sets of algebraiccurves. To characterize the topology of the regionsand their bound**in**g curves they use a numerical celldecomposition algorithm, whose output is a descriptionof the stable grasp regions, their bound**in**g curves,and their adjacency relationships.Dur**in**g this step of the construction the most expensivepart of the algorithm is the computation of theextremal po**in**ts and **in**tersection po**in**ts of the graspconguration space curves. F**in**d**in**g those featuresamounts to solv**in**g square systems of polynomial equations.For this they use the cont**in**uation method, aglobal numerical technique that can nd all solutionsof systems of polynomial equations hav**in**g up to a fewthousand roots us**in**g [Mor87].Figure 3: The maximal rectangles found **in**side everyvalid region **in** the grasp space from work by Faverjonand Ponce **in** 1991.They next obta**in** maximal grasp rectangles withsides parallel to the coord**in**ate axes s**in**ce it is theserectangles that realize the local maximum size of thegrasp regions as desired. For each rectangle, they decomposethe four variables (the edges of the rectangle)**in**to two embedded optimization process **in** twovariables. The result**in**g maximal area rectangle will6

have the maximal segments on the objects boundarywhere the ngers can be positioned **in**dependentlywhile ma**in**ta**in****in**g force-closure. This results **in** a morerobust grasp planner, mak**in**g implementation for areal robot with uncerta**in**ties possible. Typical resultsfrom this algorithm are shown **in** Figures 3{4.Figure 4: The ve grasps correspond**in**g to the rectanglesof the previous gure.This method by Faverjon and Ponce is not withoutproblems and areas of possible improvement. Oneimportant area that is **in**complete is **in** determ**in****in**g abound on the number of maximal rectangles with**in**each stable grasp region. This **in** turn would providebetter methods for generat**in**g start**in**g po**in**ts forthe correspond**in**g optimization process s**in**ce it wouldm**in**imize the number of **in**itial guesses. In addition,although their technique works well **in** practice, it hasnot been proven to yield all local maximal rectangles,mak**in**g it **in**complete. However, their approachis quite elegant and general. In pr**in**ciple, it can beextended to more complicated problems such as thesynthesis of three (or more) nger grasps of possiblythree-dimensional objects. However, as before, thenumber and degree of the equations **in**volved makesuch an implementation dicult.8.3 3D Polyhedra with 3 and 4 F**in**gersPonce, Sullivan, Boissonnat, and Merlet [PSBM93]extend the approach of [Ngu88], [PF91], and [FP91]to the synthesis of three- and four-nger force-closuregrasps of three-dimensional polyhedral objects aga**in**us**in**g **in**dependent maximal grasp regions. They showthat for polyhedral objects, the sucient conditionsfor force-closure are l**in**ear **in** the unknown parameterswhich reduces the problem of comput**in**g force-closuregrasps of polyhedral objects to the problem of project**in**ga polytope onto some l**in**ear subspace.Their research uses the fact that a necessary conditionfor three po**in**ts to form a force-closure grasp isthat there exists a po**in**t **in** the **in**tersection of the planeformed by the three po**in**ts with the double-sided frictioncone at these po**in**ts. For four nger grasps theyderive results that will allow a stronger l**in**ear sucientcondition for three-nger force-closure grasps **in** thecase where gravity acts as a fourth nger. In the fournger case, they utilize the fact that a sucient conditionfor four-nger contact to be force-closure is thatthere exists a po**in**t **in** the **in**tersection of the four open**in**ternal friction cones with the tetrahedron formed bythese po**in**ts.The sucient conditions of force-closure can bewritten as a set of l**in**ear **in**equalities **in** the positionof a po**in**t x 0 and the position of three or four contactpo**in**ts x i . For polyhedral objects, each contact can beparameterized l**in**early by two variables (u i ; v i ) whichspecify its position **in** the plane of the correspond**in**gface. To obta**in** all the solutions (u i ; v i ) yield**in**g forceclosuregrasps, they show that they must elim**in**atethe po**in**t x 0 . This results **in** a projection problem.The problem is to construct the orthogonal projectionof the d-polytope onto a k-dimensional subspaceof the orig**in**al d-dimensional Euclidean space. A simpleimplementation of this projection algorithm takesO(tn) time where t is the size of the projection ofthe d-polytope and n is the number of **in**tersect**in**ghalf-spaces used to dene the polytope. They showthat this time complexity can be improved by preprocess**in**gthe hyperplanes us**in**g a recent result byMatousek and Schwarzkopf [MS92]. In general theyshow that the projection algorithm for a d-polytopeonto a k-dimensional subspace can be computed **in**time and space of O(kn + t + ) where is any positiveconstant, t is the size of the projection of theorig**in**al d-polytope, and (roughly) 2 3 3.4Aga**in** because of uncerta**in**ties **in** robotics systemsthey attempt to m**in**imize the sensitivity of a grasp toposition**in**g errors. They achieve this by nd**in**g triplesof **in**dependent contact regions as. These regions, aswe saw before, are such that for any tuple of contactpo**in**ts chosen **in** them, the correspond**in**g grasp isforce-closure. From before, we know that nd**in**g maximal**in**dependent contact regions reduces to solv**in**g al**in**ear programm**in**g problem, mak**in**g their overall solutionquite nice.7

9 Optimal GraspsFerrari and Canny [FC92] address the problem ofcomput**in**g optimal force-closure grasps of polygonalobjects by comput**in**g a maximal ball **in**cluded **in** theconvex hull of the contact wrenches. Their notion of anoptimal grasp, called quality criteria, actually focusesaround two criteria that consider the total nger forceand the maximum nger force. They p dene the magnitudeof the wrench as kwk = kF k 2 + k k 2 withthe choice for be**in**g somewhat arbitrary. The rstof the quality criteria is concerned with nd**in**g thegrasp congurations that maximize the wrench, given**in**dependent force limits (i.e. that m**in**imize the worstcaseforce applied at any po**in**t contact). The secondcriteria m**in**imizes the sum of all applied forces. S**in**cethe magnitude of the force is proportional to the totalcurrent **in** the motors and ampliers, this second criteriaresults **in** the m**in**imization of the power neededto actuate the gripper.Their major **in**sight is that **in** the wrench space,the convex hull of the nger wrenches on the objecthave important geometrical **in**terpretations. Namely,that their quality measure is simply the distance of thenearest facet of the convex hull from the orig**in**. Thismeans that optimal contacts can be synthesized **in** thewrench space by simply comput**in**g the convex hullof candidate contacts, determ**in****in**g the facet of m**in**imumdistance from the orig**in** **in** the wrench space,and choos**in**g the maximum of these distances. Thishas to be repeated for each side of a polygon wherethe number of possible congurations grows l**in**earlyas the number of edges of the polygons while the computationof the convex hull of the primitive wrenches**in** the wrench space takes constant time. This results**in** an O(n) time algorithm with n the size of the polygon.Essentially the quality criteria of the grasp isgiven by the value of the radius of the largest closedball centered **in** the orig**in** of the wrench space, conta**in**ed**in** the set of all the possible wrenches that canbe resisted by apply**in**g at most unit forces at the contactpo**in**ts. Their solution can also be applied to anythree-dimensional object.The advantage of this solution is that these qualitycriteria can be easily calculated for a wide variety ofobjects/parts **in** hopes of feed**in**g these quality valuesto a planner that will choose optimal tool/gripper selectionfor each object/part **in** an assembly task. Themotivation for us**in**g dierent grippers, each of whichis suitable for a small subset of operations, along withan automated tool-changer, is the philosophy of RISCRobotics.RISC Robotics (Reduced Intricacy **in** Sens**in**g andControl) is a class of robotics that relies on simple controland sensory **in**formation. Sensory **in**formation **in** aRISC Robotics system (1) require little or no process**in**gtime (typically s**in**gle bit digital I/O signals are used),(2) is **in**expensive (thus mak**in**g redundancy a feasibleoption), and (3) is highly accurate (typically on the orderof 70 microns). RISC Robotics follows many of thesame paradigms of RISC for computer architecture byus**in**g simple tools to build up and perform more complicatedtasks. In terms of grasp**in**g, RISC Roboticsmoves away from mult**in**gered hands because of their**in**tricate design mak**in**g them expensive, unreliable,computationally heavyweight, and dicult to controland create planners for. In general, a RISC Roboticssystem ma**in**ta**in**s a small set of simple, **in**expensive,and accurate mach**in**es that can be easily **in**terchangedand/or comb**in**ed to perform more complex tasks.10 Future and Current WorkExcit**in**g new work is ongo**in**g **in**to many of the nonholonomicconstra**in**ts **in** grasp**in**g. Recent work by Li,Murray, and Sastry [LMS93] exam**in**es roll**in**g ngercontacts and object slid**in**g constra**in**ts. Related work**in** a recent paper by Chen and Burdick [CB93] **in**vestigatesnger gates on planar objects. Essentially, mostof the past grasp**in**g research was for static grasps,whereas new work looks at various motions that canbe done once an object is be**in**g held and exactly howto plan and control such operations. Some of the motivationfor this work comes from the observation thatthe grasp used for pick**in**g up a pencil is entirely differentfrom the one we use for writ**in**g, even thoughthe geometry rema**in**s the same.There is also still excit**in**g work ongo**in**g **in**to graspplann**in**g, **in** particular plann**in**g optimal grasps. Mostof the work **in** this areas that was presented **in** this paperhave come out of research with**in** the past year. Inaddition, work **in**to formaliz**in**g a synthesis techniqueto accompany the 1992 paper [FC92] is **in** process byMirtich and Canny.11 ConclusionWe have also seen that manipulation is complex,typically **in**volv**in**g comb**in**ations of open and closedloop k**in**ematic cha**in**s, non-holonomic constra**in**ts, redundantdegrees of freedom, and s**in**gularities. In additiontheir are non-l**in**earities **in** the contact conditionsbetween soft ngers and grasped objects as wellas the actuator dynamics. As a result, many assumptionshad to be made to make the problem tractable.Yet, humans can perform such tasks easily suggest**in**gthat a simple approach exists.In addition, an **in**terest**in**g **in**sight made by8

Nguyen [Ngu88] and others, observed that larger frictioncones were produced when nger contacts wereat edges or vertices of an object. This expla**in**s whypeople grasp objects at the edges and corners and whythe contact**in**g surface of human ngers need to be softrather than hard like the ngernails. However, thisfact has yet to be properly addressed and exploited **in**the grasp**in**g community.F**in**ally, research po**in**ted out by Ferrari andCanny [FC92] as well as others, moves grasp**in**g towardsa new direction. This new research centersaround the recent movement away from anthropomorphichands towards specialized grippers and toolchangers.Such specialized tools are simpler **in** termsof k**in**ematics and controllability, and far more accuratethan a human like hands. After all, the hand hasevolved over millions of years as an organ used as muchfor sensation and communication as for manipulation.In fact, for many manufactur**in**g tasks the human handis less than ideal. When a mechanic starts to work ona mach**in**e, the rst th**in**g he reaches for is his/her toolbox,with pliers, wrenches, tweezers and work glovesto help him/her nish the job.References[Asa79][BFG85]H. Asada. Studies on Prehension and Handl**in**gby Robot Hands with Elastic F**in**gers.PhD thesis, Kyoto University, April 1979.B. Baker, S. Fortune, and E. Grosse. Stableprehension with three ngers. In Proceed**in**gsSymp. Theory of Comput**in**g, pages114{120, 1985.[Bro88] Randy C. Brost. Automatic grasp plann**in**g**in** the presence of uncerta**in**ty. InternationalJournal of Robotics Research,7(1):3{17, 1988.[CB93]I-M. Chen and J. W. Burdick. A qualitativetest for n-nger force-closure grasps onplanar objects with applications to manipulationand nger gaits. In IEEE InternationalConference on Robotics and Automation,pages 814{820, 1993.[Cut89] Mark R. Cutkosky. On grasp choice,grasp models, and design of hands formanufactur**in**g tasks. IEEE InternationalTransations on Robotics and Automation,3(3):814{820, June 1989.[FC92][FP91][HA77][Lak78][Erd86] Michael Erdmann. Us**in**g backprojectionsfor ne motion plann**in**g with uncerta**in**ty.International Journal of RoboticsResearch, pages 19{45, 1986.C. Ferrari and J.F. Canny. Plann**in**g optimalgrasps. In IEEE International Conferenceon Robotics and Automation, pages2290{2295, 1992.B. Faverjon and J. Ponce. On comput**in**gtwo-nger force closure grasps of curved 2dobjects. In IEEE International Conferenceon Robotics and Automation, pages 424{429, 1991.H. Hanafua and H. Asada. Stable prehensionby a robot hand with elastic ngers.In International Symposium of IndustrialRobotics, pages 361{368, 1977. (Tokyo).K. Lakshm**in**arayana. Mechanics of formclosure. ASME Paper, 78-DET-32, 1978.[LMS93] Zexiang Li, Richard M. Murray, andS. Shankar Sastry. Robotics: Manipulationand Plann**in**g. Pre-pr**in**t, 1993.[LMT84][Meg80][MNP90][Mor87]T. Lozano-Perez, M.T. Mason, and R.H.Taylor. Automatic synthesis of ne-motionstrategies for robots. International Journalof Robotics Research, pages 3{24, 1984.M. Megiddo. L**in**ear algorithms for l**in**earprogramm**in**g **in** d dimensions. In Proceed**in**gsFOCS Conference, October 1980.Xanthippi Markensco, Lugun Ni, andChristos H. Papadimitriou. The geometryof grasp**in**g. International Journal ofRobotics Research, 9(1):61{74, 1990.A.P. Morgan. Solv**in**g Polynomial SystemsUs**in**g Cont**in**uation for Eng**in**eer**in**g andScientic Problems. Prentice Hall, 1987.[MP89] Xanthippi Markensco and Christos H.Papadimitriou. Optimum grip of a polygon.International Journal of Robotics Research,8(2):17{29, 1989.[MS92][MSS87]J. Matousek and O. Schwarzkopf. L**in**earoptimization queries. In ACM Symposiumon Computational Geometry, pages 16{25,1992.B. Mishra, J. T. Schwartz, and M. Sharir.On the existence and synthesis of mult**in**geredpositive grips. Algorithmica, 2:541{558, 1987.9

[Nap56]J. Napier. The prehensive movements ofthe human hand. In Journal of BoneJo**in**t Surgery, volume 38B-4, pages 902{913, November 1956.[Ngu88] Van-Duc Nguyen. Construct**in**g forceclosuregrasps. International Journal ofRobotics Research, 7(3):3{16, 1988.[Pes90][PF91]Michael A. Peshk**in**. Programmed compliancefor error corrective assembly. In IEEETransations on Robotics and Automation,pages 473{482, 1990.J. Ponce and B. Faverjon. On comput**in**gthree-nger force closure grasps of polyhedralobjects. In International Conferenceon Advanced Robotics, pages 1018{1023,1991.[PSBM93] Jean Ponce, Steven Sullivan, Jean-DanielBoissonat, and Jean-Pierre Merlet. Oncharacteriz**in**g and comput**in**g three- andfour-nger force-closure grasps of polyhedralobjects. In IEEE International Conferenceon Robotics and Automation, pages821{827, 1993.[Reu75] F. Reuleaux. K**in**ematics of mach**in**ery.Dover, 1875.[SR83] J.K. Salisbury and B. Roth. K**in**ematicand force analysis of articulated mechanicalhands. In Journal of Mechanisms,Transmission, and Automation **in** Design,pages 105:35{41, 1983.[Ste13] E. Ste**in**itz. Bed**in**gt konvergente reihenund konvexe systeme i. Journal Re**in**eAngew. Mathematics, 143:128{175, 1913.[Ste86] E. Ste**in**itz. Analysis of mult**in**geredhands. International Journal of RoboticsResearch, 4(4):3{17, 1986.[Whi82] D.E. Whitney. Quasi-static assembly ofcompliantly supported rigid parts. ASMEJournal of Dynamic Systems Measurementand Control, pages 65{77, 1982.10