Placement of Robot Manipulators to MaximizeDexterityKarim Abdel-Malek - PDF document

Placement of Robot Manipulators to MaximizeDexterityKarim Abdel-Malek and Wei YuDepartment of Mechanical EngineeringThe University of IowaIowa City, IA 52242Tel. (319) 335-5676amalek@engineering.uiowa.eduPlacement of robotics manipulators involves the specification of the position andorientation of the base with respect to a predefined work environment. A generalapproach to the placement of manipulators based on the kinematic dexterity ofmechanisms is presented. In many robotic implementations, it is necessary to carefullyplan the layout of the workplace, whether on the manufacturing floor or in robot-assistedsurgical interventions, whereby it is required to locate the robot base in such a way tomaximize dexterity at or around given targets. In this paper, we pose the problem in anoptimization form without the utilization of an inverse kinematics algorithm, but ratherby employing a dexterity measure. A new dexterous performance measure is developedand used to characterize a formulation for moving the workspace envelope (and hence therobot base) to a new position and orientation. Using this dexterous measure, numericaltechniques for placement of the robot and based on a method for determining the exactboundary to the workspace are presented and implemented in computer code. Examplesare given to illustrate the techniques developed using planar and spatial serialmanipulators.Keywords: Manipulator placement, dexterity measure, reachability, manipulability. IntroductionManipulator placement in an environment such that it will perform a number of taskswith maximum dexterity is a challenging problem. In this paper, a mathematicalformulation is presented based on the kinematics of the manipulator and that does notrequire obtaining solutions to the inverse kinematics problem.There has been limited works addressing the placement problem. This is due to theoverwhelming focus given to the placing of path trajectories in a robotic environmentversus placing the robot in a pre-specified environment. In many cases, the targetscannot be displaced because of physical constraints such as weight or geometry, orbecause of inability in the case of robotic medical interventions where it is notrecommended to move the patient. Early works addressing placement for avoidinginterference between the manipulator links was reported by Pamanes-Garcia (1989) andZegloul and Pamanes-Garcia (1993), while arm reachability as the basic criterion forplacement was reported by Seraji (1995).Workspace volume (Park and Brockett 1994), reachability, and manipulability aremeasures that have been used in the past (Bergerman and Xu 1997). Even though themanipulability ellipsoid approach is the most widely used techniques, it has been shownthat the manipulability ellipsoid does not transform the exact joint velocity constraintsinto task space and so may fail to give exact dexterousness measure and optimal directionof motion in task space. Other types of dexterity measure were proposed by Youheng andKaidong (1993) and called Average Service Coefficient (ASC) and the Dexterity Effective Coefficient (DEC). The authors demonstrated that by deducing the relationformulas between the dexterity indexes and the linkage parameters of manipulators, adimensional optimum synthesis model can be obtained. Dexterous workspaces have alsobeen addressed by researchers (Wang and Wu 1992 and Yang and Haug 1991) but offersonly a general guidance for placement.A method that does not use inverse kienematics was proposed by Rastegar and Singh(1994) using a probabilistic approach for optimal positioning of task spaces within theworkspace of a manipulator.A mathematical theory for optimizing the kinematic dexterity of robotic mechanisms waspresented by Park and Brockett (1994). Using methods from Differential Geometry, thisapproach takes into account the geometric and topological structures of the joint andworkspaces.Dexterity measures based on the notions of the scaling laws of biological systems wereproposed by Sturges (1990). Values for the index of difficulty are shown to vary in thework space, and the loci of maximum dexterity that indicate the most favorabletask/effector arrangements are determined.Kim and Khosla (1991) introduced a number of dexterity measures. For example, themeasure of manipulability has an analytical expression, but it depends on the scale of amanipulator. On the other hand, the condition number is independent of the scale, but cannot be expressed analytically. These two main problems (scale dependency andanalytical expression) of previous dexterity measures were later addresses and applied inthe design and control of manipulators.Manipulability polytopes as a dexterity measure were introduced by Lee (1997) and werecompared with manipulability ellipsoids. Extending the concept of manipulabilityellipsoids to cooperating arms have been proposed and demonstrated by Bicchi andMelchiorri (1993).In this paper, we propose a numerical method for the placement of robot manipulatorsbased on maximizing the dexterity at specified target points. We will first define anumber of necessary constraints in order to impel the workspace towards target points.We will then define a cost function that is based on a new measure of dexterity at eachtarget point. The problem is then characterized in terms of a maximization function andis addressed using numerical techniques.Criteria for Impelling the WorkspaceGeneralized coordinates defined as the set of all -tuples [...] ofreal numbers that represent joint variables, where is the number of degrees of freedom(DOF) of the manipulator. Using the Denavit-Hartenberg representation method(Denavit and Hartenberg 1955), a position vector xyz can be written as(1) where is a smooth vector function defined as a subset of the Euclideanspace. Surface patches on the boundary of the workspace were delineated in previouswork (Abdel-Malek, et al. 1997 and Abdel-Malek, et al. 1999) and is summarized inAppendix A. The result of this work is exact surface patches denoted by ;,...,in closed form that characterize the boundary of the workspace, including voids if anyexist.In this work, we will consider a number of target points denoted by ;,..., thatmust be not only reached by the end-effector but with maximum dexterity (i.e., can bereached with the maximum number of orientations). To ascertain that target points are inside the workspace, additional conditions are needed. The absolute value of thedistance between a target point and the boundary should be greater than a specifiedtolerance denoted by . This will guarantee that the target points are located inside of theworkspace but not on boundary. The distance between all target points and allsurface patches ()() should be greater than a specified minimum value such as puw()()()(,)jii where 1,..., and 1,...,(2)where we have used a rotation matrix to capture the orientation and have used aposition vector to capture the position of each surface such that the generalized surfaceis now denoted by and defined as()()(,)(,,)()(,,)wwwxyzwuRuv=×+abg(3)where the six generalized variables in are used to track the workspace envelope. xyzwwwabg and where e are specified constants. If a target point satisfies both conditions of Eq.(1) and (2), then this point is internal to the workspace (i.e, have placed the workspace ina configuration such that all points are covered by the workspace).In order to impel the workspace towards the target points, we propose the following twoconstraints: (1) The target points must satisfy the constraint equation (i.e., located in theworkspace volume) and (2) The distance between the target-points and the workspacemust satisfy specified constraints in order for the target points not to be located on theboundary. These constraints are defined as follows:(1)Workspace envelope at least covering the target points (shortest distance betweentarget points): Ÿ-ˆmin(,)pqw for 1,...,(4)where the totality of points in the workspace are also translating and rotation in space(,)(,,)()(,,)wqRqv a b g xyzwwwand b is a very small positive number and subject to the inequality constraints on jointlimits asqqq for 1,...,(5)(2)Embedding the target points inside the workspace volume (a minimum distancebetween target points and surface patches). º-³puw()()(,)for 1,..., and1,...,1,...,()(6)where is the depth of the target point inside the workspace volume. There are+™+ total number of constraints. Placement of the Manipulator for Maximum DexterityIn this section, we define a cost function that is based on maximizing the dexterity attarget points. Indeed, to mathematically formulate this problem, it is necessary to use ananalytic dexterity measure at specific target points that can be manipulated. Becausedexterity measures in the literature do not account for joint limits and because of the needfor an analytical expression that can be used in the proposed optimization method, wedefine a new dexterity measure.Dexterity MeasureIn serial manipulators, joints are constrained, sometimes due to space limitation, othersstrictly by design and are usually specified by an inequality constraint of the formqqq, where is the lower limit and is the upper limit. In order to includejoint limits in the formulation, we have used a parameterization to convert inequalities on to equalities iii . Joint limits imposed in terms of inequality constraints in theform of qqq, where 1, ... , are transformed into equations by introducing anew set of generalized coordinates lll,,..., such that qqqqq=++-()()sin2727 1,...,(7)These generalized coordinates are called slack variables in the field of optimization,where lll,,...,. For any admissible configuration and , i.e., at atarget in space, the following augmented constraint equations must be satisfied ()-(8)where the augmented vector of generalized coordinates is zxqqTTTT. Bydefining a new vector =TTT (input parameters), the augmented coordinates canbe partitioned as zxz(9)The set defined by is the totality of points in the workspace that can be touched bythe end-effector of a serial robot manipulator while considering joint limits. The inputJacobian of is obtained by differentiating with respect to as(10)which is an ()() matrix, where Fq=¶ F ¶ is a matrix, is the identify matrix, and ql= is an diagonal matrix with diagonalelements as iiii l )cos. We define as the augmented Jacobian matrix.Since this Jacobian inherently combines information about the position, orientation, andjoint limits of the end-effector, it is a viable measure of dexterity. Furthermore, becauseof the simplicity in determining an analytical expression of , it is by far a simplerapproach in comparison with the widely used manipulability ellipsoid. We define thedexterity measure as (11) Note that the measure characterized by Eq. (11) takes into consideration all joint limitsand singular orientations.Geometric Significance of the New Dexterity MeasureRelationship with the singular values A singular value decomposition of the augmented Jacobian matrix can be represented by:GqUV()**111(12)where +™+()()mnmn and are orthogonal matrices, and0000000000+™+::....::()()mnmn(13)where imn,,...., are the singular values of . Substituting Eq.(12) intoEq.(11) yields det()det()(14)wheredmnsss***(15)Equation (15) indicates that the dexterity measure is the product of augmented singularvalues. When the augmented Jacobian matrix degenerates, one or more singular valueswill be zeros, then the measure is zero, i.e, some of three type singularities occurs.2. Equivalent Ellipsoid Geometry From Eq. (10), we have: d d d GorG(16)which is equivalent to (17)where and are called the augmented output vector and augmented input vector,respectively, andRURQVQ(18)From Eq. (18), we havedsdRQimiii,...,(19)When is constrained by , then satisfies the following inequality RRRR++++ˆ(20)This redefined dexterity ellipsoid is of higher dimension as to the dexterity ellipsoidgiven by . Its volume is computed from volume()/(()/)(21)From the above discussion, we draw the following conclusions:(1)The proposed dexterity measure is more accurate in describing the manipulability ofrobot than that proposed by Yoshikawa (1995), because it considers all singularities(Jacobian and others) as well as joint limits.(2)The proposed dexterity measure and that proposed by Yoshikawa (1995) can beinterpreted in the same manner in terms of singular values and ellipsoid geometry, butare not directly related. Mathematical Modeling of the Placement ProblemGiven target points (,,)iiixyz for ,,, defined in space, we introduce adexterity measure at each point denoted by , where it is necessary to place themanipulator to achieve maximum dexterity at each target point. Note, however, that thisis a multi-objective optimization problem. Therefore, we will give dexterity at each pointa weight so as to transfer the problem into a usual optimization problem that can bereadily solved.A mathematical model of the placement of the robot base subject to maximizing thedexterity at specified target points is characterized by the following optimizationproblem.Cost function Maximize ()(,)wwp(22)where ,,,, are specified weights at each target point.Subject to: (1)Target points are inside the workspace volume (Eq. 2).(2)Target points are not on the boundary of the workspace envelope (Eq. 4).(3)The manipulator is constrainedqqq(23)(4)The motion is within a finite spacewww(24) The set of constraints in Eq. (23) is to impose the unilateral constraints on each joint andthe second set (Eq. 24) to constrain the motion of the base within a finite space.The algorithm for achieving placement is shown in Fig. 1.Define target pointsReach envelope has been identifiedCost functione.g., dexterityConstraints (no need for inverse kinematics)Move boundary of workspace Satisfies Tolerance Stop Iteratew = w + Iterative algorithm to move the workspace Define robot(dimension and ranges of motion)Defined by the six generalized coordinates that characterize its position and orientationFig. 1 Algorithm for placement based on optimizationSimple ExampleConsider a planar 3DOF manipulator arm as shown in Fig. 2 consisting of 3 revolutejoints. This example will be used to illustrate the theory and will be followed by a morerealistic manipulator model of the upper extremity. There are three target points, namely, )1(=P, T]50100125[)2(=P, and )1(=P, which must betouched by the end-effector with the ability to reach these points from many directions(i.e., maximizing dexterity). z3z0 z1z2 Fig. 2 Model of the arm with three revolute jointsThe coordinates of a point on the end-effector are given by(27)For simplicity, ranges of motion on each joint are defined as The Jacobian is calculated as--+-++-+-++-+++++++++++++4224221121231212312311212312123123sinsin()sin()sin()sin()sin()coscos()cos()cos()cos()cos()qqqqqqqqqqqqqqqqqqqqqqqqqqqqResults of the workspace determination yield the following boundary curves (note thatcurves are generated because we have restricted the manipulator to planar movement.The boundary curves are defined by the following sets: ]0 ,3[ );0,,3(22pp-Îqqx 3,0[ );0,,3(22ppÎqqx, ] 3,3[ );0,0,(21pp-Îqqx ]0,3[ );,3,3(33ppp-Î--qqx ]3,0[ );,3,3(33pppÎqqx );3,3,(11pppp-Î--qqxand );3,3,(11pppÎqqx.Subsitituting the singular sets into Eq. (27) yields equations of curves shown in Fig. 3,which is the exact workspace of the planar 3DOF arm (Abdel-Malek and Yeh 1997). Weshall use and for positioning and for orienting the workspace. 4.03.0Fig. 3 The exact workspace of the arm (restricted to planar motion)As a result of the iterative algorithm, the three design variables representing the positionand orientation of the workspace are calculated as ed as Tyxa=w[]T674.0663.4714.10= and shown in Fig. 4. -10 5 15 20 -10 -5 15 20 Fig. 4 The initial and final positions of the arm The measure of dexterity at each point is calculated asD80 D040D914Note that any configuration that would have included the three points is a solution,however, the solution calculated using this method yields the position of the arm thatwould maximize the dexterity at all three points.Example: A spatial 3DOF armConsider a 3DOF spatial manipulator where the coordinates of end-effector are given bysincoscossinsinsin122212qqqqqqwhere joint limits are specified by p , , and The boundaries of the workspace consist of eight surfaces due to the singular sets listedbelow. The workspace is shown in Fig. 5. p / 2 p }s12= p / 2 p }s1 / 2p30=}s 322 / *2p38=}sp / 2p38=}s20=}s2=p} 0 5 0 5 2 4 6 5 Fig. 5 The workspace of a 3-DOF manipulatorThree target points are specified as (,,)303433(,,)303237, and(.,,.)33435358. The results of the calculations are listed in Table 2, where initial andfinal configurations of the manipulator are shown. The trace of the coordinates of theorigin of the base is plotted in Figure 6. The relative positions of workspaces and work-point are shown in Figure 7.Table 2: The original and new base positionParameters a bg Initial.000.000.000.000.000.000 Final34.02338.59437.8951.7624.342.380 Fig. 6 The trace of the coordinates of the base of the manipulator Fig. 7 Illustration of results for the placement of the 3DOF manipulatorExample: A 9DOF robot manipulatorConsider a model of the upper extremity shown in Fig. 8 and modeled as a total of 9DOF(5DOF in the shoulder, 1DOF in the elbow, and 3DOF in the wrist). Note that we have accounted for two translations of the shoulder joint and three rotational motions. Alsonote that in order to allow for realistic motion of the shoulder, we have coupled thetranslational and rotational joints by a linear equation such that the behavior of themotion of one joint is dependent upon the other. For the shoulder will translate by asmall distance when any of the rotational joints are actuated asqqq312005.()(28)qqq412004.()(29) This coupling reduces the number of DOFs to seven. q1q2q3q5q6q7q8q9 xoyo Fig. 8 A 9-DOF robot manipulatorThe D-H parameters for this arm are presented in Table 2. Table 2: DH Table for the arm q idiai a i 1 p 2q10- p 2 2- p 2q20 p 2 30+q3 4 p 2+q4 020 p 2 60+025 70+q7 p 2 8q8- p 2 p 2 000 Ranges of motion for this arm are as follows (note that the first two joints aretranslational): -ˆˆ3838qcm-ˆˆ3838qcm -ˆˆ q; -ˆˆ11823 p p q -ˆˆ q 056 p -ˆˆ q; -ˆˆ ; and -ˆˆ p It is required to place the human (as represented by the arm’s workspase) such that thefollowing three target points are touchable and dexterity is maximized.zed.)1(=P, T]50100125[)2(=P, and )1(=P.Note that this is a common problem that arises in the design of assembly lines, cells, andin any ergonomic design of workplaces.The position of a point on the end-effector is determined from the Denavit-Hartenbergformulation asxccssscccsssccscccccsssccscscssscccccsssss()()(((())())(())=-++-+-+-+-+++--+20201010513213413213124654132131243112351241321346 ()()(((())())(())=--+---+---+-++----20201010531213431213214654312132141312352413121346csscsccsscscssccccsscscssccssssccscsscsss cccccsscccccsscsscsccss((())()) =+-+--+-- 0 1 0 1 0 23234246523424235 2 2 where cossin, and . Surface patches on the boundary aredelineated and shown below in Fig. 9 (a total of 22 boundary surfaces): -2 0 2 4 X -100 -75 -50 -25 Y -2 2 4 Z -2 2 4 X -100 -75 -50 -25 Y 2 4 X -25 -60 -40 -20 Z 2 4 X -25 2 4 X -25 50 0 2 4 X -25 50 0 40 60 -25 50 2 4 Z 0 40 60 -60 -40 -20 X -50 -25 50 -2 2 4 Z -60 -40 -20 X 0 20 0 50 75 100 0 20 0 50 75 100 -20 -10 X 0 75 100 -20 -20 -10 X 0 75 100 -2 2 4 X 0 50 75 100 -20 -2 2 4 X 0 50 75 100 0 20 30 0 50 75 100 -20 0 20 30 0 50 75 100 -30 -20 -10 X 0 50 100 -20 -30 -20 -10 X 0 50 100 2 4 X 0 50 100 -20 2 4 X 0 50 100 0 100 0 20 30 -20 0 50 75 100 -2 2 4 Z -20 -20 0 75 100 -20 -10 Z -20 0 75 100 -2 2 4 Z -20 20 0 50 75 100 -10 -20 20 0 50 75 100 -25 50 -25 50 -25 50 -25 50 -25 50 0 50 75 100 -20 -10 Z -20 0 75 100 0 -20 Fig. 9 Surface patches These surface patches are combined, the 7DOF workspace is calculated, and shown inFig. 10. Note that we will use the six variables xyzwwwabg to trackthe motion of the workspace envelope. -40 0 -20 -40 -20 40 -40 -20 40 -40 -20 -20 -40 -20 -40 -20 Fig. 10 Workspace of the upper extremityAs a result of the iterative algorithm, the six design variables representing the positionand orientation of the workspace are calculated ased asT736.0463.1850.0171.94444.83235.107-=wThe measure of dexterity at each point is maximized and its value isD83D47D54The initial and final configurations of the workspace of the arm are shown in Fig. 11. Fig. 11 Initial and final configurations of the workspaceConclusionsA general optimization method for locating the base of a serial manipulator in a workenvironment while maximizing dexterity at specified target points was presented. It wasshown that it is possible to place the manipulator by effecting translation and orientationof the workspace generated in closed form and characterized by surface patches on theboundary. It was shown that the placement problem can be formulated as an optimizationproblem where the cost function is dexterity and the constraints pertain to including thetarget points in the workspace. A new dexterity measure was introduced that takes into account singular behavior andjoint limits, which is a fundamental improvement over that reported by Yoshikawa(1995).It was shown that the proposed dexterity measure can be used as a cost function in anoptimization algorithm whereby the robot workspace’s motion is tracked using sixgeneralized variables. The final position and orientation of these variables determine theplacement of the base. The method and code were demonstrated using a simple planarexample and using a spatial model of the upper extremity with 9DOFs.ReferencesAbdel-Malek, K. and Yeh, H.J., (1997), “Path Trajectory Verification for RobotManipulators in a Manufacturing Environment” IMechE Journal of EngineeringManufacture, Vol. 211, part B, pp. 547-556.Abdel-Malek, K., Adkins, F., Yeh, H.J., and Haug, E.J.(1997)“On the Determination ofBoundaries to Manipulator Workspaces,” Robotics and Computer-IntegratedManufacturing, Vol. 13, No. 1, pp.63-72.Abdel-Malek, K., Yeh, H-J, and Khairallah, N., (1999), “Workspace, Void, and VolumeDetermination of the General 5DOF Manipulator, Mechanics of Structures andMachines, 27(1), 91-117.Bergerman, M; Xu, Y, 1997, “Dexterity of underactuated manipulators”, Proceedings ofthe 1997 8th International Conference on Advanced Robotics, ICAR'97 Jul 7-9,1997, Monterey, CA, pp. 719-724.Bicchi, A.; Melchiorri, C., 1993, “Manipulability measures of cooperating arms”,Proceedings of the 1993 American Control Conference Jun 2-4, IFAC, pp. 321-325.Denavit, J. and Hartenberg, R.S., (1955), "A Kinematic Notation for Lower-PairMechanisms Based on Matrices", Journal of Applied Mechanics, vol.77, pp.215-Kim, Jin-Oh; Khosla, Pradeep, K., 1991, “Dexterity measures for design and control ofmanipulators”, Proceedings of the IEEE/RSJ International Workshop on IntelligentRobot and Systems '91, v 2, pp. 758-763.Lee, J., 1997, “Study on the manipulability measures for robot manipulators”,Proceedings of the 1997 IEEE/RSJ International Conference on Intelligent Robotand Systems. Part 3 (of 3) Sep 7-11 1997 v 3 1997 Grenoble, France, pp. 1458- Pamanes-Garcia, J.A., 1989, “A Criterion for the Optimal Placement of RoboticManipulators”, ProceedingsIFAC Information Control Problems in manufacturing, Madrid, Spain.Park, F.C.; Brockett, R.W., “Kinematic dexterity of robotic mechanisms”, InternationalJournal of Robotics Research, v 13 n 1 Feb 1994 p 1-15.Rastegar, J.; Singh, J.R., 1994, “New probabilistic method for the performance evaluationof manipulators”, ASME Journal of Mechanical Design, v 116 n 2, pp. 462-466.Seraji, H., 1995, “Reachability Analysis for Base Placement in Mobile Manipulators”,Journal of Robotic Systems, Vol. 12(1), pp. 29-43.Sturges, R.H., 1990, “Quantification of machine dexterity applied to an assembly task”,International Journal of Robotics Research, v 9 n 3 Jun 1990 pp. 49-62.Wang, J.Y.; Wu, J.K., 1992, “Computational environment for dextrous workspaceanalysis”, DE Advances in Design Automation–Proceedings of 18th Annual DesignAutomation Conference, v 44 pt 2, pp. 293-302.Yang, F.-C.; Haug, E. J., 1991, “Numerical analysis of the kinematic working capabilityof mechanisms”, DE Advances in Design Automation-Proceedings of 17th DesignAutomation Conference, v 32 pt 1, pp. 9-17.Yoshiakawa, T., 1985, Manipulability of Robotic Mechanisms”, International Journal ofRobotics Research, Vol. 4(2), pp. 3-9.Youheng, X; Kaidong, Z, 1993, “Optimum synthesis for workspace dexterity ofmanipulators”, Journal of Shanghai Jiaotong University, v 27 n 4 1993 p 40-48.Zeghloul, S., Pamanes-Garcia, J.A., 1993, “Multi-criteria optimal placement of robots inconstrained environments”, Robotica, Vol. 11, pp. 105-110.Appendix AUsing the Denavit-Hartenberg representation, the position of the end-effector can beanalytically formulated as(a1)where qqq,,..., is the vector of joint coordinates. Joint limit constraints areimposed using the transformation defined in Eq. (7) as . For any admissibleconfiguration, the following augmented constraint equations must be satisfied(a2)where the augmented vector zxqTTT . Singularity sets resulting from a row-rankdeficiency criteria must be determined to generate envelopes to the workspace. These envelopes are characterized by surface patches that exist inside, outside, and extendingboth internal and external to the workspace. The input Jacobian of is obtained bydifferentiating with respect to as(a3)which is an ()() matrix, where Fq=¶ F ¶ is a matrix, is the identify matrix, and ql= is an diagonal matrix with diagonalelements as iiii l )cos, where is defined as the augmented Jacobian matrix.The objective is to find the constant subvectors of , denoted by , which make the sub-Jacobian row rank deficient. Three singularity types are identified:(1)Jacobian singularities (called Type I) that satisfy:],Ÿ³sq3s Rank[ for some constant (a4)(2) A set characterized by a null space criteria imposed on the reduced-ordermanipulator (called Type II singular set) SNull;dim[()],sqqq for some (a5)where is the Jacobian after reducing the order of the manipulator.(3) The set defined by a combination of all constant generalized coordinates (calledType III singular set) Sqqijnij , for ,= to ; (a6)Substituting these singular sets into the position vector defined by Eq. (a1) yieldsparametric singular geometric entities (curves or surfaces) defined by ()()()()()()iiiiuu,s(a7)Intersections between these singular surfaces may exist. Moreover, these curves partitiona singular surface into a number of regions called subsurfaces. The result is theidentification of all boundary surface patches that characterize the manipulator’sworkspace.