Workspace Optimization of Mircea Badescu 3-Legged UPU and ... · Mircea Badescu Caltech Postdoctoral Scholar, Mem. ASME e-mail: [email protected] Constantinos Mavroidis

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Mircea BadescuCaltech Postdoctoral Scholar, Mem. ASME

e-mail: [email protected]

Constantinos MavroidisAssociate Professor, Mem. ASME

e-mail: [email protected]

Robotics and Mechatronics Laboratory,Department of Mechanical and Industrial

Engineering,Northeastern University,

Boston, MA 02115

Workspace Optimization of3-Legged UPU and UPS ParallelPlatforms With Joint ConstraintsIn this paper the workspace optimization of 3-legged translational UPU and orientatioUPS parallel platforms under joint constraints is performed. The workspace ofplatforms is parametrized using several design parameters that span a large ranvalues. In this paper both the unconstrained and the constrained workspaces (i.e.,space with joint limits) are used. For the workspace of each design configurationperformance indices are calculated using a Monte Carlo method: a) the workspaceume; b) the average of the inverse of the condition number and c) a Global CondIndex which is a combination of the other indices. Plots of each performance indexfunction of the design parameters are generated and optimal values for these dparameters are determined. For the optimal design, it is shown that by introducinglimits, the global isotropy of the parallel platforms is improved at the cost of workspreduction. @DOI: 10.1115/1.1667922#

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1 IntroductionIn recent years, modular robots were increasingly propose

means to develop reconfigurable and self-repairable robotictems@1#. To perform impromptu custom tasks, increase the pload to weight ratio, and, in cases of emergency, self-repair, fuinter-planetary robots and manipulation systems need to incorate modularity and self-reconfiguration capabilities. Modularbots utilize many autonomous units, or modules, that can beconfigured into a vast number of designs. Ideally, the modulesbe homogeneous, small, and self-contained. The robot can chfrom one configuration to another by manual reassembly, oritself. Self-reconfiguring robots adapt to a new environmentfunction by changing shape. Modules must interact with oneother and cooperate in order to realize self-reconfiguration. Amodular robots can repair themselves by removing and replafailed modules. Since one self-reconfigurable modular robotprovide the functionality of many traditional mechanisms, thwill be especially suited for space and planetary exploratiwhere payload mass must be kept minimum. Because they pise self-reparability and virtually limitless functionality, futurself-reconfigurable modular robots are expected to be cheapemore useful than current robot mechanisms in space mission

In this project we investigate the use of 3-legged parallel pforms as joint modules of reconfigurable robots. Parallel platforare currently being used in many applications as multi-degrefreedom systems with high rigidity, high payload to weight rathigh precision and low inertia@2,3#. These properties are alsdesired characteristics for the joint modules of reconfigurablebots. Six-legged, six degrees-of-freedom~DOF! parallel platformshave been used as joint modules of reconfigurable robots in@4#.However, the high number of DOF and of actively controlljoints per module increases complexity and cost. In additionpurely 3 DOF translational or spherical motion would requiretivation of all six module legs which means increase in eneconsumption. Lately, a special type of 3-legged parallel platfor@6# has received a lot of attention because of its simple designits pure 3-DOF translational motion with constant orientatioThis platform could be combined with 3-legged, 3 DOF paraplatforms modules with purely spherical motion so that they fohybrid kinematic chains with decoupled translation and orien

Contributed by the Mechanisms and Robotics Committee for publication inJOURNAL OF MECHANICAL DESIGN. Manuscript received Sept. 2002; revised Ju2003. Associate Editor: M. Raghavan.

Copyright © 2Journal of Mechanical Design

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tion @5#. Figure 1 shows an example of such a hybrid systemthis example a 2-arm reconfigurable robotic system is formfrom a sequence of 3 DOF translational and orientational paraplatform modules.

The 3-DOF translational parallel platform that is studied in thpaper was first proposed by Tsai@6# and was later on generalizeby Di Gregorio and Parenti-Castelli@7,8#. It is made out of twoequilateral triangular plates connected at the corners with thidentical legs, as it is shown in Fig. 2. The legs are linked toplates with universal joints. Linear actuators control the lengththe legs forming prismatic joints. Due to the existence of univerand prismatic joints at each one of the legs, it is also called 3-Uplatform. By properly orienting the axes of universal joints, tmoving plate can have only translational motion. The direct ainverse kinematics of this platform have been studied in@6,7,9#.Its static and singularity analyses have been performed in@8,10#.

The workspace of parallel platforms in general has been stuthoroughly @3,11#. For the 3-UPU platform, the isotropy anworkspace analyses have been performed in@9#. In that work, noconstraints were assumed for the leg prismatic and univejoints, which means that full rotation of the universal joints afull extension of the prismatic joints, were considered. The wospace optimization was performed using one design paramwhich is the difference between the distances from the centethe corner of the triangular base and moving plates. UsinMonte Carlo method and varying the normalized lengths oflegs from 0 to 1 an optimum design is obtained at the value0.37 for the design parameter. However, from the practical poof view, universal joints have angular limitations that will reduthe workspace of the 3-UPU platform. Also, if these platformsused as modules for a reconfigurable robot, they must be stacone on the top of the other, severely limiting their prismatic atuator stroke. From a design standpoint, universal and prismjoint constraints should be included to produce a viable reconurable robot joint module. Moreover, the legs interference wasconsidered in the above mentioned work.

Different types of spherical or orientational parallel platformhave been proposed and analyzed. Merlet described severalof 2 and 3 DOF parallel platforms that can be used as wriorientation mechanisms@3#. Tsai classified the parallel platformwith rotational DOF in spherical and spatial orientation mechnisms @2,12#. All the moving points of a spherical parallel plaform move on concentric spheres. Analysis and fabrication ospherical 3 DOF platform were presented in@13#. Optimization of

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a spherical five bar parallel linkage was done in@14#. Gosselin andAngeles used three design criteria, symmetry, workspace voland isotropy, to optimize a 3 DOF spherical platform@15#.

The 3-DOF orientational platform that is studied in this papecomposed of two tetrahedrons with unequal equilateral triangbases~Fig. 3! and three identical legs. The moving tetrahedron ha smaller triangular base than the fixed tetrahedron. The tworahedrons are connected at their tips using a spherical jointdetermines the point around which the moving tetrahedron rotaThe corners of the two triangular bases are linked using thidentical legs that are connected to the moving platform with uversal joints and to the fixed platform with spherical joints. Lineactuators control the length of the legs forming a prismatic joEven though the points of the moving tetrahedron move osphere, due to the dissimilar motion of the legs the platform cnot be called spherical@2#. The kinematics and the singularitanalysis of this platform were reported in@16,2#. To the author’sknowledge, no workspace analysis, design optimization, or iropy analysis have been performed yet for this 3-UPS platfor

In order to facilitate the development and motion planningself-reconfigurable robots with parallel platforms as modules,robot modules need to have similar motion and force capabili

Fig. 1 Two-arm reconfigurable robot with 3-DOF parallel plat-form modules

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at any one of their two ends~fixed and moving!. This means thatthe platform modules should have similar type of motions and beable to apply the same forces regardless of which end is used asthe fixed base or the moving component. Of the various types ofspherical/orientational platforms that have been proposed so far,the 3-UPS parallel platform has this motion and force equivalenceof its two ends. Therefore, this platform has been selected forfurther study in this paper.

In this paper the workspace optimization of translational 3-UPUand orientational 3-UPS parallel platforms with prismatic and uni-versal joint constraints, and legs interference is performed. For the3-UPU translational platform the workspace is parameterized us-ing two design parameters, which are the prismatic joint strokeand the difference between the distances of the center to the cor-ners of the triangular base and moving plates. For a large range ofvalues for these design parameters the workspace of the corre-sponding 3-UPU platforms is calculated. This workspace is calledin this paper the constrained workspace to distinguish it from the

Fig. 2 The 3-UPU parallel platform

Fig. 3 The 3-UPS parallel platform

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unconstrained workspace where no joint constraints are conered. For the constrained workspace of each design three pemance indices are calculated: a! the workspace volume; b! theaverage of the inverse of the condition number and c! a GlobalCondition Index which is a combination of the other two perfomance indices. Plots of each performance index as a functiothe two design parameters are generated and optimal solutionthese design parameters are identified. Finally, for the optidesign, it is shown that by introducing limits to the angles of tuniversal joints axes, the global isotropy of the parallel platformimproved. In this paper, global isotropy of a parallel platformsimply the isotropy of a parallel platform is called the averagethe condition number of the Jacobian over the workspace.

For the 3-UPS orientational platform the workspace is paraeterized using four design parameters, which are the prismjoint stroke, the difference between the distances of the centeone of the corners of the triangular plate of the base and ofmoving platform, the height of the platform in zero orientatioand the ratio of the heights of the two tetrahedrons. In a simway as with the 3-UPU platform both the constrained and uncstrained workspaces are calculated and the same three pemance indices are calculated. For determining the optimal dea Genetic Algorithm was implemented. The Global Conditiondex was chosen as the objective function for optimization. Toptimization was performed with and without imposing joint costraints. Plots of the 3-D workspace of the optimal designsgenerated. Finally, for the optimal design, it is shown thatintroducing limits to the angles of the universal and spherijoints, the isotropy of the parallel platform is improved. Moreovthe leg interference is taken into account and it is shown thatoptimal design parameters are changed.

2 Mathematical ToolsIn this section we present the mathematical/kinematic tools

were used to formulate the workspace optimization probleThese tools include the direct and inverse kinematics, the pforms’ Jacobian matrices, their condition number and the GloCondition Index. For the direct and inverse kinematics and forcalculation of the Jacobian matrices we used the solution metology proposed in@2,6#. For the Global Condition Index we usethe same definition as in@9#.

2.1 3-UPU Translational Platform. The 3-UPU parallelplatform consists of a base plate, a moving plate and three idtical limbs ~see Fig. 2!. The plates have equilateral triangulshape of different sizes. The limbs are connected to the plates2-DOF universal joints. A linear actuator controls the leg lengand forms a prismatic joint. Each universal joint is treated asrevolute joints with axes perpendicular to each other and intersing at a point. In Fig. 4,ui1 , ui2 , ui4 , andui5 are the unit vectorsalong the axes of the universal joints andui3 is the unit vectoralong the prismatic joint axis for legi. Tsai showed that by havingui1 parallel toui5 and ui2 parallel toui4 for each leg, then themoving plate would achieve pure translation in three dimensi@6#.

As seen in Fig. 4, the position vectors of pointsAi andBi withrespect to framesA andB respectively, can be written as:

Aai5@aix ,aiy,0#T, and Bbi5@bix ,biy,0#T (1)

FramesA and B are defined at the base and moving triangrespectively. Their origin is placed at the corresponding triangcenter. Theirz-axis is perpendicular to each triangle’s plane atheir x andy-axes are in the triangle’s plane. Parameterqi denotesthe length of legi. Subscripti denotes one of the legs and can tathe values 1, 2 or 3. Vectorci is defined as the difference ofai andbi . Letters in bold represent three-dimensional vectors. Suscript A or B on the left of a vector denotes the reference frawhere its coordinates are calculated. SuperscriptT on the right ofa vector denotes the transpose of a vector. Since vectorsai andbi

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are constants, vectorsci are a constant too. Vectore determinesthe position of the center of the moving plate in the base coonate system:

Ae5@ex ,ey ,ez#T (2)

For the direct kinematics the leg lengthsqi and the triangularplates’ geometry are known and the 3 coordinates of vectore aredetermined from the following 3 scalar equations~the dot ‘‘"’’ inthis equation represents the dot product of two vectors!:

e222e"ci1ci25qi

2, for i 51,2,3. (3)

For the inverse kinematics vectore and the triangular plates’ geometry are known and the limb lengths,qi are calculated from thefollowing 3 equations:

qi56A~ex2cix!21~ey2ciy!21ez2, (4)

The manipulator Jacobian matrixJ relates the end-effector velocities vectorx to the actuated joint velocities vectorq:

q5Jx (5)

For the 3-UPU platformJ is a 333 matrix and is given by Eq.~6!. ~Note: in this paper we did not consider the rotation Jacoband so the workspace may not be free of rotation singularitThese constraints are considered in the following referen@10,17,18#!.

J5@u1,3T u2,3

T u3,3T #T (6)

The condition number of the Jacobian matrix is defined by E~7!:

k5iJi"iJ21i (7)

where the dot ‘‘"’’ in this equation represents the multiplication otwo scalars, andi•i denotes the norm 2 of a matrix@19#.

Robot manipulators and their configurations wherek is equal to1 are called isotropic. In these configurations the system is abldevelop same amount of forces and velocities in all end-effecdirections. For high values ofk, there are end-effector directionwhere the manipulator can develop much higher forces or veloties than in other directions. In many applications, this is nodesirable system property because the system looses its homneity in force and velocity development. Configurations in whik has an infinite value are singular configurations. In these cfigurations there are directions in space where the end-effectoreither not move or not apply forces. Isotropy is a very local mnipulator property because it changes from configuration to cfiguration. A manipulator can have a good isotropy in one cofiguration and a bad one in another. In many robot applicatio

Fig. 4 Joint axes in the 3-UPU platform

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isotropy is an important property, and needs to be taken intocount during the robot’s design phase. Defining a global isotrindex that will be able to characterize the system isotropy inwhole workspace is an important but difficult problem to solvThe problem is that a robot system will always have singularbad isotropy configurations. By taking a type of an average ofcondition number is only a rough indication of the quality of tsystem global isotropy and says nothing about the magnitudenumber of bad isotropy configurations. Nevertheless, in this wwe will use global isotropy indices based on the average ofcondition number, as they are being used in the current literat

In this work, three performance indices will be used to charterize a robotic system’s workspace as it is described in detaSec. 3:

a! The workspace volume. Obviously, using this performanindex as the objective function, optimal designs correspondmaximum workspace volume.

b! The average of the inverse of the condition number. Tcondition number characterizes the system’s isotropy. The aveof the inverse of the condition number is calculated by summthe inverse of the condition number in every point in the wospace and then dividing it with the number of points considerethe workspace. Optimal designs correspond to values of the aage of the inverse of the condition number close to 1.

c! The Global Condition Index as it is used in@9#. An initialdefinition of a Global Performance Index was proposed in@20#.This index, defined in Eq.~8!, is the ratio of the integral of theinverse condition numbers calculated in the whole workspacevided by the volume of the workspace:

hGA5A

B, where A5E

WS 1

kDdW, and B5EW

dW, (8)

in which k is the condition number at a particular point in thworkspaceW. B is the total volume of the workspace. The GlobCondition Index is bounded as:

0,hGA,1 (9)

Isotropic systems correspond to value ofhGA equal to 1 and sys-tems with bad isotropy correspond to values ofhGA approachingzero.

The Global Condition Index used in this paper is given in E~10! and consists only of the numerator of the index defined in~8!. In reality it is calculated as the product of the average ofinverse condition number with the workspace volume. Therefoit is like a combination of the first two performance indices.

h5EWS 1

kDdW (10)

whereW andk have the same meaning as in Equation~8!.

2.2 3-UPS Orientational Platform. The 3-UPS parallelplatform consists of a fixed tetrahedron, a moving tetrahedronthree identical limbs~see Fig. 3!. The tetrahedrons have equilaeral triangular bases of different sizes. The tips of the tetrahedare connected using a spherical joint. The limbs are connectethe moving tetrahedron base with 2-DOF universal joints andthe fixed tetrahedron base with spherical joints. A linear actuacontrols the leg length, and forms a prismatic joint. Each univejoint is treated as two revolute joints with axes perpendiculareach other and intersecting at a point. In Fig. 5,ui4 is the unitvector along the prismatic joint axis for legi, ui5 andui6 are theunit vectors along the axes of the universal joint of legi. ui7 is theCBi edge of the moving tetrahedron.

A coordinate system is chosen for each tetrahedron as showFig. 6, a! and b!. FramesA and B are defined for the base anmoving tetrahedrons respectively. Their origin is placed atcommon tip of the tetrahedrons and represents the point arowhich the moving tetrahedron rotates. Theirz-axis is perpendicu-

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lar to each base triangle’s plane, and theirx and y-axes are inplanes parallel to the base triangle’s plane. Thexa/b axes are ori-ented parallel to theOA1 andEB1 lines, respectively. The frameA, Cxayaza , is rigidly connected to the fixed tetrahedron and h

Fig. 5 Joint axes in the 3-UPS platform

Fig. 6 Coordinate systems of the 3-UPS platform

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centerC at the tetrahedron’s tip. The pointO is the center of thebase triangle. Theza axis is oriented alongOC. The xa axis isparallel toOA1 and theya axis is chosen to obtain a right hanrectangular coordinate system. The frameB, Cxbybzb , is rigidlyconnected to the moving tetrahedron and has the same centthe fixed frame,A. The pointE is the center of the base triangle othe moving tetrahedron. Thezb axis is oriented along the vectoEC. The xb axis is parallel to theEB1 and yb axis is chosen toobtain a right hand rectangular coordinate system.

As seen in Fig. 6, the position vectors of pointsAi andBi withrespect to framesA andB respectively, can be written as:

Aai5@aix ,aiy ,2ha#T, and Bbi5@bix ,biy ,2hb#T (11)

The scalaraixy ~shown in Fig. 6! denotes the distance between tcenter,O, of the base triangle of the fixed tetrahedron and anyof the corners of the base triangle. The scalarbixy denotes thedistance between the center,E, of the base triangle of the movintetrahedron and any of the corners of the base triangle. Therameterci denotes the difference betweenaixy andbixy .

Parameterdi denotes the length of legi. The vectordi repre-sents the position vector of the legi (AiBi) in the base coordinatesystem:

Adi5@di ,xa ,di ,ya ,di ,za#T (12)

The scalarha denotes the height of the fixed tetrahedron and isdistance from pointO to pointC. The scalarhb denotes the heighof the moving tetrahedron and is the distance from pointE topoint C. The parameterh represents the total height of the paralplatform in ~0,0,0! orientation and is equal to the sum ofha andhb . The parameterhab is the ratio betweenha andhb .

The transformation from the moving frameB to the fixed frameA can be described by a 333 rotation matrixARB defined by az-x-z~w-u-c! Euler rotation. Given that in the initial position~0-0-0 rotation! the xa andxb axes coincide,za andzb , andya andyb are in opposite directions respectively, the resulting rotatmatrix is given by:

ARB5F cfcc2sfcusc cfsc1sfcucc 2sfsu

sfcc1cfcusc sfsc2cfcucc cfsu

susc 2succ 2cuG

(13)

where c represents the cosine ands the sine of the followingangle.

For the direct kinematics the leg lengthsdi and the triangularbases’ geometry are known. The three orientation angles ofmoving platform are determined from expanding the followithree scalar equations~see@2# for more details!:

di25ai

21bi222ai

Tbi , for i 51,2,3 (14)

For the inverse kinematics the orientation of the moving platfois known and the limb lengths,di are calculated from the follow-ing three equations:

di56Aai21bi

222aiTbi , for i 51,2,3 (15)

The manipulator Jacobian matrixJ relates the end-effector velocties vectorv to the actuated joint velocities vectord:

Jxv1Jdd, J5Jd21Jx (16)

For the 3-UPS platformJx andJd are given by:

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Jx5F ~u1,73u1,4!T

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The condition number of the Jacobian matrix is defined by E~18!

k5iJi"iJ21i (18)

wherei•i denotes the norm 2 of a matrix@19#.The same three performance indices as for the 3-UPU platf

are used to characterize the platform’s workspace.

3 Optimization Algorithms

3.1 3-UPU Translational Platform. Based on the direct ki-nematics of the 3-UPU parallel platform, shown in Eq.~3!, theplatform’s workspace depends on two geometric parameters:magnitudeci of vectorci and the strokesi of the prismatic jointwhich is directly related to each leg’s lengthqi . Due to the factthat the 3-UPU platforms have equilateral triangular basemoving plates, and assuming that all legs are using the sametuator, thenci andsi have the same value for all legs.

The strokesi of the actuator is the maximum amount of travfor each leg’s prismatic actuator. It is expressed with a percenvalue of the minimum length of the limbqmin , which in turn istaken such thatqmax is equal to 1. The fact that the leg’s maximulength is considered to be equal to 1 means that all lengthsnormalized with respect to the leg’s maximum length as itspecified from each application’s requirements. In this projectrange of values of the stroke of the actuators is considered tbetween 20% and 87.5%. These values were selected based otechnical specifications of the majority of commercially availabprismatic actuators. The minimum value of each leg’s lengthqminis equal to 1/(11si /100). So in each platform’s configuration thvalue of the leg lengthqi should be betweenqmin and 1.

The range of values forci is between 0.27 and 0.645. Platformwith values forci close to zero~i.e., platforms where the triangular base and moving plates are almost equal! present extra DOF~self-motions! that could not be controlled from the actuation mtion and such designs are not acceptable. Platforms with vlarge values forci have very small workspace.

The algorithm to calculate the platform’s workspace as a fution of the design parameters consists of the following steps:

a! The algorithm selects the values ofci and si from theiracceptable ranges of values. Forsi , 27 values in equal incrementare selected between 20% and 87.5%. Forci , 15 values in equalincrements are selected between 0.27 and 0.645.

b! A three-dimensional, rectangular shaped enclosure bodetermined that includes all points that can be reached bycenter of the moving plate. Usually, this box is selected larger tthe expected platform’s workspace to be sure that all points inplatform’s workspace are included in the enclosure box. For eset of values ofci and si a Monte Carlo method is used to randomly select 1,000,000 points inside the determined enclosbox.

c! For each selected point, the platform’s inverse kinematicsolved using Eq.~4!. In order that the selected enclosure point liin the platform’s unconstrained workspace, the calculated lenqi should be a real number and lie in the range from 0 to 1. Ifaddition the determined limb lengthqi is within the acceptablerange of motion of the actuator, i.e., betweenqmin and 1, then theselected enclosure point is within the actuator-constrained wspace.

d! Using Eq.~7!, the condition numberk is calculated at eachenclosure point found to be within the platform’s unconstrainworkspace.

e! Once the condition number of all points in the unconstrainworkspace has been calculated then the three performance inare calculated. The workspace volume is determined by multiping the volume of the enclosure box with the total number

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points inside the workspace and dividing it by the total numbepoints generated. The average of the inverse of the condition nber is calculated by dividing the sum of the reciprocal of tcondition numbers with the total number of points inside tworkspace. The Global Condition Index is determined from E~10!. Is calculated as the product of the sum of the reciprocathe condition numbers with the volume of the unconstrainworkspace and divided by the total number of points insideworkspace.

f! For each point within the constrained workspace the angof the revolute joints~each universal joint is treated as two revlute joints! at the base are determined using inverse kinemequations. If all of them fall between the allowable angular limthen the point is within the joint-constrained workspace. Thegular limits of the revolute joints were chosen such that the cdition number of the platform remains below 10 at every nonsgular configuration.

The unconstrained workspace and the calculation of its glocondition number verified the results of@6#. However, the emphasis of this work for the 3-UPU platform is the identification of thproperties of the actuator-constrained workspace and of the joconstrained workspace and our results are presented in Sec.

3.2 3-UPS Orientational Platform. Based on the plat-form’s direct kinematics, and due to the fact that all legs are eqand that the base and moving triangles are equilateral, the 3-orientational platform workspace depends on four geometricrameters: a! the differenceci betweenaixy and bixy : ci5aixy2bixy ; b! the total height of the platform considered as the sof the heights of the two tetrahedra:h5ha1hb ~see Fig. 5!; c! theratio of the heights of the two tetrahedra:hab5ha /hb ; and d! thestroke of the prismatic jointsi .

The range of values forci andsi parameters has been selectin the same way as with the 3-UPU platform. For the other tdesign parameters we considered the range of possible valufollows. The total height of the two tetrahedrons,h, has valuesbetween 0.2 and 1.0. Below 0.2 the workspace is very smallhas no practical use. Total height equal to 1 corresponds to mmum length. The ratio of the heights of the two tetrahedrons,hab ,was selected to vary between 0.33 and 3.0.

Due to the fact that the plots of all three performance indifor any value of the design parameters are highly non-monotoa Genetic Algorithm was implemented to determine the optimdesign configuration. It consists of the following steps:

a! An initial population of 100 members is randomly generateEach member contains values for the four design parametersfined above. Each parameter has an eight digit resolution wmeans that 256 different values can be chosen for each param

b! For each population member the platform’s geometricrameters are calculated using the design parameters.

c! To determine the values of the performance indices a MoCarlo method was implemented. For each configuration withtain values of the design parameters,si , ci , h, andhab , 1,728,000orientations of the moving tetrahedron are chosen randomly.w, u and c Euler angles corresponding to az-x-z rotation arerandomly chosen in the interval@2180 deg,1180 deg#.

d! For each orientation the lengths of the legs are determiusing the inverse kinematics, Eq.~15!. If all legs have lengths inthe allowed intervaldmin,d,dmax the interference of the legs withfixed and moving tetrahedrons is checked. Also, for the cstrained workspace, the joint angles are determined and verifithey are between imposed limits. Additionally, the interferenceone leg with another and the interference of each leg withcentral spherical joint are verified. If the orientation verifiesconstraints, then the Jacobian condition number,k, is determinedusing Eq.~18!.

e! For each configuration the performance indices are calated using Eq.~10!. The population member with the best objetive function is determined and saved in a file. It will be call

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‘‘the king’’ in further references. Any one of the described perfomance indices can be chosen as the objective function.

f! A mating pool is then created from the initial populatioselecting the members with better objective function values. Tfirst member of the mating pool is chosen as the member withbest value of the objective function~the king!.

g! A crossover among the members of the mating pool is dwith a 0.45 probability. At the end, the first member is replacwith the member with the best objective function~the king!.

h! A mutation with 0.05 probability is then done over the nepopulation. The first member of the population is not affected~theking!.

i! For the new population the cycle starts over. It stops whenvalue of the objective function of the king is not improved withcertain relative quantity.

The unconstrained optimization is done without imposingconstraints mentioned in step d! above.

4 Results

4.1 3-UPU Translational Platform. Figure 7 shows thevolume of the constrained workspace as a function of the actustroke,si(%), and thedesign parameter,ci . Dark red points in-dicate a large value of the volume while dark blue indicate a smvalue. From the plot it can be seen that the volume of the cstrained workspace increases as the stroke increases and avalue of theci parameter decreases.

Figure 8 shows the average value of the inverse of the condinumber as a function ofsi(%) and of ci . It can be seen thaisotropy is affected a lot byci . Isotropic designs correspond tplatforms with large values ofci . In these platforms the movingplate is much smaller than the base. Such platforms do not preself-motions~i.e., extra uncontrollable DOF! as those platformswith very smallci do. Hence, these designs are away from sinlar designs and hence they present good isotropic behavior. Gerally, the stroke does not affect isotropy as seen from FigHowever, for large values ofci , where in general isotropy is verygood, the best designs correspond to strokes between 30%40%.

Figure 9 shows the Global Condition Index as a function ofdesign parameters. Overall, the Global Condition Index increaas stroke increases. So if good isotropy, within the whole wospace, is desired combined with large workspace volume, tactuators with high strokes should be used. For these large stvalues the Global Condition Index shows a maximum for valuof the design parameterci around 0.55 and around 0.27. Thecan be considered as optimal design values forci if the GlobalCondition Index is used as the design criterion.

Fig. 7 Workspace volume of the 3-UPU platform

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The role of angular constraints for universal joints is to includthe practical limits of the universal joints and to avoid points withigh condition number. Figures 10 and 11 present the workspaof the 3-UPU parallel platform forci equal to 0.27 and strokeequal to 85% without and with angular constraints, respectiveThese values of stroke and ofci are the optimal ones based on thGlobal Condition Index shown in Fig. 9. The color of the pointreflects the value of the condition number. The dark blue poinhave small condition numbers and they are generally foundward the center of the workspace. The dark red points have hcondition numbers and they are generally found toward the btom edges of the workspace. It can clearly be seen that by imping stricter constraints on the range of motion of the universjoints, the workspace isotropy is improved while the workspavolume is slightly decreased.

4.2 3-UPS Orientational Platform. We used a programwritten in C11 to perform the optimization and MATLAB™ toplot the workspace. We run the optimization program for the costrained and unconstrained workspace.

The optimal design for the unconstrained workspace hasfollowing parameter values: stroke583.9%, ci50.602, h50.285, andhab50.344. Figure 12 shows the workspace of thoptimal unconstrained configuration. The points represent thesitions that can be reached by the center,E, of the moving tetra-

Fig. 8 Average of the inverse of the condition number of the3-UPU platform

Fig. 9 Global condition index of the 3-UPU platform

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hedron base.E describes a sphere with the center at pointC and aradiushb . To represent the twist angle,c, the radius of the spheredescribed by pointE was changed depending on the value of ttwist angle. Forc52180 deg the radius was reduced by 10%hb and for c51180 deg it was increased by 10%. A linear dpendence was used for the intermediary values. The color wwhich the points were represented reflects the value of the cotion number of the platform’s Jacobian at the specific point.value 1 of the condition number was represented with dark bcolor. Because some of the condition numbers had extremely lavalues~more than 100!, in our representation in MATLAB™ allcondition numbers with a value over 10 were reduced to 10~Thiswas used just for representation but not for optimization!. Darkred color is used to plot points with large condition numbers.

For the optimization of the constrained workspace, we imposlimits at the universal and spherical joints of the platform. Tuniversal joints were considered as two perpendicular revojoints with one common point. In this paper we present resufrom two different schemes of constrained workspaces with awithout considering the legs interference.

For the first set of constraints we set the limits of the univerjoint angles at675 deg for the revolute joint that allows the le

Fig. 10 Workspace without angular constraints „c iÄ0.27 ands iÄ85%… of the 3-UPU platform

Fig. 11 Workspace with angular constraints „c iÄ0.27 and s iÄ85%… of the 3-UPU platform

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Fig. 12 Workspace of the optimal unconstrained configurationof the 3-UPS platform

Fig. 13 Workspace of the optimal constrained configuration ofthe 3-UPS platform

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rotation on CEBi planes and685 deg for the other revolute jointThe limits for the spherical joint angles were set the same665 deg. The optimal design parameters that were obtainedstroke571%, ci50.62, h50.44 andhab51.75. In this case theaverage value of the inverse of the condition number,k, is in-creased and the workspace volume is decreased relative to

Fig. 14 Top view of the unconstrained and constrained work-spaces for fixed twist of 0.0 deg of the 3-UPS platform

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unconstrained optimal configuration. The workspace for the omal 3-UPS platform with the first set of constraints is shownFig. 13~a!.

For the second scheme we considered also constraints dpossible interference of one leg with another and possible inference of each leg with the central spherical joint. We usedsame angular limits for the universal and spherical joints. Toptimal design parameters that were obtained are: stroke570%,ci50.61,h50.48, andhab52.04. The workspace for the optima3-UPS platform with the second set of constraints is shown in F13~b!.

For the constrained workspaces we used the same represtion method as for the unconstrained workspace. From theprovided by the optimization programs, it can be seen thatconstrained workspace volume is less than half the unconstraworkspace volume. By imposing constraints on the joint angone can exclude not only many bad isotropy points, but mgood isotropy points as well. Overall, the average value ofinverse of the condition number is increased, which meansthe isotropy of the platform is improved, and the workspace vume is reduced when joint constraints are imposed. It can alsseen that the joint constraints did not affect significantly the omal value for the design parameterci . This means that for a bettedesign, the difference in size between the bases of the tetraheneeds to be very large, independently of the joint constraintsposed.

Figures 14 and 15 present the top view of the unconstrainedconstrained workspaces for 0 deg and 30 deg twist angles, restively. Here the effect of the workspace reduction due to thegular constraints is more obvious. Even though the workspashown in Fig. 12–15 are continuous, not all of the points canreached with any orientation. This is shown in Fig. 14~b!, ~c! and15~b!, ~c!. Also it can be seen that for small values of the pitangle,w, the manipulator has a bad isotropy, no matter how smthe yaw angle,u, or the twist angle,c, are.

From Figs. 13, 14, and 15 we learn that points on the wospace can be reached only for certain twist angles. Also we lthat even though some points can be reached for a large rantwist angles for some values of these we get a better isotrThese results can be used to determine a path between two otations maintaining a good isotropy if the twist angle in the intmediate positions is not critical.

5 ConclusionsIn this paper the workspace optimization of two types

3-legged parallel platforms is performed. The platforms are3-UPU with only translational DOF and the 3-UPS with onrotational DOF. Both platforms are considered to be used as jmodules in reconfigurable robots and the quality of their cstrained workspace is an important design feature.

For the 3-UPU platform the workspace is parameterized ustwo design parameters, which are the prismatic joint strokethe difference between the distances of the center to one ocorners of the triangular plate of the base and of the movplatform. Three workspace performance indices are calculatea function of the design parameters and optimal solutions for thdesign parameters are determined. These performance inquantify the workspace volume, the system isotropy and a cbination of the two properties. It is shown that for different peformance indices different parameters correspond to the bessign. The limits that the actual actuators and joints impodetermine limitations in the workspace of parallel platforms bthey can be used in the same time to exclude points withisotropy from the platform’s workspace. For the 3-UPS platfothe workspace is parameterized using four design parameterthe same way as with the 3-UPU, three workspace performaindices are calculated as a function of the design parameters.optimal design is determined for the unconstrained and cstrained situations. For the constrained workspace both situa

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AcknowledgmentsThis work was supported by a National Science Foundat

CAREER Award ~DMI-9984051!. Additional support was pro-vided by a Rutgers University Research Council Grant andNASA’s Jet Propulsion Laboratory.

Fig. 15 Top view of the unconstrained and constrained work-spaces for fixed twist of 30 deg of the 3-UPS platform

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