The Explicit Dynamic And Inertial Parameters Of PUMA 566

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The Explicit Dynamic Model and Inertial Parameters of the PUMA 566 A mtBrian Armstrong, Oussama Khatib, Joel BurdickStanford Artificial Intelligence LaboratoryStanford UniversityAbstractsize of the models generated by these programs varies widely; andthere is little consensus on the question of whether the explicitmodels can be made sufficiently compact to be used for control.Aldon and Likgeois [1984] present an algorithm for obtaining efficient dynamicmodels; but none-the-lessrecomend the use ofrecursive algorithms for real time control, claiming that the complete results are too complicated for real-time control of robots.To provide COSMOS, a dynamic model baaed manipulatorcontrol system, with an improved dynamic model, a PUMA 560arm waadiaaaaembled; the inertial propertiea of the individuallinks were meaaured; and an ezplicit model incorporating all ofthenon-zero meaaured parametera waa deriued. The ezplicit model ofthe PUMA arm has been obtained with a derivation procedurecomprised of aeveral heuristic ruleafor simplification. A aimplijied model, abbreviated from the full ezplicit model with a 1%aignijicance criterion, can be evaluated with 305 calculationa, onefifth the numberrequiredby the recuraive Newton-Euler method.Theprocedure used to derive the model i a laid out; the meaauredinertial parametera are preaented, and the model ia included in anappendiz.As we show, explicit dynamic models of manipulators thatare more computationally efficient than the alternative recursivealgorithms can be obtained. The computational cost of the RNEalgorithm, the full explicit PUMA model, and the explicit PWMAmodel abbreviated with a 1%significance criterion are presentedin Table 1. The method presented here for factoring the dynamicequations has yielded a dynamic model of the PUMA 560 armTable 1. CalculationsRequired to Compute theForces of Motion by 3 Methods.1. I n t r o d u c t i o nThe Implementation of dynamic control systems for manipulators has been hampered because the models aredifficult toderive and computationally expensive, and because the neededparameters of the manipulator are generally unavailable. Recursive methods for computing the dynamic forces have been available for several years [Luh,Walker and Paul 1980a; Hollerbach19801. Several authors have proposedand simulated the use ofRNE in control systems [Luh, Walker and Paul 1980b; Kim andShin 19851; and [Valavanis, Leahy and Sardsi 19851 have used theRNE to control a PUMA - 600 arm. The RNE algorithm has alsofound use in the computation of forward dynamics for simulation[Walker and Orin 1982; Koozekananiet al. 19831, andnominaltrajectory control [VukobratoviE and Kirfanski 19841. The RNEmeets the need for calculation of dynamic forces in these applications, but does not offer several advantages available provided byan explicit model. The explicit modelallows of the calculationdecomposition based on a significance criterion or other criteria,and provides a more direct solution for dynamic simulation.is theThetremendous size of an explicitdynamicmodelgreatest barrier to its realization.Correspondingly,aconsiderable portion of the effort spent investigating dynamic models forcontrol has been directed toward efficient formulation and automatic generation of the manipulator equations of motion. Programs for automatic generation of manipulator dynamics are reported in [Likgeois et al. 1976; Megahed and Renaud 1982; Cesareo, F. Nicolb and S. Nicosia 1984 ; Murray and Neuman 1984;Renaud 1984; Aldon and Likgeois1984; Aldon et al. 19851. TheMethodRecursive Newton-Euler1560Evaluation of the FullExplicit PUMA Model1165Evaluation of t,he AbbreviatedExplicit PUMA Model305that requires 1165 calculations (739 multiplications and 426 additions), 25% fewer than the 1560 calculations required by the 6dof RNE. With the application of a 1% sensitivity criterion, theexplicit model can be evaluated withone fifth the count of calculations required by t.he recursive algorithm. Furthermore, thisformulation of the explicit model is not optimally compact; factorizations that were discovered and employed during the modelderivation have been expanded out to present explicit expressionsfor each component of the dynamic model. Renaud and Burdickboth report automatic generation of 6 dof manipulator modelsthat are more compact than that presented here [Renaud 1984;Burdick 19851. Theirmodelsincorporatenestedfactorizations,which were not used here.The count of 1165 calculations for the full PUMA model isthe total required to evaluate the model presented in the appendixand equation (1) below. This total and other totals presented donot include the calculations required to evaluate the sines andcosines.t Financial support for the first author has been provided byHewlettPackardCo.,throughtheir FacultyDevelopmentProgram. Partial support was provided by NSF under contract MEA80-19628, and by DARPA through the Intelligent Task AutomationProject,managedby'eheAirForce MaterialsLaboratory,under a subcontract to HoneywelI, contract F 33615-82-C-5092.2. Derivation of the D y n a m i c ModelThe dynamic model used for this analysis follows from [Liegeois et d. 19761 . It is:510CH2282-2/86/0000/0510)T;o1.000 1986 E E ECalculations

a l l 11where A(q) isB(q) isC(q) isg(q) isqisristhe n X n kineticenergymatrix;the n x n(n-1)/2 matrix of Coriolis torques;the n x n matrix of centrifugal torques;then-vector of gravitytorques;then-vector of accelerations;the generalizedjoint force vector.The symbols [qq] and [q”] are notation for the n(-l)/Z-vector ofvelocity products and the n-vector of squared velocities.and[ q 2 ] are given by:[aq]The procedure used to derive the dynamicmodel entails foursteps:1. Symbolic Generation of the kinetic energy matrix andgravity vector elements by performing the summations ofeither Lagrange’s or the Gibbs-Alembert formulation.2. Simplification of the kinetic energy matrix elements bycombining inertia constants that multiply commonvariable expressions.3.Expression of the Coriolis and centrifugal matrix elementsin terms of partial derivatives of kinetic energy matrixelements; and reduction of these expressions with fourrelations that hold on these partial derivatives.4. Formation of the needed partial derivatives, expansion ofthe Coriolis and centrifugal matrix elements in termsof the derivatives, and simplification by combininginertia constants as in 2.The first step was carried out with a LISP program, namedEMDEG, which symbolically generates the dynamic model of anarticulatedmechanism.EMDEGemploysKane’s dynamic formulation [Kane 19681, and produced a result comparable in formand size to that of ARM [Murry and Neuman 1984). Three sirnplifying assumptions were made for this analysis: the rigid bodyassumption;link 6 hasbeenassumed to besymmetric, that isI,% Zyy; and only the mass moments of inertia are considered,that is I,,, Zyy and Zzz. The original output of EMDEG, including Coriolis and centrifugal terms, required 15,000 multiplicationsand 3,500 additions. This step might also have been performedwith the momentum theorem method used in [lzaguirre and Paul19851.In the second step of this procedure, the kinetic energy matrix elements are simplified by combining inertia constants thatmultiply common variable expressions. This is the greatest sourceof computational efficiency. Looking to the dynamic model of a 3dof manipulator presented in [Murry and Neuman 19841,we seethat the kinetic energy matrix element a11 is given by: a11 J 3 2 2 c o s 2 ( & 83) J a Y y sin2(82 83) JzZr &m3 2 kf3za” cos(82)cos(82 d 3 ) uzm3 cos2(e2) 2 Mzza3 cos”(82 03) a m3 c0s2(82 83)(2) 2 a2a3m3 eos(Bz) oos(82 6‘3)J p Y y sin”(62) Jz , cos2(82) 2 dzdsms 2 M z 2 a 2 cos2(&) a;mz cos2(&) d i m s dZm2 J2zz Jizz Jizz Calculations required: 37 multiplications, 18 additions.By combining inertial constants with common variable terms andexpanding sin2(82) into (1- c o s 2 ( & ) ) , equation (2) can be reduced to: 12 coa2(82) 13 cos(82)co8(82 63) z4 cos2(e2 03)(3)Calculations required: 3 multiplications, 3 additions.where ZI. d i m 3 J2 dZm3 2 d2d3m3 dgm2 J3yy J3 Jzyv Jlzr Jizz;etc.Creating Z1 through 1 4 , which are constants of the mechanism, leads to a reduction from 35 to 3 multiplications and from18 to 3 additions. Computing the constant Z1 involves 18 calculations. Since the simple parameters required for the calculationof 11 are the input to the RNE, theRNE will effectively carry outthe calculation of Z1 on every pass, producing considerable mnecessary computation. Thirty four lumped constants are neededby the full PUMA model, 8 fewer than the count of 42 simple parameters required to describe the arm.In the third step theelements of the Coriolis matrix, Qij,and of the centrifugal matrix, ci,, arewritten in terms of theChristoffelsymbols of the first kind [Corbenand Stehle 1950;Likgeois et al. 1976]* giving:b *.J. -2‘Jwhere(qk- ,p’,jj-(4)(5)* it) is the j t h velocity product in the [q4]vector, andis the Christoffel symbol.The number of unique non-zero Christoffel symbols requiredby the PUMA model can be reduced from126 to 39 with fourequations that hold on the derivatives of the kinetic energy matrixelements.The first two equationsaregeneral;thelast two arespecific to the PUMA 560. The equations are:The reduction of Equation (7) arises from the symmetry ofthe kinetic energy matrix. Equation(8) obtains because the kinetic energy imparted by the velocity of a joint is independent ofthe configuration of the prior joints. Equation(9) results fromthe symmetry of the sixth and terminal link of the PUMA arm.a d third axes ofAndequation (10) holdsbecausethesecondthePUMAarmareparallel.Of the reductionfrom 126 to 39unique Christoffel symbols, 61 eliminations are obtained with thegeneral equations, 14 more with (9)and a further 12 with (10).Step four requires differentiating the mass matrix elementswith respect to the configuration variables. The means to ableforsome*The French authors seem to assume the use of Cristoffelsymbols, while the American authors seem unaware ofthem.CorbenandStehle,inthe1950 edition of theirtext, derive the results required here; but the derivationis largely omitted from their 1960 edition.

calculation of forward dynamics for simulation, where tesselationis the step size rather than servo interval and the cost is run timerather than bounded computing power.time [Liegeois et al. 1976; MIT Mathlab Group 19831. Only thederivatives required after the simplification of step 3 need to beformed. Of the 126 derivatives possible whenn 6, 46 are requiredby the model of the PUMA arm. After the neededderivativesareformedand expandrd into t,he Christoffelsymbols,inertialconstpats that multiplycommonvariableexpressionsare againcombined.Table 2. pUB/lA 560 Dynamic Model Evaluation RateAttainable with 1OOk FLOPS.Rate of Evaluationof ConfigurationDependent TermMethodOur method of model derivation is able to simplify to managable formthe complex sum-of-product expressionsthat are produced by synlbolically carrying outthe summations of Lagrange’sequations.Simplification is ingeneral a non-deterministictaskthat growsvery rapidly with the number of terms in an equation; but theprocedurepresented is deterministic,with a costthat grows most rapidly as p2, where p is the number of sum-ofproduct expressions in the largest individual kinetic energy matzix element. Our procedure has the virtue of producing explicitexpressions or each component of t,he dynamic model: a resultthat is veryusefulfordesignanalysisand that allows straightforward simplification by application of a sensitivity criterion.Evaluation of the FullModel Each IterationEvaluation of the ConfigurationDependent T e r n once during everyfour Evaluations of the Velocityand Acceleration Dependent T e r nRate ofComputationof Torque78 ha78 hr50 hz200 haA h a 1 decomposition to be considered is that for multiprocessing,anissue likely to becomemore important. The recursiveformulations are well suitedtopipelinecomputation,butpoorlysuited to multiprocessorcomputation. For the recursivealgorithms, the number of calculations that can be performed bycooperating processors is small in relation to the volume of communication that is required. Using an explicit model the blocksof parallel computation can be made much larger, and the ratioof computation to n into configuration dependent and velocity or acceleration dependent components is particularly suitable for multiprocessing and has been implemented at the Stanford ArtificialIntelligence Laboratory [Khatib 19851.Steps 2 t.hrough 4 of the above procedure were carried outby hand, requiring five weeks of rather tedious labor. To discovererrors, the explicit solution was numerically checked against theRNE algorithm, extended to give B and C matrix elements individually in a manner similar to that of Walker and Orin. Overa range of configurations, the explicit solution of the PUMA dynamics agrees exactly with the RNE calculation. It is instructiveto observe that the RNE algorithm was coded in 5 hours, 2% ofthe time required to develop the full explicit model.3. Several Advantages Obtained from Decomposition of4. The Utility of an Explicit Model for Dynamicthe Explicit ModelSimulationThe explicit solution of PUMA dynamics shows two structuralproperties that can be used toadvantage:atremendousrangebetween the largestand the smallestcontributingtermswithin most equations, and the depend solely upon configurationof the A , 13 and C matrix elements. Using the measured PUMAparameters an abbreviated dynamic model has been formed. Thismodelis derived from the full PUMA model by eliminating allterms that are less than 1% as great as the greatest term withinthe same equation, or less than 0.1% as great as the largest constant term applicable to the same joint. All of the elements of theA, 13 and G matrices are retained: the significance test is appliedon an equation by equation basis. The reduction in required calculations achieved via the significance test is roughly a factor offour, as shown in Table 1 above.Walker and Orin have demonstrated the use of the RNE atgorithm in the calculation of forward dynamics for simulation.By taking advantage of the symmetry of the kinetic energy matrix they have reduced the model order that must be consideredin successive applications of the RNE [Walker and Orin19821.TheRNEalgorithmhasalso beenused to computedynamicsforsimulationin fields outside of robotics[Benatietal. 1980;Koozekanani et al. 19831. Presented in table 3 are the numberof calculations required to compute the elements of the kineticenergymatrixusingWalker and Orin’s method, using the fullPUMA model, using the simplified model reported in [Izaguirreand Paul 19851, and using the abbreviated (1% significance criterion) model. The analytic modelsall show a tremendous advantage over the RNE algorithm.Observing that the A, B and C matrix elements depend onlyon configuration, it is possible to decompose the calculation intoconfiguration dependent and velocity or acceleration eslowly thanvelocity or acceleration, the configuration dependent componentsmay be computed at aslower rate [Khatib 1985; Izaguirre andPaul 19551. Shown in Table 2 is the evaluation rate of the PUMA560 dynamics that can be achieved with 100,000 floating pointoperations per second, the approximate speed of a PDP-11. Inthe first case the entire model is recomputed in each pass; in thesecond case the A , B and G matrix elements are computed onlyonce forevery four iterations of the multiplication by velocity andacceleration vectors. This partitioning of the dynamic calculationreduces the pace of computing the configuration dependent termsby one third; but increases the pace of computing thevelocity andaccelerationdependenttermsby a factor of two and one half.The advantage of this decomposition applies equally well to the512Table 3. Calculations .Re uired to determinetheKineticEnergy Matrix l%ements for a PUhlA 560 Arm.MethodWalker and OrinCalculations-2 737Full Explicit Model278Izaguirre and PaulSimplified Model58Abbreviated Explicit Model255. Measurement of the PUMA 560 Dynamic ParametersThe link parametersrequiredtocalculatetheelementsofA , B , C and g in equation (1) are mass, locationof the center

* r2I Mg-of gravity and the terms of the inertia dyadic. The wrist, linkthree and link two of a PUMA 560 arm were detached in orderto measure these parameters. The massof each component wasdetermined with a beam balance; the centerof gravity was locatedby balancing each link on a knife edge, once orthogonal to eachaxis; and the diagonal terms of the inertia dyadic were measuredwith a two wire suspension.w2where IMgrThe motor and drive mechanism at each joint contributes tothe inertia about that joint an amount equal to the inertia of therotating pieces magnified by the gear ratio squared. The drivesand reduction gears were not removed from the links, so the totalmotor and drive contribution ateach joint was determined by anidentification method. This contribution is considered separatelyfrom the I,, term of the link itself because the motor and driveinertia seen through the reduction gear does not contribute to theinertial forces at the other joints in the arm. The motorswereleft installed in links two and three when the inertia of these linkswere measured, so the effect of their mass as the supporting linksmove is correctly considered. The gyroscopic forces imparted bythe rotating motor armatures is neglected in the model, but thedata presented below include armature inertia andgear ratios, sothese forces can be determined.The parametersof the wrist linkswere not directly measured.The wrist itself was not disassembled. But the needed parameterswere estimated using measurements of the wrist mass and theexternal dimensions of the individual links. To obtain the inertialterms, the wrist links were modeled as thin shells.w1*1is theinertiaabouttheaxis of rotation;is the weight of the link;is thedistance fromeachsuspensionwire to the axis of rotation;is the oscillationfrequency inradians per second;is thelength of thesupporting wires.Measurement of the Motor and Drive InertiaA parameter identificationmethod was used t o tiaincludest,heeffective motor and drive inertia and the contribution due to themass of thearm. To makethismeasurementourcontrolsystem was configured to command a motor torque proportional todisplacement, effecting a torsional spring. By measuring the period of oscillation of the resultant mass-spring system, the totalrotational inertia about each joint was determined. By subtracting the arm contributions, determinedfrom direct measurements,from the measured total inertia, the motor and drive inertial contributions were found.Measurement ToleranceA tolerance for each direct measurement was established asthe measurement was taken.Thetolerancevaluesarederivedfrom the precision or smallest graduation of the measuring instrument used, or from the repeatability of the measurement itself. The tolerances are reported wherethe data are presented.The tolerance values assigned to calculated parameters were determined by RMS combination of the tolerance assigned to eachdirect measurement contributing to the calculation. The inertiadyadic and center of gravity parameters of link 3 were measuredwith the wrist attached; the values reported for link 3 alone havebeen obtained by subtracting the contribution of the wrist fromthe total of link 3 plus wrist. Tolerance values are reported withthe values for link 3 plus wrist, as these are the original measurements.Measurement of Rotational InertiaThe two wire suspension shown in Figure1 was used to measure the I,,, Iyyand I,, parameters of links two and three *.With this arrangement a rotational pendulumis created aboutanaxisparallelto and halfwaybetween the suspension wires.The link’s center of gravity must lie on this axis. The two wiresuspension method of measuring the rotational inertia requiresknowledge of parameters that are easily measured: the mass96. The Measured PUMA 660 ParametersThe mass of links 2 through 6 of the PUMA arm are reportedin Table 4;the mass of link 1 in not included becausethat link wasnot removed fromthe base. Separately measured mass and inertiaterms are not required for link one because that link rotates onlyabout its own 2 axis .“9Table 4. Link Masses (kilograms; kO.01 1%)Figure 1. The two wire suspension used for RotationalLinkInertia Measurement.LinkLinkLinkLinkLinkof the link, thelocation of the center of gravity, the distancefromthe wire attachment points to the axis of rotation, the length ofthe wires, and the period of rotationaloscillation. The inertiaabout eachaxis is measured by configuring the link to swingabout that axis. Rotational oscillation is started by twisting andreleasing the link. If one is carefulwhenreleasing the link, itis possible to start fundamental mode oscillation withoutvisibly exciting any of the other modes. The relationship betweenmeasured properties and rotational inertia is:*Mass234*17.404.800.820.340.095*6*Link 3 withCompleteDetached WristLVrist6.042.24* Values derived from external dimensions;f25%.The positions of the centers of gravity are reported in Table 5.The dimensions rz! ry and rz refer to the x, y and z coordinatesThis method was suggested by Prof. David Powell.513

Table 5 . Centers of Gravity.(meters 3 0.003)of the center of gravity in the coordinate frame attached t o thelink. Thecoordinateframesusedareassignedby a modifiedDenavit-Hartenbrrgmethod[Craig85j. In thisvariant of theDenavit-Hartenberg method, frame i is attached to link i, andaxis 2i lies along the axis of rotation of joint i. The coordinateframe attachments are shown inFigure 2; theyarelocatedasfollows:Link 1: Z axisalong the axis of rotation, Z up; Y10.006511 Z2.Link 2: Z axisalong the axis of rotation , Z awayfromthe base; X-Y plane in the center of the link, with X toward link 3.Link 3: 23 11 22; X-Y plane is in the center of l i d 3; Y isaway from the wrist.Link 4: The origin is at the intersection of the axes of joints4 5 and 6; 24 is along the axis of rotation anddirect.ed away from link 2; Y4 11 Z3 when joint4 is in the zero position.Link 3-0.0700.014Link 3With Wrist-0.1430.014Link 4*-0.019Link 5*Link 6*0.032Wrist0-0.064* Values derived from external dimensions;1&25%.The effective torsional spring method of inertia measurementwas applied at each joint. The motorand drive inertia, Imotor,were found by subtracting the inertial contribution dueto thearm dynamics, known from direct measurements, from the totalinert,ia measured. The uncertainty in the total inertia measurement is somewhat higher at joint one becauseof the larger frictionat that joint. It was necessary to add positive velocity feedback(damping factor -0.1) to cause joint one to oscillate for severalcycles.Link 5: The origin coincides withthat of frame 4; Z5 is directed away from the base; Y5 is directed towardlink 2 when joint 5 is in the zero position.that of frame 4; whenLink 6: Theorigincoincideswithjoints 5 and 6 are in the zero position frame 6 isaligned with frame 4.Wrist : The dimensions are reported in frame 4.Table 6. Diagonal Terms of the Inertia Dyadics andEffective Motor Inertia.* Iucrtia Diadic term derived from external dimenJions; *SO%.The gear ratios,maximummotortorque,and break awaytorque for each joint of the PUMA is reported in Table 7. Themaximum motor torque and break away torque values have beentaken from data collected during our motor calibration process.The current amplifiers of the Unimate controller are drivenby12 bit D/A converters, so the nominal torque resolution can be7btained by dividing the reported maximum joint torqueby 2048.Figure 2. The PUMA 560 in the Zero Position with AttachedCoordinate Frames Shown.Table 7 . Motor and Drive ParametemTheinertiadyadicandeffect,ive motorand driveinertiaterms are reported in Table6. For each link, the coordinate framefor theinertiadyadictermsis placed at the center of gravity,5. Thetolerancesparallel to the attached frameusedinTableassigned tothesemeasurementsareshowninparenthesis.Noto! ranceis assoriatrd with the value of I,, for link one becausethis value was not dlrectly measured; it was computed backwardsfrom themeasured totaljointinertia.It is not importanttodistinguish Izzl from t,he ml x ry1' term or from the motor anddrive inertia at joint one because these contributions are neitherconfiguration dependent nor appear in any term other than 011.The total !ink 1 inertia measured by the identification method isthe sum of Izzl andin table 6 .IGear RatioMLIaxirnumTorque(rim),1iJoint IJoint 2Joint 3Joint 4Joint 5Joint 20.121.36.35.52.61.31.01.2Break Away Torque(N-m)7. ConclusiomExplicit dynamic modelsof complex ulenipulaton are attainable. The PUMA 560 arm is as complex as any 6 dof arm with aspherical wrist, yet a deterministic simplification procedure hasproduced an explicitmodel that is moreeconomical thanthe514

RNE algorithm. With the application of a stringent significancecriterion, the computational cost of the explicit model is reducedto one fifth that of the recursive alternative. The availability ofmeasured dynamic parameters provides improved accuracy inthecalculated forcesof motion and simplifies model generationby allowing one to omit zero value parameters.As automatic modelgeneration becomes available and manufacturers become awareofthe need for dynamic parameters, we expect to see increasing useof explicit models and measured parameters in the calculation ofdynamics for control.ence, San Fransisco, June 221983, pp 491 - 496.M.J.Aldon and A.Lihgeois, “Computationalaspectsinroboted. A.dynamicsmodelling,” AdvancedSoftwareinRoboticsDanthine and M. Ghradis; Elsevier Science Publishers, NorthHolland, 1984, pp. 3 - 14.G. Cesareo, F. Nicolb and S. Nicosia, “Dymir:Acodeforgenerating dynamicmodel of robots, Proc. 1984 InternationalConference on Robotics, Atlanta, Georgia, March 13-15, 1984,pp. 115 - 120.ReferencesJ.J. MurrayandC.P.Neuman,“ARM:analgebraicrobotdy1984 ce on Robotics, Atlanta, Georgia,March 13-15, 1984, pp.103 - 114.H.C. Corben and P. Stehle, Classical Mechanics; New York: Wiley, 1950.T.R.Kane,“Dynamics;” New York: Holt,Rinehartandston, Inc., 1968.- 24,Win-M. Renaud, “ A n efficient iterative analytic procedure for obtainRobotics Research,ing a robot manipulator dynamic model,”ed. M. Brady and R. Paul, MIT Press,Cambridge, 1984, pp.749 - 764.A.Li6gois.W.Khalil, J.M.Dumas,andM. Renaud,“Mathematical and computer models of interconnectedmechanicalsystems,n(Sept., 1976, Warsaw) Theory and Practice of Robot8and Manipulators; New York: Elsevier Scientific PublishingCO., 1977, pp. 5 17.M. VukobratoviC and M KirCanski, “A dynamic approach to ,”IEEETrans. on Systems, Man, and Cybernetics, vol. SMC-14, no 4,pp. 580 - 586! July/August 1984.M.J. Aldon, A. Liegeois,B. Tondu and P. Touron,“MIRE: Asoftware for computer-aided designof industria! robots,n ,Arizona, February 1985, pp. 1 6.-M. Benati, S. Gaglio, P. Morasso, V. Tagliasco and R. Zaccaria,“AnthropomorphicRobots,” BiologicalCybernetics, vol. 38,pp. 125 140, March 1980.-J.M. Hollerbach, “A recursive lagrangian formulation of manipulator dynamics and a comparative study of dynamics formulation complexity,” IEEE Trana. on Systems, Man and Cybernetica, vol. SMC-IO, no. 11, pp. 730 - 736, November 1980.-J . Burdick, Private Communication.J.J. Craig, “Introduction to Robotics:MechanicsandControl,“Reading, Massachusetts: Addison - Wesley, 1985.J.Y.S.Luh,M.W. Walker and R.P.C. Paul, “On-linecomputational scheme for mechanical manipulators,” ASME J . of Dynamic Systems, Measurement, and Control, vol. 102, pp. 69 76, June, 1980.A. Izagnirre and R.P. Paul,“Computation of theinertialandgravitational coefficientsof the dynamics equations for a robotmanipulator with a load,” Proc. 1985 International Conferenceon Robotics and Automation, St. Louis, March 25-28, 1985, pp.1024 * 1032.J.Y.S. Luh, M.W. Walker and R.P.C. Paul, “Resolved-acceleration control of mechanical manipulators,” IEEE Trans. on Au.tomatic Control, vol. AC-25, no. 3, pp 468 - 474, June 1980.J. Hollerbach,“Dynamics,”RobotMotioned. M. Brady,J . Hollerbach, T. Johnson, T. Lozano-PCrze and M. T. Mason, MIT Press, Cambridge, 1982, pp. 51 73.0. Khatib, “Real-time obstacle avoidance for manipulators andmobile robots, Proc. 1985 International Conference on Roboticsand Automation, St. Louis, March 25-28, 1985, pp. 500 - 505.-B.K. Kim and K.G. Shin, “Suboptimal control of industrial manipulators with a weighted minimum time-fuel criterion,”IEEETransactions on Automatic Control, vol. AC-30, No l.,pp. 1- 10, January 1985.1C.P. Valavanis, M.B. Leahy and G.N. Sardis, “Real-time evaluation of robotic control methods Proc. 1985 International Gonference on Robotics and Automation, St. Louis, March 25-28,1985, pp. 644 - 649.S. Megahed and M. Renaud, “Minimization of the computationtimenecessaryforthe dynamiccontrol of robotmanipulators,* Proc. of the 12th International Symposium on IndustrialParis,Robots / 6th Conference on Industrial Robot Technology,June 9 - 11, 1982, pp 469 478.-M.W. Walker and D.E. Orin, “Efficient dynamic computer simulation of robotic mechanisms,” ASME J. ofDynamic Systems,Mearurement, and Control, vol. 104, pp. 205 211, September,’1982.-0. Khatib, “Dynamiccontrol of manipulatorsinoperationalspace,” Proc. Sizth IFToMM Congress on Theory of Machinesand Mechanisms, New Delhi, December 15-20, 1983.APPENDIXThe f i l l Expressions for the Forces of Motion of

The Explicit Dynamic Model and Inertial Parameters of the PUMA 566 Am t Brian Armstrong, Oussama Khatib, Joel Burdick Stanford Artificial Intelligence Laboratory Stanford University Abstract To provide COSMOS, a dynamic model baaed manipulator control system, with an improved dynamic model, a PUMA 5

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