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The method puma alife presented here for factoring the dynamicappendiz. equations has yielded a dynamic model of the PUMA 560 arm1. I n t r o d u c t i o n Table 1. CalculationsRequired to Compute the Forces of Motion by 3 Methods.The Implementation of dynamic control systems for manip- Method Calculationsulatorshasbeenhamperedbecausethemodelsare difficult to Recursive Newton-Eulerderive andcomputationally expensive, and because the needed Evaluation of the Full 1560 Explicit PUMA Model 1165parameters of the manipulator are generally unavailable. Recur- 305 Evaluation of t,he Abbreviatedsive methods for computing the dynamic forces have been avail- Explicit PUMA Modelableforseveralyears[Luh, Walker and Paul 1980a;Hollerbach19801.

able portion of the effort spent investigating dynamic models for 2. Derivation of the D y n a m i c Modelcontrol has been directed toward efficient formulation puma athletic shoes and auto- The dynamic model used for this analysis follows from [Lie-maticgeneration of themanipulatorequations of motion. Pro- geois et d. 19761 . It is:grams for automatic generation of manipulator dynamics are re-ported in [Likgeois et al. 1976; Megahed and Renaud 1982;Ce-sareo, F. puma basket black Nicolb and S. Nicosia 1984 ; Murray and Neuman 1984;Renaud 1984; Aldonand Likgeois1984; Aldonetal. 19851. Thet Financial support for the first author has been provided byHewlettPackardCo., throughtheir FacultyDevelopmentPro-gram.

Partial support was provided by NSF under contract MEA80-19628, and by DARPA through the Intelligent Task Automa-tionProject,managedby'eheAir Force MaterialsLaboratory,under a subcontract to HoneywelI, contract F 33615-82-C-5092. 510CH2282-2/86/0000/0510)T;o1.0010986 E E Ewhere A(q) is the n X n kineticenergymatrix; a l l = 11 12 coa2(82) 13 cos(82)co8(82 63) (3) B(q) is the n x n(n-1)/2matrix of Coriolistorques; puma basket bow z4 cos2(e2 03) C(q) is the n x n matrix of centrifugaltorques; Calculations required: 3 multiplications, 3 additions. g(q) is then-vector of gravitytorques; q is thne-vector of accelerations; where ZI.= d i m 3 dZm3 2 d2d3m3 dgm2 J3yy J3== is the generalizedjoint force vector. r J2== Jzyv Jlzr Jizz; etc.

J -- ,p',jj (5) 4. Formation of the needed partial derivatives, expansion of where (qk * it) is the j t h velocity product in the [q4]vector, and the Coriolis and centrifugal matrix elements in terms of the derivatives, and simplification by combining is the Christoffel symbol. inertia constants as in 2. The number of unique non-zero Christoffel symbols required The first step was carried out with a LISP program, named by the PUMA model can be reduced from 126 to 39 with fourEMDEG, which symbolically generates the dynamic model of an equations thathold on the derivatives of the kinetic energy matrixarticulatedmechanism.EMDEGemploys Kane's puma basket heart black dynamic for- elements.

that the kinetic energy matrix element a11 is given by: Andequation (10) holdsbecausethesecond a d third axes of a11 = J322 c o s 2 ( & 83) J a Y y sin2(82 83) JzZr &m3 thePUMAarmareparallel. Of the reductionfrom 126 to 39 2 kf3za" cos(82)cos(82 d 3 ) uzm3 cos2(e2) unique Christoffel symbols, 61 eliminations are obtained with the 2 Mzza3 cos"(82 03) a$m3 c0s2(82 83) general equations, 14 more with (9)and a further 12 with (10). $2 a2a3m3 eos(Bz)<�oos(82 6 3) JpYy sin"(62) (2) Step four requires differentiating the mass matrix elements withrespect to the configurationvariables.