ÿþ d i m 3 dZm3 2 d2d3m3 dgm2 puma australia J3yy J3== is the generalizedjoint force vector. r J2== Jzyv Jlzr Jizz; etc.The symbols [ q q ] and [q"] are notation for the n(-l)/Z-vector of C eating Z1 through 1 4 , which are constants of the mecha- nism, leads to a reduction from 35 to 3 multiplications and from[aq]velocity products and the n-vectorof squared velocities. and 18 to 3 additions. Computing the constant Z1 involves 18 calcu- lations. Since the simple parameters required for the calculation[ q 2 ] are given by: of 11 are the input to the RNE, theRNE will effectively carry out The procedure used to derive the dynamimc odel entails four the calculation of Z1 on every pass, producing considerable m-steps: necessary computation.

Thirty four lumped constants are needed 1.Symbolic Generation of the kinetic energy matrix and by the full PUMA model, 8 fewer than the count of 42 simple pa- gravity vector puma shoes elements by performing the summations of rameters required to describe the arm. either Lagrange's or the Gibbs-Alembert formulation. In thethirdstepthe elements of the Coriolis matrix,Qij, 2. Simplification of the kinetic energy matrix elements by and of the centrifugal matrix, ci,, arewritten in terms of the combining inertia constants that multiply common Christoffelsymbols of puma suede the first kind [Corbenand Stehle 1950; variable expressions. Likgeois et al. 1976]* giving: 3.Expression of the Coriolis and centrifugal matrix elements b.

* The French authors seem to assume the use of Cristoffel symbols, while the American authors seem unaware ofBy combining inertial constants with common variable termsand them.CorbenandStehle,inthe 1950 edition of theirexpanding sin2(82) into (1- c o s 2 ( & ) ) , equation (2) can be re- text, derive the results required here; but the derivationduced to: is largely omitted from their 1960 edition.of gravity and the terms of the inertia dyadic.Thewrist,link where I I = M-wg2 ** r12three and link two of puma womens shoes a PUMA 560 arm were detached in order Mgto measure these parameters.

If one is carefulwhenreleasing the link, it Link 2 17.40is possible to start fundamentalmodeoscillationwithout visi- Link 3 4.80blyexcitingany of the other modes. The relationshipbetween Link 4* 0.82measured properties and rotational inertia is: Link 5* 0.34 Link 6* 0.09 * This method was suggested by Prof. David Powell. Link 3 wiCthomplete LVrist 6.04 Detached Wrist 2.24 * Values derived from external dimensions; f 2 5 % . 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 coordinates 513of the center of gravity in the coordinate frame attached t o the Table 5 . Centers of Gravity.

It was necessary to add positive velocity feedback rected away from the base; Y5 is directed toward (damping factor -0.1) to causejoint one to oscillateforseveral link 2 when joint 5 is in the zero position. cycles.Link 6: Theorigincoincideswith that of frame 4; when joints 5 and 6 arein the zeropositionframe 6 is aligned with frame 4.Wrist : The dimensions are reported in frame 4. Table 6. DiagonalTerms of the Inertia Dyadics and Effective Motor Inertia.Figure 2. The PUMA 560 in the Zero Position with Attached * Iucrtia Diadic term derived from external dimenJions; puma mens shoes *SO%. Coordinate Frames Shown.

The gear ratios,maximummotortorque, and break away torque for eachjoint of thePUMA is reported in Table 7. The maximum motor torque and break away torque values have been taken from data collected during our motor calibration process. Thecurrentamplifiers of the Unimatecontrolleraredriven by 12 bit D/A converters, so the nominal torque resolution can be 7btained by dividing the reported maximum joint torqueby 2048. Table 7 . Motor and Drive ParametemTheinertiadyadicand effect,ive motorand driveinertia I Joint I Joint 2 Joint 3 Joint 4 Joint 5 Joint ti Gear Ratio 107.36 53.69 76.01 71.91 76.73terms are reported in Table6.