Thirty four lumped constants are needed 1.Symbolic Generation of the ï»¿new balance australia kinetic energy matrix and by the full PUMA model, 8 fewer than the count of 42 simple pa- gravity vector 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 the first kind [Corbenand Stehle 1950; variable expressions. Likgeois et al. 1976]* giving: 3.Expression of the Coriolis and centrifugal matrix elements b.
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 new balance shoes derivatives of the kinetic energy matrixarticulatedmechanism.EMDEGemploys Kane's dynamic for- elements.The first two equationsaregeneral;the last two aremulation [Kane 19681, and produced a result comparable in form specific to the PUMA 560. The equations are:and size to that of ARM [Murry and Neuman 1984).
Three sirn-plifying assumptions were new balance 247 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 Z z z . The original output of EMDEG, includ-ing Coriolisand 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 ma- The reduction of Equation (7) arises from the symmetry oftrix elements are simplified by combining inertia constants that the kineticenergymatrix.
Of the new balance 574 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.Themeans to carry Jz=, cos2(82) 2 dzdsms 2 Mz2a2 cos2(&) outdifferentiationauto,naticallyhavebeenavailableforsome a;mz cos2(&) d i m s dZm2 J2zz Jizz JizzCalculationsrequired: 37 multiplications,18additions.
Themotors were mass of thearm. To makethismeasurementourcontrolsys-left installed in linkstwo and three when the inertia of these links tem was configured to command a motor torque proportional towere measured, so the effect of their mass as the supporting links displacement, effecting a torsional spring. By measuring the pe-move is correctly considered. The gyroscopic forces imparted by riod of oscillation of the resultant mass-spring system, the totalthe rotating motor armatures is neglected in the model, but the rotational inertia about each joint was determined. By subtract-data presented below include armature inertia andgear ratios, so ing the arm contributions, determinedfrom direct measurements,these forces can new balance outlet be determined.
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; *SO%. Coordinate Frames Shown.