ÿþwhere A(q) is puma shoes 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; 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.

The symbols [ q q ] and [q"] are notation for the n(-l)/Z-vector of Creating Z1 through 1 4 , which are constants of the mecha- nism, leads to a reduction from puma trainers 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 puma suede given by: of 11 are the input to the RNE.

And reduction of these expressions with four relations that hold on these partial derivatives.  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 puma sliders constants as in 2. The number of unique non-zero Christoffel symbols required The first step was carried out with a LISP program.

Which symbolically generates the dynamic model of an equations thathold on the 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 made for this analysis.

The kinetic energy ma- The reduction of Equation (7) arises from the symmetry oftrix elements are simplified by combining inertia constants that the kineticenergymatrix.Equation (8) obtains because the ki-multiply common variable expressions. This is the greatest puma uk source netic energy imparted by the velocity of a joint is independent ofof computational efficiency. Looking to the dynamic model of a 3 theconfiguration of thepriorjoints.Equation (9) resultsfromdof manipulator presented in [Murry and Neuman 19841.

Of the link, thelocation of the center of gravity, the distancefromthe wire attachment points to the axis of rotation, the length of Table 4. Link Masses (kilograms; kO.01 1%)the wires, andtheperiod of rotationaloscillation. Theinertiaabout eachaxis is measured by configuring the link to swing Link Massabout that axis. Rotational oscillation is started by twisting andreleasing the link. If one is carefulwhenreleasing the link, it Link 2 17.