During the manufacture of robotic systems differences between actual and nominal link lengths and orientations occur. In addition, errors between nominal and actual pair variables result from sensor and controller errors. Thus, when a robotic system attempts to perform a desired task using nominal motion planning schemes and nominal linkage kinematics, it will perform an actual task that is usually quite different from that desired. A procedure for determining the differences between the nominal and actual linkages parameters, pair variables, and tasks is presented. The approach can be used on all modern-day robots, as it incorporates solution techniques for the nonsquare and singular matrices that typically occur. An example using a PUMA 560 is included.

A fundamental study of noise generation in high-speed mechanical systems is undertaken. The objective being the development of modeling techniques for the prediction of mechanical system noise levels.Recently developed dynamical procedures are used to obtain the motions of linked mechanical systems with elastic elements and connection clearances. The ranges of critical system parameters are identified and classical acoustical analysis methods are used in determining the most significant acoustic sources. Detailed acoustic models are analytically developed for these significant sources. These methods and models are then used to predict the farfield radiation of a simple, yet representative, mechanical system; an elastic link with connection clearances in a nominal motion container. PACS numbers: 43.40.At, 43.50.Ed, 43.50.Jh LIST OF SYMBOLS N ;•,(n%) c o the speed of sound in air Ds, Dr.,D r the height, length, and thickness directivity functions D Young's modulus e w unit vector perpendicular to the surface under consideration and in the direction of R the motion R E Young's modulus of elasticity f,,,f•,fe generalized forces acting on the link due to Ro the bearing compliance and dissipation g(ro, to) =6(r o-ro/Co)/4•rro; free space Green's func-r o tion r o g(ro, to) =-e-J•ro/4•rro; Fourier transformation of S o g(ro, •'o) T H height of link t I acoustic intensity t o I A area moment of inertia W(Ro, t o) -LI2 K w/co; 2e/X; wavenumber •C o = O• o / C o L length of the link l characteristic size of system element Z/2 M r = rr(yo) dy o -Z13 -• r•/•'•r(•o)•y o 'YH i J-L I 2 ill2 rn i0 = (r( Yo)• i Yo dYo -L/•-rYl = f L I 2 ,, •-• /• •(Yo)*• ayo rio outward unit surface normal nto o the frequency being considered number of assumed modes arbitrary source reference point the acoustic pressure acceleration of the /th generalized element motion coordinate complex acceleration amplitude of the ith vibrational mode at frequency nto o JR t; the magnitude of R a vector from the element center of mass to the receiver a vector from the element center of mass to a point on the link surface I rol; the magnitude of r o --R-Ro link surface area thickness of link time of the received signal time of the disturbance link deformation function X s, Ys, Or nominal rigid body motion in the x, y, and 0 direction in the plane of the mechanism motion Yo transverse amplitude of the nominal container motion positive coordinate along the link relative to the link center of mass angie between rio and r o = (nKoH sine)/2 = (nK o L cos• sine)/2 = (nK o T cos½)/2 = cos-•(cosO cos•/) del operator; o/On o for points on the surface $o at Ro dirae delta function modal damping coefficients perturbed rigid body motion in the x, y, and 0 direction in the plane of the mechanism motion 0,½ acoustic receiver orientation angles 0 o angular amplitude of the nominal container motion Yo •o *(r) 551

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The primary goal of this article is to examine various aspects of the motion᎐effort method for obtaining equations of motion and constraint of multibody systems. Gauge invariant transformations are used to recast the more commonly utilized dynamic equations into a dimensional gauge invariant form. Constraint elimination techniques based on singular value decompositions are then used to recast the invariant form of dynamic system equations into orthogonal sets of motion and effort equations. Desired motions and constraining efforts are partitioned into those ideally obtainable and unobtainable through the use of effort actuators. The actuation required to achieve these ideally obtainable constraining efforts and changes in motion are then found. The dynamic system performance resulting from the use of these ideal actuations is then evaluated. The method is compared to the more traditional augmented matrix, coordinate partitioning, and inverse dynamic solution approaches. A redundantly actuated multilinked closed chain linkage is used throughout the article as an example. ⅷ Journal of Robotic Systems-1999 r sin yr cos r J J J J J ' ' p s p s p J q J ycos p s m J m s p s T Ž . CC s 61 ycos 1 m m s s ⅷ Journal of Robotic Systems-1999