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Manipulation robots belong to a class of complex, nonlinear dynamic systems. In addition, they are subjected to the constraints resulting from work-space obstacles, kinematical and physical characteristics of the mechanism itself and the actuators. Therefore, the application of optimal control theory (in energy or time optimization) leads to substantial practical difficulties, so that significant simplifications are usually performed, either in model complexity or by neglecting the existing constraints. In this paper the problem of obtaining such an optimization method, which would take into account the complete system dynamics and all the constraints is considered. The only method found to be suitable for such a complex optimization should be based on dynamic programming. In this paper an algorithm for determining optimal velocity distribution for a given manipulator tip trajectory is elaborated in detail. Practical application of the developed procedure is in off-line calculation of nominal input generalized forces (programmed control) of a nonredundant manipulator, by which the minimum of consumed energy is ensured. This is specially important for high speed motions as well as handling of heavy loads.

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SUMMARYThis paper presents a new program package for the generation of efficient manipulator kinematic and dynamic equations in symbolic form.The basic algorithm belongs to the class of customized algorithms that reduce the computational burden by taking into account the specific characteristics of the manipulator to be modelled. The output of the package is high-level computer program code for evaluation of various kinematic and dynamic variables: the homogeneous transformation matrix between the hand and base coordinate frame, Jacobian matrices, driving torques and the elements of dynamic model matrices. The dynamic model is based on the recursive Newton-Euler equations. The application of recursive symbolic relations yields nearly minimal numerical complexity. Further improvement of computational efficiency is achieved by introducing different computational rates for the terms depending on joint angles, velocities and accelerations. A comparative study of numerical complexity for several typical industrial robots is presented.

Robot operation near isotropic configurations, in which the condition number of the Jacobian matrix reaches unity, is desirable from several points of view. However, determination of all such configurations, given arbitrary robot geometry, is a rather complex problem. In this article all the isotropic configurations of planar manipulators with two, three, and four degrees of freedom are determined, together with that of a 3- DOF spatial manipulator. The solutions are obtained in the form of a fourth-order polynomial for the 3-DOF robot and an eighth-order polynomial for the 4-DOF planar manipulator, resulting in a maximum of eight sets of solutions. The condition numbers are obtained as explicit analytic functions of joint coordinates and link lengths ratios. The optimal length of the links is determined by minimizing the criterion that the condition number increase most slowly with joint angles in the vicinity of isotropic configurations.