Modeling of dynamics of manipulators with geometrical consraints as a systems with redundant coordinates

Abstract

The application of analytical mechanic methods based on introduction of redundant coordinates is studied for dynamics manipulator modeling. Such form of the motion equations is obtained for holonomic systems in the Lagrangian variables of the given work, which enables a detailed analysis of linear and nonlinear members of the perturbed motion equations. Stability of steady motions for systems with redundant coordinates is possible only in critical cases. The absence of the roots of the characteristic equation with positive real parts does not allow general solving the stability problem of such motion, but the analysis of the nonlinear members is necessary for decision of stability problem in any of such cases. We would like to attract attention to a fundamental distinction of the suggested method: the rigorous methods of nonlinear stability theory are necessary for rigorous foundation of developed modus operandi only. The qualified use of this method dispenses with full digestion of rigorous proofs and substitutions of critical cases theory. Using rigorous methods of analytical mechanics, the nonlinear stability theory, N.N. Krasovsky method solving linear-quadratic problems and previously obtained results, the procedure for a unique determination of the coefficients of the stabilizing control has been developed. For practical calculation of the coefficients, the Repin-Tretyakov procedure may be applied. The proposed method is used for solving the issue of stabilization of manipulator steady motion with geometrical constraints.