Dynamic Equations of Docking Mechanisms. Part 2. Algorithms for Kinematical Loops

Abstract

From the point of view of a docking dynamical process, a multi-loop docking mechanism, in spite of its low mass, is a more complex mechanical system than a spacecraft. An approach providing high computational efficiency of dynamic simulation algorithms for such a class of mechanisms is considered in this paper. Before simulation, a multi-loop mechanical system is transformed to a tree structure using constrain equations instead of some joints. Each loop of a docking mechanism can be partitioned to a controlled and a dependent kinematical chains with independent and dependent joint variables, and constrain equations for its replaced joint are non-singular. This paper describes an application of the generalized coordinate partitioning method (GCPM), which specifies that dependent joint accelerations of the transformed mechanical system are expressed as a function of independent ones through matrixes, which are used to reduce the dimension of dynamic equations. All loops and dependent chains of a docking mechanism are numbered from outermost to innermost. In contrast to GCPM, constrain equations are formulated in the inverse sequence, i.e. starting with the maximum loop number, and for a current loop they incorporate all joint accelerations except previous loops. So, dependent joint accelerations of a current loop are expressed as a function of joint accelerations of all next loops. It allows reducing the dimension of dynamic equations regardless of various combinations of outer and inner kinematical loops of docking mechanisms. Redundant mathematical operations with zero matrix elements can be eliminated using symbolic manipulation system. For higher computational efficiency, analytical solutions of joint coordinates constrain equations are proposed for main types of dependent kinematical chains.