WRITING OUT INERTIAL MATRICES OF HINGED TREE-SHAPED SYSTEMS

Abstract

The goal is to develop a formalism (a sequence of formal actions) for deriving formulas for calculating the Elements of the Matrix of Inertial Coefficients (EMIC) of Tree-Like Systems with Open Branches (TSOB), whose bodies form hinges between each other, i. e., rotational kinematic pairs of the fifth class. Research methods relate to mechanics of systems of bodies, system analysis, and robotics. The results of the study contain a new formalism for automatically deriving EMIC calculation formulas, i. e., coef¬ficients in the products of relative angular velocities of bodies in the expression of kinetic energy of TSOB. EMIC formulas include constant structural, geometric, and inertial parameters of the considered TSOB. These formulas are represented as quadratic forms with respect to the direction cosines between the axes of coordinate systems rigidly connected to the bodies. The effectiveness of the formalism is demonstrated by examples of manual derivation of EMICs for a three-link angular Manipulator Robot (MR) in the vertical plane, two-armed MRs with five and seven degrees of freedom on the plane and in space. For an MR in space, the problem of synthesizing its parameters, for which EMICs do not depend on body rotation angles, has been solved. Conclusion. The proposed formalism can be used to derive EMICs for typical angular MRs, as well as walking devices in single-support phase of walking or flight, e. g., to derive dynamic equations based on them in the form of Lagrange's second-order equation. Applying known methods of constraint accounting expands the use of EMICs to TSDB, i. e., to TSOBs with end-body constraints, as well as to TSDBs with variable structure, which is relevant for deriving dynamic equations for walking machines and apparatuses in different phases of motion.