Formalism of Writing Out of Manipulators Dynamic Equation
Authors:   South Ural State University, Miass, Russian Federation, Telegin A.I. 
Published: February 2022

The problem of cumbersome equations of dynamics for manipulation systems of industrial robots (manipulators) with translational and rotational joints is solved. A new formalism for writing out the equations of dynamics of manipulators by using of guide cosines is proposed. Examples of writing out equations of dynamics of manipulators with guid cosines are given. The equations of dynamics in relative angles of rotation of bodies are obtained with the help of these guide cosines by applying their properties. These manipulators have from three to six degrees of freedom. In their equations of dynamics the geometric, kinematic, static and inertial parameters are explicit. The multipliers for accelerations and products of velocities are optimal in the sense of the minimum of arithmetic operations (additions and multiplications) that are needed for their calculations in the written out equations of dynamics. JS-code and method for verification of the equations of dynamics of manipulators written in analytical form are proposed. The problem is that when the equations are written out manually, errors and oversights in the intermediate entries and the final result are possible. Therefore it is necessary to check the results of writing out for absence of errors, i.e. to perform verification of formulas for calculation of constitutive equations of dynamics. To do this, we can use software designed to calculate the generalized driving forces of manipulators, i.e. to solve the first problem of dynamics. Such software is offered as a web-application, in which JS-function is used for verification of the equations of dynamics of manipulators. The method of verification of formulas to calculate the generalized forces of gravity and multipliers (coefficients) for generalized accelerations and products of generalized velocities in the equations of dynamics is developed. An example of verification of the equations of dynamics of the universal manipulator with six degrees of freedom in space is given. Aim. The aim of research is to develop a formalism for writing out the analytical form of the equations of the manipulators’ dynamics in the guide cosines of the principal axes of the coupled body coordinate systems, whose coefficients contain the minimum number of arithmetic operations. Research methods. The methods of research refer to vector and analytic mechanics of absolutely solid systems, to vector algebra, and to systems analysis and programming in scripting languages. Results. The results contain two proved statements, in which there are the formulas and the methodology that allow us to write manually the equations of dynamics of manipulators with three and six degrees of mobility both in guiding cosines and in generalized coordinates. In both cases it is impossible to simplify the obtained equations. Conclusion. The offered analytical types of the equations of dynamics occupy several lines. By the known classical formalisms (Lagrange, Appel, Nielsen, Newton-Euler, etc.) it is practically impossible to obtain similar results because of the large number of complex mathematical operations in their implementation and the cumbersomeness of the resulting formulas.