Analytical solution of the first problem of dynamics of manipulators with rotational joints
Authors:   South Ural State University, Telegin Aleksandr I. 
Published: May 2022

The problem of complexity of derivation and the cumbersome analytical form of equations of mathematical models of controlled body systems with explicit structural, kinematic, static and dynamic parameters is solved. First of all, this applies to the equations of dynamics on which the control systems are based. Our practical experience and theoretical results indicate that solutions to this problem should be sought in two directions. Firstly, in the direction of classifying body systems and using peculiarities of the representatives of the considered classes of body systems in terms of simplifying formalisms for deriving their equations of dynamics, as well as reducing the number of mathematical operations in the analytical representation of the equations of dynamics. Second, and in the direction of choosing the parameters of the state of bodies in which the analytical scalar-coordinate types of the equations of dynamics are written down. In this connection, it should be noted that the vast majority (more than 90 %) of industrial robots, as well as special-purpose robots, have a single open branch structure in which the bodies form rotational kinematic pairs of the fifth class (rotational articulations) with each other. If in such systems the poles of the bodies are chosen on the axes of their relative rotation, the interpole distances will be constant, which greatly simplifies the solution of the above problem. Regarding the choice of state parameters explicitly included in the equations of dynamics, it should be noted that quasi-velocities, i.e. projections of absolute angular velocities of bodies on the axes of their coupled coordinate systems, are the most suitable for these purposes. The point is that in the equations of kinematics, which close the equations of dynamics to a complete set of equations for solving a problem, one can always express quasi-velocities through any other parameters, for example, relative angles of rotation of bodies and their time derivatives, guiding cosines and their derivatives, quaternions, etc. If projections of absolute angular velocities of bodies on their axes are measured, for example, by gyroscopes on bodies, and the first problem of dynamics is solved, the solution formulas contain a minimum number of addition and multiplication operations. Thus, the goal of the study is to develop a simple method for deriving the analytical form of the equations of dynamics of manipulators with rotational joints in quasi-velocity, in which geometric, kinematic, static and inertial parameters of bo¬dies are explicitly expressed. The used research methods (vector and analytical mechanics of body systems, vector algebra, system analysis and methods of identity transformations) made it possible to reduce the derivation of the equations of dynamics of manipulators to formal actions of writing them out without performing complex mathematical operations of differentiation, magnification, calculation of vector operations, etc. The results of the study contain a proof of the general vector form of the equations of dynamics of manipulators in quasi-velocity with explicitly expressed interpole distances and parameters of mass distribution of bodies. Scalar-coordinate formulas and their simple special formulas for the case of parallel rotation axes of neighboring joints are derived for writing out the moments of driving forces in joints. As a special case, the formulas for writing out the equations of dynamics of manipulators on the plane were obtained. For them, the process of writing out the equations of dynamics is reduced to the specification of the number of bodies, their geometric and inertial parameters. Conclusion. The effectiveness of the outlined methods and the obtained formulas have been demonstrated by examples of deriving the equations of dynamics of a body with a single fixed point, a gyroscope in a gimbal, and an angular manipulator with three and six degrees of freedom in space. These results allow us to expect that the number of users of the proposed methods will grow. The positive experience of using these methods in the educational process in the disciplines “Fundamentals of Mechanics of Body Systems”, “Electromechanical Systems” and “Mechatronics” justifies our expectations.