Analytical solution of the first problem of the manipulators' dynamics

Abstract

The aim is to solve the problem of deriving an explicit analytical form and the cumbersome of the equations of dynamics of body systems. The research methods refer to the mechanics of body systems and systems analysis. The research results allow to write out formulas for calculating forces and moments of forces in the joints of the systems of bodies with one open branch. It is demonstrated in
examples of writing out analytical types of equations of dynamics of industrial robot arms with three and six degrees of freedom in space. Three kinds of equations of dynamics were obtained for such manipulators. The first equations are written out in scalar-coordinate form with explicit quasi-accelerations and velocities, whose role is played by the projections of absolute angular accelerations and velocities of bodies on their connected axes. The second ones are written in vector-matrix form and are obtained from the former in
the process of replacing quasi-accelerations by relative linear and angular accelerations of bodies with
the allocation of a symmetric matrix of inertial coefficients. The third kind of equations of dynamics is obtained from the second one in the process of replacing quasi-velocities by relative linear and angular velocities of bodies. In the third form, the centrifugal, Coriolis, and gyroscopic inertial forces are clearly expressed. Gyroscopic inertial forces allow us to simplify the formula for calculating the power consumption of drives, as well as to simplify the Timofeev formula for calculating the driving forces and moments of forces that provide control of the program motion of manipulator bodies with a given quality. A technique for reusing formulas for manipulators with matching kinematic diagrams of their subsystems is demonstra­ted in the examples . Geometric, kinematic, static and inertial parameters of bodies are explicitly expressed in the equations of dynamics The multipliers for accelerations and products of velocities in the equations of dynamics are optimal in the sense of the minimum of arithmetic operations (additions and multiplications) required for their calculations. Conclusion. All analytical types of equations of dynamics are verified. They occupy several lines of text and further simplification is practically impossible.