Energies of Accelerations in Advanced Robotics Dynamics

Abstract

This paper is devoted to the presentation of new formulations on the higher order motion energies that are used in the advanced dynamic study of robots. Integral part of these mechanical systems are the mechanical robot structures, on which an application will be presented in order to highlight the importance of the higher order motion energies regarding the dynamic behavior. In current dynamic studies, the kinetic energy is used as a central function in Lagrange - Euler equations. This paper extends the study by developing the acceleration energies of first, second and third order and their implementation in differential equations of motion of third and fourth order, which gives the possibility of applying the initial motion conditions in positions, velocities and accelerations of first and second order. This leads to a more precise control on the transitory motion phases of the multibody systems, in which the robot structures are included.

Synthesis on the Acceleration Energies in the Advanced Mechanics of the Multibody Systems

“…The angles from (8) are the components of the column matrix of orientation ψ(t), and they describe, from a geometrical point of view, dihedral angles between two geometrical planes: χ 0 = {u 0 ; v 0 ; w 0 } = cst − f ixed plane ∈ 0 /{0} and χ = {u; v; w} = 0 − mobile plane ∈ {S} Physically, the three angles defined with (8) expresses a simple rotation around one of the axes of the Cartesian reference system: χ = {u; v; w}. Based on research from [10][11][12][13][14], when combining the three simple rotations, there is a result of twelve sets of orientation angles (8). Taking χ = x; y; z , the expressions of definition are further developed for the three simple rotation matrices symbolized as:…”

“…By applying (14) and (15), an expression identical with the classical Rodriguez formula is obtained [17]:…”

“…Using the research in the field of matrix exponentials [10,14], the following form for the resultant rotation matrix (18) is written as:…”

“…The novelty of the paper consists in presenting a new approach regarding the dynamic modeling of rigid bodies and multibody mechanical systems. Based on scientific literature [10][11][12][13][14][15][16][17][18][19][20], the input kinematic and dynamic parameters have been formulated. These parameters are compulsorily included in the dynamic equations of the higher order, corresponding to fast and sudden motions.…”

“…The important formulations regarding the acceleration energies of the first, second, third and higher orders are also presented in explicit and matrix forms. The expressions for the acceleration energies of the higher order were determined, among others, based on matrix exponential functions [14,19]. Although at first sight they seem difficult to put into practice, they proved to have undisputable advantages in terms of their use in the kinematic and dynamic study of complex mechanical systems.…”