Complement of Source Equation of Elastic Line

Abstract

A new notion of joint, defined in terms of the state of motor (active or locked) and type of the elastic or rigid element, gear and/or link that follows after the motor, is introduced. Special attention is paid to the motion of the flexible links in the robotic configuration. The paper deals with the relationship between the equation of elastic line equilibrium, the "Euler-Bernoulli approach" (EBA), and equation of motion at the point of elastic line tip, the "Lumped-mass approach" (LMA). The Euler-Bernoulli equations (which have for a long time been used in the literature) should be expanded according to the requirements of the motion complexity of elastic robotic systems. The Euler-Bernoulli equation (based on the known laws of dynamics) should be supplemented with all the forces that are participating in the formation of the elasticity moment of the considered mode. This yields the difference in the structure of Euler-Bernoulli equations for each mode. The stiffness matrix is a full matrix. Mathematical model of the actuators also comprises coupling between elasticity forces. Particular integral of Daniel Bernoulli should be supplemented with the stationary character of elastic deformation of any point of the considered mode, caused by the present forces. General form of the elastic line is a direct outcome of the system motion dynamics, and cannot be described by one scalar equation but by three equations for position and three equations for orientation of every point on that elastic line. The choice of reference trajectory is analyzed. Simulation results are shown for a selected robotic example involving the simultaneous presence of elasticity of the gear and of the link (two modes), as well as the environment force dynamics.

Future directions for implementation of aerial robot

“…The validity of the defined mathematical model can be confirmed due to the software package. Future researches go a step further in implementation of the features of elastic ropes (type of nonlinear dynamic elasticity as defined in [7]- [14] in the mathematical model of CPR-B.…”

The procedure for the application of a new form of Euler-Bernoulli equation and its solutions

“…The source form of the Euler-Bernoulli equation of the elastic line of beam bending has the following form: (1) is expanded in [3], [4] from several aspects in order to be applicable in a broader analysis of elasticity of robot mechanisms. In that case, the model of elastic line of the elastic link's first mode has the new form of the Euler-Bernoulli equation:…”

New interpretation of the Euler-Bernoulli equation

“…The same interpretations of source shape of EulerBernoulli equation are adopted as in paper [27], [28], [30], [31].…”

“…All marcs are used from previously published papers [27], [28], [30], [31]. Let us consider a robotic system with m links, whereby the first link contains 1 n modes, the second link 2 n modes, etc, the m -th link contains m n modes.…”

“…The LMA (used in [5],) gives the possibility to analyze the motion of the any point of each mode. We consider that EBA and LMA, are two comparable methods addressing the same problem but from different aspects [27], [28], [30], [31]. As the equation of equilibrium for the mode tip point is essentially LMA and it follows directly from the equation of flexible line obtained via EBA for the preset boundary conditions, it clearly comes out that the structures of these equations are the same (whereas the content of elements of these structures is not the same).…”