TY - JOUR

T1 - Application of a Special Form of Differential Equations to Study Movements of a Loaded Stewart Platform

AU - Zegzhda, S. A.

AU - Petrova, V. I.

AU - Yushkov, M. P.

N1 - Zegzhda, S.A., Petrova, V.I. & Yushkov, M.P. Application of a Special Form of Differential Equations to Study Movements of a Loaded Stewart Platform. Vestnik St.Petersb. Univ.Math. 53, 82–90 (2020). https://doi.org/10.1134/S1063454120010136

PY - 2020/3/26

Y1 - 2020/3/26

N2 - Abstract: We study movements of a loaded Stewart platform in this paper. A special form of differential equations is used (motion equations in excess coordinates are derived) to compose the motion equations. In this form, vectorial Lagrange equations of the first kind are compiled using differentiation with respect to the radius-vector of the center of mass of the system and the unit vectors (orts) of the main central axes of inertia of the moving body and with respect to their derivatives. These determine the position of a rigid body in space. The length invariance of the unit vectors and their orthogonality are taken into account as abstract holonomic relations imposed on vectors describing the motion of a rigid body. “Spurious oscillations” are some of the technical effects seen in the behavior of the Stewart platform in the equilibrium position. This reason for the system to leave the unstable equilibrium position is discussed in this paper. Such standard motion of the Stewart platform as vertical oscillations of the platform will be equally unstable. The simplest mechanism of the instability of such vertical movements of the platform is revealed in this work. We propose introducing classical feedbacks to obtain steady movement. The numerical solutions of the derived differential equations completely correspond to the numerical results obtained by solving the motion equations compiled using theorems on the motion of the center of mass and on the change in the kinetic moment when the system moves relative to the center of mass.

AB - Abstract: We study movements of a loaded Stewart platform in this paper. A special form of differential equations is used (motion equations in excess coordinates are derived) to compose the motion equations. In this form, vectorial Lagrange equations of the first kind are compiled using differentiation with respect to the radius-vector of the center of mass of the system and the unit vectors (orts) of the main central axes of inertia of the moving body and with respect to their derivatives. These determine the position of a rigid body in space. The length invariance of the unit vectors and their orthogonality are taken into account as abstract holonomic relations imposed on vectors describing the motion of a rigid body. “Spurious oscillations” are some of the technical effects seen in the behavior of the Stewart platform in the equilibrium position. This reason for the system to leave the unstable equilibrium position is discussed in this paper. Such standard motion of the Stewart platform as vertical oscillations of the platform will be equally unstable. The simplest mechanism of the instability of such vertical movements of the platform is revealed in this work. We propose introducing classical feedbacks to obtain steady movement. The numerical solutions of the derived differential equations completely correspond to the numerical results obtained by solving the motion equations compiled using theorems on the motion of the center of mass and on the change in the kinetic moment when the system moves relative to the center of mass.

KW - платформа Стюарта, гексапод, специальная форма уравнений движения системы тел

KW - Stewart platform

KW - dynamic simulation stand

KW - special form of motion equations

KW - feedbacks

UR - http://www.scopus.com/inward/record.url?scp=85082588607&partnerID=8YFLogxK

UR - https://link.springer.com/article/10.1134/S1063454120010136

M3 - Article

AN - SCOPUS:85082588607

VL - 53

SP - 82

EP - 90

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -