Application of a special form of differential equations to the study of motion of a loaded Stewart platform

Abstract

The paper is concerned with motions of a loaded Stewart platform. The motion equations are composed using a special form of differential equations (motion equations are derived in redundant coordinates). This form is also used to derive first-order vector Lagrange equations involving differentiation with respect to the radius vector of the system center of mass, the unit vectors of the principal central inertia axes of a motion body and with respect to these derivatives. These vectors define the position of the rigid body in space. As abstract holonomic constraint imposed on the vectors describing the motion of the rigid body, one considers the invariableness of the lengths of unit vectors and their orthogonality. We discuss one engineering effect manifested in the behavior of a Steward platform in an equilibrium state (“parasitic oscillations”). This effect is one of the root causes for the system to move away from an unstable equilibrium position. A similar instability is also present in standard vertical oscillations of a Stewart platform. A simplest mechanism of appearance of instability of such vertical motions of a platform is revealed. To guarantee a stable motion, it is proposed introduce the classical feedbacks. Numerical solutions of the differential equations thus obtained are in complete accord with the numerical results obtained in the solution of the motion equations composed using the of center mass motion theorems and taking into account results on the variation of the kinetic moment as the system moves about the center of mass.