Abstract
The article describes the solution to the Rodrigues equation for determining the volumetric accuracy of multi-axis CNC-controlled systems. An algorithm for calculating the position of the axis of a rotary kinematic pair in problems of volumetric accuracy of mechanical motion of a portal-type system with an additional pair of rotation. The algorithm is based on the analytical solution of the Rodrigues equation in the inverse problem of finding the vector of the final rotation of the known modulus from the known initial and final values of the characteristic vector of the rotated rigid body. In contrast to the well-known direct problem, where based on a finite rotation vector known in direction and magnitude, and the initial value of the characteristic vector of a body, its final value is found, the inverse problem of the Rodrigues equation is not that common due to the nonlinearity and need to solve a nonlinear coupled system of second order equations. The results of this work make it possible to expand the dimension of the space of generalized coordinates of the system analyzed for the volumetric accuracy from three to four. This is expected contribute to the development of ultra-precise systems of controlled mechanical movement. The analytical results of this study were verified by comparing with numerical solutions of the inverse problem in Maple.
Highlights
IntroductionMulti-axis systems with software-controlled movement of units and mechanisms are a an important part of modern mechanical engineering, in terms of both the technological equipment (machine tools, industrial robots) and measuring technologies (coordinate measuring machines)
Multi-axis systems with software-controlled movement of units and mechanisms are a an important part of modern mechanical engineering, in terms of both the technological equipment and measuring technologies
To solve the problem of motion correction, here it is necessary to know the true position in space of the axis of rotation of the fourth kinematic pair
Summary
Multi-axis systems with software-controlled movement of units and mechanisms are a an important part of modern mechanical engineering, in terms of both the technological equipment (machine tools, industrial robots) and measuring technologies (coordinate measuring machines). The concept of volumetric accuracy was introduced in the work of P.A. McKeown, J. The ultimate goal of the study of volumetric accuracy is the construction of algorithms for correcting the real motion of a mechanical system in a generalized space relative to another, considered under the condition of an "ideal" arrangement of the axes of the kinematic pairs of the system. It is known that, for example, with gantry three-coordinate kinematics, the axes of the telescopic pairs of the machine tool do not form an ideal orthogonal trihedron, and this discrepancy increases with the wear of the equipment
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