The analytical singularity locus equation and the determination of singularity-free zones in the workspace of the general Gough-Stewart platform.

摘要

It is well known that one of the main factors that hinders the application of parallel mechanisms is that singular configurations may exist within their workspace, which is a serious problem. Therefore, it is of primary importance to avoid the singularities in the workspace. From a design point of view, it is desirable to obtain the analytic expression of the singularity locus of a parallel mechanism; then, with a given set of structural parameters, the singularity locus can be illustrated graphically.; According to the classification given in [14], there are three types of singularities for closed-loop mechanisms, based on the properties of the Jacobian matrices of the chain. The second type of singularity is the focus of our study, i.e., the determinant of the instantaneous direct kinematics matrix is equal to zero. In this thesis, first, an expansion algorithm is developed to obtain the analytical expression for the singularity locus in the 6-dimensional Cartesian space of the general Gough-Stewart platform, i.e., a polynomial in six variables (three position variables x, y, z, and three orientation variables, psi, theta and &phis;), which consists of 2173 terms. Then, with the expression obtained here and a given set of structural parameters, the singularity locus for either constant orientation or constant position can be obtained immediately. Although the expression is rather complicated, it is possible to obtain graphical representations.; The singularity locus expression is applicable to all Cough-Stewart platform regardless of the geometric parameters. The expression developed here is of great interest for the design and analysis of Cough-Stewart platforms. It allows the designer to visualize interactively the singularity locus superimposed on the given workspace for either constant orientation or constant position or combinations of both.; The closed-loop nature of parallel mechanisms limits the motion of the platform and creates complex kinematic singularities inside the workspace. Because of the limited workspace coupled with singularities, the trajectory planning of parallel mechanisms is a difficult problem. Hence, it is highly desirable to develop an algorithm to locate the singularity-free zones in the workspace.; In this thesis, algorithms are developed for the identification of singularity-free zones in the workspace of 3-RPR planar parallel mechanisms and the general Cough-Stewart platform. Several procedures adapted to different situations are developed. With the procedures proposed in this thesis, the end-effector can be moved arbitrarily in a zone, which means that it can undergo any trajectory, and the trajectories do not have to be further checked for singularities. (Abstract shortened by UMI.)