Study On Differential Geometry Theory Of Singularities Of Parallel Manipulators And Redundant Parallel Manipulators

With the development of parallel mechanisms towards practicalization and productization, the requirements for high performance of mechanisms are more and more critical. There are abundant and complicated singularities in parallel mechanisms, which have great effects on the accuracy, stiffness and motion ability of parallel mechanisms. Deep and clear understanding of singularities of parallel mechanisms is the fundament to reduce or eliminate the effect of singularity and improve the performance of parallel mechanisms.Although many authors have done a lot of research on singularities of parallel mechanisms, there is not a systematical theory yet, many problems are still to be solved. This dissertation will do some helpful research on this area.First, the classification of singularities of parallel mechanisms is studied. Based on the analysis of topology structure of parallel mechanisms and using differential topology and differential manifolds as mathematical tools, we propose a new classification method. This method classifies singularities of parallel mechanisms into two basic types, i.e. topology singularity and parameterization singularity. This kind of classification has clear physical and mathematical meaning and fully reveals the characteristic of configuration space of parallel mechanisms.Secondly, further research is performed on parameterization singularity. The property and distribution of parameterization singularity manifolds are analyzed. According to the relation between tangent space and annihilation space of parameterization singularity manifolds, we present the new concepts of first order and second order singularity. Degenerate and non-degenerate parameterization singularities are analyzed and the sufficient and necessary conditions for degenerate singularity are derived. For necessary condition, we present a practical Morse function to judge.Thirdly, a problem related to parameterization singularity and forward kinematics is studied, i.e., whether there exists singularity-free path between different solutions of forward kinematics. We give a clear explanation to this problem using standard idea from differential geometry.In order to avoid or eliminate the effect of singularity and improve the performance of parallel mechanisms, the redundancy is applied to the parallel mechanisms based the studying of singularity.First, the classification of redundancy of parallel mechanisms is studied. According to the mechanism constraints and the choice of actuators and end-effectors, we present a new classification method, which identifies three basic types redundancy: configuration redundancy, actuator redundancy and end-effector redundancy. And singularities are analyzed for all types of redundancy. This method gives us an unambiguous understanding of redundant parallel mechanisms.Secondly, we make a deep study of actuator redundancy. Actuator redundancy is a unique redundancy for parallel mechanism and it is also the main redundancy to eliminate the singularity effects on parallel mechanisms. At first, singularity effects on accuracy, stiffness and dynamics are analyzed, and then the reasons of improving accuracy, stiffness anddynamics by introducing actuator redundancy are analyzed in detail.Finally, we make a preliminary study on the control problem of redundant parallel mechanisms. Using pseudo-inverse method, control strategies for serial mechanisms are applied to redundant parallel mechanisms. A 2-DOF redundant planar parallel mechanism is used as control example; the experiment results show that the redundant mechanism has good localization accuracy and good performance at high speeds.