Singularity Analysis And Evaluation Of Parallel Manipulators Based On Geometric Algebra

When the parallel manipulator is in or near the singularity configuration,some phenomena such as motion instability,stiffness reduction and even mechanism damage will occur.Therefore,singularity configuration should be avoided as far as possible in practical application.Although the singularity of the parallel manipulator has been widely studied,there is still a lack of effective general method which can simultaneously describe the distribution and geometric properties of the singularity of the parallel mechanism in the workspace.There is not yet an index which can quantitatively evaluate the distance or proximity degree of the parallel mechanism to the singularity configuration and has both geometric and physical significance.Aiming at the above problems,the singularities of the parallel manipulators are systematically studied by using the advantages of algebraic and geometric intuitiveness of geometric algebraic operations,and based on the singularity analysis,an evaluation index was proposed to measure the degree to which the parallel manipulator was close to the singularity configuration.The research contents and results are as follows:A method for the singularity analysis of parallel manipulators in the framework of geometric algebra is proposed.Firstly,the non-redundant constraint space of the mechanism is calculated by the dual and outer product operations on the motion space of the mechanism branch,and the singular polynomial is deduced.Then the three-dimensional singular surface of the mechanism and the two-dimensional singular locus on any cross section are given.Singular points are selected from the two-dimensional singular locus to find the corresponding singular configurations.Finally,the geometric meaning of the singular configuration is explained according to the property that the outer product of two vectors is zero.The method based on geometric algebra for the singularity analysis of parallel manipulators is not only algebraic but also geometrically intuitive.In addition,the distribution of 3D singular surfaces in the workspace of the mechanism can provide guidance for trajectory planning and singularity avoidance.An index for closeness to singularities of parallel manipulator in the framework of geometric algebra is proposed.Based on the singularity analysis,the concept of approximate singular region is proposed by setting threshold value,and the ratio of nonsingular workspace volume(workspace volume minus singular space volume)to workspace volume is calculated by Monte Carlo method,and a new index for closeness to singularities is derived.This index is dimensionless,coordinate independent and has clear geometric and physical explanations,which can be used in the optimization and path planning of parallel manipulator.In summary,the singularity analysis,establishment of index for closeness to singularity of parallel manipulators based on geometric algebra proposed in this paper have important theoretical significance for singularity avoidance,trajectory planning of the mechanism,and have a better promoting effect on the application of parallel manipulators.