Mobility Analysis Of Parallel Mechanism In The Framework Of Geometric Algebra

Mechanism is the skeleton of the machine,and mobility is the basic property of a mechanism.The aim of mobility analysis is to know the number of the degree of freedoms(DOFs)and the motion pattern,which is an essential issue in mechanical design.The mobility analysis of the complex multi closed loop mechanism,especially the parallel mechanism,has been difficult for a long time in mechanism theory.This paper studies the mobility analysis of parallel mechanism systematically in the framework of geometric algebra.The different dimensional geometric identity can be expressed and computed directly in geometric algebra.The contributions of this paper are as follows:The twist space and wrench space are described in the framework of geometric algebra,and the origin of the redundant constraint in the overconstrained parallel mechanisms are illustrated based on the relation between twist and wrench spaces.Redundant constraints of the overconstrained parallel mechanism are the intersection of the constraint wrench spaces of different limbs,however the resource of the redundant constraints comes from the limb twist spaces.When the join of the limb twist spaces is not a 6-blade,the redundant constraints occur,and the collineation of the dual of the join is the redundant constraints exactly.A method of detecting and removing the linear dependencies is proposed.When the linearly dependencies occur,the outer product equals to zero.And this property can be utilized to detect the linearly denpendencies.In this situation,the join operation of a set of vectors is the outer product after removing the linearly dependencies.Thus the procedure of detecting and removing the linear dependencies in join operation is proposed to solve this problem.The constraint-based mobility analysis method of parallel mechanism in the framework of the geometric algebra is proposed,and the symobolic expression of its platform twist space in a generic configuration is obtained.This method determines the motion space of the moving platform through the wrench space using the map of the twist space and the wrench space.Firstly,the limb wrench spaces can be derived from their limb twist spaces.Then the platform wrench space is determined by the join of all the limb wrench spaces.Finally,the symbolic expression of the platform twist space can be obtained.The constraint-based method in the framework of the geometric algebra only involves addition and multiplication.Since division is not involved in the computational process,the discussion of the denominator whether equals to zero is not needed.The mobility analysis method of parallel mechanism based on meet operator in the framework of the geometric algebra is proposed,and the closed-form expression of its platform twist space in a generic configuration is obtained.It determines the platform twist space by calculating the intersection of all the limb twist spaces directly,thus it costs less computational steps with more straightforward way.Both of the methods obtain the symbolic expression of the motion space with clearly geometric properties,and the computation does not involve division.Since the discussion of the denominator is not involved,it is convient to programme.In order to show the superiority of the geometric algebra,this paper also presents a mobility analysis method for overconstraint parallel mechanism using Grassmann-Cayley algebra.Since the shuffle product fails in applying to overconstraint parallel mechanism,the meet operation is needed to be revised to obtain the twist space on moving platform.The symbolic expressions of the twist space on the moving platform are obtained.Compared with the Grassmann-Cayley algebra method,the geometric algebra is easy to understand and the procedure of the computation is easy to implement.