Elastostatic Stiffness Modeling And Optimization Of Parallel Mechanism Based On Geometric Algebra And Strain Energy

Aiming at parallel manipulator-based machining center for high-efficiency and highprecision of large-scale structural components in aerospace and other fields,this paper systematically studies the methodology of elastostatic stiffness modeling of parallel manipulators(PMs)as well as multi-objective optimization of kinematic performance and stiffness performance.The main contributions of this paper are as follows:An analytical elastostatic stiffness modeling of PMs considering the compliance of the link and joint is presented based on the strain energy and geometric algebra.First,the geometric algebra is used to obtain the extended constraint subspaces containing the constraint forces and driving forces.Second,the analytical expression of the strain energy of link is formulated using material mechanics and that of the joint using a mapping matrix,and the strain energy of the limb can be obtained by summing the links and joints,and thus the concentrated limb stiffness matrix corresponding to the extended constraint subspace is obtained by the Castigliano's second theorem.The overall stiffness matrix is assembled based on the deformation compatibility equations,and the correctness of the analytical modeling is verified by finite element analysis(FEA)method.The revised beam element stiffness matrix of the matrix structure analysis(MSA)is established.Based on this,the analytical expressions of the stiffness matrix of the revolute joint,universal joint and ball joint in the joint space are obtained.First,aiming at the disadvantages of the traditional MSA method ignoring the influence of shear deformation on the stiffness performance of the beam,the work comprehensively considers the composite elastic deformation of the axial tensile,shear,bending and torsion of the beam,and the stiffness matrix of the cantilever beam is obtained based on the strain energy and Cartesian second theorem.Then,the stiffness matrix of the unconstrained beam is obtained based on the deformation compatibility equations.Lastly,the analytical expression of the joint stiffness matrix is obtained based on the MSA method,and the validity of the stiffness modeling is verified by the FEA modeling.A new comprehensive stiffness index(CSI)is proposed,which takes the coupling of the nondiagonal elements into consideration.This index decouples the linear and angular stiffness,which has the definite physical dimensions and clear physical meaning.The Lagrangian function is constructed to obtain the maximum and minimum stiffness at a given position along their corresponding directions.To compare the difference between the CSI and the principal diagonal stiffness index,divergence index ? is proposed,which reflects the degree of influence of the coupling effect of non-diagonal elements on the stiffness performance.The results show that the coupling of non-diagonal elements can not be ignored.Based on multi-objective optimization game algorithm,NSGA-II genetic algorithm(GA)and particle swarm optimization(PSO)algorithm,the single-objective and multi-objective optimizations of 2UPR-RPU PM are carried out.First,the volume of the regular cylindrical workspace,motion/force transmission efficiency,and stiffness performance were considered as objective functions,and the four-dimensional slice distributions are obtained.Second,the muliobjective functions are transformed into a mixed objective function by assigning weighting factors,and the optimal values of the mixed objective function under different weighting factors are obtained.Lastly,the pareto frontier of multi-objective functions of 2UPR-RPU PM is obtained based on the three algorithms,and the optimal design parameters are obtained.Above research results have a guiding role on the elastostatic stiffness modeling and multiobjective optimization design of PMs,and it is of great significance to promote the engineering application of PMs.