Kinematics Analysis And Dimensional Synthesis Of A Planar Symmetric（CRR）₂S-（3R）₂R Parallel Mechanism

A planar symmetric（CRR）₂S-（3R）₂R parallel mechanism（PM）with two translationaland two rotational degree-of-freedom（2T2R）is proposed in this work to be applied as the body structure of screening devices.The main contributions in this paper are to analyze the kinematic performances and to research the dimensional synthesis of the mechanism.The major works are as follows:degree-of-freedom analysis,kinematics analysis,analysis of the motion/force transmission performances,dimensional synthesis,virtual prototype design,and trajectory planning.Firstly,the degree-of-freedom of the mechanism is analyzed.With the screw theory as a mathematical tool,the twist screws systems of each branch chain are deduced,and then the corresponding constrained wrench system（CWS）of each branch chain can be obtained according to the screw reciprocal characteristics.The modified Grübler-Kutzbach formula is used to calculate the degrees of freedom of（CRR）₂S-（3R）₂R PM.Furthermore,the reasonability of selection for driving pairs of the proposed mechanism is analyze by means of the constrained screws theory.Secondly,the mechanism kinematics is investigated.By utilizing coordinate transformation method and constraint condition of link length,the nonlinear equations of forward displacement solutions to（CRR）₂S-（3R）₂R PM are established and transformed into an unconstrained minimizing optimization model,which is then solved by the hybrid social cognitive optimization（HSCO）algorithm.In order to verify the optimization performance of the HSCO algorithm,a comparison test of HSCO,difference evaluation（DE）and particle swarm optimization（PSO）algorithms are employed to find the forward displacement solutions,and the results indicate that the HSCO algorithm has the best comprehensive performance.The analytical expressions for the inverse displacement solution to the mechanism are derived and the numerical example is given to verify the correctness of the forward and inverse displacement solutions.On this basis,the expressions of velocity and acceleration are derived by means of the vector method.The displacement,velocity and acceleration curves of the mechanism are obtained by MATLAB.Moreover,ADAMS is used for kinematics simulation,and the maximum errors between the MATLAB theoretical calculation and the ADAMS simulation analysis are obtained.It is found that the kinematics curves are identical,which indicates that the kinematics model of（CRR）₂S-（3R）₂R PM is correct and reliable.Thirdly,the motion/force transmission performances of the mechanism are studiedcomprehensively.The deviating output axis of the moving platform can enhance the orientation capability of this mechanism.The orientation of deviating output axis of the end-effector is depicted by azimuth and tilt angles.Using screw theory as the mathematical tool,this work analyzes the transmission wrench screw（TWS）and output twist screws（OTS）by the means of the reciprocal screw method and the augmented matrix method.Then the analytical mathematical models of evaluation indices for motion/force transmission performance are established.On this basis,the definitions and calculation methods of regular effective transmission orientational workspace（RETOW）,average input translation index（AITI）and average output translation index（AOTI）in RETOW are given.The original design mechanism is taken as an example,and the transmission performance atlases of（CRR）₂S-（3R）₂R PM are plotted and the motion/force transmissibility of this PM is analyzed.Finally,a novel workspace performance evaluation index,known as regular effective position workspace（REPW）,is proposed.Fourthly,the dimensional synthesis problem of the mechanism is solved.The constrained optimization model to formulate the dimensional parameters of（CRR）₂S-（3R）₂R PM on maximizing RETOW is established.Moreover,AITI and AOTI are taken as constraints with not less than allowable values,and then the HSCO algorithm is used to solve the dimensional synthesis model.The results show that the RETOW,AITI and AOTI of the mechanism are significantly improved after optimizing the dimensional parameters.The comparative experiments of HSCO,DE and PSO algorithms for solving dimensional parameter constrained optimization problems are also given.The test experiments show that the HSCO algorithm surpasses other two algorithms in comprehensive computational performance.To overcome the defect that the workspace of the PM is small,the HSCO algorithm is applied to optimize the workspace with maximizing REPW as the objective function and good results are obtained.Finally,the virtual prototype of the mechanism is designed.The key works are the designs of non-standard parts such as fixed platform,moving platform,and link rods,etc.On this basis,trajectory planning is carried out.The expected trajectory of the moving platform is plotted in the operating space by using the 3-4-5 polynomial.On this trajectory,13 key points are selected at the interval of 0.5 s and the inverse displacement solutions to 13 key points are calculated.The inverse displacement solutions to each branch chain are fitted to the 3-4-5-6-7 polynomial.The polynomial motion law of each branch chain is used as the input（driving）displacement,and ADAMS is used for kinematics simulation,so the kinematics curves of the mechanism are plotted.The results show that the actual trajectory is basically consistent with the expected trajectory.Furthermore,according to the results of trajectory planning,the standard parts of the virtual prototype are selected,including servo motor,reducer and so on.This paper presents the theoretical research of（CRR）₂S-（3R）₂R PM from the degree-of-freedom analysis to the virtual prototype design.The HSCO algorithm is used to solve the unconstrained and constrained optimization problems in the forward displacement solution,dimensional synthesis and trajectory planning.And satisfactory results are obtained,which lays a good theoretical foundation for the subsequent engineering application of the mechanism.