2-UPR&2-RPU冗余并联机构动力学与功耗分析

Dynamics and Power Analysis of 2-UPR&2-RPU Redundant Parallel Mechanism

提出一种新型2-UPR&2-RPU结构冗余并联机构,其具有2个转动以及1个移动(2R1T)自由度。运用螺旋理论分析机构的自由度及其运动特性,并利用修正的G-K公式验证分析的准确性,得出该机构含有1个冗余支链的结论。分析机构运动的约束条件,建立各支链的闭环矢量方程,求解得出机构的位置反解。基于符号运算的微分变换法对机构的速度和加速度进行分析,得到动平台速度雅可比矩阵和2阶影响系数矩阵,在此基础上进一步求解得出各支链的雅可比矩阵。运用虚功原理构建该机构的动力学模型,然后根据冗余并联机构驱动力配置方式不唯一的特点,采用驱动力二范数的优化方法推导得出驱动力和驱动功率方程。在此基础上,使用MATLAB对所构建的数学模型编写计算程序,并通过具体数值算例对所构建的运动学及动力学模型进行仿真。最后,为了验证所构建数学模型的正确性,在ADAMS中建立机构虚拟样机并进行相同条件的仿真,将该结果与MATLAB仿真结果进行对比,两者的仿真结果基本一致,表明所构建数学模型的正确性。仿真得出各杆驱动力的最大误差分别为0.091%、1.83%、1.04%、1.40%。研究结果表明,在给定的运动规律的条件下,机构没有产生刚性冲击,在运动的起点和终点处,会产生一定的柔性冲击,整个运动过程中各杆的驱动力变化平缓,驱动力二范数解所做的总功为1143.2 J。

A novel 2-UPR&2-RPU parallel mechanism with redundance structure was proposed,which has two rotations and one movement(2R1T)degree of freedom.Based on the screw theory,the degrees of freedom and motion characteristics of the mechanism were analyzed,and the correctness of the analysis was verified by the modified G-K formula.By analyzing the constraints of each branch,the reverse solutions of position was solved by establishing the closed-loop vector equations of each branch.The velocity and acceleration of the mechanism were analyzed by the differential transformation method based on symbolic operation,then the velocity Jacobian matrix and the second order influence coefficient matrixes were obtained.On the basis of this,the Jacobian matrix of each branch was also figured out.The dynamics model of the mechanism was constructed by using the virtual work principle based on the kinematics.For the character that the distribution of the active joints’drive forces was not unique due to the redundant actuation,the driving force and driving power equation were obtained by the method of 2-norm of driving force.Based on this,a calculation program was written for the constructed mathematical model by using MATLAB,and the constructed kinematics and dynamics model were simulated through specific numerical examples.Finally,in order to verify the correctness of the mathematical model constructed,a virtual prototype of the mechanism was established in ADAMS and simulated under the same conditions.The results were compared with those of MATLAB.The simulation results of both were basically consistent,indicating the correctness of the established mathematical model and showing the max errors of driving force of each rod with 0.091%,1.83%,1.04%,1.40%,respectively.The results showed that under the given motion law,the mechanism does not produce rigid impact.At the beginning and the end of the motion,a certain flexible impact will occur.The driving force of each driving rod changes smoothly during the entire movement,and the total work by the method of 2-norm of the driving force is 1143.2 J.