一类2R1T三自由度并联机构的设计理论与方法研究
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摘要
本文密切结合开发特种加工并联运动装备的需求,系统研究了少自由度并联机构的运动学性能指标体系,并以两种两转一移三自由度并联机构为例,研究了其尺度综合、位置正解分析、精度分析与综合等问题。论文取得了如下创造性成果:
分析了雅可比矩阵构造原理,利用约束条件建立了少自由度并联机构的全速度雅可比矩阵通用模型。该模型行列数均与位形空间维数相同,包含了对末端6维误差产生影响的尺度参数信息。基于该模型,利用矩阵Rayleigh商的性质提出一种基于支链误差传递率的性能评价指标。
以3-UPS/PU和3-PUS/PU机构为研究对象,构建其全速度雅可比矩阵。在此基础上,系统研究了机构尺度综合方法。该方法首先给定任务工作空间W_t,在考虑必要的约束条件的基础上,确定设计变量的取值范围,继而采用遗传算法得到最优解。研究结果表明,对于等同的任务空间,3-PUS/PU机构的操作性能要略优于3-U(-|P)S/PU机构。
采用解析法研究了3-U(-|P)S/PU和3-(-|P)US/PU机构的位置正解问题。研究结果表明,两者的位置正解模型无论从方程结构还是维数上均相同。
利用空间矢量链分析法,研究了3-U(-|P)S/PU和3-(-|P)US/PU并联机构的误差建模问题。由误差模型可知,由PU支链虎克铰的加工和装配误差,以及套筒导轨扭角误差引起的末端误差为不可控误差。在此基础上,预估了末端执行器在工作空间内的位姿误差范围。研究结果表明,对于等同的任务空间,3-(-|P)US/PU机构的几何精度要略优于3-U(-|P)S/PU机构。
借助灵敏度分析方法,在统计意义下定量揭示出几何误差源对3-UPS/PU和3-(-|P)US/PU并联机构末端误差的影响。基于此,系统研究了机构的精度综合方法。该方法以构件制造公差为设计变量,根据灵敏度分析结果确定并简化权重指标,以设计变量加权和最大为目标函数,进而建立机构精度综合的优化数学模型,最后以误差满足设计精度且兼顾公差均衡为约束条件,采用遗传算法求解。研究结果表明,对于等同的任务空间,两种机构的构件制造公差总体相差较小。
上述研究成果不但为指导3-UPS/PU和3-PUS/PU机构的设计奠定了坚实的理论基础,而且对开发同类制造装备具有重要的参考价值。
This dissertation deals with the theory and methodology for the kinematic design of a class of 2R1T 3-DOF parallel mechanisms, including Jacobian matrix and performance index, dimensional synthesis, forward kinematics, accuracy analysis and synthesis. The following contributions have been made.
The constructing principles of Jacobian matrix are analyzed. Then a general model of full 6×6 Jacobian matrix for sub-6 parallel mechanisms is developed using the constraining conditions, which contains the dimensional information affecting the general errors of the end-effector. Based on such model, the ratio of the limb error onto the platform pose accuracy is proposed to be a performance index by using the characteristics of Rayleigh quotient.
By taking 3-U(-|P)S/PU and 3-(-|P)US/PU mechanisms as examples, a methodology for the dimensional synthesis is presented on the basis of the development of the full 6×6 Jacobian matrix. In this method, the performance index is to be optimized in a global sense in conjunction with taking into account a set of appropriate mechanical constraints. Examples are given to illustrate the routine of the optimization. The results show that the 3-PUS/PU mechanism has little better kinematic performance in comparison with the 3-UPS/PU mechanism.
Forward kinematics analysis of the 3-U(-|P)S/PU mechanism is carried out using analytical method, compared with that of the 3-PUS/PU mechanism. It shows that the forward kinematics problem of the 3-U(-|P)S/PU mechanism is similar to that of the 3-PUS/PU mechanism in terms of formulation and solution.
The error mapping functions of the 3-UPS/PU and 3-PUS/PU mechanisms are formulated which allows the geometric error sources affecting the uncontrollable errors of the movable platform to be identified by using method of vector chain. It can be concluded that the uncontrollable errors of the mobile platform are primarily dominated by the errors of the proper-constraint limb. Based on the error models, the accuracies of the mechanisms are evaluated throughout the task workspace. It shows that for the identical task workspace, the accuracy of the 3-(-|P)US/PU mechanism is little better than that of the 3-U(-|P)S/PU mechanism.
The effects of the geometric error sources on the uncontrollable errors of the 3-UPS/PU and 3-PUS/PU mechanisms are discussed by means of the sensitivity analysis. Consequently, a novel approach for accuracy synthesis is presented. Such method can be simply depicted as follows: a mathematical model of optimization for accuracy synthesis is established by taking accuracy of connecting limbs and joints as design variables, a properly simplified sensitivity index as weight and the sum of the weighed design variables as a target function, and making position and pose errors within the design criterion.The accuracy synthesis is carried out using genetic algorithm. It concludes that the 3-U(-|P)S/PU and 3-PUS/PU mechanisms have little difference on the general tolerances of the parts provided that they share identical task workspace.
The outcome of this dissertation has provided a complete theoretical package for the development of the 3-UPS/PU and 3-PUS/PU mechanisms. And it’s also suitable to the development of some others similar parallel mechanism.
分析了雅可比矩阵构造原理,利用约束条件建立了少自由度并联机构的全速度雅可比矩阵通用模型。该模型行列数均与位形空间维数相同,包含了对末端6维误差产生影响的尺度参数信息。基于该模型,利用矩阵Rayleigh商的性质提出一种基于支链误差传递率的性能评价指标。
以3-UPS/PU和3-PUS/PU机构为研究对象,构建其全速度雅可比矩阵。在此基础上,系统研究了机构尺度综合方法。该方法首先给定任务工作空间W_t,在考虑必要的约束条件的基础上,确定设计变量的取值范围,继而采用遗传算法得到最优解。研究结果表明,对于等同的任务空间,3-PUS/PU机构的操作性能要略优于3-U(-|P)S/PU机构。
采用解析法研究了3-U(-|P)S/PU和3-(-|P)US/PU机构的位置正解问题。研究结果表明,两者的位置正解模型无论从方程结构还是维数上均相同。
利用空间矢量链分析法,研究了3-U(-|P)S/PU和3-(-|P)US/PU并联机构的误差建模问题。由误差模型可知,由PU支链虎克铰的加工和装配误差,以及套筒导轨扭角误差引起的末端误差为不可控误差。在此基础上,预估了末端执行器在工作空间内的位姿误差范围。研究结果表明,对于等同的任务空间,3-(-|P)US/PU机构的几何精度要略优于3-U(-|P)S/PU机构。
借助灵敏度分析方法,在统计意义下定量揭示出几何误差源对3-UPS/PU和3-(-|P)US/PU并联机构末端误差的影响。基于此,系统研究了机构的精度综合方法。该方法以构件制造公差为设计变量,根据灵敏度分析结果确定并简化权重指标,以设计变量加权和最大为目标函数,进而建立机构精度综合的优化数学模型,最后以误差满足设计精度且兼顾公差均衡为约束条件,采用遗传算法求解。研究结果表明,对于等同的任务空间,两种机构的构件制造公差总体相差较小。
上述研究成果不但为指导3-UPS/PU和3-PUS/PU机构的设计奠定了坚实的理论基础,而且对开发同类制造装备具有重要的参考价值。
This dissertation deals with the theory and methodology for the kinematic design of a class of 2R1T 3-DOF parallel mechanisms, including Jacobian matrix and performance index, dimensional synthesis, forward kinematics, accuracy analysis and synthesis. The following contributions have been made.
The constructing principles of Jacobian matrix are analyzed. Then a general model of full 6×6 Jacobian matrix for sub-6 parallel mechanisms is developed using the constraining conditions, which contains the dimensional information affecting the general errors of the end-effector. Based on such model, the ratio of the limb error onto the platform pose accuracy is proposed to be a performance index by using the characteristics of Rayleigh quotient.
By taking 3-U(-|P)S/PU and 3-(-|P)US/PU mechanisms as examples, a methodology for the dimensional synthesis is presented on the basis of the development of the full 6×6 Jacobian matrix. In this method, the performance index is to be optimized in a global sense in conjunction with taking into account a set of appropriate mechanical constraints. Examples are given to illustrate the routine of the optimization. The results show that the 3-PUS/PU mechanism has little better kinematic performance in comparison with the 3-UPS/PU mechanism.
Forward kinematics analysis of the 3-U(-|P)S/PU mechanism is carried out using analytical method, compared with that of the 3-PUS/PU mechanism. It shows that the forward kinematics problem of the 3-U(-|P)S/PU mechanism is similar to that of the 3-PUS/PU mechanism in terms of formulation and solution.
The error mapping functions of the 3-UPS/PU and 3-PUS/PU mechanisms are formulated which allows the geometric error sources affecting the uncontrollable errors of the movable platform to be identified by using method of vector chain. It can be concluded that the uncontrollable errors of the mobile platform are primarily dominated by the errors of the proper-constraint limb. Based on the error models, the accuracies of the mechanisms are evaluated throughout the task workspace. It shows that for the identical task workspace, the accuracy of the 3-(-|P)US/PU mechanism is little better than that of the 3-U(-|P)S/PU mechanism.
The effects of the geometric error sources on the uncontrollable errors of the 3-UPS/PU and 3-PUS/PU mechanisms are discussed by means of the sensitivity analysis. Consequently, a novel approach for accuracy synthesis is presented. Such method can be simply depicted as follows: a mathematical model of optimization for accuracy synthesis is established by taking accuracy of connecting limbs and joints as design variables, a properly simplified sensitivity index as weight and the sum of the weighed design variables as a target function, and making position and pose errors within the design criterion.The accuracy synthesis is carried out using genetic algorithm. It concludes that the 3-U(-|P)S/PU and 3-PUS/PU mechanisms have little difference on the general tolerances of the parts provided that they share identical task workspace.
The outcome of this dissertation has provided a complete theoretical package for the development of the 3-UPS/PU and 3-PUS/PU mechanisms. And it’s also suitable to the development of some others similar parallel mechanism.
引文
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118 Kim S-G, Ryu J, New dimensionally homogeneous jacobian matrix formulation by three end-effector points for optimal design of parallel manipulators, IEEE Transactions on Robotics and Automation, 2003, 19(4): 731-736
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