基于微分流形的全对称少自由度并联机构基础理论研究
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摘要
本论文以李群李代数、微分流形理论为基础,综合考虑运动特性和约束特性的影响,对全对称少自由度并联机构的运动学分析、精度分析、静力与静刚度分析的完整建模进行了深入系统的研究,并试图为机构的运动学设计、精度设计和静刚度设计的公共性能指标的提炼提供理论依据。本论文的研究工作得到国家自然科学基金重点项目(50535010)的资助,是该项目理论工作的一部分。论文主要工作如下:
1)总结了描述刚体运动的李群李代数和微分流形理论基础,并分析了受约束刚体的(切空间)速度空间和力空间(余切空间)的构成。此外在微分流形理论统一框架内,分析了运动螺旋和力螺旋与切空间和余切空间之间本质关联,以及D-H法和指数积公式的等效性。以2R机构为例,对上述分析进行了详细说明。
2)根据全对称少自由度并联机构的结构特征,综合运用D-H法、位移群上的平移计算以及对偶螺旋理论给出了此类机构运动学完整建模的一般方法,得到完整6×6雅可比矩阵,包括运动子雅可比矩阵和约束子雅可比矩阵,这为后续的误差建模分析和静刚度建模分析奠定了统一的基础。最后以Tsai 3-UPU机构和SNU 3-UPU机构为例进行了实例分析。
3)首次对全对称少自由度并联机构的可控误差和不可控误差作出了基于微分流形理论的数学解释。基于位移群上的平移计算和对偶螺旋理论,提出了此类机构的一种新的完整误差建模方法,建立了动平台不可控误差、可控误差分别与各自几何误差源之间的线性映射关系。前者可用于指导零部件制造/装配的公差设计;后者可用作机构运动学标定的基础公式。最后,以Tsai 3-UPU机构为例进行了详细分析。
4)首次对全对称少自由度并联机构支链中驱动力和约束力的传递作出了基于微分流形理论的数学解释。利用虚功原理,给出了此类机构的一种静力分析和静刚度分析的完整建模方法。除了得到传统的驱动力映射和驱动静刚度映射外,还建立了约束力映射和约束静刚度映射。最后,以一般3-UPU机构为例进行了详细说明。并通过算例首次分析了约束力映射和约束刚度分析对于全对称少自由度并联机构几何参数设计的必要性。
Based on the theory of differential manifold, the theory and methodology for modeling of the fully symmetric parallel mechanisms with lower-mobility are studied by considering of the characteristics of both kinematics and constraint, including kinematics, accuracy, static and stiffness. The research intends to provide the theoretical guide for distilling a universal performance index about kinematics, accuracy, static and stiffness. The dissertation is organized as following.
1) Preliminary theory of differential manifold describing the motion of a rigid body is analyzed, which shows that the tangent and the cotangent space consist of two orthogonal subspaces respectively when some degrees-of-freedom (DOFs) of the body are limited. In the unified frame, the relationship between twist and tangent space, wrench and cotangent space, equivalent relationship between D-H method and product-of-exponentials (POE) are presented respectively. A 2R mechanism is taken as an example to illustrate the above analysis.
2) According to the fully symmetric parallel mechanisms (FSPM) with lower-mobility, a general method for kinematic modeling of the mechanisms is presented by integrating D-H method, translation operation on the Group with the theory of reciprocal screw, in which the complete 6×6 Jacobian matrix is obtained including kinematic and constraint sub-matrices. It lays the unified foundation for both error and static stiffness modeling. Tsai’s 3-UPU parallel mechanism is taken as an example to illustrate the method.
3) By using the theory of differential manifold, an explanation is presented for the controllable and uncontrollable error of the FSPM with lower-mobility. A new method is proposed to modeling error of the mechanism based on the translation operation and reciprocal screw theory, in which the linear mapping between the controllable and uncontrollable error and their source can be set up. The former can be used to instruct tolerance design of parts, and the latter mainly act as a fundamental formula. The detailed illustration is made through an example of Tsai’s 3-UPU.
4) The explanation about transmissions of actuation and constraint force of the FSPM with lower-mobility is first presented based on the theory of differential manifold. By using the theory of virtual work, an approach is proposed to obtain the complete static and stiffness, in which constraint force and constraint stiffness mapping are also obtained besides traditional actuation force and actuation stiffness mapping. It is demonstrated in detail through an example of general 3-UPU. The analysis of constraint force and constraint stiffness is indicated to be necessary for geometric design of these parallel mechanisms.
1)总结了描述刚体运动的李群李代数和微分流形理论基础,并分析了受约束刚体的(切空间)速度空间和力空间(余切空间)的构成。此外在微分流形理论统一框架内,分析了运动螺旋和力螺旋与切空间和余切空间之间本质关联,以及D-H法和指数积公式的等效性。以2R机构为例,对上述分析进行了详细说明。
2)根据全对称少自由度并联机构的结构特征,综合运用D-H法、位移群上的平移计算以及对偶螺旋理论给出了此类机构运动学完整建模的一般方法,得到完整6×6雅可比矩阵,包括运动子雅可比矩阵和约束子雅可比矩阵,这为后续的误差建模分析和静刚度建模分析奠定了统一的基础。最后以Tsai 3-UPU机构和SNU 3-UPU机构为例进行了实例分析。
3)首次对全对称少自由度并联机构的可控误差和不可控误差作出了基于微分流形理论的数学解释。基于位移群上的平移计算和对偶螺旋理论,提出了此类机构的一种新的完整误差建模方法,建立了动平台不可控误差、可控误差分别与各自几何误差源之间的线性映射关系。前者可用于指导零部件制造/装配的公差设计;后者可用作机构运动学标定的基础公式。最后,以Tsai 3-UPU机构为例进行了详细分析。
4)首次对全对称少自由度并联机构支链中驱动力和约束力的传递作出了基于微分流形理论的数学解释。利用虚功原理,给出了此类机构的一种静力分析和静刚度分析的完整建模方法。除了得到传统的驱动力映射和驱动静刚度映射外,还建立了约束力映射和约束静刚度映射。最后,以一般3-UPU机构为例进行了详细说明。并通过算例首次分析了约束力映射和约束刚度分析对于全对称少自由度并联机构几何参数设计的必要性。
Based on the theory of differential manifold, the theory and methodology for modeling of the fully symmetric parallel mechanisms with lower-mobility are studied by considering of the characteristics of both kinematics and constraint, including kinematics, accuracy, static and stiffness. The research intends to provide the theoretical guide for distilling a universal performance index about kinematics, accuracy, static and stiffness. The dissertation is organized as following.
1) Preliminary theory of differential manifold describing the motion of a rigid body is analyzed, which shows that the tangent and the cotangent space consist of two orthogonal subspaces respectively when some degrees-of-freedom (DOFs) of the body are limited. In the unified frame, the relationship between twist and tangent space, wrench and cotangent space, equivalent relationship between D-H method and product-of-exponentials (POE) are presented respectively. A 2R mechanism is taken as an example to illustrate the above analysis.
2) According to the fully symmetric parallel mechanisms (FSPM) with lower-mobility, a general method for kinematic modeling of the mechanisms is presented by integrating D-H method, translation operation on the Group with the theory of reciprocal screw, in which the complete 6×6 Jacobian matrix is obtained including kinematic and constraint sub-matrices. It lays the unified foundation for both error and static stiffness modeling. Tsai’s 3-UPU parallel mechanism is taken as an example to illustrate the method.
3) By using the theory of differential manifold, an explanation is presented for the controllable and uncontrollable error of the FSPM with lower-mobility. A new method is proposed to modeling error of the mechanism based on the translation operation and reciprocal screw theory, in which the linear mapping between the controllable and uncontrollable error and their source can be set up. The former can be used to instruct tolerance design of parts, and the latter mainly act as a fundamental formula. The detailed illustration is made through an example of Tsai’s 3-UPU.
4) The explanation about transmissions of actuation and constraint force of the FSPM with lower-mobility is first presented based on the theory of differential manifold. By using the theory of virtual work, an approach is proposed to obtain the complete static and stiffness, in which constraint force and constraint stiffness mapping are also obtained besides traditional actuation force and actuation stiffness mapping. It is demonstrated in detail through an example of general 3-UPU. The analysis of constraint force and constraint stiffness is indicated to be necessary for geometric design of these parallel mechanisms.
引文
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