并联机构的机构分析及流形多体系统研究
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摘要
高速、高精度切削与高速物流系统是目前制造领域的重要发展趋势，因此需要提
高生产、装配及物流设备的精度和速度。传统的串联机构越来越难以适应这种要求。
九十年代问世的并联机床是结构技术上的突破性进展，给已经比较成熟的数控加工技
术带来了新的方向和希望，目前已经成为机器人技术、机床结构技术、数控技术等多
学科交叉的研究热点。
并联机床基于并联机构，是个新生事物。无论在理论方面还是实践方面都非常不
成熟，需要研究的内容非常多。本文针对并联机构的构型设计、解耦运动和多体动力
学Huston-Kane方法及其在并联机构中的应用做了研究。
本文的研究基于我们和香港科技大学合作研制的KD-2000并联原型机床，但不
限于此。实际上，本文的主要研究内容不仅适用于并联机构，也适用于一般机构。
本文的主要研究内容和创新点包括：
1、综述了并联机构、并联机床的研究、应用现状，列举了目前的研究方向和研
究方法。
2、基本成体系地介绍了微分几何中的李群、李代数理论，并分析其表达刚体运
动的作用，为本文的研究打下基础。
3、图论是机构综合、机构分析的基本理论工具。本文使用关联度码这一代数图
论方法，研究了一些并联机构的拓扑同构问题，包括KD-2000／ECLIPSE、Behi
3-PRPS、Alizade 3-PRPS、Tahmasebi-Tsai等机构。
4、提出加权刚度矩阵，这一方法是 A.C Rao提出的刚度矩阵方法的发展。A.C Rao
的刚度矩阵方法基于代数图论定性比较不同机构的拓扑刚度特性，但本论文中发现它
不能比较不同拓扑类型的机构。究其原因，是因为它没有考虑机构的构件数和运动副
的实际特性。因此，本文把运动副的刚度特性数据作为权赋给拓扑图中的相应边，形
成加权刚度矩阵方法。并使用此方法比较了传统堆叠式串联机床和并联机床的拓扑刚
度特性，指出KD-2000中球铰和圆周运动副是导致机床刚度较差的主要原因。
5、得出结论：拓扑同构不能反应机构的运动输入-输出本质。机构的运动不解
耦是并联机构的本质特性，因此本文讨论运动相互间的解耦特,比而非机构的特性。
6、首次提出了解耦运动的概念、分类，指出解耦运动对应的位姿子流形构成并
联机构位姿流形的正则子流形。首次提出了机构存在解耦运动的必要条件，提出了求
机构解耦运动的综合方法与分析方法。并得出结论6自由度并联机构实现运动和实现
解耦运动的能力不受构型影响，而取决于机构的几何约束。
7、首次提出了串联度的概念，它刻划了机构与串联机构的相似程度。并由此讨
论并联机构的分类，体现了机构运动输入-输出的本质关系。解释了J．P Merlet教授
国防科学技术大学研究生院学位论文
所述的一类所谓奇异构型（exotic）具有的内在性质。
8、通过一种二自山度转动机构的设计，体现了本文在机构综合、机构分析人仰
所做的工作。
9、通过对度量张量和Riemann张量的研究，得出并联机构运动可达子空间的内
在“弯曲”性质，指出使用相对坐标系和广义坐标是研究并联机构运动学和动力学问
题的必然选择。
10、使用微分几何理论研究了多体系统理论的 Huston－Kane方法，指出多体系统
理论的Huston－Kane方法完全可以嵌入到Lagrange力学体系中，前者只是LagrangC
动力学方程的变形。
11、首次指出 Huston－Kane方法中的相对角速度加法公式、旋转矩阵的导数与李
群、李代数的对应关系，给出了更加简单的证明，同时表述了其几何及物理含义。
12、指出偏速度、偏角速度与 Lagrange力学系统的对应，即它们构成运动微分
流形的子空间的自然标架场。更加清晰地阐述了这两个概念的意义及此自然标架场在
运动微分流形上的变化规律。
13、首次指出偏速度与偏角速度构成运动相空间（辛空间）的 Lagrange于流形，
揭示了偏速度与偏角速度的本质及Huston－Kane方法对动力学问题的变换思路。
14、建立了＊＊上00O井联机床的运动学、动力学方程，研究了其工作空间。并
对比了实验数据和仿真结果。
High speed, high accuracy and high speed material transportation system are the most important trends of manufacturing industry. It's urgent to improve the speed and accuracy of these equipments. But the traditional serial chain mechanisms can't fulfill these fast changing demands. The parallel machine tool, invented in 90's of the last century, is the breakthrough of mechanism. It brings new research direction and new prospect to NC manufacturing technology. Parallel machine tool has been the research hotpot of many studies in different fields, including robot, machine tool architecture and NC technology.
Parallel machine tool, based on parallel mechanism, is a brand new concept. The theory and technology of them is far from prefect. In this dissertation, the following aspects on parallel mechanism are studied, including mechanism design, decoupling movement and Huston-Kane method in multibody system theory.
Our research work on based a parallel machine tool prototype, named KD-2000. But not confined to it. Actually the research work in this dissertation can be applied to both parallel mechanism and generic mechanism.
Subsequent research contents and innovations are presented in this dissertation.
1. The research and application of parallel mechanism and parallel machine tool at present are summarized first. We enumerate the main research subjects and research methods.
2. We give a brief but systematic introduction to differentiable geometry and Lie group, Lie algebra. Rigid body motion expressed with the notations from these theories is also introduced. They are the theoretical bases of the subsequent research contents.
3. Graph theory is the most preliminary tool for mechanism design and mechanism analysis. With the use of incidence degree code, which based on graph theory, we studied the isomorphism of a few parallel mechanisms, including KD-2000/ECLIPSE, Behi 3-PRPS, Alizade 3-PRPS, Tahmasebi-Tsai, etc.
4. We bring forward weighted stiffness matrix, which is the development of A.C Rao's stiffness matrix. It can compare the stiffness of mechanisms in a more quantitative manner. vWe use weighted stiffness matrix to compare the stiffness of traditional machine tool and parallel machine tool. And reach the conclusion that the stiffness of joints, especially spheric hinge, is the key to improve the stiffness of parallel machine tool.
5. We draw the conclusion that isomorphism of mechanism can't express its motion in-out nature. Since the coupling of motion is the nature of parallel mechanism, we investigate the decoupling of motion, instead of the motion decoupling characteristic of mechanisms.
6. The concept of decoupling motion is brought forward for the first time, together with its classification. We point out that the sub manifold corresponds to decoupling
motion is a canonical sub manifold of configuration manifold. We bring forward the necessary condition for mechanism to achieve decoupling motion. And conclusion is reached the ability of 6-DOF mechanism to implement motion and decoupling motion is decided not by its type but by its geometry constraints.
7. The concept of serial degree is brought forward for the first time. It describes the resemblance of a mechanism to serial ones. So the motion in-out nature is described. For the so called exotic chain parallel mechanism, which was mentioned by Professor J.P Merlet, its motion nature is achieved.
8. We demonstrate a mechanism 'design example of a 2-DOF rotating mechanism. It incarnates our research work on mechanism design and mechanism analysis.
9. By the research of metric tensor and Riemann tensor on Riemann manifold, we get the inherent curvature of configuration space belonging to parallel mechanism. So the relative coordinates and generalized coordinates are inevitable choice for parallel mechanism.
10. We studied the Huston-Kane method of multibody system theory by the means of differentiable geometry. And point out that Huston-Kane method can be fully imbedded into Lagrange's mechanics. Huston-Kane method is o
高生产、装配及物流设备的精度和速度。传统的串联机构越来越难以适应这种要求。
九十年代问世的并联机床是结构技术上的突破性进展，给已经比较成熟的数控加工技
术带来了新的方向和希望，目前已经成为机器人技术、机床结构技术、数控技术等多
学科交叉的研究热点。
并联机床基于并联机构，是个新生事物。无论在理论方面还是实践方面都非常不
成熟，需要研究的内容非常多。本文针对并联机构的构型设计、解耦运动和多体动力
学Huston-Kane方法及其在并联机构中的应用做了研究。
本文的研究基于我们和香港科技大学合作研制的KD-2000并联原型机床，但不
限于此。实际上，本文的主要研究内容不仅适用于并联机构，也适用于一般机构。
本文的主要研究内容和创新点包括：
1、综述了并联机构、并联机床的研究、应用现状，列举了目前的研究方向和研
究方法。
2、基本成体系地介绍了微分几何中的李群、李代数理论，并分析其表达刚体运
动的作用，为本文的研究打下基础。
3、图论是机构综合、机构分析的基本理论工具。本文使用关联度码这一代数图
论方法，研究了一些并联机构的拓扑同构问题，包括KD-2000／ECLIPSE、Behi
3-PRPS、Alizade 3-PRPS、Tahmasebi-Tsai等机构。
4、提出加权刚度矩阵，这一方法是 A.C Rao提出的刚度矩阵方法的发展。A.C Rao
的刚度矩阵方法基于代数图论定性比较不同机构的拓扑刚度特性，但本论文中发现它
不能比较不同拓扑类型的机构。究其原因，是因为它没有考虑机构的构件数和运动副
的实际特性。因此，本文把运动副的刚度特性数据作为权赋给拓扑图中的相应边，形
成加权刚度矩阵方法。并使用此方法比较了传统堆叠式串联机床和并联机床的拓扑刚
度特性，指出KD-2000中球铰和圆周运动副是导致机床刚度较差的主要原因。
5、得出结论：拓扑同构不能反应机构的运动输入-输出本质。机构的运动不解
耦是并联机构的本质特性，因此本文讨论运动相互间的解耦特,比而非机构的特性。
6、首次提出了解耦运动的概念、分类，指出解耦运动对应的位姿子流形构成并
联机构位姿流形的正则子流形。首次提出了机构存在解耦运动的必要条件，提出了求
机构解耦运动的综合方法与分析方法。并得出结论6自由度并联机构实现运动和实现
解耦运动的能力不受构型影响，而取决于机构的几何约束。
7、首次提出了串联度的概念，它刻划了机构与串联机构的相似程度。并由此讨
论并联机构的分类，体现了机构运动输入-输出的本质关系。解释了J．P Merlet教授
国防科学技术大学研究生院学位论文
所述的一类所谓奇异构型（exotic）具有的内在性质。
8、通过一种二自山度转动机构的设计，体现了本文在机构综合、机构分析人仰
所做的工作。
9、通过对度量张量和Riemann张量的研究，得出并联机构运动可达子空间的内
在“弯曲”性质，指出使用相对坐标系和广义坐标是研究并联机构运动学和动力学问
题的必然选择。
10、使用微分几何理论研究了多体系统理论的 Huston－Kane方法，指出多体系统
理论的Huston－Kane方法完全可以嵌入到Lagrange力学体系中，前者只是LagrangC
动力学方程的变形。
11、首次指出 Huston－Kane方法中的相对角速度加法公式、旋转矩阵的导数与李
群、李代数的对应关系，给出了更加简单的证明，同时表述了其几何及物理含义。
12、指出偏速度、偏角速度与 Lagrange力学系统的对应，即它们构成运动微分
流形的子空间的自然标架场。更加清晰地阐述了这两个概念的意义及此自然标架场在
运动微分流形上的变化规律。
13、首次指出偏速度与偏角速度构成运动相空间（辛空间）的 Lagrange于流形，
揭示了偏速度与偏角速度的本质及Huston－Kane方法对动力学问题的变换思路。
14、建立了＊＊上00O井联机床的运动学、动力学方程，研究了其工作空间。并
对比了实验数据和仿真结果。
High speed, high accuracy and high speed material transportation system are the most important trends of manufacturing industry. It's urgent to improve the speed and accuracy of these equipments. But the traditional serial chain mechanisms can't fulfill these fast changing demands. The parallel machine tool, invented in 90's of the last century, is the breakthrough of mechanism. It brings new research direction and new prospect to NC manufacturing technology. Parallel machine tool has been the research hotpot of many studies in different fields, including robot, machine tool architecture and NC technology.
Parallel machine tool, based on parallel mechanism, is a brand new concept. The theory and technology of them is far from prefect. In this dissertation, the following aspects on parallel mechanism are studied, including mechanism design, decoupling movement and Huston-Kane method in multibody system theory.
Our research work on based a parallel machine tool prototype, named KD-2000. But not confined to it. Actually the research work in this dissertation can be applied to both parallel mechanism and generic mechanism.
Subsequent research contents and innovations are presented in this dissertation.
1. The research and application of parallel mechanism and parallel machine tool at present are summarized first. We enumerate the main research subjects and research methods.
2. We give a brief but systematic introduction to differentiable geometry and Lie group, Lie algebra. Rigid body motion expressed with the notations from these theories is also introduced. They are the theoretical bases of the subsequent research contents.
3. Graph theory is the most preliminary tool for mechanism design and mechanism analysis. With the use of incidence degree code, which based on graph theory, we studied the isomorphism of a few parallel mechanisms, including KD-2000/ECLIPSE, Behi 3-PRPS, Alizade 3-PRPS, Tahmasebi-Tsai, etc.
4. We bring forward weighted stiffness matrix, which is the development of A.C Rao's stiffness matrix. It can compare the stiffness of mechanisms in a more quantitative manner. vWe use weighted stiffness matrix to compare the stiffness of traditional machine tool and parallel machine tool. And reach the conclusion that the stiffness of joints, especially spheric hinge, is the key to improve the stiffness of parallel machine tool.
5. We draw the conclusion that isomorphism of mechanism can't express its motion in-out nature. Since the coupling of motion is the nature of parallel mechanism, we investigate the decoupling of motion, instead of the motion decoupling characteristic of mechanisms.
6. The concept of decoupling motion is brought forward for the first time, together with its classification. We point out that the sub manifold corresponds to decoupling
motion is a canonical sub manifold of configuration manifold. We bring forward the necessary condition for mechanism to achieve decoupling motion. And conclusion is reached the ability of 6-DOF mechanism to implement motion and decoupling motion is decided not by its type but by its geometry constraints.
7. The concept of serial degree is brought forward for the first time. It describes the resemblance of a mechanism to serial ones. So the motion in-out nature is described. For the so called exotic chain parallel mechanism, which was mentioned by Professor J.P Merlet, its motion nature is achieved.
8. We demonstrate a mechanism 'design example of a 2-DOF rotating mechanism. It incarnates our research work on mechanism design and mechanism analysis.
9. By the research of metric tensor and Riemann tensor on Riemann manifold, we get the inherent curvature of configuration space belonging to parallel mechanism. So the relative coordinates and generalized coordinates are inevitable choice for parallel mechanism.
10. We studied the Huston-Kane method of multibody system theory by the means of differentiable geometry. And point out that Huston-Kane method can be fully imbedded into Lagrange's mechanics. Huston-Kane method is o
引文
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