并联机构奇异位形的微分几何理论以及冗余并联机构的研究
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摘要
随着并联机构向实用化和产品化方向发展，对并联机构的性能提出了越来越高的要求。
丰富而复杂的奇异位形是并联机构的一个重要特点，位形空间中的奇异位形对并联机构的精
度、刚度以及运动性能有着重要的影响。对并联机构奇异位形的特性进行深入的研究是减小
和消除奇异位形的影响，提高并联机构性能的基础。
并联机构的奇异位形和冗余问题已有较多的学者进行了研究，但还没有形成一个系统的
理论，还有很多的问题尚待解决，本论文将在这方面做一些有益的探索工作。
本论文将首先研究并联机构奇异位形的分类问题。采用微分拓扑和微分流形等现代数学
工具，在对并联机构位形空间的拓扑结构进行分析的基础上，提出了一种新的奇异位形的分
类方法，即把奇异位形分为拓扑奇异位形、参数化奇异位形两种类型，这种分类方法充分体
现了并联机构位形空间的特点，具有十分明确的物理和数学意义。
其次对参数化奇异位形进行了深入的研究，分析了并联机构位形空间中参数化奇异流形
的特点及分布，根据参数化流形的切空间与零化子空间的关系提出了一阶参数化奇异位形和
二阶参数化奇异位形的概念。研究了退化和非退化奇异位形的特点，给出了判断退化参数化
奇异位形的充分和必要条件，其中对必要条件，提出了一种方便实用的Morse判断函数。
研究了一个与参数化奇异位形有关的问题，即并联机构运动学正解不同解之间无奇异位
形连接路径问题，采用微分几何的基本概念解释了同一驱动关节对应的不同正解之间是有可
能有不经过驱动奇异位形的路径的原因。
研究奇异位形的目的是为了避免和消除奇异位形对机构的影响，而机构冗余是消除奇异
位形，提高机构性能的重要方法。
在冗余问题上，本论文首先研究冗余的分类问题，根据并联机构所受的约束以及驱动
器、末端执行器的选择，提出了一种新的分类方法，把并联机构的冗余分为位形空间冗余、
驱动冗余以及末端执行器冗余三种类型，并分析了各种冗余情况下的奇异位形，这种分类方
法能较好的说明各种冗余并联机构的特点。
其次对驱动冗余进行了深入的研究，驱动冗余是并联机构所特有的一种冗余，它也是并
联机构消除奇异位形影响的主要的冗余方法。分析了驱动奇异位形对并联机构精度、刚度和
动力学性能的影响。然后引入驱动冗余，详细分析了冗余后机构精度、刚度以及动力学性能
提高的原因。
最后对冗余并联机构的控制问题进行了初步研究，采用伪逆的方法把串联机构的控制方
法引入到冗余并联机构中，并对一个二自由度冗余平面并联机构进行了控制实验，结果表明
机构具有较好的定位精度以及高速性能。
With the development of parallel mechanisms towards practicalization and productization, the requirements for high performance of mechanisms are more and more critical. There are abundant and complicated singularities in parallel mechanisms, which have great effects on the accuracy, stiffness and motion ability of parallel mechanisms. Deep and clear understanding of singularities of parallel mechanisms is the fundament to reduce or eliminate the effect of singularity and improve the performance of parallel mechanisms.
Although many authors have done a lot of research on singularities of parallel mechanisms, there is not a systematical theory yet, many problems are still to be solved. This dissertation will do some helpful research on this area.
First, the classification of singularities of parallel mechanisms is studied. Based on the analysis of topology structure of parallel mechanisms and using differential topology and differential manifolds as mathematical tools, we propose a new classification method. This method classifies singularities of parallel mechanisms into two basic types, i.e. topology singularity and parameterization singularity. This kind of classification has clear physical and mathematical meaning and fully reveals the characteristic of configuration space of parallel mechanisms.
Secondly, further research is performed on parameterization singularity. The property and distribution of parameterization singularity manifolds are analyzed. According to the relation between tangent space and annihilation space of parameterization singularity manifolds, we present the new concepts of first order and second order singularity. Degenerate and non-degenerate parameterization singularities are analyzed and the sufficient and necessary conditions for degenerate singularity are derived. For necessary condition, we present a practical Morse function to judge.
Thirdly, a problem related to parameterization singularity and forward kinematics is studied, i.e., whether there exists singularity-free path between different solutions of forward kinematics. We give a clear explanation to this problem using standard idea from differential geometry.
In order to avoid or eliminate the effect of singularity and improve the performance of parallel mechanisms, the redundancy is applied to the parallel mechanisms based the studying of singularity.
First, the classification of redundancy of parallel mechanisms is studied. According to the mechanism constraints and the choice of actuators and end-effectors, we present a new classification method, which identifies three basic types redundancy: configuration redundancy, actuator redundancy and end-effector redundancy. And singularities are analyzed for all types of redundancy. This method gives us an unambiguous understanding of redundant parallel mechanisms.
Secondly, we make a deep study of actuator redundancy. Actuator redundancy is a unique redundancy for parallel mechanism and it is also the main redundancy to eliminate the singularity effects on parallel mechanisms. At first, singularity effects on accuracy, stiffness and dynamics are analyzed, and then the reasons of improving accuracy, stiffness and
dynamics by introducing actuator redundancy are analyzed in detail.
Finally, we make a preliminary study on the control problem of redundant parallel mechanisms. Using pseudo-inverse method, control strategies for serial mechanisms are applied to redundant parallel mechanisms. A 2-DOF redundant planar parallel mechanism is used as control example; the experiment results show that the redundant mechanism has good localization accuracy and good performance at high speeds.
丰富而复杂的奇异位形是并联机构的一个重要特点，位形空间中的奇异位形对并联机构的精
度、刚度以及运动性能有着重要的影响。对并联机构奇异位形的特性进行深入的研究是减小
和消除奇异位形的影响，提高并联机构性能的基础。
并联机构的奇异位形和冗余问题已有较多的学者进行了研究，但还没有形成一个系统的
理论，还有很多的问题尚待解决，本论文将在这方面做一些有益的探索工作。
本论文将首先研究并联机构奇异位形的分类问题。采用微分拓扑和微分流形等现代数学
工具，在对并联机构位形空间的拓扑结构进行分析的基础上，提出了一种新的奇异位形的分
类方法，即把奇异位形分为拓扑奇异位形、参数化奇异位形两种类型，这种分类方法充分体
现了并联机构位形空间的特点，具有十分明确的物理和数学意义。
其次对参数化奇异位形进行了深入的研究，分析了并联机构位形空间中参数化奇异流形
的特点及分布，根据参数化流形的切空间与零化子空间的关系提出了一阶参数化奇异位形和
二阶参数化奇异位形的概念。研究了退化和非退化奇异位形的特点，给出了判断退化参数化
奇异位形的充分和必要条件，其中对必要条件，提出了一种方便实用的Morse判断函数。
研究了一个与参数化奇异位形有关的问题，即并联机构运动学正解不同解之间无奇异位
形连接路径问题，采用微分几何的基本概念解释了同一驱动关节对应的不同正解之间是有可
能有不经过驱动奇异位形的路径的原因。
研究奇异位形的目的是为了避免和消除奇异位形对机构的影响，而机构冗余是消除奇异
位形，提高机构性能的重要方法。
在冗余问题上，本论文首先研究冗余的分类问题，根据并联机构所受的约束以及驱动
器、末端执行器的选择，提出了一种新的分类方法，把并联机构的冗余分为位形空间冗余、
驱动冗余以及末端执行器冗余三种类型，并分析了各种冗余情况下的奇异位形，这种分类方
法能较好的说明各种冗余并联机构的特点。
其次对驱动冗余进行了深入的研究，驱动冗余是并联机构所特有的一种冗余，它也是并
联机构消除奇异位形影响的主要的冗余方法。分析了驱动奇异位形对并联机构精度、刚度和
动力学性能的影响。然后引入驱动冗余，详细分析了冗余后机构精度、刚度以及动力学性能
提高的原因。
最后对冗余并联机构的控制问题进行了初步研究，采用伪逆的方法把串联机构的控制方
法引入到冗余并联机构中，并对一个二自由度冗余平面并联机构进行了控制实验，结果表明
机构具有较好的定位精度以及高速性能。
With the development of parallel mechanisms towards practicalization and productization, the requirements for high performance of mechanisms are more and more critical. There are abundant and complicated singularities in parallel mechanisms, which have great effects on the accuracy, stiffness and motion ability of parallel mechanisms. Deep and clear understanding of singularities of parallel mechanisms is the fundament to reduce or eliminate the effect of singularity and improve the performance of parallel mechanisms.
Although many authors have done a lot of research on singularities of parallel mechanisms, there is not a systematical theory yet, many problems are still to be solved. This dissertation will do some helpful research on this area.
First, the classification of singularities of parallel mechanisms is studied. Based on the analysis of topology structure of parallel mechanisms and using differential topology and differential manifolds as mathematical tools, we propose a new classification method. This method classifies singularities of parallel mechanisms into two basic types, i.e. topology singularity and parameterization singularity. This kind of classification has clear physical and mathematical meaning and fully reveals the characteristic of configuration space of parallel mechanisms.
Secondly, further research is performed on parameterization singularity. The property and distribution of parameterization singularity manifolds are analyzed. According to the relation between tangent space and annihilation space of parameterization singularity manifolds, we present the new concepts of first order and second order singularity. Degenerate and non-degenerate parameterization singularities are analyzed and the sufficient and necessary conditions for degenerate singularity are derived. For necessary condition, we present a practical Morse function to judge.
Thirdly, a problem related to parameterization singularity and forward kinematics is studied, i.e., whether there exists singularity-free path between different solutions of forward kinematics. We give a clear explanation to this problem using standard idea from differential geometry.
In order to avoid or eliminate the effect of singularity and improve the performance of parallel mechanisms, the redundancy is applied to the parallel mechanisms based the studying of singularity.
First, the classification of redundancy of parallel mechanisms is studied. According to the mechanism constraints and the choice of actuators and end-effectors, we present a new classification method, which identifies three basic types redundancy: configuration redundancy, actuator redundancy and end-effector redundancy. And singularities are analyzed for all types of redundancy. This method gives us an unambiguous understanding of redundant parallel mechanisms.
Secondly, we make a deep study of actuator redundancy. Actuator redundancy is a unique redundancy for parallel mechanism and it is also the main redundancy to eliminate the singularity effects on parallel mechanisms. At first, singularity effects on accuracy, stiffness and dynamics are analyzed, and then the reasons of improving accuracy, stiffness and
dynamics by introducing actuator redundancy are analyzed in detail.
Finally, we make a preliminary study on the control problem of redundant parallel mechanisms. Using pseudo-inverse method, control strategies for serial mechanisms are applied to redundant parallel mechanisms. A 2-DOF redundant planar parallel mechanism is used as control example; the experiment results show that the redundant mechanism has good localization accuracy and good performance at high speeds.
引文
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