并联机构奇异性分析及免奇异方法研究
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摘要
并联机构以其特有的刚度大、承载能力强、累积误差小、控制精度高等优点适应了社会生产发展的需要，并在航空航天、精密机械加工、工业机器人等高科技领域得到了应用。随着并联机构的广泛应用，并联机构的奇异性问题引起了众多国内外学者的关注。奇异性是并联机构的固有特性，当机构处于奇异位置时，运动出现分岔现象，使机构的输出运动不确定。因此，如何控制机构以确定的运动通过奇异位置是实现并联机器人控制的关键。本文利用奇异性理论，对并联机构奇异位置处的奇异性规避问题进行了研究，解决了机构以确定的运动通过边界奇异位置的关键问题，并研究了与机构结构参数相关的复杂奇异性问题和有效减少机构奇异性的问题。
首次在复数域内研究机构的奇异性问题。以Watt六杆机构为例，讨论了多环机构的奇异性问题，在复数域内建立的机构分岔分析方程，形象直观，芽空间转换非常方便。基于分岔方程的多解性和奇异性理论，分析了多环机构的奇异构型和机构的分岔性态与机构参数的关系。
应用奇异性理论分析Stewart并联机构的构型分岔特性。建立了机构的分岔方程，利用Jacobian矩阵确定机构的奇异位置，根据奇异性理论的静态分岔条件来判断机构的分岔点，并将分岔方程的部分导数特性用于确定分岔点的分岔特性。提出了构型转换影响系数的概念，该系数揭示了奇异位置附近机构构型的变换规律，为机构运动控制提供了理论基础，同时也为利用开折的方法实现转向分岔点免奇异奠定了基础。
首次用数值的方法研究了对称6-6型Stewart并联机构多参数复杂奇异性问题。6-6型Stewart并联机构具有六个自由度，六个输入参数可以作为机构的分岔参数，不同的分岔参数引起机构出现不同的分岔现象。根据分岔参数组合对机构分岔的影响，将多分岔参数组合共分为八种情况，并对这八种情况下机构的奇异性和分岔点的分岔特性进行了详细讨论。分岔分析结果验证了以往研究中所出现的奇异位形，同时又发现了新的奇异位形。
首次讨论了机构的结构参数与并联机构奇异性的关系。详细讨论了Stewart并联机构动、静平台半径R_1和R_2、作动筒长度l_i和动、静平台铰链点夹角α_1和α_2对并联机构分岔特性的影响，并通过分析工作空间和稳定工作空间给出了上述参数选取的基本方法，为通过合理选择结构参数实现机构的免奇异或减少机构的奇异性奠定了理论基础。
从非线性奇异性理论出发，研究了并联机构边界奇异性的规避问题。分析了并联机构奇异位置的构型分岔特性，根据并联机构通过边界奇异位置的构型方式将边界奇异性规避问题分为三类进行了研究，分别构造了并联机构以这三种构型方式通过边界奇异位置的扰动函数，解决了机构以确定的运动通过边界奇异位置的关键问题。
最后给出了本论文的主要创新成果和有待于进一步研究的问题。
Parallel mechanism has the advantages of high structrual stiffness, highload-bearing capacity, and higher control precision, which meets the needs ofindustrial automatization.Now it has been widely used in the area of aeronautics,mechanical manufacturing, industrial robot and etc.The kinematic singularity, namely,the output movement of the mechanism is uncertain due to kenematic bifurcation atthe singular position, is the intrinsic characteristic of parallel mechanism, and this hasattracted the attention of the robotics researchers in the world.How to control themechanism to pass through the singular position with its given configuration is thekey problem for parallem mechanism.In this paper, using singularity theory, theauthor has investigated the approach of avoiding singularity at the singular positionfor parallem mechanisms, and solved the key problem that parallel mechanisms couldpass through the boundary singular position with its given configuration.The authoralso studied the relationship between the singularity and the parameters of parallelmechanisms and presented a method of improving the singularity.The mainachievements are as follows:
Singularity has been studied in complex number field for the first time.Takingthe Watt linkage as an example, the singularity for multi-loop mechanism isresearched.The bifurcation equation in complex number field has visual property, andcan be easily transformed into germ field.On the basis of the multi-solution of thebifurcation equation, the singularity theory is applied to analyze the singularconfigurations and bifurcation characteristic connecting with the structural parametersof mechanisms.
The bifurcation equation of Stewart parallel mechanism is established foranalyzing its bifurcation characteristic based on the singularity theory.The Jacobinmatrix is used to identify the singular points, and the static bifurcation condition isused to identify the bifurcation points.The partial differential of the bifurcationequation is applied to analyze its bifurcation characteristic at a bifurcation point.The conception of configuration transforming coefficient is presented in order to revealthe principle of the configuration transforming in the neighborhood of a singularposition.Therefore, it establishes theoretical basis for controlling the movement ofparallel mechanism and for avoiding the turning bifurcation point on the base of theunfolding theory.
Numerical method is applied to study the complex singularity for symmetrical6-Dof Stewart parallel mechanisms with multi-parameters bifurcation.Six inputparameters can be selected as the bifurcation parameters.Different composition of thebifurcation parameters would lead to different bifurcation phenomena.Eight cases areobtained for symmetrical Stewart parallel mechanisms with multi-parametersbifurcation.The corresponding singularity and bifurcation characteristics of theparallel mechanisms are investigated, respectively.The results verify the singularconfigurations presented in other literature and also give some new singularconfigurations for Stewart parallel mechanisms.
The relationship between the composition of parallel mechanisms and itssingularity is studied for the first time.The key parameters of Stewart parallelmechanisms include: the radiuses R~1 and R~2 for the fixed base frame and themovable platform respectively, the actuators length l~i, and the anglesα~1andα~2these parameters' influence on bifurcation behavior has been disscussed, and themethod of selecting appropriate parameters is presented according to the workspaceand the singularity-free workspace, which can improve singularity of the movement.
The method of avoiding the boundary singularity is investigated for the parallelmechanisms based on the singularity theory.The configuration bifurcationcharacteristic is analyzed at the bifurcation point.The problem of'avoiding boundarysingularity is divided into three instances to be researched based on the configurationafter passing through boundary singular position.Then three types of perturbationfunctions are constructed for these three instances.Therefore, the problem that theparallel mechanisms can pass through the boundary singular position with its givenconfiguration has been solved.
Finally, the main creative achievements are presented and some of the openproblems in the field of Stewart mechanisms, which are likely to be research interest in the future, are recommended.
首次在复数域内研究机构的奇异性问题。以Watt六杆机构为例，讨论了多环机构的奇异性问题，在复数域内建立的机构分岔分析方程，形象直观，芽空间转换非常方便。基于分岔方程的多解性和奇异性理论，分析了多环机构的奇异构型和机构的分岔性态与机构参数的关系。
应用奇异性理论分析Stewart并联机构的构型分岔特性。建立了机构的分岔方程，利用Jacobian矩阵确定机构的奇异位置，根据奇异性理论的静态分岔条件来判断机构的分岔点，并将分岔方程的部分导数特性用于确定分岔点的分岔特性。提出了构型转换影响系数的概念，该系数揭示了奇异位置附近机构构型的变换规律，为机构运动控制提供了理论基础，同时也为利用开折的方法实现转向分岔点免奇异奠定了基础。
首次用数值的方法研究了对称6-6型Stewart并联机构多参数复杂奇异性问题。6-6型Stewart并联机构具有六个自由度，六个输入参数可以作为机构的分岔参数，不同的分岔参数引起机构出现不同的分岔现象。根据分岔参数组合对机构分岔的影响，将多分岔参数组合共分为八种情况，并对这八种情况下机构的奇异性和分岔点的分岔特性进行了详细讨论。分岔分析结果验证了以往研究中所出现的奇异位形，同时又发现了新的奇异位形。
首次讨论了机构的结构参数与并联机构奇异性的关系。详细讨论了Stewart并联机构动、静平台半径R_1和R_2、作动筒长度l_i和动、静平台铰链点夹角α_1和α_2对并联机构分岔特性的影响，并通过分析工作空间和稳定工作空间给出了上述参数选取的基本方法，为通过合理选择结构参数实现机构的免奇异或减少机构的奇异性奠定了理论基础。
从非线性奇异性理论出发，研究了并联机构边界奇异性的规避问题。分析了并联机构奇异位置的构型分岔特性，根据并联机构通过边界奇异位置的构型方式将边界奇异性规避问题分为三类进行了研究，分别构造了并联机构以这三种构型方式通过边界奇异位置的扰动函数，解决了机构以确定的运动通过边界奇异位置的关键问题。
最后给出了本论文的主要创新成果和有待于进一步研究的问题。
Parallel mechanism has the advantages of high structrual stiffness, highload-bearing capacity, and higher control precision, which meets the needs ofindustrial automatization.Now it has been widely used in the area of aeronautics,mechanical manufacturing, industrial robot and etc.The kinematic singularity, namely,the output movement of the mechanism is uncertain due to kenematic bifurcation atthe singular position, is the intrinsic characteristic of parallel mechanism, and this hasattracted the attention of the robotics researchers in the world.How to control themechanism to pass through the singular position with its given configuration is thekey problem for parallem mechanism.In this paper, using singularity theory, theauthor has investigated the approach of avoiding singularity at the singular positionfor parallem mechanisms, and solved the key problem that parallel mechanisms couldpass through the boundary singular position with its given configuration.The authoralso studied the relationship between the singularity and the parameters of parallelmechanisms and presented a method of improving the singularity.The mainachievements are as follows:
Singularity has been studied in complex number field for the first time.Takingthe Watt linkage as an example, the singularity for multi-loop mechanism isresearched.The bifurcation equation in complex number field has visual property, andcan be easily transformed into germ field.On the basis of the multi-solution of thebifurcation equation, the singularity theory is applied to analyze the singularconfigurations and bifurcation characteristic connecting with the structural parametersof mechanisms.
The bifurcation equation of Stewart parallel mechanism is established foranalyzing its bifurcation characteristic based on the singularity theory.The Jacobinmatrix is used to identify the singular points, and the static bifurcation condition isused to identify the bifurcation points.The partial differential of the bifurcationequation is applied to analyze its bifurcation characteristic at a bifurcation point.The conception of configuration transforming coefficient is presented in order to revealthe principle of the configuration transforming in the neighborhood of a singularposition.Therefore, it establishes theoretical basis for controlling the movement ofparallel mechanism and for avoiding the turning bifurcation point on the base of theunfolding theory.
Numerical method is applied to study the complex singularity for symmetrical6-Dof Stewart parallel mechanisms with multi-parameters bifurcation.Six inputparameters can be selected as the bifurcation parameters.Different composition of thebifurcation parameters would lead to different bifurcation phenomena.Eight cases areobtained for symmetrical Stewart parallel mechanisms with multi-parametersbifurcation.The corresponding singularity and bifurcation characteristics of theparallel mechanisms are investigated, respectively.The results verify the singularconfigurations presented in other literature and also give some new singularconfigurations for Stewart parallel mechanisms.
The relationship between the composition of parallel mechanisms and itssingularity is studied for the first time.The key parameters of Stewart parallelmechanisms include: the radiuses R~1 and R~2 for the fixed base frame and themovable platform respectively, the actuators length l~i, and the anglesα~1andα~2these parameters' influence on bifurcation behavior has been disscussed, and themethod of selecting appropriate parameters is presented according to the workspaceand the singularity-free workspace, which can improve singularity of the movement.
The method of avoiding the boundary singularity is investigated for the parallelmechanisms based on the singularity theory.The configuration bifurcationcharacteristic is analyzed at the bifurcation point.The problem of'avoiding boundarysingularity is divided into three instances to be researched based on the configurationafter passing through boundary singular position.Then three types of perturbationfunctions are constructed for these three instances.Therefore, the problem that theparallel mechanisms can pass through the boundary singular position with its givenconfiguration has been solved.
Finally, the main creative achievements are presented and some of the openproblems in the field of Stewart mechanisms, which are likely to be research interest in the future, are recommended.
引文
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