并联机构若干基本问题的研究
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摘要
随着科技的发展，并联机构日益受到关注，其应用也越来越广泛。与串联机构不同，并联机构的发展只有几十年的历史。并联机构的一些基本问题还没有完全解决，这阻碍了并联机构的发展和应用。6-SPS并联机构是一种典型的并联机构。本文对6-SPS并联机构的运动学正解、奇异性和柔索驱动并联机构作了深入的研究和探讨。
运动学正解是并联机构的基本问题之一，同时也是研究并联机构其它问题的基础。6-SPS并联机构的正解方程是关于位置和姿态参数的高度耦合的非线性方程组，难以进行解析求解。本文通过分析动平台位置与姿态变量之间的耦合关系、运用Gr bner基算法，得到了15个只含有3个变量的相容方程。基于这15个相容方程，运用正交补方法以及逐步消去高次项的方法，最终将6-SPS并联机构的运动学正解问题表达为一元高次代数方程。研究结果表明，一般6-SPS并联机构的运动学正解问题可以表达为一元20次代数方程，虽然仍然为20次的代数方程，但是比之已有方法计算速度大为提高；用多种方法研究了对称6-SPS并联机构的运动学正解问题，最终将其表达为一元14次的代数方程；3/6-SPS并联机构和3/3-SPS并联机构的运动学正解问题均可以表达为一元8次代数方程。提出了一种椭圆型6-SPS并联机构并将其运动学正解问题表达为一元14次代数方程。
奇异性分析也是并联机构研究的基本问题之一。奇异性分析在工作空间分析、轨迹规划以及控制系统设计研究中是无法回避的问题。如何给出奇异轨迹的解析表达式一直是众多学者研究并联机构的重点。本文提出了一种研究6-SPS并联机构奇异性的新方法。旋转矩阵采用四元数描述，并将矢量坐标扩展为4维形式，通过分析动平台位置与姿态变量之间的耦合关系以及四元数的性质，得到了采用8个二次方程式表达的运动方程。本文进一步应用该运动方程导出了新型的并联机构雅可比矩阵，通过对该新雅可比矩阵求取行列式，得到了奇异轨迹的解析表达式。该方法适用于所有类型的6-SPS并联机构。
本文对运动学正解和奇异性的关系进行了初步的研究。研究发现，在奇异位形上，运动学正解将出现重根；也就是机构的运动状态将发生改变，为今后进一步深入研究奠定了基础。
柔索驱动并联机构是近年来发展起来的新型的并联机构。本文对柔索驱动并联机构的工作空间、刚度等问题进行了研究并搭建了理论分析平台。由于柔索只能承受拉力、而不能施加压力的特性，在研究工作空间时必须遵循拉力为正的原则。应用正交补的方法进行工作空间的研究。借鉴工业机器人的研究方法，研究了柔索驱动并联机构的刚度问题。
With the development of science and technology, parallel mechanisms have been paid more and moreattention, and the application of parallel mechanisms is more and more extensively. Different from theserial mechanism, it is only a few decades that the parallel mechanisms have been developed. Somefundamental problems of the parallel mechanisms have not been fully solved so far, which hinders thedevelopment and application of parallel mechanisms.6-SPS parallel mechanism is a typical parallelmechanism. The forward kinematics analysis and the singularity analysis of the6-SPS parallelmechanism, the cable-driven parallel mechanism have been deeply researched and discussed in thisthesis.
The forward kinematics analysis is one of the fundamental problems of parallel mechanisms, andwhat’s more, it is the basis of researching some other problems of parallel mechanisms. The forwardkinematic equations are highly coupled nonlinear equations about the position and orientationvarables, and it is difficult to be solved analytically. Through analyzing the coupling relationshipsamong the position and orientation variables of the moving platform, appliying the Gr bner basisalgorithm,15four order compatible algebraic equations, which containing three orientation variables,are obtained. Based on the15compatible equations, using the orthogonal complement method andeliminating the higher terms step by step, the problem of the forward kinematics analysis of the6-SPSparallel mechanisms can be expressed as higher order algebraic equations in the end. The results showthat, the forward kinematics analysis of the general6-SPS parallel mechanisms can be expressed as a20order algebraic equation. Although it is still20order equation, it is more quickly and practicable.The forward kinematics analysis of the symmetrical6-SPS parallel mechanisms has beeen studied byseveral methods, and each of them can express this problem as a14order algebraic equation. Theforward kinematics analysis of both3/6-SPS and3/3-SPS parallel mechanisms can be expressed as a8order algebraic equation. This thesis presents a kind of Elliptical6-SPS parallel mechanisms, ofwhich the forward kinematics analysis can also be expesed as a14order algebraic equation.
Singularity analysis is one of the fundamental problems of parallel mechanisms too. Thisproblem can not been avoided in the research on workspace, motion planning and the design ofcontrolling system. How to express the singularity locus in an analytical form is the research emphasisfor many researchers for a long time. This thesis presents a new method for the singularity analysis ofthe6-SPS parallel mechanisms. The rotation matrix is been described by quaternion, and both the rotation matrix and the coordinates vectors have been expanded to four dimensional forms. Throughanalyzing the coupling relationship between the position variables and the orientation variables,utilizing properties of the quaternion, eight equivalent equations are obtained. A new kind of Jacobianmatrix is derived from those equations, and the analytical expression of the singularity locus isobtained by calculating the determinant of the new Jacobian matrix. The singularity analysis of allkinds of6-SPS parallel mechanisms can be solved by this analytical method.
In this thesis, the relationship of forward kinematics and singularity has been studiedpreliminarily. The research shows that, the forward kinematics would have multiple roots if theparallel mechanisms are placed on singular configurations. It means that tha state of mtion will bechanged. And it forms a basis for deeply study in the future.
The cable-driven parallel mechanism is a new type of parallel mechanism which has beendeveloping in recent years. Workspace, stiffness and other issues have been studied in the thesis. Atheoretical analysis platform has been constructed. Since a cable can only sustains pull rather thanpressure, the principle of keeping the tension positive should be satisfied while analyzing theworkspace of the cable-driven parallel mechanisms. The workspace of the cable-driven parallelmechanisms has been analyzed by the orthogonal complement method. Use the research methods ofindustrial robots for reference, the stiffness of cable-driven parallel mechanism has been studied inthis thesis.
运动学正解是并联机构的基本问题之一，同时也是研究并联机构其它问题的基础。6-SPS并联机构的正解方程是关于位置和姿态参数的高度耦合的非线性方程组，难以进行解析求解。本文通过分析动平台位置与姿态变量之间的耦合关系、运用Gr bner基算法，得到了15个只含有3个变量的相容方程。基于这15个相容方程，运用正交补方法以及逐步消去高次项的方法，最终将6-SPS并联机构的运动学正解问题表达为一元高次代数方程。研究结果表明，一般6-SPS并联机构的运动学正解问题可以表达为一元20次代数方程，虽然仍然为20次的代数方程，但是比之已有方法计算速度大为提高；用多种方法研究了对称6-SPS并联机构的运动学正解问题，最终将其表达为一元14次的代数方程；3/6-SPS并联机构和3/3-SPS并联机构的运动学正解问题均可以表达为一元8次代数方程。提出了一种椭圆型6-SPS并联机构并将其运动学正解问题表达为一元14次代数方程。
奇异性分析也是并联机构研究的基本问题之一。奇异性分析在工作空间分析、轨迹规划以及控制系统设计研究中是无法回避的问题。如何给出奇异轨迹的解析表达式一直是众多学者研究并联机构的重点。本文提出了一种研究6-SPS并联机构奇异性的新方法。旋转矩阵采用四元数描述，并将矢量坐标扩展为4维形式，通过分析动平台位置与姿态变量之间的耦合关系以及四元数的性质，得到了采用8个二次方程式表达的运动方程。本文进一步应用该运动方程导出了新型的并联机构雅可比矩阵，通过对该新雅可比矩阵求取行列式，得到了奇异轨迹的解析表达式。该方法适用于所有类型的6-SPS并联机构。
本文对运动学正解和奇异性的关系进行了初步的研究。研究发现，在奇异位形上，运动学正解将出现重根；也就是机构的运动状态将发生改变，为今后进一步深入研究奠定了基础。
柔索驱动并联机构是近年来发展起来的新型的并联机构。本文对柔索驱动并联机构的工作空间、刚度等问题进行了研究并搭建了理论分析平台。由于柔索只能承受拉力、而不能施加压力的特性，在研究工作空间时必须遵循拉力为正的原则。应用正交补的方法进行工作空间的研究。借鉴工业机器人的研究方法，研究了柔索驱动并联机构的刚度问题。
With the development of science and technology, parallel mechanisms have been paid more and moreattention, and the application of parallel mechanisms is more and more extensively. Different from theserial mechanism, it is only a few decades that the parallel mechanisms have been developed. Somefundamental problems of the parallel mechanisms have not been fully solved so far, which hinders thedevelopment and application of parallel mechanisms.6-SPS parallel mechanism is a typical parallelmechanism. The forward kinematics analysis and the singularity analysis of the6-SPS parallelmechanism, the cable-driven parallel mechanism have been deeply researched and discussed in thisthesis.
The forward kinematics analysis is one of the fundamental problems of parallel mechanisms, andwhat’s more, it is the basis of researching some other problems of parallel mechanisms. The forwardkinematic equations are highly coupled nonlinear equations about the position and orientationvarables, and it is difficult to be solved analytically. Through analyzing the coupling relationshipsamong the position and orientation variables of the moving platform, appliying the Gr bner basisalgorithm,15four order compatible algebraic equations, which containing three orientation variables,are obtained. Based on the15compatible equations, using the orthogonal complement method andeliminating the higher terms step by step, the problem of the forward kinematics analysis of the6-SPSparallel mechanisms can be expressed as higher order algebraic equations in the end. The results showthat, the forward kinematics analysis of the general6-SPS parallel mechanisms can be expressed as a20order algebraic equation. Although it is still20order equation, it is more quickly and practicable.The forward kinematics analysis of the symmetrical6-SPS parallel mechanisms has beeen studied byseveral methods, and each of them can express this problem as a14order algebraic equation. Theforward kinematics analysis of both3/6-SPS and3/3-SPS parallel mechanisms can be expressed as a8order algebraic equation. This thesis presents a kind of Elliptical6-SPS parallel mechanisms, ofwhich the forward kinematics analysis can also be expesed as a14order algebraic equation.
Singularity analysis is one of the fundamental problems of parallel mechanisms too. Thisproblem can not been avoided in the research on workspace, motion planning and the design ofcontrolling system. How to express the singularity locus in an analytical form is the research emphasisfor many researchers for a long time. This thesis presents a new method for the singularity analysis ofthe6-SPS parallel mechanisms. The rotation matrix is been described by quaternion, and both the rotation matrix and the coordinates vectors have been expanded to four dimensional forms. Throughanalyzing the coupling relationship between the position variables and the orientation variables,utilizing properties of the quaternion, eight equivalent equations are obtained. A new kind of Jacobianmatrix is derived from those equations, and the analytical expression of the singularity locus isobtained by calculating the determinant of the new Jacobian matrix. The singularity analysis of allkinds of6-SPS parallel mechanisms can be solved by this analytical method.
In this thesis, the relationship of forward kinematics and singularity has been studiedpreliminarily. The research shows that, the forward kinematics would have multiple roots if theparallel mechanisms are placed on singular configurations. It means that tha state of mtion will bechanged. And it forms a basis for deeply study in the future.
The cable-driven parallel mechanism is a new type of parallel mechanism which has beendeveloping in recent years. Workspace, stiffness and other issues have been studied in the thesis. Atheoretical analysis platform has been constructed. Since a cable can only sustains pull rather thanpressure, the principle of keeping the tension positive should be satisfied while analyzing theworkspace of the cable-driven parallel mechanisms. The workspace of the cable-driven parallelmechanisms has been analyzed by the orthogonal complement method. Use the research methods ofindustrial robots for reference, the stiffness of cable-driven parallel mechanism has been studied inthis thesis.
引文
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