并联机器人运动学正解新算法及工作空间本体研究
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摘要
半个世纪以来,并联机器人得到深入的研究和广泛的应用。但在其研究领域内,包括运动学、动力学、工作空间、奇异特性等方面,仍然存在一些难题。作为21世纪100个交叉科学难题之一,stewart并联机构的正向运动学问题,正在受到日益广泛的关注。并联机器人在实际应用中,如空间探索、海洋开发、原子能应用、军事、抢险救灾、宏微操作等,都需要精准、快速、稳定的运动学位置解(包括逆解和正解)算法,需要经过优化的动力学控制策略和更为精确和详尽的工作空间描述等。深入地研究并联机构的运动学、动力学和工作空间,特别是并联机器人的位置正解,无论在理论方面还是实用方面,都是十分必要的。
并联机器人运动学正解数值算法、动力学和工作空间的研究存在下述问题:①运动学位置正解缺少简便、快速、且保证收敛的统一的数值算法。②不同支链构成的并联机构之间的关系及相互转化方面的研究较少。③缺少包括正解逆解和交叉解的统一算法,缺少对正解输入条件的等效代换的研究,缺少对多并联机构并联系统的研究。④对工作空间的研究不够全面、不够系统,且有关空间利用效率的研究较少。本文主要工作内容包括:①深入地研究并联机构运动学位置正解的数值算法现状;②提出一种适于所有的并联机构的正解数值算法;③提出一种适于并联机构运动学、动力学等的间接解法;④构建并联机构的工作空间简单本体。本文的主要创新点包括:
(1)提出并定义了理想并联机构的概念,提出了非理想并联机构的节肢化分析和综合方法(简称节肢化方法)。理想并联机构是支链由广义移动副组成,且支链在动平台、基础平台上的连接点与广义移动副的轴线重合的并联机构。节肢化分析方法将复杂并联机构分解为若干个节肢,包括主节肢、传递节肢、驱动节肢。其中主节肢是一个理想并联机构,可以使用已知的各种有关理想并联机构的知识、方法、程序进行求解。然后把多个节肢串联起来,形成一个整体进行综合研究。所以,节肢化方法包括分析和综合两个方面,是分析和综合的统一。节肢化分析方法是一种间接地求解复杂并联机构的运动学(包括位置解、速度、加速度等)、动力学、工作空间、奇异位形等诸问题的新方法。非理想并联机构的节肢化分析,可以从一个新的角度研究并联机构,可以分享理想并联机构的许多研究成果。
(2)归纳总结出了一种适用于所有并联机构的新的正解数值算法——几何迭代法。几何迭代法是基于并联机构的结构逆解和泛几何相似性假设、以某个初值开始,逐步逼近原始输入,并最终求得并联机构正解的满意解的迭代方法。采用实证和理论分析的方法证实了几何迭代法的可行性、稳定性和可靠性。几何迭代法主要包括泛几何相似性假设、并联机构的数学模型和迭代过程三部分内容。
几何迭代法的泛几何相似性假设包括:①泛几何相似性假设:假设几何迭代的过程中并联机构的图形具有泛几何相似性,泛几何相似性就是并联机构的运动过程中,在两个不同的时刻,并联机构的结构形状具有的相似性。②铰支点图形相似假设:假设动平台各个实际铰支点形成的图形,与迭代过程中各个铰支点形成的图形结构相似;③参考点图形相似假设:假设迭代变量(参考点)与各个迭代铰支点的相对位置和各个真实参考点与真实铰支点的相对位置相似。在迭代过程的每一步,都遵循上述相似性假设。
建立数学模型:首先进行自由度计算和自由度组合分析,然后确定几何逆解公式,最后给定迭代初值和确定理想并联机构的新的结构参数。
迭代计算过程由迭代初值(中立位置的位姿参数)开始,之后用几何逆解公式求出并联机构的结构解(例如上平台铰支点);用原始输入或基本输入(例如,已知的杆长)修正相关铰支点的位置;利用修正后的铰支点坐标,依据相似性假设,综合确定一个新的上平台平面;由新的上平台平面得到一组新的位姿数据,代替原来的迭代初值;判断迭代是否符合精度要求,确定继续迭代或结束程序。
新算法物理模型清晰,编程简单,通用性强,编程工作量小,迭代收敛速度较快,可以达到任意的计算精度,精度可控,稳定、可靠。新算法有一个通用的初值,保证了算法运行的可靠性。新算法彻底避开了非线性方程组,不需要导数运算及Jacobi矩阵的求逆运算。总体上讲,几何迭代法优于牛顿迭代法或与牛顿迭代法相当。
几何迭代法适用于所有的并联机构。几何迭代法可以完成正解、逆解和交叉解的任务。几何迭代法还适于多并联机构的并联系统的求解,用于计算复杂多面体和变几何并联机构。
(3)构建了较为完整的机器人工作空间本体。从本体论的角度分析了机器人的工作空间,扩展了工作空间的概念外延和内涵。
本文研究工作的成效和意义如下:①理想并联机构的概念和节肢化分析方法在理想与非理想并联机构之间架起了一个桥梁,为分析、研究非理想并联机构的运动学、动力学、工作空间、奇异特性等提供了一个新思路。②几何迭代法,彻底避开了非线性方程组,为所有的并联机构的正解提供了一个新的选择。新算法已经成功地应用到多种不同类型的并联机构计算。新算法为使用数值方法求解强非线性方程组提供了一个新思路,为求解交叉解、等效输入解等提供了一个新方法。几何迭代法整体上降低了正解的解算难度,为并联机构的普及和广泛应用奠定了基础。③运用本体论建立的工作空间本体,不但拓宽了机器人工作空间方面的研究领域,而且为机器人的优化设计提供了理论依据。例如,机构的各种扰动空间、机构的空间利用率、空间规整性等新内容,可资应用并有待进一步深入研究。
For half a century, parallel robot has been thoroughly studied and widely used. However, there are still some problems in their research fields, such as kinematics, dynamics, workspace, singularity, etc. Forward solution of stewart platform which is one of the 100 interdisciplinary sciences puzzles of the 21st century is being increasingly widespread concerned. The accurate, fast, stable kinematic position solution algorithm are needed, and optimized dynamics control strategies,more precise and detailed description of the workspace are needed too in practical applications, such as space exploration, ocean development, atomic energy applications, military, disaster relief, etc. it is very necessary either in theory or in practice to research in-depth on kinematics, dynamics and working space of parallel mechanism, especially the forward position solution about the parallel robot.
The study of the forward kinematics numerical solution,dynamics and workspace of parallel robot issues:①The method of forward kinematics solution lack simple, rapid and general numerical algorithm which ensure that iteration is convergent.②There is less research on the relationship between the parallel mechanisms composed of different limb.③There is a lack of general algorithm of the positive solutions ,inverse solution and cross solution; lack of the study of equivalent substitution to input conditions of the forward solutions; also lack of the research on multiple parallel mechanism with relationship in parallel④The research on workspace is not comprehensive and systematic enough, and the study about efficiency of space utilization is poor.
The main contents of this dissertation include:①reviewed current situation of the numerical algorithm, the forward kinematics solution of the parallel mechanism;②presented a forward kinematics numerical solution which is applicable for all kinds of parallel mechanisms;③presented a indirect method suitable for parallel mechanisms dynamics;④Establish a simple ontology about workspace of the parallel mechanism .
The novelties of the dissertation include:
(1) Put forward and defined the concept of ideal parallel mechanism, put forward arthropodization method of the non-ideal parallel mechanism .Ideal parallel mechanism is a parallel mechanism whose branch is a straight line limb .Arthropodization method includes two aspects: analysis and synthesis, and decomposed the complex parallel mechanism into a number of arthropods which are connected in series, including the main arthropod, transmission arthropod, drive arthropods. Then connect the plurality of arthropod in series to form a whole .Each arthropod is a simple parallel mechanism. According to the known conditions, analyses various arthropods, and the links them. Finally gets a comprehensive solution of non-ideal parallel mechanism.
Arthropodization method is a kind of new method to indirectly solve the problems such as kinematics, dynamics, workspace, singularity of the non-ideal parallel mechanism. Arthropod analysis method of non-ideal parallel mechanism could study the parallel mechanism from a new perspective and can share a lot of research results of the ideal parallel mechanism. The concept of ideal parallel mechanism can popular and apply the geometric iteration method to all of the parallel mechanism, and also can be used to study the kinematics, dynamics, workspace and singularity of the non ideal parallel mechanism.
(2) Presented a new forward numerical algorithm method suitable for all parallel mechanism----geometric iteration method. The feasibility, stability and reliability of the geometric iteration method are confirmed using empirical and theoretical analysis method. Geometric iteration method is mainly composed of three parts: the pan similarity hypothesis, the mathematics model of the parallel mechanism and the iterative process.
The similarity hypotheses of Geometric iteration method include three parts:①Universal geometrical similarity hypothesis of the parallel mechanism: Hypothesize that the graph of parallel mechanism has universal geometric similarity hypothesis in the geometric iteration process;②Graph similarity hypothesis of the hinge pivot: Hypothesize that the graphics formed by actual hinged fulcrums of the moving platform (flat or three-dimensional graphics, referred to as the true hinge pivot graphics) is similar to the graphics formed by the hinged fulcrum in the iterative process (referred to as iterative hinged fulcrum graphic) structural similarity;③Graph similarity hypothesis of the reference point : Hypothesize that the iteration variable (reference point) is similar to the relative position of each iteration hinge pivot and the real reference point is similar to the relative position of the true hinge pivot. Every step of the iterative process follows the similarity hypothesis.
The establishment of mathematical model is to transform a given parallel mechanism into an ideal parallel mechanism, then calculate the number of the DOF and analysis the combination of the DOF; Establish a formula of the geometry inverse solution based on the basic coordinate system; Finally determine the iterative initial value and new structure parameters of the ideal parallel mechanism.
The iterative calculation process begin from iterative initial value (home position parameters), then find out the platform hinge pivot position by geometrical inverse solution formula; Use basic input (e.g., known limb’s length ) to amend relevant hinge pivot position; Use the modified hinged fulcrum coordinate to comprehensive determine a new platform on a plane according to thepan-similarity hypothesis; Use the new platform plane to get a new set of position data, to replace the original iterative initial value; Judge if the iteration meets accuracy requirements, determine continuing the iteration or ending the program.
Of novel algorithm, the physical model is clarity, the programming is simple, the workload of programming is small, the iterative convergence speed is high. It can achieve arbitrary precision and the precision is controllable, stable and reliable.An universal Initial value which is a Parameter in home position assures the reliability of algorithm operation.
The novel algorithm completely avoids nonlinear equations, and does not need the derivative operation and the Jacobi matrix inverse operation. Generally speaking, geometric iteration method is better than the Newton-Raphson method or as well as the Newton-Raphson method.
Geometric iteration method can be applied to all of the parallel mechanism. And can complete the task of positive solution, inverse solution and cross solution. Geometric iteration method is suitable for the solution of multiple parallel mechanism and parallel system, it is also used to calculate the complex polyhedron and variable geometry parallel mechanism.
(3) Establish a simple ontology about workspace of the parallel mechanism. Analyse the robot workspace from the perspective of ontology. Extend the connotation and extension of the concept of the workspace.
This paper has the effects and significance as follows:
①The arthropodization analysis methods set up a bridge between the ideal and non ideal parallel mechanism, and provided a new train of thought for the analysis of kinematics, dynamics, workspace and singularity characteristics of the non ideal parallel mechanism.②Geometric iteration method witch completely avoided the nonlinear equations provided a new choice for all forward solutions of the parallel mechanism. The novel algorithm has been successfully applied to a variety of different types of parallel mechanism, and provided a new way to solve nonlinear equations with the numerical methods, and provided a new method to solve the intersection solution, and equivalent input solutions. It reduced overall difficulty of the forward solution and lay a foundation for the popularization and application of the parallel mechanism.③Established a ontology about workspace of robot. Not only widened the research field of the robot working space, but also offered the theoretical foundation for optimization design. For example, various disturbance space of the mechanism, space utilization rate of the mechanism, spatial regularity and other new contents, which can be used and need further study.
并联机器人运动学正解数值算法、动力学和工作空间的研究存在下述问题:①运动学位置正解缺少简便、快速、且保证收敛的统一的数值算法。②不同支链构成的并联机构之间的关系及相互转化方面的研究较少。③缺少包括正解逆解和交叉解的统一算法,缺少对正解输入条件的等效代换的研究,缺少对多并联机构并联系统的研究。④对工作空间的研究不够全面、不够系统,且有关空间利用效率的研究较少。本文主要工作内容包括:①深入地研究并联机构运动学位置正解的数值算法现状;②提出一种适于所有的并联机构的正解数值算法;③提出一种适于并联机构运动学、动力学等的间接解法;④构建并联机构的工作空间简单本体。本文的主要创新点包括:
(1)提出并定义了理想并联机构的概念,提出了非理想并联机构的节肢化分析和综合方法(简称节肢化方法)。理想并联机构是支链由广义移动副组成,且支链在动平台、基础平台上的连接点与广义移动副的轴线重合的并联机构。节肢化分析方法将复杂并联机构分解为若干个节肢,包括主节肢、传递节肢、驱动节肢。其中主节肢是一个理想并联机构,可以使用已知的各种有关理想并联机构的知识、方法、程序进行求解。然后把多个节肢串联起来,形成一个整体进行综合研究。所以,节肢化方法包括分析和综合两个方面,是分析和综合的统一。节肢化分析方法是一种间接地求解复杂并联机构的运动学(包括位置解、速度、加速度等)、动力学、工作空间、奇异位形等诸问题的新方法。非理想并联机构的节肢化分析,可以从一个新的角度研究并联机构,可以分享理想并联机构的许多研究成果。
(2)归纳总结出了一种适用于所有并联机构的新的正解数值算法——几何迭代法。几何迭代法是基于并联机构的结构逆解和泛几何相似性假设、以某个初值开始,逐步逼近原始输入,并最终求得并联机构正解的满意解的迭代方法。采用实证和理论分析的方法证实了几何迭代法的可行性、稳定性和可靠性。几何迭代法主要包括泛几何相似性假设、并联机构的数学模型和迭代过程三部分内容。
几何迭代法的泛几何相似性假设包括:①泛几何相似性假设:假设几何迭代的过程中并联机构的图形具有泛几何相似性,泛几何相似性就是并联机构的运动过程中,在两个不同的时刻,并联机构的结构形状具有的相似性。②铰支点图形相似假设:假设动平台各个实际铰支点形成的图形,与迭代过程中各个铰支点形成的图形结构相似;③参考点图形相似假设:假设迭代变量(参考点)与各个迭代铰支点的相对位置和各个真实参考点与真实铰支点的相对位置相似。在迭代过程的每一步,都遵循上述相似性假设。
建立数学模型:首先进行自由度计算和自由度组合分析,然后确定几何逆解公式,最后给定迭代初值和确定理想并联机构的新的结构参数。
迭代计算过程由迭代初值(中立位置的位姿参数)开始,之后用几何逆解公式求出并联机构的结构解(例如上平台铰支点);用原始输入或基本输入(例如,已知的杆长)修正相关铰支点的位置;利用修正后的铰支点坐标,依据相似性假设,综合确定一个新的上平台平面;由新的上平台平面得到一组新的位姿数据,代替原来的迭代初值;判断迭代是否符合精度要求,确定继续迭代或结束程序。
新算法物理模型清晰,编程简单,通用性强,编程工作量小,迭代收敛速度较快,可以达到任意的计算精度,精度可控,稳定、可靠。新算法有一个通用的初值,保证了算法运行的可靠性。新算法彻底避开了非线性方程组,不需要导数运算及Jacobi矩阵的求逆运算。总体上讲,几何迭代法优于牛顿迭代法或与牛顿迭代法相当。
几何迭代法适用于所有的并联机构。几何迭代法可以完成正解、逆解和交叉解的任务。几何迭代法还适于多并联机构的并联系统的求解,用于计算复杂多面体和变几何并联机构。
(3)构建了较为完整的机器人工作空间本体。从本体论的角度分析了机器人的工作空间,扩展了工作空间的概念外延和内涵。
本文研究工作的成效和意义如下:①理想并联机构的概念和节肢化分析方法在理想与非理想并联机构之间架起了一个桥梁,为分析、研究非理想并联机构的运动学、动力学、工作空间、奇异特性等提供了一个新思路。②几何迭代法,彻底避开了非线性方程组,为所有的并联机构的正解提供了一个新的选择。新算法已经成功地应用到多种不同类型的并联机构计算。新算法为使用数值方法求解强非线性方程组提供了一个新思路,为求解交叉解、等效输入解等提供了一个新方法。几何迭代法整体上降低了正解的解算难度,为并联机构的普及和广泛应用奠定了基础。③运用本体论建立的工作空间本体,不但拓宽了机器人工作空间方面的研究领域,而且为机器人的优化设计提供了理论依据。例如,机构的各种扰动空间、机构的空间利用率、空间规整性等新内容,可资应用并有待进一步深入研究。
For half a century, parallel robot has been thoroughly studied and widely used. However, there are still some problems in their research fields, such as kinematics, dynamics, workspace, singularity, etc. Forward solution of stewart platform which is one of the 100 interdisciplinary sciences puzzles of the 21st century is being increasingly widespread concerned. The accurate, fast, stable kinematic position solution algorithm are needed, and optimized dynamics control strategies,more precise and detailed description of the workspace are needed too in practical applications, such as space exploration, ocean development, atomic energy applications, military, disaster relief, etc. it is very necessary either in theory or in practice to research in-depth on kinematics, dynamics and working space of parallel mechanism, especially the forward position solution about the parallel robot.
The study of the forward kinematics numerical solution,dynamics and workspace of parallel robot issues:①The method of forward kinematics solution lack simple, rapid and general numerical algorithm which ensure that iteration is convergent.②There is less research on the relationship between the parallel mechanisms composed of different limb.③There is a lack of general algorithm of the positive solutions ,inverse solution and cross solution; lack of the study of equivalent substitution to input conditions of the forward solutions; also lack of the research on multiple parallel mechanism with relationship in parallel④The research on workspace is not comprehensive and systematic enough, and the study about efficiency of space utilization is poor.
The main contents of this dissertation include:①reviewed current situation of the numerical algorithm, the forward kinematics solution of the parallel mechanism;②presented a forward kinematics numerical solution which is applicable for all kinds of parallel mechanisms;③presented a indirect method suitable for parallel mechanisms dynamics;④Establish a simple ontology about workspace of the parallel mechanism .
The novelties of the dissertation include:
(1) Put forward and defined the concept of ideal parallel mechanism, put forward arthropodization method of the non-ideal parallel mechanism .Ideal parallel mechanism is a parallel mechanism whose branch is a straight line limb .Arthropodization method includes two aspects: analysis and synthesis, and decomposed the complex parallel mechanism into a number of arthropods which are connected in series, including the main arthropod, transmission arthropod, drive arthropods. Then connect the plurality of arthropod in series to form a whole .Each arthropod is a simple parallel mechanism. According to the known conditions, analyses various arthropods, and the links them. Finally gets a comprehensive solution of non-ideal parallel mechanism.
Arthropodization method is a kind of new method to indirectly solve the problems such as kinematics, dynamics, workspace, singularity of the non-ideal parallel mechanism. Arthropod analysis method of non-ideal parallel mechanism could study the parallel mechanism from a new perspective and can share a lot of research results of the ideal parallel mechanism. The concept of ideal parallel mechanism can popular and apply the geometric iteration method to all of the parallel mechanism, and also can be used to study the kinematics, dynamics, workspace and singularity of the non ideal parallel mechanism.
(2) Presented a new forward numerical algorithm method suitable for all parallel mechanism----geometric iteration method. The feasibility, stability and reliability of the geometric iteration method are confirmed using empirical and theoretical analysis method. Geometric iteration method is mainly composed of three parts: the pan similarity hypothesis, the mathematics model of the parallel mechanism and the iterative process.
The similarity hypotheses of Geometric iteration method include three parts:①Universal geometrical similarity hypothesis of the parallel mechanism: Hypothesize that the graph of parallel mechanism has universal geometric similarity hypothesis in the geometric iteration process;②Graph similarity hypothesis of the hinge pivot: Hypothesize that the graphics formed by actual hinged fulcrums of the moving platform (flat or three-dimensional graphics, referred to as the true hinge pivot graphics) is similar to the graphics formed by the hinged fulcrum in the iterative process (referred to as iterative hinged fulcrum graphic) structural similarity;③Graph similarity hypothesis of the reference point : Hypothesize that the iteration variable (reference point) is similar to the relative position of each iteration hinge pivot and the real reference point is similar to the relative position of the true hinge pivot. Every step of the iterative process follows the similarity hypothesis.
The establishment of mathematical model is to transform a given parallel mechanism into an ideal parallel mechanism, then calculate the number of the DOF and analysis the combination of the DOF; Establish a formula of the geometry inverse solution based on the basic coordinate system; Finally determine the iterative initial value and new structure parameters of the ideal parallel mechanism.
The iterative calculation process begin from iterative initial value (home position parameters), then find out the platform hinge pivot position by geometrical inverse solution formula; Use basic input (e.g., known limb’s length ) to amend relevant hinge pivot position; Use the modified hinged fulcrum coordinate to comprehensive determine a new platform on a plane according to thepan-similarity hypothesis; Use the new platform plane to get a new set of position data, to replace the original iterative initial value; Judge if the iteration meets accuracy requirements, determine continuing the iteration or ending the program.
Of novel algorithm, the physical model is clarity, the programming is simple, the workload of programming is small, the iterative convergence speed is high. It can achieve arbitrary precision and the precision is controllable, stable and reliable.An universal Initial value which is a Parameter in home position assures the reliability of algorithm operation.
The novel algorithm completely avoids nonlinear equations, and does not need the derivative operation and the Jacobi matrix inverse operation. Generally speaking, geometric iteration method is better than the Newton-Raphson method or as well as the Newton-Raphson method.
Geometric iteration method can be applied to all of the parallel mechanism. And can complete the task of positive solution, inverse solution and cross solution. Geometric iteration method is suitable for the solution of multiple parallel mechanism and parallel system, it is also used to calculate the complex polyhedron and variable geometry parallel mechanism.
(3) Establish a simple ontology about workspace of the parallel mechanism. Analyse the robot workspace from the perspective of ontology. Extend the connotation and extension of the concept of the workspace.
This paper has the effects and significance as follows:
①The arthropodization analysis methods set up a bridge between the ideal and non ideal parallel mechanism, and provided a new train of thought for the analysis of kinematics, dynamics, workspace and singularity characteristics of the non ideal parallel mechanism.②Geometric iteration method witch completely avoided the nonlinear equations provided a new choice for all forward solutions of the parallel mechanism. The novel algorithm has been successfully applied to a variety of different types of parallel mechanism, and provided a new way to solve nonlinear equations with the numerical methods, and provided a new method to solve the intersection solution, and equivalent input solutions. It reduced overall difficulty of the forward solution and lay a foundation for the popularization and application of the parallel mechanism.③Established a ontology about workspace of robot. Not only widened the research field of the robot working space, but also offered the theoretical foundation for optimization design. For example, various disturbance space of the mechanism, space utilization rate of the mechanism, spatial regularity and other new contents, which can be used and need further study.
引文
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