机构运动分析的几何代数新方法研究

发布时间：2018-05-11 17:26

本文选题：机构运动学 + 几何代数　；参考：《北京邮电大学》2014年博士论文

【摘要】：机构运动分析在机械设计中具有重要作用：一方面,机构运动学正解为机械设计完成后的机构性能进行验证,验证是否满足设计要求；另一方面,机构运动学反解为机械控制提供控制程序。几何代数方法是机构运动学分析的重要方法,目前仍然具有巨大的研究潜力,传统机构和新类型机构运动学分析的几何代数方法,都有待深入研究。所以本论文提出了机构运动学分析的几何代数方法研究的课题,一方面研究传统串联和并联机构运动学分析的几何代数新方法,另一方面研究球面机构和变胞机构等新类型机构运动分析的几何代数方法。主要研究内容和创新成果如下： (1)研究并提出了串联机构运动学分析的几何代数新方法—D-H四元数变换方法。给出了点映射的四元数描述方法和相邻连杆间运动的D-H四元数变换方法。建立了运动学分析的D-H四元数变换的矩阵演算方法,构造出了机器人机构学中经典的D-H齐次矩阵。证明了D-H四元数变换方法与D-H齐次变换矩阵方法的运动学分析结果是一致的,从而从理论上证明了D-H四元数变换方法的正确性。在相邻连杆运动的D-H四元数变换公式基础上进一步推广,给出了任意个连杆串联机构运动学分析的D-H四元数变换方法。串联机构的逆运动学分析是机构学中的难点问题,本论文将D-H四元数运动学方程式分离为位置和姿态2个方程式,这2个方程式可构造出含有7个方程的方程组,使方程数量满足了4R以上串联机构运动学反解的要求。为了降低方程组的求解难度,采用取姿态方程中三角函数的一半组成新的姿态方程的方法,将方程次数降低为原来的一半。通过正运动学分析和逆运动学分析的实例,验证了所提出方法的正确性和有效性。所提出的串联机构运动学分析的D-H四元数变换新方法,不但避免了复杂的矩阵运算,而且运动学方程较矩阵方法有所减少,同时也具有步骤清晰、容易通过数学机械化实现、几何意义明确和计算简单的优势,是一种正确且有效的串联机构运动学分析的新方法。 (2)研究并提出了并联机构运动学分析的共形几何代数新方法。首先,给出了平面并联机构运动学分析的共形几何代数建模方法,并给出了一种改进的Sylvester结式消元法,即冗余因子消去法,克服了结式消元法容易产生增根的不足,能够得出非线性方程组的准确解。然后,提出了空间并联机构运动学分析的共形几何代数分析方法,这种方法集几何表示和运算为一体,只通过共形几何代数的描述和运算即可建立运动学分析模型,不需要复杂的矩阵运算。 (3)研究并提出了球面机构运动分析的几何代数方法。首先,研究了球面并联机构运动分析的几何代数方法,给出了建立球面并联机构运动分析数学模型的四元数和球面几何方法,并且给出了将数学模型消元简化的方法,解决了球面并联机构运动分析的建模复杂、表示动平台位姿的直接变量求解困难等难点问题。然后,研究了球面剪叉可展机构运动分析的几何代数方法,基于螺旋理论分析了由任意个球面剪叉单元和任意个支链构成的球面剪叉可展机构的自由度特性,并基于球面几何学理论对球面剪叉可展机构进行了运动分析,编制了运动分析软件,实现了对这类机构运动的自动分析计算。基于本论文所提出的方法可揭示球面剪叉可展机构的运动特性,为设计出满足一定运动特性要求的一系列的球面剪叉可展机构产品提供了理论基础。 (4)研究并提出了变胞机构运动分析的几何代数方法。首先,研究了变胞机构运动特性分析方法,提出了李群李代数和旋量代数两种分析方法。然后,研究了变胞机构运动学分析方法,提出了并联变胞机构运动学分析的几何代数方法。变胞机构的特殊之处是构态可变,以动平台上的坐标原点描述动平台的位置,以欧拉角描述动平台的姿态,给出了动平台上任意一点在定坐标系中位置的四元数表达式,进而提出了一种建立并联变胞机构运动学分析的统一数学模型的方法。所提出的并联变胞机构运动学分析的几何代数方法,可对并联变胞机构在不同构态时进行正运动学和逆运动学分析。本论文研究了机构运动分析的几何代数新方法,提出了串联机构运动学分析的D-H四元数变换新方法,并提出了并联机构运动学分析的共形几何代数新方法,也提出了球面机构和变胞机构运动分析的几何代数方法。所提出的机构运动分析方法,不但具有几何表示与代数运算为一体、几何意义明确、计算简单且易于采用数学机械化实现等优势,而且经过验证是正确和有效的方法。本论文所提出的机构运动分析的几何代数新方法,丰富和发展了机构运动学理论。
[Abstract]:The kinematic analysis of mechanism plays an important role in mechanical design. On the one hand, the kinematics of mechanism is proved to verify the performance of mechanism after the mechanical design is completed, to verify whether the design meets the design requirements; on the other hand, the inverse kinematics of the mechanism provides the control program for the mechanical control. At present, it still has great potential for research. The geometric algebra method of the kinematic analysis of traditional and new types of mechanism needs to be studied deeply. So this paper puts forward the research topic of geometric algebra of mechanism kinematics analysis. On the one hand, it studies the new method of geometric algebra of the kinematic analysis of the traditional series and parallel mechanisms, and the other one. The geometric algebra methods for kinematic analysis of new mechanisms such as spherical mechanisms and metamorphic mechanisms are studied. The main contents and innovations are as follows:
(1) the new method of geometric algebra of the kinematic analysis of the series mechanism, D-H four element transformation method, is studied and proposed. The four element number description method of point mapping and the D-H four transformation method of the motion between adjacent connecting rods are given. The matrix representation method of the D-H four element transformation of the kinematic analysis is established. The D-H homogeneous matrix of the code proves that the D-H four element number transformation method is in agreement with the kinematic analysis results of the D-H homogeneous transformation matrix method. Thus, the correctness of the D-H four element transformation method is proved theoretically. On the basis of the D-H four element transformation formula of the adjacent connecting rod motion, a series of connecting rod series mechanism is further extended. The D-H four element transformation method of dynamic analysis. The inverse kinematics analysis of the series mechanism is a difficult problem in the mechanism. This paper divides the D-H four element number kinematics equation into 2 equations of position and attitude, and these 2 equations can construct the equation group containing 7 equations, so that the square range satisfies the kinematic inverse of the series mechanism above 4R. In order to reduce the difficulty of solving the equations, a new attitude equation is used to reduce the number of equations by taking half of the trigonometric function in the attitude equation. The correctness and effectiveness of the proposed method are verified by an example of the positive kinematics analysis and inverse kinematics analysis. The proposed series mechanism is proved. The new method of D-H four element transformation for kinematic analysis not only avoids the complex matrix operation, but also reduces the kinematic equation compared with the matrix method. At the same time, it also has clear steps, easy to realize through mechanization, clear geometric meaning and simple calculation, and is a new and effective new method for kinematic analysis of series mechanism. Method.
(2) a new method of conformal geometric algebra for kinematic analysis of parallel mechanism is studied and proposed. First, a conformal geometric algebraic modeling method for kinematic analysis of planar parallel mechanism is given. An improved Sylvester node elimination method, that is, redundancy elimination method, is given. The exact solution of the nonlinear equations is obtained. Then, a conformal geometric algebra analysis method for kinematic analysis of spatial parallel mechanism is proposed. This method combines geometric representation and operation as one. The kinematic analysis model can be established only through the description and operation of conformal geometric algebra, and no complex matrix operation is needed.
(3) the geometric algebra method of the motion analysis of spherical mechanism is studied and proposed. First, the geometric algebra method of the kinematic analysis of a spherical parallel mechanism is studied. The four elements and spherical geometric methods for establishing the mathematical model of the kinematic analysis of a spherical parallel mechanism are given, and the method of simplifying the mathematical model is given, and the spherical parallel connection is solved. The modeling of the kinematic analysis of mechanism is complicated, which indicates the difficulty of solving the direct variable of the moving platform. Then, the geometric algebraic method of the motion analysis of a spherical shear fork is studied. Based on the helix theory, the free degree characteristics of the deployable mechanism of a spherical shear fork composed of any spherical shear fork unit and any branch chain are analyzed. Based on the spherical geometry theory, the motion analysis of the spherical shear fork mechanism is carried out, and the motion analysis software is compiled to realize the automatic analysis and calculation of the movement of this kind of mechanism. Based on the method proposed in this paper, the motion characteristics of the deployable mechanism of the spherical shear fork can be revealed, in order to set up a series of requirements to meet the requirements of certain motion characteristics. The spherical shear fork deployable mechanism provides a theoretical basis.
(4) the geometric algebraic method for kinematic analysis of variable cell mechanisms is studied and proposed. First, the analysis method of motion characteristics of variable cell mechanisms is studied, and two analytical methods of Li Qunli algebra and spin algebra are proposed. Then, the kinematic analysis method of the variable cell mechanism is studied, and the geometric algebra method of the kinematics analysis of the parallel cell mechanism is proposed. The special feature of the mechanism is that the structure is variable, the position of the dynamic platform is described by the origin of the coordinate on the moving platform, the attitude of the dynamic platform is described by the Euler angle, and the four element number expression of any point on the fixed coordinate system is given, and a method of establishing a unified mathematical model for establishing the kinematics analysis of the parallel cell mechanism is proposed. The geometric algebra method of kinematic analysis of parallel metamorphic mechanisms can be used to analyze the kinematic and inverse kinematics of parallel metamorphic mechanisms in different configurations.
In this paper, a new method of geometric algebra for kinematic analysis of mechanism is studied. A new method of D-H four element transformation for kinematic analysis of a series mechanism is proposed. A new method of conformal geometric algebra for kinematic analysis of a parallel mechanism is proposed. A geometric algebra method for the kinematic analysis of a spherical mechanism and a cell mechanism is also proposed. The analysis method not only has the advantages of geometric representation and algebraic operation, geometric meaning clear, simple calculation and easy to use mathematical mechanization, but also has been proved to be correct and effective. The new method of geometric algebra for kinematic analysis of mechanism proposed in this paper enriches and develops mechanism kinematics theory.

【学位授予单位】：北京邮电大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TH112

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