本文選題：機構運動學 + 幾何代數��； 參考：《北京郵電大學》2014年博士論文

【摘要】：機構運動分析在機械設計中具有重要作用：一方面,機構運動學正解為機械設計完成后的機構性能進行驗證,驗證是否滿足設計要求；另一方面,機構運動學反解為機械控制提供控制程序。幾何代數方法是機構運動學分析的重要方法,目前仍然具有巨大的研究潛力,傳統(tǒng)機構和新類型機構運動學分析的幾何代數方法,都有待深入研究。所以本論文提出了機構運動學分析的幾何代數方法研究的課題,一方面研究傳統(tǒng)串聯和并聯機構運動學分析的幾何代數新方法,另一方面研究球面機構和變胞機構等新類型機構運動分析的幾何代數方法。主要研究內容和創(chuàng)新成果如下： (1)研究并提出了串聯機構運動學分析的幾何代數新方法—D-H四元數變換方法。給出了點映射的四元數描述方法和相鄰連桿間運動的D-H四元數變換方法。建立了運動學分析的D-H四元數變換的矩陣演算方法,構造出了機器人機構學中經典的D-H齊次矩陣。證明了D-H四元數變換方法與D-H齊次變換矩陣方法的運動學分析結果是一致的,從而從理論上證明了D-H四元數變換方法的正確性。在相鄰連桿運動的D-H四元數變換公式基礎上進一步推廣,給出了任意個連桿串聯機構運動學分析的D-H四元數變換方法。串聯機構的逆運動學分析是機構學中的難點問題,本論文將D-H四元數運動學方程式分離為位置和姿態(tài)2個方程式,這2個方程式可構造出含有7個方程的方程組,使方程數量滿足了4R以上串聯機構運動學反解的要求。為了降低方程組的求解難度,采用取姿態(tài)方程中三角函數的一半組成新的姿態(tài)方程的方法,將方程次數降低為原來的一半。通過正運動學分析和逆運動學分析的實例,驗證了所提出方法的正確性和有效性。所提出的串聯機構運動學分析的D-H四元數變換新方法,不但避免了復雜的矩陣運算,而且運動學方程較矩陣方法有所減少,同時也具有步驟清晰、容易通過數學機械化實現、幾何意義明確和計算簡單的優(yōu)勢,是一種正確且有效的串聯機構運動學分析的新方法。 (2)研究并提出了并聯機構運動學分析的共形幾何代數新方法。首先,給出了平面并聯機構運動學分析的共形幾何代數建模方法,并給出了一種改進的Sylvester結式消元法,即冗余因子消去法,克服了結式消元法容易產生增根的不足,能夠得出非線性方程組的準確解。然后,提出了空間并聯機構運動學分析的共形幾何代數分析方法,這種方法集幾何表示和運算為一體,只通過共形幾何代數的描述和運算即可建立運動學分析模型,不需要復雜的矩陣運算。 (3)研究并提出了球面機構運動分析的幾何代數方法。首先,研究了球面并聯機構運動分析的幾何代數方法,給出了建立球面并聯機構運動分析數學模型的四元數和球面幾何方法,并且給出了將數學模型消元簡化的方法,解決了球面并聯機構運動分析的建模復雜、表示動平臺位姿的直接變量求解困難等難點問題。然后,研究了球面剪叉可展機構運動分析的幾何代數方法,基于螺旋理論分析了由任意個球面剪叉單元和任意個支鏈構成的球面剪叉可展機構的自由度特性,并基于球面幾何學理論對球面剪叉可展機構進行了運動分析,編制了運動分析軟件,實現了對這類機構運動的自動分析計算�；诒菊撐乃岢龅姆椒ǹ山沂厩蛎婕舨婵烧箼C構的運動特性,為設計出滿足一定運動特性要求的一系列的球面剪叉可展機構產品提供了理論基礎。 (4)研究并提出了變胞機構運動分析的幾何代數方法。首先,研究了變胞機構運動特性分析方法,提出了李群李代數和旋量代數兩種分析方法。然后,研究了變胞機構運動學分析方法,提出了并聯變胞機構運動學分析的幾何代數方法。變胞機構的特殊之處是構態(tài)可變,以動平臺上的坐標原點描述動平臺的位置,以歐拉角描述動平臺的姿態(tài),給出了動平臺上任意一點在定坐標系中位置的四元數表達式,進而提出了一種建立并聯變胞機構運動學分析的統(tǒng)一數學模型的方法。所提出的并聯變胞機構運動學分析的幾何代數方法,可對并聯變胞機構在不同構態(tài)時進行正運動學和逆運動學分析。 本論文研究了機構運動分析的幾何代數新方法,提出了串聯機構運動學分析的D-H四元數變換新方法,并提出了并聯機構運動學分析的共形幾何代數新方法,也提出了球面機構和變胞機構運動分析的幾何代數方法。所提出的機構運動分析方法,不但具有幾何表示與代數運算為一體、幾何意義明確、計算簡單且易于采用數學機械化實現等優(yōu)勢,而且經過驗證是正確和有效的方法。本論文所提出的機構運動分析的幾何代數新方法,豐富和發(fā)展了機構運動學理論。
[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.

【學位授予單位】：北京郵電大學
【學位級別】：博士
【學位授予年份】：2014
【分類號】：TH112

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