空間機構與柔順機構的運動學分析和綜合
發(fā)布時間：2018-08-07 18:27
【摘要】：機構的運動學分析和綜合是機器人機構學研究中最基礎也是最重要的部分,不但為機構的設計奠定基礎,而且為機器人機構的實際應用提供理論支持。本文以實現(xiàn)機構運動學的數(shù)學機械化為目的,對空間機構和柔順機構運動學中的一些難點、熱點問題進行研究,主要的研究內容和創(chuàng)新成果如下：(1) 以一般6-4A Stewart臺體型并聯(lián)機構位置正解為研究對象,首先通過構型變換得到新的等效機構；然后使用由重心坐標推導出的含有Cayley-Menger行列式的三邊測量法公式對等效機構建模,建立等效機構的基本約束方程組；接著通過矢量回路關系和變量替換將8個約束方程轉換為含有5個變量的5個基本約束方程；然后用矢量消元法對其中4個(含有3個相同變量)約束方程進行消元,推導出一個含有其余兩個變量的方程；最后將矢量消元后得到的方程與余下的一個約束方程聯(lián)立,構造一個10×10的S ylvester結式,獲得該問題的一元32次方程,完成了該問題的數(shù)學機械化求解。此方法是基于幾何不變量進行建模求解,其結果更簡單有效,易于程序實現(xiàn)。(2) 提出了一種求解一般5-5B Stewart臺體型并聯(lián)機構位置正解問題的代數(shù)求解法。首先通過構型變換得到了新的等效機構,然后基于幾何不變量建立該問題的基本約束方程組,接著使用矢量消元法對得到的基本約束方程組消元,推導出三個含有兩個未知量的運動學約束方程；再使用計算機代數(shù)系統(tǒng),利用符號運算,提取出兩個約束方程的最大公因式；最后,使用第三個約束方程和最大公因式,構造出一個10×10的Sylvester結式矩陣,獲得該問題的一元24次方程。該方法的創(chuàng)新之處在于對基本約束方程的消元步驟,其整個求解過程都是以符號形式完成的,從而實現(xiàn)了該問題的數(shù)學機械化求解。(3) 改進了一般6-6Stewart臺體型并聯(lián)機構位置正解問題的代數(shù)求解。應用Cayley公式描述旋轉矩陣,建立了一般6-6Stewart臺體型并聯(lián)機構的運動學約束方程組；接著通過變量替換和線性消元將6個運動學約束方程轉換成4個含有四個變元的多項式方程；然后為了使其中一個變量優(yōu)先消去,利用變量替換,將其次數(shù)提高,應用分次逆字典序下的Grobner基法求上述4個多項式方程的約化基,得到16個約化基；最后從16個約化基中選取10個,構造一個10×10的Sylvester結式,獲得該問題的一元40次方程。該方法的優(yōu)點在于構造的結式尺寸較小,從而提高了計算速度。(4) 解決了球面四桿機構五位置剛體導引的全部實數(shù)解求解問題。首先使用Dixon結式消元法得到球面四桿機構五位置剛體導引問題的一元六次方程。接著,基于施圖姆定理,推導出了該問題存在全部實數(shù)解的充分必要條件。考慮到球面四桿機構存在曲柄的條件和無回路缺陷的條件,構造了兩個目標函數(shù),然后使用自適應遺傳算法,優(yōu)化得到相應情況下的球面四桿機構尺寸。這種通過代數(shù)消元,獲得一元高次方程,然后再基于施圖姆定理求其全部實數(shù)解的方法,可以為很多其他機構運動學問題提供一種新的研究思路。(5) 提出了空間三彈簧系統(tǒng)靜力逆分析問題的封閉解析解。首先基于幾何約束條件和靜力平衡條件,推導出一個特殊的三元四次方程組；接著使用Dixon結式消元法,通過去掉線性相關行和列,構造出一個20×20的結式矩陣；然后通過分析,進一步去掉線性相關行和列,將上述20×20的矩陣約化為18×18的結式矩陣；最終得到一個46次的單變量多項式。進一步分析可知,其中24個解是退化解,余下的22個解才是該空間三彈簧系統(tǒng)的有效解。本文提出的代數(shù)消元法是依據(jù)該問題解析幾何法求解的思路,揭示了該問題的固有幾何特性與代數(shù)消元法間的聯(lián)系。(6) 提出了柔順機構自由度的計算準則。本文使用柔順機構柔度矩陣的特征運動旋量和特征力旋量分解去判定柔順機構的自由度。首先,證明了柔度特征值具有坐標不變性。接著,引入特征長度的概念,使得移動柔度特征值和轉動柔度特征值具有相同的單位�；谌岫忍卣髦档纳鲜鰞蓷l性質,本文提出了柔順機構的兩條自由度判定準則。同時,針對串聯(lián)開環(huán)柔順機構和閉環(huán)柔順機構,本文給出了兩條選擇特征長度的指導性步驟,并且驗證了特征長度的魯棒性。接著本文給出了判定任意柔順機構自由度的一般步驟。最后,本文給出兩個實例驗證所提出方法的有效性。本文提出的柔順機構自由度計算準則,不僅可以給出機構的自由度數(shù),而且還可以了解自由度的具體分布,其結果與使用旋量法對傳統(tǒng)剛性機構的自由度計算有相似之處。綜上,本論文對空間機構和柔順機構的運動學中部分尚未解決的熱點和難點問題進行了研究,并在以下四個方面取得了開創(chuàng)性的研究成果。本論文提出了基于幾何不變量對機構運動學分析建模的新方法；首次完成了一般5.-5B Stewart臺體型并聯(lián)機構位置正解的數(shù)學機械化求解；首次提出把代數(shù)消元法和施圖姆定理相結合的方法求解機構運動學分析或綜合問題的全部實數(shù)解；以及首次提出基于柔順機構柔度矩陣的分解判定柔順機構自由度的方法。此外,本論文提出了依據(jù)解析幾何法的分析過程,進行非線性方程組的代數(shù)消元求解,為求解非線性方程組提供了一種新的思路。
[Abstract]:Kinematics analysis and synthesis of mechanism is the most fundamental and most important part in the study of robot mechanism. It not only lays the foundation for the design of mechanism, but also provides theoretical support for the practical application of robot mechanism. This paper aims at realizing the mathematical mechanization of mechanism kinematics, and one of the kinematics of space mechanism and compliant mechanism. Some difficult and hot issues are studied. The main research content and innovation results are as follows: (1) the new equivalent mechanism is obtained by the configuration transformation of the general 6-4A Stewart parallel mechanism, and then the formula with the Cayley-Menger determinant derived by the center of gravity coordinates is used. The equivalent mechanism is modeled and the basic constraint equations of the equivalent mechanism are set up, and then the 8 constraint equations are converted into 5 basic constraint equations with 5 variables through the vector loop relationship and variable substitution. Then, 4 of them (including 3 same variables) are eliminated by vector elimination, and the other two is derived. The equation of a variable is obtained. Finally, the equation obtained by the vector elimination is combined with the remaining constraint equation, and a 10 * 10 S ylvester equation is constructed to obtain the one dollar and 32 order equations of the problem, and the mathematical mechanization of the problem is solved. The method is based on the geometric invariants to solve the problem, and the result is more simple and effective and easy to be used. The realization of the program. (2) an algebraic solution for solving the positive solution of the general 5-5B Stewart parallel mechanism is proposed. First, the new equivalent mechanism is obtained by the configuration transformation. Then the basic constraint equations of the problem are established based on the geometric invariants, and then the basic constraint equations are eliminated by using the vector quantity elimination method. Three kinematic constraint equations with two unknown quantities are derived, and the maximum common factor of two constraint equations is extracted by using the computer algebra system and the symbolic operation. Finally, a 10 * 10 Sylvester node matrix is constructed with third constraint equations and the maximum common factor, and the one dollar and 24 equations of the problem are obtained. The innovation of the method lies in the elimination of the basic constraint equations. The whole solution process is completed in the form of symbols, thus realizing the mathematical mechanization of the problem. (3) the algebraic solution of the positive solution of the position of the general 6-6Stewart parallel mechanism is improved. The rotation matrix is described by the Cayley formula. The kinematic constraint equations of the general 6-6Stewart parallel mechanism are used to transform the 6 kinematic constraint equations into 4 polynomial equations with four variables by variable substitution and linear elimination. The Grobner base method is used to obtain the reductive basis of the 4 polynomial equations, and 16 reductants are obtained. Finally, 10 of the 16 reductants are selected and a 10 * 10 Sylvester form is constructed to obtain the one dollar 40 equation of the problem. The advantage of this method is that the structure is smaller in size, and the calculation speed is improved. (4) the spherical four pole machine is solved. All real number solutions of the five position rigid body guidance are solved. First, the one element and six order equation of the five position rigid body guidance of a spherical four bar mechanism is obtained by using the Dixon node elimination method. Then, based on the Stum theorem, the full and necessary pieces of all real number solutions of the problem are derived. Considering the bar of the spherical four rod mechanism, the bar has a crank bar. Two objective functions are constructed and the adaptive genetic algorithm is used to optimize the size of the spherical four bar mechanism in the corresponding case. This method can obtain a high order equation by algebraic elimination and then the method of finding all the real number solution based on the Stum theorem, which can be learned for many other institutions. A new research idea is provided. (5) a closed analytic solution for the static inverse analysis of space three spring system is proposed. First, a special three element four order equation group is derived based on the geometric constraint conditions and static equilibrium conditions, and then a 20 x 20 is constructed by using the Dixon node elimination method by removing the linear correlation rows and columns. By analyzing, the linear correlation rows and columns are further removed and the above 20 x 20 matrix is reduced to a 18 * 18 node matrix. Finally, a single variable polynomial of 46 times is obtained. Further analysis shows that 24 solutions are degenerate solutions, and the remaining 22 solutions are effective solutions for the three spring system in the space. The algebraic elimination method is based on the solution of the analytic geometric method of the problem. The relation between the inherent geometric characteristics of the problem and the algebraic elimination method is revealed. (6) the calculation criterion of the degree of freedom of the compliant mechanism is proposed. This paper uses the characteristic motion rotation of the compliant mechanism and the decomposition of the characteristic force to determine the degree of freedom of the compliant mechanism. First, it is proved that the flexibility characteristic value has the coordinate invariance. Then, the concept of characteristic length is introduced to make the moving flexibility characteristic value and the rotational flexibility characteristic value have the same unit. Based on the above two properties of the flexibility characteristic value, this paper presents the two degree of freedom criteria for the compliant mechanism. And the closed loop compliant mechanism, this paper gives two guiding steps for selecting the characteristic length, and verifies the robustness of the feature length. Then this paper gives the general steps to determine the degree of freedom of arbitrary compliant mechanisms. Finally, two examples are given to verify the effectiveness of the proposed method. The criterion is not only to give the degree of freedom of the mechanism, but also to understand the specific distribution of the degree of freedom. The results are similar to the calculation of the degree of freedom of the traditional rigid mechanism by the use of the method of rotation. In this paper, a new method of modeling the kinematic analysis of mechanisms based on geometric invariants is proposed in the four aspects. The mathematical mechanization of the positive solution of the position of the general 5.-5B Stewart parallel mechanism is solved for the first time. The method of combining the algebraic elimination method and the Stum theorem is proposed for the first time. All real number solutions of kinematic analysis or synthesis of mechanisms are solved, and the method of determining the degree of freedom of compliant mechanisms based on the decomposition of compliant mechanism is proposed for the first time. In addition, this paper presents an analytical process based on analytic geometry to solve nonlinear equations and provide a solution for nonlinear equations, and provides a solution for solving nonlinear equations. A new way of thinking.
【學位授予單位】：北京郵電大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：TH112
本文編號：2170945
[Abstract]:Kinematics analysis and synthesis of mechanism is the most fundamental and most important part in the study of robot mechanism. It not only lays the foundation for the design of mechanism, but also provides theoretical support for the practical application of robot mechanism. This paper aims at realizing the mathematical mechanization of mechanism kinematics, and one of the kinematics of space mechanism and compliant mechanism. Some difficult and hot issues are studied. The main research content and innovation results are as follows: (1) the new equivalent mechanism is obtained by the configuration transformation of the general 6-4A Stewart parallel mechanism, and then the formula with the Cayley-Menger determinant derived by the center of gravity coordinates is used. The equivalent mechanism is modeled and the basic constraint equations of the equivalent mechanism are set up, and then the 8 constraint equations are converted into 5 basic constraint equations with 5 variables through the vector loop relationship and variable substitution. Then, 4 of them (including 3 same variables) are eliminated by vector elimination, and the other two is derived. The equation of a variable is obtained. Finally, the equation obtained by the vector elimination is combined with the remaining constraint equation, and a 10 * 10 S ylvester equation is constructed to obtain the one dollar and 32 order equations of the problem, and the mathematical mechanization of the problem is solved. The method is based on the geometric invariants to solve the problem, and the result is more simple and effective and easy to be used. The realization of the program. (2) an algebraic solution for solving the positive solution of the general 5-5B Stewart parallel mechanism is proposed. First, the new equivalent mechanism is obtained by the configuration transformation. Then the basic constraint equations of the problem are established based on the geometric invariants, and then the basic constraint equations are eliminated by using the vector quantity elimination method. Three kinematic constraint equations with two unknown quantities are derived, and the maximum common factor of two constraint equations is extracted by using the computer algebra system and the symbolic operation. Finally, a 10 * 10 Sylvester node matrix is constructed with third constraint equations and the maximum common factor, and the one dollar and 24 equations of the problem are obtained. The innovation of the method lies in the elimination of the basic constraint equations. The whole solution process is completed in the form of symbols, thus realizing the mathematical mechanization of the problem. (3) the algebraic solution of the positive solution of the position of the general 6-6Stewart parallel mechanism is improved. The rotation matrix is described by the Cayley formula. The kinematic constraint equations of the general 6-6Stewart parallel mechanism are used to transform the 6 kinematic constraint equations into 4 polynomial equations with four variables by variable substitution and linear elimination. The Grobner base method is used to obtain the reductive basis of the 4 polynomial equations, and 16 reductants are obtained. Finally, 10 of the 16 reductants are selected and a 10 * 10 Sylvester form is constructed to obtain the one dollar 40 equation of the problem. The advantage of this method is that the structure is smaller in size, and the calculation speed is improved. (4) the spherical four pole machine is solved. All real number solutions of the five position rigid body guidance are solved. First, the one element and six order equation of the five position rigid body guidance of a spherical four bar mechanism is obtained by using the Dixon node elimination method. Then, based on the Stum theorem, the full and necessary pieces of all real number solutions of the problem are derived. Considering the bar of the spherical four rod mechanism, the bar has a crank bar. Two objective functions are constructed and the adaptive genetic algorithm is used to optimize the size of the spherical four bar mechanism in the corresponding case. This method can obtain a high order equation by algebraic elimination and then the method of finding all the real number solution based on the Stum theorem, which can be learned for many other institutions. A new research idea is provided. (5) a closed analytic solution for the static inverse analysis of space three spring system is proposed. First, a special three element four order equation group is derived based on the geometric constraint conditions and static equilibrium conditions, and then a 20 x 20 is constructed by using the Dixon node elimination method by removing the linear correlation rows and columns. By analyzing, the linear correlation rows and columns are further removed and the above 20 x 20 matrix is reduced to a 18 * 18 node matrix. Finally, a single variable polynomial of 46 times is obtained. Further analysis shows that 24 solutions are degenerate solutions, and the remaining 22 solutions are effective solutions for the three spring system in the space. The algebraic elimination method is based on the solution of the analytic geometric method of the problem. The relation between the inherent geometric characteristics of the problem and the algebraic elimination method is revealed. (6) the calculation criterion of the degree of freedom of the compliant mechanism is proposed. This paper uses the characteristic motion rotation of the compliant mechanism and the decomposition of the characteristic force to determine the degree of freedom of the compliant mechanism. First, it is proved that the flexibility characteristic value has the coordinate invariance. Then, the concept of characteristic length is introduced to make the moving flexibility characteristic value and the rotational flexibility characteristic value have the same unit. Based on the above two properties of the flexibility characteristic value, this paper presents the two degree of freedom criteria for the compliant mechanism. And the closed loop compliant mechanism, this paper gives two guiding steps for selecting the characteristic length, and verifies the robustness of the feature length. Then this paper gives the general steps to determine the degree of freedom of arbitrary compliant mechanisms. Finally, two examples are given to verify the effectiveness of the proposed method. The criterion is not only to give the degree of freedom of the mechanism, but also to understand the specific distribution of the degree of freedom. The results are similar to the calculation of the degree of freedom of the traditional rigid mechanism by the use of the method of rotation. In this paper, a new method of modeling the kinematic analysis of mechanisms based on geometric invariants is proposed in the four aspects. The mathematical mechanization of the positive solution of the position of the general 5.-5B Stewart parallel mechanism is solved for the first time. The method of combining the algebraic elimination method and the Stum theorem is proposed for the first time. All real number solutions of kinematic analysis or synthesis of mechanisms are solved, and the method of determining the degree of freedom of compliant mechanisms based on the decomposition of compliant mechanism is proposed for the first time. In addition, this paper presents an analytical process based on analytic geometry to solve nonlinear equations and provide a solution for nonlinear equations, and provides a solution for solving nonlinear equations. A new way of thinking.
【學位授予單位】：北京郵電大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：TH112
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