空间机构与柔顺机构的运动学分析和综合

发布时间：2018-08-07 18:27

【摘要】：机构的运动学分析和综合是机器人机构学研究中最基础也是最重要的部分,不但为机构的设计奠定基础,而且为机器人机构的实际应用提供理论支持。本文以实现机构运动学的数学机械化为目的,对空间机构和柔顺机构运动学中的一些难点、热点问题进行研究,主要的研究内容和创新成果如下：(1) 以一般6-4A Stewart台体型并联机构位置正解为研究对象,首先通过构型变换得到新的等效机构；然后使用由重心坐标推导出的含有Cayley-Menger行列式的三边测量法公式对等效机构建模,建立等效机构的基本约束方程组；接着通过矢量回路关系和变量替换将8个约束方程转换为含有5个变量的5个基本约束方程；然后用矢量消元法对其中4个(含有3个相同变量)约束方程进行消元,推导出一个含有其余两个变量的方程；最后将矢量消元后得到的方程与余下的一个约束方程联立,构造一个10×10的S ylvester结式,获得该问题的一元32次方程,完成了该问题的数学机械化求解。此方法是基于几何不变量进行建模求解,其结果更简单有效,易于程序实现。(2) 提出了一种求解一般5-5B Stewart台体型并联机构位置正解问题的代数求解法。首先通过构型变换得到了新的等效机构,然后基于几何不变量建立该问题的基本约束方程组,接着使用矢量消元法对得到的基本约束方程组消元,推导出三个含有两个未知量的运动学约束方程；再使用计算机代数系统,利用符号运算,提取出两个约束方程的最大公因式；最后,使用第三个约束方程和最大公因式,构造出一个10×10的Sylvester结式矩阵,获得该问题的一元24次方程。该方法的创新之处在于对基本约束方程的消元步骤,其整个求解过程都是以符号形式完成的,从而实现了该问题的数学机械化求解。(3) 改进了一般6-6Stewart台体型并联机构位置正解问题的代数求解。应用Cayley公式描述旋转矩阵,建立了一般6-6Stewart台体型并联机构的运动学约束方程组；接着通过变量替换和线性消元将6个运动学约束方程转换成4个含有四个变元的多项式方程；然后为了使其中一个变量优先消去,利用变量替换,将其次数提高,应用分次逆字典序下的Grobner基法求上述4个多项式方程的约化基,得到16个约化基；最后从16个约化基中选取10个,构造一个10×10的Sylvester结式,获得该问题的一元40次方程。该方法的优点在于构造的结式尺寸较小,从而提高了计算速度。(4) 解决了球面四杆机构五位置刚体导引的全部实数解求解问题。首先使用Dixon结式消元法得到球面四杆机构五位置刚体导引问题的一元六次方程。接着,基于施图姆定理,推导出了该问题存在全部实数解的充分必要条件。考虑到球面四杆机构存在曲柄的条件和无回路缺陷的条件,构造了两个目标函数,然后使用自适应遗传算法,优化得到相应情况下的球面四杆机构尺寸。这种通过代数消元,获得一元高次方程,然后再基于施图姆定理求其全部实数解的方法,可以为很多其他机构运动学问题提供一种新的研究思路。(5) 提出了空间三弹簧系统静力逆分析问题的封闭解析解。首先基于几何约束条件和静力平衡条件,推导出一个特殊的三元四次方程组；接着使用Dixon结式消元法,通过去掉线性相关行和列,构造出一个20×20的结式矩阵；然后通过分析,进一步去掉线性相关行和列,将上述20×20的矩阵约化为18×18的结式矩阵；最终得到一个46次的单变量多项式。进一步分析可知,其中24个解是退化解,余下的22个解才是该空间三弹簧系统的有效解。本文提出的代数消元法是依据该问题解析几何法求解的思路,揭示了该问题的固有几何特性与代数消元法间的联系。(6) 提出了柔顺机构自由度的计算准则。本文使用柔顺机构柔度矩阵的特征运动旋量和特征力旋量分解去判定柔顺机构的自由度。首先,证明了柔度特征值具有坐标不变性。接着,引入特征长度的概念,使得移动柔度特征值和转动柔度特征值具有相同的单位。基于柔度特征值的上述两条性质,本文提出了柔顺机构的两条自由度判定准则。同时,针对串联开环柔顺机构和闭环柔顺机构,本文给出了两条选择特征长度的指导性步骤,并且验证了特征长度的鲁棒性。接着本文给出了判定任意柔顺机构自由度的一般步骤。最后,本文给出两个实例验证所提出方法的有效性。本文提出的柔顺机构自由度计算准则,不仅可以给出机构的自由度数,而且还可以了解自由度的具体分布,其结果与使用旋量法对传统刚性机构的自由度计算有相似之处。综上,本论文对空间机构和柔顺机构的运动学中部分尚未解决的热点和难点问题进行了研究,并在以下四个方面取得了开创性的研究成果。本论文提出了基于几何不变量对机构运动学分析建模的新方法；首次完成了一般5.-5B Stewart台体型并联机构位置正解的数学机械化求解；首次提出把代数消元法和施图姆定理相结合的方法求解机构运动学分析或综合问题的全部实数解；以及首次提出基于柔顺机构柔度矩阵的分解判定柔顺机构自由度的方法。此外,本论文提出了依据解析几何法的分析过程,进行非线性方程组的代数消元求解,为求解非线性方程组提供了一种新的思路。
[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.
【学位授予单位】：北京邮电大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：TH112

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