【作者】 黄昔光；

【导师】 廖启征；

【作者基本信息】 北京邮电大学 ， 机械电子工程， 2008， 博士

【摘要】 结合目前机构运动学中部分尚未解决的热点、难点问题,本文以四元数、倍四元数、矩阵论和机器人机构学为基础,采用数值计算探索,代数法求解,数值计算验证的研究方法,对机构运动学中若干问题及其代数法进行理论研究,解决了平面机构和空间机构运动学中一些难题。主要研究内容与创新成果如下:(1)解决了给定固定铰链位置的平面铰链四杆机构五精确点轨迹综合代数解问题。构造了23×23的Sylvester结式,首次获得了该问题的一元36次方程及其全部封闭形式的解析解;并使用同伦连续法对同一个数字算例进行数值验证,结果表明,该问题在复数域内最多有36组解。(2)采用倍四元数的复指数形式,改进了一种空间6R串联机构位置逆解新算法。基于倍四元数建立了空间6R串联机构位置逆解数学模型,构造了6×6的Dixon结式,不需要提取任何公因式,直接获得了该机构位置逆解的一元16次方程及其全部封闭解。(3)改进了传统并联机构中构型复杂、求解难度较大的空间一般6-6型平台以及空间一般6-6型台体2种并联机构的位置正解代数解问题。分别构造了10×10、18×18的Sylvester结式,获得了位置正解的一元20次、40次方程及其全部解析解,其结式尺寸较小。同时,从理论上阐明了存在多个不同的结式可以获得两种机构的位置正解代数解。(4)解决了台体型5SPS-1CCS、台体型4SPS-2CCS两种新型并联机构位置正解的解析解问题。应用Mourrain簇理论对数学模型解的个数进行了理论分析,得出两种新并联机构位置正解上限分别为160、320;并应用同伦连续法进行数值计算,获得了两种机构的全部位置正解,结果表明两种机构位置正解上限是可以达到的。同时得出结论,把一般6-SPS台体并联机构中的1(或2)条运动链SPS改变为CCS后,可以使并联机构位置正解数目增加到原来的2倍(或4倍)。(5)提出了一种求解空间6-6型平台并联机构以及对称结构6-6型平台并联机构位置正解的符号消元算法。基于计算机符号运算,分别利用计算机代数系统中的分次字典序Groebner基算法求解约化Groebner基和本论文提出的算法求解非约化Groebner基,推导出2组均含15个多项式方程并可直接用于求解空间6-6型平台并联机构位置正解的方程组,实现了该方程的数学机械化推导。与计算机代数系统中提供的算法相比,基于新算法推导出的非约化Groebner基表达形式更加简洁,运算效率更高。同时,运用该方程组,解决了对称6-6型平台并联机构位置正解代数解问题,获得了该问题的一元14次方程及其全部28组封闭解。更多还原

【Abstract】 The dissertation presents a research on some of unsolved focus issues in mechanism kinematics by using quaternion, double quaternions, the theory of matrix and kinematics of robotic mechanisms. The method is pre-solved using numerical method, and then solved using algebraic method, verified using numerical method. The main research works can be described as follows.(1) The problem of synthesizing a planar 4-bar linkage with given two fixed pivots such that the coupler curve passes through five given points is considered with the Groebner-Sylevester hybrid approach. A 36th degree univariate equation is obtained directly from the determinate of a 23×23 Sylvester’s matrix. The same result can be obtained with continuation method. A numerical example is given and it verifies that the problem has at most 36 solutions in the complex field.(2) A new algorithm for the inverse kinematics of 6R mechanism is improved using the complex exponent form of double quaternions. Based on double quaternions, the mathematical model of the inverse kinematics of 6R mechanism is obtained. Then a 6×6 resultant matrix is obtained using linear elimination and Dixon resultant. A 16th degree univariate equation is obtained from the determinant of the matrix and all closed-form solutions are obtained. The proposed algorithm is very easy and simple to program, which supplies a new method for the inverse kinematics of serial mechanism.(3) To obtain the closed-form solutions of the forward kinematics of the general 6-6 Stewart platform and the general 6-6 Stewart mechanism, the Groebner-Sylevester hybrid approach is presented. The reduced Groebner basis under degree lexicographic ordering for the closed-form equations is evaluated with computer algebra. The 10×10 and 18×18 Sylvester matrixes can be constructed respectively, which are relatively small in size. A 20th degree univariate equation and a 40th degree univariate equation are obtained from the determinants of the matrixes. The proposed algorithm operates simply and rapidly to lead to the same univariate equation via various resultant. And the direct kinematics problems of all kinds of Stewart platform are solved completely using algebraic method.(4) A generalized Stewart platform 5SPS-1CCS and a generalized Stewart platform 4SPS-2CCS are put forward and their direct kinematics analysis are discussed based on quaternion. In order to obtain the number of solutions of this kind mechanism, Mourrain variety is used. The results show that their direct kinematics problems of the parallel mechanisms have at most 80 and 160 solutions, respectively. In addition, all the 80 and 160 solutions have been obtained in an example by using homotopy continuation method. The example shows that the upper bounds of 80 and 160 solutions can be reached in the general case. A result has been proved that the numbers of the direct kinematics of the new parallel mechanisms are twice or fourth as the Stewart mechanism if one or two SPS chain(s) of the Stewart mechanism is/are replaced by CCS chain(s).(5) An algebraic elimination method for the direct kinematics analysis of the general 6-6 Stewart platform and the symmetrical 6-6 Stewart platform is presented. Based on computer symbolic computation, 15 equations, by which direct kinematics analysis of the general 6-6 Stewart platform can be solved, are constructed and the method of mechanization is presented. Then, the direct kinematics of the symmetrical 6-6 Stewart platform is solved using the 15 constructed equations. A 14th univariate equation of the problem and their 28 closed-form solutions are obtained. The result shows that the presented method is universal.更多还原