【作者】 魏锋；

【导师】 魏世民；

【作者基本信息】 北京邮电大学 ， 机械电子工程， 2017， 博士

【摘要】 机构的运动学分析与综合是机器人机构学研究中最基础也是最重要的部分,其不仅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撑。机构分析与综合的求解方法主要有数值法(例如迭代法和同伦连续法)和代数法。迭代法需要在计算前给定一个初值,往往不能得到全部解,而同伦法虽然能够给出问题的全部解,但计算可能有发散路径等,会给计算带来负担。代数法求解不需要初值,可获得全部解,并且单变量的输入输出方程对于运动学的其他方面如工作空间分析、奇异位置分析等都有很大的理论价值,可进行诸多机构学问题的研究。因此,本文基于代数法针对目前机构运动学分析与综合中一些难点、热点问题进行理论研究,主要研究内容和取得的创新成果如下:(1)采用共形几何代数方法对一般6-3 Stewart并联机构位置正解问题进行建模和求解。将6-3 Stewart并联机构进行构型转换为等效的2RPS-2SPS型机构,采用共形几何代数对等效机构建立两个基本约束方程。首先根据距离约束建立一个2元2次方程;然后根据机构特征,利用四球交于一点的方法建立另外一个2元4次方程。对这两个方程采用Sylvester结式消元获得一元16次方程,完成了该问题的数学机械化求解。相比现有的求解方法,本文的方法建模简单,并且求解时只需一次消元便可得到一元16次方程。(2)改进了一般6-6 Stewart台体型并联机构位置正解问题的代数求解方法。采用共形几何代数建立该问题的数学模型,通过变量替换将约束方程转换为7个方程,利用反对称矩阵的特点再次进行变量代换,通过Groebner基消元,将这7个约束方程约化为7个含4变量的方程;通过提高某变量的次数,增加其被消元的权重,再次利用Groebner基法,推导出16个含有4变量的多项式方程,应用推导出的多项式构造9×9的Sylvester结式,从而获得一般6-6 Stewart台体型并联机构位置正解的一元40次方程。该方法相较于现有文献存在的方法,因推导出的结式尺寸相对较小,从而能有效提高该问题的求解效率,降低其计算负担。(3)提出了平面铰链四杆机构五精确点轨迹综合的代数求解方法。根据已知参数,将求解问题分为4种类型。根据杆长条件采用平面位移矩阵建立统一的运动约束方程,对Groebner基法进行改进,提出了新的项序(分组分次逆字典序),并在此基础上对约束方程求Groebner基,结合Sylvester结式对方程约化与求解,获得一元高次方程及其全部代数解,并得出类型1存在36组解,类型2不含退化解存在48组解,类型3存在92组解,类型4不含退化解存在66组解。(4) 提出了 Stephenson-Ⅲ型平面六杆机构五精确点轨迹综合代数求解方法。将Stephenson-Ⅲ型平面六杆机构拆分为一个二级杆组和一个四杆机构,先对二级杆组五精确点综合,再对四杆机构精确点综合。采用矩阵约束法建立该问题的数学模型,使用Groebner基和Sylvester结式(GS法)相结合的代数方法进行求解,最终获得一元高次方程及其全部代数解。(5)提出了球面四杆机构五精确点轨迹综合代数求解方法。基于球面空间转移矩阵建立了该问题的设计方程,使用Groebner基和Sylvester结式(GS法)相结合的代数方法进行求解,最终获得一元高次方程及其全部代数解。该方法为进一步采用代数法对其他类型球面机构轨迹综合问题的研究提供了参考。更多还原

【Abstract】 The kinematic analysis and synthesis of the mechanisms is one of the most fundamental and most important topics in the research of robot mechanisms, which does not only lay a foundation for the design of mechanisms, but also provides theoretical support to the engineeringapplication of robots. There are two ways to solve the kinematic analysis and synthesis of the mechanisms: numerical and algebraic methods. Thenumerical methods consist of traditional numerical method (such as interval iterative method and optimization method) and homotopy continuation method. For the tradition numerical method, a good initial value are required and it does not find all solutions; The homotopy continuation method can find all solutions and does not require the initial value, however, it will lead to the computational burden due to the divergent path during the calculation procedure. The solution process of the algebraic method is complex and difficult, but it does not require the initial value, and can obtain all solutions. Moreover, the input-output closed-form univariate polynomial equation can provide more information for workspace analysis and singularity analysis of mechanisms and has highly theoretical values based on which many kinematic problems will be solved easily.The dissertation develops the research on the hot and difficult problems of analysis and synthesis of the mechanisms using algebra method. The main contents and contributions can be summarized as follows:(1) A new algorithm for the forward displacement analysis (FDA)of a general 6-3 Stewart platform based on conformal geometric algebra(CGA) is presented. Firstly, a 6-3 Stewart platform structure is changed into an equivalent 2RPS-2SPS structure. Then, two kinematic constraint equations are established based on CGA, i.e., a 2th-degree equation with two unknown variables is built according to the distance of two points and the other 4th-degree equation with two unknown variables is built according to the point characteristic four balls intersect in CGA. A 16th-degree univariant polynomial equation is derived from the aforementioned two equations by the Sylvester resultant elimination and the mathematical mechanization of this problem is implemented.Compared with the previously reported method, The novelty of the proposed method lies in that the problem is modeled based on CGA and is solved based on single elimination, as a result, the solution procedure is simpler and more efficient and readily to program.(2) The algebraic solution method is improved for the forward displacement analysis (FDA) of a general 6-6 Stewart mechanism (i.e.,the connection points of the moving and fixed platforms are not restricted to lie in a plane). The kinematic constraint equations are built using conformal geometric algebra (CGA). The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Groebner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Groebner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9×9 Sylvester resultant matrix,which is smaller in size than those presented previously in the literature.Therefore the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.(3) The algebraic solution method is presented for five precision points path synthesis of planar four-bar linkage. This problem can been divided into four types in term of the input parameters. A unified formulation for the four types is built based on the planar displacement matrix. Next, the corresponding resultant matrix is constructed based on Groebner bases generated by applying the new term ordering (the groups graded reverse lexicographic ordering) for four types. Then, a high-degree univariate polynomial equation is accordingly obtained.Finally, all closed-form solutions are obtained. And it is concluded that type Ⅰ has 36 solutions, type Ⅱ has 48 excluding 16 degenerate solutions,type Ⅲ has 92 solutions and type Ⅳ has 66 solutions excluding 16 degenerate solutions.(4) An algebraic solution for five precision points path synthesis of Stephenson-Ⅲ planar six-bar linkage is presented. The Stephenson-Ⅲplanar six-bar linkage is decomposed into two parts: a dyad and a four-bar linkage. To synthesize the two parts, the dyad first then the four-bar linkage, the kinematic constraint equations are formulated based on displacement matrix. The equations are solved with the Groebner-Sylvester (GS) hybrid approach, in which a high degree univariate equation together with all its closed form solution is obtained.(5) An algebraic elimination method for five precision points path synthesis of spherical four-bar linkage is presented. Firstly, the kinematic constraint equations of path synthesis are formulated based on the spherical space displacement matrix; Next, the equations are solved using Groebner-Sylvester (GS) hybrid approach, and a high degree univariate equation is accordingly obtained; Finally, all closed-form solutions are obtained. The proposed method in this paper can also be used to solve synthesis problems of other kinds of spherical linkages.更多还原