A Study Of Geometric Algebro Method On Some Issues For Kinematic Analysis And Synthesis Of The Mechanisms

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 ? has 36 solutions, type ? has 48 excluding 16 degenerate solutions,type ? has 92 solutions and type ? has 66 solutions excluding 16 degenerate solutions.(4) An algebraic solution for five precision points path synthesis of Stephenson-? planar six-bar linkage is presented. The Stephenson-?planar 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.