A Study On Some Issues For Kinematics Of Mechanism And Their Algebraic Methods

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.