Research On New Methods Of The Geometric Algebra For The Kinematics Analysis Of Mechanisms

The kinematics analysis of mechanisms plays an important role in the mechanical design. On the one hand, the forward kinematics can verify whether the mechanism performance meets design requirements after the mechanical design is completed. On the other hand,the inverse kinematics can provide control programs for the mechanical control. Geometric algebra methods are important methods to conduct kinematics analysis of mechanisms, which still have great research potential at present. Especially some areas need further research such as the establishment of geometric algebra methods for the kinematics analysis of traditional mechanisms and novel mechanisms. Therefore the subject of researching on geometric algebra methods for the kinematics analysis of mechanisms is put forward in this paper. On the one hand, the paper researches the establishment of new geometric algebra methods for the kinematics analysis of traditional serial mechanisms and parallel mechanisms. On the other hand, the paper researches the establishment of geometric algebra methods for the kinematics analysis of novel spherical mechanisms and metamorphic mechanisms. The main contents and innovations of this paper are as follows.(1) The new geometric algebra method is proposed based on studying the kinematics analysis of serial mechanisms, which is named D-H quaternion transformation method in the paper. The point mapping representation with quaternion is given. And a D-H quaternion transformation method for the motion between adjacent linkages is proposed. Then the matrix operation method of D-H quaternion transformations is shown in detail. And the classical D-H homogeneous transformation matrix in robotics is constructed by the proposed D-H quaternion transformation method. The results of the D-H quaternion transformation method and the D-H homogeneous transformation matrix method are consistent, implying that the proposed method is correct in theory. Based on the D-H quaternion transformation formula of adjacent linkage motions, the D-H quaternion transformation method for kinematics analysis of serial mechanisms with any number of linkages is further proposed. The inverse kinematics analysis of serial mechanisms is a difficult problem in research area of mechanisms. Two equations of position and posture are obtained by separating the kinematics equation of D-H quaternion transformation, which can construct equation set with seven equations. Then the number of equations can meet the requirement of the inverse kinematics of serial mechanisms with more than four degrees of freedom. In order to lower the difficulty in solving equations, the degree of new posture equation set is reduced to half by taking out half of trigonometric functions in the original posture equations. The effectiveness and correctness of the proposed method are further proved through an example of forward kinematics and inverse kinematics analysis. The proposed D-H quaternion transformation method is a new method of kinematics analysis for serial mechanisms. It avoids complex matrix operations, and the number of kinematics equations is less than the matrix method. What’s more, the D-H quaternion transformation method also has four advantages including clear procedure, easy mathematical mechanization, explicit geometrical meanings and simple calculations. The proposed new method is correct and effective for kinematics analysis of serial mechanisms.(2) The new method of conformal geometric algebra is proposed based on studying the kinematics analysis of parallel mechanisms. Firstly, a modeling approach of conformal geometric algebra for the kinematics analysis of planar parallel mechanisms is presented. An improved method for Sylvester resultant elimination is presented, which is named redundant factor elimination method. The presented method can overcome the disadvantages of extraneous roots with resultant elimination and can obtain exact roots of the nonlinear equations. Secondly, the method of conformal geometric algebra for kinematics analysis of spatial parallel mechanisms is proposed. This method sets geometric representation and computing as a whole, just by the description and arithmetic of conformal geometric algebra to create kinematics analysis model, which does not require complex matrix operations.(3) The geometric algebra method is proposed based on studying the kinematics analysis of spherical mechanisms. Firstly, the geometric algebra method of kinematics analysis for spherical parallel mechanisms is studied. For spherical parallel mechanisms, kinematics analysis is companied with complex modeling, difficulties in solving direct variables representing the position and posture of mobile platform, etc. To solve those problems, the quaternion and spherical geometry method for establishing mathematical kinematics analysis model of spherical parallel mechanisms and the elimination and simplification method are proposed. Secondly, the geometric algebra method of kinematic analysis for spherical scissors deployable mechanisms is studied. Based on screw theory, degree-of-freedom characteristics of spherical scissors deployable mechanisms with any number of spherical scissors elements and branch chains are analyzed. Based on spherical geometry, kinematic analysis of the spherical scissors deployable mechanism is conducted. The software of kinematic analysis of spherical deployable mechanisms is developed, which achieves automatic kinematic analysis and calculation for such mechanisms. Kinematic characteristics of spherical scissors deployable mechanisms can be revealed based on the proposed method in this paper. The proposed method lays a theoretical foundation for designing a range of spherical scissors deployable mechanisms to meet certain kinematic characteristics requirements.(4) The geometric algebra method is proposed based on studying the kinematics analysis of metamorphic mechanisms. Firstly, the method of kinematic characteristic analysis of metamorphic mechanisms is studied. Two analysis methods including the Lie group Lie algebra and screw algebra are proposed. Secondly, the method of kinematics analysis for metamorphic mechanisms is studied. The geometric algebra method of kinematics analysis for parallel metamorphic mechanisms is proposed. The metamorphic mechanism is a novel mechanism, which has the characteristics of multi-configuration. In this paper, with the origin of coordinates on the moving platform describing its position and Euler angles describing its posture, the quaternion expression of an arbitrary point’s location on the moving platform in the fixed coordinate system is given, and thus a method to establish the unified mathematical model for kinematics analysis of the parallel metamorphic mechanism is presented. Forward kinematics and inverse kinematics analysis of the parallel metamorphic mechanism in different configurations can be conducted based on the proposed geometric algebra method of kinematics analysis for parallel metamorphic mechanisms.New geometric algebra methods for kinematics analysis of mechanisms are studied in this paper. The new method of D-H quaternion transformation for the kinematics analysis of serial mechanisms is proposed. And the new method of conformal geometric algebra for the kinematics analysis of parallel mechanisms is put forward. Geometric algebra methods for the kinematics analysis of spherical mechanisms and metamorphic mechanisms are also presented. The proposed methods for kinematics analysis of mechanisms are correct and effective, which have some advantages such as setting geometric representation and algebraic operation as a whole, explicit geometrical meanings, simple calculations and easy mathematical mechanization. The proposed new methods of geometric algebra for kinematics analysis of mechanisms in this paper enrich and develop the kinematics theory of mechanisms.