A Study Of Geometric Algebro Method On Some Issues For Kinematrics Of Mechanisms

The kinematics analysis of mechanisms is one of the main and basic topics in the research of robot mechanisms, which gives theoretical support to the practical application of robots. The mathematical models of robot mechanisms are complex, and are solved with many mathematical tools. To improve the real-time and accuracy of robot control, the dissertation presents a research on simplifying the mathematical model for kinematics, dynamics, trajectory planning and working space. In this paper, the mechanism kinematics are studied with Geometric algebra combined with the present hot and difficult problems, the main research works can be described as follows:(1) The Chebyshev approximation theory is applied to kinematics analysis of the body with screw. In the paper, based on RSSH, first of all, the mathematical model is built, and kinematic equation is obtained and secondly Chebyshev polynomials are used to substituting the sine and cosine functions of the equation and finally a higher order equation is obtained and solved. It is a new algorithm for solving equation in space mechanism with screw.(2) Based on a planar parallel mechanism, a forward kinematics model is built and solved by geometric algebra (conformal geometric algebra). Six results about position and pose are eventually obtained by the Maple program. Finally, comparing the ones by Wu method, the effectiveness and accuracy of this method are verified. And it does good for finding a unified mathematical tool in planar parallel mechanism analysis.(3) Conformal Geometric Algebra (CGA) and Dixon resultant are introduced to solve the inverse analysis of serial mechanisms especially for the general 6R robot. Firstly, homogeneous transformation matrix is expressed with Conformal geometric algebra (CGA) and then kinematics equations of 6R robot with Conformal geometric algebra (CGA) form is built; Secondly, the resultant is obtained by using linear elimination and Dixon elimination to eliminate 5 variables; Finally, a 16th order equation is properly derived from the resultant. The algorithm also can be applied to the inverse kinematics analysis of the others with 16 roots, such as 1P5R, so it is universal in some extent.(4) The paper presents that the cosine and sine functions of the rotation angle are changed into complex exponential functions and two complex bases of quaternion (plane rotation) are derived in view of quaternion being applied to express the plane rotation. Then the two bases are used to change quaternion and dual quaternion (3D rotation) into complex form. For the quaternion used in 3D rotation, four complex bases are derived and 8 bases are derived for dual quaternion. Finally given to the algorithms of the multiply, The formulae are derived for the conversion of the quaternion between real number form and complex number form. According to solving the inverse analysis of the 6R robot by the method, it is proved that the order of the expansion of the Dixon resultant is not 24th, but 16th. And thus the 16th order equation can be derived.(5) The paper presents another modeling method based on the quaternion and dual quaternion complex forms above:quaternion and dual quaternion matrix forms. It will be introduced to model the inverse kinematics analysis of the serial 6R robot, And five variables are eliminated by the elimination of linear and two-time applications lexicographic Groebner base, a 16th degree equation contained a variable is obtained. This method does not any roots.