Kinematics Analysis Of Mechanisms Based On Conformal Geometric Algebra

The research on kinematics analysis, which is the fundament of motion control, has theoretical value and engineering significance, and has been the research emphasis of mechanisms for a long time. Conformal geometric algebra (CGA) is a new mathematical tool that can provide not only unified presentation of geometric objects, such as points, lines, planes, circles and spheres, but also universal spinor representation of space transformation. It has fulfilled direct geometric computation using geometric language without coordinates, and provides a new solution for a series of technical problems.This paper uses conformal geometric algebra to the kinematics analysis of the serial, parallel and spatial mechanism, exploring the application of CGA in kinematics analysis.The kinematics of PUMA robot, Stanford robot and 1P5R robot are analyzed. Firstly, the positions of all the joint points are determined with the help of the outer product of basic geometric entities; then, the angles of each joint are computed by the inner product of CGA. For PUMA robot and Stanford robot, all the solutions are calculated directly by geometric computation; as to 1P5R robot, a 12th degree univariate equation is obtained, and a comparison with the general method is made, proved that this method has high solving efficiency.The forward displacement of 3-RPR mechanism and 3-RPS mechanism are analyzed by CGA. For 3-RPR mechanism, a sixth degree input and output equation is obtained by linear eliminations. For 3-RPS mechanism, the 16th degree equation is obtained with simple geometric operations.For spatial linkage mechanism, the Spatial Five-Link RRSRR mechanism is presented as an example. At start, all the "R" joint points are determined by input and output parameters. Then, four spheres are builded with those points, and the "S" joint point is the outer product of the four spheres. After that, a 4th degree input-output polynomial equation with a single unknown is derived. Finally, a comparison with the general method is made.To conclude, the kinematics model based on CGA is brevity, clear, intuitive and efficient. It does not depend on coordinates, and it can avoid the use of matrices and complex computations for nonlinear, multivariable equations.