Displacement Analysis Of Parallel Mechanisms Based On Conformal Geometric Algebra

Robot kinematics is the key to motion control and one of the difficulties in the field of robotics research.In this paper,conformal geometric algebra(CGA)is applied to the kinematics of parallel robots.It follows the principle of plane to space and simple to complex.Several typical parallel mechanism kinematics solutions are completed,and various types of parallel mechanisms are obtained.The variable input-output polynomial equation obtains the closed solution of kinematics.The solution process is geometrically intuitive and avoids the use of rotation matrices and complex algebraic eliminations.The application of CGA in the kinematics of parallel robots is explored and a new idea for the forward kinematics analysis of parallel robots is provided.For the planar parallel mechanism,the position analysis of the 3-DOF planar 3-RPR parallel robot is completed.The coordinates of the joints on the moving platform are obtained by the CGA motion operator,the outer product of the geometry and the duality,then use the four spheres to intersect at one point to find the last joint coordinate,and the equation is established by the coefficient relation expression of the joint coordinates in the CGA.Then the method is improved,the 6th-degree input-output equation is obtained directly,and all six sets of closed solutions are obtained,which realizes the CGA whole process modeling and solving of the problem.For the 4-4 space parallel mechanism,the joint coordinates on the moving platform are obtained by the motion operator and the outer product and the duality of the geometry entities,Then use the four spheres to intersect at one point to find the last joint coordinate.The equation is established by the coefficient relation expression of the joint coordinates in the CGA,and the 20th-degree univariate polynomial equations are obtained by simplification.For the space 6-4 platform parallel mechanism,the motion coordinates and the outer product of the geometry and the dual joint are used to find the joint coordinates on the moving platform.The three-sided measurement formula derived from the centroid coordinates and derived from some Cayley-Menger determinants is derived.The last joint coordinate is obtained,and the distance constraint equation is established by the moving platform and the base joint coordinates,and the input and output 32th-degree univariate polynomial equations are obtained by direct simplification,and all six closed solutions are obtained.For the 5-5 space parallel mechanism,the joint coordinates on the moving platform are obtained by the motion operator and the outer product and the duality of the geometry,and then the four joints are used to find the last two joint coordinates.Two equations are established by the coefficient relation expression of the joint coordinates in the CGA,and finally the two equations are transformed into a univariate input-output equation by the Sylvester eliminant.