Displacement Analysis Of Robot Mechanisms Based On Conformal Geometric Algebra

Displacement analysis is the most fundamental and difficult task of kinematics,which provides the fundament for speed analysis,acceleration analysis and mechanism control.Base on conformal geometric algebra,the displacement analysis of planar five-bar mechanisms,spatial serial robot mechanisms and spatial parallel mechanisms have been solved.For the planar five-bar mechanisms,the displacement of 5R and 4R1P mechanism are analyzed by CGA.the positions of two joint point are obtained by using rotation and translation in CGA,then the positions of end joint point can be determined with the help of the outer product of basic geometric entities.In the inverse displacement analysis,the joint position is obtained by the outer product of geometric entities,then can get the inputs of the driving joint.Finally,two mechanisms are verified by numerical examples,and the efficiency of the algorithm is given.For the spatial serial robot mechanisms,the displacement of two robot mechanism are analyzed by CGA.The forward displacement analysis can be solved by the description of rigid transformation under CGA framework.In the inverse displacement analysis,the geometry such as planes and spheres are constructed.All the joint points are determined with the help of the outer product of those geometric entities.The lines,and planes can be constructed by the joint points.the angles of each joint are computed by the inner product of CGA.Finally,two robots are verified by numerical examples,and the efficiency of the algorithm is given.For the spatial parallel mechanisms,the forward displacement of RRR-3S,RRP-3S,RPP-3S and PPP-3S mechanisms are analyzed by CGA.The RRR-3S and RRP-3S mechanisms use the outer product of the geometry and the distance construction to obtain the high degree equation.The RPP-3S and PPP-3S mechanisms first get a joint point by the outer product of four spheres,then use the nature that the inner product of a point and itself is zero to obtain the high degree equation.All those equation have no extraneous roots.Several mechanisms have been verified by numerical examples,and the efficiency of the algorithm are compared with the the general methods.The displacement analysis based on CGA is brevity,it not only has the geometrical intution but also avoid the use of matrices and complex computations for nonlinear,multivariable equations,and it has high solving efficiency.