Research On Kinematic Analysis Of Robotic Mechanisms Based On Conformal Geometric Algebra

In this paper,taking the planar basic kinematic chains(Baranov Trusses)as the research object,using the geometric intuition of conformal geometric algebra(CGA)and its calculation rules,the displacement analysis of the Baranov trusses are investigated and a new geometric modeling and algebraic solution method for this problem is proposed.The main research contents and innovative achievements are as follows:(1)Under the frame of CGA,in terms with the expression of the basic geometric elements such as point,line,plane,sphere and point pair and the operation rules of inner product,outer product,geometric product and duality,the formulation for the displacement analysis of the planar four-bar basic kinematic chain is derived by using the area sign of the triangle and the derivation is free of coordinate system.For the displacement analysis of the five-link Baranov truss,introducing one variable,the sixdegree polynomial equation with one variable is directly derived by using the formulation of the planar four-bar basic kinematic chain,and the resultant elimination is not required during the derivation process of the characteristic equation.(2)The displacement analysis of three kinds of seven-link Baranov trusses is studied based on CGA.Firstly,the first constraint equation is established by using the formulation for the displacement analysis of the planar four-bar basic kinematic chain contained in the seven-link Baranov truss,and then the second one is derived by using the inner product of two points according to the distance relationship under the frame of CGA.Finally,a high order degree univariate equation without extraneous roots and roots loss is derived by one-step resultant elimination.Compared with the traditional complex-vector method,the proposed method has the following advantages:1)the number of constraint equations is reduced to one,2)only one-step elimination procedure is required.In addition,the derivation of two equations are concise and coordinate-invariant and has intrinsic geometric intuition.Numerical examples show that the configuration of Baranov truss is independent of the selection of the base.(3)The displacement analysis of 6th-29th nine-link Baranov trusses based on CGA is revisited.According to the number of the planar four-bar basic kinematic chain contained in the nine-link Baranov truss,the geometric modeling and solution method of the displacement analysis are divided into two cases.For the first case,the 6th,7th,11th,12th,14th,15th,16th,17th and 22nd nine-link Baranov trusses are included,and they contain two planar four-bar linkages.And therefore two constraint equations can be directly established according to the formulation of the planar four-bar basic kinematic chain,and then a univariate polynomial equation can be obtained by onestep elimination process based on the aforementioned two equations.For the second case,the 8th-10th,13th,18th-21st,23rd-29th nine-link Baranov trusses are included,and they contain only one planar four-bar linkage.Firstly,the first constraint equation is established based on the formulation of the planar four-bar basic kinematic chain,and then the other two equations are established by using the inner product of two points according to the distance relationship under the frame of conformal geometric algebra.Finally,two-step eliminations for the three equations are required and an univariate higher-order equation is obtained.Compared with the existed complexvector method,the number of constraint equations is reduced from 4 to 2 or 3 and the number of elimination steps is reduced from 3 to 1 or 2.And therefore the proposed method can simplify the solution process,reduce the size of the constructed resultant matrix and enhance the computational efficiency.In this paper,the geometric modeling and algebraic solution procedures for the displacement analysis of the 6th,9th,16th,18th,25th and 28th nine-link Baranov trusses are given,and the correctness and effectiveness of the proposed method are verified by numerical examples.