A Research On Kinematic Solution Method Of Parallel Mechanism Based On Conformal Geometric Algebra

The kinematics analysis of the mechanism is the most basic and important part in the research of robot mechanisms,which can provide necessary theoretical support for the practical application of robot mechanisms.From the perspective of geometry,the general 6-3 Stewart parallel mechanism,the three degree-of-freedom(DOF)spatial parallel mechanism with 3-R(P)S structure,the 3-RPR planar parallel mechanism and the general 6-4A Stewart parallel mechanism are regarded as the research object.The conformal geometric algebra theory is used as the kinematic modeling method to study some difficult and hot issues in the kinematic analysis of the mechanism.The main research contents and innovative achievements are summaried as follows:(1)Combining conformal geometric algebra with Gr?bner basis elimination method,a derivation and simplification method of Gr?bner basis under the framework of conformal geometric algebra is proposed.Using the algorithm of conformal geometric algebra,two 2-blades in conformal geometric algebra are generated into the scalars.The polynomials generating Gr?bner bases are constructed with the help of the scalars.The implicit expression between the objective function and the polynomial is found by using Gr?bner basis elimination method.This method is a derivation and simplification process completely by computer operation.It can easily complete the derivation and simplification of the objective function.Its simplification speed is fast and easy to program.It can be extended to the derivation and simplification process under the framework of other geometric algebra.(2)Based on the determinant form of the inner product algorithm in conformal geometric algebra,the solution method of the forward placement solution of the general 6-3 Stewart parallel mechanism is improved.With the help of the inner product algorithm and the two formulas between 2-blade and 1-blade,the geometric constraint equations of the parallel mechanism were derived.The 16th-degree univariate polynomial equation of the parallel mechanism was deduced directly by Euler substitution.The effectiveness of the proposed method is verified by two numerical examples.This method does not need algebraic elimination and the whole solution process is carried out in symbolic form,which can provide a new idea for solving the more complex forward solution of the parallel mechanism.(3)Based on the geometric constraint relationship between two point pairs,a geometric modeling and calculation method was proposed to solve the forward placement solution problem of 3-DOF spatial parallel mechanism with 3-R(P)S structure.By studying the geometric constraint relation between the point and the point pair,the geometric constraint relation between two point pairs was formulated for the first time.The univariate high degree polynomial equation of the forward placement solution problem of this kind of parallel mechanism was derived by Euler substitution.The unified symbolic solution of 3-DOF spatial parallel mechanism with 3-R(P)S structure is obtained for the first time.At the same time,the whole process of modeling and calculation does not need algebraic elimination.(4)Based on the geometric constraint relation that four straight lines intersect at one point,a geometric modeling and calculation method was put forward to solve the forward displacement analysis problem of 3-RPR planar parallel mechanism.Based on the conformal geometric algebraic framework,the geometric constraint relation of four straight lines intersecting one point was elaborated.The geometric constraint equation of 3-RPR planar parallel mechanism was established.The univariate polynomial equation of the problem is directly derived by using Euler substitution.The modeling process has good geometric intuition,which provides a new idea for solving the kinematics problems of other complex mechanisms.Compared with other modeling methods for forward placement analysis of 3-RPR planar parallel mechanism,the modeling method proposed in this paper is a complete geometric modeling process.(5)Based on the geometric constraint relation that four straight lines intersect a plane at one point,the algebraic solution method for solving the forward placement problem of general 6-4A Stewart parallel mechanism is improved.Under the conformal geometric algebraic framework,the geometric constraint relation of four straight lines intersecting a plane at one point is formulated.A geometric constraint equation of the mechanism is published.Five polynomial equations are established by using the bar length formula in conformal geometry algebra.Based on the above six polynomial equations,a 5×5 Sylvester equation is constructed by using Sylverster elimination method and Grobner basis method under the graded reverse lexicographical order.The 32th-degree univariate polynomial equation of the problem is obtained.In this method,the geometric relation of four straight lines and one plane intersecting at one point is applied to the analysis of the forward position of the 6-4A Stewart parallel mechanism.The modeling process has good geometric intuition.At the same time,the Sylvester resultant constructed by this method is the smallest one in the known literature.