Parallel Mechanism Configuration Synthesis And Barranov Trusses Position Analysis Based On Geometric Algebra

This paper presents a theoretical study of several configurations of mechanism kinematics and structures based on geometric algebraic theory.The kinematics of three kinds of seven-link Baranov trusses and the configuration synthesis of parallel mechanisms with 3R1 T kinematic properties are used to solve the kinematic and structural problems of mechanisms under the framework of geometric algebra.The main research contents and innovations are as follows:(1)The configuration synthesis of the mechanism with 3R1 T kinematic characteristics is proposed using the virtual kinematic chain method in the geometric algebraic framework.In the synthesis process,the constraint space of the parallel mechanism is first determined and assigned to the branch chain;then the branch chain and the virtual kinematic chain are constructed into a single-loop kinematic chain to determine the specific configuration of the branch chain;afterwards,the resulting branch chain configuration is assembled to obtain the parallel mechanism with the desired motion;finally,the active joint is determined by correlation between the constraint spaces.Compared with the traditional method of configuration synthesis,the synthesis in the framework of geometric algebra can avoid the construction of matrices,and the correlation between motion spaces or constraint spaces can be easily determined by using the nature of the product of operations in geometric algebra,without the need for deterministic calculations.(2)The kinematic analysis of three kinds of seven-link Baranov trusses is solved,and the 14 th,16th and 18 th degree univariate input-output equations are obtained for three different configurations.By using translation and rotation operators in geometric algebra and the formulation of the geometry,the equations are established based on the distance between points,which can directly derive the univariate polynomial input-output equation directly without any complex algebraic modelling and nonlinear multivariable elimination computations as most current approaches do.After obtaining the equations,numerical examples of the corresponding configurations are substituted into the obtained one-dimensional higher order equations,respectively,and conform to each other with the results calculated using other methods.Compared with traditional methods such as vector method and D-H matrix method,the algorithm based on geometric algebra is geometrically intuitive in the solution process,does not require matrix operations,has one and only one unknown involved in the process of establishing the equation,and finally can directly obtain the higher order equation in one element without elimination calculations,and the roots obtained after the solution are not generated by increasing roots.