We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fin...We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fine focus at the origin for this class of equations is at most 6. Key words quartic polynomial Poincare equation - center - fine focus - order CLC number O 175. 12 Foundation item: Supported by the National Natural Science Foundation of China (19531070)Biography: TIAN De-sheng (1966-), male, Ph. D candidate, research direction: qualitative theory of differential equation.展开更多
Most parallel manipulators have multiple solutions to the direct kinematic problem.The ability to perform assembly changing motions has received the attention of a few researchers.Cusp points play an important role in...Most parallel manipulators have multiple solutions to the direct kinematic problem.The ability to perform assembly changing motions has received the attention of a few researchers.Cusp points play an important role in the kinematic behavior.This study investigates the cusp points and assembly changing motions in a two degrees of freedom planar parallel manipulator.The direct kinematic problem of the manipulator yields a quartic polynomial equation.Each root in the equation determines the assembly configuration,and four solutions are obtained for a given set of actuated joint coordinates.By regarding the discriminant of the repeated roots of the quartic equation as an implicit function of two actuated joint coordinates,the direct kinematic singularity loci in the joint space are determined by the implicit function.Cusp points are then obtained by the intersection of a quadratic curve and a cubic curve.Two assembly changing motions by encircling different cusp points are highlighted,for each pair of solutions with the same sign of the determinants of the direct Jacobian matrices.展开更多
文摘We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fine focus at the origin for this class of equations is at most 6. Key words quartic polynomial Poincare equation - center - fine focus - order CLC number O 175. 12 Foundation item: Supported by the National Natural Science Foundation of China (19531070)Biography: TIAN De-sheng (1966-), male, Ph. D candidate, research direction: qualitative theory of differential equation.
基金partially supported by the National Natural Science Foundation of China(Grant Nos.U1813221 and 52075015).
文摘Most parallel manipulators have multiple solutions to the direct kinematic problem.The ability to perform assembly changing motions has received the attention of a few researchers.Cusp points play an important role in the kinematic behavior.This study investigates the cusp points and assembly changing motions in a two degrees of freedom planar parallel manipulator.The direct kinematic problem of the manipulator yields a quartic polynomial equation.Each root in the equation determines the assembly configuration,and four solutions are obtained for a given set of actuated joint coordinates.By regarding the discriminant of the repeated roots of the quartic equation as an implicit function of two actuated joint coordinates,the direct kinematic singularity loci in the joint space are determined by the implicit function.Cusp points are then obtained by the intersection of a quadratic curve and a cubic curve.Two assembly changing motions by encircling different cusp points are highlighted,for each pair of solutions with the same sign of the determinants of the direct Jacobian matrices.
