Discrete Kinematic Geometry Of Mechanism

With support of National Natural Science Foundation under Grant No.51275067, classic kinematic geometry for finite separated positions is expanded to discrete kinematic geometry for multiple positions under minimax metric based on theory of continuous kinematic geometry, which can provide theoretical basis for kinematic synthesis of linkages and precision analysis and design of machines.Firstly, curvature theories and corresponding higher order properties of trajectories traced by points and lines are briefly introduced through kinematic invariants of centrodes and axodes, and relative components of points and lines in the moving frame. Based on fixed and quasi-fixed line/plane conditions, vector relation of characteristic point of curve enveloped by line in planar motion to the instant center and that of characteristic line of ruled surface enveloped by plane in spherical and spatial motion to the instant screw axis are obtained through envelope conditions. The kinematic invariants of the centrodes and axodes are taken to denote relative curvature of the envelope curve and construction parameters of the envelope surface. Through these curvature invariants, geometrical properties of the envelope curves and ruled surfaces are analyzed to reveal geometrical characteristics of the family of lines and planes. A RCCC linkage with given parameters is taken as an example to explain how to obtain the curvature theories of the trajectories and envelope figures of geometrical elements through the kinematic invariants of the centrodes and axodes.The minimax metric is introduced to evaluate global geometric properties of discrete curves. The saddle line and saddle circle of discrete trajectory traced by any point of rigid body with multiple given positions are firstly defined, which are totally determined respectively by three and four characteristic points on the trajectory and correspond to characteristic positions of the rigid body. The saddle line error and saddle circle error are taken to describe global proximity of the discrete trajectory and straight line and circle. Based on the distributions of the characteristic points, the orientation and position of the distribution line, and position and radius of the distribution circle can be derived, as well as the algebraic equations of the distribution line and circle errors. Essentially, the saddle line and circle error functions are respectively comprised of distribution line error subfunctions for three positions and distribution circle error subfunctions for four positions, thereby boundary characteristics of different surface patches of the error surfaces, and geometric features of saddle sliding point with minimum saddle line error and saddle circle point with minimum saddle circle error are analyzed to reveal kinematic characteristics of the discrete motion of the rigid body.Then the saddle spherical circle of spherical discrete trajectory curve is defined under minimax metric and the corresponding saddle spherical circle error is taken to describe global proximity of the discrete trajectory and spherical circle. Position and dimension of the distribution spherical circle of any spherical discrete trajectory can be obtained through the distributions of the four characteristic points on the trajectory, and then algebraic equations of the distribution and saddle spherical circle errors can be derived. No matter how many discrete positions of the rigid body are prescribed, the distribution spherical circle error for four positions can express the saddle spherical circle error for multiple positions, thereby the composition properties of the saddle spherical circle error and the distribution characters of the points with minimum errors can be analyzed.In order to disclose global geometric properties of spatial discrete trajectory curves, the saddle spherical surface and cylindrical surface are defined. Based on the distributions of characteristic points on the discrete trajectory, positions and dimensions of the distribution spherical surface and cylindrical surface are expressed by the discrete motion parameters. In non-degenerate case, the distribution spherical surface error for five positions and distribution cylindrical surface error for six positions can be respectively taken to denote the saddle spherical surface error and saddle cylindrical surface error for multiple positions, and reveal the composition properties of the errors. In a word, theories of discrete kinematic geometry are established under different minimax metrics from the point of view of global geometric properties of the discrete curves.