Planar kinematics has been studied systematically based on centrodes, however axodes are underutilized to set up the curvature theories in spherical and spatial kinematics. Through a spherical adjoint approach, an axode-based theoretical system of spherical kinematics is established. The spherical motion is re-described by the adjoint approach and vector equation of spherical instant center is concisely derived. The moving and fixed axodes for spherical motion are mapped onto a unit sphere to obtain spherical centrodes, whose kinematic invariants totally reflect the intrinsic property of spherical motion. Based on the spherical centrodes, the curvature theories for a point and a plane of a rigid body in spherical motion are revealed by spherical fixed point and plane conditions. The Euler-Savary analogue for point-path is presented. Tracing points with higher order curvature features are located in the moving body by means of algebraic equations. For plane-envelope, the construction parameters are obtained. The osculating conditions for plane-envelope and circular cylindrical surface or circular conical surface are given. A spherical four-bar linkage is taken as an example to demonstrate the spherical adjoint approach and the curvature theories. The research proposes systematic spherical curvature theories with the axode as logical starting-point, and sets up a bridge from the centrode-based planar kinematics to the axode-based spatial kinematics.
The position synthesis of planar linkages is to locate the center point of the moving joint on a rigid link, whose trajectory is a circle or a straight line. Utilizing the min-max optimization scheme, the fitting curve needs to minimize the maximum fitting error to acquire the dimension of a planar binary P-R link. Based on the saddle point programming, the fitting straight line is determined to the planar discrete point-path traced by the point of the rigid body in planar motion. The property and evolution of the defined saddle line error can be revealed from three given separate points. A quartic algebraic equation relating the fitting error and the coordinates is derived, which agrees with the classical theory. The effect of the fourth point is discussed in three cases through the constraint equations. The multi-position saddle line error is obtained by combination and comparison from the saddle point programming. Several examples are presented to illustrate the solution process for the saddle line error of the moving plane. The saddle line error surface and the contour map presented to show the variations of the fitting error in the fixed frame. The discrete kinematic geometry is then set up to disclose the relations of the separate positions of the rigid body, the location of the tracing point on the moving body, and the position and orientation of the saddle line to the point-path. This paper presents a new analytic geometry method for saddle line fitting and provides a theoretical foundation for position synthesis.
