Algebraic Method for Exact Synthesis of One-Degree-of-Freedom Linkages With Arbitrarily Prescribed Constant Velocity Ratios

This paper addresses the synthesis of one degree-of-freedom (1DOF) linkages that can exactly transmit angular motion between coplanar axes (i.e., parallel axes or intersectant axes) with arbitrarily prescribed constant velocity ratios. According to motion polynomials over dual quaternions and pure rolling models between two circles, an algebraic approach is presented to precisely synthesize new 1DOF linkages with arbitrarily prescribed constant velocity ratios. The approach includes four steps: (a) formulate a characteristic curve occurred by the pure rolling, (b) compute the motion polynomial of the minimal degree that can generate the curve, (c) deal with the factorization of the motion polynomial to construct an open chain, and (d) convert the open chain to a 1DOF linkage. Using this approach, several 1DOF planar, spherical, and spatial linkages for angular motion transmission between parallel axes or intersectant ones are constructed by designating various velocity ratios. Taking the planar and spherical linkages with a constant 1:2 velocity ratio as examples, kinematics analysis is implemented to prove their motion characteristics. The result shows that the generated linkages indeed can transmit angular motion between two coplanar axes with constant velocity ratios. Meanwhile, three-dimensional (3D)-printed prototypes of these linkages also demonstrate such a conclusion. This work provides a framework for synthesizing linkages that have great application potential to transmit motion in robotic systems that require low inertia to achieve reciprocating motion with high speed and accuracy.