Algebraic synthesis of single-loop 7R spatial mechanisms generating exact straight-line trajectories

The synthesis of straight-line mechanisms remains a challenging and active topic in kinematics. This paper presents the algebraic synthesis of a family of novel single-loop 7R spatial mechanisms capable of generating exact straight-line trajectories. Based on the motion polynomial over dual quaternions, a three-step algebraic synthesis approach is proposed for constructing one-degree-of-freedom (1-DOF) single-loop 7R spatial mechanisms with exact straight-line motion. In this process, 2R and 5R serial chains are systematically formulated from an algebraic perspective and then integrated to form single-loop 7R spatial mechanisms. Using the proposed method, several new single-loop 7R spatial mechanisms are synthesized by assigning the lengths of straight-line trajectories and transforming the associated motion polynomials. Kinematics analysis, together with virtual simulations and 3D-printed prototypes, is carried out to validate the motion characteristics of the generated mechanisms. The results demonstrate that the generated mechanisms can trace the expected exact straight-line trajectories. This work provides a systematic basis for further exploration of single-loop 7R mechanisms for other prescribed special-path motions.