Optimization-free dimensional synthesis of planar 2R-3R parallel mechanisms based on a one-dimensional algebraic manifold

Planar 2R-3R parallel mechanisms typically rely on dimensional optimization to determine certain link lengths. However, under simultaneous position and orientation constraints, existing methods are limited by the number of links and cannot directly solve for all dimensional parameters. To address this, we propose an optimization-free method based on a one-dimensional algebraic manifold. It enables direct computation of all local joint coordinates. Starting from planar kinematic mappings, the proposed approach reveals the geometric relationship between submanifolds of the planar special Euclidean group (SE(2)) and algebraic manifolds. It further characterizes the motion mapping of planar 2R-3R mechanisms. A two-dimensional algebraic manifold is first established using inner-product constraints. Then, line homogeneous coordinate equations are introduced to reduce it to a one-dimensional algebraic manifold, which yields a solvable model for dimensional synthesis. A computational procedure based on singular value decomposition (SVD) is developed using the null-space structure. Finally, a case study of a sit-to-stand (STS) rehabilitation mechanism is presented to validate the feasibility of the proposed optimization-free approach.