Projective synthesis of planar compliant mechanisms for isotropic pure translational motion

Translational stiffness isotropy is a desirable property in planar compliant mechanisms for precision applications, as it mitigates directional bias and reduces cross-axis coupling under in-plane loading. Recent projective-geometry formulations describe planar kinetostatics through the ellipse of elasticity, enabling compact synthesis procedures based on conic decomposition. However, isotropy enforced at a single output point does not necessarily guarantee a pure-translational response of the moving body, which may still exhibit parasitic rotations. This paper implements mechanism-level projective synthesis to the design of planar compliant mechanisms featuring isotropic pure translation. The key idea is to prescribe a primary circle of elasticity, which encodes translational isotropy at the output port, and implies an improper displacements pole. The primary circle is then decomposed through a top-down sequence of series and parallel arrangements using weighted self-polar triangles. The procedure yields secondary ellipses consistent with the assigned topology, while preserving reciprocity and positive definiteness at each step. Finally, the lowest-level ellipses are materialized into circular-arc flexures via closed-form relations, linking arc geometry and flexural rigidity to the corresponding ellipse of elasticity. A representative closed-chain mechanism is designed. Finite element analyses and experimental tests are conducted to validate the theoretical predictions.