Curve-Based Optimization to Synthesize Planar Compliant Structures with B-Spline Based Curvatures

Compliant structures enable mechanical functions through elastic deformation and find application in precision engineering and robotics. While existing synthesis methods like pseudo-rigid-body models require design experience and topology optimization is computationally expensive, graph-based approaches offer efficient analysis and optimization via adjoint gradients. This work extends the graph-based method from [14] by introducing cubic B-spline curvature parametrization for Euler–Bernoulli beam elements. The spline-based model increases deformation path flexibility while preserving analytical gradient computation through adjoint integration and spline basis function sensitivities. Structures are represented as graphs with nodes as boundary/transition conditions and edges as spline-parameterized beams, enabling efficient backpropagation of design sensitivities. The approach is demonstrated on a cross-axis flexible pivot (CAFP), optimizing for $\pm$60° rotation about a virtual pivot with minimized radial reaction force $F_r$. An Adam optimization reduces $F_r$ from roughly between 60 - 100 N in the initial design to 2 - 25 N across the motion range, while maintaining an absolute and relative pivot-radius error of about 0.315 mm or 2.1%. ... mehrCompared to the original piecewise-linear beams, the spline extension maintains gradient computation times for 4 beams ($\approx$ 0.2 ms slower). The method provides a computationally efficient tool for compliant mechanism synthesis, with potential for hyperparameter tuning and hybrid stochastic optimization to further improve robustness.