A study of constrained Bézier fitting curve with tangent continuity for quadruped walking robot gaits

Abstract

Smooth and stable gait generation is critical for quadruped robots operating in unstructured environments. This paper introduces a constrained Bézier fitting framework. It enforces tangent continuity (𝐶1 continuity) at the gait cycle junction. This continuity addresses the tangential discontinuities that commonly arise in unconstrained Bézier trajectories. The method formulates foot-trajectory design as a constrained least-squares problem solved via Lagrange multipliers, enabling the control points to simultaneously satisfy interpolation targets and matched-tangent conditions. The resulting curves retain the geometric flexibility of classical Bézier parametrizations while producing well-behaved velocity profiles suitable for legged locomotion. These trajectories are integrated into an impedance-controlled leg model and evaluated in the MuJoCo simulator. Simulation results indicate noticeable reductions in torque spikes and improvements in tracking accuracy when compared to unconstrained Bézier and spline baselines, with representative trials showing reductions on the order of 40% and tracking improvements of approximately 25%. The proposed approach combines mathematical rigor with practical applicability, providing an efficient and reliable solution for high-performance quadruped gait planning.