Temporal Learning for End-Effector Position Estimation under Aerodynamic Disturbances in Aerial Continuum Manipulation

This paper investigates temporal neural networks for \mbox{end-effector} position \mbox{estimation} of an aerial continuum manipulator (ACM) operating under aerodynamic effects induced by the unmanned aerial vehicle (UAV). An experimental dataset is collected under stationary (\mbox{rotor-off}) and \mbox{free-hovering} conditions across continuum robot (CR) configurations and UAV altitudes, providing \mbox{end-effector} position measurements with and without aerodynamic residuals. To establish a nominal framework, \mbox{strain-parameterized} kinematic models with progressively richer strain bases are evaluated to balance model complexity and prediction accuracy. The selected nominal model then serves as the baseline for 3D position residual estimation using a \mbox{closed-form} \mbox{continuous-time} (CfC) neural network, with a multilayer perceptron (MLP) and a gated recurrent unit (GRU) used for comparison. On unseen test experiments, the CfC achieves an RMSE of $22.00\pm1.70~\mathrm{mm}$ over five random seeds, compared with $36.38\pm3.58~\mathrm{mm}$ for the MLP and $27.72\pm2.92~\mathrm{mm}$ for the GRU, corresponding to reductions of $39.52\%$ and $20.62\%$, respectively. These results demonstrate the effectiveness of \mbox{continuous-time} learning for \mbox{end-effector} position estimation under aerodynamic disturbances relative to static and \mbox{discrete-time} learning methods.