Robust Triaxial Contour Tracking of Maglev Planar Motor via Fixed-Time Higher Order Sliding Mode CEC With Third-Order Taylor Expansion-Based CEE

To enhance the triaxial contour trajectory tracking performance of maglev planar motors (MPMs), this article proposes an analytical contouring error estimation (CEE) method based on third-order Taylor expansion and designs a contouring error controller (CEC) using the fixed-time high-order sliding mode (FxT-HOSM) algorithm with modified feedback. Initially, a state-space model of the MPM incorporating concentrated disturbances in each degree of freedom (DOF) is derived using a static decoupling control algorithm. Subsequently, accounting for the curvature and torsion of 3-D reference trajectories, a third-order Taylor expansion-based CEE algorithm is introduced to enable accurate real-time estimation and compensation. Furthermore, the CEC design based on the FxT-HOSM principle comprises: 1) a nonsingular fixed-time sliding surface (NFxTSM); 2) a fixed-time high-order reaching law; 3) a fixed-time contouring error differentiator; and 4) a fixedtime disturbance observer. This combination ensures highprecision contour tracking while mitigating chattering. Ul-timately, experimental results under various spatial trajectories with varying curvature and torsion confirm that the proposed FxT-HOSM controller achieves excellent performance in both contouring and tracking errors.