Leader-Follower Formation-Based Leader Velocities Estimation Via Adaptive-τ Tustin Differentiator

This work develops a leader–follower formation for differential-drive mobile robots with an explicit estimator for the leader's velocities. Firstly, a leader robot position-variance-adaptive, under-rate-limited differentiator provides forward speed and yaw-rate estimate for the leader velocities, using exponentially weighted increment variances that drive a convex one-sided mapping to provide adaptive time constants for this estimator. Secondly, the control stack couples three layers: first layer, a nonlinear body-frame kinematic law for leader tracking, with orientation corrections handled via a smooth bounded function sin(eθ)/eθ to eliminate singularities near zero error. Second layer, a formation-preserving follower controller based on geometric Partial Feedback Linearization (PFL) that uses the estimated leader velocities; and last layer is a dynamic-level type-2 fuzzy adaptation with self-adjusting Gaussian membership functions to handle model nonlinearities. Controller gains for the leader and PFL of followers are tuned via the Secretary Bird Optimization Algorithm (SBOA). Rigorous Lyapunov-based stability analysis is provided to guarantee asymptotic convergence and boundedness of the closed-loop system. Simulations show enhanced leader path RMSE over existing work by 85.3%, while follower tracking RMSE was enhanced by 79.9% and 77.5%, respectively. These reduced the leader velocity estimation error by 0.00182 for angular velocity and by 0.00071 for linear velocity relative to the actual one. Robustness tested under mass uncertainty by 300%, and the disturbance of torque control signals reached 200%.