Offline-Certified Robust Block Backstepping for Perturbed Underactuated Mechanical Systems

This paper develops certifiable enhancements to a block-backstepping controller for n-DOF underactuated mechanical systems in the Spong normal form. Two offline-computable tests, each reducible to a finite algebraic computation over a prescribed compact domain, are established: a minimum singular-value margin certifying uniform invertibility of the decoupling matrix, and a Jacobian Hurwitz condition certifying local exponential stability of the zero dynamics—the latter derived via an implicit zero-manifold parametrisation. Robustness overlays are introduced, comprising a bi-power term on the velocity-error channel and a matched-disturbance observer. Together, they supply an explicit observer transient budget and a practical entrance bound to(ε)+Tent for ∥z˙1∥2. No finite-time convergence of the complete closed-loop state is claimed. The resulting workflow permits offline tuning and verification prior to deployment. A simulation on a high-fidelity cart–pendulum model is consistent with those bounds; it is not a substitute for the offline certificates.