Parameter-Dependent LMI Synthesis for Semi-Global Differential ISS Trajectory Tracking of Nonholonomic Mobile Robots Under Multiplicative Wheel Slip

Abstract

This paper presents a parameter-dependent linear matrix inequality (LMI) framework for trajectory tracking of nonholonomic mobile robots subject to severe multiplicative wheel slip on variable-terrain surfaces. The sampled convex formulation, augmented with grid-to-continuum residual certification, simultaneously establishes semi-global differential input-to-state stability, a prescribed exponential decay rate, regional pole placement, and a gain-bounded feedback proxy for actuator-limited operation. A central contribution is an explicit upper bound on the additive disturbance induced by bounded multiplicative slip in the Kanayama error coordinates, bridging the physical slip mechanism and the convex synthesis paradigm. The auxiliary gain matrix and inverse storage metric are parameterized affinely in the reference velocities, while the storage metric inherits nonlinear dependence through pointwise matrix inversion. Stability is established via a cascade analysis combining variational contraction, forward invariance, slip-induced disturbance bounds, and dissipation-based trajectory reconstruction. Numerical validation compares three controllers across six reference trajectories, six disturbance classes, and a 60-second variable-terrain test featuring six severe slip patches with bidirectional slip ratios reaching +/-50%, replicated on two geometries. Supplementary studies address Gaussian sensor noise, compound stress-testing, and embedded-platform computational feasibility. Across 100 Monte-Carlo runs the proposed controller achieves complete trajectory containment within the certified envelope. On the variable-terrain scenario, peak tracking error is reduced by 12% against the fixed-gain LMI baseline and 49% against the manual baseline, with the constant-gain baseline infeasible at the prescribed decay rate.