A Relaxed Quadratic-Program-based Framework for Trajectory Tracking of Unicycle Robots with Singularity Avoidance

This paper presents a framework for safe navigation of a unicycle point robot to a goal position in an environment populated with obstacles from almost any admissible state, considering input limits. We introduce a novel QP formulation to create a Cinfinity-smooth vector field with reduced total bending and total turning. Then we design an analytic, non-linear feedback controller that inherently satisfies the conditions of Nagumo's theorem, ensuring forward invariance of the safe set without requiring any online optimization. We have demonstrated that our controller, even under hard input limits, safely converges to the goal position. Simulations confirm the effectiveness of the proposed framework, resulting in a twice faster arrival time with over 50% lower angular control effort compared to the baseline.

Input--output feedback linearization fails at decoupling singularities, where the decoupling matrix loses rank, the relative degree is lost, and the linearizing control becomes unbounded. This paper develops a singularity-robust trajectory-tracking controller for square nonlinear control-affine systems that tracks through isolated decoupling singularities with bounded control. The method recasts tracking as real-time arc-length homotopy continuation, equivalently a continuous-time Newton/Davidenko flow, and replaces the inverse decoupling matrix by the least-norm Moore--Penrose solution of an augmented matrix $A=[Λ\mid b]$, where $b$ is the homotopy direction. A transversality condition $w^T b \ne 0$, with $w$ in the left null space of the decoupling matrix, keeps the augmented matrix full row rank through a generic rank-one loss. The resulting flow agrees with feedback linearization away from the singular set, tracks with $O(1/k)$ error, and re-locks after each crossing. The theory also characterizes the reflection-versus-branch-crossing dichotomy at Whitney folds and relates the reflection case to a Filippov sliding mode. Extensions cover dynamic relative-degree-one minimum-phase systems and arbitrary relative degree via filtered-error reduction. Simulations include a redundant 2-DOF manipulator, relative-degree-one and relative-degree-two plants, and a dual-active-bridge series-resonant DC/DC converter, where the method performs bounded inversion across buck/boost and resonance singularities while preserving zero-voltage soft switching.

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.