# A Gradient Dynamics-Based Singularity Avoidance Method for Backstepping Control of Underactuated TORA Systems

## TL;DR

This paper introduces a new control method to avoid singularities in TORA systems using gradient dynamics and backstepping.

## Contribution

The novel integration of gradient dynamics into backstepping control resolves singularity issues in underactuated TORA systems.

## Key findings

- The proposed method avoids division by zero in the control design.
- Stability analysis proves bounded signals and convergence of tracking errors.
- Simulations show improved transient performance compared to existing methods.

## Abstract

In this paper, a gradient dynamics-based control method is proposed to directly tackle the singularity problem in the backstepping control design of the TORA system. This method is founded upon the construction of an energy-like positive function, which includes an auxiliary variable in terms of the intermediate virtual control law. On this basis, a gradient dynamics is created to obtain a new virtual control command, which is capable of making the auxiliary variable gradually approach zero, thereby mitigating the issue of division by zero. The core innovation is the integration of the gradient dynamics into the recursive backstepping design to overcome the singularity problem and stabilize the system at the equilibrium quickly. In addition, it rigorously proves that all the signals in the closed-loop control system are uniformly ultimately bounded, and the tracking errors converge to a small neighborhood around zero through a Lyapunov-based stability analysis. Comparative simulations demonstrate that the proposed approach not only avoids the singularity issue, but also achieves a better transient performance over other methods.