Robust Moving Horizon Estimation for Autonomous Agricultural Vehicles With GNSS Outliers Using a Robust Loss Function

We propose a Moving Horizon Estimator (MHE) for autonomous agricultural vehicles to handle GNSS outliers, a common issue in farming. To improve robustness, we replace the standard <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathrm{L_{2}}$</tex-math></inline-formula> stage cost with a loss function based on the square of the derivative of the General Adaptive Robust Loss (GARL). The GARL framework, controlled by parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha \in [1,\ 2)$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$c &gt; 0$</tex-math></inline-formula>, balances between quadratic and outlier-resistant behavior. By using the derivative, we avoid singularities at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha = 0$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha = 2$</tex-math></inline-formula>, simplifying tuning and ensuring stable optimization within MHE. This approach retains the flexibility of GARL while narrowing the design space to a singularity-free regime. We prove robust stability under standard assumptions. Simulations show our method outperforms <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathrm{L_{2}}$</tex-math></inline-formula>-based MHE and state-of-the-art methods, rejecting GNSS outliers. Field experiments validate its practical effectiveness.