Outlier-Robust Nonlinear Moving Horizon Estimation using Adaptive Loss Functions

Nestor Deniz; Guido Sanchez; Fernando Auat Cheein; Leonardo Giovanini

cs.ROApr 6, 2026

Outlier-Robust Nonlinear Moving Horizon Estimation using Adaptive Loss Functions

Authors: Nestor Deniz, Guido Sanchez, Fernando Auat Cheein, Leonardo Giovanini

Organizations: Instituto de Investigación en Senales, Sistemas e Inteligencia Computacional, sınc(i), UNL-CONICET, Santa Fe, Argentina · Harper Adams University, Engineer Department, Edgmond, Newport TF10 8NB

Abstract

In this work, we propose an adaptive robust loss function framework for MHE, integrating an adaptive robust loss function to reduce the impact of outliers with a regularization term that avoids naive solutions. The proposed approach prioritizes the fitting of uncontaminated data and downweights the contaminated ones. A tuning parameter is incorporated into the framework to control the shape of the loss function for adjusting the estimator's robustness to outliers. The simulation results demonstrate that adaptation occurs in just a few iterations, whereas the traditional behaviour $\mathrm{L_2}$ predominates when the measurements are free of outliers.

Figures & tables

Fig. 1 : Adaptive robust loss functions

\rho(r,\alpha,c)

and

\varphi(r,\alpha,c)

for different values of

\alpha

Fig. 2 : Normalized estimation error reduction per iteration and mean estimation error reduction under scenario (i) (top) and scenario (ii) (middle).

Fig. 3 : Measured, true, and estimated tractor’s

x

and

y

coordinates for one trial of the experiments. The bottom panel illustrates the trajectory followed by a tractor pulling a trailer.

Method	Uniform Noise			Normal Noise
	MS. Est. Err. ( $\Psi$ )	MS. Pos. Err. ( $\Delta$ )	Mean Time ( $\eta$ )	MS. Est. Err. ( $\Psi$ )	MS. Pos. Err. ( $\Delta$ )	Mean Time ( $\eta$ )
$\text{MHE}_{\text{prop}}$ (M=10)	0.1708	0.1154	0.3711	0.1644	0.1106	0.3744
$\text{MHE}_{\text{prop}}$ (M=3)	0.1711	0.1154	0.1090	0.1649	0.1106	0.1144
$\text{MHE}_{\text{grid s.}}$ (M=3)	0.1885	0.1284	0.0732	0.1624	0.1093	0.0733
$\text{MHE}_{\alpha\ \text{fixed}}$	0.1944	0.1326	0.0290	0.1962	0.1337	0.0307

TABLE I : Comparison of MHE methods under different noise distributions computed over

1000

trials

Moving Horizon Estimation

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Moving Horizon Estimation for Quadrotors: An $\mathcal{L}_1$ Adaptive Optimizer Approach

Thinh Nguyen, Minkyung Kim, Sandeep Banik, Jinrae Kim+1

University of Illinois Urbana-Champaign, Urbana, IL 61801 · Department of Mechanical Science and Engineering, The Grainger College of Engineering

Moving Horizon Estimation Robot State Estimation

Open paper AskPDF arXiv

Moving Horizon Estimation (MHE) is a state estimation method based on finite-horizon optimization that can offer higher accuracy at the cost of increased computation compared to Kalman filter-based approaches. We present a linear smoothing MHE formulation as a dense Quadratic Program (QP), and a solver consisting of a continuous-time Newton's method augmented with the

\mathcal{L}_1

Adaptive Optimizer (

\mathcal{L}_1

-AO). While MHE is inherently time-varying, conventional approaches treat it as a sequence of independent, time-invariant problems and employ iterative solvers at each time step, which can be both inaccurate and computationally burdensome. In contrast, time-varying solvers track the optimal solution with fewer iterations by exploiting the temporal evolution of the problem, thereby reducing the computational load. In this research, we enhance both the performance and efficiency of MHE through a time-varying solver with an

\mathcal{L}_1

-AO augmentation that compensates for the prediction inaccuracy, which is common in practice due to noisy sensors and the lack of prior knowledge of the system. Simulation results on a quadrotor platform show that the

\mathcal{L}_1

-AO-augmented approach solves the MHE optimization problem more efficiently than the baseline time-invariant solver and achieves higher estimation accuracy under challenging conditions, compared with both the Extended Kalman Filter and the standard MHE.

Abstract

Moving Horizon Estimation

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