Moving Horizon Estimation for Quadrotors: An L1 Adaptive Optimizer Approach
Authors: Thinh Nguyen, Minkyung Kim, Sandeep Banik, Jinrae Kim, Naira Hovakimyan
Organizations: University of Illinois Urbana-Champaign, Urbana, IL 61801 · Department of Mechanical Science and Engineering, The Grainger College of Engineering
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 L1 Adaptive Optimizer (L1-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 L1-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 L1-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.
Figures & tables
n =
number of system states
m =
number of control inputs
p =
number of system outputs (measurements)
Q,R,P0 =
estimator weighting matrices
Vf =
FIE objective function
V =
MHE objective function
N =
MHE horizon length
Table 1
Figure 1: Estimation errors evolution in an uncertainty-free scenario
Figure 2: Mean and standard deviation of estimation errors over 100 simulations with poor initial guesses
Figure 3: Estimation errors evolution given unknown process noise ( Q=10−2I12 )
Table 1: Comparison of root mean squared error for different values of Q and Moving Horizon Estimation horizon length N with unknown process noise, during a 30 -second long simulation ( 3001 data points). Note that the root mean squared errors, calculated using ( 19 ), are unitless.
Figure 4: Comparison between estimates of the linear position p and velocity q of the Extended Kalman Filter and the L1 Adaptive Optimizer-based Moving Horizon Estimation given unknown process noise ( Q=10−2I12 )
Figure 5: Comparison between estimates of the Euler angles Θ and angular velocity Ω of the Extended Kalman Filter and the L1 Adaptive Optimizer-based Moving Horizon Estimation given unknown process noise ( Q=10−2I12 )
N
OSQP
L1 Adaptive Optimizer
10
11.16
4.34
20
38.53
18.67
50
233.22
88.25
100
1373.45
405.30
Table 2: Mean computation time per time step (in milliseconds) for Moving Horizon Estimation with different horizon lengths N .
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 L2 predominates when the measurements are free of outliers.
Nestor Deniz, Guido Sanchez, Fernando Auat Cheein +1
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
This paper introduces an estimation and control framework for dynamic landing of multi-rotor uncrewed aerial vehicles on moving platforms. The proposed method integrates nonlinear model predictive control with a real-time minimum-jerk trajectory planner that enforces a prescribed touchdown time, enabling consistent timing during the terminal descent. To enhance robustness in the presence of time-varying sensing quality, we utilize an adaptive unscented kalman filter that updates the process and measurement noise statistics online. In addition, we provide a reference feasibility analysis showing that minimum-jerk references induce bounded thrust and torque commands under standard tracking hypotheses. The proposed framework is evaluated in simulation and hardware experiments, and it is shown to achieve repeatable landings and improved platform velocity prediction accuracy relative to EKF/UKF-based methods.
Mohammadreza Izadi, Zeinab Shayan, Steven Waslander +1
Autonomous Vehicles Laboratory, Department of Aerospace Engineering, Toronto Metropolitan University, Toronto, Canada · University of Toronto Institute for Aerospace Studies, University of Toronto, Toronto, Canada
In environmental monitoring as well as emergency response applications such as wildfires, wind velocity measurement is essential. Quadrotor UAVs have become popular platforms for wind velocity estimation due to their maneuverability, compact size, and cost-effectiveness. Numerous studies use the Extended Kalman Filter (EKF) to estimate the wind velocity based on the quadrotor dynamic model. However, most of them use hovering quadrotors only for wind estimation, others use a near-linear trajectory to estimate near-constant velocities. Furthermore, EKF performance is constrained by its reliance on linearized approximations of the nonlinear quadrotor dynamics around current states, limiting accuracy in highly nonlinear scenarios, including windy conditions. This study proposes the use of an Unscented Kalman Filter (UKF), a nonlinear estimator to provide accurate wind estimations while maintaining the trajectory of the quadrotor UAV. The quadrotor is modeled on the Special Euclidean group SE(3) and the approach is evaluated through numerical simulations using a geometric controller to maintain quadrotor flight paths. The results indicate that as the nonlinearity of the simulation increases, the UKF consistently outperforms the EKF. This demonstrates the potential of the UKF as a reliable estimator for highly nonlinear scenarios, capable of maintaining the trajectory with minimal deviation while providing accurate wind velocity estimations.
Hiranya Udagedara, Mahdis Bisheban
Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Canada