<figure><img src="image_1.png" /></figure>

HAL
open science

# Ultra-wideband Sensor Based Robust Model Predictive Control

Théo Le Terrier, Marie Babel, Vincent Drevelle

► To cite this version:

Théo Le Terrier, Marie Babel, Vincent Drevelle. Ultra-wideband Sensor Based Robust Model Predictive Control. SWIM 2025 - Summer Workshop on Interval Methods, Jun 2025, Rennes, France. pp.1-4. hal-05383932

HAL Id: hal-05383932

https://hal.science/hal-05383932v1

Submitted on 26 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Ultra-wideband Sensor Based Robust Model Predictive Control

Théo Le Terrier¹, Marie Babel¹ and Vincent Drevelle¹

¹ Univ Rennes, Inria, CNRS, IRISA
{theo.le-terrier, vincent.drevelle, marie.babel}@irisa.fr

**Keywords:** Intervals, Robotics, Model Predictive Control, Ultra-WideBand

## Introduction

We propose a robust nonlinear model predictive controller (NMPC) leveraging ultra-wideband (UWB) positioning for autonomous control of a power wheelchair (PW). Our goal is to drive the PW to a target pose. NMPC is well-suited for PW control, as it naturally handles actuator limitations and user comfort constraints. At each UWB measurement epoch, an optimal control problem is solved to generate a control sequence, of which only the first value is applied to the PW. The system evolution is predicted over a receding horizon using a nonlinear model.

An indoor UWB positioning system typically consists of fixed anchors and mobile tags. Distances are measured using two-way ranging, which calculates the time of flight of signals between tags and anchors. These measurements are influenced by various factors, including signal noise, non-line-of-sight (NLOS) conditions, multipath propagation, inherent biases in radio modules, and inaccuracies in anchor positioning. Combined with model uncertainties, these errors can lead to inaccurate pose estimates and potential constraint violations, such as collisions. To ensure robust constraint satisfaction, we employ set-membership state estimation, which computes an outer approximation of all states consistent with UWB measurements and propagates this set through
uncertain dynamics over the receding horizon. The control sequence
is then optimized to guarantee that all possible trajectories remain
within the constraints.

An algorithm solving the NMPC problem in a validated way, using
interval analysis, has been proposed in [1]. A similar approach is the
tube-based MPC [2], [3], in which an ancillary feedback controller is
used to ensure that the actual state remains within an invariant tube
around the trajectory computed by solving the nominal NMPC prob-
lem. To guarantee constraint satisfaction for all possible poses within
the tube, state and input constraints are appropriately tightened.

In this work, we propose a robust NMPC framework that avoids the
offline computation of both the ancillary controller and the associated
invariant set, which is a complex task for general nonlinear systems
[4].

<figure><img src="image_3.png" /><figcaption>Figure 1: System Overview. Each UWB tag (green) measures its dis-
tance to each anchor (yellow). Range measurements can be affected
by multipath or NLOS propagation.</figcaption></figure>
## System

The PW pose is denoted $\mathbf{x} = (x, y, \theta)^T$, where $(x, y)$ are the coordinate of the PW's center of rotation, and $\theta$ is the heading angle. The two control inputs $\mathbf{u}$ related to the PW are its linear velocity $v$ and its angular velocity $\omega$. The nonlinear discrete-time system of the PW is

$$ \mathbf{x}_{k+1} = \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k) \qquad (1) $$

$$ \mathbf{y}_k = \mathbf{g}(\mathbf{x}_k, \mathbf{v}_k) \qquad (2) $$

where $\mathbf{f}$ is the system model and $\mathbf{g}$ is the observation model, and $\mathbf{v}_k$ is the measurement disturbance.

## Robust NMPC

At each epoch $k$, UWB range measurements are used to estimate the PW pose. Assuming unknown but bounded measurement uncertainties, the set of all PW poses consistent with the UWB measurements can be conservatively over-approximated using set representations $\hat{\mathcal{X}}_{y_k}$. From the previous time step $k-1$, we have the set of all PW poses that are consistent both with the UWB measurements recorded at epoch $k-1$, and with the propagated sets of feasible poses from earlier epochs $\hat{\mathcal{X}}_{k-1}$. The optimized control sequence is denoted $\mathbf{u}^*_{k-1} = \{\mathbf{u}^*_{0|k-1}, \mathbf{u}^*_{1|k-1}, \dots, \mathbf{u}^*_{N-1|k-1}\}$, with $\cdot_{i|k-1}$ the $i^{th}$ control inputs from sampling time $k-1$. The set $\hat{\mathcal{X}}_{k-1}$ is propagated forward using model (1) while accounting for bounded uncertainties in the applied control inputs, resulting in a predicted set $\hat{\mathcal{X}}_{k|k-1}$. In the following, this is done using the natural interval extension for box representation. The natural inclusion function is denoted $[\mathbf{f}]$. The set of all possible states at epoch $k$ is given by $\hat{\mathcal{X}}_k = \hat{\mathcal{X}}_{y_k} \cap \hat{\mathcal{X}}_{k|k-1}$.

Let $\mathcal{X}_{safe}$ denotes the set of feasible states and $\mathcal{U}$ the set of admissible control inputs. The uncertainty set $\hat{\mathcal{X}}_k$ is forward propagated to tighten the NMPC states constraints. As a result, the PW trajectory is guaranteed to satisfy state constraints, despite the presence of pose
estimation uncertainty. The NMPC problem is formulated such as

$$
\begin{align*}
\{\mathbf{x}_k^*, \mathbf{u}_k^*\} &= \arg\min_{\{\mathbf{x}_{i|k}, \mathbf{u}_{i|k}\}} \sum_{i=0}^{N-1} \ell(\mathbf{x}_{i|k}, \mathbf{u}_{i|k}) + V_f(\mathbf{x}_{N|k}) \\
\text{s.t.} \quad & \mathbf{x}_{0|k} = \hat{\mathbf{x}}_k \\
& \mathcal{X}_{0|k} = \hat{\mathcal{X}}_k \\
& \mathbf{x}_{i+1|k} = \mathbf{f}(\mathbf{x}_{i|k}, \mathbf{u}_{i|k}), \quad i = 0, \dots, N-1 \\
& \mathcal{X}_{i+1|k} = [\mathbf{f}](\mathcal{X}_{i|k}, \mathbf{u}_{i|k}) \quad i = 0, \dots, N-1 \\
& \mathbf{x}_{i|k} \in \mathcal{X}_{safe} \oplus \mathcal{E}_{i|k}, \quad i = 1, \dots, N \\
& \mathbf{u}_{i|k} \in \mathcal{U}, \quad i = 0, \dots, N-1
\end{align*}
$$

where $\ell(\cdot)$ is the stage cost, $\oplus$ is the Pontryagin set difference. The set $\mathcal{E}_{i|k} = \mathcal{X}_{i|k} \ominus \mathbf{x}_{i|k}$ is the possible deviation between the nominal state and the actual PW pose at prediction step $i$, from sampling time $k$. $V_f(\cdot)$ is the terminal cost.

## References

[1] J. Alexandre dit Sandretto. *Reliable Nonlinear Model-Predictive Control Via Validated Simulation*. 2018 Annual American Control Conference (ACC), 2018

[2] V. T. H. Le, C. Stoica, D. Dumur T. Alamo, and E. F. Camacho. *Robust tube-based constrained predictive control via zonotopic set-membership estimation*. International Conference on Decision and Control and European Control Conference, 2011

[3] D. Q. Mayne, E. C. Kerrigan, E. J. Van Wyk, and P. Falugi. *Tube-based robust nonlinear model predictive control*. International journal of robust and nonlinear control, 2011

[4] S. Yu, C. Maier, H. Chen and F. Allgöwer. *Tube MPC scheme based on robust control invariant set with application to Lipschitz nonlinear systems*. Systems & Control Letters, 2013