# Efficient Attitude Estimation for Mobile Wheeled Robots Using Elevation Maps

Charifou Orou Mousse, Mohamed Benrabah, Dieumet Denis, François Marmoiton, Roland Chapuis

Marmoiton, Roland Chapuis

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Charifou Orou Mousse, Mohamed Benrabah, Dieumet Denis, François Marmoiton, Roland Chapuis. Efficient Attitude Estimation for Mobile Wheeled Robots Using Elevation Maps. 12th IFAC Symposium on Intelligent Autonomous Vehicles, May 2025, Phoenix Arizona, United States. hal-05088429v2

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# Efficient Attitude Estimation for Mobile Wheeled Robots Using Elevation Maps

Charifou Orou Mousse ** Mohamed Benrabah *

Dieumet Denis ** François Marmoiton * Roland Chapuis *

* Université Clermont Auvergne, Centre National de Recherche Scientifique, Clermont Auvergne INP, Institut Pascal UMR6602, F-63000 Clermont-Ferrand France (e-mail: firstname.lastname@uca.fr).

** Sherpa Engineering, 333 Avenue Georges Clemenceau, 92000 Nanterre France (e-mail: f.lastname@sherpa-eng.com)

**Abstract:** Ensuring the dynamic stability of a wheeled robot in off-road navigation requires continuous state monitoring to anticipate risks such as slipping, skidding, or overturning. While proprioceptive sensors can monitor the robot's state, fully predicting it remains challenging since its variables depend on robot-terrain interaction. This work aims to propose a direct geometric method to estimate the robot's attitude by approximating wheel-ground contact with circles and leveraging an elevation map. Our approach is intuitive, straightforward to implement, precise, and computationally efficient. Experimental results, both in simulation and using a real robotic platform, validate its effectiveness.

Copyright © 2025 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

**Keywords:** Off-road navigation, Attitude Estimation, Direct Geometric Method, Elevation Map, Wheel-Ground Contact

## 1. INTRODUCTION

As autonomous vehicles increasingly operate in rural environments, the risk of damage is heightened due to the nature of the terrain, which can cause instabilities such as skidding, slipping, and even rollovers. To mitigate these risks, many systems are equipped with sensors that actively assess the entire state of the vehicle, including its angular attitude, enabling to prevent potential hazards through adjustments to vehicle controls. Such technology relies on Inertial Measurement Units (IMUs), Global Positioning Systems (GPS), odometers, and data fusion algorithms to provide precise 3D absolute localization of the system (see Vaganay et al. (1993)). In the best-case scenario, these technologies are reliable and deliver instantaneous knowledge of the vehicle's state. However, their primary limitation lies in their reactive nature, meaning that by the time alerts are issued, the damage may have already occurred. To achieve a more proactive capability, a widely explored area of research focuses on the estimation of the attitude of mobile robots. More specifically, this involves predicting the angular state of a robot on a given terrain. This predictive approach aims to anticipate and mitigate risks before they manifest, thereby enhancing the safety and stability of autonomous vehicles in challenging environments.

The ambition behind our present work is to propose an efficient geometric method for estimating the attitude of a wheeled robot based on an elevation map. Our approach offers the advantages of accuracy, low computational cost, and ease of implementation. It can serve as a valuable tool for stability analysis of vehicles operating in unstructured environments, where predictive knowledge of the robot's

attitude is critical. In agricultural applications, for instance, this method can simulate an inertial sensor and compensate for potential signal losses. We present real-world results obtained using a robotic platform named *Barakuda*, equipped with a 3D LiDAR for mapping and a localization module that fuses IMU and GPS data to generate 3D pose ground truth.

<figure><img src="image_2.png" /><figcaption>Fig. 1. A canonical real-world scenario involving a speed bump used to evaluate our approach with real data</figcaption></figure>

## 2. RELATED WORKS

Predicting the position and attitude of a robot based on a model of the environment is a topic of interest in tracked robotics due to the complex architecture of these types of
robots, which sometimes include flippers, requiring a more precise assessment of their interaction with the environment. The trivial solution, which relies on using dynamic models, is costly because it requires the integration of differential equations that account for contact forces, friction, longitudinal slippage, and lateral skidding, which are parameters that depend on the terrain. This problem, also known as robot settling can be formulated as determining a mapping $\mathcal{F}$ such that, given the robot's SE(2) position $\mathbf{p}$ and an environmental model $M$, it estimate the robot's altitude $\hat{h}$ and attitude $\hat{\theta}$, to get a full SE(3) pose:

$$ \mathcal{F}(\mathbf{p}, M) = (\hat{h}, \hat{\theta}) \qquad (1) $$

Works addressing this problem can be classified into four groups: Iterative Geometric Methods (IGM), Quasi-statics Physics Engines simulation, Deep learning techniques and, Direct Geometric Methods (DGM).

IGM principle assumes that the robot starts suspended in the air, positioned close enough to the terrain to ensure rapid convergence, and then falls toward the surface under the effect of gravitational acceleration. A series of geometric evaluations are then performed iteratively until the robot's stabilization condition is met. A pioneering work in this field is that of Brunner et al. (2015), which relies on a uniform bicubic B-spline surface model to represent the environment. The pose estimation process is formulated as a Linear Complementarity Problem (LCP), and its output is used to compute a tip-over metric, the Force-Angle Stability Measure (FASM) proposed by Papadopoulos and Rey (1996), for planning purposes. Ma and Shiller (2019) similarly propose an approach for wheeled robots based on a B-spline patch interpolated over a 3D point cloud. The contact forces between the wheels and the terrain are computed using Singular Value Decomposition (SVD), and their equilibrium is verified to terminate the process and output a final pose estimate. By leveraging the concept of a robot heightmap, which is differentiated from the terrain heightmap, in Fabian et al. (2020) the pose and ground contact points are computed iteratively until a final attitude estimate is obtained. To our knowledge, Oehler and von Stryk (2023) presents the best results in the field, setting the state-of-the-art in terms of computation time. This work uses a voxel-based Euclidean Signed Distance Field (ESDF) representation. Although these iterative techniques have the reputation of being more reliable in terms of physical modeling, they require approximations to determine the wheel-ground contact distance. This is because they attempt to simulate all the physics involved in a robot landing, whereas in practice, the robot moves along the ground without fully lifting off. In addition, each study in this field relies on its own environmental model, making it difficult to compare computational performance and hindering the establishment of a unified perception framework.

Approaches based on physics engines, such as Open Dynamics Engine(ODE), PhysX, or Bullet (Collins et al. (2021)), are among the most accurate methods. They operate on the same principles as IGM by solving complex physical interactions. Physics engines are commonly used for developing stability metrics based on estimated pose (see Norouzi et al. (2013)). Their primary challenge lies in

achieving online simultaneous mapping and attitude prediction. In fact, a complete and realistic representation of the environment is required when using these simulators.

The involvement of artificial intelligence (AI) in addressing this specific problem remains relatively limited and recent. Although deep learning-based solutions have demonstrated significant potential, they focus on holistic applications rather than specifically targeting pose estimation. That said, some research (See Golroudbari and Sabour (2023); Brossard et al. (2020)) has explored AI-driven methods for filtering noisy inertial data and fusing it with deep learning techniques to achieve accurate attitude estimates. A notable advancement in this field is the work of Agishev et al. (2024), who developed a physics-aware self-supervised learning model. This approach uses visual data to estimate robot-terrain interaction forces and predict future trajectories based on control input sequences. While this architecture marks significant progress, it tackles a broader range of challenges beyond attitude estimation, resulting in a complex and multifaceted solution.

DGM are faster since they do not simulate the robot falling to the ground. They are the best compromise for the trade-off between online mapping, planning and precision efficiency. Yu et al. (2021) proposed a 3D vehicle model that incorporates terrain topology within a model predictive controller (MPC). Under certain assumptions, the vehicle's attitude is derived from the normal vector of the terrain function, which is then used to refine a tire model within the predictive framework. The environments used in this work are canonical and their functions are known. In Thakker et al. (2021), the robot's footprint is projected onto the ground, and a plane is fitted to the included point cloud to extract the robot's attitude. A traversability criterion is then defined as a combination of three metrics: tip-over stability, and the detection of both negative and positive obstacles. Fan et al. (2021) also utilize a robot footprint fitting plane to extract attitude angles and define orientation constraints. These geometric approaches exploit the entire region covered by the robot's envelope without considering the fact that the actual contact surfaces between the robot and the ground are at the wheel level. Furthermore, they do not provide a qualitative analysis of their attitude estimation methods. Jordan and Zell (2017) provide a fast and accurate method for pose estimation using an elevation image constructed from an RGB-D points cloud, taking into account wheel geometry. The elevation image is generated instantaneously without temporal integration and has a fine spatial resolution of 0.75 cm. The output of their algorithm consists of two normals defining the possible orientations of the robot and the more probable normal is not known.

### 3. SYNTHESIS AND CONTRIBUTIONS

The difficulty in comparing and selecting among most of the cited works primarily arises from the non-uniformity of environmental representations. IGM requires a representation that continuously indicates the distance between the robot and the ground surface, while approaches based on physical engines demand a realistic model of the environment. Moreover, these two families of approaches that simulate a robot being dropped from a certain height
are particularly relevant for tracked robots, given the variability of their contact surface with the environment. However, in the case of wheeled robots, which operate at higher speeds, computational efficiency is crucial to enable real-time planning, making DGM more suitable for such applications. Finally, to standardize these DGM, the 2.5D elevation map, which is now widely used in off-road navigation, should serve as a common representation of the environment. Unfortunately, only certain direct geometric techniques make use of elevation map models and in most of their applications, the problem itself is either roughly modeled by considering the entire footprint of the robot on the map or it is partially addressed by, for example, outputting candidate normal vectors for orientation.

One of the strengths of our approach lies in the use of an elevation map similar to those developed by Fankhauser et al. (2018), with a resolution of 10 cm, suitable for mapping large, unstructured environments. We propose a **polygonal wheel-ground contact model** rather than considering the entire robot footprint. Our estimation approach is **intuitive**, as it relies on Principal Component Analysis (PCA). Our approach has the advantage of being **low-cost** in terms of computing resources and time, and is both **complete and precise**, providing roll and pitch attitude angles with an accuracy of between 1 and 5°. We validate our method through simulation tests in various canonical scenarios and demonstrate its **robustness and performance in real-world experiments**.

## 4. ATTITUDE ESTIMATION

In most studies (Biris et al. (2011); Djohor et al. (2012)) on the contact patch of vehicles (the area where the wheel meets the ground), the pressure distribution of the wheels on the ground is commonly modeled as either a **circle** or a **rounded rectangle**. Adopting this approach, we have chosen to approximate the footprint of the robot's wheels using these shapes (see Fig. 2). Initially, we conducted our approach using both models to compare results and make a decision. For the sake of brevity, we present here the principle based on circles model, which is more inclusive. The circle's radius is taken to be the length of the wheel cylinder. We assumed a rigid vehicle model with no suspension considerations.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Wheel-Ground contact models: circle-based or rounded-rectangle</figcaption></figure>

Let $\mathbf{p} = [x, y, z, \alpha, \beta, \gamma]$ be the SE(3) pose of the robot, and $\mathbf{p}_{2D} = [x, y, \gamma]$ its SE(2) pose. From $\mathbf{p}_{2D}$, we determine the wheel contact points $[\mathbf{w}_{\mathrm{fl}}, \mathbf{w}_{\mathrm{fl}}, \mathbf{w}_{\mathrm{fr}}, \mathbf{w}_{\mathrm{rr}}]$ based on the robot's geometry (track width and wheelbase).

Given the contact surfaces from these points, we validate the pose $\mathbf{p}_{2D}$ by checking their correspondence with the elevation map. We need a minimum of three valid contact surfaces. According to our hypothesis, where the wheels are rigidly attached to the map, if two wheels are in contact with non-informed areas (gaps in the elevation map), the robot's attitude becomes undefined. A contact surface is considered valid if it covers at least half of the contact polygon (circle or rounded rectangle) area. This is verified by counting the number of measured cells, knowing that each cell has a surface area of $r \text{ cm}^2$, with $r$ being the resolution of our elevation map. Once at least three contact surfaces are valid, the best-fit plane for these surfaces can be computing using Principal Component Analysis (PCA).

A $3 \times N$ matrix $X$ is formed, containing the $N$ points $\mathbf{p}_i \in \mathbb{R}^3$ within all the contact surfaces, with coordinates $(x_i, y_i, z_i)$, where $x_i$ and $y_i$ represent the point's 2D position in the map frame, and $z_i$ represents its elevation value:

$$X = \begin{bmatrix} x_1 & x_2 & \dots & x_N \\ y_1 & y_2 & \dots & y_N \\ z_1 & z_2 & \dots & z_N \end{bmatrix} \in \mathbb{R}^{3 \times N}$$

Given the mean vector of the matrix $X$:

$$\bar{\mathbf{m}} = \frac{1}{N} \sum_{i=1}^{N} \mathbf{p}_i \qquad (2)$$

We compute the simplified covariance matrix $S \in \mathbb{R}^{3 \times 3}$ as follows:

$$S = \frac{1}{N} X X^T - \bar{\mathbf{m}} \bar{\mathbf{m}}^T \qquad (3)$$

To determine the principal directions of this distribution, we perform eigen decomposition of $S$, obtaining three eigenvalues $\lambda_1, \lambda_2, \lambda_3$ and their corresponding eigenvectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$, where each eigenvector is a 3D column vector in $\mathbb{R}^3$:

$$S \mathbf{v}_i = \lambda_i \mathbf{v}_i, \quad \text{for } i \in \{1, 2, 3\} \qquad (4)$$

These eigenvectors are ordered by their corresponding eigenvalues, which quantify the variance along each direction. The first principal component aligns with the direction of maximum variance, while the **last principal component, associated with the least variance**, is expected to be the normal vector $\mathbf{n}$ to the best-fitting plane when the points approximately lie on a plane:

$$\mathbf{n} = \mathbf{v}_{\arg\min_{i}(\lambda_i)} \qquad (5)$$

This normal is then oriented based on the robot $\mathbf{p}_{2D}$ yaw $\gamma$ state, which represents a **clockwise rotation** about $\gamma$ around the $z$-axis:

$$\mathbf{n} = \begin{bmatrix} n_x \\ n_y \\ n_z \end{bmatrix} = \begin{bmatrix} \cos \gamma & \sin \gamma & 0 \\ -\sin \gamma & \cos \gamma & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} n'_x \\ n'_y \\ n'_z \end{bmatrix} \qquad (6)$$

where $(n'_x, n'_y, n'_z)$ are the original components of the normal vector, and $(n_x, n_y, n_z)$ are the rotated components after applying the yaw rotation.

Roll $\alpha$ and pitch $\beta$ can be derived as follows:

$$\begin{bmatrix} n_x \\ n_y \\ n_z \end{bmatrix} = \begin{bmatrix} \sin \beta \\ -\cos \beta \sin \alpha \\ \cos \alpha \cos \beta \end{bmatrix} \qquad (7)$$
$$ \alpha = \tan^{-1}(-n_y/n_z) \qquad (8) $$

$$ \beta = \tan^{-1} \left( n_x / \sqrt{n_y^2 + n_z^2} \right) \qquad (9) $$

# 5. SIMULATION AND EXPERIMENTAL RESULTS

## 5.1 Simulation experiments

We first validate our method in simulation, using a Husky robot (see Fig. 2) in two Gazebo environments: a canonical speed bump with a 15° ascent angle, and a more realistic natural environment, a lake border with non-uniform relief, provided by Martinez et al. (2020) (see Fig. 3). To eliminate any correlation between the mapping process and the attitude estimation problem, at this stage, the elevation map is precomputed for each world.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Environments: Speed bump (left) and Lake (right)</figcaption></figure>

In the Speed bump environment, the robot operates to follow the red trajectory, then it goes back on the bumper descent and spins on it as described in Fig. 3.

<figure><img src="image_5.png" /><figcaption>Fig. 4. Attitude Estimation during Speed Bump Traversal</figcaption></figure>

The graph presented in Fig. 4 illustrates the results of the scenario conducted in the Speed Bump environment. In green, we have the ground truth provided by the simulator, in blue the estimation results using the rounded rectangle-based (R) model, and in red those obtained with the circle-based (C) model. The lower graph highlights the pitch variation. As expected, when the robot follows the red trajectory, during the front ascent of the speed bump, the pitch value decreases gradually, reaching approximately -15°. Conversely, when it starts descending the bump, this value increases up to approximately 15°. Both models produce results that are visually close to the ground truth. During this maneuver, the robot's roll values remained constant, as indicated by the upper graph in Fig. 4. When the robot spins on the front ascent of the bump, a roll variation can be observed, reaching approximately ±10°. In this case as well, we have results validated by the ground truth.

To validate the method in simulation, we conducted a generic run in the Lake environment, traversing areas with slopes ranging up to 20°. The results obtained are shown in Fig. 5, and they confirm those obtained in the bumper scenario by producing a similar variation in the estimated attitude compared to the ground truth from the simulator. The zoomed-in views of the graphs show an identical trend between the curves of the two models.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Attitude Estimation during the lakeside course</figcaption></figure>

For a clearer statistical analysis, Table 1 shows the RMSE (Root Mean Square Error), maximum, and standard deviation values for Husky roll and pitch estimation across both scenarios.

<table><caption>Table 1. Husky attitude estimation errors statistics</caption><thead><tr><th rowspan="2">Scenarios</th><th colspan="3">Roll (α) °</th><th colspan="3">Pitch (β) °</th></tr><tr><th>RMSE</th><th>Max</th><th>Std</th><th>RMSE</th><th>Max</th><th>Std</th></tr></thead><tbody><tr><td>Bump(R)</td><td>0.254</td><td>4.167</td><td>0.95</td><td>0.784</td><td>4.696</td><td>2.501</td></tr><tr><td>Bump(C)</td><td>0.277</td><td>4.411</td><td>1.067</td><td>0.746</td><td>4.358</td><td>2.344</td></tr><tr><td>Lake(R)</td><td>0.531</td><td>1.303</td><td>0.348</td><td>0.718</td><td>1.767</td><td>0.669</td></tr><tr><td>Lake(C)</td><td>0.559</td><td>1.392</td><td>0.36</td><td>0.744</td><td>1.812</td><td>0.725</td></tr></tbody></table>

From these results, we observe that our method produces estimations with a mean error of less than 1° and a maximum error of less than 5°. We also notice that the rectangle-based model does not show a significant difference in terms of precision while being computationally four times more expensive than the circle-based approach, which achieves a computation time of approximately 15µs per pose. Another point in favor of the circle-based model is its genericity, as it can be used even for Ackermann-steered vehicles. Indeed, the wheels of this type of vehicle can have orientations relative to the vehicle's base, and circles are invariant under such rotations. In fact, rounded rectangles could be used, but they are harder to transform, as they require at least four points to be handled. Our comparative results favor the use of the circle-based approach, which we adopt for the real data experiments.

## 5.2 Real Experiments

These experiments involved reproducing the canonical environment with a speed bump. The robotic platform used, Barakuda (see Fig. 1), is a 650 kg Unmanned Ground Vehicle (UGV) manufactured by the French company *Shark Robotics*. It is a skid-steering robotic model. We equipped the platform with a 64-ring LiDAR sensor for mapping, along with a localization module that takes GPS and IMU data as inputs and provides a 3D pose, which we
use to compare our results. The elevation map is created simultaneously from the LiDAR point clouds, implying that errors in map creation affect the attitude estimation process. A machine with a 13th Gen Intel® Core™ i7-13800H × 20 processor is used to compute the estimated angles.

In this setup, we tested different scenarios, including frontal ascents at various angles achieved by adding pallets to the horizontal part of the bump. Specifically, results were obtained for configurations with **one** (which produces a slope of approximately 7 degrees) and **two** pallets assembled (which produces a slope of approximately 14 degrees).

<figure><img src="image_7.png" /><figcaption>Fig. 6. Pitch variation during speed bump frontal ascent with one (up) and two (down) pallets</figcaption></figure>

As expected, during the frontal ascent, we observed variations only in pitch (see Fig. 6), similarly to the simulation results. The horizontal part of our speed bump was deliberately designed to be shorter than the vehicle's wheelbase, allowing us to observe a sinusoidal variation, with the robot descending as soon as it mounts the bump.

Additionally, we tested side-hill ascents, where two wheels on the same side of the robot, either **left**(L) (see Fig. 7) or **right**(R) (see Fig. 8), were on the bump, while the other two remained on the ground.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Attitude Estimation during side-hill **left** ascent</figcaption></figure>

In both scenarios, we observe significant variations in roll, up to approximately ±12°, either positive (R) or negative (L), and small variations in pitch.

In all of these experiments, the variation in attitude angles follows the same trend as the localization module's fused data. Errors due to map creation can be observed mainly in flat areas, especially after the descent of the bump in pitch variation (see Fig 7 and 8), where some artifacts may appear on the map due to the robot's movement and are not updated when the robot gets too close.

<figure><img src="image_9.png" /><figcaption>Fig. 8. Attitude Estimation during side-hill **right** ascent</figcaption></figure>

Table 2 shows the RMSE, maximum, and standard deviation values for *Barakuda* roll and pitch estimation across these scenarios. The errors obtained are of the same order of magnitude as those in simulation.

<table><caption>Table 2. *Barakuda* attitude estimation errors statistics</caption><thead><tr><th rowspan="2">Scenarios</th><th colspan="3">Roll (α) °</th><th colspan="3">Pitch (β) °</th></tr><tr><th>RMSE</th><th>Max</th><th>Std</th><th>RMSE</th><th>Max</th><th>Std</th></tr></thead><tbody><tr><td>Frontal(1)</td><td>-</td><td>-</td><td>-</td><td>0.972</td><td>4.669</td><td>3.182</td></tr><tr><td>Frontal(2)</td><td>-</td><td>-</td><td>-</td><td>1.214</td><td>5.201</td><td>2.975</td></tr><tr><td>Side-hill(L)</td><td>0.956</td><td>3.520</td><td>2.204</td><td>0.789</td><td>2.330</td><td>1.085</td></tr><tr><td>Side-hill(R)</td><td>0.684</td><td>2.304</td><td>0.911</td><td>0.734</td><td>1.885</td><td>0.690</td></tr></tbody></table>

## 6. CONCLUSION

This work enabled the estimation of a wheeled robot's attitude using only an elevation map and a 2.5D position as inputs. Although our proposed approach is inherently dependent on the quality of the perception data used to generate the elevation map, it provides sufficiently accurate results and computational efficiency for real-time applications. The accuracy of the map is an issue that should be addressed separately, for example, by using ground segmentation techniques in images.

In addition to attitude estimation, we also aim to extend our technique to estimate the robot's altitude (z). This can be achieved by considering the average elevation of the median plane of the wheel contact surfaces and the height of the center of gravity relative to a flat horizontal plane. The main limitation of our method stems from the assumptions of a simplified robot model with two degrees of freedom, although this remains valid under many conditions. Future imminent work includes leveraging this method to predict motion primitives in a local environment and defining a traversability criterion based on these estimated states, as part of the development of a path planner.

## ACKNOWLEDGEMENTS

This material is based upon work supported by Sherpa Engineering and the ANRT (Association nationale de la recherche et de la technologie) with a Cifre (Conventions industrielles de formation par la recherche) fellowship granted to the main author of this paper.
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