⚡ electrical engineering

Pitot-Aided Attitude and Air Velocity Estimation with Almost Global Asymptotic Stability Guarantees

This paper proposes a cascade observer architecture that integrates IMU and Pitot tube measurements with magnetometer data to achieve almost global asymptotic stability for attitude and air velocity estimation in fixed-wing UAVs under mild excitation conditions.

Original authors: Melone Nyoba Tchonkeu, Soulaimane Berkane, Tarek Hamel

Published 2026-06-09

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CC BY 4.0

Original authors: Melone Nyoba Tchonkeu, Soulaimane Berkane, Tarek Hamel

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a fixed-wing drone (UAV) flying through the sky. To fly safely, it needs to know two critical things: which way is up (its attitude) and how fast it's moving through the air (air velocity).

Usually, drones use a "brain" called an IMU (Inertial Measurement Unit) that acts like a human's inner ear, sensing rotation and acceleration. However, the IMU has a blind spot: it can't tell the difference between gravity pulling the drone down and the drone actually accelerating forward. If the drone banks or speeds up, the IMU gets confused about which way is "down," leading to a loss of balance.

This paper presents a clever solution to fix that confusion using a simple tool called a Pitot tube (a tube that measures airspeed, like the one on a real airplane) and a magnetometer (a digital compass).

Here is the breakdown of their solution using simple analogies:

1. The Two-Stage "Detective" System

The authors didn't build one giant, complex calculator. Instead, they built a two-stage detective team that works in a chain (a "cascade").

Stage 1: The "Tilt & Speed" Detective (The Riccati Observer)
- The Problem: The IMU is confused by the drone's movement. The Pitot tube only tells the drone how fast air is hitting the front of the plane, but not the sides or top. It's like trying to guess the wind speed by only feeling the wind on your nose while running.
- The Trick: The detective uses math (a "Riccati filter") to combine the IMU's spinning data with the Pitot tube's speed data. It realizes that if the plane is turning, the air hitting the nose must be coming from a specific angle.
- The Result: This stage figures out the tilt (which way is down) and the full air velocity (speed in all directions), even with just one tube.
- The Catch: If the plane flies perfectly straight and level for too long, the math gets "lazy" and stops learning. The plane needs to wiggle or turn (excitation) to keep the math sharp.
Stage 2: The "Full Orientation" Detective (The SO(3) Observer)
- The Problem: The first stage knows the tilt, but it doesn't know which way is "North" (the yaw angle). It's like knowing you are leaning forward but not knowing if you are facing North, South, East, or West.
- The Solution: This stage brings in the magnetometer (compass). It takes the tilt information from Stage 1 and fuses it with the compass reading.
- The Result: Now the drone knows its full 3D orientation (roll, pitch, and yaw) with extreme reliability.

2. The "Zero-Sideslip" Superpower

The paper highlights a special trick for fixed-wing drones. In a well-coordinated flight (like a smooth turn in a real airplane), the drone doesn't slide sideways; it flies straight into the wind.

The Analogy: Imagine the first detective is guessing the wind speed. Without help, it's guessing wildly. But if you tell the detective, "Hey, we know for a fact the wind isn't hitting the side of the plane at all," the detective's job becomes much easier.
The Paper's Claim: By mathematically forcing the "side wind" to be zero (a "pseudo-measurement"), the system becomes much more stable. It allows the drone to figure out its speed and tilt even when it isn't moving as wildly as before.

3. The Proof: Real Flight Data

The authors didn't just do this on a computer. They tested it on a real drone (a MakeFlyEasy Fighter VTOL) flying in windy conditions.

The Test: They compared their new system against the drone's standard built-in computer.
The Result:
- Without the "Zero-Sideslip" trick: The system got confused about the side-to-side and up-and-down speed, making big errors (like guessing the wind was 12 m/s off).
- With the trick: The system stayed sharp. The errors dropped significantly, and the drone's estimate of its speed and tilt matched the reality almost perfectly.

The Bottom Line

The paper claims to have built a navigation system that is mathematically guaranteed to work (almost) everywhere, not just in perfect conditions.

It uses a Pitot tube (speed) and IMU (movement) to figure out tilt and speed.
It uses a Compass to figure out the final direction.
It uses a clever math trick (assuming no sideways slip) to make the system much more reliable.
It proved this works on a real drone flying in the wind, showing that the "tilt" and "speed" estimates stay accurate even when the drone is doing aggressive maneuvers.

In short, they taught the drone how to "feel" the wind and "know" which way is up, even when its internal gyroscope gets confused, using a two-step process that is mathematically rock-solid.

Technical Summary: Pitot-Aided Attitude and Air Velocity Estimation with Almost Global Asymptotic Stability Guarantees

Problem Statement
Reliable attitude estimation for fixed-wing Unmanned Aerial Vehicles (UAVs) is challenging in Global Navigation Satellite System (GNSS)-denied or highly dynamic environments. Standard approaches relying on Inertial Measurement Units (IMUs) often infer gravity direction from accelerometers, an approximation that fails during sustained forward flight or aggressive maneuvers where non-gravitational accelerations are significant. While Velocity-Aided Attitude (VAA) estimation offers a solution, many existing methods assume full velocity measurements (inertial or body-fixed), which is restrictive for small UAVs. Fixed-wing UAVs typically measure air velocity via a single-axis Pitot tube aligned with the longitudinal body axis. Previous attempts to combine Pitot and IMU data have either relied on poorly known aerodynamic parameters or resulted in observers with only local convergence guarantees. Furthermore, single-axis Pitot measurements suffer from axial symmetry, leaving the yaw angle unobservable without an inertial heading reference, thus restricting estimation to a reduced attitude (tilt).

Methodology
The authors propose a two-stage cascade observer architecture that separates aerodynamic-aided tilt estimation from full attitude estimation, utilizing IMU data, at least one Pitot tube measurement, and magnetometer data.

Stage 1: Tilt and Air Velocity Estimation (Riccati Observer)
- The system defines the "tilt" as the gravity direction expressed in the body-fixed frame ( $z = R^\top e_3$ ).
- By combining the air velocity dynamics and the tilt dynamics, the authors formulate a Linear Time-Varying (LTV) system where the augmented state includes the body-fixed air velocity vector ( $V_a$ ) and the tilt vector ( $z$ ).
- A continuous-time Riccati/Kalman-type observer is designed to estimate this augmented state. The observer uses the Pitot measurement (scalar projection of $V_a$ ) as the output.
- The stability of this stage relies on the Uniform Observability (UO) of the LTV system. The authors prove that under mild excitation conditions (specifically, persistent excitation of the direction of the Pitot probe relative to the vehicle orientation), the estimation error dynamics are Globally Uniformly Exponentially Stable (GES).
- To address the observability limitations of a single-axis Pitot tube (which requires complex yaw and pitch motions for full excitation), the authors introduce a zero-sideslip pseudo-measurement ( $V_{a,2} = 0$ ). This is valid for coordinated flight and effectively augments the measurement vector, relaxing the excitation requirements to sustained yaw motion only.
Stage 2: Full Attitude Estimation (Nonlinear Observer on SO(3))
- The estimated tilt vector ( $\hat{z}$ ) from Stage 1 is fused with magnetometer measurements ( $m_B$ ) in a nonlinear observer defined on the Special Orthogonal group $SO(3)$.
- This observer, based on geometric filtering principles, resolves the yaw ambiguity inherent in Pitot-only sensing.
- The correction term utilizes the estimated tilt and the known inertial magnetic field vector to drive the attitude estimate.

Key Contributions

Cascade Architecture: The paper introduces a unified framework that decouples air velocity/tilt estimation from full attitude estimation, allowing for rigorous stability analysis of the interconnected system.
AGAS Guarantees: Unlike previous Pitot-based approaches that offered only local convergence, this work establishes Almost Global Asymptotic Stability (AGAS) for the overall cascade structure. This is achieved by satisfying the UO condition in the first stage and leveraging the topological properties of the $SO(3)$ observer in the second.
Zero-Sideslip Augmentation: The authors demonstrate that incorporating a zero-sideslip constraint as a pseudo-measurement significantly improves the observability properties of the system, reducing the reliance on aggressive maneuvers for state convergence.
Rigorous Analysis: The paper provides a detailed proof of uniform observability for the augmented LTV system and establishes the stability of the equilibrium points, showing that the set of unstable equilibria is measure-zero.

Experimental Results
The proposed observer was validated using real flight data from a MakeFlyEasy Fighter VTOL UAV equipped with an IMU, a forward-aligned Pitot tube, and a magnetometer. The experiments were conducted in a windy environment to test robustness. Two configurations were compared:

Configuration 1: Utilized the zero-sideslip pseudo-measurement.
Configuration 2: Relied solely on the single-axis Pitot measurement.

Findings:

Configuration 1 demonstrated superior performance, with air velocity estimation errors converging to approximately 3 m/s during significant pitch variations. The condition number of the observability Gramian remained low, indicating robust observability.
Configuration 2 exhibited large root-mean-square errors (approx. 12 m/s lateral, 9 m/s vertical) and poor numerical conditioning, confirming the loss of observability when excitation is limited.
In coordinated flight segments, all estimation errors in Configuration 1 remained bounded and converged to zero, validating the theoretical AGAS claims. The results confirm that the pseudo-measurement acts as an effective mechanism to restore observability under weak excitation.

Significance and Claims
The paper claims to extend existing Pitot-assisted estimation frameworks beyond local convergence results. By providing a unified framework for full air velocity and attitude estimation with rigorous stability guarantees, the work addresses a critical gap in UAV navigation for GNSS-denied environments. The authors assert that their approach effectively overcomes the limitations of single-axis Pitot sensing without requiring full velocity measurements or complex aerodynamic models. While the theoretical analysis assumes bounded, slowly time-varying wind and does not explicitly estimate sensor biases, the experimental validation in a windy environment demonstrates the observer's practical robustness. The authors identify the joint estimation of wind components and inertial sensor biases as a natural direction for future work.

1. The Two-Stage "Detective" System

2. The "Zero-Sideslip" Superpower

3. The Proof: Real Flight Data

The Bottom Line

Technical Summary: Pitot-Aided Attitude and Air Velocity Estimation with Almost Global Asymptotic Stability Guarantees

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