💻 computer science

Motor Angular Speed Preintegration for Multirotor UAV State Estimation

This paper introduces Motor Angular Speed LiDAR Odometry (MAS-LO), a novel state estimation algorithm that preintegrates motor speeds to replace degraded IMU data, achieving significantly higher position and velocity accuracy with lower latency compared to state-of-the-art inertial methods.

Original authors: Matěj Petrlík, Filip Novák, Robert Pěnička, Martin Saska

Published 2026-06-19

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CC BY 4.0

Original authors: Matěj Petrlík, Filip Novák, Robert Pěnička, Martin Saska

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

The Big Problem: The "Shaky Hand"

Imagine you are trying to draw a straight line while someone is vigorously shaking your hand. That is what a drone (UAV) faces when it tries to figure out where it is and how fast it is moving.

Most drones use a tiny sensor called an IMU (Inertial Measurement Unit) to guess their movement. It's like a very sensitive accelerometer that feels every bump and shake. However, because the drone's propellers are spinning at high speeds, they create vibrations. These vibrations are like that "shaky hand," drowning out the true movement data with noise. The result? The drone's estimate of its position gets messy and inaccurate, especially when flying fast or close to obstacles.

The New Idea: Listening to the Engine, Not the Shaking

The authors of this paper came up with a clever workaround. Instead of trying to measure the movement directly with a shaky sensor, they decided to listen to the motors.

Think of a car. If you know exactly how hard the engine is working (the RPMs) and you know the weight of the car, you can calculate how fast the car is accelerating. You don't need to feel the bumps in the road to know the engine is pushing hard.

Similarly, the drone's electronic controllers (ESCs) know the exact speed of every motor. The authors built a mathematical model that says: "If Motor A is spinning at X speed and Motor B is at Y speed, the drone must be accelerating in this specific direction."

Because these motor speeds are measured electronically, they are perfectly smooth and vibration-free. It's like listening to a clear radio station instead of a static-filled one.

The "Pre-Integration" Trick: The Shortcut

Calculating a drone's position from scratch every single time a new motor speed is reported would be like trying to solve a giant math puzzle from the very beginning every time you take a step. It would be too slow for a computer to handle in real-time.

To solve this, the team used a technique called Preintegration.

The Analogy: Imagine you are walking. Instead of counting every single step you've taken since you left your house to know how far you are, you just count the steps you took since the last time you checked your map.
How it works: The system groups many motor speed measurements together into a single "delta" (a small change). This allows the drone to update its position quickly without redoing all the heavy math every time.

The New System: MAS-LO

The authors combined this motor-speed trick with a LiDAR (a laser scanner that maps the surroundings). They created a new system called MAS-LO (Motor Angular Speed LiDAR Odometry).

LiDAR acts like a pair of eyes, seeing the walls and trees to correct the drone's drift.
MAS (Motor Speed) acts like a very smooth, vibration-free internal compass that tells the drone exactly how it is moving between the "eye" checks.

The Results: A Clearer Picture

The team tested their new system against the current "gold standard" (a system called LIO-SAM that uses the shaky IMU). Here is what they found:

Position: The new system was 28% more accurate at knowing where the drone was.
Speed: It was 65% better at knowing how fast the drone was moving.
Lag: The data arrived 14% faster, which is crucial for agile flying where split-second decisions matter.
Robustness: Even if the team guessed the wrong numbers for the drone's weight or size, the system still worked well.

The Trade-off: The new system was slightly less accurate at guessing the drone's rotation (tilting left/right) compared to the old system, but it was vastly superior at tracking straight-line movement and speed.

The Bottom Line

The paper proves that you don't need a vibration-dampening sensor to fly a drone precisely. By using the clean, digital data from the motors themselves, the drone can "feel" its movement much more clearly than it ever could with a traditional sensor. This makes indoor flights in tight spaces (like rescue missions in collapsed buildings) safer and more reliable.

The authors have made their code and the flight data they collected available to the public so others can build upon this "vibration-free" way of flying.

Technical Summary: Motor Angular Speed Preintegration for Multirotor UAV State Estimation

Problem Statement
Precise state estimation is fundamental for tight feedback control, enabling agile and obstacle-near flights for Unmanned Aerial Vehicles (UAVs). While state-of-the-art methods typically fuse slow pose measurements (e.g., from LiDAR or cameras) with high-frequency inertial measurements from an Inertial Measurement Unit (IMU), these IMU-based approaches face significant limitations when deployed on UAVs. Specifically, IMU measurements are degraded by high-amplitude vibrations from spinning propellers, which obscure the true acceleration trajectory and severely impact estimation accuracy. Furthermore, inertial methods are highly sensitive to the calibration of extrinsic and intrinsic parameters, and the low-cost Micro-Electro-Mechanical System (MEMS) IMUs commonly used in UAVs exhibit high measurement noise and bias drift. Additionally, obtaining angular accelerations directly for advanced controllers (such as Incremental Nonlinear Dynamic Inversion) is difficult, often requiring numerical differentiation of noisy IMU data.

Methodology
The authors propose a novel state estimation approach that replaces IMU preintegration with the preintegration of Motor Angular Speed (MAS) measurements. The core pipeline involves:

Data Acquisition: The method utilizes angular speed measurements ( $w_i$ ) reported by Electronic Speed Controllers (ESCs) driving the UAV motors. These measurements are inherently vibration-free and precise compared to IMU data.
Propulsion Modeling: The MAS measurements are passed through a UAV propulsion model. This model converts individual motor speeds into linear and angular forces and torques acting on the UAV's Center of Gravity (COG), accounting for rotor positions, spin directions, thrust coefficients, and drag coefficients.
Preintegration: To avoid the computational inefficiency of integrating accelerations for every new measurement and to ensure independence from the initial state, the authors derive a preintegration scheme on the tangent space of the UAV state manifold. This aggregates multiple MAS measurements between two time steps into a single "delta" measurement ( $\Delta x$ ), representing the relative change in state (position, orientation, velocity, angular velocity).
Factor Graph Optimization: The preintegrated measurements form a "MAS Factor," a binary constraint between two consecutive states in a factor graph. This factor is combined with a LiDAR Odometry (LO) factor (derived from a LiDAR scan matching algorithm) and bias factors to estimate the full UAV state. The authors implement this within the GTSAM framework, creating a new algorithm termed Motor Angular Speed LiDAR Odometry (MAS-LO).
Bias Handling: To compensate for model inaccuracies (e.g., unmodeled aerodynamic effects, ground effect, mass distribution offsets) and sensor biases, the system includes a random walk bias factor for linear and angular accelerations.

Key Contributions

MAS Preintegration Scheme: A novel derivation of preintegration for motor angular speeds operating on the UAV state manifold, allowing for efficient state propagation without re-computation upon linearization.
MAS Factor: The design of a binary factor based on preintegrated MAS that can directly replace the IMU factor in existing factor-graph-based localization frameworks.
MAS-LO Algorithm: The integration of the MAS factor with LiDAR odometry to create a complete state estimation algorithm that does not rely on an IMU for state propagation.
Open-Source Release: The authors open-source the MAS preintegration code, the MAS factor, the full MAS-LO algorithm, and six new datasets containing MAS, LiDAR, IMU, and RTK ground truth data for various flight maneuvers.

Experimental Results
The proposed method was evaluated against the state-of-the-art LIO-SAM (LiDAR-Inertial Odometry Smoothing and Mapping) algorithm using six outdoor flight datasets (Hover, Forward, Lateral, Vertical, Rectangle, and Loop) with RTK ground truth.

Accuracy: MAS-LO demonstrated a 28% reduction in Absolute Translation Error (ATE) and a 65% reduction in Absolute Velocity Error (AVE) compared to LIO-SAM across the datasets.
Trade-off: This improvement in linear estimation came at the cost of a 20% increase in Absolute Rotation Error (ARE), attributed to the double integration of angular accelerations in the MAS model.
Noise and Lag: Analysis showed that MAS-derived accelerations have significantly lower noise amplitude than IMU data (even filtered IMU data). Furthermore, MAS preintegration exhibited a 14% lower measurement lag (0.063s vs. 0.073s for IMU), a critical factor for agile flight control.
Robustness: A sensitivity study revealed that the MAS-LO algorithm is robust to incorrect parameter values (e.g., thrust coefficients, body dimensions), with performance remaining reasonable even when parameters deviated significantly from their true values.

Significance
The paper claims that by leveraging precise, vibration-free motor speed measurements, it is possible to eliminate the need for IMUs in the state propagation loop, thereby removing the primary source of noise and calibration sensitivity in UAV localization. The method provides a more accurate estimation of linear position and velocity, which are crucial for agile flight and obstacle avoidance. The authors position this work as a foundational step toward combining the complementary strengths of MAS and IMU factors in future hybrid state estimation systems, and as a viable alternative for visual-inertial systems where IMU calibration and noise are limiting factors. The open-source nature of the work is intended to encourage further development of MAS-based methods within the robotics community.