💻 computer science

RAGE-XY: RADAR-Aided Longitudinal and Lateral Forces Estimation For Autonomous Race Cars

This paper presents RAGE-XY, an enhanced real-time estimation framework for autonomous race cars that utilizes onboard IMUs and RADARs to accurately infer vehicle velocity, slip angles, and both longitudinal and lateral tire forces by incorporating online RADAR calibration and a tricycle vehicle model.

Original authors: Davide Malvezzi, Nicola Musiu, Eugenio Mascaro, Francesco Iacovacci, Marko Bertogna

Published 2026-04-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Davide Malvezzi, Nicola Musiu, Eugenio Mascaro, Francesco Iacovacci, Marko Bertogna

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 you are driving a race car at 150 mph. You are leaning hard into a turn, the tires are screaming, and the car is fighting to stay on the track. To drive safely and fast, the car's computer needs to know three things instantly:

How fast is it actually going? (Not just what the speedometer says, but how it's moving sideways too).
How much grip do the tires have? (Are they about to slide like a banana peel, or are they holding tight like glue?)
What forces are pushing and pulling the car? (Is the engine pushing hard, or are the brakes fighting back?)

Usually, to get this information, you'd need expensive, fragile sensors like laser cameras or high-end GPS systems that cost more than a house. But this paper introduces RAGE-XY, a "super-smart guesser" that uses only the standard, cheap sensors already found on most cars (like the ones in your phone or a regular car: an accelerometer and radar sensors).

Here is how RAGE-XY works, explained with some everyday analogies:

1. The "Blindfolded" Driver and the Radar

Think of the car's radar sensors as a driver wearing a blindfold who can only "feel" the wind hitting their face.

The Problem: If the driver's blindfold is slightly crooked (misaligned), they will think the wind is coming from the side when it's actually coming from the front. This makes them think they are drifting sideways when they aren't. In a race car, even a tiny crooked sensor can cause the computer to think the car is sliding when it's actually driving straight, leading to a crash.
The RAGE-XY Fix: The paper adds a special "auto-calibration" feature. Imagine the car drives down a perfectly straight, empty highway for a few seconds. The computer says, "Okay, if I'm driving straight, the wind should only be hitting my face, not my ears." It then automatically twists the mental map of the radar sensor until the data makes sense. This fixes the "crooked blindfold" in real-time, ensuring the car knows exactly where it is going.

2. The "Tricycle" vs. The "Unicycle"

The original version of this system (called RAGE) looked at the car like a unicycle (a single-track model). It assumed the left and right wheels were one single wheel in the middle. This is fine for gentle driving, but race cars are aggressive. They twist, turn, and accelerate hard.

The Upgrade: RAGE-XY upgrades the mental model to a tricycle. It realizes that the two back wheels are separate and can do different things.
Why it matters: Race cars have a special gear called a Limited-Slip Differential. Think of this like a smart referee in a tug-of-war. If one back wheel starts to slip (lose grip), the referee instantly sends more power to the other back wheel that still has grip. The old system couldn't see this happening. RAGE-XY can now "feel" how that power is being split between the left and right back wheels, giving a much more accurate picture of what the car is doing.

3. The "Moving Horizon" Detective

How does the computer calculate all this? It uses a method called Moving Horizon Estimation.

The Analogy: Imagine you are a detective trying to solve a crime, but you only have a 10-second video clip of the scene. You look at the last 10 seconds of footage, try to figure out what happened, and then you slide your window forward one second. You drop the oldest second and add the newest one.
The Benefit: This allows the computer to look back at recent history to smooth out "noise" (like bumps in the road or sensor glitches) while still reacting instantly to new events. It's like watching a movie in real-time but constantly re-evaluating the last few seconds to make sure your guess about the plot is right.

4. The "Magic Formula" for Grip

To know how much grip the tires have, the system uses a mathematical recipe called the Pacejka Magic Formula.

The Analogy: Think of tire grip like a rubber band. If you stretch it a little, it pulls back hard. If you stretch it too far, it snaps (the tire slides). The computer uses this formula to guess exactly how "stretched" the rubber band is based on how fast the car is turning and how hard it's accelerating.
The Innovation: RAGE-XY doesn't just use a static recipe; it learns on the fly. If the tires get hot or the road gets wet, the system adjusts its "recipe" parameters in real-time, just like a chef tasting a sauce and adding more salt as needed.

The Result

The authors tested this on a real autonomous race car (the EAV-24) in Abu Dhabi, driving at speeds up to 150 mph with extreme turns.

Without RAGE-XY: The car might think it's sliding when it isn't, or miss a chance to accelerate because it thinks the tires are slipping.
With RAGE-XY: The car knows exactly how much force is on every tire, even if the sensors are slightly crooked or the road conditions change.

In short: RAGE-XY is like giving a race car a super-intelligent co-pilot that can "feel" the road through cheap sensors, automatically fixes its own vision if it gets crooked, and understands the complex physics of how power is split between wheels, all while driving at breakneck speeds.

1. Problem Statement

Autonomous race cars require precise knowledge of vehicle dynamics (velocity, slip angles, and tire forces) to execute aggressive maneuvers safely. However, direct measurement of lateral velocity and tire slip angles typically requires expensive, high-precision instrumentation (e.g., optical sensors, high-end GNSS/INS) that is often unavailable or impractical for standard onboard systems.

Existing model-based estimation techniques often rely on simplified kinematic models or require pre-identified tire parameters that do not adapt to changing conditions (e.g., tire wear, temperature, road surface). Furthermore, many hierarchical estimation approaches suffer from error propagation between sequential stages. The specific challenges addressed in this work include:

Estimating both longitudinal and lateral tire forces simultaneously using only standard sensors (IMU and RADAR).
Compensating for sensor misalignment (installation errors and mechanical tolerances) which severely degrades lateral velocity estimation, especially at high speeds.
Modeling complex drivetrain dynamics, specifically the Limited-Slip Differential (LSD), to accurately distribute rear longitudinal forces.

2. Methodology

The proposed framework, RAGE-XY, is an extension of the original RAGE estimator. It utilizes a Moving Horizon Estimation (MHE) approach to jointly infer the vehicle state and tire forces in real-time.

A. System Architecture & State Estimation

Framework: The problem is formulated as an MHE optimization, minimizing a cost function based on residuals from prior information, system dynamics, and sensor measurements. This allows the system to handle asynchronous and delayed RADAR data effectively.
State Vector: Includes longitudinal/lateral velocities ( $v_x, v_y$ ), accelerations ( $a_x, a_y$ ), yaw rate ( $r$ ), and IMU biases ( $b_x, b_y, b_r$ ).
Dynamics Model:
- Lateral Dynamics: Uses a single-track (bicycle) model combined with the Pacejka Magic Formula to estimate lateral tire forces ( $F_{y,f}, F_{y,r}$ ) based on slip angles and vertical loads.
- Longitudinal Dynamics: Extends the model to a tricycle representation to capture rear axle force distribution. It incorporates angular momentum conservation for the driven wheels and the vehicle body.
- LSD Modeling: Introduces a split coefficient ( $\gamma$ ) and a slack variable ( $\Delta M$ ) to model torque redistribution caused by the Limited-Slip Differential, rather than assuming equal force distribution.

B. Sensor Fusion & Inputs

Sensors: The system fuses data from an IMU (accelerations, yaw rate) and multiple RADARs (Doppler velocity, azimuth, elevation).
Input Reconstruction: Instead of treating IMU measurements as ground truth, the framework treats accelerations and yaw rate as inputs to be estimated alongside biases. This filters out noise and external disturbances (e.g., road bumps).
Online RADAR Calibration: A critical component is the ROLEQ estimator. During straight-line driving, the system assumes lateral velocity is negligible. It uses this constraint to solve for the optimal rotation correction ( $\Delta R$ ) between the RADAR frame and the vehicle body frame, compensating for pitch and yaw misalignment in real-time.

C. Force Estimation Logic

Longitudinal Forces: The total rear longitudinal force ( $F_{x,r}$ ) is derived by compensating the measured longitudinal acceleration for resistive forces (aerodynamic drag, rolling resistance, cornering forces, and road slope).
Force Distribution: The total force is split between left and right rear wheels using the estimated LSD coefficient ( $\gamma$ ), allowing for the estimation of individual wheel forces despite the lack of direct wheel torque sensors.

3. Key Contributions

Unified Joint Estimation: Unlike hierarchical methods that estimate states and forces sequentially, RAGE-XY jointly estimates the full vehicle state vector and both lateral and longitudinal tire forces in a single optimization step, reducing error propagation.
Online RADAR Calibration: Introduces a real-time calibration module that corrects for sensor misalignment (rotation errors), significantly improving lateral velocity accuracy without requiring manual recalibration.
Tricycle Model with LSD: Extends the vehicle model from a single-track approximation to a tricycle model, enabling the estimation of rear longitudinal force distribution via a lightweight Limited-Slip Differential model.
Adaptive Parameter Identification: The framework includes online identification of lateral tire model parameters (Pacejka coefficients), allowing adaptation to changing grip conditions, tire wear, and thermal effects.
Low-Cost Sensor Dependency: Demonstrates that high-fidelity state and force estimation is achievable using only standard IMUs and RADARs, avoiding the need for expensive optical or high-precision GNSS systems.

4. Results and Validation

The method was validated through two primary channels:

High-Fidelity Simulation:
- Calibration Test: In a simulation with a 15° pitch and 5° yaw misalignment, the uncalibrated estimator showed significant bias in lateral velocity. The online calibration activated after 5 seconds of straight-line driving and rapidly converged the estimate to the ground truth, validating the robustness of the calibration module.
- Performance: The system successfully estimated states and forces under extreme conditions (up to 70 m/s and 28 m/s² lateral acceleration).
Real-World Experiments:
- Platform: Validated on the EAV-24 autonomous race car (Unimore Racing Team) during the Abu Dhabi Autonomous Racing League (A2RL).
- Outcome: The system demonstrated improved accuracy and robustness in estimating both lateral and longitudinal dynamics in real-world racing scenarios, handling high-speed maneuvers and varying track conditions.

5. Significance

This work represents a significant advancement in autonomous racing perception and control. By enabling the estimation of longitudinal tire forces (crucial for traction control and braking) alongside lateral forces using only standard, low-cost sensors, RAGE-XY lowers the barrier to entry for high-performance autonomous vehicle development.

The introduction of online sensor calibration addresses a major practical hurdle in deploying autonomous systems, where mechanical tolerances and vibrations often degrade sensor alignment over time. Furthermore, the unified MHE approach provides a more robust and physically consistent estimation framework compared to sequential or purely data-driven methods, making it highly suitable for safety-critical, aggressive driving applications.