⚡ electrical engineering

Global Exponential Stabilization of the Kinematic Bicycle Model of a Car in Polar Coordinates

This paper proposes a novel control strategy that achieves global exponential stabilization of the kinematic bicycle model for car-like vehicles by transforming the system into polar and range-normalized coordinates to bypass Brockett's obstruction, thereby enabling smooth feedback laws that generate realistic, human-like parking maneuvers.

Original authors: Velimir Todorovski, Kwang Hak Kim, Alessandro Astolfi, Miroslav Krstic

Published 2026-07-30

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Velimir Todorovski, Kwang Hak Kim, Alessandro Astolfi, Miroslav Krstic

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 trying to teach a robot to park a car. You might think, "Easy! Just tell the robot to drive forward, turn the wheel, and stop." But here's the catch: a car is a bit stubborn. Unlike a shopping cart that can slide sideways or a drone that can hover in place, a car can only move in the direction its wheels are pointing. It cannot slide sideways, and it cannot turn while standing still. In the world of physics and engineering, this is called a "nonholonomic" system. For decades, mathematicians have known that you can't simply write a smooth, steady set of rules to park such a car perfectly if you try to do it using standard map coordinates (like a grid on a piece of paper). It's like trying to push a heavy box up a slippery hill with a rope that keeps snapping; the math says it's impossible to do it smoothly without the car getting stuck or wobbling forever.

This is where the story gets interesting. Scientists have been trying to find a way to make these cars park as smoothly as a human driver does—gliding in, adjusting the angle, and sliding into the spot without jerky, robotic movements. The big question has been: How do we get a car to park perfectly using only the car's own sensors and a computer brain, without needing a human to take over or a super-complex computer to calculate every possible move? The answer lies in changing the way we look at the problem. Instead of seeing the car on a flat grid, what if we looked at it from the perspective of a radar gun or a lighthouse, measuring how far away the car is and the angle it's coming from? This paper dives deep into that idea, using a special kind of math to trick the car into parking itself naturally.

The Paper's Big Idea: Parking Like a Human, Thinking Like a Mathematician

In this paper, the authors tackle the tricky problem of parking a car-like vehicle (specifically, a "kinematic bicycle model," which is a fancy way of describing a car that moves like a bike with two wheels) at low speeds. They want to create a set of instructions (a control law) that will guide the car from anywhere on the road into a perfect parking spot, stopping exactly where it needs to be with the wheels straight.

The authors start by pointing out a major roadblock. If you try to park a car using standard X and Y coordinates (like a graph on a math test), you run into a mathematical wall. A famous rule in physics, known as Brockett's condition, basically says you can't use a smooth, unchanging set of rules to park this kind of car perfectly. If you try, the car will either spin in circles, shake back and forth, or take forever to stop. Most existing solutions try to work around this by making the rules change over time or by using jagged, "on-off" switches, but these often result in parking maneuvers that look weird and robotic, not like how a human driver would do it.

The Solution: A New Way to See the World

To get around this wall, the authors decided to stop looking at the car on a flat map and start looking at it through "polar coordinates." Imagine you are standing at the center of a dartboard. Instead of saying the dart is "3 inches right and 4 inches up," you say it is "5 inches away at a 53-degree angle." This change in perspective is crucial. In this new view, the mathematical wall disappears, allowing for smooth rules.

However, just switching to angles and distance wasn't enough. The authors realized that to make the car park like a human, the math needed to understand how the car slows down and turns as it gets closer to the spot. A human driver doesn't just drive straight and then spin the wheel at the last second; they start turning early and slow down gradually.

To capture this, the team invented something they call "range-normalized coordinates." Think of this as a special pair of glasses that makes the car's speed and turning angle look different depending on how close it is to the parking spot.

When the car is far away, the "glasses" make the car look fast and the turning angle look small.
As the car gets closer to the spot, the "glasses" automatically adjust so that the car's speed and angle are measured relative to the tiny distance left.

This is the secret sauce. It forces the car to naturally slow down and align its wheels before it reaches the spot, just like a human driver does. If the car is far away, it can drive fast. But as it gets close, the math demands that it slow down and straighten out, because the "distance left" is getting tiny.

The Magic Trick: Backstepping

Once they had this new way of looking at the car, the authors used a mathematical technique called "backstepping." Imagine you are trying to balance a broom on your hand. You don't just look at the top of the broom; you look at the handle, then your wrist, then your shoulder, adjusting each part to keep the whole thing upright. Backstepping works the same way. The authors built their parking rules step-by-step:

First, they figured out how to control the distance to the spot.
Then, they added rules to control the angle, making sure the car turns at the right time.
Finally, they tied it all together with rules for speed and acceleration.

By doing this, they created a set of smooth, continuous instructions that tell the car exactly how much to accelerate and how much to turn the wheel at every single moment.

What They Found

The authors proved mathematically that their new rules work. They showed that no matter where the car starts (as long as it's not already exactly on the spot but facing the wrong way, which is a rare edge case), the car will eventually park itself.

Global Exponential Stability: This is a fancy way of saying the car will get to the spot quickly and smoothly, and the closer it gets, the more perfectly it settles in. It doesn't just wander around; it zooms toward the goal and stops.
Human-Like Maneuvers: When they simulated the car parking, the results looked exactly like a human driver. The car would drive forward, turn the wheel to angle itself, and then back up or drive forward to slide perfectly into the spot. It didn't look like a robot shaking back and forth; it looked like a smooth, confident parking job.
The Catch: The paper notes that if the car is already sitting in the exact parking spot but facing the wrong direction, the math says it needs to back up a little bit first to fix its angle. This is actually true for real drivers too! You can't just spin a car in place to face the right way; you have to move it.

The Proof

The authors didn't just guess this would work; they wrote a rigorous mathematical proof showing that their system is stable. They also ran computer simulations to show what the car would actually do. In these simulations, they tested the car with different starting positions and speeds. The car successfully parked itself in every scenario, following smooth paths that looked very natural. The inputs (the commands to the gas pedal and steering wheel) were also smooth, meaning the car wouldn't jerk or shake.

Why This Matters

This paper is a big deal because it's the first time someone has successfully used this specific type of math (polar coordinates with range-normalization) to create a smooth, global parking solution for a car model that isn't "driftless" (meaning the car has momentum and acceleration, not just instant speed). Previous methods either couldn't handle the car's momentum or resulted in jerky, unnatural movements.

By showing that you can park a car smoothly using only feedback (the car's own sensors) and without needing complex, time-changing rules, this work opens the door for better self-driving cars. It suggests that if we teach robots to "see" the world in terms of angles and relative distances, rather than just X and Y coordinates, they might learn to drive and park much more like humans do. The authors also mention that this approach could be used for other tricky vehicles, like tractor-trailers, in the future.

In short, the paper shows that by changing the way we look at the problem—from a flat map to a radar-like view—and by teaching the car to respect the "distance left" as it slows down, we can finally get a robot to park a car as smoothly as a human driver.

Technical Summary: Global Exponential Stabilization of the Kinematic Bicycle Model in Polar Coordinates

Problem Statement
The kinematic bicycle model is the standard representation for car-like vehicles at parking speeds. However, stabilizing this system presents a fundamental theoretical challenge known as Brockett's condition, which precludes the existence of smooth, static, time-invariant feedback laws that can asymptotically stabilize nonholonomic systems in Cartesian coordinates. While existing literature offers solutions using time-varying or discontinuous feedback, these often result in oscillatory trajectories, slow convergence, or limited regions of attraction. Furthermore, standard approaches often rely on driftless models where velocity is directly actuated, whereas realistic parking scenarios involve longitudinal acceleration and steering angle as inputs, rendering the system non-driftless and incompatible with chained-form transformations. Additionally, existing methods struggle to reproduce the smooth, human-like parking maneuvers observed in practice.

Methodology
The authors propose a control design that bypasses topological obstructions by transforming the system from Cartesian to polar coordinates, augmented by a novel set of "range-normalized" coordinates. The methodology proceeds through the following steps:

Coordinate Transformation: The system is transformed into polar coordinates $(\rho, \delta, \gamma)$ , where $\rho$ is the distance to the target, $\delta$ is the polar angle, and $\gamma$ is the line-of-sight (LoS) angle. This formulation avoids the singularity issues inherent in Cartesian stabilization for nonholonomic systems.
Strict-Feedback Formulation: The dynamics are rewritten to expose a strict-feedback structure. This involves introducing bounded auxiliary functions ( $\psi_1, \psi_2$ ) and an auxiliary variable $z$ to decouple trigonometric terms.
Range-Normalized Coordinates: To encode the geometry of human-like parking—where the vehicle aligns its heading as it approaches the target rather than after reaching it—the authors introduce normalized states: $\bar{\delta} = \delta/\rho$ , $\bar{z} = z/\rho$ , and $\bar{v} = \hat{v}/\rho$ . These coordinates ensure that angular errors and velocity scale appropriately with the remaining distance $\rho$ .
Backstepping Design: A nonconventional backstepping design is applied to the transformed system. The control laws for longitudinal acceleration ( $a$ ) and steering angle ( $\phi$ ) are constructed to stabilize the range-normalized system.
Stability Analysis: A Control Lyapunov Function (CLF) is defined for the normalized states. The authors prove that the closed-loop system achieves global exponential stability in the transformed coordinates.

Key Contributions

First of its Kind: This work presents the first globally exponentially output-stabilizing control laws for the kinematic bicycle model in polar coordinates that rely solely on feedback to generate human-like parking trajectories.
Smooth Feedback Laws: Unlike previous methods that require discontinuous or time-varying feedback, this approach yields smooth, static feedback laws.
Handling Non-Driftless Dynamics: The design addresses the specific case where longitudinal acceleration and steering angle are the inputs, avoiding the limitations of driftless models that cannot be converted to chained form.
Range-Normalization: The introduction of range-normalized coordinates allows the controller to naturally handle the singularity at the target ( $\rho=0$ ) and enforce the physical intuition that a vehicle must slow down and align as it nears the parking spot.

Results

Global Exponential Output Stability: The proposed control laws achieve global exponential stability for the polar coordinates $(\rho, \delta, \gamma, v)$ and the normalized states.
Cartesian Attractivity: In Cartesian coordinates, the system exhibits global uniform Lagrange asymptotic stability (GULAS). This means trajectories are globally bounded and converge exponentially to the equilibrium, though the equilibrium is not Lyapunov stable in the Cartesian variables due to the inherent nonholonomic constraints.
Small Control Property: The control inputs (acceleration and steering) satisfy the small control property, remaining bounded and converging to zero as the system approaches the target, although transient values may be large for specific initial conditions (e.g., high speed close to the target).
Simulation: Numerical simulations of parallel parking maneuvers demonstrate that the generated trajectories closely resemble human driving behavior, with smooth transitions and appropriate speed profiles.

Significance and Claims
The paper claims to provide a geometric solution to the parking problem that overcomes the Brockett obstruction without resorting to discontinuous or time-varying controls. By leveraging the geometry of polar coordinates and the specific structure of range-normalized variables, the authors demonstrate that it is possible to achieve global exponential stabilization in a transformed domain that maps to realistic, smooth parking maneuvers in the physical domain. The work establishes a foundation for future safety extensions, optimality-based designs utilizing the derived CLF, and the application of similar control laws to more complex systems like tractor-trailers. The authors emphasize that this is a theoretical advancement in continuous-time geometric control, distinct from discrete optimization-based methods like Model Predictive Control.

The Paper's Big Idea: Parking Like a Human, Thinking Like a Mathematician

Technical Summary: Global Exponential Stabilization of the Kinematic Bicycle Model in Polar Coordinates

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