⚡ electrical engineering

Finite-Time Barrier Function-Based Sliding Mode Control for Trajectory Tracking of Autonomous Sailboat

This paper proposes a novel finite-time barrier function-based sliding mode control strategy that ensures rapid, safe, and accurate trajectory tracking for autonomous sailboats by guaranteeing finite-time convergence and explicitly constraining system states within predefined bounds despite nonlinear dynamics and environmental disturbances.

Original authors: Mohammad Reza Jalili, Ahmad Reza Vali, Mohammad Ali Alirezapouri, Hossein Nourmohammadi

Published 2026-07-20

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohammad Reza Jalili, Ahmad Reza Vali, Mohammad Ali Alirezapouri, Hossein Nourmohammadi

Original paper licensed under CC BY 4.0 (https://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 steer a very stubborn, invisible boat across a stormy ocean, but you can't see the wind and the waves are trying to push you off course. This is the daily reality for autonomous sailboats. Unlike motorboats that have powerful engines to fight back, sailboats are like dancers; they must use the wind to move, which makes them incredibly energy-efficient but also very sensitive to the weather. If a gust hits them unexpectedly, they can spin out of control or even capsize. To keep them safe, scientists use something called Sliding Mode Control. Think of this as an invisible, super-strong leash that constantly yanks the boat back onto its intended path whenever it tries to drift. However, traditional leashes are a bit slow; they only promise to get the boat back on track "eventually," which in the world of fast-moving storms might be too late.

This paper tackles that "eventually" problem. The researchers wanted to create a control system that doesn't just wait for the boat to settle down, but forces it to snap back to the correct path in a guaranteed, short amount of time. They also wanted to make sure the boat never gets too close to the edge of a cliff (or in this case, a dangerous tilt). By combining a "fast-reaching" strategy with a "safety barrier," they built a digital autopilot that is both lightning-fast and incredibly careful, ensuring the sailboat stays safe even when the weather gets wild.

The Race Against the Wind: A New Way to Steer

In the world of autonomous sailboats, the wind is both the engine and the enemy. It pushes the boat forward, but it also pushes it sideways or makes it roll over. The boat has two main "steering wheels": the sail (which catches the wind) and the rudder (which turns the boat in the water). The goal is to keep the sail at the perfect angle and the boat heading in the right direction, even when the wind suddenly changes or waves hit.

The authors of this paper, Mohammad Reza Jalili and his team from Malek Ashtar University of Technology, realized that the old ways of steering these boats were too slow. Imagine you are driving a car, and you see a pothole. If your steering system only promises to fix your path "eventually," you might crash before it happens. The team wanted a system that says, "I will fix your path in exactly 1.2 seconds, no matter what."

To do this, they invented a new type of Sliding Mode Control (SMC). You can think of SMC as a bouncer at a club who constantly checks if you are on the "sliding surface" (the perfect path). If you step off, the bouncer pushes you back. The problem with old bouncers is that they push gently at first and only get aggressive as you get closer to the line, which takes a long time. The team's new method uses a "Fast Reaching Law." This is like a bouncer who sees you stepping off the line and immediately grabs you with a firm, calculated yank to get you back on track instantly. They proved mathematically that this new "yank" gets the boat back on the right path much faster than previous methods.

But speed isn't everything; safety is crucial. If the boat tilts too far, it could flip over. To prevent this, the team added a Barrier Function. Imagine a magical, invisible wall surrounding the boat. As the boat gets closer to this wall (the danger zone), the wall gets stronger and stronger, pushing the boat back before it ever touches the edge. This ensures that even if the wind is crazy, the boat never tilts beyond a safe limit. The beauty of their design is that this "wall" doesn't need to know exactly how hard the wind is blowing; it just reacts to how close the boat is to the danger zone.

The Results: Faster, Smoother, Safer

The team didn't just write equations; they tested their idea in a computer simulation using MATLAB/Simulink. They created a virtual sailboat and threw some tough scenarios at it, including a sudden wind disturbance at 8 seconds into the test.

Here is what happened:

Speed: The new controller was a clear winner. When they measured how long it took for the boat to stop wobbling and get back on course (the "rise time"), the new method was 39.4% faster than the old standard method. For the sail angle, it took just 1.2 seconds to settle, compared to 2 seconds for the older method.
Accuracy: The boat didn't just get there fast; it got there precisely. The "RMS error" (a fancy way of saying "how far off the path the boat was on average") dropped by 38.9%. The new controller kept the boat much closer to the perfect line.
Smoothness: Sometimes, trying to steer too fast makes the controls jerk around wildly (like a car shaking). The team's "Barrier Function" helped smooth this out. The controls remained steady and didn't waste energy on jerky movements.
Safety: The simulation showed that the boat's path stayed strictly within the "safe zone" defined by the barrier function, proving that the boat would not capsize even during the storm.

The researchers compared their new Finite-Time Barrier Function-Based Sliding Mode Control against a well-known older method called ERLSMC. The results were in the table they provided: the new method won in almost every category, from how fast it reacted to how little energy it wasted.

What This Means

This paper suggests that we can make autonomous sailboats much more reliable. By using a control system that guarantees a fast return to the path and strictly enforces safety limits, these boats could potentially operate for longer missions in rougher weather without human help. The authors note that while their results are based on simulations, the math behind it is solid, and the "fast-reaching" law is proven to be strictly quicker than the older methods they tested against.

They didn't claim to have solved every problem in the world of sailing robots, but they did show a significant step forward. They proved that you can have both speed and safety in the same control system. As they mentioned in their conclusion, the next step for this technology might be to use it for groups of sailboats working together or to teach the boat to learn about the water conditions in real-time, but for now, this new "fast and safe" steering method is a major upgrade for the future of autonomous sailing.

Technical Summary: Finite-Time Barrier Function-Based Sliding Mode Control for Trajectory Tracking of Autonomous Sailboat

Problem Statement
Autonomous sailboats face significant control challenges due to their inherent nonlinear dynamics, underactuated nature (relying on wind and rudder for propulsion and steering), and high sensitivity to unknown environmental disturbances. While Sliding Mode Control (SMC) is a robust technique for handling uncertainties, conventional SMC guarantees only asymptotic convergence, meaning the system theoretically reaches the desired state only as time approaches infinity. Furthermore, existing Finite-Time Sliding Mode Control (FTSMC) designs often fail to explicitly enforce safety constraints during the transient reaching phase. In real-world scenarios, overshoot or state violations can lead to critical failures such as capsizing or collisions. There is a noted gap in the literature regarding control frameworks that simultaneously ensure rapid finite-time convergence and rigorous state constraints for the highly nonlinear dynamics of autonomous sailboats.

Methodology
The authors propose a novel Finite-Time Barrier Function-Based Sliding Mode Control (FTBF-SMC) strategy tailored for simultaneous heading and sail opening angle control. The methodology is structured around three core components:

System Modeling: The sailboat is modeled as a 4-DOF rigid body system (surge, sway, roll, and yaw) using a state-space representation that accounts for aerodynamic and hydrodynamic forces. The model assumes constant added mass coefficients, rigid foils for sails and rudders, and linear damping, while treating external disturbances as bounded additive terms.
Novel Fast-Reaching Law: To address the convergence speed limitations of standard FTSMC, the authors integrate a new fast-reaching law based on finite-time terminal sliding mode concepts. This law utilizes a specific mathematical formulation involving a sigmoid function and logarithmic terms to analytically guarantee a strictly bounded convergence time, which is proven to be shorter than that of Extended Reaching Laws (ERL).
Barrier Function Integration: To ensure operational safety, a barrier function is embedded within the SMC framework. This function acts as a virtual boundary that approaches infinity as the system state nears predefined unsafe limits. The control law is designed in two cases:
- Outside the boundary layer ( $|s| \geq \epsilon$ ): The controller uses the fast-reaching law to drive the system toward the sliding surface.
- Inside the boundary layer ( $|s| < \epsilon$ ): A discontinuous control law utilizing the barrier function is employed to prevent the sliding variable from violating the safety bounds.
- A key feature of this approach is that it manages disturbances without requiring a priori knowledge of their upper bounds, as the barrier function inherently generates large control inputs to pull the system back when it nears the boundary.

Key Contributions
The paper outlines three primary contributions:

Development of a Novel Fast-Reaching Law: A new law is derived that guarantees strictly bounded convergence time, analytically demonstrating superior performance over conventional exponential and extended reaching laws.
Safety-Enforcing Control Design: The seamless integration of a barrier function into the SMC design enforces hard state constraints, ensuring the system remains within safe operational bounds without relying on simplistic assumptions about disturbance magnitudes.
Rigorous Stability Analysis: The authors provide a comprehensive stability proof using Lyapunov direct method and finite-time stability lemmas, establishing that the closed-loop system errors converge to the equilibrium point in finite time despite model uncertainties.

Results
The proposed controller was validated through MATLAB/Simulink simulations using a 4-DOF sailboat model with parameters typical of autonomous vessels. The simulations included external wind disturbances and compared the proposed FTBF-SMC against a standard Terminal SMC, a Terminal SMC with an enhanced reaching law, and the Extended Reaching Law SMC (ERLSMC) from existing literature.

Convergence Speed: The proposed method demonstrated significantly faster convergence. Specifically, the rise time for the sail opening angle error improved by approximately 39.4% compared to the ERLSMC benchmark.
Tracking Accuracy: The Root Mean Square (RMS) error for the sail opening angle was reduced by 38.9%, and the heading error RMS was reduced by 24.4% compared to the reference method.
Control Effort and Chattering: The control inputs remained smooth and bounded, effectively mitigating the high-frequency chattering often associated with discontinuous SMC. The proposed method achieved these performance gains without increasing actuator demand, showing a 13.8% reduction in control effort for the sail actuator compared to ERLSMC.
Safety: The simulation results confirmed that the tracking errors consistently remained within the predefined barrier function bounds throughout the operation, validating the safety constraints.

Significance and Claims
The paper claims that the proposed FTBF-SMC scheme successfully addresses the trade-off between convergence speed and chattering reduction while explicitly guaranteeing safety constraints. By combining a fast-reaching law with barrier functions, the method offers a robust solution for autonomous sailboats operating in uncertain environments. The authors assert that their approach provides a more favorable balance between speed, accuracy, and control smoothness than existing methods. The study concludes that this framework is a significant step forward for autonomous marine navigation, with future work suggested to extend the methodology to multi-agent formations and incorporate adaptive neural networks for real-time hydrodynamic coefficient estimation.

The Race Against the Wind: A New Way to Steer

The Results: Faster, Smoother, Safer

What This Means

Technical Summary: Finite-Time Barrier Function-Based Sliding Mode Control for Trajectory Tracking of Autonomous Sailboat

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