⚡ electrical engineering

PCT-Based Trajectory Tracking for Underactuated Marine Vessels

This paper proposes a novel trajectory tracking control strategy for underactuated marine vessels that utilizes polar coordinate transformations to simplify the system model and introduces an exponential modification of orientation to address inherent singularities and strict-feedback limitations.

Original authors: Ji-Hong Li

Published 2026-04-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ji-Hong Li

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 steer a large, heavy boat across a lake. You have a steering wheel (rudder) and an engine (thrust), but you have a problem: you can't push the boat sideways. If you try to move directly sideways, the boat just slides a little and then turns. This makes steering very tricky, especially if you need to follow a specific path perfectly.

This paper is about a new "smart steering system" designed to help these underpowered boats follow a path exactly, even when things get mathematically messy. Here is the breakdown using simple analogies:

1. The Problem: The "Sideways Slide"

Most boats are underactuated. Think of it like a shopping cart with a broken wheel that only rolls forward and turns, but never slides sideways.

The Challenge: You want the boat to follow a line drawn on a map. But because the boat can't move sideways, if you try to correct your path too sharply, the boat might get stuck or spin out of control.
The Old Way: Previous methods tried to fix this by assuming the boat is always moving forward fast. If the boat stops or moves too slowly, the math breaks down (like a GPS losing signal when you stop moving).

2. The Solution: Changing the Map (Polar Coordinates)

The authors decided to stop looking at the boat's position using a standard grid (like a city map with X and Y streets). Instead, they switched to a Polar Coordinate system.

The Analogy: Imagine you are a radar operator. Instead of saying "The boat is 5 blocks North and 3 blocks East," you say, "The boat is 6 miles away at a 45-degree angle."
Why do this? It simplifies the math. It turns a complicated 3-part puzzle (forward, sideways, turning) into a simpler 2-part puzzle (distance to target, angle to target).

3. The "Ghost Target" Trick (EMO)

Here is the cleverest part. When using the radar view, there is a dangerous spot: if the boat is exactly on the target, the "angle" becomes undefined (it's like asking "which way is North if you are standing on the North Pole?"). The math explodes.

To fix this, the authors invented EMO (Exponential Modification of Orientation).

The Analogy: Imagine you are playing a game of "Hot and Cold" to find a hidden treasure. Instead of aiming directly at the treasure (which causes the angle to glitch), you aim at a "Ghost Target" slightly to the side.
How it works: You steer toward this Ghost Target. As you get closer to the real treasure, the Ghost Target slowly slides over to join the real one. This way, you never get stuck at the "North Pole" of the math. You glide smoothly into the perfect spot without ever hitting a singularity.

4. The "Stop-Start" Problem (Surge Speed)

The old math required the boat to never stop moving forward. If the boat slowed down too much, the steering system would fail. The authors wanted to fix this so the boat could stop or move slowly. They proposed two ways to do it:

Method 1: The "Safety Buffer" (Relaxed Condition)
- Analogy: You know your boat has a limit on how fast it can slide sideways (drift). If you promise to keep your forward speed just a little bit faster than your maximum possible drift, you are safe.
- Result: This is a practical rule. It doesn't require the boat to be super fast, just fast enough to overcome its own wobble.
Method 2: The "Virtual Wall" (Control Barrier Functions)
- Analogy: Imagine an invisible wall around the "danger zone" where the boat stops moving. The computer is programmed to treat this wall like a physical barrier. If the boat tries to slow down too much, the computer automatically slams the engine to push it back to safety.
- Result: This is a high-tech, mathematical guarantee that the boat will never stop, even if the wind or waves try to push it.

5. The Results: A Smooth Ride

The authors ran computer simulations (like a video game test) to see if this worked.

The Test: They made the boat follow a winding path, sometimes moving fast, sometimes slow.
The Outcome: The new system worked perfectly. The boat followed the path tightly, even when the math got tricky.
The Catch: Even with the new "Ghost Target" trick, the boat still needs to maintain a minimum speed to stay stable. If the boat tries to move too slowly (like trying to drive a car in first gear on a steep hill), the system struggles. But for normal sailing speeds, it's a huge improvement.

Summary

This paper is like inventing a new navigation app for boats that:

Changes the map to make the math easier.
Uses a "Ghost Target" to avoid getting stuck at the center of the path.
Adds safety rules so the boat doesn't need to be a speedboat to work, just fast enough to stay steady.

It allows underpowered, tricky boats to follow complex paths with the precision of a guided missile, without crashing into mathematical dead ends.

1. Problem Statement

The paper addresses the trajectory tracking problem for underactuated marine vessels. These systems are characterized by having only two control inputs (surge force $\tau_u$ and yaw moment $\tau_r$ ) to regulate motion in three degrees of freedom (surge $u$ , sway $v$ , and yaw $r$ ).

Key challenges identified in the literature include:

Non-minimum phase dynamics: When the vessel's lift effect is non-zero, standard backstepping techniques struggle.
Coordinate Singularities: Traditional approaches using polar coordinates introduce singularities where the polar angle is undefined (at the origin) or where control gains vanish (when surge speed $u=0$ ).
Restrictive Assumptions: Previous works (e.g., [6], [21]–[24]) required the assumption that the vessel's surge speed $u(t)$ must remain strictly positive ( $u(t) > 0$ ) for all time to avoid singularities in the sideslip angle calculation and control gain inversion. This limits the applicability of controllers during low-speed maneuvers or stops.

2. Methodology

The proposed solution integrates Polar Coordinate Transformations (PCTs) with a novel Exponential Modification of Orientation (EMO) and Control Barrier Functions (CBFs).

A. System Reformulation via PCTs

The author transforms the original two-input-three-output second-order model (in Cartesian coordinates) into a two-input-two-output feedback system using two sequential polar coordinate transformations:

Body-Fixed PCT: Converts surge/sway velocities $(u, v)$ into total speed $u_l$ and sideslip angle $\psi_a$ .
Navigation-Fixed PCT: Converts position errors $(x_e, y_e)$ into radial distance $p_e$ and azimuth angle $\psi_b$ .

This reduces the tracking error dynamics to a form suitable for backstepping but introduces the risk of singularities when $\cos(\psi_l - \psi_b) = 0$ or when $u_l = 0$ .

B. Exponential Modification of Orientation (EMO)

To handle the singularity caused by $\cos(\psi_l - \psi_b) = 0$ (where the virtual control input becomes ineffective), the paper introduces the EMO concept.

Instead of tracking the reference trajectory $(u_{ld}, \psi_{ld})$ directly, the controller steers the vessel to track a modified trajectory $(u^m_{ld}, \psi^m_{ld})$ .
The modification is designed such that:
1. The position error $p_e$ converges exponentially to zero.
2. As $p_e \to 0$ , the modified trajectory asymptotically approaches the original reference.
This ensures that the term $\cos(\psi^m_{ld} - \psi_b)$ never vanishes, thereby avoiding the singularity in the recursive controller design.

C. Relaxing the Positive Surge Speed Assumption

The paper proposes two distinct methods to address the critical assumption that $u(t) > 0$ (required to define the sideslip angle and ensure invertibility of the control matrix):

Method 1: Relaxed Condition (Analytical Approach)
- Instead of assuming $u(t) > 0$ a priori, the paper derives a condition based on the boundedness of the sway velocity ( $|v(t)| \leq v_M$ ).
- By ensuring the reference surge speed $u_{ld}$ is sufficiently large relative to the sway bound and control gains, it is mathematically proven that $u(t)$ will eventually become and remain positive after a finite time $t_c$ .
- This approach is less restrictive than previous works but still requires a minimum forward speed threshold.
Method 2: CBF-Based Technique (Constraint-Based Approach)
- This method completely removes the assumption of strictly positive surge speed by incorporating a Control Barrier Function (CBF).
- A barrier function $h(\nu) = u - \delta$ is defined to enforce a safety margin $\delta > 0$ .
- The control input $\tau_u$ is computed by solving a Convex Quadratic Program (QP) that minimizes the deviation from the nominal control law while satisfying the CBF constraint ( $\dot{h} \geq -\alpha(h)$ ).
- This guarantees $u(t) > \delta$ for all time, effectively eliminating the singularity without requiring a pre-defined high-speed reference, though it introduces computational overhead and potential conservatism.

3. Key Contributions

EMO Framework: Extension of the Asymptotic Modification of Orientation (AMO) to an Exponential version (EMO), guaranteeing exponential stability of the closed-loop system in the absence of uncertainties.
Singularity Mitigation: A rigorous solution to the polar coordinate singularity at the origin and the control gain singularity at zero surge speed.
Relaxation of Surge Speed Constraints:
- Provided a theoretical proof that strictly positive surge speed is not a hard requirement if sway speed is bounded and reference speed is sufficiently high.
- Developed a CBF-based QP formulation to completely eliminate the positive surge speed assumption, allowing for more flexible maneuvering.
Unified Controller: A general backstepping controller applicable to both minimum-phase and non-minimum phase vessel dynamics.

4. Simulation Results

Numerical simulations were conducted on an underactuated marine vessel model with:

Nonlinear dynamics (hydrodynamic damping, added mass).
Unmatched uncertainties (model errors, disturbances).
Reference Trajectories: Including straight lines and curved paths with varying speeds.

Key Findings:

Tracking Performance: Both methods achieved exponential convergence of position and orientation errors to a small compact set.
Low-Speed Performance: When the reference surge speed was reduced (e.g., $u_{ld} = 1.3$ $u_{l d} = 1.3$ m/s):
- Method 1 (Relaxed Condition) still performed well, suggesting the derived sufficient condition (Eq. 30) was conservative.
- Method 2 (CBF) failed to track effectively at very low speeds due to the strict enforcement of the safety margin $\delta$ , highlighting a trade-off between safety guarantees and tracking agility at low speeds.
Control Effort: Method 2 required higher control effort compared to Method 1 when the system was near the safety boundary.

5. Significance and Conclusion

This paper significantly advances the state-of-the-art in underactuated marine vessel control by:

Removing theoretical bottlenecks: It challenges the long-standing necessity of assuming strictly positive surge speed, a condition that often limits practical application in docking or low-speed operations.
Enhancing Robustness: The integration of EMO and CBFs provides a robust framework against singularities and uncertainties.
Practical Trade-offs: The study highlights the trade-off between the analytical simplicity of Method 1 (which requires a minimum speed threshold) and the rigorous safety of Method 2 (which guarantees speed positivity but may struggle with low-speed tracking or high computational load).

The proposed strategy offers a unified, theoretically rigorous, and practically adaptable solution for trajectory tracking, paving the way for more autonomous and flexible marine operations. Future work suggested includes refining the EMO scheme to better attenuate noise sensitivity when position errors are small.