💻 computer science

Optimal Excitation Trajectories for System Identification of Underwater Vehicles

This paper proposes and experimentally validates a methodology for identifying underwater vehicle dynamics by designing optimal excitation trajectories using Bezier curves to maximize data quality while respecting safety and dynamic constraints.

Original authors: Fotis Panetsos, Kostas J. Kyriakopoulos

Published 2026-09-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Fotis Panetsos, Kostas J. Kyriakopoulos

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

Deep beneath the surface, where sunlight fades and pressure mounts, underwater vehicles are tasked with some of humanity's most delicate and dangerous work. From inspecting aging oil rigs and mapping shipwrecks to monitoring coral reefs and maintaining underwater cables, these machines are the eyes and hands of scientists and engineers in an environment that is hostile to human life. For these robots to operate effectively, they must be able to move with precision and predictability. However, the ocean is a fluid, shifting medium that resists motion in complex ways. To navigate it safely, a vehicle needs a precise internal map of its own physics—a mathematical understanding of how it responds to its own thrusters, how the water pushes back, and how its weight and buoyancy interact. Without this accurate model, a robot might drift off course, fail to hold its position, or misjudge a critical maneuver. The challenge lies in creating this model: the vehicle's behavior changes depending on its cargo, its sensors, and even the specific water conditions, making pre-programmed guesses unreliable.

To solve this, researchers at New York University Abu Dhabi have developed a new way to teach underwater robots about themselves. Instead of relying on expensive, time-consuming tests in massive industrial tanks or complex computer simulations that often miss the mark, the team designed a method where the robot performs a specific, carefully crafted series of movements to reveal its own dynamic secrets. By analyzing how the robot reacts to these specific motions, the team can calculate the exact physical properties that govern its movement. This process, known as system identification, allows the robot to build a personalized model of its own behavior, which can then be used to control it more accurately or to predict its future position with high confidence.

The core of this new approach lies in the design of the movement itself. In the past, engineers often tested underwater vehicles using simple, repetitive motions, like moving back and forth in a straight line or spinning in a circle. While these basic tests provide some data, they often fail to capture the complex, coupled nature of how a vehicle moves in three dimensions. When a robot turns, it doesn't just rotate; it might also drift sideways or change its depth slightly due to the physics of the water. Simple tests often miss these subtle interactions. The researchers realized that to get a complete picture, the robot needed to be "excited" in a more sophisticated way—a sequence of movements that would stress every part of its physical system just enough to reveal its true nature without pushing it to the point of failure.

To achieve this, the team turned to a mathematical tool called Bézier curves. Imagine a flexible wire that a designer can bend and shape by pulling on a few specific points; the curve between those points follows a smooth, predictable path. The researchers used this concept to design a trajectory, or a path, for the robot to follow. By adjusting the positions of these control points, they could create a smooth, continuous motion that was guaranteed to stay within safe boundaries, such as not hitting the walls of the testing tank or exceeding the speed limits of the motors. This method offered a flexibility that older techniques lacked, allowing them to craft a path that was not just a simple loop, but a complex, multi-stage journey designed specifically to probe the robot's physics.

The researchers then set up an optimization problem, essentially asking a computer to find the single best set of control points that would generate the most informative data. The goal was to create a path that would make the robot's movements as distinct and measurable as possible, ensuring that the data collected could be used to solve for the vehicle's physical parameters with high accuracy. They imposed strict safety rules on this optimization: the robot had to start and stop gently, stay within the dimensions of a laboratory water tank, and move at speeds that its motors could actually achieve. The resulting trajectory was a 280-second journey divided into five distinct segments. In the first four segments, the robot was guided to focus on one type of movement at a time—moving forward, moving sideways, rising, and turning—while keeping other movements minimal. In the final segment, all constraints were relaxed, allowing the robot to move freely in all directions simultaneously. This structure ensured that the researchers could first identify the basic, independent behaviors of the robot and then capture the more complex interactions that happen when all movements occur at once.

The team tested this methodology using a VideoRay Defender, a small, torpedo-shaped underwater vehicle equipped with seven thrusters. They placed the robot in a laboratory water tank measuring 6.3 meters by 3.7 meters by 1.8 meters. To track the robot's position with extreme precision, they placed a board with special visual markers on the bottom of the tank and used a downward-looking camera on the robot to watch them. As the robot followed the optimized path, the team recorded its position, speed, and the commands sent to its motors. They then used this data to calculate the robot's dynamic parameters, such as its mass, how the water resisted its motion, and how its weight was distributed.

The results were striking. When the researchers used the new model to predict how the robot would move on a completely different path, the predictions matched the actual measurements with remarkable accuracy. The model successfully forecasted the robot's speed and direction across a variety of maneuvers, including smooth, pre-planned paths and even sudden, jerky movements commanded by a human operator via a joystick. The only area where the model showed a slight discrepancy was in the turning motion, or yaw, which the researchers attributed to the physical tether connecting the robot to the surface, a factor that added unexpected drag and resistance. Despite this, the overall performance demonstrated that the identified model could reliably capture the vehicle's behavior.

This work represents a significant step forward in making underwater robots more autonomous and reliable. By replacing guesswork and generic testing with a structured, data-driven approach to understanding a vehicle's own physics, the researchers have provided a tool that can be applied to any modular underwater vehicle. Because the method relies on the robot's own sensors and a standard controller, it can be easily adapted for vehicles that carry different payloads or operate in different environments. The team envisions extending this work to include the full six degrees of freedom of movement, including the rolling and pitching motions that were simplified in this study, and eventually testing these models in the open ocean. Ultimately, this approach offers a way to ensure that the machines exploring our oceans are not just following commands, but truly understand how they move through the water.

Technical Summary: Optimal Excitation Trajectories for System Identification of Underwater Vehicles

Problem Statement
Accurate dynamic models are essential for the autonomy, control, and state estimation of underwater vehicles (UVs). While traditional methods for obtaining these models—such as captive model tests in towing tanks, Computational Fluid Dynamics (CFD) simulations, and analytical estimations—are often time-consuming, expensive, or reliant on difficult-to-obtain inertial properties, System Identification (SI) offers a practical alternative using onboard sensor data. However, existing SI approaches for UVs frequently rely on large datasets from extended field trials or basic input signals (e.g., step or sinusoidal inputs) with heuristically selected parameters. These methods often assume decoupled motion or known centers of mass, limiting the validity of the identified models to basic maneuvers and failing to capture the agile, coupled dynamic behavior of modern UVs. Furthermore, the selection of excitation trajectories is often neglected, despite its critical impact on the numerical conditioning of the parameter estimation problem.

Methodology
The authors propose a structured methodology to design optimal excitation trajectories specifically for the system identification of UVs. The core of the approach involves parameterizing the vehicle's pose trajectory using Bézier curves. This choice allows for the representation of smooth, continuous, and differentiable motion profiles while facilitating the direct enforcement of constraints through the manipulation of control points.

The methodology proceeds as follows:

Dynamic Modeling: The vehicle's dynamics are modeled for four degrees of freedom (DoF): surge, sway, heave, and yaw. The model is linearized with respect to a set of base parameters ( $\pi$ ), which include entries from the inertia matrix, the hydrodynamic damping matrix, and the weight-buoyancy difference.
Trajectory Parameterization: The excitation trajectory is defined as a multi-segment Bézier curve. Each segment is characterized by a set of control points. The trajectory is differentiated to obtain velocity and acceleration profiles required for the regressor matrix.
Constraint Formulation: To ensure dynamical feasibility and safety, constraints are imposed directly on the control points using the convex hull property of Bézier curves. These include:
- Pose Constraints: Ensuring the vehicle remains within the water tank boundaries.
- Velocity and Acceleration Constraints: Preventing infeasible maneuvers.
- Continuity Constraints: Enforcing $C^2$ continuity between segments for smooth transitions.
- Boundary Constraints: Ensuring the trajectory starts and ends at rest.
Optimization Problem: An optimization problem is formulated to minimize the condition number of the regressor matrix ( $\bar{Y}$ ), which is a key metric for the quality of the Least Squares (LS) solution. The optimization is subject to the linear equality and inequality constraints derived from the Bézier properties. The problem is solved using Sequential Quadratic Programming (SQP).
Experimental Validation: The optimized trajectory is executed by a VideoRay Defender vehicle in a laboratory water tank. A PID controller tracks the trajectory while maintaining near-zero roll and pitch. Dynamic parameters are estimated using LS. The identified model is then validated by forward simulating the vehicle's velocity and comparing it against measured data from both the identification trajectory and unseen cross-validation trajectories.

Key Contributions

Systematic Trajectory Design: Unlike prior works that rely on large datasets or basic inputs, this paper introduces a systematic framework for generating excitation trajectories tailored to UVs using Bézier curves.
Coupled Dynamics Identification: The framework enables the reliable identification of coupled 4-DoF dynamic parameters without relying on simplifying assumptions like fully decoupled motion.
Constraint Handling: The method leverages the properties of Bézier curves to enforce safety and operational constraints directly on control points, significantly reducing the computational complexity of the optimization compared to methods requiring constraints at every time sample.
Experimental Verification: The approach is experimentally validated in a controlled environment, demonstrating that the identified parameters can accurately predict vehicle velocity across various maneuvers, including those not used during the identification phase.

Results
The proposed methodology was validated using a VideoRay Defender in a 6.3m $\times$ 3.7m $\times$ 1.8m water tank.

Optimization: The optimization process yielded a trajectory with a regressor matrix condition number of 160.10. The trajectory consisted of five segments, where the first four sequentially excited individual DoFs (surge, sway, heave, yaw) and the final segment excited all DoFs simultaneously to capture coupled terms.
Identification Accuracy: Using the data from the optimized trajectory, dynamic parameters were estimated via LS. Forward simulation of the identified model showed a strong correlation with measured body velocities.
Cross-Validation: The identified model was tested against 15 distinct reference trajectories and a 10-minute teleoperation experiment involving abrupt joystick commands. In all cases, the simulated velocities closely matched the measured data.
- The $R^2$ values for surge, sway, and heave were high, indicating accurate prediction.
- The yaw rate exhibited lower $R^2$ values and wider distribution, which the authors attribute to the significant influence of the tether on yaw dynamics and unmodeled propeller-hull interactions.
Generalizability: The model successfully generalized to unseen trajectories and more demanding teleoperation scenarios, confirming the robustness of the identified parameters.

Significance and Claims
The paper claims that the proposed methodology provides a robust and structured approach to system identification for underwater vehicles, overcoming the limitations of heuristic input selection and decoupled assumptions. By generating dynamically feasible and safe excitation trajectories that maximize data quality, the method enables the reliable estimation of dynamic parameters using standard Least Squares techniques.

The authors emphasize that this approach is particularly advantageous for modular UVs, where payload configurations frequently change, altering the vehicle's dynamics. The methodology requires only robust state feedback and a conventional controller (e.g., PID), making it adaptable to various underwater platforms and workspace environments. The work establishes a foundation for future extensions, including full 6-DoF identification, open-sea validation, and the integration of identified models into fault detection and diagnosis schemes.

Technical Summary: Optimal Excitation Trajectories for System Identification of Underwater Vehicles

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