Fixed-Time Cooperative Trajectory Tracking Control of Multi-AUV Formation under Unknown Disturbances

The ocean is a vast, unpredictable workspace where machines must operate without human hands to guide them. For decades, scientists have relied on single underwater robots to explore these depths, but a single machine has limits; it can carry only so much equipment, run out of power, or fail if a critical part breaks. To overcome this, engineers have turned to groups of robots working together, known as formations. The idea is simple: if one robot fails, others can take over, and together they can cover a much larger area. However, keeping a group of these machines moving in a precise shape while navigating the chaotic underwater world is incredibly difficult. The water is never still; hidden currents push and pull, and the robots themselves are heavy, complex machines that do not always react exactly as their computer models predict. Traditional methods of controlling these groups often struggle to correct errors quickly enough, or they react so harshly to disturbances that they shake the robots' motors, causing wear and tear.

A team of researchers at Shandong Jiaotong University has developed a new way to guide these underwater groups that solves both the speed and the shaking problems. They created a control system that allows multiple autonomous underwater vehicles to track a specific path and maintain their formation shape, even when hit by unknown, changing currents. The core of their solution is a two-part strategy. First, they built a virtual "sensor" inside the computer that can instantly guess what the water is doing to the robots, even though the robots cannot see the currents directly. Second, they designed a steering command that uses this guess to cancel out the push of the water before it can throw the robots off course. Unlike older methods that take a long time to correct a mistake or require the motors to jerk violently to fight the water, this new approach brings the robots back to their correct path in a guaranteed, short amount of time, regardless of how far off course they started. The researchers tested this idea in computer simulations with groups of robots forming triangles, hexagons, and arrows, and the results showed that the new method kept the formation tight and smooth while older methods struggled with delays and shaking.

The challenge of controlling these underwater groups lies in the nature of the ocean itself. Water is a fluid that moves in complex, random patterns, creating forces that are hard to predict. When a robot tries to move, it must fight against this resistance, and if a sudden gust of current hits it, the robot's position changes. In the past, control systems tried to react to these changes after they happened. This is like trying to steer a car by looking only at where you have already been; by the time you realize you are drifting, you are already far off the road. Other systems tried to be more aggressive, applying maximum force to correct the error immediately. While this might work in theory, in practice, it causes the robot's thrusters to vibrate rapidly, a phenomenon known as chattering. This vibration is like a car engine revving up and down uncontrollably; it wastes energy, creates noise, and can eventually break the mechanical parts. The researchers found that these traditional approaches were either too slow to be useful in fast-changing environments or too rough to be safe for the hardware.

To fix this, the team designed a system that anticipates the problem rather than just reacting to it. They started by creating a detailed mathematical model of how the robots move, separating the motion into two layers: the position and direction the robot is facing, and the physical forces that drive it. They realized that to control the group effectively, they needed to know exactly how much the water was pushing on the robot at any given moment. Since they could not measure the invisible currents directly, they built a special observer, which is a type of computer program that runs alongside the main control system. This observer constantly compares what the robot is supposed to be doing with what it is actually doing. By analyzing the difference, it can estimate the strength and direction of the hidden current in a fraction of a second.

The key innovation here is the speed and certainty of this estimation. Older observers took a long time to figure out the current, and their accuracy depended on how close the robot started to its target. If the robot was far off course, the observer took even longer to catch up, which was dangerous in a fast-moving environment. The new observer, however, is designed to reach its estimate in a fixed amount of time, no matter how large the initial error is. It is as if the observer has a built-in clock that guarantees it will know the wind speed within a specific number of seconds, whether the car is drifting slightly or spinning wildly. This allows the control system to act immediately, using the observer's guess to apply a counter-force that neutralizes the current before it can disrupt the formation.

Once the system knows what the water is doing, it applies a second layer of control to keep the robots in their formation. The researchers used a method where one robot acts as the leader, and the others follow it, maintaining a specific distance and angle relative to the leader. This is similar to a flock of birds, where each bird knows where to be in relation to the one in front. The control law ensures that even if the leader turns sharply or speeds up, the followers adjust their paths smoothly to stay in the correct shape. The system uses a special type of math that prevents the robots from overshooting their target or oscillating back and forth. Instead of jerking the thrusters to correct a mistake, the system applies a smooth, continuous force that guides the robot gently back to the line.

The researchers tested their system in a series of computer simulations to see how it would perform under difficult conditions. They programmed the robots to follow complex paths, including circular routes and shapes that combined triangles and circles. They also simulated scenarios where the robots started from very different positions, some far away from the intended path and others close to it. In every case, the new control system brought the robots to their target formation and kept them there. The simulations showed that the robots could track the path with high precision, and the errors in their position disappeared quickly. When they compared their new method to a traditional control strategy, the difference was clear. The traditional method caused the robots to shake and vibrate as they tried to correct their course, and it took much longer for the errors to settle down. The new method, by contrast, produced smooth, steady movements and corrected errors almost immediately.

One of the most important findings was how well the system handled the unknown disturbances. The researchers introduced random, changing currents into the simulation to mimic the real ocean. The observer was able to track these changing forces with remarkable accuracy, matching the actual strength of the current almost perfectly. This allowed the control system to cancel out the disturbance effectively, keeping the robots on course without needing to increase the power of their motors. The result was a system that was both robust against the unpredictable ocean and gentle on the robot's hardware. The simulations also showed that the system worked well for different group sizes and shapes. Whether the robots were arranged in a triangle, a hexagon, a pentagon, or an arrow shape, the control law kept them coordinated and stable.

The researchers concluded that their approach offers a significant improvement for underwater robotics. By combining a fast, accurate observer with a smooth control law, they created a system that can handle the messy reality of the ocean without damaging the equipment. The simulations proved that the robots could converge to their desired formation in a guaranteed time, independent of where they started. This means that in a real-world mission, a group of robots could be deployed from a ship in a scattered pattern, and they would quickly organize themselves into a precise formation to begin their work, even if the water was rough. While the study was conducted entirely in a computer environment, the results suggest that this method could be a powerful tool for future underwater exploration, allowing groups of robots to work together more efficiently and reliably than ever before. The team noted that future work will aim to test these ideas on actual physical robots and expand the system to handle movement in three dimensions, preparing for the complex challenges of the deep sea.