⚡ electrical engineering

Trajectory tracking of bionic fish based on a complex-valued central pattern generation network

This paper presents a robust trajectory tracking system for a bionic fish robot that integrates a complex-valued central pattern generator network, driven by gyroscope feedback to control fins and tail, with a model predictive control-based position controller under an extended Featherstone dynamics model to effectively navigate straight and circular paths despite external disturbances.

Original authors: Yong Zhang, Beibei Zhang, Yi Ding, Beiping Hou

Published 2026-08-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yong Zhang, Beibei Zhang, Yi Ding, Beiping Hou

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

In the fluid, shifting world of underwater navigation, robots face a challenge that has long stumped engineers: how to move with the grace and resilience of a living creature. Unlike a submarine that relies on rigid propellers and fixed rudders, a fish swims by coordinating the rhythmic undulation of its body and fins, a motion controlled by a biological rhythm generator that allows it to adapt instantly to currents and obstacles. This biological rhythm, known in science as a central pattern generator, acts as an internal clock for movement, producing the repeating signals needed to drive muscles without requiring constant conscious thought. For decades, researchers have tried to copy this system in machines, hoping to build robots that can navigate turbulent waters as effortlessly as a tuna or a carp. The goal is not just to mimic the shape of a fish, but to replicate the neural intelligence that allows it to maintain balance and follow a path even when the water pushes back.

A team of researchers at the Zhejiang University of Science and Technology has taken a significant step toward this goal by designing a new control system for a robotic fish they call BF-1. Their approach combines a sophisticated mathematical model of the robot's physical movement with a neural network that mimics the fish's internal rhythm. The core of their innovation is a "complex-valued" network, a type of digital brain that can process information in a way that allows for more flexible adjustments than traditional systems. Instead of simply reacting to disturbances, this system anticipates them by using signals from gyroscopes—sensors that measure how the robot tilts and turns—to instantly tweak the timing and strength of the robot's tail and fin movements. By integrating this rhythmic control with a predictive algorithm that constantly calculates the robot's position, the researchers created a system that can keep the robot on course even when the water becomes choppy.

The robot itself, BF-1, is a sleek machine with a plastic shell, an airtight cabin housing its electronics, and a bionic tail driven by a waterproof motor. It also features two pectoral fins on its sides, which work in tandem with the tail to steer and stabilize. The researchers tested their system in an outdoor water environment, subjecting the robot to two distinct conditions: calm, steady water and turbulent, disturbed water. In the calm conditions, the robot moved smoothly along a straight line, its tail and fins oscillating in a steady, coordinated rhythm. However, the true test came when the water flow suddenly became turbulent, causing the robot to tilt and drift off course. In these moments, the gyroscopes detected the change in angle and immediately sent signals to the central pattern generator. The system responded by altering the phase and amplitude of the signals driving the tail and fins, effectively resetting the robot's rhythm to counteract the force of the water.

The results of these experiments showed that the new control system worked effectively. When the water flow changed from gentle to turbulent at the five-second mark of the experiment, the robot's body angles initially shifted, but the control system quickly corrected the deviation. The robot was able to return to a stable state and continue following its intended straight path without losing its way. The researchers observed that the left and right pectoral fins moved in a synchronized manner to maintain balance, while the tail adjusted its power to push against the current. This coordination allowed the robot to track a straight line with high precision, even as the water pushed against it. The system did not just react to the disturbance; it used a predictive model to estimate the robot's future position and adjust its speed and direction proactively, ensuring that the tracking error remained minimal.

To further test the versatility of their design, the team also guided the robot through a circular trajectory under similar turbulent conditions. Turning a circle requires a different kind of coordination, where the tail must generate the power to rotate the body while the fins adjust to keep the turn smooth. The complex-valued network successfully managed this by adjusting the phase delay between the different joints, allowing the robot to spin and follow the circular path. Even as the water flow interfered with the robot's motion, causing it to tilt, the system compensated by adjusting the frequency and strength of the fin movements. The robot maintained its circular path, demonstrating that the control system could handle not just straight-line tracking but also complex maneuvers in a dynamic environment.

The researchers emphasize that their success relies on the specific combination of the rhythmic central pattern generator and the predictive position controller. Traditional control methods, such as those based on simple feedback loops, often struggle when the water becomes too turbulent, leading to unstable movements or a failure to converge on the desired path. By contrast, this new system uses a dynamic model of the robot's physics to estimate unknown disturbances and feed that information back into the control loop. This allows the robot to maintain a robust performance, adapting its internal rhythm to the external environment in real time. The study concludes that this approach offers a stable and accurate way to navigate bionic fish robots, proving that they can continue moving in their original direction even when faced with significant water flow disturbances. While the current experiments focused on surface-level movement, the researchers note that future work will need to address the challenges of deeper diving, where the dynamics of the water and the robot's behavior may change further. For now, however, the work demonstrates a clear path toward robots that can swim with the adaptability and resilience of their biological counterparts.

Technical Summary: Trajectory Tracking of Bionic Fish Based on a Complex-Valued Central Pattern Generation Network

Problem Statement
Controlling bionic fish robots in aquatic environments presents significant challenges due to the flexibility and coupling of their structures, as well as the presence of unknown external disturbances such as water flow. Traditional control methods, including standard PID algorithms and basic Model Predictive Control (MPC), often struggle with stability and convergence when facing dynamic water currents. Specifically, traditional MPC feedback can be slow to converge under unknown disturbances, and dynamic modeling using Lagrange formulas often suffers from low computational efficiency due to algebraic constraints in the inertia matrix. The core challenge addressed in this work is achieving stable trajectory tracking (both straight and circular) for a bionic fish robot while maintaining balance and adapting to external perturbations in real-time.

Methodology
The authors propose a novel trajectory tracking control system for a bionic fish robot (designated BF-1) that integrates three primary components: a Complex-Valued Central Pattern Generator (CG-CPG) network, a Model Predictive Control (MPC) based position controller, and an extended Featherstone dynamics model.

Robot Design (BF-1): The prototype features a cylindrical body with a bionic tail and left/right pectoral fins, driven by waterproof servomotors. It utilizes a gyroscope to measure body angles in the Y and Z axes, which serve as feedback for the control system.
Complex-Valued CPG (CG-CPG) Network:
- Unlike traditional real-valued CPGs, this system employs complex-valued mutual parameters to generate periodic signals with adjustable amplitude and frequency.
- Feedback Mechanism: Gyroscope signals in the Z-axis stimulate mutual parameters to control the tail for turning, while Y-axis signals act as feedback to adjust the CPG units controlling the pectoral fins for balance.
- Function: The network generates rhythmic signals for the tail and fins, allowing for phase resetting and the generation of stable limit cycles in response to external stimuli.
Position Controller (MPC):
- To improve robustness and reduce tracking errors, an MPC algorithm is employed.
- This controller calculates the desired velocity and velocity increments ( $\Delta v$ ) to provide a smooth, varying speed reference for the CG-CPG network.
- It minimizes a cost function involving weighting matrices ( $Q, R, P$ ) to predict and correct trajectory deviations.
Featherstone Dynamics Model:
- To handle the computational complexity of inverse dynamics in real-time, the authors utilize the Featherstone articulated-body algorithm.
- This approach models the robot as a hierarchical tree structure of rigid bodies, efficiently calculating joint accelerations and dynamic characteristics.
- The model estimates unknown disturbances and feeds back the actual position and dynamic characteristics to the position controller, ensuring the system converges to reference values.

Key Contributions

Complex-Valued CPG Architecture: The introduction of complex-valued mutual parameters allows for more flexible control of phase, amplitude, and frequency differences compared to real-valued CPGs, facilitating rapid adaptation to water flow disturbances.
Integrated Control Framework: The paper presents a unified system where the CG-CPG network handles low-level gait generation and balance, the MPC position controller manages high-level trajectory tracking and error reduction, and the Featherstone model provides efficient dynamic feedback.
Disturbance Rejection: The system is specifically designed to maintain stability and trajectory tracking under external perturbations (e.g., turbulent water flow) by utilizing gyroscope feedback to reset CPG phases and adjust actuator outputs.

Experimental Results
Experiments were conducted on the BF-1 robot in an outdoor water environment with controlled flow disturbances. The study evaluated performance on two trajectory types:

Straight Trajectory Tracking: During periods of calm water followed by turbulent flow (starting at 5 seconds), the system successfully maintained a straight path. When disturbances caused body angle deviations in the Y and Z axes, the CG-CPG network adjusted the tail and pectoral fin amplitudes and frequencies. The robot quickly returned to a stable state and continued moving forward along the reference line.
Circular Trajectory Tracking: The robot successfully tracked a circular path under water flow interference. The CG-CPG network coordinated the tail (for turning power) and pectoral fins (for balance) to follow the reference circle. Despite body tilting caused by currents, the robot maintained the circular trajectory.

In both scenarios, the integration of the Featherstone dynamics model allowed for rapid calculation of body acceleration and position feedback, enabling the position controller to adjust the CG-CPG parameters effectively.

Significance and Claims
The authors claim that the proposed control system effectively improves the adaptability of bionic fish robots to trajectory tracking tasks in disturbed environments. The study demonstrates that:

The combination of a complex-valued CPG network with an MPC-based position controller offers superior robustness compared to traditional methods.
The extended Featherstone dynamics model successfully converges the robot's dynamic characteristics to reference values and provides necessary feedback for the controller.
The system enables the bionic fish to maintain its original direction during water flow disturbances and successfully execute complex maneuvers like circular tracking through the coordination of fins and tail.

The paper concludes that this approach validates the effectiveness of the tracking performance under external disturbances, though it notes that future work will need to address scenarios involving deeper diving where auxiliary control laws might fail.

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