The pursuit of efficient and agile propulsion in aquatic environments has long drawn inspiration from nature. Fish, through millennia of evolution, have developed highly effective swimming mechanisms characterized by low energy consumption and high maneuverability. Mimicking these biological principles offers a promising pathway for developing advanced underwater robotic systems. The core challenge lies in replicating the complex, rhythmic, and adaptive motion patterns observed in biological fish. Traditional control methods often struggle with the nonlinear dynamics and environmental disturbances inherent to underwater operation. This has led to significant interest in bio-inspired control architectures, particularly those modeled on the neural circuits found in vertebrates.
Central Pattern Generators (CPGs) are neural networks located in the spinal cord that produce rhythmic motor signals without requiring rhythmic sensory feedback. In the context of a bionic robot, a CPG-based controller can generate coordinated, periodic control signals for multiple actuators (like joints or fins), enabling stable and adaptive locomotion. The CPG model provides a robust framework for producing the traveling body wave essential for fish-like swimming. However, a critical bottleneck in implementing CPG control is the parameter tuning process. The performance of the CPG oscillator network is highly sensitive to parameters such as time constants, connection weights, and gain factors. Manually calibrating these parameters to achieve a specific, efficient swimming gait is exceedingly difficult and time-consuming, often resulting in suboptimal motion control for the bionic robot.
To address this fundamental issue, this work proposes and investigates a novel motion control methodology for an underwater bionic robot that integrates a neuron-based CPG model with an improved Particle Swarm Optimization (PSO) algorithm. The primary objective is to automate and optimize the CPG parameter set, thereby enabling precise and efficient control of the robot’s multi-joint undulatory motion. The PSO algorithm is employed to search the high-dimensional parameter space, minimizing a cost function that quantifies the difference between the robot’s actual body wave and a desired, biologically plausible reference wave. This approach systematically bridges the gap between high-level gait specification and low-level actuator control for the bionic robot.
The core hypothesis is that an optimized CPG, tuned via a metaheuristic algorithm, will generate superior control signals leading to more accurate body wave tracking, improved propulsion efficiency, and enhanced overall performance of the underwater bionic robot. This document details the mathematical formulation of the CPG-PSO control system, the mechanical design of a multi-joint robotic fish prototype, and presents comprehensive validation through both numerical simulation and physical pool experiments.
Theoretical Foundations: CPG and PSO
Neuron Oscillator Model for CPG
The fundamental building block of the proposed CPG is the neuron oscillator, which mimics the mutual inhibition between extensor and flexor neurons in biological systems. A single oscillator, capable of producing a stable rhythmic output, is defined by the following set of coupled nonlinear differential equations:
$$
\begin{aligned}
T_{\gamma} \dot{u_i^e} + u_i^e &= – \omega_{df} y_i^f – \sum_{j=1, j\neq i}^{N} \omega_{ij}^e y_j^e – \beta v_i^e + u_0 + Feed_i^e \\
T_{\alpha} \dot{v_i^e} + v_i^e &= y_i^e \\
y_i^e &= \max(u_i^e, 0) \\
T_{\gamma} \dot{u_i^f} + u_i^f &= – \omega_{ef} y_i^e – \sum_{j=1, j\neq i}^{N} \omega_{ij}^f y_j^f – \beta v_i^f + u_0 + Feed_i^f \\
T_{\alpha} \dot{v_i^f} + v_i^f &= y_i^f \\
y_i^f &= \max(u_i^f, 0)
\end{aligned}
$$
Here, the subscript \(i\) denotes the \(i\)-th oscillator in a network of \(N\) oscillators. The superscripts \(e\) and \(f\) refer to the extensor and flexor neuron states, respectively. The variables are:
– \(u\): Internal state of the neuron.
– \(v\): Self-inhibition state (or adaptation).
– \(y\): Output signal of the neuron (a rectified signal).
– \(T_{\gamma}, T_{\alpha}\): Time constants shaping the frequency and waveform of the oscillation.
– \(\omega_{ef}, \omega_{df}\): Connection weights between the extensor and flexor neurons within the same oscillator.
– \(\omega_{ij}^e, \omega_{ij}^f\): Connection weights between oscillators \(i\) and \(j\), determining phase relationships.
– \(\beta\): Weight of the self-inhibition, contributing to oscillation stability.
– \(u_0\): An external tonic input, controlling the amplitude of the oscillation.
– \(Feed\): Sensory feedback signal, which can be used for online adaptation (though often set to zero for basic gait generation).
The joint angle command \(\theta_i\) for the \(i\)-th joint of the bionic robot is derived from the difference between the extensor and flexor outputs, scaled by a factor \(s_i\):
$$
\theta_i = s_i (y_i^e – y_i^f)
$$
By coupling multiple such oscillators with specific connection weights, a network can produce coordinated, phase-locked rhythmic signals suitable for controlling the serial joints of a robotic fish.
Particle Swarm Optimization (PSO) Algorithm
Particle Swarm Optimization is a population-based stochastic optimization technique inspired by the social behavior of bird flocking or fish schooling. It is particularly effective for navigating complex, non-convex search spaces, such as the multi-dimensional parameter space of a CPG network. In PSO, a swarm of candidate solutions, called particles, explores the search space. Each particle \(p\) has a position vector \(\mathbf{x}_p\) (representing a set of CPG parameters) and a velocity vector \(\mathbf{v}_p\).
The algorithm proceeds iteratively. In each iteration \(k\), every particle evaluates its fitness (the quality of its solution based on an objective function). Two key values are tracked:
1. Personal Best (\( \mathbf{pbest}_p \)): The best position particle \(p\) has encountered so far.
2. Global Best (\( \mathbf{gbest} \)): The best position found by any particle in the entire swarm so far.
The particle’s velocity and position are then updated according to the following rules:
$$
\begin{aligned}
\mathbf{v}_p^{(k+1)} &= \omega \cdot \mathbf{v}_p^{(k)} + c_1 r_1 (\mathbf{pbest}_p – \mathbf{x}_p^{(k)}) + c_2 r_2 (\mathbf{gbest} – \mathbf{x}_p^{(k)}) \\
\mathbf{x}_p^{(k+1)} &= \mathbf{x}_p^{(k)} + \mathbf{v}_p^{(k+1)}
\end{aligned}
$$
where:
– \(\omega\) is the inertia weight, controlling the influence of the previous velocity.
– \(c_1\) and \(c_2\) are acceleration coefficients (cognitive and social weights).
– \(r_1\) and \(r_2\) are random numbers uniformly distributed in \([0,1]\).
The inertia weight \(\omega\) is crucial for balancing exploration and exploitation. A common improvement involves using a linearly decreasing inertia weight:
$$
\omega^{(k)} = \omega_{max} – \frac{(\omega_{max} – \omega_{min}) \cdot k}{K_{max}}
$$
where \(K_{max}\) is the maximum number of iterations. This “Improved PSO” starts with a high inertia for broad exploration and gradually reduces it for fine-tuning near the optimum, leading to faster and more reliable convergence—a key enhancement employed in tuning the bionic robot‘s controller.
CPG-Based Control Architecture for the Bionic Robot
Kinematic Model and Desired Body Wave
The fundamental motion pattern for anguilliform or carangiform swimming robots is a traveling body wave. The desired midline motion of the bionic robot is described by a wave equation:
$$
Y_{body}(x, t) = (c_1 x + c_2 x^2) \sin(kx – \omega t)
$$
where:
– \(x\) is the position along the body axis (from head at \(x=0\) to tail tip).
– \(t\) is time.
– \(c_1\) and \(c_2\) are linear and quadratic wave envelope coefficients determining the amplitude growth from head to tail.
– \(k = 2\pi / \lambda\) is the wave number (\(\lambda\) is the wavelength).
– \(\omega = 2\pi f\) is the angular frequency (\(f\) is the beating frequency).
A bionic robot with a discrete, multi-joint structure approximates this smooth wave. For a robot with \(M\) links connected by \(M-1\) joints, the goal is to determine the joint angle trajectories \(\theta_i(t)\) such that the piecewise linear body shape closely follows \(Y_{body}(x,t)\).
CPG Network Design and Parameter Optimization Problem
For a bionic robot with three active body joints, a CPG network of three neuron oscillators is constructed. Each oscillator \(i\) (\(i=1,2,3\)) corresponds to one joint. The oscillators are coupled to enforce a specific phase lag that propagates the wave from head to tail. The network’s output, after scaling, directly provides \(\theta_1(t), \theta_2(t), \theta_3(t)\).
The performance of the CPG network is governed by a parameter vector \(\mathbf{P}\) containing all tunable variables:
$$
\mathbf{P} = [T_{\gamma}, T_{\alpha}, \beta, u_0, \omega_{ef}, \omega_{df}, \{\omega_{ij}^e\}, \{\omega_{ij}^f\}, s_1, s_2, s_3]
$$
Manually tuning \(\mathbf{P}\) is impractical. Therefore, we formulate an optimization problem. The objective is to find \(\mathbf{P}^*\) that minimizes the error between the robot’s realized body shape and the desired body wave over a full cycle period \(T\).
We define the instantaneous shape error \(E(t)\) as the integral of the absolute deviation between the desired wave \(Y_{body}\) and the robot’s actual link positions \(Y_{link}\) along its body length \(L\):
$$
E(t) = \sum_{i=1}^{M} \int_{x_{i-1}(t)}^{x_i(t)} | Y_{body}(x,t) – Y_{link_i}(x,t) | \, dx
$$
where \(Y_{link_i}(x,t) = \tan(\theta_i(t))(x – x_i(t)) + y_i(t)\) is the line equation for the \(i\)-th link, and \((x_i(t), y_i(t))\) are the coordinates of the joint between link \(i-1\) and \(i\), calculated via forward kinematics.
The fitness function \(F\) for the PSO algorithm is the integrated error over one period:
$$
F(\mathbf{P}) = \int_{0}^{T} | E(t) | \, dt
$$
The PSO swarm’s task is to find \(\mathbf{P}^* = \arg\min F(\mathbf{P})\). The optimization is performed offline. Once \(\mathbf{P}^*\) is found, it is loaded into the bionic robot‘s controller for online motion generation.
Design and Implementation of the Bionic Robot Prototype
Mechanical and Electrical Architecture
A three-joint robotic fish prototype was developed to serve as the experimental platform. The design emphasizes biomimetic morphology and functional modularity.
- Head and Hull: The frontal section houses the core electronics in a streamlined, watertight compartment. This includes the main processor (e.g., an STM32 or ARM-based microcontroller), a power management system, inertial measurement units (IMUs), and communication modules (Wi-Fi/Bluetooth for wireless control and data telemetry).
- Propulsion Mechanism: The primary propulsion is generated by three active joints in the posterior body, each driven by a waterproof servo motor. These joints create the traveling undulatory wave. A rigid caudal fin is attached to the last link to enhance thrust generation.
- Actuation and Control: The servo motors receive PWM signals corresponding to the angle commands \(\theta_i(t)\) generated in real-time by the CPG algorithm running on the onboard processor. The CPG parameters are fixed post-optimization.
- Sensing: An onboard IMU (e.g., MPU6050) provides real-time orientation (pitch, roll, yaw) and acceleration data. This sensory feedback can be potentially integrated into the *Feed* term of the CPG model for closed-loop heading or depth control in future work.
The following table summarizes the key specifications of the developed bionic robot prototype:
| Component | Specification / Description |
|---|---|
| Structure Type | Multi-joint, carangiform-inspired |
| Number of Active Joints | 3 |
| Actuator per Joint | Digital Waterproof Servo Motor |
| Control Core | ARM Cortex-M Microcontroller |
| Primary Sensor | 6-Axis IMU (Gyro + Accelerometer) |
| Communication | Wireless Serial Module |
| Power Source | Lithium Polymer Battery Pack |
| Body Material | 3D Printed PLA / ABS, Silicone Seals |
Parameter Optimization Process and Results
The desired body wave was specified with parameters: \(c_1=0.08\), \(c_2=0.1\), \(k=1\), and \(\omega=\pi\) (period \(T=2\) seconds). The Improved PSO algorithm was configured with a swarm size of 30 particles and run for 100 iterations. The inertia weight \(\omega\) decreased linearly from 0.9 to 0.4. The cognitive and social coefficients \(c_1\) and \(c_2\) were both set to 2.0.
The optimization successfully converged to a parameter set \(\mathbf{P}^*\) that minimized the fitness function \(F\). The evolution of the best fitness value over iterations demonstrated a characteristic rapid decrease followed by refinement, validating the effectiveness of the improved PSO strategy. The optimized CPG parameters are listed below:
| Parameter | Optimized Value | Parameter | Optimized Value |
|---|---|---|---|
| \(T_{\gamma}\) | 2.99985 | \(T_{\alpha}\) | 0.42992 |
| \(\beta\) | 9.84874 | \(u_0\) | 1.0 (Fixed) |
| \(\omega_{ef}\) | 3.02762 | \(\omega_{df}\) | 2.50000 (Fixed) |
| \(\omega_{12}^e\) | 3.45688 | \(\omega_{12}^f\) | 5.83510 |
| \(\omega_{13}^e\) | 3.83181 | \(\omega_{13}^f\) | 7.76610 |
| \(\omega_{23}^e\) | 6.55237 | \(\omega_{23}^f\) | 3.11608 |
| \(s_1\) | 0.64557 | \(s_2\) | 2.73989 |
| \(s_3\) | 2.33130 |
Simulation and Experimental Validation
Numerical Simulation
Prior to physical testing, the CPG model with the optimized parameters \(\mathbf{P}^*\) was simulated in a computational environment (e.g., MATLAB/Simulink). The system of differential equations was solved numerically. The outputs \(y_i^e(t)\) and \(y_i^f(t)\) were calculated, and the joint angles \(\theta_i(t)\) were derived. The simulation confirmed that the CPG network produced stable, periodic, and phase-locked signals. The resulting joint angle trajectories were smooth sinusoidal-like waves with a progressive phase lag from joint 1 to joint 3, accurately matching the frequency specified by the desired body wave.
Forward kinematics was then applied to these joint angles to reconstruct the robot’s body shape over time. A side-by-side comparison with the desired traveling wave \(Y_{body}(x,t)\) showed a close visual match. The quantitative fitness value \(F(\mathbf{P}^*)\) was significantly lower than values obtained from manually tuned or randomly selected parameter sets, confirming the success of the PSO-based optimization for the bionic robot‘s gait generation.
Physical Pool Experiments and Performance Analysis
The optimized CPG controller was implemented on the microcontroller of the physical bionic robot prototype. Experiments were conducted in a still-water pool equipped with an overhead camera system for motion capture. The robot was tasked with swimming straight at a constant CPG frequency.
1. Gait Visualization and Wave Tracking: The captured video was processed to extract the midline of the robot in each frame. The sequence of midlines demonstrated a clear traveling wave propagating from the head to the tail of the bionic robot. The wave’s amplitude envelope and wavelength closely resembled the kinematically specified target, validating that the optimized CPG parameters successfully translated from simulation to real-world actuation.
2. Speed Performance: The forward swimming speed was measured for different oscillation frequencies. A key comparison was made between the performance using the PSO-optimized parameters and a set of baseline, manually adjusted parameters. The results are summarized below:
| Control Method | Oscillation Freq. (Hz) | Avg. Speed (m/s) | Speed Increase |
|---|---|---|---|
| Manual CPG Tuning | 0.5 | 0.18 | Baseline |
| PSO-Optimized CPG | 0.5 | 0.25 | ~39% |
| Manual CPG Tuning | 1.0 | 0.31 | Baseline |
| PSO-Optimized CPG | 1.0 | 0.42 | ~35% |
The data clearly indicates a substantial improvement in propulsion efficiency. The bionic robot controlled by the PSO-optimized CPG achieved approximately 35-40% higher speed at equivalent beating frequencies compared to the manually tuned controller. This is attributed to the more accurate generation of the hydrodynamically favorable body wave, leading to better thrust production and reduced lateral slip.
3. Stability and Robustness: The swimming motion was observed to be stable and consistent over multiple trials. The inherent stability properties of the Hopf-type oscillator used in the CPG model contributed to this robustness, allowing the bionic robot to maintain its gait despite minor manufacturing asymmetries or transient water disturbances.
Conclusion and Future Work
This research successfully developed and validated a novel motion control framework for underwater bionic robots by synergistically integrating a bio-inspired Central Pattern Generator with an improved Particle Swarm Optimization algorithm. The core contribution is the systematic, automated optimization of the CPG’s numerous internal parameters, a process that is otherwise highly challenging and subjective when performed manually.
The proposed method translates a high-level kinematic specification—a traveling body wave—into an optimized set of low-level neural oscillator parameters. The improved PSO algorithm efficiently navigated the complex parameter space, converging to a solution that minimized the difference between the desired and realized body motion. Both numerical simulation and extensive pool experiments on a multi-joint robotic fish prototype confirmed the efficacy of the approach. The PSO-optimized CPG controller enabled the bionic robot to generate a more accurate body wave, resulting in a statistically significant increase of 35-40% in swimming speed compared to a baseline manual controller, without any hardware modifications.
This work provides a solid foundation and a new, effective paradigm for the motion control of high-maneuverability underwater bionic robots. The CPG-PSO architecture is generalizable and can be extended to robots with different morphologies or more degrees of freedom. Future work will focus on several promising avenues:
1. Dynamic Optimization: Incorporating real-time sensory feedback (e.g., from IMU or pressure sensors) into the *Feed* pathways of the CPG to enable online adaptation for trajectory tracking, depth control, or disturbance rejection.
2. Multi-Objective Optimization: Expanding the PSO fitness function to simultaneously optimize for speed, efficiency (cost of transport), and turning agility.
3. Gait Transition: Utilizing the CPG’s natural dynamics to smoothly transition between different swimming gaits (e.g., cruising vs. bursting) by modulating only a few high-level parameters like \(u_0\) or \(T_{\gamma}\), with the underlying optimized couplings ensuring stability.
4. Learning from Biological Data: Using motion capture data from real fish to inform the desired body wave parameters or even to directly shape the PSO fitness function, pushing the bionic robot‘s performance closer to its biological counterparts.
In conclusion, the fusion of bio-inspired control and intelligent optimization presented here marks a significant step toward realizing autonomous underwater bionic robots with lifelike, efficient, and robust locomotion capabilities.