Background
The advancement of biology, materials, and robotics technologies has paved the way for the creation of biologically-inspired underwater vehicles (BIUVs) that emulate the movement of aquatic animals Unlike traditional Autonomous Underwater Vehicles (AUVs) that rely on screw propellers, BIUVs utilize biomimetic fins and flippers for propulsion, enhancing their maneuverability and efficiency These innovative vehicles are versatile, serving various purposes such as marine sourcing, seabed mapping, military surveillance, environmental assessments, exploration, and scientific research Fish swimming, a result of millions of years of evolution, showcases remarkable efficiency, with some species achieving over 90% swimming efficiency compared to the 40-50% efficiency of conventional propellers Additionally, fish exhibit impressive agility, with turning radii significantly smaller than their body lengths, far surpassing the capabilities of current ships, which have larger turning radii and slower turning speeds.
The movement of fish fins and bodies offers underwater robots enhanced maneuverability compared to traditional screw propellers, allowing for precise positioning This natural inspiration drives innovative designs that improve the functionality of artificial systems in aquatic environments Additionally, understanding the underwater ecosystem is crucial for advancing the study of Bio-Inspired Underwater Vehicles (BIUVs).
The deterioration of marine life is increasingly attributed to the frequent use of noisy propellers, which disrupt aquatic environments Fish, known for their silent movement, are adversely affected by these loud sounds Consequently, engineers are compelled to innovate and create vehicles that do not rely on rotary propellers.
Biomimetic propulsion systems for swimming machines draw significant inspiration from fish movement, leading to a growing interest in robotic fish over the past two decades The primary objective of this research is to develop biomimetic fish technologies into innovative underwater vehicles that serve human needs To achieve this, researchers focus on various aspects, including the mechanical design of fish robots, materials for biomimetic propellers, actuation methods suitable for underwater environments, sensors for underwater measurements, efficient swimming control, and intelligent strategies for autonomous operations.
This dissertation explores the motion control capabilities of a propulsion module inspired by the swimming mechanics of Gymnotiform fish A key aspect of this propulsion system's flexibility is the changeover time, which has not been addressed in prior research Additionally, for robotic applications in underwater mine removal, it is crucial to identify an optimal swimming posture that maximizes thrust while minimizing disturbances to underwater sound frequencies The thesis outlines specific limitations and research constraints related to these objectives.
1- It is impossible to simulate the effects of disturbance on the marine environment
2- in the analytical calculation to focus on the thrust in the translational direction, the thesis temporarily ignores the horizontal, oblique force analysis
3- The effect of vortices and the experimental tank's narrowness is considered negligible and will develop in future studies.
Motivation
Modern warfare leaves behind explosive remnants that significantly affect the quality of life, particularly for fishermen in Vietnam's coastal regions, where the threat of underwater mines persists Traditional methods of detection and destruction are labor-intensive and risky; however, the Vietnamese Navy has recently adopted underwater robots for mine clearance, enhancing safety and efficiency in these operations.
Survey work in mining environments often faces challenges due to the mossy conditions and the presence of ocean debris As depicted in Figure 1-1, robots equipped with propellers are struggling to operate efficiently in these settings, highlighting the need for effective solutions to address these issues.
Figure 1-1 Mine underwater (source internet)
Inspired by the current state of fish robot operations, I have undertaken research focused on developing stable underwater robots with a rigid structure for effective device installation The findings presented in my thesis lay the groundwork for creating a comprehensive underwater robot designed for surveying and removing seabed mines.
Literature review
Aquatic Locomotion Modes of Fish
This section provides an overview of the swimming mechanisms utilized by fish, focusing on existing literature on aquatic biomechanisms Breder's classification scheme categorizes fish based on their swimming techniques, distinguishing between those that use body and/or caudal fins (BCF) and those that rely on median and/or paired fins (MPF) BCF movements are characterized by greater thrust and acceleration, while MPF movements, typically employed at lower speeds, offer enhanced maneuverability and propulsion efficiency The article identifies specific swimming modes for both BCF and MPF locomotion, based on the type of propulsor and the nature of the movements—oscillatory or undulatory—used to generate thrust.
Swimming locomotion has been broken down into two general types based on how quickly the movements happen[1]:
• Periodic Swimming (or steady or sustained), in which propulsive movements are repeated cyclically Periodic Swimming enables fish to cover relatively large distances at a relatively constant rate
• Voluntary (or transient) movements such as rapid acceleration, escape maneuvers, and turns Typically, millisecond-long movements are used to capture prey or evade predators
Biologists and mathematicians have traditionally focused on periodic swimming due to the complexities involved in measuring transient movements, which are harder to replicate and verify Consequently, this section emphasizes periodic swimming while also acknowledging the significance of transient movements, which enhance fish capabilities in aquatic environments This discussion utilizes BBreder's expanded nomenclature for classifying swimming movements, despite recent critiques regarding its oversimplification Breder's classification, while not the primary focus, serves as a foundational framework for understanding fish locomotion Fish typically swim by bending their bodies into waves that propel them forward, a method known as body and/or caudal fin (BCF) locomotion Alternatively, median and/or paired fin (MPF) locomotion refers to the use of middle and back fins for swimming.
The term "aired" encompasses both pectoral and pelvic fins; however, while pelvic fins assist with stabilization and steering, they do not significantly contribute to forward propulsion and lack a specific association with locomotion classifications Approximately 15% of fish families utilize non-body-caudal fin (BCF) modes for routine propulsion, whereas a larger percentage relies on BCF modes for maneuvering and stabilization Additionally, literature often distinguishes between undulatory and oscillatory motions: undulatory motions involve a wave-like passage along the propulsive structure, while oscillatory movements feature the propulsive structure swiveling at its base without forming a wave These modes should be viewed as a continuum, as oscillatory motion can evolve from undulatory movement.
Fish exhibit variations in morphology based on their unique modes of life, despite using similar propulsion methods Three optimal designs for fish morphology arise from specializations in accelerating, cruising, and maneuvering, which are closely linked to their locomotion methods While no single fish excels in all three functions, they are generally locomotor generalists that incorporate design elements from each specialization to varying degrees For further insights into the relationship between function and morphology in swimming fish, refer to sources [4] and [5].
Figure 1-2 Diagram of swimming propulsors and swimming functions
In the classification of MPF and BCF propulsion, various swimming types can be identified according to Breder's original framework These swimming modes represent a continuum rather than distinct categories, as fish can employ multiple swimming techniques simultaneously or adjust their speed The combination of median and paired fins enhances thrust, leading to smoother swimming trajectories Many fish utilize MPF mode for increased maneuverability, enabling them to transition to BCF mode at higher speeds and achieve rapid acceleration.
Figure 1-3 Swimming mode (a): BCF , (b): MPF [17]
In undulatory body-caudal fin (BCF) modes, the propulsive wave travels through a fish's body in a direction different from its overall movement and at a speed faster than the fish's swimming speed Various undulatory BCF locomotion modes exhibit unique wavelengths and amplitude envelopes, contributing to their distinct movement styles Thrust generation can occur through lift-based (vorticity) methods or by utilizing an added-mass approach, with the latter being the most commonly employed This added-mass method has a long-standing association with propulsion mechanisms, particularly in Carangiform and Subcarangiform fish, which utilize vorticity to enhance their swimming efficiency.
Figure 1-4 Gradation of BCF from (a) Anguilliform through, (b) Subcarangiform, (c)
Anguilliform swimming is characterized by large-amplitude undulations that engage the entire body, allowing for effective movement in both forward and backward directions This mode of locomotion involves a wave that is at least one full wavelength long, which helps to balance lateral forces and minimize recoil Notably, backward swimming requires increased lateral forces and enhanced body flexibility, as seen in species like eels and lampreys In contrast, the sub-carangiform mode, exemplified by trout, displays similar undulatory motions but restricts the amplitude of movement primarily to the front of the body, with increased motion towards the back.
Carangiform swimming is characterized by body undulations confined to the last third of the body length, with a stiff caudal fin providing propulsion This swimming style enables Carangiform swimmers to achieve greater speeds compared to Anguilliform or Subcarangiform swimmers However, the rigidity of their bodies limits their ability to turn and accelerate effectively Additionally, the concentration of lateral forces at the posterior increases the tendency for the body to rebound.
The Thunniform style of locomotion is the most efficient mode for aquatic environments, allowing for high cruising speeds over extended periods This swimming pattern is a pinnacle of evolution, observed in various vertebrates like teleost fish, sharks, and marine mammals, each adapted to unique habitats Notably, scombrids, such as tuna and mackerel, exemplify this mode, where over 90% of thrust is generated by the caudal fin Their streamlined bodies minimize pressure drag, while the rigid, crescent-shaped caudal fin optimizes thrust and minimizes recoil forces, reducing sideslipping Although Thunniform swimmers excel in fast swimming in calm waters, they are less effective for slow swimming, turning maneuvers, or quick acceleration in turbulent conditions.
Ostraciiform locomotion is a unique oscillatory mode of movement in fish, characterized by the pendulum-like motion of the caudal fin while the body remains rigid Fish utilizing this mode are often encased in hard bodies and rely on median and paired fin (MPF) propulsion to maneuver through their complex environments.
Caudal oscillations serve as an auxiliary locomotion method, enhancing thrust at higher speeds, maintaining body rigidity, and aiding in prey tracking In contrast, Thunniform swimmers exhibit advanced hydrodynamic adaptations that are absent in Ostraciiform movement, which, despite its similar appearance, demonstrates low hydrodynamic efficiency.
Many fish utilize undulating fins for propulsion, maneuvering, and stabilization, often relying on these systems for low-speed locomotion Certain species can actively bend their median fin rays, supported by a muscle group that provides two degrees of freedom for movement The muscular structure of paired fins is even more intricate, enabling complex movements such as individual fin ray rotations Literature reviews on teleost fin structure and properties highlight their adaptability, which has been vital in the evolution of undulatory multi-phase fin (MPF) modes.
Figure 1-5 Growth of the undulatory MPF modes [3]
Experts suggest that certain fish species, including rays, skates, and manta rays, exhibit a movement style akin to bird flight, known as the rajiform mode This mode generates thrust through vertical undulations that travel along their large, triangular pectoral fins, which are notably flexible By increasing the amplitude of these undulations from the front to the apex of the fin and then decreasing it towards the back, these fish can also flap their fins up and down to enhance their propulsion.
The Diodontiform mode propels the animal forward by utilizing its broad pectoral fins, which remain stationary during movement This unique method allows light waves to traverse the fins in two complete wavelengths, often showcasing a harmonious interplay between the waves and the fins' flapping motions.
Swimming in Amiiform mode is accomplished through undulations of a dorsal fin (typically long in base), with the body axis being maintained in many cases while swimming
African freshwater electric eels are prominent examples of unique fish species found abundantly across Africa These eels are distinguished by the absence of anal and caudal fins, featuring numerous fin rays that extend along most of their body length before tapering to a point, which can reach up to 200 rays.
The swimming mechanism of fishes
Research on the swimming mechanisms of fish has been conducted for many years, focusing on how these creatures swim efficiently in water This understanding is crucial for applying insights to the design of underwater vehicles and robots.
Recent research has examined the undulating motion of fish fins, particularly focusing on the propulsive thrust and efficiency across different swimming modes The study highlighted that the tissue fibers in cuttlefish fins can store elastic energy during bending, enabling the fins to act as harmonic oscillators, which enhances locomotion efficiency.
Research on fish morphology reveals its significant impact on swimming performance A study developed a model for carangiform swimmers, examining the mechanics of both the caudal fin and the body The findings highlight that the elongated, narrow peduncle of carangiforms positions the caudal tail fin several chord lengths from the main body, influencing the flow field around the fish and enhancing its swimming efficiency.
Researchers have created mathematical models to analyze the impact of sinusoidal inputs on fish locomotion cycles Leonard's research introduced an average-formula method to effectively describe this phenomenon, highlighting the oscillatory motions of fish fins and bodies Additionally, Li and Saimek developed a Kalman filter-based estimation technique that retrieves hydrodynamic potential from pressure measurements taken along a fish's body.
Research on fish swimming mechanisms has significantly informed the design of underwater vehicles and robots Insights gained from how fish swim efficiently enable researchers to create advanced and more effective underwater technologies.
The Development of Vertebrate Locomotion
An organism, such as a vertebrate, is a dynamic system that has evolved since its inception Despite changes in its components, the organism maintains similar functionality The process of self-organization allows the organism to grow and adapt coherently, relying on genetic, chemical, mechanical, and activity-dependent mechanisms.
All vertebrate brains develop through similar stages, with synaptogenesis influenced by the animal's activities This process occurs both prenatally and postnatally, indicating that the adult brain's connectivity is shaped by its usage Thus, brain function plays a critical role in determining its structural development.
Vertebrate embryos are known to exhibit movement before birth, a phenomenon that has gained deeper understanding in recent years Research suggests that these prenatal movements may play a crucial role in linking sensory inputs to specific muscle activity patterns within the developing nervous system These movements can be categorized into distinct stages, highlighting their significance in prenatal development.
During the pre-motile stage of species such as Xenopus laevis, gentle contact with fine hairs on the head triggers a reflex response, causing the animal to bend away from the stimulus According to Roberts, this bending behavior is attributed to a specific reflex pathway.
In the next stage of movement, the head shifts forward in an uncoordinated manner, beginning at the neck and lacking synchronization with the rest of the body Similar to swimming, a tilt to the left does not precede a turn to the right, resulting in seemingly random bending This lack of coordination indicates that the animal is functionally divided into two distinct sides, with only the head and neck being observable.
11 animal starts to move While the spinal cord is still being built, the movement begins Over time, the animal's whole body starts to bend this way
C-Bending: The forward bending of the head gives way to a bend The shape is made when the animal bends very far in one direction and then very far in the other The sides of the animal work together, which is different from early head flexion This shows that the two parts of the animal work together
S-Bending: The bending replaces the bending The letter says a lot As in the letter "S", two bend points become apparent During the "C" bending phase, both sides of the spinal cord work simultaneously During the "S" bending phase, They see the first signs that the spinal cord is starting to separate into different parts that work together
S-Wave Traveling and Swimming: Eventually, the "S" bend gives way to a moving "S" wave and a "swimming" motion In the traveling "S" wave, an "S"-shaped wave moves from the top to the bottom In short, They can infer from this that development is made up of separate events that happen in a strict order How the nervous system is set up anatomically also depends on how the body moves.
Locomotion control for elongated undulating fin
Research on bio-fish robots highlights the importance of fish species diversity and identifies key factors influencing their hydrodynamics A critical aspect is the swimming pattern, which allows these robots to execute intricate maneuvers like turning, swaying, twisting, and curving Studies on robotic fish locomotion control focus on two main categories: offline swimming gait control and online swimming gait control.
Recent studies have employed a sinusoidal-based kinematic equation to create undulating oscillatory motion in bio-fish robots, enhancing locomotion control strategies By manipulating parameters such as amplitude envelope, oscillatory frequency, and phase lag, researchers can generate diverse swimming patterns These investigations often focus on modeling specific fish swimming postures, aiming to establish practical linear swimming rules K.H Low and colleagues have contributed significantly to this field with a series of research papers on robots that mimic fish movement through undulating fins Many of these studies utilize the principles of sine wave motion for their analyses.
To enhance thrust in fish robots, tuning the fin ray parameters is essential Researchers, including Mohsen Siahmansouri et al., have advanced motion controllers by integrating phase difference angles and thrust direction through sine wave oscillators Our team is also focused on developing propulsion systems that utilize undulating fins, primarily controlled by sine wave oscillators that coordinate servo motors However, effective locomotion control for biomimetic robots necessitates more than just fixed swimming postures; it requires adaptability to aquatic environments and flexibility in control strategies Adjusting parameters like frequency and amplitude allows for smoother swimming shapes.
1.3.4.2 Online swimming gait controller used central paten generator
Orjan Ekeberg and colleagues have pioneered the use of a central pattern generator (CPG) to control fish locomotion Research indicates that the spinal nervous system, rather than the brain, coordinates the fin rays in fish The model developed by Ekeberg et al effectively embodies this concept, allowing for flexible switching of swimming postures with minimal changes to inputs from the central nervous system, while maintaining precise control over fin motion.
The early application of Central Pattern Generators (CPG) in motion control was evident in the simulation of humanoid robots and salamanders, utilizing various oscillators In 2006, Dai-bing Zhang and colleagues implemented controlled CPG in a fish robot based on Orjan Ekeberg's model, proposing a sine-cosine oscillator that they deemed more effective and flexible for fish movement compared to traditional oscillators like Matsuoka and Hofp In the same year, Daisy Lachat and her team explored CPG for the motion control of boxfish, opting to create distinct oscillators for each movement joint rather than using classical oscillators Their research aimed to demonstrate the flexibility of movements such as head turning and body waving in response to varying amplitude and frequency signals, although they did not provide detailed information on the CPG controller or the parameters essential for quality locomotion control.
In 2008, Auke Jan Ijspeert and his team outlined key principles for designing locomotion control based on Central Pattern Generators (CPG) in robotics, emphasizing specific steps to enhance the effectiveness of these systems.
The design and architecture of the Central Pattern Generator (CPG) play a crucial role, encompassing the selection of oscillators or neurons within the circuit Furthermore, it is essential to determine whether to implement position control or torque control for robotic applications.
The selection of coupling type and topology is crucial as it influences the synchronization conditions between oscillators and the resultant gaits, which include the stable phase relationships among them.
The waveforms play a crucial role in determining the trajectories of each joint angle during a cycle, influencing which movements are executed These waveforms are shaped by the limit cycle generated by the selected neural oscillator, and incorporating filters can further modify their characteristics.
Input signals play a crucial role in modulating control parameters, impacting key aspects such as frequency, amplitude, phase lags, and waveforms Additionally, feedback signals from the body influence the activity of the Central Pattern Generator (CPG), allowing it to adapt its function by accelerating or decelerating based on environmental conditions.
(5) - The fact that these five design axes are all highly interconnected presents a significant challenge when developing CPGs.These steps later become the standard procedure for developing locomotion control
Yonghui Hu and his team developed a Central Pattern Generator (CPG) network using a Mastsuka oscillator to control fish movement through a combination of pectoral, body, and caudal fins Their findings indicate that the controller enables smooth motion adjustments in frequency and amplitude, which is crucial for safeguarding the servo motor from potential damage.
In addition, the article also mentions the genetic algorithm to find the optimal swimming posture to achieve the highest speed [23]
The four-legged turtle robot, developed by Wei Zhao and his team, underwent testing for the application of an online swimming patent, showcasing its innovative design and functionality.
In 2008, researchers developed an artificial neural network utilizing Central Pattern Generators (CPG) with loop connections to coordinate the movement of a four-legged turtle robot, as illustrated in Figure 1-5 Simulations and experiments have shown that this robot can navigate smoothly by adjusting frequency and amplitude, exhibiting fluid movements similar to a real turtle, including diving, floating, and turning on the spot with remarkable flexibility.
Figure 1-6 CPG with a loop connection to control the movement of four legs turtle-like underwater robot [24]
Researchers from the Chinese Academy of Sciences have developed an innovative amphibious robot that can seamlessly transition between land and water, mimicking the movement of a fish This robot features a unique design with two wheels and a flexible body, utilizing a CPG-based motion controller to synchronize the movements of its joints and front wheels The CPG network's serial arrangement of links and associative branching enables smooth movement during transitions between environments, requiring minimal parameters from the high-level controller The introduction of a sliding controller in the second version has further enhanced the robot's capabilities, significantly increasing its turning radius while walking on land.
Figure 1-7 Configuration of the formulated CPG model (a) simplefied structure (b) CPG network configuration [25]
Chen Wang and colleagues recently introduced a novel CPG-based locomotion control method, demonstrated through a robotic fish model This innovative coupled linear oscillator system simplifies the CPG model by replacing nonlinear oscillators with linear ones, enhancing practical implementation Additionally, the model maintains satisfactory dynamic performance due to its adaptive structural parameters, offering significant advantages over existing models.
Discussion & Objective of the Disertation
Earlier studies have successfully applied Central Pattern Generators (CPG) to the locomotion control of biomimetic robots, yet most rely on trial-and-error data fitting to adjust the convergence rate—a crucial control parameter While increasing the convergence rate can shorten the processing time needed to achieve a limit cycle, it may also lead to oscillatory errors, which represent the discrepancy between the intrinsic amplitude of the CPG and its output's maximum amplitude envelope This optimization challenge remains a significant focus for researchers Some studies have utilized the Particle Swarm Optimization (PSO) algorithm to fine-tune CPG parameters, aiming to minimize the difference between the desired oscillatory waveform and the CPG's generated output, reduce control parameters, and enhance feature parameters.
Recent studies on CPG-based bio-fish robots have overlooked the optimization of the convergence rate, which is crucial for adapting swimming forms Drawing inspiration from research that utilizes reinforcement learning (RL) in conjunction with CPG, this dissertation presents an innovative approach to optimize the locomotion controller through a CPG network specifically designed for elongated undulating fins.
Central pattern generator (CPG) networks are essential for controlling the rhythmic movement of bio-robotic fish To enhance the thrust of these robotic fish, a genetic algorithm addresses the slow response time of the CPG Research has demonstrated the use of a CPG model to identify critical factors influencing propulsion and achieve undulating motion patterns Despite the success of existing mathematical models in establishing CPG-based motion controllers, enhancing propulsion force remains a significant challenge Some researchers have employed optimization algorithms to select optimal parameters, while a Hopf oscillator-based CPG network has been used for parameter synthesis to achieve the desired swimming patterns.
The integration of Andronov–Hopf oscillators with an artificial neural network (ANN) has enabled a modified central pattern generator (CPG) to facilitate unique movements in real fish Recent advancements in heuristic search techniques have significantly contributed to the optimization of CPG network parameters Notably, a genetic algorithm (GA) has been employed to generate rhythmic patterns by adjusting the weight values associated with oscillator coordination Additionally, the utilization of particle swarm optimization has been explored in related studies to further enhance CPG models.
This dissertation explores the use of differential particle swarm optimization (D-PSO) to enhance the parameter selection for Hopf oscillator-based central pattern generators (CPGs) aimed at improving propulsion While traditional particle swarm optimization (PSO) effectively identifies CPG parameters, it often faces challenges with local optima By increasing the amplitude values within the CPG network, the study demonstrates a significant improvement in the average propulsive force of the undulating fin robot, resulting in faster movement.
Outline of the Dissertation
The dissertation is presented, including six chapters:
Chapter 1: focuses on researching scientific publications in the same field to find out the contribution orientation of the dissertation
Chapter 2: Building a Motion Controller for a Specific Fish Robot Propulsion Module Model
Available on the CPG Platform Simultaneously, model the propulsion mentioned above system module
Chapter 3: Research on optimizing the specificity coefficient for the stroke switching speed of the locomotor controller built by a reinforcement learning algorithm
Chapter 4 focuses on the research aimed at identifying the optimal set of amplitude parameters for motion controllers The study emphasizes maintaining a consistent frequency while maximizing thrust using a swarm optimization algorithm This approach aims to enhance the performance and efficiency of motion controllers in various applications.
Chapter 5 focuses on evaluating the flexibility in changing swimming postures, optimizing the speed characteristic coefficient identified in Chapter 3, and measuring the thrust generated by the optimal parameters determined in Chapter 4.
Chapter 6: Conclusions and Future Research
DESIGN SWIMMING GAIT CONTROLLER AND THRUST MODELING
This chapter presents a locomotion controller for an elongated fin robot, inspired by the black Knifefish The controller utilizes a modified Central Pattern Generator (CPG) network, consisting of sixteen coupled Hopf oscillators that incorporate feedback from each fin-ray's angle With this controller, the robot can seamlessly adapt its swimming patterns Furthermore, it allows for the adjustment of key swimming pattern parameters, such as amplitude envelope and oscillatory frequency, facilitating a range of diverse swimming motions.
Elongated undulating fin description
The elongated undulating fin features sixteen interconnected fin-rays, each driven by an RC servo motor that allows for movement around a fixed rotary joint This design enables each fin-ray to act as a shaker bar with a limited sway angle, where the phase difference between adjacent fin-rays is known as the phase lag angle By adjusting kinematic parameters like amplitude envelope, oscillatory frequency, and swimming pattern, the propulsive force can be modified For forward and reverse motion, the fin can change the phase lag angle's sign, and to mitigate counter-torque, it is essential that the number of oscillation wavelengths remains even Traditional swimming gaits are typically based on sine generators.
Fish propel themselves forward by rhythmically undulating and oscillating their fins and bodies This movement is guided by Lighthill's Elongated Body Concept, which describes their kinetic method of locomotion.
[34] According to Lighthill's theory, thrust is generated by the formula:
𝑓 𝑒 (𝑥) : is the envelop equation (see Fig 2-1) f: the oscillation frequency of a body part and the speed of the traveling wave are determined by this
The Swimming Gait Based on Sine Generators focuses on identifying suitable time-dependent control functions, as illustrated in Figure 2-1, which depicts the waveform typically utilized by undulatory swimming machines.
Figure 2-2 Parallel linkage mechanisms are used to make the fish robots move
Ai amplitude is determined with the following equation:
𝑓 𝑒 (𝑥) : is the envelop equation (see Fig 2-1)
𝑙 𝑖 : is the length of fin ray
𝑥 𝑖 : is the longitudinal position of the i th joint
Instantaneous angle of a Fin ray 𝜃 𝑖 (𝑡) with the following equation:
𝜑 𝑑,𝑖 : is the phase difference between adjacent fin
To effectively control fish fins in a dynamic aquatic environment, it is essential to continuously adjust both the frequency and amplitude of the sine motion controller This involves simulating variations in amplitude and frequency to achieve optimal control performance.
The common sine generator exhibits significant drawbacks, particularly when there are abrupt changes in amplitude or frequency, leading to disrupted oscillation and resulting in unsmooth locomotion of fish robots This issue, often overlooked in existing research [36]–[41], has prompted some scientists to develop a more effective oscillation controller based on Central Pattern Generators (CPG) to enhance movement fluidity.
Swimming gait controller for elongated undulating fin base on CPGs
Oscillating neuron models
The Central Pattern Generator (CPG) is a network of oscillators that creates rhythmic patterns essential for biomimetic robots Various types of oscillators, including Wilson-Cowan, Kuramoto, Matsuoka, Amplitude-Controlled Phase, Rowat-Selverston, and Hopf, have been effectively utilized in these applications.
The research focuses on utilizing the Hopf oscillator to create a modified Central Pattern Generator (CPG) for generating rhythmic locomotion in elongated undulating fins, as it has shown superior performance and adaptability compared to other methods A typical structure of the Hopf oscillator is illustrated in Figure 2-4.
Figure 2-4 Typical structure of Hopf oscillator The dynamic of the Hopf oscillator is expressed by the following differential equation:
𝑣̇(𝑡) = 𝑘(𝐴 2 − 𝑢 2 (𝑡) − 𝑣 2 (𝑡))𝑣(𝑡) + 2𝜋𝑓𝑢(𝑡) (2-4) where 𝑢, 𝑣 are time-variant state variables of the oscillator; 𝐴 is the amplitude of the steady- state oscillation; 𝑓 is the intrinsic frequency; 𝑘 is the convergence speed to the limit cycle (𝑘 > 0)
For comparison to the traditionally sinusoidal generator, a simulation of single Hopf oscillator is conducted with the same manner as shown in Figure 2-5
Figure 2-5 Output of Hopf oscillator in abrupt change of amplitude and frequency
The Hopf oscillator demonstrates an oscillatory output that facilitates smooth transitions during sudden changes in both amplitude envelope and oscillatory frequency at any given time \( t^* \).
The Hopf oscillator demonstrates rapid convergence to a stable limit cycle, regardless of the initial conditions By manipulating the parameter \( k \) in Eq (2.4), one can control the speed at which the output approaches the limit cycle with amplitude \( A \) As \( k \) increases, the convergence becomes even faster, showcasing the oscillator's efficiency in reaching stability.
The Hopf oscillator demonstrates a rapid convergence to a limit cycle, as illustrated by simulation results featuring eight distinct initial points for each scenario, shown in Figure 2-6 This behavior occurs despite abrupt changes in amplitude and intrinsic frequency.
In this thesis, the dynamic swimming motion is analyzed using Computational Fluid Dynamics (CFD) method, specifically employing the Hopf model to simulate the swimming behavior in the simulated environment
Figure 2-6 Convergence to limit cycle of Hopf oscillator
Coupling Schemes
The modified Central Pattern Generator (CPG) network features two terminal oscillators that operate independently of adjacent oscillators The nonlinear function representing this modified CPG network is defined as Ẋ 𝑖 = 𝐹(𝑋 𝑖 ) + 𝑃 𝑖, where 𝑃 𝑖 is expressed as [𝑘(𝐴 𝑖 − 𝑢 𝑖² − 𝑣 𝑖²)𝑢 𝑖 − 2𝜋𝑓𝑣 𝑖].
𝑝 𝑣,𝑖 ] (2-5) where 𝑋 𝑖 ≜ [𝑢 𝑖 𝑣 𝑖 ] 𝑇 is the state vector of the 𝑖-th oscillator; 𝐹(𝑋 𝑖 ) represents a nonlinear function; 𝑃 𝑖 ≜ [𝑝 𝑢,𝑖 𝑝 𝑣,𝑖 ] 𝑇 is a perturbation vector
To understand the interaction between two coupled phase oscillators, begin by establishing a random phase difference between them Examine a one-way coupling scenario where oscillator one influences oscillator two, without any feedback effect In this setup, the phase of oscillator one (P1) is zero, while the phase of oscillator two (P2) remains an unknown variable that needs to be determined This coupling configuration is illustrated in Figure 2-7.
Figure 2-7 Single –directional coupling between two oscillators
The output of oscillator 2, denoted as 𝒖 𝟐, is not directly affected by the output of oscillator 1 (𝒖 𝟏) Instead, oscillator 1's output indirectly influences the state of oscillator 2.
𝒗 𝟐 of the oscillator 2, as shown in Figure 2-8 the impact that 𝒖 𝟐 can make is internally coupled with 𝒗 𝟏
In the absence of coupling between two oscillators, both oscillators begin in the same state, where 𝒖 𝟏 equals 𝒖 𝟐 When coupling is introduced, oscillator two synchronizes with oscillator one when their phases align, specifically when the phase of oscillator one is at the same point as oscillator two plus an intermediate angle 𝜹 This angle 𝜹 is used to fine-tune the phase difference 𝝋 𝒅.
"The coupling" term on oscillator two can then be defined by:
Where 𝜺 is a positive constant that determines the coupling strength
In a polar coordinate system, perturbation acting in the direction of phase generate the phase Figure 2-8 tangential to the limit cycle
Figure 2-8 Illustration of perturbation in the direction of phase angle φ
This direction for oscillator 2 (without perturbed) is given by:
The perturbation on the phase is obtained given by:
By substituting Eqs (2.7) and (2.8) into (2.9), p φ,2 can be given in terms 𝛿 and 𝜑 2 :
𝑝 𝜑,2 In a phase-locking case, two oscillators evolve with stable phase difference 𝜑 𝑑,2 :
The two oscillators are synchronized with a constant phase difference 𝜑 𝑑,2 after a short transient phase evolving (from 0 to 𝑡 0 ) Therefore, in steady-state have:
The integration of Eq (2.13) over time t can be done implicitly by integration over φ2 in the steady-state of the system [43]
If the system evolves into the steady-state since 𝜑 2 = Φ 0 ( 𝑎𝑡 𝑡 = 𝑡 0 ), then phase locking is held on after 𝜑 2 = Φ > Φ 0 ( 𝑡 > 𝑡 0 ), Thus, Eq.(2.13) can be rewritten as:
Then substitute the (2.12) into (2.14), the following result is:
By solving Eq (2.15), the intermediate angle δ is:
2) = 𝜀(𝑐𝑜𝑠𝜑 𝑑,2 + 𝑣 1 𝑠𝑖𝑛𝜑 𝑑,2 ) (2-17) The coupling terms in condition of one-directional coupling can be given by:
The interaction between two oscillators is not solely one-directional; the output of oscillator (i+1) can also impact the behavior of oscillator (i) In this scenario, oscillator (i) is coupled with oscillator (i+1) with an opposing phase difference, as illustrated in Figure 2-9 The coupling terms 𝐏 𝑖 and 𝐏 𝑖+1 can be derived in a manner similar to that used in the derivation of Eq (2.7).
Figure 2-9 Mutual coupling between two oscillators The coupling formula for two mutually coupled oscillators can be expressed as follows:
Multiple oscillatory joints may be used in the movement of an animal or a robot that looks like an animal, which needs more joints to move in a coordinated way
A single oscillator could have a lot of coupling put on it, like the one in Figure 2-10
Figure 2-6 Couplings among three oscillators Figure 2-10 shows a chain structure of three serially connected oscillators Oscillator i is influenced by the outputs of the oscillator i-1 and oscillator i+1
In this case, 𝐏 𝑖−1 , 𝐏 𝑖 and 𝐏 𝑖+1 can also be defined by:
In both invertebrates and vertebrates, joint topological couplings play a crucial role in muscle function, exhibiting both excitatory and inhibitory effects The central pattern generator (CPG) in the animal brain is a complex network of neurons To replicate this CPG for controlling a biomimetic robot, researchers simplified the coupling connections into four primary topological structures: chain coupling, radial coupling, ring coupling, and full coupling Each structure possesses unique properties that correspond to the biological traits of different species For instance, chain joints are primarily utilized to facilitate the movement of swimmers, while fully connected joints are essential for the rhythmic motion of legged robots, ensuring all legs work in harmony to adapt to environmental changes.
The long undulating fin is composed of multiple rays that move irregularly due to environmental factors, impacting only neighboring fin rays To achieve the desired undulating motion, this study develops a sequence of sixteen oscillators that apply bidirectional perturbations to excite each fin ray The interaction between each fin ray and its adjacent rays is facilitated through these perturbations, as illustrated in Figure 2-10.
Conduct oscillation simulation comparing phase reversal from -𝜋/3 to 𝜋 /3 between one-way coupling and two-way coupling, I get the results as shown below Figure 2-11
Figure 2-7 Output u of two oscillators CPG1 and CPG3 for two types of coupling
Simulation results indicate that two-way coupling for phase reversal occurs in a shorter time (9 seconds) compared to one-way coupling (13.4 seconds) Consequently, I have opted for two-way coupling to effectively control fin-ray oscillation.
Configurations of Oscillators
In both invertebrate and vertebrate organisms, topological couplings between joints enable optimal muscle function through stimulus and inhibition The central pattern generators (CPGs) in animal brains consist of complex networks with large neurons To effectively replicate these CPGs for controlling biomimetic robots, it is essential to simplify the coupling connections into four primary topological structures: chain coupling, radial coupling, ring coupling, and fully connected coupling.
• Radial coupling: radial coupling topology is a single CPG topology that affects multiple cpg peers, see Figure 2-12
Figure 2-8 Radial type CPG coupling
This articulated topology is ideal for humanoid robots, allowing both legs to move in unison The spinal cord transmits signals that coordinate the actions of each leg, although the two-pin action signal is not direct.
Ring coupling refers to the cyclical structure of Central Pattern Generators (CPGs) that operate sequentially This mechanism is particularly relevant in the context of the four-legged jumping robot, as explored in the study "Moving Control of Quadruped Hopping Robot Using Adaptive CPG Networks."
Fully connected coupling represents the most intricate form of central pattern generators (CPGs) networks, showcasing numerous biological characteristics akin to those of primates The interconnected nature of CPG units necessitates the assignment of weights to each node in the network's structure.
Figure 2-10 Fully connected coupling Chain coupling:
Chain connections are the most prevalent design for fish robots, as each node is influenced solely by its immediate predecessor and successor This structure effectively showcases the natural movement patterns of fish robots, making it an ideal choice for their functionality.
Chain coupling exists in two main types: one-way and two-way One-way chain coupling is typically utilized in fish robots that rely on body movement, while two-way chain coupling is ideal for those employing fin mechanisms along the body, as each fin ray influences the adjacent fins.
Figure 2-11 One-way chain coupling
Figure 2-16 Two-way chain coupling
Swimming gait using Multiple Coupled CPG Oscillators
From the analysis in the above sections, the suitable CPGs model for Elongated Undulating Fin Robot is determined as follows:
This article examines the use of oscillators in consumer packaged goods (CPG), specifically focusing on the motion of fins along the body The Hopf oscillator is highlighted for its exceptional stability and self-healing capabilities, making it well-suited for the challenging conditions of underwater environments.
The unique biology of fish, characterized by evenly structured caudal fins, allows for a mutual interaction among adjacent fin rays, enhancing their movement This interconnectedness makes Multiple Coupling the ideal method for simulating realistic fish-like motions.
The Elongated Undulating Fin features fin rays arranged in a long row, allowing for natural movement through a chain coupling structure The upcoming design will adhere to this chain coupling framework.
By these options, the locomotion controller for the studied fin model is presented, as shown in Figure 2-17
Figure 2-12 Chain coupling structure CPGs model for Elongated Undulating Fin
According to this structure, the fin will be in the state of self-propelled control with the parameters f, A1…A16 from the higher-order controller
For the first oscillator (𝑖 = 1), there is only perturbation from the second oscillator (𝑖 + 1), thus the perturbation of the first oscillator is given by:
𝛽(𝑣 2 𝑐𝑜𝑠𝜑 𝑑 − 𝑢 2 𝑠𝑖𝑛𝜑 𝑑 ] (2-27) where 𝛽 is the coupling strength; 𝜑 𝑑 is the phase lag angle of two adjacent oscillator
In the same manner, the sixteenth oscillator is only affected by the perturbation from the fifteenth oscillators:
For 𝑖-th (1=2,3…14,15) oscillators, the perturbation vector is given by the following:
Corresponding to various amplitudes 𝐴 𝑖 , the modified CPG network can provide different swimming patterns for the elongated undulating fin, it thus can produce different propulsive forces