Kinematic study and implementation of a bio-inspired robotic fish underwater vehicle in a Lighthill mathematical framework
© Chowdhury et al.; licensee Springer. 2014
Received: 9 July 2014
Accepted: 7 October 2014
Published: 20 November 2014
This paper has focused on the formulation of the biological fish propulsion mechanism given by Sir J. Lighthill mathematical slender body theory for a bio-inspired robotic fish. A 2-joint, 3-link multibody vehicle model biologically inspired by a body caudal fin (BCF) carangiform fish propulsion is designed. The objective is to investigate and show that a machine mimicking real fish behavior can navigate efficiently over a given distance with a good balance of speed and maneuverability. The robotic fish model (kinematics and dynamics) is integrated with the Lighthill (LH) mathematical model framework. Different mathematical propulsive waveforms are combined with an inverse kinematics-based approach for generating fish body motion. Comparative studies are undertaken among a non-LH model, a LH model, and the proposed propulsive wave models based on a distance-based performance index. Proposed LH cubic and NURB quadratic functions are found to be 16.32% and 17.94% efficient than a non-LH function, respectively. With the help of the simulation results, closed-loop experiments are done and an operating region is established for critical kinematic parameters tail-beat frequency and propulsive wavelength. The simulation and experimental plots are compared and found to be similar to the kinematic behavior study of the biological yellowfin tuna.
Biomimetics  reflects the features and capabilities of the biological evolution  of a system that could be efficiently replicated or mimicked in a human engineered system to the design of new technologies and the improvement of conventional ones. This approach has been proposed to be the answer to the improved performance and reliability for large scale complex systems by faster adaptation with dynamic environment. One of the focused technologies has been the development of autonomous underwater vehicles  as a greater part to the increasing interest in unmanned underwater surveillance and monitoring. Of particular interests are regions of the underwater environment which are unexplored and dynamic as well as underwater detection, pollution source tracking, underwater archeology, search and rescue, and so forth. The study of underwater evolution of life and its plethora of locomotion modes has long been a subject of interest to the biological community. Majority of conventional design of autonomous underwater vehicles used propellers as their principal mode of propulsion. The propeller-based locomotion , although rendered the initial answers to underwater locomotion, set issues on high-maneuverability, efficiency, and low power consumption. The scientific community and researchers also found that propeller-strikes produce greater amount of marine debris, marine creature’s mortality and shallow water ecosystem disturbances. Biomimicked or fish-like robots are expected to be quieter, more maneuverable (lesser accidents), and possibly, more energy efficient (longer missions). Undulating-finned robot can preserve undisturbed condition of its surroundings for data acquisition and exploration (stealth). The movement of fish through water without creating ripples and eddies were more reasons to choose a bio-inspired design for underwater locomotion. Considering the propulsive features  of existing fish modes, a novel propulsive mechanism that integrates fish-like swimming with modular links and fin movements has been proposed  where the modeling, simulation, and development studies of a body caudal fin (BCF)  carangiform-based prototype is built and tested. The robot will be able to implement speedy and efficient fish-like swimming. The present research work specifically identifies the usefulness of present model for purposes of both speed and maneuverability. The kinematics-based approach allows producing a dynamic body motion that can reproduce the fluid flow pressure field generating the undulatory motion of the fish. Further, the kinematics and dynamics study helped to frame the mathematical formulation of the fish body motion describing its dynamic behavior. From a robotics perspective, defining and enhancing the swimming efficiency is still a kernel issue in the study of robotic fishes. However, the fish robot swimming like a real fish does not guarantee that it would achieve the same high efficiency . The undulatory (oscillatory) nature of the fish motion has been prominently mentioned in several works . Another solution adopted by researchers - is to conduct large number of experiments and find empirical expressions to refine the body’s motion functions and fin propulsion. Due to the complex nature of the mechanical system, the paper focuses on developing a linear system model using robot dynamics derivation. The simulation environment is in MATLAB©, Simulink©, and SolidWorks©. The contributions in the present work are enumerated as following:
Mathematical input waveforms are proposed in LH framework to generate different types of (undulatory/oscillatory) body propulsive waves. Comparative study of each model with the fundamental LH quadratic wave model is done for a performance parameter based on the total trajectory length.
Each of the bio-inspired wave function combines with an inverse kinematics-led trajectory planning resulting in a bio-inspired algorithm to produce laterally compressed waveforms of the caudal region.
Integrate the present robotic fish mathematical model (kinematics and dynamics) with the proposed bio-inspired algorithms resulting from different mathematical inputs.
Finding the operating region (ORE) for the identified kinematic parameters to facilitate closed-loop control based on the characterization of the biological fish swimming model.
The underlying subject of this research is to explore the remarkable ability of mathematical (applied mechanics) and physical concepts leading to fundamental insights behind fish biology. It has been found that sophisticated arithmetic calculations can bring practical benefits in revealing the fundamental biological systems. Darwinian-inspired evolution in modeling of biological systems is a strong partner of experimental work in physical sciences based on the principles in Newtonian and quantum frames. Present research investigates into this idea. The limitation of the paper is the linearized model of the robotic fish to undertake the study. Another assumption being made is stationary fluid, and therefore, the forces acting on a single link are due to the motion of that link. This may introduce inaccuracies in the manipulator model, while the discretized algorithm is implemented in a digital computer. The paper is organized as follows. In the ‘System model’ section, the design characteristics and features of the prototype are explained with the kinematics and dynamics modeling studies of a three link robotic fish. The ‘Lighthill mathematical framework design’ section discusses the Lighthill’s existing mathematical framework and its integration with the present prototype. This section also investigates the proposal and study of the different mathematical input waveforms generating the fish undulatory and oscillatory motion. The ‘Results and discussion’ section presents the robotic fish body undulatory propulsion mechanism for a given trajectory, simulation, and experimental results with/without the Lighthill wave function as well as with different mathematical input in LH framework. Comparisons are based on a distance-based performance index. Closed-loop experiments  were made on the basis of comprehensive kinematic analysis governing the fish undulatory motion. An operating region is found for the dominant kinematic parameter TBF and propulsive wavelength each to facilitate future experiments for such bio-inspired systems. It is also verified with the reported literature for the real fish motion. In the ‘Conclusions’ the conclusions and directions for future work are discussed.
The BCF mode carangiform style  of locomotion shown in Figure 1 is approximated using a 3-link (including the pectorals attached to the head) mechanism with two actuated joints as illustrated in Figure 2. The first link as the ‘head’ and second link as the body are roughly two-thirds of the length of the entire robot. The tail of the robot is formed by the third link connected to the caudal fin. The configuration of the robotic fish and non-fixed head is illustrated. Our 3-link mechanism (n = 2) is a reasonable approximation to the BCF carangiform locomotion, and therefore, small modifications of this model should have been used in the analysis of carangiform swimming. Moreover, the caudal peduncle and fin have been rendered in to a thunniform (shark) semi-crescent structure for an efficient thrust and therefore allowing high cruising speed for longer period of time . While the kinematics study explains the geometry of the motion of robotic fish w.r.t, a fixed reference coordinate frame as a function of time, the dynamics of any rigid body - can be completely described by the translation of the centroid and the rotation of the body about its centroid. This leads to the ability to derive the actuator torques necessary to produce the tail motion that is desired. A linearized kinematic and dynamic model of the robotic fish system is developed. The present research work investigates into the system modeling as an n-joint manipulator-based mobile vehicle; the earth-fixed frame has been defined w.r.t. the fixed body reference frame. Relevantly, the two major sections of the present paper robotic fish system model are the following:
Denavit-Hartenberg (DH) kinematics model : The kinematics due to translation and rotation along the joints on the robotic fish system in fluid (water) environment.
When describing the kinematics and dynamics of the model shown in Figure 2, the interlink actuator shaft constitutes the inertial frame of reference. A local coordinate frame is assigned to each degree of freedom (DOF). The coordinate frames are assigned according to the standard Denavit-Hartenberg notation  mentioned in Appendix A.
Generalized equations of motion
Lighthill mathematical framework design
Undulatory Lighthill quadratic amplitude body wave
where ybody is the transverse displacement of body, x is the displacement along main axis, k is the body wave number (k =2π/λ), λ is the body propulsive wave length, c 1 is the linear wave amplitude envelope, c 2 is the quadratic wave amplitude envelope, and ω is the body wave frequency. Taking this body wave, the present research does kinematic analysis to determine the proper body wave parameters (c 1 , c 2 , k, ω, etc.) for a desired and efficient undulatory swimming motion. We have further compared other mathematical functions in this frame (given below) to find their suitability to replace for a better undulatory wave function for robotic fish rectilinear swimming in a two-dimensional plane.
Undulatory Lighthill cubic amplitude body wave
As the curvature of the wave primarily depends on the second derivative, it is found to be continuous here, to assist in velocity control. A major advantage of adding local intermediate points is not only to fit a smooth curvature for the harmonic motion but also flexibility to the body in physical terms. Throughout a given time period when the speed, orientation, and velocity changes gradually, these factors render the resulting trajectory smoothness. In physical terms, this smoothness means that there are lesser abrupt changes in power output for the robot’s drive system, thus reducing navigational errors and helping to moderate the robot’s energy consumption. Therefore, the function allows specifying undulatory motion that conserves both battery power while reducing travel time and minimizing navigational errors.
Non-uniform rational B-spline (NURB) quadratic and cubic body wave (tadpole-like motion)
Undulatory SINC and DIRIC body waves
It can be noted that the quadratic wave amplitude is in the denominator and is used to dampen the oscillating wave function unlike the sinc function where it is allowed to grow as the numerator.
Undulatory anguilliform body wave (EEL like/maneuvering model)
where y is the lateral position of the midline, x is the coordinate following the midline, α is the amplitude growth rate parameter , L is the body length, and λ is the propulsive wave length. Modifications were done in this equation as compared to the original equation due to the fact that the present robotic fish model is carangiform so the DOF is restricted to the body posterior (caudal) unlike anguilliform where the entire centerline participates in undulation.
The idea of testing a modified anguilliform equation in a carangiform frame is whether maneuverability can be improved in slow undulatory motion. Two changes have been accommodated in order to undertake this test. Firstly, by replacing the constant amplitude by a quadratic amplitude wave and secondly, by provisioning that α can act as both an amplitude growth and dampening factor. This mathematical expression shows that a main wave function (sine or cosine) is decomposed in two Fourier functions. The primary function is again an oscillating sine wave which is multiplied by an exponentially growing function (can be represented in Fourier series) to satisfy the boundary conditions. The continuous everywhere exponential function is used for damping the sine wave. This function also resembles the Laplace transform with open unity integral.
Results and discussion
Trajectory (geometric) points for mathematical oscillatory/undulatory propulsive waveforms
1 (start point)
9 (end point)
Trajectory (geometric) points for mathematical oscillatory/undulatory propulsive waveforms
1 (start point)
9 (end point)
The present research has focused on the relevance of Lighthill (LH) based biomimetic robotic propulsion of a proposed 2-joint, 3-link multibody vehicle model, biologically inspired by a BCF carangiform fish. The objective of this paper is to translate the BCF mode carangiform swimming behavior of a biological fish to a robotic fish to allow its energy efficient navigation over a given distance using a good balance speed and agility characteristics. The robotic fish model (kinematics and dynamics) is integrated with the LH mathematical model framework. Mathematical input waveforms are investigated in LH framework to generate posterior body undulatory movements. These functions are combined with robot inverse kinematics to generate various bio-inspired trajectories for the posterior robotic fish vehicle motion. Distance-based performance criteria for a given trajectory are proposed to do a comparative analysis for the input undulatory waveforms. Comparisons are done between a non-LH and a path defined in LH frame. Based on present kinematic model simulation of identified kinematic parameters, closed-loop experiments are done to establish operating region for two critical kinematic parameters TBF and PW. This finding also aims to facilitate future experiments for a robotic fish model. Interestingly, the robotic behavior in simulation and experiments (closed-loop) is showing swimming behavior similar to the biological fish mentioned by Lighthill slender body theory and Dewar's kinematic experiments on yellow fin tuna. The future work primarily focuses on the development of an improved inverse kinematics algorithm within the Lighthill framework. A behavior-based control strategy and its implementation for energy efficient undulatory fish motion would further strengthen the vision of the machines mimicking biology principles in a significant way.
Link and joint parameters (shown in Figure 2)
Joint angle (θ i ): the angle of rotation from the Xi−1 axis to the X i axis about the Zi−1 axis. It is the join variable if joint i is rotary.
Joint distance (d i ): the distance from the origin of the (i-1) coordinate system to the intersection of the Zi−1 axis and the X i axis along the Z i−1 axis.
Link length (a i ): the distance from the intersection of the Z i−1 axis and the X i axis to the origin of the i th coordinate system along the X i axis.
F I : inertial frame of manipulator-based system.
F B : base frame located at the center of mass of the base.
r B : position of frame F O relative to and projected onto frame F B .
r I : position of frame F O relative to and projected onto frame F O .
d i : position of frame F i relative to and projected onto frame F O .
r i : position of point on link i relative to frame F O .
i : position of point on link i relative frame F i .
b : position of point on the base relative frame F B .
b B : position of point on the base relative to frame F I .
v i : velocity of point on link i relative to frame F I .
v B : velocity of point on the base relative to frame F I .
We would like to thank Mr. Alok Agrawal of Purdue University for his key contributions for the mechanical CAD design of the robotic fish. We would also like to acknowledge the useful suggestions and feedback given by Mr. Shailabh Suman of Acoustic Research Lab NUS, Dr. Mandar Chitre of Acoustic Research Lab NUS, Dr. Pablo Alvaro Valdivia of Singapore-MIT Alliance for Research and Technology (SMART), Professor Marcelo H. Ang Jr., and Professor Xu Jianxin of Department of Electrical and Computer Engineering. We thank the Office of Defense Science and Technology Agency (DSTA) for their support of the present research.
This work is presently supported by the Defense Science and Technology Agency (DSTA), under the Ministry of Defense (Singapore), Government of Singapore under Grant R-263-000-621-232-MINDEF-DSTA for Underwater Vehicle Technology Project STARFISH.
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