Modeling and Control Design for an Autonomous Underwater Vehicle Based on Atlantic Salmon Fish

Biologically inspired autonomous underwater vehicles (AUVs) or biomimetic AUVs are made to replicate the structural and physiological features of aquatic species. Thus, incorporation of its design in AUV modelling provides higher efficiency at low speeds and improves maneuverability and controllability. This paper develops a biomimetic AUV design based on structural parameters and physiology of an adult Atlantic Salmon fish and proposes a robust control scheme for propelling the fins. For the biomimetic model design of AUV, a 3D CAD model is developed using the actual parameters of Atlantic Salmon fish. The hydrodynamic analysis is performed to calculate the effect of different angles of fin orientations on the value of drag and lift coefficients. Further, kinematic analysis of the tail propulsion system is carried out using the Denavit Hartenberg convention in the MATLAB®. Based on the obtained modeling parameters of AUV, a robust sliding mode controller is proposed for tracking the desired tail propulsion response using a DC motor under model uncertainties and disturbances. Moreover, the closed-loop asymptotic stability is also guaranteed through Lyapunov theory, which ensures the convergence of system states to the desired angular movement. Lastly, the proposed algorithm is validated using simulation results with comparative performance analysis to illustrate its efficacy.

currents and surges, create fewer wakes compared to other 92 underwater vehicles, and have quiet propulsion. The cau-93 dal fin and paired pectoral fins are primarily responsible 94 for the locomotion of fish. Based on locomotion, they are 95 classified into the median and paired pectoral fins (MPF) 96 and body caudal fins (BCF). The investigation of admiringly 97 fast and efficient maneuvering of fish can open the doors 98 for better thruster design and modeling of AUVs dynamics. 99 There is only a little research on the development of AUVs 100 inspired by fish, where fish robots use fins instead of thrusters 101 [13], [14]. 102 The motion control of a biomimetic fish is also an essential 103 aspect of such robots, and the movement of soft tail achieves 104 this propelling swimming motion in the forward direction. 105 Yaw motion can be controlled by maintaining greater fluidic 106 volume in one fish half than the other. By changing the 107 angle of attack of diving planes, the pitch can be regulated. 108 The tail's forward swimming speed influences both yaw and 109 pitch control actions. Absolute attitude readings are provided 110 via an onboard inertial measurement device with 9 degrees 111 of freedom. These measurements are then used to design 112 closed-loop attitude control for robotic fish [15]. The objec-113 tive of tracking control of a biomimetic fish is to track a route 114 in space specified by a time function starting from a prede-115 termined initial state. It can also include following a moving 116 target point [16]. Many researchers are now working on 117 controller design for path-tracking of biomimetic robotic fish, 118 primarily focused on creating an energy consumption model 119 by examining the dynamics and motion traits of bio-inspired 120 fish and determining the link between behavior and energy 121 consumption.

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Due to the complex underwater environment as well as 123 the high coupling and nonlinear properties of bio-inspired 124 robotic fish, underwater path tracking control design is more 125 tedious than other environmental control. In recent years, 126 few control strategies have been implemented for the con-127 troller design biomimetic underwater fish, such as PID con-128 trol, sliding mode control, auto disturbance rejection control, 129 fuzzy logic control, line-of-sight method, etc. The conven-130 tional PID control approach is applied in [17] for the atti-131 tude tracking problem. However, to deal with challenging 132 fish dynamics and uncertain underwater surroundings, PID 133 control is not always an appropriate method because of the 134 complexity of PID control parameter tuning. Consequently, 135 this makes it ineffective for the multi-objective optimization 136 of complex MIMO  Atlantic Salmon Fish.

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The key features of this work is discussed below:   The paper is organized as follows. Section II describes 180 the Atlantic Salmon fish system and gives its CAD design.

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Section III presents the CFD analysis of the fish model.

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In Section IV, the tail propulsion system is discussed, and 183 Section V demonstrates the Denavit-Hartenberg (D-H) rep-184 resentation for the kinematic analysis. The proposed robust 185 controller for the DC motor and its stability is presented in 186 Section VI and Section VII, respectively. Then, Section VIII 187 shows the controller performance using simulation analysis.

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The paper ends with the concluding remarks in Section IX.  Atlantic fish is prepared in the SolidWorks and is illustrated 203 in Figure 2. The anterior region of biomimetic fish is assumed and 206 designed to be rigid. The argument in favor of this assumption 207 is that the skull bone majorly constitutes it. Additionally, 208 the front half of the fish does not actively undulate. Thus, 209 whatever intrinsic biological flexibility it may have may be 210 safely disregarded for design and analysis needs without 211 significantly changing the fish dynamics. The posterior region design is more complex than the anterior 214 region. The caudal fin attached to the posterior region is 215 responsible for producing propulsion. The rear body of the 216 vehicle is joined to the caudal fin with the help of a revolute 217 joint (1 DOF). The rear body is assumed and designed to be 218 rigid as to the flexible body of the real fish. The revolute joint 219 is regarded as the active joint because it actively engages in 220 a sinusoidal motion that resembles an undulating wave and 221 passively actuates the movement of the caudal fin to produce 222 propulsion. Thus, the flexibility between the rear portion and 223 the caudal fin is provided by this joint. i.e., it has a truncated shape [28]. The assembled model is  In the present study, CFD simulation will be used to deter-  The tetrahedral mesh has been prepared with a mini-

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The idea of dynamic meshing is effectively employed in this 283 project. A steady mesh is employed in CFD simulations to 284 take that the physical geometry remains constant. Dynamic 285 meshing, on the other hand, is a necessary aspect of modelling 286 for moving geometries. The challenges faced during the 287 body's dynamic meshing include parameter selection linked 288 to cell height and solver discovered divergence (pressure cor-289 rection). Furthermore, negative cell volumes were discovered 290 when the dynamic mesh update failed multiple times, along 291 with minor issues such as simulation time step smoothing, 292 etc.

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In Figure 3, the pressure contour is shown at the mean posi-295 tion when t = 0. The posterior body (caudal fin) begins to 296 move (starting from rest) in either direction (left or right) 297 when the fish initiates the locomotion. A high-pressure region 298 is observed in the anterior region (A) and the posterior region 299 (C) of the fish. The region (B) has a lower pressure region 300 than regions A and C. Similarly, in Figure 4, the velocity 301 vector is shown at the mean position when t = 0. Velocity 302 difference can be seen in region-A, followed by region B, and 303 at the fin end (region C). In Figure 5, the velocity streamline 304 VOLUME 10, 2022    The velocity vectors vary with higher magnitudes in regions 331 A and C and lower magnitudes in region B. In Figure 9, the 332 velocity streamline is shown when the posterior region of the 333 fish starts undulating in the clockwise direction (10 degrees). 334 It can be seen that the flow is streamlined till the end of the 335 anterior body (regions A and B). It starts to separate from the 336 posterior body onwards, thus leading to flow separation and 337 vortex formation in region C.

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In Figure 10, the pressure contour is shown when the rear 339 part of the body (posterior region) of the fish starts undulating 340 in the counterclockwise direction (10 degrees). The posterior 341 97590 VOLUME 10, 2022      in the region-D near the posterior body of the fish. 354 In Figure 11, the velocity vector is shown when the posterior 355 region of the fish starts undulating in the counterclockwise 356 direction (10 degrees). The high pressure gradient creates 357 an intense velocity vector field surrounding the body. These 358 velocity vectors constantly vary with larger magnitude in 359 regions A and C and smaller magnitude in region B. In 360 Figure 12, the velocity streamline is shown when the posterior 361 region of the fish starts undulating in the counterclockwise 362 direction (10 degrees). It can be seen that the flow is stream-363 lined till the end of the anterior body (regions A and B). 364 It starts to separate from the posterior region onwards (same 365 VOLUME 10, 2022    When the rear part of the body (posterior region) moves 405 to the right completely (20 degrees), it can be seen that the 406 magnitude of pressure is higher on the left as compared 407 to the right side. In Figure 16, it is shown that the high 408 pressure area is concentrated near the rear part of the body 409 (posterior region) of the fish (regions C and D). In Figure 17, 410 velocity vector is shown. The high pressure gradient cre-411 ates an intense velocity vector field surrounding the body 412 (regions A, B, and C). These velocity vectors vary constantly 413 with larger magnitude (higher than the previous condition of 414 10 degrees in both directions) on the left side and smaller 415 magnitudes on the right side of the rear part of the body 416 (posterior region). In Figure 18, the velocity streamline is 417 shown when the posterior region of the fish starts undulating 418 in the counterclockwise direction (20 degrees). It can be seen 419 that the flow is streamline till the end of the anterior body 420 (regions A and B), and it starts to separate from the posterior 421 region onwards (region C), thus leading to flow separation 422 and vortex formation.

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It is observed that raising the body wave's amplitude causes 425 the pressure field and velocity vectors to grow proportionally 426 along the tail fin (caudal region). It is also observed that 427 as the caudal region undulates via the intermediate point to 428 the extreme position, the pressure differential between the 429 body's two sides also decreases [31]. The vortex formation 430 near the caudal region is due to the pressure difference. 431  Manoeuvrability and redundancy can be improved by further 465 addition of links for smooth wave generation [14]. However, 466 the addition of links results in increased complexity, thus 467 making the realization of the tail design challenging [32].

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The proposed structure of the vehicle is intended to have 470 2 revolute (1 DOF) joints, each powered by an individual 471 DC motor. The undulating motion required to move the 472 bio-inspired vehicle is produced by the combined motion of 473 these actuated joints. The location of the joints at particu-474 lar points in the swimming gait must be established using 475 kinematics to manage the tail motion. Hence, the D-H rep-476 resentation is implemented where each joint is assigned a 477 2-axis reference frame, as demonstrated in Figure 21. The 478 transformation matrix is given by [34] 479 i−1 From the transformation matrices in the equation above 481 (frames), the x and y coordinates for each link are obtained. 482 The homogeneous transformation matrix has the rotation 483 matrix and displacement vectors for the frames. Since our 484 concern is about the end position of the caudal fin (2D planar), 485 we will focus on displacement vectors for the positions and 486 will be neglecting the orientations from the rotation matrix, 487 as 2D orientation does not make sense. The planar fish links 488 are simulated using Denavit-Hartenberg transformation in 489 Matlab. Figure 22 shows the path traced by the caudal fin and 490 rear body.

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The designed model of the fish consists of two joints that 493 constitute the propulsion system. Separate DC motors actu-494 ate these joints. Each joint is actuated to a desired angular 495 position so that the combined motion of all joints mim-496 ics the undulation exhibited by the real Atlantic Salmon. 497 VOLUME 10, 2022  The dynamic equations of the DC motor are given by [35]: where J > 0 is the moment of inertia, θ ∈ R is the 514 Therefore, the system (7) under such model uncertainties can 515 be redefined as The third derivative system 8 can be simplified into a single 521 derivative system by using the following state transformation. 522 Let 523 x 1 = θ; x 2 =θ; x 3 =θ .
(9) 524 Therefore, 526 where x 1 , x 2 , and x 3 represent the angular position, angu-529 lar velocity, and angular acceleration of joint, respectively. 530 An effective controller is needed to turn the DC motor to 531 the proper angular position so that the combined action of 532 each joint results in an undulating movement. The controller 533 should be able to perform the desired motion while rejecting 534 the effects of total disturbances and uncertainties.

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While developing the proposed control scheme, the follow-536 ing presumptions are taken into account.

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Remark 1: The supporting argument for the boundedness 541 of the uncertainties in Assumption 2 can be explained as 542 follows. First of all, parameters A, B, and C are constituted 543 using physical parameters, such as inductance, resistance, 544 moment of inertia, coefficients of friction, and torque. Since 545 these are all finite values, the uncertainties within A, B, and C 546 are bounded. Secondly, the designed sliding mode-based con-547 trol law (15) is comprised of finite system parameters and a 548 linear combination of error and system states. Because of the 549 physical limitation of the system, these error and state values 550 are bounded. Also, the actuation of DC motor is subjected to 551 physical limitation, and they can not exceed the maximum 552 limit. Therefore, the control input has a saturation value, 553 which means u is also bounded. Lastly, it is assumed that the 554 external disturbance T d and its derivative are bounded [36]. 555 Consequently, the overall uncertainties can be considered to 556 be bounded [37].

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The objective of the proposed controller design is to achieve 559 the desired motion of fish by following the reference trajec-560 tory of fin position. In other words, the controller will actuate 561 the desired fin motion in order to track the desired path. 562 Moreover, the proposed controller must also tackle the system 563 the tracking operation [38]. 565 In view of the above statement, suppose x r be the 566 time-varying reference angular position. The error between 567 the actual and the desired angular position is given as 569 Therefore, the relative system dynamics using (10) and (11) 570 can be expressed as The sliding mode control (SMC) has been widely used for the 576 robust controller design [39] due to the property of invariance 577 against unknown disturbances and faster convergence time 578 [37], [40], [41], [42], [43], [44]. Therefore, in this paper, SMC 579 is proposed for the motion control of fins through DC motors, 580 which is subjected to model uncertainties and disturbances.

581
The structure of the sliding surface is selected as The proposed sliding mode control law is expressed as The derivative of V with respect to time yields Substituting the value ofṡ in (19) gives Substituting u from (15) inV yields where χ 1 = 2k 1 > 0 and χ 2 = √ 2k 2 > 0. Consequently, the 619 inequality (19) satisfies the finite time condition of Lemma 1. 620 Therefore, the sliding surface s converges to zero within finite 621 time. where is the Laplace variable. Since c 1 , c 2 , and c 3 are all 628 positive, Equation (20) is a Hurwitz polynomial with positive 629 coefficients. Therefore, the relative state is stable, i.e., e will 630 converge to zero asymptotically. Consequently, the angular 631 position θ will track the desired trajectory θ r . Hence, the 632 proof of Theorem 1 is completed.

Remark 2:
The key insights which can be taken from the 634 proposed investigation are as follows: 635 • A biomimetic model of the Atlantic salmon fish will 636 exhibit lower drag. Thus, it will have higher efficiency 637 than a propeller-based underwater vehicle.

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• Mimicking a real fish's locomotion can improve 639 the propulsive and maneuvering efficiencies of AUV 640 Design.

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• The hydrodynamic and Kinematic analyses give the per-642 formance verification of the biomimetic model.

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• Since it is difficult to obtain the exact parameter val-644 ues of the design model; therefore, a robust controller 645 scheme will be required.

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• A fast controller scheme, such as the SMC method, will 647 be required to achieve a quick system response with a 648 good transient and steady-state behavior. The controller 649 VOLUME 10, 2022 lar motion.

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• In summary, the biomimetic AUV system can be an effi-652 cient alternative to the existing propeller-based systems. tion (FBL) control [47].

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The uncertainties in A, B, and C parameters are considered 664 to be 10%, 15%, and 5% of the nominal value, respectively.

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The exogenous disturbance is considered as 666 T d = 0.5 sin(0.5 t).

667
The reference signal for angular movement is considered to

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The physical parameters of the DC motor, gear system, and 682 proposed controller are presented in Table 1.   performance of these three control schemes is also evaluated 699 using the measures of integral of square error (ISE), integral 700 of absolute error (IAE), integral of time square error (ITSE), 701 and integral of time absolute error (ITAE). The expression of 702 these performance measure can be seen from [49]. The val-703 ues of these performance measures are tabulated in Table 2, 704 which clearly shows that the proposed scheme has a minimum 705 value in all the error measures as compared to the other 706 two methods. Therefore, the proposed robust SMC scheme 707 performs best under multiple uncertainties and provides a 708 faster convergence with better error measures.
709 Figure 26 illustrates the sliding surface response of the 710 proposed scheme where the sliding mode phase is achieved 711 in 1s. Once the sliding phase is attained, the surface remains 712 at this mode for the whole time. The control input response 713 of these three control schemes is shown in Figure 27. Ini-714 tially, all the three control schemes have a high magnitude of 715 input response. The large input effort is required to rapidly 716 attain the desired reference signal. Once the desired angle 717    the disturbance. Moreover, compared with the comparative 723 results, the proposed SMC scheme is performing much better 724 in terms of good transient and steady-state response and error 725 performance measures.

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This paper explores the design, simulation, and mathematical 728 modelling of a biomimetic underwater vehicle, whose design 729 is inspired by an Atlantic Salmon fish. The vehicle moves 730 forward by performing an undulating motion with around half 731 of its body. Two joints are considered in the present design to 732 mimic the undulating tail movement of actual fish. Moreover, 733 a robust SMC scheme is also implemented to the given design 734 model for the fin motion of AUV using a DC motor subjected 735 to the model uncertainties and disturbances. The Lyapunov 736 analysis is carried out to show the asymptotic stability of 737 the relative angular states. Further, the effectiveness of the 738 proposed algorithm is validated using numerical analysis 739 with comparative performance. The next part of this work 740 is to design the 3-dimensional working model based on the 741 modelling analysis and execute the control algorithm.