Thrust and Swimming Speed Analysis of Fish Robot with Non-uniform Flexible Tail

Journal of Bionic Engineering 13 (2016) 73–83

Thrust and Swimming Speed Analysis of Fish Robot with Non-uniform Flexible Tail Phi Luan Nguyen, Byung Ryong Lee, Kyoung Kwan Ahn School of Mechanical Engineering, University of Ulsan, Ulsan, Korea

Abstract We present a dynamic model of a fish robot with a Non-uniform Flexible Tail (NFT). We investigate the tendencies of the thrust and swimming speed when the input driving moment changes. Based on the proposed dynamic model of the NFT, we derive the thrust estimation, equation of motion, and performance evaluation of a fish robot with a NFT. By defining the optimal stiffness of the NFT in simulation, a fish robot prototype is then designed and fabricated. A series of experiments are performed to verify the proposed model. Experiment results are in good agreement with simulation data. The results show that the thrust and swimming speed of the fish robot are proportional to the amplitude of the driving moment. There are two resonant frequencies (f = 1.4 Hz and 2.2 Hz), the maximum thrust and swimming speed (about 0.7 BL·s−1) are found to be around f = 1.4 Hz. The above results inidicate the proposed model is suitable for predicting the behavior, thrust and swimming speed of a fish robot with a NFT. Keywords: fish robot, flexible tail, dynamic modeling, swimming speed, thrust, Froude efficiency Copyright © 2016, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(14)60161-X

1 Introduction In recent years, fish robots have become the object of many studies because of the advantages including highly efficient swimming mechanisms, noiseless propulsion, and a less conspicuous wake[1]. However, most of these robots have been analyzed, controlled, and optimized using multi-joint rigid bar models[2–5,8] instead of a continuous flexible body. Most researchers have used two or three joints to model fish robots[2,4] (e.g., two or three Degrees of Freedom (DOFs)) because the complexity of these models usually increases with the number of joints. However, some studies[6–8] showed that flexible tails have better performance in terms of thrust and propulsion efficiency, and these mechanisms are simpler and more mechanically robust than a multi-joint rigid bar system. Barrett et al.[7] showed that the power to propel an actively swimming body at a Reynolds number of approximate 106 can be reduced by more than 50% compared with the power to tow the same vehicle in a rigid-straight configuration. Various methods of modeling fish robots have been Corresponding author: Kyoung Kwan Ahn E-mail: [email protected]

proposed. One method is based on dynamic equations that are usually derived by Lagrange’s principle with a simplified geometric model[5,9]. Another method is called trajectory approximation, which is based on kinematic equations. Equations that describe the kinematics of swimming were originally presented by Barrett et al.[7] and Lighthill[10], and were widely used in robotic fish research[3,4,11]. In particular, Liu and Hu[3] used a modified approximation approach to regenerate several special locomotion modes in their robotic fish. The cruise straight swim patterns were generated based on the traveling wave of Lighthill[10]. After modeling fish robots, some researchers have attempted to analyze and optimize the thrust and swimming speed. McHenry et al.[12] investigated the effects of the control variables, including body flexural stiffness, driving frequencies, and driving amplitudes, on the swimming speed in steady undulatory swimming. Their results showed that changing these control variables can control the swimming speed. In a previous study[13], we presented a dynamic model of a fish robot NFT. By approximating the geo-

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Journal of Bionic Engineering (2016) Vol.13 No.1

metric parameters of the NFT as exponential functions, an analytical solution of the governing equation was derived. This solution describes the lateral movement of the NFT as a function of material, geometrical, and actuator properties. This dynamic model successfully predicts the real lateral movement of the NFT of a fish robot. In the present work, in order to completely develop our proposed model for the purposes of control (e.g., speed control, tracking control, etc.), we developed a dynamic model, including a thrust estimation, an equation of motion, and a performance evaluation. The changing performance under the effects of stiffness of the NFT was investigated by simulation. Based on the performance evaluation, a fish robot prototype was designed and fabricated. A series of experiments that consist of verification of lateral movement of the NFT, thrust measurement using a load cell, and swimming speed measurement using a camera were set up to verify the proposed model. Simulation and experimental results were compared and analyzed. Results show that the thrust and swimming speed of the fish robot are proportional to the amplitude of the driving moment. However, the effect of the frequency of the driving moment is more complex. There are two resonant frequencies (f = 1.4 Hz and 2.2 Hz), but the maximum thrust and swimming speed (about 0.7 Body Lengths (BL) per second) occur around f = 1.4 Hz. We conclude that the proposed model is suitable for predicting the behavior, thrust, and swimming speed of a fish robot with a NFT. The remainder of this paper is organized as follows: In section 2, the lateral movement, the thrust estimation, the equation of motion, and a performance evaluation were presented. In section 3, we described the design and fabrication of the fish robot prototype, and the experiments. Simulation and experimental results are analyzed and discussed in section 4. Finally, our conclusions were given in section 5.

Sub-carangiform fish (including trout) have large, high aspect ratio tail. The posterior plays an important role in producing thrust force. The motion of this fish requires powerful muscles to generate lateral movements of the posterior. The anterior part remains in a relatively motionless state. Accordingly, the proposed modeling methodology concentrates only on the posterior of the fish robot, i.e., its tail. Fig. 1 shows a fish model with a NFT. In order to optimize the geometry of the NFT, its width profile was copied from that of a real fish, as proposed by Alvarado[14]. To minimize the recoil forces, the depth of the caudal peduncle of the NFT was reduced and the center of mass was set toward the anterior part of the fish robot[15]. Finding analytical solutions for the lateral movement y(x, t) of the NFT is generally complex. Long et al.[16] and Bahrami et al.[17] presented discrete approximations of continuous beams (such as a fish’s tail). Using their methods, a typical beam is partitioned into several continuous segments, each with a constant cross-section. Each segment has an analytical solution. By satisfying the boundary conditions, the motion of the beam was then derived. In addition, other groups[18–20] have used Computational Fluid Dynamics (CFD) to investigate the operation of fish robots. However, CFD is not suitable for control problems. El Daou et al.[21] and Alvarado et al.[14] used Euler-Bernoulli beam theory to derive the governing equations for a non-uniform fish’s body. El Daou used the trial eigenfunctions, which are the eigenfunctions of a uniform case. Although Alvarado derived a set of complex equations and differential equations to find the eigenfunctions, his method is very complicated. Similarly, we approximated (a)

y

Non-uniform flexible tail

M(t)

y(x, t)

x

2 Dynamic modeling of the fish robot 2.1 Lateral movements In order to analyze the motion of the fish robot, it is necessary to establish a dynamic model and characterize the lateral movements of the robot tail. We consider a shape similar to that of a sub-carangiform fish because of its fast swimming and high efficiency characteristics.

(b)

L (Tail's length) Anterior part

Posterior part

Fig. 1 The (a) top view and (b) side view of the fish model.

Nguyen et al.: Thrust and Swimming Speed Analysis of Fish Robot with Non-uniform Flexible Tail

the parameters of a NFT as exponential function in previous study[13], and then applied a method proposed by Suppiger and Taled[22]. As described in our previous study[13], lateral movements y(x, t) of a NFT under the effect of an input moment M(t) can be expressed as a linear combination of eigenfunctions qi(x). The coefficients ni(t) is a function of time: N

y ( x, t ) = ∑ ni (t )qi ( x),

(1)

i =0

where N is the number of eigenmodes (in our case N = 4), and ⎧q ( x) = C01 x, ⎪ Si1 x S x ⎞ ⎛ ⎪⎪ + Ci 2 sinh i1 + ⎟ ax ⎜ Ci1 cosh − ⎨ 2 2 ⎟ 2 , i = 1, 2...N ⎪qi ( x) = e ⎜ Si 2 x Si 2 x ⎜ ⎟ ⎪ + Ci 4 sin ⎜ Ci 3 cos ⎟ ⎝ ⎠ 2 2 ⎩⎪

75

based on the definition of the natural frequencies ωi and eigenmodes of the shape with respect to qi(x). The total movement, a rigid mode movement, and the first to fourth flexible mode movements of the NFT are shown in Fig. 2. As shown in Eq. (1), the total movement y(x, t) of the NFT is the sum of the rigid mode movement (i = 0) and the flexible mode movements (i = 1, …, 4). The higher flexible mode movements (i > 4) have smaller amplitudes compared to the lower case. Therefore, in our simulation, only the first four flexible modes were calculated. The coefficients ni(t) (i = 1, 2, … N), which are the function of time, are the solutions of the following set of equations:

0

(2) where C01, Ci1, Ci2, Ci3, Ci4 (i = 1, 2, … N) are coefficients, and Si1, Si2 (i = 1, 2, … N) are eigenvalues that are

⎧⎪ M i ni (t ) + Ci ni (t ) + K i ni (t ) = f i , with i = 1, 2, ... N (3) ⎨ ⎪⎩ M 0 n0 (t ) + C0 n0 (t ) = f 0 ,

where M i , Ci , K i , f i (i = 1 … N) are the modal masses, modal damping, modal stiffnesses, and modal forces, respectively[13].

Fig. 2 (a) A total movement; (b) a rigid -mode movement; (c–f) first to fourth flexible-mode movements of the NFT.

Journal of Bionic Engineering (2016) Vol.13 No.1

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curs at a frequency which is the same as the natural frequency of the NFT itself. When the frequency Ω of the external moment is equal to the natural frequency ωi, the denominator of N2i is minimized (see Eq. (9)), so the amplitude of N2i is maximized. Substituting Eqs. (2), (4) and (7) into Eq. (1), the lateral movements of the NFT can be rewritten as

Eq. (3) describes a set of second-order differential equations of n(t) functions. The solution can be written as follows: - For the rigid mode:

(

)

n0 (t ) = M 0 q0′ (0) N10 e− C0t + N 20 ,

(4)

where

⎡ Ω eC0t ⎤ ⎥, N10 = ⎢ − 2 + ⎢⎣ ( Ω + C02 ) C0 ΩC0 ⎥⎦

y ( x, t ) = M 0 q0′ (0)q0 ( x) ( N10 e−C t + N 20 ) + 0

(5)

N

∑ M q′(0)q ( x) ( N i =1

⎡ ⎤ C0 1 1 N 20 = ⎢ 2 sin(Ωt ) + 2 cos(Ωt ) ⎥ . (6) 2 2 Ω + C0 Ω ⎣ Ω + C0 ⎦

)

i =1

(7)

2 2 2 1 (ωi + 2ςωi − Ω ) sinh(ωid t ) + 2ςωiωid Ω cosh(ωid t ) 2 i

− Ω 2 ) + 4ς 2ωi2 Ω 2 2

(8) N 2i =

(ω

2 i

− Ω 2 ) sin(Ωt ) − 2ςωi Ω cos(Ωt )

(ω

2 i

− Ω 2 ) + 4ς 2ωi2 Ω 2 2

(

)

∑ M 0 qi′(0)qi ( x) N1i e−ξωit + N 2i .

.

(9)

ωi is the i-th natural frequency of the NFT, ωid is the i-th damping frequency, and ζ is the modal damping factor. In mathematical analysis, the differences between equations for the rigid mode and the flexible mode are that the coefficient K0 in the rigid mode equation is equal to zero (K0 = 0) and the coefficients Ki (i = 1 ... N) in the flexible mode equations are nonzero (Ki ≠ 0). Therefore, the solution of the rigid mode equation is simpler than the solutions of the flexible mode equations. In physical analysis, the modal mass for the rigid mode movement is equal to that of the actual NFT, whereas the modal stiffness K0 related to the same mode is zero because the NFT is unrestrained. On the other hand, the modal stiffness Ki related to the i-th flexible mode is equal to the square of the i-th natural frequency. Therefore, the solutions for the flexible mode are in form of a Lorentzian function, and this response is found in many physical situations involving resonant systems. Resonance occurs when the movement of the NFT oc-

(10)

(11)

Substituting Eqs. (6) and (9) into Eq. (11), we obtain

where

(ω

)

e−ξωi t + N 2i .

0

N

ni (t ) = M 0 qi′(0) N1i e −ξωi t + N 2i ,

ωid

1i

y ( x, t ) = M 0 q0′ (0)q0 ( x) ( N10 e−C t + N 20 ) +

- For the flexible modes:

(

i

The exponential terms will quickly decrease to zero as time increases. Eq. (10) can be rewritten as

and M(t) = M0sin(Ωt) (mN·m) is the input driving moment, Ω is the frequency of the external moment.

N1i =

0 i

y ( x, t ) = A( x)sin(Ωt ) + B( x)cos(Ωt ), ,

(12)

where 1 ⎡ ⎤ ⎢ M 0 q0′ (0)q0 ( x) Ω 2 + C 2 + ⎥ 0 ⎢ ⎥ A( x) = ⎢ N ⎥ , (13) ωi2 − Ω2 ) ( ⎢ ∑ M 0 qi′(0)qi ( x) ⎥ 2 2 2 2 2 2 ⎥ ⎢ i =1 − Ω + Ω ω ς ω 4 ( ) i i ⎣ ⎦ ⎡ ⎤ C0 1 − ⎢ M 0 q0′ (0)q0 ( x) ⎥ 2 2 Ω Ω + C0 ⎢ ⎥ B( x) = ⎢ N ⎥ . (14) 2ςωi Ω ′ (0) ( ) M q q x ⎢∑ 0 i ⎥ i 2 ⎢⎣ i =1 (ωi2 − Ω2 ) + 4ς 2ωi2Ω2 ⎥⎦

In an alternate form,

y ( x, t ) = Y ( x)sin ( Ωt + K ( x) ) ,

(15)

Y ( x) = A2 ( x) + B 2 ( x),

(16)

B( x) . A( x)

(17)

where

tan K ( x) =

2.2 Thrust estimation In Lighthill’s elongate body theory[10,23], the mean rate of work of the lateral movements is equal to the sum of the mean rate of work available for producing the

Nguyen et al.: Thrust and Swimming Speed Analysis of Fish Robot with Non-uniform Flexible Tail

mean thrust and the rate of shedding of kinetic energy of lateral fluid motions. The mean thrust T can be calculated entirely from the following conditions at the trailing edge of the caudal fin: displacement y(L,t), slope (∂y/∂x), and swimming speed U. The mean thrust is given by T=

2 ⎧⎪⎛ ∂y ⎞ 2 1 ⎛ ∂y ⎞ ⎫⎪ m( x) ⎨⎜ ⎟ − U 2 ⎜ ⎟ ⎬ , 2 ⎝ ∂x ⎠ ⎪⎭ ⎪⎩⎝ t ⎠

(18)

where m(x) is the added mass of the water per unit length of the fish (depending on the cross-sectional shape and dimension of its tail). Eq. (18) shows that the mean thrust mainly depends on the kinematic and geometric conditions at the trailing edge of the NFT (x = L). From Eq. (15), we have ∂y t ∂y x

= Y Ω sin(Ωt + K ),

(19)

x= L

= Y ′ sin(Ωt + K ) + YK ′ cos(Ωt + K ) = x= L

(20)

Y ′2 + (YK ′) 2 sin(Ωt + K + δ ),

where Y = Y(L), K = K(L), Y' = Y' (L), K' = K' (L), and YK ′ tan δ = . Y′ Finally, Eq. (18) can be rewritten as

⎧ 1 Ω 2Y 2 1 − cos(2(Ωt + K )) − ⎫ ( ) 1 ⎪⎪ 2 ⎪⎪ T = m( L ) ⎨ ⎬ 2 ⎪ 1 U 2 (Y ′2 + (YK ′) 2 ) (1 − cos(2(Ωt + K + δ )) ) ⎪ ⎩⎪ 2 ⎭⎪ 2 2 ⎛ U K′ ⎞ 1 2 2 1 = m ( L )Ω 2 Y 2 ⎜ 1 − ⎟ − U Y′ . Ω2 ⎠ 2 4 ⎝ (21) The mean thrust has a steady part and the second harmonic. The mass of the fish robot is large enough so that it does not respond to the harmonic part of the thrust. 2.3 Equation of motion When a fish robot swims, it is commonly assumed that it is possible to consider the forces exerted by the water on the fish robot in two separate parts. One force is the thrust, T, generated by the lateral movements of the fish tail, which pushes the fish robot along. The other

77

force is the drag, D, exerted by viscous shear stresses in the boundary layers on the body, which tends to slow the fish robot down. The drag is expressed as follows: D=

1 ρ Cd SU 2 , 2

(22)

where Cd is the drag coefficient, which can be determined using Ref. [24] (in our case Cd = 0.295). S is the frontal area of the fish and ρ is the density of water. The equation of motion of the fish robot is then formulated using Newton’s second law of motion:

T − D = m f U ,

(23)

where mf is the mass of the fish robot. During steady state, the fish robot swims at a constant swimming speed. Therefore, its average value can be determined as 2

⎛ ∂y ⎞ m( L ) ⎜ ⎟ ⎝ ∂t ⎠ U= = 2 ⎛ ∂y ⎞ m( L) ⎜ ⎟ + ρ Cd S ⎝ ∂x ⎠

(24)

m( L)Ω2Y 2 . m( L) (Y ′2 + (YK ′) 2 ) + ρ Cd S 2.4 Performance The Froude efficiency (propeller efficiency), which is the ratio of thrust power to the total kinetic energy added to the fluid, is calculated as follows[10]:

ηF =

TU , W

(25)

where W is the mean rate of work by the fish robot. W is defined as ⎧∂y ⎛ ∂y ∂y ⎞ ⎫ W = Um( x) ⎨ ⎜ + U ⎟ ⎬ , ∂x ⎠ ⎭ x = L ⎩ ∂t ⎝ ∂t

(26)

and W can be interpreted physically as the mean of the product of the lateral velocity ∂y/∂t at the tail trailing edge with the rate of shedding (ρVA)U of lateral momentum that is behind the trailing edge. This is based on the argument that the rate of work is equal to the swimming speed times the rate of change of momentum. Substituting Eqs. (18), (24) and (26) into Eq. (25), we obtain

Journal of Bionic Engineering (2016) Vol.13 No.1 2 ⎧⎪⎛ ∂y ∂y ⎞ ⎫⎪ ⎨⎜ + U ⎟ ⎬ ∂x ⎠ ⎪ TU 1 ⎩⎪⎝ ∂t ⎭x= L =1− ηF = 2 ⎧⎪ ∂y ⎛ ∂y W ∂y ⎞ ⎫⎪ ⎨ ⎜ + U ⎟⎬ ∂x ⎠ ⎭⎪ ⎩⎪ ∂t ⎝ ∂t x=L

=

Ω 2Y 2 − U 2 (Y ′2 + (YK ′) 2 ) Ω 2Y 2 + ΩY 2 K ′

(27)

.

The Froude efficiency ηF mainly depends on the geometry, actuator, and material properties of the NFT. The geometry of the NFT is copied from an actual fish, as previously mentioned, whereas the actuator parameters depend on the control requirements. Therefore, the material of the NFT is selected in order to optimize the Froude efficiency. The flexural stiffness is given by EI (N·m2), where E is Young’s modulus and I is the second moment of area. The flexural stiffness has a strong impact on the fish swimming performance. Changing the flexural stiffness EI changes the propulsive wavelength and wave speed, and then changes the swimming speed. It is interesting to note that live fish stiffen or relax their bodies to change the swimming speed. Our method involves simulating the relationship between Young’s modulus E and the Froude efficiency, and then selecting the optimal E in order to maximize ηF. To achieve this, the lateral movements of the NFT are first calculated based on Eq. (11). The thrust, swimming speed, and Froude efficiency are then calculated using Eqs. (21), (24) and (27), respectively. Fig. 3 shows the influence of Young’s modulus on the Froude efficiency ηF. The Froude efficiencies ηF are calculated for different values of Young’s modulus and different input frequencies. When Young’s modulus of the NFT increases to 4×105 Pa, the Froude efficiency first increases, then decreases when Young’s modulus increases from 6×105 Pa. This indicates that the optimal range of E is about 4×105 Pa to 6×105 Pa with respect to flexural stiffnesses from 8×10−6 N·m2 to 1.2×10−5 N·m2. This range is close to that of the live sunfish caudal fin[12]. Accordingly, we designed and fabricated a prototype of a fish robot. The main goal of material selection for the NFT is to maximize the Froude efficiency. A systematic selection of the best material for the fish robot begins with the Young’s moduli of the candidate materials. The NFT should have a Young’s modulus in the range of 4×105 Pa to 6×105 Pa. In addition, the NFT should be light in

weight in order to reduce power consumption and the total weight of the fish robot. In this case, PVC foam sheet (Foamex) can meet these requirements. Foamex is a lightweight material with good mechanical and insulation properties. It is easy to bend and has excellent water resistance and a smooth surface[25]. Specifications of the selected material are shown in Table 1.

3 Fish robot prototype and experiment setup 3.1 Fish robot prototype A prototype of the fish robot with an NFT was designed and fabricated as shown in Figs. 4 and 5. The dimensions of this robot are 450 mm × 50 mm × 125 mm (lengh×width×height). A servo-motor (SkyHolic DGS-1188, Hoppy Electronics Co.) is used to drive the NFT via a belt transmission. The maximum torque and speed of the motor are 100 N·cm and 7 rad·s−1 for unloaded condition, respectively. Pectoral fins were added to stabilize the motion. In addition, in order to protect the sensitive parts (i.e. the servo-motor, electric boards, batteries, etc.), thin silicon layers were inserted between the cover and the body of the fish robot. Moreover, waterproof bearings were used. Based on the analysis described in section 2, the shape of the NFT was determined and is shown in Fig. 5. A block diagram of the electric components is shown in Fig. 6. The fish robot is equipped with a

Froude efficiency

78

Fig. 3 The relationship between Froude efficiency ηF and Young’s modulus. Table 1 Specifications of the NFT Parameters

Nomenclature

Values

Young’s modulus

E

0.6 MPa

Density

ρNFT

1010 kg·m−3

Length

L

0.25 m

Nguyen et al.: Thrust and Swimming Speed Analysis of Fish Robot with Non-uniform Flexible Tail

module (Digi Xbee S2, Digi International Inc.) for communication with a host computer. The main processor of the control board is a microcontroller (ATmega128, Atmel Corporation) that receives commands from the host computer via a wireless module, and then controls the servo-motor using Pulse Width Modulation (PWM) signals. Two batteries with a total of 1500 mA·h of capacity can supply enough power to drive the fish robot. 3.2 Experiment setup 3.2.1 Verification of the lateral movements In order to identify the modal damping factors and verify the lateral movements of the model, experiments were set up as shown in Fig. 7. The fish robot with NFT was fixed in a steady position in a water tank (2 m × 0.6 m × 0.6 m). The lateral movements of the NFT were recorded in still water from the top using a Chargecoupled Device (CCD) camera (XC-HR70, Sony) and a frame grabber (Meteor-II, Matrox Electronic Systems Ltd). In our experiments, the acquisition speed was 15 frames per second (fps) and the image resolution was 1024 × 769 pixels. The movements of the trailing edge (at x = L) were calculated using Eq. (11). Then, they were compared with experimental results in the same way as our previous study[13]. Due to space limitations, only the comparison between the simulation (blue circles) and experimental (red squares) movements of the trailing edge (on which the thrust estimation mainly depends) is shown in Fig. 8. It is seen that the trajectories are almost the same. The maximum error between simulation and experiment is about 0.01 m, and the relative error is around 4% of the tail length. Generally, the proposed model clearly illustrated the actual lateral movements of the NFT of the fish robot.

3.2.2 Thrust measurements The thrust produced by the fish robot was measured using the test rig shown in Fig. 9. In this experiment, a force-sensitive resistor (FSTTM 400 series, Interlink Electronics) was used. Analog signals were first converted to digital signals using an analog-to-digital converter (ADC) function on an ATmega 128. Digital signals were sent to a computer via RS-232 communication. Fig. 10 shows the collected thrust data. Two peak forces occurred in one period, and each peak was produced in one half-cycle of a tail beat motion.

Servo motor

79

Pair fin Slewing shaft Waterproof bearing Flexible tail

Waterproof silicon layers

Gear

Fig. 4 Explosive drawing of the fish robot. Wireless module (Xbee S2) with external antena Control board

Batteries

Servo motor

Fig. 5 The fish robot.

PWM Wireless RS-232 Control board module

Servo motor

Belt

Flexible Tail

Batteries

Fig. 6 The block diagram of the electric components.

CCD camera

Xbee

Water tank

Fig. 7 The experiment setup for verifying the lateral movement of the NFT.

3.2.3 Swimming speed measurements In order to measure the swimming speed of the fish robot, experiments were set up as shown in Fig. 11. The fish robot freely swam in a long acrylic water tank (2 m × 0.6 m × 0.6 m). A camera (Bumblebee2

80

Journal of Bionic Engineering (2016) Vol.13 No.1

BB2-03S2C-25 camera, Point Grey Research, Inc., Canada[26]) captured the images of the fish robot when it swam, and sent the images to the computer. The acquisition speed was 15 fps and the image resolution was 640 × 160 pixels. In addition, a custom-designed program (see Fig. 12) handled communication with the fish robot and images processing. The communication consisted of sending commands and data such as amplitudes, frequencies, and operation time, receiving a response signal from the fish robot. During this time, the image was processed to recognize the fish robot. An algorithm was developed based on the specific color of the fish body (in this case, blue). The trajectories of the fish robot were then plotted in real time on the user interface of the program, as shown in Fig. 12. Finally, the steady-state swimming speed was determined by linearizing the relationship between the distance moved and the time taken, as shown in Fig. 13.

0.05 0.04 0.03 0.02 0.01 0.00 −0.01 −0.02 −0.03 −0.04 −0.05 0.0

Simulation Experiment Error

0.5

1.0

1.5

2.0 2.5 Time (s)

3.0

3.5

4.0

4.5

Fig. 8 The comparison of lateral movements of the trailing-edge.

4 Results and discussion 4.1 Effect of amplitude In nature, the live pumpkinseed sunfish increases its swimming speed by increasing the lateral amplitude of the rostrum, which could be an indicator of the amplitude of the driving moment[12]. Similarly, in our model, Eq. (11) shows that the lateral movement is linearly proportional to the amplitude M0 of the driving moment. Therefore, the tail-beat amplitude is also proportional to the value of M0 and, consequently, the thrust and swimming speed increase with M0. Simulation and experimental results verified this analysis. As shown in Figs. 14 and 15, the swimming speed and thrust were proportional to the amplitude of the driving moment, while the frequency of the driving moment was fixed at f = 1.5 Hz. Our experimental results of thrust and swimming speed were lower than the simulation results due to the oscillation of the fish robot head, which was not considered in simulation. Especially, at high amplitudes (M0 > 200 mN·m), the head oscillation increased. Consequently, the difference between simulation and experimental results was larger than the results at low amplitudes. In addition, there was more energy wasted in generating a vortex wake than in the case of low amplitudes. However, the thrust and swimming speed mostly increased with the amplitude of the driving moment. The maximum swimming speed at f = 1.5 Hz

Fig. 9 The test-rig for measuring thrust. Experiment 4 3 2 1 0 0.0

0.5

1.0 Time (s)

1.5

2.0

Fig. 10 The variation in thrust with time for the fish robot at the input M = 260sin(2π1.5) mN·m.

was about 0.7 BL·s−1 at A0 = 290 mN·m. It was generally accepted that our proposed model can predict the change in swimming speed when the amplitude of the driving moment changes.

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Thrust (N)

Nguyen et al.: Thrust and Swimming Speed Analysis of Fish Robot with Non-uniform Flexible Tail

Fig. 15 Variation in thrust with amplitude for the driving moment with a fixed frequency f = 1.5 Hz.

Fig. 11 Experiment setup for measuring swimming speed.

Distance (mm)

Fig. 12 The control program.

Fig. 13 Trajectory of the fish robot vs. time.

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

Simulation Experiment

0.1 0.0 50

100

150 200 Amplitude (mN·m)

250

300

Fig. 14 Variation in swimming speed with amplitude for the driving moment with a fixed frequency f = 1.5 Hz.

4.2 Effect of frequency In nature, at a low swimming speed, a real fish increases both its tail-beat frequency and tail-beat amplitude to achieve a faster swimming speed. In contrast, at a high swimming speed, a real fish only increases its tail-beat frequency to achieve faster speed[27–29]. The effect of the frequency of the driving moment is more complex than that of the amplitude. In our model, the function of time ni(t) is the solution of a second-order Ordinary Differential Equation (ODE) (Eq. (3)), so that a resonance phenomenon can occur when the frequency of the driving moment approaches the resonance frequency of the NFT. The comparisons between the simulation and experimental results of swimming speed measurements and thrust measurements are shown in Figs. 16 and 17. The swimming speed increased with the frequency up to 1.4 Hz. The maximum swimming speed was about 0.68 BL·s−1, and the maximum thrust was about 1 N at f = 1.4 Hz. When the driving frequency increased to over 1.4 Hz, the swimming speed and thrust decreased. A small peak occurred at f = 2.2 Hz, but good performance was expected at around 1.4 Hz. At the lower frequency, the fish robot swung side by side much more. Therefore, the lateral force needed to swing increased, and the thrust and swimming speed decreased. On the other hand, when the frequency increased to 1.6 Hz, the dynamic response of the NFT led to a reduction in the lateral movements of the trailing edge as well as the thrust and swimming speed. There are some differences between simulation and experimental results. For example, when the frequency changed from 1 Hz to 1.2 Hz, the swimming speed in the simulation changed faster than that in the experiment. Therefore, there was an obvious difference at f = 1.2 Hz. As a result, the complex changes in the thrust and swimming speed were not always well captured by the model. This was due to the effects of phenomena such as vortices, the boundary effect, oscillation of water in the

Journal of Bionic Engineering (2016) Vol.13 No.1

−1 Swimming speed (BL·s )

82

Thrust (N)

Fig. 16 The relationship between swimming speed and the driving frequency.

Fig. 17 The relationship between thrust and the driving frequency.

tank, and the effect of the tank wall in the real experiments, which were neglected in the proposed model. The oscillation of the water in the tank at f = 1.2 Hz was monitored, and was slightly stronger than that at f = 1.0 Hz or even 1.4 Hz. Eq. (1) predicts the lateral movements of the NFT, and its accuracy depends on approximations of parameters such as resistant forces, reactive forces, the moment of inertia, and added mass. The resistant force, for example, originates from interaction with water environments and is dependent on the aforementioned phenomena. Therefore, it was underestimated that the simulation results (thrust and swimming speed) were higher than those of the experimental results. Finally, we note that although the proposed model does not perfectly describe the operation of a real fish robot, it can predict the real behavior of swimming speed and thrust when the frequency of the driving moment changes. In comparison with other multi-joint and rigid bar fish robots, this fish robot, with a flexible tail actuated by a single actuator, is easier to control and is more mechanically robust[14,30,31]. In addition, the fish robot was completely modeled using a mathematical model, which is useful in control.

5 Conclusion We present a dynamic model with thrust estimation, an equation of motion, and a performance evaluation of

a fish robot with a NFT. By defining the optimal stiffness of the NFT in simulation, a fish robot prototype was designed and fabricated. Simulation and experiments were carried out and compared in order to investigate the tendencies of thrust and swimming speed of the fish robot. The thrust and swimming speed of the fish robot were proportional to the amplitude of the driving moment. However, the effect of the frequency of the driving moment is more complex. There were two resonant frequencies, f = 1.4 Hz and f = 2.2 Hz, but the maximum thrust and swimming speed (about 0.7 BL·s−1) occurred near f = 1.4 Hz. Our results make several contributions to the analytical approach of modeling a fish robot with a NFT. First, based on a performance evaluation (Froude efficiency), the optimal stiffness of the NFT is determined to be about 8×10−6 N·m2 to 1.2×10−5 N·m2. This range of stiffness is close to that of a live sunfish caudal fin. Secondly, simulation and experimental results are in good agreement, which verifies that the proposed dynamic model is suitable for predicting the behaviors of the thrust and swimming speed of the fish robot with a NFT. This is useful for control problems. Future works will focus on unsteady swimming modes such as acceleration and turning.

Acknowledgment This work was supported by the Human Resource Training Program for Regional Innovation and Creativity through the Ministry of Education and National Research Foundation of Korea (2015H1C1A1035547) and a part of the project titled 'R&D center for underwater construction robotics', funded by the Ministry of Oceans and Fisheries(MOF) and Korea Institute of Marine Science& Technology Promotion(KIMST), Korea.

References [1]

Sfakiotakis M, Lane D M, Davies J B C. Review of fish swimming modes for aquatic locomotion. IEEE Journal of Oceanic Engineering, 1999, 24, 237–252.

[2]

Liu Y, Chen W, Liu J. Research on the swing of the body of two-joint robot fish. Journal of Bionic Engineering, 2008, 5, 159–165.

[3]

Liu J, Hu H. Biological inspiration: From carangiform fish to multi-joint robotic fish. Journal of Bionic Engineering, 2010, 7, 35–48.

[4]

Liu J, Hu H. A 3D simulator for autonomous robotic fish.

Nguyen et al.: Thrust and Swimming Speed Analysis of Fish Robot with Non-uniform Flexible Tail

[5]

International Journal of Automation and Computing, 2004,

Ghassemi H, Maleki H. Design, fabrication and hydrody-

1, 42–50.

namic analysis of a biomimetic robot fish. Proceedings of

Gi-Hun Y, Choi W, Lee S H, Kim K S, Choi H S, Ryuh Y S.

the 10th WSEAS International Conference on Automatic

Control and design of a 3 DOF fish robot ‘ICHTUS’. IEEE

Control, Modelling& Simulation, Istanbul, Turkey, 2008,

International Conference on Robotics and Biomimetics

249–254.

(ROBIO),

Karon

Beach,

Phuket,

Thailand,

2011,

2108–2113. [6]

Yang L, Su Y, Xiao Q. Numerical study of propulsion mechanism for oscillating rigid and flexible tuna-tails.

[7]

vortex sheets. Journal of Computational Physics, 2009, 228, 2587–2603. [20] Alben S, Witt C, Baker T V, Anderson E, V. Lauder G.

Journal of Bionic Engineering, 2011, 8, 406–417.

Dynamics of freely swimming flexible foils. Physics of Fluids, 2012, 24, 051901. [21] El Daou H, Salumae T, Toming G, Kruusmaa M. A

Journal of Fluid Mechanics, 1999, 392, 183–212.

bio-inspired compliant robotic fish: Design and experiments.

Triantafyllou M S, Triantafyllou G S, Yue D K P. Hydro-

The IEEE International Conference on Robotics and

dynamics of Fishlike Swimming. Annual Review of Fluid

Automation, Saint Paul, MN, USA, 2012, 5340–5345.

Mechanics, 2000, 32, 33–53. [9]

[19] Alben S. Simulating the dynamics of flexible bodies and

Barrett D S, Triantafyllou M S, Yue D K P, Grosebaugh M A,Wolfgang M J. Drag reduction in fish-like locomotion.

[8]

83

Yu J, Liu Z, Wang L. Dynamic modeling of robotic fish using Schiehien's method. IEEE International Conference on Robotics and Biomimetics, ROBIO’06, Kunmin, China, 2006, 457–462.

[10] Lighthill M J. Note on the swimming of slender fish. Journal of Fluid Mechanics, 1960, 9, 305–317. [11] Wang H. Design and Implementation of Biomimetic Robotic Fish. Ms thesis, Mechanical and Industrial Engineering, Concordia University, Canada, 2009. [12] McHenry M, Pell C, Jr J. Mechanical control of swimming speed: stiffness and axial wave form in undulating fish models. Journal of Experimental Biology, 1995, 198, 2293–2305. [13] Nguyen P L, Do V P, Lee B R. Dynamic modeling of a non-uniform flexible tail for a robotic fish. Journal of Bionic Engineering, 2013, 10, 201–209. [14] Alvarado P V, Youcef-Toumi K. Design of machines with

[22] Suppiger E, Taleb N. Free lateral vibration of beams of variable cross section. Zeitschrift für angewandte Mathematik und Physik ZAMP, 1956, 7, 501–520. [23] Lighthill M J. Large-Amplitude Elongated-Body Theory of Fish Locomotion. Proceedings of the Royal Society of London. Series B. Biological Sciences, 1971, 179, 125-138. [24] NASA. Shape Effects on Drag, [2015-05-15], http://www.grc.nasa.gov/WWW/k-12/airplane /shaped.html. [25] PVC Foam Sheet (Foamex), http://www.par-group.co.uk/engineering-plastics/plastic-she et/pvc-foam-sheet-foamex/ [26] Point Grey, [2015], http://www.ptgrey.com//bumblebee2- firewire-stereo -vision-camera-systems [27] Bainbridge R. Caudal fin and body movement in the propulsion of some fish. Journal of Experimental Biology, 1963, 40, 23–56.

compliant bodies for biomimetic locomotion in liquid en-

[28] Zweife J R H A J R. Swimming speed, tail beat frequency

vironments. Journal of Dynamic Systems, Measurement, and

tail beat amplitude, and size in jack mackerel, Trucharzcs

Control, 2006, 128, 3–13.

symmetricas, and other fishes. Fishery Bulletin. 1971, 69,

[15] Lighthill M J. Hydromechanics of aquatic animal propulsion. Annual Review of Fluid Mechanics, 1969, 1, 413–446. [16] Long Jr J H, Adcock B, Root R G. Force transmission via

253–267. [29] Webb P W. ‘Steady’ swimming kinematics of tiger musky, an esociform accelerator, and rainbow trout, a generalist

axial tendons in undulating fish: A dynamic analysis. Com-

cruiser. Journal of Experimental Biology, 1988, 138, 51–69.

parative Biochemistry and Physiology Part A: Molecular &

[30] Anderson J M, Chhabra N K. Maneuvering and stability

Integrative Physiology, 2002, 133, 911–929. [17] Nikkhah Bahrami M, Khoshbayani Arani M, Rasekh Saleh

performance of a robotic tuna. Integrative & Comparative Biology, 2002, 42, 1026–1031.

N. Modified wave approach for calculation of natural fre-

[31] Yu J, Tan M, Wang S, and Chen E. Development of a

quencies and mode shapes in arbitrary non-uniform beams.

biomimetic robotic fish and its control algorithm. IEEE

Scientia Iranica, 2011, 18, 1088–1094.

Transactions on Systems, Man, and Cybernetics-Part B:

[18] Mohammadshahi D, Yousefi-koma A, Bahmanyar S,

Cybernetics, 2004, 34, 1798–1810.