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Abstract

This article considers the turning characteristics of robotic fish with two-degree-of-freedom pectoral fins and flexible body/caudal fin. The hydrodynamics are first established for three cases propelled by both sides of pectoral fins, flexible body/caudal fin, and composite of them. Then, the turning characteristics of such three cases are analyzed by numerical simulations and experiments. The results show that if robotic fish is cooperatively propelled by pectoral fins and flexible body, it can obtain the fast turning speed and the average turning speed is up to 0.6 rad s⁻¹. The smallest turning speed is achieved as robotic fish is only propelled by pectoral fins; however, it can turn on the spot in this case. The presented results provide the more abundant ways of turning, the better maneuverability, and the higher turning speed for the proposed robotic fish.

Keywords

Biomimetic robotic fish turning characteristics pectoral fin with two degrees of freedom multi-joint body coordinated propulsion hydrodynamic analysis

Introduction

The biomimetic robotic fish has been widely investigated in literature for many years. The early works mainly focused on the body and caudal fin (BCF) mode.¹ Since the median and/or paired fin (MPF) mode has advantage over BCF mode in maneuverability and stability, three categories of robotic fish propelled by MPF mode have been developed, namely, multi-fin oscillating, pectoral-fin flapping or/and gliding, and long-fin undulating.^{2

–10} Nanyang Technological University developed a four-fin propulsion bionic prototype based on modular design, which has high efficiency and maneuverability.³ Peking University developed a biomimetic turtle with four flapping fins, which could achieve complex three-dimensional movement including the pitch, heave, yaw, dive, and roll.⁴ Beihang University developed a biomimetic turtle with five degrees of freedom (DOFs), which could turn flexible on the spot, achieve 1 m s⁻¹ forward speed, and dive up to 30 m.⁵ Kato developed the first mechanical pectoral fin-driven bionic robotic fish, which took the black bass as the prototype.^6
–8 The Chinese Academy of Sciences developed a robotic fish with two pectoral fins and four-joint flexible body, which could turn with 72° s⁻¹. Meanwhile, its turning radius was up to 0.38 times of body length.⁹ Again, they developed a biomimetic Esox lucius with two-DOF pectoral fins and multi-joint flexible body, which could achieve fast turn with 120° s⁻¹.¹⁰

Inspired by the above works, two prototypes of biomimetic robotic fish were designed by the authors.^11
–13 Especially for the proposed robotic fish in the study by Li et al.,¹³ its pectoral fins can move forward and backward, rotate around the axis of pectoral fin, and achieve the composite motion of them. Meanwhile, it can be cooperatively driven by the two-DOF pectoral fins and three-joint flexible body together. Pectoral fins with two DOFs provide the robotic fish more maneuverability. For example, there are at least six ways for robotic fish to achieve straight swimming.¹³ Since the turning behavior is more important for fish to adapt to the environment and evade predators, this article mainly considers the turning characteristics of robotic fish. First, the hydrodynamic models are established for pectoral fins and flexible BCF, and then the turning mechanism is analyzed by numerical simulation and experiments. As a result, three turning ways are summarized, which improves the maneuverability and environmental adaptability of the robotic fish.

Kinematic modeling of robotic fish

The prototype of robotic fish is shown in Figure 1 and its propulsion mechanism is given in Figure 2. It is clearly shown that the pectoral fins are symmetrically arranged on both sides of robotic fish and the flexible body is made up of three rigid joints and a caudal fin. As shown in Figure 2, the rotating motors directly drive the pectoral fins to do oscillating rotary motion, and the swaying motors actuate the pectoral fins to move forward and backward by gear transmission. Since the rotating motor is fixed on the parts driven by swaying motors, the rotation and sway of pectoral fins are independent and thus the composite movement of them can also be achieved. The flexible body is made up of three rigid joints and the caudal fin is a passive part. In practice, the skin of flexible body is made up of rubber, and the shell of the rigid head is designed based on the streamlined appearance of codfish and made by PLA fiber. The rays and skin of pectoral fins are made by flexible carbon rods and heat-shrinkable plastic material, respectively. The technical parameters of robotic fish are shown in Table 1.

Figure 1.

Prototype of the proposed robotic fish.

Figure 2.

Overall propulsion mechanism of the biomimetic fish.

Table 1.

Technical parameters of the robotic fish.

Parameters	Description
Size	50 cm × 15 cm × 13 cm
Total weight	4.5 kg
CPU	ATMEL128 (Atmel Corporation, USA)
Work voltage	DC 6 V
Material of shell	PLA
Joint servomotor	Futaba S3003/S3102 (Futaba Corp., Japan)

CPU: Central Processing Unit; PLA: Poly Lactic Acide.

In what follows, the movement laws are given for pectoral fins. Let {X,Y,Z} and {x,y,z} denote global and local coordinate systems, respectively, U be the direction of swimming, ω be the swaying angular speed, θ_m be the backward swaying angle, θ_n be the steering angle of servomotor, and $\dot{β}$ and ${\dot{β}}_{e}$ are the angular speeds of rotating (see Figures 3 and 4). $\dot{β} = {\dot{β}}_{e}$ and θ_m = θ_n since the swing lever is fixed with servomotor, and the ratio of gear transmission is 1.

Figure 3.

Diagram of the backward and forward movements for pectoral fins.

Figure 4.

Motion diagram of mechanism for pectoral fins.

During the turning, the motion of each pectoral fin is divided into four periods as shown in Figure 5, in which ω, ω_n, γ, and ω_n1 denote the frequency of each period, α is the attack angle, β_n and β_l, respectively, denote the rotating angle of the first and second rotations, and θ_l = θ_n is the forward swaying angle. Let B denote the amplitude, θ₀ be the original angle of back stroke, and f_b and f_r are the frequencies of the backward and forward swaying, respectively. Then, the laws of movement for pectoral fins are defined as follows

Figure 5.

Diagram of the movement for pectoral fins.

\begin{array}{l} θ_{m} (t) = θ_{0} + \frac{B}{2} - \frac{B}{2} cos (π f_{b} t), 0 \leq t \leq \frac{1}{f_{b}} \\ θ_{l} (t) = \frac{B}{2} + \frac{B}{2} cos (π f_{r} t), \frac{1}{f_{b}} \leq t \leq \frac{1}{f_{b}} + \frac{1}{f_{r}} \end{array}

From equation (1), the angular speed of the backward and forward swaying can be derived as

\begin{array}{l} ω (t) = \frac{1}{2} π B f_{b} sin (π f_{b} t), 0 \leq t \leq \frac{1}{f_{b}} \\ γ (t) = - \frac{1}{2} π B f_{r} sin (π f_{r} t), \frac{1}{f_{b}} \leq t \leq \frac{1}{f_{b}} + \frac{1}{f_{r}} \end{array}

Now, the motion laws of flexible BCF are considered. Generally, the flexible body results in turning by fitting with the Lighthill curve as follows^{1,3,5,9,14,15}

h (x_{n}, t) = h_{T} g (x_{n}) sin (2 π f t)

where h(x_n,t) is the offset of each point to the centerline of the body, h_T is the swaying amplitude of flexible body in the horizontal plane, $x_{n} = \frac{x}{L}$ is the ratio of the longitudinal coordinates of the beak to the body length L, f is the swaying frequency, and g(x_n) represents the quadratic wave amplitude envelope that is always appropriated by a polynomial. For simplification, let $g (x_{n}) = 0.1 - 1.3 x_{n} + 2.2 x_{n}^{2}$ just as Romanenko¹⁶ and Yu et al.^17,15 suggested. Then, equation (3) can be discretized as

h (x_{n}, i) = h_{T} (0.1 - 1.3 x_{n} + 2.2 x_{n}^{2}) sin (2 π i / M)

where i represents discrete time, h(x_n, i) is the position of flexible body at time i, M represents the discrete resolution, and h_T has the same meaning as in equation (3).

The junction of rigid head and flexible body is at the point o, which is also the original point of local coordinate, just as shown in Figure 6. In this figure, ϕ is the swaying angle between rigid head and the centerline of flexible body, H and P are the centroids of rigid head and flexible body, respectively. Then, the kinematics of ϕ can be given as follows^14,18
–20

Figure 6.

Diagram of turning for body/caudal fin.

ϕ = ϕ_{max} sin (f t)

where $ϕ_{max} (| ϕ_{max} | \leq 60 °)$ is the maximum value of ϕ. It is noted that the swaying directions of left and right pectoral fins are opposite as the robotic fish turns. In addition, let f = 2f_b = 2f_r for simplicity.

Hydrodynamic analysis of turning

In this section, the left turning is taken as an example to analyze the hydrodynamics of turning. Following the same line, the results about the right turning can be obtained and thus omitted here. In what follows, three cases are discussed.

Case 1: Turning propelled by pectoral fins

In this case, the force condition of robotic fish is shown in Figure 7, where $F_{a}, {F'}_{n}, and {F'}_{s}$ are the additional mass force, normal force, and spanwise force, respectively, and $(x_{rp}, y_{rp}), (x_{C}, y_{C}), and θ_{rp}$ are the junction of right pectoral fin and the body, centroid of the robotic fish, and the angle between right pectoral fin and the axis X, respectively.

Figure 7.

Force analysis of pectoral fins in turning.

As shown in Figure 7, the movement of pectoral fin can be divided into two periods as the robotic fish turns left. In the first period, the right moves backwards and the left moves forward; meanwhile, both of them are perpendicular to the horizontal plane. In the second period, both of the pectoral fins are returned from the horizontal plane to the initial position in which the resistance is small and thus can be negligible. Then, by equations (1) and (2), the motion laws of right pectoral fin are derived as follows

\begin{array}{l} θ_{mrp} (t) = θ_{rp 0} + \frac{B_{rp}}{2} - \frac{B_{rp}}{2} cos (π f_{rpb} t), \\ 0 \leq t \leq \frac{1}{f_{rpb}} \\ θ_{lrp} (t) = \frac{B_{rp}}{2} + \frac{B_{rp}}{2} cos (π f_{rpr} t), \\ \frac{1}{f_{rpb}} < t \leq \frac{1}{f_{rpb}} + \frac{1}{f_{rpr}} \end{array}

where θ_mrp and θ_lrp, respectively, denote the swaying angle as the right pectoral fin strokes backward and forward, θ_ro0 is the original angle of right pectoral fin moving backward, and B_rp denotes the swaying amplitude. The frequencies of moving backward and forward are f_rpb and f_rpr, respectively. By taking the derivatives with respect to t, equation (6) becomes

\begin{array}{l} ω_{rp} (t) = \frac{1}{2} π B_{rp} f_{rpb} sin (π f_{rpb} t), 0 \leq t \leq \frac{1}{f_{rpb}} \\ γ_{rp} (t) = - \frac{1}{2} π B_{rp} f_{rpr} sin (π f_{rpr} t), \\ \frac{1}{f_{rpb}} < t \leq \frac{1}{f_{rpb}} + \frac{1}{f_{rpr}} \end{array}

where ω_rp and γ_rp are the corresponding angular speeds. Hereinafter, we analyze the force and torque generated by right pectoral fin by employing the elemental integration method. Specifically, the fan-shaped pectoral fin is divided into several slices of microelements along the direction of the expansion, and then, the force and torque of each microelement are analyzed. Finally, the force and torque of the whole pectoral fin are obtained by integrating along the direction of expansion just as shown in Figure 8, where $ω r, v_{n}, and v_{t}$ , respectively, denote linear speed, normal speed, and tangential speed of microelement P with

\begin{array}{l} v_{n} = ω r - u sin θ \\ v_{t} = u cos θ \end{array}

Let v be the vector sum of v_n and v_t and α be the attack angle, then

\begin{array}{l} v^{2} = v_{n}^{2} + v_{t}^{2} = ω^{2} r^{2} + u^{2} - 2 u ω r sin θ \\ tan α = \frac{v_{n}}{v_{t}} = \frac{ω r - u sin θ}{u cos θ} \end{array}

Again, the normal force dF_n and the spanwise force dF_s exerted by the surrounding fluid on microelement P are^21,22

\begin{array}{l} d F_{n} = \frac{1}{2} ρ C_{n} v^{2} d A \\ d F_{s} = \frac{1}{2} ρ C_{d} v_{t}^{2} d A \end{array}

where ρ is the fluid density, dA is the area of microelement P, C_n is the lift coefficient, which depends on attack angle α, and C_d is the fractional drag coefficient and its value varies with Reynolds number Re. Let μ be fluid viscous coefficient, δ be the angle of spread of fan-shaped pectoral fin, and R be the diameter of pectoral fin, then

\begin{array}{l} C_{n} = 2 - 1.55 cos (2 α - 9.82) \\ C_{d} = 1.33 \times R_{e}^{- 0.5} \\ d A = \frac{1}{2} δ r \cdot d r \\ Re = R \cdot v_{t} / μ \end{array}

Integrating the force of microelement P along with the spread direction of fins, the total normal force $F_{n^{'}}$ and the total spanwise force $F_{s^{'}}$ of right pectoral fin are as follows

\begin{array}{l} {F'}_{n} = \int_{r = 0}^{r = R} d F_{n} = \int_{r = 0}^{r = R} d (\frac{1}{2} ρ C_{n} v^{2} d A) \\ {F'}_{s} = \int_{r = 0}^{r = R} d F_{s} = \int_{r = 0}^{r = R} d (\frac{1}{2} ρ C_{d} v_{t}^{2} d A) \end{array}

In addition, the fluid additional mass force dm_a is exerted on microelement P as right pectoral fin moves backward. According to the study by Kato and Furushima⁶ and Blake,²¹ dm_a is as follows

d m_{a} = π {(\frac{c}{2})}^{2} l ρ

where c = δ ⋅ r is the chord length of the slice and l = dr is the slice length. Further, the force dF_a produced by dm_a can be obtained as follows^6,23

d F_{a} = d m_{a} \cdot (\frac{d ω}{dt} r) = (\frac{1}{4} π ρ δ^{2} r^{2} d r) \cdot (\frac{d ω}{dt} r)

By integrating the force dF_a along with the spread direction of fins

F_{a} = \int_{r = 0}^{r = R} d F_{a}

Then, as shown in Figure 8, the rotational torque T_r of right pectoral fin in the first period can be derived as follows

Figure 8.

Force analysis of infinitesimal slice for pectoral fin.

\begin{matrix} d T_{r} = [(d {F'}_{n} + d F_{a}) sin θ_{rp} + d {F'}_{s} cos θ_{rp}] x_{r} \\ x_{r} = (x_{c} - x_{rp} - x_{rp} cos θ_{rp}) \end{matrix}

Integrating dT_r along with the spread direction of fins, the left rotational torque produced by the right pectoral fin as it moves to the backward is

T_{r} = \int_{r = 0}^{r = R} d T_{r}

Similarly, the rotational torque T_l of the left pectoral fin is

T_{l} = \int_{r = 0}^{r = R} d T_{l}

Combined with equations (16) and (17), the total rotational torque M_l produced by two pectoral fins is

M_{1} = T_{r} + T_{l}

Case 2: Turning dynamics of flexible BCF

In this case, the fluid resistance is analyzed first as follows²⁴

F_{d} = - \frac{1}{2} ρ v^{2} C_{d} A

where F_d is the resistance of robotic fish, ρ is the fluid density, v is the velocity of robotic fish compared to the fluid, C_d is the resistance coefficient, and A is the area of the projection of robotic fish on a plane perpendicular to the direction of motion. The force conditions of this case are given in Figure 9, in which F_h1 and F_h2 are the lateral resistance and forward resistance exerted on the rigid head, respectively, F_t1 and F_t2 are the lateral resistance and forward resistance exerted on the flexible BCF, respectively, and C and E are the instantaneous centroid of rigid head and tail during turning, respectively. Under the assumption that the centroid and the buoyant center of robotic fish are all located on the Z-axis, these parameters are formulated as follows¹⁶

Figure 9.

Force analysis of body/caudal fin in turning.

\begin{array}{l} F_{h 1} = k_{h 1} V^{2} A_{h 1} sin \frac{ϕ}{2}, F_{h 2} = k_{h 2} V^{2} A_{h 2} cos \frac{ϕ}{2} \\ F_{t 1} = k_{t 1} V^{2} A_{t 1} sin \frac{ϕ}{2}, F_{t 2} = k_{t 2} V^{2} A_{t 2} cos \frac{ϕ}{2} \end{array}

where A_h1 and A_h2 are the areas of the side and the front of rigid head immersed in water, respectively; A_t2 and A_t2 are the areas of the side and the front of flexible BCF immersed in water, respectively; and k_h1, k_h2, k_t1, and k_t2 are the resistance coefficients associated with the swing frequencies f_k1, f_k2, f_t1, and f_t2, respectively. Assuming that ΔCHP is always isosceles triangle during turning, that is, the body of robotic fish always rotates around the whole center of gravity, it is easy to verify the following

∠ O C E = ∠ O H P = ∠ O E C = ∠ O P H = \frac{ϕ}{2}

Assuming $C (x_{1 c}, y_{1 c})$ and $E (x_{2 c}, y_{2 c})$ are the midpoint of line segments OH and OP in Figure 9, then

M_{2} = T_{h 1} + T_{t 1} - T_{h 2} - T_{t 2}, d = \frac{x_{2} c - x_{1} c}{2}

with

\begin{array}{l} T_{h 1} = F_{h 1} cos \frac{ϕ}{2} d, T_{t 1} = F_{t 1} cos \frac{ϕ}{2} d \\ T_{h 2} = F_{h 2} sin \frac{ϕ}{2} d, T_{t 2} = F_{t 2} sin \frac{ϕ}{2} d \end{array}

where M₂ is the total rotational torque coordinately produced by flexible BCF.

Case 3: Turning dynamics of coordinated propulsion by pectoral fins and flexible body together

In this case, the coordinated propulsion force and torque of turning can be obtained by combining with case 1 and case 2. Obviously, the total centripetal force F₁ produced by pectoral fins is

\begin{array}{l} F_{1} = (F_{a 1} + {F'}_{n 1}) cos θ_{rp} + {F'}_{t 1} sin θ_{rp} \\ - (F_{a 2} + {F'}_{n 2}) cos θ_{lp} - {F'}_{t 2} sin θ_{lp} \end{array}

And the total centripetal force F₂ produced by flexible body/caudal fin is

F_{2} = F_{h 1} cos \frac{ϕ}{2} + F_{t 2} sin \frac{ϕ}{2} - F_{h 2} sin \frac{ϕ}{2} - F_{t 1} cos \frac{ϕ}{2}

Then, the total centripetal force F_n exerted on the whole robotic fish is

F_{n} = F_{1} + F_{2}

Generally, the centripetal force has the form

F_{n} = \frac{m V^{2}}{R_{s}} = m ω_{s}^{2} R_{S}

where R_s and ω_s are turning radius and angular velocity, respectively. Combined with equations (18) and (21), the turning radius is

R_{s} = \frac{m V^{2}}{F_{1} + F_{2}} = \frac{F_{1} + F_{2}}{m ω_{s}^{2}}

Combined with the above, the hydrodynamic model of robotic fish is as follows

\begin{matrix} M_{1} = T_{r} + T_{l} \\ M_{2} = T_{h 1} + T_{t 1} - T_{h 2} - T_{t 2} \\ M_{s} = I \frac{d ω_{s}}{d t} = M_{1} + M_{2} \\ I = \underset{i}{Σ} m_{i} {r_{i}}^{2} = m R_{s} \end{matrix}

where M_s and I are the total torque and the moment of inertia of the whole robotic fish, respectively.

Turning characteristics analysis

Based on equations (18), (21), and (27), this section analyzes the turning characteristics of robotic fish by numerical simulation. The initial values of motion parameters are given as $θ_{lp} = 120 °, θ_{rp} = 60 °, θ = 90 °, B_{1} = 60 °, B_{2} = 90 °,$ f_b = 2.5 Hz, f_r = 2.5 Hz, f = 5 Hz, δ = 45^°, and R = 12 cm.

For case 1, the result of simulation is shown in Figure 10. It is noted that the flexible BCF bend as a C-type and keep static when the robotic fish turns. As shown in Figure 10, the robotic fish could achieve turning smoothly, and the average angular speed of turning is 0.5 rad s⁻¹.

Figure 10.

Turning angular velocity propelled by pectoral fins.

For case 2, the result of simulation is shown in Figure 11. As shown in Figure 11, both sides of pectoral fins keep parallel to the horizontal plane and are always inactive. The turning of robotic fish is also smooth and the average angular speed of turning is 0.4 rad s⁻¹, which is less than that of case 1.

Figure 11.

Turning angular velocity propelled by body/caudal fin.

For case 3, the result of simulation is shown in Figure 12. The sum of the swaying frequencies f_b and f_r is equal to the C-type swaying frequency f. As shown in Figure 12, the turning of robotic fish is also smooth and the average angular speed of turning is 0.6 rad s⁻¹, which is larger than those of case 1 and case 2.

Figure 12.

Turning angular velocity of pectoral fins and body/caudal fin.

Experiment

Experiments of robotic fish are carried out in an indoor swimming pool with 2 m × 3 m × 0.4 m. The information within the swimming tank is captured by an overhead CCD camera. The image is transmitted to a personal computer and processed by a visual tracking software platform developed to obtain the position and orientation of robotic fish in real time and the sampling period is 0.5 s. The two-dimensional trajectory of robotic fish can also be extracted and recorded for offline analysis. The motion laws of robotic fish are determined by equations (1), (2), and (5). The initial values of parameters are just the same as the numerical simulation. Figure 13 shows a photograph of the experimental environment.

Figure 13.

Experimental environment of robotic fish.

For case 1, the results of experiment are shown in Figures 14 and 15. As the Figure 15 shows, the angular speed changed periodically between 0.34 rad s⁻¹ and 0.44 rad s⁻¹ and its average is about 0.4 rad s⁻¹. Compared with the simulation results in Figure 10, the angular speed is slightly slower than 0.5 rad s⁻¹, but more stable than 0.35–0.66 rad s⁻¹. In addition, the turning radius is about 0.15 m, nearly turning on the spot.

Figure 14.

Turning experiment for bilateral pectoral fins.

Figure 15.

Turning experiment result for bilateral pectoral fins.

For case 2, the results of experiment are shown in Figures 16 and 17. As the results in Figure 17 show, the angular speed changed periodically between 0.38 rad s⁻¹ and 0.64 rad s⁻¹ and its average is nearly 0.5 rad s⁻¹. As a comparison, the average angular speed is larger than the results in Figures 11 and 15. On the other hand, the stability of turning gets worse than that in case 1.

Figure 16.

Turning experiment for body/caudal fins.

Figure 17.

Turning experiment result for body/caudal fins.

For case 3, the results of experiment are shown in Figures 18 and 19. As the results in Figure 19 show, the average angular speed is about 0.57 rad s⁻¹, the range of the angular speed is 0.3–0.78 rad s⁻¹. Obviously, the angular speed is larger than those in the above two cases and is consistent with the results of numerical simulation in Figure 12.

Figure 18.

Turning experiment for pectoral fins and body/caudal fin.

Figure 19.

Turning experiment result for pectoral fins and body/caudal fin.

Conclusion

This article investigates the turning characteristics of robotic fish proposed by Li et al.,¹³ which consists of two-DOF pectoral fins and flexible body/caudal fin. Based on the hydrodynamic models of pectoral fins and flexible body/caudal fin, three turning ways are proposed. The results of numerical simulations and experiments show that the fast turning speed is achieved as the robotic fish is cooperatively propelled by pectoral fins and flexible body/caudal fin. As robotic fish is propelled only by pectoral fins, it can turn on the spot steadily but its angular speed is the smallest. In future works, the turning characteristics of such a robotic fish will be further investigated by employing the fluid analysis and central pattern generator control method.

Footnotes

Acknowledgements

The authors would like to thank the associate editor Prof. Hai Huang and anonymous reviewers for theirs valuable advice and help.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of China (NSFC) under grant 61663020.

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