1 Introduction

Fish are often assumed to be very efficient swimmers. Understanding and adapting the underlying principlesof swimming motions of fish to the artificial swimmingmachines are greatly appealing to the academic and industrial societies.Typically,fish use the body/caudal fin(BCF) undulation for the propulsion^[1]. Different types of BCF species show significant diversity in both kinematic movement and hydrodynamic performance.The establishment of a suitable dynamic model is crucial for the purpose of optimizingthe bio-inspired design.

To our knowledge,fish have evolved swimming capabilities far superior in many ways to what has been achieved by nautical scienceand technology over millions of years in a vast and often hostile realm^[2]. Swimming mechanism of fish was studied by Taylor who adopted the static fluid theory to establish the swimmingbiological resistance theory by the quasi-static approximation methods^[3]. Lighthill^[4] proposed an elongated carangidae model emphasison reactive force between a small volume of water and the parts of the animal’s surface in contact with it. Wu^[5] proposed two-dimensional flexible plate theory which took the fish as a two-dimensional elastic thin plate. Weihs combined the large amplitudeelongated body theory (see Ref. [6]) and lift force of wing method to calculate the force in agility swimming and considered that this force was equivalent to the force of the wake vortex^{[7, 8]}. Recently,Yang et al.^[9]provided an integrated method for deforming body dynamics and unsteady fluid dynamics to investigate a modelled freely self-propelled fish.

Many researchers used digital particle image velocimetry (DPIV) to observe and investigate fish wake flow field. Lauder and Drucker^[10] used the DPIV to observe the wake flow field and explained the function of each fin. Anderson et al.^[11] found that the angle of attack was an important parameter to affect the propulsive efficiency by studying the visualization of flow. Wen et al.^[12] used DPIV measurements to illustrate the characteristicsof the flow pattern generated by the robotic swimmer of anguilliform,carangiform,and thunniform,additionally,proposed a force-feedback-controlled experimental method.

We integrate the biological behavior with the underwater bionic robot and evaluate the further possible result for reliable controlling. There are some studies focused on the control-oriented swimming modeling. Ekeberg^[13] proposed a mechanical model to simulate the forces in the water. This model is based on two hypotheses. One is that the Reynolds number is high enough that the inertial force dominates,and the other is that the fluid is stationary. A Lagrangian model for a symmetrical structure eel-like robot is presented in Ref. [14] by McIsaac and Ostrowski. Boyer et al.^[15] presented the dynamic modeling of a continuous three-dimensional (3D) swimmingeel-like robot. Ding et al.^[16] proposed a dynamic model using the Lagrangianmethod and realized backward swimming. However,all the models mentioned above have relatively complex forms,which limit their applications in engineering.

In this paper,a dynamic model based on a central pattern generators (CPGs) controlled underwater robot with multi-joint is built. In addition,the experimental research is conducted to establish a simple mathematical model between the caudal fin oscillation and the thrust force for this underwater robot. Further,the model is verified by a comparison with the numerical simulation,and the results show acceptabledifferences.

The rest of the paper is organized as follows. In Section 2,the model and CPG-based locomotion control are described. The measurement methods and the formulation of dynamic modeling for the underwater robot are described in Section 3. The simulation and experimental results are provided in Section 4. Finally,the conclusions and outline of the future work are provided in the last section.

2 Locomotion control model based on CPG

With the consideration of the difficulty in experimental studies of freely swimming animals,amulti-link bionic robot is used to realize the fish-like undulation,as shown in Fig. 1(a). The robot is modelled as a chain of four identical links serially connected by three rotational joints. To estimate the oscillation property of a fish body,Lighthill’s elongated body theory has been widely used to generate a propulsive travelling wave. Meanwhile,many other aspects such as simple time-indexed sine-based functions are used to generate travelling waves. Besides,the disadvantages of poor anti-interference capability,abrupt modifications of control parameters willlead to a discontinuousness of propulsive waves,which in turn will produce non-smooth trajectories. Thus,the CPG,the basic controller model for the robot,is developed to ensure the fundamental rhythmic movements in swimming.

The CPGs are neural networks capable of producing coordinated patterns of rhythmic ac-tivity without any rhythmic inputs from sensory feedback or higher control centers^{[17, 18]} which have been widely used in robotics recently. The controllersystem consists of three coupled linear oscillators,and the ith (i = 1,2,3) oscillators are implemented as follows:

where r_i,ϕ_i,and ψ_ji are the state variables representing the amplitude,the phase of the oscillator,and the laggingangle of ith and jth oscillators,respectively. f determines the desired frequency of the oscillators. And the variable φi(t) is the resultant burst serving as the output of ith oscillator. Nis the number of oscillators (N = 3).

The parameters,as shown in Table 1,are generated with the CPG method. All the param-eters are applied in both the experiments and numerical simulation later. Figure 1(b) plots the rhythmic CPG output signals for the three joints.

3 Experimental facilities and dynamic model 3.1 Measurement techniques

For the sake of determining the parameters used in the model,two different measurement methods,the DPIV^{[19, 20, 21, 22]} and the force transducer,are conducted to observe the flow field and calculate the thrust force F_t. Both of the measurements are carried out at a still water pool,with the assumption that the thrust force mainly comes from the relative motion between the caudal fin and thesurroundings. Firstly,the DPIV is used to measure the wake flow field generated by caudal fin,andestimate the average thrustforce. As suggested by biologists,the caudal fin midline is recommended as the best horizontal position for conducting 2-D wake measurements^[23]. The depth of 3-D vortex rings is assumed to equal the caudal fin’s height,while the measurement planes at other horizontal levels underestimate the thrust force^[24]. A general view of experimental arrangement is given in Fig. 2. The robotic fish can be controlled by the computer through the radio frequency (RF) module. The laser sheet plane of the DPIV systemis set to pass through the midline of the homocercal caudal fin of the robot. Seeding particles with median diameter of 10 μm are illuminated by a laser sheet. A complementary metal oxide semiconductor (CMOS) camera (1 024×1 024 pixels) captures the flow fields with theshooting frequency being 50 Hz. A Dell computer is used to control the whole DPIV system and store the particle images simultaneously. The dual-pulse laser (15 mJ/pulse) with a wavelength of 532 nm is expanded by a cylindrical lenses to generate a light sheet. A long-focus lens is used to adjust the thickness of waist of the light sheet around 1 mm. Due to the position of the caudal fin being includedin the particle images,the swimming behaviour of the fish can alsobe analyzed by the DPIV results.

Since the temporal resolution of DPIV is limited to 50 Hz,a single component force trans-ducer HBM Z6FC3 is used to measure the thrust of the robot swimmingin the still water,which has a sampling rate up to 5 kHz,and a sensitivity of 0.1 mN in the axial direction. Thestatic calibration is carried out before measurement. Force calibration results show that thelinearity for the forward force is better than 1%. Then,the force transducer is assem-bled vertically above the robot to measure the axial force of the robotic fish. The output of the external force transducer is recorded by a computer through a national instruments (NI) preamplifier. Synchronizing signal is used to trigger data acquisition from the force transducer and the camera. Then,images of the flow structures behind the caudal fin,synchronize with the force measurements and with the movement of fish.

3.2 Dynamic analysis

When the robot swims in water,the forces acting on the body depend on the relative movement between the body and the surroundings. At relatively low speed,the viscous force plays an important role in determining the state of motion. Whilst,for higher speed cases,inertial effects become the dominated factor^[28]. Based on the experimental results,the Reynolds number,defined as Re_l = Ul/ν,is on the order of 105,which is high enough to ignore the influence of viscous force,where l isthe body length,and U and ν are the self-propulsive speed of the robot and the kinematic viscosity of water,respectively.

In the spirit of simplification,the forces acted on the robot are divided into the drag and thrust separately,without taking the lateral forces into consideration. The drag D is commonly equivalent to the fluid resistance of a rigid body with the same Reynolds number Re and velocity of flow. Moreover,the direction of the drag force D is along the body axis from head to tail,

where ρ is the fluid density,s_d is the effective sectional area of the robot body,and C_d is the drag coefficient varying with Re,which is practically determined through experiments. Here,C_d is chosen as^[26]

Based on the elongated body theory,the thrust force is produced mainly by the caudal fin^[27]. That makes it reasonable for the thrustforce F_t to be estimated as the reaction force exerted on the caudal fin. As shown by Li and Yin^[28],the forces have the similar shape for the caudalfin models with different behaviors. Then,the thrust force can be expressed as follows:

where u is the characteristic velocity of the tail fin,s_a is the area of the tail fin,and C_t is the thrust coefficient^[25].

As indicated in the previous research,the dimensionless parameter Strouhal number domi-nates the hydrodynamic performance of fish locomotion ^{[22, 23, 28]} and is defined as follows:

where A denotes the undulating amplitude of the caudal fin tip.

According to Eq. (6),the thrust coefficient C_t can be expressed as follows:

A remarkable fact is that an unconventional definition of characteristic velocity uisused here,which is defined as the average swing speed of the tail fin. And this velocity is more reasonable in calculating the thrust force,which will be explained later in this paper.

The characteristic velocity u is the average velocity of the fin in a circle,i.e.,

Then,it is easy to derive that the characteristic velocity u is some kind of combination of the swimmingvelocity and Strouhal number,i.e.,

4 Results and discussion

Figure 3 shows the DPIV time-series of the flow field generated by the robot caudal fin at different flapping frequencies.Within a flapping cycle,the caudal fin generates a vortex ring at each flick. The wake structures shown in Fig. 3 can be attributedto double-row vortices,with the Strouhal number being 0.77 and 0.59,respectively. This is consistent with the previous experimental studies^{[29, 30]},whose results suggest that the wake generated by a robotic fish is “double-row street” when 0.325 <St < 1.025. As shown in Fig. 3(a),vortices are generated from the left side flick (towards the right) and form the vortex ring R₁,which is denoted by the dashed enveloped line. The vortex strength is getting stronger as the caudal fin moves to the right,which means that the impulse in the elliptical vortex rings shed by the robotic fish is increasing. From Fig. 3(b),the vortices 1 and 2 are generated from the right side flick (toward the left) and form the vortex ring R₁,and the vortices 3 and 4 are generated from left flick and form the vortex ring R₂. Compared with Fig. 3(a),the vortex strength of Fig. 3(b) is getting stronger since the shear strength would rise with the flicking frequency.

The vortex dynamic model,which assumes that all the impulses shed by the robot are contained in the elliptical vortex rings,can be used to analyze the wake in order to obtain the thrust force on the robot ^[12]. The flow field is formed by freezingeach vortex in its shed position. Then,the 2-D vortex dynamics theory is employed to compute the impulse from the measured wake for the thrust force determination.

During the experiments,the amplitudeof caudal fin can not always keep constant for differ-ent frequencies.For relatively low flapping frequencies,the designed amplitude of the caudal fin can be reached. However,the amplitude of caudal fin decreases as the oscillation frequency increases when the frequency exceeds a threshold. This can be explainedby restriction of the response of the steering gears and the increased hydrodynamic loads on the caudal fin. A di-mensional parameter,the product of the square of caudal fin amplitude A and the oscillation frequency,is used to characterize the relationship between these two variables. As depicted in Fig. 4,the product is found to be nearly constant when the flapping frequencyis larger than 0.7 Hz in the present experiments.

This trend can be observed for three different control parameters. This means that the real motion parameters of the underwater bionic rotor will be derived from this defined product for different control situations if the threshold frequency can be determined.

Take Case 3 for an example. The instantaneous non-dimensional thrust coefficient C_t results,which is defined in Eq. (8),are shown in Fig. 5(a). The dimensionless thrust coefficient curves for different frequencies can collapse together with almost the same shape and average value. The similar trend can be obtained from the previous results^[12]. The averaged dimensionless thrust coefficient is in the range of 0.64 ± 0.06 for different frequencies in our experiments,as shown in Fig. 5(b). The characteristic velocity is larger than the swimming velocity at the same flapping frequency. Therefore,the comparison between the normalized non-dimensional thrust coefficients of different referenced velocities is shown in Fig. 5(c). The corresponding experimental swimmingvelocities at different frequencies could be found in Fig. 8(b). This result indicates that the characteristic velocity used here is more suitable for the thrust force calculating,comparing with the self-propulsive velocity of the robot. This fact proves in an-other perspective that the Strouhal number is a dominated dimensionless parameter for the hydrodynamic performance of fish locomotion.

Raspa et al.^[29] addressed the establishment of the drag by investigating experimentally the swimming dynamics of undulating thin flexible foils. They showed that the trailing longitudinal vortices that roll-up on the lateral edges of the foils play an important role in the origin of the total drag force. The foil kinematic thrust per unit span is the same for all the aspect ratios. The aspect ratio (R) used here is the ratio between the span and the lengthof the swimmer’s tail. However,there are some differences when calculating the stand non-dimensional net thrust coefficient C_t ,which is displayed in Fig. 6(a). If we use the average swing velocity of the caudal fin instead of the self-propulsive velocity of the body as the characteristic velocity,the values of C_t become more concentrated,as shown in Fig. 6(b).

In order to gain some insight into the well-formulated relation,we estimate the forward swimming of the robot. The intrinsic undulation property of fish-like swimmingaffords the instantaneous thrust force (F_t ),drag force (F_d),and the net force (F_net),as shown in Fig. 7. The comparison of free-swimming speed between the simulation and the real robot is shown in Fig. 8.

According to the experimental results,we assume C_t can be expressed as

where constant coefficients C₀,C_a in this formulation are optimized by the least square method.

The result shows that although this model is simply divided into the drag and thrust,accept-able simulation results could be obtained. The simulated speeds and the experimental speeds show the same trends. All the simulated speeds are a little largerthan the experimental ones at the same frequency,with the discrepancy no more than 20%. The deviation is mainly caused by the intrinsicmechanical assembly of robot and the uncertainty of the water environment. What’s more,the swimmingvelocity,which is neglected for simplification purpose,will also make some contribution to the thrust force by influencing the effective angle of attack of the caudal fin.

5 Conclusions

In summary,we have proposed a simple but effective model relating the thrust force gen-eration with the caudal fin oscillation for a CPG-based locomotion control underwater bionic robot. The forces acted on the robot are dividedinto the thrustforce and the drag force for simplification. The average and instantaneous thrust coefficients have been determined by the DPIV and force transducer experiments. Both numerical simulation and physical experiments have been performed to verify effectiveness of our proposed dynamic model. The results reveal that the characteristic velocity is an effective factor to calculate the thrust force. This simple model will be useful for the design of bio-inspired robots. In the future work,we will focus on more experiments related on the bidirectional interaction of robot tail amplitude,frequency with the trailing vortex. Furthermore,a kind of locomotion optimization method will be built toimprove the ability of the robot swimmingin water.