In this paper, a new robotic fish propelled by a hybrid tail, which is actuated by two active joints, is developed. The first joint is driven by a servo motor, which generates flapping motions for main propulsion. The second joint is actuated by a soft actuator, an ionic polymer-metal composite (IPMC) artificial muscle, which directs the propelled fluid for steering. A state-space dynamic model is developed to capture the two-dimensional (2D) motion dynamics of the robotic fish. The model fully captures the actuation dynamics of the IPMC soft actuator, two-link tail motion dynamics, and body motion dynamics. Experimental results have shown that the robotic fish is capable of swimming forward (up to 0.45 body length/s) and turning left and right (up to 40 deg/s) with a small turning radius (less than half a body length). Finally, the dynamic model has been validated with experimental data, in terms of steady-state forward speed and turning speed at steady-state versus flapping frequency.

## Introduction

In the last few decades, biomimetic robotic systems, such as robotic fish, have received a growing interest from environmental monitoring, intelligence collection, and source tracking. Research has been conducted intensively in bio-inspired design, physics-based modeling, and control of robotic fish to achieve high maneuverability and propulsive efficiency. For example, the robotic fish locomotion and sensing research team has done a great amount of work on modeling, control, sensing, and bioinspired design of the robotic fish using compliant actuators and sensors [13]. Triantafyllou and Triantafyllou have spent a great deal of effort in understanding fish-like swimming [4]. Epps et al. [5] have also been studying the swimming performance of biomimic-compliant robotic fish. Since various real-life environmental tasks require different characteristics, many groups have developed different types of robotic fish [613]. To achieve high agility in water, some robotic fish, including soft-body and multiple-link tails [1417], use multiple actuators or a compliant actuation mechanism to make the body-tail more flexible. For example, Nakabayashi et al. investigated robotic fish propulsion using a fin connected with a variable-effective-length spring [18]. Nagai demonstrated the swimming performance of a small-scale mechanical fish propelled by a two-link tail actuated by a servo motor [19]. Swimming performance, in terms of forward speed, was improved by using a harder spring in the second joint. However, maneuvering capability, in terms of turning speed, with small radius was downgraded by increasing the stiffness of the second joint. In order to achieve both high speed and high maneuvering capabilities, an active second joint with varying stiffness and bending moment will be needed.

Robotic fish that employ soft actuation materials are increasingly gaining interest from researchers. Electroactive polymers, also referred to as artificial muscles, are attractive materials for creating biomimetic robots because they are flexible and lightweight, and exhibit significantly large deformation [20,21]. For this reason, using EAP soft actuators can dramatically decrease the weight of the robotic fish and enable bio-inspired designs. EAPs have different configurations. Dielectric EAPs, such as dielectric elastomers, can generate large force with fast time response but require high activation voltage [21], which limits their underwater applications. Ionic EAPs can generate large bending deformation under wet conditions, which makes them viable in robotic fish applications. Much effort has been spent on using ionic EAP only as artificial muscles in robotic fish [2224], robotic manta rays [2527], and robotic jellyfish [28,29]. However, ionic EAPs have some disadvantages in terms of their small force and slow time response, which limit the robot in achieving high-speed performance. For example, the robotic fish propelled by an ionic polymer-metal composite (IPMC) caudal fin developed by Chen et al. can only achieve about 0.125 BL/s forward swimming speed [22].

To meet the speed requirement for real-world tasks, the body-tail structure of the robotic fish needs to oscillate at a relative high frequency so that the tail can generate enough thrust force to overcome the drag force introduced by the fluid. Compared to soft actuators, traditional servo motors can generate large enough torque to drive the fish body or tail to oscillate at a relatively fast enough frequency. For example, Kopman et al. developed a robotic fish propelled by a single servo motor driven tail which can achieve 0.67 BL/s (10 cm/s) [30]. To achieve two-dimensional (2D) maneuvering capability, 2D thrust force and turning moment need to be generated by the tail. There are two approaches to generating 2D thrust force and turning moment. One approach is using a single servo to generate asymmetric flapping motion on the tail. Tan et al. studied the turning of robotic fish using one precise servo to generate asymmetric flapping [31]. They also found that the robotic fish can achieve better turning performance with a flexible caudal fin, in terms of smaller turning radius, compared to the robotic fish with a rigid caudal fin. Another approach is changing the shape of the caudal fin, while the tail is symmetrically flapping. This approach requires another actuator acting at the joint between the tail and the caudal fin. Intuitively, using a servo motor at that joint can solve the problem. However, adding another servo motor makes the caudal fin bulky and heavy, which downgrades the propulsion efficiency. Using an ionic EAP soft actuator in the second joint will make the design much simpler and lighter. For this reason, combining an ionic EAP soft actuator and servo motor in a hybrid tail is more beneficial since the hybrid tail can generate fast enough flapping motion by using a servo as well as change the thrust direction effectively by using a soft actuator.

In this paper, a robotic fish propelled by a two-joint hybrid tail is developed. The tail consists of a direct current (DC) servo motor acting on the first joint and a soft actuator acting on the second joint. The DC servo motor generates a fast enough flapping motion on the tail, which leads to a main thrust force for forward swimming. An IPMC is selected as the soft actuator in order to change the shape of the caudal fin which directs the thrust force for turning. IPMC is one type of ionic EAP that requires only low actuation voltage and can generate large deformation under wet conditions. An IPMC sample typically consists of a thin ion-exchange membrane (e.g., Nafion™), chemically plated on both sides of the surfaces with a noble metal as the electrode [32]. When voltage is applied to the IPMC, the transportation of hydrated cations and water molecules occurs within the membrane, and the associated electrostatic interactions lead to a bending motion. IPMC has been used for robotic fish by many research groups [13]. In this paper, IPMC is used only in the second joint of the hybrid tail.

Since feedback control is greatly needed for the robotic fish to achieve 2D or even three-dimensional (3D) maneuverability, it is necessary to have a reliable and practical dynamic model of the robotic fish for the design of a stable and optimal control for the robot. Modeling of a flexible IPMC beam has been studied by many researchers. For example, Chen et al. developed a steady-state speed model of a robotic fish propelled by an IPMC caudal fin [22]. Ye et al. developed a 2D dynamic model for a robotic fish propelled by multiple IPMC artificial fins, including two pectoral fins and one caudal fin [23]. The modeling of a multijointed robotic fish has been done in some research groups [18,33]. However, little modeling work has been reported on the robotic fish propelled by a hybrid tail driven by both a soft smart-material actuator and a servo motor. In this paper, a dynamic model of a robotic fish propelled by the hybrid tail is developed for control and design purposes. This model is described in state-space, which incorporates the actuation dynamics of IPMC, 2D body dynamics, and hydrodynamics of a two-link tail with a soft and active joint.

The major contributions of this paper are as follows. First, a 2D maneuverable robotic fish propelled by a two-link hybrid tail driven by a servo motor and IPMC soft actuator is developed for the first time. The servo motor is used to generate the main thrust, and the IPMC is used to generate a steering moment. Experimental results have shown that the robotic fish is capable of fast forward swimming (up to 0.45 body length/s) and quick turning (up to 40 deg/s) with a small turning radius (less than half a body length). The maneuvering capability of this robotic fish is much better than the capability of the robotic fish using multiple IPMC fins in the previous work which only demonstrated 0.067 BL/s forward speed and 2.5 deg/s turning speed [23]. Compared to the robotic fish propelled by a single servo motor, this robotic fish can achieve a smaller turning radius. For example, the reported turning radius of a single motor driven robotic fish was 2.67 BL. In this paper, both simulation and experimental results have shown that this robotic fish can achieve 0.5 BL turning radius. This hybrid design is better than the design using servo motors or ionic EAPs only. The major difference between this work and other groups' robotic fish-related research is that this hybrid design employs a two-link hybrid tail consisting of a servo motor for main propulsion and a soft actuator for steering, thus combining advantages of the servo motor for high torque output and the electroactive polymer for flexibility and easier design. This novel design can achieve fast forward speed and quick turning with a small turning radius, consequently simplifying control compared to using two pectoral fins plus one caudal fin. Second, a fully coupled dynamic model of the robotic fish is decoupled and described in state-space for control design purposes. The derived model is different from the previous model since the hybrid tail dynamics are involved. The kinematics of the tail and the dynamics of the body are decoupled by assuming an appropriate robotic fish design and operation mode. The state-space model can fully capture the hydrodynamic interactions between the tail and the fluid, actuation dynamics of the IPMC, and motion dynamics of the rigid body. The model has two separated control inputs for forward swimming and turning of the robot, which will enable advanced guidance controls such as collision avoidance control.

The remainder of this paper is organized as follows: Design of the multijointed robotic fish is described in Sec. 2. Development of the dynamic model of the robotic fish is presented in Sec. 3. Fabrication of the robotic fish is discussed in Sec. 4. Model and design verification is presented in Sec. 5. Conclusions and future work are presented in Sec. 6.

## Description of Two-Dimensional Maneuverable Robotic Fish

### Two-Dimensional Maneuvering Robotic Fish Design.

The robotic fish developed here aims to be a fully autonomous and aquatic platform for multinode sensing applications. The robot consists of a rigid body and a two-joint hybrid tail actuated by a servo motor acting on the first joint and an IPMC acting on the second joint. Figure 1(a) shows a schematic design of the robotic fish, and Fig. 1(b) shows a schematic design of the two-link tail.

The soft IPMC actuator is clipped with two gold-coated copper electrodes that are used for applying voltage to the IPMC. The voltage level depends on how fast of a turning speed is needed. The IPMC actuator is attached to a plastic passive fin to enhance propulsion. The main fish body is connected to the hybrid tail with the servo motor which drives a solid and rectangular plate to flap at a desirable frequency. The main reason for using the servo motor in the first joint is to generate high enough forward thrust. The plate is connected to a caudal fin with the IPMC actuator. The main reason for using IPMC in the second joint is to change the slope of the caudal fin, which can direct the propelled fluid to generate a 2D thrust. Since the tail has two joints and the head is larger than the tail, this robotic fish is designed for carangiform locomotion which has less head motion.

### Two-Dimensional Maneuvering Mechanism.

The two-dimensional maneuvering capability of the robotic fish can be achieved by using one multijointed tail with a caudal fin at the end, as shown in Fig. 2. The center of mass is located at point G and the body rotating axis is located at C. When no voltage is applied to the IPMC, it will stay straight, as shown in the middle image in Fig. 2. When the servo motor oscillates at a certain frequency without a bias angle, the caudal fin will also follow the flapping without a bias angle. By integrating the thrust force over one flapping period, one can obtain a mean thrust which points to the forward direction without turning moment. In this case, the robotic fish will swim forward. When the IPMC actuator is activated with a positive or negative voltage, the IPMC joint will generate either a left or right bending angle on the caudal fin. When the servo motor oscillates at a certain frequency without a bias angle, the caudal fin will follow the flapping but with a bias bending angle which is generated by the IPMC joint. A left bias bending angle will make the robot turn left, and a right bias angle will make the robot turn right. The right-turning mechanism is illustrated in the right image of Fig. 2.

## Dynamical Model

The full dynamic model of the robotic fish propelled by a single servo was developed in Ref. [28]. Since the dynamics of the fish body and kinematics of the tail were fully coupled in Ref. [30], it was impossible to develop a state-space model that can be used in advanced controls. In order to derive a state-space model of the robotic fish for control purposes, we have to simplify the kinematics of the hybrid tail. In this paper, when the kinematics of the tail is derived, only the flapping motion generated by the servo is considered and the effect from the rotation of the body is ignored. This model simplification makes the generated hydrodynamic forces and moments independent of the heading angular velocity of the fish body, which decouples the kinematics of the tail and the dynamics of the body. This model simplification is valid based on the following two conditions. The first condition is that the joint of the servo is very close to the turning axis of the fish body. Under this condition, the absolution velocity at the servo joint does not have a big component perpendicular to the fish body. In other words, under this condition, the whole fish almost rotates around the servo joint. The second condition is that the angular acceleration of the fish body is much less than the angular acceleration of the flapping tail. To meet this condition, the fish tail will operate with a large flapping amplitude and the robotic fish is designed for carangiform locomotion which has less head motion. Compared to the flapping motion of the tail, the rotating motion of the body can be ignored when deriving the hydrodynamic forces and moment of the tail.

In this section, a physics-based and control-oriented model of the dynamics of the robotic fish is presented in a 2D axis plane. The derivation of the model will be organized in the following way. First, a body dynamic model developed in Refs. [30], [34], and [35] will be introduced. The model captures the motion dynamics responding to the thrust forces in the X and Y directions and the turning moment in the Z direction. Second, the drag and lift forces acting on the body will be derived. Third, the kinematics of the hybrid tail will be described. Fourth, a hybrid tail dynamics model will provide the generated 2D thrust force and turning moment which are related to the angles in the first and second joints. Fifth, hydrodynamic forces and moments will be derived based on the simplified kinematics of the hybrid tail. Sixth, the actuation dynamics of the IPMC joint will be added into the tail dynamics, which captures the relationship between the angle of the IPMC joint and the input voltage applied to the IPMC. Finally, a physics-based and control-oriented model will be derived in state-space. The state-space model has two system inputs—the first joint angle controlled by the servo and the voltage applied to the IPMC—and two system outputs—the 2D position of the robot.

### Fish Body Dynamics.

A schematic representation of the robotic fish in a 2D planar motion is shown in Fig. 3. Let us set [X, Y, Z] to be an inertial coordinate system and [x, y, z] to be a body-fixed coordinate system. It is assumed that both the body and the tail are naturally buoyant. The entire body center of gravity is located at point G, and the body rotation is located at point C. The velocity at point C is expressed as surge (u) in the x-direction, sway velocity (v) in the y-direction, and yaw motion ($ωz$) in the z-direction.

The equations of motion dynamics can be simplified as follows [30,34,35]:
$(mb−Xu˙)u˙(t)=(mb−Yv˙)v(t)r(t)+Fx(t)$
(1)

$(mb−Yv˙)v˙(t)=−(mb−Xu˙)u(t)r(t)+Fy(t)$
(2)

$(Jbz−Nr˙)ω˙z(t)=(Yv˙−Xu˙)u(t)v(t)+Mz(t)$
(3)
where $mb$ is the body mass, $Jbz$ is the moment of inertia about the z-axis, $Fx(t)$ and $Fy(t)$ are net forces in the x-direction and the y-direction, respectively, $Mz(t)$ is the net turning moment around the z-axis, $ωz(t)$ is the angular velocity of the body, and $−Xu˙$,$−Yv˙$, and $−Nr˙$ are the constant hydrodynamic derivatives describing the effect of added mass due to the surrounding fluid, which were positive values reported in Ref. [34]. According to Ghassemi and Yari [36], for an ellipsoid shape fish body, $Xu˙$, $Yv˙$, $Nr˙$$Jbz$, and Jcm can be calculated, respectively, as follows:
$Xu˙=−k11πρwDR2L6$
(4)

$Xv˙=−k22πρwDR2L6$
(5)

$Nr˙=−k66πρwDR2L(L2+DR2)120$
(6)

$Jbz=Jcm+mbc2$
(7)

$Jcm=110mb(DR2+L2)$
(8)

where $ρw$ is the density of water, $DR$ is the diameter of the fish body, $L$ is the length of the fish body, c is the distance between G and C, $Jcm$ is the moment inertia along the axis along G, and $k11$, $k22$, and $k66$ are nondimensional added mass coefficients for an ellipsoid body.

The kinematic equations for the robot are [34]
$X˙=u cos(ψ)−v sin(ψ)$
(9)

$Y˙=v cos(ψ)+u sin(ψ)$
(10)

$ψ˙=ωz$
(11)

$|Vc|=u2+v2$
(12)

$β=atan(vu)$
(13)

where $ψ$ is the heading angle of the fish body, which is defined as the angle between the X-axis of inertial coordination and the x-axis of the fixed-body coordinate, and $β$ is the swaying angle, which is defined as the angle between the velocity $V⇀c$ and the x-axis.

### Drag and Lift Forces on Body.

Based on the body dynamics model, it is necessary to determine the net forces $Fx$ and $Fy$ and the turning moment $Mz$ in order to obtain the 2D dynamic model for the robotic fish. As shown in Fig. 3, the robot fish body is impacted by the drag force $FD$, lift force $FL$, and drag moment $MD$, which can be represented, respectively, as
$FD=12ρw|Vc|2SCD$
(14)

$FL=12ρw|Vc|2SCLβ$
(15)

$MD=−CMωz2sgn(ωz)$
(16)
where S is the wetted surface area of the robot body, $ρ$= 1000 kg/m3 is the mass density of water, and $CD$, $CL$, and $CM$ are the coefficients of drag force, lift force, and drag moment, respectively. By adding the hydrodynamic forces and moment from the hybrid tail to the body, the formulas for $Fx$, $Fy$, and $Mz$ in Eqs. (1)(3) can be derived as follows:
$Fx=T1 sin α1+T2 sin α2−FD cos β+FL sin β$
(17)

$Fy=T1 cos α1+T2 cos α2−FD sin β−FL cos β$
(18)

$Mz=MD+T1a1 cos α1+T2a1 cos α2+M1$
(19)
where $T1$ and $T2$ are the total hydrodynamic forces acting on links 1 and 2, respectively, $M1$ is the total hydrodynamic force induced bending moment on Joint 1, and $a1$ is the distance between the center of mass and the center of the first joint. For n = 1, 2, $αn$ is denoted as the angles between the nth link and the x-axis, $on$ is denoted as the centers of the nth joint, and $αc$ is the bending angle of the IPMC joint. Based on the geometry of the two-link tail, we have
$α2=α1+αc$
(20)

### Hybrid Tail Kinematics.

In the drag and lift model shown in Sec. 3.2, the net forces and moment are dependent on the hydrodynamic forces and moments acting on the hybrid tail. The hydrodynamic forces and moments of a two-link tail were captured by the model developed by Anton et al. [16], where the tail was actuated by two IPMC joints, assuming that two system inputs were the angles of two joints. In this paper, a kinematic model is developed for the new two-joint tail in which one joint is actuated by a servo motor and another joint is actuated by an IPMC. Since the angle on the IPMC joint is unknown, one needs to solve the coupled equations which consider all the hydrodynamic forces and moments, actuation dynamics of IPMC, and viscoelastic beam dynamics, to obtain the angle on the IPMC joint. There is also a bias distance between the first link and the servo joint, which is different from Anton's tail. The reason for adding a bias distance between the first link and the servo joint is to make the center of the body rotation close to the location of the first joint which satisfies the first condition for decoupling body dynamics and tail kinematics. Under that condition, the distance of CO1 in Fig. 3 should be very close to zero.

In this subsection, a part of the modeling work from Ref. [16] will be introduced first and then be modified it to capture the 2D forces and moments of the new two-link tail designed for the robotic fish. After that, the actuation model of IPMC will be introduced. Finally, the coupled equations will be solved to obtain the relationship between the system inputs (angle of servo and IPMC voltage) and the system outputs including hydrodynamic forces and moments. To make the model suitable for any trapezoid shape of links, the schematics of the tail are defined as shown in Fig. 4. Since the servo is connected to the tail with a long link, it is possible to design the length of that link to make CO1 close to zero.

Note that the passive fin shown in Fig. 1(a) was a trapezoid with a notch. To simply the analytical model of the tail, the shape of the passive fin is approximated by a trapezoidal shape. In the future work, a computational fluid dynamics analysis will be conducted to investigate how the shape of the passive fin affects swimming performance

The tail frame is defined through a set of orthonormal basis vectors ${i,j,k}$, where $i⇀$ coincides with the longitudinal axis of the first link of the tail at the neural position, $j$ is perpendicular to $i$, and in the horizontal plane, while $k⇀$ is in the upward direction. For n = 1, 2, $rn$ and $wn$ are denoted as the unit vectors along and perpendicular to the hybrid tail links, respectively. Based on the reference frame shown in Fig. 4, these vectors can be written as follows [16]:
$r1=i cos α1+j sin α1$
(21)

$w1=−i sin α1+j cos α1$
(22)

$r2=i cos(α1+αc)+j sin(α1+αc)$
(23)

$w2=−i sin(α1+αc)+j cos(α1+αc)$
(24)
One assumes that the tail flaps in an inviscid, two-dimensional flow (in the plane). Since the IPMC are short and used as joint, the hydrodynamic forces due to IPMC movement are ignored. Given the joint angle $α1(t)$ and the velocity $Vo1$ at the first joint, the velocity of any point on link 1 can be derived as
$V1(τ,t)=Vo1+τ1α˙1(t)w1, 0≤τ1≤L1+L0$
(25)
Since the body rotates about C point with the body angular velocity $ωoz$, the velocity at the first joint can be written as
$Vo1(t)=ωozCO1w0$
(26)
where CO1 is the distance between C and O1, $w0$ is the unit vector perpendicular to the vector $CO1$. With the two assumptions that $CO1≈0$ and $|ωoz|≪|α˙1(t)|$, one can rewrite Eq. (25) as
$V1(τ,t)≈τ1α˙1(t)w1$
(27)

Note that in Eq. (27), the effect of body rotation on the kinematics of the tail is ignored.

The hydrodynamic force acting on a moving rigid beam is proportional to the propelled virtual mass multiplied by the acceleration. The velocity of any point on a rigid link can be decomposed into two terms, $V⊥$ is perpendicular to the link and $V∥$ is along the link. The assumption of inviscid flows implies that the virtual mass effect introduced by movement in the $V∥$ direction is negligible. For link 1, the acceleration at this point is the time derivative of $V⇀1$ in the $w⇀1$ direction
$a1⊥(τ1,t)=d(V1⋅w1)dtw1=τ1α¨1(t)w1$
(28)
Following the same idea, the acceleration at any point of link 2 can be also derived. The velocity of any point in link 2 is
$V2(τ2,t)=τ2α˙c(t)w2+(L1+L0)α˙1(t)w1, 0≤τ2
(29)
The acceleration at that point in the $w⇀2$ direction is
$a2⊥(τ2,t)=d(V2⋅w2)dtw⇀2=(τ2α¨2(t)+χ(t))w2$
(30)
where
$χ(t)=(L1+L0)(α¨1(t)cos(αc(t))−α˙1(t)α˙c(t)sin(αc(t)))$
(31)
The width of the links can be captured by
$D(τ1)=D1−k1(τ1−L0), L0≤τ1≤L1$
(32)

$B(τ2)=B1+k2τ2, 0≤τ2≤L2$
(33)

where $k1=(D1−D2/L1)$; $k2=(B1−B2/L2)$; $D1$ and $D2$ are the beginning and ending widths of link 1, respectively; $B1$ and $B2$ are the beginning and ending widths of link 2, respectively; L1 and L2 are the lengths of links 1 and 2, respectively; and L0 is the distance between the servo joint and the left edge of link 1.

### Hydrodynamic Force and Moment.

Since an inviscid, two-dimensional flow is assumed, the hydrodynamic force acting on a moving rigid beam is proportional to the propelled virtual mass multiplied by the acceleration that is perpendicular to the rigid beam. The hydrodynamic forces per unit length acting on the links are as follows [37]:
$Fh1(τ1,t)=−ρwπ4D2(τ1)Γ(ω)α1⊥(τ1,t)$
(34)

$Fh2(τ2,t)=−ρwπ4B2(τ2)Γ(ω)α2⊥(τ2,t)$
(35)

where $Γ(ω)$ is the hydrodynamic function for rectangular beam $L0≤τ1≤L0+L1$, and $0≤τ2≤L2$. The hydrodynamic function is a dimensionless complex function of Reynold number and it can capture both added mass effect and damping effect when a cantilever beam oscillates in a viscous fluid [37]. Due to the space limitation, the analytical representation of hydrodynamic function, which can be found in Ref. [37], is not introduced here.

By integrating the hydrodynamic force density along each link, one can obtain the total hydrodynamic forces acting on the two links as
$T1(t)=−ρwπ4Γ(ω)a¨1(t)(λf+L0λl)w1$
(36)

$T2(t)=−ρwπ4Γ(ω)(α¨c(t)λb+λcχ(t))w2$
(37)
and total hydrodynamic moments acting on the two joints as
$M1(t)=−ρwπ4(Γ(ω)α¨1(t)(λe+2L0λf+L02λl)+Γ(ω)((α¨c(t)λa+χ(t)λb+α¨c(L0+L1)cos(αc(t))(λb+χ(t)λc)))$
(38)

$M2(t)=−ρwπ4Γ2(ω)(α¨c(t)λa+χ(t)λb)$
(39)

where $λa$, $λb$, $λc$, $λe$, $λf$, and $λl$ are defined in the Appendix. The detailed derivation of total hydrodynamic forces and moments is shown in the Appendix.

### Dynamics of Ionic Polymer-Metal Composite Actuator.

The remaining modeling work is focused on the actuation dynamics of the IPMC joint, which generates the joint angle $αc$. Without considering the distributed surface resistance, Chen and Tan [38] obtained the bending moment of an IPMC beam in the Laplace domain as
$Ma(s)=2α0KWke(γ(s)−tanh(γ(s)))V(s)sγ(s)+Ktanh(γ(s))$
(40)

$K ≜F2dC−keRT(1−C−ΔV)$
(41)

$γ(s)≜h2(s+K)d$
(42)
where $V(s)$ is the input voltage applied to the IPMC, $α0$ is the electromechanical coupling coefficient, d is the ion diffusion coefficient, F is Faraday's constant, $C−$ is the negative ion density, $ke$ is the effective dielectric constant of the polymer, T is the absolute temperature, R is the gas constant, $ΔV$ is the volumetric change, assuming that $1−C−ΔV≈1$, and W and h are the width and thickness of the IPMC, respectively. Based on the range of parameters, the following approximation can be made:
$γ(s)≈γ=hKd, and tanh(γ(s))≈1$
(43)
in order to simplify the model as follows:
$Ma(s)=2α0KWke(γ−1)V(s)sγ+K$
(44)
With an assumption of zero initial conditions, Eq. (43) can be converted from Laplace domain to time domain as
$γM˙a(t)+KMa(t)=2α0KWke(γ−1)V(t)$
(45)
In the IPMC joint
$αc(t)=ρ2(t)d1$
(46)
where $ρ2(t)$ is the curvature of IPMC joint and d1 is the length of the IPMC. The area moment of inertia is $I2=(1/12)Wh3$. The dynamics of the active and flexible IPMC joint can be written as
$I2α¨c(t)+ξα˙c(t)+KIαc(t)=Ma(t)+M2(t)$
(47)
where $KI$ is the spring constant of joint and $ξ$ is the damping ratio of the IPMC joint
$KI=YI2d1$
(48)
Overall, the system has two inputs: the voltage applied to the IPMC V(t) and the first joint angle $α1(t)$ controlled by the servo. To simplify the model, one can assume that the first joint angle is a sinusoid function of time
$α1(t)=Am sin(ωt)$
(49)

where Am is the amplitude and $ω$ is the angular velocity.

### State-Space Model.

For control design purposes, the dynamic model of the robotic fish needs to be described in state-space. The overall state-space model can be divided into two cascaded submodels with the headings angle $β(t)$, velocity $Vc(t)$, and angular velocity $ωz(t)$ as feedback, as shown in Fig. 5. Overall, the system's inputs are the first joint angle $α1(t)$ and the voltage applied to the IPMC $V(t)$. The system's outputs are 2D position ($X(t)$,$Y(t)$).

The first model is the caudal fin model with three system outputs: thrust forces $Fx$ and $Fy$, and turning moment $Mz$. The inputs to the caudal fin model are the first joint angle $α1(t)$, the voltage applied to the IPMC joint $V(t)$, the swaying angle $β(t)$, and the velocity of the fish body $Vc(t)$ from the output of the body dynamics. In this submodel, the state variables are defined as $x1=αc$, $x2=α˙c$, $x3=Ma$, $u1=α1$, $u2=V$, $u3=β$, and $u4=Vc$, and the outputs are defined as $y1=Fx$, $y2=Fy$, and $y3=Mz$. Based on the dynamic model derived in Secs. 2.1 and 2.2, the state-space model for the hybrid caudal fin can be written as
$x˙1=x2$
(50)

$x˙2=−ξx2−KIx1I2+ρwπ4Γ(ω)λa+x3−ρwπ4Γ(ω)χ(t)λbI2+ρwπ4Γ(ω)λa$
(51)

$x˙3=−Kx3γ+2α0KWke(γ−1)γu2$
(52)

$y1=T1 sin u1+T2 sin(x1+u1)−FD cos u3+FL sin u3$
(53)

$y2=T1 cos u1+T2 cos(x1+u1)−FD sin u3−FL cos u3$
(54)

$y3=MD+T1a1 cos u1+T2a1 cos(x1+u1)+M1$
(55)
The second submodel is the body dynamics model with $Fx$,$Fy$, and $Mz$ as the system's inputs and two 2D position ($X(t)$, $Y(t)$), swaying angle $β(t)$, velocity $Vc(t)$, and turning speed $r(t)$ as the subsystem's outputs. In this model, the state variables are defined as $x1=X$, $x2=Y$, $x3=ψ$, $x4=u$, $x5=v$, and $x6=ωz$; the system inputs are defined as $u1=Fx$, $u2=Fy$, and $u3=Mz$; and the outputs are defined as $y1=X$, $y2=Y$, $y3=β$, $y4=Vc$, and $y5=ωz$. From the dynamics model derived in Sec. 2.1, the state-space model of the body dynamics can be described as
$x˙1=x4 cos x3−x5 sin x3$
(56)

$x˙2=x5 cos x3+x4 sin x3$
(57)

$x˙3=x6$
(58)

$x˙4=mb−Yv˙mb−Xu˙x5x6+u1mb−Xu˙$
(59)

$x˙5=−mb−Xu˙mb−Yv˙x4x6+u2mb−Yv˙$
(60)

$x˙6=Yv˙−Xu˙Jbz−Nr˙x4x5+u3Jbz−Nr˙$
(61)

and the system outputs in terms of state variables are as follows: $y1=x1$, $y2=x2$, $y3=atan(x5/x4)$, $y4=(x42+x52)$, and $y5=x6$.

## Fabrication of the Robotic Fish

To validate the design and model of 2D maneuverable robotic fish, a robotic fish propelled by a hybrid tail was fabricated. The process can be divided into four steps: (1) fabrication of the IPMC artificial fin and hybrid tail structures, (2) realization of the onboard circuit, (3) construction of the fish body, and (4) assembly of the robotic fish. In step 1, the fabrication steps reported in Ref. [13] were followed to make IPMC soft actuator. In this section, the rest three fabrication steps will be introduced.

### Onboard Circuit.

To control the robotic fish remotely in a mobile sensing network, a microcontroller (particle photon) with a WiFi module was used for exchanging data between the fish and the control station personal computer (PC) and controlling the servo and IPMC joints. The Photon module received the commands from the PC though a cloud server. The robotic fish can also upload its data to the cloud server, which will enable big data research in the future. The Photon was used to generate two square wave control signals. One signal was generated to drive the IPMC joint; another signal was generated to drive the servo motor. Because the microcontroller draws only 25 mA and the output current goes through the IPMC up to 500 mA, one H-bridge driver was used to provide up to 2 A peak current output to the IPMC. The total weight of the onboard circuit and one battery weight was around 20 g. This circuit is illustrated in Fig. 6. A lithium ion polymer battery (Tenergy, 7.4 V 6000 mAh) was used to provide electricity to the robotic fish.

### Fabrication of Fish Body.

A rigid fish body was developed to house onboard circuit and battery. It was necessary for the body to have a hydrodynamic shape so that the drag force could be minimized. This body was designed using Autodesk Inventor and consisted of two shells clamped together using screws. Inside the shells were two chambers: one to house the electronic circuit and battery, and the other to provide a platform for some future underwater applications. The fish body was printed with acrylonitrile butadiene styrene material using a 3D printer. Since the density of the material is lighter than water, it was easy to make the robotic fish move near the water's surface in order to receive commands from Wi-Fi network.

### Assembly of Robotic Fish.

All the battery and circuits were zipped into a plastic bag to ensure waterproofing of the electronic components. Two gold-coated copper electrodes were placed on the rear side of the fish to provide actuation voltage signals for the pectoral fins, and one copper electrode was placed at the rear of the fish for applying a voltage signal to the caudal fin. The entire length of the fish including the hybrid tail was around 27 cm, the diameter was 8 cm, the total inside volume was around 120 cm3, and the weight was about 180 g. The dimensions of the IPMC joints and passive links were defined as previously shown in Fig. 1. Figure 7 shows the fabricated robotic fish.

## Model and Design Verification

### Free Swimming Test.

The robotic fish was tested in a 550-gallon water tank (97 in. long, 38 in. wide, and 37 in. deep). A digital camera in a Nexus smart phone was used to capture a movie of the swimming robotic fish. Figure 8 shows six snapshots of a forward swimming test. Each snapshot has taken every 4 s. The fish's forward swimming speed was controlled by changing the flapping frequency of the caudal fin. To meet the second condition described in Sec. 3, a large amplitude of flapping on the tail was adopted. A sinusoidal wave signal of 45 deg magnitude and 0.55 Hz frequency was applied to the servo motor. The IPMC joint was not actuated. The swimming speed was extracted from the collected video through an edge-detection imaging process. At 0.55 Hz, the forward swimming speed reached about 7.1 cm/s, which is about 0.26 BL/s.

Turning tests were conducted to verify the steering capability of the pectoral fin. To make a left turn, both the IPMC and the servo joints were actuated. The servo joint provided the main forward thrust force, while the bending generated by the IPMC directed the thrust force to make the fish turn right. The servo motor was oscillating under the same flapping signal as in the forward swimming test. The turning speed was extracted from the collected video through an edge-detection imaging process in which the yaw angle was detected. The right turning speed reached about 29.8 deg/s. Snapshots of the turning swimming test are shown in Fig. 9. The turning radius was measured at less than 12 cm, which was less than 0.5 body length. For opposite turning, an opposite voltage was applied to the IPMC joint, driving the caudal fin to bend in the opposite direction, in turn leading the fish to turn left, as observed in the experiment.

### System Identification.

The framework of the fish model was discussed in Sec. 3. To validate the model, all parameters needed for the model were either measured, obtained from the literature, or extracted from the experiments. First, the body data, fin shape, and hybrid tail dimensions were measured from the assembled robotic fish. Table 1 shows the dimensions of the hybrid tail. Parameters of the IPMC material were obtained from the literature [23] and are shown in Table 2. Parameters related to the fish body, which are either identified through fitting process or measured, are shown in Table 3.

Drag coefficients CD and CL and drag moment coefficient CM were extracted by fitting the model's simulation result with the experimental data. Two tests were conducted to collect the needed data for identifying these three parameters. In the first test, the IPMC joint was not activated, and the servo joint oscillated at a series of different frequencies ranging from 0.2 Hz to 2 Hz but with the same amplitude. The angle amplitude was set at 45 deg. However, due to the second-order dynamics of the servo, the amplitude decreased as the frequency increased. The forward-swimming speeds under different frequencies were extracted from the captured videos. A third-order transfer function
$G(s)=ωn2s2+2ξs+ωn211+s$
(62)
was used to capture the dynamics of the servo motor, where the natural frequency $ωn$= 6.28 rad/s and the damping ratio $ξ=$ 0.2. The drag coefficients CD and CL were adjusted to 0.06 and 4.7 in order to fit the experimental data with the simulation data. Figure 10 shows the experimental data and simulation data, indicating a reasonably good fit. The maximum forward speed reached 12 cm/s, which was about 0.45 BL/s.

In the second test, the servo oscillated at different frequencies, and a 7.3 V voltage was applied to the IPMC joint to make the fish turn. The turning angle speeds under different frequencies were extracted from the collected videos. The coefficient CM as $4.7×10−5$ was identified to fit the experimental data with the simulation data, which are shown in Fig. 11. The maximum turning speed reached 40 deg/s, which is much higher than observed in the previous study. The reason that the simulation data in forward speed and turning speed were similar is that the turning moment generated and forward force were almost proportional to the angular velocity of the servo joint. Since the dynamics of the servo was a second-order low‐pass filter. So, the angular velocity of the first joint decreased as the flapping frequency increased when the frequency went beyond the servo motor's natural frequency 6.28 rad/s. Thus, both turning speed and forward speed decreased as the flapping frequency went beyond the natural frequency.

### Design Validation.

To validate the 2D maneuverable robotic fish design, the state-space model was simulated with the identified parameters. In the simulation, the servo oscillation frequency was 0.5 Hz and the amplitude of oscillation was 45 deg. The IPMC voltage was set at −1 V, 0 V, and 1 V in three simulations, respectively. In each simulation, the robot started at the origin point (0, 0). Figure 12 shows the fish X–Y trajectory with different IPMC voltages.

When the IPMC voltage was set at zero, the robotic fish swam straight forward. When the IPMC was set at −1 V, the robotic fish made a left turn, and finally, kept counter-clockwise rotating with a 13.5 cm turning radius. Since the total length of the robot (including tail) was 27 cm, the turning radius at the steady-state was 0.5 BL. When the IPMC was 1 V, the robotic fish made a right turn, and finally, kept clockwise rotation with a 0.5 BL turning radius. Figure 13 shows the mean angular speed versus time with different IPMC voltages. It has been clearly shown that the robotic fish can achieve 2D maneuvering capabilities by controlling the IPMC voltage. Both simulation and experimental results validated the design.

## Conclusion and Future Work

This paper presents a 2D maneuverable robotic fish propelled by a hybrid tail actuated by a servo actuator and an IPMC soft actuator. The robot design was inspired by an actual biological fish, which uses a caudal fin for its main propulsion and one IPMC beam for steering. By controlling the second joint with the IPMC, the robotic fish was able to make left and right turns as well as swim forward. This paper also derives a dynamic model for the robotic fish, which is described in state-space for control design purposes. The free-swimming tests showed that the fish can reach a forward speed of up to 12 cm/s and turning speed up to 40 deg/s. With one-fin propulsion, the robot demonstrated its 2D maneuvering capability, thus proving its potential in underwater sensing network applications.

This work will be extended in several directions. First, the model will be validated using both experimental data and computational fluid dynamics simulation data. Second, the hybrid tail structure will be redesigned to make it more propulsive efficient. Third, an advanced modeling fitting process will be researched to identify the system parameters. Fourth, since the IPMC joint has not been fully utilized at this stage, the future research will be focused on how to operate the IPMC joint in order to achieve both high forward speed and good maneuvering capabilities. Finally, the developed model will be utilized to design advance controls, such as a collision avoidance control and sourcing tracking control, and implement these controls in the robotic fish to accomplish some real tasks, such as pollution source tracking and oil spill monitoring.

## Funding Data

• The National Science Foundation (Grant No. CNS #1446557; Funder ID: 10.13039/501100008982).

### Appendix

##### Derivation of Total Hydrodynamic Forces and Moments on Joint 1 and Joint 2.
The total forces acting on links 1 and 2 can be obtained by integrating the distributed force density $Fh1(τ1,t)$ and $Fh2(τ2,t)$ over the length of links 1 and 2, respectively. Based on Eqs. (32) and (34), the total force acting on link 1 can be derived as
$T⇀1(t)=∫L0L0+L1F⇀h1(τ1,t)dτ1=∫L0L0+L1−ρwπ4D2(τ1)Γ(ω)α1⊥(τ1,t)dτ1=∫L0L0+L1−ρwπ4(D1−k1(τ1−L0))2Γ(ω)τ1α¨1(t)w⇀1dτ1=−ρwπ4Γ(ω)a¨1(t)(λf+L0λl)w⇀1$
(A1)
where
$λf=D12L122−2k1D1L133+k12L144$
(A2)

$λl=D12L1−k1D1L12+k12L133$
(A3)
Based on Eqs. (33) and (35), the total force acting on link 2 can be derived as
$T⇀2(t)=∫0L2F⇀h2(τ2,t)dτ2=∫0L2−ρwπ4B2(τ2)Γ(ω)α2⊥(τ2,t)dτ2=∫0L2−ρwπ4(B2+k2τ2)2Γ(ω)(τ2α¨2(t)+χ(t))w⇀2dτ2=−ρwπ4Γ(ω)(α¨c(t)λb+λcχ(t))w⇀2$
(A4)
where
$λb=B12L222+2k2B1L233+k22L244$
(A5)

$λc=B12L2+2k2B1L222+k22L233$
(A6)
Following the same idea, the hydrodynamic force introduced bending moment acting on joint 1 can be obtained by the following integration:
$M1(t)=∫L0L0+L1r⇀1τ1×F⇀h1(τ1,t)dτ1+∫0L2(r⇀1(L0+L1)+r⇀2τ2)×F⇀h2(τ2,t)dτ2$
(A7)
With Eqs. (34) and (35), after integration, we can get
$M1(t)=−ρwπ4(Γ(ω)α¨1(t)(λe+2L0λf+L02λl)+Γ(ω)((α¨c(t)λa+χ(t)λb+α¨c(L0+L1)cos(αc(t))(λb+χ(t)λc)))$
(A8)
where
$λa=B12L233+2k2B1L244+k22L255$
(A9)

$λe=D12L133−2k1D1L144+k12L155$
(A10)
The hydrodynamic force introduced bending moment acting on joint 2 can be obtained by the following integration:
$M2(t)=∫0L2r⇀2τ2×F⇀h2(τ2,t)τ2dτ2$
(A11)
With Eq. (35), after integration, Eq. (A11) can be written as
$M2(t)=−ρwπ4Γ(ω)(α¨c(t)λa+χ(t)λb)$
(A12)

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