1

Three-Dimensional Dynamic Modeling and MotionAnalysis for an Active-Tail-Actuated Robotic Fish

with Barycentre Regulating MechanismXingwen Zheng1, Minglei Xiong1,2, Junzheng Zheng1, Manyi Wang1, Runyu Tian1,3, and Guangming Xie1,4,5

Abstract—Dynamic modeling has been capturing attention forits fundamentality in precise locomotion analyses and control ofunderwater robots. However, the existing researches have mainlyfocused on investigating two-dimensional motion of underwaterrobots, and little attention has been paid to three-dimensionaldynamic modeling, which is just what we focus on. In thisarticle, a three-dimensional dynamic model of an active-tail-actuated robotic fish with a barycentre regulating mechanismis built by combining Newton’s second law for linear motionand Euler’s equation for angular motion. The model parametersare determined by three-dimensional computer-aided design(CAD) software SolidWorks, HyperFlow-based computationalfluid dynamics (CFD) simulation, and grey-box model estimationmethod. Both kinematic experiments with a prototype andnumerical simulations are applied to validate the accuracy of thedynamic model mutually. Based on the dynamic model, multiplethree-dimensional motions, including rectilinear motion, turningmotion, gliding motion, and spiral motion, are analyzed. Theexperimental and simulation results demonstrate the effectivenessof the proposed model in evaluating the trajectory, attitude,and motion parameters, including the velocity, turning radius,angular velocity, etc., of the robotic fish.

Index Terms—Three-dimensional dynamic modeling, Newton-Euler method, computational fluid dynamics (CFD), grey-boxmodel estimation, robotic fish.

I. INTRODUCTION

IN recent years, underwater robots including varieties ofunderwater remotely operated vehicles (ROV), autonomous

underwater vehicles (AUV), and bio-inspired aquatic systems[1] have been developed and shown great potentials in pro-moting marine resource exploitation [2], [3], marine economydevelopment [4], [5], and marine ecological environment pro-tection [6], [7]. The research topics of underwater robots coverlocomotion control and optimization [8], [9], underwater navi-gation and localization [10], [11], environment perception andobject recognition [12], [13], underwater communication [14],

1Xingwen Zheng, Minglei Xiong, Junzheng Zheng, Manyi Wang,Runyu Tian, and Guangming Xie are with the State Key Laboratoryfor Turbulence and Complex Systems, Intelligent Biomimetic DesignLab, College of Engineering, Peking University, Beijing, 100871,China. {zhengxingwen, xiongml, zhengjunzheng,wangmanyi, trytian, xiegming}@pku.edu.cn.Corresponding author: G. Xie.

2Minglei Xiong is with the Boya Gongdao (Beijing) Robot TechnologyCo., Ltd., Beijing, 100084, China.

3Runyu Tian is with the China Aerodynamics Research and DevelopmentCenter, Mianyang, Sichuan, 621000, China.

4Peng Cheng Laboratory, Shenzhen, 518055, China.5Guangming Xie is with the Institute of Ocean Research, Peking University,

Beijing, 100871, China.

[15], etc. In particular, mechanism investigation of dynamicperformance of underwater robots is fundamental and criticalfor the above-mentioned researches. Besides, precise dynamicmodeling of underwater robots has always been focus anddifficulty in underwater robot research.

For dynamic modeling, the typical modeling methods in-clude Lagrangian dynamics method, Newton-Euler method,Lighthill’s elongated-body theory, Schiehlen method, etc.Basing on the Newton-Euler method, Y. Shi’s group hasbuilt a dynamic model of an AUV, and then investigateddynamic model-based trajectory tracking control of planarmotions of the AUV [16]–[18], without consideration of three-dimensional motions. J. Yu’s group has formulated a roboticfish dynamics using Schiehlen method [19] and Lagrangiandynamics method [20]. It has been demonstrated that theproposed dynamic model is efficient for seeking backwardswimming pattern of the robotic fish [20]. They have alsoproposed a data-driven dynamic modeling method in whichthe Newton-Euler formulation is applied to analyze the roboticfish dynamics, and parameters in the dynamic model areidentified using experimental data of rectilinear motion andturning motion of the robotic fish, also without investigatingthree-dimensional motions. F. Zhang’s group has establishedan analytical model for spiral motion of an underwater glidersteered by an internal movable mass block, and experimentsin the South China Sea have validated the accuracy of themodel for achieving desired spiral motion [21]. They havealso explored a dynamic model for a blade-driven glider withgliding motion [22]. However, the motion of glider is differentfrom rhythmic motion of the fin-actuated underwater robot.X. Tan’s group has explored dynamic analyses of a tail-actuated robotic fish [23]–[25] and a fish-like glider [26], [27].For the tail-actuated robotic fish, Lighthill’s large-amplitudeelongated-body theory has been combined with rigid-bodydynamics and hybrid tail dynamics for building a dynamicmodel [23]–[25]. However, only surface motion of the roboticfish has been explored. For the fish-like glider, they have built aNewton-Euler method based dynamic model for investigatingspiraling maneuver [26] and gliding motion [27]. However, thefish-like glider is just driven by displacing an internal movablemass and pumping fluids, while its tail is not active, withouta continuously varied tail angle.

The above-mentioned studies have demonstrated that dy-namic modeling is fundamental and essential for locomotionanalysis of underwater robots. However, most of the researcheshave only focused on investigating two-dimensional motions

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in horizontal plane or vertical plane. Especially for fin-actuatedunderwater robots, though there are a few preliminary worksthat have considered dynamic modeling in three-dimensionalspace [19], [20], [28], the proposed models are typicallyvalidated by limited experiments, without validation in a large-scale parameter space. Besides, for three species of underwaterrobots including active-fin-actuated underwater robot withbarycentre regulating mechanism, blade-driven underwaterrobot [22], and internal movable mass block-driven underwaterrobot [24], all of which can adjust their centers of mass, thereexist significant differences among their dynamics, because anactive-fin-actuated underwater robot with barycentre regulatingmechanism is able to generate extra rhythmic oscillation ofrobot body. However, dynamic modelling for such an under-water robot has been rarely investigated.

On the basis of the above analyses, this article mainlyfocuses on investigating three-dimensional dynamic modelingin a large-scale parameter space for an active-tail-actuatedrobotic fish with a barycentre regulating mechanism, whichhas been rarely investigated. Multiple swimming patternsincluding rectilinear motion, turning motion, gliding motion,and spiral motion are investigated. Firstly, a mathematical de-scription of the dynamic model is proposed basing on Newton-Euler method. Then multiple methods, including SolidWorkssoftware, computational fluid dynamics (CFD) simulation, andgrey-box model estimation method, are used for determiningmodel parameters. Finally, numerical simulations and massivekinematic experiments with a robotic fish prototype in a large-scale parameter space are applied to mutually validate theaccuracy of the dynamic model in predicting key features,including trajectory, attitude, velocity, etc., of the robotic fish.

The remainder of this article is organized as follows. Sec-tion II introduces the bio-inspired robotic fish. Section IIIestablishes a Newton-Euler dynamic model for the robotic fishand determines the model parameters. Section IV presentssimulation and experiment results. Section V concludes thisarticle with an outline of future work.

II. THE ROBOTIC FISH

Figure 1 (a) shows the hardware configurations of therobotic fish. Its size (Length×Width×Height) is about 29.1cm×11.6 cm×13.4 cm. It is composed of a 3D-printed shell, atail, and three compartments, including a control compartment,an engine compartment, a battery compartment, and a pressureacquisition system compartment. Figure 1 (b) shows the inte-rior of the engine compartment. Three motors, which servedifferent functions, are wrapped in the engine compartment.Specifically, motor 1 is connected with the tail. It is usedto generate propulsive force. Motor 2 is used for drivinng arotating bracket. The bracket is connected to motor 3 and acrank-slider mechanism. Motor 3 is used to drive the crank-slider mechanism mentioned above to which a weight blockis connected. Through controlling motor 2 and 3, the weightblock can move along the direction parallel to principal axisof the robotic fish and rotate about output shaft of motor 2. Bycontrolling the three motors using given frequency, amplitude,and offset parameters, the robotic fish can realize rectilinear

(a)

(b)

Fig. 1. Hardware configurations of the robotic fish. (a) CAD model ofthe robotic fish. Eleven pressure sensors named Ptop, Pbottom, P0, PLi ,and PRi (i = 1, 2, 3, 4) are mounted on the surface of the shell forestablishing an artificial lateral line system (ALLS). ALLS is used to measurethe hydrodynamic pressure variations surrounding fish body. More informationabout the ALLS can be found in our previous work [12]. (b) The diagrammaticsketch of the interior of the engine compartment. d1 indicates the distancebetween the output shaft of motor 3 and the connection point Orb of motor2 and rotating bracket. d2 indicates the distance between the output shaft ofmotor 2 and center of mass Cw of the weight block.

motion, turning motion, gliding motion, and spiral motion, asshown in Figure 2. More about motions of the robotic fish canbe in the supplementary video.

III. DYNAMIC ANALYSIS FOR THE ROBOTIC FISH

A. Definition of the Coordinate Systems

Figure 3 shows the coordinate systems of the robotic fish.OIxIyIzI , Obxbybzb, Orbxrbyrbzrb, and Otxtytzt indicate theglobal inertial coordinate system, the body-fixed coordinatesystem, the rotating-bracket-fixed coordinate system, and thetail-fixed coordinate system, respectively. The origin Ob isfixed at the intersection of horizontal section and longitudinalsection of the robotic fish, above center of mass Cm ofthe robotic fish. The longitudinal section is the symmetricalplane of the shell. The horizontal section coincides with thesymmetrical plane of the tail and is perpendicular to thelongitudinal plane. The origin Orb is fixed at the connectionpoint of motor 2 and the rotating bracket in Figure 1 (c), andexpressed as [arb, brb, crb] in Obxbybzb. The origin Ot is fixedat the connection point of the tail and the engine compartment,and expressed as [at, bt, ct] in Obxbybzb. OIxIyIzI coincideswith the initial Obxbybzb.

3

Fig. 2. Multiple three-dimensional swimming patterns of the robotic fish. (a) Rectilinear motion. (b) Gliding motion. (c) Turning motion. (d) Spiral motion.(e) The red point on fish shell means center of mass. It moves backward/forward when the weight block moves backward/forward with a distance of ∆sin gliding motion and spiral motion (lower), comparing with rectilinear motion and turning motion (upper). The tails in turning motion and spiral motionhave non-zero offsets compared to those in rectilinear motion and gliding motion. OIXIYIZI indicates the global inertial coordinate system. F indicatesthe tailed-generated propulsive force. Uk(k = r, t, g, s) indicates the movement velocity of the robotic fish. Ug is the resultant velocity of the velocity VgZIalong the axis OIZI and the velocity VgXI along the axis OIXI . Us is the resultant velocity of the velocity VsZI along the axis OIZI and the velocityVsXIYI on XI − YI plane. Rt and Rs indicates the radius in turning motion and spiral motion, respectively. ∆h indicates depth variation of the roboticfish. θ indicates pitch angle of the robotic fish.

xb

zb

yb

Ot

zt

yt

Orb

xrb

zrb

YI

OI

Vb

Vbx

Vbz

Vby

Ob

𝜔𝑏𝑥

𝜔𝑏𝑧

𝜔𝑏𝑦

ሶ𝜑

ሶ𝜓

ሶ𝜃XI

ZI

Cm

yrb xt

Fig. 3. Definitions of coordinate systems of the robotic fish.

B. Three-Dimensional Kinematic Analysis

1) Translational Motion of the Robotic Fish: The positionof the robotic fish is denoted as CI = [xI , yI , zI ]

T inOIXIYIZI . The velocity of robotic fish is denoted as VI =[VIx, VIy, VIz]

T in OIXIYIZI and Vb = [Vbx, Vby, Vbz]T in

Obxbybzb, respectively. The relationship between VI and Vbis expressed as

VI = CI = RbI · Vb (1)

where RbI is the transformation matrix from Obxbybzb toOIxIyIzI , taking the form as

RbI =

cψcθ −sψcϕ + cψsθsϕ sψsϕ + cψsθcϕsψcθ cψcϕ + sψsθsϕ −cψsϕ + sψsθcϕ−sθ cθsϕ cθcϕ

(2)

where ϕ, θ, and ψ indicate roll, pitch, and yaw angle of therobotic fish, respectively.

2) Rotational Motion of the Robotic Fish: The angular ve-locity of the robotic fish is denoted as ωb =

[ωbx , ωby , ωbz

]Tin Obxbybzb and ωI =

[ϕ, θ, ψ

]Tin OIxIyIzI . The relation-

ship between ωb and ωI is expressed as

ωI =

1 sinϕtanθ cosϕtanθ0 cosϕ −sinϕ0 sinϕ/cosθ cosϕ/cosθ

· ωb (3)

sw

Cw

d3

Weight block

Motor 3

Guideway Link rod 2

Link rod 1

𝜉3

Fig. 4. Crank-slider mechanism with the weight block. l1 and l2 indicate thelengths of the link rods. d3 indicates the distance between center of mass ofthe weight block Cw and connecting point of the weight block and link rod2. sw indicates the distance between output shaft of motor 3 and connectingpoint of the weight block and link rod 2. The masses of guideway, motor 3,link rod 1, and link 2 are all ignored.

3) Motion analysis of the weight block: As shown inFigure 1 (c), the weight block is able to rotate throughcontrolling output angle ξ2 of motor 2. Thus roll angle ϕ of therobotic fish is able to be adjusted. On the other hand, as shownin Figure 4, the distance sw is able to be adjusted throughcontrolling output angle ξ3 of motor 3. Thus the weight block

4

is able to move along the guideway, and pitch angle θ of therobotic fish is able to be adjusted. sw takes the form as

sw = sw0+ ∆d (4)

where sw0 indicates the initial value of sw, with which pitchangle and roll angle of the robotic fish are 0. ∆d is the distancebetween the weight block’s current position and its initialposition in the kinematics experiments.

The coordinate of center of mass of the weight blockCw[xCw , yCw , zCw ] is expressed in Obxbybzb, taking the formas

xCw = arb + d1 − (sw − d3)

yCw = brb + d2 · sinξ2zCw = crb + d2 · cosξ2

(5)

The coordinate of center of mass of the robotic fishCm[xCm , yCm , zCm ] takes the form as

jCm =

(Mewj +Mwj

)mtotal

(6)

where j = x, y, z, Mewj is static moment about the Objb axisfor the part apart from the weight block. Mwj is static momentabout the Objb axis for the weight block, taking the form as

Mwj = mw · jCw (7)

where mw is the mass of the weight block.The initial coordinate of center of mass of the robotic fish

is expressed as [xCm0, yCm0

, zCm0]. Besides, both pitch angle

θ and roll angle ϕ of the robotic fish are zero when the weightblock is at its initial position.

C. Three-Dimensional Force Analysis

In this part, the forces and torques acting on the tail andfish body of the robotic fish are analyzed. For the tail, the liftand drag are considered. For the fish body, we respectivelyconsider lift force and drag force in xb− zb plane and xb−ybplane, gravity, buoyancy, and impact of water flow.

𝒗𝒕

𝑭𝑫𝒕

𝑭𝑳𝒕

𝒏𝒕 𝜉1𝛼𝑡

𝑦𝑡𝑥𝑡

𝑂𝑡

𝑭𝒃𝒕𝑭𝒃𝒚

𝒕

Rotation

direction𝑭𝒃𝒙𝒕𝐶𝑃𝑡

Fig. 5. Force analysis for the tail. Cpt is center of press of the tail, and it iscoincident with center of mass of the tail.

1) Force Analysis for the Tail: For the tail of the roboticfish, its time-varying oscillating angle ξ1 is expressed as

ξ1 (t) = ξ1 +A1sin (2πf1t) (8)

where ξ1, A1, and f1 are the oscillating offset, amplitude, andfrequency of the tail, respectively.

The velocity of Cpt in Figure 5 is expressed as

vt = Vb + ωb ×ObCPt + ωt ×OtCPt (9)

where ObCpt is the vector from Ob to Cpt . It is expressedas

ObCpt=(at−rc ·cosξ1)·xb+(bt−rc ·sinξ1)·yb+ct ·zb(10)

where xb, yb, and zb are unit vector along the Obxb axis,Obyb axis, and Obzb axis in Obxbybzb, respectively. OtCpt isthe vector from Ot to Cpt , and it is expressed as

OtCpt = −rc · cosξ1 · xb − rc · sinξ1 · yb + 0 · zb (11)

ωt is the oscillating angular velocity of the tail, and it isexpressed as

ωt = ξ1 · zb = 2πf1A1cos (2πf1t) · zb (12)

The tail of the robotic fish is regarded as a rigid platewithout spanwise wave motion, which is different from fins in[29]. There are various forms of tail-generated force and torque[25], [28], [30]–[33] for different of types of tails. Here, wehave adopted forms as in [25], [28], [33], which are typicallyapplied to express torque and force caused by a rigid plate-like tail. Specifically, the lift F tL and drag F tD of the tail areexpressed as

F tλ =1

2ρ |vt|2 StCλt(|αt|) (13)

where λ = L,D. ρ is the density of water. St is the surfacearea of the tail. CLt and CLt are force coefficients which willbe determined in section III. E. αt is the angle of attack ofthe tail, which is expressed as

αt = arcsin (nt · vt) (14)

where nt is the normal vector of the tail, which is expressedas

nt = −sinξ1 · xb + cosξ1 · yb + 0 · zb (15)

Basing on the above analyses, the three-dimensional dragF tD [34] acting on the tail is expressed as

F tD = −F tDvt (16)

The three-dimensional lift F tL acting on the tail is expressedas

F tL =

{vtsinαt−nt

‖vtsinαt−nt‖ · FtL if nt · vt > 0

vtsinαt+nt

‖vtsinαt+nt‖ · FtL if nt · vt ≤ 0

(17)

Then, the tail-generated torque M tb acting on the robotic

fish is expressed as

M tb = ObCpt × (F tL + F tD) (18)

5

Ob /Cm

𝑽𝒃𝟐

𝛼𝑏2

𝑭𝑫𝟐𝒃

xb

yb𝑭𝑳𝟐𝒃

𝒏𝒃𝟐

(a)

xb

zb

𝑽𝑽𝒃𝒃𝟏𝟏

𝑭𝑭𝑳𝑳𝟏𝟏𝒃𝒃𝑭𝑭𝑫𝑫𝟏𝟏

𝒃𝒃

𝛼𝛼𝑏𝑏1

𝒏𝒏𝒃𝒃𝟏𝟏

Ob

Cm

(b)

Fig. 6. Force analysis for the fish body. (a) Force analysis for xb− zb plane.(b) Force analysis for xb − yb plane.

2) Force Analysis for Fish Body: Figure 6 shows the dragF bDi

(i = 1, 2) and lift F bLi(i = 1, 2) acting on the fish body,

of which the values are expressed as

F bDi =1

2ρ |Vbi |

2SbiCDbi (|αbi |)

F bLi =1

2ρ |Vbi |

2SbiCLbi (|αbi |)

(19)

where CDbi and CLbi are force coefficients which will bedetermined in sectionIII. E.

Vb1 = Vbx · xb + Vbz · zbVb2 = Vbx · xb + Vby · yb

(20)

Sbi(i = 1, 2) is the surface area tensor of the robotic fish. Itis defined as

Sbi = VbiT ·Ai · Vbi , (i = 1, 2) (21)

where

A1 =

[Sxx SxzSzx Szz

],A2 =

[Sxx SxySyx Syy

](22)

A1 and A2 are diagonal matrices. Sxx, Syy, and Szz indicatesthe maximum cross section area perpendicular to the axesObxb, Obyb, and Obzb. αbi(i = 1, 2) is angle of attack offish body, taking the form as

αbi = arcsin(nbi · Vbi

)(23)

nbi(i = 1, 2) is the normal vector, taking the form as

nb1 = zb,nb2 = yb (24)

Basing on the above-analyses, the three-dimensional dragF bDi

(i = 1, 2) is expressed as

F bDi= −F bDiVbi (25)

The three-dimensional lift F bLi(i = 1, 2) is expressed as

F bLi=

Vbi

sinαbi−nbi

‖Vbisinαbi−nbi‖

· F bLi if nbi · Vbi > 0

Vbisinαbi+nbi

‖Vbisinαbi+nbi‖

· F bLi if nbi · Vbi ≤ 0(26)

Besides, rotations of the robotic fish cause damping torquesMω acting on fish body, and Mω is expressed as

Mω = Cωb · ωb (27)

where Cωb is damping torque coefficient, taking the form as

Cωb = diag {Cωb1 , Cωb2 , Cωb3} (28)

In addition, the robotic fish is subjected to torque MI

caused by impact of water flow, and MI [34] is expressedas

MI = MIxb· xb +MIyb

· yb +MIzb· zb (29)

where

MIxb= 0

MIyb=

1

2ρ |Vb1 |

2Sb1CMIyb

(αb1)

MIzb=

1

2ρ |Vb2 |

2Sb2CMIzb

(αb2)

(30)

CMIyband CMIzb

are torque coefficients which will bedetermined in section III. E.

3) The Effect of Gravity and Buoyance: The gravity Fg andbuoyancy Fb of the robotic fish are expressed in Obxbybzb,taking the form as

Fg = mtotal ·RbI−1 · g (31)

Fb = −mb ·RbI−1 · g (32)

where mtotal and mb are total mass and buoyancy mass ofthe robotic fish, respectively.

The torque Mg caused by the buoyance of the robotic fishis expressed as

Mg = ObCm × Fg (33)

where ObCm is the vector from Ob to Cm, taking the formas

ObCm = xCm xb + yCm yb + zCm zb (34)

D. Newton-Euler Dynamic Model

Basing on Newton’s second law, the total force Ftotalacting on the robotic fish is expressed as{

Ftotal =dMVCm

dtFtotal = Fg+Fb+F bL1

+F bD1+F bL2

+F bD2+F tL+F tD

(35)

where M = diag {mtotal,mtotal,mtotal}. VCm indicatesvelocity of center of mass Cm of the robotic fish, taking theform as

VCm = Vb + ωb ×ObCm (36)

dVCm

dt=dVbdt

+dωbdt×ObCm+ωb×Vb+ωb×(ωb×ObCm)(37)

6

Basing on Euler’s equation, the total torque Mtotal aboutCm is expressed as{

Mtotal =dHCm

dtMtotal = Mg + Mω + M t

b + MI −ObCm × Ftotal(38)

where HCm is the moment of momentum about Cm of therobotic fish, taking the form as

HCm = Jωb + M ·ObCm × Vb (39)

dHCm

dt=Jωb + ωb × (Jωb) + M · (ωb ×ObCm)× Vb

+M ·ObCm × (Vb + ωb × Vb) (40)

J = diag {Jxx, Jyy, Jzz} is the moment of inertia about Obfor the robotic fish, taking the form as

J = Jew + Jw (41)

Jw and Jew are the moments of inertia about Ob for theweight block and the part apart from weight block, respec-tively, taking the form as

Jγ = Jγ′+mγ ∗ diag

{r2ObCγx, r

2ObCγy

, r2ObCγz

}(42)

where γ = ew,w. ’ew’ and ’w’ indicate the part apart fromthe weight block and the weight block, respectively. mew ismass of the part apart from the weight block, and mew =mtotal−mw. rObCγx, rObCγy , and rObCγx are components ofthe distance between Cm and Cγ along the Obxb axis, Obybaxis, and Obzb axis, respectively, taking the form as

r2ObCγx = y2Cγ + z2Cγ

r2ObCγy = x2Cγ + z2Cγ

r2ObCγz = x2Cγ + y2Cγ

(43)

where [xCew , yCew , zCew ] is coordinate of center of mass forthe part apart from the weight block. jCew(j = x, y, z) takesthe form as

jCew = Mewj/mew (44)

Jγ′ is the moment of inertia about Cγ for the part apart from

weight block, taking the form as

Jγ′ = diag

{J ′γxx , J

′γyy , J

′γzz

}(45)

18.3Z

XY18.2518.218.15

18.051817.9517.917.8517.817.7517.717.65

17.5517.517.45

18.1

17.6

Fig. 7. Hydrodynamic pressure variations on the surface of the fish body andtail when αb1 , αb2 , and αt are 0.

Basing on the above analyses, the concrete form of thedynamic equations (35) and (38) can be finally acquired,as shown in (46) where Fxb , Fyb , Fzb are components ofthe total force along the Obxb axis, Obyb axis, and Obzbaxis, respectively. Mxb , Myb , Mzb are components of thetotal torque about the Obxb axis, Obyb axis, and Obzb axis,respectively.

E. Determination of Model Parameters

In this part, model parameters, which include mass, di-mensions, and moment of inertia of the robotic fish, etc.are determined by three-dimensional computer-aided design(CAD) software SolidWorks, as shown in Table S1 of thesupplementary materials. We have used two robotic fish toconduct the experiments. mb1 is buoyancy mass for the roboticfish used in rectilinear motion and turning motion, while mb2

is for the robotic fish used in gliding motion and spiral motion.mb1 and mb2 are both determined by actual measurement. Liftcoefficients, drag coefficients, and impact torque coefficientsare determined by computational fluid dynamics (CFD) sim-ulation. Damping torque coefficients are determined by grey-box model estimation method.

1) Determining Force Coefficients and Torque CoefficientsUsing Computational Fluid Dynamics (CFD) Method: Specif-ically, computational fluid dynamics (CFD) simulation for fishbody and tail of the robotic fish were respectively conductedusing a CFD software called HyperFlow, which is developedby China Aerodynamics Research and Development Center(CARDC). HyperFlow is a structured/unstructured hybrid inte-grated fluid simulation software. It is able to run the structuredsolver synchronously on structured grids and unstructuredsolver on unstructured grids. Besides, it has been proved tohave good performance in multi-purpose fluid simulation [35],[36]. Figure 7 shows the hydrodynamic pressure variations ofthe tail and fish body using CFD simulation. More detailsabout the CFD simulation can be found in Section S1 ofthe supplementary materials. In the CFD simulation, anglesof attack αt, αb1 , and αb2 changed from 0 to π/6 rad withan interval of π/60 rad. Basing on the hydrodynamic pressurevariations, the lift, drag, and impact torque coefficients undercertain values of αt, αb1 , and αb2 are acquired, as shown inFigure 8. Basing on data fitting method, the quantitative equa-tions which link αt/αb1 /αb2 to coefficients mentioned abovecan be acquired, as shown in Section S1 of the supplementarymaterials.

2) Determining the damping torque coefficients using grey-box model estimation method: The damping torque coeffi-cients are determined by grey-box model estimation method[37]. In the grey-box model estimation, we recorded therectilinear velocity of the robotic fish with given oscillatingparameters, including amplitude and frequency of the tail in28 s. The input data for grey-box model were the oscillatingparameters, while the output data were the rectilinear velocity.As shown in Table S2 of the supplementary materials, werestricted ranges of the three coefficients for avoiding drift ofthe solution. The final values of the damping coefficients areshown in Table S2 of the supplementary materials. Figure 9

7

Fxb = mtotal ·[

˙Vbx − Vby · ωbz + Vbz · ωby − xCm

(ω2bz + ω2

by

)+ yCm

(ωbxωby − ˙ωbz

)+ zCm

(ωbxωbz + ˙ωby

)]Fyb = mtotal ·

[˙Vby − Vbz · ωbx + Vbx · ωbz − yCm

(ω2bx + ω2

bz

)+ zCm

(ωbyωbz − ˙ωbx

)+ xCm

(ωbxωby + ˙ωbz

)]Fzb = mtotal ·

[˙Vbz − Vbx · ωby + Vby · ωbx − zCm

(ω2by + ω2

bx

)+ xCm

(ωbzωbx − ˙ωby

)+ yCm

(ωbyωbz + ˙ωbx

)]Mxb = Jxx ˙ωbx + (Jzz − Jyy)ωbyωbz +mtotal ·

[yCm

(˙Vbz + Vbyωbx − Vbxωby

)− zCm

(˙Vby + Vbxωbz − Vbzωbx

)]Myb = Jyy ˙ωby + (Jxx − Jzz)ωbzωbx +mtotal ·

[zCm

(˙Vbx + Vbzωby − Vbyωbz

)− xCm

(˙Vbz + Vbyωbx − Vbxωby

)]Mzb = Jzz ˙ωbz + (Jyy − Jxx)ωbxωby +mtotal ·

[xCm

(˙Vby + Vbxωbz − Vbzωbx

)− yCm

(˙Vbx + Vbzωby − Vbyωbz

)](46)

0 /30 /15 /10 2 /15 /6t (rad)

0

0.1

0.2

0.4

0.5Simulated valueFitting curve

0.3

CD

t

(a)

0 /30 /15 /10 2 /15 /6t (rad)

0

0.2

0.4

0.6

0.8

1Simulated valueFitting curve

C Lt

(b)

0 /30 /15 /10 2 /15 /60.1

0.2

0.3

0.4

0.5

CD

b1

Simulated valueFitting curve

b1 (rad)

(c)

0 /30 /15 /10 2 /15 /6b1 (rad)

-0.2

0

0.2

0.4

0.6

CL b

1

Simulated valueFitting curve

(d)

0 /30 /15 /10 2 /15 /6b2

0.10.20.30.40.50.60.7

CD

b2

Simulated valueFitting curve

(rad)

(e)

0 /30 /15 /10 2 /15 /6b2

0

0.1

0.2

0.3

0.4

0.5

CL b

2

Simulated valueFitting curve

(rad)

(f)

0 /30 /15 /10 2 /15 /6b1 (rad)

0

0.005

0.01

0.015

0.02

CM

Iyb

Simulated valueFitting curve

(g)

0 /30 /15 /10 2 /15 /6b2 (rad)

0

0.005

0.01

0.015

0.02

CM

Iz

Simulated valueFitting curve

b

(h)Fig. 8. The lift, drag, and impact torque coefficients acquired by computa-tional fluid dynamics (CFD) simulation. (a) CDt . (b) CLt . (c) CDb1

. (d)CLb1

. (e) CDb2. (f) CLb2

. (g) CMIyb. (h) CMIzb

.

shows the measured velocity and simulated velocity obtainedusing the estimated coefficients. The measured velocity andsimulated velocity of the robotic fish match with a 61.45% fit.

A1=30°,f1=0.8 Hz

A1=15°,f1=1.3 Hz

A1=25°,f1=1.7 Hz

A1=20°,f1=1.9 Hz

A1=10°,f1=2.5 Hz

Fig. 9. Measured rectilinear velocity and estimated rectilinear velocityobtained using grey-box model estimation method.

IV. SIMULATIONS AND EXPERIMENTS

A. Rectilinear motion

030

0.05

25

0.1

2.0Mea

sure

d re

ctili

near

vel

ocity

of

the

robo

tic fi

sh/U

f M (m

/s)

20 1.8

0.15

1.615

0.2

105 1.0

0

0.05

0.1

0.15

0.2

Frequency/f1(Hz)

1.41.2Amplitude/A1 (°)

(a)

0

0.05

0.1

Sim

ulat

ed re

ctili

near

vel

ocity

of

the

robo

tic fi

sh/U

f S (m/s

)0.15

0.2

0

0.05

0.1

0.15

0.2

3025

Freq ncy/f1(Hz)Amplitude/A1 (°)

3025

Freq ncy/f1(Hz)Amplitude/A1 (°)

2.01.81.6

1.0 1.2 1.4ue

2015

105

(b)Fig. 10. Measured and simulated rectilinear motion velocity of the roboticfish. (a) Measured value. (b) Simulated value.

In rectilinear motion experiment, varieties of rectilinear ve-locities were obtained by changing the oscillating frequency f1and amplitude A1 of the tail, while the oscillating offset ξ1 waszero. Figure 10 shows the measured and simulated rectilinearmotion velocity Ur obtained by various combinations of f1and A1. Ur is the resultant velocity of the velocity VIx alongthe axis OIXI and the velocity VIy along the axis OIYI . Itincreases with f1 and A1. The measured and simulated Urmatch well with a coefficient of determination (R2) of 0.8898and a mean absolute error (MAE) of 0.0137 m/s. Figure 11shows the real-time attitude of the robotic fish when it wasactuated by five combinations of A1 and f1. Under eachcombination of A1 and f1, yaw angle of the robotic fishoscillates around a certain value while roll and pitch angleof the robotic fish oscillate around zero, in which case the

8

-5

0

5Pi

tch

angl

e an

d ro

ll an

gle

(°)

Simulated roll angleMeasured roll angle

0 8 13 18 23 28Time/t (s)

-35-25-15

-55

15

Yaw

ang

le o

f the

(°)

Simulated pitch angleMeasured pitch angle

Simulated pitch angleMeasured pitch angle

A1=30°, f1=0.8 HzA1=15°,f1=1.3 Hz

A1=25°,f1=1.7 Hz

A1=20°,f1=1.9 Hz

A1=10°,f1=2.5 Hz

-5

0

5

Pitc

h an

gle

and

roll

angl

e (°

)Simulated roll angleMeasured roll angle

0 8 13 18 23 28Time/t (s)

-35-25-15-55

15

Yaw

ang

le o

f the

(°)

Simulated pitch angleMeasured pitch angle

Simulated pitch angleMeasured pitch angle

-5

0

5Pi

tch

angl

e an

d ro

ll an

gle

(°)

Simulated roll angleMeasured roll angle

0 8 13 18 23 28Time/t (s)

-35-25-15-55

15

Yaw

ang

le o

f the

(°)

Simulated pitch angleMeasured pitch angle

Simulated pitch angleMeasured pitch angle

Fig. 11. Real-time attitudes of the robotic fish in rectilinear motion.

00.511.522.5XI-axis coordinate value/x (m)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

YI-a

xis c

oord

inat

e va

lue/

y (m

) f1=0.8Hz;A1=30°f1=1.3Hz;A1=15°f1=1.7Hz;A1=25°f1=1.9Hz;A1=20°f1=2.5Hz;A1=10°

Simulated trajectoryStarting point of simulated trajectoryStarting point of measured trajectoryEnd point of simulated trajectoryEnd point of measured trajectory

Fig. 12. Trajectory of the robotic fish under five combinations of A1 and f1.

robotic fish swims in a straight line. Because of the periodicaloscillation of the tail, the robotic fish body oscillates whileswimming. Thus the yaw angle, pitch angle, and roll angle ofthe robotic fish oscillate periodically with the time. It can beseen that the simulated and measured attitudes match well inthe oscillatory feature and value. A more careful inspectionrevealed that the yaw amplitude increases with the increasingA1 while the yaw rate increases with the increasing f1. Forpitch angle and roll angle, the biggest errors between theestimated values and the measured values are both less than3◦, which are small enough. The errors are results of the wavemotion of water which caused the roll motion and pitch motionof the robotic fish. The final trajectory of the robotic fishis shown in Figure 12, with a maximum error between thesimulated trajectory and measured trajectory of 0.2407 m.

B. Turning motion

In turning motion experiment, varieties of turning angularvelocities ωt and turning radii Rt were obtained by variouscombinations of oscillating offset ξ1 and frequency f1 of thetail. As shown in Figure 13 and Figure 14, the measuredvalue and simulated value of ωt match well with R2=0.7462and MAE=0.0409 rad/s, while the measured Rt matchesthe simulated Rt with a MAE=0.0657 m and an average

02.0

0.10.20.3

40

Mea

sure

d tu

rnin

g an

gula

r ve

loci

tyof

the

robo

tic f

ish/

t M

(ra

d/s)

1.5

0.4

35

0.5

30

0.6

1.0 0

0.1

0.2

0.3

0.4

0.5

0.6

Offset/ξ1 (°)

Frequency/f1 (Hz)0.5 10 15

20 25

(a)

02.0

0.10.20.3

40

Sim

ulat

ed tu

rnin

g an

gula

r vel

ocity

of th

e ro

botic

fish

/t S (r

ad/s

)

1.5

0.4

35

0.5

30

0.6

1.0 0

0.1

0.2

0.3

0.4

0.5

0.6

Offset/ξ1 (°)

Frequency/f1 (Hz)0.5

25201510

(b)Fig. 13. Measured and simulated turning angular velocity of the robotic fish.(a) Measured value. (b) Simulated value.

0.110

0.20.3

15

0.4

2.0Mea

sure

d tu

rnin

g ra

dius

of

the

robo

tic f

ish/

Rt M

(m

)

20

0.50.6

25 1.5

0.7

30 1.03540 0.5 Frequency/f1 (H

z)0.1

0.2

0.3

0.4

0.5

0.6

0.7

Offset/ξ1 (°)

(a)

0.110

0.20.3

15

0.4

2.0Sim

ulat

ed tu

rnin

g ra

dius

of

the

robo

tic f

ish/

Rt S

(m

)

20

0.50.6

25 1.5

0.7

30 1.03540 0.5 Frequency/f1 (Hz)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Offset/ξ1 (°)

(b)Fig. 14. Measured and simulated turning radius of the robotic fish. (a)Measured value. (b) Simulated value.

percentage error of 18.5913%. The ωt increases with theincreasing ξ1 and f1. The Rt decreases with the increasingξ1 and it is nearly constant with the f1. Figure 15 shows thereal-time yaw/pitch/roll rate of the robotic fish. It can be seenthat both the roll rate ωIy and pitch rate ωIy of the roboticfish oscillate around zero. The yaw rate ωIz oscillates arounda positive value when the value of ξ1 is negative, in whichcase the robotic fish turns left. While the ωIz oscillates arounda negative value when the value of ξ1 is positive, in whichcase the robotic fish turns right. A more careful inspectionreveals that the amplitude of the ωIz increases with the ξ1while decreases with the f1, while the rate of ωIz increaseswith the f1. For the amplitudes of ωIx and ωIy , they decreasewith the f1.

Yaw

/Pit

ch/R

oll

rat

e o

f th

e ro

bo

tic

fish

(°/

s)

ҧ𝜉1=30°,

f1=0.3Hz

ҧ𝜉1=20°,

f1=0.7Hz

ҧ𝜉1=35°,

f1=1.1Hz

ҧ𝜉1=10°,

f1=1.4Hz

ҧ𝜉1=-20°,

f1=1.9Hz

ҧ𝜉1=-10°,

f1=2.5Hz

Time/t (s)

Fig. 15. Real-time yaw/pitch/roll rate of the robotic fish in turning motionunder six combinations of ξ1 and f1. ωIjS and ωIjM (j = x, y, z) indicatesimulated and measured value of the yaw/pitch/roll rate, respectively.

9

C. Glidng motion

-2.00.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.230.24

Glid

ing

velo

city

of t

he ro

botic

fish

/Ug

(m/s

)

Measured velocity Simulated velocity

-1.0 -0.8 -0.6 -1.9-1.8-1.7 -1.4 -1.3-1.2 -0.4 -0.2 0d of the weight block (cm)

Fig. 16. Measured and simulated gliding velocity of the robotic fish.

In glidng motion experiment, varieties of gliding velocitiesUg were obtained by changing ∆d of the weight block. A1,f1, and ξ1 of the tail are 20◦, 2.0 Hz, and 0, respectively.Figure 16 shows the measured and simulated Ug of therobotic fish. The maximum and average percentage errorsbetween the measured and simulated Ug are 14.0507% and3.5340%, respectively. It is noteworthy that because of thedepth limitation of the water tank (only 0.8 m), the roboticfish reached the surface of the water before it reached the stateof uniform motion. So the Ug of the robotic fish for ∆d=-2.0cm, -1.9 cm, -1.8 cm, and -1.7 cm is average gliding velocityof the robotic fish in its acceleration process. While the Ug for∆d varied from -1.4 cm to 0 are velocities when the roboticfish was in uniform motion state. Comparing the Ug for ∆dfrom -1.4cm to 0, it can be seen that Ug of the robotic fishdecreases with the increasing |∆d|.

D. Spiral motion

0 2 4 6 8 12 14 16 18 2010Time/t (s)

-200

-150

-100

-50

0

50

100

150

200

Atti

tude

of t

he ro

botic

fish

in sp

iral m

otio

n (°

)

Simulated pitch angle Simulated roll angle Simulated yaw angle

Measured pitch angle Measured roll angle Measured yaw angle

Fig. 17. Real-time measured and simulated attitude of the robotic fish inspiral motion.

The spiral motion was the result of a combination of non-zero ∆d and non-zero oscillating offset ξ1 of the tail. A1 andf1 of the tail are 20◦ and 3.0 Hz, respectively. As shown

0 5 15-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Vel

ocity

of t

he ro

botic

fish

in sp

iral m

otio

n (m

/s)

10 Time/t (s)

Simulated V

Iy

Measured VIx

Simulated VIx

Measured VIySimulated VIy

20

Measured VIzSimulated VI z

Fig. 18. Real-time velocity of the robotic fish in spiral motion.

-1.6 -1.4 -1.2 -1.0 -0.6 -0.4 -0.2 00.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65Measured angular velocity Simulated angular velocity

Ang

ular

vel

ocity

of t

he ro

botic

fish

/ s (

rad/

s)

d of the weight block (cm)

Fig. 19. Measured and simulated spiral angular velocity of the robotic fish.

in Figure 17, yaw angle of the robotic fish oscillates aroundvaried values with the time, while pitch angle and the rollangle oscillate around constant values. It can be seen that thesimulated attitudes closely track the measured attitudes. Thevelocity VIx along the axis OIXI and the velocity VIy alongthe axis OIXI exhibit sine-like characteristics. The velocityVIz along the axis OIZI gradually researches a negative value,which means the robotic fish is spiralling up. Figure 19 andFigure 20 shows the measured and simulated spiral angularvelocity ωs and spiral velocity Us of the robotic fish, respec-tively. It can be seen that both the ωs and the Us barely changewith the ∆d. The maximum and average percentage error ofthe ωs are 3.6004% and 1.9323%, respectively. The maximumand average percentage error of the Us are 11.4808% and5.8953%, respectively. Figure 21 shows the measured andsimulated spiral trajectory of the robotic fish in spiral motion.The measured trajectory tracks the simulated trajectory wellwith a maximum error of 0.3974 m.

V. CONCLUSION AND FUTURE WORK

In this article, a dynamic model that accounts for multiplethree-dimensional motions, including rectilinear motion, turn-

10

-1.6 -1.4 -1.2 -1.0 -0.6 -0.4 -0.2 00.155

0.16

0.165

0.17

0.175

0.18

0.185Sp

iral v

eloc

ity o

f the

robo

tic fi

sh/Us (m

/s)

Measured velocity Simulated velocity

d of the weight block (cm)

Fig. 20. Measured and simulated spiral velocity of the robotic fish.

0.2-0.4

0

-0.2

-0.2

0

Z I-a

xis

coor

dina

te v

alue

/z (

m)

-0.4

0.2

-0.6

-0.8-0.60.4

-0.8

-0.4-0.200.6 0.2

XI -axis coordinate

value/x (m)

Measured trajectory Simulated trajectory Starting point of the measured trajectory Starting point of the simulated trajectory End point of the measured trajectory End point of the simulated trajectory

YI-axis coordinate value/y (m)

Fig. 21. Spiral trajectory of the robotic fish in spiral motion.

ing motion, gliding motion, and spiral motion, of an active-tail-actuated robotic fish with barycentre regulating mechanismwas proposed basing on Newton-Euler method. CAD softwareSolidWorks, HyperFlow based computational fluid dynamics(CFD) simulation, and grey-box model estimation methodare used for determining model parameters. Massive kine-matic experiments with robotic fish prototype and numericalsimulations demonstrate that the proposed model is capableof evaluating the trajectory, attitudes, and motion parametersincluding the linear velocity, motion radius, angular velocity,etc., for the robotic fish with small errors.

We are conducting researches on evaluating motion param-eters of the robotic fish using its onboard ALLS, and anevaluation model that links the linear velocity, angular velocity,and motion radius to the hydrodynamic pressure variations(PVs) surrounding the fish body has been preliminarily ac-quired. Using the PVs measured by the ALLS, the above-mentioned motion parameters can be evaluated by solving theevaluation model inversely. In the future work, we will inputthe ALLS-evaluated motion parameters into a dynamic model-based controller as feedback terms, for adjusting the oscillationparameters of the robotic fish, and finally realizing flow-aidedclosed-loop control for the trajectory of the robotic fish.

ACKNOWLEDGMENT

This work was supported in part by grants from the NationalNatural Science Foundation of China (NSFC, No. 91648120,61633002, 51575005) and the Beijing Natural Science Foun-dation (No. 4192026).

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Xingwen Zheng received the B.E. in Mechani-cal Engineering and Automation from NortheasternUniversity, Shenyang, China in 2015. He is currentlya PhD candidate at the Intelligent Biomimetic De-sign Lab, State Key Laboratory for Turbulence andComplex Systems, College of Engineering, PekingUniversity, Beijing, China. His current research in-terests include biomimetic robotics, lateral line in-spired sensing, and multi-robot control.

Minglei Xiong received the B.E. in North ChinaElectric Power University (NCEPU), Beijing, Chinain 2012. He is currently a PhD candidate in GeneralMechanics and Foundation of Mechanics at PekingUniversity. His current research interests includebiomimetic robotics, artificial intelligent, and gametheory.

Junzheng Zheng received the B.E. in Mechani-cal Design, Manufacturing and Automation fromHuazhong University of Science and Technology,Wuhan, China in 2017. He is currently pursuing amaster’s degree in Control Theory and Control En-gineering at Peking University. His current researchinterests include biomimetic robotics and lateral lineinspired sensing.

Manyi Wang received the B.E. in Measurement andControl Technology and Instrumentation from Na-tional University of Defense Technology, Changsha,China in 2009. He is currently pursuing a master’sdegree in Control Theory and Control Engineeringat Peking University. His current research interestsinclude biomimetic robotics and lateral line inspiredsensing.

Runyu Tian received the B.E. in Aircraft Systemand Engineering from National University of De-fense Technology, Changsha, China in 2011. He iscurrently pursuing a master’s degree in Control The-ory and Control Engineering at Peking University.His current research interests include computationalfluid dynamics, biomimetic robotics, and deep rein-forcement learning.

Guangming Xie received his B.S. degrees in bothApplied Mathematics and Electronic and ComputerTechnology, his M.E. degree in Control Theory andControl Engineering, and his Ph.D. degree in Con-trol Theory and Control Engineering from TsinghuaUniversity, Beijing, China in 1996, 1998, and 2001,respectively. Then he worked as a postdoctoral re-search fellow in the Center for Systems and Control,Department of Mechanics and Engineering Science,Peking University, Beijing, China from July 2001 toJune 2003. In July 2003, he joined the Center as a

lecturer. Now he is a Full Professor of Dynamics and Control in the Collegeof Engineering, Peking University.

He is an Associate Editor of Scientific Reports, International Journal ofAdvanced Robotic Systems, Mathematical Problems in Engineering and anEditorial Board Member of Journal of Information and Systems Science.His research interests include smart swarm theory, multi-agent systems,multi-robot cooperation, biomimetic robot, switched and hybrid systems, andnetworked control systems.