Three-Dimensional Dynamic Modeling and Motion Analysis for an Active-Tail-Actuated Robotic Fish with Barycentre Regulating Mechanism - arXiv

1

 Three-Dimensional Dynamic Modeling and Motion
 Analysis for an Active-Tail-Actuated Robotic Fish
 with Barycentre Regulating Mechanism
 Xingwen Zheng1 , Minglei Xiong1,2 , Junzheng Zheng1 , Manyi Wang1 , Runyu Tian1,3 , and Guangming Xie1,4,5

 Abstract—Dynamic modeling has been capturing attention for [15], etc. In particular, mechanism investigation of dynamic
arXiv:2006.14420v2 [cs.RO] 22 May 2021

 its fundamentality in precise locomotion analyses and control of performance of underwater robots is fundamental and critical
 underwater robots. However, the existing researches have mainly for the above-mentioned researches. Besides, precise dynamic
 focused on investigating two-dimensional motion of underwater
 robots, and little attention has been paid to three-dimensional modeling of underwater robots has always been focus and
 dynamic modeling, which is just what we focus on. In this difficulty in underwater robot research.
 article, a three-dimensional dynamic model of an active-tail- For dynamic modeling, the typical modeling methods in-
 actuated robotic fish with a barycentre regulating mechanism clude Lagrangian dynamics method, Newton-Euler method,
 is built by combining Newton’s second law for linear motion Lighthill’s elongated-body theory, Schiehlen method, etc.
 and Euler’s equation for angular motion. The model parameters
 are determined by three-dimensional computer-aided design Basing on the Newton-Euler method, Y. Shi’s group has
 (CAD) software SolidWorks, HyperFlow-based computational built a dynamic model of an AUV, and then investigated
 fluid dynamics (CFD) simulation, and grey-box model estimation dynamic model-based trajectory tracking control of planar
 method. Both kinematic experiments with a prototype and motions of the AUV [16]–[18], without consideration of three-
 numerical simulations are applied to validate the accuracy of the dimensional motions. J. Yu’s group has formulated a robotic
 dynamic model mutually. Based on the dynamic model, multiple
 three-dimensional motions, including rectilinear motion, turning fish dynamics using Schiehlen method [19] and Lagrangian
 motion, gliding motion, and spiral motion, are analyzed. The dynamics method [20]. It has been demonstrated that the
 experimental and simulation results demonstrate the effectiveness proposed dynamic model is efficient for seeking backward
 of the proposed model in evaluating the trajectory, attitude, swimming pattern of the robotic fish [20]. They have also
 and motion parameters, including the velocity, turning radius, proposed a data-driven dynamic modeling method in which
 angular velocity, etc., of the robotic fish.
 the Newton-Euler formulation is applied to analyze the robotic
 Index Terms—Three-dimensional dynamic modeling, Newton- fish dynamics, and parameters in the dynamic model are
 Euler method, computational fluid dynamics (CFD), grey-box identified using experimental data of rectilinear motion and
 model estimation, robotic fish.
 turning motion of the robotic fish, also without investigating
 three-dimensional motions. F. Zhang’s group has established
 I. I NTRODUCTION an analytical model for spiral motion of an underwater glider
 steered by an internal movable mass block, and experiments
 I N recent years, underwater robots including varieties of
 underwater remotely operated vehicles (ROV), autonomous
 underwater vehicles (AUV), and bio-inspired aquatic systems
 in the South China Sea have validated the accuracy of the
 model for achieving desired spiral motion [21]. They have
 [1] have been developed and shown great potentials in pro- also explored a dynamic model for a blade-driven glider with
 moting marine resource exploitation [2], [3], marine economy gliding motion [22]. However, the motion of glider is different
 development [4], [5], and marine ecological environment pro- from rhythmic motion of the fin-actuated underwater robot.
 tection [6], [7]. The research topics of underwater robots cover X. Tan’s group has explored dynamic analyses of a tail-
 locomotion control and optimization [8], [9], underwater navi- actuated robotic fish [23]–[25] and a fish-like glider [26], [27].
 gation and localization [10], [11], environment perception and For the tail-actuated robotic fish, Lighthill’s large-amplitude
 object recognition [12], [13], underwater communication [14], elongated-body theory has been combined with rigid-body
 dynamics and hybrid tail dynamics for building a dynamic
 1 Xingwen Zheng, Minglei Xiong, Junzheng Zheng, Manyi Wang, model [23]–[25]. However, only surface motion of the robotic
 Runyu Tian, and Guangming Xie are with the State Key Laboratory fish has been explored. For the fish-like glider, they have built a
 for Turbulence and Complex Systems, Intelligent Biomimetic Design
 Lab, College of Engineering, Peking University, Beijing, 100871,
 Newton-Euler method based dynamic model for investigating
 China. {zhengxingwen, xiongml, zhengjunzheng, spiraling maneuver [26] and gliding motion [27]. However, the
 wangmanyi, trytian, xiegming}@pku.edu.cn. fish-like glider is just driven by displacing an internal movable
 Corresponding author: G. Xie.
 2 Minglei Xiong is with the Boya Gongdao (Beijing) Robot Technology mass and pumping fluids, while its tail is not active, without
 Co., Ltd., Beijing, 100084, China. a continuously varied tail angle.
 3 Runyu Tian is with the China Aerodynamics Research and Development
 The above-mentioned studies have demonstrated that dy-
 Center, Mianyang, Sichuan, 621000, China. namic modeling is fundamental and essential for locomotion
 4 Peng Cheng Laboratory, Shenzhen, 518055, China.
 5 Guangming Xie is with the Institute of Ocean Research, Peking University, analysis of underwater robots. However, most of the researches
 Beijing, 100871, China. have only focused on investigating two-dimensional motions

2

in horizontal plane or vertical plane. Especially for fin-actuated
underwater robots, though there are a few preliminary works
that have considered dynamic modeling in three-dimensional
space [19], [20], [28], the proposed models are typically
validated by limited experiments, without validation in a large-
scale parameter space. Besides, for three species of underwater
robots including active-fin-actuated underwater robot with
barycentre regulating mechanism, blade-driven underwater
robot [22], and internal movable mass block-driven underwater
robot [24], all of which can adjust their centers of mass, there
exist significant differences among their dynamics, because an
active-fin-actuated underwater robot with barycentre regulating
mechanism is able to generate extra rhythmic oscillation of
robot body. However, dynamic modelling for such an under-
water robot has been rarely investigated. (a)
 On the basis of the above analyses, this article mainly
focuses on investigating three-dimensional dynamic modeling
in a large-scale parameter space for an active-tail-actuated
robotic fish with a barycentre regulating mechanism, which
has been rarely investigated. Multiple swimming patterns
including 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 SolidWorks
software, computational fluid dynamics (CFD) simulation, and (b)
grey-box model estimation method, are used for determining
 Fig. 1. Hardware configurations of the robotic fish. (a) CAD model of
model parameters. Finally, numerical simulations and massive the robotic fish. Eleven pressure sensors named Ptop , Pbottom , P0 , PLi ,
kinematic experiments with a robotic fish prototype in a large- and PRi (i = 1, 2, 3, 4) are mounted on the surface of the shell for
scale parameter space are applied to mutually validate the establishing an artificial lateral line system (ALLS). ALLS is used to measure
 the hydrodynamic pressure variations surrounding fish body. More information
accuracy of the dynamic model in predicting key features, about the ALLS can be found in our previous work [12]. (b) The diagrammatic
including trajectory, attitude, velocity, etc., of the robotic fish. sketch of the interior of the engine compartment. d1 indicates the distance
 between the output shaft of motor 3 and the connection point Orb of motor
 The remainder of this article is organized as follows. Sec- 2 and rotating bracket. d2 indicates the distance between the output shaft of
tion II introduces the bio-inspired robotic fish. Section III motor 2 and center of mass Cw of the weight block.
establishes a Newton-Euler dynamic model for the robotic fish
and determines the model parameters. Section IV presents
simulation and experiment results. Section V concludes this motion, turning motion, gliding motion, and spiral motion, as
article with an outline of future work. shown in Figure 2. More about motions of the robotic fish can
 be in the supplementary video.
 II. T HE ROBOTIC F ISH
 III. DYNAMIC A NALYSIS FOR THE ROBOTIC F ISH
 Figure 1 (a) shows the hardware configurations of the
robotic fish. Its size (Length×Width×Height) is about 29.1 A. Definition of the Coordinate Systems
cm×11.6 cm×13.4 cm. It is composed of a 3D-printed shell, a Figure 3 shows the coordinate systems of the robotic fish.
tail, and three compartments, including a control compartment, OI xI yI zI , Ob xb yb zb , Orb xrb yrb zrb , and Ot xt yt zt indicate the
an engine compartment, a battery compartment, and a pressure global inertial coordinate system, the body-fixed coordinate
acquisition system compartment. Figure 1 (b) shows the inte- system, the rotating-bracket-fixed coordinate system, and the
rior of the engine compartment. Three motors, which serve tail-fixed coordinate system, respectively. The origin Ob is
different functions, are wrapped in the engine compartment. fixed at the intersection of horizontal section and longitudinal
Specifically, motor 1 is connected with the tail. It is used section of the robotic fish, above center of mass Cm of
to generate propulsive force. Motor 2 is used for drivinng a the robotic fish. The longitudinal section is the symmetrical
rotating bracket. The bracket is connected to motor 3 and a plane of the shell. The horizontal section coincides with the
crank-slider mechanism. Motor 3 is used to drive the crank- symmetrical plane of the tail and is perpendicular to the
slider mechanism mentioned above to which a weight block longitudinal plane. The origin Orb is fixed at the connection
is connected. Through controlling motor 2 and 3, the weight point of motor 2 and the rotating bracket in Figure 1 (c), and
block can move along the direction parallel to principal axis expressed as [arb , brb , crb ] in Ob xb yb zb . The origin Ot is fixed
of the robotic fish and rotate about output shaft of motor 2. By at the connection point of the tail and the engine compartment,
controlling the three motors using given frequency, amplitude, and expressed as [at , bt , ct ] in Ob xb yb zb . OI xI yI zI coincides
and offset parameters, the robotic fish can realize rectilinear with the initial Ob xb yb zb .

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 ∆s
in gliding motion and spiral motion (lower), comparing with rectilinear motion and turning motion (upper). The tails in turning motion and spiral motion
have non-zero offsets compared to those in rectilinear motion and gliding motion. OI XI YI ZI indicates the global inertial coordinate system. F indicates
the 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 VgZI
along the axis OI ZI and the velocity VgXI along the axis OI XI . Us is the resultant velocity of the velocity VsZI along the axis OI ZI and the velocity
VsXI YI on XI − YI plane. Rt and Rs indicates the radius in turning motion and spiral motion, respectively. ∆h indicates depth variation of the robotic
fish. θ indicates pitch angle of the robotic fish.

 ሶ 2) Rotational Motion of the Robotic Fish: The angular ve-
 ሶ OI T
 XI locity of the robotic fish is denoted as ωb = ωbx , ωby , ωbz
 iT
 ሶ
 h
 YI in Ob xb yb zb and ωI = ϕ̇, θ̇, ψ̇ in OI xI yI zI . The relation-
 Ot Orb ZI ship between ωb and ωI is expressed as
 yt O xrb
 yrb xt b
 Vbx 
 zt z Cm
 yb
  
 Vby rb xb 1 sinϕtanθ cosϕtanθ
 ωI = 0 cosϕ −sinϕ  · ωb (3)
 Vbz 0 sinϕ/cosθ cosϕ/cosθ
 
 zb

 Vb Link rod 1 Weight block

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

 Guideway 3 d3 Link rod 2
B. Three-Dimensional Kinematic Analysis Cw
 1) Translational Motion of the Robotic Fish: The position Motor 3 sw
 T
of the robotic fish is denoted as CI = [xI , yI , zI ] in
OI XI YI ZI . The velocity of robotic fish is denoted as VI =
 T T
[VIx , VIy , VIz ] in OI XI YI ZI and Vb = [Vbx , Vby , Vbz ] in
Ob xb yb zb , respectively. The relationship between VI and Vb Fig. 4. Crank-slider mechanism with the weight block. l1 and l2 indicate the
is expressed as lengths of the link rods. d3 indicates the distance between center of mass of
 the weight block Cw and connecting point of the weight block and link rod
 2. sw indicates the distance between output shaft of motor 3 and connecting
 VI = C˙I = RbI · Vb (1) point of the weight block and link rod 2. The masses of guideway, motor 3,
 link rod 1, and link 2 are all ignored.
where RbI is the transformation matrix from Ob xb yb zb to
OI xI yI zI , taking the form as
   3) Motion analysis of the weight block: As shown in
 cψ cθ −sψ cϕ + cψ sθ sϕ sψ sϕ + cψ sθ cϕ
 Figure 1 (c), the weight block is able to rotate through
RbI = sψ cθ cψ cϕ + sψ sθ sϕ −cψ sϕ + sψ sθ cϕ  (2)
 controlling output angle ξ2 of motor 2. Thus roll angle ϕ of the
 −sθ cθ sϕ cθ cϕ
 robotic fish is able to be adjusted. On the other hand, as shown
where ϕ, θ, and ψ indicate roll, pitch, and yaw angle of the in Figure 4, the distance sw is able to be adjusted through
robotic fish, respectively. controlling output angle ξ3 of motor 3. Thus the weight block

4

is able to move along the guideway, and pitch angle θ of the The velocity of Cpt in Figure 5 is expressed as
robotic fish is able to be adjusted. sw takes the form as
 vt = Vb + ωb × Ob CPt + ωt × Ot CPt (9)
 sw = sw0 + ∆d (4)
where sw0 indicates the initial value of sw , with which pitch where Ob Cpt is the vector from Ob to Cpt . It is expressed
angle and roll angle of the robotic fish are 0. ∆d is the distance as
between the weight block’s current position and its initial
position in the kinematics experiments. Ob Cpt =(at −rc ·cosξ1 )· xˆb +(bt −rc ·sinξ1 )· yˆb +ct · zˆb(10)
 The coordinate of center of mass of the weight block
Cw [xCw , yCw , zCw ] is expressed in Ob xb yb zb , taking the form where xˆb , yˆb , and zˆb are unit vector along the Ob xb axis,
as Ob yb axis, and Ob zb axis in Ob xb yb zb , respectively. Ot Cpt is
 the vector from Ot to Cpt , and it is expressed as
 xCw = arb + d1 − (sw − d3 )
 yCw = brb + d2 · sinξ2 (5) Ot Cpt = −rc · cosξ1 · xˆb − rc · sinξ1 · yˆb + 0 · zˆb (11)
 zCw = crb + d2 · cosξ2
 ωt is the oscillating angular velocity of the tail, and it is
 The coordinate of center of mass of the robotic fish
 expressed as
Cm [xCm , yCm , zCm ] takes the form as
 ωt = ξ˙1 · zˆb = 2πf1 A1 cos (2πf1 t) · zˆb
 

 Mewj + Mwj (12)
 jCm = (6)
 mtotal
where j = x, y, z, Mewj is static moment about the Ob jb axis The tail of the robotic fish is regarded as a rigid plate
for the part apart from the weight block. Mwj is static moment without spanwise wave motion, which is different from fins in
about the Ob jb axis for the weight block, taking the form as [29]. There are various forms of tail-generated force and torque
 [25], [28], [30]–[33] for different of types of tails. Here, we
 Mwj = mw · jCw (7) have adopted forms as in [25], [28], [33], which are typically
 applied to express torque and force caused by a rigid plate-
where mw is the mass of the weight block.
 like tail. Specifically, the lift FLt and drag FD
 t
 of the tail are
 The initial coordinate of center of mass of the robotic fish
 expressed as
is expressed as [xCm0 , yCm0 , zCm0 ]. Besides, both pitch angle
θ and roll angle ϕ of the robotic fish are zero when the weight 1 2
block is at its initial position. Fλt = ρ |vt | St Cλt (|αt |) (13)
 2

C. Three-Dimensional Force Analysis where λ = L, D. ρ is the density of water. St is the surface
 area of the tail. CLt and CLt are force coefficients which will
 In this part, the forces and torques acting on the tail and be determined in section III. E. αt is the angle of attack of
fish body of the robotic fish are analyzed. For the tail, the lift the tail, which is expressed as
and drag are considered. For the fish body, we respectively
consider lift force and drag force in xb − zb plane and xb − yb αt = arcsin (nt · vˆt ) (14)
plane, gravity, buoyancy, and impact of water flow.
 where nt is the normal vector of the tail, which is expressed
 as
 
 nt = −sinξ1 · xˆb + cosξ1 · yˆb + 0 · zˆb (15)
 Rotation 
 direction Basing on the above analyses, the three-dimensional drag
 t
 FD [34] acting on the tail is expressed as
 1 
 t t
 FD = −FD vˆt (16)
 
 The three-dimensional lift FLt acting on the tail is expressed
 as
Fig. 5. Force analysis for the tail. Cpt is center of press of the tail, and it is
coincident with center of mass of the tail.
 (
 vt sinαt −nt
 kvt sinαt −nt k · FLt if nt · vˆt > 0
 FLt = vt sinαt +nt (17)
 1) Force Analysis for the Tail: For the tail of the robotic kvt sinαt +nt k · FLt if nt · vˆt ≤ 0
fish, its time-varying oscillating angle ξ1 is expressed as
 ξ1 (t) = ξ¯1 + A1 sin (2πf1 t) (8) Then, the tail-generated torque Mbt acting on the robotic
 fish is expressed as
where ξ¯1 , A1 , and f1 are the oscillating offset, amplitude, and
frequency of the tail, respectively. Mbt = Ob Cpt × (FLt + FD
 t
 ) (18)

5

 The three-dimensional lift FLbi (i = 1, 2) is expressed as
  V sinα −n
  bi bi bi
 · Fb if nbi · Vˆbi > 0
 b kVbi sinαbi −nbi k Li
 Ob /Cm 2 FLi = (26)
  Vbi sinαbi +nbi · FLb if nbi · Vˆbi ≤ 0
 xb kVbi sinαbi +nbi k i

 Besides, rotations of the robotic fish cause damping torques
 yb Mω acting on fish body, and Mω is expressed as
 
 Mω = Cωb · ωb (27)
 (a)
 where Cωb is damping torque coefficient, taking the form as
 Cωb = diag {Cωb1 , Cωb2 , Cωb3 } (28)
 Ob 1 In addition, the robotic fish is subjected to torque MI
 xb caused by impact of water flow, and MI [34] is expressed
 Cm as
 
 zb MI = MIxb · xˆb + MIyb · yˆb + MIzb · zˆb (29)
 (b) where
Fig. 6. Force analysis for the fish body. (a) Force analysis for xb − zb plane. MIxb = 0
(b) Force analysis for xb − yb plane.
 1 2
 MIyb = ρ |Vb1 | Sb1 CMIy (αb1 ) (30)
 2 b

 1 2
 2) Force Analysis for Fish Body: Figure 6 shows the drag MIzb = ρ |Vb2 | Sb2 CMIz (αb2 )
FDb
 (i = 1, 2) and lift FLbi (i = 1, 2) acting on the fish body, 2 b

 i
of which the values are expressed as CMIy and CMIz are torque coefficients which will be
 b b
 determined in section III. E.
 b 1 2 3) The Effect of Gravity and Buoyance: The gravity Fg and
 FD = ρ |Vbi | Sbi CDbi (|αbi |)
 i
 2 (19) buoyancy Fb of the robotic fish are expressed in Ob xb yb zb ,
 1 2
 FLb i = ρ |Vbi | Sbi CLbi (|αbi |) taking the form as
 2
 Fg = mtotal · RbI −1 · g (31)
where CDbi and CLbi are force coefficients which will be
determined in sectionIII. E.
 Fb = −mb · RbI −1 · g (32)
 Vb1 = Vbx · xˆb + Vbz · zˆb
 (20) where mtotal and mb are total mass and buoyancy mass of
 Vb2 = Vbx · xˆb + Vby · yˆb
 the robotic fish, respectively.
Sbi (i = 1, 2) is the surface area tensor of the robotic fish. It The torque Mg caused by the buoyance of the robotic fish
is defined as is expressed as

 Sbi = Vˆbi T · Ai · Vˆbi , (i = 1, 2) (21) Mg = Ob Cm × Fg (33)

where where Ob Cm is the vector from Ob to Cm , taking the form
     as
 Sxx Sxz S Sxy
 A1 = , A2 = xx (22) Ob Cm = xCm xˆb + yCm yˆb + zCm zˆb (34)
 Szx Szz Syx Syy
A1 and A2 are diagonal matrices. Sxx , Syy , and Szz indicates
 D. Newton-Euler Dynamic Model
the maximum cross section area perpendicular to the axes
Ob xb , Ob yb , and Ob zb . αbi (i = 1, 2) is angle of attack of Basing on Newton’s second law, the total force Ftotal
fish body, taking the form as acting on the robotic fish is expressed as
 dM VCm
 
 Ftotal =
  
 αbi = arcsin nbi · Vˆbi (23) dt (35)
 Ftotal = Fg +Fb +FL1 +FD1 +FLb2 +FD
 b b b
 2
 +FLt +FD
 t

nbi (i = 1, 2) is the normal vector, taking the form as where M = diag {mtotal , mtotal , mtotal }. VCm indicates
 velocity of center of mass Cm of the robotic fish, taking the
 nb1 = zˆb , nb2 = yˆb (24)
 form as
 Basing on the above-analyses, the three-dimensional drag VCm = Vb + ωb × Ob Cm (36)
 b
FD i
 (i = 1, 2) is expressed as
 b b ˆ dVCm dVb dωb
 FD = −FD V bi (25) = + ×Ob Cm +ωb ×Vb +ωb ×(ωb ×Ob C(37)
 m)
 i i
 dt dt dt

6

 Basing on Euler’s equation, the total torque Mtotal about Basing on the above analyses, the concrete form of the
Cm is expressed as dynamic equations (35) and (38) can be finally acquired,
  dH as shown in (46) where Fxb , Fyb , Fzb are components of
 Mtotal = dtCm the total force along the Ob xb axis, Ob yb axis, and Ob zb
 (38)
 Mtotal = Mg + Mω + Mbt + MI − Ob Cm × Ftotal axis, respectively. Mxb , Myb , Mzb are components of the
where HCm is the moment of momentum about Cm of the total torque about the Ob xb axis, Ob yb axis, and Ob zb axis,
robotic fish, taking the form as respectively.

 HCm = J ωb + M · Ob Cm × Vb (39)
 E. Determination of Model Parameters
 dHCm In this part, model parameters, which include mass, di-
 =J ω˙b + ωb × (J ωb ) + M · (ωb × Ob Cm ) × Vb
 dt mensions, and moment of inertia of the robotic fish, etc.
 +M · Ob Cm × (V˙b + ωb × Vb ) (40) are determined by three-dimensional computer-aided design
 (CAD) software SolidWorks, as shown in Table S1 of the
J = diag {Jxx , Jyy , Jzz } is the moment of inertia about Ob supplementary materials. We have used two robotic fish to
for the robotic fish, taking the form as conduct the experiments. mb1 is buoyancy mass for the robotic
 J = Jew + Jw (41) fish used in rectilinear motion and turning motion, while mb2
 is for the robotic fish used in gliding motion and spiral motion.
Jw and Jew are the moments of inertia about Ob for the mb1 and mb2 are both determined by actual measurement. Lift
weight block and the part apart from weight block, respec- coefficients, drag coefficients, and impact torque coefficients
tively, taking the form as are determined by computational fluid dynamics (CFD) sim-
 n o ulation. Damping torque coefficients are determined by grey-
 Jγ = Jγ 0 +mγ ∗ diag rO
 2
 b Cγ x
 , r 2
 , r 2
 Ob Cγ y Ob Cγ z (42) box model estimation method.
where γ = ew, w. ’ew’ and ’w’ indicate the part apart from 1) Determining Force Coefficients and Torque Coefficients
the weight block and the weight block, respectively. mew is Using Computational Fluid Dynamics (CFD) Method: Specif-
mass of the part apart from the weight block, and mew = ically, computational fluid dynamics (CFD) simulation for fish
mtotal − mw . rOb Cγ x , rOb Cγ y , and rOb Cγ x are components of body and tail of the robotic fish were respectively conducted
the distance between Cm and Cγ along the Ob xb axis, Ob yb using a CFD software called HyperFlow, which is developed
axis, and Ob zb axis, respectively, taking the form as by China Aerodynamics Research and Development Center
 (CARDC). HyperFlow is a structured/unstructured hybrid inte-
 2 2 2
 rO b Cγ x
 = yC γ
 + zC γ grated fluid simulation software. It is able to run the structured
 2
 rO = x2Cγ + zC
 2
 (43) solver synchronously on structured grids and unstructured
 b Cγ y γ
 2 solver on unstructured grids. Besides, it has been proved to
 rO b Cγ z
 = x2Cγ + 2
 yC γ have good performance in multi-purpose fluid simulation [35],
where [xCew , yCew , zCew ] is coordinate of center of mass for [36]. Figure 7 shows the hydrodynamic pressure variations of
the part apart from the weight block. jCew (j = x, y, z) takes the tail and fish body using CFD simulation. More details
the form as about the CFD simulation can be found in Section S1 of
 the supplementary materials. In the CFD simulation, angles
 jCew = Mewj /mew (44) of attack αt , αb1 , and αb2 changed from 0 to π/6 rad with
 an interval of π/60 rad. Basing on the hydrodynamic pressure
Jγ 0 is the moment of inertia about Cγ for the part apart from
 variations, the lift, drag, and impact torque coefficients under
weight block, taking the form as
 n o certain values of αt , αb1 , and αb2 are acquired, as shown in
 Jγ 0 = diag Jγ0 xx , Jγ0 yy , Jγ0 zz (45) Figure 8. Basing on data fitting method, the quantitative equa-
 tions which link αt /αb1 /αb2 to coefficients mentioned above
 can be acquired, as shown in Section S1 of the supplementary
 Z 18.3 materials.
 18.25
 Y X 18.2 2) Determining the damping torque coefficients using grey-
 18.15
 18.1 box model estimation method: The damping torque coeffi-
 18.05
 18 cients are determined by grey-box model estimation method
 17.95 [37]. In the grey-box model estimation, we recorded the
 17.9
 17.85 rectilinear velocity of the robotic fish with given oscillating
 17.8
 17.75 parameters, including amplitude and frequency of the tail in
 17.7
 17.65 28 s. The input data for grey-box model were the oscillating
 17.6
 17.55 parameters, while the output data were the rectilinear velocity.
 17.5 As shown in Table S2 of the supplementary materials, we
 17.45
 restricted ranges of the three coefficients for avoiding drift of
Fig. 7. Hydrodynamic pressure variations on the surface of the fish body and the solution. The final values of the damping coefficients are
tail when αb1 , αb2 , and αt are 0. shown in Table S2 of the supplementary materials. Figure 9

7

  h   
i
 ˙ x − Vby · ωbz + Vbz · ωby − xCm ωb2 + ωb2 + yCm ωbx ωby − ω˙bz + zCm ωbx ωbz + ω˙by
 

 
 
  F x b = m total · Vb z y
  h 
i
 ˙ y − Vbz · ωbx + Vbx · ωbz − yCm ωb2 + ωb2 + zCm ωby ωbz − ω˙bx + xCm ωbx ωby + ω˙bz
  
 

 F = m · V
 
 y total b
 
 b
  x z
 
 
  h  
i
  Fz = mtotal · V˙bz − Vbx · ωby + Vby · ωbx − zCm ωb2 + ωb2 + xCm ωbz ωbx − ω˙by + yCm ωby ωbz + ω˙bx
 
  

 b
 h y x
   i (46)
  M x = J xx ω˙
 b x + (J zz − J yy ) ω b y ω b z + m total · y C m V ˙bz + Vby ωbx − Vbx ωby − zCm Vb˙ y + Vbx ωbz − Vbz ωbx
 
  b
  h    i
 ˙ x + Vbz ωby − Vby ωbz − xCm V˙bz + Vby ωbx − Vbx ωby
 
 M = J ω ˙ + (J − J ) ω ω + m · z V
 
 y yy b xx zz b b total C b
  y z x m
  b
 
  h    i
 Mzb = Jzz ω˙bz + (Jyy − Jxx ) ωbx ωby + mtotal · xCm Vb˙ y + Vbx ωbz − Vbz ωbx − yCm Vb˙ x + Vbz ωby − Vby ωbz
 
 
 

 0.5 1
 Simulated value Simulated value
 0.4 Fitting curve 0.8 Fitting curve
 0.3 0.6
 t

 t
 CD

 CL

 0.2 0.4

 0.1 0.2

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

 (a) (b)
 0.5 0.6 A1=30°, A1=15°, A1=25°, A1=20°, A1=10°,
 Simulated value Simulated value f1=0.8 Hz f1=1.3 Hz f1=1.7 Hz f1=1.9 Hz f1=2.5 Hz
 0.4 Fitting curve 0.4 Fitting curve
 C Lb 1

 0.3
C Db 1

 0.2

 0.2 0 Fig. 9. Measured rectilinear velocity and estimated rectilinear velocity
 obtained using grey-box model estimation method.
 0.1 -0.2
 0 /30 /15 /10 2 /15 /6 0 /30 /15 /10 2 /15 /6
 b1 (rad) b1
 (rad)
 IV. S IMULATIONS AND EXPERIMENTS
 (c) (d)
 0.7 0.5 A. Rectilinear motion
 Simulated value Simulated value
 0.6 0.4
 Fitting curve Fitting curve
 0.5 0.2 0.2
 0.3
 Simulated rectilinear velocity of
 Measured rectilinear velocity of
 b2

 b2

 0.4
 the robotic fish/Uf (m/s)
 CD

 CL

 the robotic fish/Uf (m/s)
 0.2 0.2
 0.2 0.15 0.15
 0.3 0.15
 M

 S
 0.15
 0.2 0.1 0.1 0.1
 0.1 0.1
 0.1 0 0.05 0.05
 0 /30 /15 /10 2 /15 /6 0 /30 /15 /10 2 /15 /6
 0
 b2 (rad) b2 (rad) 30
 0.05 0
 30
 0.05
 25 2.0 25 2.0
 Am 20 1.8 ) Am 20 1.8
 (e) (f) plit 15 1.6 (Hz plit 15 1.6 (Hz)
 ude 10 1.2
 1.4
 cy/f 1 0 ude 10 1.2
 1.4
 cy/f 1 0
 0.02 0.02 /A1 5 1.0 uen /A1 5 1.0 uen
 Simulated value Simulated value (°) Freq (°) Freq

 0.015 Fitting curve 0.015 Fitting curve (a) (b)
 b

 Iz b
 Iy

 0.01 0.01 Fig. 10. Measured and simulated rectilinear motion velocity of the robotic
 CM

 CM

 fish. (a) Measured value. (b) Simulated value.
 0.005 0.005

 0 0 In rectilinear motion experiment, varieties of rectilinear ve-
 0 /30 /15 /10 2 /15 /6 0 /30 /15 /10 2 /15 /6 locities were obtained by changing the oscillating frequency f1
 (rad) (rad)
 b1 b2 and amplitude A1 of the tail, while the oscillating offset ξ¯1 was
 (g) (h) zero. Figure 10 shows the measured and simulated rectilinear
Fig. 8. The lift, drag, and impact torque coefficients acquired by computa- motion velocity Ur obtained by various combinations of f1
tional fluid dynamics (CFD) simulation. (a) CDt . (b) CLt . (c) CDb . (d)
 1 and A1 . Ur is the resultant velocity of the velocity VIx along
CLb . (e) CDb . (f) CLb . (g) CMIy . (h) CMIz .
 1 2 2 b b the axis OI XI and the velocity VIy along the axis OI YI . It
 increases with f1 and A1 . The measured and simulated Ur
 match well with a coefficient of determination (R2 ) of 0.8898
 and a mean absolute error (MAE) of 0.0137 m/s. Figure 11
 shows the real-time attitude of the robotic fish when it was
shows the measured velocity and simulated velocity obtained actuated by five combinations of A1 and f1 . Under each
using the estimated coefficients. The measured velocity and combination of A1 and f1 , yaw angle of the robotic fish
simulated velocity of the robotic fish match with a 61.45% fit. oscillates around a certain value while roll and pitch angle
 of the robotic fish oscillate around zero, in which case the

8

 5 A1=15°, A1=25°, A1=20°, A1=10°, 0.6 0.6

 (°) and

 Simulated turning angular velocity
 Measured turning angular velocity
 angle (°)
 f1=1.3 Hz f1=1.7 Hz f1=1.9 Hz

 of the robotic fish/ t (rad/s)
 A1=30°, f1=0.8 Hz f1=2.5 Hz

 of the robotic fish/ t (rad/s)
 0.6 0.5 0.6 0.5

 angle
 Yaw angle of the (°) Pitch angle and 55
 Pitch and 0.5 0.5
 0

 M

 S
 0.4
 roll (°)
 angle 0.4 0.4 0.4
 Simulated pitch angle Simulated roll angle 0.3
 (°)roll angle
 roll angle 0.3
 0.3 0.3
 00 Measured pitch angle Measured roll angle 0.2 0.2
 -5 Simulated pitch angle Simulated roll angle 0.1 0.2 0.1 0.2
 Simulated pitch angle Simulated roll angle
 (°)the Pitch

 0 0
 15 Measured
 Measuredpitch
 pitchangle
 angle Measured roll angle 2.0 2.0
 -5-55 Measured roll angle
 Simulated pitch angle Fre 0.1
 Fre 40
 0.1
 que 1.5
 40
 que 1.5 30 35 30 35
 1515 Measured pitch angle ncy 1.0
 /f1 20 25 /ξ1 (°) ncy 1.0 20 25 /ξ1 (°) 0
 15 e t 0 /f1 15 e t
 Offs
 -5
 Offs
 Simulated pitch angle (Hz 0.5 10 (Hz 0.5 10
 the of

 55 Simulated pitch angle ) )
 -15 Measured pitch angle
 angle

 -5 Measured pitch angle (a)
 -5
 -25 (b)
 Yaw of

 -15
 -15
 -35 Fig. 13. Measured and simulated turning angular velocity of the robotic fish.
 Yaw angle

 -25 0
 -25 8 13 18 23 28 (a) Measured value. (b) Simulated value.
 -35
 -35 Time/t (s)
 00 88 1313 1818 2323 2828
 0.7
 Time/t (s)(s)
 Time/t
 0.7

 Simulated turning radius of the
 Measured turning radius of the
 0.7 0.6 0.7 0.6

 robotic fish/Rt (m)

 robotic fish/Rt (m)
Fig. 11. Real-time attitudes of the robotic fish in rectilinear motion. 0.6 0.6
 0.5

 M
 0.5

 S
 0.5 0.5
 0.4 0.4
 0.4 0.4
 0.6 0.3 0.3

 f1=0.8Hz;A1=30° 0.2 0.3 0.2 0.3
 0.1 0.1
 Y I-axis coordinate value/y (m)

 0.5 f1=1.3Hz;A1=15° 10
 15 2.0 0.2
 10
 15 2.0 0.2
 f1=1.7Hz;A1=25°
 20 20
 Off 25 1.5 Hz) Off 25 1.5 Hz)
 set/ 30 1.0 ncy/f 1 ( set/ 30 1.0 ency/f 1 (
 ξ1 ( 35 0.1 ξ1 ( 35 0.1
 f1=1.9Hz;A1=20° °) 40 0.5 Freque °) 40 0.5 Frequ
 0.4
 f1=2.5Hz;A1=10° (a) (b)
 0.3 Fig. 14. Measured and simulated turning radius of the robotic fish. (a)
 Measured value. (b) Simulated value.
 0.2
 Simulated trajectory
 0.1
 Starting point of simulated trajectory percentage error of 18.5913%. The ωt increases with the
 Starting point of measured trajectory increasing ξ¯1 and f1 . The Rt decreases with the increasing
 0 End point of simulated trajectory
 End point of measured trajectory ξ¯1 and it is nearly constant with the f1 . Figure 15 shows the
 -0.1 real-time yaw/pitch/roll rate of the robotic fish. It can be seen
 2.5 2 1.5 1 0.5 0
 X I-axis coordinate value/x (m) that both the roll rate ωIy and pitch rate ωIy of the robotic
 fish oscillate around zero. The yaw rate ωIz oscillates around
Fig. 12. Trajectory of the robotic fish under five combinations of A1 and f1 . a positive value when the value of ξ¯1 is negative, in which
 case the robotic fish turns left. While the ωIz oscillates around
 a negative value when the value of ξ¯1 is positive, in which
robotic fish swims in a straight line. Because of the periodical case the robotic fish turns right. A more careful inspection
oscillation of the tail, the robotic fish body oscillates while reveals that the amplitude of the ωIz increases with the ξ¯1
swimming. Thus the yaw angle, pitch angle, and roll angle of while decreases with the f1 , while the rate of ωIz increases
the robotic fish oscillate periodically with the time. It can be with the f1 . For the amplitudes of ωIx and ωIy , they decrease
seen that the simulated and measured attitudes match well in with the f1 .
the oscillatory feature and value. A more careful inspection
revealed that the yaw amplitude increases with the increasing 1ҧ =30°, 1ҧ =20°, 1ҧ =35°, 1ҧ =10°, 1ҧ =-20°, 1ҧ =-10°,
A1 while the yaw rate increases with the increasing f1 . For f1=1.1Hz f1=1.4Hz f1=1.9Hz f1=2.5Hz
 Yaw/Pitch/Roll rate of the robotic fish (°/s)

 f1=0.3Hz f1=0.7Hz

pitch angle and roll angle, the biggest errors between the
estimated values and the measured values are both less than
3◦ , which are small enough. The errors are results of the wave
motion of water which caused the roll motion and pitch motion
of the robotic fish. The final trajectory of the robotic fish
is shown in Figure 12, with a maximum error between the
simulated trajectory and measured trajectory of 0.2407 m.

B. Turning motion
 In turning motion experiment, varieties of turning angular
velocities ωt and turning radii Rt were obtained by various
 Time/t (s)
combinations of oscillating offset ξ¯1 and frequency f1 of the
tail. As shown in Figure 13 and Figure 14, the measured Fig. 15. Real-time yaw/pitch/roll rate of the robotic fish in turning motion
value and simulated value of ωt match well with R2 =0.7462 under six combinations of ξ¯1 and f1 . ωIjS and ωIjM (j = x, y, z) indicate
and MAE=0.0409 rad/s, while the measured Rt matches simulated and measured value of the yaw/pitch/roll rate, respectively.
the simulated Rt with a MAE=0.0657 m and an average

9

 0.2

 Velocity of the robotic fish in spiral motion (m/s)
C. Glidng motion
 0.15
 0.24
 Gliding velocity of the robotic fish/U (m/s)

 0.1
 0.23
 g

 0.22 0.05

 0.21 0
 0.2 -0.05
 0.19
 -0.1
 0.18
 -0.15
 0.17
 0.16 -0.2
 0 5 10 15 20
 Measured velocity Time/t (s)
 0.15
 Simulated velocity Measured V Measured VIy Measured VIz
 Ix
 0.14
 -2.0-1.9-1.8-1.7 -1.4 -1.3-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 Simulated V Simulated VIy Simulated VI
 Ix z
 d of the weight block (cm)
 Fig. 18. Real-time velocity of the Simulate
 robotic fish in spiral motion.
Fig. 16. Measured and simulated gliding velocity of the robotic fish. d
 V
 y
 0.65

 Angular velocity of the robotic fish/ s (rad/s)
 In glidng motion experiment, varieties of gliding velocities I Measured angular velocity
Ug were obtained by changing ∆d of the weight block. A1 , 0.6 Simulated angular velocity
f1 , and ξ¯1 of the tail are 20◦ , 2.0 Hz, and 0, respectively.
Figure 16 shows the measured and simulated Ug of the 0.55
robotic fish. The maximum and average percentage errors
 0.5
between the measured and simulated Ug are 14.0507% and
3.5340%, respectively. It is noteworthy that because of the 0.45
depth limitation of the water tank (only 0.8 m), the robotic
fish reached the surface of the water before it reached the state 0.4
of uniform motion. So the Ug of the robotic fish for ∆d=-2.0
 0.35
cm, -1.9 cm, -1.8 cm, and -1.7 cm is average gliding velocity
of the robotic fish in its acceleration process. While the Ug for 0.3
∆d varied from -1.4 cm to 0 are velocities when the robotic -1.6 -1.4 -1.2 -1.0 -0.6 -0.4 -0.2 0
 d of the weight block (cm)
fish was in uniform motion state. Comparing the Ug for ∆d
from -1.4cm to 0, it can be seen that Ug of the robotic fish Fig. 19. Measured and simulated spiral angular velocity of the robotic fish.
decreases with the increasing |∆d|.

D. Spiral motion in Figure 17, yaw angle of the robotic fish oscillates around
 varied values with the time, while pitch angle and the roll
 200 angle oscillate around constant values. It can be seen that the
 Attitude of the robotic fish in spiral motion (°)

 Measured pitch angle simulated attitudes closely track the measured attitudes. The
 150 Measured roll angle velocity VIx along the axis OI XI and the velocity VIy along
 Measured yaw angle
 100 the axis OI XI exhibit sine-like characteristics. The velocity
 VIz along the axis OI ZI gradually researches a negative value,
 50 which means the robotic fish is spiralling up. Figure 19 and
 0 Figure 20 shows the measured and simulated spiral angular
 velocity ωs and spiral velocity Us of the robotic fish, respec-
 -50 tively. It can be seen that both the ωs and the Us barely change
 -100 with the ∆d. The maximum and average percentage error of
 Simulated pitch angle the ωs are 3.6004% and 1.9323%, respectively. The maximum
 -150 Simulated roll angle and average percentage error of the Us are 11.4808% and
 Simulated yaw angle 5.8953%, respectively. Figure 21 shows the measured and
 -200
 0 2 4 6 8 10 12 14 16 18 20 simulated spiral trajectory of the robotic fish in spiral motion.
 Time/t (s)
 The measured trajectory tracks the simulated trajectory well
Fig. 17. Real-time measured and simulated attitude of the robotic fish in with a maximum error of 0.3974 m.
spiral motion.

 The spiral motion was the result of a combination of non- V. C ONCLUSION AND FUTURE WORK
zero ∆d and non-zero oscillating offset ξ¯1 of the tail. A1 and In this article, a dynamic model that accounts for multiple
f1 of the tail are 20◦ and 3.0 Hz, respectively. As shown three-dimensional motions, including rectilinear motion, turn-

10

 0.185
 Spiral velocity of the robotic fish/Us (m/s)
 ACKNOWLEDGMENT
 Measured velocity
 0.18 Simulated velocity This work was supported in part by grants from the National
 Natural Science Foundation of China (NSFC, No. 91648120,
 0.175 61633002, 51575005) and the Beijing Natural Science Foun-
 dation (No. 4192026).
 0.17

 R EFERENCES
 0.165
 [1] R. Salazar, V. Fuentes, and A. Abdelkefi, “Classification of biological
 0.16 and bioinspired aquatic systems: A review,” Ocean Engineering, vol.
 148, pp. 75–114, Jan. 2018.
 [2] O. Khatib, X. Yeh, G. Brantner, B. Soe, B. Kim, S. Ganguly, H. Stuart,
 0.155 S. Wang, M. Cutkosky, A. Edsinger et al., “Ocean one: A robotic avatar
 -1.6 -1.4 -1.2 -1.0 -0.6 -0.4 -0.2 0
 d of the weight block (cm) for oceanic discovery,” IEEE Robotics & Automation Magazine, vol. 23,
 no. 4, pp. 20–29, Dec. 2016.
 [3] S. W. Huang, E. Chen, and J. Guo, “Efficient seafloor classification and
Fig. 20. Measured and simulated spiral velocity of the robotic fish. submarine cable route design using an autonomous underwater vehicle,”
 IEEE Journal of Oceanic Engineering, vol. 43, no. 1, pp. 7–18, Jan.
 2018.
 Measured trajectory [4] Y. S. Ryuh, G. H. Yang, J. Liu, and H. Hu, “A school of robotic fish for
 Simulated trajectory mariculture monitoring in the sea coast,” Journal of Bionic Engineering,
 Starting point of the measured trajectory
 ZI-axis coordinate value/z (m)

 vol. 12, no. 1, pp. 37–46, Mar. 2015.
 -0.8 Starting point of the simulated trajectory [5] J. E. Chang, S. W. Huang, and J. H. Guo, “Hunting ghost fishing
 -0.6 End point of the measured trajectory gear for fishery sustainability using autonomous underwater vehicles,” in
 End point of the simulated trajectory Autonomous Underwater Vehicles (AUV), 2016 IEEE/OES, Nov. 2016,
 -0.4 pp. 49–53.
 -0.2 [6] Z. Wu, J. Liu, J. Yu, and H. Fang, “Development of a novel robotic
 0 dolphin and its application to water quality monitoring,” IEEE/ASME
 Transactions on Mechatronics, vol. 22, no. 5, pp. 2130–2140, Oct. 2017.
 0.2 [7] A. Vasilijevic, D. Nad, F. Mandic, N. Miskovic, and Z. Vukic, “Coordi-
 -0.4 nated navigation of surface and underwater marine robotic vehicles for
 -0.2 ocean sampling and environmental monitoring,” IEEE/ASME Transac-
 X I val

 0 tions on Mechatronics, vol. 22, no. 3, pp. 1174–1184, Jun. 2017.
 -ax ue
 is /x (

 0.2 [8] Y. Shi, C. Shen, H. Fang, and H. Li, “Advanced control in marine mecha-
 co m

 -0.6 -0.8
 tronic systems: A survey,” IEEE/ASME Transactions on Mechatronics,
 or )

 0.4 -0.4 )
 din

 0 -0.2 al u e/ y (m vol. 22, no. 3, pp. 1121–1131, Jun. 2017.
 0.6 0.2 ordinate v
 YI-axis co
 ate

 [9] H. Li and W. Yan, “Model predictive stabilization of constrained un-
 deractuated autonomous underwater vehicles with guaranteed feasibility
Fig. 21. Spiral trajectory of the robotic fish in spiral motion. and stability,” IEEE/ASME Transactions on Mechatronics, vol. 22, no. 3,
 pp. 1185–1194, Jun. 2017.
 [10] Y. Han, B. Wang, Z. Deng, and M. Fu, “A matching algorithm based
 on the nonlinear filter and similarity transformation for gravity-aided
 underwater navigation,” IEEE/ASME Transactions on Mechatronics,
ing motion, gliding motion, and spiral motion, of an active- vol. 23, no. 2, pp. 646–654, Apr. 2018.
tail-actuated robotic fish with barycentre regulating mechanism [11] ——, “A combined matching algorithm for underwater gravity-aided
was proposed basing on Newton-Euler method. CAD software navigation,” IEEE/ASME Transactions on Mechatronics, vol. 23, no. 1,
 pp. 233–241, Feb. 2018.
SolidWorks, HyperFlow based computational fluid dynamics [12] X. Zheng, C. Wang, R. Fan, and G. Xie, “Artificial lateral line based
(CFD) simulation, and grey-box model estimation method local sensing between two adjacent robotic fish,” Bioinspiration &
are used for determining model parameters. Massive kine- biomimetics, vol. 13, no. 1, p. 016002, Nov. 2017.
 [13] A. Aggarwal, P. Kampmann, J. Lemburg, and F. Kirchner, “Haptic object
matic experiments with robotic fish prototype and numerical recognition in underwater and deep-sea environments,” Journal of field
simulations demonstrate that the proposed model is capable robotics, vol. 32, no. 1, pp. 167–185, Jan. 2015.
of evaluating the trajectory, attitudes, and motion parameters [14] W. Wang, J. Liu, G. Xie, L. Wen, and J. Zhang, “A bio-inspired elec-
 trocommunication system for small underwater robots,” Bioinspiration
including the linear velocity, motion radius, angular velocity, & biomimetics, vol. 12, no. 3, p. 036002, Mar. 2017.
etc., for the robotic fish with small errors. [15] E. R. Marques, J. Pinto, S. Kragelund, P. S. Dias, L. Madureira,
 A. Sousa, M. Correia, H. Ferreira, R. Gonçalves, R. Martins et al.,
 We are conducting researches on evaluating motion param- “Auv control and communication using underwater acoustic networks,”
eters of the robotic fish using its onboard ALLS, and an in OCEANS 2007-Europe, Jun. 2007, pp. 1–6.
evaluation model that links the linear velocity, angular velocity, [16] C. Shen, B. Buckham, and Y. Shi, “Modified c/gmres algorithm for fast
and motion radius to the hydrodynamic pressure variations nonlinear model predictive tracking control of auvs,” IEEE Transactions
 on Control Systems Technology, vol. 25, no. 5, pp. 1896–1904, Sept.
(PVs) surrounding the fish body has been preliminarily ac- 2017.
quired. Using the PVs measured by the ALLS, the above- [17] C. Shen, Y. Shi, and B. Buckham, “Path-following control of an auv: A
mentioned motion parameters can be evaluated by solving the multiobjective model predictive control approach,” IEEE Transactions
 on Control Systems Technology, vol. PP, no. 99, pp. 1–9, Jan. 2018.
evaluation model inversely. In the future work, we will input [18] ——, “Trajectory tracking control of an autonomous underwater vehicle
the ALLS-evaluated motion parameters into a dynamic model- using lyapunov-based model predictive control,” IEEE Transactions on
based controller as feedback terms, for adjusting the oscillation Industrial Electronics, vol. 65, no. 7, pp. 5796–5805, Jul. 2018.
 [19] J. Yu, L. Liu, and M. Tan, “Three-dimensional dynamic modelling of
parameters of the robotic fish, and finally realizing flow-aided robotic fish: simulations and experiments,” Transactions of the Institute
closed-loop control for the trajectory of the robotic fish. of Measurement and Control, vol. 30, no. 3-4, pp. 239–258, Aug. 2008.

11

[20] J. Yu, M. Wang, Z. Su, M. Tan, and J. Zhang, “Dynamic modeling of a Minglei Xiong received the B.E. in North China
 cpg-governed multijoint robotic fish,” Advanced Robotics, vol. 27, no. 4, Electric Power University (NCEPU), Beijing, China
 pp. 275–285, Feb. 2013. in 2012. He is currently a PhD candidate in General
[21] S. Zhang, J. Yu, A. Zhang, and F. Zhang, “Spiraling motion of un- Mechanics and Foundation of Mechanics at Peking
 derwater gliders: Modeling, analysis, and experimental results,” Ocean University. His current research interests include
 Engineering, vol. 60, no. 3, pp. 1–13, Jan. 2013. biomimetic robotics, artificial intelligent, and game
[22] Z. Chen, J. Yu, A. Zhang, and F. Zhang, “Design and analysis of theory.
 folding propulsion mechanism for hybrid-driven underwater gliders,”
 Ocean Engineering, vol. 119, pp. 125–134, May. 2016.
[23] Z. Chen, S. Shatara, and X. Tan, “Modeling of biomimetic robotic fish
 propelled by an ionic polymer–metal composite caudal fin,” IEEE/ASME
 Transactions on Mechatronics, vol. 15, no. 3, pp. 448–459, Jun. 2010.
[24] J. Wang and X. Tan, “A dynamic model for tail-actuated robotic fish with
 drag coefficient adaptation,” Mechatronics, vol. 23, no. 6, pp. 659–668,
 Aug. 2013. Junzheng Zheng received the B.E. in Mechani-
[25] ——, “Averaging tail-actuated robotic fish dynamics through force and cal Design, Manufacturing and Automation from
 moment scaling,” IEEE Transactions on Robotics, vol. 31, no. 4, pp. Huazhong University of Science and Technology,
 906–917, Aug. 2015. Wuhan, China in 2017. He is currently pursuing a
[26] F. Zhang, F. Zhang, and X. Tan, “Tail-enabled spiraling maneuver for master’s degree in Control Theory and Control En-
 gliding robotic fish,” Journal of Dynamic Systems, Measurement, and gineering at Peking University. His current research
 Control, vol. 136, no. 4, p. 041028, May. 2014. interests include biomimetic robotics and lateral line
[27] F. Zhang, J. Thon, C. Thon, and X. Tan, “Miniature underwater inspired sensing.
 glider: Design and experimental results,” IEEE/ASME Transactions on
 Mechatronics, vol. 19, no. 1, pp. 394–399, Feb. 2014.
[28] G. Ozmen Koca, C. Bal, D. Korkmaz, M. C. Bingol, M. Ay, Z. H.
 Akpolat, and S. Yetkin, “Three-dimensional modeling of a robotic fish
 based on real carp locomotion,” Applied Sciences, vol. 8, no. 2, p. 180,
 Jan. 2018.
[29] J. L. Tangorra, C. J. Esposito, and G. V. Lauder, “Biorobotic fins
 for investigations of fish locomotion,” in 2009 IEEE/RSJ International Manyi Wang received the B.E. in Measurement and
 Conference on Intelligent Robots and Systems. IEEE, 2009, pp. 2120– Control Technology and Instrumentation from Na-
 2125. tional University of Defense Technology, Changsha,
[30] D. Yun, S. Kim, K.-S. Kim, J. Kyung, and S. Lee, “A novel actuation for China in 2009. He is currently pursuing a master’s
 a robotic fish using a flexible joint,” International Journal of Control, degree in Control Theory and Control Engineering
 Automation and Systems, vol. 12, no. 4, pp. 878–885, 2014. at Peking University. His current research interests
[31] K. A. Morgansen, B. I. Triplett, and D. J. Klein, “Geometric methods include biomimetic robotics and lateral line inspired
 for modeling and control of free-swimming fin-actuated underwater sensing.
 vehicles,” IEEE Transactions on Robotics, vol. 23, no. 6, pp. 1184–
 1199, Dec. 2007.
[32] S. Chen, J. Wang, and X. Tan, “Target-tracking control design for a
 robotic fish with caudal fin,” in Proceedings of the 32nd Chinese Control
 Conference. IEEE, 2013, pp. 844–849.
[33] Y. Hu, W. Zhao, and L. Wang, “Vision-based target tracking and collision
 avoidance for two autonomous robotic fish,” IEEE Transactions on Runyu Tian received the B.E. in Aircraft System
 Industrial Electronics, vol. 56, no. 5, pp. 1401–1410, 2009. and Engineering from National University of De-
[34] E. Krause, Fluid Mechanics: With Problems and Solutions, and an fense Technology, Changsha, China in 2011. He is
 Aerodynamics Laboratory. Springer, 2005. currently pursuing a master’s degree in Control The-
[35] X. He, X. He, L. He, Z. Zhao, and L. Zhang, “Hyperflow: A ory and Control Engineering at Peking University.
 structured/unstructured hybrid integrated computational environment for His current research interests include computational
 multi-purpose fluid simulation,” Procedia Engineering, vol. 126, pp. fluid dynamics, biomimetic robotics, and deep rein-
 645–649, Dec. 2015. forcement learning.
[36] X. He, L. Zhang, Z. Zhao et al., “Validation of hyperflow in subsonic
 and transonic flow,” Acta Aerodynamica Sinica, vol. 34, no. 2, pp. 267–
 275, Apr. 2016.
[37] L. Ljung, System identification toolbox: User’s guide. Citeseer, 1995.

 Guangming Xie received his B.S. degrees in both
 Applied Mathematics and Electronic and Computer
 Technology, his M.E. degree in Control Theory and
 Control Engineering, and his Ph.D. degree in Con-
 trol Theory and Control Engineering from Tsinghua
 Xingwen Zheng received the B.E. in Mechani- University, Beijing, China in 1996, 1998, and 2001,
 cal Engineering and Automation from Northeastern respectively. Then he worked as a postdoctoral re-
 University, Shenyang, China in 2015. He is currently search fellow in the Center for Systems and Control,
 a PhD candidate at the Intelligent Biomimetic De- Department of Mechanics and Engineering Science,
 sign Lab, State Key Laboratory for Turbulence and Peking University, Beijing, China from July 2001 to
 Complex Systems, College of Engineering, Peking June 2003. In July 2003, he joined the Center as a
 University, Beijing, China. His current research in- lecturer. Now he is a Full Professor of Dynamics and Control in the College
 terests include biomimetic robotics, lateral line in- of Engineering, Peking University.
 spired sensing, and multi-robot control. He is an Associate Editor of Scientific Reports, International Journal of
 Advanced Robotic Systems, Mathematical Problems in Engineering and an
 Editorial 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, and
 networked control systems.