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J. Mar. Sci. Eng. 2024, 12, 883. https://doi.org/10.3390/jmse12060883 www.mdpi.com/journal/jmse
Article
State-Transform MPC-SMC-Based Trajectory Tracking
Control of Cross-Rudder AUV Carrying Out Underwater
Searching Tasks
Haochen Hong
1
, Zhiqiang Yang
1
, Jiawei Li
2
, Guohua Xu
1
, Yingkai Xia
2,
* and Kan Xu
3
1
School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology,
Wuhan 430074, China; hhc000608@163.com (H.H.); d202380923@hust.edu.cn (Z.Y.);
hustxu@vip.sina.com (G.X.)
2
College of Engineering, Huazhong Agricultural University, Wuhan 430070, China;
jiaweili@webmail.hzau.edu.cn
3
Wuhan Second Ship Design and Research Institute, Wuhan 430025, China; happykathy9015@foxmail.com
* Correspondence: ykxia@mail.hzau.edu.cn
Abstract: In this study, we present a novel dual-loop robust trajectory tracking framework for au-
tonomous underwater vehicles, with the objective of enhancing their performance in underwater
searching tasks amidst oceanic disturbances. Initially, a real-world AUV experiment is conducted
to validate the efficacy of a cross-rudder AUV configuration in maintaining sailing angle stability
during the diving stage, which exhibits a strong capability for straight-line sailing. Building upon
the experimental findings, we introduce a state-transform-model predictive guide law to compute
the desired velocity for the dynamics loop. This guide law dynamically adjusts the controller
across varying depths, thereby reducing model predictive control (MPC) computation while opti-
mizing timing without compromising precision or convergence speed. Subsequently, we incorpo-
rate a sliding mode controller with a prescribed disturbance observer into the velocity control loop
to concurrently enhance the robustness and convergence rate of the system. This innovative amal-
gamation of controllers significantly improves tracking precision and convergence rate, while also
alleviating the computational burden—a pervasive challenge in AUV MPC control. Finally, vari-
ous condition simulations are conducted to validate the robustness, effectiveness, and superiority
of the proposed method. These simulations underscore the enhanced performance and reliability
of our proposed trajectory tracking framework, highlighting its potential utility in real-world AUV
applications.
Keywords: cross-rudder AUV; underwater searching tasks; state-transform MPC-SMC; trajectory
tracking control
1. Introduction
The escalating demand for ocean exploration and underwater scientific research
has underscored the significance of autonomous underwater vehicles (AUVs), garnering
considerable attention from both industry and scientific research entities. The multifac-
eted capabilities of AUVs encompass a wide array of tasks, including seabed exploration
[1], underwater mapping [2], subsea facilities inspection [3], and oceanic scientific re-
search [4,5]. Leveraging the efficiency of AUVs not only mitigates the risks associated
with oceanic exploration but also enhances exploration efficacy and precision to a com-
mendable extent.
According to the conclusion in [6], the AUV sailing system is categorized as “level
2” due to the prerequisite of environmental search prior to conducting the sailing exper-
iment. Ensuring the safety and efficacy of underwater searching operations conducted
Citation: Hong, H.; Yang, Z.; Li, J.;
Xu, G.; Xia, Y.; Xu, K. State-
Transform MPC-SMC-Based
Tra jecto ry Tra cking C ontr ol of
Cross-Rudder AUV Carrying Out
Underwater Searching Tasks. J. Mar.
Sci. Eng. 2024, 12, 883.
https://doi.org/10.3390/jmse12060883
Academic Editors: Tieshan Li and
Sergei Chernyi
Received: 15 April 2024
Revised: 18 May 2024
Accepted: 21 May 2024
Published: 26 May 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(https://creativecommons.org/license
s/by/4.0/).
J. Mar. Sci. Eng. 2024, 12, 883 2 of 36
by autonomous underwater vehicles (AUVs) necessitates the development of robust tra-
jectory-tracking strategies characterized by high precision. Over the past few decades,
considerable research efforts have been dedicated to exploring diverse control method-
ologies for AUV trajectory tracking. However, given the high levels of uncertainty and
the complexity of the underwater environment [7,8], it is imperative that AUV control
strategies exhibit a requisite level of robustness and stability.
The conventional PID method offers the advantages of straightforward parameter
tuning and a stable adjustment process, making it suitable for addressing basic trajectory
tracking challenges. However, its efficacy diminishes as trajectory tracking tasks grow in
complexity, particularly within multi-input multi-output (MIMO) systems. Furthermore,
fuzzy logic control has been applied in AUV trajectory tracking control [9,10]. Neverthe-
less, in environments characterized by complexity and diverse trajectory tracking re-
quirements, frequent updates to fuzzy rules are necessary, leading to diminished ro-
bustness and applicability of the controller.
Backstepping control (BSC) is a control methodology that decomposes complex
nonlinear systems into multiple subsystems and designs Lyapunov functions. This ap-
proach has found application in AUV trajectory tracking control [11]. In [12], a back-
stepping-based integral sliding surface was introduced with the aim of eliminating the
reaching stage of the sliding surface. Additionally, to tackle disturbances and parameter
uncertainty, an adaptive extreme learning machine (AELM) was devised. Integrating
with Lyapunov-based nonlinear model predictive control (NMPC), a backstepping auxil-
iary law was introduced to expedite trajectory tracking convergence while ensuring
overall control strategy stability [13]. However, a limitation of BSC is its inability to effec-
tively suppress the adverse effects of disturbances and parameter uncertainty.
Due to the demanding and intricate operational conditions of AUVs, the imperative
consideration of robustness and strong anti-jamming capabilities arises during controller
design. Disturbances stemming from factors such as winds, waves, and ocean currents
underscore the necessity for robustness [14]. Sliding mode control (SMC) stands as a
widely employed strategy for AUVs due to its robust anti-jamming characteristics and
notable convergence amidst oceanic disturbances and parameter uncertainties. In [15], a
pioneering approach was introduced, namely the super-hyperbolic switching algorithm-
based sliding mode controller augmented with a disturbance observer, aimed at enhanc-
ing trajectory tracking for X-rudder AUVs. Additionally, a dual-loop integral sliding
mode control method was devised to statically compensate for AUV dynamics and miti-
gate model uncertainty, thereby bolstering controller robustness [16]. Furthermore, an
adaptive reaching law was developed within sliding mode control to facilitate fault tol-
erance in AUV operations, thereby improving convergence times and trajectory tracking
accuracy [17]. In [18], a combination of sliding mode feedback law and adaptive line-of-
sight was employed, effectively eliminating the need for actual attack angle and com-
pensating for pitch-tracking inaccuracies. Additionally, in reference [19], a novel chatter-
ing-free robust predefined-time sliding mode control scheme was proposed to enhance
the robustness and tracking precision of a 3-DOF ROV under matched uncertainties
conditions. In [20], a robust high-order sliding mode controller was designed to improve
the convergence speed of the AUV tracking error. However, a common challenge en-
countered in sliding mode control pertains to “chattering”, occurring when control in-
puts are non-consecutive amidst external disturbances [21,22]. Researchers have often
addressed this issue by navigating a trade-off between convergence time and chattering,
thereby compromising controller efficiency [23].
Model predictive control (MPC) presents an effective methodology for addressing
multi-constraint control challenges. This approach entails computing optimal solutions
in each step through a target function integrating various control requirements and
physical constraints. By leveraging real-time system state information, MPC resolves fi-
nite-time domain open-loop control problems and implements the initial step of the op-
timal control sequence into the real system. A novel distributed model predictive con-
J. Mar. Sci. Eng. 2024, 12, 883 3 of 36
troller utilizing Lyapunov principles, complemented by a rapid finite-time extended
state observer, was introduced to optimize velocities and control forces for AUVs across
diverse tracking tasks and disturbances [24]. Ref. [25] introduced a linear model predic-
tive control approach enhanced with a fuzzy controller to achieve constrained path fol-
lowing for median and paired fin-propelled robotic fish, resulting in improved precision
in path following. Moreover, in [26], the challenge of speed variation and obstacle avoid-
ance in multi-AUV systems was addressed by incorporating compatibility constraints in-
to MPC, thereby limiting uncertainty deviation for each AUV. Although MPC possesses
advantages, it is burdened by the substantial computational requirements needed to
solve an open-loop optimal control problem. As dimensions of decision variables, con-
trol constraints, and prediction horizons increase, computational burdens escalate, po-
tentially compromising operational efficiency. Consequently, common strategies to miti-
gate this issue include reducing decision variables and prediction horizons [27]. Fur-
thermore, it is noteworthy that the majority of current MPC research in AUVs predomi-
nantly focuses on hard constraints, often overlooking or relegating soft constraints to a
secondary consideration. However, in certain scenarios, soft constraints offer notable
advantages, such as reducing iterations, enhancing control performance, and minimiz-
ing AUV energy consumption [28].
Focusing on underwater searching tasks such as underwater mapping or underwa-
ter source searching, AUVs commonly descend to a designated depth before undertak-
ing specific tracking tasks [29,30]. The meticulously designed cross-rudder structure fa-
cilitates AUV descent to target depths without requiring directional control, thereby re-
ducing the number of decisive variables in MPC design during the diving phase. Conse-
quently, computational workload and calculation time are substantially diminished, ad-
dressing a prevalent challenge encountered in MPC implementation. As diving control
mainly centers on the vertical movement of AUVs, the movements within the horizontal
plane and lateral plane were presumed to be stable and therefore disregarded [31,32]. To
simplify the analyzing process of the diving AUV stage, the 3DOF has been selected as
offering model simplification without being divorced from reality. This has been adopt-
ed in related AUV diving control research literature [33,34]; this research, however, men-
tioned the assumption without any verification from an real-AUV experiment.
From the aforementioned discussion, it becomes evident that precise and highly ro-
bust trajectory tracking control methods are imperative for cross-rudder AUVs. To
streamline the analysis process, we chose the three-degrees-of-freedom (3DOF) AUV
model in a vertical plane as the target for simulation and experimentation during the
diving stage. Real-AUV experiments were conducted to validate this assumption. Fur-
thermore, addressing the challenge of heavy computational workload in the design of
the MPC process is crucial. To mitigate this issue, we propose a dual closed-loop control
framework that combines 3DOF MPC and 5DOF MPC in the kinematics loop. A key in-
novation lies in the introduction of a state-transform MPC controller, dynamically trans-
forming two MPCs across different depths within a kinematics loop. Simultaneously, a
sliding mode control law with a prescribed time disturbance observer is integrated to
enhance robustness, fully leveraging the computational results from MPC, and to miti-
gate the adverse effects of ocean disturbances. In the context of MPC-based trajectory
tracking control, the optimization procedure facilitates the automatic utilization of slid-
ing mode control's capabilities to generate optimal tracking control and accelerate error
convergence speed. The main contributions of this research can be summarized as fol-
lows:
(1) A real-AUV experiment was conducted wherein the cross-rudder AUV was di-
rected to dive to the target depth without lateral motion control. Upon completion
of the experiment, it was observed that the AUV successfully dived to the target
depth without lateral motion, attributed to its robust straight-line sailing ability. To
the best of the authors’ knowledge, a similar experiment has not been carried out.
J. Mar. Sci. Eng. 2024, 12, 883 4 of 36
(2) We propose a state-transform MPC guide law tailored for the cross-rudder AUV to
achieve precise trajectory tracking control within the kinematics loop. Initially, a
state-switch MPC controller is introduced to calculate the target velocity in the kin-
ematics loop. Specifically, leveraging the MPC guide law based on state transfor-
mation significantly reduces the computational workload. Additionally, to mitigate
controller overshoot, the MPC is proactively transformed prior to reaching the tar-
get depth. Comparative analysis with conventional MPC reveals superior trajectory
tracking performance and minimal control overshoot. For the dynamics loop, a slid-
ing mode controller is employed to augment velocity control effectiveness. This ve-
locity SMC with the prescribed time disturbance observer for the dynamic loop ac-
celerates velocity error convergence, fully leveraging the computational results
from MPC, and enhancing anti-jamming capability. Comparison with MPC without
sliding mode control demonstrates enhanced maneuverability and adaptability.
(3) Comparative evaluation against traditional MPC and conventional model predic-
tive controller with sliding mode controller and a disturbance observer underscores
the efficacy of the proposed control framework in alleviating computational work-
load challenges and significantly reducing optimization time. Moreover, the
framework exhibits rapid convergence, thereby enhancing the accuracy and stabil-
ity of cross-rudder AUV control.
The following sections of this paper are organized as follows. Section 2 delineates
the 5DOF-AUV model and expounds upon the trajectory tracking control problem, elu-
cidating two distinct steps involved therein. Section 3 expounds upon the design proce-
dure of the proposed control method, encompassing the state-transform MPC guide law
for the kinematic loop and the SMC with a disturbance observer for the dynamics loop
control, alongside stability analysis. In Section 4, a comparative simulation experiment is
conducted, contrasting the efficacy of various controllers. Finally, Section 5 encapsulates
the conclusions drawn from this study.
2. Modeling and Problem Formulation
2.1. Preliminaries and Nomenclatures
Lemma 1 [35]. The equilibrium point 0x= can be deemed globally finite-time stable if there
exists a Lyapunov function that fulfills the following condition:
12
() () () 0
k
Vx Vx V x
λλ
++ ≤
(1)
where 10
λ
>, 2
0
λ
>
,0< <1k. The settling time can be expressed as follows:
1
102
12
()
1ln
(1 )
k
Vx
tk
λλ
λλ
−+
≤− (2)
where 0
()Vx is the initial value of ()Vx.
Lemma 2 [36]. The following disturbance observer is able to maintain the observation error at a
defined compact set 0
Ω
in a prescribed time T. The detailed expression of estimate disturbance
is shown as follows:
00
()
ˆ()
()
t
dKCMv
t
ζ
ξζ
=+ +
(3)
where 0
K and 0
C are constant matrix.
ξ
is an auxiliary state vector and ()t
ζ
is a time-
varying function which is
J. Mar. Sci. Eng. 2024, 12, 883 5 of 36
exp[( ( ))] 1 [0, )
() exp[( ( ))] [ , )
Tt t T
ttT t T
ϖ
ζϖ
−− ∈
=−∈∞
(4)
where (0,ln(2)/ ]T
ϖ
∈,
0T>
are positive design parameters.
2.2. Frames of Reference
To facilitate motion control analysis, it is imperative to construct a coordinated sys-
tem for the AUV. This following coordinated system comprises the inertial reference
frame denoted as {E} and the body-fixed frame denoted as {I}. The transformation of co-
ordinates enables seamless interconversion between these frames, as depicted in Figure
1.
Figure 1. Coordinate system.
2.3. Modeling of the Cross-Rudder AUV
In this study, it is assumed that the vehicle exhibits neutral buoyancy and is en-
dowed with a robust hydrodynamic restoring force in the rolling direction [37,38]. Con-
sequently, to simulate the process of underwater searching, both the five-degrees-of-
freedom (5DOF) mathematical model and the three-degrees-of-freedom (3DOF) model
are employed to elucidate subsequent experiments. Drawing from references [31,39], the
equations governing AUV motion are described as follows:
(3,5)
(3,5)
iii
ii ii ii i i
i
i
η
′==
++=+=
Jv
Mv Cv Dv τ d
(5)
We s ele ct 3,5=i the subscript of the variable to distinguish the variable from the
3DOF and 5DOF models. i
η
′
is the derivative of the vehicle’s position vector. i
J
is the
rotation matrix of the AUV. i
v
is the velocity vector of the AUV. i
C
, i
D
, and
i
M
rep-
resent the addition matrix, the hydrodynamic damping matrix, and the system inertia
matrix respectively. i
τ
is defined as the force and moment from the propeller and rud-
ders.
From the AUV motion equation, the kinematics and the dynamics model of 5DOF
cross-rudder AUV [38] and 3DOF [22] cross-rudder AUV can be formulated as follows:
J. Mar. Sci. Eng. 2024, 12, 883 6 of 36
cos( )cos( ) sin( ) sin( ) cos( )
cos( ) sin( ) cos( ) sin( )sin( )
sin( ) cos( )
/cos( )
xu v w
yu v w
zu w
q
r
θ
ϕϕ
θ
ϕ
θ
ϕϕ
θ
ϕ
θθ
θ
ϕθ
=−+
=++
=− +
=
=
(6)
cos( ) sin( )
sin( ) cos( )
xu w
zu w
q
θθ
θθ
θ
=+
=− +
=
(7)
where ,,,,
θ
ϕ
xyz
and ,,
θ
x
z
represent the derivation position and orientation varia-
bles, respectively, of the 5DOF cross-rudder AUV model and 3DOF cross-rudder AUV in
the inertial reference frame. Additionally, ,, ,,uvwqr
denote the surge, sway, heave,
pitch, and yaw in the 5DOF AUV model and 3DOF AUV in {I}, respectively. Moreover,
the dynamic model of both the 5DOF and 3DOF cross-rudder AUV can be delineated as
follows [15]:
2
() ( )
( )
() ( )
() ( )
()
uwqqqrr
uu
vr u u
vur
vv rr
uv v v
wuquw
ww qq
ww
yy q
mXu Xuu X mwq Xq Xrr
Xmvr d
mYv Yvv Yrr Y mur
Yuv d
mZw Z ww Zqq Z muqZuw
d
IM
τ
τ
τ
−= +− + +
++ ++
−= + +−
+++
−= + +++
++
−
+
()
+
uq
ww qq
uw q q
zz r ur uv
vv rr
rr
qMwwMqq Muq
Muw d
I N r N v v N r r N ur N uv
d
τ
τ
=++
++
−= + + +
+
(8)
2
() ( )
() ( )
() +
uwqqquu
uu
wuquwww
ww qq
yy q uq uw q q
ww qq
mXu Xuu X mwqXq d
mZw Z ww Zqq Z muq Zuw d
IMqMwwMqqMuqMuw d
τ
τ
τ
−= +− + ++
−= + +++ ++
−= + + ++
(9)
where ,, ,,uvwqr
denote the derivation of surge, sway, heave, pitch, and yaw of AUV
in the body-fixed frame, respectively. o
A
and oo
A
(,,,,,,,,,,)AXYZKMNouvwqr==
are the hydrodynamic coefficients correspond-
ing to the direction. m denotes the mass of the cross-rudder AUV; (,,,,)
oouvwqr
τ
=
are the forces and moments exerted by the propeller and rudders, respectively; yy
I
and
z
z
I denote the vehicle moments of inertia in the body-fixed frame.
Assumption 1. The unknown oceanic disturbances (,,,,)
o
do uvwqr= are characterized as
continuous and bounded by unknown upper limits [40].
Assumption 2. The buoyancy force exerted on the AUV is equilibrated by the gravitational force
acting upon it. Furthermore, the center of buoyancy aligns exactly with the center of gravity, lo-
cated at the origin point of the AUV [34,41].
J. Mar. Sci. Eng. 2024, 12, 883 7 of 36
Assumption 3. All state variables of the kinematics and dynamics model in Equation (5)
through Equation (9) are measurable.
Remark 1. Assumption 3 stipulates the measurability of the cross-rudder AUV. In challenging
and unpredictable underwater environments, AUVs are typically outfitted with a suite of sensors
including a depth gauge, inertial navigation system (INS), Doppler velocity log (DVL), Global
Positioning System (GPS), and various others. These sensors constitute an integral component of
the AUV’s navigational system, continuously providing real-time data on key parameters such as
velocity and position.
2.4. Problem Formulation
As the mission objectives of the AUV vary significantly between the diving stages
and the searching stage, a clear definition and categorization of the tasks accordingly has
become quite essential. As depicted in Figure 2, the missions involving trajectory track-
ing of underwater searching are segmented into the diving stage and the searching
stage. In the diving stage, the AUV dives to the target depth as specified by the operator.
Upon reaching the designated depth, the AUV transitions to the searching stage. Here,
the AUV follows a predetermined search path determined by the searching area and the
location of underwater resources.
Figure 2. AUV seafloor working mission.
Based on the depiction in Figure 3, it is evident that conventional AUV configura-
tions are characterized by the presence of vertical and horizontal rudders, which afford
independent control over depth and lateral motion, respectively. Notably, these control
mechanisms operate devoid of significant coupling effects. Consequently, AUVs en-
dowed with robust straight-line sailing capabilities can effectively navigate to the de-
sired depth while concurrently maintaining a consistent sailing heading within narrow
margins. Thus, we posit the following assumption in light of the aforementioned obser-
vation.
J. Mar. Sci. Eng. 2024, 12, 883 8 of 36
Figure 3. Layout of the cross-rudder.
Remark 2. To further validate the conclusions drawn and the results obtained from the subse-
quent simulations, the 3DOF AUV model is chosen to represent the diving stage, while the
5DOF AUV model serves as the underwater searching model.
Assumption 4. As depicted in Figure 3, the distinctive structure of the cross-rudder AUV effec-
tively mitigates the coupling effect. The vertical and horizontal rudders are designated to control
the depth and sailing angle, respectively. Endowed with robust straight-line sailing ability and
stability, the cross-rudder AUV demonstrates the capability to navigate to the target depth with-
out necessitating directional control.
To validate the assumption, a real-AUV sailing experiment was conducted in a con-
trolled water pool environment. In this experiment, the dual-loop PID method was em-
ployed for depth control, while direction motion control was disabled to ascertain the
AUV’s ability to descend to the target depth and maintain a bounded sailing angle sim-
ultaneously. The control framework is depicted in Figure 4.
Figure 4. AUV-experiment control frame.
The target depths were configured at 2 m and 3 m. In the pitch transform control
PID, the
p
K
, i
K
and d
K
parameters were set to −5.6, 0, and 0. For the pitch control PID
system, the corresponding
p
K
, i
K
and d
K
PID parameters were chosen as 1.2, 0.8, and
0.2. The control effects at depths of 2 m and 3 m are presented in Figures 5 and 6, respec-
tively.
Figure 5. 3 m diving experiment.
J. Mar. Sci. Eng. 2024, 12, 883 9 of 36
Figure 6. 2 m diving experiment.
Based on the experiment data presented above, it was evident that the cross-rudder
AUV successfully descended to the target depth while maintaining the sailing angle
within narrow bounds, all without requiring directional control. This conclusion under-
scores the capability of the cross-rudder AUV, characterized by robust straight-line sail-
ing ability, to synchronize depth control and maintain the sailing angle concurrently.
With the above conclusion, the further variables are defined. To distinguish the var-
iable from the 3DOF and 5DOF models, we select the subscript of the variable. We define
the actual position state as (3,5)=
i
pi and the desired vector as (3,5)=
id
pi . We de-
fine the position state of the 5DOF AUV as T
555555
(,,, , )
θϕ
ααααα
=xyz
p, and the desired
vector T
555555
(,,, , )
θϕ
αααα α
=
d xdydzd d d
p. The desired AUV altitude 5
θ
α
d 5
ϕ
α
d can be
calculated by the following, which is shown in [42,43].
arctan( )
arctan( )
d
d
dd
d
d
d
z
x
y
y
x
θ
ϕ
α
α
−
=
+
=
(10)
Similarly, the AUV’s position and the desired position of the 3DOF vertical AUV
model are defined as T
3333
(,, )
θ
ααα
=xz
p and T
3333
(,, )
θ
ααα
=
dxdzdd
p. The 3
θ
α
d is
defined as 3
3
3
arctan( )
θ
α
−
=d
d
d
z
x
.
Therefore, considering the altitude and the position error, the AUV trajectory track-
ing error is defined as TT
53
(, , , , ), (,, )
exeyezeeeexezee
PP
θϕ θ
ααααα ααα
==
, which is
calculated by the following:
53
xe x xd
ye y yd
x
exxd
ze z zd
eezezzd
ed ed
ed
PP
θθθ θθθ
ϕϕϕ
ααα
ααα ααα
ααα ααα
ααα ααα
ααα
=−
=− =−
=−
===−
=− =−
=−
(11)
The primary objective in cross-rudder AUV underwater searching missions such as
seafloor mapping or resource searching is the design of a robust controller capable of
driving the position error vector to converge to zero in the presence of unknown oceanic
J. Mar. Sci. Eng. 2024, 12, 883 10 of 36
disturbances. The control objective can be articulated as follows: achieving error conver-
gence to zero as time approaches infinity through the utilization of the proposed control-
ler.
53
lim( , ) 0
ee
tPP
→∞ = (12)
Remark 3. From Assumption 4, it follows that if the AUV exhibits strong rectilinear navigation
capabilities, it can descend to the target depth with minimal deviation from the initial sailing an-
gle. Consequently, during the diving phase, the selection of a 3DOF vertical model is justified for
simulating the AUV’s behavior in underwater search missions. The 3DOF AUV model, as pre-
sented in reference [31], accounts for the AUV’s descent dynamics without incorporating explicit
sailing control, aligning with the context of seafloor mapping simulations in this study. The sub-
sequent controllers adhere to Assumption 4. A similar assumption has been posited in prior
works [32–34], albeit without empirical validation through real-world AUV experimentation. In
pursuit of confirming the accuracy of these assertions, real-world AUV experimentation was con-
ducted, the results of which remain pending verification.
Remark 4. Acknowledging that the experiment was conducted in a controlled test pool environ-
ment, it is important to recognize that environmental disturbances in this setting may not mirror
the intensity experienced in natural environments. While Assumption 4 offers a valuable initial
standpoint for further theoretical analysis, its applicability may necessitate refinement under spe-
cific conditions. For instance, in scenarios where the vehicle exhibits limited straight-line sailing
ability or encounters stronger natural disturbances, a more nuanced evaluation of Assumption 4
becomes imperative.
3. Controller Design
In pursuit of the control objective and to fulfill the requirements of trajectory track-
ing for cross-rudder AUVs, a dual closed-loop controller is proposed. The first loop en-
tails kinematics control, employing MPC with a transformation function to compute the
target velocity. The second loop, known as the dynamics loop, comprises a robust SMC
mechanism designed to ensure rapid convergence of the velocity state to the target ve-
locity while minimizing overshoot or maintaining it at negligible levels. The configura-
tion of the proposed controller is depicted in Figure 7.
Figure 7. Proposed method control frame.
Within the kinematics loop, the target reference (3,5)=
id
pi and predictive state
(3,5)=
i
Yi
are acquired, enabling the determination of predictive errors. Subsequently,
the target velocity (3,5)
id i=v is computed utilizing state-transformed model predic-
tive control, with controller types discerned based on the AUV’s depth. Furthermore, ve-
J. Mar. Sci. Eng. 2024, 12, 883 11 of 36
locity errors (3,5)
ie i=v are mitigated via the sliding mode controller, facilitating the
calculation of actual inputs (3,5)=τii. (|)
i
Xk lk+ is the AUV’s predictive state and
the vector (|)
i
Xkk
is the initial state to calculate the i
Y
which is the predictive position
state of the AUV.
3.1. Kinematics Predictive Model
To simulate the AUV underwater searching task, the 3DOF predictive model and
5DOF predictive model are chosen as the diving stage model and the searching stage
model, respectively. These two predictive models are derived from distinct kinematics
models. To control the motion, the kinematic model is reformulated using the LTI state-
space representation framework, integrating a sampling period T, which is demonstrat-
ed in [44,45]:
(1) () ()()Gk Gk T J k vk+= +⋅ ⋅ (13)
where ()
i
Gk , ()vik, and ()Jik are the position state, velocity state, and the rotation
matrix in thk sampling time respectively. Therefore,
T
555555
() ( (), (), (), (), ())
θϕ
ααααα
=xyz
Gk k k k k k ，T
3333
() ( (), (), ())
θ
ααα
=xz
Gk k k k ,
T
555555
() ( (), (), (), (), ())k ukvkwkqkrk=v , and T
3333
() ( (), (), ())kukwkqk=v. The
rotation matrix of 5DOF model 5()Jk and 3DOF model 3()Jk can be calculated by the
kinematics model with
55 5 55
55 5 5 5
55 5
5
33
3
cos( ( )) cos( ( )) sin( ( )) sin( ( )) cos( ( )) 0 0
cos( ( ))sin( ( )) cos( ( )) sin( ( )) sin( ( )) 0 0
() sin( ()) 0 cos( ()) 0 0
00010
00001/cos(())
cos( ( )) sin( ( )) 0
()
kk k kk
kk k kk
kk k
k
kk
k
θϕ ϕ θϕ
θϕ ϕ θϕ
θθ
θ
θθ
−
=−
=−
J
J33
cos( ( )) sin( ( )) 0
001
kk
θθ
(14)
To address this issue, an incremental version that incorporates the input vector is
defined as:
() () ( 1)
iii
kkk=−−uvv (15)
Therefore, the representation of the discrete model is illustrated with:
(1) ()() ()()
() ()
Xk AkXk Bkuk
Yk CXk
+= +
=
(16)
where ()
() (1)
i
i
i
Gk
Xk k
=
−
v,()
() ×
××
=
J
ii i
i
ii ii
ITk
Ak OI , ()
()
×
=
Ji
i
ii
Tk
Bk I, and
33 3(2-3)××
=
ii
QI O . In ()
i
A
k, ()
i
Bk and i
Q, the
I
and O stand for the unit matrix
and the zeros matrix. In accordance with the definition of model predictive control, the
prediction state sequence ()
i
X
k
of the AUV can be computed by advancing the predic-
tion model over P
N samplings with the sequence of control inputs ()
iku, which P
Nis
J. Mar. Sci. Eng. 2024, 12, 883 12 of 36
designated as the prediction horizon. To ensure precise notation, the kinematics input
()
iku and the prediction state sequence ()
i
X
k
are denoted as follows:
(|) (|)
( 1| ) ( 1| )
( ) ( 2 | ) ( ) ( 2 | )
(1|) (1|)
ii
ii
ii i i
iP iP
ukk Xkk
uk k Xk k
k uk k Xk Xk k
ukNk XkNk
++
=+ = +
+− +−
u
(17)
where (1|)(1)+− =
iP
uk l kl N
and (1|)(1)+− =
iP
Xk l kl N
represent the in-
put sequence and the prediction state at the k-th sampling time, respectively. Utilizing
the discrete model in Equation (10), the prediction state (1|)+−
i
Xk l k
at the k-th
sampling time can be computed as follows:
1
1
0
( 1|) () (|) () ()( |)( 1, )
−−−
=
+− = + + =
l
ilj
iiiiii P
j
X
kl k AkXkk Ak Bkuk jkl N (18)
where (|)
i
Xkk
is the initial state at k-th sampling time. Therefore, the prediction of
the position vectors ()
i
Yk
can be re-expressed as:
(|)
( 1| )
( ) ( 2 | )
(1|)
+
=+
+−
i
i
ii
iP
Ykk
Yk k
Yk Yk k
Yk N k
(19)
From Equations (17)–(19), the model can be re-constructed as follows:
(1) ()() ()()
iiiii
X
kAkXkBkk+= + u (20)
where 2T
() [ () () ()]=N
iii i
Ak Ak Ak Ak
and ()
i
B
k is
12
() 0 0
() () () 0
()
() () () () ()
−−
=
PP
i
ii i
i
NN
iii i i
Bk
AkBk Bk
Bk
A
kBkAk Bk Bk
(21)
Remark 5. Despite the differing degrees of freedom in the two models, both the discrete method
and the linearized method utilized remain identical. The sole disparity between the two models
lies in their state-space representation dimensions.
3.2. State-Transform MPC Design
3.2.1. Controller Design
At the core of MPC lies the determination of an optimal solution to the optimization
problem. This problem is recurrently solved to minimize a predicted performance cost
function by adjusting the current and future inputs of the system while adhering to sys-
tem constraints. Formulated by combining the target reference and the AUV state at the
J. Mar. Sci. Eng. 2024, 12, 883 13 of 36
time kth, the predicted performance cost function encompasses the state error. Upon
solving the optimization problem, the very first element from each predictive velocity
sequence is employed as the target velocity for dynamics control, while the remaining
elements are reserved for subsequent optimization. This iterative process is meticulously
repeated at each time step.
Considering the AUV depth and the target depth, the logic can be illustrated as fol-
lows:
1 depth target depth
2 depth< target depth
K
stage K
≥⋅
=⋅
(22)
Here, K denotes a constant that is less than 1, with “stage” representing the cur-
rent stage of operation. Specifically, a value of 1 corresponds to the diving stage, while 2
indicates the searching stage. The transition between stages facilitates the adjustment of
control strategies accordingly. Initiating control actions before reaching the target depth
can mitigate overshoot, as the more precise 5DOF controller is engaged.
Upon stage determination, both the 3DOF and 5DOF controllers undergo identical
computational procedures. Despite their differing degrees of freedom, both MPC con-
trollers share identical cost functions and state constraints. The primary distinction be-
tween these models lies in the dimensions they consider. Specifically, the 3DOF MPC
does not account for sway and yaw, in contrast to the 5DOF MPC.
With the above definition of the position vector ()
i
p
k and desired vector ()
id
p
k at
the kth sampling time, the stage cost of the objective function is shown as follows:
22
(|) () ( 1|)
ix iu
iid i
QQ
pk lk p k l uk l k+− + + +− (23)
where the stage cost consists of the position error quadratic objective function and the
input quadratic objective function, which is:
2
2
(|) ()
(1|)
ix
iu
iid
Q
iQ
pk lk p k l
uk l k
+− +
+− (24)
The objective function of the MPC controller at the k-th time is constructed as fol-
lows:
22
1
min( ) max( )
min( ) max( )
() ( | ) ( ) ( 1| ) ( 3,5)
( ) ( 3,5)
( ) ( 3,5)
=
=+−+++− =
≤+≤ =
≤+≤ =
P
ix iu
N
ii id i
QQ
l
ii i
ii i
Wk pk lk p k l uk l k i
uuklui
XXklXi
(25)
where i is used to distinguish the variables from the 3DOF controller and 5DOF control-
ler; 2T
=
Q
x
xQx;P
N represents the prediction horizon and control horizon; and
(1|)+−
i
uk l k denotes the system inputs at time 1k+. ix
Q and iu
Q are the weighting
matrices; min( )i
X and max( )i
X signify the upper and lower horizon of the AUV state at
time k, while min( )i
u and max( )i
u represent the upper and lower increment horizons of
the AUV state at time k-th.
The design of the cost function aims to ensure both trajectory convergence speed
and optimal utilization of the controller’s capabilities. By solving the aforementioned op-
timization problem, the optimal input sequence Δ*(, 1|)
iP
kk N k+−v is obtained, with
the first element utilized as the dynamics control input.
J. Mar. Sci. Eng. 2024, 12, 883 14 of 36
Δ() Δ*(,)
ii
kkk=vv (26)
From Equation (26), the target velocity vector at k, the time for the dynamics calcu-
lating, can be obtained as follows [46]:
() () Δ () ( 1)( 3,5)
id i i i
kk kkh== +−=vv vv (27)
By translating the state constraints min( )i
X, max( )i
X into the input constraints min( )i
u
, max( )i
u, the input and constraints can be reconstructed as follows:
max( )
min( )
max( )
min( )
() () () ()
() () () ()
()
()
ii ii i
ii ii i
ii
ii
AkX k Bk k X
AkX k Bk k X
kU
kU
+≤
−−≤−
≤
−≤
u
u
u
u
(28)
where max( )i
X and min( )i
X are the state constraints of the whole prediction vector re-
spectively. max( )i
U and min( )i
U are the input constraints of the whole prediction vector,
respectively. Considering the above constraints, the linear constraints form is shown as
follows:
()
ii i
LkH≤u (29)
where
max( )
min( )
max( )
min( )
,(3,5)
() ()
()
() ()
()
×
×
−
−
== =
−
−+
−
i
ii
i
ii
ii
ii i
i
ii i
i
U
I
U
I
LH i
XAkXk
Bk
XAkXk
Bk
(30)
3.2.2. Stability Analysis
Assumption 5 [47]. In each period, under the constraints of Equation (29), the system Equation
(25) has a feasible solution. Furthermore, only the initial optimal controller ()
iku is applied for
optimal control.
Theorem 1. If Assumption 5 holds, the system Equation (25) is stable.
Proof. Select the 5()Wk
as the candidate Lyapunov function 1
V. Define *
1()Vk
as the
minimum function of 1
V. Therefore, the *
1()Vk
is:
55
22
*
155
1
() min ( |) ( ) ( 1|)
=
=+−+++−
P
xu
N
di
QQ
l
Vk pklk p k l uk l k (31)
From the above expression, we can conclude that when *
1
0, ( ) 0==kVk , while
*
1
0, ( ) 0kVk≠>. Since the objective function in each period is larger than 0, it is obvi-
ous to draw the conclusion that *
155
() min () ()=≤V k Wk Wk. Therefore, this 5(1)+Wk
is expressed as:
J. Mar. Sci. Eng. 2024, 12, 883 15 of 36
55
55
55
22
55 5
1
22
** *
55
1
22
** *
55
(1) (1+|1) (1) (1+1|1)
= ( + | ) ( + | ) ( + 1| )
( | ) ( ) ( | )
=
=
+= + +− + + + − +
−+−
−− −
P
xu
P
xu
xu
N
di
QQ
l
N
di
QQ
l
di
QQ
Wk pk lk p k uk l k
p klk p klk u kl k
pkk p k ukk
55
22
** * *
15 5
= ( ) ( | ) ( ) ( | )−− −
xu
di
QQ
Vk pkk p k ukk
(32)
Since
55
22
** *
55
(|) () (|)−− −
x
u
di
QQ
pkk p k ukk is a quadratic objective function,
it is easy to draw the conclusion that
55
22
** *
55
(|) () (|) 0−− − ≤
xu
di
QQ
pkk p k ukk . Since
*
155
(1)min(1) (1)+= +≤ +V k Wk Wk the following conclusion is obtained:
**
151
(1) (1) ()+≤ +≤Vk Wk Vk
(33)
With the above conclusion, the Lyapunov function is monotonically decreasing and
the system (25) is stable. The objective function 3()Wk
has the same proof process,
which will not be depicted here.
From the aforementioned, the AUV employs distinct MPC control laws correspond-
ing to different operational stages determined by depth, which means they will not af-
fect the stability of each other. Only one MPC controller is adopted at one moment, and
the whole kinematic controller stability can be ensured and deducted because the task
nature does not frequently switch. Therefore, to ensure the stability of the state-
transform MPC, it suffices for each MPC to be stable. By incorporating the state-
transform controller alongside the 3DOF MPC and 5DOF MPC, the overall stability of
the kinematics closed-loop controller is guaranteed.
□
Remark 6. The state-transform MPC is engineered to expedite optimization time across all three
dimensions. In accordance with the findings outlined in [27], reducing the number of decisive
variables is a pivotal strategy for optimization time reduction. Leveraging the intrinsic character-
istics of the cross-rudder AUV, a notable reduction in optimization time is achieved through the
decreased number of decisive variables. This addresses a longstanding challenge in AUV trajecto-
ry tracking MPC.
Remark 7. To mitigate overshoot and enhance the precision of AUV trajectory tracking, a state-
transform function is integrated with the MPC. Through the utilization of the transform func-
tion, significant enhancements in tracking efficiency have been observed, particularly with the
implementation of the 5DOF MPC, for its heightened tracking/following efficacy. Consequently,
the amalgamation of the state-transform function and MPC not only bolsters efficiency but also
ensures controller stability. Additionally, the transform constant K can be adjusted to accom-
modate varying target depths.
3.3. Dynamics Controller Development
3.3.1. Controller Design
To enhance the anti-jamming ability and robustness of the cross-rudder AUV con-
troller, a sliding mode controller is introduced, utilizing the calculation outcomes from
the MPC guide law. Also, the dynamics controller encompasses both a 5DOF sliding
mode controller and a 3DOF sliding mode controller.
J. Mar. Sci. Eng. 2024, 12, 883 16 of 36
From Equation (24), defining the target velocity (3,5)=vid i and the velocity state
(3,5)
ii=v TT
5555553333
[,, , ,], [, , ]==vv
dddddddddd
uvwqr u wq and
TT
5555553333
[,, , ,], [, , ]==vvuvwqr uwq , the error vector can be calculated as follows:
T
555 55555
T
333 333
[,, , ,]
[, , ]
=− =
=− =
vvv
vvv
edeeeee
edeee
uvwqr
uwq (34)
To simplify the illustration, (3,5)=vie i represents the error vector of the 3DOF
controller and 5DOF controller, respectively. To depict each error (3,5)=vie i, we use
( 3,5)( 1,2,3 or 1,2,3,4,5)
ia
ei a== to depict the a-th row in ie
v. When 3i=,
1,2,3a=. When 5i=, 1,2,3,4,5a=. The following variables are following these rules.
In addition, a sliding mode surface ( 3,5)( 1,2,3 or 1,2,3,4,5)
ia
si a== is designed
individually to consider the error vectors as follows:
00
(())
ia
tt
r
ia ia ia ia ia ia ia
seCedtCsignee dt=+ +
(35)
where ia
C is the ath row in i
C constant Matrix and ia
r is the ath row in i
r constant Ma-
trix whose range is from 0 to 1.
From the dynamics model, the SMC law is designed as the following:
ˆ
() (3,5)
iiidiiiiii
i=−++−=τMv ε CvDvd
(36)
where ˆi
d is the estimation of the disturbances. Based on the theory in Lemma 2, the es-
timation model is depicted as follows:
0( ) 0( )
()
ˆ()
()
ii i i ii
t
KC t
ζ
ξζ
=+ +dMv
(37)
The detailed expression i
ξ
is depicted in Ref. [36]. i
ε is the sliding-mode-achieving
law which is derived as follows:
12
(()) (())
ii
rr
iiiei ieie iii ii
C C sign k k sign=+ ⋅++⋅ ⋅εv vv s ss
(38)
where 1i
k and 2i
k are two designed constant matrices. For further analysis, define ia
τ
and ia
ε
present the a-th row in i
τ and i
ε.
By employing the sliding mode control law i
τ, the velocity errors ie
v are globally
finite-time stable. The 3DOF sliding mode controller has the same design process as that
of the 5DOF sliding mode controller, and therefore will not be delineated in this section.
3.3.2. Stability Analysis
Theorem 2. The velocity tracking error (3,5)
ie i=v is globally finite-time stable with the utili-
zation of the proposed sliding mode velocity control law and the disturbance observer.
For simplicity, consider the first error vector of the 3DOF system, which is 31
e in
Equation (32). Constructing the following sliding mode surface 31
s and Lyapunov func-
tion 2
V for 31
e:
J. Mar. Sci. Eng. 2024, 12, 883 17 of 36
2
231
1
2
Vs= (39)
Differentiating the above function, the following form can be derived:
23131
Vss=
(40)
Based on the calculation results in Equation (35), 31
s
can be calculated as the fol-
lowing:
31
31 31 31 31 31 31 31
( () )
r
seCe Csignee=+ +
(41)
where 31
r stands for the first row in 3
r and 31
C denotes the first row in 3
C. Therefore,
Equation (40) is calculated as follows:
31
23131 3131 31 3131
( (()) )
r
Vse Ce Csignee=++
(42)
The dynamics model 31
e
is constructed as follows:
31 3 1 3 31 3 31 31 31 3 1
ˆ
()
d
euu
τ
=− − + + − −CD ddv
(43)
where 31
C, 31
D, 31
d, 31
ˆ
d, and 31d
v stand for the first row of 3
C, 3
D, 3
d, 3
ˆ
d, and
3d
v. Defining disturbance estimation error as 31 31 31
ˆ
=−ddd
, based on the dynamics
model, 2
V
can be further calculated as the following:
31
31
231313131 31 3131
31 31 3 31 3 31 31 3 1 31 31 31 31 31
( (()) )
= (( ) ( ( ) )
r
r
d
Vse Ce Csignee
suu CeCsignee
τ
=++
−− ++−+ +CD dv
(44)
From Equation (36), 31
τ
is calculated as
31 31 3 31 3 31 3 1 31
()
d
uu
τ
=++ −CDMvε
(45)
where, 31
ε, 31
M, and 31d
v
are the first row of 3
ε, 3
M and 31d
v
. Therefore, 2
V
can be
further calculated as follows:
31
31
31
23131 3131 31 3131
31 31 3 1 31 31 31 3 1 31 31 31 31 31
31 31 31 31 31 31 31 31 31
( (()) )
= (( ( ) )/ ( ( ) ) )
= ( / ( ( ) ) )
r
r
dd
r
Vse Ce Csignee
sCeCsignee
sCeCsignee
=++
−+ − + +
−+ + +
Mv ε d M v
εdM
(46)
Since 31 31
31 31 31 31 31 31 311 31 321 31 31
(()) (())
rr
Ce C signe e k s k signs s=+ ⋅++⋅ ⋅ε, the
above equation can be further calculated as
31 31
31
31
31+1
2 31 31 31 31 31 31 311 31 321 31 31
31 31 31 31 31 31 31
31 31 31 311 31 321 31 31
2
311 31 321 31 31 31
=( (()) (())
/ ( ( ) ) )
= ( / ( ( ) ) )
= /
rr
r
r
r
Vs Ce Csigne e ks k signs s
Ce C signe e
sksksignss
ks k s s
−− ⋅− −⋅ ⋅
+++
−−⋅ ⋅
−−⋅+
dM
dM
d
31
M
(47)
From the conclusion of Yang’s equality, we can obtain:
J. Mar. Sci. Eng. 2024, 12, 883 18 of 36
31+1
31+1
31+1
31+1 31+1
2
2 311 31 321 31 31 31 31
222
311 31 321 31 31 31 31
222
31 311 321 31 31 31 31
2
22
311 2 321 2 31 31
=/
1
( / )
22
11
= ( ) ( / )
222
1
= 2( ) 2 ( / )
22
r
r
r
rr
Vksks s
ks k s s
sk ks s
kVkV
α
α
α
αα
α
α
−−⋅+
≤− − ⋅ + +
−+ −⋅ + +
−− − +
dM
dM
dM
dM
(48)
Based on Lemma 2, we can draw the conclusion that 2
31 31
(/ )
2
α
dM
can converge to
zero in a prescribed time T. Therefore, 2
V
can be further calculated as:
31+1 31+1
22
2 311 2 321 2
1
=2( ) 2
2
rr
Vk V kV
α
−− −
(49)
Based on the conclusion from Lemma 1, 31
s can converge to zero within a pre-
scribed finite time 1
t by 31
τ
, and the expression of 1
t is shown as:
31+1 31+1
2
31+1
2
311 2
1
31+1 2
311
1
2( ) (0) 2
12
ln
1
2( )(1 ) 2
22
rr
rr
kV
tt r
k
α
α
−+
≤+
−−
(50)
From the above deduction, 31
s can converge to zero. When 31 0s=, the expressions
of 31
e
is shown as follows:
31
31 31 31 31 31 31
( () )
r
eCeCsignee=− −
(51)
Consequently, the construction of the second Lyapunov candidate function is as fol-
lows:
2
331
1
2
Ve= (52)
Combining the aforementioned calculation results and Young’s equality, the differ-
entiating results 2
V can be derived as follows:
31
31+1 31+1
33131
31 31 31 31 31 31
22
31 3 31 3
= [ ( ( ) ) ]
= 2 2
r
rr
Vee
eCe Csignee
CV CV
=
−−
−−
(53)
Based on the conclusion from Lemma 1, the velocity error 31
e can converge to zero
within a prescribed finite time by the proposed law 31
τ
, which can be calculated as the
following:
J. Mar. Sci. Eng. 2024, 12, 883 19 of 36
31+1 31+1
2
31+1
31+1 31+1
2
31+1
2
311 2
2
31+1 2
311
2
31 3 31
31+1 2
31 31
1
2( ) (0) 2
12
ln
1
2( )(1 ) 2
22
2(0)2
1
ln
2(1 ) 2
2
r
r
r
rr
r
r
kV
tt r
k
CV C
r
CC
α
α
−+
≤+
−−
+
+
−
(54)
Consequently, the conclusion can be deduced that the error 31
e convergence within
a prescribed finite time with the control law 31
τ
. Similarly, the other velocity errors
(3,5)=vie i can be proven to be finite-time stable with the corresponding control laws.
Thus, the stability of the dynamics system is established.
Taking into account the dual-loop controller discussed earlier, the overall closed-
loop stability is guaranteed when both the entire MPC controller and the dynamics ve-
locity system are asymptotically stable [42,48]. Thus, the stability of the dual-loop con-
trol framework in the diving stage and searching stage, which combines state-transform
MPC and SMC with the disturbance observer, is validated.
Remark 8. Referring to the conclusion in [27], it is evident that variations in parameters lead to
decreased tracking efficiency and slower convergence speeds. However, owing to the optimization
procedure, the controller autonomously harnesses the full extent of its control capability to gener-
ate the most effective tracking control aligned with the target velocity from MPC. Consequently,
the fusion of state-transform MPC and SMC with the disturbance observer not only alleviates the
substantial computational workload but also enhances tracking efficacy and accelerates conver-
gence speed.
4. Simulation
In this section, comparative simulation experiments are conducted to validate the
efficiency and superiority of the proposed controller. The simulation setup is outlined in
Section 4.1, followed by robustness testing and verification of the state-transform func-
tion in Sections 4.2 and 4.3, respectively. Finally, the trajectory following the experiment
is presented, comprising individual diving experiments and underwater searching tra-
jectory tracking experiments.
4.1. Simulation Introduction
Numerical simulations are conducted using both the 3DOF AUV model and the
5DOF AUV model. The REMUS AUV parameters are chosen, which are shown in Refs.
[38,48]. The main simulation parameters of 5DOF MPC and 3DOF MPC are depicted in
the following tables (Tables 1–3), respectively.
Table 1. Main simulation parameters.
0.9K= 5
p
N= 0.02
ϖ
= 0.1T=
Table 2. 5DOF controller parameters.
55555
5(10,10,10,10,10)
x
Qdiag= -4 -4 -4 -4 -4
5(10,10,10,10,10)
u
Qdiag= 50.05 (1,1,1,1,1)rdiag=⋅
T
(5)
ππ
200 200 200 2π 2π 10 10 10
20 25
MAX
X
=
T
(5)
ππ
200 200 200 2π 2π 10 10 10
20 25
MIN
X− − − −− −−−−
=
−
[
]
T
50.3 0.15 0.45 0.3 0.45C=
[
]
T
(5) 0.8 0.8 0.8 0.8 0.8
MIN
U−−−−=−
[
]
T
(5) 0.8 0.8 0.8 0.8 0.8
MAX
U= 52 0.05 (1,1,1,1,1)kdiag=⋅
J. Mar. Sci. Eng. 2024, 12, 883 20 of 36
51 0.4 (1,1,1,1,1)kdiag=⋅ 0(5) (25,25,25,25,25)Kdiag= 0(5) (1,1,1,1,1)Cdiag=
Table 3. 3DOF controller parameters.
555
3(10 ,10 ,10 )
x
Qdiag= -4 -4 -4
3(10 ,10 ,1 0 )
u
Qdiag= 30.05 (1,1,1)rdiag=⋅
T
(3)
π
200 200 2π 10 10 25
MAX
X
=
T
(3)
π
200 200 2π 10 10 25
MIN
X
=
−− − −
−−
[
]
T
30.3 0.45 0.3C=
[
]
T
(3) 0.8 0.8 0.8
MIN
U−− −=
[
]
T
(3) 0.8 0.8 0.8
MAX
U= 32 0.05 (1,1,1)kdiag=⋅
31 0.4 (1,1,1)kdiag=⋅ 0(3) (25,25,25)Kdiag= 0(3) (1,1,1)Cdiag=
The computing hardware utilized in the simulation includes the 12th Gen Intel Core
i7-12700H processor running at a clock speed of 2.30 GHz. The simulation tool employed
is MATLAB, utilizing FMINCON function for optimization. The initial state vector for all
subsequent simulation experiments is denoted by T
(0) (0.5 0 0 0 0)
η
=, and the initial ve-
locity vector is represented as T
(0) (0 0 0 0 0)v=. To evaluate the efficacy of the pro-
posed controller, we compare it against two alternative methods: the 5DOF controller
combined with both 5DOF-SMC and 5DOF-MPC, labeled as MPC-SMC, and the 5DOF-
MPC alone, labeled as MPC for brevity.
The primary performance metric employed in this study is the position tracking er-
ror, which represents the difference between the actual and desired locations in three-
dimensional space. The position tracking error at the thk sampling point is defined as
follows:
222
( ) ( ( ) ( )) ( ( ) ( )) ( ( ) ( ))
ddd
ek xk x k yk y k zk z k=−+− +− (55)
where ()
x
k,()yk , and ()zk are the actual position of AUV at the thktime. ()
d
x
k,
()
d
yk
, and ()
d
zk
are the target position of AUV at the thktime.
4.2. Robustness Test
To validate the trajectory tracking capabilities and anti-jamming resilience, robust-
ness testing is conducted using ocean current disturbances and ocean wave disturb-
ances, as described in [50]. The details of these disturbances, denoted as 1
D
and 2
D
, re-
spectively, are defined as follows:
12
0.5m/s 1.5 cos( ) m/s
0.3m/s 0.5 cos(2 ) m/s
0.5m/s cos( ) m/s
0.2m/s 1.3 cos( ) m/s
0m/s 0.2 cos( ) m/s
uu
vv
ww
qq
rr
ddt
ddt
DD
ddt
ddt
ddt
==⋅
==⋅
−
−
==
==
==⋅
−
==⋅
(56)
In the first experiment, the reference path is defined as follows:
4 sin(0.25 ) m
0.25 m
4 sin(0.25 ) 13 m
d
t
Yt
t−
⋅
=
⋅
(57)
Figure 8 illustrates the 3D-trajectory AUV tracking performance under the influence
of ocean current disturbances 1
D
and ocean wave disturbances 2
D
. As depicted in Fig-
ure 8, the AUV effectively tracks the reference path in real time despite the presence of
J. Mar. Sci. Eng. 2024, 12, 883 21 of 36
these disturbances. For a clearer assessment of the path tracking performance, Figure 9
presents the tracking performance along the YZ and XY axes with the negative effect of
the oceanic disturbances. The simulation underscores the robust anti-jamming capabili-
ties of the proposed controller.
Figure 8. The 3D trajectory tracking performance of robustness test.
(a)Following path in XY plane (b)Following path in YZ plane
Figure 9. The 2D trajectory tracking performance of robustness test.
To further assess the convergence speed of the proposed controller, we computed
the position tracking error and the error along each axis to evaluate the state conver-
gence performance. As depicted in Figures 10 and 11, the error converges to a stable
state within 3 s. These results indicate a strong ability to maintain state stability and ro-
bustness, aligning with the AUV’s requirement for strong anti-jamming capabilities. In
conclusion, the proposed control algorithm provides swift and precise trajectory track-
ing, while maintaining stability and robustness in the face of oceanic disturbances. In
complex underwater environments, the proposed method stands as a favorable choice
for AUV underwater search tasks.
J. Mar. Sci. Eng. 2024, 12, 883 22 of 36
Figure 10. The overall trajectory tracking error in robustness test.
(a)Tracking error in X axis of robustness test
(b) Tracking error in Y axis of robustness test
J. Mar. Sci. Eng. 2024, 12, 883 23 of 36
(c) Tracking error in Z axis of robustness test
Figure 11. The trajectory tracking error in the X, Y, and Z axis in robustness test.
4.3. Transform Method Test
To mitigate the overshoot of the AUV and demonstrate its superiority, a state-
transform method is proposed to complement the proposed approach, which is validat-
ed through the subsequent simulation experiment. The second experiment aims to simu-
late the scenario in which the AUV descends to the target depth and conducts the search
tasks. Precision and accuracy in controlling the motion and direction of the AUV are es-
sential for this task. Moreover, minimizing the overshoot of the controller is crucial to
enhance the effectiveness of the control system. The desired sailing path in the second
experiment is defined as the following:
12
4 m 4 sin(0.25 ) m
0 m m
20 m 4 sin(0.25 )m
dd
tt
YY
t
t
⋅⋅
==
−⋅
−−
(58)
where 1d
Y and 2d
Y denote the desired path in diving stage and searching stage, respec-
tively. The 3D tracking results and 2D tracking results in the YZ and XY axis are shown
in Figure 12 and Figure 13, respectively.
Figure 12. The 3D trajectory tracking of the transform method.
J. Mar. Sci. Eng. 2024, 12, 883 24 of 36
(a) Trajectory tracking of YZ plane of the transform
method test
(b) Trajectory tracking of XY plane of the transform
method test
Figure 13. The 2D trajectory tracking performance of the transform method.
Based on the aforementioned experiment results, the transform method demon-
strates a significant reduction in overshoot, particularly in the Z direction. By prompting
the controller to anticipate the desired trajectory, the transform method enhances both
the efficiency and accuracy of calculations, leveraging the higher precision of the 5DOF
controller. For a more intuitive understanding of tracking performance, we calculated
the position tracking error and the error in each axis to assess state convergence. The
overall tracking error and the errors in the X, Y, and Z axis are depicted in Figure 14 and
Figure 15, respectively. These figures illustrate smaller tracking errors and reduced over-
shoot, particularly evident in the Z axis, compared to the algorithm without the trans-
form method, thereby aligning with the control requirements of underwater searching
tasks.
With the presented work, the overshoot observed in the controller without the
transform method is significantly larger compared to the controller with the transform
method. This indicates that the transform method effectively enhances tracking efficien-
cy and precision. In real-world AUV underwater searching scenarios, the effectiveness of
the transform method is validated, demonstrating its feasibility and practical utility.
Figure 14. The overall trajectory tracking error in transform method test.
J. Mar. Sci. Eng. 2024, 12, 883 25 of 36
(a) Tracking error in X axis of transform method test.
(b) Tracking error in Y axis of transform method test.
(c) Tracking error in Z axis of transform method test.
Figure 15. The trajectory tracking error in the X, Y, and Z axis in transform method test .
4.4. Underwater Research Task Simulation Experiment
The final segment of this study involves simulating the effectiveness and the con-
sumed optimization time during underwater research tasks. This experiment comprises
individual diving experiments and underwater research trajectory tracking experiments.
The individual diving experiments incorporate two desired trajectories: a straight line
and a sinusoidal wave. These reference trajectories are represented by 1d
R
and 2d
R
, re-
spectively.
J. Mar. Sci. Eng. 2024, 12, 883 26 of 36
12
4 m 4 sin(0.25 ) m
0 m m
m 4 sin(0.25 )m
dd
tt
RR
t
tt
⋅⋅
==
⋅
−
−
(59)
The overall tracking results 1d
R are illustrated in Figure 16. The overall tracking er-
ror and X- and Z-axis tracking error are shown in Figure 17 and Figure 18, respectively.
The overall tracking results 2d
R are shown in Figure 19, and the overall tracking error
and tracking error in the X and Z axis are displayed in Figure 20 and Figure 21, respec-
tively. The optimization time comparison of the two situations’ results are illustrated in
Figure 22 and Figure 23, respectively.
Figure 16. The trajectory tracking of the line.
Figure 17. The tracking error of the line.
J. Mar. Sci. Eng. 2024, 12, 883 27 of 36
(a) Tracking error in X axis of straight line.
(b) Tracking error in Z axis of straight line.
Figure 18. The line tracking error in the X and Z axis.
J. Mar. Sci. Eng. 2024, 12, 883 28 of 36
Figure 19. The trajectory tracking of sine wave.
Figure 20. The overall tracking error of sine wave.
(a) Tracking error in X axis of the sine wave (b) Tracking error in Z axis of the sine wave
Figure 21. The tracking error of sine wave in the X and Z axis.
J. Mar. Sci. Eng. 2024, 12, 883 29 of 36
(a) Overall optimzing time of the straight line (b) Avergae optimizing time of the straight line
Figure 22. The optimization time of straight line.
(a) Overall optimizing time of the sine wave (b) Average optimizing time of the sine wave
Figure 23. The optimization time of sine wave.
To further determine the advantages of the proposed controller, the underwater
searching task is simulated, with the reference path in the underwater searching experi-
ment defined as the following:
12
2 m 4sin(0.5) m
0 m m
20 m 4 sin(0.5 )m
dd
tt
TT
2t
t
⋅⋅
==
−−
−−
⋅
⋅
(60)
where 1d
T and 2d
T denote the desired path in the diving stage and searching stage. The
3D- and 2D-trajectory tracking simulation results are shown in Figure 24 and Figure 25,
respectively. The tracking error results are illustrated in Figure 26 and Figure 27, respec-
tively. The optimizing time comparison between different controllers is illustrated in
Figure 28.
J. Mar. Sci. Eng. 2024, 12, 883 30 of 36
Figure 24. The 3D trajectory tracking of underwater searching.
(a) Tracking performance of YZ axes in underwater search-
ing
(b) Tracking performance of XY axes in underwater
searching
Figure 25. The 2D trajectory tracking of underwater searching.
J. Mar. Sci. Eng. 2024, 12, 883 31 of 36
Figure 26. The overall tracking error of underwater searching.
(a) Tracking error in X axis of underwater searching
(b) Tracking error in Y axis of underwater searching
(c) Tracking error in Z axis of underwater searching
Figure 27. The tracking error of underwater searching in 3-axis.
J. Mar. Sci. Eng. 2024, 12, 883 32 of 36
(a) Overall optimizing time of underwater searching (b) Average optimizing time of underwater searching
Figure 28. The optimizing time comparison.
From the aforementioned experiments, several conclusions can be drawn. Firstly,
the proposed controller exhibits higher precision and faster convergence speed. Leverag-
ing the combination of state-transform MPC and SMC with the disturbance observer, the
SMC with the disturbance observer effectively utilizes the optimization results from
MPC, resulting in enhanced convergence speed. Additionally, the incorporation of SMC
with the disturbance observer enables faster convergence and stronger robustness com-
pared to conventional MPC algorithms.
Moreover, the proposed method demonstrates a notable reduction in optimization
time across different trajectory-following scenarios. Specifically, in line-following exper-
iments, the proposed method shows an average reduction in optimization time of ap-
proximately 31% and 48% compared to alternative methods. Similarly, in the sine-wave-
following experiments, the proposed method achieves an average reduction in optimiza-
tion time of about 27% and 38% compared to alternative methods. Furthermore, in the
final simulation experiment, the proposed method exhibits an average reduction in op-
timization time of about 24% and 30% compared to alternative methods. The decrease in
optimization time highlights the effectiveness of the proposed controller in mitigating
the computational burden typically associated with MPC.
Despite the increased complexity of reference paths, the proposed method outper-
forms alternative approaches, demonstrating rapid and stable error convergence while
maintaining robustness and stability in the presence of complex oceanic disturbances. In
real-world underwater searching scenarios, the proposed method significantly enhances
control performance while meeting requirements for high robustness and strong anti-
jamming ability. Overall, the proposed method represents a more efficient control strate-
gy for various underwater application environments, offering considerable potential for
use in a wide range of AUV underwater searching tasks.
5. Conclusions
This article presents a controller that combines state-transform MPC and SMC to
enhance the precision of path-tracking, improve tracking efficiency, and reduce the op-
timization time for AUV underwater searching tasks amidst complex oceanic disturb-
ances. Initially, a real-AUV experiment validates the ability of a cross-rudder AUV with
strong straight sailing to maintain a sailing angle without direction control during the
diving stage. Building upon this, a state-transform MPC guide law algorithm is pro-
posed in the kinematics loop to compute the target velocity for dynamics calculation.
This reduction in degrees of freedom significantly reduces optimization time during the
diving stage while ensuring kinematics tracking accuracy, addressing a prevalent issue
in AUV MPC trajectory tracking control.
J. Mar. Sci. Eng. 2024, 12, 883 33 of 36
The dynamics controller employs an SMC with the disturbance observer to fully
exploit the advantages of the MPC guide law and enhance anti-jamming ability in the
presence of oceanic disturbances, thereby improving tracking precision and convergence
rate. The controller stability is demonstrated through the existence of a feasible solution
sequence of the MPC and the selection of a cost function as a Lyapunov function, estab-
lishing the asymptotic stability of the nominal system. Furthermore, by employing fi-
nite-time stable theory, the dynamics velocity tracking error is proven to converge to ze-
ro within a finite time.
A comparison of simulation results among the proposed controller, a 5DOF control-
ler, and a conventional MPC controller emphasizes the efficacy of the proposed ap-
proach. The first experiment confirms precise tracking control in the presence of diverse
oceanic disturbances. The second experiment highlights the effectiveness of the trans-
form method. The final experiment demonstrates advantages in optimization time re-
duction, tracking accuracy, and robustness.
However, limitations exist, such as reliance on AUV straight sailing ability and sen-
sor accuracy. X-rudder AUVs may not meet control requirements due to coupling effects
from different rudders. Additionally, the efficacy of the proposed control tactics may be
undermined in situations involving malfunctioning sensors. Future research directions
could explore optimization time reduction for X-rudder AUVs, fault-tolerant control un-
der sensor faults, and a combination of soft and hard functions in various task scenarios.
Additionally, real-world AUV experiments can be conducted in more complex environ-
ments, such as lakes or seas. Moreover, enhancing the robustness of MPC control for
AUVs under such conditions is a key concern, which can be addressed through the inte-
gration of a novel disturbance observer.
Author Contributions: Conceptualization, H.H. and G.X.; Data curation, Z.Y. and J.L.; Formal
analysis, H.H. and Y.X.; Funding acquisition, Y.X.; Investigation, H.H. and K.X.; Methodology, H.H.
and J.L.; Project administration, Y.X. and K.X.; Resources, K.X.; Software, H.H. and J.L.; Supervi-
sion, G.X. and Y.X.; Visualization, Z.Y.; Writing—original draft, H.H.; Writing—review and editing,
H.H., G.X. and Y.X. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by [National Natural Science Foundation of China] grant
number [52001132].
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are contained within the article.
Conflicts of Interest: Author Kan Xu was employed by the company Wuhan Second Ship Design
and Research Institute. The remaining authors declare that the research was conducted in the ab-
sence of any commercial or financial relationships that could be construed as a potential conflict of
interest.
Glossary
The primary abbreviation.
Abbreviation Definition
AUV Autonomous underwater vehicles
SMC Sliding mode controller
MPC Model predictive controller
DOF Degree of freedom
LTI Linear time-invariant
Nomenclature Definition
(3,5)
ii
η
= Position Vector
(3,5)
ii
η
′= Derivation of Position Vector
,, ,,uvwqr
Surge, Sway, Heave, Pitch, and Yaw
J. Mar. Sci. Eng. 2024, 12, 883 34 of 36
(3,5)
ii=v Velocity Vector
,,,,xyz
θ
ϕ
Position Variables in each DOF
(3,5)
ii=τ Input Vector
(3,5)
ii=d Environmental Disturbances Vector
(3,5)=
id
pi Desired Position Vector
(3,5)
i
pi= Position Vector
(3,5)
ie
Pi= Position Error Vector
(3,5)
id i=v Desired Velocity Vector
(3,5)
ii=v Velocity Vector
(3,5)=vie i Velocity Error Vector
()
iku Optimal Input Sequence
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