1 A Fuzzy Logic-based Cascade Control without Actuator Saturation for the Unmanned Underwater

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A Fuzzy Logic-based Cascade Control without

Actuator Saturation for the Unmanned Underwater

Vehicle Trajectory Tracking

Danjie Zhu, Simon X. Yang and Mohammad Biglarbegian

Abstract—An intelligent control strategy is proposed to elimi-

nate the actuator saturation problem that exists in the trajectory

tracking process of unmanned underwater vehicles (UUV). The

control strategy consists of two parts: for the kinematic modeling

part, a fuzzy logic-reﬁned backstepping control is developed to

achieve control velocities within acceptable ranges and errors

of small ﬂuctuations; on the basis of the velocities deducted by

the improved kinematic control, the sliding mode control (SMC)

is introduced in the dynamic modeling to obtain corresponding

torques and forces that should be applied to the vehicle body.

With the control velocities computed by the kinematic model and

applied forces derived by the dynamic model, the robustness and

accuracy of the UUV trajectory without actuator saturation can

be achieved.

Index Terms—backstepping control, actuator saturation, fuzzy

logic, sliding mode control, speed jump, trajectory tracking,

unmanned underwater vehicle.

I. INTRODUCTION

TO take advantages of the abundant resources embedded

in the ocean area, such as mineral resources, biological

resources and space resources, technologies that relate to the

underwater exploration have been studied for decants [1]. Due

to the complex environmental factors of the deep-water space,

such as the high pressure, invisibility or the unpredictable

obstacles, unmanned underwater vehicles (UUV) are applied

in most undermarine operation cases to guarantee the safety

and efﬁciency [2, 3]. Therefore, achieving the robustness and

accuracy of controlling the UUV to track the desired trajectory

in the desired time, is of great importance for completing the

underwater operation [4–6].

However, as the UUV cannot provide inﬁnite driving in-

puts such as torques/forces due to its underwater workspace

and limited electric power, the actuator saturation has to

be considered during the trajectory tracking process of the

vehicle[7–9], with the torques/forces constraints applied. The

actuator saturation is induced by the speed-jump problem,

which usually occurs in some conventional control methods

for trajectory tracking, like the backstepping control [10, 11].

The speed jumps negatively affects the robustness of the UUV

trajectory tracking, by introducing excessive ﬂuctuations of

velocities at initial states or other large error states during the

This work was supported by the Natural Sciences and Engineering Research

Council (NSERC) of Canada. (Corresponding author: Simon X. Yang.) The

authors are with Advanced Robotics and Intelligent System (ARIS) Laboratory

and Advanced & Intelligent Control for Vehicles (AICV) Laboratory, School

of Engineering, University of Guelph, Guelph, ON. N1G2W1, Canada (e-mail:

{danjie; syang; mbiglarb}@uoguelph.ca).

kinematic controlling procedure. In the backstepping method,

control functions for each subsystem are designed based on

the Lyapunov techniques, and generated to form the complete

control law [12]. Therefore speeds of large ﬂuctuations are

derived by the large errors accumulated from the generation

of the subsystems, where speed-jump issues are induced when

the deviation occurs.

Many methods have been used for alleviating the speed-

jump problem in the trajectory tracking for vehicles, where the

model predictive control (MPC) is one of the most commonly

applied methods. The MPC resolves the online optimization

problem at each time step and derives in-time predictions with

minimum errors [13, 14]. However, MPC usually consumes

long time in computation due to its recursive algorithm with

increasing complexity [15]. In this study, the fuzzy logic are

introduced to provide low complexity by translating the goals

in a transparent way [16]. The fuzzy logic system is used as the

function approximator to address the uncertainties, and gives

more ﬂexiable criterion for obtaining the optimized predictions

within its conceptual framework [17, 18]. It can also makes

limitation on the output data, and smoothen the kinematic

error curves derived by the conventional backstepping method

through its decision function. Compared to the MPC, the fuzzy

logic controller constructs a model that imitates the human

decisionmaking with inputs of continuous values between 0

and 1, which largely simpliﬁes the computing process [19, 20].

Practically, as UUV driving commands are directly given by

the dynamic inputs, the component that extents the kinematic

to dynamic tracking is cascaded as a part of the controller

designed in this study [21, 22]. Tiny deviation caused by

the speed-jump problem leads to the inevitable errors in

the dynamics of the tracking process, where the UUV may

produce excessive torques/forces at the jump points. The

controller designed for the dynamic model is to compute

the corresponding torques/forces that directly applied to the

vehicle to eliminate the errors created during the tracking

procedure, which offers an accurate operating instruction to

the diving vehicle [22–24]. In this paper, the sliding mode

control (SMC) is chosen to construct the complete intelligent

controller [25–30]. As one of the most basic adaptive con-

trolling strategies, the SMC is widely used due to its simple

and robust mechanism [31, 32]. A surface mode is supposed to

follow the desired tracking and keep the control outputs remain

on the surface. Once the trajectory under the control is out

of the perfect surface, the SMC will push the trajectory slide

back to the surface with addition or subtraction on the original

arXiv:2210.01706v1 [cs.RO] 4 Oct 2022

controlling equation [33, 34]. Therefore the SMC restricts the

ﬂuctuation of control outputs in an acceptable range through a

simple operation, which is highly applicable in the trajectory

tracking problems[35, 36].

Zhang and his colleagues have tried to resolve the speed-

jump problem in the trajectory tracking, but higher complexi-

ties are introduced [37]. Some researchers have achieved suc-

cessful tracking based on the fuzzy logic-reﬁned backstepping

method yet their application is based on the underactuated

surface vehicle (USV), with fewer states involved compared to

the UUV [38, 39]. Some researchers have applied synergetic

learning in their controller designed for vehicles and better

performance is obtained, but they do not consider practical

constraints of the vehicle [40]. Li has developed the fuzzy

logic-based controller that provides satisfactory tracking re-

sults even with time-varying delays or input saturation, but the

effectiveness of the algorithm on speciﬁc models such as the

UUV has not been discussed [41, 42] Wang and his colleagues

developed a fuzzy logic-based backstepping method yet it has

not been experimented under speciﬁc application scenarios,

with dynamic constraints applied [43].

Motivated by the requirement of resolving the actuator satu-

ration (thrusters’ dynamic constraints) through the elimination

of the speed jumps that exist in the conventional trajectory

control of the UUV, this paper focuses on the speed-jump as

well as the actuator saturation problems in the practical UUV

system. Due to the uncertainty of the underwater environment,

high adaptiveness and low complexity are needed to achieve

a robust trajectory tracking controller that is easy to realize.

Therefore, the fuzzy logic, the backstepping method and

the SMC are combined to construct a cascade intelligent

control. The ﬁrst two components form the kinematic velocity

controller, where the fuzzy logic system helps to resolve

the speed-jump problem of the backstepping method when

controlling the kinematic model. The SMC is constructed

as the dynamic torque controller, extending the application

of the whole design for UUV trajectory tracking in actual

cases, meanwhile the physical constraints can be introduced

in this part. Based on the shunting characteristics of this

control strategy, the outputs are bounded in a ﬁnite interval

within the vehicle’s physical constraints and results of small

ﬂuctuations are performed even when abrupt inputs are given.

The contribution of the cascade control method is supposed

to resolve the actuator saturation problem and provides satis-

factory tracking results in practical cases of UUV navigation

through a simple computation. Moreover, the problem of

navigation under uncertainties in stochastic environments is

considered due to their impacts on the vehicle motion [44, 45].

The rest of the paper is organized as follows. First, the basic

models of the UUV system are introduced, kinematic model

and dynamic model are deﬁned with their corresponding for-

mulas. The speciﬁc modeling process shows how the trajectory

tracking control works in the complex system. Next, the

fuzzy logic-reﬁned backstepping control and the sliding mode

control designed for the UUV trajectory tracking problem are

illustrated, where the mechanism and operating process are

explained in details. The ﬁnal part presents the direct results

output by the simulation, and further analysis is performed to

demonstrate the effectiveness of the reﬁned trajectory tracking

controller with dynamic constraints applied and environmental

noise involved.

II. ROBOT MODELS AND PROBLEM STATEMENT

In this section, a typical type of UUV named ”Falcon”

is studied. Its robot models and trajectory tracking problem

descriptions are given in the form of speciﬁc equations .

A. Robot models of the ”Falcon UUV”

In this subsection, the kinematic and dynamic models of

the “Falcon” UUV are given, both of which are involved in

the trajectory tracking control of a UUV. Parameters of the

“Falcon” UUV are introduced to clearly address the trajectory

tracking problem and its corresponding solution studied in this

paper.

1) Kinematic Model: The systematic analysis of UUV is

established on two basic 3D reference frames, the world-ﬁxed

frame (W), originating from a point on the surface of the earth;

and the body-ﬁxed frame, originating from the UUV body.

Directions of axes of the two reference frames are given in

Fig. 1. Among the six freedoms of the UUV, surge, sway,

heave, roll, pitch and yaw, roll and pitch can be eliminated

when establishing the trajectory model to keep a controllable

operation of the UUV during the diving process. Specially

for the UUV type “Falcon” applied in this article, the design

of the vehicle does not allow the roll and pitch movements

while only surge, sway, heave and yaw movements can be

achieved (see bold DOFs in Fig. 1) [46]. Therefore, for the

kinematic equation of “Falcon” UUV, the velocity vector vcan

be transformed into the time derivative of trajectory vector ˙p

˙p =





˙x

˙y

˙z





=J(p)v=





cos ψ−sin ψ0 0

sin ψcos ψ0 0

0 0 1 0

0 0 0 1





v

=





cos ψ−sin ψ0 0

sin ψcos ψ0 0

0 0 1 0

0 0 0 1















,(1)

where Jis a transformation matrix derived from the physical

structure of the UUV body, [u v w r]Trepresents the velocities

at the chosen four axes of the UUV (see Fig. 1) [47].

2) Dynamic Model: In an actual UUV system, several

complex and nonlinear forces such as hydrodynamic drag,

damping, lift forces, Coriolis and centripetal forces, gravity

and buoyancy forces, thruster forces, and environmental distur-

bances are acting on the vehicle. Considering the origins and

effect of the forces, a general dynamic model can be written

M ˙v +C(v)v+D(v)v+g(p) = τ,(2)

where M is the inertia matrix of the summation of rigid body

and added mass; C(v)is the Coriolis and centripetal matrix

of the summation of rigid body and added mass; D(v)is the

quadratic and linear drag matrix; g(p)is the matrix of gravity

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1AFuzzyLogic-basedCascadeControlwithoutActuatorSaturationfortheUnmannedUnderwaterVehicleTrajectoryTrackingDanjieZhu,SimonX.YangandMohammadBiglarbegianAbstractAnintelligentcontrolstrategyisproposedtoelimi-natetheactuatorsaturationproblemthatexistsinthetrajectorytrackingprocessofunmannedunderwaterveh...

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