Singularity-free Formation Path Following of Underactuated AUVs Extended Version

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Singularity-free Formation Path

Following of Underactuated AUVs:

Extended Version

Josef Matouˇs ∗Kristin Y. Pettersen ∗Damiano Varagnolo ∗,∗∗

Claudio Paliotta ∗∗∗

∗Department of Engineering Cybernetics, Norwegian University of

Science and Technology, Trondheim (name.surname@ntnu.no).

∗∗Department of Information Engineering, University of Padova, Italy.

∗∗∗ SINTEF Digital, Trondheim, Norway (claudio.paliotta@sintef.no).

Abstract This paper proposes a method for formation path following control of a ﬂeet

of underactuated autonomous underwater vehicles. The proposed method combines several

hierarchic tasks in a null space-based behavioral algorithm to safely guide the vehicles. Compared

to the existing literature, the algorithm includes both inter-vehicle and obstacle collision

avoidance, and employs a scheme that keeps the vehicles within given operation limits. The

algorithm is applied to a six degree-of-freedom model, using rotation matrices to describe the

attitude to avoid singularities. Using the results of cascaded systems theory, we prove that the

closed-loop system is uniformly semiglobally exponentially stable. We use numerical simulations

to validate the results.

Keywords: autonomous underwater vehicles, multi-vehicle systems, guidance, path following,

stability of nonlinear systems

1. INTRODUCTION

Autonomous underwater vehicles (AUVs) are being in-

creasingly used in a number of applications such as

transportation, seaﬂoor mapping, and other ocean energy

industry-related tasks. It is often advantageous to perform

such tasks with a group of cooperating AUVs. Therefore,

there is a need for algorithms that can safely guide a

formation of AUVs along a given path while avoiding

collisions with each other and obstacles, and staying within

given operation limits.

As presented in Das et al. (2016), there exists a plethora

of formation path-following methods, most of them based

on two concepts: coordinated path-following (Borhaug and

Pettersen, 2006; Ghabcheloo et al., 2006) and leader-

follower (Cui et al., 2010; Soorki et al., 2011). In the

coordinated path-following approach, each vehicle follows

a predeﬁned path separately. Formation is then achieved

by coordinating the motion of the vehicles along these

paths. In this approach, the formation-keeping error (i.e.,

the diﬀerence between the actual and desired relative

position of the vehicles) may initially grow as the vehicles

converge to their predeﬁned paths. In the leader-follower

approach, one leading vehicle follows the given path while

the followers adjust their speed and position to obtain

the desired formation shape. This latter approach tends

to suﬀer from the lack of formation feedback due to

1This work was partly supported by the Research Council of

Norway through project No. 302435 and the Centres of Excellence

funding scheme, project No. 223254.

2The authors would like to thank Aurora Haraldsen for the discus-

sions on the collision cone concept.

unidirectional communication (i.e., the leader may not

adjust its velocity based on the followers).

Another formation path-following algorithmic paradigm is

the so-called null-space-based behavioral (NSB) approach

(Arrichiello et al., 2006; Antonelli et al., 2009; Pang et al.,

2019; Eek et al., 2021), a centralized strategy that allows to

combine several hierarchic tasks. In the NSB framework,

the control objective is expressed using multiple tasks.

By combining these simple tasks, the vehicles exhibit the

desired complex behavior.

This paper aims to extend our previous NSB algorithm

(Matouˇs et al., 2022) to control a ﬂeet of AUVs. The

previous work uses a ﬁve degree-of-freedom (5DOF) AUV

model, considers only inter-vehicle collision avoidance, and

proves only the stability of the path-following algorithm.

Furthermore, the orientation of the 5DOF model was

expressed using Euler angles, which causes singularities for

a pitch angle of ±90 degrees. This work applies the NSB

algorithm to a full 6DOF model, uses rotation matrices to

describe the attitude of the vehicles to avoid singularities,

modiﬁes and extends the tasks, and proves the stability of

the combined path-following and formation-keeping tasks.

We also add a scheme that keeps the vehicles within a given

range of depths to stay within the operation limits. As

opposed to the previous work, we do not limit the analysis

to a speciﬁc low-level attitude controller. Consequently,

the new algorithm can be integrated into existing on-board

controllers. Assuming that the existing low-level controller

allows exponential tracking, we use results from cascaded

systems theory (Pettersen, 2017) to prove that the closed-

arXiv:2210.14646v3 [eess.SY] 5 Apr 2023

loop system composed by the NSB algorithm and the

low-level controller is uniformly semiglobally exponentially

stable. We verify the results in numerical simulations.

The remainder of the paper is organized as follows. Sec-

tion 2 introduces the model of the AUVs. Section 3 deﬁnes

the formation path-following problem. Section 4 describes

the proposed modiﬁed NSB algorithm. The stability of

the closed-loop system is proven in Section 5. Section 6

presents the results of the numerical simulations. Finally,

Section 7 presents some concluding remarks.

2. THE AUV MODEL

To simplify the notation, we will denote a concatenation

of vectors or scalars using angled brackets, e.g.,

hx1,...,xNi=xT

1,...,xT

NT.(1)

Let p=hx, y, zibe the position, R∈SO(3) the rotation

matrix describing the orientation, v=hu, v, withe linear

surge, sway and heave velocities, and ω=hp, q, rithe

angular velocity of the vehicle. For brevity, let us also

deﬁne the velocity vector ν=hv,ωi.

Furthermore, let Vc=hVx, Vy, Vzibe the velocities of an

unknown, constant and irrotational ocean current, given in

the inertial frame, and vc=huc, vc, wcithe ocean current

velocities expressed in the body-ﬁxed coordinate frame

vc=RTVc.(2)

We will denote the relative linear velocities of the vehicle as

vr=v−vc. We will also denote the relative surge, sway

and heave velocities as ur,vrand wr, and the relative

velocity vector as νr=hvr,ωi.

Let f=hTu,δibe the vector of control inputs, where

Tuis the surge thrust generated by the propeller, and δ

represents the conﬁguration of ﬁns. Furthermore, let Mbe

the mass and inertia matrix, including added mass eﬀects,

C(νr) the Coriolis centripetal matrix, also including added

mass eﬀects, and D(νr) the hydrodynamic damping ma-

trix. The dynamics of the vehicle in a matrix-vector form

are then (Fossen, 2011) ˙

p=Rv,(3a)

R=RS(ω),(3b)

M˙

νr+C(νr) + D(νr)νr+g(R) = Bf,(3c)

where g(R) is the gravity and buoyancy vector, Bis the

actuator conﬁguration matrix that maps the control inputs

to forces and torques, and S:R37→ so(3) is the skew-

symmetric matrix operator.

Note that (3c) describes the dynamics of a generic un-

derwater rigid body. In the remainder of this section, we

will derive a more speciﬁc model for an AUV. First, let us

present the necessary assumptions about the vehicle.

Assumption 1. The vehicle is slender, torpedo-shaped,

with port-starboard and top-bottom symmetry.

Assumption 2. The hydrodynamic damping is linear.

Assumption 3. The vehicle is neutrally buoyant, with the

center of gravity (CG) and the center of buoyancy (CB)

located along the same vertical axis.

Assumption 4. The origin of the body-ﬁxed frame is cho-

sen such that actuators produce no sway and heave accel-

eration. In other words, there exist fu, tp, tq, trsuch that

M−1Bf =hfu,0,0, tp, tq, tri.(4)

Remark. the mechanical design of typical commercial sur-

vey AUVs satisﬁes Assumptions 1 and 3. Assumption 2

is valid for low-speed missions and is often used as a

simpliﬁcation also when designing controllers for higher-

speed missions, as the higher-order damping coeﬃcients

are poorly known, and compensating for these may reduce

the robustness of the control system. The general structure

of M,C(·), D(·), and g(·) for vehicles that satisfy Assump-

tions 1–3 is shown, e.g., in Fossen (2011). In Borhaug et al.

(2007), it is shown that if a 5DOF vehicle model with port-

starboard symmetry satisﬁes Assumptions 2–3, the origin

of the body-ﬁxed coordinate frame can always be chosen

such that Assumption 4 holds. By assuming top-bottom

symmetry, the roll dynamics are decoupled from the rest of

the system. Consequently, the procedure demonstrated in

Borhaug et al. (2007) can be trivially extended to 6DOFs.

Assumption 5. The vehicle is equipped with a low-level

controller that allows exponential tracking of the surge

velocity, orientation, and angular velocity. Speciﬁcally, let

ud,Rdand ωdbe the reference signals. We deﬁne an error

X=Du−ud,logme

R,ω−e

RTωdE,e

R=RT

dR,(5)

where logm : SO(3) 7→ R3is the matrix logarithm (Iserles

et al., 2000). Note that by Assumption 4, e

Xis controllable

through the input f. Consider the closed-loop system

X=Fe

X, v, w, Vc,(6)

consisting of (3b), (3c), and the low-level controller. We

assume that e

X=0is a globally exponentially stable

(GES) equilibrium of (6).

Remark. The aim of this paper is to demonstrate that

the proposed formation path-following algorithm can be

readily implemented on vehicles with existing low-level

controllers. Consequently, the choice of a low-level velocity

and attitude controller is not discussed in this paper.

An example of a global exponential attitude tracking

controller can be found, e.g., in ?.

Note that for a complete system analysis, we need to

consider the underactuated sway and heave dynamics

explicitly. Under Assumptions 1–4, the underactuated

dynamics have the following form

˙v=Xv(ur)r+Yv(ur)vr+Zv(p)wr+ ˙vc,(7a)

˙w=Xw(ur)q+Yw(ur)wr+Zw(p)vr+ ˙wc,(7b)

where X(·), Y (·), Z(·) are aﬃne functions of the respective

variables. From (2), it follows that

vc=h˙uc,˙vc,˙wci=vc×ω,(8)

where ×denotes the vector cross product.

3. FORMATION PATH FOLLOWING

The goal is to control a ﬂeet of nAUVs so that they move

in a prescribed formation and their barycenter follows a

given path.

The prescribed path in the inertial coordinate frame is

given by a smooth function pp:R7→ R3. We assume that

pp(ξ)≡Op

Figure 1. Deﬁnition of the path angles and path-tangential

coordinate frame. Odenotes the origin of the inertial

coordinate frame, Opdenotes the origin of the path-

tangential frame.

pp(ξ)

b≡Of

f,1

f,2

f,n

Figure 2. Deﬁnition of the formation. Ofdenotes the

origin of the formation-centered coordinate frame.

the path function is C∞and regular, i.e., the function is

continuously diﬀerentiable and its partial derivative with

respect to ξsatisﬁes 



∂pp(ξ)

∂ξ 



6= 0. Therefore, for every

point pp(ξ) on the path, there exists a path-tangential

coordinate frame (xp, yp, zp) and a corresponding rotation

matrix Rp(see Figure 1).

The path-following error pp

bis given by the position of the

barycenter in the path-tangential coordinate frame

b=RT

ppb−pp(ξ),pb=1

i=1

pi.(9)

The goal of path following is to control the vehicles so that

b≡03, where 03is a 3-element vector of zeros.

To deﬁne the formation-keeping problem, we ﬁrst deﬁne

the formation-centered coordinate frame. This coordinate

frame is created by translating the path-tangential frame

into the barycenter (see Figure 2). Let pf

f,1,...,pf

f,n be

the position vectors that represent the desired formation.

From Figure 2, one can see that these vectors are constant

in the formation-centered frame. Furthermore, the mean

value of pf

f,i must coincide with the barycenter. Since the

barycenter is equivalent to the origin of the formation-

centered frame, the vectors must thus satisfy

i=1

f,i =03.(10)

The position of vehicle iin the formation-centered frame

is then given by

i=RT

p(pi−pb).(11)

The goal of formation keeping is to try to maintain pf

i≡

f,i independently of the disturbances experienced by the

agents. This problem can also be expressed in the inertial

coordinate frame as

pi≡Rppf

f,i +pb, i ∈ {1, . . . , n}.(12)

4. CONTROL SYSTEM

The AUVs must perform the goals stated in Section

3 safely, i.e., avoid collisions with other vehicles and

obstacles, and remain within a given range of depths. An

upper limit on the depth of the AUVs is needed to prevent

them from colliding with the seabed or exceeding their

depth rating. A lower limit is needed in busy environments

(e.g., harbors), where the AUVs may otherwise collide or

interfere with surface vessels.

To solve the formation path following problem, we propose

a method that combines inter-vehicle collision avoidance

(COLAV), formation keeping, line-of-sight (LOS) path

following, obstacle avoidance, and depth limiting in a

hierarchic manner using an NSB algorithm. Since the NSB

algorithm outputs inertial velocity references, we also need

a method for converting these to surge and orientation.

In this section, we ﬁrst present the NSB algorithm and

the associated tasks. We then present in Section 4.6

a strategy for converting inertial velocity references to

surge/orientation ones.

4.1 NSB algorithm

The NSB algorithm allows us to deﬁne and combine mul-

tiple tasks in a hierarchic manner. For more information,

the reader is referred to Antonelli and Chiaverini (2006).

Achieving the desired behavior requires three tasks:

COLAV, formation-keeping, and path-following. Each task

will be described in detail in Sections 4.2, 4.3, and 4.4,

while in the remainder of this subsection, we introduce

some mathematical tools instrumental for describing each

of these tasks. As we will explain in Section 4.5, ob-

stacle avoidance and depth limiting will not be deﬁned

as separate tasks but rather achieved through a mod-

iﬁcation to the path-following task. Let us denote the

variables associated with the COLAV, formation-keeping,

and path-following tasks by lower indices 1, 2, and 3,

respectively. Deﬁne the so-called task variables as σi=

fi(p1,...,pn), i ∈ {1,2,3}, and their desired values as

σd,i, i ∈ {1,2,3}.

Furthermore, let υi, i ∈ {1,2,3}be the desired velocities of

each task. In the standard NSB algorithm, υiis obtained

using the closed-loop inverse kinematics (CLIK) equation

(Antonelli and Chiaverini, 2006)

υi=J†

i˙

σd,i −Λie

σi,(13)

where Λiis a positive deﬁnite gain matrix, e

σi=σi−σd,i,

and J†

iis the Moore-Penrose pseudoinverse of the task

Jacobian

Ji=∂σd,i

∂hp1,...,pni.(14)

However, in our case, we need to modify this equation for

each task to make it applicable to underactuated AUVs.

The combined desired velocity, υNSB, is then given by

(Antonelli and Chiaverini, 2006)

υNSB =υ1+I−J†

1J1υ2+I−J†

2J2υ3,(15)

where Iis an identity matrix.

4.2 Inter-vehicle collision avoidance

Let dCOLAV be the activation distance,i.e., the distance

at which the vehicles need to start performing the evasive

maneuvers. The task variable is given by a vector of rela-

tive distances between the vehicles smaller than dCOLAV

σ1=kpi−pjk,∀i, j ∈ {1, . . . , n}, j > i,

kpi−pjk< dCOLAV.(16)

The desired values of the task are

σd,1=dCOLAV 1,(17)

where 1is a vector of ones. To ensure a faster response

to a potential collision than in Matouˇs et al. (2022), we

propose the following sliding-mode-like COLAV velocity

υ1=UCOLAV

υ1,CLIK

kυ1,CLIKk,(18)

where UCOLAV is a positive constant, k·k is the Euclidean

norm, and υ1,CLIK is the velocity vector given by (13).

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Singularity-freeFormationPathFollowingofUnderactuatedAUVs:ExtendedVersionJosefMatousKristinY.PettersenDamianoVaragnolo;ClaudioPaliottaDepartmentofEngineeringCybernetics,NorwegianUniversityofScienceandTechnology,Trondheim(name.surname@ntnu.no).DepartmentofInformationEngineering,University...

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Singularity-free Formation Path Following of Underactuated AUVs Extended Version

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