Practical formation control of multiple underactuated ships with limited sensing ranges

Robotics and Autonomous Systems 59 (2011) 457–471

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Robotics and Autonomous Systems

journal homepage: www.elsevier.com/locate/robot

Practical formation control of multiple underactuated ships with limited

sensing ranges

K.D. Do∗

School of Mechanical Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

a r t i c l e i n f o

Article history:

Received 6 August 2009

Received in revised form

22 February 2011

Accepted 7 March 2011

Available online 22 March 2011

Keywords:

Formation control

Underactuated ships

Potential functions

Transverse function approach

Smooth step functions

Lyapunov direct method

a b s t r a c t

This paper presents a constructive method to design cooperative controllers that force a group of N

underactuated ships with limited sensing ranges to perform a desired formation, and guarantee no

collisions between the ships. These ships do not have an independent actuator in the sway axis. The

desired formation is stabilized at any sufficiently smooth reference trajectories, including fixed points

and nonadmissible trajectories for the ships. The formation control design is based on several nonlinear

coordinate changes, the transverse function approach, the backstepping technique, the Lyapunov direct

method, and smooth and p-times differentiable step functions. These functions are introduced and

incorporated into novel potential functions to solve the collision avoidance problem without the need

of switchings despite the ships’ limited sensing ranges. Simulations illustrate the results.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

This paper focuses on the design of a formation control system

for underactuated ships. The goal is to obtain a desired formation

that can be stabilized at any sufficiently smooth reference trajec￾tories. Since the paper involves both the control of single underac￾tuated ships and the formation control of multiple vehicles, a brief

review of previous work in these fields is presented below to mo￾tivate contributions of the paper.

1.1. Previous work on control of underactuated ships

The problem of stabilizing an underactuated ship at a desired

reference trajectory is an important issue in many offshore appli￾cations. This goal can be achieved by solving trajectory-tracking,

path-following, path-tracking and stabilization problems [1]. The

main difficulty with controlling an underactuated ship is that only

the yaw and surge axes are directly actuated while the sway axis

is not actuated. This configuration is by far most common among

the marine surface vessels [2]. It is also known that the ships in

question are a class of underactuated mechanical systems with

nonintegrable dynamics and which are not transformable into a

driftless system [3]. An application of the Brockett theorem [4]

shows the nonexistence of time invariant, smooth, state feedback

∗ Tel.: +61 864883883; fax: +61 864881024.

E-mail address: [email protected].

control laws that are able to asymptotically stabilize an underactu￾ated ship at a fixed point. Due to numerous important applications

of underactuated ships, their motion control has received a lot of

attention from the control community.

An application of the recursive technique for standard chain

form systems [5] was used in [6] to provide a high-gain, local

exponential tracking result. By applying a cascade approach, a

global tracking result was obtained in [7]. Based on Lyapunov’s

direct method and the passivity approach, two tracking solutions

were proposed in [8]. It is noted that in [8,7,6], the yaw velocity

was required to be nonzero. This restrictive assumption implies

that a straight-line cannot be tracked. It seems that the first global

way-point tracking controller was proposed in [9] to force an

underactuated ship to track a straight-line (see also [10,11] for

robust and output feedback versions of straight-line following

controllers). In [12], a solution was proposed to solve the problem

of trajectory tracking without imposing the requirement that yaw

velocity be nonzero. In [13], a single controller was proposed to

solve both stabilization and tracking simultaneously, see also [14]

for an interesting solution on relaxing the limitation on non-zero

off-diagonal terms in the aforementioned articles. The work in [15],

see also [16] is of a particular relevance to the work presented in

this paper. The core of the work in [15] is the nontrivial coordinate

transformation that was used to transform the underactuated ship

dynamics to a convenient form. However, it is noted that this

coordinate transformation only works for spherical vessels.

In all the aforementioned papers on controlling an underactu￾ated ship, either the reference trajectories are limited or the control

0921-8890/$ – see front matter © 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.robot.2011.03.003