**1. Introduction**

98 Autonomous Underwater Vehicles

Yakimenko, O.A., Horner, D.P. & Pratt, D.G. (2008). AUV rendezvous trajectories generation

*Control and Automation, Corse, France* 

for underwater recovery, In: *Proceedings of the 16th Mediterranean conference on* 

Autonomous Underwater Vehicles (AUVs) are the most complex type of unattended marine systems, being *mobile*, with challenging dynamics and non-holonomic kinematics. They are increasingly being recognized as a keystone technology for projecting human scientific and economical interests into the deep Ocean (Papoulias et al., 1989). A recent report by Bildberg (2009) delivers the verdict of several key researchers that the AUVs are rapidly moving towards maturity.

The *autonomy* of AUVs is their key capability. They autonomously explore Ocean phenomena relevant to human scientific and economic interests. Well engineered autonomous control allows them to act robustly and predictably with regards to waves, currents, wind, sea-state and numerous other disturbances and operational conditions in nature. As a consequence, they are today being cast in the leading role in projecting human presence and human interests in the Ocean, in an increasingly diverse gamut of topics:


To increase the effectiveness, safety, availability, economics and applicability of AUVs to these and other topics of interest, this chapter proposes a *decentralized cooperative cross-layer formation-control* paradigm for entire groups of AUVs collaborating in exploration tasks. The AUVs are assumed to navigate on a common "flight ceiling" by using robust altitude controllers, based on altimeter echosounder measurements. The proposed virtual potential framework allows for the 2D organization of individual trajectories on such a "flight ceiling".

potentials alleviate some of the most distinct problems encountered by competing reactive

Formation Guidance of AUVs Using Decentralized Control Functions 101

• Lack of *reaction* to the decentralized, agent-local process of accumulating or perfecting knowledge of the environment on top of the initially imperfect situational awareness of

• Trajectory planning that is sub-optimal, or optimal based on a *hard-coded* criterion, without possibility of adjusting or restating that criterion at run-time, because the cost function is implicit in the choice of mathematical tools (such as a distinct set of curve formulations

Stemming from these considerations, we propose a scheme where each AUV in a 2D formation imbedded in the "flight ceiling" plane as previously discussed maintains a local imperfect map of the environment. Every possible map only ever consists of a finite number of

2. Obstacles which need to be circumnavigated in a safe and efficient manner, (**O***i*), ∀*i* =

3. Vertices of the *characteristic cell* of the chosen formation geometry, covered in more detail

With this in mind, let the *virtual potential* be a real, single valued function *<sup>P</sup>* : **<sup>R</sup>**<sup>2</sup> <sup>→</sup> **<sup>R</sup>**, mapping almost every attainable position of an AUV on the "flight ceiling" to a real. Let *P*-s total differential exists almost wherever the function itself is defined. *P* can be said to live on the subspace of the full-rank state-space of the AUVs, <sup>C</sup> <sup>=</sup> **<sup>R</sup>**<sup>6</sup> <sup>×</sup> **SE**3. The state-space of the AUV is composed of the Euclidean 6-space **R**<sup>6</sup> spanned by the angular and linear

Furthermore, this framework will be restricted to only those *P* that can be expressed in terms

Where *Pi* is of one of a small variety of considered function forms. Precisely, we restrict our attention to three function forms with each one characteristic of each of the three mentioned

The critical issue in the guidance problem at hand is Euclidean 2D distance (within the "flight ceiling") between pairs of AUVs in the formation, and each AUV and all obstacles. Therefore, our attention is further restricted to only such {*Pi*}⊂L(C → **R**) with L being the space of all functions mapping C to **R** whose total differential exists almost wherever each of the functions

defined by the choice of *pi*(*d*), the *isotropic potential contour generator*. Choices and design of

*n* ∑ *i*=1

<sup>0</sup> a Euclidean 2D metric across the "flight ceiling". Consequently, *Pi* is completely

∃*n* ∈ **N**| *P*<sup>Σ</sup> =

is defined on <sup>C</sup>, which can be represented as the composition *Pi* <sup>≡</sup> *pi* ◦ *di*, *pi* : **<sup>R</sup>**<sup>+</sup>

} ≡ **<sup>R</sup>**<sup>6</sup> and a full 3D, 6DOF configuration-space

*Pi* (1)

<sup>0</sup> → **R**, and

} which possesses the topology of the *Special Euclidean group*

formation guidance strategies, which are prone to the following problems:

• Reliance on the *perfect knowledge* of a map of the waterspace,

each individual agent,

used for trajectories etc.).

1... *nobs* **<sup>O</sup>***<sup>i</sup>* <sup>⊂</sup> **<sup>R</sup>**2,

in sec. 2.3.3 and 3.

 **<sup>v</sup>**<sup>T</sup> <sup>ω</sup><sup>T</sup><sup>T</sup>

} = { 

of a sum of finitely many terms:

*pi*(*d*)-s will be discussed in sec. 2.3.

velocities, {

*di* : C → **<sup>R</sup>**<sup>+</sup>

{ *<sup>x</sup>*<sup>T</sup> **<sup>Θ</sup>**<sup>T</sup><sup>T</sup>

instantiations of any of the three types of features:

} = { 

*xyz* <sup>|</sup> *ϕϑψ*<sup>T</sup>

1. A way-point that is commanded for the entire formation, *<sup>w</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>2</sup>

*uvw* <sup>|</sup> *pqr*<sup>T</sup>

*types of features* (way-point, obstacle, vertices of formation cells).

*of rank 3*, **SE**3. Function *<sup>P</sup>* therefore maps to a real scalar field over that same <sup>C</sup>.

The goal is to provide decentralized consensus-building resulting in synoptical situational awareness of, and coordinated manoeuvring in the navigated waterspace. The paradigm is formally developed and tested in a hardware-in-the-loop simulation (HILS) setting, utilizing a full-state hydrodynamical rigid-body dynamic model of a large, sea-capable, long-endurance Ocean-going vehicle. Existence of realistic, technically feasible sensors measuring proxy variables or directly the individual kinematic or dynamic states is also simulated, as is the presence of realistic, non-stationary plant and measurement noise.
