**1.1 The cooperative paradigm**

Since 1970s, robotics and control engineers have studied the cooperative paradigm. Cooperative control is a set of complete, halting algorithms and machine-realized strategies allowing multiple *individual agents* to complete a given task in a certain optimal way. This optimality results from the agents' leveraging each other's resources (e.g. manoeuvring abilities) to more effectively minimize some cost function that measures a "budget" of the entire task, in comparison to what each agent would would be capable of on their own (without the benefit of the group).

In the marine environment, such "social" leveraging is beneficial in several ways. Firstly, deployment of more AUVs significantly reduces the time needed to survey a given theater of operations. This has enormous economic repercussion in terms of conserved hours or days of usually prohibitively expensive ship-time (for the vessel that is rendering operational support to the AUV fleet). Secondly, deployment of a larger number of AUVs *diversifies the risk to operations*. In a group scenario, loss of a (small) number of AUVs doesn't necessarily preclude the achievement of mission goals. Lastly, if each of the group AUVs are furnished with adaptive-sampling algorithms, such as in the chemical plume-tracing applications (Farrell et al., 2005; Pang, 2006; Pang et al., 2003), deployment of multiple vehicles guarantees much faster convergence to the points of interest.

Cooperative control frameworks are split into *centralized* and *decentralized* strategies. A centralized cooperative control system's task is to determine the actions of each agent based on a perfectly (or as near perfectly as possible) known full data-set of the problem, which consists of the state vectors of *every agent* for which the problem is stated. The centralized system instantiates a globally optimal solution based on the assessment of momentary resource-disposition of the entire ensemble, as well as based on the total, if possibly non-ideal knowledge of the environment. The state data are usually collected by polling all agents through a communication network. After the polling cycle, the centralized system communicates the low-level guidance commands back to individual agents. This approach allows for the emergence of a global optimum in decision-making on grounds of *all obtainable information*, but heavily depends on *fault-intolerant, quality-assured, high-bandwidth communication*.

In a *decentralized* approach, such as we have chosen to present in this chapter, each agent possesses imperfect state and perception data of every other agent and of the observable portion of the environment, and *locally* decides its own course of action. The greatest issue in decentralized cooperative control is the achievement of a *consensus* between separately reasoning autonomous agents.

## **2. The virtual potentials framework**

To address the issue of reactive formation guidance of a number of AUVs navigating in a waterspace, a method based on *virtual or artificial potentials* is hereby proposed. The virtual 2 Will-be-set-by-IN-TECH

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

Since 1970s, robotics and control engineers have studied the cooperative paradigm. Cooperative control is a set of complete, halting algorithms and machine-realized strategies allowing multiple *individual agents* to complete a given task in a certain optimal way. This optimality results from the agents' leveraging each other's resources (e.g. manoeuvring abilities) to more effectively minimize some cost function that measures a "budget" of the entire task, in comparison to what each agent would would be capable of on their own

In the marine environment, such "social" leveraging is beneficial in several ways. Firstly, deployment of more AUVs significantly reduces the time needed to survey a given theater of operations. This has enormous economic repercussion in terms of conserved hours or days of usually prohibitively expensive ship-time (for the vessel that is rendering operational support to the AUV fleet). Secondly, deployment of a larger number of AUVs *diversifies the risk to operations*. In a group scenario, loss of a (small) number of AUVs doesn't necessarily preclude the achievement of mission goals. Lastly, if each of the group AUVs are furnished with adaptive-sampling algorithms, such as in the chemical plume-tracing applications (Farrell et al., 2005; Pang, 2006; Pang et al., 2003), deployment of multiple vehicles guarantees

Cooperative control frameworks are split into *centralized* and *decentralized* strategies. A centralized cooperative control system's task is to determine the actions of each agent based on a perfectly (or as near perfectly as possible) known full data-set of the problem, which consists of the state vectors of *every agent* for which the problem is stated. The centralized system instantiates a globally optimal solution based on the assessment of momentary resource-disposition of the entire ensemble, as well as based on the total, if possibly non-ideal knowledge of the environment. The state data are usually collected by polling all agents through a communication network. After the polling cycle, the centralized system communicates the low-level guidance commands back to individual agents. This approach allows for the emergence of a global optimum in decision-making on grounds of *all obtainable information*, but heavily depends on *fault-intolerant, quality-assured, high-bandwidth*

In a *decentralized* approach, such as we have chosen to present in this chapter, each agent possesses imperfect state and perception data of every other agent and of the observable portion of the environment, and *locally* decides its own course of action. The greatest issue in decentralized cooperative control is the achievement of a *consensus* between separately

To address the issue of reactive formation guidance of a number of AUVs navigating in a waterspace, a method based on *virtual or artificial potentials* is hereby proposed. The virtual

presence of realistic, non-stationary plant and measurement noise.

**1.1 The cooperative paradigm**

(without the benefit of the group).

*communication*.

reasoning autonomous agents.

**2. The virtual potentials framework**

much faster convergence to the points of interest.

potentials alleviate some of the most distinct problems encountered by competing reactive formation guidance strategies, which are prone to the following problems:


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 instantiations of any of the three types of features:


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 velocities, { **<sup>v</sup>**<sup>T</sup> <sup>ω</sup><sup>T</sup><sup>T</sup> } = { *uvw* <sup>|</sup> *pqr*<sup>T</sup>} ≡ **<sup>R</sup>**<sup>6</sup> and a full 3D, 6DOF configuration-space { *<sup>x</sup>*<sup>T</sup> **<sup>Θ</sup>**<sup>T</sup><sup>T</sup> } = { *xyz* <sup>|</sup> *ϕϑψ*<sup>T</sup> } which possesses the topology of the *Special Euclidean group of rank 3*, **SE**3. Function *<sup>P</sup>* therefore maps to a real scalar field over that same <sup>C</sup>.

Furthermore, this framework will be restricted to only those *P* that can be expressed in terms of a sum of finitely many terms:

$$\exists n \in \mathbb{N} \mid P\_{\Sigma} = \sum\_{i=1}^{n} P\_i \tag{1}$$

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

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 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>0</sup> → **R**, and *di* : C → **<sup>R</sup>**<sup>+</sup> <sup>0</sup> a Euclidean 2D metric across the "flight ceiling". Consequently, *Pi* is completely defined by the choice of *pi*(*d*), the *isotropic potential contour generator*. Choices and design of *pi*(*d*)-s will be discussed in sec. 2.3.

Fig. 1. An example of an oscillatory trajectory due to the conservativeness of the virtual

Formation Guidance of AUVs Using Decentralized Control Functions 103

Fig. 2. Example of a local minimum occurring in virtual potential guidance in a 2D

and angular velocities) there exists a lower bound *tl* after which �*xi*(*t* > *tl*) − *x*

(*j*)

*tl*) − *w*� almost always. The set {*tl*} is also dense and connected.

*i*, and the enumeration of the distinct points of convergence other than the way-point by *j* omitted for clarity) that *do not* converge to the way-point *w* or a finitely large orbit around

these uncountably many "nearby" trajectories (to be visualized as a "sheaf" of trajectories emanating from a distinct, well defined neigbourhood in **W***<sup>i</sup>* for some range of initial linear

<sup>∞</sup> (or a finitely large orbit around *it*). Therefore, for each of

(*l*)

<sup>∞</sup> �≤�*xi*(*t* >

potential system.

waterspace.

it, but rather to another point *x*

With all of the above stated, a *decentralized total control function <sup>f</sup>* : **<sup>Z</sup>** <sup>→</sup> **<sup>R</sup>**<sup>2</sup> is then defined as a sampling, repeated at sample times *<sup>k</sup>* <sup>∈</sup> **<sup>Z</sup>**0, of the 2D vector field *<sup>E</sup>* : **<sup>W</sup>***<sup>i</sup>* <sup>→</sup> **<sup>R</sup>**<sup>2</sup> over a subspace **<sup>W</sup>***<sup>i</sup>* <sup>⊆</sup> **<sup>R</sup>**<sup>2</sup> ⊂ C, the *navigable waterspace*:

$$\forall \mathbf{x} \in \mathbb{W}\_{l} \subseteq \mathbb{R}^{2} \subset \mathcal{C}\_{\prime} \; \mathbf{E}(\mathbf{x}) = -\nabla \; \mathbb{P}\_{\Sigma}(\mathbf{x}) \tag{2}$$

Where **<sup>W</sup>***<sup>i</sup>* <sup>=</sup> **<sup>R</sup>**<sup>2</sup> \ *<sup>i</sup>* **<sup>O</sup>***<sup>i</sup>* ∪ *<sup>j</sup>* **<sup>O</sup>**(*ag*) *j* contains all of **R**<sup>2</sup> to the exclusion of closed connected subsets of **<sup>R</sup>**<sup>2</sup> that represent interiors of obstacles, {**O***i*} and those that represent safety areas around all the *j*-th AUVs (*j* �= *i*) other than the *i*-th one considered. **W***<sup>i</sup>* is an open, connected subset of **R**2, the "flight ceiling", inheriting its Euclidean-metric-generated topology and always containing the way-point *w*. Sampling **E** at the specific *xi*(*k*) ∈ **W***i*, the location of the *i*-th AUV, results in *f<sup>i</sup>* (*k*), the *total decentralized control function* for the *i*-th AUV at time *k* and location *xi*.

#### **2.1 Passivity**

The decentralized total control function *f* is used as the forcing signal of an idealized dimensionless charged particle of unit mass, modeled by a *holonomic 2D double integrator*. If any AUV were able to behave in this manner, the AUV would follow an *ideal conservative trajectory* given by:

$$\mathbf{x}\_{i}(t) = \int\_{\varGamma=0}^{t} \mathbf{E}\left[\mathbf{x}\_{i}(\tau)\right] d\tau^{2} + \mathbf{x}\_{i}^{(0)} \tag{3}$$

This ideal conservative trajectory, while stable in the BIBO sense, is in general not asymptotically stable, nor convergent by construction. The simplest case when this doesn't hold is when **<sup>E</sup>**(*x*) is an *irrotational*<sup>1</sup> vector field whose norm is affine in the �*<sup>x</sup>* <sup>−</sup> *<sup>w</sup>*� 2D Euclidean distance:

$$\|\|\mathbf{E}(\mathbf{x})\|\| = e\|\|\mathbf{x} - \mathbf{z}\mathbf{w}\|\| + E\_0; e \in [0, \infty) \tag{4}$$

And whose direction is always towards *w*:

$$\forall \mathbf{x}, \; \mathbf{E}(\mathbf{x}) \cdot (\mathbf{x} - \mathbf{w}) \stackrel{\text{id}}{=} e \|\mathbf{x} - \mathbf{w}\|^2 + E\_0 \|\mathbf{x} - \mathbf{w}\| \tag{5}$$

In that case (3) can be regarded as a linear second or third order system with two of the poles in ±i. Such a system exhibits borderline-stable oscillation – a hallmark of its *conservativeness*. An example of such BIBO-stable non-convergent oscillation is given in figure 1.

Note that this analysis is irrespective of the initial condition ˙*x*<sup>0</sup> as long as (4, 5) approximate **E**(*x*) sufficiently well in some open *ε*-ball centered on *w*. However, AUVs are in general not able to actuate as ideal holonomic 2D double integrators. The introduction of *any* finite non-zero lag in the above discussion, which is sure to exist from first physical principles in a real AUV, is sufficient to cause *dissipation* and as a consequence passivity and convergence to *w*.

#### **2.2 Local minima**

In addition to the problem of passivity, the virtual potential approach suffers from the *existence of local minima*. Without further constraints, the nature of **E**(**W***i*) so far discussed doesn't preclude a dense, connected, closed state-subspace <sup>C</sup>(*j*) *<sup>i</sup>* <sup>0</sup> ⊆ **C**, containing uncountably many initial vectors {**x** (*l*) <sup>0</sup> | *l* ∈ **R**} of "related" trajectories (with the indexing by AUV denoted by

<sup>1</sup> Whose *rotor* or *curl* operator is identically zero.

4 Will-be-set-by-IN-TECH

With all of the above stated, a *decentralized total control function <sup>f</sup>* : **<sup>Z</sup>** <sup>→</sup> **<sup>R</sup>**<sup>2</sup> is then defined as a sampling, repeated at sample times *<sup>k</sup>* <sup>∈</sup> **<sup>Z</sup>**0, of the 2D vector field *<sup>E</sup>* : **<sup>W</sup>***<sup>i</sup>* <sup>→</sup> **<sup>R</sup>**<sup>2</sup> over a

subsets of **<sup>R</sup>**<sup>2</sup> that represent interiors of obstacles, {**O***i*} and those that represent safety areas around all the *j*-th AUVs (*j* �= *i*) other than the *i*-th one considered. **W***<sup>i</sup>* is an open, connected subset of **R**2, the "flight ceiling", inheriting its Euclidean-metric-generated topology and always containing the way-point *w*. Sampling **E** at the specific *xi*(*k*) ∈ **W***i*, the location of

The decentralized total control function *f* is used as the forcing signal of an idealized dimensionless charged particle of unit mass, modeled by a *holonomic 2D double integrator*. If any AUV were able to behave in this manner, the AUV would follow an *ideal conservative*

This ideal conservative trajectory, while stable in the BIBO sense, is in general not asymptotically stable, nor convergent by construction. The simplest case when this doesn't hold is when **<sup>E</sup>**(*x*) is an *irrotational*<sup>1</sup> vector field whose norm is affine in the �*<sup>x</sup>* <sup>−</sup> *<sup>w</sup>*� 2D

In that case (3) can be regarded as a linear second or third order system with two of the poles in ±i. Such a system exhibits borderline-stable oscillation – a hallmark of its *conservativeness*.

Note that this analysis is irrespective of the initial condition ˙*x*<sup>0</sup> as long as (4, 5) approximate **E**(*x*) sufficiently well in some open *ε*-ball centered on *w*. However, AUVs are in general not able to actuate as ideal holonomic 2D double integrators. The introduction of *any* finite non-zero lag in the above discussion, which is sure to exist from first physical principles in a real AUV, is sufficient to cause *dissipation* and as a consequence passivity and convergence to

In addition to the problem of passivity, the virtual potential approach suffers from the *existence of local minima*. Without further constraints, the nature of **E**(**W***i*) so far discussed doesn't

<sup>0</sup> | *l* ∈ **R**} of "related" trajectories (with the indexing by AUV denoted by

**E** [*xi*(*τ*)] *dτ*<sup>2</sup> + *x*

<sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>W</sup>***<sup>i</sup>* <sup>⊆</sup> **<sup>R</sup>**<sup>2</sup> ⊂ C, **<sup>E</sup>**(*x*) = −∇ *<sup>P</sup>*Σ(*x*) (2)

(*k*), the *total decentralized control function* for the *i*-th AUV at time *k*

(0)

�**E**(*x*)� = *e*�*x* − *w*� + *E*0; *e* ∈ [0, ∞) (4)

<sup>=</sup> *<sup>e</sup>*�*<sup>x</sup>* <sup>−</sup> *<sup>w</sup>*�<sup>2</sup> <sup>+</sup> *<sup>E</sup>*0�*<sup>x</sup>* <sup>−</sup> *<sup>w</sup>*� (5)

*<sup>i</sup>* <sup>0</sup> ⊆ **C**, containing uncountably many

*<sup>i</sup>* (3)

contains all of **R**<sup>2</sup> to the exclusion of closed connected

subspace **<sup>W</sup>***<sup>i</sup>* <sup>⊆</sup> **<sup>R</sup>**<sup>2</sup> ⊂ C, the *navigable waterspace*:

*<sup>i</sup>* **<sup>O</sup>***<sup>i</sup>* ∪

*<sup>j</sup>* **<sup>O</sup>**(*ag*) *j* 

*xi*(*t*) =

<sup>∀</sup>*x*, **<sup>E</sup>**(*x*) · (*<sup>x</sup>* <sup>−</sup> *<sup>w</sup>*) id

preclude a dense, connected, closed state-subspace <sup>C</sup>(*j*)

An example of such BIBO-stable non-convergent oscillation is given in figure 1.

 *t τ*=0

Where **<sup>W</sup>***<sup>i</sup>* <sup>=</sup> **<sup>R</sup>**<sup>2</sup> \

and location *xi*.

*trajectory* given by:

Euclidean distance:

*w*.

**2.2 Local minima**

initial vectors {**x**

(*l*)

<sup>1</sup> Whose *rotor* or *curl* operator is identically zero.

And whose direction is always towards *w*:

**2.1 Passivity**

the *i*-th AUV, results in *f<sup>i</sup>*

Fig. 1. An example of an oscillatory trajectory due to the conservativeness of the virtual potential system.

Fig. 2. Example of a local minimum occurring in virtual potential guidance in a 2D waterspace.

*i*, and the enumeration of the distinct points of convergence other than the way-point by *j* omitted for clarity) that *do not* converge to the way-point *w* or a finitely large orbit around it, but rather to another point *x* (*j*) <sup>∞</sup> (or a finitely large orbit around *it*). Therefore, for each of these uncountably many "nearby" trajectories (to be visualized as a "sheaf" of trajectories emanating from a distinct, well defined neigbourhood in **W***<sup>i</sup>* for some range of initial linear and angular velocities) there exists a lower bound *tl* after which �*xi*(*t* > *tl*) − *x* (*l*) <sup>∞</sup> �≤�*xi*(*t* > *tl*) − *w*� almost always. The set {*tl*} is also dense and connected.

Where:

to a real,

Where:

to a real, - *A*−


away from the obstacle.

*pw*(*d*) =

*∂*

∴ *pw*(*d*, *d* > *d*0) = *A*−

*<sup>∂</sup><sup>d</sup> pw*(*d*) = max(*A*<sup>−</sup>

*<sup>d</sup>* <sup>≤</sup> *<sup>d</sup>*<sup>0</sup> : *<sup>A</sup>*<sup>−</sup>

*d* > *d*<sup>0</sup> : *A*<sup>−</sup>

towards the way-point in the area of proportional attraction,

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> 5.5 <sup>6</sup> 6.5 <sup>7</sup> 7.5 −1.5

A<sup>−</sup> 0

d [m]

d0

Fig. 4. The potential contour generator of a way-point, *pw*(*d*(*x*)).

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

p*w*

towards the way-point outside the area of proportional attraction,

*p* <sup>2</sup> *<sup>d</sup>*<sup>2</sup>

<sup>0</sup> *<sup>d</sup>* <sup>−</sup> *<sup>A</sup>*<sup>−</sup> <sup>2</sup> *<sup>c</sup>* 2*A*− *p*

*<sup>p</sup> d*, *A*<sup>−</sup>

<sup>0</sup> (*d* − *d*0) + *p*<sup>0</sup>

<sup>0</sup> ); lim *d*→∞

; lim

*∂*


*<sup>p</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> \ {0} is a positive real independent parameter dictating the scale of acceleration

<sup>0</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> \ {0} is a positive real independent parameter dictating the constant acceleration


(a) Graph of *pw*(*d*) : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**. (b) Graph of *pw*(*d*(*x*)) : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>** ◦ **<sup>R</sup>**<sup>2</sup> <sup>→</sup> **<sup>R</sup>**+.

centered on the way-point that constitutes the area of proportional acceleration.

**2.3.2 Way-points**


Formation Guidance of AUVs Using Decentralized Control Functions 105


; *<sup>d</sup>*<sup>0</sup> <sup>=</sup>id

*<sup>d</sup>*→<sup>∞</sup> *pw*(*d*) = <sup>∞</sup>; lim

*<sup>∂</sup><sup>d</sup> pw*(*d*) = *<sup>A</sup>*<sup>−</sup>

<sup>=</sup> *<sup>A</sup>*<sup>−</sup> 0 *A*− *p* ; *p*<sup>0</sup> if <sup>=</sup> *<sup>A</sup>*<sup>−</sup> <sup>2</sup> 0 2*A*− *p*

> <sup>0</sup> ; lim *<sup>d</sup>*→0<sup>+</sup>

(8)

*<sup>d</sup>*→0<sup>+</sup> *pw*(*d*) = <sup>0</sup> (9)

*<sup>∂</sup><sup>d</sup> pw*(*d*) = 0 (10)

*∂*

Furthermore, there is no prejudice as to the *number* of such <sup>C</sup>(*j*) *<sup>i</sup>* <sup>0</sup> -s, i.e. there exists <sup>C</sup><sup>Σ</sup> *i* 0 ⊆ <sup>C</sup>, <sup>C</sup><sup>Σ</sup> *<sup>i</sup>* <sup>0</sup> = *<sup>j</sup>* <sup>C</sup>(*j*) *<sup>i</sup>* <sup>0</sup> . There may be multiple disjoint dense, connected, closed sets of initial conditions of the trajectory of the *i*-th AUV which all terminate in the same, or distinct local minima. The enumerator *j* may even come from **R** (i.e. there may be *uncountably many distinct local minima*, perhaps arranged in dense, connected sets – like curves or areas in **R**2).

An example of an occurrence of a local minimum is depicted in figure 2. In order to resolve local minima, an intervention is required that will ensure that either one of the following conditions is fulfilled:


Out of the two listed strategies for dealing with local minima, the authors have published extensively on strategy 2 (Barisic et al., 2007a), (Barisic et al., 2007b). However, strategy 1 represents a much more robust and general approach. A method guaranteeing <sup>C</sup><sup>Σ</sup> *i* 0 id = ∅ by designing in *rotors* will be described in sec. 2.4.

#### **2.3 Potential contour generators and decentralized control functions**

As for the potential contour generators *pi*(*d*) : **R**<sup>+</sup> <sup>0</sup> → **R**, their definition follows from the global goals of guidance for the formation of AUVs. Bearing those in mind, the potential contour generators of each *feature type*, *p*(*o*,*w*,*c*) (for obstacle, way-point and formation cell vertex, accordingly) are specified below.

#### **2.3.1 Obstacles**

Fig. 3. The potential contour generator of an obstacle, *po*(*d*(*x*)).

$$p\_o(d) = \exp\left(\frac{A^+}{d}\right) - 1; \quad \lim\_{d \to \infty} p\_o(d) = 0; \quad \lim\_{d \to 0^+} p\_o(d) = \infty \tag{6}$$

$$\frac{\partial}{\partial d}p\_o(d) = -\frac{A^+}{d^2} \exp\left(\frac{A^+}{d}\right); \quad \lim\_{d \to \infty} \frac{\partial}{\partial d}p\_o(d) = 0; \quad \lim\_{d \to 0^+} \frac{\partial}{\partial d}p\_o(d) = \infty \tag{7}$$

Where:

6 Will-be-set-by-IN-TECH

conditions of the trajectory of the *i*-th AUV which all terminate in the same, or distinct local minima. The enumerator *j* may even come from **R** (i.e. there may be *uncountably many distinct*

An example of an occurrence of a local minimum is depicted in figure 2. In order to resolve local minima, an intervention is required that will ensure that either one of the following

*ε*(*t*0) = sup �*x*(*t* > *t*0) − *x*∞| *x*0� characterizing a *ε*-ball centered on the particular *x*<sup>∞</sup> and containing all *x*(*t* > *t*0), to intervene in **E**(**W***i*) *guaranteeing* that this entire ball is outside

<sup>0</sup> ← **x**(*t*0)). Out of the two listed strategies for dealing with local minima, the authors have published extensively on strategy 2 (Barisic et al., 2007a), (Barisic et al., 2007b). However, strategy 1

global goals of guidance for the formation of AUVs. Bearing those in mind, the potential contour generators of each *feature type*, *p*(*o*,*w*,*c*) (for obstacle, way-point and formation cell

(a) Graph of *po*(*d*) : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**. (b) Graph of *po*(*d*(*x*)) : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>** ◦ **<sup>R</sup>**<sup>2</sup> <sup>→</sup> **<sup>R</sup>**+.

*∂*

*<sup>d</sup>*→<sup>∞</sup> *po*(*d*) = 0; lim

*<sup>∂</sup><sup>d</sup> po*(*d*) = 0; lim

*d*→0<sup>+</sup>

*∂*

− 1; lim

; lim *d*→∞

represents a much more robust and general approach. A method guaranteeing <sup>C</sup><sup>Σ</sup>

*local minima*, perhaps arranged in dense, connected sets – like curves or areas in **R**2).

*<sup>i</sup>* <sup>0</sup> . There may be multiple disjoint dense, connected, closed sets of initial

*<sup>i</sup>* <sup>0</sup> -s, i.e. there exists <sup>C</sup><sup>Σ</sup>

*<sup>i</sup>* <sup>0</sup>, triggering at *t*<sup>0</sup> with

*i* 0 id = ∅ by

<sup>0</sup> → **R**, their definition follows from the

*<sup>d</sup>*→0<sup>+</sup> *po*(*d*) = <sup>∞</sup> (6)

*<sup>∂</sup><sup>d</sup> po*(*d*) = <sup>∞</sup> (7)

*i* 0 ⊆

Furthermore, there is no prejudice as to the *number* of such <sup>C</sup>(*j*)

2. A halting P-complete algorithm is introduced that for every **<sup>x</sup>**<sup>0</sup> ∈ C<sup>Σ</sup>

**2.3 Potential contour generators and decentralized control functions**

*<sup>i</sup>* <sup>0</sup> is empty by construction.

(a possibly existing) new <sup>C</sup><sup>Σ</sup> � *<sup>i</sup>* <sup>0</sup> (with **<sup>x</sup>**�

designing in *rotors* will be described in sec. 2.4.

vertex, accordingly) are specified below.

As for the potential contour generators *pi*(*d*) : **R**<sup>+</sup>

*d* [m]

*po*(*d*) = exp

*<sup>∂</sup><sup>d</sup> po*(*d*) = <sup>−</sup> *<sup>A</sup>*<sup>+</sup>

Fig. 3. The potential contour generator of an obstacle, *po*(*d*(*x*)).

 *A*<sup>+</sup> *d* 

> *A*<sup>+</sup> *d*

*<sup>d</sup>*<sup>2</sup> exp

<sup>C</sup>, <sup>C</sup><sup>Σ</sup>

*<sup>i</sup>* <sup>0</sup> =

conditions is fulfilled:

1. The set <sup>C</sup><sup>Σ</sup>

**2.3.1 Obstacles**

*po*

*∂*

*<sup>j</sup>* <sup>C</sup>(*j*)



#### **2.3.2 Way-points**

$$p\_{\mathcal{W}}(d) = \begin{cases} d \le d\_0 : \quad \frac{A\_p^-}{2} d^2 \\ d > d\_0 : \quad A\_0^-(d - d\_0) + p\_0 \end{cases}; \quad d\_0 = \stackrel{\text{id}}{=} \frac{A\_0^-}{A\_p^-}; \quad p\_0 \stackrel{\text{if}}{=} \frac{A\_0^{-2}}{2A\_p^-} \tag{8}$$

$$\text{s.t. } p\_{\text{w}}(d, d > d\_0) = A\_0^- d - \frac{A\_c^{-2}}{2A\_p^-}; \quad \lim\_{d \to \infty} p\_{\text{w}}(d) = \infty; \quad \lim\_{d \to 0^+} p\_{\text{w}}(d) = 0 \tag{9}$$

$$\frac{\partial}{\partial d}p\_w(d) = \max(A\_p^-d, A\_0^-); \quad \lim\_{d \to \infty} \frac{\partial}{\partial d}p\_w(d) = A\_0^-; \quad \lim\_{d \to 0^+} \frac{\partial}{\partial d}p\_w(d) = 0 \tag{10}$$

Where:





Fig. 4. The potential contour generator of a way-point, *pw*(*d*(*x*)).

Fig. 5. The potential contour generator of a formation agent.

*a* (*o*)

*a*(*o*)

*A*<sup>+</sup>

*A*− *c dc*

proposed in (7, 10, 12) are redesigned, adding a *rotor* component:

*a*(*s*)

obstacles ∑ *i*

obstacles ∑ *i*

obstacles ∑ *i*

> vertices ∑ *i*

− ∇ *P*<sup>Σ</sup> =

=

=

**2.4 Rotor modification**

minima avoidance.

hereafter be omitted),

Where: - *a*�


+

*potential-based decentralized control functions* due to the *i*-th feature.

*<sup>i</sup>* (*x*) · *ni*(*x*) *a*(*o*) *<sup>i</sup>* (*x*)

*<sup>i</sup>* (*x*) + *<sup>a</sup>*(*w*)

*di*(*x*)<sup>2</sup> exp *<sup>A</sup>*<sup>+</sup>

*di*(*x*) exp

(20 – 21), stable local minima occur due to the *irrotationality* of the field, rot **<sup>E</sup>**(*x*) id

*<sup>i</sup>* ← *a<sup>i</sup> a*�

+ *a* (*w*)

Formation Guidance of AUVs Using Decentralized Control Functions 107

(*x*) +

<sup>1</sup> <sup>−</sup> *di*(*x*)<sup>2</sup> 2*d*<sup>2</sup> *c*

*di*(*x*)

As mentioned in sec. 2.2, the virtual potential approach to guidance is extremely susceptible to the appearance of local minima. A robust and simple approach is needed to assure local

In terms of the vector field introduced by (2), the analytical solution of which is presented in

In order to avoid irrotationality, and thereby local minima, decentralized control functions

*i* redef ←−−− *<sup>a</sup>*(*s*)

*<sup>i</sup>* is the redefined *total decentralized control function* due to the *i*-th feature (the dash will

*<sup>i</sup>* is the *stator* decentralized control function as introduced in the preceding section,

*<sup>i</sup>* (*x*) · *nw*(*x*) *a*(*w*)(*x*)

> vertices ∑ *i*

*a*(*c*)

*ni*(*x*) + min[*A*<sup>−</sup>

*<sup>i</sup>* <sup>+</sup> *<sup>a</sup>*(*r*)

+

vertices ∑ *i a* (*c*)

*<sup>p</sup> d*(*x*), *A*<sup>−</sup>

*<sup>i</sup>* (*x*) · *ni*(*x*) *a*(*c*) *<sup>i</sup>* (*x*)

*<sup>i</sup>* (*x*) (20)

*<sup>c</sup>* ] *<sup>w</sup>* <sup>−</sup> *<sup>x</sup>* �*w* − *x*�

= 0.

*<sup>i</sup>* (22)

(19)

(21)

#### **2.3.3 Cell vertices**

A good candidate potential contour generator of formation cell vertices, which behaves similar to a function *with local support*, is the *normal distribution curve*, adjusted for attractiveness (i.e. of inverted sign).

$$p\_{\mathfrak{c}}(d) = -A\_{\mathfrak{c}}^{-}d\_{\mathfrak{c}} \exp\left(1 - \frac{d^2}{2d\_{\mathfrak{c}}^2}\right); \quad \lim\_{d \to \infty} p\_{\mathfrak{c}}(d) = 0; \quad \lim\_{d \to 0^+} p\_w(d) = -A\_{\mathfrak{c}}^{-} \tag{11}$$

$$\frac{\partial}{\partial d}p\_{\mathcal{c}}(d) = \frac{A\_{\mathcal{c}}^{-}}{d\_{\mathcal{c}}}d\exp\left(1 - \frac{d^2}{2d\_{\mathcal{c}}^2}\right); \quad \frac{\partial}{\partial d}p\_{\mathcal{c}}(d)\big|\_{0} = 0; \quad \lim\_{d \to \infty} \frac{\partial}{\partial d}p\_{\mathcal{c}}(d) = 0\tag{12}$$

$$\frac{\partial^2}{\partial d^2} p\_c(d) = \frac{A\_c^-}{d\_c} \left( 1 - \frac{d^2}{d\_c^2} \right) \exp\left( 1 - \frac{d^2}{2d\_c^2} \right) \tag{13}$$

$$\text{If: } \, d\_{\text{max}} = \arg \left\{ \frac{\partial^2}{\partial d^2} p\_c(d) \stackrel{!}{=} 0 \right\} \tag{14}$$

$$d\_{\max} \stackrel{\text{id}}{=} \pm d\_{\mathcal{L}} \tag{15}$$

$$\frac{\partial}{\partial d} p\_c(d)\big|\_{d\_{\text{max}} \stackrel{\text{id}}{=} d\_c} = A\_c^- \tag{16}$$

Where:




The potential of a square formation cell surrounding an agent that figures as an obstacle is represented in figure 5.

#### **2.3.4 Reformulation in terms of decentralized control functions**

The monotonicity of (6, 8, 11) ensures that the direction of the gradient of the potential, <sup>∇</sup> *<sup>P</sup>*(*x*)/ �∇ *<sup>P</sup>*(*x*)� <sup>∈</sup> **SO**2, is always <sup>±</sup>*n<sup>i</sup>* = (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>i*)/�*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>i*�. Therefore, since (1, 2) are linear, (2) can be solved *analytically* for any finite sum of terms of the form specified by (6, 8, 11) up to the values of the independent parameters (*A*+, *A*<sup>−</sup> *<sup>p</sup>* , *A*<sup>−</sup> <sup>0</sup> , *A*<sup>−</sup> *<sup>c</sup>* , *dc*). The procedure follows:

$$\begin{aligned} -\nabla P\_{\Sigma}(\mathbf{x}) &= -\nabla \sum\_{i} P\_{i}(\mathbf{x}) \\ &= \sum\_{i} \left( -\nabla p\_{i}(d\_{i}(\mathbf{x})) \right) \end{aligned} \tag{17}$$

$$=-\sum\_{i} \frac{\partial}{\partial d\_i(\mathbf{x})} p\_i[d\_i(\mathbf{x})] \cdot \mathfrak{n}\_i(\mathbf{x}) \tag{18}$$

Equation (18) can be summarized by designating the terms in (7, 10, 12) as *a*(*o*,*w*,*c*) respectively. The terms *a* (*o*,*w*,*c*) *<sup>i</sup>* · *<sup>n</sup>i*, can likewise be denoted *<sup>a</sup>*(*o*,*w*,*c*) *<sup>i</sup>* , respectively, and represent the

Fig. 5. The potential contour generator of a formation agent.

*potential-based decentralized control functions* due to the *i*-th feature.

$$1 - \nabla P\_{\Sigma} = \sum\_{i}^{\text{obsstades}} \underbrace{a\_{i}^{(o)}(\mathbf{x}) \cdot \mathfrak{n}\_{i}(\mathbf{x})}\_{a\_{i}^{(o)}(\mathbf{x})} + \underbrace{a\_{i}^{(w)}(\mathbf{x}) \cdot \mathfrak{n}\_{\partial^{w}}(\mathbf{x})}\_{a^{(w)}(\mathbf{x})} + \sum\_{i}^{\text{vertices}} \underbrace{a\_{i}^{(c)}(\mathbf{x}) \cdot \mathfrak{n}\_{i}(\mathbf{x})}\_{a\_{i}^{(c)}(\mathbf{x})} \tag{19}$$

$$\mathbf{x} = \sum\_{i}^{\text{obsstades}} \mathbf{a}\_{i}^{(o)}(\mathbf{x}) + \mathbf{a}^{(w)}(\mathbf{x}) + \sum\_{i}^{\text{vertices}} \mathbf{a}\_{i}^{(c)}(\mathbf{x}) \tag{20}$$

$$\begin{split} \mathbf{x} &= \sum\_{i}^{\text{obsstances}} \frac{A^{+}}{d\_{i}(\mathbf{x})^{2}} \exp\left[\frac{A^{+}}{d\_{i}(\mathbf{x})}\right] \mathfrak{n}\_{i}(\mathbf{x}) + \min[A\_{p}^{-}d(\mathbf{x}), A\_{c}^{-}] \frac{\varpi - \mathbf{x}}{||\varpi - \mathbf{x}||} \\ &+ \sum\_{i}^{\text{vertices}} \frac{A\_{c}^{-}}{d\_{c}} d\_{i}(\mathbf{x}) \exp\left(1 - \frac{d\_{i}(\mathbf{x})^{2}}{2d\_{c}^{2}}\right) \end{split} \tag{21}$$

#### **2.4 Rotor modification**

As mentioned in sec. 2.2, the virtual potential approach to guidance is extremely susceptible to the appearance of local minima. A robust and simple approach is needed to assure local minima avoidance.

In terms of the vector field introduced by (2), the analytical solution of which is presented in (20 – 21), stable local minima occur due to the *irrotationality* of the field, rot **<sup>E</sup>**(*x*) id = 0.

In order to avoid irrotationality, and thereby local minima, decentralized control functions proposed in (7, 10, 12) are redesigned, adding a *rotor* component:

$$\mathbf{a}\_{i}^{(s)} \leftarrow \mathbf{a}\_{i} \qquad \mathbf{a}\_{i}^{\prime} \xleftarrow{\text{redef}} \mathbf{a}\_{i}^{(s)} + \mathbf{a}\_{i}^{(r)} \tag{22}$$

Where:

8 Will-be-set-by-IN-TECH

A good candidate potential contour generator of formation cell vertices, which behaves similar to a function *with local support*, is the *normal distribution curve*, adjusted for

; lim


*<sup>c</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> \ {0} is a positive real independent parameter dictating the scale of acceleration towards the cell vertex at the distance of maximum acceleration towards the vertex


The potential of a square formation cell surrounding an agent that figures as an obstacle is

The monotonicity of (6, 8, 11) ensures that the direction of the gradient of the potential, <sup>∇</sup> *<sup>P</sup>*(*x*)/ �∇ *<sup>P</sup>*(*x*)� <sup>∈</sup> **SO**2, is always <sup>±</sup>*n<sup>i</sup>* = (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>i*)/�*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>i*�. Therefore, since (1, 2) are linear, (2) can be solved *analytically* for any finite sum of terms of the form specified by (6, 8, 11) up

*i*

*Pi*(*x*)

*∂ ∂di*(*x*)

Equation (18) can be summarized by designating the terms in (7, 10, 12) as *a*(*o*,*w*,*c*) respectively.

*<sup>p</sup>* , *A*<sup>−</sup> <sup>0</sup> , *A*<sup>−</sup>

*<sup>d</sup>*→<sup>∞</sup> *pc* (*d*) = 0; lim

= ±*dc* (15)

<sup>0</sup> = 0; lim

*<sup>c</sup>* (16)

*<sup>c</sup>* · N (±*σ*) on a Gaussian normal distribution curve),

*d*→∞

*<sup>d</sup>*→0<sup>+</sup> *pw*(*d*) = <sup>−</sup>*A*<sup>−</sup>

*<sup>c</sup>* , *dc*). The procedure follows:

(−∇*pi*(*di*(*x*))) (17)

*pi*[*di*(*x*)] · *ni*(*x*) (18)

*<sup>i</sup>* , respectively, and represent the

*∂*

*<sup>c</sup>* (11)

(13)

(14)

*<sup>∂</sup><sup>d</sup> pc*(*d*) = 0 (12)

**2.3.3 Cell vertices**

attractiveness (i.e. of inverted sign).

*pc*(*d*) = −*A*<sup>−</sup>

*<sup>∂</sup><sup>d</sup> pc* (*d*) = *<sup>A</sup>*<sup>−</sup>

*<sup>∂</sup>d*<sup>2</sup> *pc* (*d*) = *<sup>A</sup>*<sup>−</sup>

*∂*

*∂*2

∴ *d*max = arg

*<sup>d</sup>*max id

(equivalent to the valuation of *A*−

represented in figure 5.

The terms *a*

(*o*,*w*,*c*)

*∂ <sup>∂</sup><sup>d</sup> pc* (*d*) *d*max id =*dc*

Where:


*<sup>c</sup> dc* exp

*d* exp <sup>1</sup> <sup>−</sup> *<sup>d</sup>*<sup>2</sup> 2*d*<sup>2</sup> *c* ; *∂ <sup>∂</sup><sup>d</sup> pc* (*d*) 

 <sup>1</sup> <sup>−</sup> *<sup>d</sup>*<sup>2</sup> *d*2 *c* exp <sup>1</sup> <sup>−</sup> *<sup>d</sup>*<sup>2</sup> 2*d*<sup>2</sup> *c* 

*<sup>∂</sup>d*<sup>2</sup> *pc* (*d*) !

= *A*−

= 0 

*c dc*

> *c dc*

*∂*<sup>2</sup>

function mapping a non-negative real to a real,

analogous to a Gaussian normal distribution curve).

to the values of the independent parameters (*A*+, *A*<sup>−</sup>

**2.3.4 Reformulation in terms of decentralized control functions**

− ∇ *<sup>P</sup>*Σ(*x*) = −∇ ∑

*<sup>i</sup>* · *<sup>n</sup>i*, can likewise be denoted *<sup>a</sup>*(*o*,*w*,*c*)

= ∑ *i*

= − ∑ *i*

 <sup>1</sup> <sup>−</sup> *<sup>d</sup>*<sup>2</sup> 2*d*<sup>2</sup> *c* 

> - *a*� *<sup>i</sup>* is the redefined *total decentralized control function* due to the *i*-th feature (the dash will hereafter be omitted),

> - *a*(*s*) *<sup>i</sup>* is the *stator* decentralized control function as introduced in the preceding section,


*<sup>a</sup><sup>i</sup>* <sup>=</sup> *<sup>a</sup>*(*s*)

*<sup>i</sup>* <sup>+</sup> *<sup>a</sup>*(*r*)

*<sup>i</sup>* <sup>∈</sup> **SO**<sup>2</sup> is the direction of the rotor decentralized control function,

of the superposition of the rotor and stator parts, are displayed in figure 6.


*w*


Formation Guidance of AUVs Using Decentralized Control Functions 109

The rotor decentralized control function and the *total* decentralized control function consisting

The formation introduced by the proposed framework is the *line graph occurring at the tile interfaces of the square tessellation* of **R**2, represented in figure 7. Due to a non-collocated nature of AUV motion planning, an important feature of candidate tessellations is that they be *periodic*

Each AUV whose states are being estimated by the current, *i*-th AUV, meaning *j*-th AUV, *j* � *i*) is considered to be a center of a formation cell. The function of the presented framework for potential-based formation keeping is depicted in figure 8. In an unstructured motion of the cooperative group, only a small number of cell vertices attached to *j*-th AUVs ∀*j* � *i*, if any,

*<sup>i</sup>* (*x*) id

= *r<sup>i</sup>* × *ni*(*x*),

*E* [m]

(b) A 2D display of ˆ*<sup>a</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>2</sup>

*<sup>i</sup>* and the two-term

*w*


*E* [m]

**3. Potential framework of formations**

*and regular*, which the square tessellation is.

Fig. 7. The square tiling of the plane.

(a) A 2D display of ˆ*a*(*r*) <sup>∈</sup> **<sup>R</sup>**<sup>2</sup>

Fig. 6. Direction of the rotor decentralized control function *a*(*r*)

*<sup>i</sup>* decentralized control function.

projected onto the "flight ceiling".


denoted with the superscript (s) to contrast it with the newly introduced *a*(*r*) *i* ,


The introduction of *a*(*r*) *<sup>i</sup>* establishes a non-zero rot(**E**) by design, as follows:

$$\begin{aligned} \text{rot } \mathbf{E}(\mathbf{x}) &= \sum\_{i} a\_{i}(\mathbf{x}) \neq 0 \\ &= \underbrace{\text{rot } \sum\_{i} a\_{i}^{(s)}(\mathbf{x})}\_{\stackrel{\scriptstyle \mathcal{Q}}{=} 0} + \text{rot } \sum\_{i} a\_{i}^{(r)}(\mathbf{x}) \\ &= \text{rot } \sum\_{i} a\_{i}^{(r)}(\mathbf{x}) \end{aligned} \tag{23}$$

With respect to the way-point, its potential influence on an AUV in this framework must not be prejudiced in terms of the direction of approach. If a decentralized control function of a way-point were augmented with a rotor part, the direction of *aw* would deviate from line-of-sight. The same is true of formation cell vertices. Therefore, the only non-zero rotor decentralized control functions are those of *obstacles*. As a result, (23) can be further simplified to:

$$\operatorname{rot}\,\mathbf{E}(\mathbf{x}) = \operatorname{rot}\sum\_{i}^{\text{obstacle}, \atop \text{vertices}} a\_i^{(r)}(\mathbf{x}) = \operatorname{rot}\sum\_{i}^{\text{obstacle}} a\_i^{(r)}(\mathbf{x}) \tag{24}$$

An individual *obstacle rotor* decentralized control function is defined below:

∀*i* = enum(obstacles)

$$\mathfrak{a}(\mathfrak{x}) = a\_r(\mathfrak{x})\mathfrak{a}\_r(\mathfrak{x}) \tag{25}$$

$$a\_l(\mathbf{x}) = \frac{A\_l^{(r)}}{d\_l(\mathbf{x})^2} \exp\left(\frac{A\_l^{(r)}}{d\_l(\mathbf{x})}\right) \tag{26}$$

$$\mathfrak{a}\_{\mathbf{r}}(\mathbf{x}) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ (0 & 0 & 1) \end{bmatrix} \cdot (\mathfrak{r}\_{\mathbf{i}}(\mathbf{x}) \times [\mathfrak{n}\_{\mathbf{i}}(\mathbf{x})](\mathbf{0})]^{\mathbf{T}}) \tag{27}$$

$$r\_i(\mathbf{x}) = \begin{bmatrix} \frac{\mathbf{w} - \mathbf{x}\_i}{\|\mathbf{w} - \mathbf{x}\_i\|} \end{bmatrix} (0) \cdot [\mathbf{n}\_i(\mathbf{x})](0)]^\mathrm{T} \tag{28}$$

$$r\_i(\mathbf{x}) = \begin{cases} r\_i = 1: & \mathfrak{n}\_i(\mathbf{x}) \times \left[ \frac{\mathfrak{v}}{\|\|\mathbf{v}\|} - (\frac{\mathfrak{v}}{\|\|\mathbf{v}\|} \cdot \mathfrak{n}\_i)\mathfrak{n}\_i \right](0) \right]^\mathrm{T} \\ 0 \le r\_i < 1: & \begin{bmatrix} \frac{\mathfrak{w} - \mathfrak{x}\_i}{\|\|\mathbf{w} - \mathbf{x}\_i\|} \|(0) \end{bmatrix}^\mathrm{T} \times [\mathfrak{n}\_i(\mathbf{x})](0) \mathrm{I}^\mathrm{T} \\ \text{otherwise}: & \overleftarrow{0} \end{cases} \tag{29}$$

Where:




10 Will-be-set-by-IN-TECH

*<sup>i</sup>* is the *rotor decentralized control function*, all of which are continuous real 2D vector fields over the Euclidean 2-space (mapping **R**<sup>2</sup> to itself) such that they Jacobians exist wherever each

*<sup>i</sup>* establishes a non-zero rot(**E**) by design, as follows:

� �� � id =0

+ rot ∑ *i a*(*r*) *<sup>i</sup>* (*x*)

obstacles ∑ *<sup>a</sup>*(*r*)

*a*(*x*) = *ar*(*x*)*a*ˆ*r*(*x*) (25)

�

� *<sup>v</sup>* �*v*� <sup>−</sup> ( *<sup>v</sup>*

= rot ∑ *i a*(*s*) *<sup>i</sup>* (*x*)

= rot ∑ *i a*(*r*)

With respect to the way-point, its potential influence on an AUV in this framework must not be prejudiced in terms of the direction of approach. If a decentralized control function of a way-point were augmented with a rotor part, the direction of *aw* would deviate from line-of-sight. The same is true of formation cell vertices. Therefore, the only non-zero rotor decentralized control functions are those of *obstacles*. As a result, (23) can be further simplified

*a*(*r*)

⎛ ⎝

*ri* = 1 : *ni*(*x*) ×

*<sup>i</sup>* ∈ **R** \ {0} is a positive real independent parameter dictating the scale of acceleration

0 ≤ *ri* < 1 :

otherwise : �0

*<sup>i</sup>* (*x*) = rot

*A*(*r*) *i di*(*x*)

⎦ · (*ri*(*x*) × [*ni*(*x*)

� *<sup>w</sup>*−*x<sup>i</sup>* �*w*−*xi*� � � �(0) �T

⎞

*i* ,

*<sup>i</sup>* (*x*) (23)

*<sup>i</sup>* (*x*) (24)

⎠ (26)

�(0)]<sup>T</sup> (28)

�*v*� · *<sup>n</sup>i*)*n<sup>i</sup>*

× [*ni*(*x*) � �(0)] T

�(0)]T) (27)

� � �(0) �T

(29)

denoted with the superscript (s) to contrast it with the newly introduced *a*(*r*)

*ai*(*x*) �= 0

obstacles, w.p., vertices ∑ *i*

An individual *obstacle rotor* decentralized control function is defined below:

*i di*(*x*)<sup>2</sup> exp

� *<sup>w</sup>*−*x<sup>i</sup>* �*w*−*xi*� � � �(0) � · [*ni*(*x*) �

100 010 (001)

⎤

*ar*(*x*) = *<sup>A</sup>*(*r*)

⎡ ⎣

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

*<sup>i</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> is the magnitude of the rotor decentralized control function,

*a*ˆ*r*(*x*) =

*ri*(*x*) =

*ri*(*x*) =

perpendicular to the direction of fastest flight from the obstacle,

rot **E**(*x*) = ∑

rot **E**(*x*) = rot

∀*i* = enum(obstacles)

*i*


to:

Where: - *A*(*r*)


of them is defined. The introduction of *a*(*r*) - *<sup>n</sup>i*(*x*) <sup>∈</sup> **SO**<sup>2</sup> is the unit vector in the direction of fastest flight from the *<sup>i</sup>*-th obstacle,



The rotor decentralized control function and the *total* decentralized control function consisting of the superposition of the rotor and stator parts, are displayed in figure 6.

Fig. 6. Direction of the rotor decentralized control function *a*(*r*) *<sup>i</sup>* and the two-term *<sup>a</sup><sup>i</sup>* <sup>=</sup> *<sup>a</sup>*(*s*) *<sup>i</sup>* <sup>+</sup> *<sup>a</sup>*(*r*) *<sup>i</sup>* decentralized control function.
