Real-Time Ray-Tracing for Underwater Distance Evaluation with Application to Distributed Localization of AUV Teams

Real-Time Ray-Tracing for Underwater Distance Evaluation with Application to Distributed Localization of AUV Teams G. Casalino*°, A. Caiti**°, A. Turetta*°, E. Simetti*° 

*DIST- University of Genova, Genova, Italy (e-mail: casalino, turetta, simetti @dist.unige.it) **DSEA-University of Pisa, Pisa, Italy(e-mail: [email protected]) °ISME – Interuniversity Research Centre on Integrated Systems for the Marine Environment Abstract: The paper deals with distributed acoustic localization of teams of Autonomous Underwater Vehicles (AUVs) and proposes a novel algorithm, Real-TimeRay-Tracing (RT2), for evaluating the distance between any pair of AUVs in the team. The technique, based on a modified formulation of the non-linear sound-ray propagation laws, allows efficient handling of the distorted and reflected acoustic ray paths. The proposed algorithm can be easily implemented on-board of low-cost AUVs, requiring the presence, on each vehicle, of an acoustic modem and a pair of look-up tables, a-priori built on the basis of the assumed knowledge of the depth-dependent sound velocity profile. On such a basis, every AUV can compute its distance w.r.t. to any other neighbour team member, through time-of-flight measurements and the exchanges of depth information only. Keywords: Marine Systems, Communication Control Applications and Systems, Estimation Algorithms and Theory, Localization systems, Navigation Systems, Motion Estimation, Autonomous vehicles, 

1. INTRODUCTION The paper deals with acoustic distributed localization of teams of Autonomous Underwater Vehicles (AUVs) and, more specifically, with the problem of accurately evaluating the distances between vehicles, by explicitly keeping into account the propagation anisotropy of the underwater medium (i.e. the changes in the sound velocity profile as a function of depth). The problems posed by ray distortions are well known in the underwater acoustic literature and induce significant implications on the problem of localization. Indeed, an underwater device receiving an acoustic signal cannot precisely estimate its range from the sound source, through the sole measurement of the Time-of-Flight (ToF) of the received wave, but must keep into account also the effect of the acoustic propagation laws applied to the specific sound speed profile of the considered area. Indeed, acoustic localization devices, such as USBL and LBL systems (Ultra Short & Long Base Line, see for instance Philip (2003)), can provide corrections to range estimates, through acoustic raytracing and measurements of ToF and Angle of Arrival (AoA). Despite their efficiency, there is a concrete limit in the applicability of the above systems in the framework of team cooperating AUVs. Such instruments are conceived for tracking the position of a mobile underwater device, while they do not allow the underwater device be aware of its position, which remains an information available only at a centralized level; instead, for achieving a good level of teamcooperation, it is mandatory that each team member knows its position with the maximum achievable precision. Moving

from this consideration, this paper aims to contribute to the development of algorithms and methods based on ToF measurements and depth information sharing, as recently proposed by Bahr et.al (2009). Previous treatments of the acoustic path distortion effects has been described by Caiti et.al (2005) and Kussat et.al (2005), where LBL systems are considered, and the ray correction is not performed on board the vehicle(s). Xiang et.al (2007) proposed considering straight ray paths, using as constant sound speed the weighted depth average of the true sound speed. This latter solution may be useful when propagation occurs mostly on the vertical, but it may be not accurate at oblique angles. In this paper a novel algorithm is proposed, based on a modified formulation of the sound-ray propagation laws allowing to efficiently handle the distorted and reflected acoustic ray paths and to produce accurate evaluations of distances between pairs of AUV within a team, as well as between an AUV and the buoy elements of an LBL system. Further, the implementation of the proposed RealTime Ray-Tracing (RT2) technique is very simple. It just requires the presence, on-board each vehicle, of an acoustic modem (already necessary for coordination purposes) and of a pair of look-up tables, a-priori built from the knowledge of the depth-dependent sound velocity profile. The proposed RT2 algorithm can be easily hosted even on-board of lowcost AUVs. Among the others, we can here mention the AUV “Folaga”, developed by our research team and described by Alvarez et al. (2009), on which we expect to experiment the RT2 method. Other than presenting details of the algorithm, the paper will also show how its adoption reduces the

distributed localization problem for AUV teams to its aerial counterpart, for which efficient filtering techniques for distributed localization already exist (as described by BarShalom et. al. (2001), Ristik et. al. (2004), Thrun et. al. (2005). Some of these techniques are therefore expected to be transferable to the underwater field, significantly reducing the high level of criticality generally posed by underwater medium anisotropies. The paper is organized as follows: in section two a brief review of soundray propagation theory is given, and the formulation used in this work introduced. In section three the computational blocks for the RT2 algorithm are described. In the fourth section the complete RT2 algorithm is illustrated. Section five discusses the application of RT2 in distributed localization problem. Finally concluding remarks are given 2. SOUND RAY THEORY PRELIMINARIES Let us consider a cylindrical coordinate system, with the origin at the sea surface and the z axis pointing downward toward the sea bottom. It is assumed that the sound speed in water can vary only with depth and not with range from the source (depth-dependent sound velocity profile – see Fig. 1, left). With these assumptions, acoustic propagation is represented in the plane (z–r) of depth and range, independent from the azimuth angle (Fig. 1, right). Acoustic propagation will be treated with a ray-theory approach, which is a valid approximation within the usually high (tens of kHz) transmission frequencies used in underwater localization and communication. Moreover, the assumption of a range independent environment allows to use the ray equations for a stratified media as reported in (Jensen et. al. (2000) p.176). The acoustic source is assumed to be at 0 horizontal range. Let c( z )  0 denote the depth dependent sound velocity profile function; the Snell’s refraction law is expressed as: cos  o   (1) cos   k c( z ) ; k   [ 0 , 1 / c( z )] ;   [ - ; ] c( z o )

o

o

2 2

acoustic channels. Considering (1) and the kinematic equations r  c( z ) cos  and z  c( z ) sen , we get: r  kc 2 ( z )   2 2  z   c( z ) 1  k c ( z )     arcos [kc( z )]

where the “+” sign indicates the downward ray path and the “–“ sign refers to the upward one, and where for given ( z o ; k ) the couple of equations for z are defined within the interval (2). Upon integration, each equation in (3) monotonically reaches its associated interval extreme at the finite times (see equation 3.101 from Jensen et. al (2000)): z  d   td ( zo ,k )  f ( k , z ) dz 1  zo ; f ( k , z )   zo c ( z ) 1 k 2c 2 ( z )   t ( z , k ) f ( k , z ) dz   u o    zu



c( zo )

r

c

zo

zo

o



z

z

Fig. 1: left: sample sound speed profile as a function of depth in the ocean; right: cylindrical coordinate system and acoustic ray path accordingly to Snell’s law and to the given sound speed profile of left. 1/ k 1/ k

 the grazing angle at a generic other depth z , among those

the left information we actually refer to a couple of trajectories, just as in Fig.1. Note that, since 1 / k  c( z ) , for any given starting condition ( zo ; k ) the resulting trajectories will time-evolve within the depth range, including zo : z  [ zu ( k ) , z d ( k )] ; k  [ 0 , 1 / c( zo )] (2) with zu ( k ) , zd ( k ) the abscissas of the first-encountered upper and lower intersection points that the vertical line 1 / k  c( z o ) forms with c( z ) , when seen as starting from zo . This is illustrated in Fig. 2, where for a given zo different values of 1 / k are shown. For any given k , z u ( k ) is set to zero (surface level) and z d ( k ) set to z M (bottom level) whenever the corresponding vertical segment results with one or both its end-points on one or both such extreme levels. So, depending on the shape of c( z ) , some rays may remain confined between two definite depths included between surface and bottom, giving rise to the so called

(4)



being  o the grazing angle at the ray starting depth z o , and successively traversed by the same ray (Fig.1, right). In representation (1), since ( z o ; k )  ( z o ;   o ) , when giving

(3)

zo

z

c

zd ( k )

zu ( k )

zd ( k )

1/ k

c

1/ k

c

1/ k

zo

1/k

1/ k

z

z

Fig. 2: assuming the sound speed profile of Fig. 1, left, maximum and minimum reachable depths by a given acoustic ray as a function of 1 / k In case both extrema do not coincide with the sea bottom and the water surface, it results z  0 . Such occurrences correspond to the achievement of the finite-time equilibria points z d ( k ) , zu ( k ) . However, such equilibria can be shown to be both unstable: any whatever small perturbation will force each z-motion having reached one of the extrema to

reverse its direction, yielding to periodic evolutions. This result can be obtained more rigorously including all the terms from the wave equation, in particular the dynamics for k and for   sen / c( z ) (see again Jensen et. al. (2000), p.152); this is the case usually referred in physics as total reflection. In case one or both of the extrema coincide with the surface and/or the bottom (Fig.2, middle and right) we must account for surface/bottom reflections, also imposing a periodicity to the rays capable of reaching such extrema. This is qualitatively depicted in Fig. 3, which is relevant to the case of Fig. 2, middle. 1/ k

t

c

t ,r zu za T ;R 3 3  ab ; rab

2  ab ; rab2

zb

4 4  ab ; rab

1 1  ab ; rab

zd

r

 ab ; ab

1/ k

zu zo

zu zo

 au ; au c( z )

 bd ; bd

Fig 4: ray motion in time and range as a function of depth in correspondence of a given k. The intersections of 1/k with the sound speed profile c(z) define the uppermost and downmost depths z d ( k ) , zu ( k ) ; for any given pair of depths za , zb the period in depth/range is given by the sum of the time/range components  ij ; ij , where i,j are the suffix indicating the

zd tu ru

td rd

T( k ) R ( k )

Fig.3: periodic evolution of a ray, with total reflection in the water column and reflection from the sea bottom From (4), the time period T( k ) (see Fig.3) is computed as: z

d

T ( k )  2

 f ( k ,z ) dz

(5)

 zu

Further, the ratio of the second and the first in (3) leads to: dz 1  1  k 2c 2 ( z ) dr kc( z )

(6)

which establishes a one-to-one correspondence between the pair of time-z-motions z ( zo , k , t ) with the couple of distance-z-motions z ( zo , k , r ) resulting from the integration of (6). The representation of the pair of distance-z-motions on the z , r plane also exhibits periodic behaviours, which are qualitatively similar to those of their companions time-zmotions; this explains why we have actually used the same Fig. 3 for representing both type of motions. The resulting distance-period R( k ) (by causally inverting (6) within a period, see equation 3.97 from Jensen et. al. (2000)) is: z

d

R( k ) 

 g( k , z ) dz

 zu

; g ( k , z ) 

k c( z ) 1  k 2c2 ( z )

(7)

3. RT2 HORIZONTAL DISTANCE EVALUATION Given a pair of depths za , zb , for any assigned admissible parameter k (satisfying 1 / k  max  c( za ), c( zb )  ) the data sets Ta b ( k ) , Ra b ( k ) are defined as all the possible traveltimes and their one to one associated horizontal-traveldistances connecting the depths za , zb .

depth points a, b, u, d. Noting that the horizontal-travel-distance (or equivalently the travel-time) of any ray connecting the depths za , zb is constituted by an integer number of horizontal distanceperiods (time-periods) plus one out of four possibilities of connection (as in Fig. 4, where notation is introduced), we have the following expressions for the sets Ta b ( k ) , Ra b ( k ) :

 

 Ta b ( k )  Ta b ( k )  n T ( k ) Ta b ( k )   ajb ( k ) ; j  1,.., 4 ;  j  R a b ( k )  R a b ( k )  n R( k )  R a b ( k )  ra b ( k ) ; j  1,.., 4

 

(8)

where the finite sets Ta b ( k ) and Ra b ( k ) respectively admit the following representations:   1a ,b  2  a ,b Ta b ( k )   3   a ,b  4   a ,b

  a ,b

 ra1,b   ra2,b  Ra b ( k )   3  ra ,b  4  ra ,b

  a ,b

     a ,b  2 b ,d     a ,b  2  a ,u     a ,b  2  a ,u  2 b ,d   (k )

(9a)

     a ,b  2  b ,d     a ,b  2  a ,u     a ,b  2  a ,u  2  b ,d   (k )

(9b)

From (8) and (9) it follows that, in correspondence of any given triple z a , zb , k any element of the Ta b ( k ) , Ra b ( k ) sets can be generated by knowing a finite number of parameters: the period ( T ( k ) , R( k ) ) plus the finite sets Ta b ( k ) , Ra b ( k ) . This information can be conveniently stored in a pair of lookup tables, the partial-travel-time (PTT) and partial travelleddistance (PTD) tables, constructed as follows. Assume the

sound velocity profile function c( z ) is a-priori available. Then, for any 1 / k (made gradually increasing via a suitable step increment) detect its associated segments where 1 / k  c( z ) . In correspondence of each one of them, evaluate the following integral functions (by arbitrarily considering either the “+” or the “-“ forms)

blocks, respectively accomplishing the following tasks. Given a couple of depths z a , z b , a travel-time  (travelleddistance r ) and a given k , check if the two depths can be connected at time  (with distance r ) by a ray with the given k ; and in case provide the corresponding associated travelled distance ra b ( k , ) (travel time  a b ( k , r ) ) .With reference to

z   ( k , z )   f ( k , z ) dz  z  z ; z  ( zu , z d )  z    z ( k , z )   g( k , z ) dz  z 

the above tasks, the following couple of similar algorithms can be used



(10)



where z is the abscissa of any absolute minima of c( z ) within the associated segment ( zu , z d ) (Fig. 5). Once both PTT and PTD tables have been filled with terms z ( k , z ) ,

TTC- Algorithm

TDC- Algorithm

input V a b ( k )

input Va b ( k )

n   / T ( k )

n  r / R( k )

    n T ( k )

r  r  n R( k )

if   Ta b ( k )

if r  Ra b ( k )

k valid

k valid

 z ( k , z ) , for any given triple z a , zb , k the terms appearing in

i  index of 

(9a,b) can be directly evaluated via a finite number of simple look-up operations:

ra b ( k , ) 

  a ,b ( k )   z ( k , z a )   z ( k , zb )     b d ( k )   z ( k , zb )   z ( k , z d )    a u ( k )   z ( k , z a )   z ( k , zu )

(11a)

T ( k )  2  z ( k , zu )   z ( k , zd )

i  index of  )  n R( k )

 a b ( k , r )   ai b ( k )  n T ( k ) else k invalid

4. RT2: HORIZONTAL DISTANCE EVALUATION

By composing the processing ingredients introduced in the previous section, we are now ready to describe the complete implementation of the Real-Time Ray-Tracing (RT2) algorithm for horizontal distance evaluation. An underwater vehicle B is located at depth zb , known since provided by its (12a) on-board depth-meter. It receives a signal from another vehicle A transmitting from depth level z a . Such signal is assumed encoding the time instant t a of its transmission, plus (12b) its departing depth z a . The vehicle clocks are assumed to be synchronized. It is also assumed that vehicle B can discriminate the first arrival of the signal from vehicle A; thus separating it from its subsequent, lower intensity, replicas due to multi-path. Note that these assumptions are similar to those routinely used in quite a number of existing c(z) commercial or prototypal equipments. In addition, it is further required that the vehicles have the availability of the pair of look-up-tables PTT and PTD (a-priori evaluated from the assumed knowledge of the sound profile c(z) ).

(11b)

  a b ( k )   z ( k , za )   z ( k , zb )     b d ( k )   z ( k , zb )   z ( k , z d )    a u ( k )   z ( k , z a )   z ( k , zu )

R( k )  2  z ( k , zu )   z ( k , z d )

from which follows Ta b ( k ) , Ra b ( k ) , Ta b ( k ) , Ra b ( k ) . t r

z

z

else k invalid

rai b ( k

z 1/k

Fig. 5: graphical representation of the integration in eq. (10) The PTT and PTD look-up tables, whose computation has just been described, can each now be considered as basic functional blocks that, given in input the triple z a , z b , k ,

provide as output the data vector Va b ( k )   T , Ta b R , Ra b ( k ) .

Such blocks will represent the fundamental ingredients of the complete processing system for horizontal distance evaluations. Other necessary processing components are the travel-time-checker (TTC) and travel-distance-checker (TDC)

Under the above assumptions, vehicle B can evaluate (via trivial difference) the travel time τ spent by the sound ray that, as the first one received among many possible others, has reached B starting from A. From the computed travel time, the following processing (see Fig. 6) provides a minimal set of candidate horizontal distances from vehicle A. First of all, the functional block PPT-PTD, containing the look-up tables, receives as input the couple of considered depths z a , z b . Then, at each stage i  1, 2 ,.... N of the algorithm, such a block receives the value 1 / ki , which is used for accessing the look-up tables and computing the vector Va b ( k i ) , as described in the previous section. The resulting Va b ( k i ) is then given as input to the unique TTC block, to a group of, say M, TDC blocks and to a shift buffer.

za

1 / ki

zb PT T P TD

Va b( ki )

shift buffer

i N



TTC

rh h  1 h  2

h  n

b1 ( i )

TDC

1

TDC

2

TDC

b2 ( i )

n

bn ( i )

Fig. 6: block diagram of the RT-2 algortihm The job of the TTC block is to check, at every stage, if the measured travel time  can be obtained by the data contained in the input vector Va b ( ki ) . If such is the case, the resulting distance rab ( i )  rab ( k i , ) represents one of the candidate horizontal distances compatible with the measured travel time  . Such a distance is hence given as input to the first available TDC block. At the beginning of the algorithm every TDC block is inactive, with its output set to zero. When, at a certain stage, say the p-th, an inactive TDC, say the j-th, receives a candidate distance rab ( p ) , it becomes active ( b j  1 ) and stores the value rab ( p ) . Then, at any successive i-th stage ( i  p  1,..., N ) , it receives a new vector Va b ( k i ) and checks if the associated horizontal distance rab ( p ) could have been covered by a different ray, now parameterized by the new value ki  k p . If such an occurrence is found at a later stage, say q, the considered TDC block then computes the corresponding “virtual” travel-time  ab( ( k q ,rab ( p )) and compares it with the measured time  ; two cases can happen: 1.    ab( ( k q ,rab ( p )) which implies that the considered “virtual” ray would have arrived not before than the measured one; 2.    ab( ( k q , rab ( p )) which implies that the considered “virtual” ray would have been received before the measured one, if the vehicle were at the considered horizontal distance rab ( p ) . In the first case, the block maintains its output as active and keeps on working in the same way for any successive i-th stage ( i  q  1,..., N ) . In the second case, instead, the TDC block becomes inactive ( b j  0 ) and the associated candidate horizontal distance rab ( p ) is discarded. Indeed, if the vehicle were at the horizontal distance rab ( p ) , it would have been reached by the ray parameterized by k  k q at an earlier time than  ; i.e. against the previous assumption that the first pulse is received at time  .

The overall process is then continued till the N-th stage. At that point, the set of TDC blocks with a non-zero output bit contains the set of, say m, candidate distances among which looking for the real one. In case m=1 the only valid candidate distance is obviously coincident with the real one and the procedure is completed. Otherwise a simple further attempt to reduce the set of candidates may be performed. To this aim note that, in the procedure so far described, for any computed candidate distance rab ( p ) , the associated TDC block performed consistency checks only by considering the set of vectors ( Va b ( k i ) ; i  p  1,..., N ). Since it is certainly convenient performing consistency checks w.r.t. all the possible vectors Va b ( k i ) , also the set of vectors obtained for i  1,..., p  1 should be considered for sake of completeness.

To this aim a “shift buffer” is introduced in Fig. 6, for temporarily storing the sequence of vectors Va b ( k i ) , in order to (eventually) allow the execution of all the still pending checks, once stage N has been reached. At the end of the overall process, the remaining output bits set to one indicate the resulting minimal set of horizontal distance candidates. An example where the minimal set contains more than one solution, even if unlikely from the point of view of oceanographic conditions and source-receiver geometry, is the one where receivers at different ranges, but at the same depth, may receive as first arrival a signal from the same source at the same time. However the non-uniqueness problem is not crucial in multiple vehicles localization: in fact, in order to completely localize one vehicle, range estimates from multiple vehicles are needed, to set up a trilateration scheme. The scheme itself will rule out the range solutions not corresponding to the physical situations (it may be argued that there may still be cases in which the ambiguity is not resolved with three or even more vehicles; however such cases require a combination of particular geometric symmetries in the relative position of the vehicles, together with specific environmental conditions, that their practical occurrence at sea can be ruled out remorselessly). Moreover, even with two vehicles, a change in relative position (for instance in depth) will lead to the loss of the multiple solutions not corresponding to the true one; hence the presence of spurious solution is unstable and resolvable within the trilateration phase. 4. APPLICATION TO DISTRIBUTED LOCALIZATION The framework underlying group localization is represented by a set of AUVs, each one equipped with acoustic modem. The presence at surface level of a vehicle or fixed station with GPS localization is also assumed. Any generic j-th vehicle (via modem and while moving) periodically broadcasts a pulse immediately followed by a data-stream containing (for a total of eight numbers) the information: a) Its identification code j; b) The pulse-emission instant ti j ; i  1,2 , ... c)

Its depth zi j at the pulse-emission instant ti j

d) The prediction ˆ i j/ i 1 location  i j at instant ti j e)

The variance pij/ i 1 of ˆ i j/ i  1

It is meanwhile required that any other generic k -th vehicle differing from the surface ones must be able to translate its previous prediction ˆ ik/ i 1 of horizontal location  ik (relevant to the pulse-reception instant tij  ti j and generally obtained via time-integration of its own odometric data) into the upgraded estimate ˆ ik/ i of the same location at same time. Note that we are actually requiring the production of prediction and estimates relevant to horizontal positions only, since the shared knowledge of depths straightforwardly projects the problem into the horizontal plane. The possibility offered by the RT2 technique of the previous section of directly measuring the horizontal distances between AUVs, even in presence of non-negligible ray distortions and reflections, reduces the problem to be of the same type of those generally considered within (planar) range-based localization. As a consequence most of the already existing terrestrial techniques for distributed localization are expected to be adoptable also for the underwater environment, without any further major modification. More precisely, it is deemed here that among the most appreciable advantages that could emerge from the adoption of the RT2 method, the following could reveal a particular significance: 1) In terrestrial applications, the Gaussianity of the distance measurement noise is a reasonable assumption, that motivates the adoption of parametric filtering methods, like Kalman, Extended Kalman, or even Iterated Kalman Filters [BarShalom et. al. 2001, Ristik et. al. 2004, Thrun et. al. 2005], easier to implement than non-parametric methods, as Particle Filtering [Ristik et. al. 2004, Thrun et. al. 2005, Maurelli et. al 2008]. On the contrary, neglection of ray travel path in acoustic range measurements produces systematic errors that can hardly be modelled as Gaussian. However, ray travel path is considered by the RT2 algorithm, likely removing one relevant, if not predominant, source of non-gaussianity in the measurement disturbances. In other words, the introduction of RT2 makes the assumption of Gaussian noise in horizontal distance measurement much less critical than when roughly used without keeping into account sound-wave distorting and reflecting effects. As a matter of fact, the need of avoiding the criticality posed by Gaussian assumptions, when in presence of ray propagation anisotropies and reflections, may be one of the basic motivation underlying the recent proposal of use of much more sophisticated (but also more difficult to implement) non-parametric filtering techniques. 2) Within underwater distributed group localization, one practical possibility to mitigate the problem of sound ray distortions and reflections (even those induced by possible acoustic channels) consists in keeping each vehicle sufficiently close to its neighbours. With the adoption of RT2 such type of structuring of the AUVs group is not anymore necessary. The net consequence is to allow more sparse (and then more volumetric-extended) configurations of the team.

laws can be managed in such a way to allow the preliminary construction of a pair of look-up tables. On the basis of the look-up tables, that can be updated as the environmental conditions change, a simple algorithm (the Real Time Ray Tracing – RT2) can in turn be defined, for accurately evaluating the horizontal-distance separating two vehicles, even in the presence of non-negligible sound ray distortions and reflections. Via this RT2 method, the AUVs distributed localization problem becomes the same of its terrestrial analogous one, for which consolidated, easy to implement and generally effective, parametric filtering methods already exist that could be transferred, with equal effectiveness, to the more complex underwater field. Nevertheless, the developed RT2 method does not prevent the use of nonparametric, nonlinear estimation methods. REFERENCES Alvarez, A., Caffaz, A., Caiti, A., Casalino, G., Gualdesi, L., Turetta, A., Viviani, R. (2009). Folaga: a very low cost autonomous underwater vehicle combining glider and AUV capabilities, Ocean Engineering, vol.36 No 1, pp. 24-38. Bahr, A., Leonard, J.J., Fallon, M.F. (2009). Cooperative Localization of Autonomous Underwater Vehicles, The Int. J. Robotic Research, Vol 28, No.6, pp. 714-728. Bar-Shalom, Y., Rong-Li, X., Kirubarajan, T. (2001). Estimation with Apllication to Tracking and Navigation, John Wiley, New York. Caiti, A., Garulli, A., Livide, F., Prattichizzo, D. (2005). Localization of Autonomous Underwater Vehicles by Floating Acoustic Buoys: a Set Membership Approach. IEEE J. Oceanic Engineering, Vol.30, No. 1. Jensen, F.B., Kuperman, W.A., Porter, M.B., Schmidt, H. (2000). Computational Ocean Acoustics, Springer, New York. Kussat, N.H., Chadwell, C.D., Zimmerman, R. (2005). Absolute Positioning of an Authonomous Underwater Vehicle. IEEE J. Oceanic Engineering, Vol.30, No. 1. Maurelli, F., Krupinsky, S., Petillot, Y., Salvi, J. (2008). A particle Filter Approach for AUV Localization, in Proc. IEEE Oceans 2008, Quebec city, Quebec, Canada. Philip, D.R.C. (2003). An Evaluation of USBL and SBL Acoustic Systems and the Optimization of Methods of Calibration, The Hydrographical Journal, No. 108, pp. 18-25. Ristic, B., Arulampalm, S., Gordon, N. (2004). Beyond the Kalman Filter, Artech House, Boston. Thrun, S., Burgard, W., Fox, D. (2005). Probabilistic Robotics, MIT press, Boston. Xiang, X., Xu, G., Zang, Q., Guo, Y., Huang, X. (2007). A novel Acoustic navigation Scheme for Coordinated Heterogeneous Autonomous Vehicles”, Proc. IEEE Int. Conf. Mechatronics Automation, Harbin, China.

5. CONCLUSIONS

ACKNOWLEDGMENTS

By assuming the depth dependent sound velocity profile apriori available and the AUVs equipped with acoustic modems, the paper has shown how the sound-ray propagation

This work has been partially supported by the EU-FP7 project COG3 –AUV – Cognitive Cooperative Control of AUVs, Grant Agreement No. IST – 231378.