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ADAPTIVE ON LINE PLANNING ALGORITHM FOR AUVS EXPLORATION OF UNKNOWN OCEAN ENVIRONMENTS
ADAPTIVE ON LINE PLANNING ALGORITHM FOR AUVS EXPLORATION OF UNKNOWN OCEAN ENVIRONMENTS Andrea Caiti, Andrea Munafò and Riccardo Viviani ISME, Interuniversity Ctr. Integrated Systems for the Marine Environment, c/o DSEA – University of Pisa – via Diotisalvi 2 – 56100 Pisa, Italy E-mail: [email protected] - [email protected] - [email protected]
Abstract: An adaptive on-line planning algorithm for multiple underwater vehicles in coastal oceanographic missions is introduced. The algorithm is tested for a team of autonomous underwater vehicles exploring an unknown oceanic environment in which they can execute point-wise environmental measurements. The team final objective is the construction of a three-dimensional estimated map of the measured environmental characteristics of the whole marine region. Each vehicle chooses its next sampling point in order to maximally increase the confidence level of the overall estimate of the measured quantity and in an asynchronous way with respect to the other vehicles. It is assumed that all vehicles have availability of the positions and values of the all past measurements executed by the members of the team. Simulative test cases in which the objective is the estimation of the ocean temperature over a region are presented. The algorithm performance is compared with that theoretically achievable by Rapidlyexploring Random Trees search. Copyright © 2006 IFAC Keywords: Adaptive Control, Mobile Robots, Trajectory planning and control, Underwater Robots, Vehicles.
1. INTRODUCTION The research in cooperating and planning strategies for Autonomous Underwater Vehicles (AUVs) has seen an increasing interest in the last few years, after a slower start with respect to other fields of robot cooperation. While teams of cooperating robots have been investigated since quite a long time, and there are several examples of land and airs robots cooperating in many applications, from de-mining to soccer playing, to formation control problem of teams of Unmanned Aerial Vehicles (UAV), the problem of cooperating AUVs has seen a difficult start because of the communication and localization constraints posed by the marine environment; in particular the severe bandwidth limitation of underwater communication and the absence of a GPS-like localization system makes it very hard to simply transpose general robotic techniques to the underwater scenario. Up to now, to the Authors
knowledge, the AUVs cooperation has been intended as the use of AUVs with different capabilities and measurement instrumentation within the same geographical area, so that each vehicle can fulfil a specific goal (e.g. a bottom crawling robot together with a navigation vehicle) without interaction with each other (Kermorgant, 2005). From this point of view, which does not involve interaction among the different vehicles, the cooperation is merely the exploitation of different missions running in parallel, often with ad-hoc solutions and considerations. A second kind of AUVs cooperation is referred as AUVs formation control, for mission requiring the reaching of a specific geographical location by at least one vehicle (e.g. save and rescue or mine counter measurement operations) (Spennenberg, et al., 2005; Ikeda, et al., 2005); a particularly interesting approach within this line of research has been proposed and investigated by Ögren, et al. (2004), that have considered ocean gliders with the
task of identifying the gradient of environmental oceanographic quantities. Finally the investigation on cooperating AUVs is intended as mission adaptation with respect to the specific mission task; along this line there are different examples, discussed as for acoustic scattering and propagation measurements (Schmidt and Edwards, 2002), and environmental mapping and self-localization (Fenwick et al., 2002). Popa et al. (2004) propose, along this line of research, a general methodological approach, including concurrent localization and mapping as well as adaptive oceanographic sampling. In this paper, following this last line of research and therefore within the framework of adaptive oceanographic sampling, a cooperation algorithm is considered. The algorithm is suited for a team of autonomous vehicles taking point wise environmental measurements over an ocean region, with the final objective of producing an estimated map of the measured environmental quantity over the whole region. After any measurement, each vehicle plans its next sampling point in order to maximally increase the confidence level of the overall estimate of the measured quantity. A link between the confidence level and the measured quantity permit to obtain algorithm adaptation; in particular, each measurement point is linked to a “confidence function” that indicate how far can be spatially extended the measurement in the neighbourhood of the point itself starting from the assumption that the measurement does not significantly change in proximity of the sampling point. Confidence extension depends in general on the local smoothness properties of the environmental parameter, which can be assessed on the basis of the available measurements themselves. A global confidence function over the whole region of interest can be obtained by the composition of all the confidence functions associated to the available measurements. Cooperation among the vehicles is obtained since the algorithm computes the next sampling point for each vehicle taking into account also the (known) position of the other vehicles in the team, in addition to the current confidence level and the local smoothness of the environmental map. The next sample point for each vehicle is obtained by choosing the point that maximizes the estimate confidence and that minimizes the number of sampling points required by the whole team. The algorithm is asynchronous, i.e., each vehicle can compute locally its next sampling point right after it has completed the measurement at a given point without waiting for the other vehicles or any other type of synchronization; however, the algorithm also requires communication among all the vehicles in the team through a an information relay (e.g., a shore station or a surface platform), since global knowledge of the past measurements is required. Although the proposed algorithm is fairly general and vehicle-independent, it has been devised for application to the Fòlaga vehicle (Alvarez, et al., 2004; 2005), a very low cost autonomous vehicle developed for coastal oceanographic applications, in which all the communications among the vehicles can take place at the sea surface through radio links. Experimental
testing of the proposed cooperation approach is expected by spring-summer of 2006. The paper is organized as follows: in the next section the general algorithm is described formally. In the third section, a particular instance of the algorithm is described and simulative results are reported. In all the simulations, the physical quantity to be estimated is the ocean temperature. Temperature estimation is based on supervised neural networks and Empirical Orthogonal Functions (EOFs). The cooperative behaviour of two and three AUVs in several different environmental conditions is reported. In particular, it is shown how in the case considered the proposed method allows to reduce the amount of necessary measurements, all else being equal, as compared with Rapidly-exploring Random Trees (RRT) (Tan et al., 2005). Finally, future work along this line of research is outlined and conclusions are given. 2. GENERAL ALGORITHM In this section we describe the on-line planning algorithm. The framework follows that presented in (Caiti, et al., 2005) with an important difference: while in our previous work we analyzed a synchronous algorithm, now we consider the asynchronous case, i.e., each vehicle can compute locally its next sampling point right after it has completed the measurement at a given point. Let us suppose we have the availability of n AUVs, each equipped with a sensor able to point-wise sample an oceanographic quantity θ at the geographical coordinates (x, y) over the entire depth profile, i.e. θ = θ ( x, y ) . Through the use of the EOFs we can take into account the depth dependence of the oceanographic quantity (e.g. temperature, salinity, oxygen content, etc.), as explained in the next section. As for the dependence on time, it is assumed that the quantity is stationary, i.e. the sampling timescale is smaller than the oceanographic variation time scale. Let A be the ocean area of interest, i.e. ( x, y ) ∈ A ; let
I k( j ) = {θi = θ ( xi , yi ) , i = 1...kl , l = 1...n} j
be
the
information set available to j-th vehicle at its k-th sampling step, where kl is the sampling step for the vehicles l. So not all vehicles are necessarily at the same sampling step at the same instant and the information set for each vehicle is composed by the measurements performed up that moment by the team of AUVs. Let Φ be an estimation algorithm that permits to obtain an estimate θˆk of the quantity θ over the whole region A on the basis of the current j available information I k( j ) , with j=1…n, i.e.
(
)
θˆk ( x, y ) = Φ I k( j ) , j = 1...n , ∀ ( x, y ) ∈ A . On the j
basis of the estimation algorithm Φ and the available j information set I k( j ) , j=1…n, we can calculate the estimation error ε k ( x, y ) = θ ( x, y ) − θˆk ( x, y ) in a suitable norm. Since the oceanographic quantities are the result of environmental forcing, such as heat exchange, tidal actions, costal in-flows, etc, they are
not random fields and they have a local smoothness behaviour that can be predicted a-priori or estimated through the measurements themselves. In particular we can expect to have the same temperature or salinity profile in the surrounding area of a given measurement unless we have some location-specific a-priori knowledge that this may not be so. This empirical hypothesis permits to make some assumptions on the local smoothness of the map θ ( x, y ) , in particular we can have an estimate of the upper bound on the estimation error: ( j) εˆk ( x, y ) = εˆk Φ, ( I k j , j = 1...n) such that
(
)
ε k ( x, y ) ≤ εˆk ( x, y ) , ∀ ( x, y ) ∈ A . Since this bound on the estimation error depends also on prior assumptions, we prefer to use a derived quantity that we call “confidence level”, defined as: εˆk ( x, y ) Ck ( x , y ) = 1 − . So we have the max A ε ( x, y ) confidence level which takes value 1 when the estimation error is 0 and takes value 0 when the estimation error attains its maximum admissible value, which is supposed to be known since the physical admissible values for the oceanographic quantities are necessarily bounded. Each vehicle has to choose the measurement points such that the mission ends when a desiderated confidence level is reached. The j-th vehicle on the basis of the j information available at sampling k, i.e. I k( ) , has j
now
to
( xk +1 , yk +1 )
choose ( j)
its
next
measurement
point
, j=1…n. Within the above setting, the
choice of the next sampling point is made among those that ensure that the expected confidence level is above a given threshold on the line between the current and the next sampling point; moreover the distance from the other vehicles in the team is maximized. Each vehicle chooses its next sampling points among a set B of points that belongs to a suitable ball centred in the last measurement point
( xk , yk )
( j)
and coincident with the actual position of
the vehicle j, i.e.
( xk +1 , yk +1 )
( j)
(
∈ B ( xk , yk )
( j)
)
,ρ .
The ball radius ρ is computed supposing that the local smoothness properties of the map θ in the candidate point
( xk +1 , yk +1 )
( j)
are equal to those
computed from the data obtained at the last sampling point ( xk , yk ) ; in this way the algorithm becomes ( j)
a predictor-corrector algorithm with respect to the confidence level. On the basis of this hypothesis ρ guarantees that the expected confidence level Ck +1 is greater or equal than the desired confidence level threshold δ in the segment of length ρ joining current and next measurement point, i.e. Ck +1 ( x, y ) ≥ δ , ∀ ( x, y ) ∈ ( ( xk , yk ) , ( xk +1 , yk +1 ) ) . After each vehicle has completed its (k+1)-th sampling measurement and has communicated it to all the other vehicles in the team, the information set j I k( +)1 and the confidence level Ck +1 become available. The adaptive algorithm steps within each iteration
for the j-th AUV are now summarized; starting from j the availability of the quantities, I k( j ) , Ck : 1.
Generate the next sampling point as:
( xk +1 , yk +1 )( j ) = arg
n
max
( j)
( x , y )∈B (( xk , yk )
∑ ( x, y) − ( x , y ) ρ k
, ) h≠ j
( h) 2
k
(1) with the constraint: Cˆ k +1 ( x, y ) ≥ δ , ∀( x, y ) ∈ ( ( xk , yk )( j ) , ( xk +1 , yk +1 )( j ) ) 2.
Move to the point ( xk +1 , yk +1 ) measurement θ
3.
The
(( x
k +1
, yk +1 )
( j)
( j)
(2) and execute
)
Communicate the measurement to the other vehicles; receive information from the other j vehicles, build the set I k( +)1 j and compute actual confidence level Ck +1 over the region. Ck +1 ≥ δ , algorithm terminates when
∀ ( x, y ) ∈ A .
Notice the following important characteristics of the proposal planning algorithm: The optimization algorithm is a greedy algorithm, i.e., it locally maximizes the cost function, but its global optimality, in terms of minimum number of sampling points, is not guaranteed; the algorithm naturally increases sampling where ocean quantities are most rapidly varying in space, and reduces sampling when the observed variable is almost constant; if the prior knowledge is such that the confidence function due to a single measurement point is a radially symmetric function, with maximum in the measurement point, and decreasing with the distance from the measurement point, the constraint of equation (2) is automatically satisfied by any point in the ball B(( x k , y k ) ( j ) , ρ ) . This situation may happen for instance if the oceanic dynamics are generated by only diffusive isotropic processes. The planning algorithm is asynchronous, each vehicle can decide its next sampling point, independently from all the other vehicles, using only the information communicated among the members of the team and available at the decision instant; information update must be guaranteed, however, by an information relay node, as a surface platform (like for instance an autonomous surface craft) or a shore station, or by underwater communication capabilities, as acoustic modems, of sufficient bandwidth. 3. SIMULATIVE EXAMPLE: TEMPERATURE EXTIMATION WITH A SUPERVISED NEURAL NETWORK AND EOFs The simulative scenario is composed by two and three AUVs exploring a rectangular marine area of 70 m depth and 5x4 km extension. It is assumed that the vehicles are equipped with a TD (Temperature Depth) probe, sampling the water column at one
sample per meter in depth at each required ( x, y ) position. The vertical temperature profile is coded through the use of EOFs, as commonly done in m
oceanography,
i.e., θ ( x, y, z ) =
∑ c ( x, y)ϕ ( z) , i
i
i =1
where the EOFs ϕ i ( z ), i = 1,..., m are a suitable set of orthogonal basis function defined on the basis of historical data. Each vehicle is modelled as a unicycle and has a kinematic Lyapunov controller which assures that the AUV will get to the requested point. The maximum velocity is fixed to 1 m/s for each vehicle. This information is needed to estimate the overall mission time to explore the region, using the proposed cooperation algorithm. In the simulations presented next, the confidence function C k is built in the following way: each sampling point in the set I k( j ) , j=1…n, is treated as j
the source of a diffusive isotropic process, and the confidence
function
Γi( j )
associated
to
the
( j)
is a Gaussian measurement at position ( x i , y i ) function centered in the measurement point with variance σ i( j ) . The variance is linked to the local smoothness properties of the map θ by the following empirical rule:
In the first simulation we consider an area of 5x4Km, divided in two different zones: an external one with constant temperature with depth and a circular internal region in which we find the presence of a thermocline. The paths generated by the adaptive algorithm are reported in figure 1 where it can be seen that the sampling has been increasing in the central region where ocean variability is most effective. The three AUVs have subdivided the area into three zones, one for each vehicle: the first vehicle (on the left in figure) has explored the left part of the marine area, the second AUV (to the centre in figure) has explored the internal region while the third AUV (on the right in figure) has measured within the right part of the region. We can notice that the second vehicle has executed measurements with a smaller search step because of the higher temperature variability in its region of interest. So the second vehicle explores a smaller zone than the other two vehicles but moving more slowly to sample the area with the correct accuracy. All the marine region is explored in about 3h and the confidence level reached the prescribed level in the whole region.
Circular internal region with high temperature variability
σ i( j ) = −κ ∇ z θ (( xi , y i ) ( j ) ; z ) + σ 0 , κ , σ 0 > 0 (3) Loosely speaking, the more regular the local behaviour of θ with depth, the greater the confidence with which the measurement can be extended over the neighborhood of the sampling point; the confidence is 1 (100%) at the measurement point, implying the no uncertainty in the sensor reading is considered. The global confidence function C k is build by combination of the Γi( j ) in the following way: ⎛ ⎞ C k ( x, y ) = min⎜ Γi( j ) ;1⎟ (4) ⎜ ⎟ ⎝ i j ⎠ As already mentioned, an information relay node must be used to permit the communication among the team members. This is necessary if the vehicles cannot communicate when they sail underwater, in this case a vehicle that is in immersion when another AUV is communicating the information set necessary for the cooperation loses the communication and therefore the ability to cooperate. The information relay receives from the j-
∑∑
th vehicle its last measurement θ ( x, y )
( j)
, build the
information set I k( j ) available for the vehicle using j
all measurements received from the team up to that moment, and computes the next sampling point for the vehicle in accord to the proposal adaptive algorithm. All simulations have been executed with the use of this extra communication node: although the cooperation algorithm is suitable for all kind of vehicles, it has been devised for application to the Fòlaga AUV that cannot communicate underwater.
Fig. 1. First Simulation. Path followed by three vehicles exploring a region divided in two sub regions, one of which with an high temperture variability. The sampling points are represented by the icons with the form of a thermometer. The submarines indicate the start and the final points for the vehicles. In the second simulation a 4x4 km width region was explored by two vehicles. This region is divided in two zones, an external one with the presence of a thermocline and an internal one with the presence of a temperature maximum at a depth of about 20m. At the end of the AUVs mission we have estimated the temperature over the whole region using historical data to compute the data covariance matrix and computing 4 orthogonal basis functions as eigenvectors of the same matrix. In many cases, even just 2 orthogonal basis functions are sufficient to code the vertical temperature profile with the required accuracy. Since the coefficient ci of the EOF expansions are independent, and do not depend on depth the estimated map is produced through the
use of m independent supervised neural network, of the standard backpropagation type with sigmoidal activation functions. From the estimated values cˆi ( x, y ) , i=1…m, generated by the net it is possible to produce the final estimate of the temperature m
profile: θˆ ( x, y ) = ∑ cˆi ( x, y ) ϕi ( z ) . i =1
The paths generated by the two vehicles are shown in figure 2 and also in this case it can be seen how the sampling has been increasing in the central area where the oceanic variability is most effective. In figure 3 it is shown the estimated temperature map. The marine region has been explored in 5h,33min and the confidence level is maximum in the whole region.
As a final example, we compare the adaptive sampling strategy with one single AUV with the RRT approach. Both algorithms employ the maximization of the confidence function in order to select the sequence of sampling points. A maximum of 20 samples were allowed for each method. Uniform temperature was assumed. The sampled points and the resulting confidence functions are shown in Figure 4-7, indicating that RRT does not succeed in attaining the same global confidence level. Imposing the attaining of a given confidence level and leaving free the number of sampling points has shown that RRT requires a larger number of sampling points with respect to the proposed methodology.
Circular internal region with high temperature variability
Fig. 2. Second simulation. Paths followed by two AUVs during the exploration of an area divided in two sub regions (the internal one presents an high temperature variability). The sampling points are represented by the icons with thermometer shape. The submarines indicate the start and the final points for the vehicles.
Fig. 3. Estimated temperature. Slice at constant x of the estimated temperature map, showing the precence of the maximum at the expected detph and woth the correct temperature value.
Fig. 4. RRT exploration. Sequence of sampling points generated by an RRT algorithm with 20 points limitation, constant temperature profile assumed
Fig. 5. RRT Confidence level. Global confidence function corresponding to the exploration path shown in Fig. 4.
4. CONCLUSIONS AND FUTURE WORK An original general asynchronous algorithm for mission planning and adaptation applied to multiple AUVs cooperating in the environmental exploration of an oceanic region has been described. A specific implementation of the algorithm adapted to the temperature estimation problem and assuming a simple oceanic models driven by diffusive terms has been presented and simulative results reported. Finally a comparison with an RRT algorithm is presented. We are currently working in the implementation of the scheme on a team of Fòlaga class AUVs; the experimental test at sea of the method is expected for spring summer 2006.
Fig. 6. Proposed algorithm, exploration path. One vehicle, constant temperature profile assumed (same conditions as the RRT application). Sequence of sampling points and path generated by the adaptive algorithm proposed in this paper, with 20 points limitation.
Fig. 7. Proposal algorithm, confidence function. Global confidence function corresponding to the exploration path shown in Fig. 6. ACKNOWLEDGMENTS This work has been partially supported by the Italian Ministry of University and Research, Research
Projects of National Interest, and by the European Union/Regione Liguria, PRAI FESR Initiative
REFERENCES Kermorgant, H., Architecture of AUV systems for harbour protection and mine countermeasurements, Proc. IEEE Conf. Oceans’05 Europe, Brest, France (2005). Spenneberg, D., Waldmann, C. and Babb, R., Exploration of underwater structures with cooperative heterogeneous robots, Proc. IEEE Conf. Oceans’05 Europe, Brest, France (2005). Ikeda, T., Mita, T. and Yamakita M., Formation control of autonomous underwater vehicles, Proc. 16th IFAC World Conf., Praha, Czech Rep. (2005). Ögren, P., Fiorelli, E. and Leonard, N.E., Cooperative control of mobile sensor networks: adaptive gradient climbing in a distributed environment IEEE Trans. Automatic Control, 49, 1292-1302 (2004) Popa, D.O., Sanderson, C.D., Komerska, R.J., Mupparapu, S.S., Blidberg, R.D. and Chappel, S.G., Adaptive sampling algorithms for multiple autonomous underwater vehicles, Proc. IEEE/OES AUV 2004: Autonomous Underwater Vehicle Workshop, Sebasco Estates, Maine, U.S.A. (2004). Schmidt, H. and Edwards, J.R, GOATS: multiplatform sonar concept for coastal mine countermeasures, in Schulz, A.C. and Parker, L.E. (Eds.): Multi-Robot Systems: from swarms to intelligent automata, Kluwer Academic Publ., (2002). Fenwick, J.W., Newman, P.M. and Leonard, J.J., Cooperative concurrent mapping and localization, Proc. IEEE ICRA’02 Conf. , Washington, D.C., U.S.A. (2002). Alvarez, A., Caffaz, A., Caiti, A., Casalino, G., Clerici, E., Giorgi, F., Gualdesi, L. and Turetta, A., Design and realization of a very low cost prototypal autonomous vehicle for coastal oceanographic missions, Proc. IFAC CAMS’04, Ancona, Italy (2004). Alvarez, A., Caffaz, A., Caiti, A., Casalino, G., Clerici, E., Giorgi, F., Gualdesi, L., Turetta, A. and Viviani, R., Folaga: a very low cost autonomous underwater vehicle for coastal oceanography, Proc. 16th IFAC World Conf., Praha, Czech Rep. (2005). Caiti, A., Munafò, A., Viviani, R. Cooperating autonomous underwater vehicles to estimate ocean environmental parameter, IARP workshop on underwater robotics, Genoa, Italy (2005). Tan, C.S., Sutton, R. and Chudley, J., Quasi-random, manoeuvre-based motion planning algorithm for autonomous underwater vehicles, Proc. 16th IFAC World Conf., Praha, Czech Rep. (2005).
Abstract: An adaptive on-line planning algorithm for multiple underwater vehicles in coastal oceanographic missions is introduced. The algorithm is tested for a team of autonomous underwater vehicles exploring an unknown oceanic environment in which they can execute point-wise environmental measurements. The team final objective is the construction of a three-dimensional estimated map of the measured environmental characteristics of the whole marine region. Each vehicle chooses its next sampling point in order to maximally increase the confidence level of the overall estimate of the measured quantity and in an asynchronous way with respect to the other vehicles. It is assumed that all vehicles have availability of the positions and values of the all past measurements executed by the members of the team. Simulative test cases in which the objective is the estimation of the ocean temperature over a region are presented. The algorithm performance is compared with that theoretically achievable by Rapidlyexploring Random Trees search. Copyright © 2006 IFAC Keywords: Adaptive Control, Mobile Robots, Trajectory planning and control, Underwater Robots, Vehicles.
1. INTRODUCTION The research in cooperating and planning strategies for Autonomous Underwater Vehicles (AUVs) has seen an increasing interest in the last few years, after a slower start with respect to other fields of robot cooperation. While teams of cooperating robots have been investigated since quite a long time, and there are several examples of land and airs robots cooperating in many applications, from de-mining to soccer playing, to formation control problem of teams of Unmanned Aerial Vehicles (UAV), the problem of cooperating AUVs has seen a difficult start because of the communication and localization constraints posed by the marine environment; in particular the severe bandwidth limitation of underwater communication and the absence of a GPS-like localization system makes it very hard to simply transpose general robotic techniques to the underwater scenario. Up to now, to the Authors
knowledge, the AUVs cooperation has been intended as the use of AUVs with different capabilities and measurement instrumentation within the same geographical area, so that each vehicle can fulfil a specific goal (e.g. a bottom crawling robot together with a navigation vehicle) without interaction with each other (Kermorgant, 2005). From this point of view, which does not involve interaction among the different vehicles, the cooperation is merely the exploitation of different missions running in parallel, often with ad-hoc solutions and considerations. A second kind of AUVs cooperation is referred as AUVs formation control, for mission requiring the reaching of a specific geographical location by at least one vehicle (e.g. save and rescue or mine counter measurement operations) (Spennenberg, et al., 2005; Ikeda, et al., 2005); a particularly interesting approach within this line of research has been proposed and investigated by Ögren, et al. (2004), that have considered ocean gliders with the
task of identifying the gradient of environmental oceanographic quantities. Finally the investigation on cooperating AUVs is intended as mission adaptation with respect to the specific mission task; along this line there are different examples, discussed as for acoustic scattering and propagation measurements (Schmidt and Edwards, 2002), and environmental mapping and self-localization (Fenwick et al., 2002). Popa et al. (2004) propose, along this line of research, a general methodological approach, including concurrent localization and mapping as well as adaptive oceanographic sampling. In this paper, following this last line of research and therefore within the framework of adaptive oceanographic sampling, a cooperation algorithm is considered. The algorithm is suited for a team of autonomous vehicles taking point wise environmental measurements over an ocean region, with the final objective of producing an estimated map of the measured environmental quantity over the whole region. After any measurement, each vehicle plans its next sampling point in order to maximally increase the confidence level of the overall estimate of the measured quantity. A link between the confidence level and the measured quantity permit to obtain algorithm adaptation; in particular, each measurement point is linked to a “confidence function” that indicate how far can be spatially extended the measurement in the neighbourhood of the point itself starting from the assumption that the measurement does not significantly change in proximity of the sampling point. Confidence extension depends in general on the local smoothness properties of the environmental parameter, which can be assessed on the basis of the available measurements themselves. A global confidence function over the whole region of interest can be obtained by the composition of all the confidence functions associated to the available measurements. Cooperation among the vehicles is obtained since the algorithm computes the next sampling point for each vehicle taking into account also the (known) position of the other vehicles in the team, in addition to the current confidence level and the local smoothness of the environmental map. The next sample point for each vehicle is obtained by choosing the point that maximizes the estimate confidence and that minimizes the number of sampling points required by the whole team. The algorithm is asynchronous, i.e., each vehicle can compute locally its next sampling point right after it has completed the measurement at a given point without waiting for the other vehicles or any other type of synchronization; however, the algorithm also requires communication among all the vehicles in the team through a an information relay (e.g., a shore station or a surface platform), since global knowledge of the past measurements is required. Although the proposed algorithm is fairly general and vehicle-independent, it has been devised for application to the Fòlaga vehicle (Alvarez, et al., 2004; 2005), a very low cost autonomous vehicle developed for coastal oceanographic applications, in which all the communications among the vehicles can take place at the sea surface through radio links. Experimental
testing of the proposed cooperation approach is expected by spring-summer of 2006. The paper is organized as follows: in the next section the general algorithm is described formally. In the third section, a particular instance of the algorithm is described and simulative results are reported. In all the simulations, the physical quantity to be estimated is the ocean temperature. Temperature estimation is based on supervised neural networks and Empirical Orthogonal Functions (EOFs). The cooperative behaviour of two and three AUVs in several different environmental conditions is reported. In particular, it is shown how in the case considered the proposed method allows to reduce the amount of necessary measurements, all else being equal, as compared with Rapidly-exploring Random Trees (RRT) (Tan et al., 2005). Finally, future work along this line of research is outlined and conclusions are given. 2. GENERAL ALGORITHM In this section we describe the on-line planning algorithm. The framework follows that presented in (Caiti, et al., 2005) with an important difference: while in our previous work we analyzed a synchronous algorithm, now we consider the asynchronous case, i.e., each vehicle can compute locally its next sampling point right after it has completed the measurement at a given point. Let us suppose we have the availability of n AUVs, each equipped with a sensor able to point-wise sample an oceanographic quantity θ at the geographical coordinates (x, y) over the entire depth profile, i.e. θ = θ ( x, y ) . Through the use of the EOFs we can take into account the depth dependence of the oceanographic quantity (e.g. temperature, salinity, oxygen content, etc.), as explained in the next section. As for the dependence on time, it is assumed that the quantity is stationary, i.e. the sampling timescale is smaller than the oceanographic variation time scale. Let A be the ocean area of interest, i.e. ( x, y ) ∈ A ; let
I k( j ) = {θi = θ ( xi , yi ) , i = 1...kl , l = 1...n} j
be
the
information set available to j-th vehicle at its k-th sampling step, where kl is the sampling step for the vehicles l. So not all vehicles are necessarily at the same sampling step at the same instant and the information set for each vehicle is composed by the measurements performed up that moment by the team of AUVs. Let Φ be an estimation algorithm that permits to obtain an estimate θˆk of the quantity θ over the whole region A on the basis of the current j available information I k( j ) , with j=1…n, i.e.
(
)
θˆk ( x, y ) = Φ I k( j ) , j = 1...n , ∀ ( x, y ) ∈ A . On the j
basis of the estimation algorithm Φ and the available j information set I k( j ) , j=1…n, we can calculate the estimation error ε k ( x, y ) = θ ( x, y ) − θˆk ( x, y ) in a suitable norm. Since the oceanographic quantities are the result of environmental forcing, such as heat exchange, tidal actions, costal in-flows, etc, they are
not random fields and they have a local smoothness behaviour that can be predicted a-priori or estimated through the measurements themselves. In particular we can expect to have the same temperature or salinity profile in the surrounding area of a given measurement unless we have some location-specific a-priori knowledge that this may not be so. This empirical hypothesis permits to make some assumptions on the local smoothness of the map θ ( x, y ) , in particular we can have an estimate of the upper bound on the estimation error: ( j) εˆk ( x, y ) = εˆk Φ, ( I k j , j = 1...n) such that
(
)
ε k ( x, y ) ≤ εˆk ( x, y ) , ∀ ( x, y ) ∈ A . Since this bound on the estimation error depends also on prior assumptions, we prefer to use a derived quantity that we call “confidence level”, defined as: εˆk ( x, y ) Ck ( x , y ) = 1 − . So we have the max A ε ( x, y ) confidence level which takes value 1 when the estimation error is 0 and takes value 0 when the estimation error attains its maximum admissible value, which is supposed to be known since the physical admissible values for the oceanographic quantities are necessarily bounded. Each vehicle has to choose the measurement points such that the mission ends when a desiderated confidence level is reached. The j-th vehicle on the basis of the j information available at sampling k, i.e. I k( ) , has j
now
to
( xk +1 , yk +1 )
choose ( j)
its
next
measurement
point
, j=1…n. Within the above setting, the
choice of the next sampling point is made among those that ensure that the expected confidence level is above a given threshold on the line between the current and the next sampling point; moreover the distance from the other vehicles in the team is maximized. Each vehicle chooses its next sampling points among a set B of points that belongs to a suitable ball centred in the last measurement point
( xk , yk )
( j)
and coincident with the actual position of
the vehicle j, i.e.
( xk +1 , yk +1 )
( j)
(
∈ B ( xk , yk )
( j)
)
,ρ .
The ball radius ρ is computed supposing that the local smoothness properties of the map θ in the candidate point
( xk +1 , yk +1 )
( j)
are equal to those
computed from the data obtained at the last sampling point ( xk , yk ) ; in this way the algorithm becomes ( j)
a predictor-corrector algorithm with respect to the confidence level. On the basis of this hypothesis ρ guarantees that the expected confidence level Ck +1 is greater or equal than the desired confidence level threshold δ in the segment of length ρ joining current and next measurement point, i.e. Ck +1 ( x, y ) ≥ δ , ∀ ( x, y ) ∈ ( ( xk , yk ) , ( xk +1 , yk +1 ) ) . After each vehicle has completed its (k+1)-th sampling measurement and has communicated it to all the other vehicles in the team, the information set j I k( +)1 and the confidence level Ck +1 become available. The adaptive algorithm steps within each iteration
for the j-th AUV are now summarized; starting from j the availability of the quantities, I k( j ) , Ck : 1.
Generate the next sampling point as:
( xk +1 , yk +1 )( j ) = arg
n
max
( j)
( x , y )∈B (( xk , yk )
∑ ( x, y) − ( x , y ) ρ k
, ) h≠ j
( h) 2
k
(1) with the constraint: Cˆ k +1 ( x, y ) ≥ δ , ∀( x, y ) ∈ ( ( xk , yk )( j ) , ( xk +1 , yk +1 )( j ) ) 2.
Move to the point ( xk +1 , yk +1 ) measurement θ
3.
The
(( x
k +1
, yk +1 )
( j)
( j)
(2) and execute
)
Communicate the measurement to the other vehicles; receive information from the other j vehicles, build the set I k( +)1 j and compute actual confidence level Ck +1 over the region. Ck +1 ≥ δ , algorithm terminates when
∀ ( x, y ) ∈ A .
Notice the following important characteristics of the proposal planning algorithm: The optimization algorithm is a greedy algorithm, i.e., it locally maximizes the cost function, but its global optimality, in terms of minimum number of sampling points, is not guaranteed; the algorithm naturally increases sampling where ocean quantities are most rapidly varying in space, and reduces sampling when the observed variable is almost constant; if the prior knowledge is such that the confidence function due to a single measurement point is a radially symmetric function, with maximum in the measurement point, and decreasing with the distance from the measurement point, the constraint of equation (2) is automatically satisfied by any point in the ball B(( x k , y k ) ( j ) , ρ ) . This situation may happen for instance if the oceanic dynamics are generated by only diffusive isotropic processes. The planning algorithm is asynchronous, each vehicle can decide its next sampling point, independently from all the other vehicles, using only the information communicated among the members of the team and available at the decision instant; information update must be guaranteed, however, by an information relay node, as a surface platform (like for instance an autonomous surface craft) or a shore station, or by underwater communication capabilities, as acoustic modems, of sufficient bandwidth. 3. SIMULATIVE EXAMPLE: TEMPERATURE EXTIMATION WITH A SUPERVISED NEURAL NETWORK AND EOFs The simulative scenario is composed by two and three AUVs exploring a rectangular marine area of 70 m depth and 5x4 km extension. It is assumed that the vehicles are equipped with a TD (Temperature Depth) probe, sampling the water column at one
sample per meter in depth at each required ( x, y ) position. The vertical temperature profile is coded through the use of EOFs, as commonly done in m
oceanography,
i.e., θ ( x, y, z ) =
∑ c ( x, y)ϕ ( z) , i
i
i =1
where the EOFs ϕ i ( z ), i = 1,..., m are a suitable set of orthogonal basis function defined on the basis of historical data. Each vehicle is modelled as a unicycle and has a kinematic Lyapunov controller which assures that the AUV will get to the requested point. The maximum velocity is fixed to 1 m/s for each vehicle. This information is needed to estimate the overall mission time to explore the region, using the proposed cooperation algorithm. In the simulations presented next, the confidence function C k is built in the following way: each sampling point in the set I k( j ) , j=1…n, is treated as j
the source of a diffusive isotropic process, and the confidence
function
Γi( j )
associated
to
the
( j)
is a Gaussian measurement at position ( x i , y i ) function centered in the measurement point with variance σ i( j ) . The variance is linked to the local smoothness properties of the map θ by the following empirical rule:
In the first simulation we consider an area of 5x4Km, divided in two different zones: an external one with constant temperature with depth and a circular internal region in which we find the presence of a thermocline. The paths generated by the adaptive algorithm are reported in figure 1 where it can be seen that the sampling has been increasing in the central region where ocean variability is most effective. The three AUVs have subdivided the area into three zones, one for each vehicle: the first vehicle (on the left in figure) has explored the left part of the marine area, the second AUV (to the centre in figure) has explored the internal region while the third AUV (on the right in figure) has measured within the right part of the region. We can notice that the second vehicle has executed measurements with a smaller search step because of the higher temperature variability in its region of interest. So the second vehicle explores a smaller zone than the other two vehicles but moving more slowly to sample the area with the correct accuracy. All the marine region is explored in about 3h and the confidence level reached the prescribed level in the whole region.
Circular internal region with high temperature variability
σ i( j ) = −κ ∇ z θ (( xi , y i ) ( j ) ; z ) + σ 0 , κ , σ 0 > 0 (3) Loosely speaking, the more regular the local behaviour of θ with depth, the greater the confidence with which the measurement can be extended over the neighborhood of the sampling point; the confidence is 1 (100%) at the measurement point, implying the no uncertainty in the sensor reading is considered. The global confidence function C k is build by combination of the Γi( j ) in the following way: ⎛ ⎞ C k ( x, y ) = min⎜ Γi( j ) ;1⎟ (4) ⎜ ⎟ ⎝ i j ⎠ As already mentioned, an information relay node must be used to permit the communication among the team members. This is necessary if the vehicles cannot communicate when they sail underwater, in this case a vehicle that is in immersion when another AUV is communicating the information set necessary for the cooperation loses the communication and therefore the ability to cooperate. The information relay receives from the j-
∑∑
th vehicle its last measurement θ ( x, y )
( j)
, build the
information set I k( j ) available for the vehicle using j
all measurements received from the team up to that moment, and computes the next sampling point for the vehicle in accord to the proposal adaptive algorithm. All simulations have been executed with the use of this extra communication node: although the cooperation algorithm is suitable for all kind of vehicles, it has been devised for application to the Fòlaga AUV that cannot communicate underwater.
Fig. 1. First Simulation. Path followed by three vehicles exploring a region divided in two sub regions, one of which with an high temperture variability. The sampling points are represented by the icons with the form of a thermometer. The submarines indicate the start and the final points for the vehicles. In the second simulation a 4x4 km width region was explored by two vehicles. This region is divided in two zones, an external one with the presence of a thermocline and an internal one with the presence of a temperature maximum at a depth of about 20m. At the end of the AUVs mission we have estimated the temperature over the whole region using historical data to compute the data covariance matrix and computing 4 orthogonal basis functions as eigenvectors of the same matrix. In many cases, even just 2 orthogonal basis functions are sufficient to code the vertical temperature profile with the required accuracy. Since the coefficient ci of the EOF expansions are independent, and do not depend on depth the estimated map is produced through the
use of m independent supervised neural network, of the standard backpropagation type with sigmoidal activation functions. From the estimated values cˆi ( x, y ) , i=1…m, generated by the net it is possible to produce the final estimate of the temperature m
profile: θˆ ( x, y ) = ∑ cˆi ( x, y ) ϕi ( z ) . i =1
The paths generated by the two vehicles are shown in figure 2 and also in this case it can be seen how the sampling has been increasing in the central area where the oceanic variability is most effective. In figure 3 it is shown the estimated temperature map. The marine region has been explored in 5h,33min and the confidence level is maximum in the whole region.
As a final example, we compare the adaptive sampling strategy with one single AUV with the RRT approach. Both algorithms employ the maximization of the confidence function in order to select the sequence of sampling points. A maximum of 20 samples were allowed for each method. Uniform temperature was assumed. The sampled points and the resulting confidence functions are shown in Figure 4-7, indicating that RRT does not succeed in attaining the same global confidence level. Imposing the attaining of a given confidence level and leaving free the number of sampling points has shown that RRT requires a larger number of sampling points with respect to the proposed methodology.
Circular internal region with high temperature variability
Fig. 2. Second simulation. Paths followed by two AUVs during the exploration of an area divided in two sub regions (the internal one presents an high temperature variability). The sampling points are represented by the icons with thermometer shape. The submarines indicate the start and the final points for the vehicles.
Fig. 3. Estimated temperature. Slice at constant x of the estimated temperature map, showing the precence of the maximum at the expected detph and woth the correct temperature value.
Fig. 4. RRT exploration. Sequence of sampling points generated by an RRT algorithm with 20 points limitation, constant temperature profile assumed
Fig. 5. RRT Confidence level. Global confidence function corresponding to the exploration path shown in Fig. 4.
4. CONCLUSIONS AND FUTURE WORK An original general asynchronous algorithm for mission planning and adaptation applied to multiple AUVs cooperating in the environmental exploration of an oceanic region has been described. A specific implementation of the algorithm adapted to the temperature estimation problem and assuming a simple oceanic models driven by diffusive terms has been presented and simulative results reported. Finally a comparison with an RRT algorithm is presented. We are currently working in the implementation of the scheme on a team of Fòlaga class AUVs; the experimental test at sea of the method is expected for spring summer 2006.
Fig. 6. Proposed algorithm, exploration path. One vehicle, constant temperature profile assumed (same conditions as the RRT application). Sequence of sampling points and path generated by the adaptive algorithm proposed in this paper, with 20 points limitation.
Fig. 7. Proposal algorithm, confidence function. Global confidence function corresponding to the exploration path shown in Fig. 6. ACKNOWLEDGMENTS This work has been partially supported by the Italian Ministry of University and Research, Research
Projects of National Interest, and by the European Union/Regione Liguria, PRAI FESR Initiative
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