Cooperative Control for Ocean Sampling: The Glider Coordinated Control System

Transcription

1 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1 Cooperative Control for Ocean Sampling: The Glider Coordinated Control System Dere A. Paley, Student Member, IEEE, Fumin Zhang, Member, IEEE, Naomi Ehrich Leonard, Senior Member, IEEE Abstract The Glider Coordinated Control System (GCCS) uses a detailed glider model for prediction and a simple particle model for planning to steer a fleet of underwater gliders to a set of coordinated trajectories. The GCCS also serves as simulation testbed for design and evaluation of multi-vehicle control algorithms. We describe the GCCS and present experimental results for a virtual glider deployment in Monterey Bay, California and a real glider deployment in Buzzards Bay, Massachusetts. I. INTRODUCTION Effective monitoring of the ocean enables oceanographers to mae new discoveries that affect our understanding of the environment. A critical component of sustained ocean monitoring is automatic and coordinated control of autonomous sensor platforms. Autonomous underwater gliders are unmanned submersibles characterized by high reliability and endurance. In a typical glider deployment, multiple gliders survey a compact region of interest for wees or months in order to sample the ever-changing ocean processes with adequate frequency in space and time. Feedbac controls that coordinate the glider sampling trajectories to optimally distribute measurements increase the collective survey performance [1]. To meet these objectives, we have designed the Glider Coordinated Control System (GCCS), which builds on previous experience with real-time glider coordination [2]. The GCCS also serves as a simulation testbed for development of coordinated control algorithms. Simulated, or virtual, gliders operate in realistic ocean fields that are provided as input. Accordingly, it is possible to use the GCCS to explore and test solutions to many of the challenges that come with controlling a networ of gliders in the ocean. A strong, variable flow field, which can at times be stronger even than the forward speed of a glider, is one such challenge. Other challenges include long delays in feedbac, uncertainty in communication and asynchronicity in feedbac and communication. Because a number of these challenges have not yet been fully addressed by theoretical methods, the GCCS as testbed plays an indispensable role in development. We designed the GCCS to support operation of autonomous ocean sampling networ (AOSN) [3]. These networs leverage advances in underwater robot technology to perform ocean surveys with unprecedented resolution. Since Research partially supported by ONR grants N , N , and N D. Paley supported by the NSF Graduate Research Fellowship, the Princeton Wu Graduate Fellowship, and the Pew Charitable Trust grant The authors are with the Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ ( {dpaley}{fzhang}{naomi}@princeton.edu).

2 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 2 inception of the AOSN concept, there have been several demonstrations of automated (remote) control systems for multiple underwater vehicles [4], [5], [6]. Our contribution, which we describe in this paper, is to design and demonstrate an automated control system that performs feedbac control at the level of the fleet. Since the time-scale of the fleet motion is much slower than the time-scale of the individual dynamics, we decouple the fleet trajectory design from the individual trajectory tracing. We address the trajectory design problem using decentralized feedbac control of a simplified model of glider motion. By combining the simple model with a detailed model of glider dynamics, the GCCS steers the fleet to a set of coordinated trajectories. We further demonstrate the practical use and merit of the GCCS by describing experimental results from a recent virtual deployment in Monterey Bay, California and a real ocean deployment in Buzzards Bay, Massachusetts. In Section II we give an overview of ocean sampling with underwater gliders and describe our approach for enabling collective motion of a glider fleet using cooperative control algorithms. In Section III, we describe the automated control system that implements these algorithms. In Section IV, we provide experimental results for the GCCS from both virtual and real glider deployments. II. PROBLEM BACKGROUND AND APPROACH A. Ocean Sampling with Underwater Gliders Autonomous underwater gliders soar through the ocean on a pair of fixed wings using a novel and efficient, buoyancy-driven propulsion system [7], [8]. The Slocum glider shown in Figure 1(a) is manufactured by Webb Research Corporation and owned and operated by the Woods Hole Oceanographic Institution (WHOI) in Woods Hole, Massachusetts. Another comparable glider, the Spray, is manufactured and operated by the Scripps Institution of Oceanography (SIO) in La Jolla, California. Gliders such as the Slocum and the Spray move vertically in the water by cyclically changing their buoyancy with a hydraulic pump. They convert their vertical velocity to horizontal roller-coaster motion by controlling their pitch to generate lift from a pair of fixed wings. Gliders steer either by moving an internal mass to roll and turn or by controlling an external rudder. Although they travel at low speeds (0.2 to 0.3 meters per second) relative to propeller-driven underwater vehicles (which travel at 1 to 3 meters per second), gliders are capable of much longer deployments (2 to 10 wees versus 12 to 36 hours). Gliders periodically surface to connect via satellite communication or radio frequency communication to computers on shore. Although it is possible to equip gliders with acoustic modems to communicate over short distances underwater, the majority of the gliders deployed to date do not yet have this capability. While on the surface, gliders transmit measured data and receive new waypoints, which are the coordinates of their next destinations. Also while on the surface, gliders use a Global Position System (GPS) to determine their position and to estimate the magnitude and direction of the (depth-averaged) ocean currents encountered during the previous dive. These measurements are necessary for effective navigation and planning. Gliders are ey enabling technology for the AOSN concept due to their high endurance and low cost. During deployments, gliders autonomously sample ocean properties such as temperature, salinity, and optical bacscatter at depths up to 1500 m. Gliders support a broad array of sampling objectives: improving our understanding of ocean

3 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 3 )%#) CA X Iridium )%#! +./ GCCS Planner X )0 )%#" SIO Spray Data Server GCCS Simulator )% GCCS Remote IO (00 )0!000 NJ (00 )&#*!00 MA 0 WHOI Slocum Data Server )&#$ 000 (b) The Glider!Coordinated Control System (GCCS)!0 (a) Slocum glider we04 in Buzzards Bay )&#% Fig. 1. (a) Slocum gliders are 1.5 m long and weigh 52 g. While on the surface, the tail of the vehicle, which houses antennas for satellite!!""#$!!""#%!!""#&!!""#'!!""#(!!""#)!!""#"!!""#! +,- (b) The GCCS runs on a computer in Princeton, New Jersey. communication, is elevated by an internal air bladder to achieve better reception. The three major components are the glider planner, the glider simulator, and the remote input/output (IO). The glider planner encapsulates the real-time control system, the glider simulator is a control testbed, and the remote IO module supports communication with the glider data servers. The glider data servers, which are located in Woods Hole, Massachusetts and La Jolla, Calfifornia, connect via Iridium satellite communication to the gliders as they periodically come to the surface (shown here deployed in Monterey Bay, California). dynamics by monitoring the mass and heat budget of a three-dimensional ocean volume [9]; gaining insight into the spatial and temporal dynamics of ocean biology by measuring optical bacscatter and bioluminescence [10]; and acoustically detecting and tracing pelagic fish. The design of glider sampling trajectories often maximizes a metric such as model predictive sill [11] or minimizes a metric such as objective analysis mapping error [12]. We minimize collective mapping error using coordinated control of a glider fleet [1]. We derive optimal coordinated trajectories using objective analysis (OA), also nown as optimal interpolation [12]. The OA calculations require an a priori description of the mean of a scalar field and the covariance of fluctuations around the mean, parameterized by the spatial and temporal decorrelation scales. OA provides an estimate for the average of a scalar field using using a linear combination of sensor measurements. OA also provides the mapping error of this estimate, which indicates sampling performance over space and time. We integrate the mapping error over the sampling domain to obtain the mapping performance metric as a function of time. B. Glider Coordinated Trajectories In the GCCS, feedbac control laws stabilize collective motion of a simple glider model to a set of coordinated trajectories. We design this set of trajectories, the glider coordinated trajectories (GCT), to achieve the ocean science objectives of the glider deployment. We adapt the GCT in the event of glider failure, in response to changing ocean currents, or to meet evolving scientific objectives. In typical deployments, gliders travel around closed curves in order to tae measurements along long, repeated transects. The GCT specifies the desired trac for each glider as well as the coordination and spacing between the gliders on their tracs. We refer to the spacing between gliders

4 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 4 - %(#*' / '(#' %(#** '(#" %(#*% *34-./012 '(#! '( '%#) --:;<!9 --:;<!% --:;<!& --:;<!' --80"% --80"9 --80"& --80"' --80"",56/01234 %(#*+ %(#*& %(#*( %(#* %(#%) //92(( //92") '%#$ --80"! %(#%$ '%#( -!5-67!!""#$!!""#%!!""#&!!""#" *+,-./012 %(#%! %(#%' / (/78!!"#$%!!"#$&!!"#$!!"#!$!!"#!'!!"#!%,-./01234 (a) GCT in Monterey Bay (b) GCT in Buzzards Bay Fig. 2. Glider tracs are dashed gray lines and the coordination lins are thin, solid lines connecting the gliders. (a) Four Spray gliders (labeled SIO##) travel clocwise around the 10 m by 20 m northern trac and maintain uniform separation on the trac. Three Slocum gliders (labeled we##) travel clocwise around the middle trac and three more around the southern trac; each set of three maintains uniform separation on each trac. In addition, the Slocum gliders on the middle and southern tracs synchronize with one another. (b) Two gliders travel clocwise around a 2.8 m by 5.6 m rectangular trac and maintain along-trac separation of 1/6 the curve perimeter. on the same trac as the relative curve-phase. If two or more gliders travel around the same trac, then they may maintain a fixed distance from each other (measured along the trac). For example, if two gliders stay on opposite sides of the trac then their relative curve-phase is π. If two or more gliders travel around different but similarly sized tracs, then they may synchronize their orbits so that the relative curve-phase is zero. The GCT specifies whether each glider s control effort is dedicated to steering around its assigned trac or to a balance of steering to the trac and steering to coordinate with other gliders. An essential aspect of designing GCTs is choosing the interconnection topology for glider coordination. The interconnection topology determines which pairs of gliders are coupled with each other for planning purposes; the trajectories of coupled gliders are planned jointly and depend on their relative position and direction of motion. Because gliders do not necessarily come to the surface at the same time, they are not configured to communicate directly with one another (inter-glider communication is implicitly performed by the GCCS). Nonetheless, the interconnection topology is described by a graph whose nodes represent gliders and whose edges represent (bidirectional) coordination. For example, we illustrate in Figure 2(a) the GCT and corresponding coordination graph for ten gliders in Monterey Bay. This GCT is designed to maximize the sampling performance inside and on the 20 m by 40 m bounding box (blac dashed line) that circumscribes the glider tracs by eeping the gliders uniformly

5 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 5 separated on the tracs. In addition, the six gliders on the middle and southern tracs form three synchronized glider pairs (each pair contains one glider from each trac). III. COORDINATED CONTROL SYSTEM DESIGN The GCCS implements feedbac control laws for coordination of a glider fleet. The GCCS is a modular, crossplatform software suite written in MATLAB. The three main modules of the GCCS are: the planner, which is the real-time controller; the simulator, which serves as control testbed; and the remote input/output (IO) module, which interfaces to gliders indirectly through the glider data servers. To plan trajectories for the gliders, which surface asynchronously, the GCCS uses two different models: (i) a detailed glider model with flow that is integrated to estimate glider motion and (ii) a particle model (with or without coordinated control) that is integrated to compute desired trajectories. We refer to the software that integrates these two models as the glider integrator and particle integrator, respectively. The glider integrator determines the initial conditions for the particle integrator. A. System Architecture Glider Planner: The glider planner encapsulates the multi-vehicle control algorithm. We designed the interface to the planner control subsystem to facilitate implementation of new control algorithms by separating the control subsystem from the planning subsystem. The planner uses the detailed glider model to predict glider motion underwater and the simple particle model to plan future glider trajectories. The planned trajectories originate from the position and time of the next expected surfacing of each glider. The interplay between the detailed glider model and the simple particle model is crucial to the application of the coordinated algorithm to a real system. A bloc diagram of the glider planner is shown in Figure 3(a). Planning new trajectories for all gliders occurs simultaneously; we call the sequence of steps that produces new glider trajectories a planning cycle. A planning cycle starts whenever a glider surfaces and ends when the planner generates new waypoints for all gliders. Due to operational concerns lie reduced effective speed and hazardous boat traffic, the gliders do not wait on the surface for their new waypoints. Instead, gliders use the waypoints that were computed since they last surfaced. Even so, the planning cycle executes immediately after a glider surfaces since new waypoints for all gliders are computed whenever a single glider surfaces. We now summarize the steps in the planning cycle. For each glider that has surfaced since the last planning cycle, the planner calculates inaccuracies in the predictions of effective speed, expected surface position, and expected surface time. Prediction errors are useful for gauging glider and planner performance. The planner uses the detailed glider model as described below to predict each glider s underwater trajectory and next surfacing location and time. As input to the glider model, the planner computes a surface and underwater flow forecast by fusing all recent glider flow measurements using OA. The next step in the planning cycle, described in more detail below, is to integrate the simple particle model to generate planned trajectories. The planner converts the planned trajectory of each glider to a list of waypoints, which must pass a quality control filter. Surfacing when expected is one requirement to pass

6 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 6 Glider Planner Environmental model/data latest positions, active waypoints flow predictions, bathymetry Glider Integrator η Glider Model Glider Onboard Control Glider configuration parameters R,z,ζ,p R %* %'#)& 7789.%:...$::..!:::.; <=>01?.@A?B4C>-2@./=4@5.!".D?@3 E4FG,>-5@.1HG1C510.5,.?14CD I1H5.G?10>C510.@A?B4C>-2 I16.64FG,>-5@./-1H5.'.D?@3 J=4--10.G,@>5>,-./>-.!".D?@3. Glider Coordinated Trajectories GCT Parser flow predictions Particle Integrator u Particle Model Coordinated Control r,θ r Waypoint Generator and QC +45./0123 %'#)...61""...61"! control parameters %'#(& latest positions, flow measurements active waypoints waypoints Remote Input/Output %'#(...61"% latest positions, flow measurements, active waypoints Glider Data Server waypoints.!!""#$!!""#%&!!""#%!!""#"&!!""#"!!""#!& +,-./0123 (a) Glider planner feedbac loop (b) Glider planner graphical output Fig. 3. (a) The planner creates waypoints to steer the fleet to a GCT by integrating the detailed glider model for prediction and the simple particle model for planning. The glider planner interfaces to the glider data server via the remote IO module. (b) The past, predicted and planned trajectories for three Slocum gliders we21, we22, and we23. The planned trajectories (thin solid lines) originate at each glider s next expected surface location (filled circles) and terminate at the 12 hour planning horizon (open circles). SSB stands for subsea bathymetry, or water depth. the quality control filter and failure to do so serves as an indication of potential problems with the glider or the glider data server. Figure 3(b) illustrates the output of a planning cycle produced by the GCCS for monitoring. Glider Simulator: In addition to providing a real-time controller, the GCCS serves as a simulation testbed for glider coordinated control algorithms. During a glider deployment, we use the glider simulator in real-time to predict glider motion in ocean flow forecasts. A central advantage is the ability to test strategies in the presence of strong flow and communication and feedbac constraints and uncertainties, challenges that are not yet fully addressed by theoretical methods. The glider simulator uses the detailed glider model to predict glider motion. To predict the motion of a coordinated fleet of gliders, we run the glider simulator in tandem with the glider planner. The software interface between the glider planner and simulator is identical to the interface between the glider planner and the real gliders. In fact, the simulated gliders produce identical data files as the real gliders. As a result, we use the GCCS, in conjunction with a virtual ocean model, to conduct virtual pilot experiments. Remote Input/Output: Robust networing enables the GCCS to run automatically. The remote IO module supports communication over the Internet between the glider planner and glider simulator as well as between the glider planner and the glider data servers. In addition, the remote IO module publishes real-time glider planner status for monitoring and supervision of the GCCS as shown in Figure 3(b). To support timely operator intervention, the remote IO module sends notification of software or operational errors.

7 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 7 t t ini T ini t sur T gps T com t ini z min φ z z max z Z min z uw ω p 1 R Γ 0 ω p τ ini τ uw τ sur λ (a) Glider model coordinates (b) Waypoint completion conditions Fig. 4. Notation for the detailed glider model. (a) Coordinates for time t and depth z during a single dive from left to right: t ini is the dive is the surface time, and t ini is the time at which the next dive initializes. (b) The glider position R, previous waypoint initialization time, t sur ω p 1, and current waypoint ω p in geodetic coordinates where λ and φ represent longitude and latitude, respectively. The glider satisfies the radius waypoint condition by entering the dashed circle of radius Γ 0 centered at the current waypoint. The glider satisfies the finish line waypoint condition by crossing the dashed line that passes through the current waypoint; the finish line is perpendicular to the line connecting the previous and current waypoints. B. Glider Integrator A central component of both the glider planner and simulator is the glider integrator, used to predict glider trajectories in the ocean. Predicting the glider trajectories is critical to glider planning since new trajectories are planned while gliders are underwater. We model the motion of each glider and its onboard control system under the influence of the bathymetry (water depth) and currents (water velocity). The bathymetry is important because gliders maintain a minimum altitude above the bottom. Ocean currents are important because they advect gliders and also because gliders respond to the expected currents. The glider onboard control system integrates its position, which is called the dead-reconed position, from estimates of its horizontal speed and heading. The GCCS predicts both glider position and glider dead-reconed position since these differ if, for example, the planner uses a accurate flow estimate than the glider. The detailed glider model is a discrete-time, three-dimensional, inematic model of the glider motion subject to the glider onboard pitch, heading and buoyancy control. The second-order transient effects of the onboard control are not modeled; instead we assume a fixed vertical speed and glide angle (pitch angle plus angle of attac) for both ascent and descent. Gliders move at constant speed in the direction of their desired headings and are advected by a three-dimensional flow field. We describe separately the equations of motion for horizontal position and vertical depth for the three phases of a single dive: on the surface before the dive (dive initialization); underwater during the dive; and on the surface after the dive. We describe the glider model using the notation illustrated in Figure 4(a). We index each glider by {1,..., N}, where N is the number of gliders. Let t 0 represent absolute time in the glider integrator. We denote the time

8 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 8 step and discrete-time step index by R + and n Z, respectively. The superscript sur (resp. uw ) refers to surface (resp. underwater). For glider, let z be depth, z min which the glider switches from ascending to descending), z max altitude, and z uw be bathymetry. Also for glider, let t ini be pre-dive surface duration, T gps τ ini and τ sur = [t ini, tini + T ini = [t sur, tsur + T gps be post-surface GPS duration, and T com ) denote the time interval before glider dives, τ uw be minimum inflection depth (the shallowest depth at be maximum inflection depth, Z min be minimum be dive initialization time, tsur be dive surface time, T ini be communication duration. Lastly, let = [t ini + T ini, tsur ) be during the dive, + T com ) be after the dive. For convenience, we denote the end of interval τ by τ. We use an ellipsoid E to approximate the earth geoid (shape). The glider integrator uses geodetic coordinates for the position of the th glider R = (λ, φ ) E where λ and φ are latitude and longitude. Let Γ : E E R + and η : E E S 1 be functions for computing distance and azimuth on the earth geoid. Position: The th glider position R at time t t ini is the solution to the following discrete-time model and depends on the position R, depth z 0, and waypoint index p N. We denote the p th waypoint by ω p E. Let f R 2 be the horizontal component of the th glider inertial velocity (with respect to the earth). (1) Before the dive, t τ ini = [t ini, tini + T ini), R (n + 1) = R (n) + f sur (R (n)), n = 1,..., t tini, where R (1) is the position of the glider at t ini. Since the gliders have no propulsion on the surface, the total surface velocity f sur is equal to the flow velocity on the surface. (2) During the dive, t τ uw = [t ini + T ini, tsur ), τ ini R (n + 1) = R (n) + f uw (R (n), z (n), p (n)), n =,..., t tini. To determine the glider waypoint number p (n), we integrate from the starting waypoint number using p (n) + 1 if Ψ(R (n), p (n)) p (n + 1) = p (n) otherwise where Ψ(R (n), p (n)) is a boolean waypoint completion condition. Once the glider completes a waypoint, then it heads towards the next waypoint in the list. For example, the radius waypoint condition shown in Figure 4(b) is Ψ cir (R (n), p (n)) = Γ(R (n), ω p (n) ) < Γ 0, where Γ 0 R + is the radius of the waypoint circle. In the presence of strong flow, the radius waypoint condition, combined with a heading algorithm that steers the glider directly toward the waypoint, can result in the glider turning into the flow if it overshoots the waypoint. An alternate waypoint completion condition, the finish line waypoint condition, is satisfied if the glider crosses the line that passes through the current waypoint and is perpendicular to the line connecting the previous and current waypoints. The horizontal component of the glider total underwater velocity f uw, which is the sum of the horizontal glider velocity relative to the flow and the estimated horizontal flow velocity, depends on the ocean currents and the glider onboard control system. We compute the horizontal glider speed relative to the flow using the desired vertical speed and glide angle. We assume that the orientation of the horizontal glider velocity equals the desired heading of the

9 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 9 glider, which is determined by the onboard control system using the glider estimate of the flow. The onboard control algorithms are proprietary and not described here. (3) After the dive, t τ sur = [t sur, tsur + T gps + T com ), R (n + 1) = R (n) + f sur (R (n)), n = τ uw,..., t tini. This completes the description of the horizontal glider motion model. Next we describe the vertical component. Depth: While underwater, gliders mae either a single descent and ascent or continuously descend and ascend until the maximum dive duration T max elapses. Although both configurations are used in practice, the latter rollercoaster motion minimizes time spent on the surface where the glider is vulnerable to surface currents and boats. Let g uw be the vertical component of the glider total velocity. We denote the dive direction by ζ { 1, 1}, where positive is descending. The th glider depth at time t is the solution to the following discrete-time model, which depends on the position R, depth z, and dive direction ζ. (1) Before the dive, t τ ini, z (n) = 0, n = 1,..., t tini. (2) During the dive, t τ uw, τ ini z (n + 1) = z (n) + g uw (R (n), z (n), ζ (n)), n =,..., t tini. The vertical component of the glider velocity g uw is the sum of the glider vertical velocity relative to the flow and the estimated vertical flow velocity (if available). We estimate the dive direction ζ { 1, 1} by integrating from the initial condition ζ (1) = 1 using 1 if (z (n) > z max ) (t > T max ) ( ) z (n) > z uw (R (n)) Z min ζ (n + 1) = 1 if (z (n) < ζ (n)z min ) (t < T max ) ζ (n) otherwise. In words, the glider ascends if it exceeds its maximum inflection depth, it exceeds the maximum dive duration, or its altitude is less than the minimum allowable altitude. If the glider is ascending before the end of the maximum dive duration, the dive direction reverses if the glider is shallower than the minimum inflection depth. (3) After the dive, t τ sur, z (n) = 0, n = C. Particle Integrator τ uw,..., t tini. At the core of the glider planner is the particle integrator, which generates the glider planned trajectories using closed-loop (coordinated) control of the simple particle model. Although the particle model is simple, the particle integration algorithm is complicated because gliders surface asynchronously and do not wait on the surface for new waypoints; the gliders use waypoints computed during the previous planning cycle. At the start of the particle integration step, one or more gliders has recently surfaced and all gliders are underway on their next dive. In this subsection, we define the particle model and briefly describe several coordinated control algorithms. Next we provide the particle integration algorithm. Lastly, we outline the waypoint generation and quality control.

10 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 10 Particle Model: Feedbac control of a simple glider model generates the planned trajectories. In this context, gliders are modeled by point masses (particles) confined to a plane. The motion of each particle obeys second-order Newtonian dynamics. The applied force, which is the control input, is assumed to always be perpendicular to the particle heading. This implies that the particles travel at constant speed subject to steering control. The speed used in the particle model is the effective horizontal speed of the glider, which is the horizontal speed of the glider (relative to the flow) scaled by the fraction of time spent underwater, i.e. T max /(T max + T ini + T gps + T com ). Alternative models of the particles moving in a flow field have been used; the design of coordinating control laws for gliders in the presence of a strong flow field is the subject of ongoing wor. As in the glider model, N denotes the number of particles and we index the particles by {1,..., N}. Let r = x + iy C R 2 be the particle position and ṙ = s e iθ is the particle velocity, where s R and θ T S 1 are the particle speed and heading, respectively. Using Newton s second law yields r = d dt (s e iθ ) = ṡ e iθ + s θ ie iθ (ν + s u i)e iθ where we have introduced the thrust ν = ṡ R and (gyroscopic) steering u = θ R control inputs. By assumption, ν = 0 so that s = s 0 for all. The particle model is ṙ = s 0 e iθ, θ = u, s 0 > 0, = 1,..., N. In our notation, we drop the subscript and use bold to represent a vector such as u = (u 1,..., u N ) T. Next we illustrate the types of motion that arise from the particle model and briefly describe several coordination control algorithms. In the case u = 0 we have θ (t) = θ (0) and each particle moves in a straight line along its initial heading. If u = ω 0 s 0 0 then θ (t) = θ (0) + ω 0 s 0 t and each particle moves around a circle with radius ω 0 1. The center of the circle orbited by particle is c = r +iω 1 0 eiθ. A feedbac control law presented in [13] drives all particles to orbit the same circle so that c = c 0 1, where 1 = (1,..., 1) T R N. We call this particle configuration a circular formation. Symmetric circular formations are circular formations in which the particles are arranged in symmetric patterns as they travel around the circle. A feedbac control law that isolates symmetric circular formations such as the splay state, in which the particles are uniformly spaced around the circle, is also provided in [13]. We have extended these results to the setting in which inter-particle communication is limited, directed, and time-varying [14]. Furthermore, we have derived control algorithms that stabilize formations on more general curves lie rounded rectangles suitable for oceanographic sampling [15], [16]. These controls use curvature and relative separation of particles along the desired trajectory as feedbac. During each planning cycle, the particle integrator coordinates particles that correspond to surfaced gliders with particles that correspond to underwater gliders. The particle integrator taes as input the trajectory predicted for each glider by the glider integrator as well as the desired tracs and coordination specified in the GCT. The output of the particle integrator is a new set of waypoints for each glider. The particle integrator uses a MATLAB R ODE solver to integrate trajectories from the time t 0 of the most recent glider surfacing to the planning horizon t 0 + T.

11 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 11 TABLE I PARTICLE INTEGRATOR ALGORITHM Goal: Inputs: Outputs: Integrate feedbac control algorithm using asynchronous initial conditions. Predicted trajectories up to next expected surfacing, desired tracs and coordination (GCT) New waypoints for all gliders During every planning cycle, the particle integrator performs: 1: set the integration start time t 0 to most recent glider actual surface time, t 0 = max =1,...,N t sur 2: for each particle = 1,..., N, set initial positions r (t 0 ) to glider (surface) position at time t sur, end for 3: for each particle = 1,..., N, set initial headings θ (t 0 ) according to control-specific algorithm, end for 4: call MATLAB ode solver with initial conditions r(t 0 ) and θ(t 0 ) and time span t [t 0, t 0 + T ] the following pseudo-code is executed every iteration of the MATLAB ode solver 5: for each particle = 1,..., N, 6: if t < t sur, set position r (t) and heading θ (t) to glider predicted underwater position and heading, end if 7: end for 8: compute steering control θ(t) = u(t) and particle velocity ṙ(t) using r(t) and θ(t), end for 9: for each particle = 1,..., N 10: if t < t sur, set steering control θ (t) and particle velocity ṙ(t) to zero, end if 11: end for 12: for each particle, overwrite start of planned trajectory with predicted trajectory up to expected surface time, end for 13: for each particle, generate waypoints for r (t), t > t sur, and run quality control, end for 14: for each particle, write waypoint file and push to glider data server, end for The initial position and heading of each particle is the position and heading of the glider at the next expected surfacing location and time t sur. We choose the initial heading to maximize the convergence rate of the control. Several important steps in the particle integrator algorithm occur inside the function passed to the MATLAB R ODE solver. For each glider that is predicted to have not yet surfaced by time t, we set the corresponding particle position r (t) and heading θ (t) to the predicted underwater position and heading. Then, the coordinated control algorithm computes the steering controls u(t) and velocities ṙ(t) for all particles using r(t) and θ(t). For each glider that is predicted to have not yet surfaced by t, we set the steering control u (t) and velocity ṙ (t) to zero. After the MATLAB R ode solver computes the planned trajectories, we replace the portion of each trajectory that occurs before the next expected surface time with the predicted underwater trajectory. We summarize this algorithm with a pseudo-code description in Table I. Waypoint Generation and Quality Control: We convert the glider planned trajectories to waypoints and chec them using a quality control (QC) filter. There are two alternate waypoint generation methods. In the first method, the waypoints are spaced uniformly in time assuming constant glider effective speed. In the second method, we convert portions of the planned trajectory with lower (resp. higher) curvature to fewer (resp. more) waypoints subject to a maximum (resp. minimum) spacing constraint. The latter method clusters waypoints near tight turns and spreads out waypoints along straight portions of the planned glider trajectories. To provide robustness to delays and errors incurred in satellite communication between the glider data server and the glider, each waypoint file that

12 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 12 passes the QC filter has a unique message number and an expiration date. Waypoint quality control is required for safe, automated operation of gliders. To pass the QC filter, the following criteria must be met: (1) the last glider position update must not be too old; (2) all waypoints other than those at the start of the list must be inside a prescribed bounding box; (3) the waypoints must not be shallower than the glider minimum operating depth; and (4) the waypoints must be spaced by no more (resp. less) than the maximum (resp. minimum) allowable spacing. We remove waypoints that are too shallow. Failure to meet any other requirement results in rejection of the entire waypoint list. IV. EXPERIMENTAL RESULTS We present results from two glider deployments in which the GCCS controlled multiple gliders to coordinated trajectories. These experiments further justify our approach to coordinated control of a glider fleet. First, we describe a virtual deployment in Monterey Bay in which the GCCS coordinated ten gliders to maximize sampling performance in a rectangular domain as illustrated in Figure 2(a). In this deployment, the GCCS also simulated the glider motion in a model ocean. Second, we describe a real deployment in Buzzards Bay in which two gliders maintain a prescribed spacing as they travel around a single rectangular trac as shown in Figure 2(b). A. Virtual Glider Deployment in Monterey Bay The GCCS controlled ten gliders in a two-wee-long virtual deployment in Monterey Bay in March The deployment was part of a virtual pilot exercise for the August 2006 Adaptive Sampling and Prediction (ASAP) field experiment [17]. The ocean science focus of this field experiment was to gain a better understanding of the threedimensional ocean dynamics off Point Año Nuevo, which is north of Santa Cruz, by conducting a high resolution survey with a fleet of underwater gliders and other (manned) assets. Of special interest to ASAP oceanographers was computing the mass and heat flux through the boundary of a 20 m by 40 m control volume. The virtual deployment tested the capability of the GCCS to (i) control a glider fleet according to a candidate sampling plan and (ii) respond to adaptations of this plan. We describe the initial sampling plan, the glider coordinated trajectories, in Section II-B and Figure 2(a); the control algorithm appears in [16]. During the virtual deployment, the gliders sampled a model ocean generated by the Harvard Ocean Prediction System from data collected during the 2003 Autonomous Ocean Sampling Networ (AOSN-II) field experiment [18]. The model ocean contains temperature, salinity, and three-dimensional flow velocity at 500 meter horizontal resolution with 22 vertical levels over a 35 day period starting August 6, The temperature data, which was collected by gliders during AOSN-II, has spatial and temporal temperature decorrelation scales of 22 m and 2.2 days, respectively [8]. Using these characteristic scales, we computed the OA performance of the glider fleet. Four virtual Spray gliders travelled clocwise around the northern trac and sought uniform separation as shown in Figures 5(a) and 5(b). Initially, the Spray gliders performed (simulated) Iridium communication after every dive. The resulting frequent and lengthy surfacings, combined with surface currents of 0.15 to 0.35 m/s, resulted in a low effective speed of 0.2 m/s (median value for all four Spray gliders). Consequently, strong surface currents

13 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 13 degraded the glider along-trac spacing. On the fourth day of the deployment, we reconfigured the Spray gliders to communicate on surfacing only if two or more hours had elapsed since the last communication session. In this configuration, the glider effective speed increased by 25% to 0.25 m/s and the along-trac separation recovered. We shows the glider trajectories before (Figure 5(a)) and after (Figure 5(b)) this operation change. The group of six virtual Slocum gliders formed two synchronized subgroups of three gliders each, as shown in Figures 5(a) and 5(b). During the deployment the GCCS achieved good spacing of the Slocum subgroups around each trac and good synchronization of the two subgroups. On the fourth day, we added a new trac that extends outside the sampling domain to the south and overlaps the original southern-most trac. This adaptation increases the sampling effort in a region south of the sampling domain without compromising the sampling performance in the original domain. A so-called scout Slocum glider orbited the new trac in coordination with the other Slocums. Figure 5(b) shows that this adaptation did not degrade coordination. That is, if one superimposes the original and new southern tracs, we see that the three Slocums assigned to these two tracs are uniformly spaced and, similarly, all six Slocums are still synchronized in two subgroups. In the presence of disturbances such as flow, the GCCS regulates the glider progress around the trac to achieve the desired along-trac spacing between the gliders. For example, the control algorithm used during the Monterey Bay deployment steers gliders to inside (resp. outside) lanes on each trac to speed up (resp. slow down). To evaluate the sampling performance of this algorithm, we compute the OA mapping error of the glider fleet in the sampling domain shown in Figures 5(a) and 5(b). In Figure 5(c), we plot the sampling performance measured by the mean OA error both in the interior and on the boundary of the glider sampling domains (less error is better). When we adapted the GCT, we effectively reduced the sampling effort in the original domain; this increased the average error in the domain by about 5%. However, measurements collected on the new trac fill the gap in the OA mapping error in the center of the original southern-most trac (to see this, compare Figures 5(b) and 5(a)). The GCCS supports GCT adaptations, which may improve collective mapping performance. We quantify the effect of adapting the Monterey Bay GCT by computing a new metric. We compute the percentage (of the interior or boundary) of the sampling domain that has mapping error less than a threshold of 0.5 (higher percentage is better). As shown in Figure 5(c), the percentage metric shows a degradation of the sampling performance inside the box after the adaptation but the performance on the boundary does not. That is, the interior area percentage metric exhibits downward fluctuations after the GCT was adapted. The fluctuations of the boundary percentage metric appear unaffected by switching the GCT. B. Real Glider Deployment in Buzzards Bay In collaboration with Dr. David Fratantoni of WHOI, the GCCS controlled two Slocum gliders in a GCCS sea trial in Buzzards Bay [19]. The gliders travelled clocwise around a single rectangular trac with dimensions 2.8 m by 5.6 m. In addition, the gliders sought to maintain a fixed, along-trac separation distance of 1/6 the trac perimeter, which corresponds to a curve-phase of approximately 1 radian. The Buzzards Bay GCT is shown in Figure 2(b). In this subsection, we summarize results from three days of coordinated control of the two gliders to

14 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 14 this GCT. Additional analysis and a description of the control algorithm appear in [20]. During the Buzzards Bay deployment, coordination of the gliders was difficult due to strong tidal flow. Figures 5(d) and 5(e) show the glider trajectories superimposed on gridded OA flow maps computed from their measurements. We estimated the velocity decorrelation scales to be 2.5 m (spatial) and 3 hours (temporal); that is, the ocean currents were highly variable in space and time. In Figure 5(d), the gliders have good separation, but strong northeastern flow degrades their ability to stay on the prescribed trac. In Figure 5(e), both the separation and trac-following performance is poor due to strong westward flow, which exceeded the glider effective speed. Figure 5(f) shows the gliders recover the desired curve-phase separation at the end of the third day. This experiment demonstrated both the capabilities and limitations of the GCCS in a real glider deployment with strong flow. During periods of moderate flow, we observed good system performance in terms of trac following and glider coordination. During periods of extreme flow, the system performance degraded substantially. In such situations, adaptation of the glider coordinated trajectories is often necessary. For example, the predominant flow conditions can dictate the direction of travel around the trac and even the location and orientation of the trac. Currently, GCT adaptation requires human intervention; wor is underway to automate this. In addition, are incorporating a tidal flow model to improve coordination performance in strong, oscillatory flows. V. SUMMARY AND CONCLUSIONS Autonomous underwater gliders are a reliable platform for long-duration ocean sampling with multiple vehicles. The GCCS implements real-time feedbac control of a glider fleet to a set of coordinated trajectories. During each GCCS planning cycle, we predict glider motion using a detailed, three-dimensional glider model and generate future trajectories using a simple, planar particle model. A combination of good trac planning and good real-time coordination achieves high sampling performance. We describe two GCCS demonstrations: one with ten virtual gliders in Monterey Bay and one with two real gliders in Buzzards Bay. The GCCS demonstrations justify using feedbac control of a high-level, simple model for automated, real-time trajectory planning of a fleet of autonomous vehicles. Both the Monterey Bay and Buzzards Bay deployments underscore the importance of choosing and adapting the planned glider coordinated trajectories in light of the flow conditions or fleet sampling performance. We demonstrated the value of the GCCS simulation testbed by identifying and addressing inefficiencies in the Spray glider configuration during the Monterey Bay virtual deployment. The Buzzards Bay experiment motivates our ongoing wor on improved robustness of coordinated control algorithms in adverse flow conditions. In August 2006, the GCCS coordinated six Slocum gliders for over three wees during the ASAP deployment. We are preparing a description and analysis of these promising results. ACKNOWLEDGMENT The authors would lie to than Francois Leien, Pradeep Bhatta, Dmitrijs Gurins, and Jeff Pinner (Princeton); David Fratantoni and John Lund (WHOI); Russ Davis, Jeff Sherman, and Brent Jones (SIO); Pierre Lermusiaux, Wayne Leslie, and Pat Haley (Harvard); and the entire ASAP team.

15 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 15 REFERENCES [1] N. E. Leonard, D. A. Paley, F. Leien, R. Sepulchre, D. M. Fratantoni, and R. E. Davis, Collective motion, sensor networs and ocean sampling, Proc. of the IEEE, to appear. [Online]. Available: dpaley/papers/ieeeproc.pdf [2] E. Fiorelli, N. E. Leonard, P. Bhatta, D. A. Paley, R. Bachmayer, and D. M. Fratantoni, Multi-AUV control and adaptive sampling in Monterey Bay, IEEE J. Oceanic Engineering, to appear. [Online]. Available: dpaley/papers/joeauv.pdf [3] T. B. Curtin, J. G. Bellingham, J. Catipovic, and D. Webb, Autonomous oceanographic sampling networs, Oceanography, vol. 6, no. 3, pp , [4] E. Creed, J. Kerfoot, C. Mudgal, S. Gleen, O. Schofield, C. Jones, D. Webb, and T. Campbell, Automated control of a fleet of Slocum gliders within an operational coastal observatory, in Proc. OCEANS 2003 MTS/IEEE Conf., vol. 2, 2003, pp [5] B. Schulz, B. Hobson, M. Kemp, and J. Meyer, Field results of multi-uuv missions using Ranger micro-uuvs, in Proc. OCEANS 2003 MTS/IEEE Conf., vol. 2, 2003, pp [6] S. S. Mupparapu, S. G. Chappell, R. J. Komersa,, D. R. Blidberg, R. Nitzel, C. Benton, D. O. Popa, and A. C. Sanderson, Autonomous systems monitoring and control (ASMAC) - an AUV fleet controller, in Proc IEEE/OES Autonomous Underwater Vehicles Worshop, June 2004, pp [7] R. E. Davis, C. E. Erisen, and C. P. Jones, Autonomous buoyancy-driven underwater gliders, in The Technology and Applications of Autonomous Underwater Vehicles, G. Griffiths, Ed. Taylor and Francis, 2002, ch. 3, pp [8] D. L. Rudnic, R. E. Davis, C. C. Erisen, D. M. Fratantoni, and M. J. Perry, Underwater gliders for ocean research, Marine Technology Society Journal, vol. 38, no. 1, pp , [9] D. L. Rudnic and R. E. Davis, Mass and heat budgets on the northern California continental shelf, J. of Geophysical Research, vol. 93, no. C11, pp. 14, , [10] I. Shulman, D. J. M. Jr., M. A. Moline, S. H. D. Haddoc, J. C. Kindle, D. Nechaev, and M. W. Phelps, Bioluminescence intensity modeling and sampling strategy optimization, J. of Atmospheric and Oceanic Technology, vol. 22, pp , [11] P. F. J. Lermusiaux, Adaptive modeling, data assimilation and adaptive sampling, J. of Inverse Problems, under review. [12] F. P. Bretherton, R. E. Davis, and C. B. Fandry, A technique for objective analysis and design of oceanographic experiments applied to MODE-73, Deep-Sea Research, vol. 23, pp , [13] R. Sepulchre, D. A. Paley, and N. E. Leonard, Stabilization of planar collective motion: all-to-all communication, IEEE Trans. Automatic Control, to appear. [Online]. Available: dpaley/papers/taci.pdf [14], Stabilization of planar collective motion with limited communication, IEEE Trans. Automatic Control, under review. [Online]. Available: dpaley/papers/tacii.pdf [15] F. Zhang and N. E. Leonard, Coordinated patterns on smooth curves, in Proc. IEEE Int. Conf. Net., Sens. and Cont., 2006, pp [16] D. A. Paley, N. E. Leonard, and R. Sepulchre, Collective motion of self-propelled particles: stabilizing symmetric formations on closed curves, to appear in Proc. 45th IEEE Conf. Decision and Control. [Online]. Available: dpaley/papers/cdc06.pdf [17] Adaptive Sampling and Prediction, MBARI website, [18] Adaptive Sampling and Prediction, Harvard website, asap.html, [19] D. M. Fratantoni and J. M. Lund, Glider operations in Buzzard s Bay, MA, Woods Hole Ocean. Inst., Tech. Rep. BUZZ0306, [Online]. Available: [20] F. Zhang, D. M. Fratantoni, D. A. Paley, J. M. Lund, and N. E. Leonard, Control of coordinated patterns for ocean sampling, Int. J. Control, accepted. [Online]. Available: dpaley/papers/ijc06.pdf Fig. 5. Glider positions and velocities from two GCCS deployments. Each glider is a circle with a 12 hour comet tail and a blac velocity arrow. Left column: virtual deployment in Monterey Bay. The desired tracs are thin, solid lines and the sampling domain is the dashed box that circumscribes the three tracs in (a). The bathymetry contours are 30, 400, and 1000 m. (a) Original GCT; (b) adapted GCT; (c) OA performance metrics from original (white bacground) and adapted (gray bacground) GCT. Right column: real deployment in Buzzards Bay. Two Slocum gliders orbit the same trac with desired along-trac spacing equal to 1 radian of curve-phase. The OA flow velocity is depicted by small blac arrows. The bathymetry contours are 10, 15, and 30 m. (d) The gliders maintain the desired spacing in benign flow; (e) the gliders veer off course in strong flow; (f) curve-phase performance indicates the gliders regain desired spacing (dashed line).

16 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 16 / %(#** %(#*% / "#*/789/ %(#*+,56/01234 %(#*& %(#*( %(#* //:2(( %(#%) //:2") %(#%$ %(#%! %(#%' /!!"#$%!!"#$&!!"#$!!"#!$!!"#!'!!"#!%,-./01234 (a) Monterey Bay, August 14, :00 GMT (d) Buzzards Bay, March 14, :00 GMT / %(#** %(#*% / "#*/789/ %(#*+,56/01234 %(#*& %(#*( %(#* //:2(( //:2") %(#%) %(#%$ %(#%! %(#%' /!!"#$%!!"#$&!!"#$!!"#!$!!"#!'!!"#!%,-./01234 (b) Monterey Bay, August 20, :00 GMT (e) Buzzards Bay, March 15, :00 GMT! ',* ',/ ',) ',. ',( ',- '," ',& ',! ' +!"!#$%!&''"!(!#$%!&''"!)!#$%!&''" 012+#345+#6,+733,+8/'",*+9: & ; $<=53>+#6,+733,+8!&',&+9:;?+012+#345+@+',(+#6,+733,?+012+0<=,+@+',(+#6,+733,!*!#$%!&''" &!!#$%!&''" + &"!#$%!&''" Curve!Phase Separation (rad) !1! Time (hours) startin7 Mar 14, :00 GMT (c) Monterey Bay mapping performance (f) Buzzards Bay coordination performance