In recent years, collecting scientific data from ocean environments has been increasingly undertaken by underwater gliders. For better navigation performance, the influence of flow on the navigation of underwater gliders may be significantly reduced by estimating flow velocity. However, methods for estimating flow do not always account for spatial and temporal changes in the flow field, leading to poor navigation in complex ocean environments. To improve navigation accuracy in such environmental conditions, this paper studies an approach for the real-time guidance of underwater gliders assisted by predictive ocean models. This study is motivated by glider deployments conducted from January to April 2012 and in February 2013 in Long Bay, South Carolina, where the ocean currents are characterized by strong tides and a stronger alongshore current, the Gulf Stream. The flow speed here often exceeds the forward speed of the glider. To deal with such a challenge, a computationally efficient method of depth-averaged ocean current modeling was developed. The method adjusts the ocean model based on the most recent ocean observations from gliders as feedback, and flow predictions from the model are incorporated into path planning, which produces waypoints. The entire process of flow prediction, path planning, and waypoint computation is performed off-board the gliders in real time by the glider navigation support system, the Glider-Environment Network Information System (GENIoS). This paper presents the setup and method for the glider navigation strategy applied to the Long Bay deployments. For demonstration, the performance of the method described here is compared to that of the default method implemented in the built-in glider navigation system.
The underwater glider has found applications in the monitoring and data collection of ocean environments for oil field surveys, military operations, and deep-sea and coastal research (Schofield et al. 2007; Nicholson and Healey 2008). Since the sampling performance is tightly coupled with navigation performance, underwater glider navigation is of great interest within the oceanographic research community. As part of the effort to improve the sampling and navigation performance of underwater gliders for scientific missions, existing approaches such as underwater glider control for ocean sampling (Zhang et al. 2007; Leonard et al. 2010) and underwater glider control support systems (Creed et al. 2003; Paley et al. 2008; Fernández-Perdomo et al. 2010) have been studied.
Underwater gliders primarily use the global positioning system (GPS) to identify their position on the surface of water (Meldrum and Haddrell 1994). However, since GPS signals cannot propagate through seawater, gliders cannot rely on GPS for underwater localization, so their position is estimated via dead reckoning between regular surfacing events for GPS updates. Driven by buoyancy and mass controls, underwater gliders have low speed; hence, their trajectory is strongly affected by ocean currents, causing a difference between the dead reckoning trajectory and the actual trajectory. To increase navigation accuracy in dead reckoning, some underwater gliders have the capability to compute an estimate of average flow velocity along the glider trajectory between the last and current surfacing positions and to incorporate the flow estimate into dead reckoning to navigate underwater until the next surfacing event (Merckelbach et al. 2008). We refer to this navigation method based on dead reckoning coupled with the glider-derived flow estimate as the default glider navigation method.
The default glider navigation method may help increase the accuracy of navigation in a calm water environment. However, since the glider-derived flow estimate does not account for temporal and spatial changes in the flow field, the estimated flow may differ from the actual flow experienced by the glider, causing poor navigation performance. To overcome limitations caused by insufficient knowledge of flow, we propose an approach for the real-time guidance of underwater gliders assisted by predictions from ocean models. Ocean models can be largely categorized as either physics-based or empirical (also known as statistical) (Reikard and Rogers 2011). Typically, physics-based ocean models are generated by solving geophysical fluid dynamics with given initial and boundary conditions (e.g., Luettich et al. 1992; Bleck 2002; Shchepetkin and McWilliams 2005; Haidvogel et al. 2008), while empirical models are trained using a historical dataset (e.g., Frolov et al. 2012; Chang et al. 2014). To increase the accuracy of an ocean model, real-time direct/remote oceanographic observations from satellite, ships, buoys, and gliders can be used as feedback to adjust the model via data assimilation (Kalnay 2003). Lermusiaux (2006) analyzes the uncertainties of ocean models caused by practical simplifications, inexact representations, and parameterizations of the models. In the work, model uncertainties are represented as stochastic forcings and are estimated and predicted in a Bayesian framework. Szwaykowska and Zhang (2014) discuss the navigation of marine vehicles using ocean models and study the bound of growth in error between vehicle positions in the real ocean and those in the simulated ocean based on ocean models. The method of error analysis studied there may be applied to improve the accuracy of ocean models and the performance of navigation algorithms for marine vehicles.
Our study is motivated by field experiments off the coast near Long Bay, South Carolina, from winter to early spring in 2012 and in late winter in 2013 to study persistent wintertime phytoplankton bloom formation. In the experiments, Slocum gliders collected data while maintaining desired tracks in the survey domain. Unlike other areas such as Monterey Bay, California, where previous glider navigation experiments have been conducted (Singh et al. 2001; Fiorelli et al. 2006; see special issue of Deep-Sea Research Part II, 2009, Vol. 56, Issues 3–5), ocean currents off the coast of Long Bay are not only characterized by strong tides but also influenced by the Gulf Stream, which has complex and variable circulation downstream of the Charleston Bump (Legeckis 1979; Bane 1983). This combination of tides and the Gulf Stream causes significant temporal and spatial variations in the flow field and requires a better glider navigation method than the default glider navigation method.
In recent years, there have been active studies regarding the path planning and navigation of marine vehicles. Witt and Dunbabin (2008) studied optimal path planning under strong, time-varying currents and demonstrated experimental results. Soulignac (2011) presented an optimal path planning algorithm that guarantees the existence of a path in strong current fields. Fernández-Perdomo et al. (2010) developed a path planning algorithm based on A*, a graph-based optimal path planning algorithm (Hart et al. 1968), adapted for constant surfacing times. The algorithm incorporated a regional ocean model and generated the simulated glider trajectory using predictions from the model. However, the authors did not test the algorithm for real gliders and did not provide the validation of predicted flow from the ocean model by comparing with the observed flow from a glider. Pereira et al. (2013) also incorporated predictions from ocean models into path planning. However, the main objective of their algorithms is to minimize collision risks of gliders with ships by accounting for the uncertainty in predictions from ocean models, not to improve navigation accuracy.
In this paper, we present a glider navigation strategy applied to the Long Bay deployments. First, the paper introduces a novel method of ocean current modeling for short-term forecasts to guide gliders. To effectively deal with highly variable and complex flow in the Long Bay deployments, we develop a hybrid ocean model based on flow data from a tidal ocean model and observations from gliders. The method first approximates slowly varying nontidal flow from glider-derived flow estimates and then adds rapidly varying tidal flow to the nontidal flow. The method adjusts the ocean model based on the most recent ocean observations from gliders as feedback. The adjustment is computed only in about a minute in real time on a single processor; hence, this method is computationally more efficient than typical applications of data assimilation in oceanography [e.g., Kalman filtering and variational techniques such as three-dimensional/four-dimensional variational data assimilation (3DVAR/4DVAR)]. We verify the method by comparing simulated flow with observed flow during a glider deployment.
Given flow predictions, we implement a path planning algorithm based on a flow-canceling controller to generate waypoints. We predict the trajectory of a glider under flow and generate waypoints based on the predicted trajectory, so that waypoints guide the glider to track the predicted trajectory. A similar approach to our waypoint design was previously studied in Smith et al. (2010) (see “alternate” waypoints in there). Since ocean flow varies in time and the error between predicted flow and real flow may increase over time, we run the path planning algorithm based on the latest glider and flow data and update waypoints every time a glider comes to the surface. The entire process of flow prediction, path planning, and waypoint updates is performed off-board the gliders by a glider navigation support system, the Glider-Environment Network Information System (GENIoS), automating glider navigation.
We test and verify the proposed method of glider navigation through the Long Bay deployments. To compare with our method, we also test the default glider navigation method through a series of simulated experiments. Our method guides gliders toward a target using a set of waypoints generated by path planning, but in the default glider navigation method, gliders swim toward a target without guidance. The test environment for the default glider navigation method is reconstructed from the data collected during the Long Bay deployments. Based on the experimental and simulation results, we evaluate the performance of both glider navigation methods by computing 1) the distance between the glider and the target at each surfacing event and 2) the difference between the predicted surfacing position under flow and the actual surfacing position. The results show that compared to the default glider navigation method, our method improves the performance of navigation and provides better reliability and predictability in the presence of flow with significant temporal and spatial variations.
The rest of the paper is organized as follows. Section 2 provides background information and motivation for our glider deployments and glider navigation method. Section 3 briefly introduces our glider navigation support system that was used in the deployments and the simulations. Then, we present our glider control and path planning strategy in section 4 and provide experimental and simulation results in section 5. Last, section 6 discusses conclusions and future work.
2. Background and motivation
In this section, we introduce the mechanisms of Slocum glider navigation and the description of the Long Bay deployments. Our study on glider navigation is motivated by the Long Bay experiments, in which we deployed Slocum gliders from winter to early spring in 2012 and in late winter in 2013. We present glider missions in the Long Bay deployments and the challenges in the glider operations encountered there.
a. The Slocum glider
Because of its long range and high endurance, the Slocum glider has become a robust sensor platform within the oceanographic community for long-duration ocean sampling field experiments. The Slocum glider generates forward propulsion by changing its mass distribution and buoyancy, converting its vertical velocity to horizontal motion along sawtooth underwater paths (Davis et al. 2002; Rudnick et al. 2004). The Slocum glider typically travels at low speed (the horizontal through-water speed is approximately 0.25–0.35 m s−1), yet it is capable of collecting data in the ocean for several weeks without a recharge.
The glider navigates via dead reckoning, which estimates the position of the glider using estimates of glider speed, compass heading, and flow velocity. While navigating, the glider regularly comes to the surface of water for GPS updates and data transfers to a dockserver, which is a designated onshore computer for data transmission through satellite links. Figure 1 illustrates glider navigation from the ()th diving event through the kth surfacing event to the kth diving event. Between two surfacing events, glider navigation consists of two phases: surface and subsurface. The following subevents take place during the surface phase:
b: acquisition of the first valid GPS fix
c: surface drift and data transmission
We denote the actual and dead reckoning horizontal positions of a glider at time t by and , respectively. Hereinafter, we deal with the glider and the flow in the plane. In the figure, the time associated with each subevent for the kth surfacing event is denoted by , , , or , specifying one of the subevents. While on the surface, a glider first obtains a GPS fix and transmits collected science (scientific sensing related) and flight (glider navigation related) data to the dockserver. Then, the glider receives updated mission-related instruction data (e.g., a set of waypoints) from the dockserver before beginning to dive.
Given a list of waypoints or a desired path of travel during the th surfacing phase, a Slocum glider dives at and navigates underwater by dead reckoning. The glider comes back to the surface of water at regular intervals, operationally every 4–6 h, and also once all the waypoints on the list given to the glider have been reached based on its dead reckoning estimate. Because of the influence of flow, when the glider surfaces at time , we typically observe a difference between the dead reckoning surfacing position and the GPS surfacing position . We refer to this difference as the dead reckoning error at the kth surfacing event. The GPS surfacing position is estimated at time based on the surface flow velocity computed for the time between and and the surface drift time between and (Teledyne Webb Research 2014).
Upon the kth surfacing event, a glider computes an estimate of average flow velocity along the glider trajectory based on the dead reckoning error accumulated over the th subsurface phase. This glider-derived flow estimate can be either incorporated into navigation to reduce the dead reckoning error or deactivated, so that no flow estimate is used in navigation. Let us introduce a switching signal to indicate whether the estimated flow is used for navigation. The signal , which is the default setting of the default glider navigation method, indicates that the estimated flow is used for navigation and 0, otherwise. Then, the glider-derived flow estimate at the kth surfacing event is given by
which is the previous flow estimate used to navigate over the th subsurface phase plus the new flow estimate based on the dead reckoning error accumulated during the th subsurface phase.
The glider swims in sawtooth underwater patterns during the subsurface phase, sampling most of the water column along the glider trajectory. Hence, the glider-derived flow velocity estimated in the plane may be interpreted as an estimate of the depth-averaged flow velocity over the subsurface phase. In other words, Eq. (1) is an approximation of path integral of flow over the water column that a glider experiences during one subsurface phase. The glider uses a constant flow estimate to navigate underwater until it comes back to the surface. This constant estimate does not account for temporal and spatial changes in the flow field during navigation, which may cause poor navigation performance.
We introduce the pseudocode of the default glider navigation method used in the built-in navigation system of the Slocum glider in algorithm 1 (Teledyne Webb Research 2014; Table 1). Once a glider dives into the ocean, the dead reckoning algorithm starts estimating the position of the glider by measuring the vertical speed, the pitch, and the heading of the glider. The vertical speed and the pitch are used to estimate the horizontal through-water speed. Then, the algorithm computes the glider position under flow using the glider speed, the heading, and the estimated flow at each time step with step size .
b. The Long Bay experiments and glider missions
Sustained wintertime phytoplankton blooms are observed off the coast of Long Bay in multiyear satellite chlorophyll images (Fig. 2). In most areas of the South Atlantic Bight shelf, upwelling associated with Gulf Stream frontal eddies is a major mechanism of nutrient input, driving phytoplankton growth at the shelf edge. Just upstream of the Long Bay study area, the Charleston Bump deflects the Gulf Stream offshore, effectively shutting down direct frontal eddy nutrient input to Long Bay (Lee et al. 1991). Prior in situ observations (Atkinson et al. 1996) suggest recurring input of nutrients from the upper slope to the outer shelf off Long Bay from winter to early spring, but they do not provide evidence of a physical mechanism that accounts for both the alongshelf and seasonal patterns of the winter phytoplankton blooms associated with Long Bay.
To study the mechanisms of phytoplankton bloom formation in the outer shelf to the upper-slope region near Long Bay, we deployed various oceanic sensing platforms in a field experiment (LB2012) from January to April 2012. To observe the vertically resolved property fluxes, we deployed three moored instruments at three locations on the shelf/slope: LB1 (inner shelf, 30-m depth), LB2 (shelf break, 75-m depth), and LB3 (upper slope, 175-m depth), as shown in Fig. 2. Near-continuous cross-shelf and upper-slope observations were obtained by two Slocum gliders, which collected temperature, salinity, pressure, chlorophyll fluorescence, dissolved oxygen (DO), colored dissolved organic matter (CDOM), and turbidity data. One glider (Ramses) with a maximum operable depth of 200 m operated as a virtual mooring, maintaining its position at the edge of the Gulf Stream near LB3. The other glider (Pelagia) with a maximum operable depth of 100 m conducted a cross-shelf section between LB1 and LB3 (or LB2 depending on ocean conditions). The combined dataset from both gliders help us identify the physical and biological drivers of phytoplankton bloom formation in the study domain. In a follow-up experiment (LB2013) in late February 2013, one moored profiler sampled at LB2 and one Slocum glider (Modena) with a maximum operable depth of 200 m surveyed between LB+20 (33-m depth) and LB+40 (130-m depth), which are 20 and 40 km offshore from LB1 in the cross-shore direction, respectively.
The observational domain is characterized by strong tidal and Gulf Stream currents, which can hinder glider navigation. Strong tides are largely aligned with the cross-shore direction over the shelf off Long Bay. In addition, circulation at the shelf break is dominated by a strong current called the Gulf Stream, which has highly variable and complex circulation because of eddies and filaments that develop and propagate along its shoreward front. The complex ocean dynamics in this region form steep temporal and spatial gradients in flow off the coast of Long Bay, and ocean currents here often exceed the forward speed of a glider. The typical speed of the tidal and Gulf Stream currents on the shelf is 0.2–0.3 and 0.5–0.75+ m s−1, respectively. The maximum glider-estimated current 1.67 m s−1 in the Gulf Stream is shown in Fig. 3.
3. Glider navigation support by GENIoS
In the early 2000s, the concept of the Autonomous Ocean Sampling Network (AOSN) (Curtin et al. 1993) fueled interest in automated multiglider control systems. In 2006, as part of the AOSN field experiments, various research groups took part in the Adaptive Sampling and Prediction (ASAP) experiment in Monterey Bay, California. In ASAP, homogeneous and heterogeneous fleets of two specific types of underwater gliders, Slocum and Spray gliders, were deployed. To support automated glider operations, Paley et al. (2008) developed and tested the Glider Coordinated Control System (GCCS). Feedback control in GCCS enables the automation of both individual glider navigation and fleet coordination. An individual glider navigates by simply following waypoints generated by GCCS in such a way as to maintain the coordination of a glider fleet along a sampling trajectory. The design of the sampling trajectory is determined adaptively based on a classical linear estimation technique called objective analysis, so that a fleet of gliders can sample optimally distributed measurements, improving collective survey performance.
The glider control capability of GCCS is well-verified through ASAP (Leonard et al. 2010). However, the ocean dynamics near Monterey Bay in the ASAP experiment significantly differ from those near Long Bay. Considering the strong tidal and Gulf Stream currents off Long Bay, glider control strategies of GCCS were not applicable to the Long Bay experiments. For example, in ASAP, when a glider came to the surface, GCCS started predicting the following surfacing event of the glider and generated a list of waypoints based on the predicted surfacing data. Then, the next time the glider came to the surface, GCCS sent the pregenerated waypoints to the glider. In addition, GCCS did not incorporate flow fields in path planning. These settings for waypoint generation in ASAP may decrease navigation accuracy under a strong, highly variable flow field. Therefore, navigating gliders in the Long Bay experiments required the most recent glider surfacing data and near-real-time ocean flow data.
To support the Long Bay experiments under strong, highly variable flow, we extended GCCS into GENIoS, which is composed of the following modules: glider planner, glider simulator, environmental input manager, and dockserver handler, shown in Fig. 4 [please refer to Paley et al. (2008) for details about the glider planner and the glider simulator]. Compared to GCCS, GENIoS incorporates the following two major additional features associated with the environmental input manager and the dockserver handler, respectively: 1) user-defined or general oceanic environmental inputs (e.g., oceanic general circulation models) for planning a glider trajectory and computing waypoints, and 2) a communication interface with dockservers for obtaining the most recent glider surfacing data and sending waypoints based on the recently received surfacing data in real time.
For the fast acquisition of the latest surfacing information and the fast transmission of waypoints, GENIoS incorporates the dockserver handler interface, which enables direct data communication between GENIoS and the dockserver via network protocols such as the secure shell protocol (SSH) and the secure file transfer protocol (SFTP). Figure 5 shows the mechanism of the dockserver handler. The Slocum glider communicates with glider pilots via terminal software on a remote computer. The most commonly used terminal for the data communication of the Slocum glider is the glider terminal provided by Teledyne Webb Research. Using the glider terminal, a pilot can start glider missions, check the glider or mission status, change the settings of navigational/sampling strategies, and transfer collected glider data. While a mission is running, the glider terminal logs all text outputs, including surfacing information transmitted from a glider. Using the dockserver handler interface, GENIoS continuously monitors glider log files on the dockserver. When the latest glider surfacing information is reported on the dockserver, GENIoS obtains glider-navigation-related information without waiting for the glider data files to be transmitted and processed, enabling as fast a waypoint computation as possible after a glider surfaces.
In addition to the fast acquisition of glider data and the fast transmission of waypoints, the dockserver handler interface also enables GENIoS to transmit a waypoint list to a dockserver in parallel with the data transmission between the dockserver and a glider, enabling a seamless waypoint update. During the Long Bay experiment, the entire process of waypoint computation and update required less than 2 min on average out of approximately 25 min of the total surface time until a new set of waypoints was ready to send to a glider. Moreover, the interface handles any data communication related to the status of a glider and its ongoing mission, enabling autonomous glider operation through scripted glider handling.
To more thoroughly understand the behavior of gliders in the ocean and to account for the most recent ocean observations, we developed the environmental input manager that helps GENIoS employ various real/simulated oceanic data in real time. First, for actual field experiments, it brings ocean data from the general ocean model or the ocean observation system into GENIoS for the guidance of glider navigation in real time through the Open-Source Project for a Network Data Access Protocol (OPeNDAP) framework (Cornillon et al. 2003). Thus far, we have incorporated Wave Radar (WERA), a type of HF radar; the South Atlantic Bight and Gulf of Mexico (SABGOM) model (Hyun and He 2010); the Hybrid Coordinate Ocean Model (HYCOM) (Bleck 2002); the Advanced Circulation (ADCIRC) model output (Luettich et al. 1992). Employing oceanic data through the environmental input manager enables the guidance of underwater gliders by GENIoS to be more adaptive to environmental changes predicted by oceanic data models. Second, for simulated experiments, the environmental input manager allows GENIoS to navigate a glider under a simulated field using a user-defined flow model, enabling us to analyze glider behaviors in unexpected or episodic conditions.
4. Path planning assisted by predictive ocean models
For the Long Bay deployments, we develop a new computationally efficient hybrid method of ocean current modeling. The method approximates the tidal and nontidal components of flow by using tidal ocean model ADCIRC and a glider-derived flow estimate. We test the method in the Long Bay deployments by implementing it in path planning on an onshore computer. To plan a path of a glider under flow, we design a flow-canceling controller. Using predicted flow from our ocean current model and the flow-canceling controller, our path planning algorithm computes the predicted trajectory of a glider under flow and generates waypoints based on the predicted trajectory.
a. Ocean current modeling
To increase navigation accuracy, the default navigation method of the Slocum glider computes the flow estimate based on the dead reckoning error accumulated over an underwater travel leg of a glider and navigates the glider for its next leg of underwater travel. Since ocean currents contain tidal components, there is a phase difference between real flow experienced by the glider underwater and glider-estimated flow computed prior to diving, illustrated in Fig. 6. This out-of-phase flow estimate may be acceptable if the influence of flow on the glider navigation is trivial or if a surfacing interval is small, so that the glider can update its estimated flow frequently. However, this phase difference significantly reduces navigation accuracy under a flow field with strong temporal variations. Thus, instead of using glider-derived flow estimates, incorporating predictions from ocean models into navigation may help mitigate the phase problem.
We employ the predictive tidal ocean model ADCIRC for glider navigation in the Long Bay experiment. ADCIRC solves shallow water equations by using a finite element method with a generalized wave continuity formulation in 2D and 3D. Blanton et al. (2004) introduced an implemented version of ADCIRC for the South Atlantic Bight and rigorously validated the model at multiple locations near the coast and on the shelf in the South Atlantic Bight. This implementation is a tidal database based on simulation runs using ADCIRC and provides depth-averaged flow predictions. The database stores tidal harmonic constituents of sea surface height and depth-averaged flow velocity of the barotropic tide as the amplitude and phase angle of equilibrium tidal constituents in the u and υ directions at each grid point, respectively.
Since ADCIRC is a tidal ocean model, we compute the nontidal component of flow that can be used for glider navigation along with ADCIRC. Let us denote tidal velocity generated by ADCIRC at position and time t by . Given the time interval of th subsurface phase of a glider and the surfacing position at the kth surfacing event of the glider, we define a function that computes an average tidal flow velocity along the glider trajectory by
in which and is the time step size. For simplicity, we let . Since the underwater path of a glider is unknown, average tidal flow velocity along the glider trajectory over the th subsurface phase is approximated by average tidal flow velocity at the kth surfacing position. By subtracting an average tidal flow estimate from a glider-derived flow estimate in Eq. (1), we obtain an average nontidal flow estimate at the kth surfacing event, given by .
To mitigate the phase difference problem of glider-estimated flow, we derive a hybrid model of depth-averaged ocean currents by combining an average nontidal flow estimate computed from glider-estimated flow with ADCIRC tidal flow. The method assumes that 1) the tidal component of the depth-averaged flow estimated by the glider is well approximated by the ADCIRC tidal model; and 2) the nontidal flow does not vary significantly over one surfacing interval of gliders. Given any position and time t, we compute ocean currents based on the kth surfacing data of a glider such that
in which is the nontidal flow estimate at the kth surfacing event, and is the ADCIRC tidal flow estimate at vehicle position and time t. We refer to this ocean model as the empirical hybrid ocean model (EHOM) in this paper. Since nontidal flow used in EHOM is based on the experimental data collected by a glider, this ocean model is empirical. To reduce uncertainties in the model, we can apply regression-based methods (e.g., Lermusiaux 2006; Pereira et al. 2013) to EHOM. In section 5, we show how much our navigation method just using the simple EHOM can improve the performance of glider navigation.
b. Flow-canceling algorithm
To generate the predicted trajectory of the glider, we derive a flow-canceling controller that mediates the effect of predicted flow and keeps a glider moving forward toward a target position. We use a glider model with constant through-water speed. Let us consider the motion of a glider in a two-dimensional plane. We view each glider as a Newtonian particle with a constant speed. Under flow , the motion of the glider at position and time t is described by
in which θ is the heading angle and is the constant horizontal through-water speed of the glider, and u is the glider heading control.
Consider a glider at position navigating to target position under ocean flow , as shown in Fig. 7. Let be the vector between and , and be the unit vector of . We denote the desired heading of the glider by , the desired velocity associated with by , the actual bearing under flow by , and the actual velocity under flow associated with by . Suppose we want the glider to stay on . The desired heading can be obtained using the parallelogram law, illustrated in Fig. 7, such that
in which is the effective along-track glider speed that determines the travel speed of the glider along desired path . In other words, we choose the desired heading to be
The existence of relies on the underlying technical restriction that the glider has constant through-water speed, that is, . Conversely, a glider can move forward along desired path under flow as long as . Since is a diagonal of a parallelogram with its sides and , by the properties of the parallelogram, the following relationship must be satisfied for reaching target along :
where σ is the angle between and , and δ is the angle between and . We call this relationship the matching condition for along-track motion. Using the properties of the parallelogram and the matching condition, the effective along-track glider speed is given by
in which indicates two possible solutions for caused by the symmetry of about . Angle δ is affected by desired heading , and the positive sign of the second term in Eq. (8) drives gliders to the target faster.
Because of the limited constant through-water speed of the glider, the matching condition for along-track motion is not always satisfied. For example, if the flow is too strong for a glider to move forward along the desired path [i.e., in Eq. (8)], then the condition is not met and is not well defined (i.e., ). In this case, we choose the desired heading of a glider such that the glider can stay as close to the current position as possible under the influence of flow. Suppose a glider at position with desired velocity under flow . After , the glider position is given by . To find the desired heading that minimizes the distance between and , we solve
which gives a minimizer
which represents the opposite direction of flow.
in which u is the control heading input for a glider, θ is he current heading of the glider, and is the control cycle time. If , then this controller attempts to make within ; otherwise, ( in the Long Bay experiments).
c. Path planning and waypoint generation
We introduce our path planning and waypoint generation algorithm in algorithm 2 (Table 2). In the algorithm, to predict a reference trajectory, we used the particle model with flow that is introduced in Paley (2007). Suppose a glider is predicted to dive at time and position . The algorithm predicts a glider trajectory over the length of the prediction horizon W. At the initial step of path planning, we predict ocean currents at predicted position and time using EHOM in Eq. (3). The predicted flow is used to compute the initial desired heading control input u using our controller in Eq. (10). Then, by integrating the glider motion in Eqs. (4) and (5) with time step size , we update the predicted position under . Then, we increase the path planning step by and the prediction time by . By repeating this process until reaches W, we generate the predicted glider trajectory.
Based on a predicted trajectory, we compute waypoints for a glider such that by following a series of waypoints, the glider can track the predicted trajectory. A similar idea is introduced in Smith et al. (2010), where waypoints for a predicted trajectory under flow are introduced as “alternate” waypoints. We form a series of waypoints by linear interpolation such that the estimated travel time between two neighboring waypoints meets the waypoint interval . While the default glider navigation method uses a constant flow estimate in navigation, our path planning algorithm incorporates predictions of a flow field that vary in time and space.
In this section, we provide the experimental and simulation results of our glider navigation method compared to the default navigation method. First, we describe the experimental and simulation setup. We also validate our EHOM algorithm by comparing the predicted flow generated by EHOM with the glider-estimated flow collected during the Long Bay deployments. Based on the glider data from the deployments, we generate reconstructed flow to simulate gliders driven by the default glider navigation method. Then, we evaluate the qualitative and quantitative performance of the proposed and default navigation methods.
a. Experimental and simulation setup
In the Long Bay deployments, gliders typically surfaced every 4–5 h, communicating with a dockserver via Iridium satellite communication at each surfacing event. The surface time consists of data transmission, surface dialogue display, command wait and delay, waypoint transfer and reload, and GPS updates (in the order of larger to smaller time cost). The data transmission time depends on the user-defined setup of the glider mission (e.g., the list of variables that are sent to the shore and the frequency at which they are subsampled for the mission). Our setup in the Long Bay deployments was minimal, so that the expected data transmission time was 10–12 min out of approximately 25 min of the total surface time.
While gliders were at the surface of water communicating with a dockserver, waypoints were generated using algorithm 2 (Table 2). The length of the prediction horizon was W = 12 h to provide redundant waypoints in case that the waypoint transfer fail because of communication abortion. The waypoint interval was chosen to be to account for the preferred number of yo (the vertical down/up movement of the glider) profiles and the changes of tides between waypoints. A set of waypoints was updated for every surfacing event. For predicted flow , we estimate the ADCIRC tidal component using five major constituents (M2, N2, S2, K1, O1) with the largest influence on the South Atlantic Bight shelf (Blanton et al. 2004). Figure 8 provides a true comparison between glider-derived flow estimates collected at each surfacing event during a field deployment and flow predictions generated by EHOM from the current surfacing event to the next. The figure shows that the predicted flow matches the estimated flow by the glider and that ADCIRC is a good model for estimating tidal flow. The root-mean-square (RMS) error between the predicted flow and the glider-estimated flow in x and y components are and , respectively.
To compare our navigation method with the default glider navigation method, we simulated gliders driven by the default glider navigation method. The navigation parameters for a simulated glider corresponding to each of the real gliders are determined from the experimental data collected during the deployments. All the simulated gliders dive and climb at the desired pitch angle of 26° between 3 m below the surface of water and 3 m above the bottom of water, where the overall depth ranges from 30 (LB1) to 175 m (LB3). The observed vertical speed of Ramses and Modena is 0.17 m s−1 with the horizontal through-water speed of 0.31–0.32 m s−1. The glider Pelagia was found to perform with a much slower through-water speed due to instrument configuration. Based on the experimental data, we set its vertical speed to 0.133 m s−1 with the horizontal through-water speed of 0.24–0.25 m s−1.
To simulate gliders in an ocean environment similar to that where the real gliders navigated in the Long Bay deployments, we reconstruct flow from glider-derived flow estimates collected during the deployments. The basic idea of reconstructing flow is similar to that of the EHOM algorithm. Given the surfacing data of a glider for k surfacing events, we compute estimates of average tidal flow at each surfacing event using Eq. (2). By subtracting the average tidal flow estimates from the glider-derived flow estimates , we compute nontidal flow estimates . Then, as an additional step for the reconstructed flow, by applying a fourth-order Butterworth filter to the nontidal flow estimates over the entire deployment period, we generate low-frequency flow estimates . Then, we adjust the phase of glider-derived flow estimates to reduce the phase difference between the glider-estimated flow and the real flow experienced by gliders. Since the glider-derived flow estimate collected at the ith surfacing event is average flow over the th subsurface phase, it is an estimate of flow for the time in the middle of the th subsurface phase, rather than for the time of the surfacing event. Hence, we set the time stamp of each flow estimate to be the middle of the preceding subsurface phase interval. Given a series of low-frequency flow estimates, we linearly interpolate low-frequency flow for a given time and reconstruct ocean currents by adding the low-frequency flow to ADCIRC tidal flow.
b. Navigation performance using predictive ocean models
The qualitative evaluation of the performance of navigation methods is shown in Fig. 9, which represents an ideal case with a perfect model. The figure displays the trajectories of two simulated gliders navigating a transect line (the red line) from LB2 (the bottom right) to LB1 (the top left) under a simulated flow field. The starting position is marked by a yellow star. After 14 surfacing events, both gliders managed to reach the target end point. Given waypoints generated using predicted flow via path planning, the glider driven by our method (the dark blue line with rectangles) followed the transect line very closely under flow and reached the target end point smoothly. On the contrary, the glider driven by the default navigation method (the light blue line with circles) meandered along the transect line. This result shows that our method can provide more precise control under flow.
To quantify the performance of navigation methods, we compare the navigation results of real gliders driven by our method and that of simulated gliders driven by the default navigation method at each surfacing event. In the simulation, there exist the errors and limitations of the glider model, the controller design, and the ocean model. Hence, to keep the errors from accumulating, we reset the position of simulated gliders to the same position of real gliders after one surfacing interval. Let us consider target position , a real glider driven by our method at position , and a simulated glider driven by the default navigation method at position . At each time of diving , we first set the diving position of the simulated glider to be the same as the diving position of the real glider [i.e., ]. Then, we run the simulated glider using the default navigation method for target over the same amount of time that the real glider spent underwater during the kth subsurface phase.
To evaluate the overall performance of our method compared to the default navigation method, we propose two performance metrics. Let us define as the distance between and . Then, we design the following performance metrics for real and simulated gliders:
in which the predicted surfacing positions of the real and simulated gliders are and , respectively. The metrics and are for real gliders, and and for simulated gliders. The metrics and represent the change of the distance between the target and the gliders, accumulated over one subsurface phase. The second metrics and give the error between the actual surfacing position and the predicted surfacing position. For the real glider driven by our method, we compute the predicted surfacing position from the predicted trajectory of the glider under predicted flow. For the simulated glider driven by the default glider navigation method, we compute the predicted surfacing position from the dead reckoning trajectory.
To study the navigation progress of real and simulated gliders over one subsurface phase, we examine and . Since and are the changes of the distance between the target and the gliders over one subsurface phase, we can interpret these as the navigation progress that gliders achieved toward the target over one subsurface phase. If or , then the gliders moved forward toward the target or backward away from the target, respectively; if , then the gliders stayed at the same distance from the target. In addition, represents that the real glider achieved larger progress toward the target. To display and , we plot bar graphs such that the bars for our method and the default navigation method slightly overlap to visually compare the progresses of both methods after each subsurface phase. We also examine the navigation predictability and reliability of real and simulated gliders by computing the ratio . Since the metrics and are the surfacing errors, a smaller ratio indicates that the real glider has higher predictability and reliability in terms of the surfacing error. We display via bar graphs. We show only partial segments of the results in the paper, but the rest mostly remains in the same pattern.
Figure 10 shows the performance of navigation for transect gliders Pelagia and Modena. Pelagia traversed a transect line between LB1 and LB2 () and Modena traversed a transect line between LB+20 and LB+40 (≈20 km). The top plots display of the gliders at each surfacing event. The average values of (see mean in the frame of the plots) of the real gliders driven by our method are larger than those of the simulated gliders driven by the default navigation method, which indicates that our method makes better navigation progress toward the target on average. Our method is based on a flow-canceling controller that mediates the effect of flow to move toward the target, but sometimes going with the flow would yield better progress of a glider toward the target for certain glider missions. Gliders driven by the default method always head toward the target and sometimes swim with the flow. This is why the default glider navigation method occasionally achieved larger progress than our method. This result can be improved by choosing a more sophisticated glider control algorithm that better utilizes flow. In the bottom plots, we can observe that the ratio mostly stays below one for both Pelagia and Modena, which means that our method navigates gliders under flow with higher predictability and reliability in terms of the surfacing error. Compared to Modena, Pelagia has a bit higher ratio. During the LB2012 deployment, despite the low through-water speed of Pelagia (–), the path planning system simulated Pelagia with – of the through-water speed. We assume that this overestimation of the speed was projected in the waypoints that Pelagia followed and affected the surfacing error.
In Fig. 11, Ramses kept station near LB3 as a virtual mooring at the edge of the Gulf Stream. The top plots of the figure display the performance of glider navigation based on the first metric, and , and the bottom plots the performance based on the second metric, . For the virtual mooring glider, it is difficult to judge which navigation method helps the glider navigate closer to the target based on in terms of the mean progress and which has the better predictability and reliability of navigation under flow based on . To understand this behavior, we examine the glider-estimated flow during the deployments, shown in Fig. 12. The figure shows the elementwise Euclidean norm of the flow estimated by the gliders with the mean and variance of the x and y components of the flow. While navigating toward the target, transect gliders Pelagia and Modena encountered both spatial and temporal variations () of the flow field, which we took advantage of to improve the navigation performance by employing predicted flow. However, since the region near LB3 is dominated by the Gulf Stream, which is slowly varying with respect to the surfacing interval, the temporal variability of the flow field near LB3 is very small. Moreover, since Ramses navigated to stay mostly within few kilometers from the target over the surfacing interval of 3–4 h, Ramses did not experience much spatial variability of the flow field. The variance in the x direction in Fig. 12c is somewhat big, but it is caused by just one time jump event of the flow velocity at the eighth surfacing event. As a result, under small temporal and spatial variations (mostly ), the performance difference between the proposed and default methods was not very significant for the station keeping mission near LB3.
6. Conclusions and future work
This paper presents a glider navigation strategy that improves the navigation performance of the glider by incorporating predictive ocean models. Our method uses the EHOM algorithm that models ocean currents around gliders and guides them by path planning. Using our glider navigation support system, GENIoS, the proposed method was successfully applied into the LB2012 and LB2013 experiments, in which we deployed gliders in the ocean where strong tides and the Gulf Stream current make highly variable ocean circulation. This challenging ocean condition emphasized the importance of a better glider navigation method. GENIoS also helped simulate gliders driven by the default glider navigation method under reconstructed flow. To examine the performance of real gliders driven by our method and that of simulated gliders driven by the default glider navigation method, we evaluated two proposed performance metrics. The study results demonstrated the effectiveness and benefits of the proposed method over the default glider navigation method in the presence of temporal and spatial variations. While the paper presents the proposed approach applied to underwater gliders, it can be extended to any autonomous underwater vehicles to improve their performance of navigation.
In this paper, EHOM generated predicted flow for the experiments and the simulation. However, since the algorithm relies on glider-derived flow estimates to generate the nontidal component of flow, the validity of the predicted flow is guaranteed only in the local area around gliders (i.e., the spatial variability is limited). To provide broad spatial variability, we will employ HF-radar observations, whose historic dataset can be employed to predict ocean surface currents (e.g., Frolov et al. 2012). Then, we will reevaluate the ocean current prediction algorithm and the glider navigation performance. Along with the ocean current modeling algorithm, we will improve our glider control algorithm for our future deployments.
The authors thank two anonymous reviewers for their comments. The research work is supported by ONR Grants N00014-09-1-1074 and N00014-10-10712 (YIP), and NSF Grants ECCS-0841195 (CAREER), CNS-0931576, OCE-1032285, and IIS-1319874.