Long-Baseline Ranging System for Acoustic Underwater Localization of the Seaglider Underwater Glider Laszlo Techy, Kristi A. Morgansen and Craig A. Woolsey [email protected]Department of Aeronautics & Astronautics, University of Washington Seattle WA, 98195-2400 UWAA Technical Report Number UWAATR-2010-0001 September 2010 Department of Aeronautics and Astronautics University of Washington Box 352400 Seattle, Washington 98195-2400 PHN: (206) 543-1950 FAX: (206) 543-0217 URL: http://www.aa.washington.edu
Transcript
Long-Baseline Ranging System for Acoustic UnderwaterLocalization of the Seaglider Underwater Glider
Department of Aeronautics and AstronauticsUniversity of WashingtonBox 352400Seattle, Washington 98195-2400PHN: (206) 543-1950FAX: (206) 543-0217URL: http://www.aa.washington.edu
List of Figures1 Seaglider underwater glider used in the localization experiments. . . . . . . . . . . . . . . . . . . . . 42 Round-trip travel times to the Seaglider on July 15, 2010. The figure on the left shows the raw
measurements, the figure on the right shows the data after the outliers have been removed using adifference filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Temperature and salinity variations as a function of depth in Port Susan. The data can be used tocalculate the speed of sound profile. Data collected in April, 2010. . . . . . . . . . . . . . . . . . . . 8
4 (Left) Contour plots of the positioning accuracy measured by the determinant of the covariance matrix[4]. (Right) The geometry of ranging units that was used in the ranging field experiments. . . . . . . . 9
5 (Left) The components in each node share information via serial communication. (Right) A rangingcycle is divided into three 4 second time-windows. Each node is allowed to ping only in its allocatedtime-window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6 Results of the ranging solution discussed in 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Planar position of the glider in simulations. The plots show the improved performance of the hybrid
extended Kalman filter with RTS smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Trace of the covariance matrix (left) and magnitude of the planar position error (right) in simulations
for the EKF and the EKF with RTS smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 The true and estimated values of the current speed components and glider speed in simulations. The
dashed lines are the true (constant) values, the solid lines are the EKF estimates, and the dashed-dottedlines are the EKF with RTS smoothing estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10 Ranging experiments performed in Port Susan on July 15, 2010. Relative frequency of valid responsesversus no responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
11 Ranging experiments performed in Port Susan on July 15, 2010. Figure 11b illustrates the approx-imately four hour window and corresponding three consecutive dives, where all three nodes weresimultaneously active. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
12 Underwater localization experimental results from July 15, 2010. . . . . . . . . . . . . . . . . . . . . 2213 Comparison of the current estimates as identified by the estimation algorithm to the output of compu-
tational ocean models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2314 Round-trip travel time (tRTT ) versus measured distance. The slope is 2/c; the intersection of the
regression curve with the ordinate axis gives the turn-around-time (TAT ); and the sample variance ofthe clusters yields an estimate for the range measurement variance. . . . . . . . . . . . . . . . . . . . 28
15 Range errors during a test run, where both the transponder and the glider were placed in stationary lo-cations. The round-trip times at increasing distances between the two transponders are plotted againstranging index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
16 Relative likelihood of the range measurement error. Two histograms were generated. The first onecontains all data other than some obvious outliers that have been removed manually. The secondhistogram contains only data that were within a specified range cantered around the sample mean.Those outliers were removed to consider only the random component of the residual timing error, andnot to include systematic errors resulting from multipath for example. . . . . . . . . . . . . . . . . . 29
17 Relative likelihood of the heading measurement error. Combined data set: 4408 data points. . . . . . 29
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Long-Baseline Ranging System for Acoustic UnderwaterLocalization of the Seaglider Underwater Glider
Department of Aeronautics & Astronautics, University of WashingtonSeattle WA, 98195-2400
September 2010
Abstract
This document describes a long-baseline underwater acoustic localization system that was developed to providethree-dimensional position information for the Seaglider underwater vehicle. The accurate inertial position of theglider can be used to estimate performance characteristics and to validate novel motion control and path planningstrategies in future experiments. The system consists of three acoustic transponders that are placed at known locationsat the surface of the water. Using the measured round-trip travel time of acoustic signals, slant ranges can be calculatedbetween the surface units and the glider. The range measurements from the three transponders can be used to locatethe position of the vehicle using geometrical methods or dynamic estimation algorithms. To provide smooth andaccurate position estimates, the position information can be filtered using a hybrid extended Kalman filter. If data areavailable for post-processing, the estimation accuracy can be further improved by employing batch state estimationalgorithms, such as the RTS smoothing method. Implementing such dynamic state estimators also allows to estimatethe velocity components of prevailing ocean currents and the glider’s flow relative speed.
1 IntroductionBuoyancy-driven underwater gliders are highly efficient marine vehicles that were originally developed for perform-ing oceanographic data collection missions [6, 13, 25]. Underwater gliders locomote by changing their net weight(the balance between the buoyancy force and gravity), and the location of center of gravity relative to the center ofbuoyancy [10, 11, 17]. Their flight characteristics depend on the amount of negative or positive net weight, and thelocation of movable internal masses. These vehicles are extremely efficient because they spend most of their time insteady trim flight condition, where they don’t actively expend energy. Apart from the occasional sensor measurementsand automatic scheduled self-tests, they only use energy when they change their trim flight condition — such as tran-sitioning from steady descent to ascent, or rolling to one side in order to turn. Since much of the energy is expendedduring these trim adjustments, significant energy savings are anticipated by the use of efficient maneuvering strategiesthat help achieve these desired flight conditions faster and with less control effort [17].
Theoretical study of underwater glider dynamics has revealed the potential value of novel motion control algo-rithms in improving efficiency and performance [10, 19, 16]. The motion planning algorithms fall into two categories:1) steering motion strategies that study turning motions and lateral directional dynamics [22, 20, 19]; 2) diving motioncontrol strategies, where the longitudinal trim conditions are adjusted to achieve the desired motion control objective[14, 15, 16].
Lateral Motion Control: The motion control algorithm proposed in [22] uses a feedforward control term for thelateral internal mass location that helps achieve the desired turning rate faster than a control law that only utilizesfeedback compensation for the same goal. In order to use feedforward control, the functional relationship between thelateral mass location and the resulting turn rate has to be known. Knowledge of an an accurate hydrodynamic modelallows closed-form computation for the lateral moving mass location as a function of the desired turning rate and the
3
(a) Seaglider. (b) LBL Ranging.
Figure 1: Seaglider underwater glider used in the localization experiments.
vehicle parameters. The result is obtained using first-order approximations that are valid for small turning rates, suchas the ones typical for underwater gliders [19]. If a hydrodynamic model is not available, then the desired feedforwardcontrol component can be obtained from an experimentally populated lookup table. During such experiments thecontrol parameters are held constant and recorded along with the resulting flight parameters.
Faster convergence in turning rate not only improves control responsiveness, but also reduces the amount of track-ing error in curvilinear path following applications. Precise tracking of predefined paths could open new applicationdomains for underwater gliders, such as access to shallow coastal regions, harbors, or exploration in underwatercrevices. The accuracy and responsiveness of a low-level turn rate controller allows higher-level path planning algo-rithms to treat the turn rate as the control signal [28, 29].
Longitudinal Motion Control: Informed longitudinal control scheduling allows the glider to determine the besttrim flight condition in the presence of ocean currents. Ocean currents are often of comparable magnitude (or evenmuch larger) than the flow-relative speed of most presently used underwater gliders. Such large magnitude oceancurrents may significantly alter the vehicle’s path, or even inhibit forward progress in the desired direction. Theoptimal control study presented in [14] suggests the use of current-dependent glide slope angle to maximize hori-zontal range in the presence of depth-varying ocean currents. A related study focused on the scheduling of hori-zontal pitch mass location and vertical buoyancy setting to minimize the angle of attack during transition betweensteady descent to ascent: the apogee maneuver [15]. During the apogee maneuver, the glider pumps oil into itsexternal bladder to increase its buoyancy. During this phase it unavoidably loses momentum, and may stall andlose directional stability. In that case the vehicle heading after the maneuver may be significantly different than thedesired heading angle, requiring compensation by feedback. Such compensation is costly, as it requires the shift-ing of the lateral internal moving mass. To avoid stall, an optimal control schedule has been proposed in [15].
The above motion control strategies remain to be validated in experiments. The focus of this research was todevise a set of field experiments and develop the corresponding field equipment and data processing tools to be ableto validate underwater glider motion control algorithms. To test the steering control algorithms in experiments, firsta lookup table has to be obtained that contains the glider turn rate as a function of the lateral moving mass location.Study of underwater glider dynamics revealed that — to first order — the glider speed remains unchanged as a functionof turning rate [21]. This allows to treat the longitudinal motion and lateral motion separately, and also simplifies thefield experiments: it suffices to collect turning motion data for only one given buoyancy setting. This report givesan overview of our initial results that focused on developing a long-baseline positioning system to track the path ofthe Seaglider underwater glider with high accuracy. The precise knowledge of the vehicle position during a diveenables to compute the vehicle speed, flight path angle and turning rate during a dive segment. The positioningsystem relies on acoustic round-trip travel time measurements that can be used in a spherical localization scheme toestimate the glider’s location. Since the outcome of the positioning algorithm is to be used to compute velocities
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between consecutive position measurements, the accuracy of the algorithm is crucial, as differentiation amplifiesmeasurement noise. To improve the estimation accuracy and to smoothen the derived flight parameters, an extendedKalman filter (EKF) was implemented. Dynamic filters — such as the EKF — have the ability to incorporate thedynamic motion of the glider into the estimation algorithm, and also to use different time-steps for measurementsthat don’t have the same frequency. (For example the long-baseline range measurements may become available lessfrequently than compass readings. Also, the sample time of measurements may not be uniform due to lost responses).In general, the EKF improves the estimation accuracy over less sophisticated geometric positioning methods. Inthis work the data-processing was performed post factum on previously collected experimental data. Data for theentire measurement interval is available, which allows the use of batch filtering methods. These filters provide betterestimates for the states, since all past and future measurements are taken into account. A hybrid nonlinear RTSsmoother was implemented that provides significant improvements in the estimation accuracy.
The report is organized as follows. In Section 2 we present background information on underwater acousticlocalization, the Seaglider underwater vehicle, and give an overview of the LBL system that was developed as partof this work. Sections 3- 4 describe the analytical tools that were employed to process the positioning data that wascollected during the field experiments. In Section 5 we present simulation results for the EKF and RTS smoothingmethod that proved to be very efficient tools in obtaining smooth filtered estimates of the glider path. In Section 6 wedescribe the field experiments that were performed with the Seaglider in July 2010. Section 7 concludes the paper.
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2 Background
2.1 Underwater Acoustic RangingIn underwater applications acoustic signals are primarily used for communication rather than radio signals due to poorpropagation characteristics of radio signals in water [23]. Acoustic signals can spread over long distances dependingon the frequency of the signal and the physical properties of the medium (pressure, salinity, temperature). In general,the higher the frequency, the better the accuracy of the ranging system. The price of higher accuracy is decreased range,and increased cost and power consumption. At low frequencies (7-15 kHz) the acoustic signals can spread relativelyunattenuated for miles to tens of miles. However, at these lower frequencies there is more natural and man-madenoise in the ocean, and the signals can be more difficult to detect. Accuracy can be improved using spread-spectrumsignaling — as opposed to narrow-band pings — because spread spectrum is more robust to spurious signals.
The measured round-trip travel time of an acoustic signal is given by the equation
tRTT = tTAT +2R3D
c+ ε, (1)
where tTAT is the turn-around time (the amount of time the electronics needs to detect the signal and send a response),R3D is the slant range to the vehicle, c is the speed of sound in water. The term ε represents the error between the trueand measured round-trip travel time. Since the amount of error in the slant range measurements has profound effecton the positioning accuracy, the sources of the errors need to be well understood.
The main error sources can be grouped into two categories: 1) systematic errors and 2) random errors. To achievemaximum positioning accuracy the systematic errors need to be eliminated, and the random noise components need tominimized by appropriate filtering and smoothing techniques.Systematic errors include (for a more complete list see [23] for example)
• Slant-range errors due to 1) multipath errors; 2) variations in the sound velocity with temperature, salinity, andpressure; 3) imprecise knowledge of the turn-around time (tTAT ) and; 4) acoustic ray-bending due to refraction.
• Errors due the imprecise knowledge of the baseline beacon locations (both vertically and horizontally). Thiserror may have both systematic and random components. The systematic component may include the drift ofthe acoustic beacons relative to the recorded deployment location, for example.
The random errors include
• Residual errors in the signal processing.
• Imprecise knowledge of the beacon locations due to noise in the GPS solution (in case of moving referencebeacons, such as the ones considered in this work).
• The random component of the motion of the transponders (due to small-scale fluctuations in the current speed,surface waves, etc..).
Multipath is the phenomenon when an acoustic signal is reflected off of a sound barrier — such as the ocean bottom,the water surface, a thermocline or a vertical wall — and is received at the transponder with a time delay and phaseshift. This may result in erroneous detection, depending on the amount of delay, the pulse width, and the phase shift ofthe signal. The error due to multipath often has significant magnitude, resulting in sudden large jumps in the rangingsolution. These “outliers” may be removed manually, or by use of methods based on plausibility validation in spatialor temporal domains [30]. In case they appear singularly, a simple differentiation of the consecutive measurementssuccessfully reveals outliers, as they appear as distinctive spikes. In case they appear in clusters they may be removedusing graph partitioning methods such as the one presented in [24]. Multipath is arguably one of the most significanterror sources. Treating it as part of random errors would render the noise biased, colored, and of general probabilitydistribution. If we have knowledge of the locations of the beacons and the maximum speed of the vehicle, then theobvious outliers may be easily removed from the ranging solution. Figure 2 shows results from a simple differencefilter that could successfully detect and eliminate these erroneous readings. The data set presented in Figure 2 wascollected on July 15, 2010, using the Seaglider vehicle, and three reference beacons, called nodes (see more this inSection 2.3).
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Figure 2: Round-trip travel times to the Seaglider on July 15, 2010. The figure on the left shows the raw measurements,the figure on the right shows the data after the outliers have been removed using a difference filter.
Imprecise knowledge of speed of sound in seawater results in a scale factor error when calculating the slant rangesfrom acoustic round-trip travel times. The speed of sound in seawater depends on temperature, salinity and pressure.Over the years several equations have been developed to calculate the speed of sound as a function of the physicalproperties of the medium; see [23] or [1] for a survey. Figure 3 shows the variations in speed of sound based ondata that was collected from glider field tests in April 2010. For a comparison three different equations were used tocalculate the sound profile [23]. The figure shows only minor variations in the speed of sound in Port Susan. Apartfrom increased freshwater levels at the water surface due to the melting snow in the spring and early summer, theenvironmental conditions in Port Susan show little variation year-round. For this reason, the data set in Figure 3was deemed representative, and was used for studies regarding the value of sound velocity profile compensation. Werestricted our data processing to depths below 20 meters due to poor acoustic returns near the water surface. Below20 meters depth the water exhibited relatively minor variation in temperature and salinity. Simulations were performedto estimate the error that is introduced by using a depth-averaged speed of sound profile versus a more accurate speedprofile calculated using the Mackenzie equation [18]. It was found that the error that results from using a simple depth-averaged sound profile is an order of magnitude less than the measurement accuracy of both our GPS system and theacoustic ranging system. For this reason, depth-averaged sound profile was used for the data processing methodspresented in Sections 3-4. The acoustic ray-bending phenomenon — an important error source in deep water acousticexperiments — was also found to be negligible for our application.
The error term in equation (1) can be written as the sum of two components: ε = εs + v, where εs represents thesystematic errors and v is the random noise component. After elimination of the major systematic error components,one is left with random measurement noise, which is primarily attributed to the residual timing errors in the signalprocessing. These remaining residual errors will be assumed zero-mean with known variance and white for the purposeof the positioning and filtering algorithms discussed in Sections 3-4. The probability distribution of these residualerrors can be estimated experimentally by building a sample relative likelihood histogram (see Appendix A).
2.2 SeagliderThe Seaglider (shown during a surface maneuver in Figure 1a) is a long-range long-endurance underwater gliderthat was developed at University of Washington as a collaboration between the Applied Physics Lab and School ofOceanography [6]. The vehicle can perform oceanographic data collection missions for months at a time and performhundreds of dives without needing maintenance. The glider owes both its robustness and efficiency to its cleveractuation and propulsion mechanism: the glider harvests gravitational energy for propulsion, and uses internal massshifters to change its trim flight behavior.
The glider’s mass is 51 kg: the same as the mass of the displaced water when the glider is perfectly balanced.The glider has the ability to change its volume (and hence the displaced mass) relative to this trim condition in both
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(a) Temperature and salinity variations in Port Susan. (b) Speed of sound in Port Susan calculated using equations given in [23].
Figure 3: Temperature and salinity variations as a function of depth in Port Susan. The data can be used to calculatethe speed of sound profile. Data collected in April, 2010.
directions. The full scale is approximately 850 cc of seawater: or roughly the same amount of grams in net mass:
m = mv −mw, (2)
where mv is the vehicle mass and mw is the mass of the displaced water. The vehicle outer skin has a streamlined,low-drag profile to maximize efficiency. The maximum diameter of the hull is approximately 30 cm. The wings areattached at the maximum diameter giving a total wing-span of 1 m. The vehicle length is 1.8 m without the antennamast.
The vehicle carries various navigation instruments including 1) a GPS unit to obtain a position fix at the timesit surfaces; 2) a 3D compass to measure heading and tilt angle; and 3) an acoustic pinger/altimeter to obtain depthand to be able to respond to acoustic interrogations. The low-frequency acoustic pinger on the vehicle was used toobtain range measurements between the glider and the surface acoustic transponders. The pinger has programmableinterrogation and reply frequencies in the 7-15 kHz frequency band. Whenever the glider receives an interrogation, itsends a reply, allowing the surface units to measure round-trip travel times.
The speed of the vehicle is approximately Va = 0.3 m/s, which can be influenced by a set of programmable pa-rameters. The pilot has control over the duration of the dive and the desired depth. The two parameters together specifythe vertical descent rate. The vertical speed, the vehicle pitch angle, and the hydrodynamic parameters determine theflight path angle and the horizontal speed of the vehicle.
2.3 LBL System OverviewThe precise underwater location of the Seaglider may be found using a long-baseline (LBL) acoustic localizationsystem. LBL systems are composed of several beacons in a network. The acoustic travel times between these beaconsand the glider can be used to calculate the distance to each of the beacons. If the locations of the beacons are known,then the spatial position of the glider can be calculated using a suitable geometric algorithm (static position estimation:Section 3) or a Kalman filter (dynamic position estimation: Section 4). In general, the localization can be based ontime-difference-of-arrival (TDOA) measurements, or time-of-arrival (TOA) measurements.
TDOA: Multilateration is a localization scheme where the vehicle emits a signal that is received by the nodes inthe beacon network. The time difference in these received measurements are then used to calculate the position. Thesolution lies at the intersection of hyperboloids: hence the name hyperbolic localization. Note that the exact timewhen the signal was emitted is not required, but at least four nodes are required to calculate the 3D position.
TOA: In trilateration, the position is calculated based on TOA measurements. In this localization scheme exactranges are obtained between the nodes and the vehicle, and the solution lies at the intersection of spheres: hence the
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(a) (b)
Figure 4: (Left) Contour plots of the positioning accuracy measured by the determinant of the covariance matrix [4].(Right) The geometry of ranging units that was used in the ranging field experiments.
name spherical localization. The ranges can be obtained by embedding the time information when the signal leftthe unit into the signal itself (the basis of operation for GPS), or by measuring the round-trip travel time betweenthe beacons and the vehicle. The LBL system described in this section is based on round-trip travel times betweenthe nodes and the vehicle, and the measurements can be used in spherical localization algorithms, such as the onespresented in Section 3. The round-trip travel times can be calculated to the glider, because the glider sends an acousticresponse whenever it receives and interrogation. The turn-around time on the glider is 20 ms. Since the glider canonly accept interrogations at a single predetermined frequency, the three measurements that are needed to calculatethe position are not exactly simultaneous. Since the glider moves relatively slowly, this error is comparable to GPSmeasurement accuracy. This error only affects the static positioning algorithms of Section 3. The dynamic filteringalgorithms of Section 4 can process the individual range measurements as they become available.
The geometrical placement of the LBL network has significant impact on the positioning accuracy. Chapter 3 in [4]provides a thorough discussion of the precision of underwater navigation accuracy as a function of — among others —the type of sensors that are used, and the topology of the acoustic beacon network relative to the glider. Figure 4a —reproduced after [4] — shows contour plots of the ranging accuracy in a triangular beacon network with three rangingnodes. The optimum placement of the vehicle is at the center of an equilateral triangle, so that the lines connecting thevehicle and the nodes are maximally orthogonal. Figure 4b shows the geometry that was used in the field experimentsdiscussed in this report.
The LBL system that was developed as part of this work consists of three identical nodes that were placed at knownlocations at the surface of the water. The nodes were placed on flotation devices to keep them above the water surface.To prevent them from drifting away from their deployment locations, they were anchored to the sea-floor. This measurewas taken to prevent loss of equipment, and also to keep the geometry of the beacon network unaltered during the field-experiment. The optimal locations took several days of surveying, as the bottom topography could obscure straightline-of-sight between the glider and the beacons. Each of the nodes consists of an acoustic ranging unit, GPS antenna,a computer to synchronize the ranging measurements and to log data, and batteries to power all the components. Theacoustic ranging unit consists of an AT-440 LF transponder and UDB-9000 deck box, made by Benthos. The deck boxfeatures a serial communication port over which commands can be received from a computer. The computer issuesrequests for an acoustic ping over the serial port, and the response received from the Seaglider is reported back to thecomputer upon receipt. The GPS antenna is a Garmin 17xHVS unit that reports position information in NMEA 0183compatible data format. GPS updates are reported every second over the serial port. The updates are synchronizedto GPS time: a convenient feature that is exploited to orchestrate the sequence of interrogations from different nodes.The batteries are 12V 5000 mAh NiMh battery packs. Three of these packs are used in parallel to power the laptop andan additional unit provides power for the GPS unit separately. The battery life of the system is close to 8 hours. All theelectronics are housed inside Pelican case enclosures that were fitted with waterproof connectors for communication
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(a) Node components. (b) Ranging cycle.
Figure 5: (Left) The components in each node share information via serial communication. (Right) A ranging cycle isdivided into three 4 second time-windows. Each node is allowed to ping only in its allocated time-window.
between the computer and the UDB-9000 deck box. Originally the system also contained Freewave radio modems thatwere intended for communication between the nodes and an operator station at the beach. This architecture gave thepossibility to monitor the health of the nodes and the progress of the operation from the seashore, and also the rangingcycle could be managed from the station on the beach in a master-slave polling scheme. Due to rapid deterioration ofradio signals due to reflections off of the waves, this method was found to be infeasible. A separate study showed thatraising the Freewave antenna about 20 meters above the water surface on the beach helped to recover the lost range.
In the field experiments the synchronization was performed using GPS time, which was available from the GPSunit. Each node was allocated a time window, outside which it was not allowed to ping (see Figure 5b). The pingswere issued in a round-robin fashion, where each node pinged the glider exactly once within a ranging cycle. The timeperiod of the ranging cycle is mission dependent, and should be as frequent as possible. We chose 4 second windowsize (12 seconds period for an entire ranging cycle). The ping is issued at the beginning of the time-window, andthe algorithm waits for the response not more than 4 seconds. If a response is received, the data are recorded in thelog file, with corresponding timing and GPS information. The node then goes to sleep until the beginning the nexttime-window. If no response is received within 4 seconds following the ping request, the algorithm registers it as “noresponse received”. The length of the time window was selected to give maximal resolution, while avoiding multiplenodes pinging at the same time. In order not to register a response to the previous node’s request as a valid return,the time-window has to be long enough that the pings are mutually exclusive. Based on this idea, the time-windoweffectively determines the maximum range of the beacon network: for a 4 second window size, the glider has to bewithin approximately 3 km to each of the nodes.
As discussed earlier, it is important to eliminate false detections. False positives are mostly a result of multipath.If the approximate distance of the vehicle to the nodes is known ahead of time (by informed selection of the beacongeometry), then the acoustic transponders can be configured to apply a ping lockout time (also known as inhibit time),within which the response will not be considered valid. For example, if it is known that the glider will never be within300 m from any of the beacons, a lockout time of 400 ms can be selected. Similarly, the pinger on the vehicle alsofeatures a lockout time that allows to reduce the amount of false responses. Since the ranging window size was selectedas 4 s, the glider lockout time can be specified as 3.5 s. The detection threshold level allows to specify the receivesensitivity. Smaller values will result in more false positives. Larger values will reduce responses to spurious signals,but some some of the true responses may not be considered valid. The desired value depends on the environment andapplication, and must be determined experimentally. The values used during our field experiments were determinedby performing a careful survey prior to the mission.
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3 Static Position Estimation
3.1 Spherical Localization using Nonlinear Least-SquaresThe spherical localization approach attempts to find the best estimate of the glider’s position based on three simulta-neous slant range measurements. Using the known turn-around time and depth-averaged sound velocity profile onecan find the slant ranges R3D
i ) from equation (1). Also, knowing the difference in depth between the glider and thetransponders, δz = zg − zt, one can calculate the projections of the slant ranges onto the plane, and perform thespherical localization in the plane. The depth of the transponders is recorded during the deployment of the nodes, andthe depth of the glider can be found from the glider log files during post-processing.
Ri =√
(R3Di )2 − δz2 =
√(xN − xi)2 + (yE − yi)2,
where (xN , yE)T is the position of the glider in the horizontal plane, and (xi, yi)T are the spatial position coordinates
of the nodes, where i ∈ {1, 2, 3}.Given an initial guess (xn, yn)T for the position of the glider one may write
xN = xn + δx
yE = yn + δy,
where (δx, δy)T are the unknown corrections that need to be estimated. Using the nominal in-plane distance Rni=
||(xn, yn)T − (xi, yi)T ||2 between this nominal position and the ith node, one may write
Ri ≈ Rni +xn − xiRni
δx+yn − yiRni
δy, (3)
where the higher-order-terms in the Taylor-series expansion have been neglected, and (3) yields the first order approx-imation for the in-plane range, given the initial position estimate (xn, yn)T . Define r, θ, andH so that
r =
R1 −Rn1
R2 −Rn2
R3 −Rn3
, θ =
(δxδy
), H =
xn−x1
Rn1
yn−y1Rn1
xn−x2
Rn2
yn−y2Rn2
xn−x3
Rn3
yn−x3
Rn3
Then the localization problem can be expressed as a linear least squares (LS) problem: r = Hθ + v, where v ismeasurement noise. The LS estimate (the estimate that minimizes the weighted sum of the squared errors) is
θ = (HTV −1H)−1HTV −1r,
where V is the weight matrix. If the measurement noise is normally distributed v ∼ N (0,V ), and the measurementcovariance matrix V = diag(σ2
R1, σ2R2, σ2R3
) is selected as the weight matrix, then the least squares estimate yieldsthe minimum variance estimate of the unknown parameters (Gauss-Markov theorem). The covariance matrix of theestimate is given by
Cθ = (HTV −1H)−1. (4)
The determinant of Cθ is proportional to the area of the error ellipsoid and gives a measure the estimation accuracy[4]. The value of |Cθ| was used to generate the contour plots in Figure 4a.
The algorithm provides a minimum variance estimate if the initial guess is close enough to the true position. Sincethe glider is moving fairly slowly, the previous position solution can be used for the initial guess. If no previoussolution is available, then the last known GPS location can be used to initialize the algorithm. The algorithm is usuallyimplemented in an iterative form, where the stopping criterion may be ||θi − θi−1||/||θi−1|| < εLS , where εLS is auser-specified tolerance.
3.2 Spherical Localization using modified Bancroft algorithmDrawbacks of the nonlinear least squares localization approach are the iterative nature, and that it requires an initialguess — sufficiently close to the true solution — for the algorithm to converge. Instead, one may use a direct algebraic
UWAATR-2010-0001 11
solution to estimate the position. The following discussion is extracted from [3]. The algorithm was simplified fromthe original treatment since we don’t need to estimate the GPS clock offset.
Let x0 denote the true position of the Seaglider in some Earth-fixed inertial frame, and x its estimate that wewould like to obtain. Assume that there are n ranging units, whose inertial position is denoted by si, i = 1, . . . n. Thesquared distance from x to si is
d2i = (x− si)T (x− si) = xTx− 2xTsi + sTi si.
The above equation can be also written in the following form
sTi x =1
2
(xTx+ sTi si − d2i
). (5)
Define λ = 12x
Tx and ri = 12 (sTi si − d2i ). Then equation (5) can be written as
〈si,x〉 = λ+ ri, i = 1, . . . n. (6)
Introducing the matrix of ranging unit locations A = [sT1 , sT2 , . . . , s
Tn ]T , the system of equations for all ranging units
can be written in the compact formAx = λ1 + r,
where 1 = [1, . . . , 1]T ∈ Rn, and r = [r1, r2, . . . , rn]T . Taking the weighted pseudo-inverse of the A matrix, oneobtains
x =(ATWA
)−1
ATW (λ1 + r) ,
whereW is a weight matrix. Introduce the vectors u and v:
u =(ATWA
)−1
ATW1, v =(ATWA
)−1
ATWr.
Thenx = uλ+ v.
The above equation can be squared to obtain
xTx = λ2〈u,u〉+ 2λ〈u,v〉+ 〈v,v〉,
or equivalentlyλ2〈u,u〉+ λ2(〈u,v〉 − 1) + 〈v,v〉 = 0. (7)
Equation (7) can be solved for λ. The solution is then
x = uλ+ v.
In general there will be two solutions. One of them is the true solution, the other is the mirror image of the first solutionabout the surface of the water (assuming the ranging units are all in the plane of the water surface).
3.3 Numerical ExampleThis section describes a simple numerical example to demonstrate the discussed static position estimation algorithms.Assume that the glider started a turning descent at time t = 0 from its initial position x0 = [300, 400,−200]T withrespect to an Earth-fixed inertial frame. The inertial frame’s origin was located on the surface of the water, with thex-axis pointing North, and the z-axis pointing up. The three ranging units were located at the surface of the waterat locations s1 = [−500, 0, 0]T , s2 = [500, 0, 0]T and s3 = [0, 1000, 0]T . The turn rate was ω = 0.05 rad/s, theforward speed was V0 = 2 m/s and the vertical descent rate was h = −0.2 m/s.
Also assumed was the presence of uniformly distributed noise on the ranging measurements, i.e.
ti = di + wi,
UWAATR-2010-0001 12
Figure 6: Results of the ranging solution discussed in 3.2.
where ti is the measured distance, di is the actual distance, and wi is zero-mean uniform random variable with σ2 =c2
12 = 42
12 m2. The nonlinear least-squares algorithm and the direct algebraic ranging solution were used to estimatethe position of the glider. The results can be seen in Figure 6.
The two methods yield comparable performance in simulations. The Bancroft algorithm clearly has the advantagethat it can be used when an initial guess is not available. Due to the nonlinear transformation that is involved, however,it is difficult to comment on the optimality of the algorithm. For the iterative nonlinear least squares algorithm it isknown that in case of zero-mean measurement noise with known covariance V the estimate is asymptotically unbiasedand asymptotically efficient. For this reason — and because the nonlinear least-squares method converges in practicewithin only a few steps even with rather poor initial state estimates — the nonlinear least-squares algorithm was usedto generate the static estimates of the glider position in the simulations discussed in Section 5 and the experimentaldata plots in Section 6. The corresponding solutions are labeled as “LS”.
UWAATR-2010-0001 13
4 Dynamic FilteringThe localization methods discussed in Section 3 provide a simple and robust method for identifying the position ofan underwater vehicle (for a recent effort utilizing similar tools see [12]). They are frequently used in ocean-bottommapping applications, where position resolution of up to several meters is sufficient.
To improve the estimation accuracy over that of static position estimation methods, a dynamic estimator may beemployed, that takes into account the underlying vehicle dynamics. Acoustic range measurements are inherently noisy;a filter utilizing an informed decision process that weighs the contribution of a new measurement point versus all theprevious measurements — captured in the form of a state information — provides intuitively better estimates thanan estimator that ignores the history of how the vehicle arrived in its current configuration. This concept is generallytrue, indeed, and serves as the underlying idea behind the celebrated Kalman filter. The Kalman filter has the abilityto incorporate the dynamic motion of the glider into the estimation algorithm, improving the estimation accuracy overthat of less sophisticated geometric positioning methods. The extended Kalman filter (EKF) is the generalization ofthe Kalman filter to the case where the state and/or measurement model is nonlinear. EKFs have been used in thepast to estimate AUV positions and the magnitude of prevailing currents [2, 9]. The implementation we present inSection 4.2 is hybrid EKF adopted from [5]. In addition to improving the estimation accuracy, the EKF can alsoserve as an adaptive parameter estimator for — constant or slowly varying — unknown model parameters. In thiswork we treated the current velocity and the vehicle’s flow-relative velocity as unknown parameters, and studied theEKF’s performance in estimating their numerical values. We also note that, due to the specific hardware constraints(discussed in Section 2.3), the static estimation algorithm suffers from measurement error due to the fact that the rangemeasurements cannot be issued simultaneously, but only in a round-robin fashion. A natural approach is to correct themeasurements by taking into account the motion of the glider between the consecutive range measurements. Preciselythis idea is implemented by the dynamic filter, which has the ability to process the range measurements individually,as they become available.
4.1 Vehicle ModelWe employed a simple unicycle model in the glider localization problem, described by the commonly used navigationequations. The model was extended with trivial dynamics of the velocities to allow the filter to adaptively estimatethese unknown parameters:
xN = Va cosψ + Vx (8)yE = Va sinψ + Vy (9)
ψ = u (10)Va = 0 (11)Vx = 0 (12)Vy = 0. (13)
In the above equations Va is the flow-relative speed of the glider, Vx and Vy are the North and East components ofthe current velocity vector, ψ is the heading angle measured from North, and u is the turn rate of the vehicle, the onlyexternal input in this model. The turn rate is determined by the vehicle roll angle: information that can be obtainedfrom the glider log files during the post-processing. Notice that the velocities Vx, Vy and Va are treated as states inthe model with trivial dynamics. In reality they are constant or slowly varying parameters, the values of which are tobe estimated. The EKF discussed in the next section gives estimates of these parameter values along with the vehicleposition information.
4.2 Hybrid Extended Kalman FilterTo estimate the position of the underwater glider, one may use a hybrid EKF. The discussion below follows [5],Chapter 5. The filter will be hybrid, because the states are propagated in continuous time, while the measurementsare obtained at discrete time-steps, whenever a new slant-range measurement is available. Define the state vector asx = (xN , yE , ψ, Va, Vx, Vy)T . Equations (8)-(13) can then be written in the form
x(t) = f(x(t), u(t)) +w(t), (14)
UWAATR-2010-0001 14
where w(t) ∼ N (0,W (t)) is the process noise that is assumed to be zero-mean, Gaussian and white. At discretetime intervals t = kT measurements are available defined by the output equations
y(kT ) = h(x(kT )) + v(kT ), (15)
where T is the sampling period, k ∈ Z and v(kT ) ∼ N (0,V k) is the measurement noise, which is also assumed tobe zero-mean, Gaussian and white. The measurement vector for the underwater localization problem is
h(x(kT )) =
(R(kT )ψ(kT )
), (16)
where R(kT ) is the in-plane range measurement from a corresponding node, and ψ(kT ) is the vehicle heading angleobtained from the compass. Between the measurements the state estimate is propagated in continuous time using theassumed nonlinear dynamic model without noise
˙x(t) = f(x(t), u(t)). (17)
The state covariance matrix is propagated using the linearized state matrix and using the continuous-time Kalman filterequations:
P (t) = A(x(t))P (t) + P (t)AT (x(t)) +W (t), (18)
When a new measurement arrives at time step k, the Kalman gain matrixKk is computed and corrections are appliedto the mean and covariance estimate of the state vector, in the following way. The Kalman gain matrix is
Kk = P−kH
Tk (x−
k )(Hk(x−
k )P−kH
Tk (x−
k ) + V k
)−1
. (20)
In the above equation
Hk(x−k ) ≡ ∂h
∂x
∣∣∣∣x−
k
=
(x−Nk
−x0
Rk
y−Ek−y0Rk
0 0 0 0
0 0 1 0 0 0
)∣∣∣∣∣x−
k
, (21)
where (x0, y0)T are the corresponding node locations that provided the range information, and
Rk =√
(x−Nk− x0)2 + (y−Ek
− y0)2. (22)
Then the update is applied using the equations
x+k = x−
k +Kk
(yk − h(x−
k ))
(23)
P+k =
(I −KkHk(x−
k ))P−k . (24)
The vector x−k is the a priori estimate of the state that is based on the process model (17). Equation (17) is implemented
using numerical integration between the measurement updates. The output of this integration routine is x−k : the prior
estimate of the state before the measurement arrived. The vector x+k is the a posteriori estimate of the state vector, i.e.
the estimate after the measurement has been obtained at time step k.The EKF is an extension of the Kalman filter to handle nonlinear process and/or output models. The algorithm
relies on the linearization of the nonlinear equations defined by equations (19) and (21). As long as the nominal pointabout which the linearization is performed approximates the real state very closely, the error that is introduced thisway will be negligible. In that case, the linearized model for propagating the state and covariance will be valid, andthe Kalman update equations will yield the minimum variance estimate of the states.
UWAATR-2010-0001 15
4.3 RTS SmoothingThe EKF is a sequential state estimation algorithm that uses an underlying dynamic model to propagate the state,and discrete state updates whenever a new measurement is available. It is a sequential estimator in the sense that themeasurements are processed sequentially: one at a time. All the measurements up till the latest sample time are usedrecursively to calculate the best state estimate.
If the measurements are all available for post-processing, then it is possible to use all the measurements for theentire time interval, T , to estimate the state at any given time. Estimators based on this concept are called batchestimators, to distinguish them from sequential state estimators. Such estimators give intuitively better estimates of thestates, since the measurements for all future and past times are used in the estimation algorithm. These estimators arealso called smoothers, since they have the property of providing smoothed state estimates compared to the sequentialestimators. References [5] and [26] both provide a thorough treatment on batch state estimation algorithms; here wewill only describe the hybrid nonlinear RTS smoothing algorithm presented in [5], Chapter 6.
Let xf (t) denote the output of the extended Kalman filter, and xs(t) denote the smoothed state estimate. Thesubscript ”f“ stands for forward, since the extended Kalman filter estimate is obtained by forward integration usingmeasurements up to the current time. The smoothed estimate can be obtained by integrating the dynamics in reversetime with a correction term that depends on the error between the forward state estimate and the smoothed stateestimate. The smoother gain is defined as
K(t) ≡W (t)P−1f (t), (25)
where W (t) is the process noise covariance, and P−1f (t) is the covariance matrix of the forward filter state estimate.
Then the smoother covariance and state estimates are obtained by integration in reverse time
In the above equations τ = T − t, and dx/dt = −dx/dτ .
UWAATR-2010-0001 16
5 Simulation ResultsThe methods presented in Sections 3-4 were tested in simulations. The flight of an underwater vehicle described byequations (8)- (13) was simulated on a computer. The constant glider velocity and current velocity components in thesimulations were
Va = 0.3 m/s, Vx = 0.1 m/s, Vy = −0.2 m/s.
These values were not known a priori, instead they were estimated by the hybrid extended Kalman filter. The initialvalues for the filter were selected as
Va0 = 0.2 m/s, Vx0= 0 m/s, Vy0 = 0 m/s.
During the simulations a constant turn rate was used: u = 0.9◦/s. The turn rate of the vehicle was not known for theestimator, so umodel = 0 was selected in the estimator equations. The process and measurement noise covarianceswere selected as
The measurement noise that was selected for the heading angle measurement was the same as the one estimated duringcalibration (see Appendix A). The noise variance for the range measurement was selected an order of magnitude largerthan the value estimated during calibration (Appendix A) to simulate a worst-case scenario. Selection of the processnoise covariance matrix is less trivial. In the derivation of the Kalman filter, the process noise is assumed to bezero-mean and white, which is rarely the case in practice. Nevertheless, the Kalman filter exhibits a certain amountof robustness in erroneous selection of the process covariance matrix. For this reason the introduction of fictitiousprocess noise in the Kalman filter equations has become common practice as a tuning method for the filter. Formallyjustified in [26], large process noise results in a filter that trusts the measurements more, and assumes that the model isincorrect or heavily corrupted with noise. Small process noise results in a filter that trusts the underlying model more,yielding smoother paths in navigation problems, such as the one discussed in this report.
Equations (8)-(9) represent simple translational kinematics, and are assumed to be true and noise-free. The corre-sponding variances are hence zero. The turn rate dynamics is determined by the vehicle bank angle, and in experimentsit can be obtained from the glider log files. In these simulations, the turn rate parameter was assumed to be unknown.To allow the filter to react quickly, a relatively large weight was selected for the corresponding entry in the processcovariance matrix. Most of the uncertainty in the system comes in the estimated current velocity components. Theirvalues are initially assumed to be zero. In practice they may be spatially and temporally varying. The correspondingvalues in the covariance matrix were selected after considering the trade-off between convergence time versus the sta-bility in the estimate. Smaller values tend to yield smoother convergence at a slower rate. If the glider’s flow-relativespeed is known not to vary too much during a dive, then the corresponding weight may be selected smaller than forthe other velocity components.
The simulation time was 20 minutes. A new measurement was obtained every 4 seconds: yielding a total of 300consecutive range measurements. Three ranging node locations were selected in a triangle pattern, each approximately1 km distance away from the origin, where the simulation was started. The ranging measurements were obtained fromthe nodes one at a time in a round-robin fashion: each individual node providing an updated range every 12 seconds. Inthese simulations the compass was also sampled every 4 seconds. The simulation time-step for simulating the motionof the vehicle between measurements was 50 ms. After every third range measurement a static position estimate wasalso obtained using a simple spherical localization algorithm based on the iterative nonlinear least-squares method.
Parallel to the true vehicle model, an EKF estimator was also simulated. The measurements were obtained fromthe true model output with the addition of normally distributed measurement noise v ∼ N (0,V ). The EKF estimates
UWAATR-2010-0001 17
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]
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(b)
Figure 7: Planar position of the glider in simulations. The plots show the improved performance of the hybrid extendedKalman filter with RTS smoothing.
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(P)
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(a)
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rror:
||x
− x
Hat|| [m
]
EKF
EKF+RTS
(b)
Figure 8: Trace of the covariance matrix (left) and magnitude of the planar position error (right) in simulations for theEKF and the EKF with RTS smoothing.
UWAATR-2010-0001 18
0 5 10 15 20−0.4
−0.3
−0.2
−0.1
0
0.1
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0.3
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Time [min]
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ate
d s
pe
ed
[m
/s]
va
vx
vy
Figure 9: The true and estimated values of the current speed components and glider speed in simulations. The dashedlines are the true (constant) values, the solid lines are the EKF estimates, and the dashed-dotted lines are the EKF withRTS smoothing estimates.
were then used to calculate the RTS smoothed estimate (EKF + RTS). Figures 7 - 9 show the results of these simula-tions. Planar plots of the glider position can be seen in Figure 7. As expected, the EKF estimates provide significantimprovement over the static nonlinear least-squares based estimation algorithm. The dynamic filter produces estimatesthat take into account the dynamics of the underlying physical system. Furthermore, the EKF with RTS smoothingprovides even greater improvement in the state estimates, as it uses all the measurements over the entire sampling in-terval including past and future times. Figure 8a shows the trace of the covariance matrix, and Figure 8b the estimationerror measured by ||e|| = ||x− x||. The convergence of the estimated speed components is shown in Figure 9.
In simulations, the current velocity components and the glider’s flow-relative speed converged to the true values.Setting the actual vehicle turn rate to zero would result in much poorer estimates of the speed components. Thisis due to the fact that the filter has no way of distinguishing the flow components and flow-relative components ofthe observed inertial speed. The concept is analogous to persistency of excitation in adaptive parameter estimationproblems: the glider needs to turn for the filter to be able to distinguish the current velocity components and theglider’s flow-relative speed. As long as the variation of the current speed is negligible to the rate at which the glider’smotion is excited in different inertial directions, the extended Kalman filter will converge. Simulations suggest thatthe filter converges to the true velocity components even in the case of slowly varying parameters.
UWAATR-2010-0001 19
6 Experimental ResultsThe field experiments presented in this paper took place in Port Susan on July 15-16, 2010. The location for theexperiments was chosen because of its vicinity to University of Washington campus and because of the relativelyminimal boat traffic compared to other locations in the Puget Sound. Port Susan is approximately 110 meters deep inthe middle of the bay; tidal currents are typically strong. The water features high salinity stratification, which, however,is confined to a very shallow region close to the water surface. This is due to the large amounts of freshwater enteringthe bay, creating a freshwater lens on the top of the water column. This layer of freshwater causes deterioration ofthe acoustic signal quality, and many responses are lost. For this reason, the transponders were lowered below thisfreshwater layer: to 20 meters depth.
A Seaglider was deployed in the afternoon on July 14, 2010, and recovered on July 16, 2010. The glider wasprogrammed to perform 50 minute dives to the bottom. Figure 10 shows histograms of “valid responses” versus “noresponses received” from July 15. Comparison of Figure 10a versus Figure 10b illustrates the deterioration of acousticresponses when the glider is close to the water surface. An approximately four hour portion of the entire data setrepresents a time window, where all three nodes were simultaneously active. Figure 11 shows the spatial position plotsand vertical profiles of the three consecutive dives corresponding to this four hour time window. The spatial positionplots in Figure 11a were calculated using the LS algorithm discussed in Section 3.
1 2 30
100
200
300
400
500
600
700
800
900
1000
Node
Fre
qu
en
cy
Response
No Response
(a) Frequency of valid returns and no responses: full data set.
1 2 30
100
200
300
400
500
600
700
800
Node
Fre
qu
en
cy
Response
No Response
(b) Frequency of valid returns and no responses below 20 meters depth.
Figure 10: Ranging experiments performed in Port Susan on July 15, 2010. Relative frequency of valid responsesversus no responses.
The underwater localization methods described in Section 3 and Section 4 were used to estimate the underwaterposition of the Seaglider from experimental data collected in Port Susan on July 15, 2010. The simulation environmentof Section 5 was selected to be as representative of a typical experiment as possible. The same filter parameters wereused for the experimental data processing that were identified to give good estimation results in simulations. Theresults from dive number 18 can be seen in Figure 12. The dive started at 11:56 a.m. PDT, and lasted approximately50 minutes. The data were reduced to include only the 30 minute portion that the glider spent below 20 m depth, dueto deterioration of acoustic responses near the water surface, as mentioned in Section 2.1. Figures 12a-12b show theplanar coordinates of the glider estimated position. The blue diamonds indicate the estimates obtained with the LSalgorithm. As an interesting comparison Figure 12c also shows the estimated path of the glider using dead-reckoning.Qualitative features — such as long turns — can be identified between the LBL estimates and dead-reckoning plots,but otherwise no meaningful quantitative comparison is attempted. Without additional instruments — such as acousticDoppler current profiler — the dead-reckoning algorithm lacks the ability to identify the influence of currents, whichare typically quite significant in Port Susan, and the position error grows rapidly with time.
The trace of the covariance matrix of the state estimates, tr(P ), can be seen in Figure 12d. The major contributorsof tr(P ) are the spatial position variances. From the plot it is seen that the estimation accuracy is on the order of 30 cm
UWAATR-2010-0001 20
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yE [m]
xN [m
]
Dive 18 Start
Dive 18 End
Dive 19 Start
Dive 19 End Dive 20 Start
Dive 20 End
(a) Spatial position plots of three consecutive dives performed by theSeaglider on July 15, 2010. Positions are calculated using LS estimates.
0 1 2 3 4 5 6 7-120
-100
-80
-60
-40
-20
0
Time (Hours)
Depth
(m
)
Node 1
Node 2
Node 3
(b) Glider depth versus time.
Figure 11: Ranging experiments performed in Port Susan on July 15, 2010. Figure 11b illustrates the approximatelyfour hour window and corresponding three consecutive dives, where all three nodes were simultaneously active.
and the filter converges to this value after three minutes into the experiment. The estimation covariance increasestemporarily between 8 and 10 minutes. This corresponds to the point where the glider transitions from steady descentto ascent at the bottom of the dive. During this maneuver the glider is relatively close to the ocean bottom and theobtained responses suffer from multipath. The RTS smoothing algorithm appears to provide significant improvementin terms of the position estimation accuracy, measured by the magnitude of the estimation covariance matrix.
Figure 12e shows the estimates of the glider speed and the current velocity components. Low tide occurred in PortSusan at 2:07 p.m. on July 15, 2010: over two hours after the beginning of dive number 18. The current velocityestimates in Figure 12e indicate mostly East-Southeast current direction, which is in agreement with the expected tidalmotion at that time. Figure 12f shows the estimated flow-relative glider speed obtained from independent performancecalculations. The performance calculations use the vehicle vertical descent rate, the pitch angle, and the vehiclehydrodynamic parameters to calculate the glide path angle and vehicle speed as described in [6]. There is goodagreement between the two independent speed calculations.
As mentioned in Section 5, the chosen vehicle model (8)-(13) involves the adaptive estimation of both the currentvelocities and the vehicle’s flow-relative velocity, which clearly raises observability concerns. The interested readermay find relevant information on this topic in [8] and [7]. The simulations confirmed that, as long as the system ispersistently excited, the EKF will converge if the ocean currents are constant or at most slowly varying. Such persistentexcitation is achieved by having the vehicle travel along a curved pattern such as the one shown in Figure 7, as opposedto traveling along a straight line. Given that the currents identified by the EKF show significant variability, it is difficultto assess how accurate the current estimates identified during the field experiments truly are. The accuracy could beassessed by obtaining an independent measurement with Acoustic Doppler Current Profiler (ADCP) instrument, or bycomparing the estimates to the output of computational ocean models for the given day and time. No ADCP instrumentwas at our disposal during the field experiments, and the comparison with the output of the Puget Sound PrincetonOcean Model (PSPOM) for July 15, 2010 did not yield satisfactory result. The plots illustrating this comparison can beseen in Figure 13. A time-lapse animation of the output of PSPOM revealed that the ocean currents have a much morecomplex structure than what would be expected by considering a simple diurnal tidal flow pattern. Figure 13b showsthe presence of an Ekman-spiral, and the strong shearing near the water surface due to surface winds. If the vehicleflow-relative speed was precisely known, then the filter might be able to identify the depth-dependent current structurein this dynamic environment. However, since the identification of the flow-relative speed was part of the estimationprocess, the accuracy of the estimation result is arguable. At this point we remark that the current estimation accuracycould be improved if precise knowledge of the glider’s flow-relative velocity, Vap, is available. The flow-relative speedcould be obtained from performance calculations as outlined in [6]. Such performance calculations presume accurate
UWAATR-2010-0001 21
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(a) Currents based on estimation algorithm. (b) Output of PSPOM.
Figure 13: Comparison of the current estimates as identified by the estimation algorithm to the output of computationalocean models.
knowledge of the vehicle hydrodynamic parameters. Full-scale wind tunnel studies were performed in the Universityof Washington Kirsten Wind Tunnel to support such data collection efforts [27].
UWAATR-2010-0001 23
7 Conclusions and Future WorkIn this document we described experimental underwater acoustic localization results that were performed with theSeaglider underwater glider in Port Susan in July 2010. Static and dynamic parameter estimation methods werepresented that were used for the localization. The performance of the algorithms was compared both in simulationsand using experimental data. Dynamic position estimation algorithms provide significant improvement over staticestimation methods, that don’t consider the dynamic motion of the vehicle, and only use present measurements tocalculate the position. The RTS smoothing algorithm yields accurate state estimates even in the presence of significantnoise. It was found superior to other estimation methods in simulations, and provided smooth path estimates for theexperimental data set that was collected during underwater glider field experiments.
Obtaining smooth position estimates is of great importance to compute performance parameters of underwatergliders. The estimates of speed, glide path angle, and turning rate are important flight parameters that depend onthe glider’s control configuration. Precise knowledge of the functional relationship between the control inputs andresulting flight parameters allows to devise motion planning algorithms that can yield significant improvements inefficiency and control responsiveness. Our initial goal in this effort was to develop the capability to precisely track thepath of an underwater glider in field experiments. The accuracy is crucial for using the collected data to calculate thevehicle’s performance characteristics. Careful characterization of the sources and statistical properties of the errorsin the ranging solution, and the implementation of analytical signal processing methods yielded high positioningaccuracy. However, due to the noisy and dynamic nature of our field test location, the methods were found to beinsufficient to precisely identify the vehicle’s flight characteristics.
In our future work we will use the tools described in the paper in controlled glider experiments, to obtain lookuptable of flight parameters as a function of control actuator inputs. Such experiments will have to be performed ina controlled environment, where currents are minimal, or completely absent. Our future work will also include theimplementation of efficient motion planning strategies. The algorithms will be validated using the localization system.
AcknowledgementsThe authors thank Fritz Stahr and Charlie Eriksen for their insight regarding Seaglider operations and oceanographicdata collection. The authors thank undergraduate students Jake Quenzer, Ryan Tomokiyo, Tyler Beauchamp and SamiKunze for their help with the field experiments. This research was sponsored by the Office of Naval Research underGrants No. N00014-10-1-0022 and No. N00014-08-1-0012.
TR 9407, AEAS 9501, University of Washington, Applied Physics Lab, Seattle, WA, October 1994. Chapter I.
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Appendix A: Estimating the Measurement VariancesIn order to obtain accurate state estimates it is important to know the covariance matrix, V , of the measurements.The range variance can simply be measured during calibration by recording the round-trip travel times from a setof experiments, where the distance between the transponder and the pinger is exactly known. The equation for theround-trip travel time is given by (1):
tRTT = TAT +2R3D
c+ ε,
where ε is the random residual timing error whose variance is to be estimated.Figure 14 shows the results from a calibration run, where the glider was placed at known locations with increasing
distances from the dock, and round-trip travel times were recorded. It is simple matter to fit a linear regression curve tothe data to obtain the turn-around-time and the speed of sound in water. For each of the data clusters a sample variancecan be obtained. The variances can then be combined using the relation
σ2RTT =
∑Ni=1 niσ
2i∑N
i=1 ni,
where N is the number of clusters, ni is the number of samples in the ith cluster, and σ2i is the variance of the
corresponding cluster. The result from this calibration run is σ2RTT = 1.868 ms2.
It is well known that if a random variable is transformed by a linear transformation y = ax+ b, then the varianceafter the transformation will be σ2
y = a2σ2x. The variance for the slant range measurements is then
σ2R3D = (c/2)
2σ2RTT = (1.480/2)
2σ2RTT = 1.023m2.
Note that here we used c = 1480 m/s as the speed of sound, which is more typical for seawater in Port Susan, ratherthan the c = 1440 m/s that was obtained for Lake Washington (see Figure 14).
The last step is to compute the in-plane variances. The projection onto the plane is Ri = R3Di cos γ, where γ is
the angle between plane of the transponder and the line connecting the transponder and the glider:
sin γ = δz/R3Di .
The variance is thenσ2R = cos2 γσ2
R3D .
Note that the variance vanishes when γ = ±π/2. This is a singular case when the glider is directly below or above thetransponder, and the measurement only contains information in the vertical direction.
Since the horizontal distance between the transponders and the glider is typically much larger than the verticalseparation, cos γ ≈ 1 and we finally obtain
σ2R = 1.023 m2.
Calibration for the compass can be obtained similarly. The compass reading can be recorded while the glider isstationary, and the sample mean and variance can be computed
ψ =1
n
N∑k=1
ψ(tk) (28)
σ2ψ =
1
n
N∑k=1
(ψ(tk)− ψ
)2. (29)
The heading variance was estimated to be σ2ψ = 7.6 · 10−5. The histogram for the calibration can be seen in
Figure 17. The measurement covariance matrix is then
V =
(σ2R 00 σ2
ψ
). (30)
UWAATR-2010-0001 27
0 20 40 60 80 100 120 140 160 1800
50
100
150
200
250
300
Distance from dock [m]
Round−
trip
tim
e [m
s]
1330 samples; Std: 1.3964 ms
2115 samples; Std: 0.9443 ms
1618 samples; Std: 1.8951 ms
Estimated TAT: 19.8875 ms; Estimated speed of sound: 1441.5 m/s
Figure 14: Round-trip travel time (tRTT ) versus measured distance. The slope is 2/c; the intersection of the regressioncurve with the ordinate axis gives the turn-around-time (TAT ); and the sample variance of the clusters yields anestimate for the range measurement variance.
200 400 600 800 1000 1200
85
90
95
100
Ranging Number
Rou
nd
−tr
ip tim
e [
ms]
(a)
500 1000 1500 2000
142
144
146
148
150
152
154
Ranging Number
Rou
nd
−tr
ip tim
e [
ms]
(b)
500 1000 1500
240
245
250
Ranging Number
Rou
nd
−tr
ip tim
e [
ms]
(c)
Figure 15: Range errors during a test run, where both the transponder and the glider were placed in stationary locations.The round-trip times at increasing distances between the two transponders are plotted against ranging index.
UWAATR-2010-0001 28
−15 −10 −5 0 5 10 150
0.2
0.4
0.6
0.8
1
Range error [ms]
Sam
ple
rela
tive lik
elih
ood
(a) Raw data. Combined data set: 5063 data points.
−3 −2 −1 0 1 2 30
0.5
1
1.5
Range error [ms]
Sam
ple
rela
tive lik
elih
ood
(b) Data after outliers removed. Combined data set: 4547 data points.
Figure 16: Relative likelihood of the range measurement error. Two histograms were generated. The first one containsall data other than some obvious outliers that have been removed manually. The second histogram contains only datathat were within a specified range cantered around the sample mean. Those outliers were removed to consider onlythe random component of the residual timing error, and not to include systematic errors resulting from multipath forexample.
−4 −2 0 2 40
0.2
0.4
0.6
0.8
1
Heading error [deg]
Sam
ple
rela
tive lik
elih
ood
Figure 17: Relative likelihood of the heading measurement error. Combined data set: 4408 data points.
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