1. Introduction
Autonomous underwater vehicles (AUVs) have inherent qualities that make them uniquely adapted to sampling fine-scale ocean phenomena (Bellingham 1997). Unencumbered by tethers and controlled by programmable mission specifications, AUVs possess a high degree of agility and can execute very precise surveys. Enhanced power sources are extending AUVs' operation range to the order of 1000 km. Acoustic, radio, and satellite communications have made it possible to monitor and reconfigure AUV missions from afar.
AUVs accept a variety of off-the-shelf payloads, as long as they have a reasonable power consumption and are small and light enough to be fitted. One such device is an acoustic Doppler velocimeter (ADV). Flow velocity is one of the most important physical quantities for ascertaining the ocean's state and its circulation. An ADV (SonTek 1997) measures three-dimensional water velocity using sound waves.
As illustrated in the left-hand panel of Fig. 1, an ADV is an active sonar that transmits acoustic signals and then receives their echoes reflected by sound scatterers in the water, like plankton or other floating particles. The scatterers are usually passively advected by water motion (Gordon 1996; Plimpton et al. 1997). From the measured frequency shift between emissions and echoes, flow velocity is deduced based on the Doppler principle.
An ADV has one transmitter and three receivers, as shown in Fig. 1. An echo at each receiver has a frequency shift proportional to the component of relative velocity along a line bisecting the transmission and reflection paths. Hence, three-dimensional velocity can be calculated. Acoustic beams of the transmitter and the three receivers intersect at a small sampling volume (<2 cm3, deemed a “point”) located away from the instrument base (16-cm distance for SonTek model ADVOcean, as listed in Table 1). Three-dimensional flow velocity obtained at this distant focal point can thus be considered undisturbed by the probe. As shown in Fig. 1, the ADV's local z axis is defined along the probe stem; the x axis is coplanar with one designated receiver arm; the y axis is accordingly defined by the right-hand rule. Table 1 gives the specifications (SonTek 1997) of the ADV device that we have installed in an Odyssey IIB AUV (Bellingham 1997).
It should be mentioned that an acoustic Doppler current profiler (ADCP) is another common flow velocity measurement device. It profiles velocity over a large depth range, in contrast to an ADV's point measurement. However, an ADV's spatial focus and low noise (less than a tenth of an ADCP's measurement noise at the smallest bin size) makes it uniquely suitable for experiments that require high resolution and high precision. Published applications include flow measurement near river beds (Lane et al. 1998; Bouckaert and Davis 1998), and the seabed (Kawanis and Yokosi 1997). In Voulgaris and Trowbridge (1998), Reynolds stress measured by an ADV was found to be within 1% of that obtained by laser Doppler velocimetry.
In the above applications, ADVs monitor current velocity only at spatially fixed positions. Integration of an ADV into an AUV enables point flow measurement of high resolution and high precision from a moving platform. The paper presents theoretical and experimental work in building such a system. System integration is introduced in section 2. The data processing algorithm is derived in section 3.
The Labrador Sea lies between northern Canada and Greenland. It is one of the few locations in the world where open ocean convection occurs (Marshall et al. 1998; Lilly et al. 1999). During the winter, the sea surface is subjected to intense heat flux to the atmosphere. The resulting buoyancy loss causes the surface water to sink to large depths, initiating ocean convection. In convection regimes, the water column overturns in numerous convective cells (also called plumes; Marshall and Schott 1999). Therefore, vertical flow velocity and its spatial periodicity are the key signatures of ocean convection.
Previous measurements by floats and moorings show that convection's vertical flow velocity is on the order of several centimeters per second (mainly depending on surface heat flux and latitude). For example, Fig. 20 of Marshall et al. (1998) shows rms vertical velocity of 2.3 cm s−1 measured by floats in the Labrador Sea in 1997.
During January–February 1998, researchers from the Massachusetts Institute of Technology (MIT), the Woods Hole Oceanographic Institution, and the University of Washington, made an expedition to the Labrador Sea to study ocean convection. This experiment provided a unique opportunity of testing and utilizing the AUV-borne ADV system. The challenge was to measure small vertical flows from a comparatively rapidly moving vehicle. The AUV-measured flow velocity detected convection's occurrence, as demonstrated in section 4. Computation of AUV hull's influence and a tow tank validation experiment are given in the appendixes.
2. Instrument integration
The mounting location and orientation of an ADV on an AUV must meet the following requirements: (i) avoid saturation of velocity measurement, (ii) avoid interference with other AUV instruments, (iii) keep the ADV probe out of harm's way during AUV launch and recovery, (iv) minimize corrections of the vehicle hull's influence so that the measurements are as direct as possible, (v) locate the sampling volume outside wakes as much of the time as possible, and (vi) exploit the lower (by a factor of 1/4) measurement noise along ADV's z axis than the other two axes.
According to the above requirements, the following options are ruled out. (i) At AUV's nose. The vehicle's up to 2 m s−1 speed could saturate the ADV's z velocity with the probe in the alongship direction. The ADV would also interfere with and be affected by the AUV's docking latch and the ultrashort-baseline hydrophone array that are mounted at the nose. The velocity measurement would need considerable corrections because of the ADV's closeness to the vehicle's stagnation point. (ii) At AUV's top or bottom. Either upward or downward flow velocity measurement would be in the wake, and the AUV's launch gear or the storage cart would jeopardize the ADV probe. (iii) At the horizontal-plane flank of the vehicle. The ADV probe would interfere with the vehicle's recovery hoop.
Our choice is to mount the ADV at the vehicle's largest vertical cross section, with its probe pointing 45° from the vehicle's horizontal central plane, as shown in the right-hand panel of Fig. 1. The ADV's three receiver tips reach the brink of the hull but do not protrude beyond it. This satisfies all the requirements listed above. Inside the vehicle, the ADV probe is mounted with a horizontal plate and a 45° slanted bracket.
The vehicle's heading and attitude sensor box is oriented in the alongship direction. For transformation from the ADV's to the AUV's reference frame (as will be formulated in section 3), the ADV's mounting angles need to be accurate. During installation, we use a laser pointer to ensure alignment accuracy.
The ADV works at 24 V, provided by the AUV's batteries. ADV data are read by the vehicle's computer through a serial port and saved in the vehicle's central data file. This guarantees time synchronization with other instrument data. The bandwidth of the AUV's data bus limits the ADV's data output rate to 2.5 Hz.
3. Algorithm of extracting earth-referenced flow velocity from AUV's raw measurements
The AUV-borne ADV measures flow velocity relative to the moving vehicle. Hence, to obtain the earth-referenced flow velocity, that is, the true flow velocity, we must subtract the vehicle's own velocity from the raw measurement. Another issue of concern is the vehicle hull's influence on the measurement. As shown in Fig. 1, the ADV probe's sampling volume is located about 13 cm from the vehicle's hull surface (an Odyssey IIB AUV has a length of 2.2 m, and a diameter of 0.6 m at its largest vertical cross section where the ADV is mounted). This distance is small enough to necessitate removal of the hull's influence on flow measurement. In this section, we present the algorithm of extracting the earth-referenced flow velocity.
At a normal straight-ahead flight attitude of the AUV, the ADV's sampling volume is far outside the boundary layer and also out of any wake, in accordance with the mounting criteria laid out in section 2.
Thus, for instance, if the vehicle is moving forward with velocity [1 0 0]T relative to the ambient current, the ADV's sampling volume will see a velocity of [−1.068 0 0]T due to the accelerated flow at the hull's maximum diameter.
To validate the above theoretical modeling of vehicle hull's influence on flow velocity measurement, we conducted a tow tank experiment in the David Taylor Model Basin, as detailed in appendix B. Experimental results agree well with theoretical predictions.
The earth-referenced flow velocity is extracted through the following steps.
- Transform the velocity measurement from the ADV coordinate system to the AUV coordinate system (both systems shown in Fig. 1):umADV→AUVuADVwhere uADV is the velocity vector originally measured in the ADV coordinate system.Matrixtransforms from the ADV coordinate system to the AUV coordinate system, where α = 45° is the mounting angle of the ADV probe's stem relative to the vehicle's horizontal central plane.
- Compensate for the AUV hull's influence and subtract the velocity induced by the vehicle's rotation, by applying Eqs. (2) and (5). Then the relative flow velocity in the AUV coordinate system is obtained:where the angular velocity (i.e., the vehicle's yaw/pitch/roll rate) vector Ω is measured by the AUV's KVH Digital Gyro Compass and Digital Gyro Inclinometer (KVH 1994).
- Recover the relative flow velocity in the earth coordinate system using the vehicle's heading, pitch, and roll measurements:wherewhere θh, θp, and
θ′r - Extract the earth-referenced flow velocity, that is, the true flow velocity. The vehicle's own horizontal velocity, that is, the first two components of Vearth, can be estimated by a long baseline acoustic navigation system (Bellingham 1997) in deep ocean or by acoustically tracking the ocean bottom (RDI 1996) in shallow water. The vehicle's own vertical velocity, that is, the third element of Vearth, can be estimated by differentiating its depth sensor measurement. Knowing Vearth, we can obtain the earth-referenced flow velocity Uearth from Eq. (8):When we are only concerned about the true vertical flow velocity, that is, the third component of Uearth, we only keep the third row of Eq. (10) (note that the z axis points downward, as shown in Fig. 2):To comply with the commonly adopted upward convention, the earth-referenced upward flow velocity is written as
4. Labrador Sea convection experiment
a. Data processing results of AUV-measured vertical flow velocity
AUV mission B9804107 took place at 0446 ∼ 0700 UTC 10 February 1998. The mission launch location was about 56°42′N, 52°46′W, where the (Autonomous Oceanographic Sampling Network (AOSN; Curtin et al. 1993) mooring was anchored. The AUV behaviors in this mission are illustrated in Fig. 3. The vehicle first spiraled down to 426-m depth, then it spiraled up to 250-m depth. At this depth plane, the vehicle made a “diamond” run, that is, closed a four-leg loop with 90° turns, each leg lasting for 720 s. After that, it spiraled up to 20-m depth, making an identical diamond run. At the end, the vehicle ascended to the sea surface. The vehicle's speed in level legs was about 1 m s−1.
On the vehicle, the ADV probe measured three-dimensional flow velocity, while the conductivity–temperature–depth (CTD) sensor suite measured other water properties. Those measurements are shown in Fig. 4, where potential temperature and potential density are deduced from in-situ measurements.
We apply the algorithm in section 3 to process the ADV measurements. Vertical flow velocity is a key signature of ocean convection (Schott et al. 1993, 1996). For detecting convection, we only extract the earth-referenced vertical flow velocity wearth as formulated in Eq. (12). The data processing result for the 250-m depth is shown in Fig. 5. The vehicle's own vertical velocity (middle panel) has been removed for producing wearth (top panel). The AUV's roll (bottom panel) shows when the vehicle made 90° turns. At an AUV speed of about 1 m s−1, 1 s of time on the abscissa can be approximately translated to 1 m of distance for interpreting convection cells' spatial scale. The rigorous time–space relationship of an AUV measurement is governed by the mingled spectrum principle (Zhang 2000).
In Fig. 6, we overlap wearth of the 250- and 20-m depths. A shift of 3550 s is added to the time index for the 250-m depth, during which the vehicle completed the diamond loop at the 250-m depth and ascended to the 20-m depth plane (illustrated in Fig. 3). We note that for about 600 s in Fig. 6, consistency between the two signals is pronounced.
As will be analyzed in section 4b, a convection layer is believed to exist in the upper 350-m water during this AUV mission. According to the MIT ocean convection model (Jones and Marshall 1993; Klinger and Marshall 1995), over a depth difference of 230 m and a time difference of 3550 s, vertical velocities at the same horizontal position should show strong similarity. The reason is that convective cells are vertically aligned, and the field varies very slowly as convection approaches a stationary state. Compared with 250-m depth, the 20-m depth shallow water is complicated by additional atmospheric forcing. Nevertheless, similarity between depths still offers a means for checking measurements. The consistency of wearth between the two depths as shown in Fig. 6, although only lasting for a short duration, adds to our confidence in the data processing algorithm.
To remove bias in wearth, we have corrected installation errors of the ADV probe and the heading and attitude sensor box. In AUV mission B9804107, the two errors were 2° rotation of the ADV probe and 1° pitch-up of the heading and attitude sensor box. The pitch-up angle of the heading and attitude sensor box was determined using the criterion that the mean value of convective wearth during an AUV level run should vanish. The remaining noise of wearth comes from three sources of measurement noise: (i) ADV, (ii) KVH heading/pitch/roll and rate sensor, and (iii) AUV's depth sensor. The source errors propagate into the final result wearth by following the four-step transformations in section 3. The noise sources and the result (all after 50-s smoothing) are summarized in Table 2. The standard deviation of noise in wearth is 0.7 cm s−1.
It is noted that in the Labrador Sea Experiment the AUV-borne ADV experienced low correlation coefficient and low signal-to-noise ratio, although still within operational ranges. This was probably due to low density of floating particles in the water. As a consequence, ADV's measurement noise was relatively high. Interference from an acoustic modem's transmission also added to the ADV's measurement noise during that experiment. Modem interference with the ADV was easy to remove, because it was periodic (every 28 s), and created clearly anomalous spikes in velocity measurement. Even in the presence of the 0.7 cm s−1 total noise in wearth, the AUV-based classifier successfully detects convection against internal waves, as will be discussed toward the end of this section.
b. Detection of convection's occurrence by wearth
During the 1998 Labrador Sea Experiment, meteorological data were recorded by an Improved Meteorological (IMET) system (Hosom et al. 1995) on board R/V Knorr. Prof. Peter Guest of the Naval Postgraduate School calculated the ocean surface heat flux based on the IMET measurements. During AUV mission B9804107, the heat flux was about 300 W m−2. Let us make comparisons with previous open ocean convection experiments. In the Greenland Sea Experiment (Schott et al. 1993) during the winter of 1988/89, the heat flux fluctuated between 100 and 500 W m−2, with an average value of about 250 W m−2. Ocean convection was observed during that experiment, using moored ADCPs. In an earlier Labrador Sea experiment (Lilly et al. 1999) during the winter of 1994/95, the average heat flux was about 300 W m−2. Using a moored ADCP and Profiling Autonomous Lagrangian Circulation Explorer (PALACE) floats, ocean convection was observed. The sea surface heat flux value in our Labrador Sea Experiment is close to that of the two previous experiments. We therefore have reason to expect ocean convection's occurring.
Besides surface heat loss, a vertically mixed water column is another key condition for ocean convection. Across the Labrador Sea Basin (about 600 km), a mixed water layer of depth 270∼500 m was observed by a series of CTD casts from the ship deck (data provided by Prof. Eric D'Asaro, personal communication 2000). During two different AUV missions, mixed water layers were also clearly recorded by CTD sensors on the vehicle. During AUV mission B9804107, the mixed layer was down to 350 m, as shown in Fig. 7. The 250-m depth plane in AUV mission B9804107 is within this mixed layer.
From the wearth panel in Fig. 5, convection cells' spatial periodicity of about 150 m can be observed (the AUV speed was about 1 m s−1). Using the MIT convection model (Jones and Marshall 1993; Klinger and Marshall 1995), we have designed an AUV-based classifier (Zhang 2000) to distinguish ocean processes by their spectral features. Based on wearth's spectrum, which corresponds to convective cells' periodicity “seen” by a cruising AUV, the classifier declares that the 250-m depth vertical velocity data is convective. This result is in agreement with meteorological and hydrographic expectations.
During the same experiment, Prof. Eric D'Asaro of the University of Washington deployed seven Lagrangian floats to study convection [float design can be found in D'Asaro et al. (1996)]. The floats' records confirm not only the existence of mixed layers, but also the occurrence of convection. Furthermore, the rms vertical flow velocity is found to be 2∼3 cm s−1 based on the float data (calculated by Prof. Eric D'Asaro). In the 250-m depth AUV data analyzed above, the counterpart is 2 cm s−1. Measurements by those two independent platforms are consistent.
5. Conclusions
We have enabled an AUV to make flow velocity measurement using an acoustic Doppler velocimeter (ADV). The usefulness of an ADV lies in its ability to acquire accurate measurement at a focal point. Incorporation of an ADV in an AUV provides a mobile measurement mode, which enhances coverage and responsiveness in ocean surveys. For classifying ocean processes, flow velocity is a key feature. Acquiring this measurement from a vehicle is crucial to realizing an AUV-based classifier (Zhang 2000).
The AUV-borne ADV measures flow velocity relative to the moving vehicle. We established an algorithm to extract the earth-referenced flow velocity from the raw measurement. AUV's own velocity and its hull's influence on flow measurement are removed. A tow tank validation experiment was carried out in the David Taylor Model Basin. In January/February 1998, an AUV acquired flow velocity in the Labrador Sea. Using a spectral classifier, convection is detected based on the vertical flow velocity. The results are in agreement with observations made by Lagrangian floats.
Acknowledgments
This research was funded by the Office of Naval Research (ONR) under Grants N00014-95-1-1316 and N00014-97-1-0470, and by the MIT Sea Grant College Program under Grant NA46RG0434. The authors are thankful to ONR Program Manager Dr. Thomas Curtin and MIT Sea Grant Director Prof. Chryssostomos Chryssostomidis for their support. The authors thank Mr. Cliff Goudey, Mr. Joseph Curcio, Mr. Neil Best, Dr. Bradley Moran, and Mr. Robert Grieve of MIT Sea Grant for their help in experiments. Thanks also go to Mr. Robert McCabe of the WHOI machine shop and Mr. Richard Norton of the David Taylor Model Basin for their assistance during the calibration experiment. The authors thank Prof. Peter Guest of the Naval Postgraduate School, and Prof. Eric D'Asaro and Ms. Elizabeth Steffen of the University of Washington for sharing their Labrador Sea meteorological and floats data. The authors thank Prof. John Marshall and Dr. Helen Hill for providing the MIT ocean convection model. The authors appreciate the comments and suggestions from the reviewers. Special thanks go to WHOI Senior Scientist Dr. Albert Williams III for his important help in the calibration experiment design and the Labrador Sea Experiment data processing and analysis. The first author would also like to express his gratitude to Dr. Albert Williams III and MIT Prof. John Leonard for their guidance from the Ph.D. thesis committee.
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APPENDIX A
Potential Flow Velocity Disturbances Caused by a Prolate Spheroid
The velocity potentials exterior to a prolate spheroid moving in each of the 6 degrees of freedomA1 have been compiled in Lamb (1932). Here, the steps necessary for computing the flow velocity components are shown and applied to our particular case.
Table A1 shows, for each degree of freedom, j, the expressions for the potential ϕ and its amplitudeA3 A adapted from Lamb (1932). Listed next are the the nonzero terms in the chain rule (A4) for the velocities. The seventh to ninth columns show the derivatives of ϕ with respect to (x, y, z) as needed in the chain rule. Because each of these coordinates only appears linearly in the potentials, the derivatives are compactly written as ϕ/x, etc. Finally, the derivative of ϕ with respect to ζ is shown. With the coordinate derivatives from Eqs. (A10)–(A12), any flow velocity component (u1j, u2j, u3j) due to unit motion in any degree of freedom can then be computed.A4
APPENDIX B
Calibration Experiment at the David Taylor Model Basin
To ascertain the AUV hull's influence on the flow velocity measurement, we carried out a calibration experiment in the David Taylor Model Basin before the Labrador Sea Experiment. The tow tank has a large cross section to minimize influence from the boundaries, and the carriage has a precise speed control. In the tank, a complete AUV hull equipped inside with an ADV was towed by the carriage at different attack angles under different speeds.
Experiment design
Matrix 𝗔, as given in Eq. (3), is computed by the potential flow theory described in appendix A. This matrix depicts the AUV hull's influence on the flow velocity measurement, which is induced by the vehicle's translational motion. At a series of AUV speeds, ADV-measured flow velocities are to be compared with theoretical predictions computed by Eq. (B1) using matrix A in Eq. (3). To test out various flow orientations relative to the AUV, we need to enable different combinations of the vehicle's yaw and pitch angles.
The experimental structure is illustrated in Fig. B1. The structure is composed of three parts: a rotating bracket, a wedge, and a hull platform. The upper rectangular plate of the rotating bracket attaches the whole load to the tow tank carriage. Its lower circular plate connects to the wedge via four bolts. This circular plate has multiple bolt holes to permit yaw angles of 0°, 5°, 10°, 15°, 30°, and 45°. The wedge is for realizing the AUV's pitch/roll angles of 5°, 10°, 15°, and 30°. The hull platform's upper circular plate connects to the wedge, and its lower rectangular plate attaches to the AUV's inner fairing. Its 45° slanted clamp (not visible in Fig. B1 from that perspective) holds the ADV probe. Two installation errors were found by two calibration runs (zero yaw and zero pitch with the AUV hull on, and zero yaw and zero pitch without the AUV hull): (i) 2.7° rotation of the ADV probe in the clamp, and (ii) 2.2° misalignment between the hull platform's centerline and that of the AUV.
It should be noted that in this calibration experiment, the ADV probe pointed 45° upward on the vehicle's port side, as shown in Fig. B1. In the Labrador Sea Experiment, the ADV probe pointed 45° downward on the vehicle's starboard side, as shown in Fig. 1. The purpose of the change is to facilitate field recovery of the AUV at the end of missions, which requires contacts at the upper half of the vehicle. This difference is trivial, since it is equivalent to rotating the vehicle for 180° about its alongship axis. It can be shown that this diagonal move of the ADV position from the port side to the starboard side causes no change to matrix 𝗔, but only sign flippings in 𝗧ADV→AUV [the coordinate transformation matrix given in Equation (6)]. In the 1998 Labrador Sea Experiment, the AUV had a V-shaped latch at its nose for docking to an underwater station. We also added a latch to the vehicle during the calibration experiment.
As shown in Fig. 2, the AUV coordinate system is defined such that its x axis points forward, y axis to starboard, and z axis downward. Accordingly, yaw is the angle between the AUV's alongship central vertical plane and that of the tow tank; pitch is the angle between the AUV's x axis and the horizontal plane. A plus sign of yaw means that the AUV steers to the starboard side, while for pitch it means the vehicle's nose is up. Different combinations of yaw and pitch angles as shown in Table B1 were tested. At each AUV yaw/pitch, the carriage ran successively at 1, 2, and 3 knots, each speed lasting for about 40 s. We distributed neutrally buoyant glass powder (Potters 1997) in the tank water to provide strong echoes for the ADV's measurement.
Comparison of theoretical and experimental results
Figure B2 shows the comparison of experimental results (with the AUV's latch on) and the theoretical predictions. The components υx, υy, and υz are in the ADV's coordinate system. The two installation errors have been compensated for in computations and also in plotting of Fig. B2. Each velocity point in Fig. B2 is the mean value of 600 data points (sampling rate = 25 Hz) under one constant carriage speed, which is further normalized by the carriage speed for display. With bias removed, the total measurement noise for each mean velocity is about 0.1 cm s−1.
One setting in the fourth panel of Fig. B2 was pitch = 5° (nose up) and yaw = −2.2° (both steered to the side where the ADV was mounted). We tested this doubly unfavorable orientation to determine whether wake induced by the latch had impact on flow measurement. In AUV cruises in the 1998 Labrador Sea Experiment, attack angle due to current flow was no more severe than the above setting. Tow tank experiment results in the fourth panel of Fig. B2 maintain agreement with theoretical predictions at all three speeds, not affected by the latch's wake at the above orientation.
(left) Side view of an ADV probe. (right) Cross-sectional view and side view of the ADV's mounting on the vehicle. In the upper right panel, the big circle represents the vehicle's outer fairing. In the lower right panel, the lower half of the vehicle's inner fairing is placed upside-down for ease of installation, while the outer fairing is removed
Citation: Journal of Atmospheric and Oceanic Technology 18, 12; 10.1175/1520-0426(2001)018<2038:ADVFMF>2.0.CO;2
Definition of velocity vectors in the AUV coordinate system (plan view)
Citation: Journal of Atmospheric and Oceanic Technology 18, 12; 10.1175/1520-0426(2001)018<2038:ADVFMF>2.0.CO;2
AUV behavior sequence in mission B9804107
Citation: Journal of Atmospheric and Oceanic Technology 18, 12; 10.1175/1520-0426(2001)018<2038:ADVFMF>2.0.CO;2
During AUV mission B9804107, flow velocity was measured by an ADV. Conductivity, temperature, and depth were measured by onboard CTD sensors
Citation: Journal of Atmospheric and Oceanic Technology 18, 12; 10.1175/1520-0426(2001)018<2038:ADVFMF>2.0.CO;2
(top) The earth-referenced vertical flow velocity wearth at the 250-m depth of AUV mission B9804107
Citation: Journal of Atmospheric and Oceanic Technology 18, 12; 10.1175/1520-0426(2001)018<2038:ADVFMF>2.0.CO;2
Comparison of wearth of 250- and 20-m depths, with a time shift of 3550 s corresponding to AUV's cruise time
Citation: Journal of Atmospheric and Oceanic Technology 18, 12; 10.1175/1520-0426(2001)018<2038:ADVFMF>2.0.CO;2
Profiles of potential temperature, salinity, and potential density during AUV mission B9804107
Citation: Journal of Atmospheric and Oceanic Technology 18, 12; 10.1175/1520-0426(2001)018<2038:ADVFMF>2.0.CO;2
Fig. B1. (top) Assembly diagram for the tow tank experiment, and (bottom) photo of the AUV being mounted on the carriage strut using a 5° wedge, with three ADV receiver tips visible
Citation: Journal of Atmospheric and Oceanic Technology 18, 12; 10.1175/1520-0426(2001)018<2038:ADVFMF>2.0.CO;2
Fig. B2. Comparison of the calibration experiment results and the theoretical predictions. Velocity components υx, υy, and υz are in ADV's coordinate system. Symbol color denotes velocity component of experimental data: blue—υx, green—υy, red—υz
Citation: Journal of Atmospheric and Oceanic Technology 18, 12; 10.1175/1520-0426(2001)018<2038:ADVFMF>2.0.CO;2
Specifications for SonTek ADVOcean
Measurement/estimation noise
Table A1. Potential sand velocities for flow around a spheroid
Table B1. Tested combinations of yaw and pitch angles (°)
Table B2. Relative errors between experimental results andtheoretical predictions
For brevity, subscript “AUV” has been omitted in the notation for the vehicle coordinate system. As in Fig. 2, x, y, z of the AUV are coincident with the body at the instant under consideration, but at rest in the earth coordinate system.
Compared with Lamb's formulation, we use the opposite sign for ϕ, restrict motions to unit velocity, and absorb a factor k or k2 into A. Our amplitudes therefore differ from Lamb's by a factor −1/k/(velocity amplitude) for the translational degrees of freedom, and −1/k2/(angular velocity amplitude) for the rotational degrees of freedom.
MATLAB routines that evaluate ϕj and uij, as well as examples of their use, can be downloaded from ftp://ftp.mathworks.com/pub/contrib/v5/physics/pfs/.
