2. HEADING DEVIATION CORRECTION

Though a straight course is simple for a vessel to follow, heading can be difficult to maintain when environmental forces act on the hull in an unpredictable manner. The vessel's crew may adjust engine power or alter the rudder to offset the external forces so that the desired straight-line course can be maintained. The heading of the vessel, however, may deviate from the course direction. The estimation of heading alignment error in the proposed algorithm is based on the slope of the transponder trajectory observed from the USBL transceiver. When a line survey is conducted with heading deviation, the observed transponder trajectory will be distorted, thus biasing the estimate of heading alignment error. The correction for the effect of heading deviation on USBL observations has to be made before the estimation of alignment errors.

To correct for heading deviation in the USBL measurements, the reference course of the vessel must be given first. For a vessel track which may be straight or, more commonly, curved, it can be fitted to a straight line by the use of the least-squared error approach. The fitted line is taken as the reference course. We consider a USBL line survey as depicted in Figure 1, with the reference course (dashed line), the course direction θ, and the heading deviation Δθ. Three coordinate systems, O _aX_aY_aZ_a, O_tX_tY_tZ_t, and O _hX_hY_hZ_h, are employed for heading deviation correction. The O _aX_aY_aZ_a coordinate system is an earth-fixed East-North-UP (ENU) frame with its origin on the sea surface over the top of the seabed transponder. The O _tX_tY_tZ_t coordinate system is a body-fixed reference frame for the USBL transceiver. The frame O _hX_hY_hZ_h is defined as a right-handed system such that its origin O _h is coincident with the origin O _t, its Z _h-axis is coincident with the Z _t-axis, and its Y _h-axis is along the direction of the course. When conducting a USBL line survey with heading deviation Δθ, the position vector of the transponder, P_t, is measured from the transceiver in the O _tX_tY_tZ_t coordinate system. However, the position vector of the transponder, P_h, with respect to the O _hX_hY_hZ_h coordinate system is desired for keeping the vessel's heading in line with the course direction throughout the line survey. The correction for heading deviation is therefore to transform the measured position vector P_t from the coordinate system O _tX_tY_tZ_t to O _hX_hY_hZ_h. According to the geometric scheme in Figure 1, the position vector P_t can be transformed to the O _hX_hY_hZ_h coordinate system by the following computation of coordinate transformation:

(1)$${P}_h = \left[ {\matrix{ {\cos \Delta \theta} & {\sin \Delta \theta} & 0 \cr { - \sin \Delta \theta} & {\cos \Delta \theta} & 0 \cr 0 & 0 & 1 \cr}} \right]{P}_t $$

Figure 1. Vessel's heading deviates from the course direction while conducting a USBL line survey.

3. CROSS-TRACK ERROR CORRECTION

Environmental forces (wind, wave drift, current loads, etc.) can cause loss of the ability of a vessel to maintain a straight path on a predetermined course. When a vessel is off-course, the distance from the vessel to the course is defined as the cross-track error. The proposed algorithm for alignment error estimation is designed to follow the assumption that the vessel is able to move along a predefined straight course while performing USBL line survey. Therefore, if the vessel is off the desired course while running USBL survey, the USBL observations need to be corrected for cross-track error prior to the estimation of alignment errors.

In a manner similar to the correction for heading deviation described in Section 2, the correction for cross-track error begins with the linear regression of a vessel track. Figure 2 presents a solid curve showing the curved vessel track. The least-squared straight line fit through the vessel track represented by a dashed line in Figure 2 is taken as the reference course. In the absence of heading deviation, the Y _t-axis towards the direction of the reference course and the X _t-axis is perpendicular to the reference course. Given that, as shown in Figure 2, O _t is the position of the transceiver, the fitted straight line is heading in a direction of θ, and B is the intersection of the reference course and the X _t-axis. In terms of these, the cross-track error is the distance between points O _t and B. The correction for cross-track error can be achieved by relocating the vessel's position from point O _t to point B. Thus, the observed transponder position is changed from P_t to P_c, as shown in Figure 2.

Figure 2. The vessel track and its linear fit are represented as a solid curve and a dashed line, respectively. The distance from point O _t to the reference course (dashed line) is the cross-track error.

In the determination of the corrected transponder position P_c, the coordinates of point B need to be known first. With respect to the global frame, say O _aX_aY_aZ_a, let the equation of the reference course on the X _aY_a plane be given by

(2)$$Y_a = X_a \cot \theta + b$$

where cot θ is the slope of the reference course and b is the Y _a-intercept. The unit vector of the reference course with slope of cot θ can be expressed as

(3)$${e} = \left[ {\matrix{ {\sin \theta} \cr {\cos \theta} \cr 0 \cr}} \right]$$

Let the coordinates of points O _t and B with respect to the global frame O _aX_aY_aZ_a be (x _o, y_o, 0) and (x _b, y_b, 0), respectively. Then, based on Equation (2), the distance vector from point B to point O _t is

(4)$${\bf r}_o^{(a)} = \left[ {\matrix{ {x_o - x_b} \cr {y_o - y_b} \cr 0 \cr}} \right] = \left[ {\matrix{ {x_o - x_b} \cr {y_o - x_b \cot \theta - b} \cr 0 \cr}} \right]$$

where the superscript a indicates that the vector r_o is with respect to the O _aX_aY_aZ_a frame. Because the distance vector r_o^(a) is perpendicular to the direction of the reference course, the coordinates of point B can therefore be obtained by solving the equation e· r_o^(a)=0 which gives

(5)$$\left\{ {\matrix{ {x_b = (x_o \sin \theta + y_o \cos \theta )\sin \theta - b\sin \theta \cos \theta} \cr {y_b = (x_o \sin \theta + y_o \cos \theta )\cos \theta + b\sin ^2 \theta} \cr}} \right.$$

Substitute Equation (5) into Equation (4), the vector r_o^(a) becomes

(6)$${\bf r}_o^{(a)} = \left[ {\matrix{ {(x_o \cos \theta - y_o \sin \theta + b\sin \theta )\cos \theta} \cr { - (x_o \cos \theta - y_o \sin \theta + b\sin \theta )\sin \theta} \cr 0 \cr}} \right]$$

Since the USBL measurement P_t is relative to the O _tX_tY_tZ_t coordinate system, the correction for cross-track error has to be carried out in the same frame. Thus, the vector r_o^(a) is transformed into the O _tX_tY_tZ_t frame by applying the technique of coordinate transformation:

(7)$${\bf r}_o^{(t)} = \left[ {\matrix{ {\cos \theta} & { - \sin \theta} & 0 \cr {\sin \theta} & {\cos \theta} & 0 \cr 0 & 0 & 1 \cr}} \right]{\bf r}_o^{(a)} = \left[ {\matrix{ {x_o \cos \theta - y_o \sin \theta + b\sin \theta} \cr 0 \cr 0 \cr}} \right]$$

where the superscript t indicates that the vector r_o is with respect to the O _tX_tY_tZ_t frame. As illustrated in Figure 2, the corrected USBL measurement P_c can be obtained as

(8)$${P}_c = {\bf r}_o^{(t)} + {P}_t = \left[ {\matrix{ {P_{tx} + x_o \cos \theta - y_o \sin \theta + b\sin \theta} \cr {P_{ty}} \cr {P_{tz}} \cr}} \right]$$

where P _tx, P_ty and P _tz are the X _t, Y_t, and Z _t components, respectively, of the position vector P_t.

4. FIELD EXPERIMENTAL RESULTS

The effectiveness and efficiency of the proposed algorithm for alignment error estimation is examined through a field experiment. The experiment was carried out with an off-the-shelf USBL positioning system in April 2010 off the coast of Kaohsiung Harbour, Taiwan. The USBL system has a positioning accuracy of 0·25° (better than 0·5% of slant range.) Figure 3 shows the USBL transceiver that was mounted at the end of a pole and fixed on the side of the R/V Ocean Researcher III, a vessel operated by the National Sun Yat-sen University. The reference USBL transponder was mounted on the Seafloor Acoustic Transponder System (SATS) (Chen and Wang, Reference Chen and Wang2011), as shown in Figure 4, and then deployed on the seabed at a water depth of about 300 m. In addition to the USBL positioning system, a Global Positioning System (GPS) and a gyrocompass with motion sensor were used to acquire data for the determinations of position and attitude of the USBL transceiver. The main characteristics of these instruments are summarized in Table 1.

Figure 3. The USBL transceiver was installed on an over-the-side deployment pole.

Figure 4. The Seafloor Acoustic Transponder System (SATS) was employed to carry and anchor reference USBL transponders to the seafloor.

Table 1. The main sensors and instruments employed for the field experiment.

Eight straight-line runs were carried out over the area above the seabed transponder to collect USBL observations and sensor data. The vessel tracks of the line survey are presented in Figure 5 and labeled L1-8. Sound-speed profiles of the water column were measured by taking conductivity-temperature-depth (CTD) casts over the period of line survey. Based on the collected observations of GPS and acoustic round-trip travel time with the combination of the given sound-speed profiles of the water column, the location of the seabed transponder was estimated by using the GPS/Acoustic positioning technique (Chen and Wang, Reference Chen and Wang2011). The estimates of the seabed transponder coordinates are given as (P _Tx, P_Ty, P_Tz)=(163780·57, 2489603·35, −297·09) m with respect to the Taiwan Datum 1997 (TWD97) 2-degree transverse Mercator (TM2), as shown in Figure 5. That is, the seabed transponder is at the depth of 297·09 m and the origin of the O _aX_aY_aZ_a frame is located at (163780·57, 2489603·35, 0) m in the TWD97/TM2 coordinate system. Before the correction for alignment error is made, the USBL positioning of the seabed transponder with respect to the O _aX_aY_aZ_a coordinate system is presented in Figure 6, in which the data has been segregated by marker for the eight vessel tracks. As seen from Figure 6, the USBL positioning has errors distributed around the seabed transponder with standard deviations of 23·4, 25·3, and 27·1 m in the X _a-, Y_a-, and Z _a-directions, respectively. In addition, the scatter plot of the transponder positions is distributed with particular groupings related with individual survey paths.

Figure 5. Vessel tracks L1-8 of the USBL line survey. The survey was centered on the reference seabed transponder that was at a depth of about 300 m.

Figure 6. Seabed transponder positioning obtained directly from the raw USBL observations without misalignment correction. The asterisk is the location of the seabed transponder.

4.1. Corrections for heading deviation and cross-track error

All of the transponder trajectories observed from the USBL transceiver are corrected for heading deviation and cross-track error before applying the proposed algorithm to estimate alignment errors. In the case of the transponder trajectory observed along the track L1, the vessel track with its linear fit is shown in Figure 7(a), the vessel's heading is in Figure 7(b), the cross-track error is in Figure 7(c), and the raw data of transponder trajectory observed along track L1 is in Figure 8(a). The slope of the linear fit to the vessel track L1 gives –37·25, corresponding to the reference course direction of θ=358·5°. The heading deviation Δθ at any observation point is obtained as the difference between the measured vessel heading and the calculated reference direction. Then, by using Equation (1), the transponder positions are corrected for heading deviation, and the result is presented in Figure 8(b).

Figure 7. (a) Plot of the vessel track L1 and its linear fit in the O _aX_aY_aZ_a coordinate system. (b) Heading of the vessel and the direction of the reference course derived from the linear fit of track L1. (c) Cross-track error of track L1.

Figure 8. Corrections for heading deviation and cross-track deviation of the transponder positions observed along track L1. (a) Observed transponder positions. (b) Transponder positions after correction for heading deviation. (c) Transponder positions after corrections for heading deviation and cross-track error.

Having corrected for heading deviation, the transponder trajectory is further corrected for cross-track error by using Equation (8), and the result is shown in Figure 8(c). Figure 9 presents all the eight transponder trajectories before and after the corrections for heading deviation and cross-track error. It is clear from Figure 9 that the corrections for heading deviation and cross-track error effectively reduce the distortion of the transponder trajectories, which in turn will result in a better estimation of alignment errors.

Figure 9. Transponder trajectories (a) before and (b) after the corrections for heading deviation and cross-track error.

4.2. Alignment error estimation

Having corrected for heading deviation and cross-track error, the transponder trajectories are used to estimate the alignment errors. As indicated in Part I (Chen, Reference Chen2013), the iterative process to estimate the alignment errors α, β, and γ with respect to heading, pitch and roll, respectively, is given by

(9)$$\left\{ {\matrix{ {\alpha ^{(k)} = \alpha ^{(k - 1)} + \Delta \alpha} \cr {\beta ^{(k)} = \beta ^{(k - 1)} + \Delta \beta} \cr {\gamma ^{(k)} = \gamma ^{(k - 1)} + \Delta \gamma} \cr}} \right.,\quad k = 1,\;2,\;3, \cdots $$

where k is the iteration number, and Δα, Δβ, and Δγ are increments of estimates. At each iteration, the increments Δα and Δβ are computed by

(10)$$\Delta \alpha = \cot ^{ - 1} (m_{xy} )$$

(11)$$\Delta \beta = - \tan ^{ - 1} (m_{yz} )$$

where m _xy and m _yz are the slopes of transponder trajectory on X _t-Y_t and Y _t-Z_t planes, respectively. As for the increment Δγ of roll alignment error, it can be solved from either of the following two equations:

(12)$$X_t = - d\cos \Delta \gamma - P_{Tz} \sin \Delta \gamma $$

(13)$$Z_t = - d\sin \Delta \gamma + P_{Tz} \cos \Delta \gamma $$

where d is the horizontal distance from the seabed transponder to the vessel track and −P _Tz represents the depth of the seabed transponder (i.e., P _Tz=−297·09 m).

Here we take the positioning data collected along track L1 as an example to illustrate the numerical estimation of alignment errors at the first iteration.

4.2.1 Heading misalignment estimation

Figure 10(a) presents the positioning data that has been corrected for heading deviation and cross-track error. It should be noted that the aspect ratio of the plots in Figure 10 is not equal to one for better interpretation of slope of the transponder trajectory. All iterative methods require that initial guesses be provided for each parameter. In all cases the initial guesses were taken to be the constant 0 (i.e., α ⁽⁰⁾=β ⁽⁰⁾=γ ⁽⁰⁾=0) for iteration zero (k=0). The positioning data in Figure 10(a) is fitted to a straight line by linear regression, which is shown as the solid line. The slope of the fitted line on the X _t-Y_t plane is m _xy=27·65 corresponding to heading alignment error α=2·07°. The transponder trajectory is then corrected for the estimate α=2·07° and the result is shown in Figure 10(b), in which the fitted line of the transponder trajectory on the X _t-Y_t plane becomes much steeper and its slope increases to 1·5E4.

Figure 10. Iterative alignment error correction of the positioning data collected along track L1. The circle and the solid line represent the positioning data and its linear fit, respectively. (a) Initial data without alignment error correction. (b) Result with correction for α=2·07°. (c) Result with correction for α=2·07° and β=4.52°. (d) Result with correction for α=2·07°, β=4.52°, and γ=2.09°. (e) After 5 iterations.

4.2.2 Pitch misalignment estimation

The pitch alignment error is determined based on the slope of the transponder trajectory on the Y _t-Z_t plane. In Figure 10(b), the transponder trajectory on the Y _t-Z_t plane is fitted to a straight line which has a slope of −0·079. Substituting m _yz=−0·079 into Equation (11) yields β=4.52°. The transponder trajectory corrected for the estimates α=2·07° and β=4.52° is presented in Figure 10(c), in which the linear fit of the trajectory on the Y _t-Z_t plane becomes flatter and its slope is improved from −0·079 to −0·007.

4.2.3 Roll misalignment estimation

In this case, we use Equation (12) to solve for the roll alignment error. The relevant parameters in Equation (12) include X _t, d, and P _Tz. As can be seen from Figure 10(c), the X _t coordinates of the positioning data are not identical. The value of the parameter X _t is therefore given by the average of the X _t coordinates of the transponder trajectory, which yields X _t=2.15 m. For the parameter d, it is defined as the horizontal distance from the seabed transponder to the vessel track (L1 in this case). Because the track L1 is non-straight and its linear fit is used as reference for the correction of cross-track error, the value of d is thereby the horizontal distance from the transponder to the linear fit of track L1, which yields d=8·67 m. Substituting P _Tz=−297·09, X _t=2.15, and d=8·67 into Equation (12), we obtain the roll alignment error as γ=2.09°. The transponder trajectory corrected for the estimates α=2·07°, β=4.52° and γ=2.09° is presented in Figure 10(d).

The iterative process is repeated until all estimates of three alignment errors have converged to some tolerance. Table 2 gives the history of the estimates of three alignment errors as generated by the iterative procedure. As can be seen, all estimates of α, β, and γ have converged to within 0·001 degrees after five iterations, yielding the final estimates α=1·36°, β=5·51°, and γ=2·12°. The estimation demonstrates that the rate of convergence of the algorithm is quite fast in practice. Figure 10(e) shows the corrected transponder trajectory after 5 iterations, in which the slope m _xy is −6·07E5 and the slope m _yz is −1·25E−6. As expected, on completion of alignment error correction, the magnitudes of the slopes m _xy and m _yz are close to infinity and zero, respectively.

Table 2. Iteration history for alignment error estimation with positioning data collected along track L1.

The algorithm is further tested on the positioning data collected along tracks L2-8. Table 3 summaries the final results of alignment error estimation with each dataset collected along different tracks. Note that all the estimates of roll alignment error given in Table 3 are obtained by Equation (12). In the estimation of alignment errors, our termination criterion is to stop the iteration if the increments ∆α, ∆β, and ∆γ are all less than 0·001 degrees. As can be seen from Table 3, convergence is very fast and is obtained in five iterations for all eight cases, indicating high efficiency of the proposed algorithm in the estimation of alignment errors. In the bottom two rows of Table 3 the mean and standard deviation are given, respectively, of the estimates obtained from the eight cases. The standard deviations of the estimates for different data sets are small (within 0·3 degrees), which indicates the alignment errors are robustly estimated from the positioning data. The correction for transceiver alignment error is done by the use of the mean estimates listed in Table 3 and the USBL positioning result is presented in Figure 11. As is seen, the corrected positioning data forms a concentrated point cloud around the true location of the transponder. After misalignment calibration, the error standard deviations of the USBL positioning are significantly reduced from 23·4, 25·3, and 27·1 m to 7·3, 7·0, and 5·9 m in the X _a-, Y_a-, and Z _a-directions, respectively.

Figure 11. Position of the seabed transponder as measured with and without the correction for alignment errors, in which the estimates of roll alignment error is based on Equation (12).

Table 3. Estimates of alignment errors as obtained from different positioning data collected along tracks L1-8. Note that the roll alignment error, γ, is obtained by Equation (12).

* The iteration is terminated when all estimates of α, β, and γ have converged to within 0·001 degrees.

4.3. Estimation of roll alignment error

As mentioned earlier, the determination of roll alignment error in the proposed algorithm can be carried out by solving either Equation (12) or Equation (13). That is, the roll alignment error has a coordinate dependence and can be given in terms of either X _t or Z _t:

(14)$$\gamma = \gamma ^{(x)} (X_t, d,P_{Tz} ) = \gamma ^{(z)} (Z_t, d,P_{Tz} )$$

In Part I (Chen, Reference Chen2013), we have shown that γ ^(x) is more robust to the noise in USBL measurements than γ ^(z) in the determination of roll alignment error when the condition |d|<|P _Tz| is true. In this subsection, we aimed to verify this finding with the use of data from observations of USBL positioning.

Table 4 shows the horizontal distance d from the seabed transponder to each of the eight vessel tracks. It can be observed from Table 4 that the distances, |d|, to the vessel tracks are all much less than the depth of the seabed transponder (297·09 m), i.e., |d|≪|P _Tz|. Therefore, γ ^(z) is expected to be much more sensitive to measurement noise than γ ^(x). By solving γ from Equation (13), the estimates of alignment errors with each dataset collected along different tracks are obtained and listed in Table 4. Apparently, the standard deviation of the estimates of γ ^(z) in Table 4 is much higher than that of γ ^(x) in Table 3, which is in agreement with our findings in Part I (Chen, Reference Chen2013). Moreover, imprecise estimation of roll alignment error increases the variance in the estimates of heading and pitch alignment errors as well.

Table 4. Estimates of alignment errors as obtained from different positioning data collected along tracks L1-8. Note that the roll alignment error, γ, is obtained by Equation (13).

By the use of the mean estimates listed in Table 4, the transponder positioning is corrected for the alignment errors of α=0·48°, β=6.41°, and γ=9.44° and the result is presented in Figure 12. It is clear by comparing Figure 12 with Figure 11 that the estimates of alignment errors in Table 4 are quite unreliable and no improvement is made on the accuracy of transponder positioning. A robust estimation of roll alignment error is therefore crucial in achieving effective misalignment calibration of a USBL navigation system.

Figure 12. Position of the seabed transponder as measured with and without the correction for alignment errors, in which the estimates of roll alignment error is based on Equation (13).

5. CONCLUSION

The proposed algorithm for Ultra Short Baseline (USBL) alignment calibration has been tested using the data collected from a field experiment. Effective solutions have been provided to correct the effects of vessel's heading deviation and cross-track error on the estimation of alignment errors. Therefore, with the aids of heading deviation and cross-track error corrections, the vessel trajectory for USBL calibration is not limited to a straight line but can be any arbitrary curve. Similar to the simulation results in Part I (Chen, Reference Chen2013), the filed experimental results show that the proposed algorithm has a fast convergence speed. For the eight USBL line surveys conducted, all the estimates of alignment errors have converged to within a tolerance of 0·001 degrees in five iterations. Furthermore, the variance of the estimates obtained from the eight USBL line surveys is small, indicating the robustness of the proposed algorithm on the estimation of alignment errors.

On the estimation of roll alignment error, the experimental results agree well with the theoretical predictions derived in Part I. It is shown that, when the horizontal distance from the seabed transponder to the vessel track is less than the depth of the transponder, γ ^(x) (estimation using X _t-coordinate data of USBL positioning) is more robust than γ ^(z) (estimation using Z _t-coordinate data of USBL positioning) in the determination of roll alignment error. In particular, by the use of γ ^(z), the variance of the estimate of roll alignment error increases significantly as the horizontal distance from the seabed transponder to the vessel track is close to zero. The experimental results also show that as the variance of the estimate of roll alignment error increases, so do the estimates of heading and pitch alignment errors.