The performance of the cooperative navigation proposed here could be examined by computing the trajectory-tracking error with noisy range measurements, which was with very good accuracy. The acoustic navigation system was tested as follows. It was supposed that there were four sonobuoys pre-deployed in a given square geometry of 10x10 Km in the X-Y plane, and the AUV of interesting in the coordinated team was in the depth of 500m. The sonobuoys were separately located at (0, 10000, 500), (0, -10000, 500), (10000, 10000, 500), and (10000, -10000, 500), and the AUV was locate at (0, 0, 0). The performance of the navigation method could be examined by computing the trajectory-
tracking error with noisy range measurements. Assume that the range measurements in four sonobuoys had a standard deviation of < =0.5, <22 =4, c32 =1, and <<2 =0.2. Assume also that the errors were independent. Thus the covariance matrix of the errors was R=[0.5 0 0 0; 0 4 0 0; 0 0 1 0;0 0 0 0.2]T. In fig.6, the solid track was from the true trajectory of one AUV in coordinated team moving in a spiral cord in 3D space or projected in 2D space, and the asterisks represented the corresponding tracking position estimates for the acoustic navigation system, which were with very good accuracy. Using the simultaneous LS positioning algorithm, the estimates were typically within centi-metric accuracy in northing(X-axis) and easting direction(Y-axis) of the true position. The mean length of the overall errors was 0.014171m and 0.0013966m respectively in X-axis and Y-axis, and the mean length of correction vector values was 0.2215m. The small errors give accurate measurement of the AUV position, but we should also be aware of that the actual error might be different due to the variety of noise measurements and the DGPS accuracy.
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