Building on the standalone analysis of the previous section, next we use the CRLB framework to consider design decisions inherent in combining absolute positioning (LBL) and odometry dead-reckoning (DVL+Heading). The sensing modalities are best used in concert, where the two information sources can complement each other. The CRLB framework, using the measurement model in equation (17), quantifies the benefits of this combination. The LBL and DVL sensing modalities must be matched to realize the potential of the complementary nature of these two navigation methods. Fig. 9 illustrates the constructive combination of LBL range observations, DVL velocity measurements and heading reference using a simple one-dimensional model. This comparison guides the selection of relative precision of the various sensors and the required update rate to leverage ability of absolute positioning to constrain the drift inherent to dead reckoning.

The two asymptotes in Fig. 9 are illustrative. On the right, in Region 3, we see that as odometry error is large, the overall positioning uncertainty is limited to be approximately equivalent to the absolute positioning uncertainty, indicated by a0 / ar = 1.0 when the dead-reckoning uncertainty is greater than twice the absolute uncertainty (a0 > 2.0 <rr). This could be caused by either high uncertainty in the velocity or heading measurements or large update times between absolute position updates. Conversely, the left side of the figure, Region 1, shows how precise odometry between absolute position updates links the sequential updates together. As the odometry becomes more precise the overall position uncertainty approaches the bound of ax = ar/^N, where N is the number of discrete position updates (in this case N = 100). This limiting case represents perfect odometry, where the distance between absolute reference updates is known.

Steady State LBL/DVL Position Uncertainty

Steady State LBL/DVL Position Uncertainty

Fig. 9. Based on the one-dimensional model, this figure quantifies the tradeoffs in designing a complementary positioning solution using absolute positioning (LBL) and dead-reckoning odometry (DVL+Heading). The vertical axis shows position uncertainty (ax) normalized by the absolute reference uncertainty (ffr). The horizontal axis shows the ratio of total odometry uncertainty (ff0) to absolute reference uncertainty. Designing a solution in Region 1, with — < 0.04, successfully leverages the complementary nature of the two modes of navigation.

As an illustration we present an example using representative numbers for instruments typical on modern underwater platforms. Based on Fig. 9 we would like to design the positioning solution to operate in Region 1, where the total odometry uncertainty is less then 0.04 times the absolute positioning uncertainty, i.e., g0 = avJt + d a^ < 0.1 <rr (23)

Typical vehicle instrumentation might consist of an 1,200 kHz RDI DVL3 (av = 3 mm/s), an Octans true north heading reference4 (cfy = 0.1 degrees) and Benthos LBL transponders5 (ar = 3.0 m). Furthermore we can assume a typical velocity of 1.0 m/ s for the purposes of demonstration, resulting in d = 1.0 t. Therefore,

3 1,200 kHz Workhourse Navigator Doppler velocity log by Teledyne RD Instruments.

4 6000 Series Transponders by Teledyne Benthos.

5 Octans Fiber Optic Gyroscope (FOG) by Ixsea.

Resulting in a required update rate of t < 56 seconds. Such an infrequent update rate is a consequence of the precision of the dead-reckoning solution.

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