## The CRLB for combined odometry and positioning

The CRLB framework is also capable of analyzing the tradeoffs inherent in combining observations into an integrated navigation solution. In particular, we are interested in quantifying the tradeoffs involved in combining LBL absolute positioning with DVL dead-reckoning.

To apply the CRLB framework to this case requires a measurement model including both the high update rate odometry measurements of relative distance travelled and infrequent absolute position updates. In one-dimension the absolute position measurement uncertainty is equivalent to the range uncertainty (ffr). To consider two-dimensional the odometry observation model from equations (7)-(9) we sum the two independent components of uncertainty. This simplification is similar to the notion of scalar horizontal precision discussed in the next section.

ff02 = t (J2 + d2 o$A < SXh = E[[xh - xh][xh - xh]T] (16)

Now we can create a combined one-dimensional measurement model for a set of n absolute position updates with n — 1 interspersed odometry measurements.

The additive noise vector, wc, is modelled using a zero-mean Gaussian distribution. The individual measurements are considered to be independent, resulting in a covariance matrix that based on the standalone range measurements and odomentry measurements.

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