2.7 Generalized Bearings

A different approach to iterating toward a PF solution was presented by Paradowski [18]. Given the ith azimuth AOA ? _i and the ith elevation AOA ? _i, the generalized bearing ? _i is determined as indicated in Figure 2.29. The generalized bearing is in the plane defined by triangle TS _iA and is given by

(2.227)

Figure 2.29: Generalized bearing, denoted by ? _i.

The measurements are expressed as

(2.228)

or

(2.229)

where, in the absence of measurement errors u _i = f _i( x i) = ? _i, and f _i( x _i) is the function in which the PF and LOB information is contained. The measurement noise, n _i, is characterized by

(2.230)

(2.231)

? _i and ? _j are standard deviations for the ith and jth measurements and ? _ij is the correlation coefficient between these measurements.

The gradient vector at the mth step in the iteration toward the solution is given by

(2.232)

where

(2.233)

and

(2.234)

The gradient matrix consists of rows that are the gradient vectors

(2.235)

At the mth iteration, the covariance matrix of generalized bearing errors is a diagonal matrix expressed as

(2.236)

where is the variance of the ? _i error at the mth iteration. These can be expressed in terms of the variances of the measurements as

(2.237)

where

Given that the...