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## Comparing Position Estimates

We must establish a common denominator for comparison of position estimates given measurements. We have pursued a statistical procedure for doing so. Our objective is to maximize the posterior probability of position given the bearings, $p(x, y | R)$. Position is discretized in intervals of 30 meters, agreeing with the discretization of Digital Elevation Maps. At first, we must specify a prior density for position, $p(x, y)$. Currently we use a uniform distributions to signify absence of prior knowledge. Secondly, we must specify the likelihood the bearings (the measurements) given position, $p(R | x, y)$. In practice it is difficult to determine the likelihood of a bearing given position; it is appropriate to decompose this expression in terms of the possible interpretations of the bearings:

$p(R|x, y) = ∑$I p(R|x, y, I) p(I|x, y).

The final posterior density is proportional to $p(R|x,y) p(x,y)$.

Experience with the mountain detection system has led to the conclusion that, for fixed pose and correspondences, the errors in measuring bearings are fairly independent:

i=1m p(ri|I,Γ).

We assume that when a landmark $I$i is visible from $x, y$, the errors in the bearing $r$i follow a Gaussian law. We also measure the height of the mountain associated with the bearing, and assume errors in height follow a Gaussian law. We consider that any error that exceeds 18 degrees is a mistake, that is, if a mountain at 40 degrees is associated to a bearing at 60 degrees, we consider that association a mistake. Mistakes are assumed distributed uniformly on the interval [0, 360] degrees. Every bearing is associated with all mountains within a $±18$ degrees interval; bearings that are not associated to any mountain are mistakes. Since bearings do not need to be associated to all possible mountains, the summation in expression 1 can be calculated quickly.

Next: Experiments Up: The Estimation Algorithm Previous: Searching for Position

Fabio G. Cozman