Trackability with Imprecise Localization

Imagine a tracking agent $P$ who wants to follow a moving target $Q$ in $d$-dimensional Euclidean space. The tracker has access to a noisy location sensor that reports an estimate $\tilde{Q}(t)$ of the target's true location $Q(t)$ at time $t$, where $||Q(T) - \tilde{Q}(T)||$ represents the sensor's localization error. We study the limits of tracking performance under this kind of sensing imprecision. In particular, we investigate (1) what is $P$'s best strategy to follow $Q$ if both $P$ and $Q$ can move with equal speed, (2) at what rate does the distance $||Q(t) - P(t)||$ grow under worst-case localization noise, (3) if $P$ wants to keep $Q$ within a prescribed distance $L$, how much faster does it need to move, and (4) what is the effect of obstacles on the tracking performance, etc. Under a relative error model of noise, we are able to give upper and lower bounds for the worst-case tracking performance, both with or without obstacles.

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