You Can’t Save all the Pandas: Impossibility Results for Privacy-Preserving Tracking

Abstract

We consider the problem of target tracking whilst simultaneously preserving the target’s privacy as epitomized by the panda tracking scenario introduced by O’Kane at WAFR’08. The present paper reconsiders his formulation, with its elegant illustration of the utility of ignorance, and the tracking strategy he proposed, along with its completeness. We explore how the capabilities of the robot and panda affect the feasibility of tracking with a privacy stipulation, uncovering intrinsic limits, no matter the strategy employed. This paper begins with a one-dimensional setting and, putting the trivially infeasible problems aside, analyzes the strategy space as a function of problem parameters. We show that it is not possible to actively track the target as well as protect its privacy for every nontrivial pair of tracking and privacy stipulations. Secondly, feasibility is sensitive, in several cases, to the information available to the robot at the start — conditions we call initial I-state dependent cases. Quite naturally in the one-dimensional model, one may quantify sensing power by the number of perceptual (or output) classes available to the robot. The number of initial I-state dependent conditions does not decrease as the robot gains more sensing power and, further, the robot’s power to achieve privacy-preserving tracking is bounded, converging asymptotically with increasing sensing power. Finally, to relate some of the impossibility results in one dimension to their higher-dimensional counterparts, including the planar panda tracking problem studied by O’Kane, we establish a connection between tracking dimensionality and the sensing power of a one-dimensional robot.