Path-Based Sensors: Paths as Sensors, Bayesian Updates, and Shannon Information Gathering

Abstract

Consider a sensor that reports whether or not an event has occurred somewhere along a path, but that has no conception of where along the path that event has occurred. We name this type of sensor a path-based sensor and describe the recursive Bayesian update that can be used to calculate posterior beliefs about the presence of a sensor triggering phenomenon given a path-based sensor observation. We show how the Bayesian update can be leveraged to calculate the expected Shannon information that will be gained along a particular path. We formalize two iterative information-gathering problems that result from this scenario and present path-planning algorithms to solve them. These include: 1) gathering information about the path-based sensor triggering phenomena and 2) assuming the path-based sensor triggering event is “robot destruction,” simultaneously gather information about: 1) hazards using a path-based sensor and 2) information about another environmental phenomenon using a standard sensor, such as the locations of search and rescue targets with a camera. We evaluate our methods using Monte Carlo simulations and observe that they outperform other techniques with respect to the new problems that we consider. Note to Practitioners —This work is motivated by the problem of searching for robot-destroying hazards that are otherwise invisible to the robots. That is, we can observe whether or not a robot survives a path, but, if a robot is destroyed, then we have no idea where, along the path, its destruction has occurred. A mathematically equivalent problem happens in any scenario, in which an agent is equipped with an event sensor that can only be set/triggered once, but that requires postprocessing to figure out if the sensor has been triggered or not. For example, postprocessing is needed if the determination of whether or not a biological specimen was obtained requires a manual laboratory inspection. We also consider an extension of the hazard detection problem, in which we simultaneously collect information about search-and-rescue victims using a “victim sensor,” such as a camera. In this problem, hazards indirectly affect information gathered about victims because new information about victims is lost whenever a robot is destroyed. We provide algorithms to solve these types of problems. The algorithms work even in cases with noise such that false positives and false negatives are possible. This work is useful in any application where observations take the form of a cumulative “yes” or “no” along a path.