Dynamic Sensor-Actor Interactions for Path-Planning in a Threat Field

Abstract

We consider the problem of planning the path of a vehicle, which we refer to as the actor, to traverse a threat field with minimum threat exposure. The threat field is an unknown, time-invariant, and strictly positive scalar field defined on a compact 2D spatial domain – the actor’s workspace. The threat field is estimated by a network of mobile sensors that can measure the threat field pointwise at their locations. All measurements are noisy. The objective is to determine a path for the actor to reach a desired goal with minimum risk, which is a measure sensitive not only to the threat exposure itself, but also to the uncertainty therein. A novelty of this problem setup is that the actor can communicate with the sensor network and request that the sensors position themselves such that the actor’s risk is minimized. Future applications of this problem setup include, for example, delivery (by an actor) of emergency supplies to a remote location that lies within/beyond a region afflicted by wildfire or atmospheric contaminants (the threat field). We formulate this problem on a grid defined on the actor’s workspace, which defines a topological graph $\mathcal {G}$. The threat field is assumed to be finitely parameterized by coefficients of spatial basis functions. Least squares estimates of these parameters are constructed using measurements from the sensors and the actor. Whereas edge transitions in the graph $\mathcal {G}$ are deterministic, the transition costs depend on the threat field estimates, and are deterministic but unknown. The actor and the sensors interact iteratively. At each iteration, Dijkstra’s algorithm is used to determine a minimum risk path in the graph $\mathcal {G}$ for the actor. Next, a set of grid points “near” this path are identified as points of interest. Finally, the next set of sensor locations is determined to maximize the confidence of threat field estimates on these points of interest, the threat field estimate is accordingly updated, and the iteration repeats. We explore the effect of initial sensor placement on the convergence of the iterative planner-sensor as well as discuss convergence properties with respect to the relative number of parameters and sensors available.