Resilient Distributed Diffusion for Multi-Robot Systems Using Centerpoint

In this paper, we study the resilient diffusion problem in a network of robots aiming to perform a task by optimizing a global cost function in a cooperative manner. In distributed diffusion, robots combine the information collected from their local neighbors and incorporate this aggregated information to update their states. If some robots are adversarial, this cooperation can disrupt the convergence of robots to the desired state. We propose a resilient aggregation rule based on the notion of centerpoint, which is a generalization of the median in the higher dimensional Euclidean space. Robots exchange their d-dimensional state vectors with neighbors. We show that if a normal robot implements the centerpoint-based aggregation rule and has n neighbors, of which at most (sic)n/d+1(sic)-1 are adversarial, then the aggregated state always lies in the convex hull of the states of the normal neighbors of the robot. Consequently, all normal robots implementing the distributed diffusion algorithm converge resiliently to the true target state. We also show that commonly used aggregation rules based on the coordinate-wise median and geometric median are, in fact, not resilient to certain attacks. We numerically evaluate our results on mobile multi-robot networks and demonstrate the cases where diffusion with the weighted average, coordinate-wise median, and geometric median-based aggregation rules fail to converge to the true target state, whereas diffusion with the centerpoint-based rule is resilient in the same scenario.