  Consideration of resources such as fuel, battery charge, and storage space,
is a crucial requirement for the successful persistent operation of autonomous
systems. The Stochastic Collection and Replenishment (SCAR) scenario is
motivated by mining and agricultural scenarios where a dedicated replenishment
agent transports a resource between a centralised replenishment point to agents
using the resource in the field. The agents in the field typically operate
within fixed areas (for example, benches in mining applications, and fields or
orchards in agricultural scenarios), and the motion of the replenishment agent
may be restricted by a road network. Existing research has typically approached
the problem of scheduling the actions of the dedicated replenishment agent from
a short-term and deterministic angle. This paper introduces a method of
incorporating uncertainty in the schedule optimisation through a novel
prediction framework, and a branch and bound optimisation method which uses the
prediction framework to minimise the downtime of the agents. The prediction
framework makes use of several Gaussian approximations to quickly calculate the
risk-weighted cost of a schedule. The anytime nature of the branch and bound
method is exploited within an MPC-like framework to outperform existing
optimisation methods while providing reasonable calculation times in large
scenarios.
