  Mixed Integer Linear Programs (MILPs) are often used in the path planning of
both ground and aerial vehicles. Such a formulation of the path planning
problem requires a linear objective function and constraints, limiting the
fidelity of the the tracking of vehicle states. One such parameter is the state
of charge of the battery used to power the vehicle. Accurate battery state
estimation requires nonlinear differential equations to be solved. This state
estimation is important in path planning to ensure flyable paths, however when
using MILPs to formulate the path planning problem these nonlinear equations
cannot be implemented. Poor accuracy in battery estimation during the path
planning runs the risk of the planned path being feasible by the estimation
model but in reality will deplete the battery to a critical level. To the end
of higher accuracy battery estimation within a MILP, we present here a simple
linear battery model which predicts the change in state-of-charge (SOC) of a
battery given a power draw and duration. This model accounts for changes in
battery voltage due to applied electrical load and changes in battery SOC. The
battery model is presented and then tested against alternate battery models in
numerical and in experimental tests. Further, the effect the proposed linear
model has over a simpler SOC estimation on the time-to-solve a resource
constrained shortest path problem is also evaluated, implemented in two
different algorithms. It is seen that the linear model performs well in battery
state estimation while remaining implementable in a Linear Program or MILP,
with little affect on the time-to-solve.
