Optimal trajectory planning for hot-air balloons in linear wind fields

The altitude of hot-air balloons is controlled by heating the air trapped inside the balloon and allowing the air to cool naturally. Apart from controlling the altitude, it is desirable-to utilize the wind field to position the balloon at a target location while minimizing fuel consumption. This can be posed as an optimal control problem with free end states, where the heat input to the system is the control variable. The problem is intractable because of the switching nature of the heat input and highly nonlinear state equations derived from the thermodynamic model of the balloon. In this paper we address the optimal control problem within a space of a few kilometers where we assume the wind fields to be known and linear. We simplify the dynamic model of the balloon and obtain optimal trajectories to the target location by solving a two-point boundary-value problem. By refining the simplified dynamic model, the accuracy of the optimal trajectories are improved to match well with trajectories obtained using the nonlinear model. Our approach based on simplification of the balloon dynamic model enables us to solve the intractable nonlinear optimal control problem and provides insight into the optimal trajectories, such as number of switchings of input and loss of accuracy for specific wind profiles. Except for these specific wind profiles, our approach yields accurate trajectories for the balloon and provides a solution to an important problem that has not been adequately addressed in the literature.