A minimal time optimal control for a drone landing problem

Abstract

We study a variant of the classical safe landing optimal control problem in aerospace engineering, introduced by Miele in 1962, where the target was to land a spacecraft on the moon by minimizing the consumption of fuel. A more modern model consists in replacing the spacecraft by a hybrid gas-electric drone. Assuming that the drone has a failure and that the thrust (representing the control) can act in both vertical directions, the new target is to land safely by minimizing time, no matter of what the consumption is. In dependence of the initial data (height, velocity, and fuel), we prove that the optimal control can be of four different kinds, all being piecewise constant. Our analysis covers all possible situations, including the nonexistence of a safe landing strategy due to the lack of fuel or for heights/velocities for which also a total braking is insufficient to stop the drone.

Structure of optimal control for planetary landing with control and state constraints

“…Then, the Bang-Off-Bang structure, called Max-Min-Max when the thrust is not allowed to go to zero, has been found to be optimal for numerous variants of the problem, such as the two-dimensional problem studied in [9], the problem with specified initial and final thrust studied in [21], or with throttle and thrust angle control in [22]. Recently, [8] and [14] showed that the Max-Min-Max structure is the solution of the one dimension problem, minimizing the final time for the first one and considering the effect of an atmosphere on the thrust for the latter. More generally, considering the problem of controlling a vehicle in space but not necessarily during the landing phase, [17] shows that the optimal control is a finite succession of Min and Max arcs.…”

Structure of optimal control for planetary landing with control and state constraints

“…Then, the Bang-Off-Bang structure, called Max-Min-Max when the thrust is not allowed to go to zero, has been found to be optimal for numerous variants of the problem, such as the two-dimensional problem studied in [5], the problem with specified initial and final thrust studied in [6], or with throttle and thrust angle control in [7]. Recently, [8] and [9] showed that the Max-Min-Max structure is the solution of the one dimension problem, minimizing the final time for the first one and considering the effect of an atmosphere on the thrust for the latter. However, there lack theoretical studies relating more complex formulations of the landing problem, for instance considering realistic technical and safety constraints.…”