The objective of the work presented in this Ph.D. thesis is to develop a novel method to address the aircraft-obstacle avoidance problem in presence of uncertainty, providing optimal trajectories in terms of risk of collision and time of flight. The obstacle avoidance maneuver is the result of a Conflict Detection and Resolution (CD&R) algorithm prepared for a potential conflict between an aircraft and a fixed obstacle which position is uncertain.

Due to the growing interest in Unmanned Aerial System (UAS) operations, CD&R topic has been intensively discussed and tackled in literature in the last 10 years. One of the crucial aspects that needs to be addressed for a safe and efficient integration of UAS vehicles in non-segregated airspace is the CD&R activity. The inherent nature of UAS, and the dynamic environment they are intended to work in, put on the table of the challenges the capability of CD&R algorithms to handle with scenarios in presence of uncertainty. Modeling uncertainty sources accurately, and predicting future trajectories taking into account stochastic events, are rocky issues in developing CD&R algorithms for optimal trajectories. Uncertainty about the origin of threats, variable weather hazards, sensing and communication errors, are only some of the possible uncertainty sources that make jeopardize air vehicle operations.

In this work, conflict is defined as the violation of the minimum distance between a vehicle and a fixed obstacle, and conflict avoidance maneuvers can be achieved by only varying the aircraft heading angle. The CD&R problem, formulated as Optimal Control Problem (OCP), is solved via indirect optimal control method. Necessary conditions of optimality, namely, the Euler-Lagrange equations, obtained from calculus of variations, are applied to the vehicle dynamics and the obstacle constraint modeled as stochastic variable. The implicit equations of optimality lead to formulate a Multipoint Boundary Value Problem (MPBVP) which solution is in general not trivial. The structure of the optimality trajectory is inferred from the type of path constraint, and the trend of Lagrange multiplier is analyzed along the optimal route. The MPBVP is firstly approximated by Taylor polynomials, and then solved via Differential Algebra (DA) techniques.

The solution of the OCP is therefore a set of polynomials approximating the optimal controls in presence of uncertainty, i.e., the optimal heading angles that minimize the time of flight, while taking into account the uncertainty of the obstacle position. Once the obstacle is detected by on-board sensors, this method provide a useful tool that allows the pilot, or remote controller, to choose the best trade-off between optimality and collision risk of the avoidance maneuver. Monte Carlo simu- lations are run to validate the results and the effectiveness of the method presented. The method is also valid to address CD&R problems in presence of storms, other aircraft, or other types of hazards in the airspace characterized by constant relative velocity with respect to the own aircraft.