Abstract
For a host aircraft in the abort landing mode under emergency conditions, the best strategy for collision avoidance is to maximize wrt to the controls the timewise minimum distance between the host aircraft and an intruder aircraft. This leads to a maximin problem or Chebyshev problem of optimal control. At the maximin point of the encounter, the distance between the two aircraft has a minimum wrt the time; its time derivative vanishes and this occurs when the relative position vector is orthogonal to the relative velocity vector. By using the zero derivative condition as an inner boundary condition, the one-subarc Chebyshev problem can be converted into a two-subarc Bolza-Pontryagin problem, which in turn can be solved via the multiple-subarc sequential gradient-restoration algorithm.
Optimal Trajectory. In the avoidance phase, maximum angle of attack is used by the host aircraft until the minimum distance point is reached. In the recovery phase, the host aircraft completes the transition of the angle of attack from the maximum value to that required for quasisteady ascending flight.
Guidance Trajectory. Because the optimal trajectory is not suitable for real-time implementation, a guidance scheme is developed such that it approximates the optimal trajectory results in real time. In the avoidance phase, the guidance scheme employs the same control history (maximum angle of attack) as that of the optimal trajectory so as to achieve the goal of maximizing wrt the control the timewise minimum distance. In the recovery phase, the guidance scheme employs a time-explicit cubic control law so as to achieve the goal of recovering the quasisteady ascending flight state at the final time.
Numerical results for both the optimal trajectory and the guidance trajectory complete the paper.
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Miele, A., Wang, T., Mathwig, J.A.: Optimal collision avoidance trajectory for an aircraft in abort landing. Rice University, Aero-Astronautics Report No. 353 (2005)
Miele, A., Wang, T., Mathwig, J.A.: Guidance scheme approximating the optimal collision avoidance trajectory for an aircraft in abort landing. Rice University, Aero-Astronautics Report No. 354 (2005)
Miele, A., Ciarcià, M.: Time-explicit cubic guidance scheme for an aircraft in abort landing. Rice University, Aero-Astronautics Report No. 366 (2009)
Raghunathan, A.U., Gopal, V., Subramanian, D., Biegler, L.T., Samad, T.: Dynamic optimization strategies for three-dimensional conflict resolution of multiple aircraft. J. Guid. Control Dyn. 27(4), 586–594 (2004)
Hu, J., Prandini, M., Sastry, S.: Optimal coordinated maneuvers for three-dimensional aircraft conflict resolution. J. Guid. Control Dyn. 25(5), 888–900 (2002)
Frazzoli, E., Mao, Z.H., Oh, J.H., Feron, E.: Resolution of conflicts involving many aircraft via semidefinite programming. J. Guid. Control Dyn. 24(1), 79–86 (2001)
Menon, P.K., Sweriduk, G.D., Sridhar, B.: Optimal strategies for free-flight air traffic conflict resolution. J. Guid. Control Dyn. 22(2), 202–211 (1999)
Clements, J.C.: The optimal control of collision-avoidance trajectories in air-traffic management. Transp. Res. B 33(4), 265–280 (1999)
Kuo, V.H., Zhao, Y.J.: Required ranges for conflict resolutions in air traffic management. J. Guid. Control Dyn. 24(2), 237–245 (2001)
Tornapolskaya, T., Fulton, N.: Optimal cooperative collision avoidance strategies for coplanar encounter: Merz solution revisited. J. Optim. Theory Appl. 140(2), 355–375 (2009)
Tornapolskaya, T., Fulton, N.: Synthesis of optimal control for cooperative collision avoidance for aircraft (ships) with unequal turn capabilities. J. Optim. Theory Appl. 144(2) (2010)
Miele, A., Wang, T.: Multiple-subarc sequential gradient-restoration algorithm, Part 1: Algorithm structure. J. Optim. Theory Appl. 116(1), 1–17 (2003)
Miele, A., Wang, T.: Multiple-subarc sequential gradient-restoration algorithm, Part 2: Application to a multistage launch vehicle design. J. Optim. Theory Appl. 116(1), 19–39 (2003)
Miele, A., Wang, T.: Maximin approach to the ship collision avoidance problem via multiple-subarc sequential gradient-restoration algorithm. J. Optim. Theory Appl. 124(1), 29–53 (2005)
Miele, A., Wang, T.: Fundamental properties of optimal orbital transfer. Paper IAC-03-A.7.02, 54th International Astronautical Congress, Bremen, Germany (2003)
Miele, A., Wang, T., Williams, P.N.: On feasibility of launch vehicle designs. Appl. Math. Comput. 164(2), 295–312 (2005)
Miele, A., Wang, T., Melvin, W.W.: Optimization and acceleration guidance of flight trajectories in a windshear. J. Guid. Control Dyn. 10(4), 368–377 (1987)
Miele, A., Wang, T., Melvin, W.W., Bowles, R.L.: Gamma guidance schemes for flight in a windshear. J. Guid. Control Dyn. 11(4), 320–327 (1988)
Miele, A., Wang, T., Melvin, W.W.: Penetration landing guidance trajectories in the presence of windshear. J. Guid. Control Dyn. 12(6), 806–814 (1989)
Miele, A., Wang, T., Melvin, W.W., Bowles, R.L.: Acceleration, gamma, and theta guidance for abort landing in a windshear. J. Guid. Control Dyn. 12(6), 815–821 (1989)
Pontani, M.: Differential games treated by a gradient-restoration approach. In: Buttazzo, G., Frediani, A. (eds.) Variational Analysis and Aerospace Engineering, pp. 379–396. Springer, New York (2009). Chap. 21
Miele, A., Wang, T., Ciarcià, M.: Optimal collision avoidance trajectory for an aircraft in takeoff. Aero-Astronautics Report 351, Rice University (2005)
Miele, A., Wang, T., Ciarcià, M.: Guidance scheme approximating the optimal collision avoidance trajectory for an aircraft in takeoff. Aero-Astronautics Report 352, Rice University (2005)
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Miele, A., Wang, T., Mathwig, J.A. et al. Collision Avoidance for an Aircraft in Abort Landing: Trajectory Optimization and Guidance. J Optim Theory Appl 146, 233–254 (2010). https://doi.org/10.1007/s10957-010-9669-2
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DOI: https://doi.org/10.1007/s10957-010-9669-2