Direct Method for Rapid Prototyping of Near-Optimal Aircraft Trajectories

Abstract

Adirectmethod fora real-timegenerationofnear-optimal spatialtrajectoriesofshort-term maneuversonboard a e ying vehiclewith predetermined thrust history isintroduced. Thepaperstarts with a survey about the founders of the direct methods of calculus of variations and their followers in e ight mechanics, both in Russia and in the United States. It then describes a new direct method based on three cues: high-order polynomials from the virtual arc as a reference function for aircraft’ s coordinates, a preset history of one of the controls (thrust), and a few optimization parameters. The trajectory optimization problem is transformed into a nonlinear programming problem and then solved numerically using an appropriate algorithm in accelerated scale of time. A series of examples is presented. Calculated near-optimal trajectory is compared with real e ight data, and with the solution obtainedbyPontryagin’ smaximumprinciple.Fastconvergenceofthenumericalalgorithm,whichhasbeenalready implemented and tested onboard a real aircraft, is illustrated. Nomenclature aik = polynomial coefe cients g = acceleration due to gravity J = cost function j = quantity pertaining to the jth time node m = aircraft mass N = number of nodes n = polynomial order ¯ n = relative revolutions of engine’ s rotor nx, nz = tangential and normal projections of load factor, respectively Sh = penalty function T, ¯ T = total thrust and relative thrust (fraction of maximum thrust), respectively t = time t ¤