Aerospace Science and Technology 23 (2012) 34–42 Contents lists available at SciVerse ScienceDirect Aerospace Science and Technology www.elsevier.com/locate/aescte Efficient rotorcraft trajectory optimization using comprehensive models by improved shooting methods ✩ Carlo L. Bottasso ∗ , Fabio Luraghi, Giorgio Maisano Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, 20156, Milano, Italy article info abstract Article history: Available online 9 January 2012 Keywords: Flight mechanics Rotorcraft vehicles Optimal control The present paper focuses on trajectory optimization problems for complex first-principle models of rotorcraft vehicles, accounting for the presence of slow and fast dynamic components in the solution. The maneuver optimal control problem is solved through a direct approach by means of a novel hybrid single-multiple shooting method. The capabilities of the proposed procedures are illustrated with the help of an application regarding the estimation of the H–V diagram of a tilt-rotor.  2012 Elsevier Masson SAS. All rights reserved. 1. Introduction The term trajectory optimization refers to the process of com- puting the optimal control inputs and the resulting response of a model of a vehicle, a rotorcraft in the present case, which mini- mize a cost function (or maximize an index of performance) while satisfying given constraints (which specify, for example, the vehicle flight envelope boundaries, and/or safety and procedural require- ments for a maneuver of interest) [7,11,9,10,13]. Hence, a ma- neuver can usually be given a precise definition by formulating an equivalent optimal control problem. The formulation of such a problem necessitates a vehicle model with inputs, states and out- puts, and a cost function with a list of all constraints. Clearly, the fidelity of the predictions made using this ap- proach hinges on the fidelity of the vehicle model. In fact, trajecto- ries and performance limits predicted with oversimplified models might exhibit significant discrepancies with real flight data. Fidelity improvements may be obtained by considering a more sophisti- cated description of the vehicle; the current state-of-the-art calls for analysis tools based on comprehensive approaches [1,4,20,26], which offer the ability to create hierarchical models of varying lev- els of fidelity of the various sub-systems of the vehicle. Ref. [13] illustrates a suite of trajectory optimization procedures that cater to vehicle models of varying complexity. Solution proce- dures for rotorcraft flight mechanics models have been previously described by Okuno and Kawachi [23], Carlson and Zhao [15], and Bottasso et al. [10,8]. The extension of such procedures to han- dle fine-scale aero-servo-elastic comprehensive vehicle models was first described by Bottasso et al. [11,12]. ✩ Presented at the 35th European Rotorcraft Forum, Hamburg, Germany, Septem- ber 22–25, 2009. * Corresponding author. Tel.: +39 02 2399 8315; fax: +39 02 2399 8334. E-mail address: carlo.bottasso@polimi.it (C.L. Bottasso). This work focuses on the direct multiple shooting approach [5] to the solution of maneuver optimal control problems using the so-called level 2 rotorcraft models, according to the classification of vehicle models proposed by Padfield [24]. As noted in Ref. [21], these models are seldom used in the so- lution of optimal control problems by multiple shooting methods because it is often hard to provide the required accuracy within a reasonable computation time, while avoiding numerical issues due to the complex and non-linear nature of the rotor model. The reason for this is twofold: on the one hand, a small integration time step length is needed to resolve correctly the high frequency components of the solution within a given accuracy. On the other hand, the continuity of the rotor states have to be guaranteed by satisfying the multiple shooting gluing constraints which join the successive shooting arcs. The satisfaction of such constraints can be particularly difficult and usually ends up dominating the problem. This is not surprising, since the rotor generates most of the aerody- namic forces acting on the vehicle and even small variations in its states may imply large variations in the resulting forces; this often hinders the satisfaction of the gluing constraints, thereby slowing or preventing convergence. These problems can be alleviated by using multi-time scale ar- guments. In fact, level 2 models include both slow flight mechanics scales as well as faster ones, the latter being related to rotor de- grees of freedom, including both structural (rigid and/or flexible) and aerodynamic states. To treat more effectively this class of op- timal control problems, in this work multiple shooting is used on the slow scales, and single shooting on the faster ones; this avoids the enforcement of the gluing constraints for the faster scales, thus improving the efficiency and robustness of direct multiple shooting methods when applied to complex vehicle models. The paper is organized according to the following plan. Af- ter having more precisely defined the maneuver optimal control problem in Section 2, in Section 3 the mathematical formulation 1270-9638/$ – see front matter  2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ast.2012.01.004