The objective is the development and evaluationof a fast and reliable, multiple-timescale algorithmfor the inverse simulation of rotorcraft maneuvering tasks. A recent, two-timescale approach to the solution of inverse problems of aircraft motion represents the background for devising a technique that accounts for speci c issues of rotorcraft dynamics such as the large effects of the fast, primary moment generating controls on the slow dynamics associated to the vehicle trajectory and the system being frequently nonminimum phase. Accurate solutions are obtained for the inverse simulations of the fast and slow reduced-order systems because the quasi-steady-state values of the fast controls are considered in the slow timescale. High-amplitude oscillations in the control inputs, revealed in previous applications of helicopter inverse simulation, are interpreted as due to the presence of nonobservable motions and are ruled out by the multiple timescale approach. The results show that the expected computer time reduction is realized, that the well-known dif culties of inverse methods for nding feasible solutions at convergence are practically eliminated, and, nally, that steady-state ight conditions are accurately recovered at the end of the prescribed maneuvers. Nomenclature g = gravity acceleration I x , I y , I z = moments of inertia I x z = product of inertia k 1 ; k 2 = curvature and torsion L, M , N = moment components in body axes L i , M i , N i = dimensional derivatives due to state or input i L IF , L IB = transformation matrices m = helicopter mass N = number of fast timescale intervals inside 1t n = normal unit vector R = inertial coordinate vector (north, east and down), .R N ; R E ; R D / T s = curvilinear abscissa t = time u, v, w = velocity components in body axes u = control vector V = velocity modulus V = inertial velocity vector, .V N ; V E ; V D / T X , Y , Z = force components in body axes x = state vector y = output vector ® = angle of attack, tan ¡1 (w=u) = sideslip angle, sin ¡1 (v=V ) 1t = time interval of the slow dynamics ± A = lateral cyclic stick movement, positive right ± B = longitudinal cyclic movement, positive after or nose-up ± C = collective control input, positive up ± P = pedal movement, positive left or nose-left ±t = time interval of the fast dynamics ±´= time step for the numerical integration ±¿ = time delay ² = perturbation parameter » = c.g. position in the Frenet triedron Presented as Paper 99-4112 at the ½ = air density Á; µ ; à = Euler's angles ! = angular velocity vector, . p; q; r / T Subscripts des = desired e = steady state F = fast timescale S = slow timescale Superscript 0 = displacement from trim value

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The linear multiplet, composed of the dilaton ϕ, of an antisymmetric gauge field Bμν and of a spinor χ is always present in any superstring induced N=1D=4 supergravity model. We consider its coupling to supergravity using only superspace Bianchi identities and the rheonomy approach. In this way, our results are fully general and independent from the choice of any Lagrangian, a concept which is never mentioned in this paper. We consider two situations corresponding to two different free differential algebras: (1) the case where there are no Chern-Simons terms in the Bμν field strength Hμνρ and (2) the case where such terms are included in Hμνρ. Case (2) is obviously the one chosen by string theory on the ground of anomaly cancellation. In both cases, we must solve the H-Bianchi identity using a solution of the super Poincare’ Bianchi identities as a background. Such a solution, besides the physical fields displays a certain number of auxiliary fields. The most general solution of the super Poincare’ Bianchis we have to consider corresponds to 16⊕16 off-shell multiplet which, by suitable choices can be reduced either to the so-called old minimal or to the new minimal 12⊕12 multiplet. We give the general solution of the H-Bianchi within the 16⊕16 formulation both with and without Chern-Simons terms. This is done through the D=4 analogue of Bonora-Pasti-Tonin theorem of the 10D anomaly free supergravity. By specializing our parameters, we obtain the form of the coupling in the new minimal model retrieving in this case the results of Cecotti, Ferrara and Villasante. In addition we clarify the geometrical meaning of R-symmetry showing that in the absence of Chern-Simons forms, the condition for the embedding of the linear multiplet into the Kaehler manifold [Formula: see text] spanned by the chiral multiplets (existence on [Formula: see text] of a U(1) Killing vector) is the same condition which guarantees the existence of a local Weyl transformation by means of which the 16⊕16 curvatures can be reduced to the new minimal form and the scalar complex scalar auxiliary field S can be set to zero. Finally, we discuss the arbitrariness contained in the solution of the H-Bianchi identities at the level of the (0, 3) superspace sector. We derive the D=4 analogue of the superspace cocycle which is responsible for the Grisaru-Zanon R4-terms in the D=10 case.

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