Unsteady Trim for the Simulation of Maneuvering ... - CiteSeerX

Unsteady Trim for the Simulation of Maneuvering Rotorcraft with Comprehensive Models† Carlo L. Bottasso1,∗ , Alessandro Croce2 , Domenico Leonello2 , Luca Riviello2 1

Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA 2 Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano, Italy

key words: Flight mechanics, aeroelasticity, maneuvers, comprehensive analysis, multibody dynamics, trajectory optimization, model predictive control, rotorcraft vehicles. SUMMARY We propose a methodology for extending the applicability of comprehensive analysis rotorcraft codes to the maneuvering flight regime. Our approach can be interpreted as a generalization of the classical steady flight trim procedures, implemented in all rotorcraft codes, to the problem of time dependent trim in unsteady flight. Rotorcraft maneuvers are here mathematically described in a concise yet completely general form as optimal control problems, each maneuver being defined by a specific form of the cost function and by suitable constraints on the vehicle states and controls. In principle, by solving the maneuver optimal control problem, one could determine the flight trajectory and the control time histories that fly the vehicle model along it, while minimizing the cost and satisfying the constraints. Unfortunately, optimal control problems are prohibitively expensive to solve for detailed comprehensive models of rotorcraft denoted by a large number of structural degrees of freedom and possibly very sophisticated aerodynamics. In order to make the problem computationally tractable, our formulation makes use of two models of the same vehicle. A coarse level flight mechanics model is used for solving the trajectory optimal control problem. Being based on a reduced model of the vehicle with only a few degrees of freedom, the resulting non-linear multi-point boundary value problem is computationally feasible. Next, the fine scale comprehensive model is steered in closed loop, tracking the trajectory computed at the flight mechanics level using a receding horizon model predictive controller. This amounts to a standard time marching problem for the comprehensive model, which is therefore also computationally feasible. The flight mechanics model is iteratively updated for ensuring close matching of the trajectories flown by the two models. This two-level procedure enables the simulation using comprehensive models of arbitrary complexity of highly unsteady maneuvers of possibly long duration, with general constraints on the vehicle inputs and outputs. We demonstrate the proposed approach studying the take-off of a helicopter in the one-engine failure case under Category-A certification requirements, and an obstacle avoidance problem involving a violent pull-up/pull-down.

† Under

review in: Journal of the American Helicopter Society. to: Carlo L. Bottasso, Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Dr., Atlanta, GA 30332-0150, USA. Email: [email protected]

∗ Correspondence

UNSTEADY TRIM FOR MANEUVERING ROTORCRAFT

1

INTRODUCTION Background and Motivation Modern comprehensive rotorcraft modelling tools are geared towards the evaluation of performance, vibrations, loads, stability and response of rotorcraft systems. These solution procedures provide computer implementations of high fidelity aeroelastic mathematical models of the vehicle, which are useful for solving simulation problems relevant to all phases of the design and testing processes [34, 23, 25, 18, 4]. Great progress has been made in recent years towards the comprehensive simulation of rotorcraft, mainly due to the continuous advancement in the structural dynamics and aerodynamics computational kernels. State of the art procedures are now based on non-linear dynamics formulations, such as multibody finite element based methods, which provide the ability to model in detail the most complex part of the vehicle, the rotor system. Furthermore, modern codes implement a variety of specific rotorcraft aerodynamic models, including wake and stall models, unsteady aerodynamics and multiple lifting surface interactions. Computational fluid dynamics (CFD) tools are also attracting increasing interest. These high fidelity aeroelastic mathematical models of rotorcraft systems are currently primarily focused on the analysis of the hover and forward flight regimes. For example, detailed aeroelastic models can be used for evaluating the flutter boundaries and the vibratory levels in steady trimmed flight. Specialized procedures are available to determine the constant-in-time control inputs that trim the aircraft model, typically either in wind-tunnel or free-flight modes [28]. On the other hand, the study and simulation of maneuvering flight is typically performed only with flight mechanics models [17, 12]. These models have far fewer degrees of freedom than the aeroelastic models. In fact, the vehicle is often modelled as a rigid body and the rotor is typically described using blade element theory with wake corrections. The aeroelastic and flight mechanics models are two mathematical idealizations of the same physical system, that however differ on the scale resolution. While the high fidelity aeroelastic models are able to render fine scale details of the solution, as for example the time response of each single blade, the flight mechanics models are blind to these small scales. However, they still capture the coarser scales of the physical processes involved in the gross motion of the vehicle, and in this sense are able to synthesize its flight mechanics characteristics. Helicopters and tilt-rotors perform complex, highly dynamic and often three-dimensional maneuvers, both in normal operating conditions and during emergencies. Maximum loads and other limiting factors ranging from vibrations to noise are often encountered when operating in the unsteady regime or near the boundaries of the flight envelope, as shown for example in Ref. [15] with respect to the noise emission characteristics. More often than not, the critical quantities of interest in maneuvering flight are captured only on fine scale (aeroelastic) models, rather than coarse (flight mechanics) ones. However, current comprehensive codes have very limited capabilities in their ability to fly specific maneuvers. Therefore, there is a need to extend the applicability of comprehensive codes to the unsteady flight regime. Two main areas need to be substantially improved for enabling accurate simulation of maneuvering rotorcraft: 1. Comprehensive codes need to be coupled with time-accurate airload models that can capture, among other effects, the distortion of the wake geometry caused by the maneuver, the interaction of the shed vortex filaments with the fuselage and the tail rotor, dynamic stall effects on the retreating blade, etc. The recent work of Ribera and Celi [33] goes exactly in this direction, proposing the coupling of the time-accurate free-wake model of Bhagwat and Leishman [8] with a flexible blade comprehensive code [35]. A free-wake model applicable to maneuvering flight is also described in Ref.

2

C.L. BOTTASSO ET AL.

[36], and has been included in CHARM [18]. First principle CFD approaches would also be appropriate for capturing the relevant aerodynamic effects during maneuvers, although the computing cost of fully resolved time-accurate simulations for the hundreds or thousands of rotor revolutions that take place during a typical maneuver are still prohibitive and unrealistically large to be profitably used in the design process. 2. Comprehensive codes need to be augmented with procedures for computing the controls that fly a given maneuver. In fact, these codes are primarily intended for the computation of the rotorcraft system response resulting from known time histories of the control inputs. For example, time marching simulations of an arrested descent and of a roll reversal were described in Ref. [33], in both cases using prescribed values of the stick inputs. Similarly, Brentner at al. [15] simulated a pull-up using CAMRAD [23], again with given control inputs. Simulations based on pre-assigned inputs can produce invaluable information on the transient behavior of the vehicle, and perfectly answered the specific goals of the cited references, but it is clear that this approach is somewhat limited to fairly simple maneuvers. In fact, time dependent control inputs that fly high fidelity virtual models of rotorcraft along complex, aggressive and three-dimensional maneuvers are, in general, very difficult if not impossible to determine based on simple, trial and error procedures. The use of known, experimentally measured controls might alleviate but will not solve this problem. In fact, due to inevitable inaccuracies present in even the most sophisticated simulations, the rotorcraft model will not be capable of following the trajectory flown by the actual rotorcraft during the flight test. This fact is by now well known for the case of steady forward flight: when comparing predictions with experimental data, better correlation is obtained when the model is trimmed, i.e. when the model produces the same forces and moments as the actual rotor, than when identical control inputs are used. Since comprehensive codes are primarily design and testing tools, the lack of specific maneuver modelling capabilities represents a significant limitation of the current state of the art of rotorcraft computer assisted simulation. In the present work, we try to address this problem, exclusively with respect to the latter of the two central issues mentioned above, by proposing a general procedure for the determination of the time dependent control inputs that will fly an aeroelastic rotorcraft model along a maneuver. The proposed methodology is general and is applicable to any aeroelastic rotorcraft simulation tool, although it will be here demonstrated for the multibody finite element approach of Ref. [4]. Even though it is perfectly clear that refined airload models must be a central component of any general procedure for the analysis of maneuvering rotorcraft as discussed above, here we will simply use the inflow model of Peters et al. [29, 30], and will concentrate on the sole problem of computing unsteady controls. We argue that this approach is reasonable, since, assuming that the resulting methodology is independent on the details of the airload model, then it will be applicable to more refined schemes than the one used here. In other words, if the computational procedure is modelindependent, it is always possible to increase the modelling detail to improve the accuracy when and if necessary. Solution Procedures for Maneuvering Flight In order to design computational procedures for the problem of maneuvering flight, it is first necessary to provide a way of mathematically defining maneuvers. To address this problem, we define here a maneuver as a time history of control inputs and the resulting associated time history of vehicle states that take the vehicle model from an initial state to a final one, the latter possibly known only in part, according to some criterion and while satisfying the equations of dynamic equilibrium and all the necessary input and output constraints (limits


3

on actuator authority, flight envelope boundaries, etc.). The controls that will fly a maneuver are a-priori unknown and need to be determined. A constructive way of accomplishing this goal is through the solution of an optimal control problem [16, 21]. The optimal control problem is defined in terms of a cost function, which is typically a vehicle performance index, that specifies the criterion used for flying the maneuver (minimum power, minimum time, etc.). The equations of motion of the vehicle are regarded as constraints of the problem, which is in general also subjected to various additional input and output constraints that complete the definition of the maneuver, and, for example, translate the flight envelope limitations of the aircraft and all the necessary safety and operational requirements. The solution of this optimal control problem yields the control time histories that fly the vehicle according to the prescribed criteria, together with the complete flight path. A possible example would be the determination of a 30 degree turn in minimum time (which represents the cost function, in this case), without exceeding the operational limits of the aircraft and without exceeding a given loss of altitude (which, together with the equations of motion of the vehicle, represent the problem constraints). Another cost function that is often used in flight mechanics for the purpose of defining maneuvers is some norm of the control deflections from a trim state; this situation would correspond to a maneuver with minimum “control effort”. Other possible examples of optimization cost functions will be given in the section on the numerical applications. This approach provides a general procedure for defining arbitrary maneuvers and, at the same time, computing time histories of vehicle states which are compatible with the associated control time histories. The solution of optimal control problems is however potentially expensive, especially if detailed models of high dimensionality are used. Therefore this approach, although perfectly suited for studying maneuvers in a purely flight mechanics setting as done, for example, in Refs. [17, 12], is not directly applicable per se to high fidelity aeroelastic models of rotorcraft systems. To address this issue, we propose here a Multi-Model Steering Algorithm (MMSA). This approach blends the technology of comprehensive codes with flight mechanics models. In fact, two models of the same vehicle are used: the coarse scales are represented by a “reduced” flight mechanics model, while the fine scales are captured by an aeroelastic comprehensive model of the same aircraft. The maneuver optimal control problem is solved at the coarse flight mechanics level, and it is therefore inexpensive. The vehicle controls computed as part of the solution are now used for steering the fine scale aeroelastic model. Since at the fine level the control time histories are now known from the solution of the coarse problem, the fine level solution becomes a classical forward dynamics integration, and it is therefore also of acceptable computational cost. Iterations between the coarse and fine representations of the vehicle are used to update the flight mechanics model that is “adjusted” so as to behave as closely as possible to the aeroelastic model, therefore ensuring the convergence of the trajectories flown by the two models to a common result. The matching of the two models can be done in a variety of ways. With the goal of exploring the feasibility of the overall idea of the steering process, we implement here the simplest possible form for the matching, that is based on the identification of some of the parameters of the flight mechanics equations through an optimization problem. More refined ways of matching the two models can be based on neural networks, similarly to what done for the robust adaptive real-time control of aerial vehicles [26]. This is however still work in progress, that is not covered here and will be described in detail in a forthcoming work. Since steering in open-loop is prone to instabilities, we propose a receding horizon formulation of MMSA, that can be interpreted as an application of model-based predictive control [19]. The idea in this case is to track the trajectory generated at the flight mechanics level with the aeroelastic model. To this effect, an open-loop optimal control problem is solved

4


over a finite horizon using the coarse model. The resulting control policy is implemented in the aeroelastic model for a short period of time, until a new optimization problem is solved again on a time-shifted horizon. The repeated application of open-loop optimal control brings feed-back into the system, allowing the aeroelastic model to track the computed trajectory. Here again, the coarse reduced model is progressively adapted in order to guarantee small tracking errors. In the receding horizon formulation of MMSA, the reduced flight mechanics model plays a double role: it is used at the motion planning level for producing the reference trajectory, and it is also used at the trajectory tracking level for implementing the model predictive controller. Notice that arbitrary input and output constraints can be handled in a straightforward manner at both of these stages of the solution. This decomposition of the problem in two layers, path planning and path tracking, mimics the architecture of modern control systems [20], and enables the simulation of complex maneuvers of long duration in the proximity of the flight envelope. The new procedures are demonstrated with the help of numerical applications, including the take-off of a helicopter in the one-engine failure case under Category-A certification requirements and an obstacle avoidance through a pull-up/pull-down. Conclusions on the present work and plans for future developments are discussed in the closing section. We begin the discussion by describing the flight mechanics and aeroelastic models used for this study, which are detailed next. MANEUVERING ROTORCRAFT MODELS Flight Mechanics Models of Rotorcraft Vehicles In this work we consider the two-dimensional longitudinal flight mechanics model described in detail in Ref. [12]. The model is valid for both helicopters and tilt-rotors and, being purely longitudinal, it is clearly limited to null or low turn rates. The dynamic equilibrium conditions expressed in a fixed inertial system write 1 (1) V˙ X = (Nr FXMR + FXTR + FXA ), m 1 V˙ Z = (Nr FZMR + FZTR + FZA + mg), m 1 (Nr MYMR + MYTR + MYA ), I X˙ = VX , Z˙ = VZ , q˙ =

˙ = q, Θ

(2) (3) (4) (5) (6)

where X, Z (positive downward) are the components of the position vector of the vehicle center of gravity, VX and VZ their time rates, Θ (positive nose up) is the pitch angle and q the pitch rate, while m and I are the aircraft mass and pitch moment of inertia, respectively, and g is the acceleration of gravity. FXMR , FZMR , MYMR are the components of the forces and moments generated by each of the Nr main rotors, i.e. rotors that generate thrust in the longitudinal plane of the aircraft. Similarly, FXTR , FZTR , MYTR are the force and moment components of the tail rotor. For a helicopter Nr = 1, while for a tilt-rotor Nr = 2 and FXTR = FZTR = MYTR = 0. Finally, FXA , FZA and MYA are the forces and moment generated by all other aerodynamic surfaces of the vehicle.


5

For helicopters it is necessary to include the effects of the tail rotor thrust to balance the main rotor torque, in order to accurately evaluate the total power required for flight. To this end, the helicopter model is enriched by three (approximate) algebraic equations expressed in the body attached frame (x, y, z) that enforce roll, yaw and lateral equilibrium of the vehicle: FyMR + FyTR + mg sin Φ = 0, MzMR + MzTR + MzV = 0,

(7) (8)

MxMR + MxTR + MxV = 0,

(9)

where Φ is the aircraft bank angle. The vehicle equations of equilibrium (1–6,7–9) are augmented by a power balance equation, that writes µ ¶ QMR QTR ηMR 1 P − − rt , (10) ω˙ = Jp Nr ω ηMR ηTR where ω is the main rotor angular velocity and Jp its polar moment of inertia, P (t) is the power produced by the engine(s) at time t. QMR , QTR are the torques due to the generic main and tail rotors, respectively, and ηMR , ηTR the mechanical efficiency of their transmissions; finally, rt is the ratio between tail and main rotor rotational speeds. Clearly, for a tilt-rotor, we set QTR = 0. The rotor forces and torques are expressed in terms of the piloting controls using classical blade element theory [32, 24]. The detailed expressions of these quantities are here omitted for brevity, but can be found in Ref. [12]. These aerodynamic constitutive equations complement the equations of equilibrium and the power balance equations, and allow one to write the vehicle equations in compact form as y˙ − f (y, u, t) = 0,

(11)

where y ∈ Rns,FM is the set of flight mechanics state variables, u ∈ Rnu,FM are the flight mechanics controls, and f : Rns,FM × Rnu,FM × R+ → Rns,FM . The vehicle state vector is defined as y = (X, Z, Θ, VX , VZ , q, ω), ns,FM = 7, while the controls are represented for a helicopter by u = (θ0MR , θ0TR , A1 , B1 , P ), nu,FM = 5. For a tilt-rotor we have the additional controls δH for the horizontal stabilizer and im for the nacelle tilt, but no tail rotor collective θ0TR , so that nu,FM = 6 in this case. Multibody Modelling of Rotorcraft Vehicles In this work, the fine scale aeroelastic modelling of rotorcraft is based on the comprehensive multibody dynamics analysis code described in Ref. [4]. The formulation is cast within the framework of non-linear finite element based multibody dynamics methods, and the element library includes rigid and deformable bodies, joint elements, including unilateral contact conditions, active element models, including engine and actuator models, and sensors and controls [5, 7, 10, 11]. The finite element formulation of flexible structural elements includes both non-linear models and modal-based approaches. The formulations of beams and shells are composite-ready, i.e. they support the modelling of complex cross sections made of laminated composite materials, and are geometrically exact, i.e. they account for arbitrarily large displacements and finite rotations, but are limited to small strains [5]. Structural modes can be imported from external general finite element codes and can be connected with the rest of the multibody model [6]. The equations of equilibrium are written in a Cartesian inertial frame. Constraints are modelled using the Lagrange multiplier technique, which leads to systems of equations that are highly sparse, although not of minimal size. The crucial advantage of this approach is

6


that it can treat arbitrarily complex aircraft configurations. After spatial discretization of the flexible components using the finite element method, the equations of dynamic equilibrium can be written as d(M w) − f − c,q λ − fe = 0, dt N q˙ − w = 0,

(13)

c = 0,

(14)

(12)

where q ∈ Rnq are the generalized coordinates, w ∈ Rnq the velocities, p = M (q)w : Rnq × Rnq → Rnq the system momenta, f (q, w) : Rnq × Rnq → Rnq the discretized internal and inertial forces, fe (q, w, u, t) : Rnq × Rnq × Rnq × R+ → Rnq the external forces, that also include the effects of the system controls u ∈ Rnu , c(q, t) : Rnq × R+ → Rnc are the holonomic constraints that model the mechanical joints of the system, and finally λ ∈ R nc the associated Lagrange multipliers. The generalized coordinates are composed of linear displacements d ∈ Rnl and rotation parameters r ∈ Rnr , q = (d; r), nl + nr = nq . The notation ? = (♣; ♠) represents the stacking of the column vectors ♣ and ♠ in the column vector ?. Similarly, the generalized velocities are w = (v; ω), where v ∈ Rnl are the linear velocities and ω ∈ Rnr the angular velocities. Then matrix N is defined as N=

·

I 0

0 S

¸

,

(15)

where S is a parameterization dependent matrix that for each node in the system relates the time rates of the rotation parameters r to the angular velocities ω. The case of non-holonomic constraints is not covered here for the sake of brevity, but can be easily addressed. The code includes several solution procedures, including static analyses under steady external, aerodynamic and inertial loads, dynamic response from given initial conditions and prescribed loads and control inputs u(t), stability and flutter, and trim analyses. The numerical integration in time of the equations of motion (16,17) is based on an energy decaying scheme that ensures unconditional numerical stability in the non-linear regime, a numerical property that gives superior robustness to the procedures [11]. The software implements models for computing aerodynamic contributions to f e , both through built-in features and through interfaces to external codes. Airloads can be computed by simple lifting line models based on two-dimensional wing theory and table-look-up procedures, and include classical corrections for sweep, unsteady motion and stall. Wake effects are based on the dynamic inflow model of Peters [29, 30]. Prescribed airloads, as obtained from experimental measurements, can also be used. Furthermore, interfaces to external airloads computation modules are provided. The finite element based multibody dynamics formulation implemented in the code provides a general and flexible paradigm for the modelling of maneuvering helicopters. In particular, the modular nature of the code allows for the development of hierarchies of models providing increasing levels of detail, as required by the analysis, and has the flexibility to describe novel configurations of arbitrary topology. This formulation can accommodate variable rotor speeds, non-periodic responses and large elastic motions. Furthermore, it allows for the modelling of fuselage/rotor/tail rotor interactions in conventional rotorcraft configurations, and rotor/wing/fuselage dynamics in tilt-rotor configurations, through the coupling of fully non-linear rotor models with linearized, modal-based fuselage models.

7


Notation and Relationships Between the Models To ease the notation, it is convenient to rewrite the multibody equations (12–14) in a more compact fashion as d e ) = 0, (B ye) − b(ye, λ, u dt c(e y ) = 0, where B=

·

M 0

0 N

¸

,

b = (f + c,q λ + fe ; w).

(16) (17) (18)

The multibody state vector is now defined as ye = (w; q) ∈ Rns,MB = R2nq , and the multibody e ∈ Rnu,MB , while we recall that the flight mechanics states are noted y ∈ Rns,FM controls are u and the flight mechanics controls are indicated as u ∈ Rnu,FM . Therefore, here and in the f to distinguish quantities of the multibody model from following we will use the symbol (·) analogous quantities of the flight mechanics model. For future reference, it is important to note the following facts related to states and controls of the two models. First of all, it is clear that there are in general many more multibody than flight mechanics states, i.e. Rns,MB À Rns,FM . From the multibody states ye it is always possible to compute some quantities yeFM ∈ Rns,FM , that have the same physical meaning of the flight mechanics states y (recall, representing the rigid body generalized positions, generalized velocities and the rotor angular velocity). The actual form of this mapping from the multibody to the flight mechanics models will depend on the specific details of the former, but here it will suffice to formally indicate this operation as yeFM = S(ye). (19)

For example, if the fuselage is modelled as a rigid body in the multibody model, then the operator S(·) will be a simple boolean identification of the fuselage rigid body degrees of freedom within the state vector ye. On the other hand, if a more refined flexible fuselage model is used, then the same operator will compute some form of average position, orientation and velocity of the vehicle. Later on we will use the derived states yeFM to verify whether or not the multibody and flight mechanics models fly the same maneuver, i.e. to check whether yeFM ≈ y. Regarding the controls, it should be noted that their number in the two models will typically be the same, i.e. Rnu,MB = Rnu,FM . Nonetheless, the controls in the two models might have a different physical meaning. For example, the main rotor collective θ0MR that appears among e to the the flight mechanics controls u might correspond in the multibody control vector u linear translation of an actuator connected to the swash-plate. Here again, it is not possible to specify this mapping further without knowing the specific details of the models. It will suffice, however, to simply formally write this mapping between the controls as e = C(u). u

(20)

This operator will be useful later on for applying the controls computed at the flight mechanics level to the aeroelastic multibody model.

COMPUTATION OF ROTORCRAFT TRAJECTORIES USING FLIGHT MECHANICS MODELS In this work, we base the definition of a rotorcraft maneuver on the formulation of an optimal control problem, as discussed in the introduction. Numerical solution procedures for computing

8


rotorcraft trajectory optimization problems are described in detail in Ref. [12], and are more concisely reviewed in the present section. Formulation of the Optimal Control Problem for Rotorcraft Trajectories The flight mechanics equations of dynamic equilibrium and accompanying kinematic equations for a rotorcraft were discussed previously, and write y˙ − f (y, u, p, t) = 0.

(21)

Note that, compared to the form of the same equations that we gave in (11), we have added here the dependence of the equations on some parameters p ∈ Rnp . These parameters represent the vehicle inertial properties and the aerodynamic rotor coefficients that enter into the definition of the flight mechanics equations. In standard flight mechanics applications these parameters are carefully computed and tuned by the analyst in order to provide close matching of the model behavior with experimental data. Later on, we will use these parameters as free degrees of freedom, that will be optimized by MMSA in order to guarantee close matching between the trajectories flown by the flight mechanics and aeroelastic models. The rotorcraft maneuver is defined over the temporal domain Ω = (T0 , T ) ⊂ R with boundary Γ = {T0 , T }, where the final time T is often unknown. The problem of optimal control is to determine the controls u, the states y and possibly the final time T that minimize a cost function Z T ¯ L(y, u, t) dt, (22) J = φ(y, u, t)¯ + T

T0

subject to the state equations (21) and to various possible additional constraints, as required by the problem at hand. The exact nature of the constraint conditions depends on the maneuver. For example, they might specify initial and/or final conditions, or might provide operational and flight enveloped limits. Collectively, a specific form of the cost function (22) and of the accompanying constraints effectively defines a certain maneuver. In general, the constraints can be classified as boundary conditions ψ(y(T0 )) ∈ [ψ0min , ψ0max ],

(23)

ψ(y(T )) ∈ [ψTmin , ψTmax ],

(24)

g(y, u, t) ∈ [gmin , gmax ],

(25)

constraints on states and controls

integral conditions on states and controls Z h(y, u, t) dt ∈ [hmin , hmax ],

(26)

T

and upper and lower bounds y ∈ [ymin , ymax ], u ∈ [umin , umax ].

(27) (28)

For generality, all these conditions are expressed as inequality constraints in the form x ∈ [xmin , xmax ], i.e. xmin ≤ x ≤ xmax . Equality constraints are enforced by simply selecting xmin = xmax . In certain cases, the total range Ω can be broken into p sequential phases (sub-domains), T0 < T1 < . . . < Tp−1 < Tp ≡ T , p ≥ 1. The generic phase i is defined on the interval


9

Ωi = [Ti , Ti+1 ], i = 0, . . . , p − 1, while Tj , j = 1, . . . , p − 1, is the generic internal event. In each phase, different system governing equations and/or constraints might apply. Furthermore, the final time and the event locations might be unknown. An example involving a multi-phase problem will be discussed in the section on numerical applications. For clarity, we consider here a single phase problem, and drop the phase-dependent notation. However, all the following discussion is easily extended to the more general multi-phase case. Transformation of the Optimal Control Problem into a Parameter Optimization Problem Via the Finite Element Method The governing equations of optimal control could be obtained by first augmenting the cost function J with the equations of motion (21) and the constraints (23–26) through the use of Lagrange multipliers, and then imposing the stationarity of the augmented cost [16]. The resulting two-point boundary value problem could be solved with a suitable discretization method. This is however nor necessary nor desirable. In fact, practical methods for optimal control are currently based on a discretization process that renders the problem finite-dimensional, followed by a parameter optimization problem. Therefore, instead of deriving first the equations of optimal control and then discretizing them with a suitable numerical method, the process is in effect reversed: first one discretizes the equations of equilibrium, the constraints and the cost function, so that the problem from infinite dimensional becomes finite dimensional; next, one optimizes the discrete problem. This approach, called the direct method, is simpler than the one based on the derivation of the optimal control equations, since these never need to be computed. Furthermore, this approach also presents a number of numerical advantages which generally make it more robust than other strategies [3, 2]. The temporal discretization of the problem can be obtained with initial value (shooting and multiple shooting) or boundary value techniques. In this work we use the latter approach, since it can be used even if the system to be controlled is inherently unstable. This is important in the present context, since rotorcraft vehicles are in fact usually unstable. Using a boundary value approach, elements (time steps) are assembled in order to cover the whole temporal domain. For this purpose, here we use the Discontinuous Petrov-Galerkin (DPG) finite element method [14]. This formulation enjoys desirable numerical properties, being of maximal order, algebraically stable and symplectic for Hamiltonian problems [9]. The lowest order member of this family of methods is the familiar second-order (implicit symmetric) mid-point rule. To define the discrete equations, we let Th be a grid of Ω, K denoting a generic element. More precisely, we consider a partition T0 ≡ t0 < t1 < . . . < tn−1 < tn ≡ T composed of n ≥ 1 intervals T i = [ti , ti+1 ] of size hi , i = 0, . . . , n − 1. Since T itself can be unknown in general, it is convenient to introduce a mapping of time onto a fixed domain parameter s, i.e. s : (T0 , T ) 7→ (0, 1); for example, we can choose s = t/(T − T0 ), 0 ≤ s ≤ 1, so that the generic step length is now hi = (T − T0 )(si+1 − si ), i = 0, . . . , n − 1. Using the finite element method, the infinite dimensional solution fields y(t) and u(t) are approximated with functions yh and uh chosen within suitable finite dimensional spaces. The same functions restricted to the generic element K are noted yh |K and uh |K . The functions yh and uh can be expressed in terms of finite element shape functions and of discrete (nodal) finite element degrees of freedoms yd and ud . In the following, having chosen a member of the finite element family and hence having chosen the shape functions, we will regard y h and uh as sole functions of the nodal values, so that the functional dependencies y h = yh (yd ), uh = uh (ud ) are understood. By applying the DPG finite element method, the system governing equations (21) are

10


transformed into a set of residual equations over each grid element, i.e. ξh (yh |K , uh |K , p, T ) = 0,

∀K ∈ Th ,

(29)

equations that can be collectively written as ξh (yh , uh , p, T ) = 0 on Th .

(30)

The functional dependence of these equations on T reflects the fact that the time step size h K is unknown for an unknown final time problem, as previously recalled. Through this process, the system dynamic equations (21) are transcribed into constraints of the parameter optimization problem. All other problem constraints and bounds, equations (23–26), (27,28), are expressed in terms of the same finite dimensional functions y h and uh and are appended to equation (29) as additional linear, non-linear or bound constraints, as appropriate. All these constraint conditions can be collectively written as ϕh (yh , uh , p, T ) ∈ [ϕmin , ϕmax ] on Th .

(31)

Finally, the cost function J defined in (22) is expressed in terms of the discrete problem unknowns, yielding the discrete objective Jh = Jh (yh , uh , T ). This procedure defines a finitedimensional non-linear programming (NLP) problem [21]: Z T min Jh , Jh = φ(yh , uh , t)|T + L(yh , uh , t) dt, (32) yd ,ud ,T T0 s.t.: ϕh (yh , uh , p, T ) ∈ [ϕmin , ϕmax ]. Using this approach, the optimality conditions of the discrete NLP problem converge to the optimality conditions of the optimal control problem as the grid is refined (h → 0) and the number of discrete optimization variables goes to infinity (n → ∞) [22, 2]. Additional Implementation Issues A refinement procedure is used for “boot-strapping” the solution, alleviating the need for accurate initial guesses. At first, a rough initial guess is associated to a crude grid, and the corresponding NLP problem is solved. The computed solution is then projected onto a finer grid, and used as initial guess for the subsequent NLP problem. The procedure is continued until sufficient grid refinement has been achieved to yield converged results. This procedure is used because on coarse grids fine scale details of the solution are not captured; this will usually imply a faster convergence of the NLP problem, especially if the initial guess is poor, i.e. the tentative solution is far from the converged one. If a fine grid is used starting from a poor initial guess, the fine details captured by the grid will tend to slow down or even prevent convergence. An additional robustness issue is related to the scaling of unknowns in the numerical optimization procedures. These are in fact notoriously sensitive to badly scaled problems. To address this issue, we rewrite the governing equations (21) as ¯˙ − f¯(y, ¯ u, ¯ p, ¯ t) = 0, y

(33)

¯ u ¯ and p¯ are scaled states, controls and parameters, respectively, that are defined as where y, y¯ = S y y, ¯ = S u u, u p¯ = S u p,

(34) (35) (36)


11

and where −1 −1 −1 f¯(·, ·, ·, t) = S y f (S y ·, S u ·, S p ·, t).

(37)

diag(Siy ),

i = 1, . . . , ns,FM , is a (diagonal) matrix of In the previous relations, S y = weights that scale the state variables with respect to one another. Similarly, S u = diag(Siu ), i = 1, . . . , nu,FM is the analogous scaling matrix for the controls, and S p = diag(Sip ), i = 1, . . . , np the scaling matrix for the parameters. The scaling coefficients are chosen so as to obtain states and controls that are all approximatively of order O(1). Without proper corrective actions, the control time histories computed through the optimal control problem often tend to show a somewhat rough (e.g., bang-bang) behavior, jumping from one saturation bound to the other. This implies infinite or very high, and therefore unrealistic, actuation speed and actuation power. This is due to the fact that the flight mechanics models are quasi-steady in the controls, i.e. the controls u are purely algebraic variables that lack proper dynamics. This lack of modelling detail is desirable in many flight mechanics applications, and is justifiable on the grounds of a time scale separation argument. At the same time, the procedures are now blind to the intrinsic limitations of real actuators, such as for example limited control velocities, limited actuation power, etc. In order to ensure smooth computed control time histories, techniques for incorporating approximate knowledge on the actuator dynamics can be used, as proposed in Ref. [12]. The most straightforward way of obtaining smooth controls is by using control velocities as part of the optimization constraints and objective function. This derived field can be obtained through a Galerkin projection [12].

THE MULTI-MODEL STEERING ALGORITHM The multibody and flight mechanics models of the vehicle described in the previous section are now combined into a single algorithm, whose final goal is to compute the controls that steer the multibody model according to a user-specified criterion. The two models interact according to the following iterative scheme: 1. At first, the discrete maneuver optimal control problem (32) is solved using the flight mechanics model of the aircraft. The solution is not expensive to compute, since a coarse model with relatively few degrees of freedom is used. This problem yields the control time histories uh that fly the maneuver, and the associated time history of flight mechanics states yh . 2. The computed controls are now used for conducting a forward dynamics simulation of the multibody model of the same aircraft, equations (16,17), which can yield information on solution features that are beyond the level of detail of flight mechanics models. This phase of the solution procedure is a simple forward dynamics simulation that once again does not imply prohibitively large computational costs even for high fidelity models of the aircraft. 3. Since the two models differ in the level of detail, the trajectory flown by the finer scale (multibody) model will, in general, differ from that computed for the coarse scale (flight mechanics) model. To remedy this deficiency, the coarse scale model parameters p, equation (21), are now modified through a parameter identification process, so that the difference between the two trajectories can be minimized. Therefore, goal of this step in the procedure is to determine a coarse model with global flight mechanics characteristics that closely match those of the detailed multibody model. Since this parameter identification problem is a small scale optimization whose unknowns are

12


some of the flight mechanics parameters of the coarse model, this procedure is also inexpensive. It should be noted that this step could be accomplished in a variety of ways. More in general, this step is concerned with extracting a “reduced” model from the aeroelastic one that has as few degrees of freedom as possible and yet captures the coarser scales of the solution, i.e. the flight mechanics characteristics of the vehicle. 4. The updated flight mechanics model is now used for a second solution of the optimal control problem, which in turn yields updated control settings and trajectory. Iterations between fine and coarse models through parameter identification is continued until a desired level of accuracy is obtained. To make things more precise, let us consider the algorithm of Fig. 1. The flow of the algorithm is also symbolically depicted in Fig. 2. The flight mechanics (FM, for short, in Fig. 1) model is denoted by the states y and the controls u. The multibody (MB) model is denoted by the e . Discrete values of these quantities on their corresponding grids states ye and the controls u are indicated with the subscript (·)h . The iterative algorithm tries to make the time histories of the flight mechanics states y throughout the maneuver approximately equal to the corresponding states yeFM derived from the multibody model in equation (19), when the two models are actuated by the same controls. Hence, goal of the algorithm is to make the error Z T k yh − yeFM,h k dt (38) ε = T0 Z T k yh k dt T0

as small as possible. Here k · k is some dimensionally consistent norm, or simply the 2-norm if one uses the non-dimensionalization expressed by equation (33). At first (line 1 of the algorithm), an iteration counter k is initialized to zero. Next, the main loop starts and the iteration counter is incremented. At the first iteration, the statements labelled from 4 to 5 are skipped; we will come back to them later on. At line 7, the trajectory optimization problem is solved using the flight mechanics model on the grid T h . This problem is reported here for convenience, and writes Z T min Jh , Jh = φ(yh , uh , t)|T + L(yh , uh , t) dt, (39) yd ,ud ,T T0 s.t.: ϕh (yh , uh , p, T ) ∈ [ϕmin , ϕmax ]. At the first iteration, the previously described boot-strapping procedure is used, while for k > 1 the initial guess is provided by the solution for k−1. The solution to the trajectory optimization problem yields the state and control degrees of freedom yd and ud , and possibly T if the problem has a free final time. These controls, which were computed from the flight mechanics model solving an optimal control problem, will now be used for steering the multibody model, which amounts to solving an initial value problem. This is accomplished in two steps. eh , First, one needs to map the flight mechanics controls uh into the multibody controls u since the two sets of controls might correspond to different degrees of freedom in the two models. This is achieved at line 8 of the algorithm using equation (20). Second, it should be noted that the flight mechanics controls uh are computed on the grid Th . This grid might in general be different from the grid used by the forward dynamics multibody solver, that we indicate here as Teh . More precisely, Teh is a grid of Ω, with T0 ≡ t˜0 < t˜1 < . . . < t˜m−1 < t˜m ≡ T , and composed of m ≥ 1 intervals Tei = [t˜i , t˜i+1 ] of size


13

˜ i , i = 0, . . . , m − 1. Since the multibody solver has to resolve much finer solution scales, it h ˜ will be much smaller than the sizes h. Furthermore, note is expected that the element sizes h that T is now known, since it was computed through the trajectory optimization problem. e h , which depend on uh through equation (20), will now be projected The multibody controls u onto the multibody grid Teh (line 9 of the algorithm). We formally indicate this operation as e h |Teh = P(e uh |Th ), u

(40)

eh ), yeh = MB(ye(T0 ), u

(41)

where P(·) is an interpolation operator that maps quantities computed on the grid T h onto the grid Teh . The exact form of this operator depends on the numerical methods used at the two levels. The multibody solver can now proceed with the integration of the equations of motion e 0 ) and steered by the controls u e h . This is indicated at (16,17), starting from initial values y(T line 10 of the algorithm with the notation where yeh is the resulting time history of the multibody states on the grid Teh . Note that in reality, since the multibody solver might be using adaptive time stepping for efficiency, the grid projection (line 9) is carried out simultaneously with the time stepping (line 10), rather than in a sequential fashion as indicated. From the state time history yeh one can now evaluate yeFM,h using equation (19), in order to compare these quantities with the corresponding values computed on the flight mechanics model, yh . This is accomplished at line 11 of the algorithm. The flight mechanics states derived from the multibody model, yeFM , are then projected from the Teh (multibody) to the Th (flight mechanics) grid. This is obtained with the inverse grid mapping operator, P −1 , at line 12 of the algorithm. The mismatch between the two trajectories is now computed on T h using equation (38) at line 13 of the algorithm. The resulting error ε measures how closely the two models behave in terms of gross rigid body motion along the maneuver. If the error ε is greater than a given tolerance, the next iteration is initiated. This time, before solving the optimal control problem at line 7 of the algorithm, we solve a parameter identification problem (line 5). The idea is to adjust the parameters p appearing in the flight mechanics model in order to reduce the error in the flown trajectories yh and yeFM,h . This new problem is formulated as follows: Z T p p k yh − yeFM,h k dt, min Jh , Jh = (42) yd ,p T0 s.t.: ξh (yh , uh , p, T ) = 0. Here yeFM,h , uh and T are known quantities, held fixed at their previous values, while the unknowns are represented by the parameters p and the state degrees of freedom y d . The solution to this problem yields a new estimate of the parameters entering the definition of the flight mechanics equations, so that the trajectory mismatch with respect to the previous iteration is minimized. Based on the new parameters, that define a new flight mechanics model, the trajectory optimization problem is now solved a second time (line 7 of the algorithm). The procedure is continued until convergence between the two models, or until a prescribed maximum value kmax of iterations is reached. The effect of the iterations on the trajectories flown by the two models is symbolically depicted in Fig. 3. The use of an optimization-based procedure in this work is mainly justified by the fact that, from a software stand point, its implementation is straightforward. In fact, minor modifications to the trajectory optimization software easily handle even the parameter identification case, by

14


simply including among the problem unknowns also some of the flight mechanics parameters. This approach was here taken with the specific goal of allowing a first assessment of the idea of steering an aeroelastic model using trajectory optimization performed on a reduced model. At the same time, it is quite clear that the parameter identification approach suffers from some limitations. First of all, the procedure relies heavily on the appropriate choice of the flight mechanics parameters that are chosen for the identification, a choice that should be made with care and insight. But even so, it is also clear that the degrees of freedom that are available for matching the two models, the identified parameters, are very few; furthermore, the functional dependency of the flight mechanics equations on the parameters can not be changed, while only their value will be affected by the identification process. For all these reasons, it should be clear that very small solution tolerances in terms of ε should not be expected, since the two models can not in general be made exactly equal. More sophisticated procedures for computing reduced models can be inspired by the work on adaptive control of uncertain systems, which relies on the ability of neural networks of approximating a smooth function with an arbitrary tolerance. We hope to be able to report on these ideas in a near future.

THE MULTI-MODEL STEERING ALGORITHM FOR UNSTABLE SYSTEMS The formulation of MMSA described in the previous section can be used for steering an aeroelastic model of a stable vehicle and only for short durations. However, helicopters and tilt-rotors in helicopter mode are unstable aircrafts. Hence, even if the flight mechanics and aeroelastic models were identical, the aeroelastic forward-in-time solution (line 10 of Fig. 1) would quickly diverge from the flight mechanics one. To understand what happens in the case of unstable vehicles, we should recall that we use a boundary value approach for solving the optimal control problem at the flight mechanics level. This means that, since we do not march forward in time as one would do using, for example, a multiple shooting method, instabilities of the vehicle can not manifest themselves during this phase of the solution. On the other hand, the aeroelastic problem is here formulated using an initial value approach, and it is solved with a time marching process. Hence, since the vehicle is steered in open-loop without any feed-back, even minor perturbations to the aeroelastic model will ensure the manifestation of the vehicle instability during the forwardin-time solution phase. Sources of perturbations will be, among others, any differences between the two models, approximations due to the finite precision arithmetic of computer operations, finite tolerances used for the solution of non-linear problems at both levels, and interpolations of the controls between the flight mechanics and the aeroelastic grids. In order to introduce feed-back in the steering process, we formulate here a Receding Horizon (RH) version of MMSA. The basic idea is to try to “track” with the aeroelastic model the time history of vehicle states computed at the flight mechanics level. This should be contrasted with the previous approach, where the flight mechanics controls where used for the open-loop steering of the aeroelastic model. The tracking problem is defined over a short interval (window) of time, whose duration is smaller than the characteristic time scales of the instability. The control policy is computed by solving an optimal control problem, starting from the state reached by the vehicle at the end of the interval. The use of the reached state as the initial condition for computing a new set of controls, brings feed-back into the steering process, stabilizing the vehicle. As the simulation proceeds, the window shifts forward in time, as depicted in Fig. 4. At the end of the simulation, we iterate between the two levels using the previously discussed parameter identification procedure in order to ensure convergence between the flown trajectories.


15

This approach is known in the controls literature as model predictive or receding horizon control [27, 19]. As a matter of fact, we can interpret here the aeroelastic model as the “real” system to be controlled, the reduced flight mechanics model as the plant non-linear model, while the trajectory generated by solving the global optimal control problem is the tracking output for the receding horizon controller. The outer iteration that updates the reduced model can be interpreted as an adaptive step that corrects the plant model. The idea of using a receding horizon approach is particularly attractive here, since it allows for the feed-back stabilization of the system without the need to develop additional computational tools and without the need to derive linearized models of the vehicle. In fact, we just need to slightly rearrange the sequence of operations to account for the shifting window, but the basic computational kernels remain the flight mechanics optimal control solution and the aeroelastic forward-in-time procedure. Furthermore, a model predictive controller can handle in a straightforward manner actuator limitations and other input and output constraints, which are of critical importance here and that would be difficult to account for with other approaches. A detailed description of the algorithm is given in Fig. 5. The algorithm is composed of two main loops. The external loop (lines 2–24) controls the parameter identification problem (line 5). At the first iteration, statement 5 is skipped, and, at line 9, the optimal control problem is solved using the flight mechanics model on the grid Th (T0 , T ). This optimization problem writes Z T

min Jh ,

yd∗ ,u∗ d ,T

s.t.:

Jh = φ(yh∗ , u∗h , t)|T +

T0

L(yh∗ , u∗h , t)dt,

ϕh (yh∗ , u∗h , p, T ) ∈ [ϕmin , ϕmax ], yh∗ (T0 ) = yT0 ,

(43)

where, with respect to the analogous problem of Fig. 1, we have introduced the notation y h∗ and u∗h to denote, respectively, the “to-be-tracked” states and associated controls. The algorithm now proceeds with an internal loop (lines 10-21) that introduces feed-back in the steering process. At first (lines 8-9) an initial time TI and a final time TW for the tracking problem are initialized, and, as the algorithm proceeds, are progressively incremented (lines 19-20). These values define the time interval for the tracking problem of line 11, which writes: Z TW t t M (yh , yh∗ , uh ) dt, min Jh , Jh = yd ,ud TI (44) s.t.: ϕh (yh , uh , p) ∈ [ϕmin , ϕmax ], yh (TI ) = yTI . Note that the initial conditions on the states, yh (TI ) = yTI , can either be the initial conditions at T0 when TI = T0 , or the final computed multibody states at TF in all other cases (line 18). Furthermore, M (yh , yh∗ , uh ) is the integrand term of the tracking cost function J t . There is some freedom in the definition of this term; a solution that works well in practice is to use a term that penalizes the norm of the distance between the two trajectories, k y h − yh∗ k, plus a second term that penalizes the control effort or the control velocities. Specific examples will be given later on in the section on applications. The algorithm now proceeds similarly to the stable-system case. The solution to the tracking problem yields the state and control degrees of freedom yd and ud . At this point, the controls uh = uh (ud ) are used for steering the multibody model in the interval [TI , TF ]. To this effect, the multibody controls are computed from the flight mechanics ones (line 13) and projected onto the multibody grid (line 14). The multibody solver can now proceed with the integration of the equations of motion in the interval [TI , TF ], starting from initial values yeTI , and steered e h . The values yeTI are either the initial conditions at T0 , or the states reached by the controls u at the end of the the previous integration (line 17).

16


Lines 22–23 of the external loop control the convergence of the parameter identification problem. Once a multibody solution yeh becomes available, it is compared with the flight mechanics solution yh in terms of the derived states yeFM,h , computed at line 16. This comparison is accomplished by first projecting these values from the Teh (TI , TF ) to the Th (TI , TF ) grid (line 22), and then computing the mismatch ε between the two trajectories on Th (TI , TF ) (line 23). If the error ε is greater than a given tolerance, a new iteration in the outer loop is initiated and the counter k is incremented. This time, before solving the trajectory optimization problem at line 7 of the algorithm, we solve a parameter identification problem (line 5), whose solution yields new parameters p that minimize the trajectory mismatch with respect to the previous iteration. The trajectory optimization problem can now be solved a second time (line 7 of the algorithm), and the whole procedure is continued until convergence.

NUMERICAL APPLICATIONS Continued Helicopter Take-Off under Category-A Certification Requirements We consider the take-off of a multi-engine helicopter under Category-A certification requirements, as described in Ref. [1]. The maneuver starts with a climb to a take-off decision point (TDP). In case of an engine failure during the climb, the vehicle must be able to safely land within a given rejected take-off (RTO) distance, touching the ground without exceeding given values of the horizontal and vertical velocities. The vehicle must be able to safely survive even an engine failure occurring after the TDP; this situation is termed a continued take-off (CTO). The TDP for a rotorcraft is therefore a concept somewhat similar to the V 1 of a fixed wing airplane, and, in the CTO case, the rotorcraft must continue the acceleration, clearing the ground with a given safety margin, eventually reaching a take-off safety speed V TOSS , a height of at least 11 meters above the take-off point and a positive rate of climb. Figure 6 illustrates this maneuver. These requirements are met by first accelerating the vehicle forward with a dive, immediately after the engine loss. At the end of the dive, a pull-up is used to re-establish a positive rate of climb. This is made possible by the fact that the power required diminishes rapidly with forward speed in the proximity of the hover condition. Once a positive rate of climb has been achieved, part of the remaining available power can be spent to accelerate the main rotor back to its nominal speed, as required by the norms for safety reasons. A detailed treatment of both the RTO and the CTO cases is discussed in Ref. [12], using flight mechanics vehicle models and an optimal control formulation of maneuvering flight. Here, we use MMSA for analyzing a CTO maneuver with an aeroelastic model. The coarse flight mechanics model is identical to the one used in Ref. [12], while the fine aeroelastic model is a multibody representation of the same vehicle, as schematically depicted in Fig. 7. A higher level of modelling detail is limited to the sole rotor system, while the fuselage is rendered here as a single rigid body even in the aeroelastic representation of the vehicle. Since the flight mechanics model is two-dimensional, the fuselage of the aeroelastic model is constrained to slide on a vertical plane, as shown in the figure. The effect of the tail rotor on the vehicle power balance is modelled only at the level of the flight mechanics model, as expressed by equations (7–9) and (10), but it is accounted for also at the aeroelastic level. This is achieved by integrating in time the rotor angular velocity ω as obtained from the flight mechanics model, in order to compute the shaft rotation ψ of the multibody model at each instant of time. The shaft rotation is achieved by prescribing the relative rotation ψ at the revolute joint labelled A in the figure. The rotor blades are modelled


17

using geometrically exact non-linear beam elements, while the flap, lag and pitch hinges are represented by revolute joints. For simplicity, in this example we do not model the control linkages. Given the flight mechanics and multibody models, the mapping between their respective e = C(u), can be readily established. The multibody controls are defined as controls, u e = (θ1 , θ2 , θ3 , θ4 , ψ), i.e. the pitch of each blade at their root and the rotor azimuthal angle. u The relationships between these aeroelastic model controls and the flight mechanics controls u = (θ0MR , θ0TR , A1 , B1 , P ) are as follows: ³ ³ π´ π´ − B1 sin ψ + (i − 1) , i = 1, . . . , 4, (45) θi (ψ) = θ0MR − A1 cos ψ + (i − 1) 2 2 Z t ω(u) dt. (46) ψ (t) = T0

Notice that there is no control in the multibody model that directly corresponds to the tail rotor collective θ0TR since, as previously discussed, the tail rotor has not been included in the fine model. Similarly, there is no control that directly corresponds to the power control setting P . The effects of the tail thrust, and hence of the power absorbed by the tail rotor, are accounted for by equation (10) and define the shaft rotation ψ through equation (46). The aerodynamic modelling at the aeroelastic level is quite crude and it is based on twodimensional strip theory with the inflow correction of Peters [30]. Lifting lines are associated with the rotor blades, and the lift, drag and moment characteristics of the blade profiles are given in tabular form as functions of the local angle of attack and Mach number. The formulation of the lifting lines accounts for the effects of sweep, twist, offset of the aerodynamic center, unsteadiness, radial drag and tip losses. Lifting lines are also used for modelling the aerodynamic characteristics of the horizontal tail. For the purposes of our analysis, we assume that the power failure takes place at t = T 0 = 0 s, and we do not consider here any pilot reaction time. Therefore, the optimal maneuver is computed starting from a hovering trim condition. The maximum power available from the engines is given by the expression Pmax (t) = P1 + (P2 − P1 )e−t/τ1 + P2 e−t/τ2 .

(47)

The value P1 = 1750 hp is the maximum one-engine power available in emergency conditions, P2 = 1119.4 hp is the one-engine power required in hover, and τ1 = 0.667 s, τ2 = 0.333 s are time constants for the engine transients. The power available from the engines, P (t), is constrained as P (t) ∈ [0, Pmax (t)] for t ∈ [0, T ]. The CTO problem can be formulated as a multi-phase optimal control problem on the domain Ω = (T0 , T ) with an unknown internal event at time T1 , T1 ∈ [T0 , T ], and unknown final time T . Following Ref. [12], we write the cost function as Z T³ ´ 1 J = Z(T1 ) + (T − T1 ) + w (48) B˙ 12 + θ˙02MR dt. T − T 0 T0 The first cost term enforces the minimum altitude loss problem and has an impact on both phases. The second cost term is introduced to give a strategy for the conduction of the maneuver between T1 and T . Finally, the third and last term penalizes the temporal rates of the longitudinal cyclic and of the collective pitch, w being a tunable weighting factor. This last term has the purpose of computing smoother control time histories, and the rationale for its introduction is discussed in detail in Ref. [13]. A value of w = 100 is used for this problem. The problem is subjected to the following boundary and internal conditions: 1. Hover initial conditions at T0 : X(0) = Z(0) = 0 m, Θ(0) = −0.1803 deg, VX (0) = VZ (0) = 0 m/s, q(0) = 0 deg s−1 and ω(0) = 207 rpm.

18


2. Minimum altitude condition at T1 : VZ (T1 ) = 0 m/s. 3. Exit conditions at T : altitude Z(T ) = 0 m, horizontal and vertical speed V X (T ) = VTOSS = 20 m/s, VZ (T ) = −1 m/s, pitch rate q(T ) = 0 ◦ s−1 , and final rotor angular velocity ω(T ) = 207 rpm. 4. Maximum power available from the engines as given by equation (47). The duration of the two phases of the problem are constrained as T1 ∈ [1, 8] s, T −T1 ∈ [1, 15] s. For facilitating the exit condition, we impose for t ∈ [T1 , T ] the following bounds on the horizontal and vertical velocities: VX ∈ [15, 25] m/s, VZ ∈ [−1, 0] m/s. Finally, throughout the maneuver we impose bounds on the angular velocity ω ∈ [187, 207] rpm, on the controls θ0MR ∈ [−2 ◦ , 15 ◦ ], θ0TR ∈ [−5 ◦ , 25 ◦ ], A1 ∈ [−8 ◦ , 8 ◦ ], B1 ∈ [−14 ◦ , 14 ◦ ], and on the control rates θ˙0MR , θ˙0TR , B˙ 1 , A˙ 1 ∈ [−20 ◦ s−1 , 20 ◦ s−1 ]. The optimal control solution over the interval [T0 , T ] was computed by first using a flight mechanics grid Th of 20 equal elements. In fact, on very coarse grids the convergence of the optimization problem is typically very fast, even when a poor initial guess is used. The 20element solution was then projected onto a finer grid of 40 elements, and used as initial guess for the subsequent NLP problem. The procedure was continued up to a final grid of 80 elements. For this problem, we found by trial and error that the slope of the cL -α curve of each blade and the moment of inertia Jq of the vehicle where the identification parameters p with the largest effect on the convergence between the two models. The cost function of the tracking problem, equation (44), was defined as Jt = w

Z

TW TI

¡

¢ (X − X ∗ )2 + (Z − Z ∗ )2 + (Θ − Θ∗ )2 dt+ Z

TW

TI

¡

¢ (VX − VX∗ )2 + (VZ − VZ∗ )2 + (q − q ∗ )2 dt+ Z

TW

TI

³

B˙ 12 + θ˙02MR

´

dt,

(49)

with w = 100, and where the starred quantities denote the to-be-tracked states, while the last integral term penalizes large control rates. Bounds on states, controls and control rates were also enforced during the tracking problem, which is one of the key features of this predictive control scheme. The time step length between successive activations of the RH controller, ∆T C (see algorithm of Fig. 5), was initially set equal to 1 s, which is a quite high value, barely smaller than the characteristic time of the short period mode of the vehicle. In Fig. 8 we compare the pitch attitudes computed with this value of ∆TC (Fig. 8, top) with those computed with a smaller one, ∆TC = 0.2 s (Fig. 8, bottom). The lines marked with the 4 symbols correspond to the flight mechanics level solution, while the solid lines correspond to the aeroelastic level one. In both cases a value of ∆Twindow = 2 s was used for the tracking problem window. As expected, with ∆TC = 1 s the controlled system can not effectively track pitch and pitch rate, which are clearly the most critical states being associated with the unstable mode of the vehicle. On the other hand, a value of ∆TC = 0.2 s guarantees a good overall matching between the two models. Two parameter identification iterations are required to ensure convergence between the models. Figures 9, 10, 11 and 12 show, respectively, the converged trajectories, pitch angles, pitch rates and vehicle airspeed. Here again, 4 lines indicate flight mechanic level solutions, and solid lines aeroelastic level results. For all these quantities, good overall matching between the two models can be observed, even for pitch and pitch rate. As expected, the maneuver is flown by first diving in order to exchange potential and kinetic energy. The dive is then


19

followed by a pull-up to clear the ground and gain a positive rate of climb, while restoring a safe rotor speed. The feed-back control nature of the RH formulation of MMSA can be appreciated by observing Fig. 13. The solid and dash-dotted lines correspond to the trajectories (top) and to the pitch attitude (bottom) of the aeroelastic models with and without, respectively, RH control. The 4 lines, already shown in Figs. 9 and 10, correspond to the trajectory and to the pitch attitude of the flight mechanics model. As expected, without any feed-back control, the steered vehicle quickly diverges from the reference trajectory. Optimal Helicopter Obstacle Avoidance Maneuver We consider now an All Engines Operative (AEO) maneuver for a helicopter flying in proximity of the ground in a hostile environment. The vehicle is in straight level flight at 30 m/s and must avoid an obstacle of 30 m of height, going back to its original low altitude flight condition in minimum time, in order to minimize its exposure to, for example, enemy fire. The overall maneuver is therefore composed of a violent pull-up followed by a similarly violent pull-down, as shown in Fig. 14. The problem can be formulated as a multi-phase optimal control problem on the domain Ω = (T0 , T ), with unknown final time T . The unknown internal event T1 , T1 ∈ [T0 , T ], corresponds to the instant where the vehicle passes over the obstacle. The cost function for this problem can be written as Z T³ ´ 1 (50) B˙ 12 + θ˙02MR dt. J =T +w T − T 0 T0 The first term enforces the minimum time condition, while the second term, as in the previous example, penalizes high cyclic and collective rates. The weight w is the tunable factor, as described in Ref. [13], which for this problem is assumed to be w = 100. The cost function for the tracking problem is formulated as in the previous example, as given by equation (49). The initial conditions at time T0 for the optimization problem are given by the trim conditions at 30 m/s. Throughout the maneuver we impose bounds on the angular velocity ω ∈ [207, 207] rpm, on the controls θ0MR ∈ [−5 ◦ , 20 ◦ ], θ0TR ∈ [−10 ◦ , 30 ◦ ], A1 ∈ [−12 ◦ , 12 ◦ ], B1 ∈ [−20 ◦ , 20 ◦ ], and on the control rates θ˙0MR , θ˙0TR , B˙ 1 , A˙ 1 ∈ [−16 ◦ s−1 , 16 ◦ s−1 ]. Furthermore, the maximum available power is limited as Pmax = 2500 hp, and the power rate is limited according to P˙ (t) ≤ 500 hp s−1 . The solution was computed on a grid of 80 time elements, generated by successive uniform refinement of an initial 20 element grid. Figures 15–17 show, respectively, the trajectories, the pitch angles and the vehicle airspeed, before (top) and at the end of a cycle of three parameter identifications (bottom). For this problem the identification parameters p were selected as the slope of the c L -α curve of the blades and the cD0 value of the main rotor. Here again, solid lines marked with the 4 symbol indicate flight mechanic level solutions, and dash-dotted lines aeroelastic level results. For all these quantities, good overall matching between the two models can be observed for the converged values of the parameter identification procedure. Figure 18 shows a zoom of the first 5 seconds of the time history of the vehicle pitch angle as computed at line 7 (4 line), 11 (solid lines) and 15 (dash-dotted line) of the algorithm of Fig. 5. The plot was produced for the initial values of the parameters, i.e. before the identification, and shows the tracking solutions that connect the aeroelastic level time history with the reduced model one. Finally, Fig. 19 shows the time history of engine power computed for the flight mechanics problem. The feed-back control nature of the RH formulation of MMSA can be appreciated by observing Fig. 20. The solid and dash-dotted lines correspond to the trajectories (top) and to

20


the pitch attitude (bottom) of the aeroelastic models with and without, respectively, the RH controller. The 4 lines, already shown in Figs. 15 and 16, correspond to the trajectory and to the pitch attitude of the flight mechanics model. As expected, without any feed-back control, the steered vehicle quickly diverges from the flight mechanics model. Finally, we compute a similar obstacle avoidance maneuver with a larger initial and final airspeed of 50 m/s. Figure 21 shows for this new problem the same quantities that were given in Fig. 20. Again the RH version of MMSA produces a better correspondence between the trajectories flown by the two models. However, with respect to the previous case, a closer match is observed also without the RH feed-back. This is due to the higher vehicle airspeed of this problem: the presence of the horizontal tail, in fact, increases the pitch stability of the vehicles so that the RH controller does not play such an important role as in the previous case.

CONCLUSIONS It is well known that steady flight analysis of rotorcraft requires good trim procedures in order to produce accurate results. In this work, we have argued that similar time dependent trim processes are required for maneuvering flight simulation. We have here proposed a methodology that meets this need by blending aeroelasticity, flight mechanics, trajectory optimization and optimal control, and that provides a general and flexible paradigm. The algorithms here described expand the applicability of comprehensive aeroelastic codes towards the unsteady flight regimes, moving beyond the sole steady flight case. The ability to study maneuvering flight could be of some importance in certain applications, especially for high-performance aggressive vehicles, since maximum loads, vibratory levels, noise and other critical design parameters are often encountered in this flight regime. The methodology was demonstrated on specific flight mechanics and aeroelastic models, but it is general and could be applied to models other than the ones here described. The work conducted so far seems to be promising. The preliminary examples that were presented indicate that the multi-model procedures are indeed capable of computing control time histories that fly an aeroelastic model along unsteady maneuvers. In particular, the use of a coarse flight mechanics model allows one to solve a trajectory optimization problem without incurring in overwhelming computational costs. Furthermore, the receding horizon formulation of the algorithm provides a relatively straightforward way of dealing with the inherently unstable nature of rotorcraft vehicles, while accounting for output and input constraints. Improvements to this methodology could and should be pursued in several areas. From the modelling point of view, more sophisticated unsteady aerodynamic models should be used in maneuvering flight for accounting for the distortion processes undergone by the wake. Here again a hierarchical approach of increasing modelling complexity could be used, for example employing the reduced order models of dynamic wake distortion of Prasad et al. [31] at the flight mechanics level, while the time-accurate free-wake models of Bhagwat and Leishman [8] could be used at the aeroelastic level. Furthermore, numerical experience has shown that care and insight into the models are necessary when solving the parameter identification problem. The choice of which parameters should be identified and which should be kept fixed to their nominal values in order to get the best matching between the fine and coarse models, can not probably be totally automated. More experience with the use of these procedures on different maneuver modelling problems could provide some general guidelines in the near future. It is also clear that closure between the two levels could be obtained by other means. In particular, we hope to be able to report in the near future on improved neural network-based adaptive procedures for the determination of the reduced flight mechanics model that eliminate the need to perform the parameter identification altogether, with potential gains in simplicity,


21

robustness and generality.

ACKNOWLEDGEMENT

Alessandro Croce, Domenico Leonello and Luca Riviello acknowledge the hospitality of the D. Guggenheim School of Aerospace Engineering of the Georgia Institute of Technology while working on the research described in this paper.

REFERENCES 1. Advisory Circular 29-2C, Certification of Transport Category Rotorcraft, Federal Aviation Administration, Department of Transportation, 1999. 2. Betts, J.T., Practical Methods for Optimal Control Using Non-Linear Programming, SIAM, Philadelphia, 2001. 3. Betts, J.T., “Survey of Numerical Methods for Trajectory Optimization,” Journal of Guidance, Control and Dynamics, 21, 1998, pp. 193–207. 4. Bauchau, O.A., Bottasso, C.L., and Nikishkov, Y.G., “Modeling Rotorcraft Dynamics with Finite Element Multibody Procedures,” Mathematics and Computer Modeling, 33, 2001, pp. 1113–1137. 5. Bauchau, O.A., Bottasso, C.L., and Trainelli, L., “Robust Integration Schemes for Flexible Multibody Systems,” Computer Methods in Applied Mechanics and Engineering, 192, 2003, pp. 395–420. 6. Bauchau, O.A., and Rodriguez, J., “Coupled Rotor-Fuselage Analysis with Finite Motions Using Component Mode Synthesis,” Journal of the American Helicopter Society, 2003, under review. 7. Bauchau, O.A., Rodriguez, J., and Bottasso, C.L., “Modeling of Unilateral Contact Conditions with Application to Aerospace Systems Involving Backlash, Freeplay and Friction,” Mechanics Research Communications, 28, 2001, pp. 571–599. 8. Bhagwat, M.J., and Leishman, J.G., “Stability, Consistency and Convergence of Time Marching FreeVortex Rotor Wake Algorithms,” Journal of the American Helicopter Society, 46, 2001, pp. 59–71. 9. Bottasso, C.L., “A New Look at Finite Elements in Time: A Variational Interpretation of Runge-Kutta Methods,” Applied Numerical Mathematics, 25, 1997, pp. 355–368. 10. Bottasso, C.L., and Bauchau, C.L., “Multibody Modeling of Engage and Disengage Operations of Helicopter Rotors,” Journal of the American Helicopter Society, 46, 2001, pp. 290–300. 11. Bottasso, C.L., and Bauchau, O.A., “On the Design of Energy Preserving and Decaying Schemes for Flexible, Nonlinear Multibody Systems,” Computer Methods in Applied Mechanics and Engineering, 169, 1999, pp. 61–79. 12. Bottasso, C.L, Croce, A., Leonello, D., and Riviello, L., “Optimization of Critical Trajectories for Rotorcraft Vehicles,” Journal of the American Helicopter Society, 2003, under review. 13. Bottasso, C.L, Croce, A., Leonello, D., and Riviello, L., “Rotorcraft Trajectory Optimization with Realizability Considerations,” Journal of Aircraft, 2004, under review. 14. Bottasso, C.L., Micheletti, S., and Sacco, R., “The Discontinuous Petrov-Galerkin Method for Elliptic Problems,” Computer Methods in Applied Mechanics and Engineering, 191, 2002, pp. 3391–3409. 15. Brentner, K.S., Perez, G., Bres, G.A., and Jones, H.E., “Toward a Better Understanding of Maneuvering Rotorcraft Noise,” 58th American Helicopter Society Annual Forum, Montreal, Canada, June 11–13, 2002. 16. Bryson, A.E., and Ho, Y.C., Applied Optimal Control, Wiley, New York, 1975. 17. Carlson, E.B., and Zhao, Y.J., “Optimal Short Takeoff of Tiltrotor Aircraft in One Engine Failure,” Journal of Aircraft, 39, 2001, pp. 280–289. 18. “CHARM (Comprehensive Hierarchical Aeromechanics Rotorcraft Model),” http://www.continuum-dynamics.com/products/charm/index.html, Continuum Dynamics, Inc., NJ. 19. Findeisen, R., Imland, L., Allg¨ ower, F., and Foss, B.A., “State and Output Feedback Nonlinear Model Predictive Control: An Overview,” European Journal of Control, 9, 2003, pp. 190–206. 20. Frazzoli, E., Dahleh, M.A., and Feron, E., “A Hybrid Control Architecture for Aggressive Maneuvering of Autonomous Aerial Vehicles,” in System Theory: Modeling, Analysis and Control, Djaferis, T.E., and Schick, I.C. (eds.), SECS 518, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. 21. Gill, P.E., Murray, W., and Wright, M.H., Practical Optimization, Academic Press, London and New York, 1981. 22. Hull, D.G., “Conversion of Optimal Control Problems into Parameter Optimization Problems,” Journal of Guidance, Control and Dynamics, 20, 1997, pp. 57–60. 23. Johnson, W., “CAMRAD II, Comprehensive Analytical Model of Rotorcraft Aerodynamics and Dynamics,” Jonhson Aeronautics, Palo Alto, CA, 1992–97. 24. Johnson, W., Helicopter Theory, Dover Publications, New York, 1994. 25. Johnson, W., “Technology Drivers in the Development of CAMRAD II,” American Helicopter Society Aeromechanics Specialists’ Conference, San Francisco, CA, January 1994.

22


26. Kim, B.S., and Calise, A.J.,“Nonlinear Flight Control Using Neural Networks,” Journal of Guidance, Control, and Dynamics, 20, 1997, pp. 26–33. 27. Kouvaritakis, B., and Cannon, M. (eds.), Nonlinear Predictive Control: Theory and Practice, Institution of Electrical Engineers, UK, 2001. 28. Peters, D.A., and Barwey, D., “A General Theory of Rotorcraft Trim,” Mathematical Problems in Engineering, 2, 1996, pp. 1–34. 29. Peters, D.A., Karunamoorthy, S., and Cao, W.M., “Finite State Induced Flow Models. Part I: TwoDimensional Thin Airfoil,” Journal of Aircraft, 32, 1995, pp. 313–322. 30. Peters, D.A., and He, C.J., “Finite State Induced Flow Models. Part II: Three-Dimensional Rotor Disk,” Journal of Aircraft, 32, 1995, pp. 323–333. 31. Prasad, J.V.R., Zhao, J., and Peters, D.A., “Helicopter Rotor Wake Distorsion Models for Maneuvering Flight,” 28th European Rotorcraft Forum, Bristol, UK, September 17–20, 2002. 32. Prouty, R.W., Helicopter Performance, Stability, and Control, R.E. Krieger Publishing Co., Malabar, 1990. 33. Ribera, M., and Celi, R., “Simulation Modeling of Unsteady Maneuvers Using a Time Accurate Free Wake,” 60th Annual Forum of the American Helicopter Society, Baltimore, MD, June 7–10, 2004. 34. Rutkowski, M., Ruzicka, G.C., Ormiston, R.A., Saberi, H., and Jung, Y., “Comprehensive Aeromechanics Analysis of Complex Rotorcraft Using 2GCHAS,” Journal of the American Helicopter Society, 40, 1995, pp. 3–17. 35. Theodore, C.R., and Celi, R., “Helicopter Flight Dynamic Simulation with Refined Aerodynamic and Flexible Blade Modeling,” Journal of Aircraft, 39, 2002, pp. 577–586. 36. Wachspress, D.A., Quackenbush, T.R., and Boschitsch, A.H., “First-Principles Free-Vortex Wake Analysis for Helicopters and Tiltrotors,” 59th Annual Forum of the American Helicopter Society, Phoenix, AZ, May 2003.


1. 2. 3. 4.

5. 6.

7. 8. 9. 10. 11. 12. 13.

k=0 do k =k+1 if (k > 1) Solve parameter identification problem  Z T  p p k yh − yeFM,h k dt min Jh , Jh = on Th yd ,p T0  s.t.: ξh (yh , uh , p, T ) = 0 end if Solve trajectory optimization problem  Z T  min Jh , Jh = φ(yh , uh , t)|T + L(yh , uh , t) dt on Th y ,u ,T T0  d d s.t.: ϕh (yh , uh , p, T ) ∈ [ϕmin , ϕmax ] e h = C(uh ) u Map FM controls into MB ones e h |Teh = P(e u uh | T h ) Project controls onto MB grid e h ) on Teh yeh = MB(ye(T0 ), u Solve initial value MB problem yeFM,h = S(yeh ) Compute FM states from MB ones yeFM,h |Th = P −1 (e yFM,h |Teh ) Project FM states onto FM grid Z T k yh − yeFM,h k dt T0 ε= on Th Evaluate error Z T k yh k dt T0

14.

while ((ε ≥ tol) and (k ≤ kmax )) Figure 1. The multi-model steering algorithm.

23

24


Flight mechanics level:

Identification

Steering

Aeroelastic level:

Figure 2. The multi-model steering algorithm.

25


Iteration 1

Flight mechanics level:

Iteration 2 Iteration i Converged solution

MMSA

Iteration 1 Aeroelastic level: Iteration 2 Iteration i Converged solution Figure 3. The multi-model steering algorithm: effect of iterations between coarse and fine levels.

26


Dtwindow Tracked trajectory

Predictive solution

DtC Figure 4. Receding horizon formulation of the multi-model steering algorithm.


1. 2. 3. 4.

5. 6.

27

k=0 do k =k+1 if (k > 1) Solve parameter identification problem (yields new parameters p)  Z T  p p min Jh , Jh = k yh − yeFM,h k dt on Th (T0 , T ) yd ,p T0  s.t.: ξh (yh , uh , p, T ) = 0 end if Solve trajectory optimization problem for t ∈ [T0 , T ] (yields new trajectory ∗ , flown with new estimate of the parameters, p) to be tracked, yh

7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23.

   

min Jh , ∗ ∗

yd ,ud ,T

  

Jh = φ(yh∗ , u∗h , t)|T +

Z

T T0

L(yh∗ , u∗h , t) dt

TI = T0 TW = TI + ∆Twindow do

yd ,ud

s.t.:

Th (T0 , T )

Initial time for tracking and steering Terminal time for tracking problem

Solve trajectory tracking problem for t ∈ [TI , TW ]  Z TW  t t  M (yh , yh∗ , uh ) dt  min Jh , Jh =

  

on

ϕh (yh∗ , u∗h , p, T ) ∈ [ϕmin , ϕmax ] yh∗ (T0 ) = yT0

s.t.:

TI

ϕh (yh , uh , p) ∈ [ϕmin , ϕmax ] yh (TI ) = yTI TF = TI + ∆TC e h = C(uh ) u e h |Teh (TI ,TF ) = P(e uh |Th (TI , TF )) u e h ) on Teh (TI , TF ) yeh = MB(yeTI , u

yeFM,h = S(yeh ) e F) yeTI = y(T yTI = yeFM (TF ) T I = TF TW = TI + ∆Twindow while (TW < T ) yeFM,h |Th (T0 ,TW ) = P −1 (e yFM,h |Teh (T0 ,TW ) ) Z TW k yh − yeFM,h k dt T0 on Th (T0 , TW ) ε= Z TW k yh k dt

(yields new control policy, uh )

on

Th (TI , TW )

Terminal time for steering problem Map FM controls into MB ones Project controls onto MB grid Solve initial value MB steering problem for t ∈ [TI , TF ]with initial conditions yeTI

Compute FM states from MB ones New initial conditions for MB states New initial conditions for FM states

Reset initial time Update tracking problem terminal time Project FM states onto FM grid

Evaluate error

T0

24.

while ((ε ≥ tol) and (k ≤ kmax )) Figure 5. Receding horizon multi-model steering algorithm.

28


TDP

VTOSS Z(T1)

Figure 6. Helicopter continued take-off procedure.


29

Rigid body Beam Revolute joint

Blade

Flap, lag, and pitch hinges B: qi

Hub Shaft

Horizontal lifting surface

Fuselage

A: y

Planar joint constraining motion to a vertical plane

Figure 7. Schematic representation of the helicopter multibody model. The effect of the tail rotor on the power balance equation is taken into account at the level of the flight mechanics model. Dimensions not to scale. One single blade shown, for clarity.

30


5

0

Pitch Attitude [deg]

−5

−10

−15

−20

−25

−30

0

2

4

6

8

10

12

14

16

Time [sec]

5

0


−5

−10

−15

−20

−25

−30

0

2

4

6

8

10

12

14

16

Time [sec]

Figure 8. Critical Take-Off. Pitch angles for the flight mechanics (4 lines) and aeroelastic (solid lines) models using the RH formulation of MMSA with two different values of ∆TC : 1 s (top), 0.2 s (bottom).


31

0

Yposition [m]

−2

−4

−6

−8

−10

0

50

100

150

200

250

Xposition [m]

Figure 9. Critical Take-Off. Trajectory flown by the flight mechanics (4 lines) and aeroelastic (solid lines) models at the last iteration of RH-MMSA.

32


5

0


−5

−10

−15

−20

−25

−30

0

2

4

6

8

10

12

14

16

Time [sec]

Figure 10. Critical Take-Off. Pitch angle for the flight mechanics (4 lines) and aeroelastic (solid lines) models at the last iteration of RH-MMSA.


33

15

10

Pitch Rate [deg/s]

5

0

−5

−10

−15

−20

−25

0

2

4

6

8

10

12

14

16

Time [sec]

Figure 11. Critical Take-Off. Pitch rate for the flight mechanics (4 lines) and aeroelastic (solid lines) models at the last iteration of RH-MMSA.

34


22 20 18

Vehicle Airspeed [m/s]

16 14 12 10 8 6 4 2 0

0

2

4

6

8

10

12

14

16

Time [sec]

Figure 12. Critical Take-Off. Airspeed for the flight mechanics (4 lines) and aeroelastic (solid lines) models at the last iteration of RH-MMSA.


35

0

−40

Y

position

[m]

−20

−60

−80

−100

0

50

100

150

200

Xposition [m]

250

5

0


−5

−10

−15

−20

−25

−30

0

2

4

6

8

Time [sec]

10

12

14

16

Figure 13. Critical Take-Off. Trajectories (top) and pitch angles (bottom) for the flight mechanics model (4 lines), the aeroelastic model with RH-MMSA (solid lines) and the aeroelastic model with MMSA (dash-dotted lines).

36


Figure 14. Helicopter obstacle avoidance problem.

37


40

Iteration 1

35

30

Yposition [m]

25

20

15

10

5

0

0

50

100

150

200

250

300

Xposition [m] 40

Iteration 4

35

30

Yposition [m]

25

20

15

10

5

0

0

50

100

150

200

250

300

Xposition [m]

Figure 15. Helicopter obstacle avoidance problem. Trajectories flown by the flight mechanics (4 lines) and aeroelastic (solid lines) models using RH-MMSA and three parameter identification iterations.

38


30

Iteration 1


20

10

0

−10

−20

−30 0

1

2

3

4

5

6

7

8

9

6

7

8

9

Time [sec]

30

Iteration 4


20

10

0

−10

−20

−30 0

1

2

3

4

5

Time [sec]

Figure 16. Helicopter obstacle avoidance problem. Pitch angles for the flight mechanics (4 lines) and the aeroelastic (solid lines) models using RH-MMSA and three parameter identification iterations.


39

35

Iteration 1


30

25

20

15

0

1

2

3

4

5

6

7

8

9

6

7

8

9

Time [sec]

35

Iteration 4


30

25

20

15

0

1

2

3

4

5

Time [sec]

Figure 17. Helicopter obstacle avoidance problem. Vehicle airspeed for the flight mechanics (4 lines) and the aeroelastic (solid lines) models using RH-MMSA and three parameter identification iterations.

40


15


10

5

0

−5

−10

−15 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time [sec]

Figure 18. Helicopter obstacle avoidance problem. Effects of the tracking problem on the time history of the pitch angles. 4 lines: flight mechanics solution; dash-dotted line: aeroelastic solution; solid thin line: flight mechanics solution of tracking problem.


41

2600

2400

2200

Power [hp]

2000

1800

1600

1400

1200

1000 0

1

2

3

4

5

6

7

8

9

Time [sec]

Figure 19. Helicopter obstacle avoidance problem. Power vs. time for the flight mechanics model.

42


40

35

30

Yposition [m]

25

20

15

10

5

0

0

50

100

150

200

250

300

Xposition [m] 50

40


30

20

10

0

−10

−20

−30

−40

0

1

2

3

4

5

6

7

8

9

Time [sec]

Figure 20. Helicopter obstacle avoidance problem. Trajectories (top) and pitch angles (bottom) for the flight mechanics model (4 lines), the aeroelastic model with RH-MMSA (solid lines) and the aeroelastic model with open-loop MMSA (dash-dotted lines).


43

60

40

Yposition [m]

20

0

−20

−40

−60

0

100

200

300

400

500

600

700

800

900

1000

Xposition [m] 20 15 10


5 0 −5

−10 −15 −20 −25 −30 0

2

4

6

8

10

12

14

16

18

Time [sec]

Figure 21. Helicopter obstacle avoidance problem. Trajectories (top) and pitch angles (bottom) for the flight mechanics model (4 lines), the aeroelastic model with RH-MMSA (solid lines) and the aeroelastic model with MMSA (dash-dotted lines) at high speed.

Unsteady Trim for the Simulation of Maneuvering ... - CiteSeerX

Unsteady Trim for the Simulation of Maneuvering ... - CiteSeerX

Suggest Documents

numerical simulation of unsteady cavitating flows - CiteSeerX

Simulation of Unsteady Gas2Particel2Flows including

simulation of helicopter noise in maneuvering flight

Simulation of the Maneuvering Behavior of Ships in ...

Numerical Simulation of Unsteady Flow and Aerodynamic ... - CiteSeerX

A numerical simulation of unsteady flow in a two ... - CiteSeerX

Simulation of three-dimensional unsteady flow in ... - CiteSeerX

simulation modeling of unsteady maneuvers using a time ... - CiteSeerX

Simulation of Ship Maneuvering Behavior Based on the Modular ...

Generative Graphical Models for Maneuvering Object ... - CiteSeerX

Missile-Borne SAR Raw Signal Simulation for Maneuvering Target

Numerical simulation of unsteady cavitation in liquid

Simulation of unsteady incompressible flows using temporal ...

Numerical Simulation for the Unsteady MHD Flow and Heat ... - PLOS

Numerical Simulation of Gate Control for Unsteady Irrigation ... - MDPI

Coordinated Maneuvering of Automated Vehicles in ... - CiteSeerX

TRAUMA RISK MANAGEMENT (TRiM) - CiteSeerX

Design and Maneuvering Simulation of an Autonomously ... - TU Berlin

two-dimensional simulation of the unsteady flow ...

numerical simulation of the 2d unsteady flow around a ...

Experimental study and unsteady simulation of the Flindt draft tube ...

numerical simulation of the unsteady aerodynamics in an axial counter

unsteady flow simulation in hydraulic machinery

CFD Simulation of a Sailing Boat Hull in Maneuvering ...