An acceleration measurements-based approach for helicopter nonlinear flight control using Incremental Nonlinear Dynamic Inversion

Control Engineering Practice 21 (2013) 1065–1077

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An acceleration measurements-based approach for helicopter nonlinear flight control using Incremental Nonlinear Dynamic Inversion P. Simplício n, M.D. Pavel, E. van Kampen, Q.P. Chu Delft University of Technology, Faculty of Aerospace Engineering, Kluyverweg 1, 2629 HS Delft, The Netherlands

art ic l e i nf o

a b s t r a c t

Article history: Received 18 December 2011 Accepted 12 March 2013 Available online 12 May 2013

Due to the inherent instabilities and nonlinearities of rotorcraft dynamics, its changing properties during flight and the engineering difficulties to predict its aerodynamics with high levels of fidelity, helicopter flight control requires the application of special strategies. These strategies must allow to cope with the nonlinearities of the system and assure robustness in the presence of inaccuracies and changes in configuration. In this paper, a novel approach based on an Incremental Nonlinear Dynamic Inversion is applied to simplify the design of helicopter flight controllers. With this strategy, by employing the feedback of acceleration measurements to avoid the need for information relative to any aerodynamic change, the control system does not need any model data that depends exclusively on its states, thus enhancing its robustness to model uncertainties. The overall control system is tested by simulating two tasks with distinct agility levels as described in the ADS-33 helicopter handling qualities standard. The analysis shows that the controller provides an efficient tracking of the commanded references. Furthermore, with the robustness properties verified within the range of inaccuracies expected to be found in reality, this novel method seems to be eligible for a potential practical implementation to helicopter vehicles. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Helicopter Flight control Nonlinear control Incremental Nonlinear Dynamic Inversion Pseudo-Control Hedging

1. Introduction Helicopters are generally reliable flying machines, capable of fulfilling missions impossible with fixed-wing aircraft, most notably rescue operations. These missions, however, often lead to high and sometimes excessive pilot workload. The excessive pilot workload for helicopters indicates that, even modern helicopters, often have poor Handling Qualities (HQs) (Padfield, 1998). This is mainly due to the fact that helicopters are highly nonlinear and complex dynamic systems, inherently unstable by nature, with strong coupled inter-axis behavior which makes piloting a very demanding job. Therefore, to assure safety and effectiveness in helicopter operation, these vehicles are enhanced with feedback control systems which can go from simple mechanical stabilization devices to Automatic Flight Control Systems (AFCSs) (Prouty & Curtiss, 2003; Stiles, Mayo, Freisner, Landis, & Kothmann, 2004). Precise control and carefree HQs in future helicopter designs may only be achieved with control laws that balance the

n

Corresponding author. Tel.: +351 968207255. E-mail addresses: [email protected] (P. Simplício), [email protected] (M.D. Pavel), [email protected] (E. van Kampen), [email protected] (Q.P. Chu). 0967-0661/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2013.03.009

conflicting requirements of stability and maneuverability. This means that helicopter flight control requires strategies that allow to cope with the nonlinearities of the system while providing robustness in the presence of inaccuracies due to changes in configuration and to the inability to characterize its aerodynamics with high levels of fidelity (Pavel, 2001). As the latter uncertainties are generally substantial or unknown, an adaptive control architecture may be required. This is, in fact, the most common strategy of the past few years (Hovakimyan, Kim, Calise, Prasad, & Corban, 2001; Lee, Ha, & Kim, 2005; Leitner, Calise, & Prasad, 1998; Moelans, 2008): a Nonlinear Dynamic Inversion (NDI) (also referred to as Feedback Linearization technique) of an approximate model (linearized at a pre-specified trim condition) together with adaptive elements to compensate for the inversion error. In general, further developments consider the same type of architecture, but introduce some improvements in the structure of the dynamic inversion (Johnson & Kannan, 2005) or in the adaptive laws (Zeng & Zhu, 2006). Adaptive control systems are however limited in terms of practical applicability, not only due to their complex high-order architectures, but also due to flight certification issues because (1) it is difficult to prove that the controller will never “learn” incorrectly, causing harm to the vehicle, and (2) it is also hard to

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Nomenclature rotor coning angle w.r.t. the disc plane (rad) rotor longitudinal disc-tilt w.r.t. the disc plane (positive for a backward tilt) (rad) b1 rotor lateral disc-tilt w.r.t. the disc plane (positive for a tilt to the right) (rad) CD blade drag force coefficient (–) C Lα lift curve slope (rad−1) C Mfus , C Nfus fuselage moment coefficients (–) CT, CH, CS, CQ rotor forces and torque moment coefficients (–) eβ flapping hinge offset (m) f ¼ ½f x f y f z T force vector (N) f, h generic nonlinear vector fields F0 fuselage equivalent drag area (m2) Ftr fin blockage factor (-) g Earth's gravitational acceleration (m/s2) g, G control effectiveness vector field/matrix h, l coordinates of the helicopter main components (m) J helicopter inertia matrix (kg m2) K; K 1 ; K 2 ; K 3 generic diagonal transfer matrices Kfus correction coefficient for the moments of the fuselage (–) L lift force (N) Le, Me eccentricity moments (N m) m total mass of the helicopter (kg) mbl blade mass (kg) m ¼ ½mx my mz T moment vector (N m) p ¼ ½x y zT position relative to the North-East-Down system (m) Q rotor torque moment (N m) R rotor radius (m) Rfus fuselage resultant drag force (N) S surface area (m2) T, H, S rotor thrust, longitudinal and lateral drag forces (N) T transformation matrix from the body-fixed to the North-East-Down coordinate system (–) u generic system input vector u′ ¼ ½θ1s θ1c θ0tr T control inputs for the rate controller (rad) v ¼ ½u v wT body-fixed linear velocity vector (m/s) vE ¼ ½V x V y V z T velocity relative to the North-East-Down system (m/s) V airspeed (m/s) V fusM , V fusN equivalent volumes of the fuselage (m3) x generic state vector y generic system output vector z−1 unit delay operator α, β angle of attack and sideslip angle (rad) γ lock number (–) γs rotor shaft tilt angle (rad) a0 a1

prove that it is able to recover from a failure in adaptation (Johnson & Calise, 2000). In order to overcome these shortcomings, this paper derives the application of a novel technique known as Incremental Nonlinear Dynamic Inversion (INDI) to helicopter flight control. The INDI (also referred to as modified, simplified or sensorbased NDI) has been recently adopted for fixed-wing aircraft flight control (Bacon, Ostroff, & Joshi, 2001; Chen & Zhang, 2008; Sieberling, Chu, & Mulder, 2010). By computing incremental commands instead of the total control inputs and employing acceleration feedback to extract the information relative to aerodynamic changes, the controller does not need any model data that

Δ generic uncertain term θ ¼ ½ϕ θ ψT attitude angles of the helicopter relative to the North-East-Down system (rad) θ0 collective pitch of the main rotor (positive for collective lever up) (rad) θ1s longitudinal cyclic pitch of the main rotor (positive for stick forward) (rad) θ1c lateral cyclic pitch of the main rotor (positive for stick to the right) (rad) θ0tr collective pitch of the tail rotor (positive for pedal to the right) (rad) θtw linear blade twist (rad) λ ¼ ½λ0 λ0tr T normalized induced inflow of both rotors (–) μ ¼ ½μx μy μz T normalized airspeed w.r.t. the rotor plane (–) ν; νh ; νrm generic virtual control input, pseudo-control hedge and feedforward term νβ normalized flapping frequency (–) ρ air density (kg/m3) s rotor solidity/Standard deviation of the relative aerodynamic uncertainty (–) τ generic time constant (s) ω ¼ ½p q rT body-fixed angular velocity vector (rad/s) Ω rotor rotational speed (rad/s) Ω transformation matrix from ω to θ_ (–) Subscripts 0 att b com dp fus ht mr n nav r rm rot tr vt

previous instant of time/built-in parameter attitude loop body-fixed reference frame commanded reference in the disc plane fuselage horizontal tail main rotor nominal conditions navigational loop W.r.t. the rotor plane reference model rotational loop tail rotor vertical tail

Superscripts _ ^ Gl

first-order time derivative normalized value estimated value Glauert theory

depends exclusively on the states of the system. Being significantly less depending on the model, the INDI has a simpler design and is less sensitive to model mismatch. As a pure nonlinear controller, the INDI presents also the advantage of being able to cope with the nonlinearities of the system without requiring a gain-scheduling approach (Slotine & Li, 1991), known for its inflexible and tedious design. To the best of authors’ knowledge, for helicopters, a 6-axis complete INDI-based controller has not been yet developed. Only Howitt (2005) explored the potential of this approach to control a model-scale experimental rotor rig facility in order to counter air resonance and allow for intrinsic carefree handling protection of

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hub moment limits. However, the controller developed in Howitt (2005) is only able to control the helicopter roll and pitch angles and was only tested for hovering flight conditions. In this paper, the INDI control methodology is adopted to design a generic autopilot for a single main rotor and tail rotor helicopter to achieve mission-tailored HQs for two maneuvers of the ADS-33 standard (Anonymous, 2000). In addition, a PseudoControl Hedging (PCH) technique (Johnson & Calise, 2000; Johnson & Kannan, 2005; Lam et al., 2005) is applied to alleviate the requirements associated to multiple time scale separations and to cope with saturation effects of the actuators. The paper does not embody the concept of rotor state feedback in the INDI control law design. This is done in order to understand first the interaction between the multiple feedback loops of the rigid body. A future extension of this method will include the rotor as an integral part of the rigid body dynamic system to be controlled. The main contribution of this paper is therefore the application of the INDI methodology to helicopter flight control, which involves a more complex dynamic model than for fixed-wing aircraft due to the rotor dynamics. This includes the development of a novel algorithm for helicopter vertical control (Section 3.3). Furthermore, the PCH technique, which has only been applied to generic NDI controllers, was derived and tested for the INDI-based strategy. This approach envisages the achievement of enhanced helicopter flight controllers without recourse to extremely complex control architectures and adaptive algorithms. In addition, by coping more efficiently with rotorcraft nonlinear effects and interaxis couplings, it is expectable that the proposed method allows to increase helicopter flight envelope. This paper is organized as follows. Section 2 summarizes the required control theory and Section 3 presents its application to the helicopter nonlinear model. The results obtained from the simulation of common maneuvers and from robustness tests are shown and briefly discussed in Section 4 and the main conclusions are drawn in Section 5. An appendix is also available with the equations needed for the implementation of the helicopter model.

where f and h are vector fields in Rn and Rm , respectively, and G is a n  m control effectiveness matrix. The procedure to obtain the feedback linearization for the inversion of the system consists of consecutive timedifferentiations of y until an explicit dependence on u appears. To each derivative, a new state vector is associated and the derivative of the last state vector is given by a nonlinear expression (the virtual control) to complete the transformation. If r timedifferentiations are required, r:m ≤n is known as the total relative degree of the system. Moreover, if r:m on, there are n−r:m degrees of internal dynamics, unobservable to the input–output linearization and which must be Bounded-Input Bounded-Output (BIBO) stable in the region of interest to assure the effectiveness of the controller (Enns et al., 1994). Assuming now hðxÞ ¼ x, the first-order time-derivative of y is given by y_ ¼ x_ ¼ f ðxÞ þ GðxÞu

2.1. Nonlinear Dynamic Inversion (NDI) The Nonlinear Dynamic Inversion (NDI) was developed in the late 1970s to provide control of nonlinear systems, being applicable to a class of systems known as feedback linearizable (Slotine & Li, 1991). It allows to generate a control input using a state diffeomorphism such that, when applied to the system, all the relations between a virtual control and the outputs of the system are reduced to simple integrators. For the resulting linear system, a single linear control law can be adopted without the need for gainscheduling to tune the controller for different conditions of the nonlinear system. A detailed explanation of this technique is presented, for example, in Slotine and Li (1991) or Enns, Bugajski, Hendrick, and Stein (1994). To exemplify the working principle of the NDI, consider a system of order n with the same number m of inputs u and outputs y and affine in the control inputs. Furthermore, the outputs coincide typically to the control variables and are assumed to be physically similar (for instance, three attitude angles). The extension of the theory to more complex systems is rather straightforward. This type of system can be mathematically represented by x_ ¼ f ðxÞ þ GðxÞu y ¼ hðxÞ

u ¼ G−1 ðxÞðν−f ðxÞÞ

ð3Þ

Besides performing the linearization of the system, this input also allows to decouple the responses of the control variables since each component of ν only depends on the same component of x. The NDI relies on an accurate description of f and G to cancel all the nonlinearities of the system. Nevertheless, if inaccuracies exist, the exact cancellation of the nonlinearities becomes impossible. To illustrate this situation, consider that the functions above are composed by a nominal part which is known (f n and Gn ) plus an uncertain term (Δf and ΔG). Hence x_ ¼ f n ðxÞ þ Δf ðxÞ þ ðGn ðxÞ þ ΔGðxÞÞu

ð4Þ

Unless some type of system identification is applied, only the nominal parts are known by the controller, u ¼ G−1 n ðxÞðν−f n ðxÞÞ, and the application of this input to the real system described by (4) yields ð5Þ

where I nn is the n  n identity matrix. As it can be seen, the linear relation x_ ¼ ν is only recovered for Δf ðxÞ ¼ ΔGðxÞ ¼ 0. Otherwise, the closed-loop system is not linearized anymore, degrading its performance when a linear control law is used to generate ν and thus compromising the stability of the system. This important drawback was the main motivation to develop a more robust version of the NDI, generally known as Incremental NDI. 2.2. Incremental Nonlinear Dynamic Inversion (INDI) As the name indicates, instead of determining the total vector u directly, the Incremental Nonlinear Dynamic Inversion (INDI) is based on the computation of the required control increment at a given moment with respect to the conditions of the system in the instant of time immediately before (Chen & Zhang, 2008). To do so, let (2) be approximated by the first-order terms of its Taylor series expansion around the conditions of the system at that instant (denoted by the subscript 0) _ x_ 0 þ x≈

∂ ½f ðxÞ þ GðxÞux0 ;u0 ðx−x0 Þ þ Gðx0 Þðu−u0 Þ ∂x

ð6Þ

For very small time increments (high sampling frequencies of the controller) and assuming that u can change significantly faster than x (so that x≈x0 even if u≠u0 ), the assumption x−x0 ¼ 0 can be made. Considering this, (6) is further simplified into _ x_ 0 þ Gðx0 Þðu−u0 Þ x≈

ð1Þ

ð2Þ

Since an explicit dependence on u was already found, the linear relation ν ¼ x_ can be imposed if det GðxÞ≠0 by selecting:

−1 x_ ¼ Δf ðxÞ−ΔGðxÞG−1 n ðxÞf n ðxÞ þ ðI nn þ ΔGðxÞG n ðxÞÞν

2. Basic principles of the NDI, INDI and PCH

1067

ð7Þ

and the linear relation ν ¼ x_ can again be imposed for det Gðx0 Þ≠0

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xrm , given by

with −1

u ¼ G ðx0 Þðν−x_ 0 Þ þ u0

ð8Þ

At this point, the main advantage of the INDI can already be identified: the control law for u does not depend on f anymore, meaning that the controller is completely insensitive to the part of the model that only depends on the states of the system. More precisely, this information is replaced by online measurements (or estimates) of the state derivative x_ 0 and the effectiveness of the controller is dictated by the accuracy of the sensors (or filtering processes). The control input u0 has also to be accurately known. The robustness evaluation of this control strategy follows the same procedure applied for the NDI. Assuming ideal sensors, all the model inaccuracies lie in G (uncertainties in f are reflected in x_ 0 ) and the real system is described as x_ ¼ x_ 0 þ ðGn ðx0 Þ þ ΔGðx0 ÞÞðu−u0 Þ

ð9Þ

_ As only the nominal part is known, u ¼ G−1 n ðx0 Þðν−x 0 Þ þ u0 , replacing this control law in (9) yields −1 _ x_ ¼ −ΔGðx0 ÞG−1 n ðx0 Þx 0 þ ðI nn þ ΔGðx0 ÞG n ðx0 ÞÞν

ð11Þ

where C ¼ I nn þ ΔGðx0 ÞG−1 n ðx0 Þ. Therefore, when the sampling _ frequency of the controller is high enough, the result x≈ν still holds, meaning that uncertainties in the control effectiveness matrix G do not significantly affect the INDI-based control loop and no robust control design is needed in this case.

2.3. Pseudo-Control Hedging (PCH) The NDI/INDI theory was derived without any consideration on the dynamics of the physical actuators of the system and, in the case of a multi-loop controller, it neglects the limitations imposed by the potential lack of separation between their bandwidths. If these effects are not taken into account, the performance of the overall controller may be severely degraded and the stability of the system may even be put at risk. To overcome the problem of actuator (or inner loop) dynamics, an adaptation technique known as Pseudo-Control Hedging (PCH) was pioneered in Johnson and Calise (2000), being further developed in Lam, Hindman, Shell, and Ridgely (2005) and Johnson and Kannan (2005). As explained in the latter reference, the PCH automatically moves (hedges) the signals sent to the controller in the opposite direction by an estimate of the amount νh the plant did not move due to the dynamics of the actuators. This prevents the continued effort to track the original commanded references when saturation effects are experienced. In order to implement the hedging of the commanded signals mentioned above, a first-order Reference Model (RM) is adopted. This RM has a saturation filter to keep the desired references from being physically unfeasible and it is especially useful to compute the derivatives of the commanded variables, which can be used by the controller as feedforward terms νrm ¼ K rm ðsat xcom −xrm Þ

1 ðνrm −νh Þ s

ð13Þ

When no saturations occur, νh ¼ 0 and the RM behaves as a low-pass filter with bandwidth K rmi for the i-th component of xcom . Otherwise, νh corresponds simply to the difference between the required virtual control vector that generates the commanded control input ucom and the virtual control associated with the real values of the physical inputs u, known from a model of the actuators or directly measured. Regarding again the system of the previous subsections, the _ is given by (2) for the NDI case and thus virtual control (ν ¼ x) νh ¼ ½f ðxÞ þ GðxÞucom −½f ðxÞ þ GðxÞu ¼ GðxÞðucom −uÞ

ð14Þ

For an INDI-based control law, the virtual control corresponds to (7) and the pseudo-control hedge is νh ¼ ½x_ 0 þ Gðx0 Þðucom −u0 Þ−½x_ 0 þ Gðx0 Þðu−u0 Þ ¼ Gðx0 Þðucom −uÞ

ð15Þ

ð10Þ

Once again, the relation x_ ¼ ν is only obtained for ΔGðx0 Þ ¼ 0. Nevertheless, for very high sample rates (i.e., 100 Hz in the present application), the difference between two consecutive measure_ and ments of the state vector derivative can be neglected, x_ 0 ≈x, (10) can be rewritten as _ C x≈Cν

xrm ¼

ð12Þ

where K rm is a diagonal gain matrix and sat xcom imposes the desired limitation into the commanded references. The signal sent to the control system corresponds to the state vector of the RM,

3. Helicopter controller design using the NDI, INDI and PCH techniques The NDI, INDI and PCH techniques described in the previous section will next be applied to the design of an overall helicopter flight controller. As helicopter example, the paper considers the Messerschmitt–Bölkow–Blohm (MBB) Bö-105, a light, multipurpose utility helicopter, characterized by its hingeless rotor (Hohenemser, 1974). The simulation model adopted in the paper is a generic nonlinear eight Degree Of Freedom (8-DOF) system, representing the 6-DOF of the body motion plus 2-DOF to account for the quasi-steady dynamic inflow induced by the main and tail rotors. This system is based on an in-house generic helicopter model developed at Delft University of Technology (Pavel, 2001). The helicopter is modeled by subdividing it into its main components and adding the contribution of each part to the general system of forces and moments (see Appendix A). The control inputs of the main rotor (MR) are the collective pitch of is blades θ0 and the longitudinal θ1s and lateral θ1c cyclic pitch; for the tail rotor (TR), the control input is simply its collective pitch θ0tr . The dynamics of the actuators associated with these deflections is assumed to be constrained by limitations in terms of position and rate. As explained in the introduction, the control design conducted in this paper do not include the rotor flapping states (flapping angles, rates and accelerations) as feedback variables. Therefore, the flapping dynamics is only considered through the so-called disctilt motion, i.e., the steady-state flapping angles of the rotor (Eqs. (A.17)–(A.19)). Assuming a rigid body with constant mass m and inertia J and a flat, non-rotating Earth with a uniform gravity field with acceleration g, the Equations Of Motion (EOM) that describe the 8-DOF helicopter model w.r.t. an inertial system are expressed in a generalized form as: 2 3 −sin θ 1 6 7 v_ ¼ f þ g 4 sin ϕ cos θ 5−ω  v ð16Þ m cos ϕ cos θ p_ ¼ Tv

ð17Þ

_ ¼ J −1 ðm−ω  JωÞ ω

ð18Þ

θ_ ¼ Ωω

ð19Þ

P. Simplício et al. / Control Engineering Practice 21 (2013) 1065–1077

λ_ ¼



C T −C Gl T τ λ0

C T tr −C Gl T tr τλ0;tr

T ð20Þ

where v ¼ ½u v wT represents the body-fixed linear velocity of the helicopter, p ¼ ½x y zT the position of its Center of Gravity (CG) in the North-East-Down (NED) reference frame, ω ¼ ½p q rT the body-fixed angular velocity, θ ¼ ½ϕ θ ψT the orientation angles of the helicopter relative to the NED reference frame and λ ¼ ½λ0 λ0tr T contains the nondimensional inflow of both main and tail rotors. This dynamic inflow is modeled with Glauert theory by means of two time constants, τλ0 and τλ0;tr , representing their quasi-steady variation in time. Furthermore, f and m are, respectively, the total force and moment acting on the helicopter; CT and C Gl T are the thrust coefficients predicted with the blade element method and with Glauert theory. Appendix A presents in detail the equations adopted to compute these variables. Denoting cos α and sin α by cα and sα , matrix T represents the coordinate conversion from the body-fixed to the NED reference frame 2 3 cψ cθ cψ sθ sϕ −sψ cϕ cψ sθ cϕ þ sψ sϕ 6 7 T ¼ 4 sψ cθ sψ sθ sϕ þ cψ cϕ sψ sθ cϕ −cψ sϕ 5 ð21Þ −sθ cθ sϕ cθ cϕ and Ω is the transformation matrix from ω to θ_ 2 3 1 sin ϕ tan θ cos ϕ tan θ 60 cos ϕ −sin ϕ 7 Ω¼4 5 0 sin ϕ=cos θ cos ϕ=cos θ

The most relevant limitations of the control system proposed in this paper are the following: (1) the robustness of the INDI methodology is only assured for high controller sample rates, as explained in Section 2.2, (2) all the states of the system (and some state derivatives) have to be known to perform the dynamic inversion, which may require additional sensors or estimation algorithms and (3) a formal stability proof of the multiple time scale control architecture is not straightforward. Nonlinear controllers with a stability-based design do exist using, for example, Lyapunov theory (Haddad & Chellaboina, 2008), but the INDI is rather a performance-oriented strategy that allows to directly enforce the desired HQs in the closed-loop system. 3.1. Rate controller As the name indicates, the control variables of this inner loop are the angular rates of the helicopter yrot ¼ hrot ðxÞ ¼ ω

The overall control system proposed in this paper consists of a three loops architecture, i.e., a rate, an attitude and a navigational controller, to track commands in terms of ground velocities (V x ; V y ; V z ) and yaw angle (ψ) (see Simplício, 2011). These loops perform the inversion of the rotational dynamics, kinematics and translational dynamics of the helicopter and, between them, the existence of a time scale separation is assumed. This type of assumption is often carried out for flight dynamics and control applications. Between two loops, the parameters associated with the outer loop (slow dynamics) are treated as constants by the inner loop (fast dynamics) and its dynamic inversion is performed assuming that the states controlled in the inner loop achieve their commanded values instantaneously. The fast variables are thus used as control inputs to the slow dynamics. A simplified architecture of the multi-loop overall control system is schematized in Fig. 1. In this figure, vectors u and x contain respectively the control inputs and state variables introduced throughout this section. From this figure, it is possible to observe that not all the control inputs are used to provide tracking of the inner loop control variables. Since the cyclic pitch and the collective of the tail rotor are moment generators while the collective of the main rotor is primarily a force effector, the latter is not used to control the angular rates of the vehicle. Instead, its command signal is generated in the navigational loop. The following subsections present in more detail the internal structure of the three loops.

ð23Þ

In order to apply the INDI technique, this equation has to be time-differentiated until an explicit dependence on the control inputs appears. The first-order derivative corresponds to the rotational dynamics given by (18), which can be recast as _ ¼ f ðxÞ þ gðx; uÞ y_ rot ¼ ω

ð22Þ

1069

ð24Þ

where f ðxÞ is the control independent part of the model and gðx; uÞ comprises moment terms that depend on the control inputs directly, being known as control effectiveness function. For the case of a fixed-wing aircraft, this function is normally affine in the control inputs and the system is simplified into the form of (1). For helicopters however, the most complex part of the model is contained in gðx; uÞ due to the aerodynamics of the rotors gðx; uÞ ¼ J −1 ½mmr ðx; uÞ þ mtr ðx; uÞ

ð25Þ

Since this equation expresses the influence of the controls in the _ Followsystem, the rotational virtual control is defined as νrot ¼ ω. ing the same procedure presented in Section 2.2 and taking into account that only a vector u′ ¼ ½θ1s θ1c θ0tr T is used for rotational control, the command signal sent to these actuators is given by −1 ∂gðx; uÞ _ 0 Þ þ u′0 u′com ¼ ðνrot −ω ð26Þ ∂u′ x0 ;u0 Comparing this equation with (8), it is possible to verify that, in the case of a system that is not affine in the controls, the information contained in matrix GðxÞ is simply replaced by the Jacobian matrix of the control effectiveness function gðx; uÞ with respect to the different control inputs. For the model under analysis, the derivative term of mtr could be determined analytically but, because of its complexity, the Jacobian of mmr was computed with central finite differences. Furthermore, due to the fact that angular accelerometers are still not common today, the _ 0 were estimated with backward finite angular accelerations ω differences and first-order low-pass filters were introduced before the actuators to attenuate high frequency oscillations caused by numerical noise.

Fig. 1. Simplified architecture of the overall flight control system.

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Fig. 2. Schematic of the rate control system based on INDI and PCH.

The control law (26) linearizes and decouples the response of the helicopter from νrot to ω. As shown in Sieberling et al. (2010), if a proportional controller with diagonal gain matrix K 1 is applied to generate the virtual control from the tracking error, νrot ¼ K 1 ðωcom −ωÞ, the closed-loop transfer function for each component i of the angular rate vector is given by K 1i ωi ðsÞ ¼ ωicom ðsÞ s þ K 1i

ð27Þ

Hence, the stability of this control loop is ensured if K 1i 4 0, for i ¼ 1; 2; 3, and the time constants of the responses are directly enforced by K −1 1 . In addition, a PCH layer as the one derived in Section 2.3 is also implemented in order to protect the system against undesirable effects from the dynamics of the actuators. It makes use of a first-order RM described by (12) and (13) to adjust the signals commanded to the controller and to provide a feedforward term νrmrot to increase the tracking performance. The internal structure of the rate control system as described above is depicted in Fig. 2. At this point, the overall system is still slightly unstable due to the existence of unbounded internal dynamics, but this problem will be automatically solved with the introduction of external control loops. In the end, five degrees of internal dynamics will remain in the system. Three are associated with the translational kinematics (17) which are bounded since T is orthonormal and the translational dynamics will be stabilized. The remaining two degrees concern the rotor induced inflows (20). In steady operation, these internal dynamics are stable since the thrust coefficients from the blade element method and Glauert theory coincide, yielding λ_ ¼ 0. For a more thorough stability assessment, an analysis using Lyapunov theory would be required (Haddad & Chellaboina, 2008).

straightforward for det Ω ¼ cos θ≠0 ωcom ¼ Ω−1 νatt

ð30Þ

Note that this inversion is performed according to the standard NDI technique since, as there is not a part of the model that does not depend on ω and as Ω can normally be determined very accurately, the application of the INDI would not bring any advantage in this case. The virtual control can again be generated by a proportional controller νatt ¼ K 2 ðθcom −θÞ and, in order to alleviate the time scale separation requirements, allowing the bandwidth of the control loops to be closer, a combined analysis is performed to select the linear gains of both loops. Taking into account that a first-order response with bandwidth K 1 was imposed to the inner loop, the closed-loop transfer function for each Euler angle i is K 1i K 2i θi ðsÞ ¼ 2 θicom ðsÞ s þ K 1i s þ K 1i K 2i

ð31Þ

It corresponds thus to a second-order response for each axis and the diagonal matrices K 1 and K 2 are chosen to impose the desired natural frequencies and damping ratios, ensuring also the stability of the system. In order to further prevent undesirable interactions between the two loops, an additional PCH layer is introduced to adjust the angular references to the capabilities of the rate controller. The working principle of the current PCH layer is exactly the same as the one presented in the previous subsection and the pseudo-control hedge for this case is νhatt ¼ Ωðωcom −ωÞ

ð32Þ

3.3. Navigational controller This control loop aims to track references in terms of ground velocities

3.2. Attitude controller

ynav ¼ hnav ðxÞ ¼ p_ ¼ vE ¼ ½V x V y V z T

The attitude controller is constructed externally to the rate controller and uses commands in terms of angular rates to track the desired attitude angles of the vehicle. A time scale separation is assumed between the two loops and the dynamics of the inner loop are neglected, just like the inner loop INDI control law neglected the dynamics of the actuators. The output vector of the attitude controller corresponds thus to the attitude angles

by using the pitch and roll angles as control inputs for the horizontal components and the collective of the main rotor for the vertical speed. Once again, the existence of a time scale separation is assumed between translational and rotational dynamics, hence this outer loop assumes that the attitude angles are exactly what they are commanded to be. In addition, as depicted in Fig. 1, the commanded value for the yaw angle needs to be provided externally or, for a flight with no sideslip, computed as ψ com ¼ arctan2 ðV ycom =V xcom Þ. For horizontal control, an approach based on an approximate dynamic inversion proposed in Prasad and Lipp (1993) is adopted. To do so, consider the equation of the translational dynamics written in the NED reference frame 2 3 2 3 2 3 fx 0 V_ x 6_ 7 1 6f 7 607 7 _y nav ¼ 6 V ð34Þ 4 y 5 ¼ m T4 y 5 þ 4 5 fz g V_ z

yatt ¼ hatt ðxÞ ¼ θ

ð28Þ

and following the same procedure of the previous subsection, when this vector is differentiated with respect to time, differential equation (19) is obtained: y_ att ¼ θ_ ¼ Ωω

ð29Þ

Since a dependence on the control inputs (the angular rates) has already appeared, the virtual control is defined as νatt ¼ θ_ and the control law for the commanded angular rates comes

ð33Þ

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As a dependence on the control inputs (the attitude angles) appears through matrix T, define the virtual control νnav ¼ ½νx νy νz T ¼ ½V_ x V_ y V_ z T and, assuming jf z j b jf x j; jf y j since it corresponds practically to the thrust produced by the main rotor, the following expressions can be obtained: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fz ≈− ν2x þ ν2y þ ðνz −gÞ2 ð35Þ m θcom ≈arctan

νx cos ψ com þ νy sin ψ com νz −g

ð36Þ

ϕcom ≈arcsin

νx sin ψ com −νy cos ψ com f z =m

ð37Þ

The horizontal control is thus achieved by changing the attitude of the helicopter in such a way that the thrust vector points in the required direction. In (36) and (37), the trim values of θ and ϕ shall be summed up to reduce the steady-state error. Furthermore, note the existence of a singularity in (36) for νz ¼ g, meaning that the system must never demand a downward acceleration larger or equal than 1 g. When the virtual control is generated with a proportional control law, νnav ¼ K 3 ðvEcom −vE Þ, the closed-loop system becomes of third-order and, as presented for the attitude controller, a combined analysis of the three loops was considered to impose the desired characteristics in the responses of the overall closedloop system and to ensure its stability. The vertical tracking is provided by changing the magnitude of the thrust force and, with this objective, a control law for the collective of the main rotor θ0 is derived based on INDI. To do so, note that the influence of θ0 in V_ z ¼ νz is contained in the third component of (34) through f mr (the contribution of the main rotor to the force vector f ). Therefore, assuming again x≈x0 for small time increments, the following incremental inversion law is obtained: 02 1−1 3T −sin θ  B6 C 7 ∂f mr ðx; uÞ C ðνz −V_ z Þ þ θ0 θ0com ¼ mB ð38Þ  0 0 @4 cos θ sin ϕ 5 ∂θ0 x0 ;u0 A cos θ cosϕ where again the subscript 0 denotes the conditions of the system at the instant of time immediately before. As in the rate controller, central finite differences are used to compute the derivative term in (38) and the vertical acceleration V_ z0 is estimated using measurements from the accelerometers and information about the attitude of the helicopter. Furthermore, for the choice of the last entry of K 3 , it is important to have in mind that the closedloop for vertical velocity control is of first-order. Another PCH layer is also introduced in the overall navigational controller to cope with limitations relative to a potentially weak time scale separation between loops and to the dynamics of the collective of the main rotor. With the current INDI-based laws, the internal structure of this control loop is similar to the previous ones. Fig. 3 shows the benefits of a control law based on INDI and PCH for a simple example. It represents the responses to a doublet input in V zcom with three different control laws for θ0com : a linear one available in Lee et al. (2005), the INDI approach (38) and the same INDI law with PCH. The time constant imposed on this control channel is of 0.1 s and the simulation started with the helicopter in hover. As depicted, the linear approach looses accuracy when the system moves away from the initial trim conditions, introducing tracking errors even in steady-state. These errors are eliminated with the INDI strategy, but this response is still affected with significant overshoots due to saturation effects of the collective actuator. A satisfactory tracking is only achieved when the PCH is added to the INDI approach, eliminating the

Fig. 3. Vertical velocity responses with different control laws for the collective.

Table 1 Characteristics of the responses selected for the overall control system. Control variable

Rot. natural frequency (rad/s)

Rot. damping ratio

Trans. time constant (s)

Vx Vy Vz ψ

2.5 2.5 – 4.0

0.8 0.8 – 0.8

0.2 0.2 0.4 –

referred overshoots while also reducing the settling time of the response.

4. Simulation results The INDI control system developed in this paper will be next used as an autopilot capable of achieving precise maneuvering for ADS-33 mission task elements (Anonymous, 2000). As demonstrated, the INDI methodology allows to decouple the responses of the helicopter associated with different axes and to directly enforce the desired characteristics (HQs) to each one. The characteristics selected for each control variable are registered in Table 1. For the following simulations, the computations relative to the dynamics of the model, control system and sensors were sampled at 100 Hz which, as mentioned in Section 2.2, is an adequate value for current helicopter systems. Only the Global Positioning System (GPS) receiver estimates the ground velocities at 20 Hz. 4.1. The combined bob-up/bob-down with acceleration/deceleration This maneuver consists of a combination of horizontal and vertical path segments. The bob-up starts at 2000 ft with a cruise speed of 15 m/s and the helicopter is commanded to decelerate to hover, climb at a constant rate of 4 m/s, hover at an higher altitude and accelerate back to 15 m/s and maintain this speed to prepare the bob-down maneuver. Similarly, for the bob-down, the initial speed is of 15 m/s, the vehicle is commanded to decelerate to hover, descend at 4 m/s and finish the maneuver in hovering flight approximately at the initial altitude. During the entire maneuver, the helicopter is commanded to keep flying North with no sideslip angle. The responses obtained with this maneuver are depicted in Fig. 4.

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Fig. 4. Simulation of the combined bob-up/bob-down with acceleration/deceleration. (a) Commanded (dashed line) and actual (solid line) ground velocity and attitude. (b) Control inputs of the autopilot.

As depicted in Fig. 4(a), the commanded velocities are properly tracked, presenting fast evolutions without oscillations and with a reduced steady-state error. As expected intuitively, the horizontal velocity is mainly controlled by changing the commanded pitch angle of the helicopter since it does not involve changes in direction. The required attitude angles computed by the navigational loop are also properly tracked by generating the necessary commands in terms of angular rates (not shown in the plot). The autopilot inputs can be read from Fig. 4(b). The deceleration ( ) is achieved by gradually pulling back the longitudinal stick. As the helicopter tends to climb, the collective is lowered, which requires an input in the right pedal to correct the yaw motion. In order to eliminate the lateral velocity, a change to the left in the roll angle is demanded and achieved with the lateral cyclic input. The collective is then adjusted to track the desired vertical velocity with a first-order response ( , ) and the remaining inputs are again used to compensate the moments and balance the rotorcraft. Little cross-couplings between the controls exist in these regions, as required by the ADS-33. In , the acceleration is initiated by pulling forward the longitudinal cyclic and increasing the collective and, in , and , the maneuver is repeated to decelerate the helicopter and obtain the vertical descent. 4.2. The pirouette maneuver The pirouette is a more aggressive maneuver, often used to check the ability to accomplish precision control of simultaneous rotational and heave motion. The maneuver is initiated from a stabilized hover ( ) at 10 ft over a point on a circumference with a

30 m radius, with the nose pointed at its center. Then, a uniform lateral translation is accomplished while keeping the nose of the rotorcraft pointing towards the trajectory center until returning to the starting position ( ). After a short transition hover ( ), the 3601 turn is performed in the opposite direction ( ) and the maneuver is finished again in hovering flight ( ). If one complete circumference is to be flown in Δt seconds, a lateral velocity V ¼ 60π=Δt has to be maintained and the references commanded to the system are generated according to   2π t m=s V zcom ðtÞ ¼ 0 m=s V xcom ðtÞ ¼ V sin Δt  V ycom ðtÞ ¼ 7 V cos

2π t Δt

 m=s

ψ com ðtÞ ¼ ∓

360 t deg Δt

The results obtained with Δt ¼ 40 s are presented in Fig. 5. From Fig. 5(a), it is possible to observe that the commanded components of the velocity are followed very efficiently, with a very small tracking error and without undesirable oscillations. Furthermore, it can be visualized that the vertical velocity is practically unaffected by the motion of the helicopter, indicating that the evolutions associated with the different axes are satisfactorily decoupled. In fact, for this trial, all the performance requirements advised in the ADS-33 standard for the pirouette were met: a desired radial error below 3 m, altitude error below 90 cm, heading error below 101 and also an adequate time to achieve a stabilized hover below 10 s. With respect to the attitude angles, the commanded yaw is tracked only with a lag of around 0.4 s and changes in roll are used to control the lateral velocity.

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Fig. 5. Simulation of the pirouette maneuver. (a) Commanded (dashed line) and actual (solid line) ground velocity and attitude. (b) Control inputs of the autopilot.

Since all the responses are very smooth, the control effort to perform this maneuver is quite simple, as depicted in Fig. 5(b). The pirouette to the right is initiated by applying right pedal followed by right lateral cyclic pitch ( to ). In the beginning, some compensations in longitudinal cyclic and collective are required, but afterwards all controls are kept constant. The transition to hover is made by applying left pedal and left lateral cyclic ( to ) and the pirouette to the left is executed in a similar manner ( to ). 4.3. Robustness tests This section will now assess whether the proposed helicopter controller is robust to modifications on the nominal conditions of its model. The nominal response corresponds to the results obtained from the simulation of the slalom maneuver of the ADS-33. This maneuver is initiated with the helicopter flying North without sideslip and involves two turns at constant speed and altitude. In the following figures, this response is depicted with a solid line and, besides the control inputs, only the evolutions of Vy and ψ are shown as the effect of disturbances on the remaining variables is less noticeable. 4.3.1. Aerodynamic uncertainties One of the main difficulties in helicopter modeling is to accurately describe the very complex processes in the rotor that generate the aerodynamic forces and moment. As already shown theoretically, the performance of INDI-based controllers is not significantly affected by model uncertainties. However, for the proposed method, part of the information about the control effectiveness is lost when considering this function only through

its Jacobian matrix and when finite differences are used to determine some of its entries. To analyze the influence of these effects, inaccuracies in the model of the main rotor are introduced simultaneously in the aerodynamic coefficients according to C^ n ¼ C n ð1 þ εÞ. Cn is the real value of the coefficient, n can assume each of the rotor forces or moment (Eqs. (A.7)–(A.10)) and ε has a normal distribution with zero mean and standard deviation s (2s is approximately the maximum relative uncertainty of the model). Due to this stochastic evolution, several trials were performed for different values of s. The comparison between the nominal response of the system and the trial associated with the worst case for each s is presented in Fig. 6. For a maximum inaccuracy of 100% ðs ¼ 0:5Þ, the performance of the system was not affected and all the trials were performed exactly as in the nominal conditions. For 200% ðs ¼ 1:0Þ, the response of the helicopter kept practically unchanged, but occasional control oscillations appeared. These oscillations are due to the influence of extremely incorrect control effectiveness approximations on the computation of the required control actions. Finally, for a maximum inaccuracy of 400% ðs ¼ 2:0Þ, the performance of the system became severely degraded. If severe model inaccuracies are not persistent, the system is still able to recover the nominal evolution but, otherwise, it may result in the crash of the helicopter. Note however that, in reality, it is not expectable to find aerodynamic models with relative inaccuracies above 200%.

4.3.2. Tail rotor malfunction The insensitivity of the INDI to modeling errors also allows to use the exact same controller for the helicopter when its configuration changes, for example, in the presence of structural

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Fig. 6. Response of the system with uncertainties in the aerodynamic model.

Fig. 7. Response of the system with malfunctions in the tail rotor.

damages or system failures (Bacon et al., 2001). In the latter case, the presence of the PCH is also beneficial. The following simulation was carried out in order to evaluate the influence of a malfunction of the tail rotor. This malfunction was introduced by assuming that its collective actuator is stuck in different positions and the corresponding pitch is kept at −81, 01 and 201. The results are depicted in Fig. 7. It is possible to verify that velocity tracking, despite being degraded, can be achieved quite efficiently for all the malfunctions. The responses of the yaw angle suffered a more severe degradation since this angle is directly influenced by the tail rotor, but the controller was always able to guarantee the stability of the system over the entire maneuver. As expected, the degradation in performance is more intense when the difference between the fixed position of the actuator and the values assumed in the nominal condition is larger. Despite not presented in this paper, additional reliability tests were performed in Simplício (2011). It was verified that the overall system is only slightly affected by unaccounted inertia mismatches with a magnitude within the range expectable to find in reality and especially if the components of this matrix are

underestimated. Regarding the existence of a time delay between the control system and the actuators, it was proved that the behavior of the helicopter remains acceptable with lags up to 50 ms. It was also concluded that sensor dynamics (measurement noise and time delays) introduce high frequency oscillations in the control inputs but, as their amplitude is practically negligible, the responses of the system remain unaffected. Finally, the performance of the system was tested for lower sample rates of the controller. In this case, not only the INDI assumption x≈x0 looses strength, the angular accelerations are also estimated less accurately. The results showed that for frequencies of 60 Hz, the response of the system was already quite degraded, especially in terms of oscillations in the control inputs.

5. Conclusions The application of the INDI technique proposed in this paper may represent an important contribution for the development of improved helicopter flight control laws since it allows to cope with the nonlinearities of the model while providing robustness in the

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presence of inaccuracies and without the need for a complex control structure or an adaptive element. The overall navigational controller developed is mainly based on this technique and employs a PCH methodology to provide a dynamical filtering of the commands such that the demanded signals are within the capabilities of the system. The multi-loop control system was tested by performing different maneuvers of the ADS-33 regulations and the results obtained showed that it is able to efficiently stabilize the system and track the desired ground velocities with the performance characteristics imposed by the flight control laws. In terms of robustness, the most relevant observation is related to the fact that this control system is insensitive to model uncertainties within the range expected to find in reality. This property is notably useful since the accurate identification of helicopter high-fidelity models is extremely difficult and costly. The proposed approach has thus the potential to serve as a basis for a helicopter flight-by-wire control system. If this strategy is adopted, a simplified model may be enough to compute an approximate Jacobian of the control effectiveness function and efficiently control the vehicle.

Appendix A. Helicopter model For the development of the nonlinear 8-DOF helicopter model adopted throughout this paper, the following assumptions are made:

 Aerodynamic forces and moments are calculated using the blade element theory.

 The tail rotor is modeled as an actuator disc.  The fuselage and the tails are modeled with linear aerodynamics.

 The dynamic inflows of both rotors are included in the model

           

as state variables and can be described as quasi-steady dynamic inflows by means of time constants with values between 0.1 and 0.5 s. Rotor disc-tilt dynamics (often the so-called flapping dynamics) is neglected and only steady-state rotor disc-tilt motion is considered. The rotor is modeled with an equivalent flapping hinge offset. The lead-lag motion of the blades is neglected. There are no pitch-flap or pitch-lag couplings. The flapping and flow angles are small. Gravitational forces are small compared to aerodynamic, inertial and centrifugal forces. The rotor angular velocity is constant and anticlockwise. No reverse flow regions or tip losses are considered. The airflow is incompressible and its density is computed according to the International Standard Atmosphere (ISA). The blades are rectangular, with a linear twist. The blades have a uniform mass distribution. The blade elastic, aerodynamic, control and CG axes coincide.

As mentioned in Section 3, the general forces and moments acting on the helicopter result from the sum of the contributions of its components

Fig. A1. Helicopter forces and moments on its main components. (a) Left view (b) Bottom view, (c) Aft view.

the following derivation, the reader is referred to Pavel (2001), Bramwell, Done, and Balmford (2001) and Johnson (1980).

A.1. Main rotor Using cα and sα to represent cosα and sin α, respectively, the total force and moment generated by the main rotor are given by 2 32 3 −H dp sa1r þγ s sb1r sa1r þγ s cb1r ca1r þγ s 6 7 6 0 cb1r −sb1r 54 Sdp 7 f mr ¼ 4 ðA:3Þ 5 −T dp −sa1r þγs ca1r þγ s sb1r ca1r þγ s cb1r 2

Le

3 2

l

3

6 7 6 7 mmr ¼ 4 M e 5−4 l1 5  f mr Q dp h

ðA:4Þ

f ¼ f mr þ f tr þ f fus þ f ht þ f vt

ðA:1Þ

The angles a1r þ γ s ¼ a1 −θ1s þ γ s and b1r ¼ b1 þ θ1c as well as the displacements l, l1 and h are shown in Fig. A1. Le and Me correspond to the moments on the rotor hub resulting from the hinge offset of the blades

m ¼ mmr þ mtr þ mfus þ mht þ mvt

ðA:2Þ

Le ¼ Ω2 Reβ mbl sin b1r

ðA:5Þ

M e ¼ Ω2 Reβ mbl sinða1r þ γ s Þ

ðA:6Þ

Fig. A1 allows to more clearly visualize these contributions. The forces and moments are expressed in the body-fixed reference frame fxb ; yb ; zb g and the equations required for their computation are now presented. For a more insightful understanding on

The main rotor thrust Tdp, longitudinal Hdp and lateral Sdp drag forces and torque moment Qdp are expressed by their non-

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dimensional coefficients T dp ¼ ρπR2 ðΩRÞ2 C T dp

ðA:7Þ

H dp ¼ ρπR2 ðΩRÞ2 C Hdp

ðA:8Þ

Sdp ¼ ρπR2 ðΩRÞ2 C Sdp

ðA:9Þ

2

2

Q dp ¼ ρπR ðΩRÞ RC Q dp

"

! # μ −λ0tr 1 μ2xtr þ θ0tr þ ztr 3 2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ2xtr þ ðμztr −λ0tr Þ2

ðA:23Þ

ðA:24Þ

The fuselage participates in the motion with ðA:12Þ

   i CD   ð1 þ 4:7μ2x Þ−C T dp ðμsin αdp −λ0 Þ−C Hdp μx ¼s 8

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjμjcos αdp Þ2 þ ðjμjsin αdp þ λ0 Þ2

f fus ¼ −Rfus ½cos βfus cos αfus sin βfus cos βfus sin αfus T

ðA:25Þ

mfus ¼ ρπR2 ðΩRÞ2 R½0 C Mfus C Nfus T

ðA:26Þ

The angles αfus and βfus are also illustrated in Fig. A1. The aerodynamic force is calculated through an equivalent drag area F0

ðA:14Þ

The thrust coefficients according to the blade element method and to Glauert theory are respectively given by    sC Lα 1 μ2x 1 þ μ2x μ p μ −λ0 CT ¼ þ θ0 þ θtw þ x þ z ðA:15Þ 3 2 4 2 8 2

Rfus ¼ 12ρV 2 F 0

ðA:27Þ

and, applying a correction factor Kfus that depends on the dimensions of the fuselage, the moment coefficients are given by  2 V 1 K V α ðA:28Þ C Mfus ¼ ΩR πR3 fus fusM fus  C Nfus ¼

 V 2 1 K V β ΩR πR3 fus fusN fus

ðA:29Þ

ðA:16Þ

The rotor disc-tilt angles are determined from the solution of the system     γ 4 2 4 2 2 4 þ μx − μx θ1s a0 ¼ 2 θ0 ð1 þ μ2x Þ þ ðμz −λ0 Þ þ μx p þ θtw 3 3 5 3 3 8νβ ðA:17Þ a1 ¼

str C Lα;tr 2

A.3. Fuselage

ðA:13Þ

C Gl T ¼ 2λ0

C T tr ¼

C Gl T tr ¼ 2λ0tr

   sC Lα 1 μ μ2 q θtw − μx a0 θ0 þ −a0 x þ b1 x − ¼ 2 4 3 4 4  μ a1 −μz þ λ0 μ2 þ 1 þ a0 a1 x −3a0 μx ðμx a1 −μz þ λ0 Þ þ b1 x 3 2

C Q dp

where

ðA:22Þ

ðA:11Þ

  μ sC Lα μ2 μ ðμ −λ0 Þ θtw a1 x þ μx ðμz −λ0 Þ θ0 þ x z C Hdp ¼ sC D x þ 2 4 4 2    b1 μx a0 a0 b1 μ pðμz −λ0 Þ − − þ ða20 þ a21 Þ x þ þq 2 4 3 3 2 C Sdp

T tr ¼ ρπR2tr ðΩtr Rtr Þ2 C T tr

ðA:10Þ

in which C T dp ¼ C T

blockage factor F tr ¼ 1−3Svt =ð4πR2tr Þ was introduced. The thrust Ttr can also be expressed through its coefficient

8 ν2β −1 b1 1 γ 1− μ2x 2   8 16 3 μx θ0 þ 2μx ðμz −λ0 Þ þ p− q þ 2θtw μx − 1 þ μ2x θ1s 3 γ 2 þ 1 2 1− μx 2

A.4. Horizontal tail The influence of the horizontal tail can be considered only through its lift force Lht and the moment it creates due to the displacement lht f ht ¼ ½0 0 −Lht T

ðA:30Þ

mht ¼ −Lht ½0 lht 0T

ðA:31Þ

The lift force can expressed as Lht ¼ 12ρV 2ht Sht C Lα;ht αht

ðA:32Þ

where the local velocity and angle of attack are respectively given by ðA:18Þ

8 ν2β −1 b1 ¼ − a1 1 γ 1 þ μ2x 2   4 16 1 1:33μx =jμz −λ0 j μx a0 þ q− p þ 1 þ μ2x θ1c þ λ0 3 γ 2 1:2 þ μx =jμz −λ0 j þ 1 1 þ μ2x 2

V 2ht ¼ u2 þ ðw þ qlht Þ2 αht ¼ arctan

  w þ qlht þ αht 0 u

ðA:33Þ ðA:34Þ

A.5. Vertical tail ðA:19Þ

A.2. Tail rotor The force and moment produced by the tail rotor are f tr ¼ ½0 T tr F tr 0T

ðA:20Þ

mtr ¼ T tr F tr ½htr 0 −ltr T

ðA:21Þ

The displacements htr and ltr are depicted in Fig. A1 and the fin

Similarly to the horizontal tail, the contribution of the vertical tail is f vt ¼ ½0 −Lvt 0T

ðA:35Þ

mvt ¼ −Lvt ½hvt 0 −lvt T

ðA:36Þ

with: Lvt ¼ 12ρV 2vt Svt C Lα;vt βvt

ðA:37Þ

V 2vt ¼ u2 þ ðv þ phvt −rlvt Þ2

ðA:38Þ

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 βvt ¼ arctan

v þ phvt −rlvt u

 þ βvt 0

ðA:39Þ

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