Università degli Studi di Napoli Federico II. Tecniche di Simulazione di Volo

Transcription

1 Università degli Studi di Napoli Federico II Dipartimento di Progettazione Aeronautica Appunti di Tecniche di Simulazione di Volo Ing. Agostino De Marco Anno Accademico

2 Contents 3 Mathematical models 3.1 Introduction Basic concepts Mathematical modeling versus stability and control analysis Elements of the model Aerodynamic model Engine model Undercarriage model Mass model Atmosferic model Control system model Equations of motion General equations Small perturbation equations Aircraft orientation Kinematic relationships for Euler angles The direction cosine matrix The quaternion approach Axes and frames of reference Incidence angles Direction angles i

3 Chapter3 Mathematical models 3.1 Introduction As stated earlier in these notes, to a simulation engineer, a faithful simulation requires the following three elements Model: a complete model, preferably expressed mathematically, of the response of the aircraft to all inputs, from the pilot and from the environment; Solver: a means of solving these equations in real-time, or in other words of animating the model; Response feedback: a way of presenting the output of this solution to the pilot by means of mechanical, visual and aural responses. None of these has yet been completely solved nowadays, if judged by a strict engineering criterium. Another question is whether or not these requirements have to be completely met Basic concepts The principal task, that of modelling the dynamic behaviour of the flight vehicle, is an extension of the well known basic principles of Flight Mechanics. However, simu- 3 1

4 Mathematical models Introduction 3 2 lation must deal with more than just airframe aerodynamics and broader aspects of mathematical modelling will be outlined here. The nature of the problem is that the motion of a flying vehicle is governed by equations of motion of the generic form ẍ = F/m, θ = M/I (3.1) where, respectively ẍ, F, m are the vehicle acceleration, applied force and mass, and θ, M, I are the angular acceleration, applied moment and moment of inertia. A more specific form of the equations of motion must be established at this point, but one can state that the basic mathematical model of the vehicle under simulation is embodied in the definition of the terms F and M in the eq.s (3.1). For an atmospheric flying vehicle, this mathematical model is primarily the relationship between the air reactions (aerodynamic forces and moments) and the motion of the airplane relative to the air. This is currently known as the aerodynamic model of the aircraft. Other external forces and moments arise from engine thrust and landing gear contact with the ground. Definition of all these force and moment components is the key to a realistic description of an aircraft s flight characteristics. The performances of an aircraft and its dynamic behaviour can then be calculated for a wide range of flight conditions, according to the amount of data available for the evaluation of F (at varying airplane attitudes). Simulation is fundamentally the generation of those forces and solution of the equations of motion. The term mathematical model is thus growing in complexity and extent, and is also applied to other features of the aircraft such as its control system, and many internal systems. For each feature being modelled, it is necessary to formulate the model so that the behaviour of the whole system, i. e. the response of the system to a stimulus, may be calculated. A model is only useful if it can actually be implemented for some application. So far, the term model has not properly separated the concept from the implementation. This may usefully be done, with the aid of some definitions given below.

5 Mathematical models Introduction 3 3 Figure 3.1: The simulation space and the definition of models. The so-called simulation space is divided into three basic elements as depicted in fig Starting with Reality, a conceptual model, described via equations or other governing relationships, is obtained by analysis. Implementation via computer programming yields a Computer Model which, through simulation, may be related to Reality. The credibility of the conceptual model is then evaluated by procedures which test the adequacy of the model to provide an acceptable level of agreement with reality, while the computer model, in the form of an operational computer program, is confirmed as an adequate representation of the conceptual model by procedures of verification. Finally, model validation demonstrates that the computer model possesses a satisfactory range of accuracy in comparison with reality and consistent with its intended application. In some aircraft simulation applications, the mathematical models may be defined

6 Mathematical models Introduction 3 4 by one agency, such as the aircraft manufacturer, for implementation by another, a simulator manifacturer. A third party, such as a regulatory authority, may require that the original model be implemented unchanged. Verification would be performed by the simulator manifacturer using his own internally-generated checks, and validation by comparison with real-life data redorded from flight tests. In creating or deriving mathematical models, it is important that the modeller (the person or staff doing the modelling) has a clear idea of what the model is for, and that he states this together with his definition of his model. It is important because the purpose of the model influences its form and quality. Many systems are strictly governed by equations which may be extremely complex but which may often be simplified in the interests of obtaining practical solutions and yet still retain sufficient realism for the task at hand. A practical solution in the present context means one of adequate accuracy which can be achieved in real-time during simulation. Real-time here refers to a solution in which the calculation of a systen s behaviour over a given elapsed time interval t can be achieved in the same t or less of computing time. Adequate accuracy means that the real-time solution yelds the steady-state performance of the vehicle and its transient behaviour with accuracy which is acceptable and sufficient for the role of the simulation Mathematical modeling versus stability and control analysis Deriving mathematical models for simulation has some similarities with the basic techniques of stability and control theory but there are also some differences. Similarities include mathematical notation, systems of axes and basic nomenclature. Differences, however are significant, and include the need for a wide range of speed, aircraft configuration and manoeuvre real-time solution, as defined earlier. The classical approach to stability and control analysis is to start with the complete equations of motion and make assumptions that enable the equations to be linearised about some local point of equilibrium (a trimmed state). Once linear equations are

7 Mathematical models Introduction 3 5 available, a host of techniques can be applied to the analysis of the system to investigate the stability of the motion following a disturbance from trim. It will provide answers to such questions as will the aircraft return to the original trimmed state and with what sort of transient behaviour? While such techniques have been, and are, invaluable in the design process, only small disturbances from the equilibrium state are permitted before the model is invalid. Small disturbances means, say, 10% escursions in speed or not more than 5 degrees in angle of attack. Many simulation tasks wether for training or for research and development, demand a wide range of flight speed or require large changes in aircraft configuration in order to accomplish that task. A complete sortie, from take-off to landing, is an extreme example. Real-time solution is necessary because, in simulation, there is usually a man in the control loop (pilot-in-the-loop simulations), who must be supplied with information in a timely fashion so that he can use his normal control techniques and strategies. Figure 3.2: The simulation space and the definition of models.

8 Mathematical models Elements of the model Elements of the model As stated, the primary elements of a conceptual/mathematical model for flight simulation are those which directly produce the values of external forces acting on the airframe, namely aerodynamics, engine thrust and undercarriage. Secondary elements of a model contribute significantly to the core, and include a control system and the model of atmospheric environment. All these features are summarized in fig Aerodynamic model The aerodynamic model has to reproduce the dominant features of the forces and moments acting on the aircraft: lift, drag the main contributors to the aircraft performance, and moments about all three axes, through which pilot exercises control. Mathematical models normally consider the physical components of the vehicle, such as, for conventional aircrafts, wings, body (i.e. the fuselage), and tail, and build mathematical expressions for the contributions made by these components and their mutual interferences. A possible expression is the following L aero = L Wing + L Body + L Wing Body + L Tail (3.2) for the total aerodynamic lift L aero (or simply L) of the aircraft, where the term L Wing Body, usually estimated with semi-empirical formulas, takes into account the interference effects on the wing lift due to the presence of the fuselage, the remaining terms being calculated as the wing, fuselage and tail were isolated. Similar formulas are implemented for the other aerodynamic force components and moments. Additional featuresare included to represent such influences as ground effect, significant during take-off and landing. In these situations, forces acting on the aircraft differ markedly from those applicable to the same configuration in free air, due to the presence of the ground close to the aircraft constraining the lower airflow to follow the runway. This can resul in increased wing lift and possibly in a sharp increase in

9 Mathematical models Elements of the model 3 7 nose-down pitching moment close to the ground. Both these effect are important as they influence the pilot s ability to perform a safe and smoth landing. Aerodynamic models are normally static or quasi-static, based on the assumption that the airflow can re-establish steady conditions in a time scale much shorter than the characteristic time scale of the aircraft manoeuvre itself. Thus the time variable t does not appear explicitly in the mathematical model, but the aircraft s transient behaviour is, of course, a function of time Engine model An engine model, embedded a simulator s conceptual model, must produce the correct value of steadt thrust to correspond with pilot s demand through his power lever. Usually, thrust is modelled as a net thrust: the net propulsive force is directly calculated, having allowed for intake momentum drag. More comprehensive models will reproduce gross thrust and momentum drag as separate entities, taking the difference explicitly rather than implicitly. These latter models are applied in cases where the simulated aircraft is capable of flying at high angles of attack or can vector its engine exhaust nozzles in some way, as the thrust vector and the momentum drag vector may then deviate by a large angle. Interference between the engine exhaust flow and the tailplane may further affect the available applied forces. An additional example of the need for detailed engine modelling can occur with the simulation of a VSTOL (Vertical or Short Take-Off and Landing) aircraft, which in the hover, must use a proportion of the airflow through the engine to generate control forces by blowing out at the extremities. Over-active pilots, trying to keep the hovering of these type of aircraft, end up with a descending motion as a result of their control activities. These situation must be simulated correctly. Pilot s ability to control the aircraft also depend on the time needed by the engine to produce the request level of thrust. An engine can not change its thrust instantaneously: burning more fuel to increase thrust must first accelerate the rotating machinery of the engine and this takes time. The timescale varies in the range of few tents of

10 Mathematical models Elements of the model 3 8 a second to several seconds. This implies, besides the static, performance-orientated thrust modelling, the need for an adequate modelling of the growth or decay of thrust with time. Engine modelling is an area where a wide variety of techniques is used, according to the fidelity required to the engine dynamic response. A description of some engine models adopted in flight simulation practice will be given elsewhere in these notes Undercarriage model The simulation of take-off or landing require some means to produce the force which supports the aircraft in contact with the ground. This force is given by the undercarriage model. The real undercarriage is a complex mechanical, hydraulic and in some cases pneumatic assembly, which can be modelled in a variety of ways. A model of a spring coupled with a damper can provide the ground reaction in a simple way. An extra damping can be added for the recoil phase after the impact in landing. Detailed models are necessary in some simulation tasks, which demand the reproduction of tyres and brakes, anti-skid systems and nose-wheel steering Mass model The discussion so far has been concerned with the contributions to the external forces, broken down as F = F aero + F engine + F undercarriage (3.3) The other item which also need modelling, recalling the generic equations (3.1), is the set of vehicle s mass properties. These comprises the its mass, its moment of inertia and center of gravity, usually all function of payload, fuel state, and geometry, e. g. wing sweep, or stores configuration Atmosferic model A full atmospheric model is needed to represent a number of properties. The starting point is the variation of density and temperature with altitude, as defined by the

11 Mathematical models Elements of the model 3 9 International Standard Atnosphere (ISA). Non-standard ambient ambient conditions may also be needed, such as in cases when the training for hot day is of interest or when design checks have to be accomplished. These properties are usually known as atmospheric statics as opposed to wind environments, or atmospheric dynamics, which are needed in terms of mean surface wind for take-off and landing, and winds at altitude. Atmospheric dynamics, i. e. turbulence associated with wind, affect performance on a route schedule, for example, or can cause distraction and disturbance on pilot in a precise task, or sometimes gives rise to a design case for regulation authority or crew ride comfort. Methods of modelling atmospheric turbulence are nowadays quite advanced as a consequence of active research in the real world environment. Extreme phenomena, such as wind shear, responsible for several major accidents, are now modelled and are a legitimate item to be simulated, either for research into aircraft design or for training pilots to recognise and counter it effectively. Many training tasks demand a full range of metereological phenomena to be simulated, including weather radar returns and runway state Control system model The control system of an aircraft also needs modelling. It is interposed, in the real aircraft, between the pilot s control stick and the vehicle control surfaces, and consists typically of a system of levers, cables and bell-cranks, terminating in a hydraulicactuated jack. The mechanical deficiencies of these system have to be modelled, such as cable stretch, control jams etc. Current military aircraft and projected future civil transport aircraft are replacing mechanical connectionc by electrical ones, coupled with computer-driven systems shaping the pilot s demands and processing feedback signals. These developments make the modelling needs growing dramatically as more and more functions are included in the flight control computer. In this respect, incorporating real flight hardware in the simulator may be the best solution.

12 Mathematical models Equations of motion Equations of motion To the general discussion of mathematical models given above, follows here a presentation of some explicit mathematical formulations. Equation of motion will be outlined in this section while some forms of data representation and managing will be discussed later General equations The following exposition assumes that the reader is familiar with the simpler concepts of aerodynamics and the basic theory of aircraft stability and control. The mathematical formulation used here adopts the classical notation of alphabetical triads to represent the three components of force, moment, linear and angular velocity. These are illustrated in fig. 3.7 for the conventional right-handed orthogonal axes with origin fixed in the aircraft. No particular set of vehicled carried axes is yet assumed. The equations of motion of a rigid aircraft flying in still air, when referred to any system of axes fixed in the aircraft and rotatong with it, are W g ( u + qw rv) = X + W g g x W g ( v + ru pw) = Y + W g g y (3.4) W g (ẇ + pv qu) = Z + W g g z I xx ṗ I xy (q 2 r 2 ) I zx (ṙ + pq) I xy ( q rp) (I yy I zz ) qr = L I yy q I zx (r 2 p 2 ) I xy (ṗ + qr) I yz (ṙ pq) (I zz I xx ) rp = M I zz ṙ I xy (p 2 q 2 ) I yz ( q + rp) I zx (ṗ qr) (I xx I yy ) pq = N (3.5) These are sometimes known as the total force equations, as opposed to the small perturbation equations of classical stability and control.

13 Mathematical models Equations of motion 3 11 Precisely, the first three equations (3.4) are the force equations while the second three (3.5) are the moment equations in their complete form. The full non-linear set equations is sufficient to cope with general, large-scale motion of an aircraft and is soluble in these form by numerical integration techniques only once the forcing term X, Y, Z, L, M, and N on the right-hand side can be prescribed at each time step. The moment equations (3.5) are complete in the sense that all the cross-inertia terms have been retained. If, as for most cases, it can be assumed that the aircraft has a symmetrical distribution of mass with respect to the fore-and-aft plane of synnetry, then I yz = I xy = 0 and the equations become more compact I xx ṗ I zx (ṙ + pq) (I yy I zz ) qr = L I yy q I zx (r 2 p 2 ) (I zz I xx ) rp = M (3.6) I zz ṙ I zx (ṗ qr) (I xx I yy ) pq = N Such an assumption is reasonable for conventional aircraft, such as a transport or executive aircraft, but a fast-jet aircraft could well have, either intentionally or a result of a failure, an asymmetric distribution of stores beneaths its wing, in which case the complete equations (3.5) are necessary. It is well known the major influence that inertia distribution can have on the flying qualities of a fast-jet aircraft and must be included with care in a simulator s matematical model. Regarding the force equations (3.4), while simple in appearence, they pose some practical difficulties in their solution. These difficulties are associated firstly with product terms like qw, rv, etc. which, although nominally second-order, can become dominant in large-scale, vigorous manoeuvres, secondly, with the still air assumption. Some simulator mathematical models are implemented according to a formulation referred to a earth-based frame, to eliminate both the above difficulties. In an inertial frame of reference, the product terms disappear, as (i) the axis system is not rotating and (ii) the treatment of winds is simpler.

14 Mathematical models Equations of motion Small perturbation equations In many simulated situations, there is no alternative to use of the total force equations and fortunately modern computer techniques now permit this. However, in some cases it may still be desiderable to deal with a linearized form of the equations of motion, since they become amenable to treatment by a wide range of analytical techniques. Such procedures are important at a design stage of a new aircraft and also in research, when experiments are conducted to establish data on the fundamental handling qualities of aircraft based on simplified situations and models. If the aircraft s motion deviates to only a small extent from a datum, trimmed state, then the well-known small perturbation equations of motion are applicable. The datum state is also called equilibrium state in which flight quantities, say f assume a value f eq, and evolve in time being perturbed by a small variation f, which becomes the unknown, viz. f = f eq + f. Below are reported, as an example of formulation, the linearized equations of longitudinal motion. Assuming level flight (θ eq = 0), using aerodynamic axes (such that w eq = 0), and omitting some derivatives that are commonly neglected, one can write the following equations in matrix form u u x u x w x q x θ d w w = dt z u z w z q z θ + q q m u m w m q m θ θ θ x η z η m η 0 η (3.7) where η (sometimes denoted also with δ e ) is the elevator deflection, and the coefficients, or aerodynamic stability derivatives are constants matrix elements for a given aircraft configuration and datum flight condition, reported here in their concise form. A corresponding set of equations can be written for the latero-directional motion, but are not reported here.

15 Mathematical models Equations of motion 3 13 The small perturbation equations are rarely used in full-blown simulations, but can have relevance in research studies of handling qualities or to preliminary studies of a new design, when fundamental properties of an aircraft s dynamic behaviour may be examined Aircraft orientation The aircraft s orientation in space is given by its attitude, which may be defined in a number of ways. The most common method consists in defining a sequence of three angles, known as the Euler attitude angles. Starting with a set of axes, with origin C fixed in the aircraft, initially aligned with those of a fixed inertial, or anyway datum, reference frame Ox 0 y 0 z 0, and with C O, the first set of axes is brought into alignment with the body-fixed axes C xyz by rotating the first set successively through each angle in turn. The usual trio of angles consists of: ➀ the heading or azimuth or yaw angle ψ, ➁ the inclination or pitch angle θ, and ➂ the roll angle φ. The sequence of rotations is illustrated in fig In fig. 3.3(b) the intermediate axes set between successive rotations are labeled x 0 y 0 z 0, x 1 y 1 z 1, x 2 y 2 z 2, xyz. The sequence is significant, as a different order from that defined here will produce a different final orientation. With the definitions given here, the attitude angles can take the following range of values: π < ψ +π, π 2 < θ +π, π < φ +π (3.8) Kinematic relationships for Euler angles Given these definitions of Euler attitude angles, derivation of the angles in practice requires a relationship between the rates of change of the attitude angles and the components of angular velocity of the aircraft s body axes. When the latter is referred

16 Mathematical models Equations of motion 3 14 (a) Euler angles as a definition of aircraft orientation with respect to a datum frame Ox 0 y 0 z 0. Frame C xyz is body-fixed and represents the aircraft. (b) The sequence of rotations bringing the datum frame into alignment with the body-fixed axes. The classical color sequence Red-Green-Blue is applied accordingly. Figure 3.3: Euler attitude angles (φ, θ, ψ) and rotation sequence.

17 Mathematical models Equations of motion 3 15 to the body-fixed frame these relationships take the form φ 1 sin φ tan θ cos φ tan θ p θ = 0 cos φ sin φ q sin φ cos φ ψ 0 cos θ cos θ r (3.9) the inverse being p q = r 1 0 sin θ 0 cos φ sin φ cos θ 0 sin φ cos φ cos θ φ θ ψ (3.10) Note that φ p only when θ = 0 and θ q only when φ = 0. Equations (3.9), also known as gimbal equations are widely used, but do contain a singularity, when the x body axis becomes vertical (θ = ±90 ), tan θ and 1/ cos θ (= ± ) make the explressions for φ and ψ indeterminate. The situations in which this occur are clear. If such manoeuvres are avoided, then the equations may be used without difficulty. For the simulation of aircraft which perform aerobatics or similar gross manoeuvres, an alternative formulation is required so that the derivation of the aircraft attitude angles is always trouble-free The direction cosine matrix Transformation of variables between pairs of reference systems is often required in order, for example, to reformulate the equations of motion in a form suitable for the specific treatment of some terms. The expression providing the airplane center of mass velocity components referred to the body axes as a function of the corresponding components in earth axes is reported

18 Mathematical models Equations of motion 3 16 below. It is put in a form similar to the classical x y = z l 1 l 2 l 3 m 1 m 2 m 3 n 1 n 2 n 3 x 0 y 0 z 0 (3.11) where {x} = {x, y, z} T represents a generic vector quantity, (of velocity, say, or force) with components referred to body axes, and {x} 0 = {x 0, y 0, z 0 } T, the same entity but with components in the earth axes. finally the matrix in eq. (3.11) is denoted here as [R] is known as direction cosine matrix. The inverse relationship, giving earth axes components in terms of body axes components, is x 0 y 0 z 0 l 1 m 1 n 1 x = l 2 m 2 n 2 y l 3 m 3 n 3 z (3.12) in which the transformation matrix is written as the transpose of [R], as it is also its inverse: [R] T = [R] 1. When direction cosines are expressed in terms of the Euler attitude angles one has [R] = cos θ cos ψ cos θ sin ψ sin θ sin φ sin θ cos ψ cos φ sin ψ sin φ sin θ sin ψ + cos φ cos ψ sin φ cos θ cos φ sin θ cos ψ + sin φ sin ψ cos φ sin θ cos ψ sin φ cos ψ cos φ cos θ (3.13) The direction cosine may be defined and derived in other ways, but the transformation matrices written above are still needed and used.

19 Mathematical models Equations of motion The quaternion approach A practical alternative to the use of Euler equations for defining the orientation of an aircraft, which avoids singularities when pitch attitude reaches 90 degrees, is to use the so-called four parameter method or quaternion approach. It is well known that when a frame of axes Oxyz may be brought into coincidence with a reference frame Ox 0 y 0 z 0 by a single rotation D about a fixed axis in space, making angles A, B, C with reference axes Ox 0, Oy 0, Oz 0, respectively. These four parameters A, B, C, D can then define the orientation of the frame Oxyz with respect to the reference frame. It can be shown that the transformation matrix R relating a vector (x 0, y 0, z 0 ) in the reference frame to its representation in the new frame (x, y, z) according to (3.11) is given by l 1 l 2 l 3 e e 2 1 e 2 2 e (e 1 e 2 + e 0 e 3 ) 2 (e 1 e 3 e 0 e 2 ) m 1 m 2 m 3 = 2 (e 1 e 2 e 0 e 3 ) e 2 0 e e 2 2 e (e 2 e 3 + e 0 e 1 ) n 1 n 2 n 3 2 (e 0 e 2 + e 1 e 3 ) 2 (e 2 e 3 e 0 e 1 ) e 2 0 e 2 1 e e 2 3 (3.14) where e 0 = cos D 2 e 1 = cos A sin D 2 (3.15) e 2 = cos B sin D 2 e 3 = cos C sin D 2 are a new set of four parameters called quaternions. It can be shown that they are constrained by the equation e e e e 2 3 = 1 (3.16) Furthermore, it can be shown that the following espression are valid for quaternion

20 Mathematical models Equations of motion 3 18 rates ė 0 = 1 2 (e 1p + e 2 q + e 3 r) ė 1 = 1 2 (e 0p + e 2 r e 3 q) ė 2 = 1 2 (e 0q + e 3 p e 1 r) (3.17) ė 3 = 1 2 (e 0r + e 1 q e 2 p) These equations provide the mean to generate the quadruplet (e 0,..., e 3 ) from the body axis components of angular velocity p, q, r. Considering the constraint expressed by eqs. (3.16) above, one of the values obtained from time integration of eqs. (3.17) is effectively redundant. Implementation in a computer code of these equations has to take into account this constraint. One of the most widely diffused way to do this is to rewrite the quaternion rates as ė 0 = 1 2 (e 1p + e 2 q + e 3 r) + k λ e 0 ė 1 = 1 2 (e 0p + e 2 r e 3 q) + k λ e 1 ė 2 = 1 2 (e 0q + e 3 p e 1 r) + k λ e 2 (3.18) ė 3 = 1 2 (e 0r + e 1 q e 2 p) + k λ e 3 where k is a constant chosen such that k t 1 for an integration time step t and λ = 1 ( ) e e e e 2 3 (3.19) This is known as algebraic constraint method.

21 Mathematical models Equations of motion 3 19 Finally, expressed in terms of the three Euler angles, the four quaternions are e 0 = cos ψ 2 cos θ 2 cos φ 2 + sin ψ 2 sin θ 2 sin φ 2 e 1 = cos ψ 2 cos θ 2 sin φ 2 sin ψ 2 sin θ 2 cos φ 2 (3.20) e 2 = cos ψ 2 sin θ 2 cos φ 2 + sin ψ 2 cos θ 2 sin φ 2 e 3 = cos ψ 2 sin θ 2 sin φ 2 + sin ψ 2 cos θ 2 cos φ 2 Espressions (3.20) are necessary to derive initial values for (e 0,..., e 3 ) when attitude angles (ψ, θ, φ) are known. In the context of aircraft simulation, the direction cosines are still required for the transformation of variables from body axes to earth axes (and vice versa), and equation (3.11) may be used to derive them from quaternion parameters. The Euler angle derivatives are also required for displays on pilot s instruments and elsewhere. A suitable selection from eq. (3.14) gives sin θ = l 3 =2 (e 0 e 2 + e 1 e 3 ) =: θ = sin 1 ( l 3 ) (3.21) Since, by definition, the elevation angle θ of the x- body axis above the horizontal plane lies in the range ± π 2, π 2 θ +π 2 (3.22) the inverse sine process will then yeld a unique value of θ. Next, from eq. (3.14), one has cos ψ = l 1 /cos θ (3.23) and since cos θis always positive for the defined range of θ, then yelds cos θ sin ψ = l 2 (3.24) sgn [sin ψ] = sgn [l 2 ] (3.25)

22 Mathematical models Equations of motion 3 20 and hence ψ = cos 1 (l 1 /cos θ) sgn [l 2 ] (3.26) Similarly cos φ = n 3 /cos θ (3.27) φ = cos 1 (n 3 /cos θ) sgn [m 3 ] (3.28) To summarize the two approaches to aircraft orientation, the Euler technique is simple, widely known and widely used. It has the weakness consisting in the singularity at zenith and nadir. The quaternion technique is more complex and less familiar but is robust and will cope with all kinds of manoeuvres. Here, for the sake of comparison, are reported the solution sequences in each case: Euler angle approach Quaternion approach ➊ calculate body rates calculate body rates ➋ calculate Euler rates calculate quaternion rates ➌ calculate Euler angles calculate quaternions ➍ calculate direction cosines calculate direction cosines ➎ apply transformation matrix ➀ calculate Euler angles ➁ apply transformation matrix Axes and frames of reference In Flight Simulation, and generally in the study of Flight Mechanics problems, it is necessary to define and take into account some ets of axes as frames of reference (or coordinate frames) for many purposes and reasons. They are necessary to define the position of a point in space or to quantify the orientation of a body relative to another one. The general need is that of characterizing the motion of a body in terms of its

23 Mathematical models Equations of motion 3 21 position and orientation versus time. Of course by saying orientation of a body one means the orientation of a reference frame tied to it. Moreover, equations of motions of a body subject to external action are written down with respect to a specified coordinate frame, usually chosen to simplify the specification of particular groups, such as the aerodynamic forces and moments; or to take advantage of circumstances in which some terms vanish. The definition of reference frames is required also when there is the need to read or measure a vectorial quantity in terms of its scalar counterparts and to transform these entities from a frame to another one. There are many possible axis systems, but for simulation they fall into two broad classes: ➀ inertial or, under certain hypoteses earth axes, and ➁ body axes. Of particular interest also is the class of wind axes, which will be described below. Wind axes, as well as some earth axes, are vehicle-carried, i.e. they all have a moving origin superimposed to the vehicle s center of mass, but they are not rotating with the body, like a body axis system rigorously does. They are defined with respect to the relative wind, or to the trajectory in general, or to the local geographic position. All these frames are orthogonal, right-handed coordinate frames. Generally, the earth is supposed to be a sphere having its center of mass superimposed to its geometric center. Inertial frame {Oxyz} i This is a reference frame needed in every dynamics problem, explicitly defined or lurking implicitly in the background of theoretical formulations. An inertial frame is defined as a frame which is non-accelerated with respect to the fixed stars. In Navigation, a coordinate frame that has origin at the center of the earth and is non-rotating relative to the stars can be considered an inertial frame for measurements made in the vicinity of the earth. Earth frames The following earth frames are of interest in Flight Mechanics and Flight Simulation.

24 Mathematical models Equations of motion 3 22 Earth-Center Frame {Oxyz} ec This frame has its origin at the earth s center. Its axes are fixed in the earth. This frame is thus rotating with respect to the inertial frame {Oxyz} i previously defined. The z ec -axis points North and is supposed coincident with the earth s angular velocity vector, while the plane {Oxy} ec is equatorial. The axes of this plane can be arranged such that this earth frame and the inertial frame are superimposed at some initial time t = 0. The angular rate of the earth-centered frame, i.e. the angular rate of the earth relative to the inertial frame, is denoted by ω E. This frame is needed when the earth s rotation has to be necessarily taken into account, as in problems concerning, for instance, inertial navigation or space trajecoties. Tangent Frame {Oxyz} t This is an earth-fixed frame with origin near the vehicle. Its axes are aligned with the north, east, and down directions. Here down, or vertical, means the normal to the earth s surface. The north axis is in the direction of the projection of the earth s angular velocity vector into the local horizontal plane (this plane being perpendicular to the down direction). The east direction completes the right-handed orthogonal set. The definition of the tangent frame follows the so-called North-East-Down (NED) convention. Sometimes the origin of the tangent frame is chosen so that it is superimposed to the vehicle mass center at the start of a particular manoeuvre under study, or at its projection on the ground. When the earth is supposed to be spherical, the tangent frame is also a geocentric frame, in the geodesic jargon. The z t -axis has the same direction of the geocentric position vector from the earth s center to the origin O t, and opposite versus. The origin of a tangent frame is positioned, relative to the earth-center frame, through its geographic coordinates, i.e. the two angles: λ t, latitude of O t, and µ t latitude of O t. Vehicle-Carried Vertical Frame {Oxyz} v A simply defined moving reference frame is the one whose origin is fixed at the vehicle s center of mass C (whose origin is carried

25 Mathematical models Equations of motion 3 23 by the vehicle and follows it), but has its three axes aligned in such a way that the z v -axis is continuously directed vertically and the plane {Oxy} v is locally tangent to a sphere having radius R E + h, where R E is the earth s radius and h is the vehicle s altitude. For atmospheric flight one has h R E. Note that the vehicle-carried vertical frame is a frame moving with the body but is not a body frame (cfr. below). This frame follows the NED convention, i.e. the x v -axis points to the north and the y v -axis points to the east. Since the orientations of the tangent frame and of the vehicle-carried vertical frame are defined similarly, the latter can be transported over the former by two consecutive rotations in the sense of latitude and longitude, respectively. This means that the the moving origin of the vehicle carried vertical frame is positioned, at the generic time instant, in a point in space having { latitude: λv = λ t + λ longitude: µ v = µ t + µ where µ is a rotation around the earth-centered frame s z ec -axis and λ is a rotation around an axis which is parallel to the tangent frame s y t -axis and passing through the earth s center. The angular rate of the vehicle-carried vertical frame with respect to the inertial frame is usually denoted by ω v. It is useful to point out that in inertial navigation problems the earth s surface is supposed to be an ellipsoid. This is an analytically defined surfacewhich is an approximation to the mean sea-level gravity equipotential surface, also known as geoid. When the earth is supposed even spherical, the reference frame used for navigation coincides with the geocentric frame and the tangent frame is aligned with a geographic frame at a fixed location on the earth. The relative orientation of earth frames defined so far can be derived from fig Datum-Path Earth/Vehicle-Carried Frame {Oxyz} 0 In flight dynamics applications, another adopted definition for earth axes exists, useful when short term motion is of

26 Mathematical models Equations of motion 3 24 interest, and perfectly adequate. It is a reference frame moving with the earth, or with the aircraft, known as datum-path earth frame, or datum-path vehicle-carried frame. Flight above a flat earth is still assumed, initially straight and level. Straight and level flight implies a motion in a horizontal plane at a constant altitude, and whatever the subsequent motion of the airplane might be, the attude is determined with respect to the horizontal. Referring to the tangent frame defined above, a horizontal plane {Oxy} 0 is defined, being parallel to the plane {Oxy} t. The only difference is that the x 0 -axis points in the arbitrary initial direction of flight rather than to the north. The z 0 -axis points down as in the earth frame. It is only necessary to place the origin O 0 in the atmosphere at the most convenient point. Sometimes this reference frame is considered as moving with the aircraft by assuming its origin coincident with the vehicle mass center. But, unlike the vehicle-carried vertical frame, the vehicle-carried datum-path frame x 0 -axis gives constantly a unique reference direction, which is the initial flight direction, while keeping its {Oxy} 0 plane horizontal and its {Oxz} 0 plane vertical. It provides an inertial frame for short term airplane motion when the flight speed varies under the hypoteses of the small perturbation theory. Practical usage of earth frames When atmospheric flight only is of interest, it is usual to measure aeroplane motion with reference to an earth fixed framework like the tangent frame. Clearly the plane {Oxy} t defines the local horizontal plane, thus the flight path of an airplane flying in the atmosphere in the vicinity of the reference point O t may be completely described by its coordinates in the axis system, assuming a flat earth, where the vertical is tied to the gravity vector. This model is quite adequate to localized flight, where the effects of earth s rotation on vehicle motion are negligible, and the tangent frame becomes an inertial, or fixed, frame for the particular problem under study or motion under simulation. The widely used notation for the tangent frame in such cases is {Oxyz} e, which stands for earth frame, fig The earth frame is best

27 Mathematical models Equations of motion 3 25 suited to navigation and performances applications, and to flight simulations where flight path trajectories are of primary interest. For investigations involving trans-global navigation. the earth frame described above is inappropriate, the earth-centered ec frame being preferred. Body axes Body axes consist of an axis system fixed in the aircraft, and normally with origin C at its center of gravity. Body axes move in space and rotate with the aircraft. For a specific set of body axes it essential to know how the orientation of the axis system is defined with respect to the aircraft. Geometric-body axes {Oxyz} b Among possible body axis systems, geometric-body axes {Oxyz} b, or simply {Oxyz}, are usually those defined such that the x-axis is aligned with a geometric feature, such as the fuselage reference line (FRL) or a wing datum plane (WDP). Often such axes are known just as body axes. Both kinds of reference may be used for one mathematical model of the aircraft, for example WDP to define the aerodynamic forces and FRL for all other uses, including axis transformations. Tipically the inclination of the FRL with respect to the WDP is a small angle (one or two degrees). Principal-body axes {Oxyz} P An alternative set of body axes is the set of principalbody axes, defined such that the x P -axis is aligned with the principal inertia axis, thus making the cross-products of inertia I zx zero. This simplifies the lateral equations of motion, as can be seen by inspecting eq. (3.6), and is useful in theoretical studies, but is not particularly helpful in simulation, as the location of the principal axis could change with aircraft configuration. Geometric-body axes offer sufficient advantage in giving constant moments of inertia. Wind axes The following definitions introduce wind axes as they are generally accepted in the aeronautical practice. The subtle differences that arise between different

28 Mathematical models Equations of motion 3 26 Figure 3.4: Earth reference frames. ec: earth-centered frame; t: tangent frame; v: vehicle-carried vertical frame.

29 Mathematical models Equations of motion 3 27 Figure 3.5: Earth frame in flat earth hypothesis. types of wind frames of reference, which sometimes are referred to with the same name in different contexts, are also pointed out in this section. General Wind-axis Frame {Oxyz} w A general wind-axis system is defined such that the origin of a rectangular Cartesian system is at the center of gravity of the aircraft, having the x w -axis pointing into the direction of the oncoming free-stream velocity vector. The general wind z w -axis is more rigorously defined with unique reference to the trajectory of the aircraft mass center. it is defined as the intersection of the plane locally normal to the trajectory and the vertical plane containing x w. The remaining axis y w completes the system. According to this definition, the geometric-body axis z b is at an angle ν, known as the bank angle, with respect to the vertical plane {Oxz} w. The relative orientation

30 Mathematical models Equations of motion 3 28 of the earth, body, and wind frames defined here is illustrated in fig Figure 3.6: Relative orientation of the general wind axes, earth axes, and body axes. UK Wind-axis Frame {Oxyz} w According to this convention, the x w -axis is defined as in the general case. The z w -axis, instead, lies this time in the aircraft plane of symmetry, is perpendicular to the x w -axis and is directed generally downward. The y w -axis completes the system and is thus coincident with the body y b -axis. In many theoretical problems airplane motion is in the geometric plane of symmetry (no yawing motion) so that the x w -axis also lies in the plane of symmetry. The system then is termed the simplified wind-axis system, but is not of practical interest in flight simulation applications.

31 Mathematical models Equations of motion 3 29 Aerodynamic-axis Frame {Oxyz} a Aerodynamic axes, also called aerodynamic-body axes, have their origin at the aircraft center of gravity and have the x a -axis aligned with the projection of the velocity vector, on to the plane of symmetry. The z a -axis belongs to the aircraft plane of symmetry, the y a -axis completing the axis system accordingly. Note that the latter axis coincides with the body axis y b. According to this definition, aerodynamic axis x a is therefore at an angle α relative to the geometric body axis x b. Moreover, x a is an angle β relative to the geometric wind axis x w, i.e. to the direction of the relative wind. In the analysis of short-term motion, aerodynamic axes are also known as stability axes in USA, corresponding to the UK wind axes, if the x a -axis is the projection of the velocity vector on to the plane of symmetry in a datum flight condition Incidence angles Unlike attitude angles, which define the orientation of an aircraft with respect to a set of earth axes, incidence angles define the direction of the airflow relative to the vehicle s body, and are needed in the derivation of the aerodynamic forces acting on the airframe. Denoted with V r the opposite of the relative-wind velocity vector, the angle of attack α is the angle between ➀ the projection of V r on to the aircraft plane symmetry and ➁ the body x-axis. Note that the definitions given above for the reference frames consider the velocity vector of the aircraft mass center in a fixed reference frame. Nothing is said about the local wind. One could also redefine the wind frames starting from the direction of the total relative-wind velocity vector, including the local atmospheric wind. In that case α is the angle between the body axis x b and the aerodynamic axis x a. The sideslip angle formed by ➀ velocity vector V r with ➁ the aircraft s symmetry plane. It is the angle between the wind axis x w and aerodynamic axis x a. If the velocity components are known in the body axis system, the incidence angles

32 Mathematical models Equations of motion 3 30 Figure 3.7: The body- and aerodynamic- frames of references. The three components of force, moment, linear and angular velocity. Aerodynamic angles.

33 Mathematical models Equations of motion 3 31 Figure 3.8: The Euler angles (ψ, θ, φ) with respect to the definition of general wind axes, earth axes, and body axes.

34 Mathematical models Equations of motion 3 32 are defined as (fig. 3.7 and 3.9) where V 2 r tan α = w u, sin β = v V r (3.29) = u 2 + v 2 + w 2. The ranges of variations of α and β are π < α +π, π 2 < β +π 2 (3.30) Direction angles Direction angles define the direction of the flight path velocity vector V with respect to a fixed reference frame, e.g. the earth axes. The angles, shown in fig. 3.9, are the climb angle γ and the track angle χ. The climb angle is that formed by the wind axis x w with the horizontal plane {Oxy} e, positive for a climbing path. The track angle is that formed by the wind axis x w with the reference vertical plane {Oxz} e, i.e. the angle between the local tangent to the ground track and the northpointing axis x e. It is positive for an east-travelling aircraft. The difference between track angle and heading ψ is called drift and is due to the presence of winds along the flight path. The summation of the relative wind due to the motion of the aircraft with respect to the earth reference frame, and of the local atmospheric wind, given or modelled in terms of wind velocity components in earth axes, gives the instantaneus relative wind experienced by the aircraft, the same relative wind which defines the incidence angles introduced above. If the flight path velocity vector is V C, i.e. the velocity of the aircraft mass center C, its components in earth axes u e, v e, w e are u C e = V C cos γ cos χ ve C = V C cos γ sin χ ve C = V C sin γ Incidence angle, attitude angles and direction angles can be related but the expressions are complex and omitted here. In the absence of winds, in steady, straight

35 Mathematical models Equations of motion 3 33 symmetric (i.e. wings-level) flight, it can be shown that γ = θ α (3.31) but it must be emphasized that this helpful relationship is only true in the restricted but common circumstances quoted. Figure 3.9: Direction angles. Climb γ, track χ.