A Simulink Toolbox for Flight Dynamics and Control Analysis

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FDC 1.4 – A S

IMULINK

Toolbox for

Flight Dynamics and Control Analysis

DRAFT VERSION 7

Marc Rauw

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The Flight Dynamics and Control Toolbox for MATLABand SIMULINKis distributed via the Internet, together with this FDC report, which is available in PostScript and PDF format. If you wish to make a hard-copy of the report, it is advised to download the PostScript version and send it to a PostScript printer or a software PostScript interpreter such as GHOSTSCRIPT(see http://www.ghostscript.com for more information).

This document was created with MIK_{TEX in L}A_{TEX 2ε format. The vector figures were drawn in TEX}_CAD_{(part of}

theEM_{TEX package for MS-DOS), M}ATLAB, SIMULINK, and MAYURADRAW. The screenshots were created with IRFANVIEWand the wonderfulJPEG2PStool. The Postscript graphics were prepared with the DVI-driverDVIPS, GHOSTSCRIPTand GSVIEW; the PDF version was created with GHOSTSCRIPTand ADOBEACROBAT.

The Flight Dynamics and Control Toolbox has been released under and is subject to the terms of the Open Soft-ware License, version 2.0, a copy of which has been included in appendix F.

This report has been released under and is subject to the terms of the Common Documentation License, ver-sion 1.0, the terms of which are incorporated in appendix G. Please read it before using this material - your use of this material signifies your agreement to the terms of the License.

Instead of a list: All trademarks mentioned in this document are registered to whoever it is that owns them.

To contact the author, please send an e-mail to: [email protected]. For the most actual information about the FDC toolbox, see the Dutchroll homepage: http://www.dutchroll.org or http://www.dutchroll.com.

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Chapter 1 Flight control system development

During the last decades, active flight control technology has dramatically changed the way aircraft are designed and flown. Flight control systems with mechanical linkages have been replaced by full authority, ‘fly-by-wire’, digital control systems. As a consequence, the flying qualities of modern aircraft are largely determined by a set of control laws in the heart of a computer system.

Modern computer assisted control system design (CACSD) software provides a wide variety of user-friendly analytical tools that can assist in flight control system

(FCS) design and analysis. A typical example is the MATLAB/ SIMULINKsoftware

environment from The Mathworks, which offers advanced modelling and simula-tion capabilities and easy access to control system design tools. The Mathworks and other vendors offer several specialized toolboxes for a wide range of control system design methods.

A prerequisite for successful FCS design is the availability of sophisticated math-ematical models of the airplane, the environment it has to operate in, and elements from the control system itself. The FDC toolbox tries to offer such models for the

MATLAB / SIMULINK environment, and several tools and utilities to access those

models. This chapter provides an overview of flight control systems in general, and offers some insight in the FCS design process, in order to put the objectives behind the FDC toolbox into the right perspective.

1.1 Automatic flight control systems

A major factor in Wright brothers’ success in achieving the first powered flight in De-cember 1903 has been the emphasis they placed on making their aircraft controllable by the pilot, rather than inherently stable. However, the difficulties of controlling early aircraft and the progress toward longer flight times, expanded flight-envelopes, and bigger aircraft created a need for the development of power-driven aerodynamic control surfaces, stability augmentation systems, and ‘automatic pilot systems’. This evolution of flight control systems has been parallelled by the development of closed-loop control theories and major achievements in computer technologies [33].

Since the term ‘automatic flight control systems’ can be used to describe a wide variety of controllers, categorizing them into different categories can be beneficial. The first class of automatic controllers are the Stability Augmentation Systems (SAS).

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2 Chapter 1. Flight control system development

These systems are used to damp and stabilize high-frequency rotational modes of the aircraft, making it easier for pilots to control the aircraft. Common types of SAS are roll dampers, pitch dampers, and yaw dampers.

If an augmentation system is intended to control a rotational mode and to pro-vide the pilot with a particular type of response to the control inputs, it is known as a Control Augmentation System (CAS) [33]. Examples are controllers for the roll rate, pitch rate, and normal acceleration of an aircraft. CAS systems that are coupled to a conventional control system with a direct mechanical linkage between the control actuators and the control column, are often called ‘control wheel steering systems’ (CWS). If the connection is achieved by means of electrical (or optical) wires, such controllers are usually called ‘fly-by-wire systems’.

Although the slow modes of an aircraft (phugoid and spiral) are more easily con-trollable by a pilot, it still is undesirable for a pilot to have to pay continuous atten-tion to controlling these modes, especially during longer flights. Therefore, an auto-matic controller is often needed to provide ‘pilot relief’. An autopilot is an autoauto-matic control system that provides both pilot relief functions and special functions such as automatic landing. Typical autopilot functions are attitude hold, altitude hold, turn-coordination, heading select, automatic approach, and VOR navigation. Modern aircraft also allow the autopilot to be coupled to a flight management system (FMS), which offers flight path optimisation and wide-area navigation through the use of inertial reference system and global positioning system guidance and navigation.

In summary, we can conclude that the main function of an FCS is to contribute to the safe, comfortable, and economic operation of the aircraft, such that the in-tended flight missions can be accomplished and unexpected events can be handled. A modern FCS consists of three main components [24]: (i) sensors, which provide the flight control computers information on air data, inertial data, and cockpit data, (ii) the flight control computers themselves, which are used to evaluate the flight con-trol laws, and (iii) the actuation systems of the aircraft concon-trol surfaces and throttles. Feedback control is used to provide tight pilot command tracking, to attenuate ex-ternal disturbances such as gusts and turbulence, and to provide robustness against modelling errors.

Fly-by-wire systems allow the pilot to control aircraft states, in contrast to the con-ventional direct control of aerodynamic control surfaces and engines. This offers enhanced flexibility for the control laws, which provides new opportunities to in-crease the overall safety level. For example, error-tolerant control laws can provide flight envelope protection, and help the pilot to successfully achieve critical flight manoeuvres and to recover from unusual attitudes. The use of a modern FCS can also be beneficial from an economic point of view: fuel consumption can sometimes be reduced by relaxing static stability, counteracted by the application of active con-trol, and the weight of fly-by-wire systems is often lower than that of conventional control systems. Furthermore, fly-by-wire systems make it possible to give differ-ent aircraft of differdiffer-ent sizes almost the same ‘feel’, allowing the creation of a large aircraft family that will significantly reduce pilot training costs.

Most importantly, modern flight control systems have contributed to improved dynamical behaviour. Certain military aircraft cannot be flown without a stability augmentation system; they use the open-loop instability, which is related to the

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1.2. The FCS design cycle 3

agility of the aircraft, to obtain better performance and improved manoeuvrability of the closed-loop system. For civil aircraft, active flight control systems can provide gust suppression and auto-trimming, in order to achieve improved ride quality.

A disadvantage of these techniques is that the costs and efforts involved to de-velop an advanced FCS have escalated over the years. The danger exists that the economical benefits described above are nullified by higher design and maintenance costs, while the increasing complexity of modern flight control systems can poten-tially have a negative effect on safety. Clever use of modern FCS design techniques and optimisation of the design tools will be necessary to counteract these disadvan-tages [24].

1.2 The FCS design cycle

Regardless of the level of complexity, it is always possible to divide the control sys-tem design process into a number of distinct phases. A practical division is given in ref.[26] (a reference, which, albeit somewhat outdated with respect to the available computer hardware and software tools, is still valuable for this discussion):

1. Establish the system purpose and overall system requirements. System pur-pose can be equated with mission or task definitions, while the system require-ments can be separated in (i) operational requirerequire-ments, derived from the func-tions needed to accomplish the mission phases and (ii) implied requirements, de-rived from the characteristics of the interconnected components of the control system and the environment in which they operate.

2. Determine the characteristics of unalterable elements, command-signals, and external disturbances. The characteristics of some parts of the system cannot easily be changed by the designer. Often the vehicle itself, its control surface

actuators and sometimes also its sensors are ‘unalterable’.1 Moreover, the

struc-ture of the commands and disturbances is a direct consequence of the mission requirements and the environment in which the control system has to operate. 3. Evolve competing feasible systems, i.e. determine the basic block diagrams.

Usually, there are more ways to achieve the requirements, e.g. with different sensed motion quantities and/or the application of different control theories. Then it is possible to evolve competing candidate systems for selection on the basis of certain desirable properties.

4. Select the ‘best’ system. The competing designs can be compared on the basis of (i) design qualities, which include dynamic performance (speed of response, bandwidth, etc.) and physical characteristics (volume, weight, power consump-tion, etc.), and (ii) design quantities, which include safety, reliability, maintain-ability, cost, etc. An optimum system is one that has some ‘best’ combination of these features.

1_{If the FCS is designed for an all-new aircraft, the selection of the hardware (sensors, actuators,}

computers, etc.) must be included in the FCS design and analysis instead of taking the hardware as being unalterable. In this report, all hardware is considered to be given, so we will concentrate on issue of developing appropriate control laws to make a given aircraft fly a certain mission.

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5. Study the selected system in detail. The selected system must be evaluated for all normal and abnormal operating conditions. At each step in the FCS valida-tion the assumpvalida-tions made earlier in the FCS design process must be checked for validity. If necessary, a new iteration of the design should be started from the point where the wrong assumption was made.

This scheme reflects the FCS design process within a manufacturing environment. In a research context, the design process may have to be modified somewhat, as the research and manufacturing tasks are clearly different: whereas the task of research is to determine what is required and to produce a clear and comprehensive defini-tion of the requirements, the manufacturing task is to make and deliver a reliable and effective product [35]. Consequently, the first FCS design phase in particular will often be different in a research environment, because the system requirements are often poorly understood or may even be the objective of the research itself. In addition, the design tools may still be immature, and their development may itself be an objective of the research. See for instance the development of the FDC toolbox in the context of the Beaver autopilot studies.

The design, simulation, and implementation of control laws within a research context will be similar to the manufacturing task, although more flexibility of the tools will sometimes be required. For instance, it should be possible to rapidly alter the control laws within the flight control computers of the aircraft in order to evaluate different solutions to typical control system design problems with a minimum of programming efforts. In the research environment, step 4 does not necessarily need to include the selection of a ‘best’ system since it may be useful to evaluate competing control solutions, if necessary even all the way up to evaluation in real flight, just to gain more knowledge about their advantages and disadvantages. A typical example of this can be found in ref.[24], which treats a GARTEUR design challenge of robust flight control systems.

Finally, the requirements with respect to fail-safety of the FCS may be less restric-tive in a research context than for manufacturing. For instance, during the autopi-lot design project for the DHC-2 Beaver laboratory aircraft, a single flight control computer (a ‘luggable’ 80286 Compaq PC, coupled to a 16-bit ROLM-1603 general-purpose computer that handled the I/O functions; see refs.[6], [28], [37], and [38]) was used, whereas production aircraft normally apply multiple FCCs that constantly monitor eachother’s command signals.

1.3 Taking a closer look at the FCS design cycle

Let’s explore the individual design phases a little further. Figure 1.1 visualizes the first step in the FCS design process: the definition of the mission, which imposes re-quirements upon the shape of the flight-path and the velocity along this flight-path. This translates into the control problem depicted in figure 1.2: how to generate ap-propriate deflections of aerodynamic control surfaces or changes in engine power or thrust such that this mission can be fulfilled.

The classical approach to the FCS design problem is to start with the complete set of non-linear equations of motion, and then make assumptions which enable these equations to be linearized about some local equilibrium point. Getting familiar with

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1.3. Taking a closer look at the FCS design cycle 5

Mission

Flight-path ( )t

Control Surface Deflections ( )δ t

Figure 1.1:Definition of the aircraft’s mission

Specify Control Problem

Design Control

Laws Achieve SpecifiedBehaviour

Model

Real World

Figure 1.2:The general flight control problem

the dynamical behaviour by means of trimming, stability and control analysis, and nonlinear open-loop simulations (for stable aircraft), and understanding the influ-ences of the modelling assumptions is very important, and linear simulation of the aircraft model may also be required at this stage [24].

The next step is to define the controller architecture. In the initial phase of the FCS design, control system design tools based upon linear system theory can be applied to linearized models of the aircraft and its subsystems; modern CACSD software provides the required computer support. Although the linear control system design and analysis techniques will provide insight in the essential behavior of the FCS, only relatively small deviations from the equilibrium state are permitted before the results start to deviate from those of the real aircraft.

Luckily the main purpose of many FCS control laws is to keep the deviations from the equilibrium state as small as possible, e.g. in order to maintain a certain alti-tude or heading, but there are other control laws which require large deviations from nominal values, e.g. selecting a new reference heading or altitude which differs

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con-6 Chapter 1. Flight control system development

siderably from the original value. For this reason, detailed nonlinear simulations will be required, in order to validate (and possibly enhance) the results from the linear analysis and design phase. Gain scheduling functions need to be implemented, to ensure that the FCS will work well over the compete range of the flight-envelope for which it is designed, taking into account a suitable safety margin. Also, sig-nal limiters will be required to deal with certain physical limitations (e.g. the max-imum deflection of control surfaces) and for safety reasons. If a robust design is to be achieved, much attention should be given to understanding nonlinearities and model uncertainties.

This off-line analysis, which should cover a wide range of velocities and altitudes and all possible aircraft configurations, can be performed on a single PC or

work-station, using an integrated design and simulation environment such as MATLAB/

SIMULINK. ‘Off-line’ in this context means that the analysis does not have to be per-formed in real-time and does not yet include piloted flight-simulation. In a later stage the control system will have to be evaluated in a real-time pilot-in-the-loop simulation environment, to enable test-pilots to assess the handling qualities of the automatically controlled aircraft. In particular, the pilot–aircraft interaction should be examined thoroughly, especially if the pilot will be actively involved in the air-craft control loop (e.g. for fly-by-wire systems).

Based upon these results it is possible to select the best solution if there are more feasible methods to fulfill the mission requirements. If the results of the on-line and off-line analysis are completely satisfactory the next step in the design process will be the implementation of the control laws in the flight control computer(s) of the aircraft. The actuators and sensors in the aircraft must be installed (if that hasn’t been done already), thoroughly tested, and calibrated. For some purposes, e.g. aircraft certification, it may even be useful to test the complete control system in an ‘Iron Bird’ test-stand arrangement with hardware-in-the-loop simulation capabilities.

Special care has to be taken to prevent errors due to the conversion of control laws when moving from one design phase to the next. In particular, the transfer of control laws from the simulation environment to the flight control computers of the aircraft needs to be carefully verified, to ensure that the control laws used in flight

match those analyzed on the ground.1In order to reduce the risks of such conversion

errors, it would be beneficial to be able to couple at least the complete FCC software, but preferably also its hardware, to the real-time flight-simulator and/or off-line de-sign environment.

After successfully concluding the simulations and ground tests of the hardware and FCC software, the FCS can be evaluated in real flight. In an ideal world this phase would only be a straightforward verification of the previous results, but in practice it will often be necessary to return to a previous stage in the FCS development for fixing errors or fine-tuning the control laws. It also may be necessary to update the mathematical models if deficiencies in these models are found during the in-flight

1_{During the Beaver autopilot studies some dramatic examples of conversion errors were}

encoun-tered, luckily for a large part before the flight-tests were started. Still, some minor errors remained undis-covered until the actual flight-tests! References [28], [37], and [38] provide ample food for thought for future FCS projects.

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1.4. FCS design environments 7

evaluations. Quantitative results from the flight tests need to evaluated on-ground to confirm the correct control behavior. Obviously, the off-line CACSD environment that was used for the FDC design can also provide the required computer support for this post-flight analysis.

The iterative nature of this FCS development cycle should be acknowledged: at any stage in the process, the discovery of a fault, design error, or previously unrecog-nized uncertainty might require the return to a previous design stage. Figure 1.3 summarizes the FCS design process. It illustrates the different design stages from ref.[26] and the more detailed divisions presented in refs.[12], [31] and [35], and it clearly acknowledge the iterative nature of the whole process. On the left-hand side of the figure the models and tools (software and hardware environments) are shown, while the right-hand side shows the design stages themselves.

1.4 FCS design environments

It is obviously very important to make the transitions between the different develop-ment phases as straightforward as possible, in order to reduce the number of tran-sition errors which inevitably will arise if the tools for the different phases are not compatible (Murphy’s Law), and also to reduce the time needed for the FCS develop-ment.

For this reason, there is a need for an integrated software environment that pro-vides full access to all required mathematical models, as well as a large range of linear and nonlinear control system design and simulation tools, preferably from a single PC or workstation. The software tools for the off-line analysis should be able to effectively communicate with eachother, the tools for real-time pilot-in-the-loop simulations, and with the flight control computers of the actual aircraft. Access to the mathematical models and the control systems should be made as easy as possible, using a modern (graphical) user-interface.

Refs.[12] and [31] present some practical examples of integrated FCS design envi-ronments. These papers particularly emphasize the need for multidisciplinary design in which aerodynamic, structural, propulsive, and control functions are considered all together. This is important, because modern flight controllers may excite struc-tural modes of the aircraft and interact with the control-actuator dynamics, and be-cause of the increasing need to integrate flight controls with engine controls and load-alleviation functions. Interactions between the aerodynamic, propulsive, and structural models must be taken into account, especially for modern aircraft that em-ploy extensive use of composite materials (resulting in greater aero-elastic coupling) and relaxed static stability.

The Flight Dynamics and Control toolbox that is presented in this report attempts

to bring the required tools and models together in the MATLAB / SIMULINK

envi-ronment. It basically provides easy access to the flight dynamics models and other

relevant dynamic models, allowing the FCS designer to use MATLAB (if necessary

including a selection of the available control system design tools) for the initial FCS

development, and use SIMULINKfor the subsequent nonlinear off-line simulations.

The toolbox itself provides an analytical software tool to trim aircraft models for

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de-8 Chapter 1. Flight control system development M S F A A M S F Linearized models SIMULINK model-library with nonlinear aircraft model Flightsimulator model-library with nonlinear aircraft model Linear FCS design Nonlinear FCS validation & fine-tuning Evaluation in real-time piloted flight-simulation Implementing FCS hardware and software Evaluation in real flight Trimming & Linearization Future: automatic transfer of models?

Update sensor/actuator models

Update aero/engine models

Current scope of the FDC envi-ronment Future enhance-ments M S F A = = = = MATLAB SIMULINK Real-time flightsimulator Actual aircraft

Figure 1.3:FCS design cycle using MATLABand SIMULINK

scriptions of aircraft models and perform digital flight simulation. Several other

MATLABtoolboxes, from The Mathworks and others, can be used to perform a

mul-titude of analytical operations on the linear equations.

1.5 FCS design and the FDC toolbox

The current scope of the FDC toolbox is shown in figure 1.3, above the dashed line on

the right. Starting from the SIMULINKmodel library, it is possible to obtain linearized

models. Using designated control system design tools from other MATLABtoolboxes,

controllers can be developed for the resulting linear systems. The resulting control laws can be evaluated in SIMULINK, by means of nonlinear simulations; the required models are again obtained from the FDC model library.

However, the FDC toolbox does not (yet) offer any help to simplify the subse-quent design steps, being evaluation in a real-time piloted flight simulator and

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eval-1.5. FCS design and the FDC toolbox 9

}

Off-line

}

Real-time

Graphical SIMULINK system Graphical SIMULINK system

Flight Control Computer

Flight Simulator Control Laws (block-diagram) Control Laws (C-code) Control Laws

(C-code) _{Control Laws}

(C-code)

C-code generator

Figure 1.4:‘Portable’ control laws

uation in real flight. If the designer wants to move his control laws to these next levels, he or she will still have to convert the control laws and the dynamic models manually to the flightsimulator and/or the flight control computers of the airplane; the FDC toolbox does not provide any assistance for that task. This is obviously still a major obstacle in the FCS development process, because it increases the chance of encountering conversion errors, as explained earlier.

The figure suggests that automatic transfer of complete simulation models from

the SIMULINKenvironment to a flightsimulator might be feasible. However, it would

probably not be easy to ensure the integrity of such automatically converted models. A less ambitious, but perhaps more realistic proposal would be to focus on auto-matic conversion of the control laws only, using e.g. autoauto-matic code-generation

soft-ware to translate SIMULINKmodels into a high-level programming language like C.

Dedicated interface-subroutines that will optimize communications between the off-line SIMULINK-environment and the on-off-line flightsimulator environment could help streamline these conversions. This concept of ‘portable’ flight control laws has been illustrated in figure 1.4, assuming that the language C is used to implement the flight control laws in the flightsimulator and FCCs.

The Mathworks offers several solutions which could be helpful for this purpose,

such as the REALTIMEWORKSHOPand the REALTIMEWORKSHOP EMBEDDED

CO-DER, the latter of which targets time embedded processors, DSP boards, real-time operating systems, and PCs. Other useful third-party tools that could be ap-plied to interface with the FDC models are STATEFLOW, which can e.g. be used to

cre-ate very complex mode-controller systems, the DIALS& GAUGESBLOCKSET, which

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and the VIRTUALREALITY TOOLBOX, which provides three-dimensional animation

facilities for SIMULINKmodels. There are also several third-party products available

that allow SIMULINK models to be coupled to specific experimental hardware

se-tups. By integrating MATLAB, SIMULINK, and other tools that allow easy interfacing with external simulation devices or FCC’s, quick prototyping of flight control laws becomes feasible, and the transitions between off-line simulations, real-time simula-tions, and actual flight could be streamlined enormously.

It is not likely that these functions will soon be integrated in future versions of the FDC toolbox, no matter how exciting those prospects may be. Any future work on this software will probably mainly be concentrated on optimizing and improving the existing tools and models, and widening the application areas of the toolbox within its current scope. However, it is hoped that this far-reaching vision of FDC’s future will inspire others to develop their own add-ons or variants of the software, so that maybe one day some parts of this vision will be realized.

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Chapter 2 Rigid body equations of motion

Before we can start building the mathematical model of the aircraft, some fundamen-tal knowledge about the equations of motion is needed. In this chapter, the equations of motion of a rigid aircraft will be derived and expressed in the state-space form; a summary of these state equations will be given in section 2.5. The next chapter will build upon these equations to construct the nonlinear six-degree-of-freedom aircraft

model. 1

2.1 General rigid-body equations of motion

The aircraft equations of motion will be derived from Newton’s laws, which state the connection between force and motion. We start by deriving the general force and moments equations for a rigid body and defining the relations for the angular mo-mentum. This results in six ordinary differential equations, representing the linear and angular accelerations in the body-fixed reference frame.

2.1.1 General force equation for a rigid body

Consider a mass point δm that moves with time-varying velocity V under the influ-ence of a force F. Both V and F are measured relatively to a right-handed orthogonal reference frame OXYZ. This reference frame may be moving with a constant linear velocity, relative to the fixed position of the stars (a.k.a. ‘inertial space’), but it may not accelerate or rotate. Applying Newton’s second law yields:

δF=δm·V˙ (2.1)

Applying this equation to all mass points of a rigid body and summing all contribu-tions across this body yields:

∑

δF=

∑

δmdV

dt =

d

dt

∑

Vδm (2.2)

Let the center of gravity of the rigid body have a velocity Vc.g. with components

u, v, and w along the X, Y, and Z-axes of the right-handed reference frame. The

velocity of each mass point within the rigid body then equals the sum of Vc.g. and

1_{The derivation of the rigid-body equations has been extensively discussed in literature. This}

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12 Chapter 2. Rigid body equations of motion

the velocity of the mass point with respect to this center of gravity. If the position of the mass point with respect to the c.g. is denoted by the vector r, the following vector equation is found: V=Vc.g.+˙r (2.3) therefore:

∑

Vδm=

∑

(Vc.g.+˙r)δm=mVc.g.+ d dt

∑

rδm (2.4)

In this equation, m denotes the total mass of the rigid body. In the center of gravity we can write:

∑

r δm=0 (2.5)

so the equation for the total force F acting upon the rigid body, becomes:

F=m ˙Vc.g. (2.6)

2.1.2 General moment equation for a rigid body

The moment δM, which is measured around the center of gravity, is equal to the time-derivative of the angular momentum of the mass point relative to the c.g.:

δM= d dt(r×V)δm= (˙r×V)δm+ (r×V˙)δm (2.7) where: ˙r= V−Vc.g. (2.8) and: (r×_V˙)δm=r×δF=δMc.g. (2.9)

In this equation δMc.g.denotes the moment of the force δF about the center of gravity.

The angular momentum of the mass point relative to the c.g. will be denoted by δh,

which is defined as: δh ≡ (r×V)δm. Writing this out yields:

δMc.g.=δ ˙h− (V−Vc.g.) ×Vδm=δ ˙h+Vc.g.×Vδm (2.10)

The contributions of all mass points are once again summed across the whole rigid body, yielding:

∑

δMc.g.=

∑

δ ˙h+Vc.g.×

∑

Vδm (2.11)

The equation for the resulting moment Mc.g.about the c.g. then becomes:

Mc.g.= ˙h (2.12)

where h denotes the resulting angular momentum of the body about the center of gravity.

2.1.3 Angular momentum around the center of gravity

Consider a rigid body with angular velocityΩ, with components p, q, and r about

the X, Y, and Z axes of the right-handed reference frame respectively:

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2.1. General rigid-body equations of motion 13 symbol definition Ixx ∑(y2+z2)δm Iyy ∑(x2+z2)δm Izz ∑(x2+y2)δm Jxy ∑ xy δm Jxz ∑ xz δm Jyz ∑ yz δm

Table 2.1:Moments and products of inertia

where i, j, and k are unity vectors along the X, Y, and Z-axes. The total velocity vector of a mass point of a rigid body that both translates and rotates becomes:

V=Vc.g.+Ω×r (2.14)

hence, the angular momentum of the rigid body about the c.g. can be written as:

h≡

∑

δh =

∑

r× (Vc.g.+Ω×r)δm=

∑

r×Vc.g.δm+

∑

r× (Ω×r)δm

(2.15) The first term of the right hand side of equation (2.15) is equal to zero:

∑

rδm

×Vc.g.=0 (2.16)

and for the second term we can write:

∑

r× (Ω×r)δm =

∑

Ω(r·r) −r(Ω·r) δm =

∑

Ωkrk2−r(Ω·r) δm

(2.17) Substitution of r =i x+j y+k z, (2.16), and (2.17) in equation (2.15) yields:

h=Ω

∑

(x2+y2+z2)δm−

∑

r(px+qy+rz)δm (2.18)

The components of h along the X, Y, and Z axes will be denoted as hx, hy, and hz

respectively, yielding:

hx = p∑(y2+z2)δm−q∑ xy δm−r∑ xz δm

hy = −p∑ xy δm+q∑(x2+z2)δm−r∑ yz δm

hz = −p∑ xz δm−q∑ yz δm+r∑(x2+y2)δm

(2.19) The summations appearing in these equations are defined as the inertial moments

and products about the X, Y, and Z axes respectively; see table 2.1.1 Using these

definitions, equations (2.19) can be written in vector notation as a product of the

inertia tensor I with the angular velocity vectorΩ:

h=I·_Ω (2.20)

where I is defined as:

I=   Ixx −Jxy −Jxz −Jyx Iyy −Jyz −Jzx −Jzy Izz   (2.21)

1_{The summations across the body actually have to be written as integrals, but that further}

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If the airplane is symmetrical, Jxyand Jyzare both identically zero, which would

fur-ther simplify the inertia tensor. Although this assumption is often made in literature, this report will consider the more general case where the aircraft does not necessarily have to be symmetrical.

So far, in this analysis of angular motion we have neglected the angular momentum of any spinning rotors, assuming that the airplane is a single rigid body. However, these effects can actually be quite significant in practice. For example, a number of World War I aircraft had a single ‘rotary’ engine that had a fixed crankshaft and rotat-ing cylinders; the gyroscopic effects caused by the angular momentum of the engine gave these aircraft tricky handling characteristics. The effects are also noticeable in maneuvering flight of a small jet with a single turbofan engine on the longitudinal axis [33].

Each rotor has an angular momentum relative to the body axes. This can be computed from equation (2.20) by interpreting the moments and products of inertia therein as those of the rotor with respect to the axes parallel to the airplane body-axes and origin at the rotor mass center. The angular velocities in equation (2.20) are interpreted as those of the rotor relative to the airplane body axes [15]. If the

resultant relative angular momentum of all rotors is called h0, with components h0_x,

h0_y, and h0_z in FB, which are assumed to be constant, the total angular momentum of

an airplane with spinning rotors can be obtained by simply adding h0 to the angular

momentum of the airframe:

h=I·Ω+h0 (2.22)

The additional terms in the angular momentum cause certain extra terms, known as gyroscopic couples, to appear in the moment equations, as we will see later.

2.1.4 Resulting general equations of motion for a rigid body

When we choose a reference frame fixed to the body (OXYZ=OXBYBZB) the inertial

moments and products from the equations (2.19) become constants. The reference

frame itself then rotates with angular velocityΩ. For an arbitrary position vector r

with respect to the body reference frame we can then write: ˙r= ∂r

∂t+Ω×r (2.23)

Applying equation (2.23) to the general force and moment equations for a rigid body, (2.6) and (2.12), we find: F=m ∂Vc.g. ∂t +Ω×Vc.g. (2.24) and: Mc.g. = ∂h ∂t +Ω×h = ∂ ∂t I·Ω+h 0 +Ω× I·Ω+h0 = = ∂ ∂t(I·Ω) +Ω× (I·Ω) +Ω×h 0 _(2.25)

The last term in this equation, Ω×h0, contains the gyroscopic couples, which take

into account the effect of spinning rotors. Notice that this derivation assumes the resulting angular momentum of the rotors to be constant.

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2.1. General rigid-body equations of motion 15 symbol definition |I| IxxIyyIzz−2JxyJxzJyz−IxxJyz2−IyyJxz2−IzzJxy2 I1 IyyIzz−Jyz2 I2 JxyIzz+JyzJxz I3 JxyJyz+IyyJxz I4 IxxIzz−Jxz2 I5 IxxJyz+JxyJxz I6 IxxIyy−Jxy2 Pl I1 / |I| Pm I2 / |I| Pn I3 / |I| Ppp −(JxzI2−JxyI3) / |I| Ppq (JxzI1−JyzI2− (Iyy−Ixx)I3) / |I| Ppr −(JxyI1+ (Ixx−Izz)I2−JyzI3) / |I| Pqq (JyzI1−JxyI3) / |I| Pqr −((Izz−Iyy)I1−JxyI2+JxzI3) / |I| Prr −(JyzI1−JxzI2) / |I| Ql I2 / |I| Qm I4 / |I| Qn I5 / |I| Qpp −(JxzI4−JxyI5) / |I| Qpq (JxzI2−JyzI4− (Iyy−Ixx)I5) / |I| Qpr −(JxyI2+ (Ixx−Izz)I4−JyzI5) / |I| Qqq (JyzI2−JxyI5) / |I| Qqr −((Izz−Iyy)I2−JxyI4+JxzI5) / |I| Qrr −(JyzI2−JxzI4) / |I| Rl I3 / |I| Rm I5 / |I| Rn I6 / |I| Rpp −(JxzI5−JxyI6) / |I| Rpq (JxzI3−JyzI5− (Iyy−Ixx)I6) / |I| Rpr −(JxyI3+ (Ixx−Izz)I5−JyzI6) / |I| Rqq (JyzI3−JxyI6) / |I| Rqr −((Izz−Iyy)I3−JxyI5+JxzI6) / |I| Rrr −(JyzI3−JxzI5) / |I|

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The vector-equations (2.24) and (2.25) form the basis for the development of the gen-eral rigid-body dynamic model. In order to make these equations usable for control system design and analysis, flight simulation, system identification, and other ana-lytical tasks, they need to be rewritten in nonlinear state-space format. This can be achieved by moving the time-derivatives of the linear and angular velocities to the left-hand side of these equations:

∂Vc.g. ∂t = F m−Ω×Vc.g. (2.26) ∂Ω ∂t = I−1 Mc.g.−Ω×I·Ω−Ω×h0 (2.27) These equations can be written-out into their components along the body-axes, using the following definitions for Vc.g.,Ω, F, and Mc.g.:

Vc.g. = i u + j v + k w Ω = i p + j q + k r F = iFx + jFy + kFz Mc.g. = iL + jM + kN This yields: ˙u = Fx m −qw+rv ˙v = Fy m +pw−ru (2.28) ˙ w = Fz m −pv+qu and: ˙p = Pppp2+Ppqpq+Pprpr+Pqqq2+Pqrqr+Prrr2+PlL+PmM+PnN+ ˙p0 ˙q = Qppp2+Qpqpq+Qprpr+Qqqq2+Qqrqr+Qrrr2+QlL+QmM+QnN+ ˙q0 ˙r = Rppp2+Rpqpq+Rprpr+Rqqq2+Rqrqr+Rrrr2+RlL+RmM+RnN+˙r0 (2.29)

The last terms in equations (2.29) express the effects of the gyroscopic couples: ˙p0 = qh0_z−rh0_y

˙q0 = rh0_x−ph0_z (2.30)

˙r0 = ph0_y−qh0_x

Ppp, Ppq, . . . , Rnare inertia coefficients, which are derived from the matrix

multipli-cations involving the inertia tensor I; they have been listed in table 2.2.

Equations (2.28) and (2.29) describe the motions of any rigid body relatively to the Earth if the following four restrictive assumptions are made:

1. the body is assumed to be rigid during the motions considered (attached spin-ning rotors are allowed, provided these are accounted for in the moment equa-tions),

2. the mass of the body is assumed to be constant during the time-interval in which its motions are studied,

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2.1. General rigid-body equations of motion 17

3. the Earth is assumed to be fixed in space, i.e. its rotation is neglected, and 4. the curvature of the Earth is neglected.

The latter two assumptions allow us to assume that the inertial reference frame in which the motions of the rigid body are considered is fixed to the Earth. If the equa-tions are to be applied to a moving vehicle, the description of the vehicle motion un-der assumptions 3 and 4 are accurate for relatively short-term guidance and control analysis purposes only. The assumptions do have practical limitations when very long term navigation or extra-atmospheric operations are of interest [26]. Ref.[33] contains a more elaborate model for around-the-Earth navigation.

In order to simplify the notations for the remainder this report, the velocity vector

Vc.g.will from this point onwards shortly be denoted as V. The body-axes

compo-nents of this vector are u, v, and w, respectively, and the length of this vector is denoted as V. Likewise, the subscript c.g. will be omitted from the moment vector

Mc.g.in the remainder of this report.

In the next chapter, the dynamics of airplanes will be described in terms of the rigid-body equations of motion which we just derived. Figure 2.1 gives a graphical

representation of the external forces and moments (Fx, Fy, Fz, L, M, and N), and

the linear and rotational velocity components of the airplane (u, v, w, p, q, and r) in relation to its body-fixed reference frame. The orientation of the body-axes in this figure conforms to the definition from section A.7.1 of appendix A. The figure also shows the graphical representation of the airspeed vector V, the angle of attack

α, and the sideslip angle β, which define the orientation of the flight-path axes in

relation to the aircraft’s body-axes.

M, q N, p F , w F , v F , u L, r Y -axis c.g. X -axis Z -axis x y z B B B α β V

Figure 2.1:Orientation of the linear and angular velocity components, external forces and moments, angle of attack, and sideslip angle in relation to the body-fixed refer-ence frame of the aircraft.

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2.2 Expressing translational motions in flight-path axes

Although it seems logical to express translational velocities in terms of the body-axes velocity components u, v, and w, it is often more convenient to use the true airspeed V, angle of attack α, and sideslip angle β instead when considering aerodynamic problems. The latter variables express the translational motions in relation to the flight-path axes (the airspeed vector V coincides with the flight-path or wind-axis

XW, while α and β define the orientation of the flight-path reference frame FW in

relation to the body-fixed reference frame FB; see section A.7.1 of appendix A).

2.2.1 Advantages of relating translations to flight-path axes

Since V, α, and β can be expressed in terms of u, v, and w, and vice-versa, it is possible to use either set of variables for the solution of the equations of motion. However, V,

α, and β are usually better suited for simulation tasks, for two reasons:

1. From a physical point of view it is logical to express the aerodynamic forces and moments in terms of V, α, and β. For simulation purposes, we want the linear force equations to be re-written as a set of explicit ordinary differential equations (ODEs), moving all time-derivatives to one side of the equations and all other terms to the other side. This may be difficult to achieve, since the aero-dynamic forces and moments may depend on the time-derivatives of the angle of attack and sideslip angle, while ˙α and ˙β themselves will not not available un-til after the force equations have been evaluated. In practice it is often possible to assume a linear relationship between these time-derivatives and the aerody-namic forces, which makes it relatively easy to convert the force equations to explicit ODEs (this has been demonstrated in section 3.4 for the Beaver model), provided the translational equations are written in terms of α and β instead of v and w.

2. A higher accuracy of the numerical computations can be achieved by relating the translational motions to the flight-path axes. For agile aircraft having an up-per limit of the pitch rate q of about 2 rad s−1_{, and flying at high airspeeds (e.g.}

V = 600 m s−1), the term u q in equation (2.28) may become as large as 120 g!

On the other hand, the factor Fz/m, which represents the normal acceleration

due to the external force along the ZB-axis (primarily gravity and aerodynamic

lift) has an upper-limit of only a few g’s. Hence, ‘artificial’ accelerations of much greater magnitude than the actual physical accelerations of the aircraft are introduced in the equations for u, v, and w, because of the high rotation rates of the body-axes. In practice this means less favourable computer scal-ing and hence poorer accuracy for a given computer precision if the simulation model is based upon body-axes velocity components [16].

Because of said advantages, we will later treat V, α, and β as state variables in the resulting nonlinear state-space model of the aircraft, while u, v, and w (and their time-derivatives) will be treated as output variables. The first three variables are re-quired to solve the equations of motion; the other ones provide useful additional information, that allows us to determine e.g. airplane drift due to wind and atmos-pheric turbulence.

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2.2. Expressing translational motions in flight-path axes 19 V -X D L -Za a α

Figure 2.2:Relationship between the aerodynamic forces in flight-path and body-axes

2.2.2 Expressing forces and velocities in terms of flight-path axes

Transforming the forces and velocities from body to flight-path axes is quite straight-forward (see ref.[14] for a detailed description). From figure 2.1, we can observe that the body-axes velocity components are equal to:

  u v w  =V   cos α cos β sin β sin α cos β   (2.31) Hence: V = pu2₊_v2₊_w2 _(2.32) α = arctan w u (2.33) β = arctan v √ u2₊_w2 (2.34)

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Aerodynamic forces and moments are commonly expressed in terms of aerodynamic

lift L, drag D, and sideforce Y, which are aligned along the flight-path axes ZW (in

negative direction, i.e. upwards), XW, and YW, respectively. However, for simulation

purposes, it is more convenient to use the body-axes force-components Xa, Ya, and

Za instead. The relationship between these variables has been shown in figure 2.2.

The body-axes components can be derived from the flight-path axes components by means of the following axis-transformation:

  Xa Ya −Za  =   −cos α 0 sin α 0 1 0 sin α 0 cos α  ·   D Y L   (2.35)

Notice the minus sign for the aerodynamic force component along the ZB-axis, which

is due to the fact that the positive ZB-axis points downwards. 2.2.3 Derivation of the ˙V-equation

From equation (2.32) we can deduce that: ˙

V= u˙u+v˙v+ww˙

V (2.36)

Substituting the definitions (2.31) for u, v, and w, and cancelling terms yields: ˙

V= ˙u cos α cos β+ ˙v sin β+w sin α cos β˙ (2.37)

If we substitute equations (2.28) for ˙u, ˙v, and ˙w, the terms involving the vehicle rota-tional rates p, q, and r turn out to be identically zero, which becomes obvious after again substituting equation (2.31) for u, v, and w. The resulting equation becomes:

˙

V= 1

m Fxcos α cos β+Fysin β+Fzsin α cos β

(2.38)

2.2.4 Derivation of the ˙α-equation

Differentiating equation (2.33) with respect to the time yields: ˙α= uw˙ − ˙uw

u2₊_w2 (2.39)

Using equation (2.31) we can re-write the denominator of this equation:

u2+w2 =V2−v2=V2(1−sin2β) =V2cos2β (2.40)

Substituting the u and w-relations from equation (2.31) and equation (2.40) into equa-tion (2.39) yields:

˙α= w cos α˙ − ˙u sin α

V cos β (2.41)

Substituting equations (2.28) for ˙u and ˙w, and rewriting terms yields:

˙α= 1

V cos β 1

m(−Fxsin α+Fzcos α) +pv cos α+qu cos α+qw sin α−rv sin α

(2.42) Using equations (2.31) for u, v, and w, we find:

˙α= 1

V cos β 1

m(−Fxsin α+Fzcos α)

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2.3. Equations of motion in nonsteady atmosphere 21

2.2.5 Derivation of the ˙β-equation

Differentiating equation (2.34) with respect to the time yields: ˙β= ˙v(u

2₊_v2_{) −}_v₍_u_˙u₊_w_w_˙₎

V2√_u2₊_w2 (2.44)

From equations (2.31) the following relations can be derived: u2+w2 = V2cos2β

uv = V2sin β cos β cos α

vw = V2sin β cos β sin α (2.45)

These values substituted in equation (2.44) yield: ˙β= 1

V(−˙u cos α sin β+ ˙v cos β−w sin α sin β˙ ) (2.46)

Substituting equations (2.28) for ˙u and ˙w yields: ˙β= 1

V 1

m(−Fxcos α sin β+Fycos β−Fzsin α sin β) +qw cos α sin β +

−rv cos α sin β+pw cos β−ru cos β+pv sin α sin β−qu sin α sin β

(2.47) If we substitute equations (2.31), many terms can be cancelled and we find:

˙β= 1

V 1

m(−Fxcos α sin β+Fycos β−Fzsin α sin β)

+p sin α−r cos α (2.48)

2.3 Equations of motion in nonsteady atmosphere

The equations of motion are valid only if the body-axes velocity components are mea-sured with respect to a non-rotating system of reference axes having a constant trans-lational speed in inertial space. Under the assumptions 3 and 4 from section 2.1.4, it is possible to select a reference frame that is fixed to the surrounding atmosphere

as long as the wind velocity vector Vw is constant. In that case, the components u,

v, and w of the velocity vector V express the aircraft’s velocity with respect to the surrounding atmosphere.

If the wind velocity vector Vwis not constant during the time-interval over which

the motions of the aircraft are studied it is not possible to fix the reference frame to the surrounding atmosphere. This happens for instance during the approach and landing of an aircraft, because the wind velocity changes with altitude. Again using assumptions 3 and 4 of section 2.1.4, the most obvious choice of the reference frame

in this case turns out to be the Earth-fixed reference frame FE[19].

Let Ve be the velocity with respect to the Earth, Va the velocity with respect to the

surrounding atmosphere, and Vwthe wind velocity with respect to the Earth. Then

we can write:

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or:

ue = ua+uw

ve = va+vw (2.50)

we = wa+ww

where ue, ve, and we are the components of V, ua, va, and wa are the components of Va, and uw, vw, and wware the components of Vw, all measured along the body-axes

of the aircraft. The force equations now become:

F=m ∂Ve ∂t +Ω×Ve (2.51) Rewriting equation (2.51) yields:

∂Ve

∂t =

F

m−Ω×Ve (2.52)

For the individual components along the body-axes we thus find: ˙ue = Fx m −qwe+rve ˙ve = Fy m +pwe−rue (2.53) ˙ we = Fz m −pve+que

In order to compute the aerodynamic forces and moments, it is necessary to know the

values of Va (the true airspeed), α, and β.1 In a manner analogous to the derivation

of ˙V, ˙α, and ˙β in section 2.2, we can find expressions for the time-derivatives of Va, αa, and βa:

˙

Va =

1

m Fxcos α cos β+Fysin β+Fzsin α cos β

+

− (qww−rvw+ ˙uw)cos α cos β+ (pww−ruw− ˙vw)sin β+

− (pvw−quw+w˙w)sin α cos β (2.54)

˙αa = 1

V cos β 1

m(−Fxsin α+Fzcos α) +

− (pvw−quw+w˙w)cos α+ (qww−rvw− ˙uw)sin α

+

+q− (p cos α+r sin α)tan β (2.55)

˙βa = 1

V 1

m(−Fxcos α sin β+Fycos β−Fzsin α sin β) +

+ (qww−rvw+ ˙uw)cos α sin β+ (pww−ruw− ˙vw)cos β+ + (pvw−quw+w˙w)sin α sin β

+p sin α−r cos α (2.56)

The subscript a has been used here for reasons of clarity only, allowing us to make a clear distinction between the equations for a steady atmosphere and the equations

1_{Notice that expressions (2.31) to (2.34) remain valid if V}

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2.4. Kinematic relations 23

for a nonsteady atmosphere. Since V, α, and β are always determined relatively to the surrounding atmosphere, these subscripts will be omitted again in the remainder of this section.

These expressions differ from equations (2.38), (2.43), and (2.48) in that they in-clude additional terms that depend on the wind velocity components and their time-derivatives. These terms can be expressed in terms of corrections of the external force components along the aircraft’s body-axes, Fx, Fy, and Fz, This means that we can

ap-ply equations (2.38), (2.43), and (2.48) for steady as well as nonsteady atmosphere, as long as the force components are properly corrected for the wind, if necessary. These corrections will be summarized later in section 3.3.4.

2.4 Kinematic relations

So far we have derived differential equations for the true airspeed, angle of attack, sideslip angle, and the rotational velocity components. However, to solve the equa-tions of motion it is also necessary to know the attitude of the aircraft relatively to the Earth, because some contributions to the external forces and moments depend upon those quantities. We also need to know the altitude of the aircraft, because the air pressure, temperature, and density change with altitude, affecting both the aerodynamic and propulsive forces and moments. And finally, we want to be able to track the flight path relative to the Earth, e.g. for simulations of navigational tasks. The orientation of the airplane relative to the Earth is given by a series of three con-secutive rotations, the Euler angles ψ, θ, and ϕ, see figure A.2 from appendix A. As shown in section A.7.3, the rotations can be expressed by three transformation matri-ces T_Ψ, T_Θ, and T_Φ. It is possible to express the total angular velocity of the aircraft expressed in terms of the derivatives with respect to time of the Euler angles:

  p q r  =   ˙ϕ 0 0  + TΦ z }| {   1 0 0 0 cos ϕ sin ϕ 0 −sin ϕ cos ϕ     0 ˙θ 0  + +   1 0 0 0 cos ϕ sin ϕ 0 −sin ϕ cos ϕ   | {z } TΦ   cos θ 0 −sin θ 0 1 0 sin θ 0 cos θ   | {z } T_Θ   0 0 ˙ ψ   (2.57)

This can be written as:   p q r  =   1 0 −sin θ

0 cos ϕ sin ϕ cos θ

0 −sin ϕ cos ϕ cos θ

    ˙ϕ ˙θ ˙ ψ   = T_R   ˙ϕ ˙θ ˙ ψ   (2.58)

where TRis the matrix that transforms angular velocities in the Earth-fixed axis

sys-tem into body-axes angular velocities. Consequently, the time-derivatives of the

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the following kinematic relations: ˙

ψ = q sin ϕ+r cos ϕ

cos θ

˙θ = q cos ϕ−r sin ϕ (2.59)

˙ϕ = p+ (q sin ϕ+r cos ϕ)tan θ = p+ψ˙sin θ

The position of the aircraft with respect to the Earth-fixed reference frame is given by the coordinates xe, ye, and ze. To find these coordinates, we need to resolve the

body-axis velocity vector in the Earth-bound reference system FE:

  ˙xe ˙ye ˙ze  = T_B_→_E·   ue ve we   (2.60)

where TB→E = TB→V = TV→B−1is the transformation matrix from FB to FE, see the

definition in section A.7.3 of appendix A. This results in the following equations: ˙xe = {uecos θ+ (vesin ϕ+wecos ϕ)sin θ}cos ψ− (vecos ϕ−wesin ϕ)sin ψ

˙ye = {uecos θ+ (vesin ϕ+wecos ϕ)sin θ}sin ψ+ (vecos ϕ−wesin ϕ)cos ψ

˙ze = −uesin θ+ (vesin ϕ+wecos ϕ)cos θ (2.61)

In practice, the altitude of the aircraft is a more useful property than the coordinate

ze. The relationship between the time-derivatives of H and zeis simple:

˙

H= −˙ze (2.62)

Notice that the positive ZE-axis points downwards.

The state equations (2.59) have the advantage of using physically meaningful varia-bles, and they express the airplane’s attitude using the minimum number of three first-order differential equations. However, it should be noted that these equations also have some significant disadvantages.

First of all, a division by zero occurs if the pitch angle reaches plus or minus 90◦. Although in digital simulations, this exact number has an almost zero probability of occurrence, this may still cause significant computational errors in the vicinity of this singularity. Second, the Euler angles may integrate up to values outside the normal

±90◦range of pitch and the normal±180◦range of roll and yaw angles, which may

make it difficult to determine the attitude uniquely. And finally, the equations are linear in p, q, and r, but nonlinear in terms of the Euler angles themselves [33].

There are several other ways besides the Euler angles to represent the orientation of a rotated coordinate frame. These methods, which involve four, five, or even six variables instead of the three Euler angles, aim to avoid the singularity of the Euler angle representation, and maximize the speed of computer processing in navigation applications. The most common of these methods is the so-called quaternion represen-tation, which uses four variables. For a detailed discussion about this method, refer to ref.[33]. In this report, we will limit ourselves to the Euler angle representation, because it is still the most commonly used method for aircraft simulations.

A Simulink Toolbox for Flight Dynamics and Control Analysis

FDC 1.4 – A S

IMULINK

Toolbox for

Flight Dynamics and Control Analysis

Contents

Chapter 1

Flight control system development

1.1

Automatic flight control systems

1.2

The FCS design cycle

1.3

Taking a closer look at the FCS design cycle

1.4

FCS design environments

1.5

FCS design and the FDC toolbox

}

}

Chapter 2

Rigid body equations of motion

2.1

General rigid-body equations of motion

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

2.2

Expressing translational motions in flight-path axes

2.3

Equations of motion in nonsteady atmosphere

2.4

Kinematic relations