A Generic Model of Aircraft Dynamics JOHN W.C. ROBINSON FOI, Swedish Defence Research Agency, is a mainly assignment-funded agency under the Ministry of Defence. The core activities are research, method and technology development, as well as studies conducted in the interests of Swedish defence and the safety and security of society. The organisation employs approximately 1000 per- sonnel of whom about 800 are scientists. This makes FOI Sweden’s largest research institute. FOI gives its customers access to leading-edge expertise in a large number of fi elds such as security policy studies, defence and security related analyses, the assessment of various types of threat, systems for control and management of crises, protection against and management of hazardous substances, IT security and the potential offered by new sensors. FOI Defence Research Agency Phone: +46 8 555 030 00 www.foi.se SE-164 90 Stockholm Fax: +46 8 555 031 00 FOI-R--3185--SE ISSN 1650-1942 April 20122 John W.C. Robinson A Generic Model of Aircraft Dynamics Titel En generisk modell for¨ flygplansdynamik Title A Generic Model of Aircraft Dynamics Report no FOI-R--3185--SE Month April Year 2012 Pages 50 ISSN ISSN-1650-1942 Customer Swedish Armed Forces Project no E36500 Approved by Lars Hostbeck¨ Head, Information- and Aeronautical Systems Division Information- and Aero Systems FOI Swedish Defence Research Agency This work is protected under the Act on Copyright in Literary and Artistic Works (SFS 1960:729). Any form of reproduction, translation or modification without permission is prohibited. 2 FOI-R--3185--SE Abstract This report describes a generic aircraft model which is based on a simplifica- tion of the full rigid body equations of motion most often used for realistic simulations. The model is applicable to piloted simulations as well as modeling of autonomous behavior of aircraft when there is a path planner och behav- ior generator present. For the latter types of applications a simple velocity vector following autopilot is included in the model . Particular emphasis has been put on making visible the various assumptions used when obtaining the model. This makes it easy to adapt the model to other applications, such as bank-to-turn operated missiles and aircraft with nonstandard configurations. Keywords Flight mechanics, Flight dynamics, Aircraft model 3 FOI-R--3185--SE Sammanfattning Rapporten beskriver en generisk flygplansmodell som baseras p˚aen f¨orenkling av den fulla stelkroppsdynamiken som oftast anv¨andsvid realistiska simulering- ar. Modellen ¨artill¨ampbarb˚adef¨orpilotstyrda simuleringar och modellering av autonomt beteende i de fall d˚adet finns en planeringsfunktion eller be- teendegenerator tillg¨anglig.F¨orde senare typerna av till¨ampningarfinns det inkluderat i modellen en enkel autopilot som klarar att f¨oljacommandon f¨or hastighetsvektorn. Speciell vikt har lagts vid att synligg¨orade olika antagan- den som anv¨ants vid framtagandet av modellen. Detta g¨ordet l¨attatt anpassa modellen till andra till¨ampningar,s˚asom\bant-to-turn"-opererade missiler och flygplan med ickestandard konfiguration. Nyckelord Flygmekanik, Flygplansmodell 4 FOI-R--3185--SE Contents 1 Introduction 7 1.1 Outline ................................ 7 1.2 Notation ............................... 7 1.3 Acknowledgment .......................... 8 2 Summary of model 9 2.1 Aircraft dynamics .......................... 9 2.1.1 Definitions ......................... 9 2.1.2 Pitch channel ........................ 11 2.1.3 Roll channel ........................ 12 2.1.4 Yaw channel ........................ 12 2.1.5 Velocity ........................... 12 2.1.6 Computational dependencies ............... 12 2.1.7 Summary of assumptions ................. 12 2.2 Autopilot ............................... 13 2.2.1 Guidance law ........................ 13 2.2.2 Orientation command generator ............. 14 3 Rigid Body Mechanics 15 3.1 The Newton-Euler equations ................... 15 3.2 Motion in an Earth fixed frame .................. 16 3.3 Aerodynamic coordinates ..................... 16 3.3.1 Wind axes .......................... 17 3.4 Equilibrium points ......................... 18 4 Simplifying the Equations I 19 4.1 Simplified force equation ...................... 19 4.1.1 Sideslip angle ....................... 19 4.1.2 Angle of attack ....................... 19 4.1.3 Velocity ........................... 20 4.2 Simplified moment equation .................... 20 4.2.1 Roll channel ........................ 21 4.2.2 Pitch channel ........................ 22 4.2.3 Yaw channel ........................ 22 4.3 Simplified nonlinear pitch plane model .............. 22 5 Simplifying the Equations II 25 5.1 Linearized model of the pitch plane dynamics .......... 25 5 FOI-R--3185--SE 5.1.1 Aerodynamic equilibrium ................. 25 5.1.2 Aerodynamic coefficients ................. 26 5.2 The short period approximation .................. 26 5.2.1 State space representation ................ 27 5.3 Closed Loop System ........................ 28 5.4 Load factor ............................. 28 5.4.1 Aerodynamic load factor .................. 28 5.4.2 Load factor (total) ..................... 30 (gt) 5.4.3 Computation of the term F_훼 ............... 31 5.5 Velocity ............................... 31 5.6 Load Limiter ............................. 32 5.6.1 Predictive limiting ...................... 32 6 Autopilot 35 6.1 Velocity direction following ..................... 35 6.1.1 Guidance law ........................ 35 6.1.2 Orientation command generator ............. 37 A Relative motion 47 A.1 Relative velocity .......................... 47 Bibliography 49 6 FOI-R--3185--SE 1 Introduction In this report we develop a generic mathematical model of an aircraft flying in coordinated flight.1 The model is indented to serve as a platform for simula- tions as well as performance assessments and can be used in (human) piloted simulations or in an autopilot guided mode. It is is based on linearized equa- tions of motion in all three axes (a 6 degree-of-freedom model) and can be made to represent a large number of aircraft by changing the (relatively) few parameters in it. An underlying assumption of this is that the aircraft to be modeled is either \conventional" in its (open loop, bare airframe) dynamics, or, in case of an aircraft with a stability augmenting or dynamics synthesizing control system, has been rendered \conventional" in its closed loop behavior (i.e. with flight control system engaged). 1.1 Outline In the next chapter we present a summary of the model which can be used as a reference and lookup when implementing it and in the following chapters we present a detailed derivation of it. 1.2 Notation We use standard mathematical notation where (real or complex) scalars are marked with ordinary typeface, like m, and vectors in Rn and matrices in Rn×m are indicated by bold typeface, like x and A. Unless otherwise indi- cated, vectors are considered as column vectors and the elements of vectors and matrices are indicated by subscripts. Transposition of a vector or ma- trix is indicated by a superscript T and the norm (always the 2-norm) of a vector is marked as k · k. When the elements of a vector are explicitly listed T together they are enclosed by square brackets like x = [x1; x2] . A square n×n diagonal matrix D 2 R with the diagonal elements d1; : : : ; dn is indicated as D = diag(d1; : : : ; dn). Superscripts within parentheses are often used to enumerate individual el- (a) ements of a family of vectors or scalars, like f . The linear subspace of Rn spanned by m vectors x(1);:::; x(m) 2 Rm is denoted by [x(1);:::; x(m)] and the orthogonal complement, i.e. the set fy 2 Rn j yT x(j) = 0; j = 1; : : : ; mg, is denoted [x(1);:::; x(m)]?. Unit length vectors in R3 pointing along the (pos- itive direction of) coordinate axes in coordinate systems will occur below and they are denoted as ej where the subscript j 2 f1; 2; 3g refer to the coordi- nate axis in question. More generally, for any vector x 2 Rn n 0 we define 3 ex = (1=kxk)x. The cross product between vectors in R is marked with ×. The imaginary unit is marked i. Time differentiation of a (time differen- tiable) vector x is marked with the dot notation as x_ . Often the dependence in a certain function on some variables is suppressed in the notation to avoid cluttering the presentation, but when this is the case it should be clear from the context. In particular, the time argument t implicit in many quantities is almost never written out. 1The model is also applicable to missiles which are operated in bank-to-turn mode. In (Robinson, 2010) a similar model is developed which is applicable for skid-to-turn operated missiles. The material presented here builds on material in (Robinson, 2010). 7 FOI-R--3185--SE 1.3 Acknowledgment The contents of the report has been much influenced by Mr. Peter Str¨omb¨ack and Mr. Emil Salling, both at FOI. Mr. Salling implemented and tested the model in the framework Merlin (Salling et al., 2009) with the aid of Mr. Str¨omb¨ack. The latter also did reference implementations of parts of the model (including the autopilot) in Matlab which resulted in the correction of several errors in the manuscript and improvements of the formulations used. For this the author is very grateful. The author is also indebted to Dr. Petter Ogren¨ at FOI for help with proofreading of the manuscript. Dr. Ogren¨ suggested several changes to the manuscript which increased clarity and readability. 8 FOI-R--3185--SE 2 Summary