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Quasi-LPV modelling of a projectile’s behaviour in flight
Proceedings,18th IFAC Symposium on System Identification Proceedings,18th IFAC Symposium on System Identification July 9-11, 2018. Stockholm, Sweden on System Identification Proceedings,18th IFAC Symposium July 9-11, 2018. Stockholm, Sweden on Proceedings,18th IFAC Symposium System Identification online at www.sciencedirect.com July 9-11, 2018. Stockholm, Sweden Available Proceedings,18th IFAC Symposium July 9-11, 2018. Stockholm, Sweden on System Identification July 9-11, 2018. Stockholm, Sweden
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IFAC PapersOnLine 51-15 (2018) 1080–1085
Quasi-LPV modelling of a projectile’s Quasi-LPV modelling of a projectile’s Quasi-LPV modelling of a projectile’s Quasi-LPV modelling of a projectile’s behaviour in flight behaviour in flight Quasi-LPV modelling of a projectile’s behaviour in flight behaviour in flight ∗,∗∗,∗∗∗ ∗ ∗ behaviour in flight
Dawid Machala ∗,∗∗,∗∗∗ Simona Dobre ∗ Marie Albisser ∗ Dawid Dobre Albisser ∗,∗∗,∗∗∗ Simona ∗ Marie ∗∗,∗∗∗ ∗∗,∗∗∗ Dawid Machala Machala Simona DobreGilson Albisser ∗∗ Floriane Collin Marion ∗,∗∗,∗∗∗ ∗ Marie ∗∗,∗∗∗ ∗∗,∗∗∗ Dawid Machala Simona Dobre Marie Albisser ∗ Floriane Collin Marion Gilson ∗∗,∗∗∗ ∗∗,∗∗∗ ∗,∗∗,∗∗∗ ∗ Floriane Collin Marion Gilson Dawid Machala Simona DobreGilson Marie Albisser ∗∗,∗∗∗ ∗∗,∗∗∗ Floriane Collin Marion ∗∗,∗∗∗ ∗∗,∗∗∗ ∗ Floriane CollinInstitute Marion Gilson Research of Saint Louis (ISL), 5 Rue du ∗ French-German ∗ French-German Research Institute of Saint Louis (ISL), 5 Rue du Research Institute of SaintFrance Louis (ISL), 5 Rue du G´ e n´ e ral Cassagnou, 68301, ∗ French-German French-German Research Institute of Saint Louis (ISL), 5 Rue du G´ een´ 68301, France ∗ ∗∗ Universit´ G´ n´eeral ral Cassagnou, Cassagnou, 68301, France ee de Lorraine, CRAN, UMR 7039, BP 70239, French-German Research Institute of Saint Louis (ISL), 5 54506 Rue du ∗∗ G´ e n´ e ral Cassagnou, 68301, France Universit´ de Lorraine, CRAN, UMR 7039, BP 70239, ∗∗ e deG´eLorraine, CRAN, UMR 7039, BP 70239, 54506 54506 Vandoeuvre-l` e s-Nancy, France n´ e ral Cassagnou, 68301, France ∗∗ Universit´ e de Lorraine, CRAN, UMR France 7039, BP 70239, 54506 Vandoeuvre-l` ees-Nancy, ∗∗ Universit´ Vandoeuvre-l` s-Nancy, France CRAN, UMR 7039, France Universit´e ∗∗∗ de CNRS, Lorraine, CRAN, UMR 7039, BP 70239, 54506 ∗∗∗ Vandoeuvre-l` es-Nancy, France CRAN, UMR 7039, France ∗∗∗ CNRS, CRAN, UMR 7039, France Vandoeuvre-l` es-Nancy, France ∗∗∗ CNRS, CNRS, CRAN, UMR 7039, France ∗∗∗ CNRS, CRAN, UMRis7039, France Abstract: The behaviour of a projectile in flight usually described by the Newton-Euler Abstract: The behaviour of a projectile in flight is usually described Newton-Euler Abstract:ofThe behaviour of aanonlinear projectilemodel in flight is usually described by by the the Newton-Euler equations motion, through structure. Its transformation into aa quasi-LPV Abstract:ofThe behaviour of aanonlinear projectilemodel in flight is usually described by the Newton-Euler equations motion, through structure. Its transformation into equations ofrequires motion, through aato nonlinear model structure. Itsonly transformation into a quasi-LPV quasi-LPV form often the model be simplified, or analysed as a set of locally linearized Abstract: The behaviour of projectile in flight is usually described by the Newton-Euler equations ofrequires motion,the through a to nonlinear model structure. Itsonly transformation into a quasi-LPV form often model be simplified, or analysed as a set of locally linearized form oftenofrequires the modela Consequently, to be simplified, orregion analysed only asofathe set model of into locally linearized approximations of the system. the of validity is reduced. In equations motion, through nonlinear model structure. Its transformation a quasi-LPV form often requires thesystem. model Consequently, to be simplified, orregion analysed only asofathe set model of locally linearized approximations of the the of validity is reduced. In approximations of the system. Consequently, theor region of validity ofabased the model is reduced. In this paper, the aim is to develop a less restrictive modelling approach, on two main steps. form often requires the model to be simplified, analysed only as set of locally linearized approximations of the system. Consequently, the region of validity of the model is reduced. In this paper, the aim is to develop a less restrictive modelling approach, based on two main steps. this paper, the aim is to develop a less restrictive modelling approach, based on two main which steps. Firstly, the nonlinear model is described w.r.t. a so-called non-rolling reference frame, approximations of the system. Consequently, the region of validity of the model is reduced. In this paper, the aim is to develop a less restrictive modelling approach, based on twoframe, main which steps. Firstly, the nonlinear model is described w.r.t. a so-called non-rolling reference Firstly, the nonlinear model is described w.r.t. amodelling so-called non-rolling reference frame, which allows to reduce the model complexity while preserving its accuracy. Then, a quasi-LPV model this paper, the aim is to develop a less restrictive approach, based on two main steps. Firstly,tothe nonlinear modelcomplexity is described w.r.t. a so-called non-rolling reference frame, model which allows reduce the while preserving its Then, aa quasi-LPV allows tothe reduce theanmodel model complexity while preserving its accuracy. accuracy. Then, quasi-LPV model is developed using analytical function substitution method, where the scheduling variables Firstly, nonlinear model is described w.r.t. a so-called non-rolling reference frame, which allows to reduce theanmodel complexity while preserving its accuracy. Then, a quasi-LPV model is developed using analytical function substitution method, where the scheduling variables is developed using anmodel analytical function substitution method, whereThen, the scheduling variables correspond to the values which are measurable or observable during real-world experiments. allows to reduce the complexity while preserving its accuracy. a quasi-LPV model is developed using an analytical function substitution method, where the scheduling variables correspond to the values which are measurable or during real-world experiments. correspond the values which are measurable or observable observable during real-world experiments. Consequently, a model whose validity is not restricted to an equilibrium manifold is created. is developedto an analytical function substitution method, where the scheduling correspond tousing values which are measurable or observable during real-world experiments. Consequently, athe model whose validity is not not restricted to an an equilibrium equilibrium manifold is variables created. Consequently, a model whose validity is restricted to manifold is created. The validation of the quasi-LPV model developed in a non-rolling frame, w.r.t. the nonlinear correspond to the values which are measurable or observable during real-world experiments. Consequently, aofmodel whose validity is developed not restricted to an equilibrium manifold isnonlinear created. The validation the quasi-LPV model in a non-rolling frame, w.r.t. the The validation ain ofmodel the quasi-LPV model in a to non-rolling frame, w.r.t. theisnonlinear one standard frame, is performed through a series of simulations. Consequently, whoserolling validity is developed not restricted an equilibrium manifold created. The developed validation in of a quasi-LPV model developed in a non-rolling frame, w.r.t. the nonlinear one developed aathe standard rolling frame, is through of simulations. one developed in standard rolling frame, is performed performed through aa series series of w.r.t. simulations. The validation of the quasi-LPV model developed in a non-rolling frame, the nonlinear one developed in a standard rolling frame, is performed through a series of simulations. © 2018, IFAC (International Federation of Automatic Control) Hosting byaElsevier Ltd. All rights reserved. one developed a standard rolling frame, is performed through series of simulations. Keywords: LPVinmethodologies, methodologies, model reduction, complex systems. Keywords: LPV model reduction, complex systems. Keywords: LPV methodologies, model reduction, complex systems. Keywords: LPV methodologies, model reduction, complex systems. Keywords: LPV methodologies, model reduction,tions, complex systems. 1. INTRODUCTION INTRODUCTION and could lead to aa quasi-LPV model with almost 1. tions, and could to model with 1. INTRODUCTION tions, and could lead lead to scheduling a quasi-LPV quasi-LPV modelSuch witha almost almost all state variables being signals. model 1. INTRODUCTION tions, and could lead to scheduling a quasi-LPV modelSuch witha almost all state variables being signals. model all state variables being scheduling signals. Such a model would not be practical, especially since the complexity of 1. on INTRODUCTION tions, and could lead to a quasi-LPV model with almost Modelling based the laws of physics is an art of all state variables being especially schedulingsince signals. Such a model would not be practical, the complexity of Modelling based on the laws of physics is an art of would not be practical, especially since the complexity of an LPV controller increases exponentially with the number Modelling based on the laws of physics is an art of all state variables being scheduling signals. Such a model compromise: complex nonlinear equations, large numnot be practical, especially since the complexity of an LPV controller increases exponentially with the number Modelling based on nonlinear the laws ofequations, physics islarge an art of would compromise: complex numan LPV controller increases exponentially with the number of scheduling signals [Kwiatkowski and Werner (2005)]. A compromise: complex nonlinear equations, large numwould not be practical, especially since the complexity of ber of state variables, and identifiability problems can Modelling based on the laws of physics is an art of an LPV controller increases exponentially with the number of scheduling signals [Kwiatkowski and Werner (2005)]. A compromise: complex nonlinear equations,problems large number of state state variables, variables, and identifiability identifiability can of scheduling signals [Kwiatkowski and Werner (2005)]. A usual solution is to simplify the nonlinear model. It has ber of and problems can an LPV controller increases exponentially with the number be expected. Potential simplifications are invaluable. In compromise: complex nonlinear equations, large numof scheduling signals [Kwiatkowski and Werner (2005)]. A usual solution is to simplify the nonlinear model. It has ber expected. of state variables, and identifiability problems can be Potential simplifications are invaluable. In usual solution is to simplify the nonlinear model. It has been done through e.g. decoupling the nonlinear model be expected. Potential simplifications arehave invaluable. In of scheduling signals [Kwiatkowski and Werner (2005)]. A aerospace, different research communities developed ber of state variables, and communities identifiability problems can solution is to e.g. simplify the nonlinear model. It has been done through decoupling the nonlinear model be expected. Potential simplifications arehave invaluable. In usual aerospace, different research developed been so-called done through decoupling the nonlinear model into longitudinal and aerospace, different research communities have developed usual solution is to e.g. simplify thelateral-directional nonlinear model. modes It has various methods to cope with the complexity of the model. be expected. Potential simplifications are invaluable. In been done through e.g. decoupling the nonlinear model into so-called longitudinal and lateral-directional modes aerospace, different research have developed various methods to cope cope with communities the complexity complexity of the the model. [Shin into so-called longitudinal lateral-directional modes (2005)], assuming aa and small incidence angle of the various methods to with the done through e.g. decoupling the nonlinear model In control applications, the usual approachhave isof tothe usemodel. linear been aerospace, different research communities developed into so-called longitudinal and lateral-directional modes [Shin (2005)], assuming small incidence angle of various methods to cope with the complexity of model. In control applications, the usual approach is to use linear [Shin (2005)], assuming a and small incidence angle of the the projectile [Ghersin and Sanchez Pena (2002)], or analysing In control applications, the usual approach isof tothe usemodel. linear into so-called longitudinal lateral-directional modes parameter varying (LPV) models, which encapsulate the various methods to cope with the complexity [Shin (2005)], assuming a small incidence angle of the projectile [Ghersin and Sanchez Pena (2002)], or analysing In control applications, themodels, usual approach is to use linear parameter varying (LPV) which encapsulate the projectile [Ghersin and Sanchez Pena (2002)], or analysing only the high-frequency effects of the forces and moments parameter varying (LPV) whichand encapsulate the [Shin (2005)], assuming a small incidence angle of the nonlinearities in aa quasi-linear quasi-linear structure allow the use In control applications, themodels, usual approach isallow to usethe linear [Ghersin and Sanchez Pena (2002)], or analysing only high-frequency effects of the forces moments parameter varying (LPV) models, whichand encapsulate the projectile nonlinearities in structure use only the the high-frequency effects of the (2002)], forces and and moments [Pfifer (2012); Tan et al. (2000)]. nonlinearities in a quasi-linear structure and allowsystems the use projectile [Ghersin and Sanchez Pena or analysing of autopilot design methods dedicated for linear parameter varying (LPV) models, which encapsulate the only the high-frequency effects of the forces and moments [Pfifer (2012); Tan et al. (2000)]. nonlinearities in a quasi-linear structurefor andlinear allowsystems the use [Pfifer (2012); Tan et al. (2000)]. of autopilot design design methods dedicated dedicated of autopilot methods for systems the high-frequency of the forces and moments [Al-jiboory etdesign al.a (2017); (2017); Fleischmann et al.linear (2016); Shin nonlinearities in quasi-linear structureet and allow theShin use only [Pfifer (2012); Tan etare al.effects (2000)]. These assumptions not always appropriate. For exof autopilotet methods dedicated for linear systems [Al-jiboory al. Fleischmann al. (2016); These assumptions are not always appropriate. For ex[Al-jiboory etdesign al. (2017); Fleischmann et al.linear (2016); Shin [Pfifer (2012); Tan et al. (2000)]. (2005)]. However, in aerodynamic identification applicaof autopilot methods dedicated for systems These assumptions are projectile’s not alwaysmotion appropriate. For exa fast-spinning cannot be de[Al-jiboory et al. (2017); Fleischmann et al. (2016); Shin ample, (2005)]. However, in aerodynamic identification applicaThese assumptions are projectile’s not alwaysmotion appropriate. For example, a fast-spinning cannot be de(2005)]. However, inmodel aerodynamic identification applications, a nonlinear is often used [Albisser et al. [Al-jiboory et al. (2017); Fleischmann et al. (2016); Shin ample, assumptions a fin-stabilised fast-spinning projectile’s motion cannotto bewide decoupled, projectiles can be subject (2005)].a However, inmodel aerodynamic identification applications, nonlinear is often used [Albisser et al. These are not always appropriate. For example, a fast-spinning projectile’s motion cannot be decoupled, fin-stabilised projectiles can be subject to wide tions, a nonlinear model is often used [Albisser et al. (2017), Vitale and Corraro (2012)]. It remains a challeng(2005)]. However, in aerodynamic applicafin-stabilised projectiles can be subject tobewide variations of the incidence angle, and the low frequency tions, aVitale nonlinear model is often identification used [Albisser et al. coupled, (2017), and Corraro (2012)]. It remains a challengample, a fast-spinning projectile’s motion cannot decoupled, fin-stabilised projectiles can be subject to wide of the angle, and the low (2017), and Corraro (2012)]. It remains a challenging research prospect to develop develop a model model structure which tions, aVitale nonlinear model is often used [Albisser et al. variations variations the incidence incidence angle, and thesubject low frequency frequency effects of aaof pull are important a high(2017), Vitale and Corraro (2012)]. It remains a challenging research prospect to structure which coupled, fin-stabilised projectiles can be to wide variations ofgravitational the incidence angle, and the lowfor frequency effects of gravitational pull are important for a ing research prospect to develop aa model structure which would be useful for both the aerodynamic coefficients (2017), Vitale and Corraro (2012)]. It remains a challengeffectsre-entry of aofgravitational pull are important for a highhighdrag vehicle [Albisser (2015)]. Additionally, the ing research prospect to develop a model structure which variations would be useful for both the aerodynamic coefficients the incidence angle, and the low frequency of a gravitational pull are important for a highdrag vehicle [Albisser (2015)]. Additionally, the would be useful both the aerodynamic coefficients identification and for the to subsequent autopilot design. It has effects ing research prospect develop aautopilot model structure which drag re-entry re-entry vehicle [Albisser (2015)]. Additionally, the LPV model design is usually preceded by selecting a set would be useful for both the aerodynamic coefficients identification and the subsequent design. It has effects of a gravitational pull are important for a highdrag re-entry vehicle [Albisser (2015)]. Additionally, the LPV model design is usually preceded by selecting a set identification and the subsequent autopilot design. It has been proposed that an LPV model structure could be used would be useful for both the aerodynamic coefficients model design is [Albisser usually preceded by selecting a the set of equilibrium points around which the analysis will be identification and the subsequent autopilot design. Itused has LPV been proposed that an LPV model structure could be drag re-entry vehicle (2015)]. Additionally, LPV model design is usually preceded byanalysis selecting a set of equilibrium points around which the will be been proposed that an subsequent LPV(2015)]. model autopilot structure couldpaper beItused for this purpose [Albisser The following is identification and the design. has of equilibrium points around which thebyanalysis will be performed. Such approach has been recently presented by beenthis proposed that an LPV(2015)]. model structure couldpaper be used for purpose [Albisser The following is LPV model design is usually preceded selecting a set of equilibrium points around which the analysis will be Such approach has been recently presented by for this purpose [Albisser (2015)]. The following paper is performed. dedicated to the development of an identification-oriented been proposed that an LPV model structure could be used performed. Such approach has been recently presented by [Berger et al. (2017)], where a frequency-based identificafor this purpose [Albisser (2015)]. The following paper is of dedicated to the development of an identification-oriented equilibrium points around which the analysis will be performed. Such approach has been recently presented by [Berger et al. (2017)], where a frequency-based identificadedicated to the development of an identification-oriented LPV model ofthe a projectile projectile in flight. flight. for this purpose [Albisser (2015)]. following paper is [Berger et al. (2017)], where a been frequency-based identification of locally linearised models of a jet airplane has been dedicated toof development of anThe identification-oriented LPV model a in performed. Such approach has recently presented by [Berger et al. (2017)], where a frequency-based identificaof linearised models of aa jet has LPV model a projectile in flight. dedicated toof development of an identification-oriented tion tion of locally locally linearised models jet airplane airplane has been been performed. However, in the case of ballistic free-flight testLPVprojectile’s model ofthe a in-flight projectile in flight. [Berger et al. (2017)], where a frequency-based identificaThe behaviour, represented by aa nontion of locally linearised models a jet airplane has been performed. However, in the case of ballistic free-flight testThe projectile’s in-flight behaviour, represented by nonLPV model of a projectile in flight. performed. However, in the case of ballistic free-flight testing, control input is not available [Albisser et al. (2017)] The projectile’s in-flight behaviour, represented by a nontion of locally linearised models a jet airplane has been linear model, is is in-flight described by the Newton-Euler of performed. However, in the case of ballistic free-flight testing, control input is not available [Albisser et al. (2017)] The projectile’s behaviour, represented bylaws a nonlinear model, described by the Newton-Euler laws of ing, control input is not available [Albisser et al. (2017)] and an off-equilibrium analysis is required. Additionally, linear model, is described by the Newton-Euler laws of performed. However, in the case of ballistic free-flight testmotion [Zipfel (2007)]. Its transformation into an LPV The projectile’s in-flight behaviour, represented by a noning, control input is not available [Albisser et al. (2017)] and an off-equilibrium analysis is required. Additionally, linear model, described the Newton-Euler laws of and motion [Zipfelis(2007)]. (2007)]. Its by transformation into an an LPV an off-equilibrium analysis is required. Additionally, lack of direct control over experimental parameters remotion [Zipfel Its transformation into LPV ing, control input is not available [Albisser et al. (2017)] model is not straightforward. Inter-state coupling and linear model, is described by the Newton-Euler laws of an off-equilibrium analysis is required.parameters Additionally, lack of direct control over experimental remotion is[Zipfel (2007)]. Its transformation into an LPV model not straightforward. straightforward. Inter-state coupling coupling and and lack ofthe direct control over experimental parameters restricts use of local identification methods. model is[Zipfel not Inter-state and and an off-equilibrium analysis is required. Additionally, trigonometric relations appear in most of the state equamotion (2007)]. Its transformation into an LPV ofthe direct control over experimental parameters restricts use of local identification methods. model is not relations straightforward. Inter-state coupling and lack trigonometric appear in most of the state equastricts the use of local identification methods. trigonometric relations appear in most of the state equalack of direct control over experimental parameters remodel is not relations straightforward. coupling and stricts the use ofa local identification methods. In this paper, quasi-LPV modelling approach avoidtrigonometric appear inInter-state most of the state equaIn this paper, a quasi-LPV modelling approach avoidThis work is in part financed by the French Defense Procurement stricts the use of local identification methods. trigonometric appear inFrench most Defense of the state equa- ing In this paper, a quasi-LPV modelling and approach avoidThis work is inrelations part financed by the Procurement the aforementioned simplifications equilibriumAgency (DGA, Direction G´ en´ erale de l’Armement). This work is in part financed by the French Defense Procurement In this paper, a quasi-LPV modelling and approach avoiding the aforementioned simplifications equilibriumAgency (DGA, Direction G´ en´ erale de l’Armement). ing the aforementioned simplifications and equilibriumThis work is in part financed by the French Defense Procurement In this paper, a quasi-LPV modelling approach avoidAgency (DGA, Direction G´ en´ erale de l’Armement). ing the aforementioned simplifications and equilibriumThis work is in part financed by the French Defense Procurement Agency (DGA, Direction G´ en´ erale de l’Armement). ing the aforementioned simplifications and equilibriumAgency (DGA, Direction G´ en´ erale de l’Armement).
Copyright © 2018, 2018 IFAC 1080Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018 IFAC 1080 Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 1080Control. Copyright © 2018 IFAC 1080 10.1016/j.ifacol.2018.09.050 Copyright © 2018 IFAC 1080
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oriented analysis, is introduced. The proposed approach is based on a premise that complexity of the system’s dynamic equations depends on a selected reference frame: the usual choice is the Variable-Roll (VR) frame, rigidly fixed on the projectile’s centre of mass and subject all its motions [Zipfel (2007); Wernert et al. (2010)]. Alternatively, the model may be developed in a so-called Fixed-Roll frame (FR), in which the frame is fixed on the projectile’s centre of mass, but does not experience the rolling motion [Wernert et al. (2010)]. Such a method origins from the 1960s [Barnett (1962)] and was initially used to reduce computation complexity of flight trajectory calculations. Interestingly, the FR equations are well-adapted for quasiLPV modelling. The approach is applied on the example of a spin-stabilized projectile, with scheduling signals and environmental conditions corresponding to these of the real-world experiments performed on ballistic test ranges [Dobre et al. (2016)]. The analysis has been intended as a preliminary study to investigate modelling prospects, before the sensitivity analysis and parameter identification steps. The paper is organised as follows. The nonlinear model is presented in Section 2, followed by its transformation into an FR frame. The proposed quasi-LPV model is developed in Section 3. In Section 4, the results of quasiLPV and nonlinear model simulations are compared. The main conclusions are drawn in Section 5. 2. NONLINEAR MODEL The model of in-flight dynamics is a nonlinear state-space continuous-time structure M: x(t) ˙ = f (x(t), C(x(t), pa )), x(0) = x0 M: (1) y(t) = g(x(t))
where the state vector x ∈ R12 has components corresponding to the translational velocities (u, v, w), the rotational velocities (p, q, r), the position of the centre of gravity of the vehicle w.r.t. Earth (xE , yE , zE ), and the roll, the pitch and the yaw angles (φ, θ and ψ, respectively). The aerodynamic coefficients C are described by polynomial functions whose coefficients are the aerodynamic coefficient parameters pa . These polynomial descriptions depend on the linear and angular velocities, and on the Mach number M and the total angle of attack αt , which are defined by: u V M= αt = arccos (2) a V where a is√the speed of sound, and V is the total velocity, i.e. V = u2 + v 2 + w2 . The output vector y represents the measurements obtained by available sensors and state observers, presented thereafter. The analysed projectile is equipped with three types of sensors: gyroscopes, magnetometers, and accelerometers. Additionally, the in-flight measurements are supplemented by an on-ground Doppler radar. The gyroscopes measure angular velocities of the projectile, thus p, q, r are considered as known. Accelerometer’s signal correspond to the specific force acting on the projectile which could be used for estimation of at least some of the linear velocities [Peter et al. (2012), Colgren (2001)]. The magnetometers measure a projection of the Earth’s magnetic field on the sensor
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axes, which allows to obtain the values of the roll angle φ [Changey et al. (2014)]. The Doppler radar measures the total velocity V of the projectile. Moreover, both the initial and terminal positions, corresponding to the gun placement and impact point, are known. Consequently, 4 out of 12 state variables in the rolling frame, i.e. p, q, r, φ can be considered to be measurable. Furthermore, the total velocity V is measured, and will be considered as an exogenous scheduling variable. 2.1 VR nonlinear model The nonlinear model dynamic equations are based on Newton’s and Euler’s laws of motion. In the VR frame, they have the following form: u˙ 0 −r q u −sθ q¯S Cx v˙ = r 0 −p v + Cy + g sφcθ (3) m C w˙ −q p 0 w cφcθ z p˙ 0 −r q p Cl −1 q˙ = J − r 0 −p J q + q¯Sd Cm (4) r˙ −q p 0 r Cn where g is the gravitational acceleration, m is the projectile’s mass, J is the inertia tensor, and S and d are the reference surface and diameter, respectively. Additionally, s. and c. correspond to trigonometric sine and cosine, respectively. The projectile is assumed to be axisymmetric, hence J = diag(Ix , Is , Is ), where Ix , Is are the longitudinal and lateral moments of inertia. These physical properties are measured for each projectile and considered known. The dynamic pressure is denoted by q¯ = 12 ρV 2 , where ρ is the air density, also assumed to be known. The coefficients Cx , Cy , Cz and Cl , Cm , Cn correspond to the aerodynamic forces and moments, respectively, acting along x, y, z axis. The evolution of the projectile’s position and attitude can be described by the following kinematic relations: φ˙ 1 sφtθ cφtθ p θ˙ = 0 cφ −sφ q (5) 0 sφc−1 θ cφc−1 θ r ψ˙ x˙ E cθcψ sφsθcψ − sψcφ cφsθcψ + sφsψ u y˙ E = cθsψ sφsθsψ + cψcφ cφsθsψ − sφcψ v z˙E −sθ sφcθ cφcθ w (6) where t. corresponds to trigonometric tangent. 2.2 Equivalence of VR and FR frames The FR reference frame is defined by a rotation around the x body axis by the roll angle φ. It can be described using the following transformation equations [Fischer and Hathaway (1988)] as: u 1 0 0 u v = 0 cφ −sφ v (7) 0 sφ cφ w w p 1 0 0 p q = 0 cφ −sφ q (8) 0 sφ cφ r r
where the superscript · indicates a variable expressed w.r.t. the FR frame. The Euler angles and positions are equivalent regardless of the selected frame.
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2.3 FR nonlinear model
3.1 Decomposition of the nonlinear remainder term
Based on (7)-(8) and through a series of algebraic transformations, the nonlinear model equations in the FR frame are describedas follows: u 0 −r q u˙ v˙ = r 0 r tθ v w˙ w −q −r tθ 0 −sθ q¯S Cx Cy + g 0 + (9) m C cθ
The suggested decomposition approach is based on algebraic transformations, as in [Tan et al. (2000), Pfifer (2012), Qin et al. (2015)]. Alternatively, a numerical decomposition could be applied [Shin (2005)], but the physical insight into the nonlinearity would be reduced.
z
p p˙ 0 −r q Cl q˙ = J −1 − r 0 r tθ J q + q¯Sd Cm Cn r r˙ −q −r tθ 0 (10) u x˙ E cθcψ −sψ sθcψ y˙ E cθsψ cψ sθsψ v (11) −sθ 0 cθ z˙E w 1 0 q θ˙ . (12) = 0 c−1 θ r ψ˙
Firstly, the force coefficients Cy , Cz can be expressed as coefficients of the normal force slope CN a and Magnus force slope Cypa . Similarly, the moment coefficients Cl , Cm , Cn can be developed into the following coefficients: roll damping Clp , pitch moment coefficient slope Cmα , pitch damping Cmq , and Magnus moment coefficient slope Cnpa [Albisser (2015)]. With the exception of the axial force coefficient Cx , the linear and angular velocities appear in the equations through linear dependence, which can be used in the decomposition of the remainder term: Cx = −CX (u, V ) v d w Cypa (V ) p Cy = −CN a (V ) + V 2V V d w v Cypa (V ) p Cz = −CN a (V ) − V 2V V d Clp (V )p Cl = 2V d w Cmq (V )q Cm = Cmα (V ) + V 2V v d Cnpa (V ) p + 2V V d v Cmq (V )r Cn = −Cmα (V ) + V 2V w d Cnpa (V ) p + 2V V
The roll angle can be calculated a posteriori [Wernert et al. (2010)] as: (13) φ˙ = p + ψ˙ sθ or t
φ(t) =
p (τ ) + r (τ )tθ(τ )dτ.
(14)
0
The signals measured in the rolling frame can be transformed to the FR frame through (7)-(8). 3. QUASI-LPV MODEL DEVELOPMENT The quasi-LPV model proposed in this paper is developed using a function substitution approach. This method allows to represent the off-equilibrium behaviour of a projectile [Shin (2005)], and has several applications in aerospace, e.g. in autopilot design [Shin et al. (2002), Tan et al. (2000), Pfifer (2012)] and aircraft fault detection [Marcos and Balas (2004)]. The standard derivation of the function substitution approach, as described in [Pfifer (2012)], is based on the assumption that the nonlinear model has the following form: x˙ = A(x1 , ρ)x + B(x1 , ρ)u + f (x1 , ρ) (15) where x1 and ρ are the scheduling states and exogenous scheduling signals, respectively. The nonlinear function f , called the remainder term, is decomposed into a form linear w.r.t. x1 , such as: (16) f (x1 , ρ) = A1 (x1 , ρ)x. Consequently, a quasi-LPV model can be obtained: (17) x˙ = [A(x1 , ρ) + A1 (x1 , ρ)]x + B(x1 , ρ)u. The nonlinear model equations (9)-(12) are already in a quasi-LPV form, with the exception of gravitational and aerodynamic components. Consequently, the forces and moments equations are considered to be the remainder term (called also a decomposition function [Marcos and Balas (2004)]).
(18)
where each Ci (u, V ) has a polynomial description dependent on the measurable signals: Ci (u, V ) = Ci1 + Ci2 s2 αt + Ci3 where s2 αt = (1 −
V V2 + Ci4 2 a a
(19)
u2 V 2 ).
The nonlinear remainder term then has the following form, described through auxiliary functions X1 , ...X7 : q¯S Cx − gsθ = X1 (u, V, zE )sθ ) − g(zE m q¯S Cy = X2 (V, zE )w p + X3 (V, zE )v m q¯S Cz + gcθ = −X2 (V, zE )v p + X3 (V, zE )w m + g(zE )cθ q¯Sd Cl = X4 (V, zE )p Ix q¯Sd Cm = X5 (V, zE )w + X6 (V, zE )q Is + X7 (V , zE )v p q¯Sd Cn = −X5 (V, zE )v + X6 (V, zE )r Is + X7 (V, zE )w p where:
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X1 (u, V, zE ) = −0.5m−1 ρ(zE )V 2 SCX (u, V ) )SdCypa (V ) X2 (V, zE ) = 0.25m−1 ρ(zE −1 X3 (V, zE ) = −0.5m ρ(zE )V SCN a (V ) X4 (V, zE ) = 0.25ρ(zE )V Sd2 Clp (V ) )V SdCmα (V ) X5 (V, zE ) = 0.5ρ(zE X6 (V, zE ) = 0.25ρ(zE )V Sd2 Cmq (V ) )Sd2 Cnpa (V ). X7 (V, zE ) = 0.25ρ(zE
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Additionally, based on (2) and (7), the following equivalences can be stated: αt = αt and V = V .
Consequently, all the nonlinear remainder terms have been developed to a form which either depends on the measurable variables, or can be directly assigned to nonmeasurable state variable. As a result, the quasi-LPV model of the following form has been obtained: x˙ = A(u , V, p , q , r , θ, ψ, zE )x (22) with the full matrix A presented in (A.1).
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Fig. 1. Evolution of the roll angle φ (signal evolution enlarged). 10-5
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Fig. 2. Evolution of the pitch rate q (signal evolution enlarged).
Difference
4. SIMULATION RESULTS The quasi-LPV model has been tested through a series of simulations in order to assess its accuracy w.r.t. the existing nonlinear model results, obtained under the same conditions.
There are several potential sources of error that were investigated. Firstly, inaccurate calculations of the aerodynamic coefficients would result in increasingly incorrect forces and moments obtained in the simulation, regardless of the frame being used. Secondly, the aerodynamic coefficients
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Both models have been initialised with the state vector values corresponding to real-world ballistic test campaigns of spin-stabilised projectiles. The initial Mach number presented a variation range from 0.88 to 1.77 and spin rate from 608 to 1217 rad/s. Simulations were conducted for a flat-fire scenario: the initial velocity vector is almost parallel to the Earth’s surface. The polynomial descriptions of the aerodynamic coefficients for such experimental conditions are known, and were used in both models’ descriptions. The simulation results consist of the quasiLPV model data in FR frame, and of the nonlinear model data in VR frame, denoted in the following figures as ’qLPV’ and ’Nonlinear’, respectively. To allow direct comparison, the quasi-LPV results were transformed into the VR frame, using an inverse of the (7)-(8) transformation.
q-LPV
50
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Only X1 and the gravity components are not in a linear dependence w.r.t. a measurable state variable. Hence, they need to be decomposed arbitrarily, by multiplication and division by any non-zero measurable signal, such as the spin rate p. By integrating the spin rate in the decomposition, the following decomposition of X1 may be obtained: 1 (21) X1 (u, V, zE ) = X1 (u, V, zE ) · · p. p The gravity terms are treated in the same way. The linear velocity u, pitch angle θ and yaw angle ψ are simulated, knowing their initial values. Moreover, θ˙ and ψ˙ equations in FR frame depend only on q , r signals, which are known through (8) since q, r, φ are measurable.
[deg]
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2018 IFAC SYSID July 9-11, 2018. Stockholm, Sweden
0.87 0.865 0.86 0.855 0.85 0
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Fig. 3. Evolution of the Mach number M . depend on the linear and angular velocities, whose accuracy must also be evaluated. Finally, the measurable linear and angular velocities are sensitive to erroneous roll angle φ, since it is used for their transformations between the reference frames. Error of the φ angle remained negligible for the tested scenarios (less than 10−6 deg difference between models, see Fig. 1). Consequently, the measurable signals could be accurately transformed between the frames. It is illustrated by the pitch rate error evolution: the difference between the signal evolution in VR and FR frames is four orders of magnitude smaller than the signal itself (Fig. 2).
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The quasi-LPV model accuracy has been assessed through comparison with a nonlinear model simulation. The differences between the two models were negligible for the tested scenarios, with the misalignment being of several orders of magnitude smaller than the signals values. It is predicted, that the differences between models stem from the numerical inaccuracies of the ode solvers being used. The future analysis aims to perform the convergence and stability assessment of the numerical solvers, in order to verify their accuracy and provide insight into the possible sources of differences present in the model. Afterwards, a global sensitivity analysis of the model outputs will be performed, based on the obtained global quasi-LPV model, and with the final goal of identifying the aerodynamic coefficients.
0
0
1
Fig. 6. Evolution of the pitch damping coefficient Cmq.
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t [s]
Fig. 4. Evolution of the total angle of attack αt . Difference
0 -2
3
0.6
Cx coefficient
10-7
2
Difference
Difference [deg]
2018 IFAC SYSID July 9-11, 2018. Stockholm, Sweden 1084
2
2.5
3
t [s]
Fig. 5. Evolution of the axial force coefficient CX . The polynomial descriptions of the coefficients in (19) are functions of M and αt . For both reference frames, these variables are in good coherence: the errors are increasing over time, but their magnitude is of at least four orders smaller than the signals (see Fig. 3-4). Good coherence of evolutions of the roll angle, the measurable signals, the total angle of attack, and the Mach number have resulted in accurate calculations of the aerodynamic coefficients. Behaviour of the error is the same as previously: slowly increasing, but of negligible magnitude. It is depicted by the trajectory of the axial force coefficient CX (Fig. 5), and the pitch damping coefficient Cmq (Fig. 6). Additionally, the expected computation time reduction has indeed been observed. For the initial spin rate p0 = 608 rad/s, and 3s of the flight time, the simulation lasted 15s for the nonlinear model in the VR frame, and 0.8s for the quasi-LPV model in FR frame. 5. CONCLUSIONS This paper investigates the prospects of using a reference frame transformation in developing a quasi-LPV model of a projectile’s in-flight behaviour. The analysis have been dedicated to an example of a spin-stabilised projectile. Firstly, the projectile’s equation of motion have been transformed into a non-rolling reference frame, which has simplified the model equations and reduced computation time. Secondly, the quasi-LPV model has been obtained using the analytic function substitution method.
REFERENCES Al-jiboory, A., Zhu, G.G., Swei, S.S.M., Su, W., and Nguyen, N.T. (2017). LPV Model Development for a Flexible Wing Aircraft. In 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Albisser, M. (2015). Identification of aerodynamic coefficients from free flight data. Ph.D. thesis, University of Lorraine. Albisser, M., Dobre, S., Berner, C., Thomassin, M., and Garnier, H. (2017). Aerodynamic Coefficient Identification of a Space Vehicle from Multiple Free-Flight Tests. Journal of Spacecraft and Rockets, 54(2), 426–435. Barnett, B. (1962). Trajectory equations for a six-degreeof-freedom missile. Technical report, Picatinny Arsenal. Berger, T., Tischler, M.B., Hagerott, S.G., Cotting, M.C., Gray, W.R., Gresham, J., George, J., Krogh, K., D’Argenio, A., and Howland, J. (2017). Development and Validation of a Flight-Identified Full-Envelope Business Jet Simulation Model Using a Stitching Architecture. AIAA Modeling and Simulation Technologies Conference. Changey, S., Pecheur, E., and Brunner, T. (2014). Attitude Estimation of a projectile using Magnetometers and Accelerometers: Experimental Validation. IEEE/ION PLANS Symposium 2014, 1168–1173. Colgren, R. (2001). Method and System for Estimation and Correction of the Angle of Attack and Sideslip Angle from Acceleration Measurements. U.S. Patent No. 6,273,370.
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The quasi-LPV model has the following form:
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0 00
q
) − g(zE )sθ X1 (u , V, zE 0 0 p X3 (V, zE ) r tθ + X2 (V, zE )p 0 0 0 g(z )cθ E −r tθ − X2 (V, zE )p X3 (V, zE ) 0 0 p 0 0 X4 (V, zE ) 0 0 Ix r r tθ X7 (V, zE )p X6 (V, zE ) X5 (V, zE ) − r − 0 Is p Ix q r tθ 0 X6 (V, zE ) X7 (V, zE )p q + ) −X5 (V, zE Is p 0 0 0 1 0 0 0 0 0 c−1 θ −sψ sθcψ 0 0 0 cψ sθsψ 0 0 0 0 cθ 0 0 0
000
0 0 0 u v 0 0 0 w 0 0 0 p q 0 0 0 · r θ 0 0 0 ψ x 0 0 0 E yE 0 0 0 0 0 0 zE 0 00
(A.1)
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−sθ
u˙ r v˙ −q w˙ p˙ 0 q˙ r˙ 0 = θ˙ ˙ 0 ψ x˙ 0 E y˙ 0 E cθcψ z˙E cθsψ
−r
Appendix A. QUASI-LPV MODEL EQUATION
0
Dobre, S., Albisser, M., Berner, C., Bernard, L., and Grignon, C. (2016). Identification of the Aerodynamic Coefficients of a 155mm Artillery Shell Based on Free Flight Data. In 29th International Symposium on Ballistics. Fischer, M. and Hathaway, W. (1988). Aeroballistic Research Facility Data Analysis System. Technical report, Eglin Air Force Base, USA. Fleischmann, S., Theodoulis, S., Laroche, E., Wallner, E., and Harcaut, J.P. (2016). A Systematic LPV/LFR Modelling Approach Optimized for Linearised Gain Scheduling Control Synthesis. AIAA Modeling and Simulation Technologies Conference. Ghersin, A.S. and Sanchez Pena, R.S. (2002). LPV control of a 6-DOF vehicle. IEEE Transactions on Control Systems Technology, 10(6), 883–887. Kwiatkowski, A. and Werner, H. (2005). Parameter Reduction for LPV Systems via Principal Components Analysis. IFAC Proceedings Volumes, 38(1), 596–601. Marcos, A.E. and Balas, G.J. (2004). Development of Linear Parameter Varying Models for Aircraft. Journal of Guidance Control and Dynamics, 27(2), 218–228. Peter, F., Leit˜ ao, M., and Holzapfel, F. (2012). Adaptive Augmentation of a New Baseline Control Architecture for Tail-Controlled Missiles Using a Nonlinear Reference Model. In AIAA GNC Conference. Pfifer, H. (2012). Quasi-LPV Model of a NDI-Controlled Missile Based on Function Substitution. In AIAA GNC Conference. doi:10.2514/6.2012-4970. Qin, W., Liu, J., Liu, G., He, B., and Wang, L. (2015). Robust parameter dependent receding horizon H control of flexible air-breathing hypersonic vehicles with input constraints. Asian Journal of Control, 17(2), 508–522. Shin, J.Y. (2005). Quasi-Linear Parameter Varying Representation of General Aircraft Dynamics Over NonTrim Region. Technical report, National Institute of Aerospace, USA. Shin, J.Y., Balas, G.J., and Kaya, A.M. (2002). Blending Approach Of Linear Parameter Varying Control Synthesis For F-16 Aircraft. Journal of Guidance Control and Dynamics, 25(6). Tan, W., Packard, A.K., and Balas, G.J. (2000). QuasiLPV Modeling and LPV Control of a Generic Missile. In Proceeding of the American Control Conference. Vitale, A. and Corraro, F. (2012). Identification from flight data of the italian unmanned space vehicle. In 16th IFAC Symp. on Sys. Iden., volume 45. Wernert, P., Theodoulis, S., and Morel, Y. (2010). Flight Dynamics Properties of 155 mm Spin-Stabilized Projectiles Analyzed in Different Body Frames. In AIAA AFM Conference. doi:10.2514/6.2010-7640. Zipfel, P.H. (2007). Modeling and Simulation of Aerospace Vehicle Dynamics. AIAA Inc., Blacksburg, 2nd edition.
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IFAC PapersOnLine 51-15 (2018) 1080–1085
Quasi-LPV modelling of a projectile’s Quasi-LPV modelling of a projectile’s Quasi-LPV modelling of a projectile’s Quasi-LPV modelling of a projectile’s behaviour in flight behaviour in flight Quasi-LPV modelling of a projectile’s behaviour in flight behaviour in flight ∗,∗∗,∗∗∗ ∗ ∗ behaviour in flight
Dawid Machala ∗,∗∗,∗∗∗ Simona Dobre ∗ Marie Albisser ∗ Dawid Dobre Albisser ∗,∗∗,∗∗∗ Simona ∗ Marie ∗∗,∗∗∗ ∗∗,∗∗∗ Dawid Machala Machala Simona DobreGilson Albisser ∗∗ Floriane Collin Marion ∗,∗∗,∗∗∗ ∗ Marie ∗∗,∗∗∗ ∗∗,∗∗∗ Dawid Machala Simona Dobre Marie Albisser ∗ Floriane Collin Marion Gilson ∗∗,∗∗∗ ∗∗,∗∗∗ ∗,∗∗,∗∗∗ ∗ Floriane Collin Marion Gilson Dawid Machala Simona DobreGilson Marie Albisser ∗∗,∗∗∗ ∗∗,∗∗∗ Floriane Collin Marion ∗∗,∗∗∗ ∗∗,∗∗∗ ∗ Floriane CollinInstitute Marion Gilson Research of Saint Louis (ISL), 5 Rue du ∗ French-German ∗ French-German Research Institute of Saint Louis (ISL), 5 Rue du Research Institute of SaintFrance Louis (ISL), 5 Rue du G´ e n´ e ral Cassagnou, 68301, ∗ French-German French-German Research Institute of Saint Louis (ISL), 5 Rue du G´ een´ 68301, France ∗ ∗∗ Universit´ G´ n´eeral ral Cassagnou, Cassagnou, 68301, France ee de Lorraine, CRAN, UMR 7039, BP 70239, French-German Research Institute of Saint Louis (ISL), 5 54506 Rue du ∗∗ G´ e n´ e ral Cassagnou, 68301, France Universit´ de Lorraine, CRAN, UMR 7039, BP 70239, ∗∗ e deG´eLorraine, CRAN, UMR 7039, BP 70239, 54506 54506 Vandoeuvre-l` e s-Nancy, France n´ e ral Cassagnou, 68301, France ∗∗ Universit´ e de Lorraine, CRAN, UMR France 7039, BP 70239, 54506 Vandoeuvre-l` ees-Nancy, ∗∗ Universit´ Vandoeuvre-l` s-Nancy, France CRAN, UMR 7039, France Universit´e ∗∗∗ de CNRS, Lorraine, CRAN, UMR 7039, BP 70239, 54506 ∗∗∗ Vandoeuvre-l` es-Nancy, France CRAN, UMR 7039, France ∗∗∗ CNRS, CRAN, UMR 7039, France Vandoeuvre-l` es-Nancy, France ∗∗∗ CNRS, CNRS, CRAN, UMR 7039, France ∗∗∗ CNRS, CRAN, UMRis7039, France Abstract: The behaviour of a projectile in flight usually described by the Newton-Euler Abstract: The behaviour of a projectile in flight is usually described Newton-Euler Abstract:ofThe behaviour of aanonlinear projectilemodel in flight is usually described by by the the Newton-Euler equations motion, through structure. Its transformation into aa quasi-LPV Abstract:ofThe behaviour of aanonlinear projectilemodel in flight is usually described by the Newton-Euler equations motion, through structure. Its transformation into equations ofrequires motion, through aato nonlinear model structure. Itsonly transformation into a quasi-LPV quasi-LPV form often the model be simplified, or analysed as a set of locally linearized Abstract: The behaviour of projectile in flight is usually described by the Newton-Euler equations ofrequires motion,the through a to nonlinear model structure. Itsonly transformation into a quasi-LPV form often model be simplified, or analysed as a set of locally linearized form oftenofrequires the modela Consequently, to be simplified, orregion analysed only asofathe set model of into locally linearized approximations of the system. the of validity is reduced. In equations motion, through nonlinear model structure. Its transformation a quasi-LPV form often requires thesystem. model Consequently, to be simplified, orregion analysed only asofathe set model of locally linearized approximations of the the of validity is reduced. In approximations of the system. Consequently, theor region of validity ofabased the model is reduced. In this paper, the aim is to develop a less restrictive modelling approach, on two main steps. form often requires the model to be simplified, analysed only as set of locally linearized approximations of the system. Consequently, the region of validity of the model is reduced. In this paper, the aim is to develop a less restrictive modelling approach, based on two main steps. this paper, the aim is to develop a less restrictive modelling approach, based on two main which steps. Firstly, the nonlinear model is described w.r.t. a so-called non-rolling reference frame, approximations of the system. Consequently, the region of validity of the model is reduced. In this paper, the aim is to develop a less restrictive modelling approach, based on twoframe, main which steps. Firstly, the nonlinear model is described w.r.t. a so-called non-rolling reference Firstly, the nonlinear model is described w.r.t. amodelling so-called non-rolling reference frame, which allows to reduce the model complexity while preserving its accuracy. Then, a quasi-LPV model this paper, the aim is to develop a less restrictive approach, based on two main steps. Firstly,tothe nonlinear modelcomplexity is described w.r.t. a so-called non-rolling reference frame, model which allows reduce the while preserving its Then, aa quasi-LPV allows tothe reduce theanmodel model complexity while preserving its accuracy. accuracy. Then, quasi-LPV model is developed using analytical function substitution method, where the scheduling variables Firstly, nonlinear model is described w.r.t. a so-called non-rolling reference frame, which allows to reduce theanmodel complexity while preserving its accuracy. Then, a quasi-LPV model is developed using analytical function substitution method, where the scheduling variables is developed using anmodel analytical function substitution method, whereThen, the scheduling variables correspond to the values which are measurable or observable during real-world experiments. allows to reduce the complexity while preserving its accuracy. a quasi-LPV model is developed using an analytical function substitution method, where the scheduling variables correspond to the values which are measurable or during real-world experiments. correspond the values which are measurable or observable observable during real-world experiments. Consequently, a model whose validity is not restricted to an equilibrium manifold is created. is developedto an analytical function substitution method, where the scheduling correspond tousing values which are measurable or observable during real-world experiments. Consequently, athe model whose validity is not not restricted to an an equilibrium equilibrium manifold is variables created. Consequently, a model whose validity is restricted to manifold is created. The validation of the quasi-LPV model developed in a non-rolling frame, w.r.t. the nonlinear correspond to the values which are measurable or observable during real-world experiments. Consequently, aofmodel whose validity is developed not restricted to an equilibrium manifold isnonlinear created. The validation the quasi-LPV model in a non-rolling frame, w.r.t. the The validation ain ofmodel the quasi-LPV model in a to non-rolling frame, w.r.t. theisnonlinear one standard frame, is performed through a series of simulations. Consequently, whoserolling validity is developed not restricted an equilibrium manifold created. The developed validation in of a quasi-LPV model developed in a non-rolling frame, w.r.t. the nonlinear one developed aathe standard rolling frame, is through of simulations. one developed in standard rolling frame, is performed performed through aa series series of w.r.t. simulations. The validation of the quasi-LPV model developed in a non-rolling frame, the nonlinear one developed in a standard rolling frame, is performed through a series of simulations. © 2018, IFAC (International Federation of Automatic Control) Hosting byaElsevier Ltd. All rights reserved. one developed a standard rolling frame, is performed through series of simulations. Keywords: LPVinmethodologies, methodologies, model reduction, complex systems. Keywords: LPV model reduction, complex systems. Keywords: LPV methodologies, model reduction, complex systems. Keywords: LPV methodologies, model reduction, complex systems. Keywords: LPV methodologies, model reduction,tions, complex systems. 1. INTRODUCTION INTRODUCTION and could lead to aa quasi-LPV model with almost 1. tions, and could to model with 1. INTRODUCTION tions, and could lead lead to scheduling a quasi-LPV quasi-LPV modelSuch witha almost almost all state variables being signals. model 1. INTRODUCTION tions, and could lead to scheduling a quasi-LPV modelSuch witha almost all state variables being signals. model all state variables being scheduling signals. Such a model would not be practical, especially since the complexity of 1. on INTRODUCTION tions, and could lead to a quasi-LPV model with almost Modelling based the laws of physics is an art of all state variables being especially schedulingsince signals. Such a model would not be practical, the complexity of Modelling based on the laws of physics is an art of would not be practical, especially since the complexity of an LPV controller increases exponentially with the number Modelling based on the laws of physics is an art of all state variables being scheduling signals. Such a model compromise: complex nonlinear equations, large numnot be practical, especially since the complexity of an LPV controller increases exponentially with the number Modelling based on nonlinear the laws ofequations, physics islarge an art of would compromise: complex numan LPV controller increases exponentially with the number of scheduling signals [Kwiatkowski and Werner (2005)]. A compromise: complex nonlinear equations, large numwould not be practical, especially since the complexity of ber of state variables, and identifiability problems can Modelling based on the laws of physics is an art of an LPV controller increases exponentially with the number of scheduling signals [Kwiatkowski and Werner (2005)]. A compromise: complex nonlinear equations,problems large number of state state variables, variables, and identifiability identifiability can of scheduling signals [Kwiatkowski and Werner (2005)]. A usual solution is to simplify the nonlinear model. It has ber of and problems can an LPV controller increases exponentially with the number be expected. Potential simplifications are invaluable. In compromise: complex nonlinear equations, large numof scheduling signals [Kwiatkowski and Werner (2005)]. A usual solution is to simplify the nonlinear model. It has ber expected. of state variables, and identifiability problems can be Potential simplifications are invaluable. In usual solution is to simplify the nonlinear model. It has been done through e.g. decoupling the nonlinear model be expected. Potential simplifications arehave invaluable. In of scheduling signals [Kwiatkowski and Werner (2005)]. A aerospace, different research communities developed ber of state variables, and communities identifiability problems can solution is to e.g. simplify the nonlinear model. It has been done through decoupling the nonlinear model be expected. Potential simplifications arehave invaluable. In usual aerospace, different research developed been so-called done through decoupling the nonlinear model into longitudinal and aerospace, different research communities have developed usual solution is to e.g. simplify thelateral-directional nonlinear model. modes It has various methods to cope with the complexity of the model. be expected. Potential simplifications are invaluable. In been done through e.g. decoupling the nonlinear model into so-called longitudinal and lateral-directional modes aerospace, different research have developed various methods to cope cope with communities the complexity complexity of the the model. [Shin into so-called longitudinal lateral-directional modes (2005)], assuming aa and small incidence angle of the various methods to with the done through e.g. decoupling the nonlinear model In control applications, the usual approachhave isof tothe usemodel. linear been aerospace, different research communities developed into so-called longitudinal and lateral-directional modes [Shin (2005)], assuming small incidence angle of various methods to cope with the complexity of model. In control applications, the usual approach is to use linear [Shin (2005)], assuming a and small incidence angle of the the projectile [Ghersin and Sanchez Pena (2002)], or analysing In control applications, the usual approach isof tothe usemodel. linear into so-called longitudinal lateral-directional modes parameter varying (LPV) models, which encapsulate the various methods to cope with the complexity [Shin (2005)], assuming a small incidence angle of the projectile [Ghersin and Sanchez Pena (2002)], or analysing In control applications, themodels, usual approach is to use linear parameter varying (LPV) which encapsulate the projectile [Ghersin and Sanchez Pena (2002)], or analysing only the high-frequency effects of the forces and moments parameter varying (LPV) whichand encapsulate the [Shin (2005)], assuming a small incidence angle of the nonlinearities in aa quasi-linear quasi-linear structure allow the use In control applications, themodels, usual approach isallow to usethe linear [Ghersin and Sanchez Pena (2002)], or analysing only high-frequency effects of the forces moments parameter varying (LPV) models, whichand encapsulate the projectile nonlinearities in structure use only the the high-frequency effects of the (2002)], forces and and moments [Pfifer (2012); Tan et al. (2000)]. nonlinearities in a quasi-linear structure and allowsystems the use projectile [Ghersin and Sanchez Pena or analysing of autopilot design methods dedicated for linear parameter varying (LPV) models, which encapsulate the only the high-frequency effects of the forces and moments [Pfifer (2012); Tan et al. (2000)]. nonlinearities in a quasi-linear structurefor andlinear allowsystems the use [Pfifer (2012); Tan et al. (2000)]. of autopilot design design methods dedicated dedicated of autopilot methods for systems the high-frequency of the forces and moments [Al-jiboory etdesign al.a (2017); (2017); Fleischmann et al.linear (2016); Shin nonlinearities in quasi-linear structureet and allow theShin use only [Pfifer (2012); Tan etare al.effects (2000)]. These assumptions not always appropriate. For exof autopilotet methods dedicated for linear systems [Al-jiboory al. Fleischmann al. (2016); These assumptions are not always appropriate. For ex[Al-jiboory etdesign al. (2017); Fleischmann et al.linear (2016); Shin [Pfifer (2012); Tan et al. (2000)]. (2005)]. However, in aerodynamic identification applicaof autopilot methods dedicated for systems These assumptions are projectile’s not alwaysmotion appropriate. For exa fast-spinning cannot be de[Al-jiboory et al. (2017); Fleischmann et al. (2016); Shin ample, (2005)]. However, in aerodynamic identification applicaThese assumptions are projectile’s not alwaysmotion appropriate. For example, a fast-spinning cannot be de(2005)]. However, inmodel aerodynamic identification applications, a nonlinear is often used [Albisser et al. [Al-jiboory et al. (2017); Fleischmann et al. (2016); Shin ample, assumptions a fin-stabilised fast-spinning projectile’s motion cannotto bewide decoupled, projectiles can be subject (2005)].a However, inmodel aerodynamic identification applications, nonlinear is often used [Albisser et al. These are not always appropriate. For example, a fast-spinning projectile’s motion cannot be decoupled, fin-stabilised projectiles can be subject to wide tions, a nonlinear model is often used [Albisser et al. (2017), Vitale and Corraro (2012)]. It remains a challeng(2005)]. However, in aerodynamic applicafin-stabilised projectiles can be subject tobewide variations of the incidence angle, and the low frequency tions, aVitale nonlinear model is often identification used [Albisser et al. coupled, (2017), and Corraro (2012)]. It remains a challengample, a fast-spinning projectile’s motion cannot decoupled, fin-stabilised projectiles can be subject to wide of the angle, and the low (2017), and Corraro (2012)]. It remains a challenging research prospect to develop develop a model model structure which tions, aVitale nonlinear model is often used [Albisser et al. variations variations the incidence incidence angle, and thesubject low frequency frequency effects of aaof pull are important a high(2017), Vitale and Corraro (2012)]. It remains a challenging research prospect to structure which coupled, fin-stabilised projectiles can be to wide variations ofgravitational the incidence angle, and the lowfor frequency effects of gravitational pull are important for a ing research prospect to develop aa model structure which would be useful for both the aerodynamic coefficients (2017), Vitale and Corraro (2012)]. It remains a challengeffectsre-entry of aofgravitational pull are important for a highhighdrag vehicle [Albisser (2015)]. Additionally, the ing research prospect to develop a model structure which variations would be useful for both the aerodynamic coefficients the incidence angle, and the low frequency of a gravitational pull are important for a highdrag vehicle [Albisser (2015)]. Additionally, the would be useful both the aerodynamic coefficients identification and for the to subsequent autopilot design. It has effects ing research prospect develop aautopilot model structure which drag re-entry re-entry vehicle [Albisser (2015)]. Additionally, the LPV model design is usually preceded by selecting a set would be useful for both the aerodynamic coefficients identification and the subsequent design. It has effects of a gravitational pull are important for a highdrag re-entry vehicle [Albisser (2015)]. Additionally, the LPV model design is usually preceded by selecting a set identification and the subsequent autopilot design. It has been proposed that an LPV model structure could be used would be useful for both the aerodynamic coefficients model design is [Albisser usually preceded by selecting a the set of equilibrium points around which the analysis will be identification and the subsequent autopilot design. Itused has LPV been proposed that an LPV model structure could be drag re-entry vehicle (2015)]. Additionally, LPV model design is usually preceded byanalysis selecting a set of equilibrium points around which the will be been proposed that an subsequent LPV(2015)]. model autopilot structure couldpaper beItused for this purpose [Albisser The following is identification and the design. has of equilibrium points around which thebyanalysis will be performed. Such approach has been recently presented by beenthis proposed that an LPV(2015)]. model structure couldpaper be used for purpose [Albisser The following is LPV model design is usually preceded selecting a set of equilibrium points around which the analysis will be Such approach has been recently presented by for this purpose [Albisser (2015)]. The following paper is performed. dedicated to the development of an identification-oriented been proposed that an LPV model structure could be used performed. Such approach has been recently presented by [Berger et al. (2017)], where a frequency-based identificafor this purpose [Albisser (2015)]. The following paper is of dedicated to the development of an identification-oriented equilibrium points around which the analysis will be performed. Such approach has been recently presented by [Berger et al. (2017)], where a frequency-based identificadedicated to the development of an identification-oriented LPV model ofthe a projectile projectile in flight. flight. for this purpose [Albisser (2015)]. following paper is [Berger et al. (2017)], where a been frequency-based identification of locally linearised models of a jet airplane has been dedicated toof development of anThe identification-oriented LPV model a in performed. Such approach has recently presented by [Berger et al. (2017)], where a frequency-based identificaof linearised models of aa jet has LPV model a projectile in flight. dedicated toof development of an identification-oriented tion tion of locally locally linearised models jet airplane airplane has been been performed. However, in the case of ballistic free-flight testLPVprojectile’s model ofthe a in-flight projectile in flight. [Berger et al. (2017)], where a frequency-based identificaThe behaviour, represented by aa nontion of locally linearised models a jet airplane has been performed. However, in the case of ballistic free-flight testThe projectile’s in-flight behaviour, represented by nonLPV model of a projectile in flight. performed. However, in the case of ballistic free-flight testing, control input is not available [Albisser et al. (2017)] The projectile’s in-flight behaviour, represented by a nontion of locally linearised models a jet airplane has been linear model, is is in-flight described by the Newton-Euler of performed. However, in the case of ballistic free-flight testing, control input is not available [Albisser et al. (2017)] The projectile’s behaviour, represented bylaws a nonlinear model, described by the Newton-Euler laws of ing, control input is not available [Albisser et al. (2017)] and an off-equilibrium analysis is required. Additionally, linear model, is described by the Newton-Euler laws of performed. However, in the case of ballistic free-flight testmotion [Zipfel (2007)]. Its transformation into an LPV The projectile’s in-flight behaviour, represented by a noning, control input is not available [Albisser et al. (2017)] and an off-equilibrium analysis is required. Additionally, linear model, described the Newton-Euler laws of and motion [Zipfelis(2007)]. (2007)]. Its by transformation into an an LPV an off-equilibrium analysis is required. Additionally, lack of direct control over experimental parameters remotion [Zipfel Its transformation into LPV ing, control input is not available [Albisser et al. (2017)] model is not straightforward. Inter-state coupling and linear model, is described by the Newton-Euler laws of an off-equilibrium analysis is required.parameters Additionally, lack of direct control over experimental remotion is[Zipfel (2007)]. Its transformation into an LPV model not straightforward. straightforward. Inter-state coupling coupling and and lack ofthe direct control over experimental parameters restricts use of local identification methods. model is[Zipfel not Inter-state and and an off-equilibrium analysis is required. Additionally, trigonometric relations appear in most of the state equamotion (2007)]. Its transformation into an LPV ofthe direct control over experimental parameters restricts use of local identification methods. model is not relations straightforward. Inter-state coupling and lack trigonometric appear in most of the state equastricts the use of local identification methods. trigonometric relations appear in most of the state equalack of direct control over experimental parameters remodel is not relations straightforward. coupling and stricts the use ofa local identification methods. In this paper, quasi-LPV modelling approach avoidtrigonometric appear inInter-state most of the state equaIn this paper, a quasi-LPV modelling approach avoidThis work is in part financed by the French Defense Procurement stricts the use of local identification methods. trigonometric appear inFrench most Defense of the state equa- ing In this paper, a quasi-LPV modelling and approach avoidThis work is inrelations part financed by the Procurement the aforementioned simplifications equilibriumAgency (DGA, Direction G´ en´ erale de l’Armement). This work is in part financed by the French Defense Procurement In this paper, a quasi-LPV modelling and approach avoiding the aforementioned simplifications equilibriumAgency (DGA, Direction G´ en´ erale de l’Armement). ing the aforementioned simplifications and equilibriumThis work is in part financed by the French Defense Procurement In this paper, a quasi-LPV modelling approach avoidAgency (DGA, Direction G´ en´ erale de l’Armement). ing the aforementioned simplifications and equilibriumThis work is in part financed by the French Defense Procurement Agency (DGA, Direction G´ en´ erale de l’Armement). ing the aforementioned simplifications and equilibriumAgency (DGA, Direction G´ en´ erale de l’Armement).
Copyright © 2018, 2018 IFAC 1080Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018 IFAC 1080 Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 1080Control. Copyright © 2018 IFAC 1080 10.1016/j.ifacol.2018.09.050 Copyright © 2018 IFAC 1080
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oriented analysis, is introduced. The proposed approach is based on a premise that complexity of the system’s dynamic equations depends on a selected reference frame: the usual choice is the Variable-Roll (VR) frame, rigidly fixed on the projectile’s centre of mass and subject all its motions [Zipfel (2007); Wernert et al. (2010)]. Alternatively, the model may be developed in a so-called Fixed-Roll frame (FR), in which the frame is fixed on the projectile’s centre of mass, but does not experience the rolling motion [Wernert et al. (2010)]. Such a method origins from the 1960s [Barnett (1962)] and was initially used to reduce computation complexity of flight trajectory calculations. Interestingly, the FR equations are well-adapted for quasiLPV modelling. The approach is applied on the example of a spin-stabilized projectile, with scheduling signals and environmental conditions corresponding to these of the real-world experiments performed on ballistic test ranges [Dobre et al. (2016)]. The analysis has been intended as a preliminary study to investigate modelling prospects, before the sensitivity analysis and parameter identification steps. The paper is organised as follows. The nonlinear model is presented in Section 2, followed by its transformation into an FR frame. The proposed quasi-LPV model is developed in Section 3. In Section 4, the results of quasiLPV and nonlinear model simulations are compared. The main conclusions are drawn in Section 5. 2. NONLINEAR MODEL The model of in-flight dynamics is a nonlinear state-space continuous-time structure M: x(t) ˙ = f (x(t), C(x(t), pa )), x(0) = x0 M: (1) y(t) = g(x(t))
where the state vector x ∈ R12 has components corresponding to the translational velocities (u, v, w), the rotational velocities (p, q, r), the position of the centre of gravity of the vehicle w.r.t. Earth (xE , yE , zE ), and the roll, the pitch and the yaw angles (φ, θ and ψ, respectively). The aerodynamic coefficients C are described by polynomial functions whose coefficients are the aerodynamic coefficient parameters pa . These polynomial descriptions depend on the linear and angular velocities, and on the Mach number M and the total angle of attack αt , which are defined by: u V M= αt = arccos (2) a V where a is√the speed of sound, and V is the total velocity, i.e. V = u2 + v 2 + w2 . The output vector y represents the measurements obtained by available sensors and state observers, presented thereafter. The analysed projectile is equipped with three types of sensors: gyroscopes, magnetometers, and accelerometers. Additionally, the in-flight measurements are supplemented by an on-ground Doppler radar. The gyroscopes measure angular velocities of the projectile, thus p, q, r are considered as known. Accelerometer’s signal correspond to the specific force acting on the projectile which could be used for estimation of at least some of the linear velocities [Peter et al. (2012), Colgren (2001)]. The magnetometers measure a projection of the Earth’s magnetic field on the sensor
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axes, which allows to obtain the values of the roll angle φ [Changey et al. (2014)]. The Doppler radar measures the total velocity V of the projectile. Moreover, both the initial and terminal positions, corresponding to the gun placement and impact point, are known. Consequently, 4 out of 12 state variables in the rolling frame, i.e. p, q, r, φ can be considered to be measurable. Furthermore, the total velocity V is measured, and will be considered as an exogenous scheduling variable. 2.1 VR nonlinear model The nonlinear model dynamic equations are based on Newton’s and Euler’s laws of motion. In the VR frame, they have the following form: u˙ 0 −r q u −sθ q¯S Cx v˙ = r 0 −p v + Cy + g sφcθ (3) m C w˙ −q p 0 w cφcθ z p˙ 0 −r q p Cl −1 q˙ = J − r 0 −p J q + q¯Sd Cm (4) r˙ −q p 0 r Cn where g is the gravitational acceleration, m is the projectile’s mass, J is the inertia tensor, and S and d are the reference surface and diameter, respectively. Additionally, s. and c. correspond to trigonometric sine and cosine, respectively. The projectile is assumed to be axisymmetric, hence J = diag(Ix , Is , Is ), where Ix , Is are the longitudinal and lateral moments of inertia. These physical properties are measured for each projectile and considered known. The dynamic pressure is denoted by q¯ = 12 ρV 2 , where ρ is the air density, also assumed to be known. The coefficients Cx , Cy , Cz and Cl , Cm , Cn correspond to the aerodynamic forces and moments, respectively, acting along x, y, z axis. The evolution of the projectile’s position and attitude can be described by the following kinematic relations: φ˙ 1 sφtθ cφtθ p θ˙ = 0 cφ −sφ q (5) 0 sφc−1 θ cφc−1 θ r ψ˙ x˙ E cθcψ sφsθcψ − sψcφ cφsθcψ + sφsψ u y˙ E = cθsψ sφsθsψ + cψcφ cφsθsψ − sφcψ v z˙E −sθ sφcθ cφcθ w (6) where t. corresponds to trigonometric tangent. 2.2 Equivalence of VR and FR frames The FR reference frame is defined by a rotation around the x body axis by the roll angle φ. It can be described using the following transformation equations [Fischer and Hathaway (1988)] as: u 1 0 0 u v = 0 cφ −sφ v (7) 0 sφ cφ w w p 1 0 0 p q = 0 cφ −sφ q (8) 0 sφ cφ r r
where the superscript · indicates a variable expressed w.r.t. the FR frame. The Euler angles and positions are equivalent regardless of the selected frame.
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2.3 FR nonlinear model
3.1 Decomposition of the nonlinear remainder term
Based on (7)-(8) and through a series of algebraic transformations, the nonlinear model equations in the FR frame are describedas follows: u 0 −r q u˙ v˙ = r 0 r tθ v w˙ w −q −r tθ 0 −sθ q¯S Cx Cy + g 0 + (9) m C cθ
The suggested decomposition approach is based on algebraic transformations, as in [Tan et al. (2000), Pfifer (2012), Qin et al. (2015)]. Alternatively, a numerical decomposition could be applied [Shin (2005)], but the physical insight into the nonlinearity would be reduced.
z
p p˙ 0 −r q Cl q˙ = J −1 − r 0 r tθ J q + q¯Sd Cm Cn r r˙ −q −r tθ 0 (10) u x˙ E cθcψ −sψ sθcψ y˙ E cθsψ cψ sθsψ v (11) −sθ 0 cθ z˙E w 1 0 q θ˙ . (12) = 0 c−1 θ r ψ˙
Firstly, the force coefficients Cy , Cz can be expressed as coefficients of the normal force slope CN a and Magnus force slope Cypa . Similarly, the moment coefficients Cl , Cm , Cn can be developed into the following coefficients: roll damping Clp , pitch moment coefficient slope Cmα , pitch damping Cmq , and Magnus moment coefficient slope Cnpa [Albisser (2015)]. With the exception of the axial force coefficient Cx , the linear and angular velocities appear in the equations through linear dependence, which can be used in the decomposition of the remainder term: Cx = −CX (u, V ) v d w Cypa (V ) p Cy = −CN a (V ) + V 2V V d w v Cypa (V ) p Cz = −CN a (V ) − V 2V V d Clp (V )p Cl = 2V d w Cmq (V )q Cm = Cmα (V ) + V 2V v d Cnpa (V ) p + 2V V d v Cmq (V )r Cn = −Cmα (V ) + V 2V w d Cnpa (V ) p + 2V V
The roll angle can be calculated a posteriori [Wernert et al. (2010)] as: (13) φ˙ = p + ψ˙ sθ or t
φ(t) =
p (τ ) + r (τ )tθ(τ )dτ.
(14)
0
The signals measured in the rolling frame can be transformed to the FR frame through (7)-(8). 3. QUASI-LPV MODEL DEVELOPMENT The quasi-LPV model proposed in this paper is developed using a function substitution approach. This method allows to represent the off-equilibrium behaviour of a projectile [Shin (2005)], and has several applications in aerospace, e.g. in autopilot design [Shin et al. (2002), Tan et al. (2000), Pfifer (2012)] and aircraft fault detection [Marcos and Balas (2004)]. The standard derivation of the function substitution approach, as described in [Pfifer (2012)], is based on the assumption that the nonlinear model has the following form: x˙ = A(x1 , ρ)x + B(x1 , ρ)u + f (x1 , ρ) (15) where x1 and ρ are the scheduling states and exogenous scheduling signals, respectively. The nonlinear function f , called the remainder term, is decomposed into a form linear w.r.t. x1 , such as: (16) f (x1 , ρ) = A1 (x1 , ρ)x. Consequently, a quasi-LPV model can be obtained: (17) x˙ = [A(x1 , ρ) + A1 (x1 , ρ)]x + B(x1 , ρ)u. The nonlinear model equations (9)-(12) are already in a quasi-LPV form, with the exception of gravitational and aerodynamic components. Consequently, the forces and moments equations are considered to be the remainder term (called also a decomposition function [Marcos and Balas (2004)]).
(18)
where each Ci (u, V ) has a polynomial description dependent on the measurable signals: Ci (u, V ) = Ci1 + Ci2 s2 αt + Ci3 where s2 αt = (1 −
V V2 + Ci4 2 a a
(19)
u2 V 2 ).
The nonlinear remainder term then has the following form, described through auxiliary functions X1 , ...X7 : q¯S Cx − gsθ = X1 (u, V, zE )sθ ) − g(zE m q¯S Cy = X2 (V, zE )w p + X3 (V, zE )v m q¯S Cz + gcθ = −X2 (V, zE )v p + X3 (V, zE )w m + g(zE )cθ q¯Sd Cl = X4 (V, zE )p Ix q¯Sd Cm = X5 (V, zE )w + X6 (V, zE )q Is + X7 (V , zE )v p q¯Sd Cn = −X5 (V, zE )v + X6 (V, zE )r Is + X7 (V, zE )w p where:
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Difference [deg]
X1 (u, V, zE ) = −0.5m−1 ρ(zE )V 2 SCX (u, V ) )SdCypa (V ) X2 (V, zE ) = 0.25m−1 ρ(zE −1 X3 (V, zE ) = −0.5m ρ(zE )V SCN a (V ) X4 (V, zE ) = 0.25ρ(zE )V Sd2 Clp (V ) )V SdCmα (V ) X5 (V, zE ) = 0.5ρ(zE X6 (V, zE ) = 0.25ρ(zE )V Sd2 Cmq (V ) )Sd2 Cnpa (V ). X7 (V, zE ) = 0.25ρ(zE
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10-6
5 0 -5 0
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2
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2.9
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3
400 Nonlinear
Additionally, based on (2) and (7), the following equivalences can be stated: αt = αt and V = V .
Consequently, all the nonlinear remainder terms have been developed to a form which either depends on the measurable variables, or can be directly assigned to nonmeasurable state variable. As a result, the quasi-LPV model of the following form has been obtained: x˙ = A(u , V, p , q , r , θ, ψ, zE )x (22) with the full matrix A presented in (A.1).
300
200
250
150 100
2.75
2.85
Difference [rad/s]
Fig. 1. Evolution of the roll angle φ (signal evolution enlarged). 10-5
1 0 -1 0
0.5
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2
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2.85
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-0.05 Nonlinear
q-LPV
2.95
3
t [s]
Fig. 2. Evolution of the pitch rate q (signal evolution enlarged).
Difference
4. SIMULATION RESULTS The quasi-LPV model has been tested through a series of simulations in order to assess its accuracy w.r.t. the existing nonlinear model results, obtained under the same conditions.
There are several potential sources of error that were investigated. Firstly, inaccurate calculations of the aerodynamic coefficients would result in increasingly incorrect forces and moments obtained in the simulation, regardless of the frame being used. Secondly, the aerodynamic coefficients
2.8
t [s]
10-9
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1
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2
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3
0.88 Nonlinear
q-LPV
0.875
Mach number
Both models have been initialised with the state vector values corresponding to real-world ballistic test campaigns of spin-stabilised projectiles. The initial Mach number presented a variation range from 0.88 to 1.77 and spin rate from 608 to 1217 rad/s. Simulations were conducted for a flat-fire scenario: the initial velocity vector is almost parallel to the Earth’s surface. The polynomial descriptions of the aerodynamic coefficients for such experimental conditions are known, and were used in both models’ descriptions. The simulation results consist of the quasiLPV model data in FR frame, and of the nonlinear model data in VR frame, denoted in the following figures as ’qLPV’ and ’Nonlinear’, respectively. To allow direct comparison, the quasi-LPV results were transformed into the VR frame, using an inverse of the (7)-(8) transformation.
q-LPV
50
Pitch rate q [rad/s]
Only X1 and the gravity components are not in a linear dependence w.r.t. a measurable state variable. Hence, they need to be decomposed arbitrarily, by multiplication and division by any non-zero measurable signal, such as the spin rate p. By integrating the spin rate in the decomposition, the following decomposition of X1 may be obtained: 1 (21) X1 (u, V, zE ) = X1 (u, V, zE ) · · p. p The gravity terms are treated in the same way. The linear velocity u, pitch angle θ and yaw angle ψ are simulated, knowing their initial values. Moreover, θ˙ and ψ˙ equations in FR frame depend only on q , r signals, which are known through (8) since q, r, φ are measurable.
[deg]
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Roll angle
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0.87 0.865 0.86 0.855 0.85 0
0.5
1
1.5
2
2.5
3
t [s]
Fig. 3. Evolution of the Mach number M . depend on the linear and angular velocities, whose accuracy must also be evaluated. Finally, the measurable linear and angular velocities are sensitive to erroneous roll angle φ, since it is used for their transformations between the reference frames. Error of the φ angle remained negligible for the tested scenarios (less than 10−6 deg difference between models, see Fig. 1). Consequently, the measurable signals could be accurately transformed between the frames. It is illustrated by the pitch rate error evolution: the difference between the signal evolution in VR and FR frames is four orders of magnitude smaller than the signal itself (Fig. 2).
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10-5
5 0 -5 0
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1
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0
q-LPV
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3
Cmq coefficient
-22.6
0.4
[deg]
0.5
-22.4
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t
0.3 0.2 0.1
-22.8 -23 Nonlinear
-23.2
q-LPV
-23.4
0 0
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1
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2
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-23.6
3
0
t [s]
-5 0.5
1
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2
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3
0.123 Nonlinear
q-LPV
0.122 0.121 0.12 0.119 0.118 0.117 0
0.5
1
1.5
1.5
2
2.5
3
The quasi-LPV model accuracy has been assessed through comparison with a nonlinear model simulation. The differences between the two models were negligible for the tested scenarios, with the misalignment being of several orders of magnitude smaller than the signals values. It is predicted, that the differences between models stem from the numerical inaccuracies of the ode solvers being used. The future analysis aims to perform the convergence and stability assessment of the numerical solvers, in order to verify their accuracy and provide insight into the possible sources of differences present in the model. Afterwards, a global sensitivity analysis of the model outputs will be performed, based on the obtained global quasi-LPV model, and with the final goal of identifying the aerodynamic coefficients.
0
0
1
Fig. 6. Evolution of the pitch damping coefficient Cmq.
10-8
5
0.5
t [s]
Fig. 4. Evolution of the total angle of attack αt . Difference
0 -2
3
0.6
Cx coefficient
10-7
2
Difference
Difference [deg]
2018 IFAC SYSID July 9-11, 2018. Stockholm, Sweden 1084
2
2.5
3
t [s]
Fig. 5. Evolution of the axial force coefficient CX . The polynomial descriptions of the coefficients in (19) are functions of M and αt . For both reference frames, these variables are in good coherence: the errors are increasing over time, but their magnitude is of at least four orders smaller than the signals (see Fig. 3-4). Good coherence of evolutions of the roll angle, the measurable signals, the total angle of attack, and the Mach number have resulted in accurate calculations of the aerodynamic coefficients. Behaviour of the error is the same as previously: slowly increasing, but of negligible magnitude. It is depicted by the trajectory of the axial force coefficient CX (Fig. 5), and the pitch damping coefficient Cmq (Fig. 6). Additionally, the expected computation time reduction has indeed been observed. For the initial spin rate p0 = 608 rad/s, and 3s of the flight time, the simulation lasted 15s for the nonlinear model in the VR frame, and 0.8s for the quasi-LPV model in FR frame. 5. CONCLUSIONS This paper investigates the prospects of using a reference frame transformation in developing a quasi-LPV model of a projectile’s in-flight behaviour. The analysis have been dedicated to an example of a spin-stabilised projectile. Firstly, the projectile’s equation of motion have been transformed into a non-rolling reference frame, which has simplified the model equations and reduced computation time. Secondly, the quasi-LPV model has been obtained using the analytic function substitution method.
REFERENCES Al-jiboory, A., Zhu, G.G., Swei, S.S.M., Su, W., and Nguyen, N.T. (2017). LPV Model Development for a Flexible Wing Aircraft. In 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Albisser, M. (2015). Identification of aerodynamic coefficients from free flight data. Ph.D. thesis, University of Lorraine. Albisser, M., Dobre, S., Berner, C., Thomassin, M., and Garnier, H. (2017). Aerodynamic Coefficient Identification of a Space Vehicle from Multiple Free-Flight Tests. Journal of Spacecraft and Rockets, 54(2), 426–435. Barnett, B. (1962). Trajectory equations for a six-degreeof-freedom missile. Technical report, Picatinny Arsenal. Berger, T., Tischler, M.B., Hagerott, S.G., Cotting, M.C., Gray, W.R., Gresham, J., George, J., Krogh, K., D’Argenio, A., and Howland, J. (2017). Development and Validation of a Flight-Identified Full-Envelope Business Jet Simulation Model Using a Stitching Architecture. AIAA Modeling and Simulation Technologies Conference. Changey, S., Pecheur, E., and Brunner, T. (2014). Attitude Estimation of a projectile using Magnetometers and Accelerometers: Experimental Validation. IEEE/ION PLANS Symposium 2014, 1168–1173. Colgren, R. (2001). Method and System for Estimation and Correction of the Angle of Attack and Sideslip Angle from Acceleration Measurements. U.S. Patent No. 6,273,370.
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The quasi-LPV model has the following form:
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0 00
q
) − g(zE )sθ X1 (u , V, zE 0 0 p X3 (V, zE ) r tθ + X2 (V, zE )p 0 0 0 g(z )cθ E −r tθ − X2 (V, zE )p X3 (V, zE ) 0 0 p 0 0 X4 (V, zE ) 0 0 Ix r r tθ X7 (V, zE )p X6 (V, zE ) X5 (V, zE ) − r − 0 Is p Ix q r tθ 0 X6 (V, zE ) X7 (V, zE )p q + ) −X5 (V, zE Is p 0 0 0 1 0 0 0 0 0 c−1 θ −sψ sθcψ 0 0 0 cψ sθsψ 0 0 0 0 cθ 0 0 0
000
0 0 0 u v 0 0 0 w 0 0 0 p q 0 0 0 · r θ 0 0 0 ψ x 0 0 0 E yE 0 0 0 0 0 0 zE 0 00
(A.1)
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−sθ
u˙ r v˙ −q w˙ p˙ 0 q˙ r˙ 0 = θ˙ ˙ 0 ψ x˙ 0 E y˙ 0 E cθcψ z˙E cθsψ
−r
Appendix A. QUASI-LPV MODEL EQUATION
0
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2018 IFAC SYSID July 9-11, 2018. Stockholm, Sweden