MATHEMATICAL MODELS FOR CONTROL OF AIRCRAFT AND SATELLITES THOR I. FOSSEN

Transcription

1 MATHEMATICAL MODELS FOR CONTROL OF AIRCRAFT AND SATELLITES THOR I. FOSSEN Professor of Guidance, Navigation and Control D E C N U S T June 216 4th edition Copyright c Department of Engineering Cybernetics, NTNU. 1st edition published February nd edition published January rd edition published April th edition published June 216.

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3 Contents Figures iii 1 Introduction 1 2 Aircraft Modeling Definition of Aircraft State-Space Vectors Body-Fixed Coordinate Systems for Aircraft Rotation matrices for wind and stability axes Aircraft Equations of Motion Kinematic equations for translation Kinematic equations for attitude Rigid-body kinetics Sensors and measurement systems Perturbation Theory (Linear Theory) Definition of nominal and perturbation values Linearization of the rigid-body kinetics Linear state-space model based using wind and stability axes Decoupling in Longitudinal and Lateral Modes Longitudinal equations Lateral equations Aerodynamic Forces and Moments Longitudinal aerodynamic forces and moments Lateral aerodynamic forces and moments Propeller Thrust Aerodynamic Coeffi cients Aerodynamic forces and moments Drag coeffi cient Lift coeffi cient Sideforce coeffi cient Rolling moment coeffi cient Pitching moment coeffi cient Yawing moment coeffi cient Equations of motion including aerodynamic coeffi cients Standard Aircraft Maneuvers i

4 ii CONTENTS Dynamic equation for coordinated turn (bank-to-turn) Dynamic equation for altitude control Aircraft Stability Properties Longitudinal stability analysis Lateral stability analysis Design of flight control systems Satellite Modeling Attitude Model Euler s 2nd Axiom applied to satellites Skew-symmetric representation of the satellite model Actuator Dynamics Design of Satellite Attitude Control Systems Nonlinear quaternion set-point regulator Matlab Simulation Models Boeing Longitudinal model Lateral model F-16 Fighter Longitudinal model F2B Bristol Fighter Lateral model NTNU student cube-satellite

5 List of Figures 1.1 Sketch showing a modern fighter aircraft (Stevens and Lewis 1992) Definition of aircraft body axes, generalized velocities, forces, moments and Euler angles (McLean 199) Definition of stability and wind axes for an aircraft (Stevens and Lewis 1992) Control inputs for conventional aircraft. Notice that the two ailerons can be controlled by using one control input: δ A = 1/2(δ AL + δ AR ) Lift and drag as a function of angle of attack Aircraft longitudinal eigenvalue configuration plotted in the complex plane The NUTS CubeSat project at NTNU- Courtesy to Schematic drawing of the Bristol F.2B Fighter (McRuer et al 1973) iii

6 iv LIST OF FIGURES

7 Chapter 1 Introduction This booklet is a collection of lecture notes used in the course TTK419 Guidance and Control of Vehicles, which is given at the Department of Engineering Cybernetics, NTNU. The kinematic and kinetic equations of a marine craft (ships, high-speed craft and underwater vehicles) can be modified to describe aircraft and satellites by minor adjustments of notation and assumptions. Consequently, the vectorial notation introduced in the two Wiley textbooks: Fossen, T. I. (1994). Guidance and Control of Ocean Vehicles Fossen, T. I. (211). Handbook of Marine Craft Hydrodynamics and Motion Control is used to describe the aircraft and satellite equations of motion. Figure 1.1: Sketch showing a modern fighter aircraft (Stevens and Lewis 1992). The booklet is organized according to: Chapter 2: Aircraft Modeling Chapter 3: Satellite Modeling Chapter 4: Matlab Simulation Models 1

8 2 CHAPTER 1. INTRODUCTION Other useful references on flight control are: Blakelock, J. H. (1991). Aircraft and Missiles (John Wiley & Sons Ltd.) Etkin, B. and L. D. Reid (1996). Dynamics of Flight: Stability and Control Wiley & Sons Ltd.) (John Fortescue, P., G. Swinerd and J. Stark (211). Spacecraft Systems Engineering (John Wiley & Sons Ltd.) Hughes, P. C. (1986). Spacecraft Attitude Dynamics (John Wiley & Sons Ltd.) McLean, D. (199). Automatic Flight Control Systems (Prentice Hall Inc.) McRuer, D., D. Ashkenas and A. I. Graham (1973). Aircraft Dynamics and Automatic Control (Princeton University Press) Nelson, R. C. (1998). Flight Stability and Automatic Control (McGraw-Hill Int.) Roskam, J. (1999). Airplane Flight Dynamics and Automatic Flight Controls (Darcorporation) Schmidt, D. K. (212). Modern Flight Dynamics (McGraw-Hill Int.) Stevens, B. L. and F. L. Lewis (23). Aircraft Control and Simulation (John Wiley & Sons Ltd.) Information about the graduate coursesttk419 Guidance and Control of Vehicles and TK819 Advanced Topics in Guidance and Navigation are found on the Wiki-pages URL: URL: Thor I. Fossen Trondheim 3 April 213

9 Chapter 2 Aircraft Modeling This chapter gives an introduction to aircraft modeling. The equations of motion are linearized using perturbation theory and the final results are state-space models for the longitudinal and lateral motions. The models can be used for aircraft simulation and design of flight control systems. 2.1 Definition of Aircraft State-Space Vectors The aircraft generalized velocity vector is defined according to (see Figure 2.1): ν := η := U V W P Q R X E Y E Z E, h Φ Θ Ψ = = longitudinal (forward) velocity lateral (transverse) velocity vertical velocity roll rate pitch rate yaw rate Earth-fixed x-position Earth-fixed y-position Earth-fixed z-position (axis downwards), altitude roll angle pitch angle yaw angle (2.1) (2.2) Forces and moments are defined in a similar manner: X longitudinal force Y transverse force Z L := vertical force roll moment M pitch moment N yaw moment (2.3) Comment 1: Notice that the capital letters L, M, N for the moments are different from 3

10 4 CHAPTER 2. AIRCRAFT MODELING Figure 2.1: Definition of aircraft body axes, generalized velocities, forces, moments and Euler angles (McLean 199). those used for marine craft that is, K, M, N. The reason for this is that L is reserved as length parameter for ships and underwater vehicles. Comment 2: For aircraft it is common to use capital letters for the states U, V, W, etc. while it is common to use small letters for marine craft. 2.2 Body-Fixed Coordinate Systems for Aircraft For aircraft it is common to use the following body-fixed coordinate systems: BODY axes, superscript b STABILITY axes, superscript s WIND axes, superscript w The axis systems are shown in Figure 2.2 where the angle of attack α and sideslip angle β are defined as: tan(α) := W U (2.4) sin(β) := V V T (2.5) where V T = U 2 + V 2 + W 2 (2.6)

11 2.2. BODY-FIXED COORDINATE SYSTEMS FOR AIRCRAFT 5 Figure 2.2: Definition of stability and wind axes for an aircraft (Stevens and Lewis 1992). is the total speed of the aircraft. Aerodynamic effects are classified according to the Mach number: M := V T a (2.7) where a = 34 m/s = 1224 km/h is the speed of sound in air at a temperature of 2 o C on the ocean surface. The following terminology is used for varying speed: Subsonic speed M < 1. Transonic speed.8 M 1.2 Supersonic speed 1. M 5. Hypersonic speed 5. M An aircraft will break the sound barrier at M = 1. and this is clearly heard as a sharp crack. If you fly at low altitude and break the sound barrier, windows in building will break due to pressure-induced waves Rotation matrices for wind and stability axes The relationship between vectors expressed in different coordinate systems can be derived using rotation matrices. The BODY axes are first rotated a negative sideslip angle β about the z-axis. The new coordinate system is then rotated a positive angle of attack α about the new y-axis such that the resulting x-axis points in the direction of the total speed V T. The first rotation defines the WIND axes while the second rotation defines the STABILITY

12 6 CHAPTER 2. AIRCRAFT MODELING axes. This can be mathematically expressed as: cos(β) sin(β) p w = R z, β p s = sin(β) cos(β) p s (2.8) 1 cos(α) sin(α) p s = R y,α p b = 1 p b (2.9) sin(α) cos(α) The rotation matrix becomes: Hence, R w b = R z, β R y,α (2.1) p w = R w b p b (2.11) p w = p w = cos(β) sin(β) sin(β) cos(β) 1 cos(α) sin(α) 1 sin(α) cos(α) cos(α) cos(β) sin(β) sin(α) cos(β) cos(α) sin(β) cos(β) sin(α) sin(β) sin(α) cos(α) p b (2.12) p b (2.13) This gives the following relationship between the velocities in body and wind axes: v b = Consequently, U V W = (R w b ) v w = R y,αr z, β V T = U = V T cos(α) cos(β) V = V T sin(β) W = V T sin(α) cos(β) V T cos(α) cos(β) V T sin(β) V T sin(α) cos(β) (2.14) (2.15) 2.3 Aircraft Equations of Motion Kinematic equations for translation The kinematic equations for translation and rotation of a body-fixed coordinate system BODY with respect to a local geographic coordinate system NED (North-East-Down) can be expressed in terms or the Euler angles: Ẋ E Ẏ E ŻE = R n b U V W = R z,ψ R y,θ R x,φ U V W (2.16)

13 2.3. AIRCRAFT EQUATIONS OF MOTION 7 Expanding this expression gives: Ẋ E cψ sψ Ẏ E = sψ cψ 1 ŻE Ẋ E Ẏ E ŻE = cθ sθ 1 sθ cθ 1 cφ sφ sφ cφ cψcθ sψcφ + cψsθsφ sψsφ + cψcφsθ sψcθ cψcφ + sφsθsψ cψsφ + sθsψcφ sθ cθsφ cθcφ U V W U V W (2.17) Kinematic equations for attitude The attitude is given by: P Q = R Φ + R x,φ Θ + R x,φr y,θ Ψ (2.18) which gives: Φ Θ Ψ = 1 sφtθ cφtθ cφ sφ sφ/cθ cφ/cθ P Q R, cθ (2.19) Rigid-body kinetics The aircraft rigid-body kinetics can be expressed as (Fossen 1994, 211): m( ν 1 + ν 2 ν 1 ) = τ 1 (2.2) I CG ν 2 + ν 2 (I CG ν 2 ) = τ 2 (2.21) where ν 1 := [U, V, W ] T, ν 2 := [P, Q, R] T, τ 1 := [X, Y, Z] T and τ 2 := [L, M, N] T. It is assumed that the coordinate system is located in the aircraft center of gravity (CG). The resulting model is written: M RB ν + C RB (ν)ν = τ RB (2.22) where [ mi3 3 O M RB = 3 3 O 3 3 I CG ] [ ms(ν2 ) O, C RB (ν) = 3 3 O 3 3 S(I CG ν 2 ) ] (2.23) The inertia tensor is defined as (assume that I xy = I yz = which corresponds to xz plane symmetry): I x I xz I CG := I y (2.24) I xz I z The forces and moments acting on the aircraft can be expressed as: τ RB = g(η) + τ (2.25)

14 8 CHAPTER 2. AIRCRAFT MODELING where τ is a generalized vector that includes aerodynamic and control forces. The gravitational force f G = [ mg] T acts in the CG (origin of the body-fixed coordinate system) and this gives the following vector expressed in NED: mg sin(θ) [ ] mg cos(θ) sin(φ) g(η) = (R n fg b ) = mg cos(θ) cos(φ) O 3 1 (2.26) Hence, the aircraft model can be written in matrix form as: or in component form: M RB ν + C RB (ν)ν + g(η) = τ (2.27) m( U + QW RV + g sin(θ)) = X m( V + UR W P g cos(θ) sin(φ)) = Y m(ẇ + V P QU g cos(θ) cos(φ)) = Z I x P I xz (Ṙ + P Q) + (I z I y )QR = L (2.28) I y Q + I xz (P 2 R 2 ) + (I x I z )P R = M I z Ṙ I xz P + (I y I x )P Q + I xz QR = N Sensors and measurement systems It is common that aircraft sensor systems are equipped with three accelerometers. If the accelerometers are located in the CG, the measurement equations take the following form: a xcg = X m = U + QW RV + g sin(θ) (2.29) a ycg = Y m = V + UR W P g cos(θ) sin(φ) (2.3) a zcg = Z m = Ẇ + V P QU g cos(θ) cos(φ) (2.31) In addition to these sensors, an aircraft is equipped with gyros, magnetometers and a sensor for altitude h and wind speed V T. These sensors are used in inertial navigation systems (INS) which again use a Kalman filter to compute estimates of U, V, W, P, Q and R as well as the Euler angles Φ,Θ and Ψ. Other measurement systems that are used onboard aircraft are global navigation satellite systems (GNSS), radar and sensors for angle of attack. 2.4 Perturbation Theory (Linear Theory) The nonlinear equations of motion can be linearized by using perturbation theory. This is illustrated below.

15 2.4. PERTURBATION THEORY (LINEAR THEORY) Definition of nominal and perturbation values According to linear theory it is possible to write the states as the sum of a nominal value (usually constant) and a perturbation (deviation from the nominal value). Moreover, The following definitions are made: X δx Y δy τ := τ + δτ = Z L + δz δl M δm N δn Total state = Nominal value + Perturbation, ν := ν + δν = U V W P Q R + u v w p q r (2.32) Similar, the angles are defined according to: Θ Φ := Ψ Θ Φ Ψ + θ φ ψ (2.33) Consequently, a linearized state-space model will consist of the following states u, v, w, p, q, r, θ, φ and ψ Linearization of the rigid-body kinetics The rigid-body kinetics can be linearized by using perturbation theory. Equilibrium condition If the aerodynamic forces and moments, velocities, angles and control inputs are expressed as nominal values and perturbations τ = τ + δτ, ν = ν + δν and η = η + δη, the aircraft equilibrium point will satisfy (it is assumed that ν = ): This can be expanded according to: C RB (ν )ν + g(η ) = τ (2.34) m(q W R V + g sin(θ )) = X m(u R P W g cos(θ ) sin(φ )) = Y m(p V Q U g cos(θ ) cos(φ )) = Z (I z I y )Q R P Q I xz = L (2.35) (P 2 R 2 )I xz + (I x I z )P R = M (I y I x )P Q + Q R I xz = N

16 1 CHAPTER 2. AIRCRAFT MODELING Perturbed equations The perturbed equations that is, the linearized equations of motion are usually derived by a 1st-order Taylor series expansion about the nominal values. Alternatively, it is possible to substitute (2.32) and (2.35) into (2.27) and neglect higher-order terms of the perturbed states. This is illustrated for the first degree of freedom (DOF): Example 1 (Linearization of surge using perturbation theory) m[ U + QW RV + g sin(θ)] = X (2.36) m[ U + u + (Q + q)(w + w) (R + r)(v + v) + g sin(θ + θ)] = X + δx This can be written: sin(θ + θ) = sin(θ ) cos(θ) + cos(θ ) sin(θ) θ small sin(θ ) + cos(θ )θ (2.37) Since U = and m(q W R V + g sin(θ )) = X (2.38) Equation (2.36) is reduced to: m[ u + Q w + W q + wq R v V r vr + g cos(θ )θ] = δx (2.39) If it is assumed that the 2nd-order terms wq and vr are negligible, the linearized model becomes: m[ u + Q w + W q R v V r + g cos(θ )θ] = δx (2.4) Linear state-space model for aircraft If all DOFs are linearized, the following state-space model is obtained: m[ u + Q w + W q R v V r + g cos(θ )θ] = δx m[ v + U r + R u W p P w g cos(θ ) cos(φ )φ + g sin(θ ) sin(φ )θ] = δy m[ẇ + V p + P v U q Q u + g cos(θ ) sin(φ )φ + g sin(θ ) cos(φ )θ] = δz I x ṗ I xz ṙ + (I z I y )(Q r + R q) I xz (P q + Q p) = δl (2.41) I y q + (I x I z )(P r + R p) 2I xz (R r + P p) = δm I z ṙ I xz ṗ + (I y I x )(P q + Q p) + I xz (Q r + R q) = δn This can be expressed in matrix form as: M RB ν + N RB ν + Gη = τ (2.42)

17 2.4. PERTURBATION THEORY (LINEAR THEORY) 11 where M RB = N RB = G = m m 3 3 m I x I xz 3 3 I y I xz I z mr mq mw mv mr P W U mq mp mv mu I xz Q (I z I y )R I xz P (I z I y )Q 3 3 (I x I z )R 2I xz P (I x I z )P 2I xz R (I y I x )Q (I y I x )P + I xz R I xz Q mg cos(θ ) 3 3 mg cos(θ ) cos(φ ) mg sin(θ ) sin(φ ) mg cos(θ ) sin(φ ) mg sin(θ ) cos(φ ) In addition to this, the kinematic equations must be linearized Linear state-space model based using wind and stability axes An alternative state-space model is obtained by using α and β as states. If it is assumed that α and β are small such that cos(α) 1 and sin(β) β, Equation (2.15) can be written as: U = V T U = V T V = V T β W = V T α Furthermore, the state-space vector: u β x = α p = q r β = V V T α = W V T (2.43) surge velocity sideslip angle angle of attack roll rate pitch rate yaw rate (2.44) is chosen to describe motions in 6 DOF. The relationship between the body-fixed velocity vector: ν = [u, v, w, p, q, r] T (2.45) and the new state-space vector x can be written as: ν = Tx = diag{1, V T, V T, 1, 1, 1, 1}x (2.46)

18 12 CHAPTER 2. AIRCRAFT MODELING where V T >. If the total speed is V T = U = constant (linear theory), it is seen that: α = 1 V T ẇ (2.47) β = 1 V T v (2.48) V T = (2.49) If nonlinear theory is applied, the following differential equations are obtained: α = UẆ W U U 2 + W 2 (2.5) β = V V T V V T V 2 T cos β (2.51) V T = U U + V V + W Ẇ V T (2.52) In the linear case it is possible to transform the body-fixed state-space model: to where ν = Fν + Gu (2.53) ẋ = Ax + Bu (2.54) A = T 1 FT, B = T 1 G (2.55) For V T this transformation is much more complicated. The linear state-space transformation is commonly used by aircraft manufactures. An example is the Boeing B-767 model (see Chapter 4). 2.5 Decoupling in Longitudinal and Lateral Modes For an aircraft it is common to assume that the longitudinal modes (DOFs 1, 3 and 5) are decoupled from the lateral modes (DOFs 2, 4 and 6). The key assumption is that the fuselage is slender that is, the length is much larger than the width and the height of the aircraft. It is also assumed that the longitudinal velocity is much larger than the vertical and transversal velocities. In order to decouple the rigid-body kinetics (2.41) in longitudinal and lateral modes it will be assumed that the states v, p, r and φ are negligible in the longitudinal channel while u, w, q and θ are negligible when considering the lateral channel. This gives two sub-systems:

19 2.5. DECOUPLING IN LONGITUDINAL AND LATERAL MODES Longitudinal equations Kinetics: m m I y m[ u + Q w + W q + g cos(θ )θ] = δx m[ẇ U q Q u + g sin(θ ) cos(φ )θ] = δz (2.56) u ẇ q + mq mw mq mu I y q = δm mg cos(θ ) mg sin(θ ) cos(φ ) u w q + θ = δx δz δm (2.57) Kinematics: θ = q (2.58) Lateral equations Kinetics: m I x I xz I xz I z m[ v + U r W p g cos(θ ) cos(φ )φ] = δy I x ṗ I xz ṙ + (I z I y )Q r I xz Q p = δl (2.59) I z ṙ I xz ṗ + (I y I x )Q p + I xz Q r = δn v ṗ ṙ + mw mu I xz Q (I z I y )Q (I y I x )Q I xz Q mg cos(θ ) cos(φ ) v p r + φ = δy δl δn (2.6) Kinematics: [ φ ψ ] = [ 1 tan(θ ) 1/ cos(θ ) ] [ p r ] (2.61)

20 14 CHAPTER 2. AIRCRAFT MODELING Figure 2.3: Control inputs for conventional aircraft. Notice that the two ailerons can be controlled by using one control input: δ A = 1/2(δ AL + δ AR ). 2.6 Aerodynamic Forces and Moments In the forthcoming sections, the following abbreviations and notation will be used to describe the aerodynamic coeffi cients: X index = Y index = Z index = X L index index = L index Y index M index = M index Z index N index = N index In order to illustrate how control surfaces influence the aircraft, an aircraft equipped with the following control inputs will be considered (see Figure 2.3):

21 2.6. AERODYNAMIC FORCES AND MOMENTS 15 δ T Thrust Jet/propeller δ E Elevator Control surfaces on the rear of the aircraft used for pitch and altitude control δ A Aileron Hinged control surfaces attached to the trailing edge of the wing used for roll/bank control δ F Flaps Hinged surfaces on the trailing edge of the wings used for braking and bank-to-turn δ R Rudder Vertical control surface at the rear of the aircraft used for turning A conventional aircraft has control surfaces such as ailerons δ A, elevators δ E, flaps δ F and rudders δ R. A positive deflection of the control surfaces are to give a negative aerodynamic moment on the aircraft. Some of the control surfaces, like the ailerons, deflects simultaneously in an asymmetric manner. If an aircraft has two ailerons δ AL and δ AR, the aileron deflection is computed by: δ A = 1 2 (δ A R δ AL ) (2.62) Other control surfaces, such as elevators, deflect symmetrically. If an aircraft has two elevators δ EL and δ ER, the elevator deflections become: δ E = 1 2 (δ E R + δ EL ) (2.63) Linear theory The number of aerodynamic coeffi cients is reduced if linear theory is assumed. The control inputs and aerodynamic forces and moments can be written as: τ = M F ν N F ν + Bu (2.64) where M F is aerodynamic added mass, N F is aerodynamic damping and B is a matrix describing the actuator configuration including the force coeffi cients. The actuator dynamics is modeled by a 1st-order system: u = T 1 (u c u) (2.65) where u c is commanded input, u is the actual control input produced by the actuators and T = diag{t 1, T 2,..., T r } is a diagonal matrix of positive time constants. Substitution of (2.64) into the model (2.42) gives: (M RB + M F ) ν + (N RB + N F )ν + Gη = Bu M ν + Nν + Gη = Bu (2.66) The matrices M and N are defined as M = M RB + M F and N = N RB + N F. The linearized kinematics takes the following form: The resulting state-space models are: η = J ν ν + J η η (2.67)

22 16 CHAPTER 2. AIRCRAFT MODELING Linear state-space model with actuator dynamics η ν u = J η J ν M 1 G M 1 N M 1 B T 1 η ν u + T 1 u c (2.68) Linear state-space model neglecting the actuator dynamics [ η ν ] [ J = η M 1 G J ν M 1 N ] [ η ν ] [ + M 1 B ] u (2.69) Longitudinal aerodynamic forces and moments McLean [9] expresses the longitudinal forces and moments as: dx dz dm = X u Xẇ X q Z u Zẇ Z q M u Mẇ M q u ẇ q + X u X w X q Z u Z w Z q M q M w M q X δt X δe X δf Z δt Z δe Z δf M δt M δe M δf u w q δ T δ E δ F + (2.7) which corresponds to the matrices M F, N F and B in (2.64). If the aircraft cruise speed U = constant, then δ T =. Altitude can be controlled by using the elevators δ E. Flaps δ F can be used to reduce the speed during landing. The flaps can also be used to turn harder for instance by moving one flap while the other is kept at the zero position. This is common in bank-to-turn maneuvers. For conventional aircraft the following aerodynamic coeffi cients can be neglected: X u, X q, Xẇ, X δe, Z u, Zẇ, M u (2.71) Hence, the model for altitude control reduces to: dx dz dm = X q Z q Mẇ M q u ẇ q + X u X w Z u Z w Z q M q M w M q u w q + X δe Z δe M δe δ E (2.72) If the actuator dynamics is important, aerodynamic coeffi cients such as X δt, X δe,... must be included in the model Lateral aerodynamic forces and moments The lateral model takes the form [9]:

23 2.7. PROPELLER THRUST 17 dy dl dn = Y v Yṗ Yṙ L v Lṗ Lṙ N v Nṗ Nṙ v ṗ ṙ + Y v Y p Y r L v L p L r N v N p N r v p r + Y δa L δa N δa Y δr L δr N δr [ δa δ R (2.73) which corresponds to the matrices M F, N F and B in (2.64). For conventional aircraft the following aerodynamic coeffi cients can be neglected: This gives: Y v, Y p, Yṗ, Y r, Yṙ, Y δa,l v, Lṙ, N v, Nṙ (2.74) ] dy dl dn = Lṗ Nṗ v ṗ ṙ + Y v L v L p L r N v N p N r v p r + Y δr L δa N δa L δr N δr [ δa δ R ] (2.75) 2.7 Propeller Thrust The thrust from a propeller can be modeled as: where T Thrust force [N] K t Thrust coeffi cient ρ Density of air [kg/m 3 ] n Propeller angular velocity [rad/s] D Propeller diameter [m] T = K t ρn 2 D 4 (2.76) Assuming that the trust force acts in the horizontal plane of {b}, and parallel with the x b -axis, it can be expressed in {b} as: X t T ft b = Y t Z t = (2.77) If the line of action of the thrust is not directed through CO, it will in addition generate an applied moment given by L t m b t = M t = r b t/b ft b = S(r b t/b)ft b N t = T r tz T r ty (2.78)

24 18 CHAPTER 2. AIRCRAFT MODELING where r b t/b = [r t x, r ty, r tz ] is the location of the propeller with respect to CO. Sometimes the effects due to the rotating mass caused by the propeller must be considered as an applied moment. This is because the equations of motion has been derived by assuming that the aircraft is a rigid body with no internal moving parts. The applied moment due to the rotating mass caused by the propeller is given by: m b p = i d dt hb p (2.79) where h b p is the angular momentum of the rotating mass, and is given in {b} as: h b p = I p Ω p (2.8) where I p is the inertia of the rotating mass and Ω p is the angular velocity. Assuming the angular velocity of the rotating mass is constant: m b p = L p M p N p = i d dt hb p = b d dt hb p + ω b b/n h b p = ω b b/n h b p = S(ω b b/n)h b p = I p Ω p R I p Ω p Q (2.81) The total forces and moments due to thrust, τ t are: [ τ t = f b t S(r b t/g )f b t + S(ω b b/n )hb p ] (2.82) 2.8 Aerodynamic Coeffi cients Different methods can be used to estimate the aerodynamic coeffi cients such as wind tunnel experiments or system identification based on logged experimental data from the aircraft. The aircraft equations of motion can be quite simple if the aircraft is not highly maneuverable. The more maneuverable aircraft, the more coeffi cients are needed to accurately describe the aerodynamic forces and moments. The aerodynamic coeffi cients are usually linear at small angles of angle of attack and sideslip. The most important parameters used to describe an airfoil are: b c c S AR = b 2 /S Wing span (tip to tip) Wing chord (varies along span) Mean aerodynamic chord Total wing area Aspect ratio

25 2.8. AERODYNAMIC COEFFICIENTS 19 Aircraft with delta-shaped wings often have a low-aspect ratio, AR. A low-aspect ratio permits very high roll rates, but have a greatly reduced lift-over-drag. High lift-over-drag gives an aircraft that has effective cruise performance, that is passenger jets. The aerodynamic forces and moments are often proportional to the freestream mass density, ρ a, and the square of the freestream airspeed, V T. It is thus convenient to define the dynamic pressure, q, which is later used to calculate the aerodynamic forces and moments: Aerodynamic forces and moments q = 1 2 ρ av 2 T (2.83) The aerodynamic forces and moments acting on an aircraft can be parametrized as: X a C X fa b = Y a Z a = qs C Y C z (2.84) m b a = L a M a N a = qs bc l cc m bc n (2.85) where the aerodynamic coeffi cients C X and C Z often are replaced by expressions that are functions of the drag coeffi cient C D and the lift coeffi cient C L (see Figure 2.4). Positive drag and lift coeffi cients are directed along the x s and z s stability axes, respectively, such that: C D C L = R s b = C X C Z cos(α) sin(α) 1 sin(α) cos(α) Hence, the aerodynamic forces can then be rewritten as: X a fa b = Y a = qs C Y + qsr b s Z a = qs C D cos(α) + C L sin(α) C Y C D sin(α) C L cos(α) The generalized aerodynamic forces become: [ f b τ a = a m b a ] C X C Z C D C L (2.86) (2.87) (2.88) The aerodynamic coeffi cients C i, i = D, Y, L, l, m, n, are in general functions of angle of attack α, sideslip angle β, Mach number M, altitude h, control surface deflections δ S and the thrust coeffi cient: T C = T (2.89) qs D

26 2 CHAPTER 2. AIRCRAFT MODELING where S D is the area of the disc swept out by a propeller blade. Therefore, the aerodynamic coeffi cients are functions: C i = C i (α, β, M, h, δ S, T C ) (2.9) where i = D, Y, L, l, m, n. Consequently, the aerodynamic coeffi cients may have complicated dependences, which are hard to model with great accuracy. If the aircraft has restricted flight modes, thts is restricted to low Mach numbers and small angles of attack and sideslip, the aerodynamic coeffi cients may be greatly simplified. The complicated dependency may then be broken into a sum of simpler linear terms Drag coeffi cient The drag of an aircraft is one of the most important features to model. By reducing drag, the fuel consumption is reduced, thus giving higher range, and in addition higher maximum cruise speed. The total drag of an aircraft is usually taken as the sum of the following drag components: Parasite drag = friction drag + form drag Induced drag (effect of wing-tip vortices) Wave drag (effect of shock waves) where Friction drag also known as skin friction, describes the friction of the fluid against the surface of the aircraft. The friction drag is proportional to the area in contact with the fluid, also known as the wetted area. Form drag is the pressure drag caused by flow separation at high alpha. Induced Drag also known as vortex drag, is the pressure drag caused by the tip vortices of a finite wing when it is producing lift. Wave drag is the pressure drag caused if shock waves are present over the surface of the aircraft. The wave drag becomes significant at Mach numbers M >.4 for a fighter aircraft, where the flow reaches supersonic speed. The friction drag often varies parabolically with the lift coeffi cient, while the induced drag can often be modeled as: C Di = C2 L (2.91) πear where e is the effi ciency factor, close to unity, and AR is the aspect ratio. The total drag of an aircraft takes the form: C D (C L, M) = k(m) (C L C LDM ) 2 + C DM (M) (2.92)

27 2.8. AERODYNAMIC COEFFICIENTS 21 where k(m) is a proportionality constant that varies with the Mach number. In addition, the drag coeffi cient may be a function of the control surfaces and landing gear. A more general equation for the drag coeffi cient is: C D = C D (α, β, M, h) + C D (M, δ E ) + C D (M, δ R ) + C D (δ F ) + C D (gear) + (2.93) A linear approximation of this expressions is: Lift coeffi cient C D = C D + C Dα α + C DδE δ E + C DδR δ R + C DδF δ F (2.94) The lift coeffi cient, C L, is in general mainly determined by the fuselage, wings and their interference effects. It is also a function of Mach and the angle of attack. The lift coeffi cient is usually quite linear with a until near stall, where it drops sharply, and then may rise again before falling to zero at large angles of attack (see Figure 2.4). A general equation for the lift coeffi cient is given in C L = C L (α, β, M, T C ) + C L (δ F ) + C Lge (h) (2.95) where C Lge (h) is the increment of lift due to ground effect. A typical simple linear model structure for the lift coeffi cient is given in C L = C L + C Lα α + C LδF δ F (2.96) Sideforce coeffi cient The sideforce coeffi cient, C Y, is in general mainly determined by sideslip and rudder deflection. It is often quite linear in sideslip angle β, and a positive sideslip will give a negative sideforce. A typical expression for the sideforce coeffi cient is: C Y = C Y (α, β, M)+ C YδR (α, β, M, δ R )+ C YδA (α, β, M, δ A )+ b 2V T [ CYp (α, M)P + C Yr (α, M)R ] A typical simple linear model structure for the sideforce coeffi cient is given in Rolling moment coeffi cient (2.97) C Y = C Y + C Yβ β + C YδR δ R + C YδA δ A (2.98) The rolling moment coeffi cient, C l, is in general mainly determined by sideslip and by control action of the ailerons and the rudder. It is often quite linear in sideslip angle and a positive sideslip will give a negative rolling moment. A typical expression for the rolling moment coeffi cient is: C l = C l (α, β, M)+ C lδa (α, β, M, δ A )+ C lδr (α, β, M, δ R )+ b [ Clp (α, M)P + C lr (α, M)R ] 2V T (2.99)

28 22 CHAPTER 2. AIRCRAFT MODELING Figure 2.4: Lift and drag as a function of angle of attack. A linear model for the rolling moment coeffi cient is: b b C l = C l + C lβ β + C lδa δ A + C lδr δ R + C lp P + C lr R (2.1) 2V T 2V T Pitching moment coeffi cient The pitching moment coeffi cient, C m, is in general mainly determined by angle of attack α and by control action of the elevators. It may be quite linear in angle of attack, and a positive angle of attack will give a negative pitching moment. A typical model for the pitching moment coeffi cient is: C m = C m (α, M, h, δ F, T C ) + C mδe (α, M, h, δ E ) + c ( Cmq (α, M, h)q + C m 2V α (α, M, h) α ) + x R T c C L + C mthrust (δ T, M, h) + C mgear (h) (2.11) The term x C R c L is used to correct for any x-displacement of the aircraft CG from the aerodynamic data reference position. The second last term represents the effect of the engine thrust vector not passing through the aircraft CG, while the last term represents the moment due to the landing-gear doors and the landing gear. A linear model for the pitching moment coeffi cient takes the form: c C m = C m + C mα α + C mδe δ e + C mq Q (2.12) 2V T

29 2.9. STANDARD AIRCRAFT MANEUVERS Yawing moment coeffi cient The yawing moment coeffi cient, C n, is in general mainly determined by the sideslip angle β and by control action of the rudders. It is quite linear in sideslip β for small angles of attack α, where a positive sideslip β will give a positive yawing moment. A general equation for the yawing moment coeffi cient is given by: C n = C n (α, β, M, T C ) + C nδr (α, β, M, δ R ) + C nδa (α, β, M, δ A ) + b 2V T [ Cnp (α, M)P + C nr (α, M)R ] (2.13) A linear model for the yawing moment coeffi cient is gven by: b b C n = C n + C nβ β + C nδr δ R + C nδa δ A + C np P + C nr R (2.14) 2V T 2V T Equations of motion including aerodynamic coeffi cients The resulting equations of motion including the aerodynamic coeffi cients are: P I xz I x Ṙ I xz I z U = V R W Q g sin(θ) + T m + qs m C X (2.15) V = W P UR + g cos(θ) sin(φ) + qs m C Y (2.16) Ẇ = UQ V P + g cos(θ) cos(φ) + qs m C Z (2.17) Ṙ = I z I y QR + I xz P Q + qbs C l (2.18) I x I x I x Q = I x I z P R I xz (P 2 R 2 ) + T r t z I y I y I y P = I y I x I z P Q I xz I z QR T r t y I z 2.9 Standard Aircraft Maneuvers The nominal values depends on the aircraft maneuver. For instance: 1. Straight flight: Θ = Q = R = 2. Symmetric flight: Ψ = V = 3. Flying with wings level: Φ = P = Constant angular rate maneuvers can be classified according to: 1. Steady turn: R = constant 2. Steady pitching flight: Q = constant 3. Steady rolling/spinning flight: P = constant + I p I y Ω p R + qs c I y C m (2.19) I p I z Ω p Q + qsb I z C n (2.11)

30 24 CHAPTER 2. AIRCRAFT MODELING Dynamic equation for coordinated turn (bank-to-turn) A frequently used maneuver is coordinated turn where the acceleration in the y-direction is zero ( β = ), sideslip β = and zero steady-state pitch and roll angles that is, β = β = (2.111) Θ = Φ = (2.112) Furthermore it is assumed that V T = U = constant. Since β = β = and β = V/U it follows that V = V =. This implies that the external forces δy =. From (2.28) it is seen that: m[ V + UR W P g cos(θ) sin(φ)] = Y (2.113) m[(u + u)(r + r) (W + w)(p + p) g cos(θ) sin(φ)] = Y (2.114) Assume that the longitudinal and lateral motions are decoupled that is, u = w = q = θ =. If perturbation theory is applied under the assumption that the 2nd-order terms ur = pw =,we get: m(u R + U r W P W p g sin(φ)) = Y (2.115) The equilibrium equation (2.35) gives the steady-state condition: Substitution of (2.116) into (2.115) gives: or m(u R W P ) = Y (2.116) m(u r W p g sin(φ)) = (2.117) r = W U p + g U sin(φ) (2.118) The aircraft is often trimmed such that the angle of attack α = W /U =. This implies that the yaw rate can be expressed as: r = g U sin(φ) φ small g U φ (2.119) which is a very important result since it states that a roll angle angle φ different from zero will induce a yaw rate r which again turns the aircraft (bank-to-turn). With other words, we can use a moment in roll, for instance generated by the ailerons, to turn the aircraft. The yaw angle is given by: ψ = r (2.12) An alternative method is of course to turn the aircraft by using the rear rudder to generate a yaw moment. The bank-to-turn principle is used in many missile control systems since it improves maneuverability, in particular in combination with a rudder controlled system.

31 2.9. STANDARD AIRCRAFT MANEUVERS 25 Example 2 (Augmented turning model using rear rudders) Turning autopilots using the rudder δ R as control input is based on the lateral state-space model. The differential equation for ψ is augmented on the lateral model as shown below: v ṗ ṙ φ ψ = a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 31 a 32 a v p r φ ψ + b 11 b 21 b 31 δ R (2.121) Example 3 (Augmented bank-to-turn model using ailerons) Bank-to-turn autopilots are designed using the lateral state-space model with ailerons as control inputs, for instance δ A = 1/2(δ AL + δ AR ). This is done by augmenting the bank-toturn equation (2.119) to the state-space model according to: v ṗ ṙ φ ψ = a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 31 a 32 a 33 1 g U v p r φ ψ + b 11 b 21 b 31 δ A (2.122) Dynamic equation for altitude control Aircraft altitude control systems are designed by considering the equation for the vertical acceleration in the center of gravity expressed in NED coordinates; see (2.28). This gives: a zcg = Ẇ + V P QU g cos(θ) cos(φ) (2.123) If the acceleration is perturbed according to a zcg = a z + δa z and we assume that Ẇ =, the following equilibrium condition is obtained: Equation (2.123) can be perturbed as: a z = V P Q U g cos(θ ) cos(φ ) (2.124) a z + δa z = ẇ + (V + v)(p + p) (Q + q)(u + u) g cos(θ + θ) cos(φ + φ) (2.125) Furthermore, assume that the altitude is changed by symmetric straight-line flight with horizontal wings such that V = Φ = Θ = P = Q =. This gives: a z + δa z = ẇ + vp q(u + u) g cos(θ) cos(φ) (2.126)

32 26 CHAPTER 2. AIRCRAFT MODELING Assume that the 2nd-order terms vp and uq can be neglected and subtract the equilibrium condition (2.124) from (2.126) such that: δa z = ẇ U q (2.127) Differentiating the altitude twice with respect to time gives the relationship: ḧ = δa z = U q ẇ (2.128) If we integrate this expression under the assumption that ḣ() = U θ() w() =, we get: ḣ = U θ w (2.129) The flight path is defined as: γ := θ α (2.13) where α = w/u. This gives the resulting differential equation for altitude control: ḣ = U γ (2.131) Example 4 (Augmented model for altitude control using ailerons) An autopilot model for altitude control based on the longitudinal state-space model with states u, w(alt. α), q and θ is obtained by augmenting the differential equation for h to the state-space model according to: u ẇ q θ ḣ + a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a U u w q θ h + b 11 b 12 b 21 b 22 b 31 b 32 [ δt δ E ] (2.132) 2.1 Aircraft Stability Properties The aircraft stability properties can be investigated by computing the eigenvalues of the system matrix A using: For a matrix A of dimension n n there is n solutions of λ. det(λi A) = (2.133)

33 2.1. AIRCRAFT STABILITY PROPERTIES Longitudinal stability analysis For conventional aircraft the characteristic equation is of 4th order. Moreover, (λ 2 + 2ζ ph ω ph λ + ω 2 ph)(λ 2 + 2ζ sp ω sp λ + ω 2 sp) = (2.134) where the subscripts ph and sp denote the following modi: Phugoid mode Short-period mode The phugoid mode is observed as a long period oscillation with little damping. In some cases the Phugoid mode can be unstable such that the oscillations increase with time. The Phugoid mode is characterized by the natural frequency ω ph and relative damping ratio ζ ph. The short-period mode is a fast mode given by the natural frequency ω sp and relative damping factor ζ sp. The short-period mode is usually well damped. Classification of eigenvalues Conventional aircraft: Conventional aircraft have usually two complex conjugated pairs of Phugoid and short-period types. For the B-767 model in Chapter 4, the eigenvalues can be computed using damp.m in Matlab: a = [ ] damp(a) Eigenvalue Damping Freq. (rad/sec) i i i i From this is seen that ω ph =.596, ζ ph =.17, ω sp = and ζ sp = Tuck Mode: Supersonic aircraft may have a very large aerodynamic coeffi cient M u. This implies that the oscillatory Phugoid equation gives two real solutions where one is positive (unstable) and one is negative (stable). This is referred to as the tuck mode since the phenomenon is observed as a downwards pointing nose (tucking under) for increasing speed.

34 28 CHAPTER 2. AIRCRAFT MODELING Figure 2.5: Aircraft longitudinal eigenvalue configuration plotted in the complex plane. A third oscillatory mode: For fighter aircraft the center of gravity is often located behind the neutral point or the aerodynamic center that is, the point where the trim moment M w w is zero. When this happens, the aerodynamic coeffi cient M w takes a value such that the roots of the characteristic equation has four real solutions. When the center of gravity is moved backwards one of the roots of the Phugoid and shortperiod modes become imaginary and they form a new complex conjugated pair. This is usually referred to as the 3rd oscillatory mode. The locations of the eigenvalues are illustrated in Figure Lateral stability analysis The lateral characteristic equation is usually of 5th order: λ 5 + α 4 λ 4 + α 3 λ 3 + α 2 λ 2 + α 1 λ + α = (2.135) λ(λ + e)(λ + f)(λ 2 + 2ζ D ω D λ + ω 2 D) = (2.136) where the term λ in the last equation corresponds to a pure integrator in roll that is, ψ = r. The term (λ + e) is the aircraft spiral/divergence mode. This is usually a very slow mode. Spiral/divergence corresponds to horizontally leveled wings followed by roll and a diverging spiral maneuver. The term (λ+f) describes the subsidiary roll mode while the 2nd-order system is referred to as Dutch roll. This is an oscillatory system with a small relative damping factor ζ D. The natural frequency in Dutch roll is denoted ω D. If the lateral B-767 Matlab model in Chapter 4 is considered, the Matlab command damp.m gives:

35 2.11. DESIGN OF FLIGHT CONTROL SYSTEMS 29 a = [ damp(a) Eigenvalue Damping Freq. (rad/sec) i i In this example we only get four eigenvalues since the pure integrator in yaw is not include in the system matrix. It is seen that the spiral mode is given by e =.143 while the subsidiary roll mode is given by f = Dutch roll is recognized by ω D = and ζ D = Design of flight control systems A detailed introduction to design of flight control systems are given by [9], [13], [1] and [14]. The control system are based on linear design techniques using linearized models similar to those discussed in the sections above.

36 3 CHAPTER 2. AIRCRAFT MODELING

37 Chapter 3 Satellite Modeling When stabilizing satellites in geostationary orbits only the attitude of the satellite is of interest since the position is given by the Earth s rotation. Figure 3.1 shows the NUTS CubeSat project at NTNU. 3.1 Attitude Model Euler s 2nd Axiom applied to satellites The rigid-body kinetics (2.21) gives: I CG ω + ω (I CG ω) = τ (3.1) were ω = [p, q, r] and I CG = I CG > is the inertia tensor about the center of gravity given by: I x I xy I xz I CG = I xy I y I yz (3.2) I xz I yz I z Both the Euler angles and quaternions can be used to represent attitude. Moreover, q = T q (q)ω (3.3) Figure 3.1: The NUTS CubeSat project at NTNU- Courtesy to 31

38 32 CHAPTER 3. SATELLITE MODELING where q = [η, ε 1, ε 2, ε 3 ], θ = [Φ, Θ, Ψ] and T θ (θ) = T q (q) = 1 2 θ = T θ (θ)ω (3.4) 1 sin(φ) tan(θ) cos(φ) tan(θ) cos(φ) sin(φ) sin(φ)/ cos(θ) cos(φ)/ cos(θ) ε 1 ε 2 ε 3 η ε 3 ε 2 ε 3 η ε 1 ε 2 ε 1 η, cos(θ) (3.5) (3.6) The satellite has three controllable moments τ = [τ 1, τ 2, τ 3 ], which can be generated using different actuators. Magnetic actuators are described in Section Skew-symmetric representation of the satellite model The dynamic model (3.1) can also be written: I CG ω (I CG ω) ω = τ (3.7) where we have used the fact that a b = b a. Furthermore, it is possible to find a matrix S(ω) such that S(ω) = S (ω) is skew-symmetric. With other words: The matrix S(ω) must satisfy the condition: I CG ω + S(ω)ω = τ (3.8) S(ω)ω (I CG ω) ω (3.9) A matrix satisfying this condition is: I yz q I xz p + I z r I yz r + I xy p I y q S(ω) = I yz q + I xz p I z r I xz r I xy q + I x p (3.1) I yz r I xy p + I y q I xz r + I xy q I x p 3.2 Actuator Dynamics Magnetic actuators are used to control the NUTS cube-satellite, which is a student satellite project at NTNU. The magnetic actuators are cheap, solid-state and energy-effi cient. Due to variations of the magnetic field for a moving satellite, modeling and control may be challenging. An analogy could be an airplane moving through a various density atmospheres, suddenly leaving you with limited control of either roll, pitch or yaw. When applying an electrical current through the windings of the coils, a magnetic dipole moment is generated. When this field reacts with the magnetic field surrounding the Earth, a magnetic torque is generated, which makes the satellite move. Let the Earth s magnetic

39 3.2. ACTUATOR DYNAMICS 33 field in BODY coordinates be denoted by B b = [B x, B y, B z ] while the magnetic field m b = [m x, m y, m z ] set up by the actuators is: m b = N xa xv x R x N ya yv y R y N za zv z R z (3.11) = K coil V (3.12) where K coil = N xa x R x N ya y R y N za z R z, V = V x V y V z (3.13) and for n = x, y, z, N n A n V n R n is the number of turns with copper thread in the coil is the area of the coil is the voltage over the coil is the resistance of the coil Then, the torque becomes: where S( B b ) = τ = m b B b = S( B b )m b (3.14) B z B y B z B x B y B x (3.15) For control purposes, we usually assign τ, however, the satellite only accepts voltage signals as input, and not torque. It is straightforward to find the magnetic field m b due to the actuators: m b = τ B b (3.16) In view of (3.13), one may find the relation between voltages and the torque vector: V = K 1 coil ( τ B b ) (3.17) The matrix K coil, usually referred to as the control allocation matrix, is a constant matrix. Note that B b will change throughout the orbit. According to (3.14), τ b always lies in the plane perpendicular to B b. Suppose that τ d is the torque that the controller produces. It is required that τ d lies in the plane perpendicular

40 34 CHAPTER 3. SATELLITE MODELING to B b. It implies that τ only represents the component of τ d that is perpendicular to B b. To avoid unnecessary use of electricity, the length of τ d is scaled down by 1/ B b 2. Accordingly, m b = τ = 1 B b 2 (τ d B b ) (3.18) 1 B b 2 (τ d B b ) B b (3.19) Let γ be the angle between τ d and B b. The magnitude of τ is given by: τ = 1 B b 2 τ d B b sin(γ) B b = τ d sin(γ) (3.2) 3.3 Design of Satellite Attitude Control Systems Design of nonlinear trajectory-tracking control systems for Hamiltonian systems in the form (3.8) is quite common in the literature. One example is the nonlinear and passive adaptive attitude control system of Slotine and Di Benedetto [12]. An alternative representation is proposed by Fossen [3], which simplifies the representation of the regressor matrix. Trajectory-tracking controllers require that the model parameters are known or estimated using parameter adaptation. For set-point regulation is possible to derive controllers that do not require model parameters. This is presented below Nonlinear quaternion set-point regulator The classical PD-controller (Wen and Kreutz-Delago [16], Fjellstad and Fossen [5]) for vehicle attitude regulation uses feedback from the imaginary part ε = [ ε 1, ε 2, ε 3 ] of the unit quaternion error q = [ η, ε 1, ε 2, ε 3 ] where η denotes the real part of the unit quaternion. An extension to trajectory tracking in 6 DOF is found in Fjelllstad and Fossen [6]. Let q d = constant be constant (set-point regulation). This means that the desired attitude must be specified as a unit quaternion q d, which usually is computed by specifying the desired Euler angles. The attitude error dynamics q can be expressed in terms of the unit quaternions according to: η = 1 2 ε ω (3.21) ε = 1 2 [ ηi 3 + S( ε)] ω (3.22) Consider the Lyapunov function candidate (Fjellstad and Fossen, 1994): V = 1 2 ω I CG ω + 2k p h( η) (3.23) where h( η) = 1 η (3.24)

41 3.3. DESIGN OF SATELLITE ATTITUDE CONTROL SYSTEMS 35 and k p > is a constant. The unit quaternion error is computed by: [ ] [ ] ηd ε q = d η ε d η d I 3 3 S(ε d ) ε (3.25) Differentiation of V with respect to time gives: Since S(ω) = S (ω), it follows that: V = ω T I CG ω + 2k p h η η = ω ( S(ω)ω + τ ) + k p ε ω = ω ( S(ω)ω + τ + k p ε) (3.26) ω S(ω)ω = ω (3.27) This suggests that the control input τ should be chosen as a PD controller: τ = k p ε K d ω (3.28) where K d >. Hence, V = ω K d ω (3.29) The closed-loop system has two equilibrium points ω = and ε = η = ±1. The equilibrium point corresponding to η = 1 is asymptotically stable while η = 1 is unstable. However, it is possible to modify the design to obtain a globally asymptotically stable (GAS) control law by choosing h( η) = 1 η (3.3) which for τ = k p sgn(η) ε K d ω (3.31) gives one stable equilibrium point ω = and ε = η = 1. As pointed out by Mayhew, Sanfelice and Teel [15] the global attractivity property is not robust to arbitrarily small measurement noise and hence they propose a hybrid feedback control law to overcome this problem. As explained by (3.11) (3.13), the input signal is voltage V of the coil. Thus, we need to find the relation between τ and V. In practical situations, scaling is also required so as to avoid unnecessary electricity load. Therefore, one may derive the following input signal: V = K 1 coil ( τ B b ) = k p B b 2 K 1 coil ( ε B b ) k d B b 2 K 1 coil ( ω B b ) (3.32)