Dynamic Modeling and Flight Control Simulation of a ... - AIAA ARC

AIAA 2008-6620

AIAA Guidance, Navigation and Control Conference and Exhibit 18 - 21 August 2008, Honolulu, Hawaii

Dynamic Modeling and Flight Control Simulation of a Large Flexible Launch Vehicle Wei Du∗ and Bong Wie† Iowa State University, Ames, IA 50011-2271

Mark Whorton‡ NASA Marshall Space Flight Center, Huntsville, AL 35812 This paper presents the preliminary study results of dynamic modeling and flight control simulation of large flexible launch vehicles such as the Ares-I Crew Launch Vehicle. A complete set of six-degrees-of-freedom dynamic models of the Ares-I, incorporating its propulsion, aerodynamics, guidance and control, and structural flexibility, is developed. NASA’s Ares-I reference model and the SAVANT Simulink-based program are utilized to develop a Matlab-based simulation and linearization tool for an independent validation of the performance and stability of the ascent flight control system of the Ares-I. A linearized state-space model as well as a non-minimum-phase transfer function model (which is of typical for flexible vehicles with non-collocated actuators and sensors) is validated for the ascent flight control design and analysis of the Ares-I.

Nomenclature a b (cx , cy , cz ) CA CY β CN 0 CN α CM rβ CM p0 CM pα CM yβ C B/I C B/S C D Fbase (Faero.xb , Faero.yb , Faero.zb ) (Frkt.xb , Frkt.yb , Frkt.zb ) (Frcs.xb , Frcs.yb , Frcs.zb ) (Ftotal.xb , Ftotal.yb , Ftotal.zb ) (Ftotal.xi , Ftotal.yi , Ftotal.zi ) (gx , gy , gz )

speed of sound (= 1117 ft/s at sea level in the standard atmosphere) reference body length = 12.16 ft components of the center of gravity location in the structure reference frame with its origin at the top of vehicle = (220.31, 0.02, 0.01) ft at t = 0 axial force coefficient side force curve slope with respect to β normal force coefficient at zero angle of attack normal force curve slope with respect to α rolling moment curve slope with respect to β pitching moment coefficient at zero angle of attack pitching moment curve slope with respect to α yawing moment curve slope with respect to β direction cosine matrix of the frame B with respect to the frame I direction cosine matrix of the frame B with respect to the frame S lateral (side) force drag (axial) force base drag force body-axis components of aerodynamic force body-axis components of solid rocket booster force body-axis components of RCS force body-axis components of total force inertial components of total force inertial components of the gravitational acceleration

∗ Ph.D. Student, Space Systems and Control Laboratory, Department of Aerospace Engineering, [email protected], Student Member AIAA. † Vance Coffman Endowed Chair Professor, Space Systems and Control Laboratory, Department of Aerospace Engineering, 2271 Howe Hall, Room 2355, (515) 294-3124, [email protected], Associate Fellow AIAA. ‡ Chief, Guidance Navigation and Mission Analysis Branch, [email protected], Associate Fellow AIAA.

1 of 25 American Institute of Aeronautics and Astronautics Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

(~i, ~j, ~k) (~is , ~js , ~ks ) ~ J, ~ K) ~ (I, ~ e) (I~e , J~e , K J2 J3 J4 (Kp , Ki , Kd ) m M N p0 (p, q, r) (q1 , q2 , q3 , q4 ) (q1c , q2c , q3c , q4c ) (q1e , q2e , q3e , q4e ) Q Re Rp ~r r S T T0 (Taero.xb , Taero.yb , Taero.zb ) (Trkt.xb , Trkt.yb , Trkt.zb ) (Trcs.xb , Trcs.yb , Trcs.zb ) U (u, v, w) ~ V ~rel V ~w V ~m V Vm (Vm.xb , Vm.yb , Vm.zb ) (x, y, z) Xa Xg Xcp (α, β) δy δz µ φ Φ Φ ™ (φ, θ, √) ρ η ≥ ≠ = diag(ωi ) ω ~ ~ ω ~ e = ωe K

basis vectors of the body-fixed reference frame B basis vectors of the structure reference frame S basis vectors of the Earth-centered inertial reference frame I basis vectors of the Earth-fixed equatorial rotating reference frame E Earth’s second order zonal coefficient = 1.082631 × 10−3 Earth’s third order zonal coefficient = −2.55 × 10−6 Earth’s fourth order zonal coefficient = −1.61 × 10−6 proportional, integral, and derivative gains of a PID controller vehicle’s mass Mach number normal force local atmospheric pressure body-axis components of ω ~ attitude quaternions commanded attitude quaternions attitude-error quaternions dynamic pressure = 0.5ρVm2 Earth’s equatorial radius = 20,925,646 ft = 6378.13 km Earth’s polar radius = 20,855,486 ft = 6356.75 km vehicle’s position vector magnitude of ~r reference area = 116.2 ft2 total thrust total vacuum thrust body-axis components of aerodynamic torque body-axis components of solid rocket torque body-axis components of RCS torque Earth’s gravitational potential ~ body-axis components of V vehicle’s inertial velocity vector vehicle’s velocity vector relative to the Earth-fixed reference frame velocity vector of the wind air stream velocity vector magnitude of the air stream velocity body-axis components of vehicle’s air stream velocity vector inertial components of vehicle’s position vector ~r aerodynamic reference point location in the structure frame = 275.6 ft gimbal attach point location in the structural frame = 264.47 ft center of pressure (cp) location in the structure frame angle of attack and sideslip angle pitch gimbal angle (rotation about the body y-axis) yaw gimbal angle (rotation about the body z-axis) Earth’s gravitational parameter = 1.407644176 × 1016 ft3 /s2 Earth’s geocentric latitude Earth’s geodetic latitude flexible-mode influence matrix at gimbal attach point flexible-mode influence matrix at instrument unit location three Euler angles for a rotational sequence of C1 (φ) ← C2 (θ) ← C3 (√) atmospheric density (= 0.002377 slug/ft2 at sea level) flexible-mode state vector damping ratio of the flexible modes = 0.005 undamped natural frequency matrix of the flexible modes angular velocity vector of the vehicle angular velocity vector of the Earth where ωe = 7.2921 × 10−5 rad/s 2 of 25 American Institute of Aeronautics and Astronautics

V K Ke z

Structure Frame

ks

is js

k

i

cg

Body Frame

r j

Equator Inertial Frame Vernal! Equinox! Direction

I

x

Greenwich! longitude = 0! latitude = 0

I e Earth Frame

Je y

J

~ J, ~ K}, ~ Figure 1. Illustration of the Earth-centered inertial reference frame with {I, the Earth-fixed reference ~ e }, the structure reference frame with {~is , ~js , ~ks }, and the body-axis reference frame with frame with {I~e , J~e , K {~i, ~j, ~k}.

I.

Introduction

This paper describes the preliminary results of developing a complete set of coupled dynamic models and a Matlab-based simulation program for the Ares-I Crew Launch Vehicle (CLV), incorporating its propulsion, aerodynamics, guidance and control, and structural flexibility. The Ares-I CLV is a large, slender, and aerodynamically unstable vehicle. It is a two-stage rocket with a solid-propellant first stage derived from the Shuttle Reusable Solid Rocket Motor/Booster and an upper stage employing a J-2X engine derived from the Saturn J-2 engines. NASA’s Ares-I CLV reference model and the SAVANT Simulink-based program [1-3] were utilized to develop a Matlab-based simulation and linearization tool for an independent validation of the performance and stability of the ascent flight control system of the Ares-I CLV. Various dynamic models of launch vehicles available in the literature [4-8] were also utilized in this study. Currently, the robust control design and stability analysis of the ascent flight control system of the Ares-I CLV are of practical interest [9-11]. In support of NASA’s ascent flight control design efforts for the Ares-I CLV, we have also investigated the fundamental principles of flight control analysis and design for launch vehicles in a companion paper [12]. Nonlinear coupled dynamic simulation results of an ascent flight control system, which directly controls the inertial attitude-error quaternions and also employs non-minimum-phase filters, will be presented for a reference model of the Ares-I Crew Launch Vehicle. A linearized state-space model as well as a non-minimumphase transfer function model of the Ares-I (which is of typical for flexible vehicles with non-collocated actuators and sensors) will also be presented for the ascent flight control design and analysis [12].

II.

Reference Frames and Rotational Kinematics

Various reference frames and rotational kinematics, which are essential for describing the six-degrees-offreedom dynamic model of launch vehicles, are briefly discussed in this section. A.

Earth-Centered Inertial Reference Frame

~ J, ~ K} ~ has its origin at the Earth The Earth-centered inertial (ECI) frame with a set of basis vectors {I, center as illustrated in Fig. 1. The x-axis points in the vernal equinox direction and the z-axis points in the direction of the north pole. 3 of 25 American Institute of Aeronautics and Astronautics

The position vector ~r of a launch vehicle is then described as ~ ~r = xI~ + y J~ + z K

(1)

The inertial velocity and the inertial acceleration of a launch vehicle become respectively ~ = x˙ I~ + y˙ J~ + z˙ K ~ V

(2)

~˙ = x ~ V Ï~ + y¨J~ + z¨K

(3)

For example, if an inertial position vector of the Ares-I at liftoff from Kennedy Space Center (with longitude of 80.6-deg west and latitude of 28.4-deg north) is given by ~ = −8.7899E4I~ − 1.8385E7J~ + 9.9605E6K ~ (ft) ~r(0) = x(0)I~ + y(0)J~ + z(0)K

(4)

then, the inertial velocity vector of the vehicle at liftoff is obtained as ~ (0) = x(0) ~ =ω V ˙ I~ + y(0) ˙ J~ + z(0) ˙ K ~ e × ~r(0) = 1340.65I~ − 6.41J~ (ft/sec)

(5)

~ is the angular velocity vector of the Earth and ωe = 7.2921 × 10−5 rad/s which corresponds where ω ~ e = ωe K to 360 deg per a sidereal day of 23 h 56 min 4 s. B.

Earth-Fixed Equatorial Reference Frame

~ e }, with its origin at the Earth The geocentric equatorial rotating frame with a set of basis vectors {I~e , J~e , K ~ center, is fixed to the Earth (Fig. 1). Its basis vector Ie is in the Greenwich meridian direction. This ~ ≡K ~ e with an angular velocity ωe . Earth-fixed frame rotates about K C.

Body-Axis Reference Frame

The body-axis reference frame with its basis vectors {~i, ~j, ~k} is fixed to the vehicle’s body as illustrated in Fig. 1. Its origin is located at the instant center of gravity. The ~i-axis is along the vehicle’s longitudinal axis. The ~k-axis perpendicular to the ~i-axis points downward while the ~j-axis points rightward. ~ is then expressed as The inertial velocity vector V ~ = u~i + v~j + w~k V

(6)

The angular velocity vector ω ~ of the launch vehicle is also expressed as ω ~ = p~i + q~j + r~k

(7)

The inertial acceleration vector is then described as ~˙ = (u˙~i + v˙~j + w˙ ~k) + ω ~ V ~ ×V D.

(8)

Structure Reference Frame

A structure reference frame with its basis vectors {~is , ~js , ~ks } and with its origin located at the top of vehicle is also employed in the SAVANT program. The locations of center of gravity, gimbal attach point, and aerodynamic reference point and other mass properties are defined using this structure frame. However, Euler’s rotational equations of motion will be written in terms of the body-fixed frame with its origin at the center of gravity. Because ~is = −~i, ~js = ~j, and ~ks = −~k, we have   −1 0 0   C B/S =  0 1 0  (9) 0 0 −1 where C B/S is the direction cosine matrix of the frame B with respect to the frame S. 4 of 25 American Institute of Aeronautics and Astronautics

E.

Euler Angles and Quaternions

The coordinate transformation to the body frame B from the inertial frame I is described by three Euler angles (φ, θ, √). For a rotational sequence of C1 (φ) ← C2 (θ) ← C3 (√), we have   cos θ cos √ cos θ sin √ − sin θ   C B/I =  sin φ sin θ cos √ − cos φ sin √ sin φ sin θ sin √ + cos φ cos √ sin φ cos θ  (10) cos φ sin θ cos √ + sin φ sin √ cos φ sin θ sin √ − sin φ cos √ cos φ cos θ which is the direction cosine matrix of the body frame B relative to the inertial frame I. However, the three Euler angles (φ, θ, √) do not actually represent the vehicle’s roll, pitch, and yaw attitude angles to be used for attitude feedback control. The rotational kinematic equation for three Euler angles (φ, θ, √) is given by      φ˙ cos θ sin φ sin θ cos φ sin θ p 1   ˙    = (11) cos φ cos θ − sin φ cos θ   q   θ   0 cos θ ˙ √ 0 sin φ cos φ r

The inherent singularity problem of Euler angles can be avoided by using quaternions. The rotational kinematic equation in terms of quaternions (q1 , q2 , q3 , q4 ) is given by      q˙1 0 r −q p q1      p q   q2   q˙2  1  −r 0 (12)  =     q˙3  2  q −p 0 r   q3  q˙4 −p −q −r 0 q4 subject to the constraint: follows: q1 q2 q3 q4

q12 + q22 + q32 + q42 = 1. The quaternions are related to the three Euler angles as

= sin(φ/2) cos(θ/2) cos(√/2) − cos(φ/2) sin(θ/2) sin(√/2) = cos(φ/2) sin(θ/2) cos(√/2) + sin(φ/2) cos(θ/2) sin(√/2) = cos(φ/2) cos(θ/2) sin(√/2) − sin(φ/2) sin(θ/2) cos(√/2) = cos(φ/2) cos(θ/2) cos(√/2) + sin(φ/2) sin(θ/2) sin(√/2)

(13)

The coordinate transformation matrix to the body frame from the inertial frame in terms of quaternions is given by   1 − 2(q22 + q32 ) 2(q1 q2 + q3 q4 ) 2(q1 q3 − q2 q4 )   CB/I =  2(q1 q2 − q3 q4 ) 1 − 2(q12 + q32 ) 2(q2 q3 + q1 q4 )  (14) 2(q1 q3 + q2 q4 ) 2(q2 q3 − q1 q4 ) 1 − 2(q12 + q22 ) We also have

CI/B = [CB/I ]−1



 1 − 2(q22 + q32 ) 2(q1 q2 − q3 q4 ) 2(q1 q3 + q2 q4 )   = [CB/I ]T =  2(q1 q2 + q3 q4 ) 1 − 2(q12 + q32 ) 2(q2 q3 − q1 q4 )  2(q1 q3 − q2 q4 ) 2(q2 q3 + q1 q4 ) 1 − 2(q12 + q22 )

III.

(15)

The 6-DOF Equations of Motion

The six-degrees-of-freedom (6-DOF) equations of motion of a launch vehicle consist of the translational and rotational equations. The translational equation of motion of the center of gravity of a launch vehicle is simply given by ~˙ = F~ mV

(16)

where F~ is the total force acting on the vehicle. Using Eq. (3) and Eq. (8), we obtain the translational equation of motion of the form ~ ~ = F x Ï~ + y¨J~ + z¨K (17) m 5 of 25 American Institute of Aeronautics and Astronautics

or

~ ~ = F u˙~i + v˙~j + w˙ ~k + ω ~ ×V m The Euler’s rotational equation of motion of a rigid vehicle is ~˙ = T~ H

(18)

(19)

~ is the angular momentum vector and T~ is the total external torque about the center of gravity. where H The angular momentum vector is often expressed as ~ = Iˆ · ω H ~

(20)

where ω ~ = p~i + q~j + r~k is the angular velocity vector and Iˆ is the vehicle’s inertia dyadic about the center of gravity of the form [8]    ~i Ixx Ixy Ixz ≥ ¥    Iˆ = ~i ~j ~k  Ixy Iyy Iyz   ~j  (21) ~ Ixz Iyz Izz k The rotational equation of motion is then given by

Iˆ · ω ~˙ + ω ~ × Iˆ · ω ~ = T~

(22)

where ω ~˙ = p˙~i + q˙~j + r˙~k is the angular acceleration vector. A.

Aerodynamic Forces and Moments

Aerodynamic forces and moments depend on the vehicle’s velocity relative to the surrounding air mass, called the air speed. It is assumed that the air mass is static relative to the Earth. That is, the entire air mass ~m is then described rotates with the Earth without slippage and shearing. The air stream velocity vector V by ~m = V ~rel − V ~w = V ~ −ω ~w V ~ e × ~r − V (23)

~rel is the vehicle’s velocity vector relative to the Earth-fixed reference frame, V ~w is the local disturwhere V ~ is the inertial velocity of the vehicle, and ω bance wind velocity, V ~ e is the Earth’s rotational angular velocity vector, and ~r is the vehicle’s position vector from the Earth center. The matrix form of Eq. (23) in    Vm.xb     Vm.yb  =  Vm.zb

the body frame is   u 0  B/I  − C v   ωe w 0

−ωe 0 0

    0 x Vw.xb     0   y  −  Vw.yb  0 z Vw.zb

(24)

where (Vm.xb , Vm.yb , Vm.zb ) are the body-axis components of the vehicle’s air stream velocity vector. Note ~ and ω ~ that ~r = xI~ + y J~ + z K ~ e = ωe K. The aerodynamic forces are expressed in the body-axis frame as D = CA QS − Fbase C = CY β βQS N = (CN 0 + CN α α)QS

(25)

where the base force Fbase is a function of the altitude, the aerodynamic force coefficients are functions of Mach number, and Vm M= = Mach number (26) a 1 Q = ρVm2 = dynamic pressure (27) 2 6 of 25 American Institute of Aeronautics and Astronautics

Ω

æ Vm.zb α = arctan = angle of attack Vm.xb Ω æ Vm.yb β = arcsin = sideslip angle Vm The speed of sound a and the air density ρ are functions of the altitude h. Furthermore, we have Faero.xb = −D Faero.yb = C Faero.zb = −N The aerodynamic moments about the center of    Taero.xb 0 cz −cy     Taero.yb  =  −cz 0 −Xa + cx Taero.zb cy Xa − cx 0

(28) (29)

(30)

gravity are also expressed in the body-axis frame as     Faero.xb CM rβ QSb     (31)   Faero.yb  +  (CM p0 + CM pα α)QSb  Faero.zb Cmyβ βQSb

where Xa = 275.6 ft is the aerodynamic reference point in the structure frame, (cx , cy , cz ) is the center of gravity location in the structure reference frame with its origin at the top of vehicle. At t = 0, we have (cx , cy , cz ) = (220.31, 0.02, 0.01) ft. The aerodynamic moment coefficients are functions of Mach number. B.

Gravity Model

The J4 gravity model used in the SAVANT program is described as ∑ µ ∂ µ ∂ µ ∂∏ Re2 3 1 Re 5 3 Re2 35 15 3 2 3 4 2 C1 = −1 + 2 3J2 sin φ − + 4J3 sin φ − sin φ + 5J4 2 sin φ − sin φ + r 2 2 r 2 2 r 8 4 8 (32) where φ is the Earth’s geocentric latitude. µ ∂ µ ∂ Re 15 3 Re2 35 15 2 3 C2 = J2 (3 sin φ) + J3 sin φ − + 2 J4 sin φ − sin φ (33) r 2 2 r 2 2 The inertial components of the gravitational acceleration are       gx x/r 0 −z/r µ        gy  = 2 C1  y/r  − C2  z/r 0 r gz z/r −y/r x/r

  y/r −y/r   −x/r   x/r  0 0

(34)

The mathematical models used in the SAVANT program for computing the vehicle’s altitude h are summarized as Re − Rp 1 f= = (35) Re 298.257 tan φ (1 − f )2 µ ∂2 µ ∂2 cos Φ sin Φ A= + Re Rp p x2 + y 2 cos Φ z sin Φ B=− − Re2 Rp2 µ ∂2 z x2 + y 2 C= + −1 Rp Rp2 sµ ∂ 2 B B C h=− − − A A A tan Φ =

(36) (37) (38) (39)

(40)

where f is the Earth’s flatness parameter, φ is the geocentric latitude, and Φ is the geodetic latitude (which is commonly employed on geographical maps). 7 of 25 American Institute of Aeronautics and Astronautics

C.

Rocket Propulsion Model

The rocket thrust is simply modeled as T = T0 + (pe − p0 )Ae

(41)

where T is the total thrust force, T0 = |m|V ˙ e the total vacuum thrust, pe the nozzle exit pressure, p0 the local atmospheric pressure (a function of the altitude), m ˙ the propellant mass flow rate, Ve the exit velocity, and Ae the nozzle exit area (= 122.137 ft2 ). The body-axis components of the thrust force are     Frkt.xb T     (42)  Frkt.yb  =  −T δz  Frkt.zb T δy where δy and δz are the pitch and yaw gimbal deflection angles, respectively. Gimbal deflection angles are assumed to be small (with δmax = ±10 deg). The body-axis components of the rocket thrust-generated torque are      Trkt.xb 0 cz −cy Frkt.xb      (43) 0 −Xg + cx   Frkt.yb   Trkt.yb  =  −cz Trkt.zb cy Xg − cz 0 Frkt.zb where Xg = 296 ft is the gimbal attach point location in the structure frame. The body-axis components of the roll control torque from the reaction control system (RCS) are     Trcs.xb Trcs      Trcs.yb  =  0  Trcs.zb 0 D.

(44)

Guidance and Control

The commanded quaternions (q1c , q2c , q3c , q4c ) computed by the guidance system is used to generate the attitude-error quaternions (q1e , q2e , q3e , q4e ) as follows [8]:      q1e q4c q3c −q2c −q1c q1      q1c −q2c   q2   q2e   −q3c q4c (45)  =    q3e   q2c −q1c q4c −q3c   q3  q4e q1c q2c q3c q4c q4

where the attitude quaternions (q1 , q2 , q3 , q4 ) are computed by numerically integrating the following kinematic differential equations:      q˙1 0 r −q p q1      p q   q2   q˙2  1  −r 0 (46)  =     q˙3  2  q −p 0 r   q3  q˙4 −p −q −r 0 q4 The simplified control laws of the ascent flight control system are then described as Trcs = −Kpx (2q1e ) − Kdx p Z δy = −Kpy (2q2e ) − Kiy (2q2e )dt − Kdy q δz = −Kpz (2q3e ) − Kdz r

(47) (48) (49)

An integral control is added to the pitch control channel. The terms (2q1e , 2q2e , 2q3e ) are the roll, pitch, and yaw attitude errors, respectively. This quaternion-error feedback control is in general applicable for an arbitrarily large angular motion of vehicles [8]. Feedback of Euler-angle errors (φ − φc , θ − θc , √ − √c ) is not applicable here because the Euler angles employed in this paper (also used in the SAVANT program) are defined with respect to the Earth-centered inertial reference frame, not with respect to the so-called pitch plane or a navigation reference frame of launch vehicles [4-8, 11]. 8 of 25 American Institute of Aeronautics and Astronautics

Figure 2. Flexible mode shapes and sensor locations of the Ares-I Crew Launch Vehicle [1]. rate-gyro blending is not considered for the Ares-I.

E.

Currently,

Flexible-Body Modes

A flexible-body model of launch vehicles is often expressed as η¨ + 2≥≠η˙ + ≠2 η = ΦT Frkt

(50)

where Frkt = (Frkt.xb , Frkt.yb , Frkt.zb )T and Φ is the flex-mode influence matrix at the gimbal attach point. Sensor measurements including the effects of the flexible bending modes are modeled as   2q1e   eattitude =  2q2e  + ™η (51) 2q3e   p   erate =  q  + ™η˙ (52) r where ™ is the flex-mode influence matrix at the instrument unit location (Fig. 2). F.

A Summary of the 6-DOF Equations of Motion

Total force expressed in the body  Ftotal.xb   Ftotal.yb Ftotal.zb

frame:  

     Faero.xb Frkt.xb Frcs.xb         =  Faero.yb  +  Frkt.yb  +  Frcs.yb  Faero.zb Frkt.zb Frcs.zb

Total force expressed in the inertial frame:     Ftotal.xi Ftotal.xb    I/B   Ftotal.yi  = C  Ftotal.yb  Ftotal.zi Ftotal.zb 9 of 25

American Institute of Aeronautics and Astronautics

(53)

(54)

Table 1. Initial conditions at liftoff

State variables x˙ y˙ z˙ x y z p q r q1 q2 q3 q4

Initial values 1340.65 −6.41 0 −8.7899 × 104 −1.8385 × 107 9.9605×106 3.4916×10−5 6.4018×10−5 0 0.3594 −0.6089 −0.3625 0.6072

Units ft/s ft/s ft/s ft ft ft rad/s rad/s rad/s

Translational equation in the inertial frame:       x ¨ Ftotal.xi gx 1        y¨  =  Ftotal.yi  +  gy  m z¨ Ftotal.zi gz Rotational equation in  Ixx Ixy   Ixy Iyy Ixz Iyz

the body  Ixz  Iyz   Izz

    

IV.

q˙1 q˙2 q˙3 q˙4

frame:   p˙ 0 −r   q˙  = −  r 0 r˙ −q p  Taero.xb  +  Taero.yb Taero.zb 



 1   =   2

(55)

   q Ixx Ixy Ixz p    −p   Ixy Iyy Iyz   q  0 Ixz Iyz Izz r      Trkt.xb Trcs.xb       +  Trkt.yb  +  Trcs.yb  Trkt.zb Trcs.zb

0 r −q −r 0 p q −p 0 −p −q −r

p q r 0

    

q1 q2 q3 q4

    

(56)

(57)

Simulation Results of the Ares-I Crew Launch Vehicle

A set of initial conditions for the Ares-I is provided in Table 1. The corresponding initial conditions for Euler angles are: (φ, θ, √) = (90, -28.61, -90.28) deg. The inertia matrix about the body frame with its origin at the center of gravity at t = 0 is     Ixx Ixy Ixz 1.2634E6 −1.5925E3 5.5250E4     2 (58)  Ixy Iyy Iyz  =  −1.5925E3 2.8797E8 −1.5263E3  slug-ft Ixz Iyz Izz 5.5250E4 −1.5263E3 2.8798E8 Flexible-body mode shape matrices (with 6 flexible modes) are

10 of 25 American Institute of Aeronautics and Astronautics



0.000000272367963 0.000000174392026 −0.000000347086527  Φ =  −0.000364943105155 0.006281028219530 0.000491932740239 0.006281175443849 0.000364891432306 −0.006260333099131



 −0.000000266173427 0.000000329288262 −0.000000369058169  −0.006259406451949 −0.000542750533582 −0.007673360355205  −0.000491798506301 0.007676195145027 −0.000542218216634

0.002287263504447 −0.003936428406260 −0.002315093057592  ™ =  −0.193164818571118 −0.011222878633268 −0.253963647069578 −0.011222390194941 0.193169876159476 −0.019955470041040

 0.003866041325542 0.002898204936058 0.005033045198158  −0.019956097043432 −0.130518411476224 0.009224310239736  × 10−3 0.253971718426341 −0.009226496087855 −0.130493158175449

(59)

(60)

The first three bending mode frequencies are: 6 rad/s, 14 rad/s, and 27 rad/s. The damping ratio is assumed as ≥ = 0.005. The simulation results of a test case for a Matlab-based simulation program are shown in Fig. 3-19. These results are identical to those obtained using the SAVANT program for the same test case. However, these simulation results were for a preliminary reference model of the Ares-I available to the public, not for the most recent model of the Ares-I with properly updated, ascent flight guidance and control algorithms. Our study purpose was to develop a Matlab-based simulation tool for an independent validation of the performance and stability of NASA’s ascent flight control system design for the Ares-I. The center of pressure (cp) location shown in Fig. 11 was computed as Xcp = Xa −

(CM p0 + CM pα α)b CN 0 + CN α α

(61)

where Xcp is the distance to the cp location from the top of vehicle and Xa is the distance to the aerodynamic reference point from the top of vehicle (i.e., the origin of the structure reference frame).

V. A.

Linear Rigid-Body Model

Nonlinear 6-DOF Equations

Translational equations in the body frame:            u˙ 0 −r q u gx Faero.xb T 1  1          B/I   v˙  = −  r 0 −p   v  + C  gy  +  Faero.yb  +  −T δz  m m w˙ −q p 0 w gz Faero.zb T δy where

CB/I Rotational equation in  Ixx Ixy   Ixy Iyy Ixz Iyz



 1 − 2(q22 + q32 ) 2(q1 q2 + q3 q4 ) 2(q1 q3 − q2 q4 )   =  2(q1 q2 − q3 q4 ) 1 − 2(q12 + q32 ) 2(q2 q3 + q1 q4 )  2(q1 q3 + q2 q4 ) 2(q2 q3 − q1 q4 ) 1 − 2(q12 + q22 )

the body  Ixz  Iyz   Izz

frame:   p˙ 0 −r   q˙  = −  r 0 r˙ −q p  Taero.xb  +  Taero.yb Taero.zb

   q Ixx Ixy Ixz p    −p   Ixy Iyy Iyz   q  0 Ixz Iyz Izz r      Trkt.xb Trcs.xb       +  Trkt.yb  +  Trcs.yb  Trkt.zb Trcs.zb


(62)

(63)

(64)

7

x 10 1.015

Z Position (ft)

1.01

1.005

1

0.995 !1.838 !1.84 7

x 10

2 !1.842

1 !1.844

Y Position (ft)

5

0 !1.846

!1

x 10 X Position (ft)

Figure 3. Vehicle’s position (x, y, z) in the Earth-centered inertial reference frame.

B.

Linearization Results ∆u˙ = r0 ∆v − q0 ∆w − w0 ∆q + v0 ∆r + (2gy q2c + 2gz q3c )∆q1 + (−4gx q2c + 2gy q1c − 2gz q4c )∆q2 1 +(−4gx q3c + 2gy q4c + 2gz q1c )∆q3 + (2gy q3c − 2gz q2c )∆q4 + m ∆Faero.xb

(65)

∆v˙ = −r0 ∆u + p0 ∆w + w0 ∆p − u0 ∆r + (2gx q2c − 4gy q1c + 2gz q4c )∆q1 + (2gx q1c + 2gz q3c )∆q2 1 T +(−2gx q4c − 4gy q3c + 2gz q2c )∆q3 + (−2gx q3c + 2gz q1c )∆q4 + m ∆Faero.yb + (− m )∆δz

(66)

∆w˙ = q0 ∆u − p0 ∆v − v0 ∆p + u0 ∆q + (2gx q3c − 2gy q4c − 4gz q1c )∆q1 + (2gx q4c + 2gy q3c − 4gz q2c )∆q2 1 T +(2gx q1c + 2gy q2c )∆q3 + (2gx q2c − 2gy q1c )∆q4 + m ∆Faero.zb + ( m )∆δy (67)      Ixx 0 0 ∆p˙ b1      (68)  0 Iyy 0   ∆q˙  =  b2  0 0 Izz ∆r˙ b3

where the relatively small products of inertia are ignored, and

b1 = [r0 Iyy − Izz r0 ]∆q + [Iyy q0 − q0 Izz ]∆r + ∆Taero.xb + (−cy T )∆δy + (−cz T )∆δz + ∆Trcs

(69)

b2 = [−r0 Iyy + Izz r0 )]∆p + [−(Ixx p0 + p0 Izz ]∆r + ∆Taero.yb + (cx − Xg )T ∆δy

(70)

b3 = [q0 Ixx − +Iyy q0 +]∆p + [(Ixx p0 − p0 Iyy ]∆q + ∆Taero.zb + (Xg − cx )(−T )∆δz

(71)

∆q˙1 =

1 (q4c ∆p − q3c ∆q + q2c ∆r + r0 ∆q2 − q0 ∆q3 + p0 ∆q4 ) 2 12 of 25 American Institute of Aeronautics and Astronautics

(72)

4500 4000

Relative Velocity (ft/sec)

3500 3000 2500 2000 1500 1000 500 0

0

20

40

60 Time (sec)

80

100

120

Figure 4. Relative velocity Vrel with respect to the Earth-fixed reference frame.

4

14

x 10

12

Altitude (ft)

10

8

6

4

2

0

0

20

40

60 Time (sec)

80

100

Figure 5. Altitude h.


120

4.5 4 3.5

Mach Number

3 2.5 2 1.5 1 0.5 0

0

20

40

60 Time (sec)

80

100

120

100

120

Figure 6. Mach number M .

700

Dynamic Pressure (psf)

600

500

400

300

200

100

0

0

20

40

60 Time (sec)

80

Figure 7. Dynamic pressure Q in units of lb/ft2 .


6

2.2

x 10

2

Rocket Weight (lb)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4

0

20

40

60 Time (sec)

80

100

120

100

120

Figure 8. Vehicle’s total weight mg.

6

3.5

x 10

3

Thrust (lb)

2.5

2

1.5

1

0.5

0

0

20

40

60 Time (sec)

80

Figure 9. Total thrust T .


8000 6000 4000

Base Force (lb)

2000 0 !2000 !4000 !6000 !8000 !10000 !12000

0

1

2

3

4 5 Altitude (m)

6

7

8

9 5

x 10

Figure 10. Base force Fbase as a function of altitude in units of meters.

0 cx cp

Location (ft)

!50

!100

!150

!200

!250

0

20

40

60 Time (sec)

80

100

120

Figure 11. Locations of the center of pressure (cp) and the center of gravity (cg) from the top of vehicle.


Vw.xb (ft/sec)

20 10 0 !10

0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60 Time (sec)

80

100

120

Vw.yb (ft/sec)

20 0 !20 !40

Vw.zb (ft/sec)

50 0 !50 !100

Figure 12. Disturbance wind velocity components in the body frame.

q1c

0.6 0.4 0.2 0

0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60 Time (sec)

80

100

120

q2c

0 !0.5 !1

q3c

0.5 0 !0.5

Figure 13. Commanded quaternions (q4c not shown).


p (rate/sec)

0.2 0 !0.2

0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60 Time (sec)

80

100

120

q (rate/sec)

0.04 0.02 0 !0.02

r (rate/sec)

0.02 0.01 0 !0.01

Figure 14. Body angular rates (p, q, r).

! (deg)

140 120 100 80

0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60 Time (sec)

80

100

120

" (deg)

!20 !40 !60

# (deg)

0 !50 !100

Figure 15. Euler angles (φ, θ, √). These Euler angles (φ, θ, √) do not actually represent the vehicle’s roll, pitch, and yaw attitude angles for attitude feedback control.


0.04 q1e

0.02 0 !0.02

0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60 Time (sec)

80

100

120

0.04 q2e

0.02 0 !0.02 0.02

q3e

0.01 0 !0.01

Figure 16. Attitude-error quaternions (q4e not shown).

!1 (inch)

20 10 0 !10

0

20

40

60

80

100

120

0

20

40

60

80

100

120

0

20

40

60 Time (sec)

80

100

120

!2 (inch)

2 0 !2 !4

!3 (inch)

1 0 !1

Figure 17. Flexible modes η1 , η2 , and η3 . Note that the higher-frequency components of flexible-mode vibrations cannot be seen in this plot because the flexibl modes are persistently but slowly excited by gimbal contol deflection.


Angle of Attack (deg)

5

0

!5

Angle of Sideslip (deg)

!10

0

20

40

60

80

100

120

0

20

40

60 Time (sec)

80

100

120

5

0

!5

Yaw Gimbal Angle (deg)

Pitch Gimbal Angle (deg)

Figure 18. Angle of attack α and angle of sideslip β.

6 4 2 0 !2

0

20

40

60

80

100

120

0

20

40

60 Time (s)

80

100

120

1

0.5

0

!0.5

Figure 19. Pitch gimbal angle δy and yaw gimbal angle δz .


Ixx (slug!ft2)

6

2

x 10

1 0

0

20

40

60

80

100

120

20

40

60

80

100

120

20

40

60 Time (sec)

80

100

120

Iyy (slug!ft2)

8

3

x 10

2 1

0

Izz (slug!ft2)

8

3

x 10

2 1

0

Figure 20. Moments of inertia about the center of gravity.

∆q˙2 = ∆q˙3 =

1 (q3c ∆p + q4c ∆q − q1c ∆r − r0 ∆q1 + p0 ∆q3 + q0 ∆q4 ) 2

(73)

1 (−q2c ∆p + q1c ∆q + q4c ∆r + q0 ∆q1 − p0 ∆q2 + r0 ∆q4 ) 2

(74)

1 (−q1c ∆p − q2c ∆q − q3c ∆r − p0 ∆q1 − q0 ∆q2 − r0 ∆q3 ) 2 Linearization of the aerodynamic forces and moments:      ∆Faero.xb 0 0 0 ∆u       ∆Faero.yb  =  0 CY β Q0 S/Vm 0   ∆v  ∆Faero.zb 0 0 −CN α Q0 S/Vm.xb ∆w ∆q˙4 =

    ∆Taero.xb 0 cz −cy ∆Faero.xb      0 −Xa + cx   ∆Faero.yb   ∆Taero.yb  =  −cz ∆Taero.zb cy Xa − cx 0 ∆Faero.zb    0 0 0 ∆u    + 0 0 CM pα Q0 Sb/Vm.xb   ∆v  0 Cmyβ Q0 Sb/Vm 0 ∆w

(75)

(76)



Linearization of α and β:

∆α =

(77)

1 ∆w Vm.xb

(78)

1 ∆v Vm

(79)

∆β =


Linearization of the thrust force and moment:     ∆Frkt.xb 0      ∆Frkt.yb  =  −T ∆δz  ∆Frkt.zb T ∆δy      ∆Trkt.xb 0 cz −cy ∆Frkt.xb      0 −Xg + cx   ∆Frkt.yb   ∆Trkt.yb  =  −cz ∆Trkt.zb cy Xg − cx 0 ∆Frkt.zb

(80)

(81)

Quaternion errors:

     C.

q1e q2e q3e q4e





    =  

q4c −q3c q2c q1c

q3c q4c −q1c q2c

−q2c q1c q4c q3c

−q1c −q2c −q3c q4c

    

∆q1 ∆q2 ∆q3 ∆q4

    

(82)

Linear State-Space Equations

A linearized state-space model of a rigid vehicle is described by x˙ = Ax + Bu y = Cx

(83)

where x = (∆u, ∆v, ∆w, ∆p, ∆q, ∆r, ∆q1 , ∆q2 , ∆q3 , ∆q4 )T , u = (∆Trcs , ∆δy , ∆δz )T , (2q1e , 2q2e , 2q3e , p, q, r)T ,   A11 A12 A13   A =  A21 A22 0  0 A32 A33   0 r0 −q0   A11 =  −r0 CY β Q0 S/(mVm ) p0  q0 −p0 −CN α Q0 S/(mVm.xb )   0 −w0 v0   A12 =  w0 0 −u0  −v0 u0 0 A13

y =

(84)

(85)

(86)



 (2gy q2c + 2gz q3c ) (−4gx q2c + 2gy q1c − 2gz q4c ) (−4gx q3c + 2gy q4c + 2gz q1c ) (2gy q3c − 2gz q2c )   =  (2gx q2c − 4gy q1c + 2gz q4c ) (2gx q1c + 2gz q3c ) (−2gx q4c − 4gy q3c + 2gz q2c ) (−2gx q3c + 2gz q1c )  (2gx q3c − 2gy q4c − 4gz q1c ) (2gx q4c + 2gy q3c − 4gz q2c ) (2gx q1c + 2gy q2c ) (2gx q2c − 2gy q1c ) (87)   0 0 0 h  x )CN α Q0 S  −1    − (−Xa +c V m.xb   i Ixx 0 0 0 0   C Q Sb 0 M pα     + A21 =  0 Iyy 0   (88) Vm.xb  h   (Xa +cx )CY β Q0 S 0 0 Izz   − Vm  0  i 0 Cmyβ Q0 Sb + Vm A22



Ixx  = 0 0

0 Iyy 0

−1  0 0   0   −r0 Iyy + Izz r0 Izz q0 Ixx − Iyy q0

r0 Iyy + Izz r0 0 Ixx p0 − p0 Iyy


 Iyy q0 − q0 Izz  −Ixx p0 + p0 Izz  0

(89)

A32 =



1   2 



Ixx  B2 =  0 0  0 0   0 0   0 0 C=  0 0    0 0 0 0

VI.

q4c q3c −q2c −q1c

−q3c q4c q1c −q2c



q2c −q1c q4c −q3c

0 r0 −q0 p0 1 0 p0 q0  −r0 A33 =  2  q0 −p0 0 r0 −p0 −q0 −r0 0   B1   B =  B2  0   0 0 0   B1 =  0 0 −T /m  0 T /m 0 −1  0 0 1 −cy T   Iyy 0   0 (cx − Xa )T 0 Izz 0 0 0 0 0 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 2q4c 0 −2q3c 0 2q2c 0 0 0 0 1 0

2q3c 2q4c −2q1c 0 0 0

   

    

 −cz T + cy T  0  (cx −Xa )T  −2q2c −2q1c  2q1c −2q2c   2q4c −2q3c    0 0   0 0  0 0

(90)

(91)

(92)

(93)

(94)

(95)

Linear Flexible-Body Model

The linear state-space equation of the Ares-I including the flexible-body modes is described by ˆ + Bu ˆ x˙ = Ax ˆ y = Cx where x = (∆u, ∆v, ∆w, ∆p, ∆q, ∆r, ∆q1 , ∆q2 , ∆q3 , ∆q4 , η1 , η2 , η3 , η4 , η5 , η6 , η˙ 1 , η˙ 2 , η˙ 3 , η˙ 4 , η˙ 5 , and   A 0  ˆ = A   0 I 0 −≠2 −2≥≠   B  ˆ = B 0   ΦT mB1   0 0 0 0 0 0 2q4c 2q3c −2q2c −2q1c   ™ 0   0 0 0 0 0 0 −2q3c 2q4c 2q1c −2q2c    0 0 0 0 0 0 2q2c −2q1c 2q4c −2q3c    ˆ C=  0 0 0 1 0 0 0 0 0 0     0 ™   0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 23 of 25 American Institute of Aeronautics and Astronautics

(96) η˙ 6 )T ,

(97)

(98)

(99)

Po(e!*ero Ma/ 30

5maginary 12is

20 10 0 !10 !20 !30 !20

!15

!10

!5

0 0ea( 12is

5

10

15

20

Figure 21. Pitch transfer function model of a reference model of the Ares-I CLV.

A pitch transfer function model of an Ares-I reference model at t = 60 sec is found as 2q2e (s) −0.8589(s + 0.043)(s + 3.68)(s − 3.75)(s2 − 35s + 512)(s2 + 35s + 512) = δy (s) (s + 0.69)(s − 0.015)(s − 0.66)(s2 + 0.06s + 36)(s2 + 0.14s + 202)(s2 + 0.27s + 738)

(100)

The poles and zeros of this pitch transfer function are illustrated in Fig. 21. Such a pole-zero pattern is of typical for flexible vehicles with non-collocated actuator and sensor. Flight control design and analysis of the Ares-I, based on this pitch transfer function model, is presented in [12].

VII.

Conclusions

A set of dynamic models of the Ares-I Crew Launch Vehicle, incorporating its propulsion, aerodynamics, guidance and control, and structural flexibility, has been described in this paper. The preliminary results of developing a Matlab-based simulation and linearization program by utilizing NASA’s SAVANT Simulinkbased program have been discussed. The study purpose was to develop an independent validation tool for the performance and stability analysis of the ascent flight control system of the Ares-I. A linearized model of the Ares-I was obtained as a test case of an independent validation of the ascent flight control design and analysis of the Ares-I.

References [1] Whorton, M., Hall, C., and Cook, S., “Ascent Flight Control and Structural Interaction for the Ares-I Crew Launch Vehicle,” AIAA 2007-1780, April 2007. [2] Betts, K. M., Rutherford, R. C., McDuffie, J., Johnson, M. D., Jackson, M., and Hall, C., “Time Domain Simulation of the NASA Crew Launch Vehicle,” AIAA 2007-6621, August 2007. [3] Betts, K. M., Rutherford, R. C., McDuffie, J., Johnson, M. D., Jackson, M., and Hall, C., “Stability Analysis of the NASA Ares-I Crew Launch Vehicle Control System,” AIAA 2007-6776, August 2007. [4] James, R. L., “A Three-Dimensional Trajectory Simulation Using Six Degrees of Freedom with Arbitrary Wind”, NASA TN D-641, 1961. [5] Harris, R. J., “Trajectory Simulation Applicable to Stability and Control Studies of Large Multi-Engine Vehicles”, NASA TN D-1838, 1963. 24 of 25 American Institute of Aeronautics and Astronautics

[6] Greensite, A. L., “Analysis and Design of Space Vehicle Flight Control Systems,” Spartan Books, New York, 1970. [7] Zipfel, P. H., Modeling and Simulation of Aerospace Vehicle Dynamics, AIAA Education Series, 1998. [8] Wie, B., Space Vehicle Dynamics and Control, AIAA Education Series, 1998. [9] Jang, J.-W., Bedrossian, N., Hall, R., Norris, H., Hall, C., and Jackson, M., “Initial Ares-I Bending Filter Design,” AAS 2007-059, February 2007. [10] Jang, J.-W., Bedrossian, N., and Hall, R., “Ares-I Bending Filter Design Using a Constrained Optimization Approach,” AIAA 2008-6289, AIAA Guidance, Navigation, and Control Conference, Honolulu, Hawaii, August 18-21, 2008. [11] Hall, C., Lee, M., Jackson, M., West, M., Whorton, M., and Brandon, J., “Ares-I Flight Control System Overview,” AIAA 2008-6287, AIAA Guidance, Navigation, and Control Conference, Honolulu, Hawaii, August 18-21, 2008. [12] Wie, B., Du, W., and Whorton, M., “Analysis and Design of Launch Vehicle Flight Control Systems,” AIAA 2008-6291, AIAA Guidance, Navigation, and Control Conference, Honolulu, Hawaii, August 18-21, 2008.
