Launch vehicles: Attitude Dynamics & control

Transcription

1 Launch vehicles: Attitude Dynamics & control Dr. A. Pechev Thrust vector control Measurements Attitude control Stabilising the pitch motion 0

2 Launch Vehicles Guidance Navigation and Control The motion of a rigid body in space is specied by two sets of parameters: 1) position and velocity of the center of mass (six state variables plus rate of change of mass) 2) attitude and attitude motion (six more state variables) Position and velocity describe the translational motion of the center of the mass of the launch vehicle with respect to an inertial reference frame. In the previous lectures we derived the equations describing this motion. The attitude and the attitude motion describe the orientation of the rocket in space. This is the subject of this lecture. 1

3 Motion of Center of Mass in Polar coordinates v c = ksin(γ) r 2 γ = kcos(γ) v c r 2 ν = v c cos(γ) r ṙ ḣ = v csin(γ) ṁ = T I sp D m + 1 m T cos(ϕ), D = 1 2 C daρv 2 c + L v c m + v ccos(γ) r + 1 v c m T sin(ϕ) 2

4 Inertial Motion in body coordinates v x = v xv y r + T cos(φ) m 0 ṁt v y = v2 y r µ + T sin(φ) r 2 m 0 ṁt ṙ = v y ẋ = v x 3

5 Launch Vehicle Thrust Vector Control For attitude control we need to manipulate the direction of the thrust (Thrust Vector Control, TVC). This is possible either by: a) having three or four motors that can throttled to produce dierential thrust b) having capabilities to gimbal the motors 4

6 In the gure above T x = (T 1 T 3 )l T z = (T 2 T 4 )l T = R 2 (ϕ) (T 1 T 3 )l 0 (T 2 T 4 )l R 2 (ϕ) is the rotation matrix about the longitudinal axis. rocket, ϕ is maintained zero. For a non-spinning 5

7 Thrust Vector Control: gimbaling Gimbaling the booster provides another mechanism for determining the direction of the thrust. Saturn V, for example, has ve motors, where the four outer motors can be gimbaled by separate hydraulic actuators to get thrust components normal to the longitudinal axis. All three degrees of freedom, pitch (θ), yaw (ψ) and roll (ϕ), can be controlled by gimbaling the motors. Saturn V is aerodynamically unstable. The Space Shuttle's Solid Rocket Booster also has mechanisms that allow gimbaling the nozzle for thrust vector control. 6

8 Saturn V: Engine congurations 7

9 Launch vehicle measurements Inertial Stabilised Platform: a device used for measuring acceleration and attitude of launch vehicles. 8

10 Saturn V: measurements ST-124 consists of: three gimbal mechanisms attached on gas bearings with servo motors and resolvers (encoders). three integrating accelerometers for measuring the motion of the vehicle for the purpose of the guidance. three-axis gyros for measuring attitude rates and angles; Angles are fed back through control loops to the servo motors for maintaining zero attitude error. The accumulated in the gimbal angles correspond to the attitude of the vehicle. 9

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12 Launch vehicle Center of Mass and Pressure 11

13 Bending Modes 12

14 Bending Modes 13

15 Eects of wind One of the most important tasks of the ght system design of a launch vehicle is to reduce the aerodynamic lateral loads during atmospheric ght. 14

16 Launch vehicle Attitude In the gure: γ is the ight path (from local horizon to velocity) α is the angle of attack (from local horizon to roll axis) θ = γ + α is the rocket's pitch angle All discussions below consider the pitch motion; The results however apply to yaw motion. 15

17 Attitude Dynamics/Kinematics 16

18 The Euler's Equations Euler's equations is the equivalent of Newton's second law of motion for rotation about the center of mass (i) dh dt = T where T is the sum of all external torques. Remember this is a vector equation, i.e. h = h x i + h y j + h z k and T = T x i + T y j + T z k This representation is in an inertial frame. Using the fact that in body-xed frame (b) rotating with an angular velocity ω (i) dh dt = (b) dh dt + ω h Euler's equation becomes (b) dh dt + ω h = T or using the fact that (h = Iω) I (b) dω dt + ω Iω = T, or equivalently I ω + ω Iω = T 17

19 The Euler's Equations Using principal axes, I = I I I 3 we can reduce I ω + ω Iω = T to a set of three scalar equations I 1 ω x + (I 3 I 2 )ω y ω z = T x I 2 ω y + (I 1 I 3 )ω x ω z = T y I 3 ω z + (I 2 I 1 )ω x ω y = T z T = T x i + T y j + T z k above is the sum of the control torque (from actuators) and all disturbance torques. For small angles 18

20 θ = ω x φ = ω y ψ = ω z For small angles and small angular velocities θ = 1 I 2 T x φ = 1 I 1 T y ψ = 1 I 3 T z 19

21 Launch vehicle Pitch dynamics For this description we can write the rotational equation of motion about center of mass I θ = T l 1 sinδ + Nl 2 where l 1 and l 2 as distance between the center of mass and thrust/center of pressure (as in gure) I is the pitch moment of inertia, δ is the gimbal angle (angle from roll axis to rocket thrust T ) N = 1/2C L Aρv 2 ccos(α) is the normal aerodynamic force. How to use this equation? 20

22 Launch vehicle Attitude For constant turn-over maneuver, a constant pitch angle is maintained, i.e. θ = const. Thus θ = 0 and from pitch equation of motion 0 = T l 1 sinδ + Nl 2 we can calculate the gimbal angle that is required to maintain this constant pitch, turn-over maneuver. ( ) δ = sin 1 Nl2 T l 1 In this equation, the aerodynamic force N, the thrust T, and l 1,2 are functions of time and their values have to be constantly updated during the maneuver in order to calculate the gimbal oset δ. 21

23 Control of pitch motion Start with I θ = T l 1 sinδ + Nl 2 For controller design we set N = 0 (The eect of N = 0 is later evaluated by studying the feedback system). This reduces the dynamic equation to I θ = T l 1 sinδ Assigning a state vector x = [θ, θ] we can write the linear state-space equation (with the assumption that sin(δ) δ) ẋ = Ax + Bu with A = [ ],B = [ 0 l 1 T/I ] 22

24 Control of pitch motion analysis of open-loop response To analyse the open-loop response, we calculate the poles of the system (eigenvalues of A) 1) Construct the characteristic equation p(s) = det(si A) = det = det ([ s 1 0 s ]) = s 2 ([ s 0 0 s ] [ ]) = This gives two poles at the origin, i.e. s 1 = s 2 = 0. Since the poles are not in the left-hand side of the s-plane, the system is unstable and requires feedback control. 23

25 Pole placement design Given the state-space model of the system ẋ = Ax + Bu The state-feedback controller has the form u = Kx The closed-loop system then becomes ẋ = Ax + B( Kx) = (A BK)x The stability of this closed-loop system is determined from the location of the closed-loop poles or the eigenvalues of (A BK). The closed-loop poles are the solution to the following characteristic equation p(s) = det(si (A BK)) = 0, where s = σ + jω is the complex frequency (Laplace variable). For the design we specify a given desired closed-loop pole locations, i.e. we give 24

26 p = s n + a 1 s (n 1) + a 2 s (n 2) a n, where a 1, a 2,... are known! We then nd the closed-loop gains by comparing p(s) = p(s) collecting equal in power terms coecients and solving for unknown controller gains. 25

27 Pole placement design for pitch motion Start with ẋ = Ax + Bu with A = For given [ ] [ 0,B = 1 p = (s ( 2 j2))(s ( 2 + j2)) = s 2 + 4s + 8 Using ] K = [k 1, k 2 ], the closed-loop characteristic equation is [ s 0 p(s) = det(si (A BK)) = det( [ ] 0 s 0 1 ] [ 0 1 ( 0 0 ] [k 1 k 2 ])) = s 2 k 2 s k 1 Comparing p(s) with p(s) we calculate k 1 = 8 and k 2 = 4. 26

28 Bending Modes 27

29 Modelling exibility For modelling exibility in the rocket, one approach (for linear design) is to consider that the rigid body is constructed by a series of rigid bodies coupled by spring/damper links. A mass-damper system can be modelled by a second-order transfer function F M i (s) = k i s 2 + 2ζω i + ω 2 i where k i is the modal gain, ζ is the damping coecient and ω i is the i-th exible mode frequency. 28

30 Modelling exibility For Saturn V we have approx. ω = 2π, 3.5π, 6π ). Typically, ζ << 1. Considering transfer-function representation, the overall model transforms to θ(s) = ( T l 1 I 1 s 2 + i k i s 2 + 2ζω i + ω 2 i ) δ The design of the controller can be performed using the simple model (double integrator), but the analysis must be done for the full-order model. Depending on the location of the exible modes with respect to the control-system bandwidth, we can experience quite dierent behaviour. 29

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34 Modelling slosh Modelling slosh motions is a challenging problem. The model would be highly nonlinear and dependent on tank geometry, construction, etc. For the purpose of the control system design (as a part of the Thrust Vector Control task), we can consider represent the motion of the liquid using a single slosh-pendulum. 33

35 Modelling slosh By modelling the coupled system, rigid body plus the pendulum, it can be identied that the overall closed-loop model (pitch axis) is θ(s) = ( T l1 I 1 (s 2 + ω 2 ) z) s 2 (s 2 + ωp) 2 δ where the zero and the pole of the slosh mode depend on the parameters of the tank, the mass, etc ( ) ωp 2 = ωs m m 0 + mb(b+l 1) ( I ) ωz 2 = ωp 2 ωs 2 mb l 0 m 0 + mb(b+l 1) I ωs 2 = T/(m + m 0 )l The pole-zero separation determines the magnitude of the slosh contribution. If the slosh pendulum is pivoted below the centre of the mass of the rocket, the interaction between the rigid body and the slosh mode is stable (the zero has a lower frequency component than the pole). 34

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39 Control of pitch motion 1. How is the inertial acceleration related to the body acceleration? 2. How many state variables are used to describe the attitude of the vehicle? 3. How are the inertial accelerations measured in a ying launch vehicle? 4. What principles can be used to manipulate the direction of the thrust? 5. What is the signicance of the location of the Centre Of Pressure in respect to the Centre of Mass? 6. How is lift related to the angle of attack? 7. Use the diagram on p. 15 to construct the equation describing the motion about x (pitch, θ) 8. Design a control system for the problem dened in (7) to deliver stable motion with settling time around 1 second. 38