Guidance and Control for Spacecraft Planar Re-phasing via Input Shaping and Differential Drag

Transcription

1 Università Sapienza, Dipartimento di Matematica March 5th, 014 Guidance and Control for Spacecraft Planar Re-phasing via Input Shaping and Differential Drag Riccardo Bevilacqua Rensselaer Polytechnic Institute

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3 Outline Introduction & Motivation (why along-track discrete thrust?) Finding the analytical guidance solutions for re-phasing using Input Shaping and CW equations Interpolation on relative orbit semi-major axis Atmospheric Differential Drag Adaptive Lyapunov-based controller closing the loop Numerical Simulations Conclusions Other activities at the ADAMUS lab. (ADvanced Autonomous MUltiple Spacecraft) Stop me any time if I am not clear

4 Introduction & Motivation S/C proximity maneuvers are critical for: On-orbit maintenance missions Refueling and autonomous assembly of structures in space Envisioned operations by NASA s Satellite Servicing Capabilities Office High cost of refueling calls for an alternative for thrusters as the source of the control forces (At LEO drag forces are an alternative) In general, finite, discrete, continuous thrust, especially low thrust, represents the reality of most recent spacecraft, especially CubeSats & small satellites 1 and differential drag. 1 Burges, J. D., Hall, M. J., Lightsey, E. G., "Evaluation of a Dual-Fluid Cold-Gas hruster Concept", International Journal of Mechanical and Aerospace Engineering, 6, 3-37, (01).

5 Analytical re-phasing guidance via Input Shaping (1) Re-phasing on a given orbit can be seen as relative maneuvering with respect to a virtual spacecraft ahead or behind or, there may be a real spacecraft. For short distances (few km) we may use linearized equations (coupled double integrator and harmonic oscillator): x Ax Bu, A, B, , u 1 x x y x y u y SCDV m

6 Brief comment on Clohessy-Wiltshire equations For the in-plane motion, they couple a double integrator and an harmonic oscillator. Image credit:. Alan Lovell, Steven G. ragesser, "GUIDANCE FOR RELAIVE MOION OF LOW EARH ORBI SPACECRAF BASED ON RELAIVE ORBI ELEMENS", AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Providence, Rhode Island, 004/08/16-004/08/19..

7 Analytical re-phasing guidance via Input Shaping () PHYSICAL CONSRAIN: Along-track (y) thrust only; this represents the physical layout of a single engine, gravity gradient stabilized spacecraft, or differential drag YPES OF MANEUVERS: From leader follower to leader follower From leader follower to stable relative orbit From stable relative orbit to stable relative orbit WHY ONLY HESE YPES? Input shaping can go from equilibrium to a new equilibrium We can add vibration via input shaping, but not damp it

8 Analytical re-phasing guidance via Input Shaping (3) u y A1 ft A 1 ft A 3 ft, A 3 1, A, A3 4 4 * t y fd y t0 u * fd 0, * * 0,,, u sign y y t if t t ft u sign y 1 fd y t if t t t ft f t t t f 1 t f 3 t t t 1 * 0, if t t, t*<δt Experimental example (video) y n y u t t*>δt M. Romano, B.N. Agrawal, F. Bernelli-Zazzera, "Experiments on Command Shaping Control of a Manipulator with Flexible Links", AIAA Journal of Guidance, Control, and Dynamics, Vol. 5, No., 00, pp

9 Analytical re-phasing guidance via Input Shaping (4) u y A1 ft A 1 ft A 3 ft, A 3 1, A, A3 4 4 u sign y y t if t t ft u sign y 1 fd y t if t t t ft f t t t f 1 t f 3 t t t 1 * 0, if t t * t y fd y t0 u * fd 0, * * 0,,,, x Ax Bu, A, B, x x y x y, uu y For different initial conditions, and generic Δt:? 1) (from leader-follower) ) (from stable relative orbit) x x t y t x y x x

10 Analytical re-phasing guidance via Input Shaping (5) 1) (from leader-follower) x t y s sin... c cos t t t t t t t t t s 0.5 s 0.5 s u x f 0.5s t s s 0.5s t s y x y f f f c 0.5 4c 0.5 c c c 0.5 c c u ct 0.5y0 u 1.5yfd u 1 t t t t t t t t t t t t c 0.5 c 0.5 c u 0.5 c c c c 0.5 c 0.5 u 4s 0.5 s 0.5 s s s 0.5 s s s y fd u y 0 y f x non-drifting relative orbit f x 4x y 0 f f y y x 0.5y 1.5y f f 0 fd 0 y ' 3 y 1 3 y y y fd fd fd e a b, a x x, b y y c cos... rel f f 1 1 6c 4c t 4c t c t u erel 0.5 4c t 16c t 4c 16c t c t 6ct 4ct 4ct 3 3 3yfd 3y u Leonard, C. L., Hollister, W., M., and Bergmann, E. V. Orbital Formationkeeping with Differential Drag. AIAA Journal of Guidance, Control and Dynamics, Vol. 1 (1) (1989), pp if t 0.5 e rel 0

11 Analytical re-phasing guidance via Input Shaping (6) 1) (from leader-follower) x t y e 8u cos t sin t rel y y fd, cos 1 4 t u cos t u t k, k 1,,... max t k, k 1,,... min

12 Guidance examples (lead-follow start) Orbital Parameter Desired S/C initial Semi-major axis a 6,778.1 km 6,778.1 km Eccentricity e 0 0 Inclination i deg deg Right Ascension of the Ascending Node (RAAN) deg deg Argument of Perigee 30 deg 30 deg Polar Angle 7.16 deg 7.18 deg

13 Analytical re-phasing guidance via Input Shaping (7) ) (from stable relative orbit) x t x y x x s sin... c cos x0 ct u s u s t u s t 1 1 x f 4u s t u s t u s t u s 1 u st x0 s t y x sin t f 0 4x0 c t u u c 3 y fd y c 4 c 4 c 1 u u t u t u ct 4x0 u ct 1 8u c t 4u ct 1 x0 s t u c u u c t u ct x f u c t u c 4u c t u ct u c t x0 c t 1 x0 s t u s u s t u s u s t 1 1 y f u st 4u s t x0 c t u s t 1 us t y fd u y 0 y f x non-drifting relative orbit f x 0, y y x if y ' 3y 1 3y fd 0 fd fd 0 x twait k k x0 s sin... c cos... e rel 1 3yfd 3y u 1 tan 1 0, 0,1,,... x0 c t x0 s t u s t u s t u s t u s t u s u s u st u st 3 1 x0 s t x0 c t 4u c t u c t u c t u c t u c u c u ct u ct u 3 0 Non trivial to deal with but

14 Analytical re-phasing guidance via Input Shaping (8) ) (from stable relative orbit) x t x y x x s sin... c cos... e rel 1 x0 c t x0 s t u s t u s t u s t u s t u s u s u st u st 3 1 x0 s t x0 c t 4u c t u c t u c t u c t u c u c u ct u ct u 3 ω is the highest frequency. he Nyquist Shannon sampling theorem enables capturing the nature of the curve by computing it only at Δt points spaced by a 1/4ω time distance, that is, theoretically 8π points total (i.e. at least 6), within one orbital period. his way we can interpolate and find the correct Δt for a desired e rel. 3y 3y fd u 0

15 Guidance examples (stable orbit start) Orbital Parameter Desired S/C initial Semi-major axis a 6,778.1 km 6,778.1 km Eccentricity e Inclination i deg deg Right Ascension of the Ascending Node (RAAN) deg deg Argument of Perigee 30 deg 30 deg Polar Angle 7.16 deg 7.18 deg

16 Atmospheric Differential Drag Control Differential in the aerodynamic drag is a differential in along track acceleration his differential can be used to control the relative motion of the S/C on the orbital plane only he drag differential can be generated with controllable surfaces It is assumed that that the surfaces move almost instantly (on-off control)

17 Drag Acceleration he drag acceleration experienced by a S/C at LEO is a function of: Atmospheric density Atmospheric winds Velocity of the S/C relative to the medium, Geometry, attitude, drag coefficient and mass of the S/C Challenges for modeling drag force: he interdependence of these parameters Lack of knowledge in some of their dynamics Large uncertainties on the control forces (drag forces) Control systems for drag maneuvers must cope with these uncertainties. Differential aerodynamic drag for the S/C system is given as: a 1 BCv Drel s BC CD A m

18 Nonlinear Model he dynamics of S/C relative motion are nonlinear due to J perturbation Variations on the atmospheric density at LEO Solar pressure radiation Etc. he general expression for the real world nonlinear dynamics, including nonlinearities is: a, x f x Bu x x y x y, u 0 a Drel Drel

19 Linear Reference Model he Schweighart and Sedwick model is used to create the stable reference model LQR controller is used to stabilize the Schweighart and Sedwick model he resulting reference model is described by: x y x y x A x + Bu, A A - BK, x, u d d d d d d d d d d d Kx t

20 Lyapunov Approach A Lyapunov function of the tracking error is defined as: V e Pe, e = x - xd, P 0 After some algebraic manipulation, the time derivative of the Lyapunov function is: Defining A d Hurwitz and Q symmetric positive definite, P can be found using: -Q A d P + PA d ˆ V e (A P + PA )e e P( f x - A x + Ba u - Bu ) d d d Drel d

21 Drag panels activation strategy Rearranging V yields V 0 Guaranteeing would imply that the tracking error (e) converges to zero By selecting: uˆ sign( ) sign( e PB) V 1 V e Qe ( uˆ ), uˆ 0, 1 e PBa, = -e P A x f x Bu Drel is ensured to be as small as possible. d d

22 Critical value for the magnitude of differential drag acceleration Product βû is the only controllable term that influences the behavior of V uˆ here must be a minimum value for a Drel that allows for to be negative for given values of β and δ V his value is found analytically by solving: e PBa uˆ 0 Drel Solving this expression for a Drel yields a Drel e P A dx f x Bud e PB e PB a Dcrit

23 Matrix derivatives Choosing appropriate values for the entries of Q and A d can reduce a Dcrit o achieve this, the following partial derivatives were developed a Dcrit A d Starting from the general case of the critical value e P Adx f x Bud a Dcrit e PB he Lyapunov equation was transformed into:, a Dcrit Q - Q A P + PA, A P Q, d A I A A I, P vec( P), Q vec( Q), v 4 x4 d d 4 x4 v v d P A Q 1 v v v v v v

24 Matrix derivatives Using he following derivatives were found in previous work P 1 A v 1 P v 3 I16 x16 I4 x4 1, A d Ad Av Av I U U I I U U I A d P A v v P A Q 4 x x 4 4 x 4 4 x4 1 4 x4 4 x4 P Q I16 x16 Av U16 x16 I16 x16 Av I16 x16 Qv, 1 Av Using the chain rule the desired final expressions can be found: a Dcrit Q 1 v v v ( P) v I4 x4 I4 x4 d A 1 A x f x Bud, P adcrit 1 P 1( P) 3 I4 x4 1 I4 x4 Adx f x Bud Ad Ad P I e P U ( I x ), e PB 4x 4 16x16 4x 4 ( P) ( I ) I ( I ) P I4x 4 4 x 4 e U 16x16 4 x 4 4 x 4 e P 3 e PB 4 e PB, e PBe B

25 Adaptive Lyapunov Control strategy* Using these derivatives A d and Q are adapted as follows: A da ij a dq Dcrit ij a Dcrit A sign( ) A, Q sign( ) Q dt Aij dt Qij adcrit adcrit adcrit adcrit 1 if for i, j k, l 1 if for i, j k, l Aij A kl, Q Q ij Qkl 0 else 0 else hese were designed such that: Q is symmetric positive definite A d is Hurwitz hese adaptations result in an adaptation of the quadratic Lyapunov function Initial A d and Q should be such that initial estimated differential drag (underestimated 30% w.r.t. models) is above critical value at same time. WE ASSUME HA DRAG ONLY INCREASES. *Perez, D., Bevilacqua, R., Differential Drag Spacecraft Rendezvous using an Adaptive Lyapunov Control Strategy, Acta Astronautica 83 (013) Perez, D., Bevilacqua, R., Spacecraft Maneuvering via Atmospheric Differential Drag using an Adaptive Lyapunov Controller, paper AAS at the 3rd AAS/AIAA Spaceflight Mechanics Meeting, Kauai, Hawaii, February 013.

26 Numerical Simulations Simulations were performed using an SK scenario with High-Precision Orbit Propagator (HPOP) that included: Full gravitational field model Variable atmospheric density (using NRLMSISE-00) Solar pressure radiation effects Obtain state vector from SK Propagate in SK for XX minutes Calculate matrix derivatives Run panel activation strategy Adapt matrices Q, and A d

27 NS: lead-follow start and end - 10 min left, 5 min right

28 NS: lead-follow start, relative orbit end - 10 min left, 5 min right

29 NS: relative orbit start, relative orbit end - 10 min left, 5 min right

30 Conclusions Analytical guidance solutions for short distance re-phasing Combining guidance and the adaptive Lyapunov controller makes differential drag a very effective way to steer the spacecraft without propellant Potentially implementable on any small spacecraft We are building a drag sail to create a differential drag capable CubeSat

31 PADDLES Propellant-less Atmospheric Differential Drag LEo Spacecraft Sail subsystem is patent pending

32 ADAMUS spacecraft robotics activities Credit: lanetnetwork.com actical echnology Office O Satellite Servicing Capabilities Office SSCO

33 Acknowledgements