NEW ADAPTIVE SLIDING MODE CONTROL AND RENDEZ-VOUS USING DIFFERENTIAL DRAG

Transcription

1 NEW ADAPTIVE SLIDING MODE CONTROL AND RENDEZ-VOUS USING DIFFERENTIAL DRAG Hancheol Cho Lamberto Dell Elce Gaëtan Kerschen Space Structures and Sstems Laborator (S3L) Universit of Liège, Belgium

2 Contents Rendez-vous Using Differential Drag 1. Assumptions. Maneuver Planner 3. Drag Estimation 4. Controller Design 5. Numerical Simulation

3 Rendez-vous Using Differential Drag Assumptions Onl in-plane control is considered. The target has a stable attitude. The rotation about the orbital plane of the chaser is controlled. The control output is S d (cross-sectional area eposed to the atmosphere). S d is bounded b S min S d S ma. The maneuver planner and the drag estimator are eecuted at the beginning of the maneuver. ˆ S d ŷ Chaser Target

4 Maneuver Planner Rendez-vous Using Differential Drag The maneuver planner computes the off-line optimal control strateg for the global rendezvous maneuver. It is obtained using an hp-adaptive Radau pseudospectral transcription using the software GPOPS. The dominant effects (secular J effects and short period and altitude-dependent variations of drag) are modeled in the planner. For computational efficienc and accurac, relative dnamics is epressed in terms of decomposed curvilinear variables and linear dnamics (Schweighart-Sedwick equations) is considered. Drag Estimator To estimate the drag force on the two satellites, their accurate positions are monitored for at least two orbits. The estimation is performed b minimizing the mean square error between observed and simulated mean semi-major ais (generated through high-precision propagation based on the Jacchia 71 atmospheric model).

5 Rendez-vous Using Differential Drag

6 Rendez-vous Using Differential Drag C, C, F, c est est est b, t d, c d, t h Drag Estimator Maneuver Planner S,,,,, d r r r r r Controller S d Plant

7 Ideal (Reference) Sstem where Controller Design c 5c, r r r c F, r r d, r est est Cd, cs d, r Fd, r Fd, t 1 1 chr rur r, SD Smin, Sma, ur 0,1. est Cb, t mc Real Sstem (including higher harmonics, SRP, third-bod perturbations, NRLMSISE-00) where c 5c g, c F g, F u, u 0,1. d r r d

8 Error Dnamics where Controller Design e ce c e g 5, e ce u u g, r u u u, e, e. r r r

9 Sliding Surface Controller Design Consider the sliding surface of the form: where Stabilit: e e e T and s s s e Se, c S c c c 0 S Re eig 0 0 s T. 0, c. I S

10 Phsical Interpretation Controller Design When s = 0 and ς is sufficientl small, the error on the mean component of the trajector is 0 (but oscillating) because 4c c, m e e e 0, c c c 4 c e, m e e. e c c Hence, the damping ratio ς > 0 is used to progressivel reduce the size of the relative orbit of the error vector. Furthermore, when s = 0, the error dnamics is analogous to the dnamics of the oscillator components: c c, o, o d e e F c c e ce., o, o, e c e c c e e ce., e c e c c e e ce.,

11 Lapunov Function 1 T Let V s Ps, P 0, then where V s T Ps Controller Design * P11 s P1 s L P1 s P s r u L * P1 s P s L r u L P 1 s s P L r u L P P *, Assume 0 5c * L 5c e c e e : measurable c L u g g : uncertain, L L P s P s 11 1 P s P s 1 P M P 11 1 P11 0, P11 P P1 P1 P1 P P

12 Controller Design Control Input * P 1 L Let u s s, then r P r P 1 P 1 V s s P L s s P P P P s s P L P s s P 1 1 M P. Consider the region where P1 s P s P P V s s P L P s s P 1 1 M P P1 s s P L M P 0. holds, then L M

13 Observation Adaptive Rules for the Gain Γ Controller Plant P1 s P s L M Controller L M LM / Plant P1 s s P ˆ Controller ˆ ˆ LM LM / ˆ Plant P1 s P s ˆ

14 Adaptive Rules for the Gain Γ Estimation of Γ Assume that a rough estimate ˆ is applied, and the resultant sliding variable is bounded b ˆ so that P1 / P. s ˆ s. Then, the real (unknown) gain Γ can be estimated b ˆ ˆ. : desired Adaptive Rule Let Γ 0 be the initial estimate. At each instant of time, P1 / P s s is compared with ε, and the real-time adaptive law for Γ is given b the following rules: ( k1) ( k ) P1 ( k ) ( k ), if s s, P ( k 1) ( k ) P 1 ( k ) ( k ) P1 ( k ) ( k ) min, s s /, if s s. P P

15 Control Parameters Numerical Simulation L 10, 0.005, 10, =1.01, P 3000, P 1000, P 0, P 10. M Mean elements of the target Initial gap of the chaser Semi-major ais m Eccentricit 0 Inclination 98 deg RAAN 90 deg Argument of perigee 0 deg True anomal 0 deg Along-track m Radial 100 m Target properties Ballistic coefficient m kg -1 Mass 4 kg Chaser properties Dimensions m 3 Drag coefficient.8

16 Control Parameters Numerical Simulation L 10, 0.005, 10, =1.01, P 3000, P 1000, P 0, P 10. M Global trajectories in - plane (left) and the final phase (right)

17 Control Parameters Numerical Simulation L 10, 0.005, 10, =1.01, P 3000, P 1000, P 0, P 10. M Cross-sectional area of the chaser S d

18 Control Parameters Numerical Simulation L 10, 0.005, 10, =1.01, P 3000, P 1000, P 0, P 10. M Errors in the -ais (upper) and -ais (lower)

19 Control Parameters Numerical Simulation L 10, 0.005, 10, =1.01, P 3000, P 1000, P 0, P 10. M P P s s 1 / (upper) and the gain Γ (lower) updated b the adaptive law

20 Numerical Simulation Summar The small ε is, the smaller error is. If the initial Γ 0 is too large, the control output S d is easil saturated and the errors start to oscillate with large amplitudes from the beginning of the maneuver. The response to the variation of u is slow, so the gain seems to be ended up with a larger value than the real one. How to determine δ and P ij?