Model predictive control for spacecraft rendezvous hovering phases based on validated Chebyshev series approximations of the transition matrices

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Florent BRÉHARD 1,2 Clément GAZZINO 1 prarante@laas.fr florent.brehard@ens-lyon.

fr 1 Methods and Algorithms for Control (MAC) 2 Proofs and Languages (PLUME) & Arithmetic and

9èmes Rencontres «Arithmétique de l Informatique Mathématique» October 24, 2017 This work was

1 Model predictive control for spacecraft rendezvous hovering phases based on validated Chebyshev series approximations of the transition matrices Paulo Ricardo ARANTES GILZ 1 Florent BRÉHARD 1,2 Clément GAZZINO 1 prarante@laas.fr florent.brehard@ens-lyon.fr cgazzino@laas.fr 1 Methods and Algorithms for Control (MAC) 2 Proofs and Languages (PLUME) & Arithmetic and Computing (AriC) LAAS - CNRS / University of Toulouse LIP - CNRS / ENS Lyon / UCB Lyon / INRIA 9èmes Rencontres «Arithmétique de l Informatique Mathématique» October 24, 2017 This work was supported by the FastRelax (ANR-14-CE ) project of the French National Agency for Research (ANR). Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

2 Outline 1 Orbital rendezvous 2 Objectives 3 Models 4 Results 5 Conclusions Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

3 Overview 1 Orbital rendezvous 2 Objectives 3 Models 4 Results 5 Conclusions Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

4 Orbital rendezvous Rendezvous problem Two spacecraft (follower and target) orbiting another body; Objective: to perform a sequence of maneuvers in order to steer the chaser spacecraft from a state A to another state B; Constraints: actuators saturation, fuel budget, orbit restriction, safety constraints,... B Earth Target A Follower Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

5 Rendezvous hovering phases During the hovering phases The follower spacecraft must have station-keeping abilities for different purposes: stand by for orders from the mission control; wait for a better positioning to proceed to next step; observe and collect data about the target. Target Follower Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

6 Overview 1 Orbital rendezvous 2 Objectives 3 Models 4 Results 5 Conclusions Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

7 Objectives Goal: To design a model predictive control (MPC) algorithm based on rigorous polynomial approximations (Joldes, 2011, RPAs ) of the relative trajectories. Past Future Model predictive control Reference Predicted Measured For each call of the control algorithm, one sequence of control actions is computed, but only some of them are executed. t 0 t 1 t 2 t 3 Receding horizon t 1 t 2 t 3 t 4 t 5 t 6 Trigger signal Control Algorithm Computed impulse Thrusters Applied impulse Relative movement dynamics Navigation Measured position system Real position and velocities and velocities Control feedback loop scheme Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

8 Overview 1 Orbital rendezvous 2 Objectives 3 Models 4 Results 5 Conclusions Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

9 Relative motion dynamics Space mechanics assumptions: - Keplerian relative motion; - Passive target; - Propellant thrusters; - Predefined control dates; - Inter-satellite distance. t e : eccentricity (0 < e < 1); a : semi-major axis; ν : true anomaly; µ : Earth s gravitational constant; S t : target satellite; S f : follower satellite; O : Earth s center of mass. Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

10 Relative motion dynamics (Tschauner, 1967) Linearized Tschauner-Hempel equations (physical meaning): ẍ = νż + νz + ν 2 x µ R 3 x ÿ = µ R 3 y z = 2 νẋ νx + ν 2 z + 2 µ R 3 z By performing the change of variable: Ẋ(t) = A(t)X(t) X(t) = [x(t), y(t), z(t), ẋ(t), ẏ(t), ż(t)] T. [ x(ν), ȳ(ν), z(ν) ] = (1 + e cos(ν)) [ x(t), y(t), z(t) ]. Linearized simplified Tschauner-Hempel equations: x (ν) = 2 z (ν) ȳ (ν) = ȳ(ν) z (ν) = 2 x 3 (ν) + 1+e cos ν z(ν) X (ν) = Ā(ν) X(ν) X(ν) = [ x(ν), ȳ(ν), z(ν), x (ν), ȳ (ν), z (ν)] T. Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

11 State propagation via Transition Matrix Linear "time"-variant (LTV) system: X (ν) = Ā(ν) X(ν). The evolution of an arbitrary initial state X(ν 0 ) up to ν f is given by: X(ν f ) = Φ(ν f, ν 0 ) X(ν 0 ), where Φ(ν f, ν 0 ) is the so-called State-Transition Matrix of the system. It is computed by: Φ(ν f, ν 0 ) = φ(ν f )( φ(ν 0 )) 1, where φ(ν) is the so-called Fundamental Solution Matrix of the system, satisfying: φ (ν) = Ā(ν) φ(ν), det( φ(ν)) 0. Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

12 RPA via Chebyshev series (Bréhard et al., 2017) The solution of the following IVP: Find X(ν) s.t. X (ν) = Ā(ν) X(ν), X(ν0 ) = e k = { ( e k ) l = 1, ( e k ) l = 0, l = k l k (IVP) is equivalent to the k-th column of Φ(ν 0, ν): X(ν) = Φ(ν, ν 0 ) e k = Φ :,k (ν 0, ν). Idea: compute approximations of the solutions of (IVP) on the interval [ν 0, ν f ] for k = , obtaining an approximated Φ(ν, ν 0 ) such that: Φ ij (ν 0, ν) Φ ij (ν 0, ν) ε ij (ν 0, ν f ), ν [ν 0, ν f ] and each entry Φ ij is approximated by a truncated Chebyshev series: Φij (ν 0, ν) = k n a kt k (ν), The Chebyshev family of polynomials T k (ν) is defined as: T 0 (ν) = 1, T 1 (ν) = ν, T k+2 (ν) = 2νT k+1 (ν) T k (ν), k 0. Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

Thrusters model I Spacecraft thrusters https://commons.wikimedia.org/wiki/file:htv-1_close-up_view.

13 Thrusters model I Spacecraft thrusters Thrusters disposition Instantaneous velocity change: X + (ν i ) = X(ν i ) + [ O3 I 3 ] V (ν i ) X + (ν i ) = X(ν i ) + B(ν i ) V (ν i ), where V (ν i ) = [ V x(ν i ), V y(ν i ), V z(ν i )] T R 3 is the vector representing the velocity change at ν i. Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

14 Thrusters model II Let V be the saturation threshold. The associated saturation constraint is modeled by: The fuel-consumption is given by: V x(ν i ) V, V y(ν i ) V, V z(ν i ) V. J( V ) = N i=1 V (ν i) 1. The state obtained right after a sequence of N velocity corrections is given by: X + (ν N ) = Φ(ν N, ν 1 ) X(ν 1 ) + N k=1 Φ(ν, ν k ) B(ν k ) V (ν k ). Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

15 z Space constraints I Steer the chaser satellite into the rectangular cuboid hovering region: x min x(t) x max, y min y(t) y max, z min z(t) z max, t [t N, t N+1 ] Idea: compute N velocity corrections generating a relative trajectory that remains inside the hovering region during the interval [t N, t N+1 ]. x y t 1 t2 t 3 t N-1 t N+1 t N Steering into the hovering region within N velocity corrections Polynomial approximations of [ x(ν), ȳ(ν), z(ν) ] : [ P x(ν), P T [ ] ( Φ(ν, ȳ(ν), P z(ν)] = I3 O 3 ν1 ) X(ν N 1 ) + k=1 Φ(ν, νk ) B(ν ) k ) V k. Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

16 Space constraints II Recalling the variable change X(t) X(ν): [ x(ν), ȳ(ν), z(ν) ] = (1 + e cos(ν)) [ x(t), y(t), z(t) ]. Let P (ν) be a polynomial satisfying: P (ν) > 0, P (ν) (1 + e cos(ν)) ε(νn, ν N+1 ), ν [ν N, ν N+1 ]. The real positions can then be approximated by: x(ν) P x(ν)/ P (ν), y(ν) P ȳ(ν)/ P (ν), z(ν) P z(ν)/ P (ν). The space constraints can be rewritten as: x min x(t) x max, y min y(t) y max, z min z(t) z max, t [t N, t N+1 ] P x(ν) P (ν)x min 0, P (ν)xmax P x(ν) 0 P y(ν) P (ν)y min 0, P z(ν) P (ν)z min 0, P (ν)ymax P y(ν) 0 P (ν)zmax P z(ν) 0, ν [ν N, ν N+1 ] Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

17 Formulation using polynomial non-negativity constraints Problem (Semi-infinite program) Given X(ν 1 ) R 6, N true anomaly firing instants ν 1 <... < ν N R +, a true anomaly interval [ν N, ν N+1 ] R +, find V [ V, V ] 3N solution of: argmin V s.t. J( V ) P x(ν) P (ν)x min 0, P y(ν) P (ν)y min 0, P z(ν) P (ν)z min 0, P (ν)xmax P x(ν) 0 P (ν)ymax P y(ν) 0 P (ν)zmax P z(ν) 0, ν [ν N, ν N+1 ] (P.1) Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

18 Formulation using polynomial non-negativity constraints The cone of univariate non-negative polynomials on an interval can be described by linear matrix inequalities (LMIs): Definition A symmetric real matrix Y S n(r) is semi-definite positive (Y 0) if, and only if, one of the equivalent statements is satisfied: v R n s.t. v 2 0, v T Y v 0; i {1,..., n}, λ i (Y ) 0. Theorem (Nesterov, 2000, Th. 9 and 10) Let be P (w) = n i=0 p iw i and [a, b] R. Then, P (w) 0, w [a, b] Y 1, Y 2 0 s.t. [p 0,..., p n] T = Λ (Y 1, Y 2 ), where Λ (Y 1, Y 2 ) is a bilinear operator. The sizes of Y1 ans Y 2 and the operator Λ differ for even and odd degrees of P (w). Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

19 Semi-definite program formulation Problem (Semi-infinite program) Given X(ν 1 ) R 6, N true anomaly firing instants ν 1 <... < ν N R +, a true anomaly interval [ν N, ν N+1 ] R +, find V [ V, V ] 3N solution of: argmin V s.t. J( V ) P x(ν) P (ν)x min 0, P y(ν) P (ν)y min 0, P z(ν) P (ν)z min 0, P (ν)xmax P x(ν) 0 P (ν)ymax P y(ν) 0 P (ν)zmax P z(ν) 0, ν [ν N, ν N+1 ] (P.1) Problem (Semi-definite program) Given X(ν 1 ) R 6, N true anomaly firing instants ν 1 <... < ν N R +, a true anomaly interval [ν N, ν N+1 ] R +, find V [ V, V ] 3N, Y 1w, Y 2w 0 solution of: argmin J( V ) V,Y 1w,Y 2w { p xmin = Λ (Y 1xmin, Y 2zmin ) s.t. p ymin = Λ (Y 1ymin, Y 2zmin ) p zmin = Λ (Y 1zmin, Y 2zmin ) p xmax = Λ (Y 1xmax, Y 2zmax ) p ymax = Λ (Y 1ymax, Y 2zmax ) p zmax = Λ (Y 1zmax, Y 2zmax ) (P.2) Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

20 Semi-definite program formulation Problem (Semi-infinite program) Given X(ν 1 ) R 6, N true anomaly firing instants ν 1 <... < ν N R +, a true anomaly interval [ν N, ν N+1 ] R +, find V [ V, V ] 3N solution of: argmin V s.t. J( V ) P x(ν) P (ν)x min 0, P y(ν) P (ν)y min 0, P z(ν) P (ν)z min 0, P (ν)xmax P x(ν) 0 P (ν)ymax P y(ν) 0 P (ν)zmax P z(ν) 0, ν [ν N, ν N+1 ] (P.1) Problem (Semi-definite program) Given X(ν 1 ) R 6, N true anomaly firing instants ν 1 <... < ν N R +, a true anomaly interval [ν N, ν N+1 ] R +, find V [ V, V ] 3N, Y 1w, Y 2w 0 solution of: argmin J( V ) V,Y 1w,Y 2w { p xmin = Λ (Y 1xmin, Y 2zmin ) s.t. p ymin = Λ (Y 1ymin, Y 2zmin ) p zmin = Λ (Y 1zmin, Y 2zmin ) p xmax = Λ (Y 1xmax, Y 2zmax ) p ymax = Λ (Y 1ymax, Y 2zmax ) p zmax = Λ (Y 1zmax, Y 2zmax ) (P.2) Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

21 Overview 1 Orbital rendezvous 2 Objectives 3 Models 4 Results 5 Conclusions Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

22 Simulations I Simulated scenario: Semi-major axis: a = 7011 km Eccentricity: e = 0.4 Initial true anomaly: ν 1 = 0 Number of velocity corrections: N = 3 True anomaly intervals: ν = π/4 Saturation threshold: V = 1 m/s Initial relative state [m, m/s]: [200, 150, 100, 0, 0, 0] [x min, x max, y min, y max, z min, z max] [m]: [50, 150, 25, 25, 25, 25] Degree of RPAs: 5, 7 Solving the problem step-by-step: 1. Compute RPAs of the space-transition matrix in C; 2. Formulate the optimization problem on Matlab via Yalmip (Löfberg, 2004); 3. Solve it using SDP solvers (SeDuMi, sdpt3, Mosek); 4. Evaluate the enclosures via the b4m interval arithmetic toolbox library (Zemke, 1999). Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

23 z z y z Simulations II XY Trajectory XZ Trajectory Initial point Impulses XY Trajectory Initial point Impulses XZ Trajectory x x YZ Trajectory 3D Trajectory 100 Initial point Initial point Impulses Impulses 80 YZ Trajectory 3D Trajectory Bounds y x y Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

24 x x Simulations III Certfied x-trajectory Certfied x-trajectory Trajectory Certificate Bounds Trajectory Certificate Bounds X-trajectory N=5 X-trajectory N=7 Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

25 x x Simulations IV Certfied x-trajectory Certfied x-trajectory Trajectory Certificate Bounds Trajectory Certificate Bounds X-trajectory N=5 X-trajectory N=7 Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

26 z z Simulations V 100 Certfied z-trajectory 100 Certfied z-trajectory 80 Trajectory Certificate Bounds 80 Trajectory Certificate Bounds Z-trajectory N=5 Z-trajectory N=7 Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

27 z z Simulations VI Certfied z-trajectory Trajectory Certificate Bounds Certfied z-trajectory Trajectory Certificate Bounds Z-trajectory N=5 Z-trajectory N=7 Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

28 y y Simulations VII Certfied y-trajectory Certfied y-trajectory Trajectory Certificate Bounds Trajectory Certificate Bounds Y-trajectory N=5 Y-trajectory N=7 Recalling the equations: x (ν) = 2 z (ν) z (ν) = 2 x 3 (ν) + 1+e cos ν z(ν), ȳ (ν) = ȳ(ν) ȳ(ν) = c 1 cos(ν) + c 2 sin(ν) Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

29 Overview 1 Orbital rendezvous 2 Objectives 3 Models 4 Results 5 Conclusions Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

30 Conclusions Summary: RPAs of the solutions of Ẋ(t) = A(t)X(t), X(t 0) = X 0; RPAs of the entries Φ(t, t 0); Polynomial Φ i,j(t, t 0) space constraints are described by non-negativity of polynomials (SIP); Nesterov theorem existence of semi-definite positive matrices satisfying LMIs (SDP); A posteriori certification of the obtained relative trajectory; Certification enclosure depends on the degree of the polynomial approximation. Ongoing work: FPGA-embeddable C version of the MPC strategy for space dedicated microprocessors; RPAs for more complicated dynamics taking into account non-linearities and disturbances. Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

Contributions Publication: P.R. ARANTES GILZ, F. BREHARD, C. GAZZINO, Validated semi-analytical transition matrices for linearized relative spacecraft dynamics via Chebyshev series approximations.

, 2018 Summary: Florent: validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations; Clément: fuel-optimal station keeping of a geostationary spacecraft

31 Contributions Publication: P.R. ARANTES GILZ, F. BREHARD, C. GAZZINO, Validated semi-analytical transition matrices for linearized relative spacecraft dynamics via Chebyshev series approximations. Space Flight Mechanics Meeting, AIAA Science and Technology Forum and Exposition, Jan 2018, Kissimmee, United States. 24p., 2018 Summary: Florent: validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations; Clément: fuel-optimal station keeping of a geostationary spacecraft equipped with a low-thrust propulsion system (taking into account disturbances the oblateness of the Earth, gravitational attraction of Sun and Moon and solar radiation pressure). (Arantes Gilz and Louembet, 2015),(Arantes Gilz et al., 2017) Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

32 References Arantes Gilz, P. R., Joldes, M., Louembet, C., and Camps, F. (2017). Model predictive control for rendezvous hovering phases based on a novel description of constrained trajectories. In IFAC World Congress, pages pp , Toulouse, France. Arantes Gilz, P. R. and Louembet, C. (2015). Predictive control algorithm for spacecraft rendezvous hovering phases. In Control Conference (ECC), 2015 European, pages IEEE. Bréhard, F., Brisebarre, N., and Joldes, M. (2017). Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations. Joldes, M. M. (2011). Rigorous Polynomial Approximations and Applications. Theses, Ecole normale supérieure de lyon - ENS LYON. Löfberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. In Computer Aided Control Systems Design, 2004 IEEE International Symposium on, pages IEEE. Nesterov, Y. (2000). Squared functional systems and optimization problems. In et al., J. F., editor, High performance optimization. Kluwer Academic Publishers. Tschauner, J. (1967). Elliptic orbit rendezvous. AIAA Journal, 5(6): Zemke, J. (1999). b4m: A free interval arithmetic toolbox for matlab. Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

33 Nesterov: polynomials non-negativity LMIs (I) Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

34 Nesterov: polynomials non-negativity LMIs (II) Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

35 Nesterov: polynomials non-negativity LMIs (III) Source: Nesterov results (Nesterov, 2000) on polynomials non-negativity as presented in Deaconu, G. (2013). On the trajectory design, guidance and control for spacecraft rendezvous and proximity operations. PhD thesis, Univ. Toulouse 3 - Paul Sabatier, Toulouse, France. Paulo Ricardo ARANTES GILZ MPC with approximated transition matrices October 24, / 31

Model predictive control for rendezvous hovering phases based on a novel description of constrained trajectories

Model predictive control for rendezvous hovering phases based on a novel description of constrained trajectories Paulo Ricardo ARANTES GILZ 1,2 prarante@laas.fr Mioara JOLDES 1,2 Christophe LOUEMBET 1,2