A hybrid control framework for impulsive control of satellite rendezvous

Transcription

1 A hybrid control framework for impulsive control of satellite rendezvous Luca Zaccarian Joint work with Mirko Brentari, Sofia Urbina, Denis Arzelier, Christophe Louembet LAAS-CNRS and University of Trento Seoul National University, October 7, 27 / 32

2 Introduction Example: the PRISMA mission Figure : The PRISMA mission. Photo courtesy of CNES. Proximity operations (active debris removal, reparation of a spacecraft, refueling, etc). Target/chaser. 2 / 32

3 Reference frames Modeling Orbital reference frames X o V-bar 3D Z i v X i O i Z o R-bar O o Inertial reference frame: ) I = (O i, X i, Y i, Z i Y i Y o 2D LVLH reference frame: ) O o = (O o, X o, Y o, Z o Z i X o V-bar X i Y i Z o R-bar v Y o 3 / 32

4 Modeling Linear translational motion Linear translational equations (Tschauner-Hempel) Linearization Ẋ(t) = A(t) X(t) + B F(t) derived by Tschauner-Hempel dν Denoting: dt = n ( + e cos ν) ( e 2 ) 3/2 }{{} 2 =: k 2 ρ(ν) 2 ρ(ν) ẋ(t) ẏ(t) ż(t) ẍ(t) ÿ(t) z(t) } {{ } Ẋ(t) = ν 2 + 2k 4 ρ(ν) 3 ν 2 ν ν 2 k 4 ρ(ν) 3 } ν k 4 ρ(ν) 3 {{ 2 ν } A(t) x(t) y(t) z(t) ẋ(t) ẏ(t) ż(t) } {{ } X(t) F x propc [ ] m c F y propc I 3 }{{} m c F z B propc } m {{ c } F(t) () 4 / 32

5 Modeling Linear translational motion Linear translational equations (Tschauner-Hempel) Linearization Ẋ(t) = A(t) X(t) + B F(t) derived by Tschauner-Hempel dν Denoting: dt = n ( + e cos ν) ( e 2 ) 3/2 }{{} 2 =: k 2 ρ(ν) 2 ρ(ν) ẋ(t) ẏ(t) ż(t) ẍ(t) ÿ(t) z(t) } {{ } Ẋ(t) = ν 2 + 2k 4 ρ(ν) 3 ν 2 ν ν 2 k 4 ρ(ν) 3 } ν k 4 ρ(ν) 3 {{ 2 ν } A(t) Chemical propulsion impulsive problems x(t) y(t) z(t) ẋ(t) ẏ(t) ż(t) } {{ } X(t) F x propc [ ] m c F y propc I 3 }{{} m c F z B propc } m {{ c } F(t) () Impulsive control input: u(t k ) = V(t k ) = V k = Ẋ = A(t)X [ X + = X I 3 ] u in free motion t + k across impulsive thrusts t k F(t) dt = t + k t k F propc m c dt. 4 / 32

6 Modeling Coordinate transformations Linearized translational equations (Tschauner-Hempel) Transformations of the free dynamics X(t) H(ν) X(ν) T(ν) X(ν) C(ν) ξ(ν) S(ν) ξ(ν) Ẋ(t) = A(t) X(t) H(ν) X(ν) = A(ν) X(ν). From the t-domain to the ν-domain (change of time scale). 2 X(ν) = A(ν) X(ν) T(ν) X(ν) = Ã(ν) X(ν). Provides simple periodic equations governed by true anomaly ν [, 2π] 3 X(ν) = Ã(ν) X(ν) C(ν) ξ(ν) = Ā(ν) ξ(ν). Explicit solution of free motion exists (Yamanaka-Ankersen) It provides a basis for parametrization of constant solutions Representation of periodic trajectories ξ 6 = 4 ξ(ν) = Ā(ν) ξ(ν) S(ν) ξ(ν) = Â ξ(ν). Application of Floquet-Lyapunov theory transformation of LTV periodic system into an equivalent LTI system (with a constant dynamic matrix Â). 5 / 32

7 LTI dynamics derivation Transformation T(ν) Modeling Coordinate transformations X(t) H(ν) X(ν) T(ν) X(ν) C(ν) ξ(ν) S(ν) ξ(ν) Second coordinate transformation T(ν): [ ] ρ(ν) I 3 3 X(ν) = ρ(ν) T I 3 ρ(ν) I 3 X(ν). (2) } {{ } T(ν) Simplified Tschauner-Hempel equations: x = 2 z, ỹ = ỹ, z 3 = + e cos ν z 2 x. (3) State vector: X(ν) = [ x ỹ z x ỹ z ] T. 6 / 32

8 LTI dynamics derivation Transformation C(ν) Modeling Coordinate transformations X(t) H(ν) X(ν) T(ν) X(ν) C(ν) ξ(ν) S(ν) ξ(ν) Third coordinate transformation C(ν): it allows a periodic representation of the trajectories in the original coordinates. c ν s ν s ν c ν 3esν (+ρ) esν (+ρ) ρ ρ(e ξ(ν) = 2 ) e 2 2 ecν 3 e 2 e 3s ν s ν ( + ρ) c ν ρ X(ν). 3(cν +e) cν (+ρ)+e e 2 e 2 s ν ρ 3(3ecν +e2 +2) 3ρ 2 e 2 e 2 3esν ρ e 2 }{{} C(ν) State vector: ξ(ν) = [ ξ ξ 2 ξ 3 ξ 4 ξ 5 ξ 6] T. 7 / 32

9 LTI dynamics derivation Transformation S(ν) Modeling Coordinate transformations X(t) H(ν) X(ν) T(ν) X(ν) C(ν) ξ(ν) S(ν) ξ(ν) Fourth coordinate transformation: based on Floquet-Lyapunov theory. It allows an LTI representation of the dynamics. where σ(ν) := ν 2π T t. σ(ν) ξ(ν) = ( e 2 ) 3/2 ξ(ν), } {{ } S(ν) State vector: ξ(ν) = [ ξ ξ2 ξ3 ξ4 ξ5 ξ6] T. The final model is obtained after the four coordinate changes: ξ(ν) = R(ν) X(ν) := S(ν) C(ν) T(ν) X(ν). 8 / 32

10 Modeling Coordinate transformations Linearized translational equations (Recap) Transformations of the free dynamics X(t) H(ν) X(ν) T(ν) X(ν) C(ν) ξ(ν) S(ν) ξ(ν) Equations of motion are transformed into: { ˆξ = ˆξ in free motion ˆξ + = ˆξ + ˆB(ν)u across impulsive thrusts Proposition. R(ν) is a coordinate change (bounded and with bounded inverse) and motion of X(t) is periodic if and only if ˆξ 6 =. Reference Periodic Motion can be specified in terms of periodic relative trajectory: ξ ref := [ξ ref ξ ref 2 ξ ref 3 ξ ref 4 ξ ref 5 ] T May also define the tracking error: ε = ξ ξ ref. 9 / 32

11 Modeling Examples of Periodic Reference Trajectories Reference periodic trajectories Periodic trajectories Periodic free motion ξ ref = [ ξ ref ξ ref 2 ξ ref 3 ξ ref 4 ξ ref 5 ] T. e =.9 8 e =.8 e = e =.6 e =.5 e =.4 e = e =.2 e =. z (m) z (m) e = y (m) 2 5 x (m) x (m) y (m) 5 System representation: dynamics mismatch between current state and reference orbit: ε = ξ ξ ref. / 32

12 Summary of proposed hybrid context Hybrid dynamical systems Closed-loop dynamics Tracking of ξ ref periodic trajectory chaser inside a tolerance box. System representation: error w.r.t. a desired reference orbit ε = ξ ξ ref. Linear translational model: ε = Â ε, ε + = ε + B(ν) u, ε C (free motion), ε D (across impulsive thrusts). Hybrid dynamical systems (Teel, Goebel, Sanfelice 22) Continuous-time differential equations: free motion dynamics flow set C. Discrete-time difference equations: impulses execution jump set D. Control law amounts to selecting u and D (C = closed complement of D): We propose a dynamical solution depending on timers (controller states). / 32

13 A typical evolution of a hybrid solution Hybrid dynamical systems jumps ε = Â ξ, ε C, ε = ε + B(ν) u, ε D. j + 2 ε(ν 2, j + 2) j + ε(ν, j + ) Flow during ν Jump Û 2 ε(ν 2, j + ) Jump Û j ε(ν, j) ν ν 2 time ν 2 / 32

14 Hybrid dynamical systems Timers are the states of the dynamic controller Satellite dynamics periodic of period 2π. Timer ν used to track the periodic time-varying nature of the satellite dynamics. ν + = each time it reaches 2π. Thrusters firings instants. Timer τ used to capture the information about how long we need to wait until the next impulsive control action. When τ = thrusters are fired by u = γ u( ε, ν) and next firing instant is selected τ + = γ τ ( ε, ν) D is now selected. Design problem reduced to: choose γ u and γ τ 3 / 32

15 Hybrid dynamical systems Hybrid closed-loop with controller states FLOW DYNAMICS JUMP DYNAMICS ( ε, ν, τ) C ( ε, ν, τ) D ν ( ε, ν, τ) D τ ε = Â ε, ν =, τ =, ε + = ε, ν + =, τ + = τ, ε + = ε + B(ν) γ u( ε, ν), ν + = ν, τ + = γ τ ( ε, ν). C := (R 6 [, 2π] [, 2π]) \ D D ν := R 6 {2π} [, 2π] D τ := R 6 [, 2π] {} Timer Action Set ν Captures periodicity of motion. Reset to when it reaches 2π [, 2π] τ Impulsive control actions sequencing. How long need to wait until the next impulse [, 2π] 4 / 32

16 Hybrid design problem statement Hybrid problem statement Objective We wish to (partially) meet the following specifications: } A. The chaser departs from an initial state and converges to a final state { ξ = ref ξ (reference periodic trajectory) which is uniformly globally asymptotically stable. B. The chaser tracks the reference periodic trajectory during the inspection. C. The cost J in (??) is minimized over the full maneuver: J = N V k = k= N ( vx k + v y k + v z k ). k= Goal: Design state feedback impulsive guidance laws by selecting: the firing instants ν k, k N given by the trigger law γ τ ( ε, ν); the corresponding input law γ u( ε, ν). 5 / 32

17 Hybrid guidance laws Proposed guidance laws Controller Trigger law γ τ ( ε, ν) Input law u = γ u( ε, ν) #: uniform sampling normminimizing γ τ ( ε, ν) = ν Arrival at ξ ref by means of periodic trajectories. Minimize the Euclidean norm of the error ε. #2: u.s. biimpulsive γ τ ( ε, ν) = ν Tracking of ξ ref after 2 control actions. #3: non-u.s. biimpulsive γ τ ( ε, ν) = arg min J ν [,2π] Tracking of ξ ref after 2 control actions. Minimize the cost J. #4: u.s. triimpulsive γ τ ( ε, ν) = ν Arrival at ξ ref by means of periodic trajectories. Tracking of ξ ref after 3 control actions. 6 / 32

18 A closer look at the dynamic matrices Hybrid guidance laws Hybrid equations: ε =  ε, ε C, ε + = ε + B(ν) u, ε D. Matrices  and ˆB have useful structure: Â=( e 2 ) 3/2, B(ν)= ( e 2 )s ν ( e 2 )c ν 3σρ 2 3σeρs ν e(+ρ)s ν ( e 2 ) 3/2 ( e 2 ) 3/2 ρ2 +ρ 2 k 2 ρ( e 2 ) ( e 2 )(+ρ)s ν ( e 2 )ρc ν (+ρ)c ν+e ρs ν 3ρ 2 3eρs ν where c ν := cos(ν) s ν := sin(ν) 7 / 32

19 Hybrid guidance laws Uniform sampling and norm-minimizing Control law Theorem. The reference periodic orbit ξ ref is stable (attractivity not guaranteed). The chaser moves through periodic trajectories ε + 6 =, while the rest of the states are minimized. Jump equation: [ ε + 5 ] = [ ε 5 ε 6 ] + B(ν) u u = u 6 + u 6 u 6 in charge of ε + 6 =. u 6 must min ε 5 while having no influence over u 6. Computation of the input law u = γ u( ε, ν): γ u ( ε, ν) = b 6(ν) T b 6(ν) 2 ε6 B 6 (ν) ( ) L ( B(ν) B 6 (ν) ε B(ν) b 6(ν) T b 6(ν) 2 ε6 ). 8 / 32

20 Uniform sampling bi-impulsive Control law 2 Hybrid guidance laws Theorem. The reference periodic orbit ξ ref is stable and attractive. The chaser arrives at the reference ξ ref in 2 control impulses separated by an angle ν: jumps ε(ν + ν, j + 2) =. (4) j + 2 ε(ν 2, j + 2) j + ε(ν, j + ) Flow during ν Jump Û 2 ε(ν 2, j + ) Jump Û j ε(ν, j) ν ν 2 time ν 9 / 32

21 Hybrid guidance laws jumps j + 2 ε(ν 2, j + 2) j + ε(ν, j + ) Flow during ν Jump Û 2 ε(ν 2, j + ) Jump Û j ε(ν, j) ν ν 2 time ν Thrust u at time ν : ε(ν, j + ) = ε(ν, j) + B(ν ) u. Flow during ν: ε(ν + ν, j + ) = Φ( ν) ] [ ε(ν, j) + B(ν ) u. Thrust u 2 at time ν + ν: ε(ν + ν, j + 2) = Φ( ν) [ ε(ν, j) + B(ν ) u + Φ( ν) B(ν ] + ν) u 2. 2 / 32

22 Uniform sampling bi-impulsive Control law 2 Hybrid guidance laws Theorem. The reference periodic orbit ξ ref is stable and attractive. This results from forward invariance and global uniform attractivity. The chaser arrives at the reference ξ ref in 2 control impulses separated by an angle ν: ε(ν + ν, j + 2) =. (5) Computation of the input law γ u( ε, ν) = u: ] γ u( ε, ν) = [ I ] [ B(ν) Φ( ν) B(ν + ν) } {{ } M(ν, ν) ε. (6) Conjecture: (no proof) Matrix M is invertible ν kπ, k Z. det(m(ν, ν)) ν Figure 3 : Determinant of M. 2 / 32

23 Non-uniform sampling bi-impulsive Control law 3 Hybrid guidance laws Theorem. The reference periodic orbit ξ ref is stable and attractive. This results from forward invariance and global uniform attractivity. The chaser arrives at the reference ξ ref in 2 control impulses separated by an angle ν: ε(ν + ν, j + 2) =. J (m/s) 5 The input law u = γ u( ε, ν) is the same as for the previous controller: J (m/s) 5 γ u( ε, ν) = [ I ] M(ν, ν ) ε. The trigger function is evaluated at ν kπ, k Z, which minimizes the cost: J (m/s) ν (deg) ν = γ τ ( ε, ν) = arg min [I ] M(ν, ν) ε = arg min J. ν [,2π] ν [,2π] The corresponding minimization is not (not even locally) convex: hard to implement. 22 / 32

24 Uniform sampling tri-impulsive Control law 4 Hybrid guidance laws Theorem. The reference periodic orbit ξ ref is stable and attractive. This results from forward invariance and global uniform attractivity. Take advantage of the decoupling in matrices  and B(ν) such that the control design is separated in two completely decoupled problems corresponding to the following partitions: ε,2 ε 3 5 ε 6 ε +,2 ε ε + 6 = = [f (e) 2 ] T } 2 3 {{ }  ε,2 ε 3 5 ε 6 + [ B α(ν) ] 4 B β (ν) 4 2 }{{} B(ν) ε,2 ε 3 5 ε 6, [ vb V ac ]. (7) 23 / 32

25 Guidance law #4: tri-impulsive Hybrid guidance laws  = ( e 2 ) 3/2, ( e 2 ) ρ sin(ν) ( e 2 ) ρ cos(ν) 3σρ 3 3σeρ 2 sin(ν) e( + ρ)ρ sin(ν) B(ν) = k 2 ρ 2 ( e 2 ( e 2 ) 3/2 ( e 2 ) 3/2 ρ3 + ρ 2 + 2ρ ) }{{} ( e 2 )( + ρ)ρ sin(ν) ( e 2 )ρ 2 cos(ν) K ( + ρ)ρ cos(ν) + eρ ρ 2 sin(ν) 3ρ 3 3eρ 2 sin(ν) 24 / 32

26 Periodic tri-impulsive Control law 4 Hybrid guidance laws The chaser moves through periodic trajectories ε + 6 =. ε +,2 = ε,2 + B α(ν) u b, [ ε ] = [ ε 3 5 ε 6 ] + B β (ν) u ac u ac = u 6 ac + u 6 ac. (8) u b can be computed as in the bi-impulsive method. In 2 impulses ε,2 =. u 6 ac in charge of ε + 6 =. In impulse ε 6 =. u 6 ac must have no influence over u 6 ac. In 3 impulses ε 3 5 =. 25 / 32

27 Hybrid guidance laws Actions u b and u ac computed separately Computation of input law γ u, (,2) ( ε, ν) = v b : State ε,2 becomes after 2 impulses: ε,2(ν + ν, j + 2) =. Propagation of the hybrid dynamics along the 2 impulses, executed at ν and ν + ν: γ u, (,2) ( ε, ν) = [ ] [ B,2(ν) B,2(ν + ν) ε,2. (9) ] Computation of input law γ u, (3 6) ( ε, ν) = V ac: State ε 6 becomes and remains after each impulse: ε + 6 =. Design of a control component u 6 ac that ensures periodicity ε + 6 = while u 6 ac has no influence over ε 6. Propagation of the hybrid dynamics along 3 impulses, executed at ν = ν, ν 2 = ν + ν and ν 3 = ν + 2 ν: γ u, (3 6) ( ε, ν) = ˆb ] 6(ν) ˆb ˆε6 B 6 (ν)[ ] [ B xz(ν) B xz(ν + ν) B xz(ν + 2 ν) ε3 6 6(ν) 26 / 32

28 Periodic bi-impulsive Control law 4 Hybrid guidance laws Theorem. The reference periodic orbit ξ ref is stable and attractive. The proof uses forward invariance and global attractivity. The chaser arrives at the reference ξ ref in 3 control impulses separated two angles ν and ν 2: Computation of the input law u = γ u( ε, ν) when ν = ν 2 = ν: [ ] [ ] γ u ( ε, ν) = ε(ν 3, j + 3) =. () γ u, ( 2) ( ε, ν) + γ u, (3 6) ( ε, ν) () Proposition. only if ν + ν 2 2kπ, k Z. Matrix M is invertible if and 27 / 32

29 Simulations Matlab-Simulink Hybrid simulations Simulators Approach and inspection phases for the PRISMA mission. Linear vs nonlinear simulations (J 2, atmospheric drag, saturations, dead zones). Performance indices: fuel consumption & convergence time. Tolerance box B z (m) Target satellite ztol X B ξ ref y tol x tol y (m) x (m) 28 / 32

30 Hybrid simulations Initial state: X chaser = [ ] T. Target position: X target = [ ] T. Tolerance box centered at X tol = [ 5 5] T of widths [ ] T. z (m) 4 2 Linear sim, control Nonlinear sim, control Linear sim, control 2 Nonlinear sim, control 2 Linear sim, control 4 Nonlinear sim, control 4 Initial position Target satellite Reference orbit 2 4 4, 2, y (m) 2,,5, 5 5, 4, x (m) Norm-minimizing & tri-impulsive: periodic trajectories safe. 29 / 32

31 Hybrid simulations.5 Saturated impulses.5 Saturated impulses.5 Saturated impulses v (m/s).5 v (m/s).5 v (m/s).5 Error profile Linear sim Nonlinear sim Tc, Linear sim Tc, Nonlinear sim Error profile Linear sim Nonlinear sim Tc, Linear sim Tc, Nonlinear sim Error profile Linear sim Nonlinear sim Tc, Linear sim Tc, Nonlinear sim Cost (m/s) 3 2 Cost (m/s) Cost (m/s) Error profile Linear sim Nonlinear sim Tc, Linear sim Tc, Nonlinear sim Error profile Linear sim Nonlinear sim Tc, Linear sim Tc, Nonlinear sim Error profile Linear sim Nonlinear sim Tc, Linear sim Tc, Nonlinear sim / 32

32 Hybrid simulations.5 Saturated impulses.5 Saturated impulses.5 Saturated impulses v (m/s).5 v (m/s).5 v (m/s).5 Error profile Linear sim Nonlinear sim Tc, Linear sim Tc, Nonlinear sim Error profile Linear sim Nonlinear sim Tc, Linear sim Tc, Nonlinear sim Error profile Linear sim Nonlinear sim Tc, Linear sim Tc, Nonlinear sim Consumption (m/s) for the different uniform sampling guidance laws: #: norm-min #2: 2-imp #4: 3-imp LIN NL LIN NL LIN NL * * Convergence time (nb. orbits) for the different uniform sampling guidance laws: #: norm-min #2: 2-imp #4: 3-imp LIN NL LIN NL LIN NL * * / 32

33 Conclusions & perspectives Conclusions & perspectives Conclusions Floquet-Lyapunov theory successfully applied to rendez-vous dynamics Suitable coordinate transformations provide simplified dynamics. Suggestive hybrid framework enables the use of intuitive laws (timers, periodicity). Fundamental theorems with compact attractors and data regularity allow to conclude stability from (uniform) convergence and invariance. Robustness of asymptotic stability is confirmed by perturbed simulations. Satellite rendez-vous control is an example where hybrid dynamics is useful: Perspectives Physical meaning of the parameters after the coordinate transformations: all constant except one link to relative orbital parameters? Address saturation more directly Design sub-optimal laws based on simple hybrid mechanisms (estimates of J?). 32 / 32