Interior point algorithm for the optimization of a space shuttle re-entry trajectory

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1 Interior point algorithm for the optimization of a space shuttle re-entry trajectory. Julien Laurent-Varin, CNES-INRIA-ONERA Common work with Julien Laurent-Varin J. F. Bonnans INRIA, N. Bérend ONERA, M. Haddou U. Orléans, C. Talbot CNES julien.laurent-varin@inria.fr

2 Context Ultimate aims of this work Survey Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 2

3 Context Context Ultimate aims of this work Survey A typical mission Orbital station Rendez-vous with station End of mission Booster separation Orbiter et Booster on ground Retour booster A challenge for optimal control software Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 3

4 Ultimate aims of this work Context Ultimate aims of this work Survey Efficient method for solving optimal control problems with path constraints Compute the ascent trajectory of classical launcher Compute reentry trajectory of an orbitor Compute the whole trajectory of a future space launcher (with possible rendez-vous) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 4

5 Survey Context Ultimate aims of this work Survey [1] J.T. Betts. Survey of numerical methods for trajectory optimization. AIAA J. of Guidance, Control and Dynamics, 21: , [2] J.T. Betts. Practical methods for optimal control using nonlinear programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 5

6 Problem descrition (P) Discretized Problem (DP) Optimal conditions of (DP) Optimal conditions of (P) Commutative diagram Summary Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 6

7 Problem descrition (P) Consider the optimal control problem (P) : Problem descrition (P) Discretized Problem (DP) Optimal conditions of (DP) Optimal conditions of (P) Commutative diagram Summary Min Φ(y(T )); ẏ(t) = f(u(t), y(t)), t [0, T ]; y(0) = y 0. (P ) Discretize dynamic with Runge-Kutta method Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 7

8 Discretized Problem (DP) Problem descrition (P) Discretized Problem (DP) Optimal conditions of (DP) Optimal conditions of (P) Commutative diagram Summary Min Φ(y N ); y k+1 = y k + h s k i=1 b if(u ki, y ki ), y ki = y k + h s k j=1 a ijf(u kj, y kj ), y 0 = y 0. (DP ) [1] W. Hager. Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math., 87(2): , Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 8

9 Optimal conditions of (DP) Problem descrition (P) Discretized Problem (DP) Optimal conditions of (DP) Optimal conditions of (P) Commutative diagram Summary y k+1 = y k + h k s i=1 b if(u ki, y ki ), y ki = y k + h k s j=1 a ijf(u kj, y kj ), p k+1 = p k + h k s i=1 ˆb i f y (y ki, u ki ) T p ki, p ki = p k + h k s j=1 âijf y (y kj, u kj ) T p kj, 0 = f u (y k, u k ) T p k, 0 = f u (y ki, u ki ) T p ki, y 0 = y 0, p N = Φ (y N ). Where : ˆb = b and â ij = b ib j b j a ij b i for all i, j. Partitioned RK methods (DOC) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 9

10 Optimal conditions of (P) Problem descrition (P) Discretized Problem (DP) Optimal conditions of (DP) Optimal conditions of (P) Commutative diagram Summary ẏ(t) = f(u(t), y(t)), ṗ(t) = f y (u(t), y(t)) T p(t), p(t ) = Φ (y(t )), y(0) = y 0, 0 = f u (u(t), y(t)) T p(t). (OC) Previous scheme = (partitioned) symplectic schemes for (OC) deeply described in [1] [1] E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration. Springer-Verlag, Berlin, Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 10

11 Commutative diagram Problem descrition (P) Discretized Problem (DP) Optimal conditions of (DP) Optimal conditions of (P) Commutative diagram Summary optimality conditions (P ) (OC) discretization (DP ) optimality conditions discretization (DOC) (D) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 11

12 Summary Problem descrition (P) Discretized Problem (DP) Optimal conditions of (DP) Optimal conditions of (P) Commutative diagram Summary Complete error analysis for : unconstrained problems with strongly convex Hamiltonians Existing theory for : control constraint order one state constraints (Dontchev, Hager) Open problems : Singular arcs high order state constraints Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 12

13 - I - II - III Symplectic schemes - order 4 Symplectic schemes - order 4 Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 13

14 - I - I - II - III Symplectic schemes - order 4 Symplectic schemes - order 4 For partitionned Runge-Kutta Schemes, we have order conditions based on bi-colored rooted tree. For example : Bi-coloured tree t φ(t) = 1/γ(t) s k i,j,k,l=1 j i l b i â ij a jk a jl = 1 12 Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 14

15 - II - I - II - III Symplectic schemes - order 4 Symplectic schemes - order 4 But our partitionned schemes is particular and have the following propertise : (1) ˆbi = b i â ij = b j b j b i a ji i = 1,..., s i = 1,..., s j = 1,..., s This propertise leads to computation that can be interpreted in term of graph computation. Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 15

16 - III - I - II - III Symplectic schemes - order 4 Symplectic schemes - order 4 Bi-coloured tree t φ(t) = 1/γ(t) φ(t) = φ(t 1 ) φ(t 2 ) t 1 t 2 k l j i s b i â ij a jk a jl = 1 12 s s b i b j a jk a jl b j a ji a jk a jl i,j,k,l=1 i,j,k,l=1 i,j,k,l=1 k l k l j j i i Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 16

17 Symplectic schemes - order 4 - I - II - III Symplectic schemes - order 4 Symplectic schemes - order 4 Graph Condition Graph Condition 1 b k a lk d k d l = 1 8 bi b k a ik c i d k = 5 24 c 2 j d j = 1 12 ajk d j c k = 1 24 bi a ij c i c j = 1 8 bi c 3 i = b k c k d 2 k = b 2 l d 3 l = 1 4 Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 17

18 Symplectic schemes - order 4 In the previous table, we use the usual notations d j = i b i a ij - I - II - III Symplectic schemes - order 4 Symplectic schemes - order 4 and c i = j a ij. Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 18

19 History Basic idea For our problem For our problem Combination with mesh refinement Sparsity structure Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 19

20 History History Basic idea For our problem For our problem Combination with mesh refinement Sparsity structure [1] A.V. Fiacco and G.P. McCormick. Nonlinear programming. John Wiley and Sons, Inc., New York-London-Sydney, [2] L. G. Hačijan. A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR, 244(5): , [3] N. Karmarkar. A new polynomial-time algorithm for linear programming. Combinatorica, 4(4): , Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 20

21 Basic idea Transformation of constrained problem (forget equality constraints in order to simplify presentation) History Basic idea For our problem For our problem Combination with mesh refinement Sparsity structure Logarithmic penalty Min f(x); g(x) 0 (P ) Min f(x) ε ln g(x) (P ε ) Unconstrained problem solved by Newton steps on optimality conditions, with basic safeguards Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 21

22 For our problem Min T 0 l(u(t), y(t), u p, t)dt + l f (y(t ), u p, T ); History Basic idea For our problem ẏ(t) = f(u(t), y(t), u p, t), t [0, T ]; For our problem Combination with mesh refinement Sparsity structure y i (0) = y 0 i i I 0, y i (T ) = y T i i I T a g(u(t), y(t), u p, t) b, t [0, T ]. (P ) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 22

23 For our problem History Basic idea For our problem For our problem Combination with mesh refinement Sparsity structure With Min T 0 l ε (u(t), y(t), u p, t)dt + l f (y(t ), u p, T ); ẏ(t) = f(u(t), y(t), u p, t), t [0, T ]; y i (0) = yi 0 i I 0, y i (T ) = yi T i I T. (P ε ) l ε (u, y, u p, t) := l(u, y, u p, t) ε X i [log(g i (u, y, u p, t) a i ) + log(b i g i (u, y, u p, t)) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 23

24 Combination with mesh refinement We must have ε 0 (IP parameter) and E 0 (Integration error) History Basic idea For our problem For our problem Combination with mesh refinement Sparsity structure Good News : Both can be reduced simultaneously : interpolate values at additional points. Our (somewhat arbitrary) choice : For a fixed ε, we refine mesh until E cε; c = 1/10 in our tests. Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 24

25 Sparsity structure History Basic idea For our problem For our problem Combination with mesh refinement Sparsity structure A C B D Band matrix: distributed variables, band QR available Full pieces: static parameters Solve by elimination of distributed variables Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 25

26 Modelisation of error Optimal refinement problem Maximal gain index Resolution of (ORP) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 26

27 Modelisation of error Modelisation of error Optimal refinement problem Maximal gain index Resolution of (ORP) k : index of time step k : error model (valid for h k small) estimated by variable order (symplectic) schemes. Adding (q k 1) points in k th step estimate of error becomes : e k q p k e k = C k h p+1 Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 27

28 Optimal refinement problem Modelisation of error Optimal refinement problem Maximal gain index Resolution of (ORP) Reach a specific error estimation Minimise the number of added points Integer programming problem with a single nonlinear constraints Min q N N N q k ; k=1 N k=1 e k q p k E (ORP ) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 28

29 Maximal gain index Modelisation of error Optimal refinement problem Maximal gain index Resolution of (ORP) Definition 1. Maximal marginal gain g and maximal gain index k g for which the maximum error reduction is obtained by adding only one point : (2) (3) g(q) := max k k g := argmax k e k ( 1/q p k 1/(q k + 1) p) e k ( 1/q p k 1/(q k + 1) p) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 29

30 Resolution of (ORP) Modelisation of error Optimal refinement problem Maximal gain index Resolution of (ORP) Algorithm 1. AORP For k = 1,..., N do q k := 1. End for While N k=1 e k/q p k > E do Compute k g, the maximal gain index. q kg := q kg + 1. End While This algorithm solves (ORP) in O(((E 0 /E) p + 1)N log N) operations Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 30

31 Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 31

32 Control variables µ Control variables α State variables Dynamic I Dynamic II γ Cost Constraints Result without heating constraints Result with heating constraints Result (state) Bank angle µ V Flightpath angle (γ) Angle of attack (α) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 32

33 State variables Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints z altitude V velocity λ longitude γ flightpath angle φ latitude ψ azimuth Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 33

34 Dynamic I ż = V sin γ Control variables State variables Dynamic I λ = V cos γ sin ψ (z + R e ) cos φ Dynamic II Cost Constraints Result without heating constraints φ = V cos γ cos ψ z + R e Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 34

35 Dynamic II V = D m g sin γ Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints γ = L cos µ mv + ψ = L sin µ mv cos γ + ( V 2 ) cos γ g z + R e V V z + R e cos γ sin ψ tan φ Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 35

36 Cost Max crossrange (final latitude) J := φ(t ) Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 36

37 Constraints Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Control constraints : { 0 α µ 0 Mixed control-state contraint (Heating flux) : with q(α) ρv 3.07 Q max q(α) = c 0 + c 1 α + c 2 α 2 + c 3 α 3 Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 37

38 Result without heating constraints Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints ε Integration error Time steps Newton steps Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 38

39 Result (control) Angle of attack (deg) Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Bank angle (deg) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 39

40 Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 40

41 Result with heating constraints We add the constraint of heating and we compare the results Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 41

42 Result (control) Angle of attack (deg) Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Bank angle (deg) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 42

43 Result (state) Altitude (ft) Longitude (deg) Latitude (deg) Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Velocity (ft/s) Flight path angle (deg) Azimute (deg) Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 43

44 Result (constraint:heating) 180 Heating 160 Control variables State variables Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 44

45 3D view: no heating constraints Control variables State variables 26 Dynamic I Dynamic II Cost Constraints Result without heating constraints Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints h/10^ theta phi Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 45

46 3D view: heating constraints Control variables State variables Dynamic I 26 Dynamic II Cost Constraints Result without heating constraints h/10^ Result with heating constraints Result (state) Result (constraint:heating) 3D view: no heating constraints 3D view: heating constraints theta phi 40 Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 46

47 Scenario graph Edges: variables A view after discretization Edges: cost and constraints Vertices: variables Vertices: cost and constraints Cost of factorization and solve Perspectives Some references Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 47

48 Scenario graph Scenario graph Edges: variables A view after discretization Edges: cost and constraints Vertices: variables Vertices: cost and constraints Cost of factorization and solve Perspectives Some references V set of vertices E set of edges Edges: Distributed in time cost and constraints Vertices: Junction conditions, cost and constraints Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 48

49 Edges: variables Edge E e = (i, j) y(t) IR n e : state u(t) IR m e : control π IR r e : static optimization parameters Scenario graph Edges: variables A view after discretization Edges: cost and constraints Vertices: variables Vertices: cost and constraints Cost of factorization and solve Perspectives Some references Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 49

50 A view after discretization Scenario graph Edges: variables A view after discretization Edges: cost and constraints Vertices: variables Vertices: cost and constraints Cost of factorization and solve Perspectives Some references Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 50

51 Edges: cost and constraints l e (t, y(t), u(t), π): Distributed cost ẏ(t) = f e (t, y(t), u(t), π): Dynamics g e (t, y(t), u(t), π) 0: Distributed constraints Scenario graph Edges: variables A view after discretization Edges: cost and constraints Vertices: variables Vertices: cost and constraints Cost of factorization and solve Perspectives Some references The value function over arc = integral of cost function. Feasibility = dynamics are satisfied and distributed constraints hold. Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 51

52 Vertices: variables Vertice j V Variables z j IR n j Includes copy of boundary (initial of final) state y B e and parameters π e of connected edges Possibility of other variables Scenario graph Edges: variables A view after discretization Edges: cost and constraints Vertices: variables Vertices: cost and constraints Cost of factorization and solve Perspectives Some references Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 52

53 Vertices: cost and constraints l i (z): Cost y B e = z e : Copy constraints g i (z)) = 0: Equality constraints g i (z)) 0: Inequality constraints Scenario graph Edges: variables A view after discretization Edges: cost and constraints Vertices: variables Vertices: cost and constraints Cost of factorization and solve Perspectives Some references Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 53

54 Cost of factorization and solve Scenario graph Edges: variables A view after discretization Edges: cost and constraints Vertices: variables Vertices: cost and constraints Cost of factorization and solve Perspectives Some references Arcs: O(n 2 N T ) - n number of state variables - N T average number of time steps per arc Vertices (junctions): O(n) size linear systems - Recursive elimination, starting from leaves - O(n 3 ) operations for each (elimination of) vertex Total number of operations: n 2 O( E N T + n V ) Seems to be the least possible number! Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 54

55 Perspectives Scenario graph Edges: variables A view after discretization Edges: cost and constraints Vertices: variables Vertices: cost and constraints Cost of factorization and solve Perspectives Some references Implementation of multi-arc methodology Test on various examples Analysis of convergence (difficult) - Vanishing parameter ε - Mesh refinement Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 55

56 Some references Scenario graph Edges: variables A view after discretization Edges: cost and constraints Vertices: variables Vertices: cost and constraints Cost of factorization and solve Perspectives Some references [1] J. Laurent-Varin, N. Bérend, F. Bonnans, M. Haddou, and C. Talbot. On the refinement of discretization for optimal control problems. 16 th IFAC SYMPOSIUM Automatic Control in Aerospace, june, St. Petersburg, Russia. [2] J Laurent-Varin and F Bonnans. Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control. Technical Report RR 5398, INRIA, Julien Laurent-Varin 7 th French-Latin American Congress on Applied Mathematics - 56

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