Asteroid Rendezvous Missions

Transcription

1 Asteroid Rendezvous Missions An Application of Optimal Control G. Patterson, G. Picot, and S. Brelsford University of Hawai`i at Manoa 25 June, 215 1/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

2 Missions to Asteroids Launch Mission Time (years) v (m/s) 1978 ISEE Galileo NEAR Cassini Deep Space Stardust Hayabusa Rosetta DIXI Dawn Hayabusa Osiris-Rex ARM 4. 2/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

3 Missions to Asteroids Asteroid Perigee (LD) Diameter Ida km Mathilde km Gaspra km Borrelly km Halley km Churyumov km Braille km Eros km Itokawa km 26RH km z (AU) y (AU) x (AU) 2 4 3/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

4 Granvik et al. [211] Database of 16,923 simulated minimoons At least one in orbit at any time (1-meter diameter) 1 2 z (LD).5.15 z (LD) 1 1 z (LD) y (LD) x (LD) y (LD) 2 x (LD) y (LD) x (LD) z (LD) z (LD) 1 1 z (LD) y (LD) y (LD) x (LD) 6 y (LD) x (LD) 4/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions 4 x (LD) 2

5 Asteroid 26RH 12 z (LD) September 27 4 y (LD) x (LD) 3 4 April 26 Defn. temporary capture : 1. the geocentric Keplerian energy E < 2. the geocentric distance is less than three Hill radii.3 AU. 3. at least one full revolution around Earth in co-rotating frame 5 5/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

6 Paths to solutions 6/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

7 Modern optimal control theory : the Pontryagin's Maximum Principle Consider a general optimal control problem ẋ(t) = f (t, x(t), u(t)) tf min u(.) U t f (t, x(t), u(t))dt + g(t f, x f ) x(t ) = x M M, x(t f ) = x f M 1 M In 1958, the Russian mathematician Lev Pontryagin stated a fundamental necessary rst order optimality condition, known as the Pontryagin's Maximum Principle. This result generalizes the Euler-Lagrange equations from the theory of calculus of variations. 7/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

8 Modern optimal control theory : the Pontryagin's Maximum Principle If u is optimal on [, t f ] then there exists p T x M and p R such that (p, p) (, ) and almost everywhere in [t, t f ] there holds z(t) = (x(t), p(t)) T M is solution to the pseudo-hamilonian system where ẋ(t) = H p (x(t), p, p(t), u(t)), ṗ(t) = H q (x(t), p, p(t), u(t)) H(x, p, p, u) = p f (x, u)+ < p, f (x, u) > ; maximization condition H(x(t), p, p(t), u(t)) = max v U N H(x(t), p, p(t), v) ; transversality condition p() T x() M and p(t f ) p g t (t f, x(t f )) T x(tf )M 1). /5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

9 Modern optimal control theory : the Pontryagin's Maximum Principle If u is optimal on [, t f ] then there exists p T x M and p R such that (p, p) (, ) and almost everywhere in [t, t f ] there holds z(t) = (x(t), p(t)) T M is solution to the pseudo-hamilonian system where ẋ(t) = H p (x(t), p, p(t), u(t)), ṗ(t) = H q (x(t), p, p(t), u(t)) H(x, p, p, u) = p f (x, u)+ < p, f (x, u) > ; maximization condition H(x(t), p, p(t), u(t)) = max v U N H(x(t), p, p(t), v) ; transversality condition p() T x() M and p(t f ) p g t (t f, x(t f )) T x(tf )M 1). A triplet (x, p, u) satisfying these 3 conditions is called an extremal solution. /5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

10 Modern optimal control theory : the Pontryagin's Maximum Principle Locally U = R n H u =. Assumption : The quadratic form 2 H u 2 extremal (x(t), p(t), u(t)). is negative denite along the Implicit fonction theorem In a neighborhood of u, extremal controls are feedback controls i.e smooth functions u r (t) = u r (x(t), p(t)) extremal curves are pairs (x(t), p(t)) solutions of the true Hamiltonian system ẋ(t) = Hr (x(t), p(t)), p ṗ(t) = Hr (x(t), p(t)) x where H r is the true Hamiltonian function dened by H r (x, p) = H(x, p, u r (x, p)). 9/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

11 Geometric conjugate times Question : How can we verify if an extremal z(t) = (x(t), p(t)) provides a solution to the optimal control problem? /5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

12 Geometric conjugate times Question : How can we verify if an extremal z(t) = (x(t), p(t)) provides a solution to the optimal control problem? Use a second order optimality condition /5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

13 Geometric conjugate times Question : How can we verify if an extremal z(t) = (x(t), p(t)) provides a solution to the optimal control problem? Use a second order optimality condition The variational equation δz(t) = d H r (z).δz(t) is called the Jacobi equation along z. One calls a Jacobi eld a nontrivial solution J(t) of the Jacobi equation along z. t is said to be vertical at time t if dπ x(z(t)).j(t) =. /5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

14 Geometric conjugate times Question : How can we verify if an extremal z(t) = (x(t), p(t)) provides a solution to the optimal control problem? Use a second order optimality condition The variational equation δz(t) = d H r (z).δz(t) is called the Jacobi equation along z. One calls a Jacobi eld a nontrivial solution J(t) of the Jacobi equation along z. t is said to be vertical at time t if dπ x(z(t)).j(t) =. A time t c is said to be geometrically conjugate if there exists a Jacobi eld vertical at and t c. In which case, x(t c), is said to be conjugate to x(). /5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

15 Geometric conjugate times Denote exp t( H r ) the ow of the Hamiltonian vectoreld H r. One denes the exponential mapping by exp x,t(p()) Π x(z(t, z )) = x(t, q, p ) where z(t, z ), with z() = (x(), p()) is the trajectory of H such that z(, z ) = z. 1/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

16 Geometric conjugate times Denote exp t( H r ) the ow of the Hamiltonian vectoreld H r. One denes the exponential mapping by exp x,t(p()) Π x(z(t, z )) = x(t, q, p ) where z(t, z ), with z() = (x(), p()) is the trajectory of H such that z(, z ) = z. Proposition : Let x M, L = T x M and L t = exp t( H r )(L ). Then L t is a Lagrangian submanifold of T M whose tangent space is spanned by Jacobi elds starting from L. Moreover q(t c) is geometrically conjugate to x if and only if exp x,t c is not a immersion at p. 1/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

17 Geometric conjugate times Denote exp t( H r ) the ow of the Hamiltonian vectoreld H r. One denes the exponential mapping by exp x,t(p()) Π x(z(t, z )) = x(t, q, p ) where z(t, z ), with z() = (x(), p()) is the trajectory of H such that z(, z ) = z. Proposition : Let x M, L = T x M and L t = exp t( H r )(L ). Then L t is a Lagrangian submanifold of T M whose tangent space is spanned by Jacobi elds starting from L. Moreover q(t c) is geometrically conjugate to x if and only if exp x,t c is not a immersion at p. calculating a conjugate point is equivalent to verifying a rank condition. 1/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

18 Second order optimality condition Assume the strong regularity condition : (S) the control u is of corank 1 on every subinterval of [, t f ]. 2/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

19 Second order optimality condition Assume the strong regularity condition : (S) the control u is of corank 1 on every subinterval of [, t f ]. Theorem : Let tc 1 be the rst conjugate time along z. The trajectory q(.) is locally optimal on [, tc 1 ) in L topology ; if t > tc 1 then x(.) is not locally optimal on [, t]. 2/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

20 Application to space Earth-Moon space transfers Objective : use optimal control theory to compute optimal space transfers in the Earth-Moon system time-minimal space transfers energy-minimal space transfers First question : How to model the motion of a spacecraft in the Earth-Moon system? Neglect the inuences of other planets The spacecraft does not aect the motion of the Earth and the Moon Eccentricity of orbit of the Moon is very small (.5) 13/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

21 The Earth-Moon-spacecraft system The motion of the spacecraft in the Earth-Moon system can be modelled by the planar restricted 3 body problem. Description : The Earth (mass M 1 ) and Moon (mass M 2 ) are circularly revolving aroud their center of mass G. The spacecraft is negligeable point mass M involves in the plane dened by the Earth and the Moon. Normalization of the masses : M 1 + M 2 = 1 Normalization of the distance :d(m 1, M 2 ) = 1. Moon.5 Earth.5 G.5.5 spacecraft Figure : The circular restricted 3-body problem. The blue dashed line is the orbit of the Earth and the red one is the orbit of the Moon. The trajectory of spacecraft lies in the plan deined by these two orbits. 14/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

22 The Rotating Frame Idea : Instead of considering a xed frame {G, X, Y }, we consider a dynamic rotating frame {G, x, y} which rotates with the same angular velocity as the Earth and the Moon. rotation of angle t substitution ( X Y ) = ( cos(t)x + sin(t)y sin(t)x + cos(t)y ) simplies the equations of the model Figure : Comparision between the xed frame {G, X, Y } and the rotating frame {G, x, y}. 15/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

23 Equations of Motion In the rotating frame dene the mass ratio µ = M 2 M 1 +M 2 the Earth has mass 1 µ and is located at ( µ, ) ; the Moon has mass µ and is located at (1 µ, ) ; Equations of motion where V : is the mechanical potential { ẍ 2ẏ x = V x ÿ + 2ẋ y = V y V = 1 µ ϱ µ ϱ 3 2 ϱ 1 : distance between the spacecraft and the Earth ϱ 1 = (x + µ) 2 + y 2 ϱ 2 :distance between the spacecraft and the Moon ϱ 2 = (x 1 + µ) 2 + y 2. 16/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

24 Hill Regions There are 5 possible regions of motion, know as the Hill regions Each region is dened by the value of the total energy of the system Figure: The Hill regions of the planar restricted 3-body problem Toplogy/Shape of the regions is determined with respect to the total energy at the equilibrium points of the system 17/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

25 Equilibrium points Critical points of the mechanical potential = V Points (x, y) where V = x y Euler points : colinear points L 1,L 2, L 3 located on the axis y =, with x , x , x Lagrange points : L 4, L 5 which form equilateral triangles with the primaries. Figure: Equilibrium points of the planar restricted 3-body problem 18/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

26 The controlled restricted 3-Body problem Control on the motion of the spacecraft? Thrust/Propulsion provided by the engines of the spacecraft control term u = (u 1, u 2) must be added to the equations of motion controlled dynamics of the spacecraft { ẍ 2ẏ x = V + x u1 ÿ + 2ẋ y = V + u2. y Setting q = (x, y, ẋ, ẏ) bi-input system q = F (q) + F 1(q)u 1 + F 2(q)u 2 where F (q) = 2q 4 + q 1 (1 µ) 2q 3 + q 2 (1 µ) F 1(q) = q 3 q 4 q 1 +µ ((q 1 +µ) 2 +q2 2) 3 2 q 2 ((q 1 +µ) 2 +q 2 2 ) 3 2 q 3, F 2(q) = µ µ q 4 q 1 1+µ ((q 1 1+µ) 2 +q2 2) 3 2 q 2 ((q 1 1+µ) 2 +q 2 2 ) 3 2, 19/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

27 Time-minimal problem Objective : minimizing transfer time between geostationary orbit O G and a circular parking orbit O L around the Moon when low-thrust is applied. solve the the optimal control problem q = F (q) + ɛ(f 1(q)u 1 + F 2(q)u 2), ɛ > tf min u(.) BR 2 (,1) t dt q() O G, q(t f ) O L. where ɛ = bound on the control= maximum thrust allowed 2/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

28 Strategy This optimal control problem problem can not be solved analytically Highly non-linear Singularities Approximate low-thrust optimal solutions Apply Pontryagin's Maximum Principle (necessary conditions) Turn the pseudo-hamiltonian system into a true Hamiltonian system (Implicit function Theorem) Use shooting method to compute extremal curves of the problem Check local-optimality of these extremals by using second order optimality condition 1/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

29 Strategy This optimal control problem problem can not be solved analytically Highly non-linear Singularities Approximate low-thrust optimal solutions Apply Pontryagin's Maximum Principle (necessary conditions) Turn the pseudo-hamiltonian system into a true Hamiltonian system (Implicit function Theorem) Use shooting method to compute extremal curves of the problem Check local-optimality of these extremals by using second order optimality condition compute the rst conjugate time along each extremal and very that it is greater than the transfer time 1/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

30 Simple shooting method Idea : Writing boundary and transversality conditions in the form R(z(), z(t f )) =, 22/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

31 Simple shooting method Idea : Writing boundary and transversality conditions in the form R(z(), z(t f )) =, the boundary value problem from the Pontryagin's Maximum Principle becomes { ż = H r (z(t)) R(z(), z(t f )) =. 22/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

32 Simple shooting method Idea : Writing boundary and transversality conditions in the form R(z(), z(t f )) =, the boundary value problem from the Pontryagin's Maximum Principle becomes { ż = H r (z(t)) R(z(), z(t f )) =. The initial condition x is xed. 22/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

33 Simple shooting method Idea : Writing boundary and transversality conditions in the form R(z(), z(t f )) =, the boundary value problem from the Pontryagin's Maximum Principle becomes { ż = H r (z(t)) R(z(), z(t f )) =. The initial condition x is xed. Dene the shooting function as the mapping E : p R(z, z tf ). 22/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

34 Simple shooting method Idea : Writing boundary and transversality conditions in the form R(z(), z(t f )) =, the boundary value problem from the Pontryagin's Maximum Principle becomes { ż = H r (z(t)) R(z(), z(t f )) =. The initial condition x is xed. Dene the shooting function as the mapping Goal : to determine a zero of E E : p R(z, z tf ). 22/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

35 Simple shooting method Idea : Writing boundary and transversality conditions in the form R(z(), z(t f )) =, the boundary value problem from the Pontryagin's Maximum Principle becomes { ż = H r (z(t)) R(z(), z(t f )) =. The initial condition x is xed. Dene the shooting function as the mapping Goal : to determine a zero of E Remark : u r (x, p) is smooth E : p R(z, z tf ). 22/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

36 Simple shooting method Idea : Writing boundary and transversality conditions in the form R(z(), z(t f )) =, the boundary value problem from the Pontryagin's Maximum Principle becomes { ż = H r (z(t)) R(z(), z(t f )) =. The initial condition x is xed. Dene the shooting function as the mapping Goal : to determine a zero of E Remark : u r (x, p) is smooth E is smooth E : p R(z, z tf ). 22/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

37 Simple shooting method Idea : Writing boundary and transversality conditions in the form R(z(), z(t f )) =, the boundary value problem from the Pontryagin's Maximum Principle becomes { ż = H r (z(t)) R(z(), z(t f )) =. The initial condition x is xed. Dene the shooting function as the mapping Goal : to determine a zero of E Remark : u r (x, p) is smooth E is smooth E : p R(z, z tf ). one can use a Newton type algorithm. 22/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

38 Continuation method Problem : To ensure the convergence of the Newton method, we need a precise guess for the initial condition p we are searching. 23/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

39 Continuation method Problem : To ensure the convergence of the Newton method, we need a precise guess for the initial condition p we are searching. Idea : Consider H r as the element H 1 of a family (H λ ) λ [,1] of smooth Hamiltonians. 23/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

40 Continuation method Problem : To ensure the convergence of the Newton method, we need a precise guess for the initial condition p we are searching. Idea : Consider H r as the element H 1 of a family (H λ ) λ [,1] of smooth Hamiltonians. build a one-parameter family (E λ ) λ [,1] of shooting functions such that the shooting function associated with E is easy to solve. 23/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

41 Continuation method Problem : To ensure the convergence of the Newton method, we need a precise guess for the initial condition p we are searching. Idea : Consider H r as the element H 1 of a family (H λ ) λ [,1] of smooth Hamiltonians. build a one-parameter family (E λ ) λ [,1] of shooting functions such that the shooting function associated with E is easy to solve. Scheme : 23/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

42 Continuation method Problem : To ensure the convergence of the Newton method, we need a precise guess for the initial condition p we are searching. Idea : Consider H r as the element H 1 of a family (H λ ) λ [,1] of smooth Hamiltonians. build a one-parameter family (E λ ) λ [,1] of shooting functions such that the shooting function associated with E is easy to solve. Scheme : 1. Setting λ =, one computes the extremal z(t) on [, t f ] starting from z() = (q, p ) using a simple shooting method. 23/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

43 Continuation method Problem : To ensure the convergence of the Newton method, we need a precise guess for the initial condition p we are searching. Idea : Consider H r as the element H 1 of a family (H λ ) λ [,1] of smooth Hamiltonians. build a one-parameter family (E λ ) λ [,1] of shooting functions such that the shooting function associated with E is easy to solve. Scheme : 1. Setting λ =, one computes the extremal z(t) on [, t f ] starting from z() = (q, p ) using a simple shooting method. 2. One chooses a discretization = λ, λ 1,..., λ N = 1 such that the shooting function is solved iteratively at λ i+1 from λ i. 23/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

44 Continuation method Problem : To ensure the convergence of the Newton method, we need a precise guess for the initial condition p we are searching. Idea : Consider H r as the element H 1 of a family (H λ ) λ [,1] of smooth Hamiltonians. build a one-parameter family (E λ ) λ [,1] of shooting functions such that the shooting function associated with E is easy to solve. Scheme : 1. Setting λ =, one computes the extremal z(t) on [, t f ] starting from z() = (q, p ) using a simple shooting method. 2. One chooses a discretization = λ, λ 1,..., λ N = 1 such that the shooting function is solved iteratively at λ i+1 from λ i. 3. One builds up a sequence p,...,pn with zeros of shooting functions E λ,...,e λn. 23/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

45 Continuation method Problem : To ensure the convergence of the Newton method, we need a precise guess for the initial condition p we are searching. Idea : Consider H r as the element H 1 of a family (H λ ) λ [,1] of smooth Hamiltonians. build a one-parameter family (E λ ) λ [,1] of shooting functions such that the shooting function associated with E is easy to solve. Scheme : 1. Setting λ =, one computes the extremal z(t) on [, t f ] starting from z() = (q, p ) using a simple shooting method. 2. One chooses a discretization = λ, λ 1,..., λ N = 1 such that the shooting function is solved iteratively at λ i+1 from λ i. 3. One builds up a sequence p,...,pn with zeros of shooting functions E λ,...,e λn. 4. p N is the zero that we wanted to determine. 23/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

46 Theoretical justication of the continuation method Theorem :For each λ, the exponential mapping expx λ,t f is of textitmaximal rank if and only if the point x 1 = expx λ,t f (p()) is non-conjugate to x. Moreover, solutions of the parametrized shooting equation contain a smooth curve which can be parametrized by λ and the derivative E λ can be computed integrating the Jacobi equation. convergence of the smooth continuationis guaranteed if there is no conjugate point along any extremal curve z λi. 24/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

47 Application of the Pontryagin's Maximum Principle Time-minimal problem min u BR 2 (,ɛ) tf t dt 25/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

48 Application of the Pontryagin's Maximum Principle Time-minimal problem min u BR 2 (,ɛ) tf t dt Maximization condition 25/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

49 Application of the Pontryagin's Maximum Principle Time-minimal problem min u BR 2 (,ɛ) tf t dt Maximization condition if (H 1, H 2), we have u i = H i, with H i (q, p) =< p, F i (q) > H H2 2 25/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

50 Application of the Pontryagin's Maximum Principle Time-minimal problem min u BR 2 (,ɛ) tf t dt Maximization condition if (H 1, H 2), we have u i = H i, with H i (q, p) =< p, F i (q) > H H2 2 Normal case p 25/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

51 Application of the Pontryagin's Maximum Principle Time-minimal problem min u BR 2 (,ɛ) tf t dt Maximization condition if (H 1, H 2), we have u i = H i, with H i (q, p) =< p, F i (q) > H H2 2 Normal case p Normalization p = 1 25/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

52 Application of the Pontryagin's Maximum Principle Time-minimal problem min u BR 2 (,ɛ) tf t dt Maximization condition if (H 1, H 2), we have u i = H i, with H i (q, p) =< p, F i (q) > H H2 2 Normal case p Normalization p = 1 one obtains the true Hamiltonian H r (z) = 1 + H (z) + (H 2 1 (z) + H2 2 (z)). 25/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

53 Application of the Pontryagin's Maximum Principle Time-minimal problem min u BR 2 (,ɛ) tf t dt Maximization condition if (H 1, H 2), we have u i = H i, with H i (q, p) =< p, F i (q) > H H2 2 Normal case p Normalization p = 1 one obtains the true Hamiltonian H r (z) = 1 + H (z) + Continuation on ɛ (maximal thrust) (H 2 1 (z) + H2 2 (z)). 25/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

54 Application of the Pontryagin's Maximum Principle Time-minimal problem min u BR 2 (,ɛ) tf t dt Maximization condition if (H 1, H 2), we have u i = H i, with H i (q, p) =< p, F i (q) > H H2 2 Normal case p Normalization p = 1 one obtains the true Hamiltonian H r (z) = 1 + H (z) + Continuation on ɛ (maximal thrust) (H 2 1 (z) + H2 2 (z)). Compute Earth-L 1 trajectories prior to Earth-Moon trajectories 25/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

55 Earth-L 1 time-minimal trajectories (a) Fixed frame Figure: Thrust=1N. Red curve : time-minimal transfer to L 1. Green curve : orbit of the Moon. 26/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

56 Earth-L 1 time-minimal trajectories (a) Fixed frame Figure: Thrust=.8N. Red curve : time-minimal transfer to L 1. Green curve :orbit of the Moon. 27/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

57 Earth-Moon time-minimal trajectories (a) Rotating frame (b) Fixed frame Figure: Thrust=1.Red curve : time-minimal transfer to a circular orbit around the Moon. Green curve :orbit of the Moon. Blue curve : circular parking orbit around the Moon 28/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

58 Earth-Moon time-minimal trajectories (a) Rotating frame (b) Fixed frame Figure: Thrust=.8. Red curve : time-minimal transfer to a circular orbit around the Moon. Green curve :orbit of the Moon. Blue curve : circular parking orbit around the Moon 29/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

59 Energy-minimal problem Objective : minimizing energy-cost of a transfer between geostationary orbit O G and a circular parking orbit O L around the Moon. 3/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

60 Energy-minimal problem Objective : minimizing energy-cost of a transfer between geostationary orbit O G and a circular parking orbit O L around the Moon. minimizing the L 2 -cost of the control u along the transfer 3/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

61 Energy-minimal problem Objective : minimizing energy-cost of a transfer between geostationary orbit O G and a circular parking orbit O L around the Moon. minimizing the L 2 -cost of the control u along the transfer solve the the optimal control problem q = F (q) + F 1(q)u 1 + F 2(q)u 2 tf min u(.) R 2 t u1 2 + u2dt 2 q() O G, q(t f ) O L. 3/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

62 Application of the Pontryagin's Maximum Principle Energy-minimal problem min u R 2 tf t u u 2 2dt 31/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

63 Application of the Pontryagin's Maximum Principle Energy-minimal problem min u R 2 Maximization condition tf t u u 2 2dt 31/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

64 Application of the Pontryagin's Maximum Principle Energy-minimal problem min u R 2 Maximization condition u i = H i tf t u u 2 2dt 31/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

65 Application of the Pontryagin's Maximum Principle Energy-minimal problem min u R 2 Maximization condition u i = H i Normal case p tf t u u 2 2dt 31/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

66 Application of the Pontryagin's Maximum Principle Energy-minimal problem min u R 2 Maximization condition u i = H i Normal case p Normalization p = 1 2 tf t u u 2 2dt 31/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

67 Application of the Pontryagin's Maximum Principle Energy-minimal problem min u R 2 Maximization condition u i = H i Normal case p Normalization p = 1 2 one obtains the true Hamiltonian tf t u u 2 2dt H r (z) = H (z) (H2 1 (z) + H 2 2 (z)). 31/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

68 Application of the Pontryagin's Maximum Principle Energy-minimal problem min u R 2 Maximization condition u i = H i Normal case p Normalization p = 1 2 one obtains the true Hamiltonian Continuation on µ (mass ratio) tf t u u 2 2dt H r (z) = H (z) (H2 1 (z) + H 2 2 (z)). 31/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

69 Application of the Pontryagin's Maximum Principle Energy-minimal problem min u R 2 Maximization condition u i = H i Normal case p Normalization p = 1 2 one obtains the true Hamiltonian Continuation on µ (mass ratio) tf t u u 2 2dt H r (z) = H (z) (H2 1 (z) + H 2 2 (z)). Compute Earth-L 1 trajectories prior to Earth-Moon trajectories 31/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

70 Earth-L 1 energy minimal trajectories (a) Rotating frame (b) Fixed frame Figure: µ=. Red curve : energy-minimal transfer to L 1. Green curve : orbit of the Moon. 32/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

71 Earth-L 1 energy minimal trajectories (a) Rotating frame (b) Fixed frame Figure: µ = e 2. Red curve : energy-minimal transfer to L 1. Green curve : orbit of the Moon. 33/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

72 Earth-L 1 energy minimal trajectories (a) µ= (b) µ = e 2 Figure: Control norms 34/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

73 Earth-Moon energy minimal trajectories (a) Rotating frame (b) Fixed frame Figure: µ=. Red curve : energy-minimal transfer to a circular orbit around the Moon. Green curve :orbit of the Moon. Blue curve : circular parking orbit around the Moon 35/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

74 Earth-Moon energy minimal trajectories (a) Rotating frame (b) Fixed frame Figure: µ = e 2. Red curve : energy-minimal transfer to a circular orbit around the Moon. Green curve :orbit of the Moon. Blue curve : circular parking orbit around the Moon 36/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

75 Earth-Moon energy minimal trajectories (a) µ= (b) µ = e 2 Figure: Control norms 37/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

76 Gravitational model : 3 bodies Szebehely [1967] Circular restricted three-body problem ẍ = 2ẏ V x ÿ = 2ẋ V y z = V z where V is the potential energy function : and V = x 2 + y µ + µ µ(1 µ) + ρ 1 ρ 2 2 ρ 1 = (x + µ) 2 + y 2 + z 2 ρ 2 = (x 1 + µ) 2 + y 2 + z 2 38/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

77 Gravitational model : 4 bodies Simó et al. [1995] Circular restricted four-body problem 1.5 to Sun Sun at (r s cos θ, r s sin θ) with mass µ s ẍ = 2ẏ V 4 x ÿ = 2ẋ V 4 y z = V 4 z y (LD) Spacecraft ( q(t), m(t) ) Earth θ(t) Moon 1.5 where x (LD) V 4 = V V s, V s(t) = µs µs ρ s rs 2 (x cos θ + y sin θ) ρ s = (x r s cos θ) 2 + (y r s sin θ) 2 + z 2, µ s = , r s = 389.2, θ = ω s =.925 θ(t) = ω st + θ 9/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

78 Notation and drift equations State variables : Uncontrolled system : Ẋ = F(X) = ( X(t) = q(t) m(t) 2ẏ + x (1 µ) x+µ ρ 3 1 ) = 2ẋ + y (1 µ) y ρ 3 1 (1 µ) z ρ 3 1 s(t) v(t) m(t) ẋ ẏ ż µ x 1+µ ρ 3 2 µ y ρ 3 2 µ z ρ 3 2 µ s x r s cos θ ρ 3 s µ s y r s sin θ ρ 3 s µ s z ρ 3 s µs r 2 s µs r 2 s cos θ sin θ 4/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

79 Control system Control variables : u(t) = Set of admissible controls : { U = u : R B R 3(, 1) : u 1(t) u 2 (t) u 3 (t), u 1, u U } u measurable, u(t) = ᾱ(t) for a sequence of times t 1 t 2 t 3 t 4 t f { 1, t [, t1 ] [t ᾱ(t) = 2, t 3 ] [t 4, t f ], otherwise } 1 u t 1 t 2 t 3 t 4 t f t i, i = 1..4 are called switching times 41/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

80 Minimal fuel problem We can now state the minimal cost problem : tf min u(t) dt u U Ẋ = F(X) + ( ) Tmax 3 D m i=1 u i Fi T maxβ u F4 q() = qdprt, m() = m q(t f ) = qrdvz, θ(t f ) = θ rdvz D is the problem data D = (t f, qdprt, qrdvz, θ rdvz ) /5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

81 Pontryagin maximum principle (Pontryagin [1962]) Let (X, u) be an optimal pair for ( ) D. Then there exists an absolutely continuous function P( ), called the costate vector, and a constant p, (P, p ) (, ), such that for a.e. t we have : Pseudo-Hamiltonian Equations : Ṗ = H X, where H(t, P, X, u) = p u + P, Ẋ. Maximization Condition : Transversality Condition : H(t, P(t), X(t), u(t)) = P() T X() M, Ẋ = H P max v B R 3 (,1) H(t, P(t), X(t), v) P(t f ) T X(tf )M f A solution (P, X, u) is called an extremal. 43/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

82 V at rendezvous (m/s) z (LD) y (LD) x (LD) /5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

83 V at continuation points (m/s) z (LD) y (LD) x (LD) /5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

84 Visualization research Further questions : Why were some transfers better than others? What characterizes a good candidate asteroid? What some general features that allow low cost transfers? This prompted exploratory work using visualization and statistical analysis. 46/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

85 Visualization demo 47/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

86 Observations and hypotheses Potential predictors based on visualization and talks with industry : Average energy Average velocity Planarity Lunar planarity Z-displacement Distance from barycenter Eccentricity Average curvature 48/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

87 Statistical results The most important factors were : Average energy Lunar planarity Variance of distance from barycenter [include other graphs here] 49/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions

88 Conclusions The following are fair predictors of good transfers : Low average energy relative to the departure energy (from earth-moon L 2) Minimoon trajectories that lie mostly in the lunar plane Roughly circular geocentric orbits For mission design, this implies that parking an array of spacecraft at various energy levels could be a viable solution to maximize low cost interceptions. Furthermore, prioritizing minimoons with roughly moon-like orbits could help reduce fuel costs. This has implications for missions like NASA's ARM as well. 5/5 G. Patterson, G. Picot, and S. Brelsford Asteroid Rendezvous Missions