- Email: [email protected]
First ACT global trajectory optimisation competition: Result found at Alcatel Alenia Space
Acta Astronautica 61 (2007) 753 – 757 www.elsevier.com/locate/actaastro
First ACT global trajectory optimisation competition: Result found at Alcatel Alenia Space Thierry Dargenta,∗ , Vincent Martinotb a Alcatel Alenia Space, 100 Bd du Midi, BP 99, 06322 Cannes La Bocca Cedex, France b Alcatel Alenia Space, 26, av. J.-F. Champollion, BP 1187, 31037 Toulouse Cedex 1, France
Available online 25 April 2007
Abstract In the frame of the first ACT competition on global trajectory proposed by ESA [1], Alcatel Alenia Space has provided a solution based on an internal optimal control Tool T_3D. The method proposed to solve the problem is based on the utilisation of the minimum principle. The necessary condition of optimality for the orbit transfer problem is derived, a boundary problem is formulated and presented. The effort of formulation has been restricted to direct transfers. Finally, a synthesis of the trajectories found in the frame of the competition and after is presented to illustrate the capacity of the method and its limitations. © 2007 Elsevier Ltd. All rights reserved.
1. Introduction In the last recent years a significant progress has been made in optimal control orbit transfers using low-thrust electrical propulsion for interplanetary missions. The system objective is always the same: decrease the transfer duration and increase the useful satellite mass. The optimum control strategy determination to perform the transfer uses sophisticated mathematical tools. Alcatel Alenia Space has developed such types of tools to perform trajectory design as requested by mission analysis. The problem objective is launching from the Earth a satellite, transfer and impact it to asteroid 2001 TW229 in an optimal way defined by a cost function J [1].
∗ Corresponding author.
E-mail addresses: [email protected] (T. Dargent), [email protected] (V. Martinot). 0094-5765/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2007.03.010
The proposed solution is based on T_3D, an internal tool providing solutions for the optimal orbit transfer problem [2]. The theory behind this tool is the application of the maximal principle and the utilisation of continuation and smoothing techniques. The satellite dynamics is a two-body model and relies on an equinoctial formulation of the Gauss equation. This choice has been made for numerical purpose. In order to handle the classical problem of co-state variables initialisation, problems simpler than the actual one has been used to initialise the solution. To solve the problem, the cost criteria J has been introduced in the software by application of the maximal principle and the necessary conditions of optimality have been derived: J = mf × |Urel · vast |.
(1)
Solutions without flybys have been looked for. In the time frame given for the competition, the best-found
754
T. Dargent, V. Martinot / Acta Astronautica 61 (2007) 753 – 757
solution is a direct flight from Earth to asteroid 2001 TW229 without flyby. At impact the cost criteria J reaches the value of J = 330, 385.78 kg km2 /s2 .
The necessary condition of optimality is jJ = 0, which after the variation calculation gives x˙ = f (x, u, t)
(4)
n differential equations, 2. General description of the method The method used to solve the problem proposed by ESA is based on the minimum principle and the utilisation of smoothing and continuation technique for the numerical resolution. A lot of work has been performed in this field since the 1950s as illustrated in Refs. [3–8]. Alcatel Alenia Space has introduced such types of methods in a tool called T_3D with the objective to perform this analysis quickly. T_3D tackles the optimal control problem of orbit transfer with continuous thrust propulsion system by solving a two-point boundary values problem in minimum time transfer and minimum fuel consumption and fixed time. In the frame of the competition we have introduced the cost criteria defined by ESA [1]. Between the different techniques of optimisation, the choice has been made on “indirect” methods by using the maximum principle. Such an approach is called “indirect” because, rather than trying to solve directly the problem, we try to solve the equations given the necessary conditions of optimality. The advantages are a reduced number of optimisation parameters, which will give a rather quick calculation of the solution. The main drawback is the need of a good initial guess of the optimisation parameters to allow the convergence of the solver. We introduce the cost function J . The general form of the cost function augmented by the constraint can be written as: J = [x(tf ), tf ] + T [x(tf ), tf ] tf + L[x, u, t] + T (f (x, u, t) − x) ˙ dt, t0
(5)
n differential equations of the co-state dynamics. The command u(t) is determined by the relation jH = ju
jf ju
T
+
jL ju
T =0
(6)
m algebraic relations, and the boundary conditions xk (t0 ) given at t0 or k (t0 ) = 0 n boundary conditions on the state at t0 , j j + T (7) T (tf ) = jx jx t=tf n boundary conditions on the co-state at tf , j j j j + + T f +L =0 + T jt jx jx jt t=tf
(8)
one boundary condition on the final time if final time is free, [x(tf ), tf ] = 0
(9)
q boundary conditions on the state at tf . T_3D satellite dynamic is a two-body model. The tool uses the equinoctial formulation of the Gauss equation with parameters: p = a|1 − e2 |,
(2)
where [x(tf ), tf ] = mf × |Urel . vast | is the part of the cost function at arrival, here the cost criteria defined in [1], [x(tf ), tf ] are the boundary conditions at arrival, here to be at asteroid position at final time, L[x, u, t] is the part of the cost function along the trajectory, for example, 1 in case of minimum time problem, but 0 in the frame of the competition, f (x, u, t) − x˙ is the dynamic constraint equation along the trajectory. We define the Hamiltonian by H (x, u, , t) = L(x, u, t) + T f (x, u, t).
T T ˙ = − jf − jL jx jx
(3)
ex = e × cos( + ), ey = e × sin( + ), i cos(), hx = tan 2 i sin(), hy = tan 2 l = + + .
(10)
The use of the semi-rectum letus p instead of the semimajor axis allows the tool to work either with elliptic or hyperbolic orbits (Fig. 1).
T. Dargent, V. Martinot / Acta Astronautica 61 (2007) 753 – 757
755
Fig. 1. Orbit parameter and local orbital frame.
3. Application of the method to the problem
Fig. 2. Trajectory 1 for 2001 TW229.
The problem proposed by ESA was to impact asteroid 2001 TW229 with a cost criteria (11). The first difficulty of the used method is to have a first guess of the initial and final conditions. In the time frame of the competition, the effort was put on direct fly to simplify the initialisation process. Multiple flyby has not really been considered because of the combinatorial effort to be done for choosing the strategy. The analysis of the cost criteria has allowed to define reasonable rules on the position and the geometry of impact: impact at asteroid perihelion and maximise the satellite arrival mass. These two rules define the date of arrival and tell you to maximise the transfer time in the bound of the problem to minimise gravity loss during trust arcs. The numerical resolution relies on solving a boundary value problem where the Lagrange multiplier of the dynamics are unknown at t0 and the constraints of rendezvous and optimality on Lagrange multipliers have to be verified at the end of the trajectory (Fig. 2). By manipulating the cost function vast |, J = mf × |Urel . vast − vsat ). vast | J = mf × |(
(11)
and the rendezvous constraints sat − X ast , (x(tf ), tf ) = X
(12)
we obtain the constraint function to be solved ⎤ ⎡ ast sat − X X
⎢ sat ⎥ jX j vsat ⎥ ⎢ T . vast null ⎥ = 0, ⎢ +m× jx jx ⎦ ⎣ m − ( vast − vsat ). vast
(13)
tf
where x are the state variables constructed with the equinoctial parameters and satellite mass: x = [p, ex , ey , hx , hy , l, m]T , sat and vsat the satellite position and velocity vectors, X Xast and vast the asteroid position and velocity vectors. To manage the initialisation a first run has been performed to cross the asteroid with a minimum time problem then a new run with the cost criteria initialisation with the “min time” problem has been performed. We found that the general trend was to decrease the perihelion as much as possible, violating the 0.2 AU constraint put on the problem. This issue has been managed by introducing an intermediate fly path constraint at the place where the 0.2 AU constraint was violated. A third run was performed where we solved a mass maximum problem with an impact in a tangential way of the asteroid at its perigee and with a satellite perihelion of 0.2 AU. The maximum time allowed was used (multiple run cases with multiple asteroid perihelion in time window).
756
T. Dargent, V. Martinot / Acta Astronautica 61 (2007) 753 – 757
A fourth run was performed with the cost criteria and one intermediate fly path constraint at 0.2 AU. The run was initialised by the best case of the third run. This fourth run is the solution provided to ACT for the competition. 4. Competition results The proposed solution has the following features. Departure date Arrival date Time of flight Vinf departure Vinf arrival Satellite initial mass Satellite final mass Total thrust duration J = mf × |Urel . vast |
21/04/2026 00:00:00 04/02/2056 22:56:54 29.79 years 2.5 km/s 15.43 km/s 1500 kg 873.97 kg 106,584.3 h 330, 385.78 kg km2 /s2
The trajectory has been computed with respect to the constraint on the minimum allowed heliocentric distance of 0.2 AU and can be seen in Fig. 3. The thrust history of the trajectory can be seen in Fig. 4. Without this constraint the trend is to reduce the perigee as much as possible and perform a large inclination change to inverse the trajectory. 5. Post-competition results The analysis of the other teams’ results on direct transfer (in particular team 9) has shown that the global optimisation was not found because the trajectory presents many local optimums and the path constraint of 0.2 AU does help for an automatic scanning by doing a random initialisation. Rework has been done in this direction: we have removed the fly path constraint and introduced a constraint of 0.2 AU on perigee as a terminal condition (and not as a fly path constraint).
Evolution of satellite sun distance in AU satellite sun distance in AU
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
2000
4000
6000
8000
10000
12000
Time in Day from the 21/04/2026
Fig. 3. Sun distance history. Thrust law 0.045
on/off Thrust in N
0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0
2000
4000
6000
8000
travel days from 21/04/2026
Fig. 4. Motor switching function history.
10000
12000
T. Dargent, V. Martinot / Acta Astronautica 61 (2007) 753 – 757
757
The interesting thing of these 2 trajectories—shown in Figs. 5 and 6—is the fact that they have almost the same departure and arrival dates, nearly the same cost criteria but the trajectory and the switching strategy are quite different. It illustrates the difficulty for the method to find global optimisation. 6. Conclusion
Fig. 5. Rework trajectory 2 for 2001 TW229.
Fig. 6. Rework trajectory 3 for 2001 TW229.
We have performed multiple starts with different initialisation directly with the final cost criteria and played with the smoothing parameter of the software to facilitate the convergence of the solver. We have extracted 2 of the best solutions found: Departure date Arrival date Time of flight Vinf Departure Vinf arrival Satellite initial mass Satellite final mass Total thrust duration J = mf × |Urel . vast |
20/04/2010 14:24:00 30/12/2034 02:24:00 24.694 years 2.5 km/s 28.32 km/s 1500 kg 850.92 kg 109,830 h 358, 275.6 kg km2 /s2
Departure date Arrival date Time of flight Vinf Departure Vinf arrival Satellite initial mass Satellite final mass Total thrust duration J = mf × |Urel . vast |
20/04/2010 16:46:34 16/01/2035 16:46:34 24.74 years 2.5 km/s 23.45 km/s 1500 kg 904.13 kg 101,450 h 362, 951.6 kg km2 /s2
Concentrating on direct Earth–asteroid trajectories, many solutions have been found with Alcatel Alenia Space internal optimal control tool T_3D, some of them after the competition by reworking on trajectories proposed by other teams (in particular thanks to teams 9, 1 and 8). At the end, the best direct Earth–Asteroid solution has a cost criteria of 362, 951.6 kg km2 /s2 and has been obtained by putting the 0.2 AU constraint at terminal condition. For sure, with regards to the topology of the solution space, better solutions exist and there is no insurance of global optimality. The use of minimum principle coupled with smoothing technique is a good method to find optimal control bang-bang trajectories in orbit transfer. It allows to find precise local solutions. However, for global optimality, the method has to be improved by adding an upper level dedicated tool to search for promising first guesses. References [1] D. Izzo, 1st ACT global trajectory optimisation competition: problem description and summary of results, Acta Astronautica 2007, this issue, doi:10.1016/j.actaastro.2007.03.003. [2] T. Dargent, V. Martinot, An integrated tool for low thrust optimal control orbit in interplanetary trajectories, in: ISSD, Munich, 2004. [3] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957. [4] A.E. Bryson Jr., Y.-C. Ho, Applied Optimal Control: Optimisation Estimation and Control, Hemisphere, Washington DC, 1975. [5] V. Béletski, Essai sur le mouvement des corps cosmiques, le vol interplanétaire: petite poussée grand objectif, Edition MIR, Moscou, 1977–1986, pp. 69–305. [6] S. Geffroy, Généralisation des techniques de moyennation en control optimal application aux problèmes de transfert et de rendez-vous orbitaux à poussée faible, Ph.D. Thesis, ENSEEIHT CNRS, 1997. [7] R. Bertrand, Optimisation de trajectoires interplanétaires sous hypothèses de faible poussée, Ph.D. Thesis, LAAS CNRS, 2001. [8] R. Bertrand, R. Epenoy, CNES technical note no. 147, December 2002, p. 36.
First ACT global trajectory optimisation competition: Result found at Alcatel Alenia Space Thierry Dargenta,∗ , Vincent Martinotb a Alcatel Alenia Space, 100 Bd du Midi, BP 99, 06322 Cannes La Bocca Cedex, France b Alcatel Alenia Space, 26, av. J.-F. Champollion, BP 1187, 31037 Toulouse Cedex 1, France
Available online 25 April 2007
Abstract In the frame of the first ACT competition on global trajectory proposed by ESA [1], Alcatel Alenia Space has provided a solution based on an internal optimal control Tool T_3D. The method proposed to solve the problem is based on the utilisation of the minimum principle. The necessary condition of optimality for the orbit transfer problem is derived, a boundary problem is formulated and presented. The effort of formulation has been restricted to direct transfers. Finally, a synthesis of the trajectories found in the frame of the competition and after is presented to illustrate the capacity of the method and its limitations. © 2007 Elsevier Ltd. All rights reserved.
1. Introduction In the last recent years a significant progress has been made in optimal control orbit transfers using low-thrust electrical propulsion for interplanetary missions. The system objective is always the same: decrease the transfer duration and increase the useful satellite mass. The optimum control strategy determination to perform the transfer uses sophisticated mathematical tools. Alcatel Alenia Space has developed such types of tools to perform trajectory design as requested by mission analysis. The problem objective is launching from the Earth a satellite, transfer and impact it to asteroid 2001 TW229 in an optimal way defined by a cost function J [1].
∗ Corresponding author.
E-mail addresses: [email protected] (T. Dargent), [email protected] (V. Martinot). 0094-5765/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2007.03.010
The proposed solution is based on T_3D, an internal tool providing solutions for the optimal orbit transfer problem [2]. The theory behind this tool is the application of the maximal principle and the utilisation of continuation and smoothing techniques. The satellite dynamics is a two-body model and relies on an equinoctial formulation of the Gauss equation. This choice has been made for numerical purpose. In order to handle the classical problem of co-state variables initialisation, problems simpler than the actual one has been used to initialise the solution. To solve the problem, the cost criteria J has been introduced in the software by application of the maximal principle and the necessary conditions of optimality have been derived: J = mf × |Urel · vast |.
(1)
Solutions without flybys have been looked for. In the time frame given for the competition, the best-found
754
T. Dargent, V. Martinot / Acta Astronautica 61 (2007) 753 – 757
solution is a direct flight from Earth to asteroid 2001 TW229 without flyby. At impact the cost criteria J reaches the value of J = 330, 385.78 kg km2 /s2 .
The necessary condition of optimality is jJ = 0, which after the variation calculation gives x˙ = f (x, u, t)
(4)
n differential equations, 2. General description of the method The method used to solve the problem proposed by ESA is based on the minimum principle and the utilisation of smoothing and continuation technique for the numerical resolution. A lot of work has been performed in this field since the 1950s as illustrated in Refs. [3–8]. Alcatel Alenia Space has introduced such types of methods in a tool called T_3D with the objective to perform this analysis quickly. T_3D tackles the optimal control problem of orbit transfer with continuous thrust propulsion system by solving a two-point boundary values problem in minimum time transfer and minimum fuel consumption and fixed time. In the frame of the competition we have introduced the cost criteria defined by ESA [1]. Between the different techniques of optimisation, the choice has been made on “indirect” methods by using the maximum principle. Such an approach is called “indirect” because, rather than trying to solve directly the problem, we try to solve the equations given the necessary conditions of optimality. The advantages are a reduced number of optimisation parameters, which will give a rather quick calculation of the solution. The main drawback is the need of a good initial guess of the optimisation parameters to allow the convergence of the solver. We introduce the cost function J . The general form of the cost function augmented by the constraint can be written as: J = [x(tf ), tf ] + T [x(tf ), tf ] tf + L[x, u, t] + T (f (x, u, t) − x) ˙ dt, t0
(5)
n differential equations of the co-state dynamics. The command u(t) is determined by the relation jH = ju
jf ju
T
+
jL ju
T =0
(6)
m algebraic relations, and the boundary conditions xk (t0 ) given at t0 or k (t0 ) = 0 n boundary conditions on the state at t0 , j j + T (7) T (tf ) = jx jx t=tf n boundary conditions on the co-state at tf , j j j j + + T f +L =0 + T jt jx jx jt t=tf
(8)
one boundary condition on the final time if final time is free, [x(tf ), tf ] = 0
(9)
q boundary conditions on the state at tf . T_3D satellite dynamic is a two-body model. The tool uses the equinoctial formulation of the Gauss equation with parameters: p = a|1 − e2 |,
(2)
where [x(tf ), tf ] = mf × |Urel . vast | is the part of the cost function at arrival, here the cost criteria defined in [1], [x(tf ), tf ] are the boundary conditions at arrival, here to be at asteroid position at final time, L[x, u, t] is the part of the cost function along the trajectory, for example, 1 in case of minimum time problem, but 0 in the frame of the competition, f (x, u, t) − x˙ is the dynamic constraint equation along the trajectory. We define the Hamiltonian by H (x, u, , t) = L(x, u, t) + T f (x, u, t).
T T ˙ = − jf − jL jx jx
(3)
ex = e × cos( + ), ey = e × sin( + ), i cos(), hx = tan 2 i sin(), hy = tan 2 l = + + .
(10)
The use of the semi-rectum letus p instead of the semimajor axis allows the tool to work either with elliptic or hyperbolic orbits (Fig. 1).
T. Dargent, V. Martinot / Acta Astronautica 61 (2007) 753 – 757
755
Fig. 1. Orbit parameter and local orbital frame.
3. Application of the method to the problem
Fig. 2. Trajectory 1 for 2001 TW229.
The problem proposed by ESA was to impact asteroid 2001 TW229 with a cost criteria (11). The first difficulty of the used method is to have a first guess of the initial and final conditions. In the time frame of the competition, the effort was put on direct fly to simplify the initialisation process. Multiple flyby has not really been considered because of the combinatorial effort to be done for choosing the strategy. The analysis of the cost criteria has allowed to define reasonable rules on the position and the geometry of impact: impact at asteroid perihelion and maximise the satellite arrival mass. These two rules define the date of arrival and tell you to maximise the transfer time in the bound of the problem to minimise gravity loss during trust arcs. The numerical resolution relies on solving a boundary value problem where the Lagrange multiplier of the dynamics are unknown at t0 and the constraints of rendezvous and optimality on Lagrange multipliers have to be verified at the end of the trajectory (Fig. 2). By manipulating the cost function vast |, J = mf × |Urel . vast − vsat ). vast | J = mf × |(
(11)
and the rendezvous constraints sat − X ast , (x(tf ), tf ) = X
(12)
we obtain the constraint function to be solved ⎤ ⎡ ast sat − X X
⎢ sat ⎥ jX j vsat ⎥ ⎢ T . vast null ⎥ = 0, ⎢ +m× jx jx ⎦ ⎣ m − ( vast − vsat ). vast
(13)
tf
where x are the state variables constructed with the equinoctial parameters and satellite mass: x = [p, ex , ey , hx , hy , l, m]T , sat and vsat the satellite position and velocity vectors, X Xast and vast the asteroid position and velocity vectors. To manage the initialisation a first run has been performed to cross the asteroid with a minimum time problem then a new run with the cost criteria initialisation with the “min time” problem has been performed. We found that the general trend was to decrease the perihelion as much as possible, violating the 0.2 AU constraint put on the problem. This issue has been managed by introducing an intermediate fly path constraint at the place where the 0.2 AU constraint was violated. A third run was performed where we solved a mass maximum problem with an impact in a tangential way of the asteroid at its perigee and with a satellite perihelion of 0.2 AU. The maximum time allowed was used (multiple run cases with multiple asteroid perihelion in time window).
756
T. Dargent, V. Martinot / Acta Astronautica 61 (2007) 753 – 757
A fourth run was performed with the cost criteria and one intermediate fly path constraint at 0.2 AU. The run was initialised by the best case of the third run. This fourth run is the solution provided to ACT for the competition. 4. Competition results The proposed solution has the following features. Departure date Arrival date Time of flight Vinf departure Vinf arrival Satellite initial mass Satellite final mass Total thrust duration J = mf × |Urel . vast |
21/04/2026 00:00:00 04/02/2056 22:56:54 29.79 years 2.5 km/s 15.43 km/s 1500 kg 873.97 kg 106,584.3 h 330, 385.78 kg km2 /s2
The trajectory has been computed with respect to the constraint on the minimum allowed heliocentric distance of 0.2 AU and can be seen in Fig. 3. The thrust history of the trajectory can be seen in Fig. 4. Without this constraint the trend is to reduce the perigee as much as possible and perform a large inclination change to inverse the trajectory. 5. Post-competition results The analysis of the other teams’ results on direct transfer (in particular team 9) has shown that the global optimisation was not found because the trajectory presents many local optimums and the path constraint of 0.2 AU does help for an automatic scanning by doing a random initialisation. Rework has been done in this direction: we have removed the fly path constraint and introduced a constraint of 0.2 AU on perigee as a terminal condition (and not as a fly path constraint).
Evolution of satellite sun distance in AU satellite sun distance in AU
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
2000
4000
6000
8000
10000
12000
Time in Day from the 21/04/2026
Fig. 3. Sun distance history. Thrust law 0.045
on/off Thrust in N
0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0
2000
4000
6000
8000
travel days from 21/04/2026
Fig. 4. Motor switching function history.
10000
12000
T. Dargent, V. Martinot / Acta Astronautica 61 (2007) 753 – 757
757
The interesting thing of these 2 trajectories—shown in Figs. 5 and 6—is the fact that they have almost the same departure and arrival dates, nearly the same cost criteria but the trajectory and the switching strategy are quite different. It illustrates the difficulty for the method to find global optimisation. 6. Conclusion
Fig. 5. Rework trajectory 2 for 2001 TW229.
Fig. 6. Rework trajectory 3 for 2001 TW229.
We have performed multiple starts with different initialisation directly with the final cost criteria and played with the smoothing parameter of the software to facilitate the convergence of the solver. We have extracted 2 of the best solutions found: Departure date Arrival date Time of flight Vinf Departure Vinf arrival Satellite initial mass Satellite final mass Total thrust duration J = mf × |Urel . vast |
20/04/2010 14:24:00 30/12/2034 02:24:00 24.694 years 2.5 km/s 28.32 km/s 1500 kg 850.92 kg 109,830 h 358, 275.6 kg km2 /s2
Departure date Arrival date Time of flight Vinf Departure Vinf arrival Satellite initial mass Satellite final mass Total thrust duration J = mf × |Urel . vast |
20/04/2010 16:46:34 16/01/2035 16:46:34 24.74 years 2.5 km/s 23.45 km/s 1500 kg 904.13 kg 101,450 h 362, 951.6 kg km2 /s2
Concentrating on direct Earth–asteroid trajectories, many solutions have been found with Alcatel Alenia Space internal optimal control tool T_3D, some of them after the competition by reworking on trajectories proposed by other teams (in particular thanks to teams 9, 1 and 8). At the end, the best direct Earth–Asteroid solution has a cost criteria of 362, 951.6 kg km2 /s2 and has been obtained by putting the 0.2 AU constraint at terminal condition. For sure, with regards to the topology of the solution space, better solutions exist and there is no insurance of global optimality. The use of minimum principle coupled with smoothing technique is a good method to find optimal control bang-bang trajectories in orbit transfer. It allows to find precise local solutions. However, for global optimality, the method has to be improved by adding an upper level dedicated tool to search for promising first guesses. References [1] D. Izzo, 1st ACT global trajectory optimisation competition: problem description and summary of results, Acta Astronautica 2007, this issue, doi:10.1016/j.actaastro.2007.03.003. [2] T. Dargent, V. Martinot, An integrated tool for low thrust optimal control orbit in interplanetary trajectories, in: ISSD, Munich, 2004. [3] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957. [4] A.E. Bryson Jr., Y.-C. Ho, Applied Optimal Control: Optimisation Estimation and Control, Hemisphere, Washington DC, 1975. [5] V. Béletski, Essai sur le mouvement des corps cosmiques, le vol interplanétaire: petite poussée grand objectif, Edition MIR, Moscou, 1977–1986, pp. 69–305. [6] S. Geffroy, Généralisation des techniques de moyennation en control optimal application aux problèmes de transfert et de rendez-vous orbitaux à poussée faible, Ph.D. Thesis, ENSEEIHT CNRS, 1997. [7] R. Bertrand, Optimisation de trajectoires interplanétaires sous hypothèses de faible poussée, Ph.D. Thesis, LAAS CNRS, 2001. [8] R. Bertrand, R. Epenoy, CNES technical note no. 147, December 2002, p. 36.