Solving Multiobjective Optimal Control Problems in Space Mission Design using Discrete Mechanics and Reference Point Techniques

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1 Solving Multiobjective Optimal Control Problems in Space Mission Design using Discrete Mechanics and Reference Point Techniques Sina Ober-Blöbaum, Maik Ringkamp, and Garlef zum Felde Abstract Many different numerical methods have been developed to compute trajectories of optimal control problems on the one hand and to approximate Pareto sets of multiobjective optimization problems on the other hand. However, so far only few approaches exist for the numerical treatment of the combination of both problems leading to multiobjective optimal control problems. In this contribution we combine the optimal control method Discrete Mechanics and Optimal Control (DMOC) with reference point techniques to compute Pareto optimal control solutions. The presented approach is verified by determining a multiobjective optimal transfer for a space mission. Due to the approximation of the Pareto set, structurally different kind of mission trajectories can be detected which would not have been found by local single-objective optimal control methods and provide important information for a mission designer. I. INTRODUCTION Optimal control problems arise in a variety of relevant applications ranging from space mission design over robotics to economical problems. The goal is to determine state and control trajectories that are optimal w.r.t. a prescribed objective function such as minimal energy consumption, maximal comfort or maximal resources. However, many technical applications require the optimization of not only one but several conflicting objectives at the same time leading to multiobjective optimal control problems. Typically, these objectives are conflicting such that no unique optimum can be determined. Rather, one is interested in determining a set of compromise solutions. In this contribution, we are particularly interested in space mission design problems for which on the one hand a minimal amount of fuel consumption is required and on the other hand short mission times are desirable. To solve such multiobjective optimal control problems we combine reference point techniques from multiobjective optimization with local optimal control methods in order to approximate the set of optimal compromises. Thus, in the next two subsections, we give a brief introduction into these two research fields as well as an overview of common numerical methods. This contribution was partly developed and published in the course of the Collaborative Research Centre 614 Self-Optimizing Concepts and Structures in Mechanical Engineering funded by the German Research Foundation (DFG) under grant number SFB 614. S. Ober-Blöbaum, M. Ringkamp and G. zum Felde are with the Department of Mathematics, University of Paderborn, Warburger Str. 1, 3398 Paderborn, Germany {sinaob, ringkamp, dasgarf}@math.upb.de A. Multiobjective Optimization Optimization problems with several conflicting objective functions are called multiobjective optimization problems and the solution set of all optimal compromises is called the Pareto set 1. Multiobjective optimization is an active research area which provides a huge variety of different methods to compute these Pareto sets. One can classify these methods in global and local methods, where the global methods stepwise iterate a complete set of parameters to an approximation of the Pareto set. Contrary to global methods, local methods use local information of single Pareto points to compute nearby solutions. Well known representatives of global methods are for example evolutionary algorithms (e.g. [1], [2]) or subdivision methods (e.g. [3], [4]). Typical local methods are the weighted sum method (e.g. [5], [1], [6]), continuation methods (e.g. [7], [8]) or reference point methods (e.g. [5], [9]). In this article we focus on approximating these sets for the special case that the considered objectives originate from an optimal control problem. Therefore a high-dimensional multiobjective optimization problem arises what made up our decision to use a local method, in particular we use the reference point method. B. Optimal Control Whereas in multiobjective optimization one searches for Pareto optimal parameters, in optimal control one searches for optimal trajectories, which are solutions of a dynamical system given by a differential equation. Common approaches, such as direct methods (for an overview of different methods we refer to [1]), are based on a time-discretization of the trajectories and the differential equation such that the time-dependent functions are represented by a sequence of discrete state and control parameters that are approximations to the trajectories. In the end one is faced with a highdimensional nonlinear constrained (multiobjective) optimization problem for which generally the knowledge of good initial guesses is crucial for the computation of optimal solutions. We consider special kinds of dynamical systems, namely those systems that can be derived from a variational principle. In particular, we are interested in Lagrangian systems which comprise e.g. mechanical, but also electrical or mechatronical systems. In order to solve optimal control problems for those systems, we use DMOC (Discrete Mechanics and Optimal Control [11]), a technique that relies on a 1 Named after the economist Vilfredo Pareto,

2 direct discretization of the variational formulation of the dynamics of the system. Based on the discretization the problem is transformed into a finite dimensional constrained optimization problem. DMOC has been successfully applied to problems in different fields, e.g. systems in space mission design ([12], [13]), or constrained multi-body dynamics ([14], [15]). The principal approach can be extended to the case of optimal control problems with multiple objectives. C. Multiobjective Optimal Control The numerical treatment of multiobjective optimal control problems is a relatively young research field. One of first works combining methods of direct optimal control and multiobjective optimization is e.g. [16], in which shooting techniques for solving optimal control problems are combined with normal boundary methods from multiobjective optimization. In particular, multiobjective optimal control problems are of high interest in designing low-thrust trajectories with short transfer times in space mission design (see e.g. [17] and references therein). In [17] a multi-agent collaborative search is introduced to optimize three different orbit transfers, where the transfer arcs are parametrized by several single parameters such as their radii and angles (in polar coordinates) and transfer times. For the special case of computing multiple gravity assist low-thrust trajectories, in [18] a multiobjective evolutionary algorithm is presented, where the trajectories are modeled through a shaping approach based on exponential sinusoids ([19]) parametrized by special shaping parameters. Thus, in both mentioned works, only single parameters rather than time-dependent trajectories have to be determined which typically results in problems of moderate dimension. Our principal approach to solve a general multiobjective optimal control problem is to transform it into a highdimensional nonlinear multiobjective optimization problem by the discretization of the system as described before. To solve this problem numerically we use methods from multiobjective optimization that are appropriate to handle such high-dimensional problems. In particular, in this contribution we combine reference point methods with DMOC. The local nature of the reference point method provides good initial guesses for the computation of all Pareto optimal trajectories. D. Outline The remaining part of this contribution is organized as follows. In Section II the multiobjective optimal control problem for a mechanical system is formulated and the relevant definitions regarding Pareto optimality are introduced. In Section III the numerical optimal control method DMOC is described that allows for a transformation of the infinitedimensional multiobjective optimal control problem into a finite-dimensional constrained multiobjective optimization problem by applying principles from discrete variational mechanics. This problem is solved by reference point techniques that are introduced in Section IV for general multiobjective optimization problems. In Section V the presented approach is demonstrated by means of a spacecraft transfer in the Planar Circular Restricted Three Body Problem for which Pareto optimal trajectories with minimal amount of fuel and minimal mission times are computed. We conclude with an outlook to future work in Section VI. II. MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM Let M be an n-dimensional configuration manifold with tangent bundle T M and cotangent bundle T M. Consider a mechanical system with time-dependent configuration vector q(t) M and velocity vector q(t) T q(t) M, t [, t f ], whose dynamical behavior is described by the Lagrangian L : T M R. Typically, the Lagrangian L consists of the difference of the kinetic and potential energy. In addition, a force f : T M U T M depending on a time-dependent control parameter u(t) U R m influences the system s motion. The aim is to move the mechanical system on a curve q(t) M, t [, t f ], from an initial state (q, q ) to a final state (q t f, q t f ) under the influence of f(q, q, u) such that the curves q and u minimize a given objective functional J(q, q, u) = C(q(t), q(t), u(t)) dt (1) with C : T M U R k. Here, the objective functional J involves k single objective functionals given as J(q, q, u) = (J 1 (q, q, u),..., J k (q, q, u)) T with J i (q, q, u) = C i (q(t), q(t), u(t)) dt, i = 1,..., k, and C(q, q, u) = (C 1 (q, q, u),..., C k (q, q, u)) T. Note, that also the final time t f can be an optimization variable that has to be determined, making the objective functional dependent on t f as well. At the same time, the motion q(t) has to satisfy the Lagrange-d Alembert principle, which requires that δ L(q(t), q(t)) dt+ f(q(t), q(t), u(t)) δq(t) dt = (2) for all variations δq with δq() = δq(t f ) =. The principle (2) is equivalent to the forced Euler-Lagrange equations d dt q L(q, q) L(q, q) = f(q, q, u), (3) q which provide as system of differential equations the equations of motion. Thus, we are faced with the following Problem 2.1. Problem 2.1: A multiobjective optimal control problem for a mechanical system is given by min J(q, q, u) (4) q( ), q( ),u( ),t f subject to the forced Euler-Lagrange equations (3) and subject to initial r(q(), q()) = and optionally final conditions r(q(t f ), q(t f )) =. The minimization of the vector valued functional J(q, q, u) is defined by the partial order < p on R k. Let v, w R k, then the vector v is less than w (v < p w), if v i < w i for all i {1,..., k}. The relation p is defined

3 analogously. By this relation, we can introduce the concept of dominance and Pareto optimality (see also [5]). Definition 2.1: (i) A point (q, q, u, t f ) of Problem 2.1 is called admissible if it satisfies the constraints (3) and the initial and final conditions. (ii) An admissible point (q, q, u, t f ) is dominated by an admissible point (q, q, u, t f ) if J(q, q, u ) p J(q, q, u) and J(q, q, u) J(q, q, u ), otherwise (q, q, u, t f ) is non-dominated by (q, q, u, t f ). (iii) An admissible (q, q, u, t f ) is called (Pareto) optimal if there exists no (q, q, u, t f ) which dominates (q, q, u, t f ). (iv) The set of all Pareto optimal points (or Pareto optimal solutions) (q, q, u, t f ) is called the Pareto set and its image under J the Pareto front. The goal is to determine the Pareto set, i.e. those state and control trajectories and those final times t f that are Pareto optimal for Problem 2.1 III. DISCRETE MECHANICS AND OPTIMAL CONTROL (DMOC) The multiobjective optimal control problem 2.1 is numerically solved using a direct discretization approach [2], [11]. The state space T M is replaced by M M and a path q : [, t f ] M by a discrete path q d : {, h, 2h,..., Nh = t f } M, with time step h and N a positive integer such that q k = q d (kh) is an approximation to q(kh). Similar, the control function u : [, t f ] U is replaced by a discrete control function u d : {, h, 2h,..., Nh = t f } U, approximating the control on each interval [kh, (k + 1)h] by a discrete control u k (writing u k = u d ((k )h)). Via an approximation of the action integral in (2) by a discrete Lagrangian L d : M M R, L d (q k, q k+1 ) and discrete forces f k δq k +f + k δq k 1 (k+1)h kh (k+1)h kh L(q(t), q(t)) dt, (5) f(q(t), q(t), u(t)) δq(t) dt, (6) where the left and discrete forces f ± k now depend on (q k, q k+1, u k ) we obtain the discrete Lagrange-d Alembert principle (7). This requires to find discrete paths {q k } N k= such that for all variations {δq k } N k= with δq = δq N =, one has N 1 δ k= N 1 L d (q k, q k+1 ) + k= f k δq k + f + k δq k+1 =, (7) which is equivalent to the forced discrete Euler-Lagrange equations D 2 L d (q k 1, q k ) + D 1 L d (q k, q k+1 ) + f + k 1 + f k = (8) for k = 1,..., N 1, and where D i denotes the derivative w.r.t. the i-th argument. In the same manner we obtain via an approximation of the objective functional (1) the discrete objective function J d (q d, u d, h), such that we can formulate the following problem. Problem 3.1: The discrete constrained multiobjective optimization problem is given by N 1 min J d(q d, u d, h) = C d (q k, q k+1, u k, h) (9) q d,u d,h k= subject to the discretized boundary conditions and the forced discrete Euler-Lagrange equations (8). Here, we have J d = (J d,1,..., J d,k ) T and C d = (C d,1,..., C d,k ) T, where J d,i and C d,i are approximations to J i and C i, respectively, with i = 1,..., k. Note, that in the discrete setting the optimization of the time t f can be modeled by introducing the time step h as additional optimization variable, which is the reason, why the discrete objective functions J d and C d are explicitly dependent on h. 2 To ensure a positive step size and solutions of desired accuracy this variable as to be bounded as < h h max. For a fixed number of N + 1 discretization points the final time is given by t f = Nh. The number of the optimization parameters x = (q d, u d, h) with q d = (q,..., q N ), u d = (u,..., u N 1 ) as well as the number of the equality constraints (the forced discrete Euler-Lagrange equations summarized as g(x) = ) of this nonlinear multiobjective optimization problem depend on the discrete time grid that is used for the approximation. To meet accuracy requirements of the approximated trajectories (for a detailed convergence analysis dependent on the quadrature rules used in (5) and (6) we refer to [11]), typically a fine grid which corresponds to a small time step h is chosen, which leads to a high number of optimization parameters and equality constraints, whereas the number of objective functions J d is independent on the time step. To handle such high-dimensional multiobjective optimization problems, the reference point algorithm presented in Section IV is appropriate since it works in the lowdimensional image space rather than in the high-dimensional parameter space. IV. MULTIOBJECTIVE OPTIMIZATION In this section we first recall the general formulation of a multiobjective optimization problem. Furthermore, we present a reference point optimization method to approximate Pareto sets and describe how this algorithm is applied to the discrete constrained multiobjective optimization problem stated in Problem 3.1. A. Multiobjective Optimization Problem Without loss of generality a multiobjective optimization problem can be viewed as a minimization problem, but in contrary to a (single) objective problem, the objective function is vector-valued. More precisely: Problem 4.1: A multiobjective optimization problem is given by min {F (x)}, S := {x x S Rn g(x) =, a(x) }, (1) 2 Note, that the explicit h-dependence of the discrete Lagranigan L d is neglected for simplicity.

4 with functions F : R n R k, k > 1, g : R n R m and a : R n R q. The method presented in the following subsection is developed to compute an approximation of the Pareto set especially for a high dimension n. To give a mathematically exact definition for this set in the case of a general multiobjective minimization problem the following definitions are appropriate: Definition 4.1: (i) Let v, w R k. Then the vector v is less than w (v < p w), if v i < w i for all i {1,..., k}. The relation p is defined analogously. (ii) A vector y S is dominated by a vector x S (in short: x y) with respect to Problem 4.1 if F (x) p F (y) and F (x) F (y). (iii) A point x S is called Pareto optimal or a Pareto point if there is no y S which dominates x. Sometimes also its corresponding point in image space F (x) is called a Pareto point. (iv) The Pareto set P is defined as the set of all Pareto points and the corresponding set in the image space F (P) is called the Pareto front. B. Reference Point Optimization In case that the multiobjective optimization problem originates from an optimal control problem the dimension of the parameter space is typically high. Thus we decided to use a reference point method ([9], [21]) to approximate the Pareto set. These methods come with the drawback that they always need at least one first solution at the beginning. However, they have advantages in high dimensions: Because of their local behavior good initial guesses are provided during the computation of further Pareto optimal points. We used a method related to the one in [9]. In principle, the method generates unreachable targets (or reference points) T R k in the image space of F. Where each of them is used for the following distance minimization: min F (x) T, S := {x x S Rn g(x) =, a(x) }. (11) Standard minimization algorithms like SQP ([22]) as e.g. implemented in NAG 3 (Numerical Algorithms Group) or Ipopt 4 can be applied to solve these single objective minimization problems. This yields a set P of minima which approximates the whole Pareto set in the convex case, and a continuous connected (local) part of it in the nonconvex case. To be more precisely, for a given multiobjective optimization problem 4.1 a metric d : R k R k R and an optimal point x P our reference point optimization algorithm works as follows: P := {x } F P := {F (x )} for i =,..., M do for j = 1,..., k do Choose Tj i Rk near F (x i ), s. t. Tj i p F (x i ) Solve x j := arg min x S d(f (x), T i j ) if d(f (x j ), T i j ) > then x P +1 := x j P := P {x P +1 } F P := F P {F (x P +1 )} end if end for end for Here, M N is a predefined maximum number of steps, P is the resulting set that approximates P (at least in parts) and F P is the corresponding front. The if -condition assures that the chosen targets are not reachable at least for the chosen local minimizer. So far, the choice of the targets Tj i for the point x i is formulated in a general manner. Different possibilities how to generate good targets are proposed in e.g. [9]. One of the possibilities is to generate new targets Tj i along the shifted tangent space on the Pareto front F (P) in F (x i ). More precisely, let T i be the target for which the given x i is a minimizer (Fig. 1 (a)). First compute Ti F (xi) the normal vector N F (xi)m := T i F (x i) and use it to compute vectors v 1,..., v k 1 with span(v 1,..., v k 1 ) = T F (xi)m = (N F (xi)m) (Fig. 1 (b)). Further, the targets can be computed as follows: k 1 Tj i := F (x i ) + α l,i,j v j + λ j N F (xi)m, j = 1,..., k, l=1 (12) with coefficients α l,i,j R and λ j > such that the resulting targets Tj i (Fig.1 (c)) are below the Pareto front but close enough to F (x i ) such that x i is a good initial point for the following distance minimization, and the resulting set of points (Fig. 1 (d)) gives a good approximation of the Pareto set and its front. C. Using Reference Point Methods for Multiobjective Optimal Control Problems For the multiobjective optimization problem stated in Problem 3.1 we have x = (q d, u d, h) R (N+1)nq+Nnu+1, where N + 1 is the number of time grid points, n q is the dimension of the configuration q, and n u the dimension of the control parameter u. Thus, the objective function F corresponds to J d : R (N+1)nq+Nnu+1 R k, where k is the number of objective functionals. The forced discrete Euler-Lagrange equations (8) and the boundary constraints can be summarized as equality constraints g(x) = with g : R (N+1)nq+Nnu+1 R (N 1)nq+r, where r is the number of the boundary constraints. Inequality constraints a(x) can e.g. occur, if limits on configuration, control variables and the step size are enforced. With this notation we have a problem formulation as in Problem 4.1 and the reference point optimization algorithm can directly be applied. To reduce iteration numbers and computation time of the nonlinear optimizer, the actual Pareto optimal solution always provides the initial guess for the next target minimization to compute a nearby Pareto point rather than choosing the same initial guess for all target optimizations.

5 (a) (b) Due to the conservative nature of the gravitational forces, the equations of motion can be derived by the Lagrangian of the model. It reads ([23], [24]) ((v x y) 2 + (v y + x) 2) where L(x, y, v x, v y ) = µ + µ + 1 µ (1 µ), r 1 r 2 2 r 2 1 = (x + µ) 2 + y 2 r 2 2 = (x + µ 1) 2 + y 2 are the distances of the moving mass from the two primaries and µ is the mass parameter of the system. The Euler- Lagrange equations provide the planar equations of motion for the uncontrolled spacecraft as ẋ = v x (c) Fig. 1. Principal functioning of the reference point method (the black curve is the unknown Pareto front, black dots are Pareto points and the gray dots are targets): (a) The given Pareto point and its given target. (b) The computed normal vector and its orthogonal complement. (c) The shifted tangent space and the new targets. (d) The new computed Pareto points. V. MULTIOBJECTIVE OPTIMAL SPACECRAFT TRANSFER A typical multiobjective problem is the trajectory design for space missions. Due to limited amount of fuel, such trajectories have to be fuel-efficient, however, especially for manned spacecraft, it is important to accomplish a space mission in minimal time. To demonstrate the multiobjective optimal control approach, we compute a Pareto optimal transfer of a spacecraft in the Sun-Earth Planar Circular Restricted Three Body Problem (PCR3BP) ([23], [24]). A. Sun-Earth PCR3BP Model In this section we first briefly introduce the PCR3BP model and summarize its main dynamical properties following [23]. We refer to [23], [24] for a detailed description. The PCR3BP model describes the motion of a small body in the gravity fields of two other massive bodies (the primaries) with different masses. The dynamics are described in a reference frame centered in the common center of mass of the two primaries and rotating with them. Moreover, a set of non-dimensional units is chosen such that the unit of distance is the distance between the two primaries, the unit of mass is the sum of the primaries masses and the unit of time is such that the angular velocity of the primaries around their common center of mass is unitary. The coordinates in this rotating non-dimensional reference frame are indicated with q = (x, y) for the positions and q = (v x, v y ) for the velocities. (d) ẏ = v y v x = 2v y + x (1 µ) x + µ r 3 1 µ x + µ 1 r 3 2 (13) v y = 2v x + y (1 µ) y r1 3 µ y r2 3. For the Sun-Earth system, Sun and Earth are the primaries, where the Sun represents the main attractor and the Earth the gravitational perturbation. The mass parameter of this system is µ = This system has five equilibria, the so-called libration points [25]. As in [23], we focus on the two collinear equilibria close to the Earth called L 1 and L 2 with L 1 being located between the two primaries (see Fig. 2). Both equilibria have an unstable nature, thus the planar dynamics in their neighborhoods involve a center and a saddle component. The center component allows the existence of periodic solutions around these points, which are known as Lyapunov orbits ([24]) and represent a family of solutions orbiting a massless point in the rotating frame. In addition to these periodic solutions, the saddle dynamical component allows the existence of ballistic trajectories moving to/from these Lyapunov orbits. These trajectories are known as the stable/unstable manifolds ([24]) of the periodic orbits. In Fig. 2 the two libration points of interest, two Lyapunov orbits around these and the associated invariant manifolds in the region of the Earth are represented as computed in [23]. The intersection of the stable and unstable manifold of two periodic orbits provides a trajectory moving for free from one periodic solution to the other, also known as Heteroclinic Connection in space mission design. In many works (see e.g. [26], [27]), manifolds and intersections of manifolds have been used to design non conventional space paths. In particular, in [23] both the stable and the unstable manifolds have been computed up to some fixed intersection (thick black line in Fig. 2). The resulting trajectory provided as first guess solution to compute fuel-minimal transfers from the L 1 to the L 2 orbit using DMOC. In this contribution, we are not only interested in fuel-minimal, but also in time-minimal solutions, i.e. we want to approximate all

6 making use of the reference point algorithm described in Section IV. Here, the distance of the objective values to the targets is optimized subject to the forced discrete Euler- Lagrange equations and boundary constraints. Fig. 2. Earth neighbourhoods in the Earth-Sun CR3BP (µ = ): the two Earth-close equilibria (L 1 and L 2 ), the associated Lyapunov periodic orbits and their stable (red) and unstable (black) manifolds [23]. Pareto optimal solutions w.r.t. minimal fuel consumption and minimal mission time. B. The Multiobjective Optimal Control Problem For optimally controlling a spacecraft from the L 1 orbit to the L 2 orbit, we assume that the spacecraft is controlled via two time-dependent translational control forces, one for each degree of freedom of the spacecraft, i.e. the control forces are simply f(u(t)) = u(t) = (u x (t), u y (t)) R 2. (14) By applying the Lagrange-d Alembert principle, these controls correspond to additional accelerations on the right hand side of the third and fourth equation of (13). The multiobjective optimal control problem as stated in Problem 2.1 consists of minimizing the two-dimensional objective functional J = (J 1, J 2 ) according to minimal control effort (as indicator for minimal fuel consumption) and minimal mission time as J 1 (q, q, u) = u(t) 2 dt and J 2 (q, q, u) = 1 dt = t f subject to the controlled equations of motion and boundary constraints. As initial and final constraints we assume prescribed initial and final states on the L 1 and L 2 orbit, respectively, with q() = ( , ), p() = ( , ) and q(t f ) = (1.8, ), p(t f ) = ( , 1.15) with p being the momenta p = L q (q, q). A discretized version of this multiobjective optimal control problem can be formulated by discretizing the objective functional and applying the discrete Lagrange-d Alembert to derive the forced discrete Euler-Lagrange equations as described in Section III. For the discretization we employ the midpoint rule and use 2 discretization points with a maximal time step h max =.3. The resulting constrained nonlinear multiobjective optimization problem is solved by C. Numerical Results In the first step, a single optimization of control effort only (i.e. only J 1 is considered) yields the first Pareto point. For this optimal control problem the trajectory along the unstable and stable manifolds computed in [23] provides a good initial guess. To roughly approximate the remaining Pareto front based on the first Pareto point, only few and relatively widely distributed targets are chosen in a way as described in the algorithm in Section IV-B, where the actual solution provides the initial guess for the next target minimization to compute a nearby Pareto point. As indicated by different colors (red, magenta, blue, green) in Fig. 3 the computed trajectories can be divided into four different types also yielding four different components of the Pareto front. The algorithm detected points in all components by starting from the initial Pareto point. However, for a better covering of the single components, the algorithm was started for all four components individually again with a target distribution of higher density such that in the end four dense components as shown in Fig. 3 have been obtained. Since the different components intersect, a non-dominance test of all computed points is required to filter out the actual Pareto points which are illustrated in Fig. 4. Note, that the entire magenta Pareto set is dominated by the blue one. As general result, it can be seen as expected, that the control effort increases for decreasing mission time. Furthermore, four types of trajectories with qualitatively different dynamics have been detected. In Fig. 6 all trajectories that correspond to the points in Fig. 3 are depicted, whereas in Fig. 5 only four different trajectories are chosen which belong to different points of the Pareto front as indicated with black stars in Fig. 3. The trajectories corresponding to red points in Fig. 3 have long mission times and low control effort. These trajectories show the same qualitative behavior as a transfer along the unstable and stable manifolds as shown in Fig. 6 in red. Blue points in Fig. 3 correspond to blue trajectories in Fig. 6. These trajectories first approximately follow the unstable manifold of the L 1 orbit, however, after passing the Earth the spacecraft is directly transfered to the L 2 orbit without following its stable manifold. The magenta trajectories show symmetric behavior: For these the spacecraft first directly flows close to Earth and then approximately follows the stable manifold of the L 2 orbit. Due to their symmetry, the last two types of solutions have approximately the same control efforts and the same mission times. Both types lead to shorter mission times and a higher control effort compared to the red trajctories. The fourth type of trajectory (green Pareto points in Fig. 3 and green trajectories in Fig. 6) do not follow any stable and unstable manifold anymore. Such a transfer costs more amount of fuel but reduces the mission time significantly. In Fig. 7 the control trajectories for the colored trajectories

7 6 8 x mission time 3 2 y control effort x Fig. 3. Earth-Sun PCR3BP: Computed points of the multiobjective optimization problem with objectives control effort and mission time. The control effort (x-axis) increases for decreasing mission time (y-axis). The four different colors correspond to four qualitatively different trajectories. Black crosses correspond to chosen trajectories shown in Fig. 5. Fig. 5. Earth-Sun PCR3BP: Some chosen Pareto optimal trajectories for minimal control effort and time-minimal transfer from a periodic orbit around L 1 (black orbit on the left) to a periodic orbit around L 2 (black orbit on the right). Red trajectories correspond to high mission times and low control effort; green trajectories correspond to small mission times and high control effort; the blue and magenta trajectories are solution between the first two mission time 3 2 y control effort Fig. 4. Earth-Sun PCR3BP: Pareto front of multiobjective optimization problem (after non-dominance test). Only three different types of trajectories remain. in Fig. 5 are shown. Here, the solid trajectories correspond the control in the x-component, whereas the dashed lines are the controls in y-component for the red, green, magenta, and blue trajectories in Fig. 5, respectively. VI. CONCLUSIONS In this contribution we presented a numerical framework for the approximation of Pareto sets of multiobjective optimal control problems. To this end, the local optimal control method DMOC is combined with a reference point method such that the Pareto optimal trajectories can iteratively be determined. The reference point method is advantageous since on the one hand high-dimensional problems can be handled quite easily and on the other hand since its local nature provides good initial guesses for the computation of the next Pareto optimal solution. For a spacecraft transfer x Fig. 6. Earth-Sun PCR3BP: All computed trajectories for minimal control effort and time-minimal transfer from a periodic orbit around L 1 to a periodic orbit around L 2. between two periodic orbits in the PCR3PB the Pareto set was computed. The Pareto front consists of several connected components and thus qualitatively different Pareto optimal solutions could be detected. The knowledge about the existence of these different trajectories is important for designing space missions for which the requirements on mission time and fuel consumption might vary. Thus, it is not clear in advance that a mission with slightly stricter fuel consumption should be qualitatively similar to a mission for which a higher amount of fuel is available. In the future we want to investigate different approaches for the solution of multiobjective optimal control methods to reveal advantages and disadvantages of the presented approach, in particular for the case of high-dimensional problems as they typically occur for optimal control prob-

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