IAC-10.C1.7.5 DIRECT TRANSCRIPTION OF LOW-THRUST TRAJECTORIES WITH FINITE TRAJECTORY ELEMENTS

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1 IAC-1.C1.7.5 DIRECT TRANSCRIPTION OF LOW-THRUST TRAJECTORIES WITH FINITE TRAJECTORY ELEMENTS Federico Zuiani PhD Candidate, Department of Aerospace Engineering, University of Glasgow, UK, Massimiliano Vasile Reader, Department of Mechanical Engineering, University of Strathclyde, UK Alessandro Palmas Graduate Student, Department of Aerospace Engineering, Politecnico di Torino, Turin Italy, Giulio Avanzini Assistant Professor, Department of Aerospace Engineering, Politecnico di Torino, Turin Italy, This paper presents novel approach to Low-Thrust trajectory design, based on a first order approximate analytical solution of Gauss planetary equations. The analytical solution is fairly accurate and could be employed for a fast propagation of perturbed Keplerian motion. Thus, it has been integrated in a Direct Transcription Method based one Finite Perturbed Elements in Time (FPET). It has been proved that this method is suitable for the solution of boundary value transfer problem by few iterations of a gradient based local optimizer. This allows in turn to include the FPET Transcription method into a global optimization tool to solve orbital transfer problem where both total V and transfer time need to be minimised with respect to departure and arrival dates. To prove the concept, two transfer problems have been investigated, the first being a direct rendez-vous transfer between Earth and Mars; the second is a spiralling orbit rising for a low Earth orbit. In both cases the results obtain confirmed the soundness of the proposed approach. I. INTRODUCTION The design of a low-thrust (LT) trajectory requires the definition of the thrust profile that satisfies a twopoint boundary value problem. The scope of this work is to provide a computationally efficient way to determine a good approximated solution to this problem with a more accurate representation of the control profile compared to other approximations. In the literature the problem has been tackled in a number of different ways generally classified in two families: indirect methods and direct methods. Indirect methods translate the design of a low-thrust trajectory into the solution of an optimal control problem and derive explicitly the associated first order optimality conditions. The first order optimality conditions are a system of mixed differential-algebraic equations (DAE). Shooting, multiple-shooting, collocation and approximated analytical approaches have been proposed to solve the DAE system and satisfy the boundary conditions. Direct methods, instead, do not derive the optimality conditions but transcribe the differential dynamic equations of motion into a system of algebraic equations and then solves a nonlinear programming problem. Numerical integration and collocation techniques have been proposed to transcribe the differential dynamic equations. Direct methods are generally computationally intensive while indirect methods can display some convergence problem. Both require some form of first guess solution. In the past decade, some low-fidelity approximation techniques have been proposed to generate the first guess solution. However, the use of these low-fidelity solutions is not always straightforward. Sims and Flanagan 1 first proposed a fast direct method based on the transcription of a lowthrust trajectory into a multi-burn transfer to generate a medium-fidelity solution at a low computational cost. This has also been used as a basis for global optimization tools 2,3. Recently, Sukhanov et. al. 4 proposed a method in which multi-revolution LT trajectories are divided into sub-arcs and on each of them an linearised optimal control problem is solved. In this paper, a direct method is presented where the trajectory is decomposed into a number of finite elements. Gauss planetary equations are solved over IAC-1-C1.7.5 Page 1 of 11

2 each element by means of a perturbative approach, for constant thrust modulus and direction. The trajectory is assumed to be an ε-variation of a Keplerian arc, where ε is a small acceleration term due to the low-thrust. A very fast transcription of the trajectory into a nonlinear programming problem is thus obtained, the accuracy of which is controlled by the number of elements, assuming that every trajectory element remains a first order epsilon-variation of a Keplerian arc. We will show how this approach can be used for fast multiobjective optimization of long low-thrust spirals where both the mass of propellant and the transfer time need to be minimized. II. MODEL AND PROBLEM DEFINITION The equations of perturbed two body motion are usually expressed in terms of Cartesian coordinates or, alternatively, in Keplerian orbital elements but it is also possible to rewrite them in terms of non-singular, equinoctial elements 5. This is a particular formalization of the 6-element state vector which is not affected by singularities for orbit types like those with zeroinclination (undefined line of nodes) or zero-eccentricity (undefined periapsis). This is achieved by replacing e, i, Ω, ω and θ with derived quantities: a P1 e sin ( ω = Ω + ) P2 = e cos( Ω + ω) i X = Q1 = tan sin Ω [1] 2 i Q2 = tan cos Ω 2 L = ( Ω + ω) + ϑ The equations of perturbed two body motion are expressed in terms of a set of 6 Gauss variational equations in Equinoctial parameters: 2 da 2a = ( P2 sin L P1 cos L) cos cos dt h ε β α p + ε cos β sinα r dp1 r p = cos L ε cos β cosα dt h r p + P sin L ε cos β sinα r P ( Q cos L Q sin L) ε sin β } dp2 r p = cos L ε cos β cosα dt h r p + P sin L ε cos β sinα r P ( Q cos L Q sin L) ε sin β 3 dt a h dq1 r = ( 1+ Q1 + Q2 ) sin L ε sin β dt 2h dq2 r = ( 1+ Q1 + Q2 ) cos L ε sin β dt 2h dl µ r = Q cos L Q sin L ε sin β 1 2 where ε, α and β are modulus, azimuth and elevation of the perturbing acceleration in the radial-tangential reference frame. In a typical LT transfer problem, one wants to find the control law conditions: } [2] which satisfies the boundary ( t ) X = X [3] X( t f ) = X f With t f =t +ToF, where ToF is the time of flight. At the same time, the aim is usually that of minimizing the total V of the transfer: V t f = ε t dt [4] t III. THE PERTURBATIVE APPROACH The system of nonlinear ordinary differential equations (ODE s) expressed in [2] can be easily integrated by means of a numerical approach, but this is rather demanding from the computational point of view, especially when massive computation on many different candidate control profiles is necessary, as in the framework of the solution of orbit transfer optimization problems. On the converse, a perturbative approach, which employs low-thrust non-dimensional acceleration as the perturbation parameter, can be used in order to analytically obtain an efficient, first-order accurate approximate solution for the evolution of orbital parameters under the action of a constant perturbing force in the radial-tangential reference frame 6. Gauss's variational equations [2] are rewritten using true longitude L as the independent variable instead of the time, by means of the derivation chain rule, where dx dx dt = [5] dl dt dl The variable x can be any of the first 5 equinoctial orbital elements, namely, a, P 1, P 2, Q 1, and Q 2, while dt/dl is given by IAC-1-C1.7.5 Page 2 of 11

3 2 3 dt r h = = dl h P L P L ( 1+ sin + cos ) [6] This latter expression also provides for the additional equation needed for completing the model, in order to derive time as a function of L. At this point, a first order expansion for each equinoctial orbit parameter (plus time) is introduced in [7], where the zero-order terms obtained for ε = (that is, Keplerian motion) are expressed by constants of the motion, with the only exception of t, which clearly depends on L. a = a + ε a 1 P = P + ε P P = P + ε P 21 Q = Q + ε Q Q = Q + ε Q 21 [7] t = t + ε t1 The remaining first-order terms, which approximate the effects of the perturbation action on the trajectory, are determined by means of a technique based on standard perturbation theory 7, applied to the set of first order ODEs [2]. In order to analytically solve the relevant equations, the solution of a set of integrals in the form LF 1 I1 ( LF ) = dl L 3 1+ P sin L + P cos L C F L L 1 2 LF cos L I L = dl [8] ( 1+ P sin L + P cos L) 1 2 LF sin L I L = dl S F ( 1+ P sin L + P cos L) 1 2 is necessary, that must be evaluated between the initial and final angular coordinate values. A closedform solution is available in the complex domain but its derivation is not straightforward and it is omitted here for the sake of conciseness. As an example, the solution for I S is reported in [9]. Once the analytical expressions for a 1 (L), P 11 (L), P 21 (L), Q 11 (L), and Q 21 (L) are available, together with t (L) and t 1 (L), for the considered values of the initial conditions along the transfer arc, it is possible to evaluate all the parameters for a known control force (constant in magnitude and orientation in the radialtangential reference frame), both forward and/or backward in the angular coordinate L. This means that the five variation of orbital parameters and time is known as a first order approximated function of the true longitude L for low-thrust trajectory arcs. Their values can be analytically evaluated from an arbitrary set of 3 3 initial conditions and control force components, expressed in terms of magnitude and two angles. L sin L IS ( L) = dl = 2 1+ P sin L + P cos L 1 2 ( + ) P P 1 P cos L P1 ln + ( P2 1) sin L + P1 + P2 1 P1 = + 3/2 [9] ( P1 + P2 1) ( + ) P P 1 P cos L P1 ln + ( 1 P2 ) sin L + P1 + P2 1 + P1 + + ( P1 + P2 1) 3/2 ( P2 + 1) cos L + P1 sin L + P2 + 1 ( P1 + P2 1)( P2 cos L + P1 sin L + 1) The length of the arc along which it is possible to propagate the evolution of a 1, P 11, P 21, Q 11, and Q 21 depends on the requested accuracy. If the accuracy is fixed (e.g. below one percent) the approximated trajectory arc must be smaller when the ratio ε between low-thrust force and gravitational pull grows, so that in case of low-thrust propulsion applied far away from the main body (e.g. in a Heliocentric transfer scenario) the approximated arc must be reduced to 1 to 2 deg for matching the accuracy conditions. On the converse, when low-thrust is applied close to a primary body, as in a geocentric transfer, the ratio ε becomes smaller, and the considered perturbative expansion may provide acceptable results also for propagations over several orbits. IV. FINITE ELEMENTS TRANSCRIPTION The analytical solution found for Gauss planetary equations is included in a Direct Finite Perturbed Elements in Time (FPET) Transcription method. This could be regarded as a variant of the well established Sims & Flanagan Direct Transcription Method 1. The latter used a zero-order approximation of the perturbed Keplerian motion by decomposing the trajectory into n sub-arcs; each subarc was modelled with unperturbed Keplerian motion. The effect of continuous thrust was modelled with V discontinuities at the boundaries of each sub-arc. This had the advantage of being simple and computationally cheap since the solution for Keplerian motion is obviously available in closed form. In the present work, the simple Keplerian model with discrete V impulses is replaced with a perturbed Keplerian model with constant thrust along each sub-arc (see Fig. 1). IAC-1-C1.7.5 Page 3 of 11

4 Fig. 1: LT Multiple Shooting Method. Every trajectory is divided into n sub-arcs characterised by a constant thrust vector in the radialtransversal frame. The i-th arc of amplitude L i is defined by the following quantities: X m, the 6 Equinoctial parameters at the mid-point of the arc and the 3 control parameters ε, α and β. To obtain the boundary points of the Element the perturbed motion is analytically propagated backward and forward with the same amplitude ± L/2. The mid-point along the arc L was chosen as the base-point for the analytical propagation in the Finite Element for better accuracy, since the error increases superlinearly with arc amplitude. Thus, a two-sided propagation in the form + m L X = f X,, ε, α, β 2 [1] m L X = f X,, ε, α, β 2 provides a better accuracy than that obtained over the same total arclenght L by a forward expansion. The single sub-arcs are then interconnected to each other by imposing matching conditions at their boundaries. To test the accuracy of the Transcription Method with respect to the number of Finite Elements, in the following example we propagated analytically for a fixed time of 1.5 years a constant tangential acceleration -8 2 of km / s, departing from 1AU. The trajectory was subdivided with a variable number of Finite Elements and for each case, the final state computed was compared with a simple implementation of a Modified Euler Method, on the same thrust profile. The results of a numerical integration with the results of numerical propagation with MatLab ode113 were used as a reference to compute the relative error on the final state. CPU times for all three propagation methods were also considered. Fig. 2: Error on final state for heliocentric orbit propagation. Fig. 3: Error on time of flight for heliocentric orbit propagation. Fig. 4: CPU cost for heliocentric orbit propagation. These test confirmed the first-order behaviour of the error of the analytic formulation and showed that it also guarantees a good accuracy already with a relatively small number of Finite Elements. Moreover, the present IAC-1-C1.7.5 Page 4 of 11

5 test should be regarded as a worst case since it involves a high perturbation force, roughly equivalent to the thrust of.5 N on a 2 kg spacecraft, continuously active for more than a year. This means that the ratio between the perturbative acceleration and the local gravity is relatively high and thus its accuracy is affected. Finally, the method has a computational cost only marginally higher than the Modified Euler method but still 4 to 7 times lower than the numerical integration with ode113. A similar test has also been carried out propagating a perturbed LEO orbit for 5 days, roughly equivalent to 5 revolutions. Fig. 7: CPU cost for LEO propagation. Fig. 5: Error on final state for LEO propagation. Fig. 6: Error on time of flight for LEO propagation. Here the advantages of the Analytic propagation are even more evident since it completely outperforms the Modified Euler Method, being 1 to 7 times faster that the Numerical Integration with ode113. This is easily explained by the fact that in LEO the gravitational force of the Earth is many times higher than the perturbation force and thus a first order approximation of perturbed Keplerian motion is adequate. It should also be noted that the Analytic propagation is able to give very good accuracy even with only one (or even a fraction of) Finite Element per revolution. This modelling approach can then be employed to solve Orbital transfer optimization problems. The problem under consideration is that of finding a V optimal trajectory from an initial state X and a final state X f, to be covered in a certain time ToF. The problem, formulated with the proposed Transcription method, translates into: min J = u nfpet i= 1 ε t X1 X [11] + s. t. Ceq = Xi Xi+ 1, i = 2,, nfpet 1 = + Xn X FPET f nfpet ToF t i i= 1 εi ε max, i = 1,, nfpet The decision variables are the thrust vector and the first 5 Equinoctial Elements of the midpoint of each arc. Equality constraints are given by matching condition between adjacent sub-arcs; boundary conditions on the i i IAC-1-C1.7.5 Page 5 of 11

6 initial and final state and condition that the time of flight be equal to a given value. Note that continuity conditions with respect to the Longitude are already automatically satisfied since all the analytical expressions for the variation of Equinoctial elements are already parameterised with respect to L. Therefore the matching constraints apply only to the remaining five Equinoctial Elements. Observe also that in this way the total longitude L tot covered by the trajectory arc could be easily determined as Ltot = Lf L. However, given the periodicity of the Longitude it is also straightforward to model additional revolutions by simply adding multiples of 2π to this figure. This obviously influences the amplitude of the single sub-arcs, given the relationship: L tot n FPET = i= 1 L i. Until now, only a uniform mesh (with respect to L) has been considered. The use of nonuniform grids will be addressed in future work. Limits on maximum engine thrust are formalised as limits on maximum perturbative acceleration: this is not entirely correct since, in fact, while the maximum thrust is constant the maximum achievable acceleration gradually increases with time due to a gradual decrease of spacecraft mass. However as a first approximation this is more than adequate and most of all greatly simplifies the optimization process since the acceleration limit is explicitly expressed as an upper bound on decision variables. This optimization problem could be efficiently solved with a Gradient-like optimizer such as MatLab fmincon. Given n FPET sub-arcs, the problem has 8n FPET decision variables and 5 n scalar equality FPET constraints. Given its computational efficiency it is also possible to extend the field of application by using the proposed transcription method to solve a Global, Multi-Objective (MO) Optimization problem for trajectory design. As a first example, a simple direct, rendez-vous transfer problem between a celestial body A and another celestial body B is considered; the aim is to find the optimal transfers with respect to Time of Flight and total V, within a certain range of departure dates and transfer time. The optimization parameters in this case are simply the departure date T, the ToF and the number of additional revolutions around the central body. A stochastic global optimizer then generates a number of candidate decision vectors. Then, for each value of the global decision vector an internal boundary problem (analogous to the one of [11]) is solved with a local optimizer to find the feasible transfer with minimum V. Boundary conditions are given by the ephemeris of body A at T and those of body B at (T +ToF). This internal process makes the Global Optimization rather CPU expensive but this is mitigated by employing the proposed Transcription Method. V. CASE STUDIES To test the proposed Transcription Method on realistic transfer problems, two cases were considered: the first is a direct, rendezvous transfer from Earth to Mars while the second is an orbit rising from a 26 km circular Low Earth Orbit (LEO) to the International Space Station (ISS) orbit. Both problems were solved first as a simple boundary problem and subsequently as a Multi-Objective problem using Epic, a population based Memetic algorithm developed by the authors and already described in previous works 8. V.I Low-Thrust Earth-Mars Spiral For the simple boundary problem solution, the objective is that of finding a V optimal transfer between Earth and Mars departing from Earth at T = 56 MJD2, with a time of flight of days, and with 2 complete revolutions. Maximum 8 2 acceleration was set at 2.51 km / s, equivalent to a thrust of.5 N applied to a 2 kg spacecraft. Initial guess was given by a constant, tangential thrust profile of magnitude half the maximum acceleration. The orbit was modelled with 4 Finite Elements. The solution obtained has a total V of 5.64 km/s. To check solution accuracy, optimized thrust profile has been numerically integrated to calculate the final state, giving a relative error for the analytical solution of Fig. 8: Variation of Equinoctial Elements for Earth- Mars LT transfer: a. IAC-1-C1.7.5 Page 6 of 11

7 Fig. 9: Variation of Equinoctial Elements for Earth- Mars LT transfer: P 1. Fig. 12: Variation of Equinoctial Elements for Earth- Mars LT transfer: Q 2. Fig. 1: Variation of Equinoctial Elements for Earth- Mars LT transfer: P 2. Fig. 13: Optimised trajectory for Earth-Mars LT Fig. 11: Variation of Equinoctial Elements for Earth- Mars LT transfer: Q 1. Fig. 14: Acceleration modulus ε for Earth-Mars LT IAC-1-C1.7.5 Page 7 of 11

8 accuracy and CPU cost. The Epic algorithm was run for a maximum number of 8 function evaluations. Fig. 17 and Fig. 18 report the solution points in the parameter space and the Pareto front obtained respectively. Fig. 15: Acceleration azimuth α for Earth-Mars LT Fig. 17: Parameters of the solutions for MO Earth-Mars LT transfer problem. Fig. 16: Acceleration elevation β for Earth-Mars LT Regarding the Thrust profile, is it possible to recognize a typical on-off with thrusting concentrated around the pericenter and apocenter passes. The same problem was solved with DITAN, an optimization tool specifically created for the design of Low-Thrust Trajectories 9. The output solution had a total V of 5.71 km/s, confirming the consistency of the V computed with the Perturbative approach. For the Multi-Objective test, the boundaries for optimization parameters in Table 1 were considered. Lower Upper T [MJD2] ToF [days] 1 15 n rev 1 3 Table 1: boundaries for optimization parameters for MO Earth-Mars transfer problem. The trajectory was modelled with 24 Finite Elements, chosen as a good compromise between Fig. 18: Pareto front for MO Earth-Mars LT transfer problem. The Pareto front presents some discontinuities, which are due to the presence of the discontinuous variable n rev. V.II LEO-to-ISS Orbit Rising The second test case considers an hypothetic orbit transfer between the Ariane 5ATV injection orbit and the ISS orbit, requiring an altitude increase of 95 km. The boundary problem is formulated in an analogous way as the Earth-Mars case, although the ISS motion is modelled as a simple, planar, keplerian motion and the injection orbit is assumed to be coplanar to the latter. The parameters of the departure LEO are considered fixed, with the exception of the initial. Departure time is also considered to be fixed, but the Time of IAC-1-C1.7.5 Page 8 of 11

9 Flight determines the position of the rendez-vous with the ISS. It is therefore essential to define the optimal phasing between the departure from LEO and the encounter with the ISS. As in the previous section, the simple boundary problem is considered first, optimizing the V for a transfer with ϑ = 21, ToF = 3.15 days and n rev = Finite Elements were use in the optimisation. The optimised trajectory has a V of 54.9 m/s. The accuracy is 1-5, a truly remarkable result considering the high number of revolutions. It should also be noted, that convergence was also very fast, with fmincon requiring only 3 iterations. Fig. 21: Variation of Equinoctial Elements for LEO-ISS LT transfer: P 2. Fig. 19: Variation of Equinoctial Elements for LEO-ISS LT transfer: a. Fig. 22: Variation of Equinoctial Elements for LEO-ISS LT transfer: Q 1. Fig. 2: Variation of Equinoctial Elements for LEO-ISS LT transfer: P 1. Fig. 23: Variation of Equinoctial Elements for LEO-ISS LT transfer: Q 2. IAC-1-C1.7.5 Page 9 of 11

10 number of additional revolutions was fixed to 3. The Epic algorithm was run for a maximum number of 3 function evaluations. Lower Upper [rad] 2π ToF [days] Table 2: boundaries for optimization parameters for MO LEO-ISS transfer problem. Fig. 24: Acceleration modulus ε for LEO-ISS LT Fig. 27: Parameters of the solutions for MO LEO-ISS LT transfer problem. Fig. 25: Acceleration azimuth α for LEO-ISS LT Fig. 28: Pareto front for MO LEO-ISS LT transfer problem. Fig. 26: Acceleration elevation β for LEO-ISS LT The MO problem was formulated by considering two optimization parameters, and the ToF, with the lower and upper boundaries reported in Table 2. The VI. CONCLUSION In this work we have presented a novel numerical approach for Low Thrust trajectory modelling using a first-order analytical solution for Gauss s planetary in equinoctial elements. This allowed for a fast and relatively accurate propagation of perturbed Keplerian motion and was thus integrated in a Direct, Multiple Shooting Method for the solution of two-point boundary transfer problems. Given the computational efficiency of the method it was also possible to solve Global and IAC-1-C1.7.5 Page 1 of 11

11 Multi Objective Optimisation problems for LT trajectory design which involve the solution of many boundary sub-problems Sims, J.A., Flanagan, S. N.: Preliminary Design of Low-Thrust Interplanetary Missions, Paper AAS , Vavrina, M.A., Howell, K. C.: Global Low-Thrust Trajectory Optimization through Hybridization of a Genetic Algorithm and a Direct Method, AIAA/AAS Astrodynamics Specialist Conference pp , 28. Yam, C.H. and Di Lorenzo, D. and Izzo, D.: Constrained Global Optimization of Low-Thrust Interplanetary Trajectories, Proceedings of the Twelfth conference on Congress on Evolutionary Computation (CEC 21), 21. Sukhanov, A. A., and Prado, A. F. B. de A.: Optimization of Transfers under Constraints on the Thrust Direction: II, Cosmic Research, Vol. 46, No. 1, 28, pp Battin, R.H.: An introduction to the mathematics and methods of astrodynamics, AIAA Education Series, Palmas, A.: Approximation of Low Thrust Trajectory Arcs by means of Perturbative Approaches, M.Sc. Thesis, Politecnico di Torino, July 21. Kevorkian J., Cole J.D.: Multiple Scale and Singular Perturbation Methods, Springer, New York, Vasile, M., Zuiani, F.: A Hybrid Multiobjective Optimization Algorithm Applied to Space Trajectory Optimization, Proceedings of the Twelfth conference on Congress on Evolutionary Computation (CEC 21), 21. Vasile, M.: Ditan User Manual Version 5.8.2, 29. IAC-1-C1.7.5 Page 11 of 11