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Spaceplane Trajectory Optimisation with Evolutionary-Based Initialisation

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Spaceplane Trajectory Optimisation with Evolutionary-Based Initialisation Spaceplane Trajectory Optimisation with Evolutionary-Based Initialisation Christie Alisa Maddock, Edmondo Minisci Aerospace Centre of Excellence Department of Mechanical & Aerospace Engineering University of Strathclyde Glasgow G1 1XJ, United Kingdom [email protected], [email protected] Abstract— In this paper, an evolutionary-based initialisation de-orbiting manoeuvres in space. Coupled with changes in method is proposed based on Adaptive Inflationary Differential operation, such as switching from an air-breathing engine to Evolution algorithm, which is used in conjunction with a a closed-intake rocket engine or vehicle stage separations, the deterministic local optimisation algorithm to efficiently identify clusters of optimal solutions. The approach is applied to software models must be designed to handle discontinuities. an ascent trajectory for a single stage to orbit spaceplane, The design of spaceplane mission delivering a payload employing a rocket-based combine cycle propulsion system. The to a specified orbit and re-entering is in itself a multiphase problem is decomposed first into flight phases, based on user problem. Across the entire mission, assuming a returnable defined criteria such as a propulsion cycle change translating vehicle, the division is similar to an aircraft with runway to different mathematical system models, and subsequently transcribed into a multi-shooting NLP problem. Examining the take-off, ascent, operations (e.g., payload delivery), re-entry results based on 10 independent runs of the approach, it can be and runway landing. The two transatmospheric trajectories seen that in all cases the method converges to clusters of feasible can be further divided in multiple segments; the elements solutions. In 40% of the cases, the AIDEA-based initialisation defining the problem can differ though disciplinary models found a better solution compared to a heuristic approach (e.g., propulsion modes for a hybrid engine, or in a multi- using constant control for each phase with a single shooting transcription (representing an expert user). The problem was stage propulsion system), problem objectives and constraints, run using randomly generated control laws, only 2/20 cases and level of fidelity needed within the models. To allow such converged, both times with a less optimal solution compared to flexibility in the design, the simulation needs to be structured the baseline heuristic approach and AIDEA. in multiple phases, with interchangeable disciplinary models, plus mission objectives and constraints. The approach herein I. INTRODUCTION has been designed to allow each phase to define the set of One forerunner for the next generation of space access models used and all the variables involved in the parametri- vehicles are spaceplanes. Spaceplanes operate both as an sation of the controls and propagation of the trajectory. The aircraft, taking advantage of the atmosphere at low altitudes continuity between phases is guaranteed at convergence of where, for example, air-breathing engines can be used along the optimisation process by the optimiser and the problem with lifting surfaces, as a rocket at higher altitudes where the formulation through constraints functions. atmosphere is less dense, and as a spacecraft in orbit. This The development of methods presented here are part of a concept offers mass savings due to air-breathing engines, larger initiative to develop an integrated, multi-disciplinary which reduces the need for on-board oxygen, and allow more design platform for quickly assessing and optimising con- control into the flight path and therefore the orbits that can ceptual vehicle design and performance for future space be reached from a given take-off and landing site as well as access vehicles, in particular single and multi-stage reusable a wider choice in the site, or spaceport, itself. vehicles. The trajectory optimisation solves for an open loop While a single stage to orbit offers full re-usability of control scheme using a multi-phase approach for the configu- the vehicle, it is still largely in the development stage, with ration implemented with a direct multi-shooting transcription many of the key technologies still unproven and also comes method. A set of first guesses are generated using the in- with a cost of lower payload fraction [1]. Reaction Engines’ house Adaptive Inflationary Differential Evolution (AIDEA) Skylon vehicle is currently under research and development, algorithm [2], which focuses on exploration, followed by a with a proposed test flight in 2025. Multi-stage spaceplanes local optimisation of candidate solutions using a derivative offer a more immediate time to market though often at a based method for non-linear, constrained problems using cost of partial expendability of parts. Virgin Galactic and S3 a fixed time step integrator. The trajectories are then re- SOAR are two examples which use a carrier subsonic aircraft propagated with the locally optimal control laws using a to launch a smaller spaceplane. Orbital ATK Pegasus is a variable step numerical integrator with lower tolerances multi-stage rocket launched from an aircraft that has been (higher accuracy) to ensure the feasibility of the solutions. operational since 1990s. The results for the first guess using AIDEA are compared From a design point of view, these vehicles are highly against a phase-based single shooting method using an user complex systems operating across very different environ- supplied control law based on educated guessing. ments from low-altitude subsonic flight to high thermal Results are presented for a test case of conceptual single loads during high altitude re-entry, to orbital insertion and stage to orbit spaceplane optimising the ascent trajectory for a payload delivery mission to a 100 km altitude, circular, subject to equatorial orbit. The objective function is the maximisation of the payload mass delivered to an orbit, with equality xi;k = F ([ti−1;k; ti;k]; xi−1;k); constraints on the final position and velocity vectors. Using ui−1;k = ui;k; nc 0 AIDEA for the first guess generation has proved robust and x1;k = x(tn+1;k−1); effective at finding sets of feasible, locally optimal solutions u1;k = un+1;k−1; which can be used to understand the trade-offs between 0 nc performance and vehicle design. c(x(t); u(t)) ≤ 0; t 2 [t0; tf ] g(x ; un;k) ≤ 0; n;k nc II. OPTIMISATION APPROACH !(x0;1; xn;np ) = 0 The optimisation algorithms and approach are shown for i = 1; :::; n − 1, k = 1; :::; n and ∆t = t − t . below, describing the transcription method, based on direct p i;k i+1;k i;k Path constraints are evaluated at a discrete set of points based multi-shooting transcription and flight mission decomposi- n;k in time, and g(xn;k; u ) are the inequality constraints for tion, the evolutionary-based initialisation using AIDEA, and nc phase switching. the local derivative-based optimisation and refinement step. The resulting optimisation problem is usually handled via standard derivative-based methods, requiring a trial-and-error A. Transcription approach by an knowledgeable or expert user to find a The optimal control problem is transcribed into an nonlin- suitable starting point, i.e., an initial solution that can produce ear programming problem by using a multi-phase, multiple- a near-feasible solution that will allow the local deterministic shooting approach. The mission is initially divided into np optimiser to converge. In this respect, the possibility to use user-defined phases. Within each phase, the time interval is evolutionary algorithms to find a suitable starting point is further divided into n multiple shooting segments. explored in this paper. Using an evolutionary approach also has the benefit of producing multiple first guesses as well as np n−1 [k=1 [i=0 [ti;k; ti+1;k] (1) clustering information on the archived solution sets. With each interval [ti;k; ti+1;k], the control is further discre- B. Adaptive Inflationary Differential Evolution Algorithm i;k i;k tised into nc control nodes fu ; :::; u g and collocated on 0 nc The creation of the algorithm stemmed from the idea Tchebycheff points in time. to create new hybrid algorithms that take elements from Continuity constraints on the control and states are im- different approaches and use them as building blocks for posed, a new algorithm. Following this approach, one of the co- x = F ([t ; t ]; x ) authors [3] first co-proposed the Inflationary Differential i;k i−1;k i;k i−1;k for k = 1; :::; n (2) ui−1;k = ui;k p Evolution Algorithm (IDEA) in 2011, which combines dif- nc 0 ferential evolution (DE) [4] with the restarting procedure of Monotonic Basin Hopping (MBH) [5] algorithm. IDEA x = x(t ) 1;k n+1;k−1 for k = 2; :::; n (3) showed very good results when applied to problems with a u1;k = un+1;k−1 p 0 nc single or multi-funnel landscape. However, its performance was found to depend on the parameters controlling both the where F ([t ; t ]; x ) is the final state of the nu- i−1;k i;k i−1;k convergence of DE and MBH, and the inflationary stopping merical integration on the interval [t ; t ] with initial i−1;k i;k criterion used to terminate the DE search. In particular, conditions x . This approach increases the degree of i−1;k the DE performance is strongly influenced by the crossover freedom of the optimisation process reducing the sensitivity probability CR and the differential weight F whose best of the overall problem to its variables although at a cost of settings are heavily problem dependent [6]. This led to the a steep increase in the number of optimisation variables. development of Adaptive-IDEA, or AIDEA, which uses a The optimisation variables are therefore: probabilistic, Parzen-based kernel approach to automatically • The initial state vector of each shooting segment within adapt the values of both CR and F during the search process every phase (excluding the first segment of the first [2].
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