NEW TRENDS AND CHALLENGES IN OPTIMIZATION THEORY APPLIED TO SPACE ENGINEERING

In this talk we will discuss the numerical computation of high order Taylor-Fourier approximations to the stable/unstable invariant manifolds of invariant tori. As applications, we will discuss the capture of an asteroid near the L3 point of the Earth-Moon system and the transfer and control of a spacecraft near an unstable torus.

Main author: Ariadna Farrés, University of Maryland at College Park, Maryland, United States

Co-Authors: David C. Folta (1), Cassandra Webster(1), Jeremy Petersen (2)
(1) NASA Goddard Space Flight Center, Maryland, United States
(2) a.i. Solutions, Maryland, United States

Over this decade, there has been an increase on space missions taking advantage of the natural dynamics around the L1 and L2 points in the Circular Restricted Three-Body (CRTBP). Libration Point orbits (LPOs around these two equilibrium points are ideal for space weather missions like Solar and Heliospheric Observatory (SoHO) and the Space Weather Follow-On (SWFO); or deep space observations like the James Webb Space Telescope (JWST) and the Roman Space Telescope (RST). It is well known that LPOs are linearly unstable and require routine station-keeping maneuvers for a spacecraft to remain close to them.

In [1] we used the Floquet mode reference frame to compare two different station-keeping strategies and describe the geometry behind them. The first strategy was the x-axis velocity constraint at the plain crossing, which has been used by NASA in operations for missions like SoHO and JWST. This approach looks for the delta-v required to ensure that at the 4th plane crossing Vx = 0, ensuring the spacecraft will orbit around the LPO. The second strategy was the Floquet mode approach proposed by the Dynamical System group in Barcelona and takes advantage of the natural dynamics around LPOs to determine the maneuver. This approach looks for the delta-v that cancels the unstable mode, bringing the trajectory close to the stable manifold or a reference orbit. In [1] we compared both strategies using high-order techniques and saw that both approaches have many basic geometrical common features, and that when we force both strategies to find the delta-v maneuver along a fixed thrust direction, the results are the same up to first order. This allows us to use the Floquet mode approach to describe the classical station-keeping strategy used during operations.

Understanding the geometry of station-keeping strategies allows us to study the dependence between thrust direction and the delta-v magnitude and the impact of constraints in the thrust directions, which play an important role on JWST. Here we want to discuss how to use the Floquet mode interpretation to envision strategies that can help mitigate the impact of mission constraints and other perturbations on the mission orbit. We focus our study on three different missions (JWST, RST and SWFO), each one with its challenges and particularities. For instance, JWSTs trajectory is largely affected by solar radiation pressure and has restrictions on the thrust direction [2], and SWFO has large momentum unloads that impact the size of station-keeping maneuvers [3]. We will use the Floquet Modes to describe the impact of different mission constraints on the cost of station-keeping and derive strategies that can help mitigate them.

This is a joint work with Dave Folta, Cassandra Webster, and Jeremy Pettersen

[1] A. Farrés, C. Gao, J. Masdemont, G. Gomez, D. Folta, C. Webster, “Geometrical Analysis of Station-Keeping Strategies About Libration Point Orbits”, Journal of Guidance, Control, and Dynamics Vol. 45, No. 6, June 2022

[2] J. Petersen, “L2 station keeping maneuver strategy for the James Webb Space Telescope”, Proceedings of the 2019 AAS/AIAA Astrodynamics Specialist Conference. (AAS 19-806)

[3] A. Farrés, D. Folta, A. Michaels, C. Webster, “Mitigating the impact of Momentum Unloads on Station Keeping around Libration Point orbits”, Proceedings of the 2022 AAS/AIAA Astrodynamics Specialist Conference. (AAS 22-688)

Bang-bang control often characterizes optimal solutions in dynamical problems that are linear in one of the control variables. This is typically the case of space trajectory opti-mization, where the differential equations that describe the state of the spacecraft are linear with respect to the thrust magnitude, and thrust is bounded between a minimum (usually, zero) and a maximum value. The optimal solution is usually made up by a succession of maximum-thrust and minimum-thrust arcs, i.e., a bang-bang control law (singular arcs with intermediate thrust arcs are optimal only in very limited cases, typically involving atmo-spheric flight).

Bang-bang problems are particularly challenging in the context of indirect optimization method. These methods enforce necessary optimality conditions to transform the optimiza-tion problem into a boundary value problem (BVP), typically solved by a shooting method starting from a tentative solution. Pontryagin’s Maximum Principle (PMP), which states that the control variables must maximize the problem Hamiltonian, can be used to deter-mine the optimal control law. For the thrust magnitude, this means that arcs with either maximum or zero thrust alternate, according to the sign of the switching function. The discontinuities in the control variable makes the BVP solution hard, and affect the method robustness in the sense that the convergence radius is usually small and a quite accurate tentative solution is needed.

Different techniques can be envisaged to deal with these issues. Two large groups can be identified: approaches that, during integration, determine the instantaneous thrust according to the local value of the switching function (e.g., with smoothing techniques to avoid the discontinuities), versus techniques that preliminarily assume the problem switching structure and use PMP to modify the structure when required. The latter approach is considered here.

In this talk, different strategies that exploit PMP to improve convergence and the perfor-mance of optimization algorithms are presented, in the context of various space missions and in conjunction with different optimization approaches. In particular, examples concerning collaborative satellite deployment, escape maneuvers from Lagrangian points, lunar land-ing (in the context of indirect methods) and orbit transfer and formation reconfiguration (treated with pseudospectral methods) will be presented.

The exploration of the Moon, both through human and robotic missions, has emerged as a paramount objective in space engineering. Rendezvous and Docking (RVD) operations, including the critical phase of orbital phasing, play a pivotal role in enabling the deployment of the Lunar Gateway, station assembly, crew rotations, cargo deliveries, and lunar sample return, especially in the vicinity of Earth-Moon Lagrange Points (EML). While RVD in traditional two-body scenarios, such as Low Earth Orbit (LEO), Geostationary Earth Orbit (GEO), and Low Lunar Orbit (LLO), has accumulated significant experience, challenges persist in addressing orbital phasing in non-Keplerian environments, particularly around Lagrange points where standard Keplerian dynamics no longer apply.
In light of the inherent dynamical complexity of Lagrange point regions, this presentation aims to investigate strategies for achieving optimal orbital phasing in the cislunar environment. The focus will be on comparing multiple strategies to successfully execute this critical phase of RVD between a target (e.g., the Lunar Gateway) and a chaser (cargo spacecraft, crew vehicles, ascent/descent vehicles, station modules, etc.), taking into account the specific orbits of these objects. Additionally, exploration will encompass the development of semi-analytical tools to compute and model a range of orbits relevant to the Circular Restricted Three-Body Problem (CR3BP), including Halo orbits (including Near Rectilinear Halo Orbits, NRHO, a subgroup of the Halo family), Butterfly orbits, Dragonfly orbits, Disturbed Rectilinear Orbits (DRO), and quasi-periodic orbits. This approach becomes indispensable as weakly stable trajectories and chaotic dynamics demand a meticulously crafted strategy for approaching the station, making the utilization of quasi-periodic structures an innovative and promising solution.
While working to ensure the success of orbital phasing for rendezvous operations in the cislunar environment, this research adds to the ongoing discourse on optimization theory. Key optimization parameters in these endeavors include Time of Flight (TOF) and total consumption (DeltaV). These
optimization scheme typically begin with an initial guess, which can be derived from a Lambert problem in a Keplerian approximation or generated using a stack method. Subsequently, differential corrections are then applied to refine the trajectory. To further enhance results, advanced techniques based on multi-impulsivee transfers are also employed. This research paves the way for more efficient and reliable operations within the challenging realm of Lagrange points, thus advancing the capabilities of future lunar exploration missions.

Stéphanie Lizy-Destrez – Associate Professor of Space System Engineering – ISAE-SUPAERO, Toulouse
Spencer Boone – Post-doctoral researcher – ISAE-SUPAERO, Toulouse
Thomas Caleb – PhD student – ISAE-SUPAERO, Toulouse

In this talk, we will first consider the Sun-Earth-Moon Bicircular Restricted Four-Body Problem and will compute minimum-fuel low-thrust Earth-Moon trajectories by means of an indirect optimal control approach. Thanks to a massive exploration of the set of unknown variables appearing in the optimality conditions, different families of locally optimal trajectories will be identified and classified according to the dynamical structures they leverage. The latter are invariant manifolds of the two underlying Earth-Moon and Sun-Earth Circular Restricted Three-Body Problems [1].
In the second part of this talk, we will consider a full ephemeris model incorporating the gravitational influence of the Sun, the Moon and all planets of the solar system plus the solar radiation pressure perturbation. An initial velocity increment to be provided by the launcher will be computed in order to put the spacecraft on the counterpart of a stable invariant manifold defined in the Sun-Earth Circular Restricted Three-Body Problem. Then, after a coast arc, a closed-loop thrust law based on Lyapunov control theory will be designed to bring the spacecraft to the target lunar orbit [2].
[1] D. Pérez-Palau, R. Epenoy, Fuel Optimization for Low-thrust Earth-Moon Transfer via Indirect Optimal Control, Celestial Mechanics and Dynamical Astronomy, 130, 21, 2018.
[2] R. Epenoy, D. Pérez-Palau, Lyapunov-based Low-energy Low-thrust Transfers to the Moon, Acta Astronautica, Vol. 162, pp. 87-97, 2019.

In this lecture we shall present some recent results on the interplay between control and Machine Learning, and more precisely, Supervised Learning, Universal Approximation and Normaliying flows.
We adopt the perspective of the simultaneous or ensemble control of systems of Residual Neural Networks (ResNets). Roughly, each item to be classified corresponds to a different initial datum for the Cauchy problem of the ResNets, leading to an ensemble of solutions to be driven to the corresponding targets, associated to the labels, by means of the same control.
We present a genuinely nonlinear and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies.
This property is rarely fulfilled by the classical dynamical systems in Mechanics and the very nonlinear nature of the activation function governing the ResNet dynamics plays a determinant role. It allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill.
The turnpike property is also analyzed in this context, showing that a suitable choice of the cost functional used to train the ResNet leads to more stable and robust dynamics.
This lecture is inspired in joint work, among others, with Borjan Geshkovski (MIT), Domènec Ruiz-Balet (IC, London), Martin Hernandez (FAU) and Antonio Alvarez (UAM)

The talk discusses one possible model problem for certain tasks in reinforcement learning. The model provides a framework to deal with situations in which the system dynamics are not known and encodes the available information about the state dynamics that an agent has as a measure on the space of functions. In this framework, a natural question is if whether the optimal policies and the value functions converge, respectively, to an optimal policy and to the value function of the real, underlying optimal control problem as soon as more information on the environment is gathered by the agent. We provide a positive answer in the linear-quadratic case and discuss some open problems and questions.

The rapid advancements in artificial intelligence (AI) have revolutionized various fields, and the space industry is no exception. This talk provides an overview of recent developments in AI-enabled guidance and control (G&C) applications at the European Space Agency (ESA) through various research and development (R&D) studies and internal projects. The talk will begin with the motivation behind incorporating AI to enhance the accuracy and efficiency of space G&C systems. Next, the talk explores how some machine learning techniques can complement more traditional G&C strategies to enable spacecraft to autonomously navigate through complex & uncertain environments, enhance mission efficiency, and improve overall mission success rates. The talk then highlights key advancements in AI algorithms and techniques explored by ESA and the challenges encountered in implementing these techniques. The challenges range from limited computational resources, reliability & robustness, along with possible strategies to overcome them in some scenarios. Finally, the achievements, lessons learned, and future prospects for AI in G&C are presented.

The use of AI techniques in guidance control problems and space missions has gained increasing attention in recent years due to their potential to improve the performance and reliability of space systems. In this session, there will be discussed a) the different AI techniques that can be used in guidance control problems and space missions, such as reinforcement learning, deep learning, and fuzzy logic and also b) present case studies and examples of the successful application of these techniques in various space missions.

On emerging guidance control problems in the satellite industry

The development of full electric satellite platforms equipped with electric propulsion has introduced in past decade new optimal control problems for the guidance function during the low-thrust orbit transfer operations.
Three use cases are found to be especially challenging and will be presented in this talk. All these problems are still open to our best knowledge. The opportunity to solve these problems using AI techniques vs extensions to the conventional, state-of-the-art optimal control theory will be discussed.
The first use case is related to the definition of a simple approximation model with a low optimality gap for the estimation of the low-thrust Delta-V requirement between two orbits that can be used as an initial guess either with the indirect approach or with the direct approach (it is worth mentioning that no simplified formula is available in the literature for the elliptic non-coplanar transfer case).
The second use case addresses the feasibility of a closed-loop solution of the low-thrust orbit transfer problem with terminal phasing or longitude rendezvous constraint using the control Lyapunov function theory (no viable stabilizing closed-loop solution is known without chattering effect, limiting its potential application as a candidate for implementation in autonomous operations).
The third use case is related to the orbital trajectory optimization problem under an extensive set of operational constraints with prescribed limitations in the first and second-order dynamics of the control (a control input smoothing problem in terms of the rotational speed and/or rotational acceleration of the desired thrust direction, still open).

Since almost a decade many papers and studies are being directed in demonstrating that Artificial Intelligence (AI) will play a significant role in guidance and control problems for space missions; a large literature has proposed its application in various Space domains, where AI-based algorithms can enable spacecraft to autonomously perform either navigation or attitude control by processing sensor data and making real-time decisions.
Traditional on-board control engineering has demonstrated to be quite effective in almost all scenarios; yet, being based on a consolidated knowledge of the mission characteristics and of the environment situation, they are less effective in managing unpredicted situations or missions where communication delays make real-time human control impractical.
This work is mainly devoted to a general analyses on the use of AI for autonomous navigation, trajectory optimization, adaptive control and sensor fusion in a difficult predictive context, as in rendezvous and collision avoidance maneuvers.
A rendezvous maneuver in space refers to the process of bringing two spacecraft together in a controlled manner to meet at a specific location and time. It is a critical operation for various space missions, including crewed missions, satellite deployments, spacecraft servicing, and docking operations. Interest on rendezvous and docking maneuver relies also either in the possibility to capture dangerous debris or in performing in orbit servicing maneuvers to elongate a satellite’s mission lifetime.
Collision avoidance concerns predicting and mitigating the risk of collisions between space objects. This involves tracking the trajectories of satellites and debris and providing warnings and recommendations for evasive actions if a potential collision is detected. The evaluation of potential reactions to counteract collision risk from debris or to put in place space traffic management actions able to manage big constellations in a crowded orbital environment are of increasing importance (especially in the recent topic of space Situational Awareness – SSA).
Many different approaches have been proposed for the use of AI-based algorithms in the various trajectory and attitude control problems, without evidencing a conclusive methodology even when applied to “toy-models” (as it is usually the case); a sign that we are still in an evaluation epoch.
Most of the literature is focused on exploring even simple cases with very different techniques trying to put in evidence pros and contras of each approach. In the Space domain the robustness of the design and of the control strategies drive fundamental requirements; so it is of key importance to enforce the knowledge about the theoretical foundations of each AI technique in order to be able to assess its feasibility in the Space domain.
The present work introduces a discussion surveying the most promising AI-based algorithms applied to guidance control and autonomous navigation, with the aim to identify key theoretical aspects potentially able to enforce the knowledge behind some architectural choices, together with proposed related metrics.
In particular the following topics will be discussed and analyzed with reference to their foundational aspects:
• The use of Reinforcement Learning (RL) in rendezvous and proximity operations (RPO); this type of machine learning relies on the concept of learning by trial and error, where a so called agent will explore the environment in an early stage of its training, exploiting what it has learned in a later stage, trading off exploration versus exploitation in a “bounded rationality” scenario . This technique is typically difficult to be used when the environment changes continuously or becomes too complex, as could be the case of a SSA scenario, where new objects have to be identified and explicit actions have to be put in place for collision avoidance or escape maneuvers.
• The use of PINN (Physics Informed Neural Networks); the problem of how to include the physics inside an algorithm based on the use of Neural Networks (NN); two methodological approaches are compared: the one which is strongly based on an inductive approach, which tries to implicitly encapsulate the physics inside the neural network weights (in the form of invariances and symmetries) after an adequate training (to be better defined through specific metrics). The other which tries to reduce the inductive bias introducing the physics through models which constraint the output of the NN. Much theoretical effort has been devoted to his approach but which is still based on “ad hoc” choices depending on the applications.
• Mixed approaches for trajectory optimizations (e.g. multi-layer perceptrons used as value functions, surrogate modelling, deep-Q-Networks, comparison with direct transcription methods,..).

The identification of specific metrics could help in a more general evaluation of the various methods and of the different theoretical approaches, helping the design of these complex algorithmic architectures. In particular it is shown how functions and conceptual instruments, typically used in statistical physics and in information theory ,could be of help to enforce the “explanatory knowledge” behind these techniques, in order to assess their feasibility for application in space.

We will recall the main properties of no-regret algorithms in discrete and continuous time and their use in on-line framework, game learning/ variational inequalities and convex optimization.
We will explicitly describe extensions of projected gradient procedures like “mirror descent” (incremental) or “dual averaging” (cummulative) and compare their performances in the fields mentioned above.

We consider Hegselmann-Krause type models in presence of time de-lays. We are able to prove that the solutions of the Hegselmann-Krause opinion formation model with time variable time delays converge expo-
nentially to consensus without assuming the smallness of the time delay size and the monotonicity of the influence function. Also, we establish the exponential convergence to consensus for the Hegselmann-Krause model with constant time delay and possibly lack of interaction between the agents of the system.
Joint work with C. Pignotti.

Since the beginning of the space era, interplanetary probes have commonly been operated from ground. Operations are conducted by flight control and involve per- forming a number of routine tasks, mainly of scientific, systems engineering, and flight-related nature. Governing the space flight consists of determining the space- craft position, planning its trajectory, and controlling its motion. Accordingly, these activities are known as a whole as Guidance, Navigation and Control (GNC). The EXTREMA project (Engineering Extremely Rare Events in Astrodynamics for Deep-Space Missions in Autonomy) aims towards a paradigm shift on how deep-space GNC is performed, enabling CubeSats with autonomous capabilities. The project has received a consolidator grant from the European Research Council (ERC), a prestigious acknowledgment that funds cutting-edge research in Europe. This presentation is intended to give an overview of EXTREMA, highlighting the approaches, methodologies and objectives; moreover, the expected results, outcomes, and impact on future space exploration scenarios are also discussed.

In on-board computing frameworks, the optimization of resource allocation and decision-making poses significant challenges, particularly in dynamic and different environments. However, distributed (also named, decoupled) optimization techniques have emerged as a promising approach to address these challenges by facilitating collaboration among multiple agents, leveraging their local knowledge. This talk focuses on the application of distributed optimization methods within multi-agent systems operating under time-varying communication networks. Integer linear programming is utilized as a mathematical framework to concentrate our attention. Indeed, such a setting, which provides a structured representation of scheduling problems, aligns seamlessly with the objective of improving response time in earth observation missions carried out by satellite constellations. To verify the effectiveness of the proposed approach, numerical simulations are conducted. These simulations allow for a comprehensive evaluation of the mathematical model’s performance and provide valuable insights into its practical implementation.
In summary, the results shed light on the potential of decentralized optimization strategies in achieving optimal resource allocation and decision-making in the context of Earth Observation. By delving into distributed optimization techniques and their applications to real-world scenarios, this talk offers an understanding of the possibilities and advantages they hold for improving operational efficiency.

The increasing demand of performing in-space activities is opening a new era of interest both for industry and academy [1]. In this study, we consider a generic chaser spacecraft that is able to reach any target belonging to a given set of objects positioned on different Low Earth orbits (SSO) within an assigned time period. Once the chaser spacecraft is launched from Earth, it reaches a number of targets where it performs the requested services and finally returns to Earth (or performs a destructive reentry). A certain value, such as the economic return or a relative priority, is associated with each target/service. As, in general, different sequences and target selections are possible, those maximizing the overall value are searched for. Alternative optimization criteria, such as the minimization of the total time spent or the propellant consumed, can be considered.
Orbit-to-orbit transfers can be very demanding in terms of change in velocity (delta-v). Actually, propellant mass consumption together with the time availability represent major and mutually interacting limitations on the whole mission. In fact, either the transfer duration is reduced at the expense of the delta-v exerted or vice versa. The orbit transfer strategy here considered aims to exploit Earth perturbations in order to enable in-orbit servicing (IOS) commercial sustainability (IOS multi-mission approach). In addition to these substantial overall restrictions, a number of specific operational conditions may occur, such as precedence requirements between targets or temporal slots delimiting the activity initial/final times. Although time, delta-v, and mass can all be properly treated as continuous variables, the problem in question features a pivotal combinatorial nature, since a sequence of steps has to be determined on the basis of the target selection. A set of decisional variables covering all the potential target-step associations is thus implicit. In formalizing the problem, it is realistically assumed that the transfer durations and the corresponding delta-v are inversely proportional. Moreover, the rocket equation is adopted to correlate mass and delta-v variations.
A dedicated mixed integer linear programming (MILP) model is conceived, adopting piece-wise linearization and exploiting convexity wherever opportune [2]. Since real-world applications of interest are expected to give rise to very large-scale instances, a dedicated MILP-based heuristic approach, aimed at yielding satisfactory albeit not-optimal-proven solutions, is currently under study.
Keywords: In-orbit servicing, IOS, IOS mission optimization, Low Earth orbits, LEO, LEO orbit transfers, mixed integer linear programming, MILP.
MSC: 90C11-90C90-90C59.
References

[1] NASA Goddard Space Flight Center “On-Orbit Satellite Servicing Study”. Project Report, October 2010
[2] L. Wolsey, G. Nemhauser, “Integer and Combinatorial Optimization”. Wiley 1999.

Spacecraft (S/C) interplanetary trajectories are typically perturbed by some external disturbance sources, such as third-bodies attraction, possibly drag force contributions during close or aerobraking (AEB) passages, solar radiation pressure, but also by initial thrust and/or attitude errors, for instance. Similar mission scenarios related to critical entry-descent and landing (EDL) phases exist. Such external inputs are injected into the S/C nominal expected transfer or descent trajectories as errors, accumulated continuously. Consequently, the capability to both determine the number and the entities of the so-called trajectory correction manoeuvres (TCMs) to recover or compensate such errors trajectories is one of the most important engineering activity along the entire cycle of the mission design process. It may have sensible impacts on the consolidation of the overall S/C propellant mass budget, which in turn strongly affects the definition of a reliable S/C design fitting with the imposed overall mission requirements. To target specific optimum midcourse, arrival or descent trajectory conditions, the B-plane targeting method is widely used in the field of Mission Analysis. Being the B-plane targeting method based on the differential correction method introduced by Battin [1], the goal of this study is to present the results obtained integrating such procedure in the definition of a reference coordinates map(s) where each coordinate set on the B-plane represents the optimal arrival or insertion conditions that shall be targeted by the general S/C in order to further realize a direct or indirect descent and landing trajectory profiles allowing the maximization of the payload mass to be deployed over the surface of the Moon. Such type of activities are explored in the frame of the “Argonaut” space exploration programme of the European Space Agency (ESA), i.e., the European flagship Moon exploration program. Additionally, several B-plane approaches that offer analytical solutions are explored and suggested [2,3,4].
Keywords: B-Plane, landing, descent, trajectory optimisation, mission analysis
MSC: 34H05 – 9308 -90C26.
References
[1] R.H. Battin, “An Introduction to the Mathematics and Methods of Astrodynamics”. AIAA Education Series, 1987.
[2] W. Kizner, “A Method of Describing Miss Distance for Lunar and Interplanetary Trajectories”. External Publication No. 674, Jet Pro- pulsion Laboratory, California Institute of Technology, 1969.
[3] S.G. Sergeyevsky, R. Cunniff, “Interplanetary Mission Design Handbook, Part 2: Earth to Mars Ballistic Mission Opportunities 1990-2005”. Technical Report JPL Publication 82-43, Jet Propulsion Laboratory, 1983.
[4] V.M. Becerra, “Practical Direct Collocation Methods for Computational Optimal Control“. In: G. Fasano, J.D. Pintér, Eds., Modeling and Optimization in Space Engineering (Springer Optimization and Its Application, 73), pp. 33–60, 2013.

Current initiatives for launcher stage recover and reuse rely on the assertion that these stages have a significant weight regarding the launch cost. The main objective of the Guidance function is to reach the recovery site and to perform a landing maneuver satisfying all kinematic conditions that respect the launcher physical integrity, to contribute to its reuse.
In the case where flight conditions are far outside the qualified flight domain (due for example to an unexpected event occurring during the flight), the landing zone may not be physically reachable anymore. Nevertheless, it could be worth to try to recover the stage in degraded conditions, for example by shifting the landing point in the neighborhood of the nominal landing zone.
The talk describes an on-board re-targeting algorithm updating the guidance target and control, depending on real flight conditions. The Space Shuttle had a similar function, allowing to change the targeted landing site in case of a failure threatening the on-board crew. While this principle shifted the target inside a discrete set of options, the present approach allows to negotiate continuously the mission constraints, and to minimize the amplitude of the corresponding constraint relaxations.
The guidance principle takes into account limitations regarding the set of observable parameters available for guidance decision, maneuver agility bounding relating to vehicle design, and on-board computational restrictions due to avionics. The main principals of the approach are described, its behavior and performance in terms of flight control are then illustrated on the vertical landing of a reusable stage.

We revisit the classical safe landing optimal control problem in aerospace, intro-duced by Miele in the Sixties, where the target is to land a spacecraft on the moon surface at rest with minimal fuel consumption. We show that a detailed analysis in the related 3D phase space uncovers the existence of infinitely many safe landing curves, contrary to several former 2D descriptions that implicitly claim the existence of just one such curve. Our results lead to a deeper understanding of the dynamics and allow for a precise characterization of the optimal control. Such control is known to be bang-bang and our results give a full characterization of the switch position.

The problem of finding the optimal trajectories of multistage launch vehicles from ground to orbit injection implies the inclusion of severe path constraints, in particular during the atmospheric part of the ascent path. These constraints make the dynamics depending on a few parameters and open the possibility of analytic derivations, which can be used either to represent the flight in a sufficiently accurate way or to provide a good guess for iterative optimization algorithms. In particular, in the approach presented here, the atmospheric arcs of the launch vehicle depend on two parameters only: the final flight-path angle at the end of the gravity-turn arc and the duration of the coasting arc following the gravity-turn phase. These two parameters can be quickly obtained by analytical derivations. The exo-atmospheric arc of the trajectory precedes orbit injection. A semi-analytic explicit near-optimal guidance algorithm is presented that is based on the local projection of the position and velocity variables. A minimum-time problem is defined, and consists of finding the optimal thrust direction that minimizes the time of flight for the upper stage to enter the operational orbit. The optimal control problem at hand is proven to be amenable to an analytical solution. This allows translating the minimum-time problem into five nonlinear equations in five unknowns. Their numerical solution can be performed as a real-time process, because a suitable guess, related to intuitive dynamical variables, is available. This research shows that the overall performance of a launch vehicle can be predicted through the previously mentioned approach, based on analytic derivations for the ascent dynamics and near-optimal guidance for the upper stage. Keywords: multistage launch vehicles, explicit guidance.

In the life of a spacecraft the LEOP (Launch and Early Operational Phase) is one of the most critical, because the knowledge of the behavior of the S/C (SpaceCraft) platform is still uncertain. This is in particular true for the propulsive maneuvers performed during the first phases of a mission, when the performance of the spacecraft and thrusters is still unknown.
The maneuvers that modify the trajectory can be carefully planned on ground and executed by the spacecraft under operator supervision or can autonomously be computed by the on-board software. In both cases the maneuvers are traditionally computed and optimized considering specific nominal values for the spacecraft performance (nominal case), while the robustness of the designed trajectory is tested against one or more identified worst-case scenarios or Monte-Carlo simulations. It is considered an interesting topic of discussion the trajectory design improving the robustness of the obtained solution [1,2,3,4].
The problem proposed for discussion is how to find a robust control law that takes into account dramatic thruster underperformance. The insertion of a spacecraft in a highly elliptic orbit with multiple apogee burns is considered as an example and the effect of non-nominal thrust in the first apogee maneuver. The approach is extended to the problem of cooperative deployment of two satellites constellation. An indirect optimization method is proposed in order to find a robust optimal control law that guarantees performance even in such an event. [5]
Keywords: Robust Optimization, indirect optimization, highly elliptic orbit, thruster underperformance
MSC: 34H05, 93C73, 49Mxx.
References
[1] Oguri K. and McMahon J. W.: Stochastic Primer Vector for Robust Low-Thrust Trajectory Design Under Uncertainty. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS.
[2] Ozaki N., Campagnola S. and Funase R.: Tube Stochastic Optimal Control for Nonlinear Constrained Trajectory Optimization Problems. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS, Vol. 43, No. 4, April 2020
[3] Di Carlo M., Vasile M., Greco C. and Epenoy R.: Robust Optimization of Low-Thrust Interplanetary Transfers using Evidence Theory. 29th AAS/AIAA Space Flight Mechanics Meeting, Vol. 168, Pages 339-358
[4] Zavoli A. and Federici L.: Reinforcement Learning for Robust Trajectory Design of Interplanetary Missions. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS. Vol. 44, No. 8, August 2021
[5] Simeoni F., Casalino L., Amelio A.: “Indirect optimization of robust orbit transfer considering thruster underperformance “. In: G. Fasano, J.D. Pintér, Eds., Modeling and Optimization in Space Engineering – New Concepts and Approaches, pp. 349–375, 2023.

The motion planning of powered descent problems has often been treated in the deterministic optimal control framework, which provides efficient theoretical and numerical tools but lacks robustness. In this talk we rather consider robust motion planning: the goal is to design open-loop controls which optimally steer a system to a specific target region while mitigating uncertainties and disturbances which affect the dynamics. Recently, stochastic optimal control has enabled particularly accurate formulations of the problem. Nevertheless, despite interesting progresses, these problem formulations still require expensive numerical computations. In this talk, we start bridging this gap by leveraging statistical linearization. Specifically, through statistical linearization we reformulate the robust motion planning problem as a simpler deterministic optimal control problem subject to additional constraints. We rigorously justify our method by providing estimates of the approximation error, as well as some controllability results for the new constrained deterministic formulation. Finally, we apply our method to the powered descent of a space vehicle, showcasing the consistency and efficiency of our approach through numerical experiments.

This talk will focus on computational methods to optimise trajectories subject to a combination of aleatory and epistemic uncertainty. Methods to incorporate aleatory uncertainty in optimal control are well established but the treatment of epistemic uncertainty has received less attention. Epistemic uncertainty is due to a lack of knowledge, often related to the dynamic model but also to the state of an object at a given time or to the execution of a control action. The talk will present some past and recent developments to model epistemic uncertainty in trajectory optimisation using belief functions, p-boxes or a mix of the two. The talk will then introduce some elements of many-objective and belief optimal control to solve trajectory optimisation problems under epistemic uncertainty and will show how to couple robust trajectory optimisation with the, recently introduced, stochastic dynamical indicators. Finally, some possible uses of machine learning to solve specific problems in trajectory design and optimisation, will be presented.

Systems of interest to astrodynamics, for example launch vehicle trajectories, interplanetary trajectories, or spacecraft attitude maneuvers, are nonlinear systems in which it is important that the control be applied optimally. But determining such trajectories a priori, using some form of the necessary conditions of the Pontryagin Minimum Principle (PMP), the “indirect” approach, or a transcription of the problem into a NLP problem, the “direct” approach, is a computationally difficult task; not presently feasible in real time. The resulting trajectory is of course open-loop and is only guaranteed to be locally optimal. All of these astrodynamics problems can be subject to disturbances or error or uncertainties in the modeling of their dynamics, which will cause the realized trajectories and the expected trajectories to differ.
Ideally, the optimal trajectory from the instantaneous off-nominal state to the target state should be flown. As noted above, except for very unsophisticated dynamic systems a priori optimal trajectories from an off-nominal state can’t be generated in real-time, which rules out their application in feedback controllers. One alternative is an optimal feedback control that uses necessary conditions in terms of the perturbations to the states and controls, in the neighborhood of the nominal optimal trajectory. Another approach relies on learning or intelligent interpolation among a family of pre-computed extremals computed for different initial condition errors and perturbations. In both approaches the computationally expensive work is done on the ground before the system is “flown”. Both approaches are also approximate and require that the realized trajectory be within an expected neighborhood of the nominal trajectory.
A qualitatively different approach, that simultaneously determines the nominal optimal trajectory and the optimal feedback controller, is to solve the system’s associated Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). The HJB equation describes the field of extremal trajectories for an optimal control problem. The principal advantage of the HJB solution, in comparison to solutions using the PMP, is that the solution yields a map of the system value function. This is the optimal “cost-to-go” from any instantaneous state (including of course an off-nominal state the vehicle may be in by virtue of the dynamics modeling errors, initial condition errors, or perturbations previously mentioned) to the desired terminal manifold. Once the solution process for the HJB equation (i.e. for the value function J or its Kruzkov transform v, in our case) has converged, the optimal control is found by a simple minimization of an equation involving v at neighboring points, which can be done in real time or in advance by creating a kriging model of the map. Thus the solution of the HJB equation can provide a feedback control formulation, which provides stability against uncertainty. In fact, for disturbance attenuation, robust feedback controllers can be derived through the HJB formulation. An additional advantage of the HJB approach is that because the solution is based on dynamic programming the extremal trajectories are globally optimal. This is in contrast with solutions via the PMP, which are only guaranteed locally optimal. There are many advantages to solving the optimal control problem using the HJB equation; that it has been done so seldom is due to the numerical difficulties encountered.
The principal difficulty is that the space complexity of the problem is exponential with respect to the number of dimensions of the system. Bellman referred to this as the curse of dimensionality, that is, the PDE is solved numerically on a grid with dimension equal to the number of states; for accuracy the grid should have many points, thus the total number of grid points required rapidly increases with the number of state variables and degree of accuracy required. It is also necessary to evaluate the value function J (or its Kruzkov transform v) at points that are not grid points. Doing this interpolation accurately consumes large amounts of time. Moreover, the value function of the HJB equation may be nondifferentiable, which renders traditional PDE solution methods impractical. In order to deal with these issues, we have employed a newly-developed methodology for the solution of the HJB equation for general nonlinear systems that combines PDE viscosity solutions, quasi-Monte Carlo grids, and kriging regression to implement globally optimal nonlinear feedback controllers for practical applications1.
everal examples of this new paradigm for solution of the HJB equation for astrodynamics systems will be presented including launch vehicle optimal injection into orbit and a spacecraft detumbling problem with 6 states (3 angles, 3 angular velocities) and 3 (thruster) controls.
1. Chilan, C. M. and Conway, B. A., “Optimal Nonlinear Control Using Hamilton-Jacobi-Bellman Viscosity Solutions on Unstructured Grids”, J. Guidance, Control & Dynamics, 43, No. 1, 2020.

We study a variant of the classical safe landing optimal control problem in aerospace, introduced by Miele in the Sixties. Assuming that the spacecraft has a failure and that the thrust (representing the control) can act in both vertical directions, the new target be-comes to land safely by minimizing time, no matter of what the consumption is. In dependence of the initial data (height, velocity, and fuel), we prove that the optimal control can be of four different kinds, all being piecewise constant. Our analysis covers all possible situations, including the nonexistence of a safe landing strategy due to the lack of fuel or for heights/velocities for which also a total braking is insufficient to stop the spacecraft.

I will discuss second-order sufficient optimality conditions for weak local minima of the Mayer type optimal control problems with a general control constraint U in R^m. These conditions are formulated in terms of critical directions and second order tangents to U at critical directions. The obtained results are illustrated through an example of classical control constraints defined by inequalities involving functions with linearly independent gradients of active constraints, and also in the case when U is the union of two sets possibly having singularities at points of their intersection. This second case is the one where the known approaches do not apply.
The paper is written in collaboration with N. P. Osmolovskii, Polish Academy of Sciences.

Multi-arc optimal control problems regard dynamical systems governed by differential equations, subject to either discontinuities or constraints at several intermediate times. These constraints can involve both the state variables, the control variables, and the time-independent unknown parameters. The general formulation of the first-order necessary conditions for optimality leads to identifying a set of corner conditions, to enforce at the junction points between two subsequent subarcs. In their general form, these additional conditions do not lead to a closed-form solution for the jump relations of the costate variables at the junction times, except for some remarkable special cases of practical interest.To obtain a closed-form, sequential solution of the jump equations, distinct (and
coexisting) representations for the state can be profitably used, in conjunction with implicit costate transformation. In the context of indirect solution approaches, this allows identifying a parameter set that retains the reduced sizeof the typical set associated with a single-arc optimization problem. As an illustrative example, the minimum-time low-thrust orbit transfer about a single attracting body, with eclipse constraints on the available thrust, is addressed. Orbit dynamics is described with the use of modified equinoctial elements (MEE), able to mitigate the hypersensitivity issues in detecting the initial adjoint variables, which is a major difficulty when indirect algorithms are employed. Low thrust is assumed to be unavailable when the spacecraft is shadowed. The entire transfer path is split in a large number of light arcs and eclipse arcs. The corner conditions for optimality are enforced at the light-to-shadow and shadow-to-light transitions, and can be sequentially solved in closed form, using two representations for the state (MEE and spherical coordinates), in conjunction with implicit costate transformation. This approach was successfully applied to detecting low-thrust orbit transfers about both Earth and Moon, even in the presence of multiple celestial bodies responsible for shadowing. For the problem at hand, the multi-arc formulation avoids using regularization or averaging in the numerical solution process, and leads to identifying the thrust direction instantaneously, unlike averaging. The multi-arc formulation can also be regarded as a viable option for other challenging space trajectory optimization problems, such as (i) orbit transfers in the presence of multiple attracting bodies or (ii) orbit transfers with different propulsion systems or further operational constraints. Other than potential hypersensitivity issues on the detection of the unknown initial values of the adjoint variables, these problems require enforcement of the extended corner conditions for optimality, as well as the identification of a closed-form sequential solution of these conditions, possibly through a convenient choice of different state representations.