Sensitivity of Optimal Control Problems Arising from their Hamiltonian Structure

International audienceFirst-order necessary conditions for optimality reveal the Hamiltonian nature of optimal control problems. Regardless of the overwhelming awareness of this result, the implications that it entails have not been fully explored. We discuss how the symplectic structure of optimal control constrains the flow of sub-volumes in the phase space. Special emphasis is devoted to dynamics in the neighborhood of optimal trajectories and insight is gained into how errors in the initial states affect terminal conditions. Specifically, we prove that if the optimal trajectory does not satisfy a particular condition, then there exists a set of variations in the initial states yielding a greater error in norm when mapped to the terminal time through the state transition matrix. We relate this result to the sensitivity problem in solving indirect problems for optimal control

Chattering-free Sliding Mode Control for Propellantless Rendez-vous using Differential Drag

peer reviewedThis paper develops a differential drag-based sliding mode controller for satellite rendez-vous. It is chattering-free and avoids bang-bang type control to adjust the relative motion more efficiently. In spite of uncertain nonlinear perturbations and disturbances, it is shown that the in-plane relative motion between two satellites can be effectively controlled by regulating the drag difference. An adaptive tuning rule is also presented such that the errors are suppressed to lie within a desired error box. The proposed controller is simple and easy to implement in a small satellite, and numerical simulations are carried out to demonstrate its effectiveness in a high fidelity environment

Periodic Corrections in Secular Milankovitch Theory Applied to Passive Debris Removal

International audienceMost cartographic stability maps advocated for use in the new passive debris removal ideology based on orbital resonances are obtained through crude averaging methods. This means that from an operational perspective, its not clear where in the osculating space one should actually target to place the satellite on a natural disposal trajectory. It is also not obvious what effects the short-periodic terms may have on these re-entry solutions. We will derive the periodic corrections terms for the dominant perturbations affecting Earth satellites and investigate these considerations

Multi-phase averaging of time-optimal low-thrust transfers

International audienceAn increasing interest in optimal low-thrust orbital transfers was triggered in the last decade by technological progress in electric propulsion and by the ambition of efficiently leveraging on orbital perturbations to enhance the maneuverability of small satellites. The assessment of a control sequence that is capable of steering a satellite from a prescribed initial to a desired final state while minimizing a figure of interest is referred to as maneuver planning. From the dynamical point of view, the necessary conditions for optimality outlined by the infamous Pontryagin maximum principle (PMP) reveal the Hamiltonian nature of the system governing the joint motion of state and control variables. Solving the control problem via so-called indirect techniques, e.g., shooting method, requires the integration of several trajectories of the aforementioned Hamiltonian. In addition , PMP conditions exhibit very high sensitivity with respect to boundary values of the satellite longitude owing to the fast-oscillating nature of orbital motion. Hence, using perturbation theory to facilitate the numerical solution of the planning problem is appealing. In particular, averaging techniques were used since the early space age to gain understanding into the long-term evolution of perturbed satellite trajectories. However, it is not generally possible to treat low-thrust as any other perturbation (whose spectral content is well defined and predictable) because the control variables may introduce additional frequencies in the system. The talk focuses on time optimal maneuvers in a perturbed orbital environment, and it addresses two questions: (1) Is it possible to average the vector field of this problem? Optimal control Hamiltonians are not in the classical form of fast-oscillating systems. However, we demonstrate that averaged trajectories well approximate the original system if the ad-joint variables of the PMP (i.e., conjugate momenta associated to the enforcement of the equations of motion) are adequately transformed before integrating the averaged trajec-tory. We discuss this transformation in detail, and we emphasize fundamental differences with respect to well-known mean-to-osculating transformations of uncontrolled motion. (2) What is the impact of orbital perturbations and their frequencies on the controlled tra-jectory? We show that control variables are highly sensitive to small exogenous forces. Hence, even the crossing of a high-order resonance may trigger a dramatic divergence between trajectories of the averaged and original system. We then discuss how averaged resonant forms may be used to avoid this divergence. The methodology is finally applied to a deorbiting maneuver leveraging on solar radiation pressure. The presence of eclipses make the original planning problem highly challenging. Averaging with respect to satellite and Sun longitudes drastically simplifies the extremal flow yielding an averaged counterpart of the PMP conditions, which is reasonably easy to solve

Periodic and Quasi-Periodic Orbits near Close Planetary Moons

International audienceUpcoming missions toward remote planetary moons will fly in chaotic dynamic environments that are significantly perturbed by the oblateness of the host planet. Such a dominant perturbation is often neglected when designing spacecraft trajectories in planetary moon systems. This paper introduces a new time-periodic set of equations of motion that is based on the analytical solution of the zonal equatorial problem and better describes the dynamic evolution of a spacecraft subject to the gravitational attraction of a moon and its oblate host planet. Such a system, hereby referred to as the zonal hill problem, remains populated by resonant periodic orbits and families of two-dimensional quasi-periodic invariant tori that are calculated by means of numerical continuation procedures. The resulting periodic and quasi-periodic trajectories are investigated for the trajectory design of future planetary moons explorers

Multiple bifurcations around 433 Eros with Harmonic Balance Method

peer reviewedThe objective of this paper is to carry out periodic orbital propagation and bifurcations detection around asteroid 433 Eros. Specifically, we propose to exploit a frequency-domain method, the harmonic balance method, as an efficient alternative to the usual time integration. The stability and bifurcations of the periodic orbits are also assessed thanks to the Floquet exponents. Numerous periodic orbits are found with various periods and shapes. Different bifurcations, including period doubling, tangent, real saddle and Neimark-Sacker bifurcations, are encountered during the continuation process. Resonance phenomena are highlighted as well

Periodic Corrections in Secular Milankovitch Theory Applied to Passive Debris Removal

International audienceMost cartographic stability maps advocated for use in the new passive debris removal ideology based on orbital resonances are obtained through crude averaging methods. This means that from an operational perspective, its not clear where in the osculating space one should actually target to place the satellite on a natural disposal trajectory. It is also not obvious what effects the short-periodic terms may have on these re-entry solutions. We will derive the periodic corrections terms for the dominant perturbations affecting Earth satellites and investigate these considerations

Considerations on Two-Phase Averaging of Time-Optimal Control Systems

International audienceAveraging is a valuable technique to gain understanding of the long-term evolution of dynamical systems characterized by slow dynamics and fast periodic or quasi-periodic dynamics. Averaging the extremal flow of optimal control problems with one fast variable is possible if a special treatment of the adjoint to this fast variable is carried out. The present work extends these results by tackling averaging of time optimalsystems with two fast variables, that is considerably more complex because of resonances. No general theory is presented, but rather a thorough treatement of an example, based on numerical experiments. After providing a justification of the possibility to use averaging techniques for this problem ``away from resonances'' and discussing compatibility conditions between adjoint variables of the original and averaged systems, we analyze numerically the impact of resonance crossings on the dynamics of adjoint variables. Resonant averaged forms are used to model the effect of resonances and cross them without loosing the accuracy of the averaging prediction

On the convergence of time-optimal maneuvers of fast-oscillating control systems

International audienceFor a control system with one fast periodic variable, with a small parameter measuring the ratio between time derivatives of fast and slow variables, we consider the Hamiltonian equation resulting from applying Pontryagin maximum principle for the minimum time problem with fixed initial and final slow variables and free fast variable. One may perform averaging at least under normalization of the adjoint vectors and define a "limit" average system. The paper is devoted to the convergence properties of this problem as the small parameter tends to 0. We show that using the right transformations between boundary conditions of the "real" and average systems leads to a reconstruction of the fast variable on interval of times of order 1/ε where ε is the small parameter. This is only evidenced numerically in this paper. Relying on this, we propose a procedure to efficiently reconstruct the solution of the two point boundary problem for nonzero ε using only the solution of the average optimal control problem

Multi-phase averaging of time-optimal low-thrust transfers

International audienceAn increasing interest in optimal low-thrust orbital transfers was triggered in the last decade by technological progress in electric propulsion and by the ambition of efficiently leveraging on orbital perturbations to enhance the maneuverability of small satellites. The assessment of a control sequence that is capable of steering a satellite from a prescribed initial to a desired final state while minimizing a figure of interest is referred to as maneuver planning. From the dynamical point of view, the necessary conditions for optimality outlined by the infamous Pontryagin maximum principle (PMP) reveal the Hamiltonian nature of the system governing the joint motion of state and control variables. Solving the control problem via so-called indirect techniques, e.g., shooting method, requires the integration of several trajectories of the aforementioned Hamiltonian. In addition , PMP conditions exhibit very high sensitivity with respect to boundary values of the satellite longitude owing to the fast-oscillating nature of orbital motion. Hence, using perturbation theory to facilitate the numerical solution of the planning problem is appealing. In particular, averaging techniques were used since the early space age to gain understanding into the long-term evolution of perturbed satellite trajectories. However, it is not generally possible to treat low-thrust as any other perturbation (whose spectral content is well defined and predictable) because the control variables may introduce additional frequencies in the system. The talk focuses on time optimal maneuvers in a perturbed orbital environment, and it addresses two questions: (1) Is it possible to average the vector field of this problem? Optimal control Hamiltonians are not in the classical form of fast-oscillating systems. However, we demonstrate that averaged trajectories well approximate the original system if the ad-joint variables of the PMP (i.e., conjugate momenta associated to the enforcement of the equations of motion) are adequately transformed before integrating the averaged trajec-tory. We discuss this transformation in detail, and we emphasize fundamental differences with respect to well-known mean-to-osculating transformations of uncontrolled motion. (2) What is the impact of orbital perturbations and their frequencies on the controlled tra-jectory? We show that control variables are highly sensitive to small exogenous forces. Hence, even the crossing of a high-order resonance may trigger a dramatic divergence between trajectories of the averaged and original system. We then discuss how averaged resonant forms may be used to avoid this divergence. The methodology is finally applied to a deorbiting maneuver leveraging on solar radiation pressure. The presence of eclipses make the original planning problem highly challenging. Averaging with respect to satellite and Sun longitudes drastically simplifies the extremal flow yielding an averaged counterpart of the PMP conditions, which is reasonably easy to solve