There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

M. MAGGIORE University of Toronto C. MANZIE Univ. of Melbourne A. MARINO University of Salerno P. MHASKAR McMaster Univ. G. NOTARSTEFANO University of Salento P. ODGAARD Vestas Wind Systems M. OISHI University of New Mexico N. OLGAC Univ. of Connecticut T. OOMEN Eindhoven University Y. ORLOV CICESE Mexico Y. PAN National University of Singapore C. PANAYIOTOU Univ. of Cyprus G. PAPAFOTIOU ABB A. PAVLOV NTNU, Norway G. PIN Electrolux S. PIROZZI Università di Napoli H. POTA Univ. of New South Wales at ADFA C. PRIEUR CNRS E. PUNTA National Research Council, Italy N. QUIJANO Universidad de los Andes Colombia D. M. RAIMONDO University of Pavia G. A. ROVITHAKIS Univ. of Thessaloniki K. RUDIE Queen’s Univ.

Ieee transactions on control systems technology

A novel stable and safe model predictive control framework for autonomous rendezvous and docking with a tumbling target

An efficient indirect method is presented to determine fuel-optimal many-revolution low-thrust transfers in the presence of Earth-shadow eclipses. Specifically, the events of shadow entrance and exit are modeled as interior-point constraints. Following the observation that an ill-conditioned state transition matrix may occur when the spacecraft flies over the edge of the shadow, a two-level continuation scheme is introduced to generate many-revolution trajectories. The established computational framework integrates analytic derivatives, switching detection, and continuation with an augmented flowchart, which yields discontinuous bang-bang solutions and their gradients. Transfers from a geostationary transfer orbit to a geostationary orbit are simulated to illustrate the effectiveness and efficiency of the method developed. 

/pdf/indirect-optimization-of-fuel-optimal-many-revolution-low-2beu2s6k.pdf

Indirect Optimization of Fuel-Optimal Many-Revolution Low-Thrust Transfers With Eclipses

Global Optimization of Multiple-Spacecraft Rendezvous Mission via Decomposition and Dynamics-Guide Evolution Approach

This letter tackles spacecraft optimal control problems in which the cost function is defined by a sum of vector norms, in order to optimize fuel consumption while achieving sparse actuation. A model predictive control (MPC) strategy is devised for such type of problems, accounting for different spacecraft maneuvering modes. Closed-loop stability is guaranteed by a conic Lyapunov function, which is employed as a terminal cost in the formulation. A systematic method to construct such function is presented. The proposed design is compared to a standard quadratic MPC scheme on a long-range rendez-vous mission.

Sum-of-Norms Model Predictive Control for Spacecraft Maneuvering

The optimization of low-thrust, multi-revolution orbit transfer trajectories is often regarded as a difficult problem in modern astrodynamics. In this paper, a flexible and computationally efficient approach is presented for the optimization of low-thrust orbit transfers under eclipse constraints. The proposed approach leverages a new dynamic model of the orbital motion and a Lyapunov-based initial guess generation scheme that is very easy to tune. A multi-objective, single-phase formulation of the optimal control problem is devised, which provides a convenient way to trade off fuel consumption and time of flight. A distinctive feature of such a formulation is that it requires no prior information about the structure of the optimal solution. Simulation results for two benchmark orbit transfer scenarios indicate that minimum-time, minimum-fuel and mixed time/fuel-optimal instances of the control problem can be readily solved via direct collocation, while incurring a significantly lower computational demand with respect to existing techniques.

Optimal Low-Thrust Orbit Transfers Made Easy: A Direct Approach

In this paper, the trajectory planning problem for autonomous rendezvous and docking between a controlled spacecraft and a tumbling target is addressed. The use of a variable planning horizon is proposed in order to construct an appropriate maneuver plan, within an optimization-based framework. The involved optimization problem is nonconvex and features nonlinear constraints. The main contribution is to show that such problem can be tackled effectively by solving a finite number of linear programs. To this aim, a specifically conceived horizon search algorithm is employed in combination with a polytopic constraint approximation technique. The resulting guidance scheme provides the ability to identify favourable docking configurations, by exploiting the time-varying nature of the optimization problem endpoint. Simulation results involving the capture of the nonoperational EnviSat spacecraft indicate that the method is able to generate optimal trajectories at a fraction of the computational cost incurred by a state-of-the-art nonlinear solver.

Variable-Horizon Guidance for Autonomous Rendezvous and Docking to a Tumbling Target.

Model predictive control (MPC) is receiving increasing attention in space applications, as a key technology for enhancing autonomy of the flight control system. Sum-of-norms formulations are specifically suited to this context, because they allow to optimize meaningful performance figures and to promote control sparsity. This brief presents a sum-of-norms MPC scheme for linear periodically time-varying systems. Closed-loop stability is proven by suitably defining periodic sequences of terminal weights and terminal sets. The proposed solution is applied to a rendezvous case study involving periodic dynamics due to geopotential effects and solar eclipses.

Sum-of-Norms Periodic Model Predictive Control for Space Rendezvous

A periodic model predictive control (MPC) scheme is proposed for tracking halo orbits. The problem is formulated and solved in the elliptic restricted three-body problem (ER3BP) setting. The reference trajectory to be tracked is designed by using eccentricity continuation techniques. The MPC design exploits the periodicity of the tracking model and guarantees exponential stability of the linearized closed-loop system, through a suitable choice of the terminal set and weight matrices. A sum-of-norms cost function is adopted to promote fuel saving. The proposed control scheme is validated on two simulated missions in the Earth–Moon system, which, respectively, involve station keeping on a halo orbit near the L1 Lagrange point and rendezvous to a halo orbit near the L2 Lagrange point. Results illustrate the advantage of designing the reference trajectory and the periodic control directly in the ER3BP setting versus approximate solutions based on the circular restricted three-body problem (CR3BP). 

Periodic Model Predictive Control for Tracking Halo Orbits in the Elliptic Restricted Three-Body Problem

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