THIS POSITION HAS BEEN FILLED !
Proposed by INRIA and Thales Alenia Space
Funded by : Thales Alenia Space and Région Provence Alpes Côte d’Azur
The PhD student will be employed by INRIA, location: Sophia Antipolis, team McTAO
Context
With over a million orbital objects of various size referenced around the Earth, and new satellites or satellite constellations being launched more and more frequently, the risk for collisions is obviously increasing in a significant manner, in particular on Low Earth orbits (LEO). There is a clear need for some automatic process that, whenever the risk of an encounter is identified between the considered primary spacecraft, equipped with an active propulsion system, and one or more “secondary” non-cooperative objects (debris or passive satellites) is identified, with any possible encounter geometry and any relative velocity (fast or slow encounter), would plan a modification of the guidance of the primary (for instance planned beforehand for orbit transfer or station keeping, without knowledge of possible collisions) that guarantees avoidance of these collisions, on a typical 2 to 7 day time horizon.
Although this topic is somehow recent (The First International Orbital Debris Conference was held in Sugar Land in December, 2019, cf. [1]), it gave rise to some interesting work in the last 10 years, mostly considering impulsive maneuvers, i.e. high thrust. Designing low thrust avoidance maneuvers is more delicate and was less studied. Low thrust does not impeach controllability but the maneuvering time depends, of course, on the available thrust and continuous thrust makes computations more delicate. As a model, one may use the full (nonlinear) Gauss equations, taking any type of perturbations into account, as in [1], or the linearized Tschauner-Hempel model for relative movement as in [2,5], there is a trade-off between computation complexity, numerical efficiency, and accuracy. Gauss equations may also be averaged (see the rich literature on averaging Gauss equations, dating back to Edelbaum’s pioneering work in the 60s) and give rise to semi-analytic methods combining explicit approximate average integration and numerical integration as in [1]. A common feature to all methods is the need for a reliable quantitative evaluation of the collision risk, or of the “distance” to collision, combining geometry and risk estimation; they lead to notions like “Closest approach” or “Time of Closest Approach” in the analysis of the possible collision. In the framework of optimal control, these can be used either as a penalty term in the cost function or as a constraint, forcing solutions to remain under a certain threshold for the risk. A penalty (for instance a regular barrier function) can also be combined with continuation (discrete or differential) on a penalty hyper-parameter to connect the constraint-free problem with the constrained one (treating state constraints is still quite difficult in optimal control, all the more so as they are non-convex in the avoidance case). These approaches are present in [2, 4, 6]. Other types of methods must be mentioned: in [5], taking advantage of differential flatness of the linearized equations, the optimization problem (independently of the cost function) is reformulated as one parameterized by the flat outputs, and polynomial approximation for constraint and flat outputs leads to an approximate semi-definite problem; this does not allow a very large number of variables but may provide certifiable avoidance. A different heuristic for low-trust control is to design an impulsive maneuver and transpose it into a succession of continuous thrust arcs having an equivalent end result.
Objective and methodology
The focus will be on typical (long) low thrust orbital transfers for positioning a satellite, with a nominal “long term” control law decided beforehand without taking possible collisions into account, and on the design of avoidance methods based on modification of the control law, preferably on specific time slots and as independently as possible of the initial design. Some suggested directions are as follows, they have to be enriched by a complete and deep bibliography of the subject.
- Momentary interruptions of the thrust on some time-slots. This is easy to implement and automatize. It requires a combinatorial analysis to identify the number of interruptions and their schedule (or position on the orbit) in order to minimize the perturbation of the long-term reference trajectory (orbital performance loss criteria) while guaranteeing that the collision risk is down to a certain given threshold.
- Solving the minimum energy optimal control problem with collision avoidance as a state constraint is obviously the most satisfying solution, but very complex. A reliable solution, with guaranteed numerical convergence, requires some simplifications; for instance, the use of time-scale separation in the case of fast encounters will be investigated.
- Another control technique is based on artificial potentials (or control Lyapunov functions), or variants like the notion of “control barrier functions” [7] that is taylored for collision avoidance. This and other less standard solutions like the flatness based one mentionned above will require investigation too.
Solutions will be evaluated on realistic use cases provided by Thales Alenia Space, concerning low-thrust positioning on either GEO or LEO-MEO (constellations) orbits.
A background in control theory and/or dynamical systems and/or space mechanics is expected.
References
[1] Colombo, C. et al. « Introducing MISS, a new tool for collision avoidance analysis and design ». 1st Int’l Orbital Debris Conference, Sugar Land, Dec. 2019.
[2] Epenoy, R. « Fuel-Optimal Trajectories for Continuous-Thrust Orbital Rendezvous with Collision Avoidance Constraint ». J. of Guidance, Control and Dynamics, 34 (2011), no. 2, 493-503.
[3] Graziano et al. « CryoSat collision warning and low thrust avoidance maneuver strategy ». Proceedings 3 rd European Conference on Space Debris, Darmstadt, March, 2001
[4] Lee, K. et al. « Near-Optimal Guidance and Control for Spacecraft Collision Avoidance Maneuvers ». AIAA/AAS Astrodynamics Specialist Conference, Aug. 2014. erence. August 2014.
[5] Louembet, C. et al. « Collision avoidance in low thrust rendezvous guidance using flatness and positive B-splines ». American Control Conference, San Francisco, 2011.
[6] Salemme, G. « Continuous-thrust collision avoidance manoeuvres optimization ». MS thesis, Polit. Milano, 2019.
[7] Breeden, J. Et al. « High Relative Degree Control Barrier Functions Under Input Constraints ». Proceedings of the Conference on Decision and Control (CDC), Austin, 2021.
Contact:
If interested, please contact us : Jean-Baptiste Pomet, Lamberto Dell’Elce
(jean-baptiste.pomet and lamberto.dell-elce “at” inria.fr).
