Section: Application Domains
Space engineering, satellites, low thrust control
Space engineering is very demanding in terms of safe and high-performance control laws (for instance optimal in terms of fuel consumption, because only a finite amount of fuel is onborad a sattelite for all its “life”). It is therefore prone to real industrial collaborations.
We are especially interested in trajectory control of space vehicles using their own propulsion devices, outside the atmosphere. Here we discuss “non-local” control problems (in the sense of section 3.1 point 1 ): orbit transfer rather than station keeping; also we do not discuss attitude control.
In the geocentric case, a space vehicle is subject to
- gravitational forces, from one or more central bodies (the corresponding
acceleration is denoted by
- a thrust, the control, produced by a propelling
device; it is the
- other “perturbating” forces (the corresponding acceleration is denoted by
In position-velocity coordinates, its dynamics can be written as
In the case of a single attracting central body (the earth) and in a geocentric frame,
Low thrust
Low thrust means that
Why low thrust ? The common principle to all propulsion devices is to eject particles, with
some relative speed with respect to the vehicle; conservation of momentum
then induces, from the point of view of the vehicle alone, an external force, the “thrust” (and a mass decrease).
Ejecting the same mass of particles with a higher relative speed
results in a proportionally higher thrust; this relative speed (specific impulse,
Electric engines theoretically allow many more maneuvers with the same amount of particles, with the drawback that the instant force is very small; sophisticated control design is necessary to circumvent this drawback. High thrust engines allow simpler control procedures because they almost allow instant maneuvers (strategies consist in a few burns at precise instants).
Typical problems
Let us mention two.
-
Orbit transfer or rendez-vous. It is the classical problem of bringing a satellite to its operating position from the orbit where it is delivered by the launcher; for instance from a GTO orbit to the geostationary orbit at a prescribed longitude (one says rendez-vous when the longitude, or the position on the orbit, is prescribed, and transfer if it is free). In equation (1 ) for the dynamics,
is the Newtonian gravitation force of the earth (it then does not depend on time); contains all the terms coming either from the perturbations to the Newtonian potential or from external forces like radiation pressure, and the control is usually allowed in all directions, or with some restrictions to be made precise. -
Three body problem. This is about missions in the solar system leaving the region where the attraction of the earth, or another single body, is preponderant. We are then no longer in the situation of a single central body,
contains the attraction of different planets and the sun. In regions where two central bodies have an influence, say the earth and the moon, or the sun and a planet, the term in (1 ) is the one of the restricted three body problem and dependence on time reflects the movement of the two “big” attracting bodies.An issue for future experimental missions in the solar system is interplanetary flight planning with gravitational assistance. Tackling this global problem, that even contains some combinatorial problems (itinerary), goes beyond the methodology developed here, but the above considerations are a brick in this puzzle.
Properties of the control system.
If there are no restrictions on the thrust direction, i.e., in equation (1 ), if the control
Context for these applications
The geographic proximity of Thales Alenia Space, in conjunction with the “Pole de compétitivité” PEGASE in PACA region is an asset for a long term collaboration between Inria - Sophia Antipolis and Thales Alenia Space (Thales Alenia Space site located in Cannes hosts one of the very few European facilities for assembly, integration and tests of satellites).
B. Bonnard and J.-B. Caillau in Dijon have had a strong activity in optimal control for space, in collaboration with the APO Team from IRIT at ENSEEIHT (Toulouse), and sometimes with EADS, for development of geometric methods in numerical algorithms.