## Section: Application Domains

### Space engineering, satellites, low thrust control

Space engineering is very demanding in terms of safe and high-performance (for instance in terms of fuel consumption, because only a finte amount of fuel is onborad a sattelite for all its “life”) control laws. 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 ): in the geocentric case, orbit transfer rather than station keeping; also we do not discuss attitude control.

A space vehicle is subject to

- gravitational forces, from one or more central bodies (the corresponding acceleration is denoted by ${F}_{\mathrm{grav}.}$ below),

- other forces of small amplitude (the corresponding acceleration is denoted by ${F}_{2}$ below),

- a thrust, the control, produced by a propelling
device; it is the $G\phantom{\rule{0.166667em}{0ex}}u$ term below; assume for simplicity that control in all
directions is allowed, *i.e.* $G$ is an invertible matrix.

In position-velocity coordinates, its dynamics can be written as

$\ddot{x}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4pt}{0ex}}{F}_{\mathrm{grav}.}(x,t)\phantom{\rule{0.277778em}{0ex}}\left[\phantom{\frac{1}{2}}+\phantom{\rule{0.277778em}{0ex}}{F}_{2}(x,\dot{x},t)\right]\phantom{\rule{0.277778em}{0ex}}+\phantom{\rule{0.277778em}{0ex}}G(x,\dot{x})\phantom{\rule{0.166667em}{0ex}}u\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\parallel u\parallel \le {u}_{\mathrm{max}}.$ | (1) |

The second term is often neglected in the design of the control. Time-dependence reflects the movement of attracting celestial bodies if there is more than one (see below).

#### Low thrust

means that ${u}_{\mathrm{max}}$ is small, or more precisely that the maximum magnitude of $G\phantom{\rule{0.166667em}{0ex}}u$ is small with respect to the one of ${F}_{\mathrm{grav}.}$ (but is usually large compared to ${F}_{2}$). Hence the influence of the control is very weak instantaneously, and trajectories can only be significantly modified by accumulating the effect of this low thrust on a long time. Obviously this is possible only because the free system is somehow conservative. This was “abstracted” in section 3.5 .

*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, ${I}_{sp}$) is a
characteristic of the engine; the higher the
${I}_{sp}$, the smaller the mass of particles needed for the same change in the vehicle momentum.
Engines with a higher ${I}_{sp}$ are highly desirable
because, for the same maneuvers, they reduce the mass of "fuel" to be taken
on-board the satellite, hence leaving more room (mass) for the payload.
“Classical” chemical engines use combustion to eject particles, at a somehow limited speed even with very efficient fuel;
the more recent electric engines use a magnetic field to accelerate particles and eject them at a considerably higher speed;
however electrical power is limited (solar cells), and only a small amount of particles can be accelerated per unit of time,
inducing the limitation on thrust magnitude.

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, ${F}_{\mathrm{grav}.}$ is the Newtonian gravitation force of the earth (it then does not depend on time); ${F}_{2}$ 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, ${F}_{\mathrm{grav}.}$ 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 ${F}_{\mathrm{grav}.}$ in (1 ) is the one of the restricted three body problem and dependence on time reflects the movement of the two “big” attracting bodies.In the 2003 mission SMART-1, the project was to send a small observation vehicle from the Earth to the Moon using low-thrust propulsion. The vehicle was launched and reached the moon: the goal was achieved; precise reports on the control used can be found in [73] . There was no attempt to minimize fuel consumption, or transfer time, and it is not a surprise that the implemented solution is far from optimal with respect to these criteria. In a recent work in the Dijon team, and in collaboration with J. Gergaud from APO team at IRIT- ENSEEIHT (Toulouse) we have computed optimal trajectories for the Earth-Moon transfer according to the energy minimization problem, the time minimal transfer or the propellant minimization consumption. These results combine geometric optimal control and numerical simulations with adapted numerical codes. The contributions are described in G. Picot Phd thesis, (Dijon November 2010), B. Daoud (Phd thesis defended at Dijon in October 2011) and the numerical codes are developed by O. Cots (Phd thesis to be defended at Dijon in June 2012). Our previous work [29] gives a

*feedback*solution for this problem, divided in three phases, the design being based on a two-body model in the first and last phase, where the effect of the primaries is preponderant and the second phase is in the neighborhood of the L2 Lagrange point. This opens perspectives in trajectory optimization; see the recent work [32] . For a state of the art, the reader may refer to [62] or [28] .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 $u$ has dimension 3 with an invertible matrix $G$, then the control system
is “static feedback linearizable”, and a fortiori flat, see section
3.2 . However, implementing the static feedback transformation would consist in using the control to
“cancel” the gravitation; this is obviously impossible since the available thrust is very small.
As mentioned in section
3.1 , point 3 , the problem remains fully nonlinear in spite of this “linearizable” structure (However, the linear approximation around *any* feasible trajectory is controllable (a periodic time-varying linear system);
optimal control problems will have no singular or abnormal trajectories.).

#### 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.