Control Strategies for Solar Sail SMAI 2011 ` es 1 A. Jorba 2 A. Farr´ & 1 Institut de M´ eleste et de Calcul des ´ ecanique C´ Eph´ em´ erides, Observatoire de Paris ( afarres@imcce.fr ) 2 Departament de Matem` atica Aplicada i ` Analisi, Universitat de Barcelona ( angel@maia.ub.es ) 23 - 27 May 2011
Background Station Keeping + realistic model Conclusions 1 Brief Introduction to Solar Sails 2 Station Keeping Strategies Around Equilibria 3 Towards a More Realistic Model 4 Conclusions & Future Work es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 2 / 42
Background Station Keeping + realistic model Conclusions What is a Solar Sail ? • Solar Sails a proposed form of propulsion system that takes advantage of the Solar radiation pressure to propel a spacecraft. • The impact of the photons emitted by the Sun on the surface of the sail and its further reflection produce momentum on it. • Solar Sails open a wide new range of possible missions that are not accessible by a traditional spacecraft. es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 3 / 42
Background Station Keeping + realistic model Conclusions There have recently been two successful deployments of solar sails in space. • IKAROS: in June 2010, JAXA managed to deploy the first solar sail in space. • NanoSail-D2: in January 2011, NASA deployed the first solar sail that would orbit around the Earth. es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 4 / 42
Background Station Keeping + realistic model Conclusions The Solar Sail We consider the solar sail to be flat and perfectly reflecting. Hence, the force due to the solar radiation pressure is in the normal direction to the surface of the sail. The force due to the sail is defined by the sail’s orientation and the sail’s lightness number . • The sail’s orientation is given by the normal vector to the surface of the sail, � n . It is parametrised by two angles, α and δ . • The sail’s lightness number is given in terms of the dimensionless parameter β . It measures the effectiveness of the sail. Hence, the force is given by: F sail = β m s � n � 2 � � � r s ,� n . r 2 ps es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 5 / 42
Background Station Keeping + realistic model Conclusions The Dynamical Model We use the Restricted Three Body Problem (RTBP) taking the Sun and Earth as primaries and including the solar radiation pressure due to the solar sail. Z Sail n � � F Sun � F Earth X 1 − µ µ Earth Sun Y es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 6 / 42
Background Station Keeping + realistic model Conclusions Equilibrium Points (I) • The RTBP has 5 equilibrium points ( L i ). For small β , these 5 points are replaced by 5 continuous families of equilibria, parametrised by α and δ . • For a fixed small value of β , we have 5 disconnected family of equilibria around the classical L i . • For a fixed and larger β , these families merge into each other. We end up having two disconnected surfaces, S 1 and S 2 . Where S 1 is like a sphere and S 2 is like a torus around the Sun. • All these families can be computed numerically by means of a continuation method. es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 7 / 42
Background Station Keeping + realistic model Conclusions Equilibrium Points (II) Equilibrium points in the XY plane T2 T2 1 T1 0.04 T1 0.5 0.02 Z 0 Z 0 Sun Earth -0.02 -0.5 -0.04 -1 -1 -0.5 0 0.5 1 -1.02 -1.01 -1 -0.99 -0.98 -0.97 X X Equilibrium points in the XZ plane 0.04 T2 T2 0.02 0.03 0.02 0.01 0.01 0 0 Z Z Sun Earth -0.01 -0.01 -0.02 -0.03 -0.02 -0.04 -1 -0.5 0 0.5 1 -1.01 -1.005 -1 -0.995 -0.99 -0.985 -0.98 X X es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 8 / 42
Background Station Keeping + realistic model Conclusions Interesting Missions Applications Observations of the Sun provide information of the geomagnetic storms, as in the Geostorm Warning Mission. z y CME x ACE Sun Earth L 1 0 . 01 AU Sail 0 . 02 AU Observations of the Earth’s poles, as in the Polar Observer. N N Sail Sail z z Earth Earth x x L 1 L 1 Sun Sun Winter Solstice Summer Solstice S S es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 9 / 42
Background Station Keeping + realistic model Conclusions AIM of this TALK One of the main goals of our work was to understand the geometry of the phase space and how it varies when the sail orientation is changed. Then use this information to derive strategies to control the trajectory of a Solar Sail. es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 10 / 42
Background Station Keeping + realistic model Conclusions AIM of this TALK One of the main goals of our work was to understand the geometry of the phase space and how it varies when the sail orientation is changed. Then use this information to derive strategies to control the trajectory of a Solar Sail. We will: 1 describe the dynamics of a solar sail around an equilibrium point (for a fixed sail orientation) and show the effects of variations on the sail orientation on the sail trajectory and show how to use this knowledge to derive a station keeping strategy around an equilibrium point. 2 we have two different ways to use this information. We will describe both strategies and apply them to the GeoStorm Mission. 3 finally we will discuss the robustness of these strategies when we include different sources of error. es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 10 / 42
Background Station Keeping + realistic model Conclusions Station Keeping Strategies Around Equilibria es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 11 / 42
Background Station Keeping + realistic model Conclusions Station Keeping for a Solar Sail We want to design station keeping strategy to maintain a trajectory of a solar sail close to an unstable equilibrium point. Instead of using Control Theory Algorithms , we will use Dynamical System Tools to find a station keeping algorithm for a Solar Sail. The main ideas are ... • To focus on the linear dynamics around an equilibrium point and study how this one varies when the sail orientation is changed. • To change the sail orientation (i.e. the phase space) to make the system act in our favour: keep the trajectory close to a given equilibrium point. es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 12 / 42
Background Station Keeping + realistic model Conclusions Station Keeping for a Solar Sail We focus on the two previous missions, where the equilibrium points are unstable with two real eigenvalues, λ 1 > 0 , λ 2 < 0, and two pair of complex eigenvalues, ν 1 , 2 ± i ω 1 , 2 , with | ν 1 , 2 | << | λ 1 , 2 | . • To start we can consider that the dynamics close the equilibrium point is of the type saddle × centre × centre. • From now on we describe the trajectory of the sail in three reference planes defined by each of the eigendirections. ( x 2 , y 2 ) ( x 1 , y 1 ) ( x 3 , y 3 ) • For small variations of the sail orientation, the equilibrium point, eigenvalues and eigendirections have a small variation. We will describe the effects of the changes on the sail orientation on each of these three reference planes. es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 13 / 42
Background Station Keeping + realistic model Conclusions Effects of Variations on the Orientation (I) In the saddle projection of the trajectory: • When we are close to the equilibrium point, p 0 , the trajectory escapes along the unstable direction. • When we change the sail orientation the equilibrium point is shifted. es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 14 / 42
Background Station Keeping + realistic model Conclusions Effects of Variations on the Orientation (II) In the saddle projection of the trajectory: • Now the trajectory will escape along the new unstable direction. • We want to find a new sail orientation ( α, δ ) so that the trajectory will come close to the stable direction of p 0 . � � es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 15 / 42
Background Station Keeping + realistic model Conclusions Effects of Variations on the Orientation (III) In the centre projection of the trajectory: � A sequence of changes on the sail orientation implies a sequence of rotations around different equilibrium points on the centre projection, which can result of an unbounded grouth. es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 16 / 42
Background Station Keeping + realistic model Conclusions Effects of Variations on the Orientation (III) In the centre projection of the trajectory: � A sequence of changes on the sail orientation implies a sequence of rotations around different equilibrium points on the centre projection, which can result of an unbounded grouth. es, ` A. Farr´ A. Jorba (IMCCE, UB) Control Strategies for Solar Sails SMAI 2011 16 / 42
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