Control Strategies for Solar Sail SMAI 2011 ` es 1 A. Jorba 2 A. - - PowerPoint PPT Presentation

Control Strategies for Solar Sail

SMAI 2011

• A. Farr´

es1 & `

• A. Jorba2

1Institut de M´

ecanique C´ eleste et de Calcul des ´ Eph´ em´ erides, Observatoire de Paris (afarres@imcce.fr)

2Departament 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

• A. Farr´

es, `

• 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.

• A. Farr´

es, `

• 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
• rbit around the Earth.
• A. Farr´

es, `

• 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:

• Fsail = β ms

r 2

ps

• rs,

n2 n.

• A. Farr´

es, `

• 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.

1 − µ µ

• FEarth
• FSun

Sail

• n

X Y Z Earth Sun

• A. Farr´

es, `

• 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 (Li).

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 Li.

• For a fixed and larger β, these families merge into each other. We end up

having two disconnected surfaces, S1 and S2. Where S1 is like a sphere and S2 is like a torus around the Sun.

• All these families can be computed numerically by means of a continuation

method.

• A. Farr´

es, `

• 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

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 Z X Sun T2 T1

• 0.04
• 0.02

0.02 0.04

• 1.02
• 1.01
• 1
• 0.99
• 0.98
• 0.97

Z X Earth T2 T1

Equilibrium points in the XZ plane

• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.04

• 1
• 0.5

0.5 1 Z X Sun T2

• 0.02
• 0.01

0.01 0.02

• 1.01
• 1.005
• 1
• 0.995
• 0.99
• 0.985
• 0.98

Z X Earth T2

• A. Farr´

es, `

• 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.

Sun Earth x y z

0.01 AU 0.02 AU L1 ACE

Sail CME

Observations of the Earth’s poles, as in the Polar Observer.

Sun Earth x z

L1

N S Summer Solstice Sail Sun Earth x z

L1

N S Winter Solstice Sail

• A. Farr´

es, `

• 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.

• A. Farr´

es, `

• 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.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 10 / 42

Background Station Keeping + realistic model Conclusions

Station Keeping Strategies Around Equilibria

• A. Farr´

es, `

• 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.

• A. Farr´

es, `

• 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, λ1 > 0, λ2 < 0, and two pair of complex eigenvalues, ν1,2 ± iω1,2, with |ν1,2| << |λ1,2|.

• 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.

(x1, y1) (x2, y2) (x3, y3)

• 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.

• A. Farr´

es, `

• 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, p0, the trajectory escapes along

the unstable direction.

• When we change the sail orientation the equilibrium point is shifted.
• A. Farr´

es, `

• 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 p0.

• A. Farr´

es, `

• 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.

• A. Farr´

es, `

• 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.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 16 / 42

Background Station Keeping + realistic model Conclusions

Schematic idea of the Station Keeping Algorithm

We look at the sails trajectory in the reference system {x0; v1, v2, v3, v4, v5, v6}, so z(t) = x0 + Σisi(t) vi. During the station keeping algorithm:

1 when α = α0, δ = δ0:

if |s1(t)| ≥ εmax ⇒ choose new sail

• rientation α = α1, δ = δ1.

2 when α = α1, δ = δ1:

if |s1(t)| ≤ εmin ⇒ restore the sail

• rientation: α = α0, δ = δ0.

3 Go Back to 1.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 17 / 42

Background Station Keeping + realistic model Conclusions

1st Idea for finding αnew, δnew

We will choose a the position of the new equilibrium point (i.e. a new sail

• rientation) so that projection of the trajectory on the saddle will come back and

the two centre projections remain bounded ?

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 18 / 42

Background Station Keeping + realistic model Conclusions

1st Idea for finding αnew, δnew

We will choose a the position of the new equilibrium point (i.e. a new sail

• rientation) so that projection of the trajectory on the saddle will come back and

the two centre projections remain bounded ?

emax emin d p0 p1

p0 p1

The constants εmin, εmax and d will depend on the mission requirements and the dynamics around the equilibrium point.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 18 / 42

Background Station Keeping + realistic model Conclusions

Remarks

• We do not know explicitly the position of the equilibrium points p(α, δ). But

we can compute the linear approximation of this function: p(α, δ) = p(α0, δ0) + Dp(α0, δ0) · (α − α0, δ − δ0)T.

• There are some restrictions of the position of the new equilibria when we

change α and δ. We have 2 unknowns and at least 6 conditions that must be satisfied.

• We will change the sail orientation so that the position of the new fixed point

is as close as possible to the desired new equilibrium point and in the correct side in the saddle projection.

• To decide the new sail orientation we will assume that the eigenvalues and

eigendirections do not vary when the sail orientation is changed.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 19 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (RTBPS)

XY and XZ and XYZ Projections

• 0.0026
• 0.0024
• 0.0022
• 0.002
• 0.0018
• 0.0016
• 0.0014
• 0.0012
• 0.9804
• 0.9803
• 0.9802
• 0.9801
• 0.98
• 0.9799
• 0.9798
• 0.9797
• 0.9796

• 6e-06
• 4e-06
• 2e-06

• 0.9804
• 0.9803
• 0.9802
• 0.9801
• 0.98
• 0.9799
• 0.9798
• 0.9797
• 0.9796

• 0.9804
• 0.9803
• 0.9802
• 0.9801-0.98-0.9799
• 0.9798
• 0.9797
• 0.9796
• 0.0026
• 0.0024
• 0.0022
• 0.002
• 0.0018
• 0.0016
• 0.0014
• 0.0012
• 6e-06
• 4e-06
• 2e-06

Saddle × Centre × Centre Projections

• 2e-05

• 0.001
• 0.0008
• 0.0006
• 0.0004
• 0.0002

• 0.001
• 0.0008
• 0.0006
• 0.0004
• 0.0002

• 6e-06
• 4e-06
• 2e-06

• 8e-06
• 6e-06
• 4e-06
• 2e-06

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 20 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (RTBPS)

Variation of the sail orientation

0.55 0.6 0.65 0.7 0.75 0.8 20 40 60 80 100 120 140 160 180 200 alfa

• 0.004
• 0.003
• 0.002
• 0.001

0.001 0.002 0.003 0.004 20 40 60 80 100 120 140 160 180 200 delta

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 21 / 42

Background Station Keeping + realistic model Conclusions

2st Idea for finding αnew, δnew

The computation of variational equations (of suitable order) w.r.t. α and δ gives explicit expressions for the effect of different orientations (close to the reference values α = α0, δ = δ0) trajectory.

φt(x0, α0 + ha, δ0 + hd) = φt(x0, α0, δ0) + ∂φ ∂α(x0, α0, δ0) · ha + ∂φ ∂δ (x0, α0, δ0) · hd,

With this we can impose conditions on the “future” of the orbit and find

• rientations that fulfil them (or show that the condition is unattainable).
• We will define the parameters εmax, Dtmin and Dtmax that will vary for each

mission application.

• We will find αnew, δnew and dt ∈ [Dtmin, Dtmax] so that the trajectory is close

to the fixed point.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 22 / 42

Background Station Keeping + realistic model Conclusions

Remarks

We use the variational equations up to first order. Hence, we have a linear map for the different final states. As before we want the final position to be close to the stable direction, keeping small the two centre projections. One can think of different ways to solve this problem. We have seen that the best results are found if we:

• For each dt ∈ [Dtmin, Dtmax] we will find αnew and δnew such that s1 = 0 and

(s5, s6) are minimum (i.e. we are close to stable direction and one of the centres is small).

• We finally choose the dt, αnew and δnew that minimises the other centre

projection (s3, s4).

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 23 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (RTBPS)

XY and XZ and XYZ Projections

• 0.0026
• 0.0024
• 0.0022
• 0.002
• 0.0018
• 0.0016
• 0.0014
• 0.98025
• 0.9802
• 0.98015
• 0.9801
• 0.98005
• 0.98
• 0.97995
• 0.9799
• 0.97985
• 0.9798

• 5e-06
• 4e-06
• 3e-06
• 2e-06
• 1e-06

• 0.98025
• 0.9802
• 0.98015
• 0.9801
• 0.98005
• 0.98
• 0.97995
• 0.9799
• 0.97985
• 0.9798

• 0.98025
• 0.9802
• 0.98015
• 0.9801
• 0.98005
• 0.98
• 0.97995
• 0.9799
• 0.97985
• 0.9798
• 0.0026
• 0.0024
• 0.0022
• 0.002
• 0.0018
• 0.0016
• 0.0014
• 5e-06
• 4e-06
• 3e-06
• 2e-06
• 1e-06

Saddle × Centre × Centre Projections

• 0.00015
• 0.0001
• 5e-05

• 0.00015
• 0.0001
• 5e-05

• 0.0006
• 0.0004
• 0.0002

• 0.0006
• 0.0004
• 0.0002

• 5e-06
• 4e-06
• 3e-06
• 2e-06
• 1e-06

• 6e-06
• 4e-06
• 2e-06

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 24 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (RTBPS)

Variation of the sail orientation

• 3
• 2
• 1

1 2 3 4 20 40 60 80 100 120 140 160 180 200 alfa

• 0.08
• 0.07
• 0.06
• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.01 20 40 60 80 100 120 140 160 180 200 delta

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 25 / 42

Background Station Keeping + realistic model Conclusions

Results

We have applied these station keeping strategy to different mission scenarios. We show the results for the Geostorm Warning Mission. For each mission:

• We have done a Monte Carlo simulation taking a 1000 random initial

conditions.

• For each simulation we have applied the station keeping strategy for 30 years.
• We have tested the robustness of our strategy including random errors on the

position and velocity determination, as well as on the orientation of the sail at each manoeuvre. Note : All the simulations have been done using the full set of equations, we only use the linear dynamics to decide the change on the sail orientation.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 26 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (RTBPS)

Algorithm Used: Fixed Point Algorithm EType % Success ∆t (days) ∆α (deg) ∆δ (deg) E0 100.0 % 158.32 - 38.90 0.211 - 0.209 4.884e-03 - 5.040e-06 P0 100.0 % 158.44 - 38.86 0.211 - 0.209 4.879e-03 - 5.041e-06 V1 100.0 % 166.28 - 38.41 0.212 - 0.207 6.399e-03 - 2.461e-05 V2 100.0 % 233.42 - 37.01 0.219 - 0.199 2.324e-02 - 1.003e-04 V3 100.0 % 363.40 - 35.38 0.228 - 0.189 5.588e-02 - 2.179e-04 V4 79.0 % 370.85 - 28.23 0.283 - 0.101 2.123e-01 - 6.749e-04

EType stands for the kind of errors considered in each simulation: E0 = No errors, P0 = Errors on Position and Velocity only, V1, V2, V3, V4 = Errors on Position, Velocity and Sail Orientation, where V1 = 0.001◦, V2 = 0.005◦, V3 = 0.01◦, V4 = 0.05◦

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 27 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (RTBPS)

Algorithm Used: Variational Equation Algorithm EType % Success ∆t (days) ∆α (deg) ∆δ (deg) E0 100.0 % 317.88 - 2.32 2.82 - 0.109 0.160 - 0.000 P0 100.0 % 317.88 - 2.32 2.82 - 0.109 0.160 - 0.000 V1 100.0 % 321.57 - 2.32 2.82 - 0.110 0.163 - 3.78e-05 V2 100.0 % 346.67 - 2.32 2.86 - 0.104 0.292 - 1.72e-04 V3 100.0 % 361.96 - 2.32 4.09 - 0.098 0.557 - 2.38e-04 V4 100.0 % 334.56 - 2.32 4.47 - 0.041 2.598 - 4.54e-04

EType stands for the kind of errors considered in each simulation: E0 = No errors, P0 = Errors on Position and Velocity only, V1, V2, V3, V4 = Errors on Position, Velocity and Sail Orientation, where V1 = 0.001◦, V2 = 0.005◦, V3 = 0.01◦, V4 = 0.05◦

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 28 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (RTBPS)

Fixed Point Algorithm. Error type V3

XY and XZ and XYZ Projections

• 0.0026
• 0.0024
• 0.0022
• 0.002
• 0.0018
• 0.0016
• 0.0014
• 0.9802
• 0.98015
• 0.9801
• 0.98005
• 0.98
• 0.97995
• 0.9799
• 0.97985
• 0.9798

• 6e-05
• 4e-05
• 2e-05

• 0.9802
• 0.98015
• 0.9801
• 0.98005
• 0.98
• 0.97995
• 0.9799
• 0.97985
• 0.9798

• 0.9802
• 0.98015
• 0.9801
• 0.98005-0.98-0.97995
• 0.9799
• 0.97985
• 0.9798
• 0.0026
• 0.0024
• 0.0022
• 0.002
• 0.0018
• 0.0016
• 0.0014
• 6e-05
• 4e-05
• 2e-05

Saddle × Centre × Centre Projections

• 0.00015
• 0.0001
• 5e-05

• 0.00015
• 0.0001
• 5e-05

• 0.0006
• 0.0004
• 0.0002

• 0.0008
• 0.0006
• 0.0004
• 0.0002

• 6e-05
• 4e-05
• 2e-05

• 8e-05
• 6e-05
• 4e-05
• 2e-05

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 29 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (RTBPS)

Variational Equations Algorithm. Error type V3

XY and XZ and XYZ Projections

• 0.0028
• 0.0026
• 0.0024
• 0.0022
• 0.002
• 0.0018
• 0.0016
• 0.0014
• 0.0012
• 0.9804
• 0.9803
• 0.9802
• 0.9801
• 0.98
• 0.9799
• 0.9798
• 0.9797
• 0.9796

• 3e-05
• 2e-05
• 1e-05

• 0.9804
• 0.9803
• 0.9802
• 0.9801
• 0.98
• 0.9799
• 0.9798
• 0.9797
• 0.9796

• 0.9804
• 0.9803
• 0.9802
• 0.9801-0.98-0.9799
• 0.9798
• 0.9797
• 0.9796
• 0.0028
• 0.0026
• 0.0024
• 0.0022
• 0.002
• 0.0018
• 0.0016
• 0.0014
• 0.0012
• 3e-05
• 2e-05
• 1e-05

Saddle × Centre × Centre Projections

• 0.0002
• 0.00015
• 0.0001
• 5e-05

• 0.00015
• 0.0001
• 5e-05

• 0.001
• 0.0008
• 0.0006
• 0.0004
• 0.0002

• 0.001
• 0.0008
• 0.0006
• 0.0004
• 0.0002

• 4e-05
• 3e-05
• 2e-05
• 1e-05

• 4e-05
• 3e-05
• 2e-05
• 1e-05

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 30 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (RTBPS)

Variation of the sail orientation

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 20 40 60 80 100 120 140 160 180 200 alfa

• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.04 0.05 20 40 60 80 100 120 140 160 180 200 delta

• 4
• 3
• 2
• 1

1 2 3 4 20 40 60 80 100 120 140 160 180 200 alfa

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 20 40 60 80 100 120 140 160 180 200 delta

Left: ALgorithim Fixed Point, Right: Algorthim Variational Equations

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 31 / 42

Background Station Keeping + realistic model Conclusions

Including more realism to the dynamical model

There are several ways to include more realism to the dynamical model. For example,

• taking a more realistic model for the Solar Sail by including the force

produced by the absorption of the photons, the reflectivity properties of the sail material, ... .

• taking a more realistic model for the gravitational perturbations by including

the eccentricity in the Earth - Sun system. Or the gravitational attraction of

• ther bodies, i.e. the Moon, Jupiter, ... .

We have started by considering the eccentricity in the Earth - Sun system and studied the robustness of our strategies. So we take the Elliptic Restricted Three Body Problem with a Solar sail as a model.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 32 / 42

Background Station Keeping + realistic model Conclusions

In the ERTBP + Solar Sail

The fixed points that existed in the RTBP + Solar sail no longer exist in this

• model. They have been replaced by periodic orbits of same period as the Earth’s
• rbit around the Sun.

We can apply the same ideas to remain close to one of these periodic orbits and fulfil the mission requirements of the Geostorm mission or the Polar Observer. Notice that:

• for each sail orientation (α, δ) we have an unstable periodic orbit replacing

the fixed point.

• taking the Floquet modes we have a periodic reference system that will give

us a good description of the local dynamics around these periodic orbits.

• we can apply the same ideas as before considering this appropriate reference

system.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 33 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (ERTBPS)

Algorithm Used: Fixed Point Algorithm EType % Success ∆t (days) ∆α (deg) ∆δ (deg) E0 100.0 % 155.65 - 92.46 0.174 - 0.168 3.650e-03 - 4.087e-06 P0 100.0 % 170.83 - 92.61 0.177 - 0.168 4.093e-03 - 2.268e-05 V1 100.0 % 173.56 - 92.70 0.177 - 0.167 5.762e-03 - 2.256e-05 V2 100.0 % 262.21 - 95.06 0.183 - 0.160 2.478e-02 - 9.557e-05 V3 100.0 % 362.44 - 99.53 0.190 - 0.149 5.313e-02 - 1.928e-04 V4 29.0 % 351.75 - 87.30 0.242 - 0.700 2.101e-01 - 6.972e-04

EType stands for the kind of errors considered in each simulation: E0 = No errors, P0 = Errors on Position and Velocity only, V1, V2, V3, V4 = Errors on Position, Velocity and Sail Orientation, where V1 = 0.001◦, V2 = 0.005◦, V3 = 0.01◦, V4 = 0.05◦

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 34 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (ERTBPS)

• 0.00012
• 0.0001
• 8e-05
• 6e-05
• 4e-05
• 2e-05

2e-05

• 0.0001-8e-05 -5e-05 -2e-05 0

saddle

• 0.0004
• 0.0003
• 0.0002
• 0.0001

0.0001 0.0002 0.0003 0.0004 0.0005

• 0.0004-0.0002

0.0002 0.0004 centre 1

• 1.2e-05
• 1e-05
• 8e-06
• 6e-06
• 4e-06
• 2e-06

2e-06 4e-06 6e-06 8e-06 1e-05

• 1.5e-05
• 1e-05-5e-06

5e-06 1e-05 1.5e-05 centre 2

• 0.99
• 0.982
• 0.974
• 0.02
• 0.01 0 0.01

0.02

• 8e-06
• 4e-06

4e-06 8e-06 xyz proj

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 35 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (ERTBPS)

0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 20 40 60 80 100 120 140 160 180 200 alfa

• 0.004
• 0.003
• 0.002
• 0.001

0.001 0.002 0.003 0.004 0.005 20 40 60 80 100 120 140 160 180 200 delta

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 36 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (ERTBPS)

Algorithm Used: Variational Equation Algorithm EType % Success ∆t (days) ∆α (deg) ∆δ (deg) E0 100.0 % 280.56 - 64.07 2.35 - 0.213 0.12 - 0.00 P0 100.0 % 338.21 - 67.29 2.37 - 0.102 0.12 - 1.13e-05 V1 100.0 % 323.99 - 66.60 2.38 - 0.096 0.13 - 5.09e-05 V2 100.0 % 369.22 - 69.12 2.62 - 0.085 0.28 - 1.54e-04 V3 100.0 % 355.27 - 69.72 4.00 - 0.077 0.55 - 2.88e-04 V4 100.0 % 321.73 - 61.39 4.71 - 0.025 2.34 - 5.90e-04

EType stands for the kind of errors considered in each simulation: E0 = No errors, P0 = Errors on Position and Velocity only, V1, V2, V3, V4 = Errors on Position, Velocity and Sail Orientation, where V1 = 0.001◦, V2 = 0.005◦, V3 = 0.01◦, V4 = 0.05◦

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 37 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (ERTBPS)

• 0.00015
• 0.0001
• 5e-05

5e-05 0.0001 0.00015

• 0.00015
• 5e-05 0 5e-05

0.00015 saddle

• 0.0005
• 0.0004
• 0.0003
• 0.0002
• 0.0001

0.0001 0.0002 0.0003 0.0004

• 0.0004-0.0002

0.0002 0.0004 centre 1

• 1.5e-05
• 1e-05
• 5e-06

5e-06 1e-05

• 1.5e-05
• 5e-06

5e-06 1.5e-05 centre 2

• 0.99 -0.984
• 0.98 -0.974
• 0.02
• 0.01 0 0.01

0.02

• 8e-06
• 4e-06

4e-06 8e-06 xyz proj

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 38 / 42

Background Station Keeping + realistic model Conclusions

Results for the Geostorm Mission (ERTBPS)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 2.5 3 3.5 20 40 60 80 100 120 140 160 180 200 alfa

• 0.02

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 20 40 60 80 100 120 140 160 180 200 delta

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 39 / 42

Background Station Keeping + realistic model Conclusions

References

[1] Farr´ es, A. & Jorba, ` A.: Dynamical system approach for the station keeping of a solar

• sail. The Journal of Astronautical Science, 58:2, pp. 199-230, 2008.

[2] Farr´ es, A. & Jorba, ` A.: Solar sail surfing along families of equilibrium points. Acta Astronautica, Vol. 63, Issues 1-4, pp. 249-257, 2008. [3] Farr´ es, A.: Contribution to the Dynamics of a Solar Sail in the Earth-Sun System PHD Thesis, University of Barcelona, October 2009. [4] Farr´ es, A. & Jorba, ` A.: Sailing Between The Earth and Sun. Proc. of the Second International Symposium on Solar Sailing. New York, USA, 20-22 July 2010. Ed: R.Ya. Kezerashvili, pp. 177-182, 2010. [5] Farr´ es, A. & Jorba, ` A.: On the station keeping of a Solar sail in the Elliptic Sun-Earth system. Advances in Space Research. To appear in 2011.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 40 / 42

Background Station Keeping + realistic model Conclusions

Conclusions & Future Work

• We have described the linear dynamics around an unstable equilibrium point

and how it varies when the sail orientation changes.

• We have designed two station keeping strategies using this information and

applied them to a particular mission.

• We have extended these strategies to a more complex model, and we are now

working to include the effect of the other planets.

• We have discussed the robustness of these algorithms when different sources
• f errors are included. We have seen that the controllability of the sail is

strictly related to the nature of the neighbourhood of the equilibrium point.

• Notice that these strategies do not require previous planning as the decisions

are taken depending on the sails position at each time.

• A. Farr´

es, `

• A. Jorba (IMCCE, UB)

Control Strategies for Solar Sails SMAI 2011 41 / 42

Merci pour votre attention