19 June 2015

AstroNet-II International Final Conference

Tossa de Mar, Spain

UnconventionalSolar Sailing

Matteo Ceriotti

2

2cos ˆr

a n

r

,s max

g

a A

a m

Use of solar radiation pressure to generate a net force on the spacecraft

J. C. Maxwell, in his theory of electromagnetic fields and radiation (1864), showed that light has momentum and thus can exert pressure on objects.

F. Zander, “Problems of flight by jet propulsion: interplanetary flights” (1925)

Solar sailing

Introduction

19 June 2015 3Matteo Ceriotti

“Conventional” architecturesdesigned for easy stowage and deployment

NANOSAIL-D2, 4 kg, 10 m2

LightSail, 3 kg, 32 m2

Comet Halley sail (concept) IKAROS, 315 kg, 20 m diag

Introduction

Solar sailing disadvantages:• Large Area-to-Mass Ratio (AMR or β) required for sensible acceleration

o Reliability of deploying large, lightweight membranes

• High demand on attitude control

Three different architectures/concepts to:• Ease solar sailing advancement in the TRL scale

• Use solar sailing on Earth’s orbits

• Facilitate the use of solar sails on nano/pico-satellites (CubeSats)

19 June 2015 4Matteo Ceriotti

Can we overcome these issueswith unconventional spacecraft/sail design?

HYBRID SOLAR SAIL PROPULSION

Matteo CeriottiJeannette Heligers

Colin McInnes

Solar Sailing

19 June 2015

Use of solar radiation pressure to generate a net force on the spacecraft

IKAROS (JAXA)First successful

demonstrator in space

2010

Drawbacks• Low TRL• High AD2

• Force always pointing away from the Sun

Advantages• Propellantless propulsion –

continuous, potentially unlimited Δv

1900’s

NASA

Original concept by Tsiolkovsky and Tsander

LightSail3U CubeSat

The Planetary Society

2015

Matteo Ceriotti 6

Solar Electric Propulsion (SEP)

19 June 2015

NASA

Deep Space 1 (NASA)

1998

ESA

SMART-1 (ESA)

2003

Dawn (NASA)

2007

GOCE (ESA)

2009

Isp = 3500 s

Isp = 3100 s

Isp = 1540 s

Advantages• Very high specific impulse (4500+ s)• Consolidated technology (high TRL)• Low AD2 (Advancement Degree of Difficulty)

Drawbacks• Low thrust level• Mission time limited by

propellant mass on board

Isp = Up to 3300 s

Uses electric energy to accelerate ions to high

velocity

Matteo Ceriotti 7

19 June 2015

Hybrid Propulsion

Force always pointing away from the Sun

Propellant consumption

Low TRL, High AD2

Propellantless

Thrust is steerable

High TRL, Low AD2

Increased complexity

How to take advantage of both SEP and sail?Combine them on the same spacecraft:

Matteo Ceriotti 8

19 June 2015

Hybrid Propulsion Spacecraft

Sail attached to spacecraft bus• Steering of the sail is performed through

attitude control

Steerable thruster mounted on a gimbal Sail area partly covered by thin film solar

cells (TFSC)• Used to power the SEP

Controls of the spacecraft are:• Thrust vector (3 components)• Sail attitude (2 components)

Ceriotti, McInnes, “Systems design of a hybrid sail pole-sitter”, Advances in Space Research, vol. 48, n. 11, 2011Matteo Ceriotti 9

Pole-sitter

19 June 2015

J. M. Driver, “Analysis of an arctic polesitter”, Journal of Spacecraft and Rockets, vol. 17, n. 3, p. 263-269, 1980

Average acceleration over 1 year

Need of constant thrust maintain the stationary position

Stationary above the Pole at constant distance d

Acceleration required to counterbalance all other forces (Sun, Earth, centrifugal, Coriolis)

For realistic level of acceleration

• d = 2 ~ 3 million km

• a = 0.15 ~ 0.17 mm/s2

Matteo Ceriotti 10

Pola

r axis

Inertial vs. Synodic reference frame

19 June 2015

Synodic reference frame

Rotating with Sun-Earth line

As Earth goes around the Sun, polar axis describes a conical surface

Motion of spacecraft is confined to lie on cone

y x

z

Winter solstice

d

23.5

Synodic reference frame

Inertial reference frame

June

September

December

March

Ecliptic plane

Spacecraft on Earth’s rotation axis

Matteo Ceriotti 11

Optimal hybrid pole-sitter orbits

19 June 2015

Problem:

Ceriotti, McInnes, “Generation of optimal trajectories for Earth hybrid pole-sitters”, Journal of Guidance, Control, and Dynamics, vol. 34, n. 3, 2011.

Solved with Pseudo-Spectral methods

(PSOPT)

d(t)Find:

Such that:

Objective:

Subject to:

Constraint:

optimal path d(t)

optimal sail steering law and SEP thrust

Propellant mass is minimised over 1 year

Pole-sitter position is maintained

Matteo Ceriotti 12

Optimal pole-sitter orbits

19 June 2015

HybridSEP

t = 0Winter solstice

SEP acceleration

Hybrid

SEP

Matteo Ceriotti 13

Pole-sitter mission lifetime

For 100 kg payload, Lifetime (years):

19 June 2015

HybridSEP

Ariane 5

Soyuz

SEP

Soyuz 3.6

Ariane 5 4.3

Hybrid

Soyuz 5.6 (+2)

Ariane 5 6.6 (+2.3)

M. Ceriotti, J. Heiligers, C. R. McInnes, “Trajectory and spacecraft design for a pole-sitter mission”, Journal of Spacecraft and Rockets, vol. 51, n. 1, p. 311-326, 2014

Matteo Ceriotti 14

View from the pole-sitter

19 June 2015

t = 0

Winter solstice

Summer solstice0.5 deg FOV

Matteo Ceriotti 15

Displaced GEO

19 June 2015

2 0.1 0.2

h, km ρ, km α, deg a, mm/s2

±35 42164.165 0.0476 0.1861

±75 42164.147 0.1019 0.3988

±150 42164.080 0.2038 0.7976

2012

a Out-of-plane

Ix

Rx

GEO GEOr

GEOr

Ry

h Iy

I Rz z

Z-axis not to scale!

2 2 2

2

GEO

Displaced

GEO

Matteo Ceriotti 16

Displaced GEO

Can a solar sail alone provide the required acceleration?• Not for a full year

Mass budget results:

19 June 2015

h = 35 kmh = 150 km

Ex

Ez

obli

ˆsr

Ex

Ez

Summer

,mins ,mins

,maxs

,maxs

.min

,max

s obl

s obl

i

i

Winter

.min

,max

s obl

s obl

i

i

obli

ˆsr

Equatorial plane

J. Heiligers, M. Ceriotti, C. R. McInnes, J. D. Biggs, “Displaced geostationary orbit design using hybrid sail propulsion”, Journal of Guidance, Control, and Dynamics, vol. 34, n. 6, p. 1852-1866, 2011

There is an optimal AMR!

β ~= 0.1

Matteo Ceriotti 17

Earth solar sailing

How can we use solar radiation pressure only for orbit manoeuvring around the Earth?

19 June 2015 Matteo Ceriotti 18

ADDING AN ELECTRIC THRUSTER IS CHEATING!

It’s no more solar sailing!

Earth solar sailing

The majority of Earth mission scenarios require considerable variation in acceleration direction (and magnitude) throughout the orbit.

For Earth orbits, the period can be quite small (few hours)

Require fast slew manoeuvres• Difficult to achieve due to the high moments of inertia of the sail surface.

19 June 2015

Figures:Stolbunov, V., et al., Optimal law for inclination change in an atmosphere through solar sailing. Journal of Guidance, Control, and Dynamics, 2013. 36(5): p. 1310-1323.Borja, J.A. and D. Tun, Deorbit process using solar radiation force. Journal of Spacecraft and Rockets, 2006. 43(3): p. 685-687.

Matteo Ceriotti 19

Is it possible to use the sun pressure to both control the attitude and provide the acceleration to change the orbital parameters passively? Small demand on attitude actuators (or no need at all)

No propellant mass required

QUASI-RHOMBIC PYRAMID

Matteo CeriottiPatrick Harkness

Malcolm McRobb

Quasi-rhombic pyramid

Idea:• Passive attitude stabilisation

• Control of the thrust magnitude with variable geometry

The sail is mounted on a Quasi-Rhombic Pyramid (QRP) shape• Spacecraft bus at the apex

• 4 booms deployed along the slant edges

• Sail membranes filling the slant faces

19 June 2015

Reflective sail membrane

Spacecraft bus

Booms

M. Ceriotti, P. Harkness, M. McRobb, “Variable-geometry solar sailing: the possibilities of the quasi-rhombic pyramid”, Advances in Solar Sailing, M. Macdonald, Editor. 2014, Springer Berlin Heidelberg, p. 899-919.

Matteo Ceriotti 21

Quasi-rhombic pyramid

4 booms arranged in 2 opposite pairs

By simultaneously:• Increasing the flare angle of one opening boom pair

• Reducing the flare angle of the other closing pair

Will reduce the effective area exposed to the sun

No net torque on the bus

19 June 2015 Matteo Ceriotti 22

Geometry of the QRP

19 June 2015

Fully closed

Fully open

Matteo Ceriotti 23

Bevel Crux Drive (BCD)

19 June 2015

Spool

Rotatingcage

Matteo Ceriotti 24

Spacecraft sizing

19 June 2015

Spacecraft 1 2 3Boom length = base length, l = b, m 1 2 3Bus mass, kg 1 2 3Bus size, cm 10 12.6 14.4

Total mass, m, kg 1.088 2.221 3.401Equivalent flat sail area-to-mass ratio(booms open), m2/kg

0.30 0.60 0.88

Principal moments of inertia(booms open), kg m2

0.02940.02940.0163

0.2810.2810.153

1.1201.1200.612

Boom linear density 16.3 g/m

Sail membrane areal density 13.2 g/m2

Efficiency of the sail 0.85

Matteo Ceriotti 25

Attitude Dynamics

When open, fully or partially, the QRP sail is heliostable, i.e. has a stable equilibrium at sun-pointing direction.

A perturbation will induce an undampened harmonic motion

2 viscous fluid ring dampers can be used to dampen the oscillations (x and y axes)

Rings are mounted on the +x and +y bus faces, maximising fluid inertia If

19 June 2015

y

x

fx x

fx

f

fy y

fy

f

c

I

c

I

Matteo Ceriotti 26

Attitude Dynamics

Spacecraft 1

Initial displacement w.r.t. sun:• 30° in y

• 30° in x-y

19 June 2015

ω spacecraft

(body axes)

ω fluid

Matteo Ceriotti 27

Orbital Dynamics

Control law to increase semi-major axis a

Equatorial orbit

Sail fully open when travelling away from the sun

Sun rotates 1 rev/year

Due to helio-stability, sail is always sun-pointing throughout the year• This results in a quasi-null net increase in eccentricity

Similar law can be used to reduce a (e.g. for EOL disposal)

19 June 2015

Spacecraftorbit

control

Matteo Ceriotti 28

Orbital Dynamics

At lower altitudes, atmosphere plays major role

19 June 2015

Spacecraft 1 2 3Altitude gain, km/year +20 +40 +60

Semi-major axis

Eccentricity

Inclination

Matteo Ceriotti 29

WHAT HAPPENS DURING ECLIPSES?

Matteo CeriottiPatrick HarknessLeonard Felicetti

Attitude Stability Issues

Eclipses impact the stability of the spacecraft because external disturbance torques, like the gravity gradient, and residual angular velocity cannot be counterbalanced by heliostability

• The oscillation amplitude builds up at every eclipse passage leading to a unstable behaviour of the sail

19 June 2015

Spacecraftorbit

control

Eclipse

Matteo Ceriotti 31

Solution:Spin Stabilizing the Sail

It is possible to stabilize the sail by spinning it around one of its principal axis (“gyroscopic stiffness”)• Augments the stabilizing effects of the solar radiation torque, when

the sail is exposed to the sun

• Counterbalances the disturbing effects occurring during the (short) eclipse phases

19 June 2015

Spacecraftorbit

control

𝜔0

𝜔0

𝜔1

𝜔1

𝜔1𝜔1 = 𝐈1

−1𝐈0𝜔0

𝜔0

𝜔1

Matteo Ceriotti 32

Stability AnalysisYaw-Pitch Motion

19 June 2015

𝐌 𝒒 + 𝐆 𝒒 + 𝐊 + 𝐊1 +𝑲2 𝒒 = 0 𝒒 =𝛼1𝛼2

𝐌 =𝐼1 00 𝐼2

𝐆 = 𝑟00 − 𝐼1 + 𝐼2 − 𝐼3

𝐼1 + 𝐼2 − 𝐼3 0𝐊 = 𝑟0

2 𝐼3 − 𝐼2 0

0 𝐼3 − 𝐼1

𝐊2 = −4

3

1 − 𝜂

1 − 𝑐𝛼2𝑝𝐴𝐿

𝑠𝜃1 𝑐𝜃1𝑠𝜃2 + 2𝑠𝜃2𝑐𝜃2 0

0 𝑠𝜃2 𝑐𝜃2𝑠𝜃1 + 2𝑠𝜃1𝑐𝜃1

𝐊1 = −16

3

1 + 𝑐𝛼

1 − 𝑐𝛼2

32

𝜂𝑝𝐴𝐿𝑠𝜃13 𝑐𝜃2𝑠𝜃2 0

0 𝑠𝜃23 𝑐𝜃1𝑠𝜃1

Terms due to solar radiation torque

Solar radiation pressure

Face area Sail aperture anglesLength of the booms

𝐿

Linearization (small attitude deviations) for yaw-pitch motion (𝛼1, 𝛼2) with respect to the nominal configuration:

Matteo Ceriotti 33

Terms containing the moments of inertia of the system

The Routh-Hurwitz stability criterion has been applied for obtaining the stability bounds of such system:

A parametric study has been performed in order to select the best geometry of the sail in both open/close configurations, varying:

1. Boom length 𝐿 = 1𝑚, 2𝑚, 5𝑚

2. Base length 𝑏 = 𝐿[0 2]

3. Aperture angle 𝜃1 = 0 𝜋 2

4. Spin rate 𝑟0 (range: [1 10] rph)

Stability AnalysisRouth-Hurwitz Criterion

19 June 2015

𝑑𝑒𝑡 𝐌𝑠2 + 𝐆𝑠 + 𝐊 + 𝐊1 + 𝐊2 = 0

𝑠4 + 𝑎2𝑠2 + 𝑎4 = 0

𝑎2 > 0

𝑎4 > 0

𝑎22 − 4𝑎4 > 0

Stability Conditions

Matteo Ceriotti 34

Stability Regionsin Sun Exposure

L = 1m

19 June 2015

control

Sail open

Sail closed

0 20 40 60 800

0.5

1

1.5

k1

A, deg

b,

m

0 20 40 60 800

0.5

1

1.5

A, deg

k2

b,

m

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Stable

Lower bound(sail closed)

Upper bound (sail open)

Len

gth

of

bas

e b

(des

ign

)

Angle of booms 𝜃𝐴 (control variable)

Matteo Ceriotti 35

b

Stability Regionsin Eclipse

The same analysis in eclipse:

• No SRP

• Gravity Gradient

• Spin axis aligned along the local vertical direction

The stability region is reduced

19 June 2015

𝐌 𝒒 + 𝐆 𝒒 + 𝐊 + 𝐊1 +𝑲2 +𝑲gg 𝒒 = 0

control

0 20 40 60 800

0.5

1

1.5

A, deg

b,

m

0 20 40 60 800

0.5

1

1.5

A, deg

b,

m

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Stable

Gyro Stable but StaticallyUnstable

𝐾1

Matteo Ceriotti 36

b

Orbit control phases

3. Sail Totally Closed

to minimize the sun exposure

19 June 2015

0 20 40 60 800

0.5

1

1.5

A, deg

b,

m

0 20 40 60 800

0.5

1

1.5

A, deg

b,

m

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Eclipse0 20 40 60 80

0

0.5

1

1.5

k1

A, deg

b,

m

0 20 40 60 800

0.5

1

1.5

A, deg

k2

b,

m

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Sun

2. Sail Partially Closedto maintain stability during the eclipse

1. Sail Totally Opento maximize the sun exposed area

𝑏 = 1.4𝑚

Matteo Ceriotti 37

Numerical Results: attitude

Case study:• GEO, 1 year

• 𝐿 = 1 𝑚

• 𝑏 = 1.4 𝑚

• 𝑟0 = 5 𝑟𝑝ℎ

Euler angles (pitch, yaw)w.r.t. sun-pointing (nominal) direction

19 June 2015

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05

0

0.05

t, d

Eu

ler

an

gle

s,

de

g Euler angles with respect to nominal sun-pointing direction

x

y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.5

0

0.5

1

1.5

t, d

In eclipse

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

1

2

t, d

Sail Mode

Sail closed

Eclipse mode

Sail open

Attitude is perturbed at each eclipse passage but nominal direction is recovered afterwards

Matteo Ceriotti 38

Sail is sun-pointing (not inertially-pointing) throughout the year

Numerical Results: orbit

19 June 2015

0 50 100 150 200 250 300 350 4004.2

4.25

4.3

4.35

4.4

4.45

4.5

4.55

4.6x 10

4

t, d

a,

km

Semi-major axis

Sail open

Eclipse mode

Sail closed

+2000 km

Semimajor axis, 1 year

5 days

Matteo Ceriotti 39

Earth solar sailing

QRP provides a passive, self-stabilizing effect under solar radiation pressure

By varying the boom angles, it is possible to change the effective area-to-mass ratio of the spacecraft for orbit control.

By spinning the sail, it is possible to achieve stable sun-pointing throughout the year and even in eclipses

Realistic architectures appear likely to have the capability to raise or lower the orbit of CubeSat-class spacecraft by several 10’s km/year in LEO (>1000 km), 1000’s km/year in GEO

19 June 2015 Matteo Ceriotti 40

Earth solar sailing

The heliostable sail (QRP) reduces the demands on an attitude control system

Yet, it requires control of the boom flare angle (open/close)

Is it possible to envisage a completely passive system?• Heliostability

+

• Orbit control

Use of a heliostable oscillating sailto create forces alternating throughout the orbit

19 June 2015 Matteo Ceriotti 41

SYNCHRONISED ORBIT-ATTITUDE OSCILLATIONS

Matteo CeriottiPatrick Harkness

Malcolm McRobb

Oscillatory motion

Displacing the cm w.r.t. the cp:

19 June 2015

22 sinsun

Aa P

m

x

z

CP

CM

d

cm = cp

a

22 sinsun

Aa P

m

22 sinsun

AP d

I

cp

a

cm

cp

a

cm

Matteo Ceriotti 43

Oscillatory motion

Period and phase of oscillation can be (gradually) adjusted with changes in:• Geometry

o Moment of inertiao Heliostatic margino Size of the sail

• Reflectivity propertieso Photochromic materials

19 June 2015

2

sinmax

osc

t tT

2sink

t

Matteo Ceriotti 44

Oscillatory motion

In principle the oscillation plane is arbitrary• Depends on the initial attitude w.r.t. the sun

We restrict our analysis to an oscillation plane on the equatorial plane• The oscillation is maintained while the sun line rotates (yearly)

19 June 2015

ˆsr

ˆsr

ˆoscn

ˆoscn

CP

Oscillation plane

CMa

ˆosct

M. Ceriotti, P. G. Harkness, M. McRobb, “Synchronized orbits and oscillations for free altitude control”, Journal of Guidance, Control, and Dynamics, vol. 37, n. 6, p. 2062-2066, 2014

Matteo Ceriotti 45

Orbital dynamics

Lagrange variational equations:

19 June 2015

2

2

2 22 2

2

2 2

2 2

2

2

2 2

2 sin 2 1

1

11 sin 1

cos

1

sin

1 sin

1 cos

cot sin1 11 sin

1 cos 1

r

r

h

h

r

h

da e a ea a

dt nrn e

a ede e ea r a

dt na rna e

rdia

dt na e

rda

dt na e i

d ea

dt nae

r iea a

nae e na e

0

T

x x t dt

Net change of orbital parameters over 1 orbit

“slow” orbital elements are considered constant within 1 orbit (averaged approach)

“fast” variable is ν, not time…

Matteo Ceriotti 46

Orbital dynamics

Change of variable:

Change of reference frame toperifocal frame (PQh)

19 June 2015

22

2 20

22 2

2

2 30

22

2

2 30

22

2

2 30

22 2

2

2 1 sin cos

1 cos

1 sin cos 1 2 cos cos

1 cos

1 cos

1 cos

1 sin

sin 1 cos

1 1 cos sin sin co

P Q

P Q

h

h

P

e a e aa d

n e

e e a e ae d

n a e

ei a d

n a e

ea d

n a i e

e e a

n ae

2

30

s cot sin

1 cos

Q ha e i ad

e

2

0

xx d

r

θ

ˆsr

Q

P

ddt

Matteo Ceriotti 47

Orbital dynamics

19 June 2015

PQh slot E a R a

sinE e Et

n

2

sinmax

osc

t tT

2 ˆ2 sinsun

AP

m a n

12arctan tan

1 2

eE

e

Matteo Ceriotti 48

Synchronous attitude/orbit oscillations

1 full oscillation per orbit

2 full oscillations per orbit

2 full orbits per oscillation

19 June 2015

oscT T

2oscT T

2oscT T

oscT T

Matteo Ceriotti 49

Test-case orbits

Orbit a, km e i, deg Ω, deg ω, deg

CEq 16,378 0 0 0 0

EEq 16,378 0.55 0 0 0

EP 16,378 0.55 90 0 270

19 June 2015

ˆsr at t = 0

Oscillation & Equatorial plane

Ecliptic plane

ˆsr at t = 0

Oscillation & Equatorial plane

Ecliptic plane

ˆsr at t = 0

North

Ecliptic plane

Oscillation & Equatorial plane

CEq – Circular Equatorial

EEq – Elliptic EquatorialEP – Elliptic Polar

Matteo Ceriotti 50

Results: orbit CEq

19 June 2015

60 degmax

oscT T

(3.7 km in GEO)

Δa = 224 m/orbit/AMRΔe ≈ 0 (over 1 year)

Δe

Δa

a

va

v

Matteo Ceriotti 51

Results: orbit CEqSemimajor axis change, 1 year

Fully passive!

19 June 2015 52Matteo Ceriotti

+ 5,323 km

+ 440 km

a

Conclusion

Novel architectures can enable solar sail missions:• Around the Earth

o Lower cost than deep space

• For nano-satellites, where often there is limited (or no) capabilities of:

o Attitude control

o Thrust (Dv)

Low-cost, low-risk missions will be essentialfor a stepped entry of sail technologyinto future, more ambitious missions

19 June 2015 Matteo Ceriotti 53

UKube-1

Space Glasgow

@SpaceGlasgow

www.glasgow.ac.uk/space

Thank you!

matteo.ceriotti@glasgow.ac.uk

@mtcerio

Acknowledgements

Part of this work was carried out at the University of Strathclyde,funded by the European Research Council, project 227571 VISIONSPACE

I would like to acknowledge all the collaborators involved in this research

ATTITUDE DYNAMICS WITH SRP

Matteo CeriottiAndreas BorggräfeJeannette Heiligers

Colin McInnes

1 (on)

0 (off)

Electro-chromic materials

Attitude control critical for a solar sail• Large moments of inertia

Might be impossible to use reaction wheels

Classic method is cm-cp shifting through moving masses (and mechanisms)

A different solution is the use of electro-chromic materials

electro-chromic surface coatings→ Reflectivity Control Devices (RCDs)

Image: JAXA

19 June 2015 Matteo Ceriotti 56

Attitude control through SRP

Due to gravity-gradient, there are two stable attitudes in the system, with max M.o.I. aligned with local vertical

Can we create new artificial equilibria?

m1

m

2

Sun

CoM

Y

X

β2

β1

rMB eq

Introducing variable surface reflectivities on the tip masses

A. Borggräfe, J. Heiligers, C. R. McInnes, M. Ceriotti, “Attitude dynamics of the classical dumbbell problem with solar radiation pressure”, Journal of Guidance, Control, and Dynamics, vol. 38, n. 1, p. 168-173, 2015

19 June 2015 Matteo Ceriotti 57

Artificial equilibria

19 June 2015 Matteo Ceriotti 58

Artificial attitude equilibria + circular, non-Keplerian orbitArtificial attitude equilibria

Attitude manoeuvres

19 June 2015 Matteo Ceriotti 59

β1 = β2

no SRP effectPure GG

β1 != β2

Natural equilibria

Artificial equilibria

Bevel Crux Drive (BCD)

Three articulation methods are proposed:

1. Independent Stepper Motors

2. Paired Stepper Motors

3. Single Mechanically Linked Stepper Motor• Combination of bevel and elliptical gears

19 June 2015

2

2

22

22

2

sin 2

2

2cos 14cos

A

B A

A

A

b

l

b

l

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Results: orbit CEq

Orbit 2i

Orbit 2i+1

19 June 2015

θmax = 90 deg

Tosc/T = 2

Δa = 285 m/orbit/AMR

(Δa = 4.8 km in GEO)

Δa

a

v

a

v

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Results: orbit EEq

19 June 2015

θmax = 90 deg

Tosc/T = 2

Δa = 197 - 261 m/orbit/AMRΔω + ΔΩ ≈ 0

Δe ≈ 0 (over 1 year)

Δa Δω + ΔΩ

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Results: orbit EEqSemimajor axis change, 1 year

19 June 2015

+ 4,620 km

+ 374 km

a

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Results: orbit EP

19 June 2015

θmax = 90 deg

Tosc/T = 2

Δa = 0 - 249 m/orbit/AMR(depending on time of the year)

Δi = 0 - 8.4×10-6 deg/orbit/AMR(depending on time of the year)

1 year

tt

0

ΔaΔi

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Results: orbit EPInclination change, 1 year

xzzz

19 June 2015

0

+ 3.6 deg

+ 0.36 deg

+ 2032 km

+ 180 kmAlso (for 10 m2/kg):Δe = -0.074Δω = -0.5 degΔΩ ≈ 0

a

i

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