Spacecraft maneuvering via atmospheric differential drag using an adaptive Lyapunov

controller

(AAS 13-440)

David Pérez & Riccardo Bevilacqua

ADvanced Autonomous MUltiple Spacecraft LAB

Rensselaer Polytechnic Institute

bevilr@rpi.edu http://afosr-yip.riccardobevilacqua.com/

23rd AAS/AIAA Spaceflight Mechanics Meeting Kauai, Hawaii

Outline

Introduction

Linear reference model and Nonlinear Model

Lyapunov Approach

Drag devices activation strategy

Critical value for the magnitude of differential drag

Adaptive Lyapunov Control strategy

Numerical Simulations: Re-Phase, Fly-Around, and Rendezvous

See also:

Perez, D., Bevilacqua, R., “Differential Drag Spacecraft Rendezvous using an Adaptive Lyapunov Control Strategy”, Acta Astronautica 83 (2013) 196–207 http://dx.doi.org/10.1016/j.actaastro.2012.09.005.

(winner of best student paper award at 1st International Academy of Astronautics Conference on Dynamics and Control of Space Systems)

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Introduction

Differential in the aerodynamic drag produces a differential in acceleration

This differential can be used to control the relative motion of the S/C on the

orbital plane only

It is assumed that that the drag devices act instantly (on-off control)

Control systems for drag maneuvers must cope with many uncertainties

(density changes, winds, contact dynamics, etc.).

3

Linear Reference Model

The Schweighart and Sedwick model is used to create

the stable reference model

LQR controller is used to stabilize the Schweighart

and Sedwick model

The resulting reference model is described by:

K is found by solving the LQR problem

4

T

d d d d d d d d d

t

x y x y

d

d

x A x + Bu , A A - BK, x ,

u Kx

Nonlinear Model

The dynamics of S/C relative motion are nonlinear due to

J2 perturbation

Variations on the atmospheric density at LEO

Solar pressure radiation

Etc.

The general expression for the real world nonlinear dynamics, including nonlinearities is:

5

, , 0

DrelT

Drel

a

x y x y u

a

x f x Bu x

Lyapunov Approach

A Lyapunov function of the tracking error is defined

as:

After some algebraic manipulation, the time

derivative of the Lyapunov function is:

Defining Ad Hurwitz and Q symmetric positive

definite, P can be found using:

6

, 0dV Te Pe, e = x - x P

Td d

-Q A P + PA

ˆ2 ( )T T T

DrelV a u - d d d d

e (A P + PA )e e P f x - A x + B Bu

Drag panels activation strategy

Rearranging yields

Guaranteeing would imply that the tracking error (e) converges to zero

By selecting:

is ensured to be as small as possible.

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0V

V

1

ˆ ˆ 2( ), 0 ,

1

, Drel

V u u

a = -

T

T T

d d

e Qe

e PB e P A x f x Bu

ˆ ( ) ( )u sign sign Te PB

V

Critical value for the magnitude of differential drag acceleration

Product βû is the only controllable term that

influences the behavior of

There must be a minimum value for aDrel that allows

for to be negative for given values of β and δ

This value is found analytically by solving:

Solving this expression for aDrel yields

8

ˆ2V u

V

ˆ0 Drela u Te PB

Drel Dcrita a

T

d d

T T

e P A x f x Bu

e PB e PB

Matrix derivatives

Choosing appropriate values for the entries of Q and Ad

can reduce aDcrit

To achieve this, the following partial derivatives were

developed

Starting from the general case of the critical value

The Lyapunov equation was transformed into:

9

, Dcrit Dcrita a

dA Q

4 4 4 4

1

, ,

, ( ), ( ),

T

v v v

v v v

v v v

vec vec

I I

d d

d dx x

- Q A P + PA A P Q

A A A P P Q Q

P A Q

Dcrita

T

d d

T

e P A x f x Bu

e PB

Matrix derivatives

Using

The following derivatives were found in previous work

Using the chain rule the desired final expressions can be found:

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16 16 4 4

4 4 4 4 4 4 4 4 4 4 4 4

16 16 16 16 16 16 16 16

1 1

2 3 1

1 1

1 1 1

1

,

,

,

v v

v

v

Tv

v v v v

v

I I

I I I I

I I I

x x

x x x x x x

x x x

d

d

x

d

P A P

A A A

AU U U U

A

P PA U A Q A

A Q

4 4 4 4

4 4 4 4

4 4

1 1 1

3 1 1

1 1

3

4 4

1

( ),

( )

( ),

( )

TDcrit

v

Dcrit

a

a

I I

I I

I I

d d

d d

d d

x x

x x

x 16x16 x

T

T

PA A x f x Bu

Q P

P PA x f x Bu

A A P

e PU x

e PB

P

P

4 4

4 4 4 4 4 4 3( ) ( )

II I Ix

x 16x1

T T

T T

T T6 x x

e PB e Be U e P

e PB e PB

1 v v v

P A Q

Adaptive Lyapunov Control strategy

Using these derivatives Ad and Q are adapted as

follows:

These were designed such that:

Q is symmetric positive definite

Ad is Hurwitz

These adaptations result in an adaptation of the

quadratic Lyapunov function

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( ) , ( )

1 if for , , 1 if for , ,,

0 else 0 el

ij ijDcrit DcritA A Q Q

ij ij

Dcrit Dcrit Dcrit Dcrit

ij kl ij klA Q

dA dQa asign sign

dt A dt Q

a a a ai j k l i j k l

A A Q Q

se

Numerical Simulations

Simulations in STK using High-Precision Orbit Propagator (HPOP)

Full gravitational field model

Variable atmospheric density (using NRLMSISE-00)

Solar pressure radiation effects

The maneuver ended when S/C were within 10m.

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Parameter Value

Second zonal harmonic J2 1.08E-03

Radius of the Earth R (km) 6378.1363

Gravitational parameter μ (km3/sec2) 398600.4418

Target’s inclination (deg) 98

Target’s semi-major axis (km) 6778

Target’s right ascension of the ascending node (deg) 262

Target’s argument of perigee (deg) 30

Target’s true anomaly (deg) 25

Target’s eccentricity 0

vs (km/sec) 7.68

m(kg) 10

Smin surface withheld (m2) 0.5

Smax surface deployed (m2) 2.5

CDmin 1.5

CD0 2

CDmax 2.5

Parameter Rendezvous Fly-Around Re-Phase

x (km) -1 0 0

y (km) -2 -4.25 -1.9

(km/sec) 4.8E-007 0 0

(km/sec) 1.70E-04 0 0

xy

Numerical Simulations: Re-Phase

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-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15-2

-1

0

1

2

3

4

x [km]

y [km

]

No Adaptation

Start

With Adaptation

-80 -60 -40 -20 0 20 40 60 80

2940

2960

2980

3000

3020

3040

3060

x [m]

y [m

]

No Adaptation

End

With Adaptation

10m Circle

Simulated trajectory in the x-y plane

Initial relative position of 0 km in x, -1.9 km in y in the LVLH

Final relative position of 0 km in x, 3 km in y in the LVLH

Numerical Simulations: Re-Phase Control signal for both controllers

Adaptive VS Non Adaptive

Number of control switches: 107 VS 124 (13.7%, less actuation)

Maneuver time: 27 hr VS 31 hr (10.9%, less time)

14

No Adaptation

With Adaptation

0 0.2 0.4 0.6 0.8 1 1.2

-1

-0.5

0

0.5

1

Time [days]

Co

ntr

ol S

ign

al

0 0.2 0.4 0.6 0.8 1 1.2

-1

-0.5

0

0.5

1

Time [days]

Co

ntr

ol S

ign

al

Numerical Simulations: Fly-Around

Simulated trajectory in the x-y plane

Initial relative position of 0 km in x, -4.25 km in y in the LVLH

Final State: Stable orbit around Target S/C

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-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

x [km]

y [km

]

No Adaptation

Start

With Adaptation

Guidance

-150 -100 -50 0 50 100 150-300

-200

-100

0

100

200

300

x [m]

y [m

]

No Adaptation

End

With Adaptation

Guidance

Desired final stable orbit

Numerical Simulations: Fly-Around Control signal for both controllers

Adaptive VS Non Adaptive

Number of control switches: 37 VS 41 (9.8% less actuation)

Maneuver time: 13 hr for both since maneuver is stopped after a set time

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No Adaptation

With Adaptation

0 0.1 0.2 0.3 0.4 0.5

-1

-0.5

0

0.5

1

Time [days]

Co

ntr

ol S

ign

al

0 0.1 0.2 0.3 0.4 0.5

-1

-0.5

0

0.5

1

Time [days]

Co

ntr

ol S

ign

al

Numerical Simulations: Rendezvous Case 1

Simulated trajectory in the x-y plane

Initial relative position of -1km in x, -2km in y in the LVLH

Less realistic linear reference model for the rendezvous (RLQR=1.6*1018)

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-1.5 -1 -0.5 0 0.5 1-5

0

5

10

15

20

25

30

35

40

x [km]

y [km

]

No Adaptation

With Adaptation

Start

Linear Model

Guidance

-100 -50 0 50 100-200

-150

-100

-50

0

50

100

150

200

x [m]

y [m

]

No Adaptation

End

With Adaptation

Guidance

Linear Model

Numerical Simulations: Rendezvous Case 1 Control signal for both controllers

Adaptive VS Non Adaptive

Number of control switches: 124 VS 239 (37% less actuation)

Maneuver time: 49 hr VS 66 hr (25.4% less time)

18

No Adaptation

With Adaptation

0 0.5 1 1.5 2 2.5

-1

-0.5

0

0.5

1

Time [days]

Co

ntr

ol S

ign

al

0 0.5 1 1.5 2 2.5

-1

-0.5

0

0.5

1

Time [days]

Co

ntr

ol S

ign

al

Numerical Simulations: Rendezvous Case 2

Simulated trajectory in the x-y plane

More realistic linear reference model (RLQR=1.5*1017).

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-1.5 -1 -0.5 0 0.5 1 1.5-15

-10

-5

0

5

10

15

20

25

30

35

x [km]

y [km

]

No Adaptation

With Adaptation

Start

Linear Model

Guidance

-150 -100 -50 0 50 100 150-300

-200

-100

0

100

200

300

x [m]

y [m

]

No Adaptation

End

With Adaptation

Guidance

Linear Model

Numerical Simulations: Rendezvous Case 2 Control signal for both controllers

Adaptive VS Non Adaptive

Number of control switches: 36 VS 37 (2.7% less actuation)

Maneuver time: 37 hr VS 39 hr (4.9% less time)

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No Adaptation

With Adaptation

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

-1

-0.5

0

0.5

1

Time [days]

Co

ntr

ol S

ign

al

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

-1

-0.5

0

0.5

1

Time [days]

Co

ntr

ol S

ign

al

Conclusions

The adaptive Lyapunov controller enables tracking of a trajectory, the dynamics of a reference model, or simply regulating to a desired final state

Adaptation provides smoother maneuvers with less duration, less actuation, and greater control margin for the three different controller configurations studied.

The use of the general derivatives will allow for the implementation of the adaptive Lyapunov controller in maneuvers, in which a specific path is desired, consequently, opening the possibilities for many other maneuvers using differential drag, provided that they are confined to the orbital plane.

If the linear reference model is not accurate (unrealistic), the adaptive controller is capable of tune itself; thus improving its performance.

more sophisticated adaptation methods (EIGENVALUES) are expected to significantly improve the ability of the adaptive controller to perform well even when the linear reference model greatly misrepresents the actual dynamics

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