Circular orbit spacecraft control at the L4 point using...

CR3BP Model Control design Stability Simulations Domain of stability

Circular orbit spacecraft control at the L4 point usingLyapunov functions

Rachana AgrawalRavi N Banavar

Indian Institute of Technology, Bombay

June 29, 2016

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Overview

1 The Circular Restricted Three Body Problem

2 System Model and Problem Definition

3 Controller design using Lyapunov function

4 Asymptotic stability of desired orbit

5 Numerical Simulations

6 Stability Region

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The Circular Restricted Three Body Problem (CR3BP)

• Two bodies revolve around their center of mass in circular orbits underthe influence of their mutual gravitational attraction. A third body(attracted by the other two but not influencing their motion) movesunder the influence of the two revolving bodies. The restricted problemof three bodies is to describe the motion of this third body

r 1s

r2s

rscenter of mass

m2

m1

ms

O

Figure: The three body problem

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Libration (or Lagrange) points• Libration points are those positions in the CR3BP where the third

body, only influenced by gravity, remains stationary with respect tothe two massive bodies

• Five libration points in the earth-moon system - L1 to L5 - shown inthe figure.

• Collinear points - L1, L2, L3 unstable• Triangular points - L4, L5 stable

Figure: The Libration Points in the Earth-Moon system

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Stability of triangular libration points

• Stable periodic elongated bean shaped orbits exist around thetriangular points

• Very low station keeping efforts required to orbit triangular points

Figure: Orbits in the CR3BP of Earth-Moon system Credit: David A. Kring,LPI-JSC Center for Lunar Science and Exploration

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System Model

rs

rm1

rm2

rc

r 1s

r2s

rcs

O

m2

m1

ms L4

yi

xi

yb

xb

Figure: The three body system with a spacecraft in the vicinity of the L4 point

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Nomenclature

• m1,m2,ms - two primaries and spacecraft

• m1,m2, L4 form a plane of rotation - the synodic frame

• rmj - distance of mj from the centre of mass (COM) of m1m2

• rc - position vector of L4 point from the COM

• rcs - position vector of spacecraft from L4

• rjs - position vector of spacecraft from mj

• φ - angle of rotation of the synodic frame with respect to the siderealframe

• subscript b - vectors in synodic frame

• subscript i - vectors in sidereal (inertial) frame

• xi, yi are along the inertial frame

• xs, ys are along the synodic frame

• ω - angular velocity vector of the two primaries about the COM

• ω - skew symmetric matrix of ω given by

ω = φ

0 1 0−1 0 00 0 0

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System model

• Dynamics in the inertial frame

msrsi =km1ms

||r1si ||2(−r1si) +

km2ms

||r2si ||2(−r2si)

• Transforming from the sidereal (inertial) frame to the synodic(rotating) frame

ri = Rrb

• Natural dynamics of third particle

rcsb =φ2rcsb + φ2rcb − 2ωrcsb

− km1

||r1sb ||3(rcsb + r1cb)− km2

||r2sb ||3(rcsb + r2cb)

• Introducing a controller

rcsb = f(rcsb , rcsb) + ub

where f(rcsb , rcsb) denotes the natural dynamics.

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Desired circular orbit around the L4 is characterized by a two-tuple (d, Ld),where d stands for the magnitude of the desired radius of the orbit and Ld

is the desired angular momentum.

The velocity vector is perpendicular to the position vector,

rcsb · rcsb = 0

The angular momentum is constant,

rcsb × rcsb = 0

The orbital radius is of a specified magnitude,

||rcsb || = d

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Lyapunov Function

Candidate Lyapunov function

V (rcsb , rcsb) =1

2(|rcsb · rcsb |

2 + ||rcsb × rcsb −Ld||2)

+a

2(||rcsb || − d)2

where a > 0 is a tuning parameter for control design purpose

• V is chosen such that V (rcsb , rcsb) = 0 at the desired orbit

• The desired orbit is reached when rcsb · rcsb = 0, rcsb × rcsb = Ld and||rcsb || = d

• V (·, ·) > 0 for all other states

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Assumption

The radius of the orbit of the spacecraft is small compared to the distanceof the libration point from the two primaries, and hence the followingapproximations are made: ||r1sb || ≈ ||r1cb ||(1 + ε1) and||r2sb || ≈ ||r2cb ||(1 + ε2) where |εi| << 1

Dynamics at L4

At L4 we have

φ2rcb −km1

||r1cb ||3(r1cb)− km2

||r2cb ||3(r2cb) = 0

• Approximation: Using the above assumption and a binomial expansion wearrive at the following approximation

kmi

||risb ||3≈ kmi

||ricb ||3(1 + εi)3≈ kmi(1− 3εi)

||rcb ||3

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Lyapunov Function

• Using the assumption and the the dynamics at the L4 point we have

dV

dt=||rcsb ||

2(rcsb · rcsb)(φ2 +||rcsb ||

2

||rcsb ||2− km1

||r1sb ||3

− km2

||r2sb ||3)− (−2ωrcsb) · (Ld × rcsb)

+ z · (rcsb ||rcsb ||2 −Ld × rcsb)

+ ub · (rcsb ||rcsb ||2 −Ld × rcsb)

+ a(||rcsb || − d)rcsb · rcsb

||rcsb ||

where

z4=km13ε1||r1cb ||3

r1cb +km23ε2||r2cb ||3

r2cb

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Controller

• The proposed controller

ub =− β(rcsb ||rcsb ||2 −Ld × rcsb) + rcsb × η

+ qrcsb −a(||rcsb || − d)

||rcsb ||3rcsb − z

withq = −p and η = −2ω

where

p = φ2 +||rcsb ||

2

||rcsb ||2− km1

||r1sb ||3− km2

||r2sb ||3

• With the above controller the derivative of the Lyapunov function reducesto

dV

dt= −β||(rcsb ||rcsb ||

2 −Ld × rcsb)||2 ≤ 0

Therefore, the feedback system is Lyapunov stable.

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LaSalle’s Invariance Principle

ClaimConsider the positively invariant and compact set

Ω = (rcsb , rcsb) ∈ (R3 × R3) |V (rcsb , ˙rcsb) ≤ c

The largest invariant set in

E = (rcsb , rcsb) ∈ Ω|V = 0

is (rcsb , rcsb) : ||rcsb || = d, rcsb × rcsb = Ld.

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Applying LaSalle’s Invariance Principle

• Considering the orbits in set E we have

rcsb · rcsb = 0 (1)

Ld = rcsb × rcsb (2)

• Equations (1) and (2) imply that the orbits lying in the limit set E arecircular with desired angular momentum.

• However, we need to prove that radius of orbit is the desired one.

• Substituting the expression for controller in the system dynamics and usingthe assumption made earlier, dynamics equation reduces to

rcsb =− β(rcsb ||rcsb ||2 −Ld × rcsb)− ||rcsb ||

2

||rcsb ||2rcsb

− a(||rcsb || − d)

||rcsb ||3rcsb

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• At steady state

rcsb = −||rcsb ||2

||rcsb ||2rcsb −

a(||rcsb || − d)

||rcsb ||3rcsb

• However, for a circular orbit, the acceleration should be equal to centripetalacceleration.

rcsb = −||rcsb ||2

||rcsb ||2rcsb

• Therefore||rcsb || = d

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Remarks on controller

The control law has a feedback linearization structure

ub =− βe1 − ae2 − f(rcsb , rcsb)− ||rcsb ||2

||rcsb ||2rcsb

where

• e1 = rcsb ||rcsb ||2 −Ld × rcsb

• e2 =(||rcsb

||−d)

||rcsb||3 rcsb

• f(rcsb , rcsb) is the natural dynamics of the spacecraft

• the fourth term is the desired dynamics.

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Numerical Simulations

• Saturation introduced in the controller:

u =

ub if ||ub|| ≤ umax

umaxub||ub||

if ||ub|| > umax

where ub is the proposed controller and umax is the maximum feasiblecontrol input

• For the Earth-Moon-Spacecraft three body system two cases simulated:Initial position and velocity vector -

• in the plane of rotation of primaries

• out of plane of rotation of primaries

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Case 1: In plane of rotation of primaries

Figure: Spacerafttrajectory with initialpostion and desiredorbit

Figure: Norm of thecontrol vector in theinitial few seconds

Figure: Norm of thecontrol vector duringand after orbit capture

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Case 1 (contd.)

Figure: Distance ofspacecraft from L4

Figure: Norm of theangular momentum inthe initial few seconds

Figure: Norm of theangular momentumduring and after orbitcapture

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Case 2: Out of plane of rotation of primaries

Figure: Spacerafttrajectory with initialpostion and desiredorbit

Figure: Norm of thecontrol vector in theinitial few seconds

Figure: Norm of thecontrol vector duringand after orbit capture

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Case 2 (contd.)

Figure: Distance ofspacecraft from L4

Figure: Norm of theangular momentum inthe initial few seconds

Figure: Norm of theangular momentumduring and after orbitcapture

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Stability region

• We have proved that the desired orbit is asymptotically stable using thegiven designed controller

• It is under the approximation ||risb || ≈ ||ricb ||(1 + εi)

• Need to determine region around L4 in which this assumption is reasonable

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Stability region around the L4 point

• The invariant manifolds associated with the central manifold around L3seem to play an important role in the effective stability regions around thetriangular equilibrium points L4 and L5

Figure: Orbits in the CR3BP of Earth-Moon system Credit: David A. Kring,LPI-JSC Center for Lunar Science and Exploration

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Future work

• Find the invariant manifolds around the L4 point

• The objective can be to make the spacecraft stay in a region around the L4point rather than an orbit

• Can be extended to spacecraft formations

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References

• H K Khalil. Nonlinear Systems. Pearson Education. Prentice Hall, 2002

• V Szebehely and E Grebenikov. Theory of orbits-the restricted problem ofthree bodies. Astronomicheskii Zhurnal, 46:459, 1969.

• Dong Eui Chang, D F Chichka, and J E Marsden. Lyapunov-based transferbetween elliptic keplerian orbits. Discrete and Continuous DynamicalSystems Series B, 2(1):5768, 2002.

• Gerard G omez, G Gomez C Simo J Llibre, and Regina Mart nez.Dynamics and Mission Design Near Libration Points, Vol I: Funda-mentals: the Case of Collinear Libration Points. Number v. 1 in WorldScientific monograph series in mathematics. World Scientific, 2001.

• K A Catlin and C A McLaughlin. Earth-moon triangular libration pointspacecraft formations. Journal of Spacecraft and Rockets, 44(3):660 670,2007

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References

• A Deprit and A Deprit-Bartholome. Stability of the triangular Lagrangianpoints. The Astronomical Journal 72 (2), 72:173, March 1967.

• B G Marchand and K C Howell. Control strategies for formation flight inthe vicinity of the libration points. Journal of guidance, control, anddynamics (AIAA), 28(6):12101219, 2005.

• S R Vadali, H W Bae, and K T Alfred. Design and control of librationpoint satellite formations. 14 AAS/AIAA Space Flight MechanicsConference, 2004.

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