dynamical systems and space mission design
TRANSCRIPT
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Dynamical Systems
and Space Mission Design
Jerrold Marsden, Wang Koon and Martin Lo
Wang Sang KoonControl and Dynamical Systems, Caltech
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Halo Orbit and Its Computation: Outline
In Lecture 5A, we have covered
Importance of halo orbits.
Finding periodic solutions of the linearized equations.
Highlights on 3rd order approximation of a halo orbit.
Using a textbook example to illustrate Lindstedt-Poincare method.
In Lecture 5B, we will cover
Use L.P. method to find a 3rd order approximationof a halo orbit.
Finding a halo orbit numerically via differential correction.
Orbit structure near L1
and L2
.
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Review of Lindstedt-Poincare Method
To avoid secure terms, Lindstedt-Poincare method
Notices non-linearity alters frequency to ().
Introduce new independent variable = ()t:
t = 1
= (1 + 1 + 2
2 + ). Rewrite equation using as independent variable:
q + (1 + 1 + 22 + )
2(q+ q3) = 0.
Expand periodic solution in a power series of :
q =
n=0
nqn() = q0() + q1() + 2q2() +
By substituing q into equation and equating terms in n:
q0 + q0 = 0,
q1 + q1 = q30 21q0,q2 + q2 = 3q
20q1 21(q1 + q
30) + (
21 + 22)q0,
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Review of Lindstedt-Poincare Method
Remove secular terms by choosing suitable n.
Solution of 1st equation: q0 = acos( + 0).
Substitute q0 = acos( + 0) into 2nd equation
q
1 + q1 = a3
cos3
( + 0) 21a cos( + 0)=
1
4a3 cos3( + 0) (
3
4a2 + 21)acos( + 0).
Set 1 = 3a2/8 to remove cos( + 0) and secular term.
Therefore, to 1st order of , we have periodic solution
q = acos(t + 0) +1
32 cos3(t + 0) + o(
2).
with
= (1 3
8a2
15
2562a4 + o(3).
Lindstedt-Poincare method consists insuccessive adjustments of frequencies.
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Lindstedt Poincare Method: Nonlinear Expansion
CR3BP equations can be developed using Legendre polynomial Pn
x 2y (1 + 2c2)x =
x
n3
cnnPn(
x
)
y + 2x + (c2 1)y =
yn3
cnnPn(x
)
z+ c2z =
zn3
cnnPn(
x
)
where 2 = x2+y2+z2, and cn = 3(+(1)n(1)( 1)
n+1).
Useful if successive approximation solution procedure is carriedto high order via algebraic manipulation software programs.
Pn(x
) =
x
(
2n 1
n)Pn1(
x
) (
n 1
n)Pn2(
x
).
Recall that < 1.
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Lindstedt Poincare Method: 3rd Order Expansion
3rd order approximation used in Richardson [1980]:
x 2y (1 + 2c2)x =3
2c3(2x
2 y2 z2)
+2c4x(2x2 3y2 3z2) + o(4),
y + 2x + (c2 1)y = 3c3xy 32c4y(4x2 y2 z2) + o(4),
z+ c2z = 3c3xz3
2c4z(4x
2 y2 z2) + o(4).
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Construction of Periodic Solutions
Recall that solution to the linearized equations
x 2y (1 + 2c2)x = 0
y + 2x + (c2 1)y = 0
z+ c2z = 0
has the following form
x = Ax cos(t + )
y = kAx sin(t + )z = Azsin(t + )
Halo orbits are obtained if amplitudes Ax and Az
of linearized solution are large enoug so thatnonlinear contributions makes eigen-frequencies equal ( = ).
This linearized solution ( = )is the seed for constructing successve approximations.
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Construction of Periodic Solutions
We would like to rewrite linearized equations in following form:
x 2y (1 + 2c2)x = 0
y + 2x + (c2 1)y = 0
z+ 2z = 0
which has a periodic solution with frequency .
Need to have a correction term = 2 c2for high order approximations.
z+ 2z = 3c3xz3
2c4z(4x
2 y2 z2) + z+ o(4).
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Lindstedt-Poincare Method
Richardson [1980] developed a 3rd order periodic solution
using a L.P. type successive approximations.
To remove secular terms, a new independent variable anda frequency connection are introduced via
= t.
Here,
= 1 +n1
n, n < 1.
The n are assumed to be o(Anz)
and are chosen to removesecure
terms. Notice that Az
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Lindstedt-Poincare Method
Equations are then written in terms of new independent variable
2x 2y (1 + 2c2)x =3
2c3(2x
2 y2 z2)
+2c4x(2x2 3y2 3z2) + o(4),
2y + 2x + (c2 1)y = 3c3xy
3
2c4y(4x
2 y2 z2) + o(4),
2z + 2z = 3c3xz
32
c4z(4x2 y2 z2) + z+ o(4).
3rd order successive approximation solution is a lengthy process.
Here are some highlights: Generating solution is linearized solution with t replaced by
x = Ax cos( + )
y = kAx sin( + )z = Azsin( + )
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Lindstedt-Poincare Method
Some highlights:
Look for general solutions of the following type:
x =n0
an cos n1, y =n0
bn sin n1, z =n0
cn cos n1,
where 1 = + = t + .
It is found that
1 = 0, 2 = s1A2
x + s2A2
z,which give the frequence ( = 1 + 1 + 2 + ) and theperiod T (T = 2/) of a halo orbit.
To remove all secular terms, it is also necessary to specifyamplitude and phase-angle constraint relationships:
l1A2x + l2A
2z + = 0,
= m/2, m = 1, 3.
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Halo Orbits in 3rd Order Approximation
3rd order solution in Richardson [1980]:
x = a21A2x + a22A
2zAx cos 1
+(a23A2x a24A
2z)cos21 + (a31A
3x a32AxA
2z)cos31,
y = kAx sin 1+(b21A2x b22A2z)sin21 + (b31A3x b32AxA2z)sin31,
z = mAzcos 1+md21AxAz(cos21 3) + m(d32AzA
2x d31A
3z)cos31.
where 1 = + and m = 2 m, m = 1, 3.
2 solution branches are obtained according towhether m = 1 or m = 3.
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Halo Orbit Phase-angle Relationship
Bifurcation manifests through phase-angle relationship:
For m = 1, Az > 0. Northern halo.
For m = 3, Az < 0. Southern halo.
Northern & southern halos are mirror images across xy-plane.
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Halo Orbit Amplitude Constraint Relationship
For halo orbits, we have amplitude constraint relationship
l1A2x + l2A
2z + = 0.
Minimim value for Ax to have a halo orbit (Az > 0) is|/l1|,
which is about 200, 000 km.
Halo orbit can be characterized completely by Az.
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Halo Orbit Period Amplitude Relationship
The halo orbit period T (T = 2/)
can be computed as a function ofAz.
Amplitude constraint relationship: l1A2x + l2A
2z + = 0.
Frequence connection ( = 1 + 1 + 2 + ) with
1 = 0, 2 = s1A2x + s2A
2z,
ISEE3 halo had a period of 177.73 days.
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Differential Corrections
While 3rd order approximations provide much insight,
they are insufficient for serious study of motion near L1.
Analytic approximations must be combined withnumerical techniques to generate an accurate halo orbit.
This problem is well suited to a differential corrections process,
which incorporates the analytic approximationsas the first guess
in an iterative process aimed at producing initial conditions that lead to a halo orbit.
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Differential Corrections: Variational Equations
Recall 3D CR3BP equations:
x 2y = Ux y + 2x = Uy z = Uz
where U = (x2 + y2)/2 + (1 )d11 + d12 .
It can be rewritten as 6 1st order ODEs: x = f(x),where x = (x y z x y z)T is the state vector.
Given a reference solution x to ODE,
variational equations which are linearized equations forvariations x (relative to reference solution) can be written as
x(t) = Df(x)x = A(t)x(t),
where A(t) is a matrix of the form0 I3U 2
.
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Differential Corrections: Variational Equations
Given a reference solution x to ODE,
variational equations can be written as
x(t) = Df(x)x = A(t)x(t), where
A(t) =
0 I3U 2
.
Matrix can be written
=
0 1 01 0 0
0 0 0
Matrix U has the form
U=
Uxx Uxy UxzUyx Uyy UyzUzx Uzy Uzz
,
and is evalutated on reference solution.
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Differential Corrections: State Transition Matrix
Solution of variational equations is known to be of the form
x(t) = (t, t0)x(t0),
where (t, t0) represents state transition matrix from time t0 to t.
State transition matrix reflects sensitivity of state at time t tosmall perturbations in initial state at time t0.
To apply differential corrections,need to compute state transition matrix along a reference orbit.
Since
(t, t0)x(t0) = x(t) = A(t)x(t) = A(t)(t, t0)x(t0),
we obtain ODEs for (t, t0):
(t, t0) = A(t)(t, t0),
with
(t0, t0) = I6.
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Differential Corrections: State Transition Matrix
Therefore, state transition matrix along a reference orbit
x(t) = (t, t0)x(t0),
can be computed numericallyby integrating simultaneously the following 42 ODEs:
x = f(x),
(t, t0) = A(t)(t, t0),
with initial conditions:x(t0) = x0,
(t0, t0) = I6.
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Numerical Computation of Halo Orbit
Halo orbits are symmetric about xz-plane (y = 0).
They intersect this plan perpendicularly(x = z = 0).
Thus, initial state vector take the form
x0
= (x0
0 z0
0 y0
0)T.
Obtain 1st guess for x0 from 3rd order approximations.
ODEs are integrated until trajectory cross xz-plane.
For periodic solution, desired final state vector has the formxf = (xf 0 zf 0 yf 0)
T.
While actual values for xf, zf may not be zero,
3 non-zero initial conditions (x0, z0, y0) can be usedto drive these final velocities xf, zf to zero.
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Numerical Computation of Halo Orbit
Differential corrections use state transition matrix
to change initial conditions
xf = (tf, t0)x0.
The change x0
can be determined by the difference betweenactual and desired final states (xf = xdf xf).
3 initial states (x0, z0, y0)are available to target 2 final states (xf, zf).
But it is more convenient to set z0 = 0and to use resulting 2 2 matrix to find x0, y0.
Similarly, the revised initial conditions x0 + x0
are used to begin a second iteration. This process is continued until xf = zf = 0
(within some accptable tolerance).
Usually, convergence to a solution is achieved within 4 iterations.
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Numerical Computation of Lissajous Trajectories
Howell and Pernicka [1987] used similar techniques
(3rd order approximation and differential corrections)to compute lissajous trajectories.
Gomez, Jorba, Masdemont and Simo [1991] used
higher order expansions to compute halo, quasi-halo and lissajousorbits.
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Veritcal Orbit
A vertical orbit and its 3 projections.
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Lissajous Orbits
A lissajous orbit and its 3 projections.
H l O bi
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Halo Orbits
A halo orbit and its 3 projections.
Q i H l O bit
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Quasi-Halo Orbits
A quasi-halo orbit and its 3 projections.
O bit St t d L
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Orbit Structure around L1
Poincare sections of center manifold of L1 corresponding to h =
0.2, 0.5, 0.6, 1.0.