dynamical systems and space mission design

7/29/2019 DYNAMICAL SYSTEMS AND SPACE MISSION DESIGN

1/28

Dynamical Systems

and Space Mission Design

Jerrold Marsden, Wang Koon and Martin Lo

Wang Sang KoonControl and Dynamical Systems, Caltech

[email protected]


2/28

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

.


3/28

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,


4/28

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.


5/28

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.


6/28

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).


7/28

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.


8/28

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).


9/28

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


10/28


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( + )


11/28


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.


12/28

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.


13/28

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.


14/28

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.


15/28

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.


16/28

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.


17/28

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

.


18/28

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.


19/28

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.


20/28

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.


21/28

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.


22/28

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.


23/28

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.


24/28

Veritcal Orbit

A vertical orbit and its 3 projections.


25/28

Lissajous Orbits

A lissajous orbit and its 3 projections.

H l O bi


26/28

Halo Orbits

A halo orbit and its 3 projections.

Q i H l O bit


27/28

Quasi-Halo Orbits

A quasi-halo orbit and its 3 projections.

O bit St t d L


28/28

Orbit Structure around L1

Poincare sections of center manifold of L1 corresponding to h =

0.2, 0.5, 0.6, 1.0.

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