_{1}

^{*}

In this paper, we use a Circle Restricted Three - Body Problem (CRTBP) to simulate the motion of a satellite. Then we reformulate this problem with the controller into the description using Koopman eigenfunction. Although the original dynamical system is nonlinear, the Koopman eigenfunction behaves linearly. Choosing Jacobi integral as the Koopman eigenfunction and using the zero velocity curve as the reference for control, we are allowed to combine well-developed Linear Quadratic Regulator (LQR) controller to design a nonlinear controller. Using this approach, we design the low thrust orbit transfer strategy for the satellite flying from the earth to the moon or from the earth to the sun.

Restricted three-body problems usually have a satellite which has a relatively small mass introduced into a system of two large masses orbiting circularly around their center of mass. CRTBP has many situations that it applies for in the solar system, such as a satellite traveling from earth to moon or comet entering our Sun-Earth system.

In order to solve the optimal control problem of the orbit transfer, a wide range of traditional optimal control strategy is employed. Traditional methods for optimal control of orbit transfer are mainly divided as the direct method [

However, neither direct method nor indirect method could provide a general solution methodology for a nonlinear optimal control problem [

In this paper, we use a Circle Restricted Three-Body Problem (CRTBP) to simulate the motion of a satellite. Then we reformulate this problem with controller into the description using Koopman eigenfunction. Choosing Jacobi integral as the Koopman eigenfunction and using the zero velocity curve as the reference for control, we are allowed to combine well-developed linear controller to design nonlinear control strategy. This approach is then employed to perform the optimal control on circular restricted three-body problem.

In a circular restricted three-body problem, the satellite’s mass is negligible with respect to the other two, and thus we neglect the force from the third mass acting on other two larger masses. When considering a rotating reference frame, the two co-orbiting bodies are considered stationary, and the ratios of mass and distance are considered instead of their actual value because their relative value is of the most importance. The governing equations of restricted circle three-body problem are:

d 2 x d t 2 − 2 d y d t = x − ( 1 − μ ) x − x 1 r 1 3 − μ x − x 2 r 2 3 (1)

d 2 y d t 2 − 2 d x d t = x − ( 1 − μ ) y r 1 3 − μ y r 2 3 (2)

where

r 1 = ( y 1 + μ ) 2 + y 3 2 (3)

r 2 = ( y 1 − 1 + μ ) 2 + y 3 2 (4)

and we set µ = 0.012155085 that is the mass ratio between the earth and moon. We can rewrite these formula with a substitution: y 1 = x , y 2 = x ˙ , y 3 = y , y 4 = y ˙

( y ˙ 1 y ˙ 2 y ˙ 3 y ˙ 4 ) = ( y 2 2 y 4 + y 1 − ( 1 − μ ) ( y 1 + μ ) r 1 3 − μ ( y 1 − 1 + μ ) r 2 3 y 4 − 2 y 2 + y 3 − ( 1 − μ ) y 3 r 1 3 − μ ) : = f ( y ) (5)

Jacobi’s integral is the only known conserved quantity for the circular restricted three-body problem. This integral is used to derive numerous solutions in special cases of three body problem. The Jacobi integral remains constant even though Energy and momentum are not conserved separately, and is expressed as follow in the (x, y)-coordinate system:

C = ( x 2 + y 2 ) + 2 ( 1 − μ r 1 + μ r 2 ) − ( x ˙ 2 + y ˙ 2 ) (6)

The zero-velocity surface relates to the restricted three body problem of gravity and it plays a critical role in restricted circle three-body problem. It represents a surface that a body of given energy cannot cross through, as it would have zero velocity if reaching the surface. As is shown in _{1} Lagrangian point represents the surface that the third body flying at rest from the left of L_{1} cannot cross. Thus, the third body needs to move around the largest body permanently. However, the zero velocity contour that passes through L_{2} is much more interesting and we can see that the third body

can transfer between the largest body and the second largest body. This zero velocity curve provides us the potential that controlling the Jacobean (that include both kinetic energy and potential energy) will control the orbit transfer between these two large bodies. In what follows, we utilize this difference to move the satellite from orbits around the earth to the orbits around the moon, where we denote the larger two bodies as earth and moon while treating the third body as satellite.

We write the governing equation of the three body problem in this generalized form:

d d t y ( t ) = f ( y ) (7)

And then we are going to define its corresponding Koopman eigenfunction. In 1931, B. O. Koopman [_{t} that acts to advance all measurement functions g of the state with the flow of the dynamics. For the infinite dimensional linear operator K_{t}, its eigenfunction is of the most importance and it satisfies the following equation [

d d t ϕ ( y ) = λ ϕ ( y ) (8)

The Koopman eigenfunction can be incorporated into the original governing Equation (7) using the chain rule, where it gives:

d d t ϕ ( y ) = ∇ ϕ ( y ) ⋅ y ˙ = ∇ ϕ ( y ) ⋅ f ( y ) (9)

With the definition of the Koopman eigenfunction, the eigenfunction satisfies:

∇ ϕ ( y ) ⋅ f ( y ) = λ ϕ ( y ) (10)

Furthermore, we can implement Koopman eigenfunction into a dynamical system with control. For example, we add the thrust force in three body problem acting directly on x and y direction acceleration.

d d t y = f ( y ) + B u (11)

where u is the input and B is the corresponding matrix:

B = [ 0 0 1 0 0 0 0 1 ] , u = [ u x u y ] (12)

This linear system represented in Koopman eigenfuction is:

d d t ϕ ( y ) = ∇ ϕ ( y ) ⋅ y ˙ = ∇ ϕ ( y ) ⋅ [ f ( y ) + B u ] = λ ϕ ( y ) + ∇ ϕ ( y ) ⋅ B u (13)

In this form, the orignial nonlinear system f(y) in the governing equation is transformed into a linear one, which is represented as the eigenvalue. The control input enters the dynamics of Koopman eigenfunction ϕ via an additional term which is linear in ϕ but possibly nonlinear in control.

Because the Jacobi integration is the conservation quantity of restricted three body problem, it corresponds to a Koopman eigenfunction with eigenvalue λ = 0, i.e.:

d C d t = 0 × C (14)

Thus, we use Jacobi integration of restricted three body problem as Koopman eigenfunction for the reduced order optimal control.

Although there are plenty of possible Koopman eigenfunctions, Jacobean integral is the known one that satisfies the definition of Koopman eigenfunction exactly. Setting velocity in Jacobean integral as zero, it also provides the physical meaning; i.e., zero velocity curve that serves to represent the potential energy of the system. In the following work, we are also going to utilize the zero velocity curve as the guidance for the control reference of Koopman eigenfunction.

With Koopman eigenfunction, we have transformed the dynamics in the system of ϕ , which is linear. This offers us the space to utilize the traditional linear quadratic controller [

J ( ϕ , u ) = 1 2 ∫ 0 1 ϕ T Q ϕ + u T R u (15)

where ϕ is the Koopman eigenfunction for our problem; i.e. Jacobean integral in restricted three body problem. Q is the cost matrix that weights the current state variable and R is the cost matrix that regulates the input u. The LQR provides a feedback controller represented by Koopman eigenfunction:

u = − K ϕ ϕ ( y ) (16)

which is linear in the eigenfunction, but generally nonlinear in original state y. We also consider the reference tracking, u = − K ϕ ( ϕ ( x − ϕ r e f ) ) , with a modified cost function. With this reference, we are allowed to control the Koopman eigenfunction into different levels. In our problem, the reference is determined through the zero velocity curve. Increasing the Jacobean integral corresponding to the system, we will have more total energy and thus satellite is allowed to fly to high potential energy space.

In this section, we are trying to solve a classical orbit transfer problem using the proposed Koopman reduced order control approach. To stimulate the practical application and to simplify the following description, we consider the earth to be the largest body in our restricted three body problem, the moon as the smaller planet, and our satellite as the third body. Using Equation (15), which describes a magnitude of thrust, we found a way to solve the minimal thrust orbit problem that allows satellite to transfer from the earth orbit to the moon orbit. In

As shown in _{1} and L_{2}, respectively (with a minus sign difference).

Furthermore, we modify the mass ratio µ in circle restricted three body problem to consider the sun-earth-moon system, assuming that the force of the moon and other planets are negligible. In that the orbit moving around the earth is extremely small compared to the earth-sun distance, we also show results that zoom in to the region around the earth (

As shown in _{1} and L_{2}, respectively (with a minus sign difference).

controller we design is linear for Jacobi Integral (16), but the controller is nonlinear with respective to original state variable; i.e., velocity and displacement in x and y directions.

In this paper, we use a Circle Restricted Three-Body Problem (CRTBP) to simulate the system among the earth, moon and satellite and the system among the sun, earth and satellite. Then we reformulate this problem with controller into the description using Koopman eigenfunction. Although original dynamical system is nonlinear, the Koopman eigenfunction behaves linearly. Choosing Jacobi integral as the Koopman eigenfunction and using the zero velocity curve as the reference for control, we are allowed to combine traditional Linear Quadratic Regulator (LQR) controller to design a nonlinear controller. At last, we design the low thrust orbit transfer strategy for the satellite flying from the earth to the moon or from the earth to the sun.

The author declares no conflicts of interest regarding the publication of this paper.

Tang, H.Z. (2019) Koopman Reduced Order Control for Three Body Problem. Modern Mechanical Engineering, 9, 20-29. https://doi.org/10.4236/mme.2019.91003