_{1}

In this paper, we use Port-Hamiltonian framework to stabilize the Lagrange points in the Sun-Earth three-dimensional Circular Restricted Three-Body Problem (CRTBP). Through rewriting the CRTBP into Port-Hamiltonian framework, we are allowed to design the feedback controller through ener gy-shaping and dissipation injection. The closed-loop Hamiltonian is a candidate of the Lyapunov function to establish nonlinear stability of the designed equilibrium, which enlarges the application region of feedback controller compared with that based on linearized dynamics. Results show that th e Port-Hamiltonian a pproach allows us to successfully stabilize the Lagrange points, where the Linear Quadratic Regulator (LQR) may fail. The feedback system based on Port-Hamiltonian approach is also robust against whit e noise in the inputs.

The Circular Restricted Three Body Problem (CRTBP) [_{1} and L_{2} Lagrange points with respect to the Sun and the Earth [

However, that is also of the most demanding challenge; i.e., among the five Lagrange points, three of them are unstable [_{1}, L_{2}, L_{3}), there exist stable halo orbits, which allow artificial satellite to reside. In order to design an orbit around Lagrange points, the most broadly used methods are based on linearization around a halo orbit or Lagrange point [

However, stabilization of a linear system is only able to guarantee the stability of the system against an infinitely small perturbation, but is unable to guarantee the stability against a finite disturbance. Disturbances for the Earth-Moon Lagrange points mission under CRTBP model include the eccentricity of the Earth- Moon orbit, the gravity from the other bodies [

Port-Hamiltonian system [

In this research, we use the Port-Hamiltonian system to reformulate the Sun- Earth three-dimensional CRTBP, which retains the original nonlinear dynamics in contrast to the linear approximation. Then, we designed the feedback control law to stabilize the unstable Lagrange points by taking input and dissipation as two other actors into consideration. The closed-loop Hamiltonian serves as the candidate of Lyapunov’s function, which helps to establish the nonlinear stability of the open-loop unstable Lagrange points. This research provides further possibility in Lagrange point launching mission.

In Circular Restricted Three Body Problem (CRTBP), it requires the largest two objects of the three have a significantly larger mass compared to the third one; i.e., M 1 ≫ M 3 and M 2 ≫ M 3 . The largest two objects are in circular orbits centered at their center of mass. We consider the third object, M 3 , moving in three-dimensional (3D) space; i.e., 3D-CRTBP. In order to solve this problem, we need to use mathematical method to approximate this situation. We set the center of mass of the system as the origin and we set the position of M 1 at ( − μ , 0 ) , position of M 2 at ( 1 − μ , 0 ) , where we define μ = M 2 / M and M = M 1 + M 2 . Then, the dynamics of the third body can be described as [

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

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

d 2 z d t 2 = − ( 1 − μ r 1 3 − μ r 2 3 ) z (3)

where

r 1 = ( x + μ ) 2 + y 2 + z 2 (4)

r 2 = ( x − 1 + μ ) 2 + y 2 + z 2 (5)

We set μ = 3.003490055444426 × 10 − 6 , which is the mass ratio in the Sun-Earth CRTBP system (see

Lagrange points are points in systems on which the third object could reach equilibrium state. According to [_{1}, L_{2}, and L_{3} can be computed using algebraic equations:

x − ( 1 − μ ) x + μ r 1 3 − μ x − 1 + μ r 2 3 = 0 (6)

y = 0 (7)

In CRTBP, Jacobian integral is the only conserved variable, which is widely used to derive solutions in special cases, and is expressed as what follows in the (x, y)-coordinate system:

C = ( x 2 + y 2 ) + 2 ( 1 − μ r 1 + μ r 2 ) − ( x 2 + y 2 + z 2 ) (8)

Conservation of energy (kinetic energy and potential energy) is shown by Jacobian integral in CRTBP under synodic coordinates description, in which the dissipation caused by the air drag is not included in the model.

L_{1} | L_{2} | L_{3} | L_{4} | L_{5} | |
---|---|---|---|---|---|

x | 0.990026583427689 | 1.010034125978081 | 1.000001251454178 | 0.499996996509945 | 0.499996996209945 |

y | 0 | 0 | 0 | 0.866025403784439 | 0.866025403784439 |

If we use the Hamiltonian proportional to the Jacobian integral:

H = − C 2 = 1 2 ( x ˙ 2 + y ˙ 2 + z ˙ 2 ) − 1 2 [ 2 ( 1 − μ r 1 + μ r 2 ) + ( x 2 + y 2 ) ] (9)

We can reformulate the CRTBP into Port-Hamiltonian system description:

d d t x = [ J − R ] ∇ x H ( x ) + B u (10)

where x = [ x y z x ˙ y ˙ z ˙ ] T are state variables and u = [ u x u y u z ] T are inputs representing thrust force on x, y and z directions. In order to map inputs to state

variables, we define B = [ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ] T . J is a skew-symmetric matrix,

which represents the energy conserving part, while matrix R is a symmetric positive semi-definite matrix, which represents the energy dissipation:

J = [ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 − 1 0 0 0 2 0 0 − 1 0 − 2 0 0 0 0 − 1 0 0 0 ] T , R = 0 6 × 6 (11)

Since CRTBP does not consider energy dissipation such as air drag, R is a zero matrix. This causes the conservation of energy (Hamiltonian or Jacobian integral used here), which is the fact reflected in the following equation:

d H ( x ) d t = ∇ x H ( x ) ⋅ d x d t = ∇ x H ( x ) ⋅ [ J − R ] ∇ x H ( x ) = ∇ x H ( x ) ⋅ [ − R ] ∇ x H ( x ) = 0 (12)

Instead of controlling the Hamiltonian, we are focused on reshaping the Hamiltonian in CRTBP, where the equilibrium points x * = [ x * y * z * 0 0 0 ] T are stable. Thus, we designed a closed-loop Hamiltonian H d ( x ) :

H d ( x ) = 1 2 [ x ˙ 2 + y ˙ 2 + z ˙ 2 + ( x − x * ) 2 + ( y − y * ) 2 + ( z − z * ) 2 ] (13)

with the minimal position:

x * = arg min { H d ( x ) } (14)

assuming that the closed loop system is also a Port-Hamilton system

d x d t = [ J − R ] ∇ x H d (15)

Then, we design the feedback control law as B is full column rank but not full row rank:

u E S ( x ) = ( B T B ) − 1 B T [ J − R ] ∇ x H a ( x ) (16)

H a ( x ) = H d ( x ) − H ( x ) (17)

where the difference between close-loop system and open-loop system is shown by

B ⊥ [ J − R ] ∇ x H a ( x ) = 0 (18)

Under this control law, it can be shown that the closed-loop Hamiltonian is a Lyapunov function satisfying:

H d ( x ) = 0 , x = x * (19)

H d ( x ) > 0 , ∀ x ≠ x * (20)

d H d ( x ) d t = ∇ x H d ( x ) ⋅ [ − R ] ∇ x H d ( x ) = 0 , ∀ x ≠ x * (21)

According to Lyapunov stability theorem, the equilibrium x of the closed-loop system is stable.

Previous energy shaping process utilizes a state feedback control law to shape the closed-loop Hamiltonian (energy) with a stable equilibrium at x * . However, the closed-loop Hamiltonian also remains a constant; i.e.,

d H d ( x ) d t = ∇ x H d ( x ) ⋅ [ − R ] ∇ x H d ( x ) = 0 . We may also design the equilibrium

point as asymptotically stable. This can be achieved through dissipation injection, which modifies the R matrix in the closed-loop system. Aside from the energy shaping control law, we also implement a state feedback control representing dissipation injection:

u D I ( x ) = − K d B T ∇ x H d ( x ) (22)

where K d is positive definite. With u ( x ) = u E S ( x ) + u D I ( x ) , we have the closed-loop system as:

d x d t = [ J − R d ( x ) ] ∇ x H d (23)

with a closed-loop dissipation matrix: R d ( x ) = R ( x ) + B ( x ) K d B T .

Thus, the closed-loop Hamiltonian evolves like:

d H d ( x ) d t = ∇ x H d ( x ) ⋅ [ − R d ] ∇ x H d ( x ) ≤ 0 , ∀ x ≠ x * (24)

The closed-loop Hamiltonian H d ( x ) also satisfies H d ( x ) = 0 , x = x * and H d ( x ) > 0 , ∀ x = x * .

As the union of complete trajectories contained entirely in the set has no trajectory except the trivial one x ( t ) = 0 for t ≥ x * , we obtain the asymptotic stability of the closed-loop system according to LaSalle’s invariance principle [

Figures 2-4 shows the orbit of the third body without background noise on x-y, x-z, and y-z plane, respectively. The blue dashed lines in these figures indicated the controlled orbits using (a) Linear Quadratic Regulator (LQR) controller compared with (b) Port-Hamiltonian controller. The results show that the Port- Hamiltonian controller allows us to drive the third body to the target equilibrium position, while the LQR controller fail. The LQR controller is designed [_{1} Lagrange point:

d x d t = A x + B u where A is the Jacobian of ( J − R ) ∇ H ( x ) evaluated at the

equilibrium point x * ; i.e. the first Lagrange point L_{1} in this case. Using the Linear Quadratic Regulator (LQR) controller, we obtain the control law as u LQR = − K ( x − x * ) , where K minimizes the cost function J LQR = x 0 s ∞ T Q x + u T R u with Q = I 6 × 6 and R = I 3 × 3 .

Figures 5-7 shows the orbit of the third body with background noise on x-y, x-z, and y-z plane, respectively. This background noise is used to model the disturbance including eccentricity of the orbit, the gravity from the other bodies [_{1} Lagrange point in the Sun-Earth CRTBP. However, the approach based on the Port-Hamiltonian successfully drive the third body to the targeting position.

In this paper, we use Port-Hamiltonian framework to stabilize the Lagrange points in the Circular Restricted Three-Body Problem (CRTBP). Through exploring the energy (Jacobi integral) conserving property in CRTBP, this problem is rewritten into Port-Hamiltonian framework, where we use the Jacobi integral as the Hamiltonian. Then, we design the feedback controller through energy-shaping and dissipation injection to stabilize the L_{1} Lagrange point of the Sun-Earth CRTBP. The closed-loop Hamiltonian is designed as the candidate of the Lyapunov function to establish nonlinear stability of the designed equilibrium, which enlarges the application region of feedback controller compared with that based on linearized dynamics. Results show that the Port-Hamiltonian approach allows us to successfully stabilize the Lagrange points, where the Linear Quadratic Regulator (LQR) may fail. Adding the white noise into the inputs, the designed feedback controller based on Port-Hamiltonian approach also allows us to stabilize the Lagrange points, which demonstrates the robustness against the background noise of the designed feedback control.

Special thanks to Prof who offer us tremendous support and offer the idea of project. Special thanks to offering all the material, labor, technology, and all the equipment that our team needs to finish the project.

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

Yan, H.T. (2020) Port-Hamiltonian Based Control of the Sun-Earth 3D Circular Restricted Three-Body Problem: Stabilization of the L_{1} Lagrange Point. Modern Mechanical Engineering, 10, 39-49. https://doi.org/10.4236/mme.2020.103005