CR3BP with Low-Thrust (CR3BP-LT)

Definition

CR3BP-LT is an extension of the standard Circular Restricted Three-Body Problem (CR3BP) that adds continuous low-thrust acceleration terms to the spacecraft's equations of motion, used for studying orbit transfer problems with electric propulsion, ion propulsion, and other low-thrust propulsion systems[1]. This model was formally proposed and systematically applied in the A2PPO research by Ul Haq et al. in 2026.

Equations of Motion

In the normalized rotating coordinate system, the equations of motion for CR3BP-LT are:

\begin{cases} \ddot{x} = -\left[(1-\mu)\frac{x+\mu}{r_1^3} + \mu\frac{x-(1-\mu)}{r_2^3}\right] + 2\dot{y} + x + \frac{\tilde{T}_{\max}}{\tilde{m}} u_x \\[0.8em] \ddot{y} = -\left[(1-\mu)\frac{y}{r_1^3} + \mu\frac{y}{r_2^3}\right] - 2\dot{x} + y + \frac{\tilde{T}_{\max}}{\tilde{m}} u_y \\[0.8em] \ddot{z} = -\left[(1-\mu)\frac{z}{r_1^3} + \mu\frac{z}{r_2^3}\right] + \frac{\tilde{T}_{\max}}{\tilde{m}} u_z \end{cases}

where $r_1 = \sqrt{(x+\mu)^2 + y^2 + z^2}$ and $r_2 = \sqrt{(x-1+\mu)^2 + y^2 + z^2}$ are the distances from the spacecraft to Earth and Moon respectively, $-2\dot{y}$ and $+2\dot{x}$ are Coriolis force terms, and $u = (u_x, u_y, u_z)$ is the dimensionless thrust control vector satisfying $\|u\| \leq 1$ .

Mass evolution follows the Tsiolkovsky rocket equation:

\dot{\tilde{m}} = -\frac{\tilde{T}_{\max}}{\tilde{c}} \|u\|

where $\tilde{c} = I_{sp} g_0 T^*/L^*$ is the dimensionless exhaust velocity.

Key Parameters

Characteristic normalization parameters for CR3BP-LT:

Parameter	Symbol	Earth-Moon Value
Mass ratio	$\mu$	0.01215
Characteristic length	$L^*$	$3.844 \times 10^8$ m
Characteristic time	$T^*$	375,132 s
Characteristic acceleration	$g_0$	9.80665 m/s²

CR3BP-LT vs. Standard CR3BP

Property	Standard CR3BP	CR3BP-LT
Energy conservation	Jacobi constant conserved	Continuous low-thrust breaks conservation
Dynamics	Integrable (conservative)	Non-conservative, highly nonlinear
Orbit characteristics	Periodic/quasi-periodic orbits	Transfer trajectories freely designable
Computational cost	Lower	Higher (requires mass equation integration)

The CR3BP-LT model degenerates to the standard CR3BP when thrust is zero ( $\tilde{T}_{\max} = 0$ ), at which point the Jacobi constant is conserved again.

Application in Trajectory Optimization

The CR3BP-LT model is the core environment for deep reinforcement learning methods such as A2PPO applied to low-thrust trajectory optimization. This model:

Retains the essential complexity of three-body dynamics — chaotic characteristics, manifold structures, energy variations, etc.
Introduces continuous thrust control — enabling continuous acceleration/deceleration through the dimensionless thrust vector $u$
Maintains computational affordability — compared to high-fidelity Ephemeris models, CR3BP-LT can support the millions of integrations required for large-scale RL training

References

[1] Ul Haq I U, Dai H, Du C. Autonomous low-thrust trajectory optimization in cislunar space via attention-augmented reinforcement learning[J]. Aerospace Science and Technology, 2026.
[2] Du C, Song L, Zhang J, et al. A novel calculation method for low-thrust transfer trajectories in the Earth-Moon restricted three-body problem[J]. Aerospace Science and Technology, 2024.