Indirect Trajectory Optimization for Cislunar Trajectories

Objective

Find the sequence of thrusts $\pmb{u}(t)\forall t \in [t_I,t_F]$ to take a spacecraft from an orbit about Earth at time $t_I$ to an orbit about the $\mathcal{L}_2$ Lagrange point at time $t_F$, while minimizing the total amount of expended fuel $m(t_F)$.

Dynamics

The dynamics of a spacecraft in the circular restricted three-body problem (CRTBP) are characterised in scaler form by the ordinary differential equations

$$\begin{align} \dot{x} &= v_x \\ \dot{y} &= v_y \\ \dot{z} &= v_z \\ \dot{v}_x &= x - \frac{(1-\mu)(x+\mu)}{r_1^3} - \frac{\mu (x+\mu-1)}{r_2^3} + 2 v_y + \frac{u T \hat{u}_x}{m} \\ \dot{v}_y &= y - \frac{(1-\mu)y}{r_1^3} - \frac{\mu y}{r_2^3} - 2 v_x + \frac{u T \hat{u}_y}{m} \\ \dot{v}_z &= - \frac{(1-\mu)z}{r_1^3} - \frac{\mu z}{r_2^3} + \frac{u T \hat{u}_z}{m} \\ \dot{m} &= -\frac{u T}{I_{sp} g_0} \\ r_1 &= \sqrt{(x+\mu)^2+y^2+z^2}\\ r_2 &= \sqrt{(x+\mu-1)^2+y^2+z^2} \end{align}$$

where $[x,y,z,v_x,v_y,v_z,m]^\intercal$ describes the state of the spacecraft and $\pmb{u} = u [\hat{u}_x,\hat{u}_y,\hat{u}_z]^\intercal$ is the control to be chosen along the spacecraft’s trajectory, where the throttle $u \in [0,1]$ and thrust direction $\sqrt{\hat{u}_x^2 + \hat{u}_y^2 + \hat{u}_z^2} = 1$. The parameters inherent to the problem are:

1. Maximum thrust: $T$
2. Specific impulse: $I_{sp}$
3. Earth’s sea-level gravity: $g_0$

Cost Function

The desire for most spacecraft trajectories is to reduce fuel consumption. Such as it is, an optimal trajectory should minimize the homotopic path cost

$$\mathcal{J} = \frac{T}{I_{sp} g_0} \int_{t_0}^{t_f} (u - \alpha u (1-\alpha)) dt$$

from the initial time $t_i$ to the specified final time $t_f$. The parameter $\alpha$ is implemented in order to avoid numerical convergence difficulties associated with the discontinuous nature of mass optimal bang-bang control. In practice, the trajectory is initially optimized with the homotopy parameter $\alpha = 1$ for convergence ease, after which the trajectory is optimized for iteratively smaller values until $\alpha=0$, corresponding to a mass path cost.

Optimal Control

The dynamical system is a Hamiltonian System, and its Hamiltonian is

$$\begin{multline} \mathcal{H} = \lambda_x v_x + \lambda_y v_y + \lambda_z v_z \\ + \lambda_{v_x} \left( x - \frac{(1-\mu)(x+\mu)}{r_1^3} - \frac{\mu (x+\mu-1)}{r_2^3} + 2 v_y + \frac{u T \hat{u}_x}{m} \right) \\ + \lambda_{v_y} \left(y - \frac{(1-\mu)y}{r_1^3} - \frac{\mu y}{r_2^3} - 2 v_x + \frac{u T \hat{u}_y}{m} \right) \\ + \lambda_{v_z} \left(\frac{(1-\mu)z}{r_1^3} - \frac{\mu z}{r_2^3} + \frac{u T \hat{u}_z}{m} \right) - \lambda_m \frac{u T}{I_{sp} g_0} \\ + \frac{T}{I_{sp} g_0} (u - \alpha u (1-\alpha)) \end{multline}$$

Nonlinear Parameter Optimization

Determine the decision vector

$$[\lambda_x, \lambda_y, \lambda_z, \lambda_{v_x}, \lambda_{v_y}, \lambda_{v_z}, \lambda_m, t_f] \Bigr\rvert_{t_0}$$