Space Objects Classification via Light-Curve Measurements Using Deep Convolutional Neural Networks

Abstract

This work presents a data-driven method for the classification of light curve measurements of Space Objects (SOs) based on a deep learning approach. Here, we design, train, and validate a Convolutional Neural Network (CNN) capable of learning to classify SOs from collected light-curve measurements. The proposed methodology relies on a physics-based model capable of accurately representing SO reflected light as a function of time, size, shape, and state of motion. The model generates thousands of light-curves per selected class of SO, which are employed to train a deep CNN to learn the functional relationship. between light-curves and SO classes. Additionally, a deep CNN is trained using real SO light-curves to evaluate the performance on real data, but limited training set. The CNNs are compared with more conventional machine learning techniques (bagged trees, support vector machines) and are shown to outperform such methods, especially when trained on real data.

Appendix A: Angular Velocity Determination Method

This appendix provides a summary of the angular velocity determination method discussed in Ref. [28]. This work uses three possible control profiles, Sun pointing, Nadir pointing, and spin-stabilized. For each of these profiles the SO is control led to point to the Sun direction, the Nadir, or in a spin-stabilized configuration. For the spin-stabilized configuration, the desired angular velocity is chosen to be perpendicular to the orbital plane. When pointing in Sun direction or the Nadir direction we must compute the desired angular velocity to track these directions. If we assumed that $\tilde {\textbf {b}}_{j_{k}}$ and $\tilde {\textbf {b}}_{j_{k+1}}$ represent the j^th pointing directions at time step. k and k + 1, then the goal is to estimate the angular velocity, ω_k, from these directions. Both the Sun pointing and Nadir directions pointing are determined from the SO trajectories. The ω_k is then estimated using a finite difference method which is outline below [28]. Ċonsider the following unit-vector measurement model at time t_k:

$$ \tilde{\textbf{b}}_{j_{k}} = A_{k} \textbf{r}_{j} + \textbf{v}_{j_{k}} $$

(36)

where $\tilde {\textbf {b}}_{j_{k}}$ is the j^th pointing vector in the inertia frame and is r_j the same pointing vector in the body frame. The attitude matrix mapping from inertial to the body frame is denoted by A_k. The goal is to determine the rate of change of this attitude matrix or the angular velocity. Taking the difference between successive measurements of Eq. 36 gives

$$ \tilde{\textbf{b}}_{j_{k+1}}-\tilde{\textbf{b}}_{j_{k}} = [A_{k+1} - A_{k}] \textbf{r}_{j} + \textbf{v}_{j_{k+1}}-\textbf{v}_{j_{k}} $$

(37)

We assume that the body angular velocity ω is constant between t_k and t_k+ 1, and ignore terms higher than first order in ωΔt. With these assumptions the following first-order approximation can be used [26]:

$$ A_{k+1} \approx \left[I_{3\times 3} -{\Delta} t [\boldsymbol{\omega}_{k} \times] \right] A_{k} $$

(38)

In this case ω_k is the average velocity, but this becomes less of a problem as the sampling interval decreases. Substituting Eq. 38 into Eq. 37 gives

$$ \tilde{\textbf{b}}_{j_{k+1}}-\tilde{\textbf{b}}_{j_{k}} = -{\Delta} t [\boldsymbol{\omega}_{k} \times] A_{k} \textbf{r}_{j} + \textbf{v}_{j_{k+1}}-\textbf{v}_{j_{k}} $$

(39)

Our goal is to determine an angular velocity just using $\tilde {\textbf {b}}_{j_{k}}$ and $\tilde {\textbf {b}}_{j_{k+1}}$. This is accomplished by solving Eq. 36 in terms of A_kr_i and substituting the resultant into Eq. 39, which yields

$$ \frac{1}{\Delta t} [ \tilde{\textbf{b}}_{j_{k+1}}-\tilde{\textbf{b}}_{j_{k}} ] = [\tilde{\textbf{b}}_{j_{k}} \times] \boldsymbol{\omega}_{k} + \textbf{w}_{j_{k}} $$

(40)

where $\textbf {w}_{j_{k}}$ is the new effective measurement noise vector but for this work we assume this is zero. Note that Δt will have finite values, since discrete-time measurements are assumed. Equation 40 can now be cast into a linear least-squares form for all measurement vectors, which leads to

$$ \begin{array}{@{}rcl@{}} \hat{\boldsymbol{\omega}}_{k} &= \frac{1}{\Delta t} \left[{\sum}_{j=1}^{n_{k}} [\tilde{\textbf{b}}_{j_{k}} \times]^{T} R_{j_{k}}^{-1} [\tilde{\textbf{b}}_{j_{k}}\times]\right]^{-1}\\ & {\sum}_{j=1}^{n_{k}} [\tilde{\textbf{b}}_{j_{k}} \times]^{T} R_{j_{k}}^{-1} (\tilde{\textbf{b}}_{j_{k+1}} -\tilde{\textbf{b}}_{j_{k}}) \end{array} $$

(41)

where $\hat {\boldsymbol {\omega }}_{k}$ is the estimate of ω_k. For this work we assume $R_{j_{k}}^{-1}=I_{3 \times 3}$.