Space debris in Earth’s orbit is rapidly increasing, posing significant collision risks to spacecraft and threatening their normal operations. Monitoring and predicting their orbits are essential. Space-based optical monitoring, unconstrained by geographical constraints, can enhance observation coverage and frequency of space debris. However, it provides only angular measurements, making orbit determination challenging, especially for initial orbit determination under short-arc observations. To improve convergence and computational efficiency, we construct an extended objective function model and propose an initial orbit determination algorithm using the adaptive-moment-estimation (Adam) optimization in the range-range difference solution space.

We introduce an extended objective function model that considers proximity between predicted and observed values and evaluates eccentricity when the orbit exists and the observation constraints are met (within the admissible region). Outside this region, the objective function ensures its value does not exceed boundaries and has no local minimum. This design aims to achieve two objectives: 1) Both the objective function and its derivative can be calculated at any point in the range-range difference solution space, thereby facilitating optimization methods based on the first derivative. 2) The solution will only converge to permissible region extrema. In addition to pre-processing angular measurements, the proposed method comprises four steps: first, set weight factors, hyper-parameters and threshold values for the cost function; second, calculate the initial value in the range-range difference space; third, perform iterative updates following the Adam optimization algorithm while evaluating the optimization objective function using the extended objective function introduced herein; finally, based on predefined convergence criteria, decide whether to continue or terminate the iterations and subsequently output the result.

Simulation experiments confirm the method’s effectiveness, adaptability to various orbit types, initial value sensitivity, computational efficiency, convergence, and accuracy. The results (Fig. 5) indicate good performance for geostationary earth orbit (GEO), medium earth orbit (MEO), and low earth orbit (LEO) debris optical measurements. Sensitivity to initial values is low (Fig. 6), but appropriate initial values reduce iterations (Table 4). The Adam optimization algorithm outperforms stochastic-gradient-descent (SGD), Momentum, and adaptive-gradient (AdaGrad) algorithm (Fig. 7). The elapsed time (Table 5) associated with the proposed method across various arc segments spans from tens of milliseconds to a few seconds. This performance generally surpasses that of the admissible region particle swarm optimization algorithm, which also guarantees convergence. For a specific Leo optical surveillance platform observing a GEO target under consistent observation intervals (3 s), accuracy improves with longer observation arcs. The root mean square errors (Table 7) for the position at intermediate observation points measure 49.82, 34.73, and 16.37 km, respectively. Conversely, with a fixed observation arc length of 3 min, the root mean square errors (Table 8) for the initial orbit determination results at the mid-observation for 3, 6, and 9 s amount to 34.73, 51.65, and 66.24 km, correspondingly.

To enhance the convergence and computational efficiency of initial orbit determination in space-based optical surveillance, we have developed an extended loss function model and introduced an initial orbit determination algorithm that utilizes Adam optimization to find the optimal solution within the range-range difference space. The method’s adaptability to various orbit types and initial values, along with its algorithm efficiency, convergence, and accuracy, have been rigorously assessed. The results show that our approach is well-suited for the initial orbit determination of space debris in GEO, MEO, and LEO. While convergence to an acceptable solution is achievable even under stringent initial conditions, at the expense of increased iteration, we advocate for the proposed initial settings to ensure high efficiency. The initial orbit determination error of the proposed method is statistically analyzed. The root mean square error for the position at the mid-observation epoch is on the order of 10 km, and the root mean square error for the semi-major axis of the orbit is on the order of 100 km, in the context of space-based optical surveillance with an angular measurement error of 2 arc seconds.