The collision probability computation of space objects plays an important role in space situational awareness, particularly for conjunction assessment and collision avoidance. Early works mainly relied on Monte Carlo simulations to predict collision probabilities. Although such simulations are accurate when a large number of samples are used, these methods are perceived as computationally intensive, which limits their application in practice. To overcome this limitation, many approximation methods have been developed over the past three decades. This paper presents a comprehensive review of existing space-object collision probability computation methods. The advantages and limitations of different methods are analyzed and a systematic comparison is presented. Advice regarding how to select a suitable method for different short-term encounter scenarios is then provided. Additionally, potential future research avenues are discussed.

A novel simplified parametric model for long-duration impulsive orbit rendezvous is proposed. Based on an existing fast estimation method, the optimal impulses and trajectory can be expressed by only ten parameters whose initial values can be easily determined. Then, these parameters are used to predict orbital deviations with a target orbit. A simple correction process is designed to sequentially update the parameters based on the J₂ perturbed analytical dynamic equations of circular orbits. Finally, an iteration loop is formed to obtain the precise parameters and optimal trajectory. The simulation results confirm that the simplified parametric optimization method can be applied to elliptical orbits of small eccentricity and adapts well to both analytical and high-precision dynamics. The deviations could always converge within five iterations and the calculation was more efficient than the existing methods.

This paper presents the method created by the National University of Defense Technology (NUDT) team in the 10th China Trajectory Optimization Competition, which entails a 3-year observation mission of 180 regions on Jupiter. The proposed method can be divided into three steps. First, a preliminary analysis and evaluation via an analytical method is undertaken to decide whether the third subtask of the mission, i.e., exploring the Galilean moons, should be ignored. Second, a near-optimal orbit for magnetic field observation is designed by solving an analytical equation. Third, a set of observation windows and their sequence are optimized using a customized genetic algorithm. The final index obtained is 354.505, ranking second out of all teams partaking in the competition.

The usage of state transition tensors (STTs) was proved as an effective method for orbital uncertainty propagation. However, orbital maneuvers and their uncertainties are not considered in current STT-based methods. Uncertainty propagation of spacecraft trajectory with maneuvers plays an important role in spaceflight missions, e.g., the rendezvous phasing mission. Under the effects of impulsive maneuvers, the nominal trajectory of a spacecraft will be divided into several segments. If the uncertainty is piecewise propagated using the STTs one after another, large approximation errors will be introduced. To overcome this challenge, a set of modified STTs is derived, which connects the segmented trajectories together and allows for directly propagating uncertainty from the initial time to the final time. These modified STTs are then applied to analytically propagate the statistical moments of navigation and impulsive maneuver uncertainties. The probability density function is obtained by combining STTs with the Gaussian mixture model. The proposed uncertainty propagator is shown to be efficient and affords good agreement with Monte Carlo simulations. It also has no dimensionality problem for high-dimensional uncertainty propagation.

Rendezvous on the geostationary orbit (GEO) is much more complex than that on the low earth orbit and has a higher critical requirement for safety performance. This paper presents a safe scenario design method for GEO rendezvous proximity missions where the safety constraint of a collocated satellite is considered. A recently proposed quantitative index considering trajectory uncertainty is introduced to analyze the safety performance of the scenario parameters including the V-bar keeping positions and the fly-by trajectory radius. Furthermore, an exhaustive analysis is performed to find the dangerous regions of the V-bar keeping positions and the appropriate semi-major axis of the fly-by ellipse, considering the safety requirements of both the target and the collocated satellite. A geometry method is then developed for designing a feasible and suboptimal safe rendezvous scenario. The method is tested by designing four rendezvous scenarios with $\pm$V-bar approach directions respectively in the situations with and without one collocated satellite. Safety performance and velocity increments of the scenarios are compared and a conclusion is reached that the collocated satellite has a significant influence on the scenario design.