Advances in Deep Space Probe Navigation

1. Introduction

Deep space exploration, generally referring to the space exploration activities targeting the Moon and more distant extraterrestrial celestial bodies, stands as a critical indicator of a nation’s comprehensive capabilities and technological prowess. Conducting deep space exploration is beneficial for breaking through the space technology necessary for human survival and development and expanding human activity space. Within this ambitious endeavor, effective navigation serves as the cornerstone for ensuring mission success, enabling spacecraft to traverse the vast and unpredictable cosmic frontier. As humanity’s reach into the cosmos continues to expand—from lunar to interplanetary voyages and beyond—the demand for real-time, high-precision, and highly reliable navigation solutions has grown exponentially.

In this Special Issue, we have collected original research and review papers that reflect the latest advancements in the navigation theories and methods for deep space probes.

2. An Overview of Published Articles

This Special Issue contains six papers that focus on the following aspects of recent progress in aerospace field: (1) interstellar voyage propulsion systems; (2) celestial navigation and positioning technology; and (3) optimization algorithm applications. The following is a brief summary of the published articles.

2.1. Interstellar Voyage Propulsion System

The periodic a1nd active comet 29P/Schwassmann-Wachmann 1 is an interesting but unexplored small celestial body that has been regarded as an object of the Centaur group. Ref. [1] analyzes the optimal transfer towards the periodic comet of a solar sail-based spacecraft. A classical (heliocentric) orbit-to-orbit transfer is studied from an optimal viewpoint, by finding the spacecraft trajectories that minimize the flight time for a given value of the solar sail characteristic acceleration, that is, the typical performance parameter of a photonic sail. Research has found that high-performance solar sails can fly for over 10 years, while high-performance sails can fly for less than 10 years. The calculation results of the two-dimensional simplification method are similar to those of the three-dimensional model, but the calculation time is significantly reduced.

An Electric Solar Wind Sail (E-sail) is a propellantless propulsion concept that extracts momentum from the high-speed solar wind stream to generate thrust. Ref. [2] analyzes the performance of an E-sail in obtaining the transition from prograde to retrograde motion. A mechanism for flipping a circular heliocentric orbit through a two-dimensional propulsion trajectory is proposed, and the optimal trajectory and flight time under different propulsion scenarios is studied. The results indicate that the trajectory can be divided into two symmetrical parts, with the spacecraft reaching a stationary state at the aphelion. Direct transfer and single solar wind assisted transfer each have their own characteristics, with the latter being able to shorten flight time, but considering the limitation of perihelion distance.

Ref. [3] studies the optimal control law of a spacecraft equipped with a Solar Wind Ion Focusing Thruster (SWIFT) in three-dimensional (3D) heliocentric orbit transfer. A thrust vector model suitable for 3D scenes is proposed. The optimal guidance law is obtained by solving optimization problems, and it is verified in simplified Earth–Venus and Earth–Mars transfer missions. The results show that the flight time of 3D transfer has increased compared to two-dimensional cases, and the transfer performance is closely related to the design parameter K.

2.2. Celestial Navigation and Positioning Technology

X-ray pulsar-based navigation (XNAV) is a navigation method that estimates the position and velocity of a spacecraft using the X-ray radiation from pulsars. Flight experiments on the Insight-Hard X-ray Modulation Telescope (Insight-HXMT) and Neutron Star Interior Composition Explorer (NICER) have successfully verified the feasibility of using XNAV for a single spacecraft. Ref. [4] derives a pulsar-based navigation method that uses the pulse phase delay between spacecraft for spacecraft in formation. Moreover, a direct estimation method for pulse phase delay, which is independent from the pulsar template, is proposed. The verification of simulation data of the Crab pulsar and real data of the same pulsar obtained from Insight-HXMT and NICER show that the proposed method is feasible, and the proposed direct estimation method has higher computational efficiency, reducing CPU time costs by about 36.99%.

Ref. [5] proposes a dynamic phase comparison algorithm for planar direction finding on a high-speed moving satellite radio receiver, treating the moving antenna as equivalent to single-baseline array antennas. Based on a phase interferometer algorithm, this algorithm adjusts the baseline length according to the frequency measurement module and the satellite’s high-speed motion to avoid phase ambiguity indirectly. By integrating the traditional amplitude comparison algorithm based on orthogonal dipole antennas, a dynamic fusion direction-finding method is proposed. Simulation shows that the proposed algorithm covers a broader range of direction finding, achieves higher accuracy, and can effectively solve the problem of angle ambiguity.

2.3. Optimization Algorithm Applications

The Oryctolagus cuniculus, also known as the European rabbit, is named for its ability to dig complex cave systems. To improve the global optimization ability and convergence speed of the swarm intelligence algorithm, Ref. [6] proposed a new swarm intelligence optimization algorithm, namely the Oryctolagus cuniculus algorithm (OCA). OCA simulates the collective behavior of Oryctolagus cuniculus and applies it to the inversion method of the asteroid spectrum reflectance template combined with empirical mode decomposition (EMD) to screen intrinsic mode functions (IMFs) and optimize template combinations. The experimental results show that compared with artificial rabbit optimization, the proposed algorithm has a faster rate of convergence and better solution, effectively screens the reflectance template, and improves the Doppler difference velocimetry accuracy. In addition, the application of the Oryctolagus cuniculus algorithm to the knapsack problem also results in effective performance.

3. Exploring Further Advances in Deep Space Probe Navigation

This section will introduce several recent noteworthy studies outside of this Special Issue for scholarly consideration. Although they may not fully showcase all the progress of deep space probe navigation, they all have some commendable attributes.

3.1. Pulsar Navigation

The recent flight experiments with the Neutron Star Interior Composition Explorer (NICER) and Insight-Hard X-ray Modulation Telescope (Insight-HXMT) have demonstrated the feasibility of X-ray pulsar-based navigation (XNAV) in space. Ref. [7] proposes a fast algorithm for on-orbit estimation of the pulse phase of the Crab pulsar called X-ray pulsar navigaTion usIng on-orbiT pulsAr timiNg (XTITAN), which is computationally efficient and has the potential to be employed for onboard computation. Ref. [8] proposes an adaptive grid search (AGS) method for estimating the pulse phase and the Doppler frequency.

Epoch folding is a classical period estimation method in the time domain. In order to reduce the computational complexity, Ref. [9] improves the fast folding algorithm through segment correlation and amplitude accumulation, which is based on the post-order traversal of a binary tree. In order to overcome the time-varying frequency problem caused by the acceleration of spacecraft in the process of photon time-of-arrival data, Ref. [10] proposes an improved method called nonuniform epoch folding (NUEF). The pulsar’s directional error and the onboard clock error are two types of systematic errors that seriously reduce navigation accuracy. To solve this problem, Ref. [11] proposes a star angle/double-differenced pulse time of arrival (SA/DDTOA) integrated navigation method.

3.2. Celestial Spectral Velocimetry

Intelligent celestial spectral velocimetry emerges as a key research focus for enhancing measurement accuracy and real-time performance. Ref. [12] introduces the SRGAN-LSTM, which combined semi-supervised regression generative adversarial networks (SRGANs) with long short-term memory (LSTM) networks. Ref. [13] integrates the Gaussian genetic algorithm (GGA) and empirical mode decomposition (EMD), enhancing celestial spectral reflectivity template construction by optimizing intrinsic mode function (IMF) combinations, thus reducing the effects of asteroid absorption. In a follow-up investigation, the authors proposed the Oryctolagus cuniculus algorithm (OCA) [14] and Anelosimus eximius colony algorithm (AECA) [6], which simulated the behaviors of rabbit and spider populations, respectively. These algorithms provided improved convergence and optimization performance compared to traditional genetic algorithms (GAs) in celestial spectral reflectivity inversion. To correct the velocity error of the SINS, a DVS-aided SINS integrated navigation method is explored, where the Doppler velocities of the Sun and two stars are used as measurements [15].

3.3. Other Novel Deep Space Probe Navigation Methods

StarNAV is a novel celestial navigation method that utilizes relativistic effects, which mostly provides the velocity information of the spacecraft. Ref. [16] proposes a star angle modified with relativistic effects (SAMRE)/StarNAV integrated navigation method. The measurement model of SAMRE is established by considering relativistic effects in the measurement model of star angle. Ref. [17] proposes a time dilation-modified TDTOA/StarNAV (TDM-TDTOA/StarNAV) integrated navigation method. The effect of time dilation on navigation accuracy is analyzed and compensated in the measurement model of time differential time of arrival (TDTOA). Ref. [18] explores a new navigation method using multi-path solar panel-reflected solar oscillations. Solar disk velocity difference is another novel celestial navigation measurement, which can be obtained by four spectrometers installed on the four corners of a quadrangular pyramid [19]. Ref. [20] proposes an augmented-state Sun direction/solar disk velocity difference integrated navigation, which greatly suppresses the effect of spectrometer installation error on navigation accuracy.

3.4. Advanced Filtering Algorithm

Adaptive algorithms can identify the statistical characteristics of noise online, dynamically adjust filtering parameters, and adapt to changes in noise. Ref. [21] proposed a variational Bayesian implicit unscented Kalman filter (VBIUKF) method and applied it to celestial navigation using time delay measurement. Ref. [22] proposes an adaptive unscented Kalman filter based on sequential state difference (AUKF-SSD) to address the issue of divergence in spacecraft navigation systems and enhance navigation accuracy. Ref. [23] proposes a variable-dimensional adaptive IMM strong tracking filtering algorithm (VAIMM-STEKF) to estimate the spacecraft’s position, velocity, and maneuvering acceleration state. Ref. [24] presents a novel parallel Q-learning extended Kalman filter (PQEKF) to implement the measurement bias calibration. Ref. [25] developed an improved Q-learning-based extended Kalman filter (IQEKF) to obtain the accurate motion state estimate of both the spacecraft and the space targets based on the LOS direction measurements.

3.5. Spacecraft Control

Self-learning control emerges as a new approach by incorporating a self-learning process of historical control information directly into the controller [26]. Therefore, the control algorithm is simplified, consisting merely of a learning term and an updating term. The algorithm was investigated in depth in Refs. [27,28], which also presented methods for varying learning intensity and applications on spacecraft formation control.

Performance-adjustable prescribed performance control has more general performance tuning capabilities, and its application to nonlinear multi-agent systems was originally proposed in Ref. [29]. In the following studies, the authors further extended this method to spacecraft attitude control systems [30,31] and spacecraft attitude–orbit control systems [32]. This performance-adjustable approach is also applied to event triggering mechanisms in Ref. [33], which implements the trade-offs of sampling rate, convergence accuracy, and convergence rate for spacecraft attitude–orbit control systems [32].

Ref. [34] proposes a minimum-time capture control method for the test mass release phase of drag-free spacecraft, which quickly captures the test mass to the cage center of the inertial sensor. Ref. [35] proposes a novel distributed control method for surrounding a noncooperative target that has maneuverability through spacecraft formation.

3.6. Orbital Dynamics and Orbital Design

Electrodynamic tether (EDT) is a key prospective technique for space debris removal without the use of propellants. Ref. [36] proposes a novel electrodynamic multi-tether (EMT) system to overcome the disadvantages of the classical EDT system. Aiming at the interception problem of noncooperative evader spacecraft adopting a random maneuver strategy in the one-to-one orbital pursuit–evasion problem, an interception strategy with a decision-making training mechanism for the pursuer based on deep reinforcement learning is proposed [37]. Ref. [38] investigates the distribution of the semi-major axis and eccentricity of IOD solutions in a pool and find that choosing the solution with the maximum kernel density in the distribution is a much better way to determine the final solution from the pool.

4. Conclusions

This Special Issue contributes to deep space exploration technology by analyzing the performance of various propulsion systems, developing innovative celestial navigation algorithms, and investigating optimal estimation algorithms for autonomous spacecraft positioning. These advancements provide theoretical foundations and technical references for achieving real-time, high-precision, and highly reliable celestial navigation in deep space missions. Overall, these advancements aim to advance the technological frontier of deep space exploration, enabling more ambitious extraterrestrial missions.