Models and performance of deep space navigation for the gravity assist phase

Abstract

Introduction:

Deep space exploration, as a critical means for humanity to understand and investigate the evolutionary history of the universe, has entered a phase of rapid development. A single deep space navigation mode struggles to meet the high-precision and high-reliability requirements for long-duration mission operations in highly dynamic and extreme environments. Hence, multi-mode resilient navigation will be the inevitable future approach for deep space exploration.

Methods:

The study proposes a set of deep space resilient navigation models specifically for gravity assist phase, including X-ray pulsar navigation, Very Long Baseline Interferometry (VLBI) navigation, celestial body navigation, and combined navigation using these methods. Various navigation filtering algorithms applicable to deep space models are introduced, and the performance of the navigation models under different conditions is analyzed through simulation. For six gravity assist phases with the flight sequence proceeding as Earth, Venus, Venus, Earth, Jupiter, Neptune and heliosphere bottom, three navigation filtering methods, which are Extended Kalman Filter (EKF), Game theory-based H∞ filter, and Robust Student’s t Kalman Filter (RSTKF), exhibit consistent performance for X-ray pulsar navigation. Besides, ground-based VLBI observations can be utilized throughout the gravity assist phase and exhibit superior navigation performance compared to X-ray pulsar navigation.

Results:

The performance of the celestial body navigation is significantly worse than that of X-ray pulsar and VLBI navigation, with the accuracy degrading as mission distance increases. The positioning errors of the resilient navigation approach are within 10 km across all tested scenarios. Moreover, the feedback-free resilient navigation demonstrates slightly better performance than the fusion feedback mode, though the latter maintains high-frequency output consistency with astronomical navigation subsystems.

Discussion:

These results provide the significant theoretical support and practical reference for advancing deep space navigation.

1 Introduction

Deep space exploration serves as a crucial method for humanity to understand and explore the evolutionary history of the universe, while simultaneously serving as a critical strategic Frontier through which global powers bolster their technological and scientific capabilities (Zhang et al., 2024). Since the launch of the first lunar probe for Pioneer 0 in the 1950s, deep space exploration has persisted to the present day, with the scope of exploration progressively expanding from Earth-Moon space to the boundary of the solar system. For instance, China’s independently developed Chang'E series of lunar probes has successfully achieved soft landings and carried out lunar surface exploration missions, with plans to realize the first Chinese manned lunar landing before 2030 (Wang et al., 2022). Nowadays, China continuously advances its strategic planning for deep space exploration, encompassing Mars exploration, near-Earth asteroid exploration, and Jupiter exploration, with the aim of enhancing its international competitiveness in the field of deep space exploration. It can be observed that deep space exploration has entered the phase of rapid development (Su et al., 2026a).

Currently, navigation technologies adopted in international deep space exploration missions primarily fall into two categories: ground-based tracking, telemetry, and command (TT&C) navigation (Huang et al., 2014; Liu Q. et al., 2010) and celestial autonomous navigation (Christian and Lightsey, 2012; Wang et al., 2016; Yang et al., 2016). Ground-based TT&C navigation relies on technologies such as the Global Navigation Satellite System (GNSS) and Very Long Baseline Interferometry (VLBI), whereas celestial autonomous navigation, serving as the core method for probe autonomous navigation, enables autonomous determination of probe state through continuous observation of celestial body and X-ray pulsar (Liu and Jin, 2018). Nowadays, international deep space missions predominantly employ multi-source integrated navigation strategies, which can be categorized into two types: one dominated by ground-based TT&C supplemented by optical autonomous navigation, and the other transitioning from near-Earth space operations based on ground-based TT&C toward progressively adopting celestial autonomous navigation (Su et al., 2026b). The notable advantage of these strategies lies in reducing reliance on ground support and enhancing the autonomy of deep space navigation capability (Turan et al., 2022).

During the gravity assist phase of deep space navigation, the precise adjustment of spacecraft velocity and orbit is achieved by leveraging the slingshot effect of planetary gravitational fields. This approach not only saves fuel but also significantly enhances orbital maneuver efficiency and extends mission range for deep space exploration (Hu et al., 2026). Current study on gravity assist phase of deep space navigation is primarily oriented toward meeting deep space exploration requirements, with a focus on technologies such as the VLBI network and celestial body navigation. VLBI navigation technology is highly mature and capable of supporting various deep space missions, though it has limitations in the number of concurrent users it can support and requires relatively long orbit determination times. During the Chang'E 5 mission, VLBI navigation exhibits transfer orbit accuracy better than 1 km and lunar orbit determination accuracy better than 200 m (Kong and Zhang, 2022). Celestial autonomous navigation methods primarily include celestial angle measurement navigation, celestial range measurement navigation, and celestial velocity measurement navigation (Liu et al., 2017).

Starting from Deep Space 1, NASA began attempting on-orbit validation of fully autonomous celestial angle measurement navigation technology (Ma et al., 2014; Mastrodemos et al., 2005). The ESA is also actively investigating celestial autonomous navigation technology by the ground validation (Mancuso, 2004). Moreover, as the field of deep space exploration continues to expand, reliance on ground-based tracking and navigation systems will no longer be sufficient to achieve high-precision navigation in regions far from Earth (Su, 2023). Traditional astrometric techniques also have inherent limitations. X-ray pulsar navigation, as an emerging autonomous celestial navigation technology, is applicable not only to near-Earth objects but also to deep space exploration missions (Emadzadeh and Speyer, 2010; Sheikh et al., 2006; Shemar et al., 2016). Reichley et al. (1970) suggests that periodic measurement of phase signals can be employed to determine the position of probe, which firstly indicating that X-ray pulsars is a means to determine both the position and time of a probe in orbit.

Deep space probe typically operates under long-endurance, high-dynamic and extreme environment conditions (Liu et al., 2018). Hence, the single deep space navigation mode is no longer adequate to meet the stringent requirements for high precision and reliability. Kang et al. (2017) developed an integrated navigation system by combining Doppler velocity measurements with X-ray pulsar navigation. This system exhibits strong resilience to spectral distortion effects and achieves improved accuracy in both position and velocity estimation. Xiong et al. (2013) introduces a federated filtering framework incorporating a dynamic vector-form information sharing algorithm. By leveraging eigenvalues of the error covariance matrix and singular values of the observability matrix, the approach overcomes the limitations of scalar-coefficient-based methods through variable-specific weight assignment. This adaptive strategy enhances navigation precision in high-dynamic environments by enabling global information optimization. The development trend of deep space navigation inevitably involves the integration of multiple navigation modes to form a resilient navigation framework.

To address the unique characteristics and orbit dynamics of deep space exploration, a resilient deep space navigation model is proposed specifically for the gravity assist phase. The model integrates X-ray pulsar navigation, VLBI navigation, celestial body navigation, and their resilient integration models. Various navigation filtering algorithms compatible with the models are validated to analyze the performance of the deep space navigation system under different environments and conditions through the simulation.

2 Resilient deep space navigation frame

The infrastructure for resilient deep space navigation primarily relies on deep space navigation constellations. Position information is essential for various deep space missions, including safe navigation of deep space probe, deep space environment monitoring, and deep space scientific research. Deep space navigation assurance mainly depends on celestial navigation, ground-based VLBI and X-ray pulsar navigation. To establish a controllable deep space navigation infrastructure, rational deployment of deep space navigation constellations is required, along with the construction of corresponding functional models and filter models (Yang et al., 2023). Figure 1 depicts the deep space navigation infrastructure.

FIGURE 1

The functional model serves as the foundation for integrating deep space navigation information. A rational resilient functional model should establish appropriate observation models tailored to specific environments to eliminate various systematic errors. We introduce resilient functional models including the X-Ray pulsar navigation, VLBI navigation, and celestial body navigation model. Theoretically, the methods can exhibit different levels of deep space navigation performance. The filter model acts as a metric for quantifying the uncertainty of various deep space navigation sensing information, as the uncertainty of navigation data from the same source varies across different environments. We present several filtering methods applicable to deep space navigation stochastic models, including the Extended Kalman Filter (EKF), the game theory-based filter, the Robust Student’s t Kalman Filter (RSTKF), and the Federated filter.

3 Deep space navigation model

The section begins with the dynamical model with the Sun as the central celestial body, and then introduces the deep space navigation model, including the X-Ray pulsar navigation, VLBI navigation, celestial body navigation and navigation of their combination.

3.1 Dynamical model with the Sun as the central celestial body

During the gravity assist phase of the deep space probe exploration mission, the orbital dynamics model can be regarded as a perturbed two-body problem with the Sun as the central gravitational body. In addition to the central gravitational force exerted by the Sun, the probe is affected by perturbative forces from the nearby planets, Earth, other celestial bodies, and solar radiation pressure. The dynamical model of the probe orbit can be expressed as Equation 1 (Jordan, 1987):where and are the probe position and velocity vectors in the heliocentric ecliptic inertial coordinate system. is the gravitational constant of the Sun. is the position vector from the Sun to the probe. is the gravitational constant of the ith perturbing planet. is the position vector from the probe to the ith perturbing planet. is the position vector from the Sun to the ith perturbing planet in the inertial system, i.e., . N is the number of perturbing planets. is the vector of other unmodeled perturbative accelerations.

To explore a more accurate dynamic model for probes during the gravity assist phase, the gravitational effect of Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune are considered, in addition to the Sun as the central celestial body. According to the dynamic model, the high-precision orbital dynamics model incorporating the eight planets can be expressed as Equation 2:where , , , , , , , and are the gravitational constants for the Sun, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune, respectively. is the position vector from the Sun to the probe. , , , , , , and are the position vector from Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune to the probe. , , , , , , and is the position vector from Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune to the Sun. Converting the vectors to the component forms and selecting the heliocentric ecliptic inertial coordinate system, the dynamic model of the probe can be written as Equation 3 (Yan et al., 2018):where (x, y, z), (x_mer, y_mer, z_mer), (x_ven, y_ven, z_ven), (x_e, y_e, z_e), (x_m, y_m, z_m), (x_j, y_j, z_j), (x_sat, y_sat, z_sat), (x_ura, y_ura, z_ura) and (x_nep, y_nep, z_nep) denote the coordinates of the probe, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune, respectively. The corresponding coordinates are obtained from the planetary ephemeris DE441.

3.2 X-ray pulsar navigation model

Figure 2 illustrated the principle of X-ray pulsar navigation. In the Solar System barycenter (SSB) inertial frame, the product of the difference between the time when a pulsar reaches the SSB and the time when the pulsar arrival is measured on the probe, multiplied by the speed of light c, equals the projection of the probe's position vector relative to the SSB onto the unit vector along the pulsar’s line of sight, which can be expressed as Equation 4 (Liu J. et al., 2010):where is directly measured by the probe’s onboard receiving apparatus, is accurately predicted based on the pulsar phase model. By fusing measurement data from a minimum of four pulsars, the three-dimensional spatial coordinates of the deep space probe are fully determinable.

FIGURE 2

The time reference of the above equation must be in a unified coordinate system. The time provided by the deep space probe clock meets the requirement only when the clock is stationary relative to the SSB and at the same gravitational potential. However, since the deep space probe is moving, for the purpose of probe navigation, the time measured by the moving clock on the probe must undergo coordinate transformation to be unified and converted into the SSB coordinate system. The coordinate transformation method can refer to Tao et al. (2024).

Then, by assuming the motion state vector of the probe in the SSB inertial frame , the X-ray pulsar navigation measurement equation can be expressed as Equation 5:where denotes the measurement noise.

3.3 VLBI navigation model

Radio signals in plane wavefronts from extragalactic radio sources are received at two separated tracking stations. Following independent signal conditioning processes performed under distinct local clock references at each station, the signals undergo analog-to-digital conversion synchronized and baseband transformation. The interferometric time delay between station pairs designated as the VLBI group delay is composed of the relativistic geometric delay, clock synchronization residuals, atmospheric delays and instrumental delays. Figure 3 depicts the principle of VLBI navigation. The angular separation between the arrival directions of the spacecraft’s signal wavefronts can be formally expressed as Equation 6 (Tornatore et al., 2014):where denotes the angle between the baseline vector and the source vector, c denotes the speed of light, B denotes the baseline length between two ground-based tracking stations. Hence, the accuracy of VLBI navigation is determined by the baseline length and the time delay measurement accuracy. According to the principle of single point positioning, three independent baselines can be used to perform the probe positioning.

FIGURE 3

3.4 Celestial body navigation

The probe can utilize navigation information from the Sun, Earth and celestial bodies. For the navigation mode based on solar information, two types of observations are employed including the sun line-of-sight direction vector and the probe’s radial velocity relative to the Sun. The observation scheme of the celestial observation autonomous navigation system primarily involves measuring reference celestial bodies, with measurement values converted into corresponding line-of-sight vectors. The principle of celestial body navigation is illustrated in Figure 4. The observations of the celestial body navigation can be divided into three parts, that are solar line-of-sight direction vector observation, radial velocity observation, and celestial navigation line-of-sight vector observation.

FIGURE 4

3.4.1 Solar line-of-sight direction vector observation

The solar sensor determines the orientation of the solar vector in the probe body coordinate frame by sensing the angle between the solar vector and the probe. Combined with the attitude transformation matrix, it enables the acquisition of the solar line-of-sight direction vector relative to the probe in the heliocentric inertial coordinate system. According to the attitude transformation, the observation equation for the ideal solar line-of-sight direction vector in the heliocentric inertial coordinate system can be expressed as Equation 7 (Cui et al., 2010):

Incorporating the measurement errors of the solar sensor, the observation model for the solar line-of-sight direction vector can be expressed as Equation 8:where denotes the probe state vector, denotes the measurement noise.

3.4.2 Radial velocity observation

Due to the relative motion between the light source and the probe, spectral lines exhibit a shift relative to their emitted wavelengths. A spectrometer or spectrograph utilizes visible light from solar radiation as the incident spectrum. Through filters, polarizers, and electro-optic modulators, it reveals the polarized light shift components influenced by gas magnetic fields, thereby enabling the measurement of radial velocity. The radial velocity of the probe in the heliocentric inertial coordinate system can be expressed as Equation 9 (Fränz and Harper, 2002):

Incorporating measurement errors, the radial velocity observation model becomes (Equation 10):where denotes the measurement noise.

3.4.3 Celestial navigation line-of-sight vector observation

The observation model for the line-of-sight vector of other celestial navigation bodies relative to the probe is Equation 11 (Bella et al., 2022):

Accounting for measurement errors, the observation equation for the celestial navigation line-of-sight vector is Equation 12:where denotes the observation error of the line-of-sight vector. Totally, the solar line-of-sight vector, the probe’s radial velocity, and the celestial line-of-sight vectors form a seven-dimensional measurement vector for the celestial body navigation.

4 Deep space navigation algorithm

This section discusses the deep space navigation algorithms, including EKF, the game theory-based filter, RSTKF, and federated filter.

4.1 EKF

The Kalman filter assumes that the mathematical model of the physical system is linear. However, both orbital dynamics models and certain measurement models exhibit nonlinearity in deep space exploration missions. While the Kalman filter cannot directly address nonlinear systems, if the system’s nonlinearity is weak, the nonlinear functions in the state equation and measurement equation can be approximated through the first-order Taylor series expansion, thereby linearizing the model and enabling the use of the standard Kalman Filter framework. This linearization approach gives rise to the EKF, which is applicable to nonlinear state estimation problems in solar system margin exploration (Terejanu, 2008). The EKF algorithm still comprises the processes of time update and measurement update. Suitable for weakly nonlinear systems under Gaussian noise assumptions, the EKF represents the simplest nonlinear filtering method.

Time update in Equation 13:

The equation replaces the linear state evolution model with a nonlinear state equation, retaining significant nonlinear terms to reduce linearization errors. denotes the Jacobian matrix of the state transition function.

Measurement update in Equation 14:

The equation computes measurement predictions using the nonlinear measurement function rather than linearized equations, minimizing linearization errors, and then calculates the Kalman gain and update the error covariance matrix. The EKF maintains computational efficiency comparable to the standard Kalman filter while providing satisfactory performance for weakly nonlinear systems.

4.2 Game theory-based filter

Game theory-based filter addresses model uncertainties and external disturbances in filter systems by incorporating the norm into the filter problem, aiming to construct a filter that minimizes the norm from disturbance inputs to filtering error outputs. The approach makes no assumption about the spectral characteristics of disturbance signals and minimizes estimation variance under the worst-case disturbance scenarios. Building upon the filter, two variant methods have been developed: a game theory-derived approach and an augmented-dimensional approach (Xu and Lam, 2007). The filter implementation proceeds as follows (Benkessirat et al., 2021):

System definition in Equation 15:

Filter algorithm in Equation 16:

Constraints in Equation 17:

The EKF denotes a special case of the filter when γ→ , corresponding to unbounded cost functions. The game theory-based filter accounts for additional uncertainties and noise disturbances affecting estimation results, though at the cost of higher computational complexity. Conversely, the augmented-dimensional filter maintains relatively low computational complexity but requires system dimension expansion (Lim, 2014).

4.3 RSTKF

The RSTKF models the one-step prediction and measurement likelihood probability density functions using Student’s t-distribution, decomposes them into a Gaussian-Gamma hierarchical structure, and ultimately employs variational Bayesian learning inference to solve for the posterior estimates of states and parameters (Hao et al., 2024). The robustness of RSTKF manifests in the variational measurement update stage, where it adaptively adjusts the one-step prediction error covariance matrix and measurement noise covariance matrix based on derivations from variational Bayesian techniques (Huang et al., 2017). The RSTKF workflow is as follows:

Time is updated as Equation 18:

Variational measurement is updated as Equation 19:

The variable i loops between 0 and N-1 until the following condition is satisfied as Equation 20.where and denote the prior degrees of freedom parameters for the one-step prediction and measurement likelihood, respectively. Prior to the variational iteration, the parameters are initialized as , , , .

4.4 Federated filter

Carlson (1990) proposed the federated filter employing a two-stage hierarchical structure for integrated navigation. In the framework, the primary filter directly receives system outputs while simultaneously distributing measurement values to constituent sub-filters, with subsystem outputs restricted to their corresponding local filters. Local state estimates and their associated error covariance matrices from individual sub-filters are transmitted to the master filter for fusion with the master filter own estimates, yielding an optimal global state estimate. The consolidated global estimate and its corresponding error covariance matrix undergo amplification before being fed back to sub-filters, enabling adaptive reset of local state estimates. The iterative process facilitates fault-tolerant information fusion across heterogeneous subsystems with varying reliability characteristics. Based on different information allocation strategies, the federated filtering algorithm has four implementation forms: feedback-free mode, fused feedback mode, zero-reset mode, and variable-scaling mode. Considering the characteristics of deep space probes such as long distance from the Sun and wide field of view with low probability of celestial body occlusion, we designed a hybrid navigation system for deep space navigation that integrates X-ray pulsar, celestial body, and VLBI observations. We categorize X-ray pulsar and celestial body navigation as astronomical navigation. Due to the significant difference in measurement cycles between astronomical navigation and VLBI, we developed two hybrid navigation systems based on the architectures of federated Kalman filters, that are feedback-free and fused feedback modes.

Figure 5 illustrates the schematic diagram of the astronomical+VLBI integrated navigation system under feedback-free mode and fused feedback modes. For the feedback-free mode, VLBI subsystem operates at a much lower measurement frequency compared to the astronomical subsystem, and its measurements exhibit significant latency. The astronomical subsystem utilizes pulsar phase data to derive frequency measurements, solar line-of-sight vectors, celestial body line-of-sight vectors, and radial velocity relative to the Sun. The parameters enable real-time estimation of the probe’s position and velocity. Meanwhile, the VLBI subsystem estimates delayed navigation information using interferometric frequency measurements, which are then extrapolated to the current time via orbital dynamics models. The integrated navigation solution is obtained by fusing the astronomical subsystem’s real-time estimates with the VLBI subsystem’s extrapolated values. The fused estimate serves as the navigation output. In this mode, both subsystems operate independently, with the latest filter outputs from each subsystem serving as initial values for the next fusion cycle. While providing high-precision navigation, the output frequency remains low.

FIGURE 5

For the fused feedback mode, information fusion occurs at the VLBI subsystem’s interferometric measurement frequency. The key difference from feedback-free mode lies in the feedback mechanism, by which the fused state estimate is fed back to the subsystems. Since VLBI measurements are inherently delayed, real-time joint estimation holds no practical value for this subsystem. Hence, the fusion results are fully proportionally fed back only to the pulsar subsystem. Additionally, the astronomical subsystem outputs both its real-time state estimates and the fused results as the system’s navigation solution. The configuration enables high-frequency continuous navigation output. However, as astronomical navigation accuracy degrades with increasing cruise orbit altitude, the fused feedback mode exhibits worse navigation performance compared to the feedback-free mode (Li et al., 2017).

5 Gravity assist orbit simulation

When a probe approaches deep space, it often utilizes planetary gravity assist maneuvers to reduce the escape launch energy from Earth and the maneuver energy required for mid-course orbit adjustments, thereby conserving fuel consumption. Due to the significant mass, major solar system planets can provide substantial energy for probes and are frequently selected as gravity assist bodies in deep space exploration missions.

The gravity assist models can be divided into two categories. The first applies velocity-aligned impulses at the periapsis of hyperbolic trajectories relative to the gravity assist body to satisfy velocity matching conditions before and after the gravity assist, which is known as Maneuvering Gravitational Assist (MGA) (Macdonald and Mcinnes, 2005). Since this model connects different gravitational assist bodies through single Lambert arc segments, it only requires determining variables such as launch time, gravitaty assist time, and arrival time to define the complete transfer trajectory. Its advantages lie in fewer optimization variables, while its limitation is the relatively limited coverage of feasible trajectories. The second category, termed Gravitational Assist with Additional Deep Space Maneuver (MGA-1DSM), involves no maneuvers within the gravity influence sphere of the assisting planet followed by a single deep space maneuver after exiting the influence sphere (Stracquadanio et al., 2011). The model connects different gravity assist phases through concatenated Keplerian orbit propagation arcs and Lambert arcs at maneuver points. Its advantages include higher model accuracy and broader trajectory coverage, though it requires more optimization variables.

The study proposes the gravity assist phase of deep space exploration by integrating the advantages of both approaches to minimize optimization variables while satisfying mission requirements. Specifically, all transfer arcs from Earth launch to the final gravity assist phase employ the periapsis maneuver strategy from the MGA model. For the final gravity assist, it adopts the pure gravity assist approach from the MGA-1DSM model. Post-final gravity assist transfer to the deep space exploration is achieved through sequential Keplerian orbit propagation, a deep space maneuver, and subsequent Keplerian orbit propagation to generate the complete trajectory.

The probe departed from Earth and utilized five gravity assist phases to ultimately reach the bottom of the heliosphere through the simulation. Figure 6 illustrates the gravity assist model for the gravity assist phases, with the flight sequence proceeding as Earth, Venus, Venus, Earth, Jupiter, Neptune and heliosphere bottom.

FIGURE 6

Assuming the probe departs on 19 January 2025, at 12:39:18, it completes five deep space phases before ultimately arriving at the bottom of the heliosphere on 30 September 2049, at 20:59:47. Table 1 summarizes the departure times and flight durations for each mission phase. The gravity assist trajectory diagram is illustrated in Figure 7. This mission employs a high-precision gravitational model to simulate planetary interactions during each flyby, ensuring trajectory accuracy within predicted values. Advanced adaptive control algorithms dynamically adjust the spacecraft’s approach vectors in real-time to optimize gravitational momentum transfer efficiency. The heliospheric boundary detection system utilizes solar wind particle spectrometry and magnetic field mapping technologies to precisely identify the heliopause transition zone.

FIGURE 7

6 Performances of deep space navigation for the gravity assist phase

The section analyzes the performance of deep space navigation for the gravity assist phase, including X-Ray pulsar navigation, VLBI navigation, celestial body navigation and deep space resilient navigation.

6.1 X-ray pulsar navigation

For the deep space navigation of the gravity assist phase, X-ray pulsar with assumed observations of B1821-24, B0531+21, B1937+21, and B1509-58 are considered. The probe effective area is set to 1 m². Based on the signal-to-noise ratio, background radiation flux, observation period, and other parameters of X-ray pulsar radiation signals, the ranging accuracies along the line-of-sight directions for these four pulsars are determined to be 109 m, 325 m, 344 m, and 1807 m, respectively. The sampling rate of the X-ray pulsar signal is configured as 300 s. Navigation was performed for six gravity assist phases, including Earth-Venus, Venus-Venus, Venus-Earth, Earth-Jupiter, Jupiter-Neptune, and Neptune-heliosphere bottom. A 10-body dynamical model is adopted for the kinematic modeling.

Due to positioning errors in X-ray pulsar, navigation performance degrades with the increasing heliocentric distance. The covariance matrix of the measurement noise was therefore adjusted according to the heliocentric distance variations. To simplify this process, the matrix was set in Arcs based on heliocentric distance, in which Arcs 1-3 use 1 astronomical unit (AU) heliocentric distance, Arcs 4-5 use 20 AUs, and Arc 6 used 50 AUs for the first 2000 days and 100 AUs thereafter. Figures 8, 9 illustrate the corresponding X-ray pulsar positioning and velocity errors, with average root mean square (RMS) errors also provided. The results indicate that the three filtering methods yield comparable navigation performance for X-ray pulsar navigation. During Earth-Venus, Venus-Venus, and Venus-Earth gravity assist phases, navigation performance remains relatively high, and X-ray pulsar navigation can generally meet the requirements. However, positioning errors deteriorate for Earth-Jupiter and Jupiter-Neptune Arcs. During Neptune-heliosphere bottom Arc, navigation performance further deteriorates with increasing heliocentric distance, with average errors approaching 300 km. At this time auxiliary measurements are needed for enhanced navigation.

FIGURE 8

FIGURE 9

6.2 VLBI navigation

For probe utilizing VLBI-only navigation during the gravity assist phase, the rapid velocity of the probe and the small size of planets make it less likely for radio signals to be occluded by planets or other small celestial bodies. Additionally, VLBI signals are not blocked by the Sun, enabling continuous ground-based VLBI measurement throughout the phase. During Arcs 1-3 in gravity assist phase, the communication delay remains under 30 min. In Arcs 4 and 5, as the probe moves farther from Earth, the delay increases to several hundred minutes. Ultimately, during the probe’s journey toward the heliopause, the delay reaches thousands of minutes.

To evaluate the performance of VLBI-only navigation during the gravity assist phase, a 10-body dynamical model was employed. Positioning and velocity errors for VLBI were set at 10 km and 0.2 m/s, respectively, with a filtering cycle matching the VLBI measurement interval of 300 min. The simulation spanned the entire gravity assist phase, with initial positioning and velocity errors set to 200 km and 5 m/s, respectively. Figures 10, 11 depict the positioning and velocity errors for VLBI-only navigation. The results demonstrates that positioning errors remain below 10 km across all the gravity assist phases. Notably, VLBI navigation outperforms X-ray pulsar navigation during the gravity assist phase.

FIGURE 10

FIGURE 11

6.3 Celestial body navigation

The section discusses the deep space navigation performance of probe during gravity assist phases using only celestial body observation information. Based on previous analyses, the available navigation information contains solar line-of-sight vectors, line-of-sight vectors, and probe radial velocity. During different gravity assist phases, the distances between the probe and planets vary from millions of kilometers to tens of AUs. The navigation performance improves as the probe approaches celestial bodies. We first analyzed the distance variations between the probe and planets during the gravity assist phase, providing critical reference for selecting celestial body targets. Based on it, we evaluated navigation performance using single celestial body observation models during each gravity assist phase.

The probe-to-celestial-body distance significantly determines deep space navigation performance when using celestial observations. Based on the model in Figures 6, 12 illustrates the distance between the probe and planets during each gravity assist phase. We assume optimal observation conditions occur when the selected navigation celestial body is closest to the probe. As shown in Figure 13, in Arc 1, Earth is used for the first 57 days, followed by Venus thereafter. In Arc 2, Venus is used for the first 126 days, Earth from days 127-334, and Venus again afterward. In Arc 3, Venus is used for the first 53 days, followed by Earth. In Arc 4, Earth is used for the first 65 days, Mars from days 65-223, and Jupiter afterward. In Arc 5, Jupiter is used for the first 152 days, Mars from days 152-1,300, and Neptune thereafter. In Arc 6, Neptune is used throughout the entire phase.

FIGURE 12

FIGURE 13

The 7-dimensional measurement vector comprises the solar line-of-sight vector, probe line-of-sight vectors, and the probe radial velocity. The measurement errors for the solar line-of-sight vector and planetary line-of-sight vectors are set to and . respectively. The measurement error for the probe radial velocity relative to the Sun is set to . The sampling rate is set to 300 s, with simulation duration consistent with previous analysis. Initial positioning errors with 200 km and velocity errors with 5 m/s are applied. Figures 13, 14 depict the positioning and velocity errors of the celestial body navigation. It is evident that celestial navigation exhibits significantly lower performance compared to X-ray pulsar and VLBI navigation. As distance increases, the performance of celestial body navigation continues to degrade.

FIGURE 14

6.4 Deep space resilient navigation

Based on the previous analysis, the probe remains capable of receiving ground-based VLBI signals throughout the gravity assist phase, with relatively small delays observed in Arc 1 and Arc 5. The latency increases significantly to over a thousand minutes only during the transition from Neptune to the heliosphere bottom. For celestial observations, continuous and fully autonomous navigation information is ensured by consistently selecting the nearest celestial body with great visibility. During the gravity assist phase, the heliocentric distance remains around 1 AU from Arcs 1 to 3. From Arcs 4 to 6, the increasing heliocentric distance gradually reduces the performance of X-ray pulsar navigation, though it still provides stable autonomous navigation references for most of the time. Hence, the gravity assist phase continues to employ a flexible integrated navigation system based on ground-based VLBI, X-ray pulsar, solar line-of-sight vector measurement, celestial body line-of-sight vector measurement, and relative solar radial velocity measurements. The section analyzes the results of two resilient navigation models for the feedback-free and the fused feedback modes, while also presenting navigation results using astronomical-only and VLBI-only navigation. To verify the applicability of the navigation models, simulations were conducted under both Gaussian measurement noise and time-varying measurement noise conditions.

To verify the resilient navigation performance of the astronomical+VLBI combined navigation system during the gravity assist phase under Gaussian noise conditions, the X-ray pulsar selection, celestial body selection, sensor sampling frequency, and measurement error settings for all navigation methods remain consistent with Sections 6.2–6.4. The simulation covers the entire gravity assist phase duration, with initial positioning and velocity errors set to 200 km and 5 m/s for each Arc. When assuming the probe is subject to time-varying and complex environmental disturbances, measurement values have a 5% probability of encountering outliers, with noise amplitude 100 times that of normal noise. The noise configuration here is refined to test the outcomes under a hypothetical scenario, ensuring rigorous validation of the assumed conditions (Su et al., 2026a). Figures 15, 16 illustrate the resilient positioning and velocity errors under Gaussian conditions, whereas Figures 17, 18 depict the results under time-varying measurement noise conditions. Simulation results for both feedback-free and fused feedback modes of the resilient navigation system during the gravity assist phase indicate that the integrated navigation errors remain within 10 km throughout the entire process, meeting the requirements for high precision, long endurance, and high-frequency navigation information output in deep space exploration. The resilient navigation of feedback mode achieves slightly higher accuracy than the fused feedback mode. Under Gaussian measurement noise conditions, the accuracy of EKF theoretically outperforms the game theory-based filter and RSTKF. However, since X-ray pulsar navigation errors vary with heliocentric distance, which is a parameter difficult to estimate accurately in real-time during navigation, the simulation only provides a coarse selection method for measurement noise covariance matrices by analyzing heliocentric distance variations. Consequently, the game theory-based filter and RSTKF may outperform EKF in certain Arcs. Under time-varying measurement noise conditions, RSTKF models the time-varying noise as a heavy-tailed distribution, achieving robustness to outliers. Hence, RSTKF demonstrates significantly higher navigation accuracy than the other two filters in the simulation.

FIGURE 15

FIGURE 16

FIGURE 17

FIGURE 18

7 Conclusion

The study provides pivotal technical support for deep space missions, advancing autonomous navigation capabilities in harsh space conditions. The framework marks a significant step toward fully autonomous and resilient deep space navigation systems, aligning with global advancements in deep space exploration technology.

References

• 1

  BellaS.AndreisE.FranzeseV.PanicucciP.TopputoF. (2022). Line-of-sight extraction algorithm for deep-space autonomous navigation. Adv. Astronautical Sci.177, 1–18.

  • Google Scholar
  • View reference in article

• 2

  BenkessiratS.BoustiaN.NachidaR. (2021). A new collaborative filtering approach based on game theory for recommendation systems. J. Web Engineering20, 303–326. 10.13052/jwe1540-9589.2024

  • CrossRef
  • Google Scholar
  • View reference in article

• 3

  CarlsonN. A. (1990). Federated square root filter for decentralized parallel processes. IEEE Trans. Aerosp. Electron. Syst.26, 517–525. 10.1109/7.106130

  • CrossRef
  • Google Scholar
  • View reference in article

• 4

  ChristianJ. A.LightseyE. G. (2012). Onboard image-processing algorithm for a spacecraft optical navigation sensor system. J. Spacecr. Rockets49, 337–352. 10.2514/1.A32065

  • CrossRef
  • Google Scholar
  • View reference in article

• 5

  CuiP.ChangX.CuiH. (2010). “Autonomous orbit determination of deep space probe based on the Sun line-of-sight vector,” in 2010 3rd International symposium on systems and control (Harbin, China: IEEE), 8-10 June 2010, 540–544. 10.1109/ISSCAA.2010.5632558

  • CrossRef
  • Google Scholar
  • View reference in article

• 6

  EmadzadehA. A.SpeyerJ. L. (2010). Relative navigation between two spacecraft using X-ray pulsars. IEEE Trans. Control Syst. Technol.19, 1021–1035. 10.1109/tcst.2010.2068049

  • CrossRef
  • Google Scholar
  • View reference in article

• 7

  FränzM.HarperD. (2002). Heliospheric coordinate systems. Planet. Space Sci.50, 217–233. 10.1016/s0032-0633(01)00119-2

  • CrossRef
  • Google Scholar
  • View reference in article

• 8

  HaoP.KarakusO.AchimA. (2024). RKFNet: a novel neural network aided robust Kalman filter. arXiv Preprint, arXiv:240316756. 10.1016/j.sigpro.2024.109856

  • CrossRef
  • Google Scholar
  • View reference in article

• 9

  HuJ.YangH.LiS.BaoyinH. (2026). Robust design of low-thrust gravity-assist trajectories via reinforcement learning. J. Guid. Control, Dyn.49, 1–16. 10.2514/1.g009427

  • CrossRef
  • Google Scholar
  • View reference in article

• 10

  HuangY.ChangS.LiP.HuX.WangG.LiuQ.et al (2014). Orbit determination of Chang’E-3 and positioning of the lander and the rover Chinese science. Bulletin59, 3858–3867. 10.1007/s11434-014-0542-9

  • CrossRef
  • Google Scholar
  • View reference in article

• 11

  HuangY.ZhangY.LiN.WuZ.ChambersJ. A. (2017). A novel robust student’s t-based Kalman filter. IEEE Trans. Aerosp. Electron. Syst.53, 1545–1554. 10.1109/taes.2017.2651684

  • CrossRef
  • Google Scholar
  • View reference in article

• 12

  JordanJ. F. (1987). Navigation of spacecraft on deep space missions. J. Navigation40, 19–29. 10.1017/S0373463300000266

  • CrossRef
  • Google Scholar
  • View reference in article

• 13

  KangZ.XuZ.LiuJ.LiN. (2017). Doppler velocity measurement based on double measurement model and its integrated navigation. J. Astronautics38, 964–970. 10.3873/j.issn.1000-1328.2017.09.009

  • CrossRef
  • Google Scholar
  • View reference in article

• 14

  KongJ.ZhangY. (2022). Orbit determination ability of Chang’e 5 based on CDSN tracking data. J. Astronautics43, 183. 10.3873/j.issn.1000-1328.2017.09.009

  • CrossRef
  • Google Scholar
  • View reference in article

• 15

  LiZ.GaoJ.WangJ.YaoY. (2017). PPP/INS tightly coupled navigation using adaptive federated filter. GPS Solutions21, 137–148. 10.1007/s10291-015-0511-z

  • CrossRef
  • Google Scholar
  • View reference in article

• 16

  LimJ. (2014). A tutorial–game theory-based extended H infinity filtering approach to nonlinear problems in signal processing. Digit. Signal Process.34, 1–15. 10.1016/j.dsp.2014.07.010

  • CrossRef
  • Google Scholar
  • View reference in article

• 17

  LiuJ.JinS. (2018). Evaluation of Mars probe positioning using X-ray pulsars, celestial, gravity-aided and ground-based measurements Planetary and Space. Science163, 14–34. 10.1016/j.pss.2018.06.014

  • CrossRef
  • Google Scholar
  • View reference in article

• 18

  LiuJ.MaJ.TianJ.-W.KangZ.-W.WhiteP. (2010). X-ray pulsar navigation method for spacecraft with pulsar direction error. Adv. Space Res.46, 1409–1417. 10.1016/j.asr.2010.08.019

  • CrossRef
  • Google Scholar
  • View reference in article

• 19

  LiuQ.ChenM.GoossensS.KikuchiF.MatsumotoK.PingJ.et al (2010). Applications of same-beam VLBI in the orbit determination of multi-spacecrafts in a lunar sample-return mission. Sci. China Phys. Mech. Astronomy53, 1153–1161. 10.1007/s11433-010-4009-y

  • CrossRef
  • Google Scholar
  • View reference in article

• 20

  LiuJ.WeiE.JinS. (2017). Mars cruise orbit determination from combined optical celestial techniques and x-ray pulsars. J. Navigation70, 719–734. 10.1017/s0373463316000874

  • CrossRef
  • Google Scholar
  • View reference in article

• 21

  LiuJ.WeiE.JinS.LiuJ. (2018). Absolute navigation and positioning of mars rover using gravity-aided odometry. J. Navigation71, 530–546. 10.1017/s0373463317000893

  • CrossRef
  • Google Scholar
  • View reference in article

• 22

  MaP.BaoyinH.MuJ. (2014). Autonomous navigation of Mars probe based on optical observation of Martian moon. Opt. Precis. Eng.22, 863–869. 10.3788/ope.20142204.0863

  • CrossRef
  • Google Scholar
  • View reference in article

• 23

  MacdonaldM.McinnesC. R. (2005). Spacecraft planetary capture using gravity-assist maneuvers. J. Guidance, Control, Dynamics28, 365–369. 10.2514/1.11866

  • CrossRef
  • Google Scholar
  • View reference in article

• 24

  MancusoS. (2004). “Vision based GNC systems for planetary exploration,” in Proceedings of the dynamics and control of systems and structures in space (DCSSS) 6th conference (Citeseer).

  • Google Scholar
  • View reference in article

• 25

  MastrodemosN.KubitschekD. G.SynnottS. P. (2005). Autonomous navigation for the deep impact mission encounter with comet tempel 1. Space Sci. Rev.117, 95–121. 10.1007/s11214-005-3394-4

  • CrossRef
  • Google Scholar
  • View reference in article

• 26

  ReichleyP.DownsG.MorrisG. (1970). Time-of-arrival observations of eleven pulsars. Astrophysical J. Lett.159, L35–L40. 10.1086/180473

  • CrossRef
  • Google Scholar
  • View reference in article

• 27

  SheikhS. I.PinesD. J.RayP. S.WoodK. S.LovelletteM. N.WolffM. T. (2006). Spacecraft navigation using X-Ray pulsars. J. Guid. Control, Dyn.29, 49–63. 10.2514/1.13331

  • CrossRef
  • Google Scholar
  • View reference in article

• 28

  ShemarS.FraserG.HeilL.HindleyD.MartindaleA.MolyneuxP.et al (2016). Towards practical autonomous deep-space navigation using X-Ray pulsar timing. Exp. Astron.42, 101–138. 10.1007/s10686-016-9496-z

  • CrossRef
  • Google Scholar
  • View reference in article

• 29

  StracquadanioG.La FerlaA.De FeliceM.NicosiaG. (2011). “Design of robust space trajectories,” in International conference on innovative techniques and applications of artificial intelligence (Springer), 341–354.

  • Google Scholar
  • View reference in article

• 30

  SuK. (2023). BDS pseudorange biases handling: simultaneous estimation, daily magnitude and analytical comparison of the BDS PPP models. Adv. Space Res.71, 4681–4690. 10.1016/j.asr.2023.01.019

  • CrossRef
  • Google Scholar
  • View reference in article

• 31

  SuK.JiaoG.JinS.YangY.GaoP.WeiJ.et al (2026a). Resilient deep space navigation models for the cruise phase. Adv. Space Res77 (12), 12618–12633. 10.1016/j.asr.2026.03.087

  • CrossRef
  • Google Scholar
  • View reference in article

• 32

  SuK.JiaoG.YangY.JinS.YanZ.GaoP.et al (2026b). Detection of GPS and BDS flex power for mitigating its impact on satellite-based services. Satell. Navig.7, 2. 10.1186/s43020-026-00190-3

  • CrossRef
  • Google Scholar
  • View reference in article

• 33

  TaoW.ZhangJ.HuH.ZhangJ.SunH.ZengZ.et al (2024). Intelligent navigation for the cruise phase of solar system boundary exploration based on Q-learning. EKF Complex and Intelligent Syst.10, 2653–2672. 10.1007/s40747-023-01286-y

  • CrossRef
  • Google Scholar
  • View reference in article

• 34

  TerejanuG. A. (2008). Extended Kalman Filter Tutorial (University at Buffalo), 27, 6.

  • Google Scholar
  • View reference in article

• 35

  TornatoreV.HaasR.CaseyS.DuevD.PogrebenkoS.CalvésG. M. (2014). “Direct VLBI observations of global navigation satellite system signals,” in Earth on the Edge: Science for a Sustainable Planet. Editors RizosC.WillisP. (Berlin, Heidelberg: Springer Berlin Heidelberg), 247–252.

  • Google Scholar
  • View reference in article

• 36

  TuranE.SperettaS.GillE. (2022). Autonomous navigation for deep space small satellites: scientific and technological advances. Acta Astronaut.193, 56–74. 10.1016/j.actaastro.2021.12.030

  • CrossRef
  • Google Scholar
  • View reference in article

• 37

  WangM.ZhengX.ChengY.ChenX. (2016). Scheme and key technologies of autonomous optical navigation for Mars exploration in cruise and capture phase. Geomatics Inf. Sci. Wuhan Univ.41, 434–442. 10.13203/j.whugis20150293

  • CrossRef
  • Google Scholar
  • View reference in article

• 38

  WangC.LiL.ZhangA.ZhangS.HouD.XuZ.et al (2022). The solar wind and particle radiation environment on the surface of the Moon—new observations from Chang’E-4. J. Deep Space Explor.9, 239–249.

  • Google Scholar
  • View reference in article

• 39

  XiongZ.ChenJ.-H.WangR.LiuJ.-Y. (2013). A new dynamic vector formed information sharing algorithm in federated filter. Aerosp. Sci. Technol.29, 37–46. 10.1016/j.ast.2013.01.006

  • CrossRef
  • Google Scholar
  • View reference in article

• 40

  XuS.LamJ. (2007). Reduced-order filtering for singular systems. Syst. and Control Lett.56, 48–57. 10.1016/j.sysconle.2006.07.010

  • CrossRef
  • Google Scholar
  • View reference in article

• 41

  YanJ.YangX.YeM.LiF.JinS.JinW.et al (2018). New tracking modes and performance for Mars spacecraft orbit determination and lander positioning. Planet. Space Sci.163, 5–13. 10.1016/j.pss.2018.01.008

  • CrossRef
  • Google Scholar
  • View reference in article

• 42

  YangB.YuX.MiaoJ. (2016). A method of starlight navigation using the infrared horizon sensor. J. Astronautics37, 1089. 10.3873/j.issn.1000-1328.2016.09.008

  • CrossRef
  • Google Scholar
  • View reference in article

• 43

  YangY.RenX.JiaX.SunB. (2023). Development trends of the national secure PNT system based on BDS. Sci. China Earth Sci.66, 929–938. 10.1007/s11430-022-1069-7

  • CrossRef
  • Google Scholar
  • View reference in article

• 44

  ZhangZ.TangY.ZhengZ.QiaoD.ChenH.ZhangT. (2024). Research on artificial intelligence technology for solar system boundary exploration missions. J. Deep Space Explor.11, 523–537. 10.15982/j.issn.2096-9287.2024.20220100

  • CrossRef
  • Google Scholar
  • View reference in article