With the increasing demand for high-altitude high-speed vehicles in complex adversarial environments, autonomous navigation under conditions of GNSS signal unavailability or damaged ground infrastructure has become a critical technological bottleneck. Existing navigation means—including visual navigation, terrain matching, inertial navigation, celestial navigation, and geomagnetic navigation—each have inherent limitations, making it difficult to simultaneously satisfy requirements for all-weather operation, high accuracy, and long endurance. Inspired by the use of relative position vectors for absolute navigation in spacecraft formation flying, this paper extends this principle to high-altitude high-speed vehicles that are significantly influenced by non-conservative forces such as engine thrust and aerodynamic forces. However, it is necessary to utilize accelerometer measurements from inertial navigation to subtract the effects of non-conservative accelerations, so that the gravitational acceleration information—which reflects the absolute position—can be extracted from the relative position vectors. Nevertheless, the influence of multi-source uncertainties—including accelerometer noise and bias, relative position measurement errors, and star tracker attitude errors—on navigation accuracy remains unclear, and how to achieve high-precision autonomous navigation under this framework remains a key issue to be resolved.
In a recent study published in Space: Science & Technology, the research team from the School of Astronautics, Beihang University, proposed an absolute navigation method that integrates relative position vectors, inertial navigation, and star trackers. The study formulates the state vector in the Earth-Centered Earth-Fixed (ECEF) frame, comprising position errors, velocity errors, and accelerometer biases, and employs the relative position vector errors between vehicles as measurements. An extended Kalman filter (EKF) model incorporating the Earth's rotational effects and a high-order gravitational field is established, and the system observability is verified through singular value decomposition of the observability matrix. Simulation results demonstrate that under a baseline scenario with a formation spacing of 400 km and a flight altitude of 50 km, the proposed method achieves three-dimensional positioning accuracy better than 200 m over a 150-minute mission, whereas a comparative scheme relying solely on inertial navigation and star trackers diverges to the order of several kilometers within 10 minutes, confirming the critical role of relative position vector measurements. Analysis of influencing factors indicates that accelerometer measurement noise exerts the most significant impact on navigation accuracy; larger formation spacing and higher relative position and attitude measurement precision yield superior navigation performance, while flight altitude in the range of 50–200 km has a relatively minor effect. This research provides a feasible technical solution for autonomous navigation of high-altitude high-speed vehicle formations in GNSS-denied environments, offering significant engineering application value for enhancing the survivability and mission execution capability of vehicles in complex adversarial scenarios.
First, this paper focuses on the autonomous navigation problem of high-altitude high-speed vehicles in GNSS-denied environments and proposes a novel method that utilizes inter-vehicle relative position vectors for absolute navigation. As illustrated in Fig. 1, high-altitude high-speed vehicle formations operating in long-endurance missions may encounter complex scenarios wherein satellite navigation signals are unavailable or ground infrastructure is damaged, necessitating autonomous navigation capabilities that do not rely on external information. Existing navigation means—including visual navigation, terrain matching, inertial navigation, celestial navigation, and geomagnetic navigation—each suffer from inherent limitations, rendering them insufficient to simultaneously satisfy the requirements for all-weather operation, high accuracy, and long endurance in complex adversarial environments. Inspired by the principle of using relative position vectors for absolute navigation in spacecraft formation flying, this paper extends this concept to high-altitude high-speed vehicles that are significantly affected by non-conservative forces such as engine thrust and aerodynamic forces. Fig. 2 presents the complete navigation framework: inter-vehicle relative position vectors are obtained through laser ranging and optical direction finding, non-conservative accelerations are measured by accelerometers, and attitude information is provided by star trackers; these three data sources are fused via an extended Kalman filter to output estimates of position, velocity, and accelerometer biases. The core of this method lies in: deriving the relative acceleration by differencing the relative position vectors, then subtracting the non-conservative accelerations measured by accelerometers, thereby extracting the gravitational acceleration information that reflects the absolute position, which is subsequently used to iteratively solve for the absolute position. This framework enables the vehicle formation to achieve autonomous navigation and positioning without requiring any external signal input.
Second, the paper establishes the system state equation and observation equation, and verifies the feasibility of the proposed navigation scheme through observability analysis. In view of the characteristics of high-altitude high-speed vehicles subject to non-conservative forces, the study adopts the inertial navigation equations in the Earth-Centered Earth-Fixed (ECEF) frame to formulate the state model. The state vector comprises position errors, velocity errors, and accelerometer biases, while the relative position vector errors between vehicles are employed as measurements, leading to the construction of an extended Kalman filter (EKF) model. To verify system observability, the paper constructs the observability matrix based on the output and its time derivatives, and analyzes its rank and condition number via singular value decomposition (SVD). Fig. 3 presents the observability analysis results; under the assumption of a central gravitational field, the minimum singular value is on the order of 10⁻⁷ and the condition number is approximately 10⁷, indicating that the system exhibits good observability, and the adoption of a high-order gravitational field model can further improve observability performance. Table 1 lists the flight parameters for the baseline simulation scenario: flight altitude of 50 km, velocity of 2.0 km/s, formation spacing of 400 km, and simulation duration of 150 minutes. In the filter, the relative position vector measurements are sampled at 1 Hz, with a range error of 1 m and a direction error of 3 arcseconds; the star tracker three-axis measurement errors are 3, 3, and 10 arcseconds; the accelerometer bias is 30 μg with a random walk noise of 10 μg/√Hz; and the initial position and velocity errors are 200 m and 1 m/s, respectively. The core of this filtering framework lies in indirectly sensing the Earth's gravitational field information through the differential measurements of relative position vectors, thereby enabling absolute positioning without any external reference.
Finally, this paper validates the effectiveness of the proposed method through multiple sets of numerical simulations and systematically analyzes the influencing factors. Fig. 4 presents the position error estimation results under the baseline scenario; the navigation solution converges rapidly within the first hour, with the three-dimensional positioning error stabilizing below 200 m. Over the final three hours, the root-mean-square errors along the three axes are 127.5 m, 90.4 m, and 82.6 m, respectively, yielding a total three-dimensional error of 176.8 m. For comparison, Fig. 5 shows the navigation results under identical conditions but without incorporating relative position vector measurements, relying solely on inertial navigation and star trackers; the error diverges to the order of several kilometers within 10 minutes, confirming the critical role of relative position vector measurements in enhancing navigation accuracy. In terms of influencing factor analysis, the effects of formation spacing, flight altitude, accelerometer noise, relative position vector noise, and star tracker noise on navigation performance are respectively examined. The results indicate that accelerometer measurement noise exerts the most significant impact: when the noise increases from 10 μg/√Hz to 30 μg/√Hz, the three-dimensional error grows from 177 m to 317 m. Larger formation spacing yields superior navigation performance, with errors of 237 m at 200 km spacing and 146 m at 800 km spacing. Increased relative position vector direction errors and star tracker noise both lead to degraded navigation accuracy, whereas flight altitude in the range of 50–200 km has a relatively minor effect on navigation precision. This research provides a feasible technical solution for autonomous navigation of high-altitude high-speed vehicle formations in GNSS-denied environments, offering significant engineering application value for enhancing the survivability and mission execution capability of vehicles in complex adversarial scenarios.
