Navigation in Space by X-ray Pulsars

One of the main requirements for any space mission is to navigate the spacecraft. Current space navigation methods highly depend on ground-based operations. To achieve more autonomy and also to augment the current available navigation solutions, we are interested in utilization of celestial-based navigation systems. Such systems use signals emitted from celestial sources located at great distances from Earth. Because of their special characteristics, X-ray pulsars are potential celestial candidates for navigation.

In this chapter, we present an overview of spacecraft navigation using X-ray pulsars. In Sect. 2.2, we present a concise treatment of current navigation methods being utilized for space missions. Section 2.3.1 describes why employing celestial-based navigation techniques is desirable for space missions. We introduce different types of pulsars in Sect. 2.3.2. Section 2.3.3 explains why X-ray pulsars are interesting candidates to be used for navigation purposes. A short history of pulsar-based navigation is given in Sect. 2.3.4.

In this chapter, we present an overview of the X-ray pulsar-based navigation system. We provide the mathematical tools needed to analyze the proposed system. We also characterize the epoch folding procedure.

As explained in Chap. 3, the navigation system measurement is obtained through estimation of the time delay between the received signals. The delay estimation problem plays the most important role in the navigation system. In this chapter, we study this problem in more detail.

In this chapter we offer our first approach for estimation of the pulse delay. The proposed approach is to retrieve the photon intensity functions on each detector via epoch folding to obtain the empirical rate functions, i.e., \(\breve{{\lambda }}_{k}(t)\). Then, the empirical intensities will be used for estimation of the initial phase on each detector.

An important point regarding the pulse delay estimation techniques introduced in Chap. 5 is that they need to employ the epoch folding procedure which needs the exact knowledge of the spacecraft velocities. Furthermore, it was shown that the proposed epoch folding-based estimators are not asymptotically efficient. To address these problems, we propose another method in this chapter. This approach is based on direct use of the measured TOA sets, \(\{{t}_{i}^{(1)}\}_{i=1}^{{M}_{1}}\) and \(\{{t}_{i}^{(2)}\}_{i=1}^{{M}_{2}}\). Because the measured TOAs are being used directly, it is not necessary for the estimators to have access to the velocity data. Nonetheless, the effect of imprecise velocity data on the performance of the pulse delay estimator is studied. Computational complexity study is also performed.

In this chapter, a recursive algorithm is formulated which can be used to find the relative navigation solution between the two spacecraft. The navigation system is equipped with IMUs which provide the spacecraft acceleration data. The dynamics of relative position between the two spacecraft and a model of the IMU accuracy are utilized for developing the navigation algorithm. The measurements, which are obtained by time tagging the photons, are modeled as a linear function of the projected relative position onto the unit direction pointing to the pulsar plus the measurement noise. The measurement noise variance is selected based on how well the pulse delay is estimated. Then, by applying a Kalman filter, the relative position, the relative velocity, the relative bias between accelerometers, and the differential time between clocks are estimated, and the steady state estimation error covariance is obtained. The effect of different system parameters on the achievable accuracy of relative position estimation is investigated. In particular, the effect of different values of IMU uncertainty, measurement noise variance, and number of pulsars used for estimation are considered.

This book has proposed a new approach for navigation of spacecraft in space employing X-ray pulsar measurements. The presented approach is applicable for both absolute and relative navigation problems. Pulsars emitting in the X-ray band were chosen because of their stable period and their geometric distribution in the sky map. The main advantage of using X-ray pulsars for navigation is that relatively small size detectors can be used for detection of the X-ray photons on board a spacecraft. This facilitates the spacecraft design procedure.