International Journal of Aerospace Engineering

Volume 2015 (2015), Article ID 714302, 14 pages

http://dx.doi.org/10.1155/2015/714302

## Decentralized State Estimation Algorithm of Centralized Equivalent Precision for Formation Flying Spacecrafts Based on Junction Tree

^{1}College of Mechatronics Engineering and Automation, National University of Defense Technology, No. 109, Deya Street, Changsha, Hunan 410073, China^{2}The Australian Centre for Field Robotics, University of Sydney, Rose Street Building J04, Sydney, NSW 2006, Australia

Received 29 April 2015; Revised 28 July 2015; Accepted 9 August 2015

Academic Editor: Paolo Tortora

Copyright © 2015 Mengyuan Dai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

As centralized state estimation algorithms for formation flying spacecraft would suffer from high computational burdens when the scale of the formation increases, it is necessary to develop decentralized algorithms. To the state of the art, most decentralized algorithms for formation flying are derived from centralized EKF by simplification and decoupling, rendering suboptimal estimations. In this paper, typical decentralized state estimation algorithms are reviewed, and a new scheme for decentralized algorithms is proposed. In the new solution, the system is modeled as a dynamic Bayesian network (DBN). A probabilistic graphical method named junction tree (JT) is used to analyze the hidden distributed structure of the DBNs. Inference on JT is a decentralized form of centralized Bayesian estimation (BE), which is a modularized three-step procedure of receiving messages, collecting evidences, and generating messages. As KF is a special case of BE, the new solution based on JT is equivalent in precision to centralized KF in theory. A cooperative navigation example of a three-satellite formation is used to test the decentralized algorithms. Simulation results indicate that JT has the best precision among all current decentralized algorithms.

#### 1. Introduction

Formation flying has been recognized as an enabling technology for many future space missions [1], for example, the synthetic aperture radar (SAR) [2], and the stellar interferometry [3]. Compared with conventional space missions based on a single spacecraft, formation flying has several advantages: lower price, robustness after spacecraft failures, modularity, and good flexibility for reconfigurations [4]. The concept of formation flying is driven by the idea that a fleet of small spacecrafts could form a virtual spacecraft of unlimited scale which may serve as a substitution of a monolithic spacecraft [4]. As the size of the formation becomes larger, the guidance, navigation, and control (GNC) task becomes more and more onerous due to the increased size of state and measurement data [5]. Thus it becomes a necessity to develop decentralized algorithms to balance the computational loads among spacecrafts.

This paper takes the navigation task as an example to study the decentralized state estimation algorithms. There are generally two kinds of sensors used for formation navigation: the proprioceptive sensors and the exteroceptive sensors [6]. The first kind of sensors is used for single platform measurements to estimate the state of local spacecraft, for example, GPS receivers [4, 5, 7]. The second kind of sensors is used for interformation measurements to estimate the states of local spacecraft and its neighbors, for example, optical devices [8, 9] and radio frequency (RF) devices [10, 11]. With interformation measurements and communications, the navigation resources at different spacecrafts may be shared among the formation. This kind of navigation is also known as cooperative navigation [12]. As a typical navigation application for formation flying spacecrafts, this paper studies formation flying missions using RF range augmented GPS sensors, which are introduced by Park [4] and Ferguson and How [5]. Though the sensors are specially appointed in this paper, the technique used in designing decentralized algorithms may inspire other cooperative navigation missions.

To the state of the art, almost all the decentralized estimation algorithms for formation flying are based on distributing centralized EKF through simplifying the measurement models or decoupling the interformation measurements through approximations. In fact, as interformation measurements make the states of spacecrafts strongly coupled, the simplification and decoupling will result in suboptimal estimations, which means that the precision of these algorithms is inferior to EKF. From the perspective of probabilistic graphical models in artificial intelligence, Lauritzen introduced a method for modeling and local computing of exact mean and covariance in dynamic systems based on JT [13]. The most remarkable advantage of the JT algorithm to other decentralized algorithms is that the JT algorithm has equivalent precision to centralized EKF [14]. Thus it is possible to develop a new decentralized cooperative navigation algorithm for formation flying spacecrafts.

This paper is arranged in 5 sections. In Section 2, the model for cooperative navigation is introduced. A brief introduction of existing decentralized solutions is provided in Section 3. And Section 4 introduces the new decentralized solution as a series of local computations and messages passing on JT. Simulation results in Section 5 show that the new solution has centralized equivalent precision which is better than the decentralized algorithms reviewed.

#### 2. Model of Cooperative Navigation for Formation Flying

This paper takes the formation flying mission based on RF range augmented GPS as an example. The GPS pseudorange is used for position estimate at local spacecraft, while the RF range devices are used for relative motion measurements between spacecrafts. As the dynamic model and the measurement model are both nonlinear, some partial derivatives are needed for linearization. The partials used in the numerical simulation of this research are listed in Appendix A.

##### 2.1. Dynamic Model

The state of each spacecraft is defined as the Keplerian orbital elements. Let be the Keplerian orbital elements of satellite at time step , , where is the semimajor axis, is the eccentricity, is the inclination angle, is the right ascension of ascending node, is the argument of perigee, and is the mean anomaly at reference time. Without considering system inputs such as orbit control, the dynamic model of satellite is given bywhere is the noise and , is a Gaussian distribution of mean and variance .

Linearize (1) at using first order Tyler expansion, and we havewhere is the state transition matrix and

##### 2.2. Measurement Model

There are two kinds of measurements in the formation flying example: single platform measurement based on GPS pseudorange and interformation measurement based on RF range.

###### 2.2.1. Single Platform Measurement

For simplicity, the time delay for signal transmission and errors such as hardware delay, ionosphere delay, multipath, relativistic effect, and clock error are not considered. Let be the position of observed GPS satellite given by the broadcast ephemeris, and let be the position of spacecraft at time step . The single platform measurement based on GPS pseudorange, , is given bywhere means the length of vector and is the measurement noise, .

In general, the single platform measurement function may be written as

Linearize (5) at using first order Tyler expansion, and we havewhere is the measurement matrix and

###### 2.2.2. Interformation Measurement

Similar to single platform measurement, let and be the position of spacecraft and at time step , respectively. The interformation measurement between spacecrafts and , , is given bywhere is the measurement noise, .

In general, the single platform measurement function may be written as

Linearize (9) at and using first order Tyler expansion, and we havewhere and are the measurement matrix and

#### 3. Review of Decentralized Algorithms

To the best knowledge of the authors, most current decentralize solutions are based on centralized EKF. Generally, there are two types of decentralized algorithms for formation flying spacecraft: the full order extended Kalman filter (FOEKF) and the reduced order extended Kalman filter (ROEKF) [5]. In FOEKF, each spacecraft runs a full order estimator which estimates the full states of the system, but only based on local measurements related to the very spacecraft. In ROEKF, each spacecraft runs a reduced order estimator which only processes the state of itself. The reduction of the order of the filter is based on decoupling the measurement models. As FOEKF and ROEKF all require simplifications and approximations of the EKF model, they all suffer from suboptimal precision which is inferior to centralized EKF. To compensate the precision loss, a direct method is to use iterations, which leads to the iterated cascade extended Kalman filter (ICEKF) [5].

##### 3.1. Centralized EKF

Consider a formation of spacecrafts. Let be the state of the whole formation at time step , . Without considering the system input, the linearized model of cooperative navigation could be rewritten as follows.

*Dynamic Model.*

*Measurement Model.*where is the state transition matrix, is the noise of the dynamic model, is the vector of measurements, is the measurement matrix, and is the noise of measurements. The procedure of EKF could be described as a sequence of time update and measurement update.

*Time Update.*

*Measurement Update.*where mean and covariance are the best estimates of system state at time step .

##### 3.2. FOEKF

Considering a local FOEKF estimator on spacecraft , it only processes the measurements related to spacecraft , which means that the measurement , measurement matrix , and measurement noise are subsets of , , and in EKF, respectively. The measurement update of FOEKF at spacecraft is as follows:

Given that the RF range and Doppler measurements are both nonlinear (the linearization requires the states of other spacecrafts), spacecrafts doing relative measurements must communicate with each other exchanging information of mean and covariance of its own state. Because only part of the measurements is considered at each spacecraft, FOEKF is suboptimal. Meanwhile, as FOEKF runs a full order estimator at each spacecraft, the computational load may not be well balanced.

##### 3.3. ROEKF

###### 3.3.1. Original ROEKF

ROEKF is a further simplification of FOEKF. Taking spacecraft for example, suppose the states of other spacecrafts are known, and the measurement model between spacecrafts and could be simplified as

Notice that is known. Let ; the ROEKF measurement model could be rewrite aswhere is the set of all the measurements related with spacecraft and is the measurement matrix related with .

Thus the local estimator at spacecraft only needs to estimate its own state, and . The measurement update of FOEKF at spacecraft is as follows:

Compared with FOEKF, the computational load is reduced remarkably in ROEKF, because ROEKF only updates the state of local spacecraft. However, as only estimates instead of real states of other spacecrafts could be obtained, the errors of state estimation of other spacecrafts are not considered in ROEKF, rendering a further precision loss than FOEKF. We could use a proper increase to the measurement noise, the matrix, to absorb such errors. This leads to the increased extended Kalman filter (IREKF) [5].

###### 3.3.2. IREKF

Considering the measurement , the measurement noise caused by using estimates of the state of spacecraft could be estimated by the following equation:

Notice that only estimates of the states of other spacecrafts could be obtained, which means that we could only use to approximate . Thus the increase to the measurement noise of is given by .

##### 3.4. ICEKF

Iteration is a straight forward method to compensate the precision loss caused by approximations. ICEKF could serve as a general scheme to improve FOEKF and ROEKF. In ICEKF, the local estimation at each spacecraft is passed to other spacecrafts for the linearization of measurement functions through a cascade communication chain. Figure 1 gives an example of ICEKF for a three-spacecraft formation compared with EKF. Although ICEKF reduces the computational load while obtaining a better precision than FOEKF or ROEKF, the introduction of iteration will result in an increase in the communicational load.