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 threestep 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 threesatellite formation is used to test the decentralized algorithms. Simulation results indicate that JT has the best precision among all current decentralized algorithms.
Formation flying has been recognized as an enabling technology for many future space missions [
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 [
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 [
This paper is arranged in 5 sections. In Section
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
The state of each spacecraft is defined as the Keplerian orbital elements. Let
Linearize (
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.
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
In general, the single platform measurement function may be written as
Linearize (
Similar to single platform measurement, let
In general, the single platform measurement function may be written as
Linearize (
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) [
Consider a formation of
Considering a local FOEKF estimator on spacecraft
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.
ROEKF is a further simplification of FOEKF. Taking spacecraft
Notice that
Thus the local estimator at spacecraft
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
Considering the measurement
Notice that only estimates of the states of other spacecrafts could be obtained, which means that we could only use
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
ICEKF compared with centralized EKF [
The decentralized solutions reviewed in the previous section are all based on distributing centralized EKF through simplifications or decoupling, rendering suboptimal estimations with inferior precisions to EKF. This section proposes a general scheme for the design of decentralized algorithms from another point of view: analyzing the hidden decentralized structure of EKF with the aid of probabilistic graphical models.
From the perspective of BE, the dynamic model and the measurement model used in EKF are all conditional Gaussian (CG), which could be interpreted by the probabilistic graphical model of DBN. Previous researches have demonstrated that inference on DBN could be distributed by message passing on a decentralized data structure named JT [
The algorithm of JT contains two parts: symbolic calculation and mathematical calculation. Figure
Flowchart of JT.
BE is the inference engine on JT. Let
There are three basic mathematic operations in BE: multiplication, division, and marginalization. Interested readers may refer to Appendix
In the probabilistic graph theory, DBN is used to visualize the relationship of conditional distribution between random variables. Since the dynamic model and the measurement model of EKF each define a CG distribution, the time update and the measurement update of EKF (or BE) may be represented by DBN.
Figure
Example of DBN.
DBN holds the joint distribution of all the variables in
JT is first introduced by Lauritzen [
The Kruskal algorithm and the Prim algorithm are the most common algorithms to generate JT. Interested readers may refer to [
The probabilistic distribution defined by a set of random variables is assigned as a potential to the first clique which contains the set. If a clique is assigned with more than one potential, the multiplication of these potentials is treated as the potential for the clique; if there is no distribution assigned to a clique, the potential of this clique is initialized as standard normal.
A measurement is deployed as an evidence to the first clique which contains all the variables related to the very measurement.
Figure
Example of JT.
Inference on JT is driven by a series of local computations on cliques and messages passing between adjacent cliques [
A message to a clique is defined as a distribution (or potential) passed from an adjacent clique. If clique
In fact, the operation of receiving message is a distributed implementation of the time update in centralized BE (or EKF).
Take clique
The operation of collecting evidence may be seen as a distributed implementation of the measurement update in centralized BE (or EKF).
After updating the potential of a clique based on messages and evidences, the next operation is to generate the message passed to the next clique. The message generated is based on the separator between adjacent cliques. Considering clique
Consider the example of JT in Figure
Spacecraft 1:
Spacecraft 2:
Spacecraft 3:
The message generated by spacecraft 3,
The effectiveness of the decentralized solution proposed is validated by the simulation of a threesatellite formation. The Keplerian elements of the formation satellites are listed in Table
Keplerian orbital elements of a satellite formation.
Sat number 







1  8400.000000  0.005000  30.000000  120.000000  0.000000  90.000000 
2  8400.000000  0.004743  29.996649  119.988151  1.783138  88.250782 
3  8400.000000  0.004743  30.003353  120.011847  1.762616  88.250782 
Relative coordinate system and ECI coordinate system. The relative coordinate system (or the Hill frame) is represented by
Relative motion between satellites.
Every spacecraft in the formation is equipped with a GPS receiver as well as RF range and communication devices. All noised are treated as Gaussian. The standard deviation for GPS measurement and RF range are 0.3 m and 0.1 m, respectively. The estimations of the absolute state of satellite 3 using different algorithms are listed as follows.
Centralized EKF is setup as a standard to evaluate the precision of other algorithms. As there are no further simplifications or decoupling, the precision of centralize EKF is better than all the decentralized solutions reviewed in Section
Error of absolute position estimation for satellite 3 using EKF.
All decentralized algorithms are compared with centralized EKF. For FOEKF, because only local measurements related with each satellite are considered, the simplification of measurement model will result in suboptimal estimations. Figure
Error of absolute position estimation of satellite 3 using FOEKF.
The core idea that differs ROEKF from FOEKF is that the states transferred from other spacecrafts are assumed to be precise, and therefore the interformation measurement functions could be decoupled. After decoupling, the measurement function is only related to the state of the local spacecraft. This assumption will result in a further precision loss of ROEKF compared with FOEKF. Figure
Error of absolute position estimation for satellite 3 using ROEKF.
Notice that the states transmitted from other spacecrafts are not precise, in other words, with an uncertainty represented by the covariance. In IREKF, such uncertainty is considered as a proper bumpup to the noise of measurements. Figure
Error of absolute position estimation for satellite 3 using IREKF.
A common method to compensate the precision loss caused by simplification or linearization is to use iterations. ICEKF is based on the same intuition. Figure
Error of absolute position estimation for satellite 3 using FOEKFbased ICEKF.
Figure
Error of absolute position estimation for satellite 3 using JT.
In order to provide an intuitive comparison of different algorithms, Table
RMSE of all algorithms at the end of simulation.
Algorithm  EKF  FOEKF  ROEKF  IREKF  ICEKFFOEKF  JT 

RMSE (m)  0.0248  0.0739  28.1360  0.0690  0.0310  0.0248 
While analyzing the computational complexity of algorithms, the operation of multiplication of two real numbers is defined as a unit element. Recall the procedure of centralized EKF in Section
The computational complexity of decentralized algorithms could be estimated using similar methods. For FOEKF, suppose
Because centralized BE is equivalent to centralized EKF, the total computational complexity of JT is the same as centralized EKF. For spacecraft
The decentralized state estimation algorithms for formation flying spacecraft studied in this paper could be categorized as two main groups: decomposition through simplification and decoupling and decomposition through structural analysis of models. In the first category, irrelevant measurements are ignored and interformation measurements are decoupled at each spacecraft, rendering suboptimal estimates with inferior precisions to centralized EKF. All algorithms reviewed in Section
Recall the dynamic model in Section
Recall the measurement model in Section
Consider two random vectors,
Equation (
The joint distribution of
If
If the joint distribution of
The authors declare no conflict of interests.
All authors contributed equally to this work.
This work was supported by the National Natural Science Foundation of China (Grant no. 61203200).