The Federated Filtering Algorithm based on the Asynchronous Multisensor System

Abstract: This paper is concerned about the asynchronous multisensor dynamic system and an asynchronous fusion algorithm is proposed, which is depended on the federated Kalman filter. Each sub filter is updated on the rate that is associated with the local sensors. Then outputs of sub filters are propagated to the next fusion time. These propagated values are to be combined with the time update value of the main filter and the fusion results are fed back to each sub filter. This algorithm effectively solves the problem that sub filters may not able to provide data to be fused at the fusion time for the asynchronous sensor system. The detailed process of the new algorithm is introduced. The simulation results for the satellite autonomous navigation system is given to demonstrate its validity.


Introduction
Data fusion is a new technology in the information science field.It uses multiple information sources or multiple sensors to get, process and synthesize information.The purpose is to get more accurate and more complete estimation and judgement [1,2].With the development of modern science and technology, the structure of various devices is more and more complex and it's common to use devices which have different sampling time and different sampling period.In particular, many navigation sensors work asynchronously because of their operation mode, range of action, inherent delay and system transmission delay.
Most researches about the multi sensor information fusion aim at the synchronous situation.But considering that many devices work asynchronously, it's necessary to research asynchronous information fusion algorithm.Alouani and Rice proposed a suboptimal asynchronous fusion algorithm in paper [3] based on the linear minimum mean square error.But this algorithm artificially rewrites the measurements and leads the noises in the new system to be correlated.Aiming at the shortcoming, in paper [4] Wang Jie proposed an asynchronous fusion algorithm focused on the measurement noises which are correlated after discretization.In paper [5] Wen Chenglin proposed a distributed prediction fusion algorithm.In a filtering period, all the sensors do time updating and measurement updating according to their respective measurements and do one step prediction to the fusion time after the last measurement updating.Then weighted fusion is implemented under the condition that the trace of the fusion variance matrix should be the minimum.However, in the algorithm, each sensor filters independently throughout and the result after fusion will not be used to correct the sub filters.
To deal with the asynchronous data fusion problem, a new algorithm based on the federated filter is discussed in the paper.Each sub filter is dedicated to a seperate sensor.In one filtering period, the estimated values of sub filters are obtained by the time updating and measurement updating depended on their own measurements respectively.Then the optimal state estimate of each sub filter is propagated from the last measurement time to the next fusion time.Finally, all the estimated values are combined with the predicated values of the main filter and the results are fed back to each sub filter.In this paper, the system description is presented in Section II, the algorithm is introduced in Section III, and the simulation result is given in Section IV.Section V contains conclusion.

System description
A satellite autonomous system composed by the ultraviolet sensor and the receiver of GNSS is considered.The direction vector pointing to the earth center and the apparent radius are provided by the ultraviolet sensor.The apparent radius is the included angle of the tangent between the edge of the earth and the ultraviolet sensor and the line between the earth's core and the ultraviolet sensor.The pseudo range and pseudo range rate are provided by the receiver of GNSS.These data are used as the measurement data.Based on the Earth Centered Inertial Frame, the measurement equation can be written as follows, arcsin ( ) ) where [ , , ] T x y z = r is the position vector from the satellite to the earth's core.
[ , , ] T s s s s x y z = r is the position vector from the earth's core to the navigation satellite.

[ , , , , ]
  is the observation noise.Considering that there may have more than one navigation satellite which is visible, the observation dimension is not fixed and depends on the number of the navigation satellite.
Let the state vector be [ , , , ] is the velocities of the satellite, u t δ and u f δ are the clock correction and cock drift.The effects of J2 perturbation and the solar and lunar gravitational perturbation are considered in the dynamical model.The state equation is where (3) The system process noise w , is the gaussian white noise with zero mean and covariance Q. s µ is the solar gravitational constant and m µ is lunar gravitational constant.
[ , , ] T x y z = r are the position vectors of the solar and lunar in the inertial coordinate system.

Algorithm description
The federated filtering algorithm is a decentralized processing algorithm which uses two stage parallel structure.The navigation devices can make up multiple subsystems and use a main filter to fuse the data of each subsystem.According to the principle of information distribution, by reasonably distributing the state information to the main filter and the sub filters, it's effective to eliminate the correlation of all the sub filters and make local estimations independently in the sub filters [6,7].The basic flow Figure is illustrated in Figure 1.The reference system is represented by the dynamic equation of the satellite.
When the sensors operate asynchronously, some sensor may not provide data at the fusion time.Aiming at solving the problem that the sampling time is not synchronize and the sampling period is not consistent, this paper designs a new algorithm named Federated Predictive Filtering Algorithm.Its core idea is to use the predicted values of the sub filter from last measurement updating time to fusion time to be the contribution values of the sub filter at fusion time.
Assuming that the number of the sub filters is N, which depends on the number of navigation sensors, and the filtering period is T.
During the time period , the number of measurements for the ith sub filter is represented by i n .Letting i n t be the time that the last measurement is taken.
The steps of the algorithm are as follows: (1) The sub filters and main filter are reset by the fused value at time k t , i.e., Where The following condition must be satisfied, Where the factor 1 N γ + is associated to the main filter.
(2) The process noise covariance values of the sub filters and the main filter are assigned by the common process noise covariance value.By variance amplification, it's effective to make local estimations independently in the sub filters, i.e., (3) The estimated values of (5) The predicted values of sub filters are combined with those of the main filter by the following fusion algorithm,

P t T t T t T t T P t T t t T t P t T t t T t P t T t t +T t
(5) Repeat the above steps.

Simulation
Based on the established model, the validity of the algorithm is tested by simulating and analyzing.The simulation time is set to 86400s and the filtering period is set to 0.9s.The sampling periods of the ultraviolet sensor and the receiver of GNSS are respectively set to 1.1s and 1s.The sensor specification is shown in the Table 1.The mean values of the error and rootmean square error(RMSE) are chosen to represent the algorithm precision.The simulation results of 50 Monte Carlo simulation are shown in Figure 2 to Figure 5.   From the above figures, it can be verificated that the accuracy of the asynchronous federated filtering algorithm can reach the accuracy of the synchronous federated filtering algorithm.It is shown that the new algorithm can usefully deal with the asynchronous problem and can still reach high accuracy.So the new asynchronous algorithm is feasible.

Conclusion
This paper designs a new asynchronous federated filtering algorithm based on an actual project.The new algorithm can be used in the case that the sampling time is not synchronize or the sampling period is not consistent.Through the simulation results based on the new algorithm and the synchronous algorithm, the conclusion can be gotten that the new asynchronous algorithm is feasible.

Figure 1 .
Figure 1.The basic flow Figure of federated filter algorithm

Figure 2 .Figure 3 .
Figure 2. The mean value of position error of 50 asynchronous simulation

Figure 4 .Figure 5 .Figure 6 .
Figure 4.The mean value of velocity error of 50 asynchronous simulation

Table 1 .
The sensor specification