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This paper deals with the problem of state estimation for the transfer alignment of airborne distributed position and orientation system (distributed POS). For a nonlinear system, especially with large initial attitude errors, the performance of linear estimation methods will degrade. In this paper, a nonlinear smoothing algorithm called the unscented particle smoother (UPS) is proposed and utilized in the off-line transfer alignment of airborne distributed POS. In this algorithm, the measurements are first processed by the forward unscented particle filter (UPF) and then a backward smoother is used to achieve the improved solution. The performance of this algorithm is compared with that of a similar smoother known as the unscented Rauch-Tung-Striebel smoother. The simulation results show that the UPS effectively improves the estimation accuracy and this work offers a new off-line transfer alignment approach of distributed POS for multiantenna synthetic aperture radar and other airborne earth observation tasks.

Airborne synthetic aperture radar (SAR) is an important tool for earth observation. Multiantenna SAR can achieve higher accuracy of interferometric processing and ground moving target indication through multibaseline interferometry technology (i.e., interferometric SAR), and it has become an important research direction of radar remote sensing [

Airborne distributed POS is generally composed of four parts: a high-precision POS (main system), a few low-precision inertial measurement units (IMU, subsystem), a POS computer system (PCS), and postprocessing software [

It should be noted that distributed POS is a nonlinear system. In some emergency situations, the distributed POS is expected to try to shorten the preparation time on the ground and even start up in flight, which will bring the uncertain errors and the misalignment of initial attitude is not small. These make the nonlinearity of the system to be further increased. Thus, the performance of transfer alignment based on the linear model and linear estimation method will degrade [

The unscented Kalman filter (UKF) is a typical nonlinear filtering method. It uses a deterministic sampling approach named as sigma points to propagate nonlinear systems and has been discussed in many literatures for the inertial/satellite integration system [

The above UKF and PF are based on the idea of forward filtering, where only the observation information of the current moment and the previous time are used. The other methods, such as smoothing estimation, can make full use of all the observation information to estimate the state of each moment, and is particularly suitable for off-line data processing. Since the smoother uses more observations than the filter, although it cannot be used in real time, the accuracy of the optimal smoothing algorithm is theoretically higher than that of the KF [

For the airborne earth observation system, there are two modes of image processing: real time and off-line. In the off-line cases, there is no high requirement for fast data processing in real-time imaging mode. The smoother can be employed to provide a better solution for off-line image processing. Motivated by these, a smoother called unscented particle smoother (UPS) is proposed in this paper, which is then applied to the off-line transfer alignment of airborne distributed POS. The UPS is made by taking the advantages of the UKF, PF, and smoothing. In the UPS, UKF is used to generate the proposed distributions to overcome the shortage of particle degradation in PF. Then, the forward-backward smoother using all observations is further combined together to obtain higher estimation accuracy. The UPS can be applied to nonlinear and non-Gaussian noise systems as well. There is no need to keep still to obtain the initial motion information of the main system and subsystem before flying off.

The rest of this paper is organized as follows. In Section

The proposed UPS has two structures: a forward filter and a backward smoother. The forward filter is UPF used to obtain the filtering state estimation. In UPF, the proposed density is determined by UKF, which not only solves the problem of the degradation of particles but also enables particles to get the latest a posteriori information of the measurement vector when they are updated, which is helpful for particles to move toward the area with high likelihood. Then, the backward smoother is conducted after the forward filter to modify the importance weights to achieve the smoothed state estimation.

Suppose the discrete form of the

Based on (

Initialization (

For

Importance sampling step (for

Update the particles with the UKF:

Calculate sigma points

Time update (propagate particle into future)

Measurement update (incorporate new observation)

Importance sampling step.

For

Update and normalize the importance weights.

For

For

Compute the filtering state estimation

Using the importance weights stored in the forward filtering to conduct the backward smoothing recursion.

Set

Resampling

For

Compute the smoothed state estimation

The principle diagram of UPS is shown in Figure

The principle diagram of UPS.

The design of UPS for airborne distributed POS is given in this section. Firstly, the definition of coordinate frames is introduced.

Figure

The block diagram of transfer alignment in distributed POS based on UPS.

In Figure

The inertial navigation error equations are the basis of the mathematical model for transfer alignment. According to the definitions above, the nonlinear SINS error model of the subsystem based on angle error, which includes attitude error equation, velocity error equation, position error equation, and inertial senor constant error equation, is given by [

The nonlinear terms of attitude and velocity error equations are

During the transfer alignment, there are flexure angles and rigid misalignment angles between the main system and the subsystem. These angles cannot be measured accurately and the flexure angles vary with the time. Therefore, the flexure angles and rigid misalignment angles should be modeled and estimated.

The models of rigid misalignment angles and flexure angles are shown in the following equations:

On the basis of models established in Subsections

Then, the continuous-time system model can be given by

The differences of velocity and attitude between the main system and the subsystem solutions are considered as the measurement vector. The measurement vector can be given as

The measurement vector can be expressed linearly by the state vector shown in (

Here, the attitude matrix of main system

The UPS for transfer alignment of airborne distributed POS postprocessing can be summarized into two parts: the forward filtering solution and the backward smoothing solution.

Figure

The data flaw of UPS used in airborne distributed POS.

In order to analyze the performance of the proposed UPS, numerical simulation is provided and the comparison with the URTSS using the simulated data of a distributed POS based on one flight trajectory is presented as well.

In this subsection, a typical “S + U” shaped flight trajectory of airborne earth observation is designed. Figure

Plane trajectory with S-shaped maneuver plus U-shaped flight.

Parameters of flight trajectory are listed as follows: initial latitude is 40°, longitude is 116°, and height is 500 m; initial flight velocity is 100 m/s; and initial heading angle, pitch angle, and roll angle are 40°, 0°, and 0°, respectively. Firstly, the aircraft flies 100 seconds at constant velocity; secondly, the aircraft turns 60° clockwise (100 seconds) and then turns 60° anticlockwise (100 seconds); and thirdly, the aircraft flies 400 seconds at constant velocity, finally turns 180° clockwise (100 seconds), and continually flies 400 seconds at constant velocity. Total flight time is 1300 s.

A trajectory generator is used to generate the theoretical data of the scheduled flight trajectory, which includes position, velocity, attitude, and the output data of gyros and accelerometers. The real outputs of the main system are obtained by adding the corresponding measurement noise to the theoretical position, velocity, and attitude. Then, the theoretical outputs of gyros and accelerometers are converted by rigid misalignment angles and flexure angles, and the constant noise and random noise are added to be the inertial sensor outputs of the subsystem. Meanwhile, the motion parameter benchmarks of subsystem can be obtained through transforming theoretical position, velocity, and attitude by flexure angle.

The measurement noises of the main system at heading, pitch, roll, and velocity are

The rigid misalignment angle of the subsystem relative to the main system is

Flexure angle on

Flexure angle rate on

The differences between theoretical motion parameters of subsystem and their estimates obtained from estimation, called estimate errors, are used to assess and compare the performance of UPS and URTSS.

Figures

Attitude estimate errors of UPS and URTSS.

Velocity estimate errors of UPS and URTSS.

Position estimate errors of UPS and URTSS.

Baseline estimate errors of UPS and URTSS.

Means of RMSE and STD values of estimate errors.

Parameter | URTSS | UPS | |||
---|---|---|---|---|---|

RMSE | STD | RMSE | STD | ||

Attitude error | Heading (°) | 3.7874 | 1.2914 | 0.4959 | 0.0217 |

Pitch (°) | 0.5166 | 0.1218 | 0.0766 | 0.0781 | |

Roll (°) | 0.8238 | 0.0371 | 0.0569 | 0.0568 | |

Velocity error | East (m/s) | 0.0264 | 0.0245 | 0.0133 | 0.0134 |

North (m/s) | 0.0403 | 0.0414 | 0.0225 | 0.0148 | |

Up (m/s) | 0.0125 | 0.0124 | 0.0224 | 0.0211 | |

Position error | Latitude (m) | 2.1444 | 0.0990 | 1.0077 | 0.0529 |

Longitude (m) | 1.7280 | 0.0878 | 0.0510 | 0.0495 | |

Height (m) | 0.1078 | 0.0987 | 0.1294 | 0.0126 | |

Baseline error | (m) | 0.7039 | 0.0532 | 0.0014 | 0.0008 |

For attitude estimation, from Figure

For velocity estimation, Figure

For position estimation, Figure

Besides, the space distance between the main system and subsystem (i.e., the baseline between two antennas of SAR) is a very important parameter for interference precision of interferometric SAR and the higher the measurement accuracy of the baseline, the higher the object’s digital elevation model accuracy of the interferometric SAR. Here, the baseline error is calculated by theoretical position and real position output of the main system and subsystem. From Figure

From the simulation result, we can see that the estimation accuracy of motion parameters of the proposed method is higher than that of the URTSS as a whole.

In this paper, a nonlinear smoother called UPS has been proposed to deal with the off-line transfer alignment estimation of airborne distributed POS. In the UPS, UKF, PF, and smoothing are combined together to provide the respective advantages from each of them to improve estimation accuracy. The validity of the proposed algorithm is verified by simulation test and comparison with URTSS on the estimation accuracy of motion parameters and baseline. The simulation results show that UPS can improve estimation accuracy and has more adaptability to the maneuver than that of URTSS. This method is a better choice for off-line transfer alignment in distributed POS with large misalignment and will greatly improve the resolution of flexible baseline interferometric SAR imaging. The next work is that the practical flight experiment based on long flexible baseline will be implemented to validate the performance of this method.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported in part by the National Natural Science Foundation of China (Grant nos. 61473020 and 61721091), International (Regional) Cooperation and Communication Project (Grant no. 61661136007), and Fundamental Research Funds for the Central Universities.