Research ArticleUnscented Particle Smoother and Its Application to TransferAlignment of Airborne Distributed POS

Xiaolin Gong ,1,2 Xiaorui Zheng,1 Xing-Gang Yan ,3 and Zhaoxing Lu 1,4

1School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing 100191, China2Science and Technology on Inertial Laboratory, Beijing 100191, China3School of Engineering and Digital Arts, University of Kent, Canterbury CT2 7NT, UK4Xi’an Institute of Hi-tech, Xi’an 710025, China

Correspondence should be addressed to Xiaolin Gong; gongxiaolin@buaa.edu.cn

Received 30 January 2018; Accepted 6 August 2018; Published 30 September 2018

Academic Editor: Hikmat Asadov

Copyright © 2018 Xiaolin Gong et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

This paper deals with the problem of state estimation for the transfer alignment of airborne distributed position and orientationsystem (distributed POS). For a nonlinear system, especially with large initial attitude errors, the performance of linearestimation 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 measurementsare first processed by the forward unscented particle filter (UPF) and then a backward smoother is used to achieve theimproved solution. The performance of this algorithm is compared with that of a similar smoother known as the unscentedRauch-Tung-Striebel smoother. The simulation results show that the UPS effectively improves the estimation accuracy andthis work offers a new off-line transfer alignment approach of distributed POS for multiantenna synthetic aperture radar andother airborne earth observation tasks.

1. Introduction

Airborne synthetic aperture radar (SAR) is an important toolfor earth observation. Multiantenna SAR can achieve higheraccuracy of interferometric processing and ground movingtarget indication through multibaseline interferometry tech-nology (i.e., interferometric SAR), and it has become animportant research direction of radar remote sensing [1].Usually, these antennae are installed on both sides of thewing of the aircraft, and the motion parameters of eachantenna are needed to complete the motion compensation[2]. As a result, airborne distributed position and orientationsystem (distributed POS) is developed, which is used toprovide the time and space reference information for allpoints of multiantenna SAR or other requirements [3].

Airborne distributed POS is generally composed offour parts: a high-precision POS (main system), a fewlow-precision inertial measurement units (IMU, subsys-tem), a POS computer system (PCS), and postprocessing

software [4, 5]. The high precision main system is usuallyinstalled in the cabin or underpart of the aircraft and per-forms the inertial/satellite integrated estimation with theposition and velocity of global navigation satellite system(GNSS) as the measurement information. The subsystem,which is placed as close as possible to the measurementcenter of the load, is used to measure the accelerationand angular velocity which are sent to the PCS. In thePCS, the high precision motion information of main systemis used as the reference signal to achieve the motion informa-tion of subsystem by transfer alignment technology. Thenature of transfer alignment is to estimate and compensatethe calculation error of motion parameters of the subsystemby using the high precision information of the main sys-tem [6]. It shows that the estimation method is crucialfor distributed POS.

It should be noted that distributed POS is a nonlinearsystem. In some emergency situations, the distributed POSis expected to try to shorten the preparation time on the

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 3898734, 13 pageshttps://doi.org/10.1155/2018/3898734

ground and even start up in flight, which will bring theuncertain errors and the misalignment of initial attitude isnot small. These make the nonlinearity of the system to befurther increased. Thus, the performance of transfer align-ment based on the linear model and linear estimationmethod will degrade [7].

The unscented Kalman filter (UKF) is a typical nonlinearfiltering method. It uses a deterministic sampling approachnamed as sigma points to propagate nonlinear systemsand has been discussed in many literatures for the inertial/satellite integration system [8–10]. UKF is a representativefilter to overcome the flaw of truncated error in extendedKalman filter (EKF) [11]. However, the UKF method is anapproximation of linear minimum variance estimationand does not apply to general non-Gaussian distributions[12, 13]. Another popular solution strategy for general fil-tering problems is particle filter (PF) which is based on therecursive Bayesian estimation and suitable for the nonlin-ear and non-Gaussian distributed system [14]. The esti-mated accuracy of PF is higher than the accuracy ofusing EKF or UKF alone [7]. PF uses a set of samplingpoints extracted from the posterior probability to expressits distribution. Choosing a reasonably recommendedprobability density is the core of PF. The closer the recom-mended density selected is to the true density, the betterthe filter effect is. Otherwise, it is worse or even divergent[7, 15]. The UKF is able to more accurately propagate themean and covariance of the Gaussian approximation tothe state distribution than the EKF. So PF is combinedwith UKF and formed a new PF called unscented particlefilter (UPF) [15], in which the UKF is used for proposaldistribution generation and the performance of the filteringalgorithm is improved by taking into account the influenceof the measured values on the state estimation.

The above UKF and PF are based on the idea of for-ward filtering, where only the observation information ofthe current moment and the previous time are used. Theother methods, such as smoothing estimation, can makefull use of all the observation information to estimate thestate of each moment, and is particularly suitable for off-line data processing. Since the smoother uses more observa-tions than the filter, although it cannot be used in real time,the accuracy of the optimal smoothing algorithm is theoret-ically higher than that of the KF [16]. The unscentedRauch-Tung-Striebel smoother (URTSS) was first proposedby Särkkä in 2008 [17]. This smoother takes the benefitover unscented transformation as well as the smoothingiterative characteristics, which makes the estimation accu-racy much better than that of the UKF. In ref. [18], theURTSS is applied to the off-line integrated estimation ofthe inertial/satellite integration system and shows an obvi-ous accuracy advantage over extended R-T-S smoother inattitude estimation. In particle smoothing, ref. [19] pro-posed a particle smoothing algorithm which is a directextension of the PF. This method has the advantage of fixedinterval smoothing, but it cannot express the state variablesin the past very well. Then, the forward-backward smootherand two-filter smoother based on PF are proposed [20].This kind of smoothing algorithm needs to carry on the

particle filter first and store the particles with weightselected in the approximate process and then perform thesmoothing process. However, there is still the problem ofparticle degeneracy.

For the airborne earth observation system, there aretwo modes of image processing: real time and off-line. Inthe off-line cases, there is no high requirement for fastdata processing in real-time imaging mode. The smoothercan be employed to provide a better solution for off-lineimage processing. Motivated by these, a smoother calledunscented particle smoother (UPS) is proposed in thispaper, which is then applied to the off-line transfer alignmentof airborne distributed POS. The UPS is made by taking theadvantages of the UKF, PF, and smoothing. In the UPS,UKF is used to generate the proposed distributions to over-come the shortage of particle degradation in PF. Then, theforward-backward smoother using all observations is furthercombined together to obtain higher estimation accuracy. TheUPS can be applied to nonlinear and non-Gaussian noisesystems as well. There is no need to keep still to obtain theinitial motion information of the main system and subsystembefore flying off.

The rest of this paper is organized as follows. InSection 2, the UPS are proposed. In Section 3, thedesign of UPS for airborne distributed POS is given indetail. In Section 4, the performance of the UPS methodis demonstrated and compared with that of the URTSS inestimation accuracy by a simulation test. Finally, conclusionsare given in Section 5.

2. Unscented Particle Smoothing Algorithm

The proposed UPS has two structures: a forward filterand a backward smoother. The forward filter is UPFused to obtain the filtering state estimation. In UPF,the proposed density is determined by UKF, whichnot only solves the problem of the degradation of par-ticles but also enables particles to get the latest a pos-teriori information of the measurement vector whenthey are updated, which is helpful for particles to movetoward the area with high likelihood. Then, the back-ward smoother is conducted after the forward filter tomodify the importance weights to achieve the smoothedstate estimation.

Suppose the discrete form of the nx-state system equationand the observation equation is as follows:

xk = f xk−1, k − 1 + Γk−1wk−1,

yk = h xk, k + vk,1

where xk ∈ Rnx represents the state vector of the system attime k and yk ∈ R

ny denotes the measurement vector at timetk. The functions f and h describe state and measurementmodels, respectively [8, 21]; Γk−1 is the state noise distribu-tion matrix, and wk−1 and vk represent the state and mea-surement noises, respectively, which can be described bythe corresponding probability density functions p wk−1

2 International Journal of Aerospace Engineering

and p vk . In this work, wk−1 and vk are assumed to theuncorrelated white Gaussian noise. The mean value ofwk−1 and vk is zero and their variance matrices are Qk−1and Rk, respectively [8, 22].

Based on (1), the process of UPS is describedas follows.

Step 1. Initialization (k = 0) [7, 15]. Suppose the initial statevariable x0 ~ p x0 , and the covariance matrix is P0.

For i = 1, 2,… ,N , draw the states (particles) x i0 from the

prior probability density p x0 and set x i0 ~N x i

0 , P i0 ,

where x i0 = x i

0 and P i0 = P0. N is the number of

particles.

Step 2. Importance sampling step (for i = 1, 2,… ,N ,k = 1, 2, 3,… , T).

(1) Update the particles with the UKF:

(i) Calculate sigma points

χik−1 0 = x

ik−1,

χik−1 j = x i

k−1 + nx + λ P ik−1

j

 j = 1, 2,… , nx ,

χik−1 j = x i

k−1 − nx + λ P ik−1

j

 j = nx + 1, nx + 2,… , 2nx,2

where the sigma point set S = χ ik−1 j ,W

l′j ;

j = 0, 1,… , 2nx, l′ ∈ m, c is composed of

the sigma points χ ik−1 j and their respective

mean (m) and covariance (c) weights W l′j .

λ = α2 nx + κ − nx denotes a scaling parame-ter. The parameter α determines the spreadof samples that is usually set between 10−4and 1. The constant κ is typically set to 0or 3 − nx where nx is the state dimension.The corresponding weights are given by:

W m0 =

λ

nx + λ,

W c0 =

λ

nx + λ+ 1 − α2 + β ,

W mj =W c

j =0 5

nx + λ j = 1, 2,… , 2nx,

3

where the parameter β is used to incorpo-rate prior knowledge of the distribution (forGaussian distribution, β = 2 is optimal).

(ii) Time update (propagate particle into future)

χik∣k−1 j = f χ

ik−1 j , k − 1 ,

x ik∣k−1 = 〠

2nx

j=0W m

j χik∣k−1 j ,

P ik∣k−1 = 〠

2nx

j=0W c

j χik∣k−1 j − x i

k∣k−1

χik∣k−1 j − x i

k∣k−1T+ Γk−1QkΓTk−1,

γik∣k−1 j = h χ

ik∣k−1 j , k ,

y ik∣k−1 = 〠

2nx

j=0W m

j γik∣k−1 j , 4

where P ik∣k−1 denotes the prior error covariance.

(iii) Measurement update (incorporate new obser-vation)

P iykyk

= 〠2nx

j=0W c

j γik∣k−1 j − y i

k∣k−1

γ ik∣k−1 j − y i

k∣k−1T+ Rk,

P ixkyk

= 〠2nx

j=0W c

j χik∣k−1 j − x i

k∣k−1

γik∣k−1 j − y i

k∣k−1T,

K ik = P i

xk ykP iyk yk

− 1,

x ik = x i

k∣k−1 + K ik yk − y i

k∣k−1 ,

P ik = P i

k∣k−1 − K ik Pykyk

K ik

T, 5

where y ik∣k−1 denotes the predicted measurement,

P iyk yk

denotes the covariance of predicted mea-

surement error, and P ixkyk

denotes the cross-

covariance matrix between x ik∣k−1 and y i

k∣k−1.

K ik denotes the filter gain.

3International Journal of Aerospace Engineering

(2) Importance sampling step.

For i = 1, 2,… ,N , sample x ik ~ q x i

k ∣ x i0 k−1, y1 k =

N x ik , P i

k and set x ik ≜ x i

0 k−1, xik and P i

0 k ≜P i0 k−1, P

ik .

(3) Update and normalize the importance weights.

For i = 1, 2,… ,N , evaluate the importance weightsup to a normalizing constant:

w ik =w i

k−1

p yk ∣ xik p x i

k ∣ x ik−1

q x ik ∣ x i

0 k−1, y1 k

6

For i = 1, 2,… ,N , normalize and store the impor-tance weights:

w ik =w i

k 〠N

i=1w i

k

−1

7

Compute the filtering state estimation xFk :

x Fk = 〠

N

i=1w i

k x ik 8

Step 3. Using the importance weights stored in the forwardfiltering to conduct the backward smoothing recursion.

Set w iT∣T =w i

T∣T . For k = T − 1,… , 1 and i = 1, 2,… ,N ,evaluate

w ik∣T =w i

k+1∣T 〠N

j=1w j

k+1∣T

p x jk+1 ∣ x

ik

∑Nm=1w

mk p x j

k+1 ∣ xmk

9

(1) Resampling

For x ik (i = 1, 2,… ,N), Using sequential impor-

tance resampling (SIR) [15] to complete resam-

pling to obtain N random samples x′ i0 k, and set

w ik =w i

k = 1/N .

(2) Compute the smoothed state estimation xSk

x Sk =

1N〠N

i=1x′ ik 10

The principle diagram of UPS is shown inFigure 1. In summary, there are two processes: a for-ward filtering process and a backward smoothingprocess. The measurements are first processed by

the forward filter and the importance weight w ik

should be stored in the forward filter stage. Andthen, a separate backward smoothing pass is usedto modify the importance weights for obtaining thesmoothing solution.

3. UPS Design for Airborne Distributed POS

The design of UPS for airborne distributed POS is givenin this section. Firstly, the definition of coordinate framesis introduced. i and e denote the earth-center inertialframe and the earth-centered earth-fixed frame, respec-tively. The navigation frames of the main system and sub-system are both topocentric frames (the z-axis parallel tothe upward vertical, the x-axis pointing eastward, andthe y-axis pointing northward), represented by n and ns,respectively. The body frames of the main system and sub-system are denoted by b and bs. A detailed description ofthese coordinate frames is available in [18, 23].

Figure 2 show the block diagram of transfer alignment indistributed POS based on UPS.

In Figure 2, the strapdown navigator calculates theattitude (heading, pitch, and roll), velocity, and positionexpressed in ns-frame by using the angular rates and

tk + 1tk + 1|kTime

Forward filter(UPF)

Backward smoother

tk

ˆ Sxkˆ

~

Sxk + 1

ˆ (i)xk + 1|k

ˆ (i)xk + 1 ,

(i)wk|T

(i)wk

~ˆ (i)xk ,

ˆ (i)xk|k − 1

(i)wk − 1

tk|k − 1

Figure 1: The principle diagram of UPS.

4 International Journal of Aerospace Engineering

Transfer alignment based on UPS

Angular rates

Accelerations Time

AttitudeVelocity

Time

Navigatorcorrection

Filtercorrections

Estimatederrors

Strapdownnavigator

Closed-looperror controller

Backwardsmoothing

PositionVelocityAttitude

Subsystem

Mainsystem

Forwardfiltering

Feed forwarderror controller

Lever armcompensation

Improved position,velocity, and

attitude

Imagingsensor

Figure 2: The block diagram of transfer alignment in distributed POS based on UPS.

accelerations of subsystem. Since the strapdown navigation isan integration process, any errors of the accelerometers andgyroscopes will integrate into slowly growing errors onvelocity, position, and attitude. The differences of velocityand attitude between the main system and the strapdownnavigator are used in the forward filtering to estimate thegrowing velocity, position, and attitude errors in the strap-down navigator. The closed-loop error controller is used tocorrect the result of strapdown navigator by using theestimation of forward filtering. After the end of the forwardfilter, the backward smoother is performed to compensatethe estimation of forward filtering, and the improved estima-tion results are used to correct the result of strapdownnavigator in the feed forward error controller. The improvedattitude, velocity, and position are outputted to the imagingsensor at last.

3.1. Nonlinear Inertial Navigation Error Model. The inertialnavigation error equations are the basis of the mathematicalmodel for transfer alignment. According to the definitionsabove, the nonlinear SINS error model of the subsystembased on angle error, which includes attitude error equation,velocity error equation, position error equation, and inertialsenor constant error equation, is given by [24]:

ϕns = I − Cpns

ωnsins

+ δωnsins

−Cnsbsεbs ,

δVns = I −Cnsp f ns − 2δωns

ie + δωnsens

×Vns − 2δωnsie + δωns

ens× δVns + Cns

b ∇bs ,

δL =δVN

RM +H−

VN

RM +H 2 δH,

δλ =sec LRN +H

δVE +VE sec L tan L

RN +HδL −

VE sec LRN +H 2 δH,

δH = δVU,

εbsc = 0,

∇bsc = 0, 11

where ϕns = ϕE ϕN ϕUT denotes the attitude error

vector in ns-frame, the subscripts E,N, andU denote the east,north, and up components, respectively; ωns

insdenotes the

rotation velocity of the ns-frame relative to the i-frameexpressed in ns-frame with error δω

nsins; Cns

bsdenotes the

coordinate transformation matrix from the bs-frame tothe ns-frame; εbs denotes the gyro drift of subsystem in

bs-frame, which consists of random constant drift εbsc and

the Gaussian white noise ωbsε with ε

bsc = ε

bsx ε

bsy ε

bsz

Tand

ωbsε = ω

bsεx ω

bsεy ω

bsεz

T; Vns = VE VN VU

T denotes the

velocity vector in ns-frame with error δVns =δVE δVN δVU

T; f ns = f E f N f UT denotes the

specific force measured by the accelerometers of subsystemexpressed in ns-frame; ωns

ie denotes the rotation velocity ofthe e-frame relative to the i-frame expressed in ns-frame witherror δωns

ie ; ωnsens denotes the rotation velocity of the ns-frame

relative to the e-frame expressed in ns-frame with error δωnsens ;

∇bs denotes the accelerometer bias of subsystem in bs-frame,

which consists of random constant bias ∇bsc and the

Gaussian white noise ωbs∇ with ∇

bsc = ∇bs

x ∇bsy ∇bs

zTand ω

bs∇ =

ωbs∇xωbs∇yωbs∇z

T; secL=1/cosL. The symbols δL, δλ, and δH

denote the latitude error, longitude error, and height error,respectively; and RM and RN denote the meridian andtransverse radius of curvature, respectively.

The nonlinear terms of attitude and velocity error equa-tions are I −Cp

nsωnsins

and I − Cnsp f ns in (11), respectively,

and the nonlinear matrix Cpns= Cns

pTis expressed as follows:

5International Journal of Aerospace Engineering

3.2. Model of Rigid Misalignment Angle and Flexure Angle.During the transfer alignment, there are flexure angles andrigid misalignment angles between the main system and thesubsystem. These angles cannot be measured accurately andthe flexure angles vary with the time. Therefore, the flexureangles and rigid misalignment angles should be modeledand estimated.

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

ρj = 0,

θj + 2βjθj + β2j θj = ηj j = x, y, z,

13

where ρj denotes the rigid misalignment angle of subsystem;θ j denotes the flexure angle and described by the second-order Markov process [25]; βj = 2 146/τj and τj is the corre-lation time; x, y, and z are the axes of bs-frame; ηj denotes the

Gaussian white noise with covariance Qη j= 4β3

jσ2j ; and σ2j

denotes the covariance of θ j; Qη = QηxQηy

Qηz

T.

3.3. Design of the Smoother for Transfer Alignment ofDistributed POS

3.3.1. State Equation. On the basis of models established inSubsections 3.1 and 3.2, let

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

x = f x, t +G t WI t , 15

where the nonlinear function f x, t is determined byerror model of transfer alignment in Subsections 3.1and 3.2, the vector WI t represents the system noise.

W1 = ωbsεx ω

bsεy ω

bsεz ω

bs∇x

ωbs∇y

ωbs∇z

ηx ηy ηzT

is

the zero-mean Gaussian white noise vector with covarianceQI which consists of covariance Qε of gyro random drift,covariance Q∇, of accelerometer random bias, and Qη.

3.3.2. Measurement Equation. The differences of velocityand attitude between the main system and the subsystemsolutions are considered as the measurement vector. Themeasurement vector can be given as

y t = δψ δθ δγ δVE′ δVN′ δVU′T, 16

where δψ, δθ, and δγ denote the differences of headingangle, pitch angle, and roll angle between the main system

and the subsystem; δVE′, δVN′ , and δVU′ denote the differ-ences of the east, north, and up velocities between themain system and the subsystem accounting for the leverarm between the main system and subsystem [25].

The measurement vector can be expressed linearly by thestate vector shown in (14). Hence, the linear measurementmodel can be given by

y t =Hx t + v t , 17

where the measurement noise v t = vδψ vδθ vδγ vδVE′

vδVN′vδVU′

T is the white noise vector of the solution of the

main system with zero mean and covariance matrix R. Themeasurement matrix H t can be given as

H t =H1 03×3 03×9 H2 H3 03×303×3 I3×3 03×9 03×3 03×3 03×3 6×24

18

Cpns=

cos ϕN cos ϕU − sin ϕE sin ϕN sin ϕU cos ϕN sin ϕU + sin ϕE sin ϕN cos ϕU −cos ϕE sin ϕN

−cos ϕE sin ϕU cos ϕE cos ϕU sin ϕE

sin ϕN cos ϕU + sin ϕE cos ϕN sin ϕU sin ϕN sin ϕU − sin ϕE cos ϕN cos ϕU cos ϕE cos ϕN

12

x = x1 x2T,

x1 = ϕE ϕN ϕU δVE δVN δVU δL δλ δH εbsx εbsy εbsz ∇bsx ∇bs

y ∇bsz

T,

x2 = ρx ρy ρz θx θy θz θx θy θzT

14

6 International Journal of Aerospace Engineering

Here, the attitude matrix of main system Cnb is denoted as

Ta. Ta, H1, H2, and H3 are as follows:

where φm, θm, and γm denote the heading, pitch, and roll ofthe main system, respectively.

The UPS for transfer alignment of airborne distrib-uted POS postprocessing can be summarized into twoparts: the forward filtering solution and the backwardsmoothing solution.

Figure 3 shows the data flaw of UPS used in airbornedistributed POS and the relation between the state vectorestimation of forward filtering xFk and backward recursionxSk at time k and k + 1. In forward filtering solution, the filterresults are applied to the nominal trajectory during each fil-tering step and will be reset to zero after the correction ofstrapdown navigation (velocity, position, and attitude). Thefunction of the backward smoothing solution is to compen-sate for the estimation of the forward filter stored duringthe forward filtering process. The filtering and smoothingare calculated at the main system rate. And after each

estimation points, a strapdown inertial navigation algorithmis performed to provide the navigation information.

4. Simulation and Analysis

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

4.1. Design of Simulation. In this subsection, a typical “S+U”shaped flight trajectory of airborne earth observation isdesigned. Figure 4 shows the plane trajectory and trajectoryparameters, respectively. In Figure 4, the AB and CD segmentcan be considered as the imaging segment.

Cnb =

cos γm cos φm − sin γm sin θm sin φm −cos θm sin φm sin γm cos φm + cos γm sin θm sin φm

−cos γm sin φm + sin γm sin θm cos φm cos θm cos φm sin γm sin φm − cos γm sin θm cos φm

−sin γm cos θm sin θm cos γm cos θm

, 19

H1 =

Ta12 Ta

32

Ta12 2

+ Ta22 2

Ta22 Ta

32

Ta12 2

+ Ta22 2 −1

−Ta

22

1 − Ta32 2

Ta12

1 − Ta32 2

0

Ta21 Ta

33 − Ta31 Ta

23

Ta33 2

+ Ta31 2

Ta31 Ta

13 − Ta11 Ta

33

Ta33 2

+ Ta31 2 0

,

H2 =H3 =

Ta12 Ta

23 − Ta13 Ta

22

Ta12 2

+ Ta22 2 0

Ta11 Ta

22 − Ta12 Ta

21

Ta12 2

+ Ta22 2

Ta33

1 − Ta32 2

0 −Ta

31

1 − Ta32 2

−Ta

31 Ta32

Ta33 2

+ Ta31 2 1 −

Ta32 Ta

33

Ta33 2

+ Ta31 2

20

7International Journal of Aerospace Engineering

Parameters of flight trajectory are listed as follows: initiallatitude is 40°, longitude is 116°, and height is 500m; initialflight velocity is 100m/s; and initial heading angle, pitchangle, 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 turns60° anticlockwise (100 seconds); and thirdly, the aircraft flies400 seconds at constant velocity, finally turns 180° clockwise(100 seconds), and continually flies 400 seconds at constantvelocity. Total flight time is 1300 s.

4.1.1. Data Generation. A trajectory generator is used togenerate the theoretical data of the scheduled flight trajec-tory, which includes position, velocity, attitude, and theoutput data of gyros and accelerometers. The real outputsof the main system are obtained by adding the correspondingmeasurement noise to the theoretical position, velocity, andattitude. Then, the theoretical outputs of gyros and acceler-ometers are converted by rigid misalignment angles andflexure angles, and the constant noise and random noise areadded to be the inertial sensor outputs of the subsystem.Meanwhile, the motion parameter benchmarks of subsystemcan be obtained through transforming theoretical position,velocity, and attitude by flexure angle.

4.1.2. Parameters of the Main System and Subsystem. Themeasurement noises of the main system at heading, pitch,roll, and velocity are 0 005°(1σ), 0 0025°(1σ), 0 0025°(1σ), and 0 005m/s (1σ), respectively. Both gyro constantdrift and random drift of subsystem are 0 01°/h. Both

accelerometer constant bias and random bias of the sub-system are 50μg.

4.1.3. Other Parameters. The rigid misalignment angle ofthe subsystem relative to the main system is ρ =0 5∘ 0 5∘ 0 5∘ T, and the lever arm between the mainsystem and subsystem is rbm = 5m 0 1m 0 1m T. Formultiantenna SAR, these antennae are fixed under bothsides of the wing. So the flexure angle rotated aroundthe vertical axis is large, while the flexure angles aroundthe other two axes are small. Accordingly, the correlationtime of the second-order Markov processes is selected as2 s, 5 s, and 2 s, and the covariances of flexure angles are0.01, 0.15, and 0.01, respectively. The curves of flexureangle and flexure angle rate selected in this simulationare shown in Figures 5 and 6. All the initial attitude errorsare 40∘. The particle number is 100. The data update rateof the subsystem is 100Hz and the filter frequency is 1Hz.

4.2. Simulation Results and Analysis. The differences betweentheoretical motion parameters of subsystem and theirestimates obtained from estimation, called estimate errors,are used to assess and compare the performance of UPSand URTSS.

Figures 7–10 show the estimate error curves of UPS andURTSS, including the estimate errors of attitude, velocity,position, and baseline. The means of root mean square error(RMSE) and standard deviation (STD) values of motioninformation estimate errors in imaging segments AB and

116 116.2 116.4 116.6 116.840

40.1

40.2

40.3

40.4

40.5

Longitude (°)

Latit

ude (

°)

A

B

D

C

Figure 4: Plane trajectory with S-shaped maneuver plus U-shaped flight.

Filter solution

True trajectory Smoother solution

Strapdown navigation

k k + 1

Strapdown navigation Strapdown navigation Time

ˆ Sxk + 1ˆ F

xk + 1ˆ Sxkˆ F

xk

Figure 3: The data flaw of UPS used in airborne distributed POS.

8 International Journal of Aerospace Engineering

CD are calculated and shown in Table 1. The calculationformulas of RMSE and STD are given as follows:

RMSE =∑n′

i=1 ei2

n′,

STD =∑n′

i=1 ei − e 2

n′,

21

where ei denotes the ith estimate error, n′ denotes thenumber of estimate error, and e denotes the mean of allestimate error.

For attitude estimation, from Figure 7 and Table 1, wecan see that there is a large and obvious trend item in theheading estimation error of URTSS, and the estimationerror of URTSS on pitch and roll is prone to be affectedby maneuver and their value are bigger than UPS evenafter 180° clockwise. Compared with URTSS, when the

0 40,000 80,000 130,000Time (0.01 s)

0 40,000 80,000 130,000Time (0.01 s)

0 40,000 80,000 130,000Time (0.01 s)

−0.2

0

0.2

x-a

xis fl

exur

ean

gle r

ate (

°/s)

−2

0

2

y-a

xis fl

exur

ean

gle r

ate (

°/s)

−0.2

0

0.2

z-a

xis fl

exur

ean

gle r

ate (

°/s)

Figure 6: Flexure angle rate on x-axis, y-axis, and z-axis.

0 40,000 80,000 130,000

−0.2

0

0.2

Time (0.01 s)

0 40,000 80,000 130,000Time (0.01 s)

0 40,000 80,000 130,000Time (0.01 s)

x-a

xis

flexu

re an

gle (

°)

−4−2

024

y-a

xis

flexu

re an

gle (

°)

−0.2

0

0.2

z-a

xis

flexu

re an

gle (

°)

Figure 5: Flexure angle on x-axis, y-axis, and z-axis.

9International Journal of Aerospace Engineering

initial misalignment angle is 40°, the attitude estimation errorof UPS is significantly reduced and the error curves are morestable both on AB and CD segments; the estimation accuracy

on heading and pitch of UPS is obviously better than that ofURTSS, and only the STD on roll of UPS is slightly lowerthan that of URTSS.

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

−0.5

0

0.5

East

velo

city

estim

ate e

rror

(m/s

)

−0.50

0.51

1.5

Nor

th v

eloc

ityes

timat

e err

or (m

/s)

−0.4

−0.2

0

0.2

Up

velo

city

estim

ate e

rror

(m/s

)

Time (s)

URTSSUPS

Figure 8: Velocity estimate errors of UPS and URTSS.

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

−20

0

20

Hea

ding

estim

ate

erro

r (°)

Roll

estim

ate

erro

r (°)

Time (s)

−5

0

5

−5

0

5

Pitc

h es

timat

eer

ror (

°)

URTSSUPS

Figure 7: Attitude estimate errors of UPS and URTSS.

10 International Journal of Aerospace Engineering

For velocity estimation, Figure 8 and Table 1 showthat the estimation accuracy on east velocity and northvelocity of UPS is better than that of URTSS and theerror curves of URTSS are prone to be affected bymaneuver, especially in the initial ‘S’ shape flight, theestimation error of velocity of URTSS is unstable andvery large; the estimation accuracy on up velocity ofUPS is lower than that of URTSS.

For position estimation, Figure 9 and Table 1 show thatthe estimation accuracy on the latitude and longitude ofUPS is better than that of URTSS, and the estimation accu-racy on the height of UPS is lower than that of URTSS, whichcorresponds to the up velocity error result. The estimationerror of latitude and longitude of URTSS in the initial ‘S’

shape flight (from 0 s to 300 s) is very large due to the largevelocity error in this stage.

Besides, the space distance between the main system andsubsystem (i.e., the baseline between two antennas of SAR) isa very important parameter for interference precision ofinterferometric SAR and the higher the measurementaccuracy of the baseline, the higher the object’s digitalelevation model accuracy of the interferometric SAR. Here,the baseline error is calculated by theoretical position andreal position output of the main system and subsystem.From Figure 10 and Table 1, we can see that the baselineerror curve of UPS is more stable and the baselineestimation accuracy of UPS is better than that of URTSSboth on RMSE and STD.

0 200 400 600 800 1000 1200 1400−25

−20

−15

−10

−5

0

Base

line

estim

ate e

rror

(m)

Time (s)

URTSSUPS

0 200 400 600 800 1000 1200 1400−1.5

−1−0.5

00.5

Partial magnification

Figure 10: Baseline estimate errors of UPS and URTSS.

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

Time (s)

URTSSUPS

0

10

20

Latit

ude

estim

ate e

rror

(m)

0

10

20Lo

ngitu

dees

timat

e err

or (m

)

−0.5

0

0.5

Hei

ght

estim

ate e

rror

(m)

Figure 9: Position estimate errors of UPS and URTSS.

11International Journal of Aerospace Engineering

From the simulation result, we can see that the estimationaccuracy of motion parameters of the proposed method ishigher than that of the URTSS as a whole.

5. Conclusion

In this paper, a nonlinear smoother called UPS has beenproposed to deal with the off-line transfer alignment estima-tion of airborne distributed POS. In the UPS, UKF, PF, andsmoothing are combined together to provide the respectiveadvantages from each of them to improve estimation accu-racy. The validity of the proposed algorithm is verified bysimulation test and comparison with URTSS on the estima-tion accuracy of motion parameters and baseline. Thesimulation results show that UPS can improve estimationaccuracy and has more adaptability to the maneuver thanthat of URTSS. This method is a better choice for off-linetransfer alignment in distributed POS with large misalign-ment and will greatly improve the resolution of flexiblebaseline interferometric SAR imaging. The next work isthat the practical flight experiment based on long flexiblebaseline will be implemented to validate the performanceof this method.

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (Grant nos. 61473020 and61721091), International (Regional) Cooperation and Com-munication Project (Grant no. 61661136007), and Funda-mental Research Funds for the Central Universities.

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Table 1: Means of RMSE and STD values of estimate errors.

ParameterURTSS 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

12 International Journal of Aerospace Engineering

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