# Dynamic Lever Arm Error Compensation of POS Used for Airborne Earth Observation

Authors: Lu Zhaoxing; Fang Jiancheng; Gong Xiaolin; Li Jianli; Wang Shicheng; Wang Yun
Journal: International Journal of Aerospace Engineering
 (2018)

---

## Abstract

The position and orientation system (POS) is widely applied in airborne Earth observation, which integrates the strapdown inertial navigation system (SINS) and global positioning system (GPS) to provide high-accuracy position, velocity, and attitude information for remote sensing motion compensation. However, for keeping the appointed direction of remote sensing load, the inertial measurement unit (IMU) and remote sensing load will be driven to sweep by the servo machine. The lever arms among IMU, GPS, and remote sensing load will be time varying, and their influence on the measurement accuracy of POS is serious. To solve the problem, a dynamic lever arm error compensation method is proposed, which contains the first-level lever arm error compensations between IMU and GPS and the second-level lever arm error compensation between POS and remote sensing load. The flight experiment results show that the proposed method can effectively compensate the dynamic lever arm error and achieve high measurement accuracy for POS.


---

## Body

## 1. Introduction

Airborne Earth observation is a kind of high technology which makes the aircraft as the platform and uses remote sensing load to acquire a wide-range, high-accurate, and multilayered space-time information of global surface and deep Earth [1, 2]. The ideal motion for precise airborne Earth observation is moving along a straight line with constant speed. However, the aircraft will inevitably deviate remote sensing load from the ideal motion trajectory due to the effects of gust, air turbulence, and the error of flight control. The position and orientation system (POS) is becoming crucial as an important way for remote sensing motion compensation [3–5]. POS is a dedicated strapdown inertial navigation system- (SINS-)/global positioning system- (GPS-) integrated system and can provide navigation information with excellent short-term accuracy and long-term stable accuracy characters, which gather the advantages of both SINS and GPS [6, 7].As a multisensory system, POS is composed of inertial measurement units (IMUs) and global positioning system and needs to transfer the solutions to the measuring center of the remote sensing load. There exist some spatial differences among sensors. It makes the sensed motion information different from each other, which is called the lever arm effect. Without compensation, the lever arm effect will result in some serious measurement errors [8]. Over the past few years, many researches have been carried out to reduce the lever arm errors. The mechanization and compensation methods of the lever arm between IMU and GPS are analyzed in [9, 10]. In [11], multilevel lever arm errors have been analyzed and compensated, including the inner lever arm caused by the displacement of the sensing center of an accelerometer from the center of inertial sensing assembly, the first-level lever arm between IMU and GPS, and the second-level lever arm between POS and remote sensing load. However, the direct calibration of lever arm is not always available. In [12], the lever arm error is extended to be a state variable of the Kalman filter, and the observability properties are studied with a control-theoretic approach, which indicates that changes in angular velocity enhance the lever arm estimate and both the motions of translation and constant angular velocity have no influence on the estimation of the lever arm. In [13], both numerical and experimental studies on the estimation of the lever arm are presented. Based on the observability analysis, the estimation precision in different stages has been analyzed in [14], and smoothing methods including Rauch-Tung-Striebel smoother and two-filter smoother are developed for precise lever arm estimation. Besides, the lever arm between the IMU and odometer is also taken into account for the land-vehicle navigation system in [15]. The lever arm between the Locata antenna and IMU is compensated for the Locata/SINS-integrated system in [16]. In the application of transfer alignment, the lever arm effects among master IMU and slave IMUs are analyzed and compensated in [17–19].All above methods deal with the rigid lever arms. However, in the imaging process of airborne Earth observation, the appointed direction of remote sensing load is adjusted in real time in order to meet specific imaging needs. Consequently, the spatial relationship between IMU, GPS, and remote sensing load will change, which is named the dynamic lever arm effect. In [20, 21], the first-level dynamic lever arm between GPS and IMU caused by an inertially stabilized platform has been analyzed and compensated. For the second-level lever arm between POS and remote sensing load, there exist the installation angle errors, and the relative attitude between POS and remote sensing load will change with the rotation of remote sensing load. So the compensation method in [20, 21] is not suitable for the second-level dynamic lever arm. To solve this problem, a new algorithm which tackles the problem of not only the first-level dynamic lever arm but also the second-level dynamic lever arm for airborne Earth observation is proposed in the paper.The paper is organized as follows. The conventional rigid lever arm error compensation method is presented in Section2. The principle of both the first- and the second-level dynamic lever arm error compensation methods is derived in detail in Section 3. In Section 4, the experiment data of two flights is utilized to demonstrate the effectiveness of the proposed algorithm. Finally, Section 5 concludes the paper.


## 2. Modeling and Compensation of Rigid Lever Arm Error

The multilevel rigid lever arm mechanism is shown in Figure1. It describes the relative spatial relation of various sensors including IMU, GPS, and imaging sensor. The first-level lever arm vector lIb refers to the spatial difference between the measuring centers of IMU and GPS in the IMU body frame, and the subscript b denotes the IMU body frame (b frame). The second-level lever arm vector lIIb refers to the spatial difference between the measuring centers of the imaging sensor and IMU in the b frame. The IMU cannot directly integrate with GPS because of lIb, and the motion parameters of POS cannot be directly used for remote sensing load because of lIIb. lIb and lIIb can be calibrated by the total station after the installation of the system, which can be expressed as lIb=xIyIzIT and lIIb=xIIyIIzIIT, respectively.Figure 1
The multilevel rigid lever arm mechanism.


### 2.1. Modeling and Compensation of First-Level Rigid Lever Arm Error

By compensating the first-level rigid lever arm error, the position and velocity measured by GPS can be directly integrated with SINS. The position difference vectorδPInt caused by lIb in the navigation frame (n frame) can be expressed as
(1)δPInt=CbntlIb,where Cbnt is the transformation matrix from the b frame to the n frame, which can be calculated by three Euler angles, heading φt, pitch θt, and roll γt, and Cbnt=CnbtT=Cnbt−1. With the position compensation, the position vector PGnt measured by GPS can be transferred to the measuring center of IMU in the n frame, which can be expressed as
(2)PIGnt=PGnt+AtδPInt,where At represents the matrix which can transform the position vector in the n frame to the Earth frame (e frame) and can be expressed as
(3)At=1RMt+Hpt000secLptRNt+Hpt0001,where Lpt and Hpt, respectively, represent the latitude and altitude measured by POS, RM and RN represent the main curvature radiuses along the meridian and the vertical plane of the meridian, respectively.According to the derivation of (1), the velocity difference vector δVIn caused by lIb in the n frame can be written as
(4)δVIn=Cbn·tlIb=Cbntωnbbt×lIb,where ωnbbt is the angular velocity vector of the b frame with respect to the n frame denoted in the b frame and can be calculated as
(5)ωnbbt=ωibbt−ωinbt=ωibbt−Cnbtωient+ωennt,where ωibbt is the angular velocity vector of the b frame with respect to the inertial frame (i frame) denoted in the b frame and can be directly measured by a gyroscope. ωient is the angular velocity of the e frame with respect to the i frame denoted in the n frame, which can be calculated by
(6)ωient=Cent⋅ωiee=−sinλptcosλpt0−sinLptcosλpt−sinLptsinλptcosLptcosLptcosλptcosLptsinλptsinLpt⋅00ωie,where ωie is the rotation rate of the Earth. ωenn is the angular rate vector of the n frame with respect to the e frame expressed in the n frame, which can be calculated as
(7)ωennt=VpNtRMt+HptVpEtRNt+HptVpEttanLptRNt+Hpt,where VpEt and VpNt are the velocities measured by POS in the east and north axes, respectively.With the velocity compensation, the velocity vectorVGnt measured by GPS can be transferred to the measuring center of IMU in the n frame, which can be expressed as
(8)VIGnt=VGnt+δVInt.


### 2.2. Modeling and Compensation of Second-Level Rigid Lever Arm Error

With the second-level rigid lever arm error compensation, the measurement solutions of POS can be transferred to the measuring center of the imaging sensor. The position difference vectorδPIInt caused by lIIb in the n frame can be expressed as
(9)δPIInt=CbntlIIb.With the position compensation, the position vectorPspnt compensated by δPIInt can be expressed as
(10)Pspnt=Ppnt+AtδPIInt=Ppnt+AtCbnlIIb.According to the derivation of (9), the velocity difference vector δVIIn caused by lIIb in the n frame can be written as
(11)δVIInt=Cbn·tlIIb=Cbntωnbbt×lIIb.Using (11), the velocity vector Vspnt of the measuring center of remote sensing load compensated by δVIInt in the n frame can be written as
(12)Vspnt=Vpnt+δVIInt=Vpnt+Cbntωnbbt×lIIb.Considering the installation angle error, the transformation matrixCnst from the n frame to the remote sensing load frame (s frame) can be expressed as
(13)Cnst=CbsCnbt,where Cbs is the invariable transformation matrix from the b frame to the s frame and can be calculated during the initial calibration, the expression of which is as follows:
(14)Cbs=cosϑzsinϑz0−sinϑzcosϑz0001cosϑy0−sinϑy010sinϑy0cosϑy1000cosϑxsinϑx0−sinϑxcosϑx,where ϑx, ϑy, and ϑz are the three rotary Euler angles from the b frame to the s frame in turn along the x, y, and z axes. Further, the heading, pitch, and roll of the remote sensing load can be calculated according to (14).Remark 1.
The laser total station is chosen for the initial calibration ofϑx, ϑy, and ϑz. Firstly, the target points are attached to the referenced position of IMU and SAR, and the coordinates of each point can be calibrated by the laser total station. Secondly, three axial unit vectors of both IMU and SAR can be calculated based on the calibration information of the target points. Thirdly, the installation error angles between IMU and SAR can be calculated according to the three axial unit vectors.


## 2.1. Modeling and Compensation of First-Level Rigid Lever Arm Error

By compensating the first-level rigid lever arm error, the position and velocity measured by GPS can be directly integrated with SINS. The position difference vectorδPInt caused by lIb in the navigation frame (n frame) can be expressed as
(1)δPInt=CbntlIb,where Cbnt is the transformation matrix from the b frame to the n frame, which can be calculated by three Euler angles, heading φt, pitch θt, and roll γt, and Cbnt=CnbtT=Cnbt−1. With the position compensation, the position vector PGnt measured by GPS can be transferred to the measuring center of IMU in the n frame, which can be expressed as
(2)PIGnt=PGnt+AtδPInt,where At represents the matrix which can transform the position vector in the n frame to the Earth frame (e frame) and can be expressed as
(3)At=1RMt+Hpt000secLptRNt+Hpt0001,where Lpt and Hpt, respectively, represent the latitude and altitude measured by POS, RM and RN represent the main curvature radiuses along the meridian and the vertical plane of the meridian, respectively.According to the derivation of (1), the velocity difference vector δVIn caused by lIb in the n frame can be written as
(4)δVIn=Cbn·tlIb=Cbntωnbbt×lIb,where ωnbbt is the angular velocity vector of the b frame with respect to the n frame denoted in the b frame and can be calculated as
(5)ωnbbt=ωibbt−ωinbt=ωibbt−Cnbtωient+ωennt,where ωibbt is the angular velocity vector of the b frame with respect to the inertial frame (i frame) denoted in the b frame and can be directly measured by a gyroscope. ωient is the angular velocity of the e frame with respect to the i frame denoted in the n frame, which can be calculated by
(6)ωient=Cent⋅ωiee=−sinλptcosλpt0−sinLptcosλpt−sinLptsinλptcosLptcosLptcosλptcosLptsinλptsinLpt⋅00ωie,where ωie is the rotation rate of the Earth. ωenn is the angular rate vector of the n frame with respect to the e frame expressed in the n frame, which can be calculated as
(7)ωennt=VpNtRMt+HptVpEtRNt+HptVpEttanLptRNt+Hpt,where VpEt and VpNt are the velocities measured by POS in the east and north axes, respectively.With the velocity compensation, the velocity vectorVGnt measured by GPS can be transferred to the measuring center of IMU in the n frame, which can be expressed as
(8)VIGnt=VGnt+δVInt.


## 2.2. Modeling and Compensation of Second-Level Rigid Lever Arm Error

With the second-level rigid lever arm error compensation, the measurement solutions of POS can be transferred to the measuring center of the imaging sensor. The position difference vectorδPIInt caused by lIIb in the n frame can be expressed as
(9)δPIInt=CbntlIIb.With the position compensation, the position vectorPspnt compensated by δPIInt can be expressed as
(10)Pspnt=Ppnt+AtδPIInt=Ppnt+AtCbnlIIb.According to the derivation of (9), the velocity difference vector δVIIn caused by lIIb in the n frame can be written as
(11)δVIInt=Cbn·tlIIb=Cbntωnbbt×lIIb.Using (11), the velocity vector Vspnt of the measuring center of remote sensing load compensated by δVIInt in the n frame can be written as
(12)Vspnt=Vpnt+δVIInt=Vpnt+Cbntωnbbt×lIIb.Considering the installation angle error, the transformation matrixCnst from the n frame to the remote sensing load frame (s frame) can be expressed as
(13)Cnst=CbsCnbt,where Cbs is the invariable transformation matrix from the b frame to the s frame and can be calculated during the initial calibration, the expression of which is as follows:
(14)Cbs=cosϑzsinϑz0−sinϑzcosϑz0001cosϑy0−sinϑy010sinϑy0cosϑy1000cosϑxsinϑx0−sinϑxcosϑx,where ϑx, ϑy, and ϑz are the three rotary Euler angles from the b frame to the s frame in turn along the x, y, and z axes. Further, the heading, pitch, and roll of the remote sensing load can be calculated according to (14).Remark 1.
The laser total station is chosen for the initial calibration ofϑx, ϑy, and ϑz. Firstly, the target points are attached to the referenced position of IMU and SAR, and the coordinates of each point can be calibrated by the laser total station. Secondly, three axial unit vectors of both IMU and SAR can be calculated based on the calibration information of the target points. Thirdly, the installation error angles between IMU and SAR can be calculated according to the three axial unit vectors.


## 3. Modeling and Compensation of Dynamic Lever Arm Error

In some special imaging process of airborne Earth observation, the appointed direction of remote sensing load will be adjusted in order to evidently improve imaging efficiency and accuracy. Consequently, the relative spatial relationships among sensors vary in real time, which is called the dynamic lever arm. The dynamic lever arm can be divided into two kinds. One kind is named the first-level dynamic lever arm, which is shown in Figure2. In this case, the remote sensing load fixed to IMU can be driven by the servo machine to sweep along the rotary axis, which changes the relative spatial vector between GPS and IMU. This kind of installation allows the IMU to directly measure the motion parameters of the remote sensing load, including the influence of random vibration on the remote sensing load.Figure 2
The first-level dynamic lever arm mechanism.The other kind is named the second-level dynamic lever arm, which is shown in Figure3. In this case, the IMU and GPS are directly assembled onto the plane body and are close to the servo machine. The rotation of the remote sensing load will change the relative spatial vector between the IMU and remote sensing load. This kind of installation makes the POS a universal measuring instrument, which can be used for various kinds of remote sensing loads such as CCD cameras and SAR.Figure 3
The second-level dynamic lever arm mechanism.


### 3.1. Modeling and Compensation of First-Level Dynamic Lever Arm Error

As shown in Figure2, LIs refers to the rigid lever arm between the GPS measuring center and the rotary axis center. LIr is the rotation lever arm between the rotary axis center and the measuring center of IMU. After system installation, a high-precision laser total station can be used to calibrate the rigid lever arm LIsb0 and the rotation lever arm LIrb0 in the initial IMU body frame (b0 frame). The transformation matrix Cb0R0t0 from the b frame to the initial rotary frame (R0 frame) can also be calculated.The servo machine turns along the rotary axis in real time. At momentt, the output angle σt of the photoelectric encoder can be measured, and the transformation matrix from the R0 frame to the R frame can be given:
(15)CR0Rt=cosσt−σ00−sinσt−σ0010sinσt−σ00cosσt−σ0.As the b frame is fixed in the R frame, the transformation matrixCbRt from the b frame to the R frame is time invariant and can be expressed as
(16)CbRt=Cb0R0t0.


#### 3.1.1. Position Error Modeling and Compensation

The rigid lever armLIsbt can be calculated as
(17)LIsbt=Cb0btLIsb0=Cb0R0t0−1CR0RtCb0R0tLIsb0.As the rotation lever armLIr is fixed in the b frame, it can be calculated at moment t as
(18)LIrbt=LIrb0.According to (17) and (18), the position difference vector δPIdnt caused by the first-level dynamic lever arm in the n frame can be expressed as
(19)δPIdnt=CbntLIbt=CbntCb0R0t0−1CR0RtCb0R0tLIsb0+CbntLIrb0.With the position compensation, the position vectorPGnt measured by GPS can be transferred to the measuring center of IMU in the n frame, which can be expressed as
(20)PIdGnt=PGnt+AtδPIdnt.


#### 3.1.2. Velocity Error Modeling and Compensation

According to the derivation of (19), the velocity difference vector δVIdn caused by the first-level dynamic lever arm in the n frame can be written as
(21)δVIdnt=ĊbntLIbt+CbntL̇Ibt=Cbntωnbbt×LIbt+Cbntσ̇bt×LIrbt=Cbntωnbbt×LIbt+CbntCb0R0t0−1σ̇Rt×LIrb0,where ωnbbt can be calculated by (5) and σ̇Rt is the angular velocity vector of the R frame with respect to the R0 frame denoted in the R frame and can be calculated as
(22)σ̇Rt=0σt−σt−t−t−0,where σt and σt−, respectively, present the output angles of the photoelectric encoder in last update time t and current update time t− along the rotary axis of the servo machine. Since the differential coefficient has a noise amplification effect, the output angles σt and σt− should be well filtered. Otherwise, the noise error of angular acceleration is insufferable.With the velocity compensation, the velocity vectorVGnt measured by GPS can be transferred to the measuring center of IMU in the n frame, which can be expressed as
(23)VIdGnt=VGnt+δVIdnt.Remark 2.
As POS is a nonlinear integrated system, nonlinear estimation algorithms such as the extended Kalman filter and unscented Kalman filter can be used for IMU/GPS integration, in which the measurement of GPS must be compensated as Section3.1.


### 3.2. Modeling and Compensation of Second-Level Dynamic Lever Arm Error

As shown in Figure3, LIIs refers to the rigid lever arm between the measuring center of IMU Ob and the rotary axis center OR and LIIr is the rotation lever arm between the rotary axis center OR and the measuring center of the remote sensing load Os. After system installation, the rigid lever arm LIIsb0 and the rotation lever arm LIIrb0 can be calculated in the b0 frame. The transformation matrix CbR0t0 from the b frame to the R0 frame and the transformation matrix CR0st0 from the R0 frame to the s frame can all be calibrated by a high-precision laser total station. The transformation matrix CRs from the R frame to the s frame and the transformation matrix CbR0 from the b frame to the R0 frame are time invariant, which can be expressed as
(24)CRst=CR0st0,CbR0t=CbR0t0.


#### 3.2.1. Position Error Modeling and Compensation

The rigid lever armLIIsbt denoted in the b frame is time invariant and can be calculated as
(25)LIIsbt=LIIsb0.The rotation lever armLIIr is fixed in the s frame and is time invariant, which is denoted in the s frame and expressed as
(26)LIIrst=LIIrst0=CR0st0CbR0t0LIIrb0.Further, the rotation lever armLIIr can be expressed in the b frame as
(27)LIIrbt=CR0btCRR0tCsRtLIIrst=CbR0t0−1CRR0tCbR0t0LIIrb0.According to (25) and (27), the position difference vector δPIIdnt caused by the second-level dynamic lever arm in the n frame can be expressed as
(28)δPIIdnt=CbntLIIbt=CbntLIIsbt+LIIrbt=CbntCbR0t0−1CRR0tCbR0t0LIIrb0+CbntLIIsb0.With the position compensation ofδPIIdnt, the position vector Ppnt measured by POS can be transferred to the measuring center of the remote sensing load in the n frame, which can be expressed as
(29)Pspnt=Ppnt+AtδPIIdnt.


#### 3.2.2. Velocity Error Modeling and Compensation

According to the derivation of (28), the velocity difference vector δVIIdn caused by the second-level dynamic lever arm in the n frame can be written as
(30)δVIInt=ĊbntLIIbt+CbntL̇IIbt=Cbntωnbbt×LIIbt+Cbntσ̇bt×LIIrbt=Cbntωnbbt×LIIbt+CbntCbR0t−1CRR0tσ̇Rt×LIIrbt,where σ̇bt and ωnbbt can be calculated by (22) and (5), respectively.With the velocity compensation, the velocity vectorVpnt measured by POS can be transferred to the measuring center of the remote sensing load in the n frame, which can be expressed as
(31)Vspnt=Vpnt+δVIIdnt.


#### 3.2.3. Attitude Error Modeling and Compensation

Because of the installation error angle and the rotation of the remote sensing load, the attitude of POS cannot be directly used for the remote sensing load. The transformation matrixCsnt from the s frame to the n frame at moment t can be calculated as
(32)Csnt=CbntCR0btCRR0tCsRt=CbntCR0bt0CRR0t−1CsR0t0,where CRR0t can be calculated by (15). Using the attitude matrix Csnt of the remote sensing load, the heading, pitch, and roll of the remote sensing load can be calculated.


## 3.1. Modeling and Compensation of First-Level Dynamic Lever Arm Error

As shown in Figure2, LIs refers to the rigid lever arm between the GPS measuring center and the rotary axis center. LIr is the rotation lever arm between the rotary axis center and the measuring center of IMU. After system installation, a high-precision laser total station can be used to calibrate the rigid lever arm LIsb0 and the rotation lever arm LIrb0 in the initial IMU body frame (b0 frame). The transformation matrix Cb0R0t0 from the b frame to the initial rotary frame (R0 frame) can also be calculated.The servo machine turns along the rotary axis in real time. At momentt, the output angle σt of the photoelectric encoder can be measured, and the transformation matrix from the R0 frame to the R frame can be given:
(15)CR0Rt=cosσt−σ00−sinσt−σ0010sinσt−σ00cosσt−σ0.As the b frame is fixed in the R frame, the transformation matrixCbRt from the b frame to the R frame is time invariant and can be expressed as
(16)CbRt=Cb0R0t0.


### 3.1.1. Position Error Modeling and Compensation

The rigid lever armLIsbt can be calculated as
(17)LIsbt=Cb0btLIsb0=Cb0R0t0−1CR0RtCb0R0tLIsb0.As the rotation lever armLIr is fixed in the b frame, it can be calculated at moment t as
(18)LIrbt=LIrb0.According to (17) and (18), the position difference vector δPIdnt caused by the first-level dynamic lever arm in the n frame can be expressed as
(19)δPIdnt=CbntLIbt=CbntCb0R0t0−1CR0RtCb0R0tLIsb0+CbntLIrb0.With the position compensation, the position vectorPGnt measured by GPS can be transferred to the measuring center of IMU in the n frame, which can be expressed as
(20)PIdGnt=PGnt+AtδPIdnt.


### 3.1.2. Velocity Error Modeling and Compensation

According to the derivation of (19), the velocity difference vector δVIdn caused by the first-level dynamic lever arm in the n frame can be written as
(21)δVIdnt=ĊbntLIbt+CbntL̇Ibt=Cbntωnbbt×LIbt+Cbntσ̇bt×LIrbt=Cbntωnbbt×LIbt+CbntCb0R0t0−1σ̇Rt×LIrb0,where ωnbbt can be calculated by (5) and σ̇Rt is the angular velocity vector of the R frame with respect to the R0 frame denoted in the R frame and can be calculated as
(22)σ̇Rt=0σt−σt−t−t−0,where σt and σt−, respectively, present the output angles of the photoelectric encoder in last update time t and current update time t− along the rotary axis of the servo machine. Since the differential coefficient has a noise amplification effect, the output angles σt and σt− should be well filtered. Otherwise, the noise error of angular acceleration is insufferable.With the velocity compensation, the velocity vectorVGnt measured by GPS can be transferred to the measuring center of IMU in the n frame, which can be expressed as
(23)VIdGnt=VGnt+δVIdnt.Remark 2.
As POS is a nonlinear integrated system, nonlinear estimation algorithms such as the extended Kalman filter and unscented Kalman filter can be used for IMU/GPS integration, in which the measurement of GPS must be compensated as Section3.1.


## 3.1.1. Position Error Modeling and Compensation

The rigid lever armLIsbt can be calculated as
(17)LIsbt=Cb0btLIsb0=Cb0R0t0−1CR0RtCb0R0tLIsb0.As the rotation lever armLIr is fixed in the b frame, it can be calculated at moment t as
(18)LIrbt=LIrb0.According to (17) and (18), the position difference vector δPIdnt caused by the first-level dynamic lever arm in the n frame can be expressed as
(19)δPIdnt=CbntLIbt=CbntCb0R0t0−1CR0RtCb0R0tLIsb0+CbntLIrb0.With the position compensation, the position vectorPGnt measured by GPS can be transferred to the measuring center of IMU in the n frame, which can be expressed as
(20)PIdGnt=PGnt+AtδPIdnt.


## 3.1.2. Velocity Error Modeling and Compensation

According to the derivation of (19), the velocity difference vector δVIdn caused by the first-level dynamic lever arm in the n frame can be written as
(21)δVIdnt=ĊbntLIbt+CbntL̇Ibt=Cbntωnbbt×LIbt+Cbntσ̇bt×LIrbt=Cbntωnbbt×LIbt+CbntCb0R0t0−1σ̇Rt×LIrb0,where ωnbbt can be calculated by (5) and σ̇Rt is the angular velocity vector of the R frame with respect to the R0 frame denoted in the R frame and can be calculated as
(22)σ̇Rt=0σt−σt−t−t−0,where σt and σt−, respectively, present the output angles of the photoelectric encoder in last update time t and current update time t− along the rotary axis of the servo machine. Since the differential coefficient has a noise amplification effect, the output angles σt and σt− should be well filtered. Otherwise, the noise error of angular acceleration is insufferable.With the velocity compensation, the velocity vectorVGnt measured by GPS can be transferred to the measuring center of IMU in the n frame, which can be expressed as
(23)VIdGnt=VGnt+δVIdnt.Remark 2.
As POS is a nonlinear integrated system, nonlinear estimation algorithms such as the extended Kalman filter and unscented Kalman filter can be used for IMU/GPS integration, in which the measurement of GPS must be compensated as Section3.1.


## 3.2. Modeling and Compensation of Second-Level Dynamic Lever Arm Error

As shown in Figure3, LIIs refers to the rigid lever arm between the measuring center of IMU Ob and the rotary axis center OR and LIIr is the rotation lever arm between the rotary axis center OR and the measuring center of the remote sensing load Os. After system installation, the rigid lever arm LIIsb0 and the rotation lever arm LIIrb0 can be calculated in the b0 frame. The transformation matrix CbR0t0 from the b frame to the R0 frame and the transformation matrix CR0st0 from the R0 frame to the s frame can all be calibrated by a high-precision laser total station. The transformation matrix CRs from the R frame to the s frame and the transformation matrix CbR0 from the b frame to the R0 frame are time invariant, which can be expressed as
(24)CRst=CR0st0,CbR0t=CbR0t0.


### 3.2.1. Position Error Modeling and Compensation

The rigid lever armLIIsbt denoted in the b frame is time invariant and can be calculated as
(25)LIIsbt=LIIsb0.The rotation lever armLIIr is fixed in the s frame and is time invariant, which is denoted in the s frame and expressed as
(26)LIIrst=LIIrst0=CR0st0CbR0t0LIIrb0.Further, the rotation lever armLIIr can be expressed in the b frame as
(27)LIIrbt=CR0btCRR0tCsRtLIIrst=CbR0t0−1CRR0tCbR0t0LIIrb0.According to (25) and (27), the position difference vector δPIIdnt caused by the second-level dynamic lever arm in the n frame can be expressed as
(28)δPIIdnt=CbntLIIbt=CbntLIIsbt+LIIrbt=CbntCbR0t0−1CRR0tCbR0t0LIIrb0+CbntLIIsb0.With the position compensation ofδPIIdnt, the position vector Ppnt measured by POS can be transferred to the measuring center of the remote sensing load in the n frame, which can be expressed as
(29)Pspnt=Ppnt+AtδPIIdnt.


### 3.2.2. Velocity Error Modeling and Compensation

According to the derivation of (28), the velocity difference vector δVIIdn caused by the second-level dynamic lever arm in the n frame can be written as
(30)δVIInt=ĊbntLIIbt+CbntL̇IIbt=Cbntωnbbt×LIIbt+Cbntσ̇bt×LIIrbt=Cbntωnbbt×LIIbt+CbntCbR0t−1CRR0tσ̇Rt×LIIrbt,where σ̇bt and ωnbbt can be calculated by (22) and (5), respectively.With the velocity compensation, the velocity vectorVpnt measured by POS can be transferred to the measuring center of the remote sensing load in the n frame, which can be expressed as
(31)Vspnt=Vpnt+δVIIdnt.


### 3.2.3. Attitude Error Modeling and Compensation

Because of the installation error angle and the rotation of the remote sensing load, the attitude of POS cannot be directly used for the remote sensing load. The transformation matrixCsnt from the s frame to the n frame at moment t can be calculated as
(32)Csnt=CbntCR0btCRR0tCsRt=CbntCR0bt0CRR0t−1CsR0t0,where CRR0t can be calculated by (15). Using the attitude matrix Csnt of the remote sensing load, the heading, pitch, and roll of the remote sensing load can be calculated.


## 3.2.1. Position Error Modeling and Compensation

The rigid lever armLIIsbt denoted in the b frame is time invariant and can be calculated as
(25)LIIsbt=LIIsb0.The rotation lever armLIIr is fixed in the s frame and is time invariant, which is denoted in the s frame and expressed as
(26)LIIrst=LIIrst0=CR0st0CbR0t0LIIrb0.Further, the rotation lever armLIIr can be expressed in the b frame as
(27)LIIrbt=CR0btCRR0tCsRtLIIrst=CbR0t0−1CRR0tCbR0t0LIIrb0.According to (25) and (27), the position difference vector δPIIdnt caused by the second-level dynamic lever arm in the n frame can be expressed as
(28)δPIIdnt=CbntLIIbt=CbntLIIsbt+LIIrbt=CbntCbR0t0−1CRR0tCbR0t0LIIrb0+CbntLIIsb0.With the position compensation ofδPIIdnt, the position vector Ppnt measured by POS can be transferred to the measuring center of the remote sensing load in the n frame, which can be expressed as
(29)Pspnt=Ppnt+AtδPIIdnt.


## 3.2.2. Velocity Error Modeling and Compensation

According to the derivation of (28), the velocity difference vector δVIIdn caused by the second-level dynamic lever arm in the n frame can be written as
(30)δVIInt=ĊbntLIIbt+CbntL̇IIbt=Cbntωnbbt×LIIbt+Cbntσ̇bt×LIIrbt=Cbntωnbbt×LIIbt+CbntCbR0t−1CRR0tσ̇Rt×LIIrbt,where σ̇bt and ωnbbt can be calculated by (22) and (5), respectively.With the velocity compensation, the velocity vectorVpnt measured by POS can be transferred to the measuring center of the remote sensing load in the n frame, which can be expressed as
(31)Vspnt=Vpnt+δVIIdnt.


## 3.2.3. Attitude Error Modeling and Compensation

Because of the installation error angle and the rotation of the remote sensing load, the attitude of POS cannot be directly used for the remote sensing load. The transformation matrixCsnt from the s frame to the n frame at moment t can be calculated as
(32)Csnt=CbntCR0btCRR0tCsRt=CbntCR0bt0CRR0t−1CsR0t0,where CRR0t can be calculated by (15). Using the attitude matrix Csnt of the remote sensing load, the heading, pitch, and roll of the remote sensing load can be calculated.


## 4. Experiment and Analysis

To validate the effectiveness, the proposed dynamic lever arm error compensation method is applied to two actual airborne SAR imaging experiments which contain the first-level dynamic lever arm and the second-level dynamic lever arm, respectively.


### 4.1. Hardware Configuration

For the first-level dynamic lever arm error compensation, the flight experiment equipment includes an experiment plane (YUN-VIII), SAR, servo machines, and the ring laser gyroscope (RLG) POS which consists of an RLG IMU, POS computer system, and carrier phase differential GPS. The RLG IMU consists of three ring laser gyroscopes with constant and random drifts (0.01°/h) and quartz mechanical accelerometers with constant and random biases (50μg) assembled in orthogonal triads. The RLG IMU is fixed on the back of the SAR antenna and can be driven to sweep by the servo machine. The experiment was carried out for SAR in Shaanxi Province, China, in April 2017, and the flight trajectory is shown in Figure 4.Figure 4
The flight trajectory of the experiment in Shaanxi Province, China, April 2017.For the second-level dynamic lever arm error compensation, the flight experiment equipment includes an experiment plane (YUN-VII), SAR, servo machines, and the fiber optic gyroscope (FOG) POS which consists of a FOG IMU, PCS, and carrier phase differential GPS. The FOG IMU consists of three fiber optic gyroscopes with constant and random drifts (0.01°/h) and quartz mechanical accelerometers with constant and random biases (50μg) assembled in orthogonal triads. The FOG POS is fixed on the plane, while the SAR antenna is installed on the servo machine and can be driven to sweep along the rotary axis. The experiment was carried out for SAR in Shaanxi Province, China, in March 2016, and the flight trajectory is shown in Figure 5.Figure 5
The flight trajectory of the experiment in Shaanxi Province, China, in March 2016.A reference GPS receiver is set up on the ground within 100 km and takes a real-time radio link to the roving GNSS receiver. The GPS can achieve carrier phase difference by combining the phase data from roving and reference receivers. The real-time kinematic precision is typically 1~2 cm plus 1 ppm of baseline separation.


### 4.2. Initial Calibration

The laser total station is an optimal instrument and can be used to calibrate the initial lever arms and transformation matrix among sensors. The measurement precision can reach submillimeter level, which is beyond the measurement precision of POS. So the calibration errors can be ignored. The initial calibration result is obtained in Tables1 and 2.Table 1
Calibration result of the experiment system in Figure4.
x
y
z
First-level dynamic lever armLIb (m)
Rigid lever armLIsb
3.547
−3.090
2.030
Rotation lever armLIrb
−0.117
0.512
0.042
Second-level rigid lever armLIIb (m)
−0.288
0.526
0.180
Initial transformation matrixCb0R0
100010001
Initial output angleσ0 (°)
57.105Table 2
Calibration result of the experiment system in Figure5.
x
y
z
First-level rigid lever armLIb (m)
−1.738
−3.503
2.537
Second-level dynamic lever armLIb (m)
Rigid lever armLIIsb
−0.008
0.553
−0.136
Rotation lever armLIIrb
0.0633
0.001
0.001
Initial transformation matrixCb0R0
0.9999−0.0012−0.00940.00120.9999−0.00210.00940.00210.9999
Initial transformation matrixCR0s
100010001
Initial output angleσ0 (°)
42.812


### 4.3. Result and Analysis

Two groups of 10-minute experiment data are processed and analyzed for the first-level dynamic lever arm error compensation of the experiment system in Figure4 and the second-level dynamic lever arm error compensation of the experiment system in Figure 5, respectively. The postprocessed result with the dynamic lever arm error compensation is chosen as the reference, because of its dramatic measurement accuracy provided by forward and reverse in time processing.


#### 4.3.1. First-Level Dynamic Lever Arm Error Compensation Result

For the first-level dynamic lever arm error compensation of the experiment system in Figure4, two experiment results are obtained. One adopts the proposed method, while the other considers the first-level dynamic lever arm a rigid one. The output angles of the photoelectric encoder used in the experiment system in Figure 4 are shown in Figure 6, from which we can learn that the dynamic change of the first-level lever arm is relatively small during the period.Figure 6
The output angles of the photoelectric encoder used in the experiment system in Figure4.The errors of RLG POS in Figure4 calculated by the real-time Kalman filtering method with the proposed lever arm error compensation method and the conventional rigid lever arm error compensation method are shown in Figures 7–9 and Table 3.Figure 7
Position error comparison of the first-level dynamic lever arm error compensation.Figure 8
Velocity error comparison of the first-level dynamic lever arm error compensation.Figure 9
Attitude error comparison of the first-level dynamic lever arm error compensation.Table 3
Measurement error comparison of the first-level dynamic lever arm error compensation.
Parameters measured by the main POS
The measurement error (RMSE)
Conventional rigid lever arm error compensation
Proposed dynamic lever arm error compensation
Position (m)
Latitude
0.9377
0.0348
Longitude
2.0680
0.0259
Height
2.4657
0.0800
Velocity (m/s)
East velocity
0.0089
0.0067
North velocity
0.0085
0.0077
Upward velocity
0.0131
0.0091
Attitude (°)
Heading
0.0382
0.0266
Pitch
0.0132
0.0115
Roll
0.0034
0.0027With the proposed first-level dynamic lever arm error compensation, the latitude, longitude, and height errors have been reduced from 0.9377 m, 2.0680 m, and 2.4657 m to 0.0348 m, 0.0259 m, and 0.0800 m, respectively. The east, north, and upward velocity errors have been reduced from 0.0089 m/s, 0.0085 m/s, and 0.0131 m/s to 0.0067 m/s, 0.0077 m/s, and 0.0091 m/s, respectively. The heading, pitch, and roll angle errors have been reduced from 0.0382°, 0.0132°, and 0.0034° to 0.0266°, 0.0115°, and 0.0027°, respectively.Remark 3.
The position accuracy is significantly improved by the proposed method, because the length of the first-level lever arm is long and the output angle is quite different from the initial angle. However, the improvements of velocity and attitude accuracy are small, because the servo was controlled to keep stable in the survey phase, and the influence of the dynamic lever arm effect on velocity and attitude is little.


#### 4.3.2. Second-Level Dynamic Lever Arm Error Compensation Result

For the second-level dynamic lever arm error compensation of the experiment system in Figure5, two experiment results are obtained. One adopts the proposed method, while the other considers the second-level dynamic lever arm a rigid one. The output angles of the photoelectric encoder used in the experiment system in Figure 5 are shown in Figure 10, from which we can learn that the servo made a greatly adjustment during 300 s–450 s.Figure 10
The output angles of the photoelectric encoder used in the experiment system in Figure5.The errors of FOG POS in Figure5 calculated by the real-time Kalman filtering method with the proposed lever arm error compensation method and the conventional rigid lever arm error compensation method are shown in Figures 11–13 and Table 4.Figure 11
Position error comparison of the second-level dynamic lever arm error compensation.Figure 12
Velocity error comparison of the second-level dynamic lever arm error compensation.Figure 13
Attitude error comparison of the second-level dynamic lever arm error compensation.Table 4
Measurement error comparison of the second-level dynamic lever arm error compensation.
Parameters measured by the main POS
The measurement error (RMSE)
Conventional rigid lever arm error compensation
Proposed dynamic lever arm error compensation
Position (m)
Latitude
0.5470
0.2921
Longitude
0.5738
0.2805
Height
0.1816
0.0544
Velocity (m/s)
East velocity
0.0230
0.0210
North velocity
0.0283
0.0255
Upward velocity
0.0140
0.0132
Attitude (°)
Heading
0.0530
0.0221
Pitch
0.1224
0.0062
Roll
7.0999
0.0045With the proposed first-level dynamic lever arm error compensation, the latitude, longitude, and height errors have been reduced from 0.5470 m, 0.5738 m, and 0.1816 m to 0.2921 m, 0.2805 m, and 0.0544 m, respectively. The east, north, and upward velocity errors have been reduced from 0.0230 m/s, 0.0283 m/s, and 0.0140 m/s to 0.0210 m/s, 0.0255 m/s, and 0.0132 m/s, respectively. The heading, pitch, and roll angle errors have been reduced from 0.0530°, 0.1224°, and 7.0999° to 0.0221°, 0.0062°, and 0.0045°, respectively.Remark 4.
The attitude accuracy is significantly improved by the proposed method, because the SAR is not fixed on IMU, and the attitude error will be insufferable without dynamic compensation. However, the improvements of position and velocity accuracy are small, because the  length of second dynamic lever arm is short, and the influence of the dynamic lever arm effect on position and velocity is little.Above all, it is validated that the proposed dynamic lever arm error compensation method can effectively compensate the dynamic lever arm error and achieve higher accuracy and more extensive application for airborne POS.


## 4.1. Hardware Configuration

For the first-level dynamic lever arm error compensation, the flight experiment equipment includes an experiment plane (YUN-VIII), SAR, servo machines, and the ring laser gyroscope (RLG) POS which consists of an RLG IMU, POS computer system, and carrier phase differential GPS. The RLG IMU consists of three ring laser gyroscopes with constant and random drifts (0.01°/h) and quartz mechanical accelerometers with constant and random biases (50μg) assembled in orthogonal triads. The RLG IMU is fixed on the back of the SAR antenna and can be driven to sweep by the servo machine. The experiment was carried out for SAR in Shaanxi Province, China, in April 2017, and the flight trajectory is shown in Figure 4.Figure 4
The flight trajectory of the experiment in Shaanxi Province, China, April 2017.For the second-level dynamic lever arm error compensation, the flight experiment equipment includes an experiment plane (YUN-VII), SAR, servo machines, and the fiber optic gyroscope (FOG) POS which consists of a FOG IMU, PCS, and carrier phase differential GPS. The FOG IMU consists of three fiber optic gyroscopes with constant and random drifts (0.01°/h) and quartz mechanical accelerometers with constant and random biases (50μg) assembled in orthogonal triads. The FOG POS is fixed on the plane, while the SAR antenna is installed on the servo machine and can be driven to sweep along the rotary axis. The experiment was carried out for SAR in Shaanxi Province, China, in March 2016, and the flight trajectory is shown in Figure 5.Figure 5
The flight trajectory of the experiment in Shaanxi Province, China, in March 2016.A reference GPS receiver is set up on the ground within 100 km and takes a real-time radio link to the roving GNSS receiver. The GPS can achieve carrier phase difference by combining the phase data from roving and reference receivers. The real-time kinematic precision is typically 1~2 cm plus 1 ppm of baseline separation.


## 4.2. Initial Calibration

The laser total station is an optimal instrument and can be used to calibrate the initial lever arms and transformation matrix among sensors. The measurement precision can reach submillimeter level, which is beyond the measurement precision of POS. So the calibration errors can be ignored. The initial calibration result is obtained in Tables1 and 2.Table 1
Calibration result of the experiment system in Figure4.
x
y
z
First-level dynamic lever armLIb (m)
Rigid lever armLIsb
3.547
−3.090
2.030
Rotation lever armLIrb
−0.117
0.512
0.042
Second-level rigid lever armLIIb (m)
−0.288
0.526
0.180
Initial transformation matrixCb0R0
100010001
Initial output angleσ0 (°)
57.105Table 2
Calibration result of the experiment system in Figure5.
x
y
z
First-level rigid lever armLIb (m)
−1.738
−3.503
2.537
Second-level dynamic lever armLIb (m)
Rigid lever armLIIsb
−0.008
0.553
−0.136
Rotation lever armLIIrb
0.0633
0.001
0.001
Initial transformation matrixCb0R0
0.9999−0.0012−0.00940.00120.9999−0.00210.00940.00210.9999
Initial transformation matrixCR0s
100010001
Initial output angleσ0 (°)
42.812


## 4.3. Result and Analysis

Two groups of 10-minute experiment data are processed and analyzed for the first-level dynamic lever arm error compensation of the experiment system in Figure4 and the second-level dynamic lever arm error compensation of the experiment system in Figure 5, respectively. The postprocessed result with the dynamic lever arm error compensation is chosen as the reference, because of its dramatic measurement accuracy provided by forward and reverse in time processing.


### 4.3.1. First-Level Dynamic Lever Arm Error Compensation Result

For the first-level dynamic lever arm error compensation of the experiment system in Figure4, two experiment results are obtained. One adopts the proposed method, while the other considers the first-level dynamic lever arm a rigid one. The output angles of the photoelectric encoder used in the experiment system in Figure 4 are shown in Figure 6, from which we can learn that the dynamic change of the first-level lever arm is relatively small during the period.Figure 6
The output angles of the photoelectric encoder used in the experiment system in Figure4.The errors of RLG POS in Figure4 calculated by the real-time Kalman filtering method with the proposed lever arm error compensation method and the conventional rigid lever arm error compensation method are shown in Figures 7–9 and Table 3.Figure 7
Position error comparison of the first-level dynamic lever arm error compensation.Figure 8
Velocity error comparison of the first-level dynamic lever arm error compensation.Figure 9
Attitude error comparison of the first-level dynamic lever arm error compensation.Table 3
Measurement error comparison of the first-level dynamic lever arm error compensation.
Parameters measured by the main POS
The measurement error (RMSE)
Conventional rigid lever arm error compensation
Proposed dynamic lever arm error compensation
Position (m)
Latitude
0.9377
0.0348
Longitude
2.0680
0.0259
Height
2.4657
0.0800
Velocity (m/s)
East velocity
0.0089
0.0067
North velocity
0.0085
0.0077
Upward velocity
0.0131
0.0091
Attitude (°)
Heading
0.0382
0.0266
Pitch
0.0132
0.0115
Roll
0.0034
0.0027With the proposed first-level dynamic lever arm error compensation, the latitude, longitude, and height errors have been reduced from 0.9377 m, 2.0680 m, and 2.4657 m to 0.0348 m, 0.0259 m, and 0.0800 m, respectively. The east, north, and upward velocity errors have been reduced from 0.0089 m/s, 0.0085 m/s, and 0.0131 m/s to 0.0067 m/s, 0.0077 m/s, and 0.0091 m/s, respectively. The heading, pitch, and roll angle errors have been reduced from 0.0382°, 0.0132°, and 0.0034° to 0.0266°, 0.0115°, and 0.0027°, respectively.Remark 3.
The position accuracy is significantly improved by the proposed method, because the length of the first-level lever arm is long and the output angle is quite different from the initial angle. However, the improvements of velocity and attitude accuracy are small, because the servo was controlled to keep stable in the survey phase, and the influence of the dynamic lever arm effect on velocity and attitude is little.


### 4.3.2. Second-Level Dynamic Lever Arm Error Compensation Result

For the second-level dynamic lever arm error compensation of the experiment system in Figure5, two experiment results are obtained. One adopts the proposed method, while the other considers the second-level dynamic lever arm a rigid one. The output angles of the photoelectric encoder used in the experiment system in Figure 5 are shown in Figure 10, from which we can learn that the servo made a greatly adjustment during 300 s–450 s.Figure 10
The output angles of the photoelectric encoder used in the experiment system in Figure5.The errors of FOG POS in Figure5 calculated by the real-time Kalman filtering method with the proposed lever arm error compensation method and the conventional rigid lever arm error compensation method are shown in Figures 11–13 and Table 4.Figure 11
Position error comparison of the second-level dynamic lever arm error compensation.Figure 12
Velocity error comparison of the second-level dynamic lever arm error compensation.Figure 13
Attitude error comparison of the second-level dynamic lever arm error compensation.Table 4
Measurement error comparison of the second-level dynamic lever arm error compensation.
Parameters measured by the main POS
The measurement error (RMSE)
Conventional rigid lever arm error compensation
Proposed dynamic lever arm error compensation
Position (m)
Latitude
0.5470
0.2921
Longitude
0.5738
0.2805
Height
0.1816
0.0544
Velocity (m/s)
East velocity
0.0230
0.0210
North velocity
0.0283
0.0255
Upward velocity
0.0140
0.0132
Attitude (°)
Heading
0.0530
0.0221
Pitch
0.1224
0.0062
Roll
7.0999
0.0045With the proposed first-level dynamic lever arm error compensation, the latitude, longitude, and height errors have been reduced from 0.5470 m, 0.5738 m, and 0.1816 m to 0.2921 m, 0.2805 m, and 0.0544 m, respectively. The east, north, and upward velocity errors have been reduced from 0.0230 m/s, 0.0283 m/s, and 0.0140 m/s to 0.0210 m/s, 0.0255 m/s, and 0.0132 m/s, respectively. The heading, pitch, and roll angle errors have been reduced from 0.0530°, 0.1224°, and 7.0999° to 0.0221°, 0.0062°, and 0.0045°, respectively.Remark 4.
The attitude accuracy is significantly improved by the proposed method, because the SAR is not fixed on IMU, and the attitude error will be insufferable without dynamic compensation. However, the improvements of position and velocity accuracy are small, because the  length of second dynamic lever arm is short, and the influence of the dynamic lever arm effect on position and velocity is little.Above all, it is validated that the proposed dynamic lever arm error compensation method can effectively compensate the dynamic lever arm error and achieve higher accuracy and more extensive application for airborne POS.


## 4.3.1. First-Level Dynamic Lever Arm Error Compensation Result

For the first-level dynamic lever arm error compensation of the experiment system in Figure4, two experiment results are obtained. One adopts the proposed method, while the other considers the first-level dynamic lever arm a rigid one. The output angles of the photoelectric encoder used in the experiment system in Figure 4 are shown in Figure 6, from which we can learn that the dynamic change of the first-level lever arm is relatively small during the period.Figure 6
The output angles of the photoelectric encoder used in the experiment system in Figure4.The errors of RLG POS in Figure4 calculated by the real-time Kalman filtering method with the proposed lever arm error compensation method and the conventional rigid lever arm error compensation method are shown in Figures 7–9 and Table 3.Figure 7
Position error comparison of the first-level dynamic lever arm error compensation.Figure 8
Velocity error comparison of the first-level dynamic lever arm error compensation.Figure 9
Attitude error comparison of the first-level dynamic lever arm error compensation.Table 3
Measurement error comparison of the first-level dynamic lever arm error compensation.
Parameters measured by the main POS
The measurement error (RMSE)
Conventional rigid lever arm error compensation
Proposed dynamic lever arm error compensation
Position (m)
Latitude
0.9377
0.0348
Longitude
2.0680
0.0259
Height
2.4657
0.0800
Velocity (m/s)
East velocity
0.0089
0.0067
North velocity
0.0085
0.0077
Upward velocity
0.0131
0.0091
Attitude (°)
Heading
0.0382
0.0266
Pitch
0.0132
0.0115
Roll
0.0034
0.0027With the proposed first-level dynamic lever arm error compensation, the latitude, longitude, and height errors have been reduced from 0.9377 m, 2.0680 m, and 2.4657 m to 0.0348 m, 0.0259 m, and 0.0800 m, respectively. The east, north, and upward velocity errors have been reduced from 0.0089 m/s, 0.0085 m/s, and 0.0131 m/s to 0.0067 m/s, 0.0077 m/s, and 0.0091 m/s, respectively. The heading, pitch, and roll angle errors have been reduced from 0.0382°, 0.0132°, and 0.0034° to 0.0266°, 0.0115°, and 0.0027°, respectively.Remark 3.
The position accuracy is significantly improved by the proposed method, because the length of the first-level lever arm is long and the output angle is quite different from the initial angle. However, the improvements of velocity and attitude accuracy are small, because the servo was controlled to keep stable in the survey phase, and the influence of the dynamic lever arm effect on velocity and attitude is little.


## 4.3.2. Second-Level Dynamic Lever Arm Error Compensation Result

For the second-level dynamic lever arm error compensation of the experiment system in Figure5, two experiment results are obtained. One adopts the proposed method, while the other considers the second-level dynamic lever arm a rigid one. The output angles of the photoelectric encoder used in the experiment system in Figure 5 are shown in Figure 10, from which we can learn that the servo made a greatly adjustment during 300 s–450 s.Figure 10
The output angles of the photoelectric encoder used in the experiment system in Figure5.The errors of FOG POS in Figure5 calculated by the real-time Kalman filtering method with the proposed lever arm error compensation method and the conventional rigid lever arm error compensation method are shown in Figures 11–13 and Table 4.Figure 11
Position error comparison of the second-level dynamic lever arm error compensation.Figure 12
Velocity error comparison of the second-level dynamic lever arm error compensation.Figure 13
Attitude error comparison of the second-level dynamic lever arm error compensation.Table 4
Measurement error comparison of the second-level dynamic lever arm error compensation.
Parameters measured by the main POS
The measurement error (RMSE)
Conventional rigid lever arm error compensation
Proposed dynamic lever arm error compensation
Position (m)
Latitude
0.5470
0.2921
Longitude
0.5738
0.2805
Height
0.1816
0.0544
Velocity (m/s)
East velocity
0.0230
0.0210
North velocity
0.0283
0.0255
Upward velocity
0.0140
0.0132
Attitude (°)
Heading
0.0530
0.0221
Pitch
0.1224
0.0062
Roll
7.0999
0.0045With the proposed first-level dynamic lever arm error compensation, the latitude, longitude, and height errors have been reduced from 0.5470 m, 0.5738 m, and 0.1816 m to 0.2921 m, 0.2805 m, and 0.0544 m, respectively. The east, north, and upward velocity errors have been reduced from 0.0230 m/s, 0.0283 m/s, and 0.0140 m/s to 0.0210 m/s, 0.0255 m/s, and 0.0132 m/s, respectively. The heading, pitch, and roll angle errors have been reduced from 0.0530°, 0.1224°, and 7.0999° to 0.0221°, 0.0062°, and 0.0045°, respectively.Remark 4.
The attitude accuracy is significantly improved by the proposed method, because the SAR is not fixed on IMU, and the attitude error will be insufferable without dynamic compensation. However, the improvements of position and velocity accuracy are small, because the  length of second dynamic lever arm is short, and the influence of the dynamic lever arm effect on position and velocity is little.Above all, it is validated that the proposed dynamic lever arm error compensation method can effectively compensate the dynamic lever arm error and achieve higher accuracy and more extensive application for airborne POS.


## 5. Conclusion

The dynamic lever arm error compensation method of POS used for airborne Earth observation has been proposed in this paper. Firstly, the conventional rigid lever arm error compensation method is presented. Secondly, the first-level and the second-level dynamic lever arm error compensation methods are derived in detail, respectively. Finally, two actual flight experiments are carried out to validate the proposed method, which include both the first-level and the second-level dynamic lever arm error compensations. The results show that the derived dynamic lever arm compensation method outperforms the conventional one. Therefore, the proposed dynamic lever arm error compensation method can effectively compensate the dynamic lever arm error and achieve higher accuracy and more extensive application for airborne POS.



---

*Source: 9464568-2018-05-09.xml*