Deformation Measuring Methods Based on Inertial Sensors for Airborne Distributed POS

This paper is focused on deformation measuring methods based on inertial sensors, which are used to achieve high accuracy motion parameters and the spatial distribution optimization of multiple slave systems in the airborne distributed Position and Orientation Systemor other purposes. In practical application, the installation difficulty, cost, and accuracy ofmeasuring equipment are the key factors that need to be considered synthetically. Motivated by these, deformation measuring methods based on gyros and accelerometers are proposed, respectively, and compared with the traditional method based on the inertial measurement unit (IMU). The mathematical models of these proposed methods are built, and the detailed derivations of them are given. Based on the Kalman filtering estimation, simulation and semiphysical simulation based on vehicle experiment show that the method based on gyros can obtain a similar estimation accuracy to the method based on IMU, and the method based on accelerometers has an advantage in y-axis deformation estimation.


Introduction
he airborne distributed Position and Orientation System (POS) has been proposed to achieve multipoint spatiotemporal motion parameters for synthetical earth observation systems with multiple remote sensing loads [1][2][3].Distributed POS can be composed of a few high precision master systems, some low precision slave systems, POS Computer System (PCS), and postprocessing sotware.Usually, the master system is a high precision integrated system of Strapdown Inertial Navigation System and Global Navigation Satellite S y s t e m [ 4 ]( a l s oc a l l e dt h em a i nP O S ) .h es l a v es y s t e mi s only an inertial measurement unit (IMU), which consists of three orthogonally mounted gyros and accelerometers, respectively, and is placed as close as possible to the location o ft h el o a d .hes l a v es y s t e m s ,a l s oc a l l e dt h es u b -I M U s , depend on the master system to transfer alignment to achieve their high accuracy motion parameters.Due to the deformation of aircrat caused by gust, turbulence, and other factors, there is a time-varying and complex lexure angle between the main POS and each sub-IMU besides the rigid misalignment angle. he schematic diagram of the measuring system and the cross section of aircrat with deformation at a certain moment are shown in Figure 1, where the grey part with dotted line is the ideal state of the wings without any deformation.It is clear to see that the premise and key of high accuracy transfer alignment is the attitude transformation, determined by lexure and misalignment angle between the master system and slave system, which can be estimated and compensatedwithhighaccuracy.
Furthermore, when there are many remote sensing loads working simultaneously, airborne distributed array antenna Synthetic Aperture Radar (SAR) is a typical example which has many subantennas on both sides of the wing; the high accuracy motion parameters of each load must be measured [5,6].Since the bearing capacity of aircrat is limited,especiallythewingsection,thereareverystringen t requirements on the weight and size of the measurement equipment, while the measurement accuracy of sub-IMU is proportional to the weight and size.It should be noted that a high accuracy sub-IMU may not be available at the location of each load, and the positions of sub-IMU and load are not 2 International Journal of Aerospace Engineering Master system Slave system Slave system Slave system Slave system

Cabin
Wing with deformation Wing with deformation always matched with each other.herefore, it is necessary to consider the arrangement optimization of the distributed POS, such that the high precision motion parameters of a l ll o a d sc a nb eo b t a i n e du s i n gt h em i n i m u ms u b -I M U s .And the arrangement optimization also requires the attitude transformation between each node [7][8][9].
At present, aircrat deformation measuring methods can be summarized into three types: strain sensor measurement, optical measurement, and inertial measurement.he strain sensor measurement can be traced back to the 1940s and later i tw a si m p r o v e db yS k o p i n s k ie ta l .[ 1 0 , 1 1 ] .I ti sak i n do f mechanical measuring method which is widely used because of its convenient operation.However, it has limitation to the aircrat material and needs many wires which will increase the load of the aircrat.Besides, the strain sensor is easily afected by the physical abrasion, temperature, and so forth [12][13][14].As for optical measurement, the Dutch National Aerospace Laboratory (NLR) used a camera to record the black and white striped pattern on the wing surface to estimate the lexible deformation [15].hen, NLR presented a noncontact optical measurement which can obtain the deformation rule [16].In addition, there are other optical measurements using visual sensors, optical iber sensors, and bionic optical sensors to measure the lexible deformation [17][18][19].All those optical measurements need the external measurement components and the beam transceivers must be intervisible, which make them not only complex to be installed, but also prone to be afected by the weather conditions.Inertial measurement is mainly based on the IMUs which are installed at the places of both main and subnodes.he diference of the navigation results between the main node and subnode, such as attitude diference and velocity diference, is utilized to estimate the lexible deformation.his procedure is known as the transfer alignment [20,21].
Compared with IMU, three gyros or three accelerometers will reduce the weight, cost, and size of the measurement equipment.Particularly, high precision accelerometers have signiicant advantages of small size, light weight, low cost, and convenient installation when compared with the IMUs and gyros.It is necessary and signiicant to study the deformation measurement using gyros or accelerometers only.herefore, this paper deduces the formulations of deformation measurement based on gyros (Gs) and accelerometers (As), respectively, and provides the mathematical modeling and algorithm design of deformation measuring method based on IMU, Gs, and As.Finally, taking the deformation measurement of the wing as an example, the measurement accuracies of these three methods are compared and analyzed by the simulation experiment based on the Kalman ilter (KF). he rest of this paper is organized as follows.In Section 2, the mathematical modeling of deformation measuring algor i t h m sb a s e do nI M U ,G s ,a n dA si sg i v e n .I nS e c t i o n3 , detailed numerical simulation and semiphysical simulation are performed.Section 4 concludes this paper.

Mathematical Modeling
he detailed mathematical modeling of the three deformation measuring algorithms based on inertial sensors is given in this section.he details of KF can be found in [22,23].
h em e a s u r e m e n ts y s t e m sa tt h em a i nn o d ea n dt h e subnode can be called the master system and slave system, respectively.he coordinate frames used in this paper are deined as follows: and denote the earth-centered inertial frameandtheearth-centeredearth-ixedframe,respectively. he navigation frames of the master system and slave system (an IMU or three gyros or three accelerometers) are deined with ---axes pointing to east-north-up (E-N-U), represented by and , respectively.he body frames of the master system and slave system are deined on the rigid body of the inertial measurement unit and are denoted by and , respectively.A detailed description of these coordinate frames is available in [21,24,25].

Mathematical Model of Deformation
Measuring Algorithm Based on IMU.his method needs three orthogonally mounted gyros and accelerometers, respectively, on each note.For distributed POS, the main POS can be used as the master system and the sub-IMU can be used as the slave system. he mathematical model for KF includes the state equation and measurement equation.
where C is the coordinate transformation matrix fromframe to -frame, and it is an orthogonal matrix; system error angle u = + ,w h e r e =[ ] T and =[ ] T represent rigid misalignment angle and lexure angle between the master system and slave system, respectively.he following are all the error equations based on the deined coordinate frames.Firstly, the inertial navigation error equation of the sub-IMU is given in (2), which includes attitude error equation, velocity error equation, position error equation, and inertial sensor constant error equation [26,27]: where the subscripts E, N, and U represent east, north, and up, respectively; T is the attitude error vector in -frame; is the rotation velocity of the -frame relative to the -frame expressed in -frame with error ; C is the coordinate transformation matrix from -frame to -frame; isthegyrorandomdritofthesla vesystem in -frame, which consists of random constant drit and Gaussian white noise [28,29] with T is the speciic force measured by the accelerometers of sub-IMU expressed in -frame; is the rotation velocity of the -frame relative to the -frame expressed in -frame with error ; is the rotation velocity of the -frame relative to the -frame expressed in -frame with error ; ∇ is the accelerometer random bias of slave system in -frame, which consists of random constant bias ∇ and Gaussian white noise T ; M and N d e n o t et h em e r i d i a na n dt r a n s v e r s er a d i u so f curvature, respectively; and denote the latitude and altitude, respectively.he symbols , ,a n d denote theerrorofthelatitude,longitude,andaltitude,respectively .sec =1/cos .
Secondly, the diferential equation of rigid misalignment angle and lexure angle is shown in the following equations: where the lexure angle is described by the second-order Markov process [30]; = 2.146/ and is the correlation time; , ,a n d represent the axis of -frame; is the Gaussian white noise with covariance 2.1.2.System State Equation.he system state equation can be described as follows: where the subscript represents the deformation measuring algorithm based on IMU; X is a 24 × 1 error state vector deined by X =[ E N U E N U ∇ ∇ ∇ ] T ; F and G are system transition matrix and system noise distribution matrix, respectively; the elements of F and G can be obtained from ( 2) and (3); system noise vector =[ T is the zero-mean Gaussian white noise vector with covariance Q which consists of covariance Q of gyro random drit, covariance Q ∇ of accelerometer random bias, and Q .

Measurement Equation.
Based on the velocity plus attitude matching method, the measurement equation is given by where measurement vector =[ T denotes the diferences of velocity between the slave system and master system ater compensation for lever arm velocity; , ,a n d denote the diferences of heading, pitch, and roll between the slave system and master system, respectively; measurement V is computed by where V is the velocity of the master system in -frame and the lever arm velocity V canbecalculatedby[30] where C is the coordinate transformation matrix fromframe to -frame; is the rotation velocity of thefra m er e la ti v et oth e-frame expressed in the -frame; r is the lever arm between the master system and slave system expressed in the -frame.

Mathematical Model of Deformation Measuring Algorithm
Based on Gs. his method needs three orthogonally mounted gyrosastheslavesystemandmainPOSasthemastersystem. he deformation is estimated by using the diference of gyros between the master system and slave system as the measurement of KF.
2.2.1.State Vector Selection.he rigid misalignment angle, lexure angle and its derivative, and gyro constant drits of the master system and slave system are selected as the state vector: where subscript represents the deformation measuring algorithm based on Gs; the symbols and ( = , , ) denote gyro random constant drit of the master system and slave system, respectively.

System State Equation. he system state equation is given by
where F and G are system transition matrix and system noise distribution matrix, respectively; system noise vector W =[ ] T is a zero-mean Gaussian white noise vector with covariance Q = Q . he diferential equation of rigid misalignment angle and lexure angle is the same as in (3). he diferential equation of gyro random constant drit is given by According to (3) and (11), F and G are given as follows: where ] , 2.2.3.Measurement Equation.he relationship between gyro outputs of the master system and slave system can be expressed by where gyro angular velocity of the master system and slave system, respectively; =[ ] T ; is the gyro random drit of the master system in -frame, which consists of random constant drit and Gaussian white noise .According to (1) and ( 14), the diference of gyro output between the master and slave systems is where B a s e do n( 1 7 ) ,t h em e a s u r e m e n te q u a t i o ni sg i v e na s follows: where measurement vector Z =Δ Ω;measuremen tma trix T is a zero-mean Gaussian white noise sequence with covariance R .

Mathematical Modeling of Deformation
Measuring Algorithm Based on As.In this method, taking three orthogonally mounted accelerometers as the slave system and main POS as the master system, the diference of accelerometers between the master system and slave system is selected as the measurement of KF to estimate the deformation.
2.3.1.State Vector Selection.he state vector is deined by where subscript represents the deformation measuring algorithm based on As; ∇ and ∇ ( = , , ) are accelerometer random constant bias of the master system and slave system, respectively.

System State Equation. he system state equation is given by
where F and G are system transition matrix and system noise distribution matrix, respectively; system noise vector W =[ ] T is a zero-mean Gaussian white noise vector with covariance Q = Q . he diferential equation of rigid misalignment angle and lexure angle is the same as in (3). he diferential equation of accelerometer random constant bias is given by ∇ he expression of F and G can be obtained from (3) and (21).Furthermore, F = F and G = G .

Measurement Equation.
he relationship between the accelerometer outputs of the master system and slave system canbeexpressedby where are the speciic forces measured by the accelerometers of the master system and slave system, respectively; ∇ is the accelerometer random bias of the master system in -frame, which consists of random constant bias ∇ and Gaussian white noise ∇ ; a is the relative acceleration between the master system and slave system caused by lever arm; a is given by According to (1) and ( 22), the diference of accelerometer output between the master system and slave system is where Substituting Basedon (26),themeasurementequationis where measurement vector Z =Δ f;m e a s u r e m e n tm a t r i x T is a zero-mean Gaussian white noise sequence with covariance R .

Simulation and Semiphysical Simulation
In order to verify the estimation efect of methods based on GsandAs,respectively,proposedinSection2,alongrodwith a master system and a slave system installed on both ends is a better way.But in this case, it is not possible to know the exact value of deformation between the master system and slave system.herefore, light simulation and vehicle semiphysicalsimulationareprovidedandcomparedwiththe method based on IMU in this section.3.1.Flight Simulation and Analysis.From the attitude transfer relationship shown in (1), it can be seen that the attitude diference between the master system and slave system, also called the system error angle, is determined by the sum of rigid misalignment angle and lexure angle.herefore, not only should the estimation accuracy of the lexure angle or the rigid misalignment angle be evaluated, but also the estimation accuracy of the system angle error should be evaluated.In connection with this, the estimation error curves of lexure angle, rigid misalignment angle, and system error angle are given in the simulation, and the estimation error of the system error angle is used to evaluate the measuring precision of each method.

Design of Simulation.
In this paper, the typical "S + U"shaped trajectory of airborne earth observation is simulated.he plane trajectory and trajectory parameters are shown in Figure 2 and Table 1, respectively.Total light time is 1300 s.AB and CD section can be regarded as the imaging section in Figure 2. Initial heading angle, pitch angle, and roll angle are 40 ∘ ,0 ∘ ,and0 ∘ , respectively.he light velocity is 100 m/s and thealtitudeis500m.hissimulationhasbeenperformedtentimes.
he measurement noise of the main POS (as the master system) at heading, pitch, roll, and velocity are 0.02 ∘ (1), 0.005 ∘ (1), 0.005 ∘ (1), and 0.03 m/s (1), respectively.Both gyro constant drit and random drit of the main POS are 0.01 ∘ /h.Both accelerometer constant bias and random bias o ft h em a i nP O Sa n ds l a v es y s t e ma r e50 g.Both gyro constant and random drit of the slave system are 0.1 ∘ /h. he misalignment angle of the slave system relative to the main POS is given as =[ 0.5 ∘ 0.5 ∘ 0.5 ∘ ] T ,a n dt h el e v e r arm between the main POS and slave system is r = [5 m 0.1 m 0.1 m] T . he data update rate of the main system and slave system is 100 Hz.For the deformation measurement of the wing, the lexure angle rotated around the vertical axis is big, while the lexure angles around the other two axes are small.Accordingly, the correlation times of the second-order Markov processes are selected as 2, 5, and 2, and the covariances of lexure angle are 0.01, 0.15, and 0.01, respectively.he curves of lexure angle and lexure angle rate areshowninFigures3and4.
Data Generation.A trajectory generator is used to generate the theoretical data of the scheduled light trajectory, which i n c l u d ep o s i t i o n ,v e l o c i t y ,a t t i t u d e ,a n dt h eo u t p u td a t ao f gyros and accelerometers.he real outputs of the main POS are obtained by adding the corresponding measurement noise to the theoretical position, velocity, and attitude.hen, t h et h e o r e t i c a lo u t p u t so fg y r o sa n da c c e l e r o m e t e r sa r e converted by rigid misalignment angles and lexure angles, and the constant noise and random noise are added to be the inertial sensor outputs of the slave system.

Simulation Results
Analysis.Figures 5-7 show the estimate error curves of deformation measuring method based on IMU, Gs, and As, respectively, including the estimate errors of rigid misalignment angle, lexure angle, and system error angle.For improving the estimate accuracy, a maneuver is added in the S-shape of the light trajectory and the system error angle estimate errors are shown in Figure 8. he details ofthemaneuverareasfollows:thelightvelocityofS-shapeis increasedequablyfrom100m/sto300m/sbetween100sand 200 s and then decreased equably to 100 m/s between 200 s and 300 s. his kind of maneuver is very easy to implement for earth observation aircrat, because it is usually necessary to makeSorothertypesofmaneuvertoimprovetheestimation precision of POS before the aircrat enters the observation area.
Besides, the Root Mean Square Error (RMSE) and Standard Deviation (STD) values of system error angle estimate errors in imaging segments AB and CD are counted and shown in Table 2, where all values are the average of ten simulations.Since the maneuver mentioned above only afectsthemethodbasedonGs(itcanbeseenfromT able2), onlythesystemerrorangleestimateerrorsofmethodbased on IMU, Gs without this maneuver, and method based on Gs with this maneuver are shown in Figure 8.
Figures 5 and 6 show that the estimation accuracies of lexure angle and rigid misalignment angle are not good at the same time and coupled with each other, while the sum of two estimation errors of system error angle shown in Figure 7 is relatively stable on each axis.
Figures 7 and 8 and Table 2 show that the deformation measurement based on IMU has the highest estimation accuracy without any additional maneuver, followed by the m e t h o db a s e do nG s ,a n dt h em e t h o db a s e do nA si st h e worst. he velocity maneuvers in S-shape can improve the estimation efect of the method based on Gs, especially  the RMSE, and obtain the estimation precision close to the method based on IMU.It is worth mentioning that although themethodbasedonAshasverypoorestimationaccuracyon -axis and -axis,theSTDandRMESofsystemerrorangle estimation error on -axis are only 3.3 and 5 ,r espectively , which is very small relative to the lexure angle on -axis shown in Figure 3.

Semiphysical Simulation and Analysis.
A real road experiment is carried out in Shahe Town, Changping District, Beijing, China.In this vehicle experiment, a high precision POS is used as the main system, whose gyros, accelerometers, position, velocity, and attitude output are recorded.Based on these data, the lexure and misalignment sets in Section 3 a r eadded ,a ndthenthetheo r eticalda tao fthesla v esys t em can be obtained.Ater considering the error of the gyro and accelerometerintheslavesystem,therealoutputofthegyros and accelerometers in the slave system can be simulated.[33]. he trajectory of the test is shown in Figure 11 and the total test time is 1500 s. Figure 12 shows the van's velocity and acceleration.
he speciications of the high-precision POS inertial sensors applied in the laser gyro-based IMU and position, velocity, and attitude postprocessing output are listed in Table 3. he inertial sensors' errors of the simulated slave system are the same as these used in Section 3. he output rate of the main POS is 100 Hz.

Semiphysical Simulation Results
Analysis.Figure 13 shows the system error angle estimate error curves of deformation measuring method based on IMU, Gs, and As.In  order to see the changing trends of errors more clearly, the error curves of the irst two methods are shown in Figure 14, where L1 and L2 are two long straight segments in the trajectory. he statistics of system error angle estimate errors ofL1andL2aregiveninT able4.F i g u r e s1 3a n d1 4a n dT a b l e4s h o wt h a tt h ee s t i m a t i o n accuracy of the method based on Gs is close to that of IMU method.In the horizontal direction, the estimation accuracy of the IMU method is more stable and the Gs method is greatly inluenced by the turning of the vehicle; in the vertical direction, since there is not any maneuver and the vehicle began to move in a straight line, the estimation accuracy of the IMU method is gradually improved with the turn maneuver, and iltering is also gradually stabilized, while the Gs method can converge quickly. he method based on As has the largest estimation error, but its STD and RMES of system error angle estimation error on -axis are no more than 30 .Overall, the vehicle experiment results and the simulation results are basically similar and consistent.

Conclusion
In this paper, the mathematical models of deformation measurement based on Gs and As are derived, respectively, and compared with the deformation measuring method based on IMU. he results of simulation and semiphysical simulation show that the measuring method based on Gs can achieve similar estimation accuracy to the method based on IMU.Since the method based on IMU has the disadvantages of l a r g es i z e ,h i g hc o s t ,a n dh e a vyw e i g h t ,w h e nt h ea c c u r a c y requirement is not very high and can be met by the method b a s e do nG so rA s ,t h em e a s u r e m e n te q u i p m e n tw i l lb e greatly simpliied.In particular, the method based on As has a good estimation accuracy on -axis, which is a better choice for the cases with large deformation only on -axis and strict   limitation on the weight, size, and cost of the measurement equipment.In view of the deformation measuring method based on Gs and As, the intrinsic relationship between the maneuver and the estimation accuracy needs further analysis and research in the future.

Figure 1 :
Figure 1: he schematic diagram of the measuring system and the cross section of aircrat.

Figure 2 :
Figure 2: Plane trajectory with S-shaped maneuver and U-shaped light.

3. 2 . 1 .
Hardware Coniguration.he van and the sensors installation are shown in Figures 9 and 10, respectively.he high-precision POS, developed by the Integrated System Research Group at Beihang University, Beijing, China, consists of a laser gyro-based IMU, a PCS, and Novatel DLV-3 GPS receiver (based on Novatel OEMV-3 receiver board) and has mobile station and base station equipment with 20 Hz output rate

Figure 14 :
Figure 14: System error angle estimate error of methods based on IMU and Gs, respectively, in semiphysical simulation.

Table 1 :
Parameters setting of simulation trajectory.

Table 2 :
Estimate errors of system error angle in simulation test ( ).

Table 4 :
Estimate errors of system error angle in experiment test ( ).