Research ArticleA Real-Time Estimation Method of Roll Angle and Angular RateBased on Geomagnetic Information

Lizhen Gao12 Yingying Zhang 1 Xiaoming Zhang12 and Yuyang Xue1

1National Key Laboratory for Electronic Measurement Technology North University of China Taiyuan 030051 China2Key Laboratory of Instrumentation Science amp Dynamic Measurement of Ministry of Education North University of ChinaTaiyuan 030051 China

Correspondence should be addressed to Yingying Zhang 19834406826163com

Received 22 August 2020 Revised 6 October 2020 Accepted 13 October 2020 Published 27 October 2020

Academic Editor Tao Zhang

Copyright copy 2020 Lizhen Gao et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the course of the guidance transformation of the rotating projectile the accurate acquisition of the roll angle and roll angle rateis very important to the attitude determination and guidance control of the rotating projectile However due to the impact of highrotation and high overload of projectile MEMS gyros have problems such as limited range saturation overload and evenperformance degradation which make the roll angle rate unable to be output normally At the same time because theMEMS gyroestimation of roll angle is in the form of angular rate integral the roll angle cannot be estimated normally if the roll angle ratecannot be accurately obtained In order to solve this problem a real-time estimation of projectile roll angle and roll rate based ongeomagnetic information under high dynamic and high overload conditions is presented Firstly according to the motioncharacteristics of the rotating projectile the motion model of the projectile is established and the roll angle and roll angle rate ofthe projectile are estimated by Kalman filtering algorithm under the conditions of high axial rotation and high overloadConsidering the high dynamic characteristics of the rotating projectile based on the Kalman filter the algorithm of the forgettingfilter with the forgetting factor is further adopted to estimate the roll angle and roll angle rate so as to reduce the error caused bythe estimation delay in the process of high-speed dynamic change Simulation data and semiphysical test results show that theaccuracy of roll angle estimated by this method reaches about 2deg in semiphysical test which is one time higher than that calculatedby the system In the semiphysical experiment the accuracy of the estimated roll rotation rate reaches 5 degs which is more than 6times higher than that obtained by direct derivation In the high dynamic stage compared with the pure Kalman filter theaccuracy of roll angle with forgetting factor estimation is improved by an order of magnitude and the accuracy of roll angle rate isimproved by 4 times which meets the desired accuracy of rotating projectile

1 Introduction

In the course of the guidance transformation of conventionalammunition there are some problems such as high rota-tional speed and high dynamic state in the flight state of theprojectile so that the gyro cannot accurately measure theangular velocity of the projectile body roll For example inthe case of a rotating projectile the maximum axial rotationspeed of the projectile can be up to 200 rs during thelaunching process and it is accompanied by a large changeIn this case of high dynamic and high rotation speed theordinary MEMS gyro is already saturated and cannotnormally measure angular rate and solve the roll angle e

acquisition of the roll angle of the projectile is the keytechnology of the control of the rotating projectile channelOnly by acquiring the roll angle of the projectile accuratelycan the angle position of the steering gear be determined soas to guide the projectile and hit the target accuratelythrough the control system of the guided ammunition Atthe same time the acquisition of the roll angular rate is toprovide damping loop for the projectile control system andfurther improve the dynamic performance of projectilecontrol which is one of the essential factors in the process ofprojectile controlerefore it is necessary to have a methodto estimate the gyro-free roll angle and roll rotation rateunder the harsh environment

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9035710 11 pageshttpsdoiorg10115520209035710

MEMS gyro is commonly used in the rolling anglemeasurement of missile unmanned aerial vehicle and otherunmanned attitude measurement In the flight environmentof high axial rotary speed and high overload this kind ofsystem has some problems such as performance degrada-tion or even failure after overload and initial alignmentdifficulty which makes it impossible to continue attitudeestimation erefore in recent years more and moresystems use geomagnetic information to obtain roll anglee rolling angle measurement scheme using geomagneticinformation has the advantages of small space occupationfast response speed good antihigh overload performance noerror accumulation over time low cost etc [1 2] and can benormally solved under high overload and high dynamicflight environment However due to the magnetic fieldcharacteristics its roll rotation angle is easily interfered bythe projectile body magnetic field and the accuracy of rollrotation is closely related to the measurement accuracy ofgeomagnetic field

In order to estimate the axial speed of the projectile byusing magnetic measurement information time-frequencydomain analysis methods such as STFT (Short-time-Four-ier-transform) [3ndash5] differential filtering [6] zero passdetection [7 8] and deformation Kalman filtering algorithmcombined with peak detection are currently employed [9]but these methods all have their own defects For examplethe STFT estimation method uses Fourier transform toestimate the angular rate In the case of high dynamics thereis a contradiction between the accuracy and real-timeperformance of this method in estimating the roll angularrate that is the longer the interval time the higher theestimation accuracy but the worse the real-time perfor-mance e shorter the interval time the better the real-timeestimation but the lower the estimation frequency accuracyDifferential filtering method is to get the angular rate in-formation by difference of magnetic measurement signale error of this method is large and the angular rate errorobtained by difference will be larger when there is a mea-surement error of magnetic signal After low-pass filtering[10] the estimation error can be reduced but the estimationresult will be delayed Similarly the rate of change obtaineddirectly by magnetic signals is also greatly affected bymeasurement errors e detection method of zero crossingis to obtain angular rate information by measuring the timeof each zero crossing through sinusoidal magnetic mea-surement signal However data updating rate of this methodis different under the influence of rotational speed that iswhen the axial rotational speed is fast the angular rateinformation updates quickly when the axial rotational speedis too slow the angular rate information updates slowly andthe information update rate cannot be determined

n addition for only using the magnetometer to obtainthe angular rate and other attitude information also throughthe establishment of the projectile dynamics model Li et alintroduces a kind of gyro angular rate estimation method[11] Natanson and others introduce a three-axis magne-tometer measurement spacecraft attitude and angular rateRTSF (real-time) sequential filtering method and Ma andXu put forward improved real-time sequential filtering

method (IRTSF) [12] Sabzevari et al use the magnetometer[13 14] to estimate the attitude in vehicle dynamics model isestablished on the basis of calculating However in theprocess of high-speed flight the force and torque includingaerodynamic force subjected to the projectile are verycomplex therefore accurate dynamic model cannot beestablished Hu et al proposed a new filtering method basedon UKF which is suitable for combined inertial navigationand GPSmeasurement [15] On the contrary the change rateof roll angle obtained directly by magnetic signal is greatlyaffected by the measurement error [16] erefore a simplefast practical and high accuracy algorithm is needed toestimate the real-time roll angle and roll angle rate of therotating projectile

In this paper the roll angle and angular rate estimationbased on geomagnetic information are estimated when theprojectilersquos torque is unknown Firstly the kinematics modelis established according to the motion characteristics of theprojectile Based on the accurate estimation of projectile rollangle and roll angle rate by the Kalman filter in view of thedynamic characteristics of projectile motion and high speedappropriate forgetting factor is added for strong tracking tochange the measurement noise and system noise in real timeimprove the adaptive ability of the algorithm [17] us theattitude information of roll angle and roll angle rate nec-essary for guidance and control of guided munitions underbad onboard environment is obtained

2 Establish Coordinate System

Geomagnetic attitude measurement system is mainly basedon the information of geomagnetic field Although geo-magnetic field is a global long-term changing magnetic fieldit changes slowly and is almost constant in a very shortperiod of time compared with the rapid launch of ammu-nition erefore geomagnetic information can be widelyused in aviation aerospace navigation and other fields ethree-axis magnetic sensor can measure the three compo-nents of geomagnetic field in each position in real time Ithas the advantages of not accumulating errors with time lowcost and high antioverload performance Using the threecomponents of the geomagnetic field at the initial time oflaunch and the information of the three components of thegeomagnetic field at any time during the projectilersquos flightthe attitude angle of the projectile can be obtained throughthe transformation of the coordinate system [18 19] Afterthat roll angle and roll rotation rate can be further esti-mated so the first thing is to establish the coordinate system

In order to obtain the motion attitude angle of theprojectile body relative to the initial moment two coordinatesystems need to be established namely the launching co-ordinate system f system (O minus XfYfZf) and the projectilebody coordinate system b system (Oprime minus XbYbZb)e launchas a benchmark coordinate system belongs to the staticcoordinate system in order to determine the initial positionof the projectile the origin is located at the centroid at thelaunch site the X-axis is from origin to destination Y-axisvertically points inside the centroid transverse section Z-axis within the centroid transverse section pointing to the

2 Mathematical Problems in Engineering

right level and coordinates measured three component

Hfx H

fy H

fz1113960 1113961

Tis the initial three-component magnetic

field e missile system belongs to the moving coordinatesystem which is used to represent the motion attitude of theprojectile at any moment of its flight relative to the initiallaunch time Its origin is located in the center of mass of theprojectile and it moves with the projectilersquos flight motione X-axis is along the direction of the projectile axis andthe Y-axis and Z-axis are located in the transverse section ofthe center of mass and rotate with the projectile around theX-axis e geomagneticfield threecomponents Hb

x Hby Hb

z1113960 1113961Tmeasured in the missile co-

ordinate system is the three-axis magnetic field value at eachposition and attitude during the flight of the projectile Bothcoordinate systems satisfy the right-hand rule as shown inFigure 1

e transformation matrix of the launching system tothe projectile system can be obtained according to theprojection relation and rotation mode and the rotationmatrix results are different with different rotation orders Inthis paper rotation is carried out according to the rotationmode of 231 that is the yaw angle rotation matrix Cψ is

obtained by first rotating about the Y-axis en rotateabout the Z-axis to get the pitch angle rotation matrix CθFinally rotate about the X-axis to get the roll rotation matrixCc e details are shown below

Cψ

cos ψ 0 minus sin ψ

0 1 0

sin ψ 0 cos ψ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Cθ

cos θ sin θ 0

minus sin θ cos θ 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Cc

1 0 0

0 cos c sin c

0 minus sin c cos c

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(1)

Finally according to the rotation order of Y-Z-Xmultiply the three rotation matrices to obtain the rotationmatrix Cb

f of the launching system to the projectile system asfollows

Cbf CxCzCy CcCθCψ

cos ψ cos θ sin θ minus sin ψ cos θ

sin ψ sin c minus cos ψ sin θ cos c cos θ cos c cos ψ sin c + sin ψ sin θ cos c

sin ψ cos c + cos ψ sin θ sin c minus cos θ sin c cos ψ cos c minus sin ψ sin θ sin c

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)

us the relationship between the three componentsHf

of the initial magnetic field in the emission system and thethree components Hb of the magnetic field in the carriercoordinate system can be obtained

Hb C

bnH

f

Hb Hb

x Hby Hb

z1113960 1113961T

Hf H

fx H

fy H

fz1113960 1113961

T

(3)

3 Estimation Algorithm

31 Magnetic Information to Solve the Roll From the abovesection it can be seen that the attitude angle of the projectilebody can be solved by using geomagnetic field informationWhen the three components of the projectile bodyrsquos mag-netic field and the three components of the projectile bodyrsquosmagnetic field are known one of the attitude angles must beknown to solve the other two attitude angles Due to the factthat the yaw angle of the guided munitions such as therotary bomb is small in the trajectory fire plane duringflight the yaw angle can be set as 0 and the followingformula can be obtained

Hbx

Hby

Hbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

cos θ sin θ 0

minus sin θ cos c cos θ cos c sin c

sin θ sin c minus cos θ sin c cos c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Hfx

Hfy

Hfz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (4)

us the formula of pitch angle and roll angle of theprojectile can be obtained as follows

θ arctanH

nx

Hnz

1113888 1113889 minus arcsinH

bx

Hnx( 1113857

2+ H

nz( 1113857

21113969

Hbx

Hnx( 1113857

2+ H

nz( 1113857

21113969

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

c arctanH

nx sin θ + H

nz cos θ( 1113857H

by minus H

nyH

bz

Hnx sin θ + H

nz cos θ( 1113857H

bz + H

nyH

by

⎡⎢⎢⎣ ⎤⎥⎥⎦

(5)

32 Estimation of Roll Angle and Roll Rate In the kinematicsof the projectilersquos external trajectory the attenuation law ofthe axial rotational speed of the projectilersquos external ballistictrajectory can be obtained according to the Roguery formula[20]

_c _c0 exp minus 0075kLD

3

At1113888 1113889 (6)

Mathematical Problems in Engineering 3

where L is the projectile length D is the projectile diameterA is the moment of inertia of the projectile pole and k is thecoefficient

us we can know that the axial rotation speed ω of theprojectile decreases exponentially with time t e rotationalaxial rotational speed of the projectile conforms to theflexible formula in the uncontrolled-free flight stage or in thesingle control period so the roll angular velocity of theprojectile can be regarded as the change of uniform de-celeration in a short time According to this characteristic aquadratic kinematics equation can be established as theequation of state

c(t) 12at2 + bt + c (7)

where a b and c are constantsSuppose in the flight process of the rotating projectile

the rotational angle rotational angle rate and rotationalangle acceleration at time T are c(t) ω(t) and a(t) re-spectively From the above formula it can be seen that theangular velocity in a short time is uniform in other wordsthe roll angle plus acceleration is a random variable and theroll angle acceleration a(t) is driven by white noise If thewhite noise is j(t) the mean value of the system noise is zeroand the covariance matrix is Q and the following kinematicrelationship can be obtained

_c(t) ω(t)

_ω(t) a(t)

_a(t) j(t)

(8)

where E[jt] 0 and E[jtjTk ] Qtδtk

en the state equation of the model can be written asfollows

_X(t) FX(t) + Gj(t) (9)

where

X(t)

c(t)

ω(t)

a(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F

0 1 0

0 0 1

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

After that the roll angle calculated by the magneticmeasurement system is taken as the observation value andthe measurement model can be obtained as follows

Zt HXt + Vt (11)

where H is the measurement matrix

H 1 0 01113858 1113859 (12)

Vt is the measurement noise the mean value is zero andthe covariance matrix is R e equivalent of E[Vt] 0 andE[VtV

Tk ] Rtδtk

When the sampling period is Ts the model is discretizedand the results are as follows

Xk ϕkkminus 1Xkminus 1 + 1113946k

kminus 1ϕ(k τ)Gj(τ)dτ (13)

where ϕkkminus 1 I + FTs + (T2S2)F2 + (T3

S3)F3 + middot middot middotWhen the sampling interval is small that is the sampling

frequency is large the higher order term can be omitted andthe one-step transfer matrix is

ϕttminus 1

1 Ts 0

0 1 Ts

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (14)

At the same time the system noise driving array can beobtained as follows

Γ 0T2s2

Ts1113890 1113891

T

(15)

en the final equation of state of the system is

Xk ϕkkminus 1Xkminus 1 + Γjkminus 1 (16)

where Xk ck ωk ak1113858 1113859T

In the process of rotating projectile launch there areabrupt changes in the projectile roll angular rate which donot conform to the model setting of short-time uniformacceleration and thus the model is not accurate enough-erefore for the estimation of roll angle and roll angle ratethe measurement noise and system noise of the modelshould be appropriately modified to improve the weight ofnew information and reduce the interference of the previousestimation results to the current time estimation

Assuming that in the angular rate mutation stage thevariance matrix of measurement noise V(k) and systemnoise W(k) are respectively

E VkVTj1113960 1113961 s

Nminus kRkδkj

E WkWTj1113960 1113961 s

Nminus k+1Qkδkj

(17)

4 Mathematical Problems in Engineering

where k and j are both a certain moment in time series N Itcan be seen that when s is 1 the noise variance matrix doesnot change When s is greater than 1 the noise variancematrix increases According to the case that the model es-timation error increases when the angular rate changes sshould be a real number slightly greater than 1 us a newone-step prediction mean square error matrix and gainmatrix are

Pkkminus 1 ϕkkminus 1Pkminus 1ϕTkkminus 1 + Γkminus 1s

Nminus kQkminus 1Γ

Tkminus 1

Kk Pkkminus 1HTk HkPkkminus 1H

Tk + s

Nminus kRk1113872 1113873

minus 1

(18)

By multiplying the one-step prediction mean squareerror of the above equation by sminus (Nminus k) we can obtain

sminus (Nminus k)

Pkkminus 1 ϕkkminus 1sminus (Nminus k)

Pkminus 1ϕTkkminus 1 + Γkminus 1Qkminus 1Γ

Tkminus 1

(19)

Assume that

Plowastkkminus 1 ϕkkminus 1 sP

lowastkminus 1( 1113857ϕT

kkminus 1 + Γkminus 1Qkminus 1ΓTkminus 1 (20)

Similarly multiply the left and right sides of the esti-mated mean square error by sminus (Nminus k) to simplify and arrangethe final model equation as follows

1113954Xlowastkkminus 1 ϕkkminus 1 lowast 1113954X

lowastkminus 1

Plowastkkminus 1 ϕkkminus 1 sPlowastkminus 1( 1113857ϕT

kkminus 1 + Γkminus 1Qkminus 1ΓTkminus 1

Klowastk Plowastkkminus 1H

Tk HkP

lowastkkminus 1H

Tk + Rk1113872 1113873

minus 1

1113954Xlowastk 1113954X

lowastkkminus 1 + K

lowastk Zk minus Hk

1113954Xlowastkkminus 11113872 1113873

Plowastk I minus KkHk( 1113857P

lowastkkminus 1

(21)

where 1113954Xlowastkminus 1 1113954X

lowastk 1113954Xlowastkkminus 1 and Klowastk respectively represent the

estimated value at time k minus 1 and time k as well as the stateprediction and gain at time k and s is the forgetting factor

It can be seen that the simplified filtering model can onlybe multiplied by a forgetting factor before the estimatedmean square error at the previous moment

4 Results and Discussion

41 Validation of Simulation Data According to the abovemodel the initial value is given for simulation verification

First a set of roll angle and roll angle rate data with angularrate mutation are generated by simulation and appro-priate measurement error is added and then the algo-rithm is used for estimation Q 22 R 0012 and s 12are set in this group of simulation data e estimatedresults and the original generated data are shown in thefigure below

In the above simulation results it can be concluded fromFigures 2 and 3 that the estimation algorithm can simul-taneously estimate the roll angle and roll angle rate of theprojectile From Figures 3ndash5 it can be seen that the angularrate mutation exists at 0 s and 5 s It can be seen fromFigures 4 and 5 that the error of the roll angle and roll anglerate estimated by the Kalman filter is large due to the

Roll rotation rate

0 02 040

1000

2000

5 52 54

ndash3600

ndash3500

ndash3400

ndash3300

ndash4000

ndash3000

ndash2000

ndash1000

0

1000

2000

3000

4000

Roll

rota

tion

rate

(degs

)

10 150 5t (s)

Generated bysimulationEstimated by pureKalman filter

Estimated by forgettingfactor algorithmEstimated by the directderivation method

Figure 3 e roll angle rate of the generated and the estimated

0 5 10 15t (s)

Roll angle

0

50

100

150

200

250

300

350

400

Roll

angl

e (deg)

The roll angle measuredEstimated by pure Kalman filterEstimated by forgetting factor algorithm

Figure 2 e roll angles of the generated and the estimated

Position at any timeprojectile body coordinate

system (b)

Initial position-launching coordinate system (b) The traget

Ballistic plane

yn

znxn

o

o

zb (yb)

yb (zb)

xb

Figure 1 e relationship between the two coordinates

Mathematical Problems in Engineering 5

constraint of the estimation model while the error of thealgorithm estimated by adding the forgetting factor will begreatly reduced e error of roll angle and roll angle rate atthe angular rate mutation is shown in the table below

It can be seen from Table 1 that in the angular ratemutation stage compared with the pure Kalman filter es-timation the estimation accuracy of the algorithm with theforgetting factor is improved by an order of magnitude and

The error of roll angle rate

0 02 04

ndash1500

ndash1000

ndash500

0

5 52 54ndash80ndash60ndash40ndash20

020

1335 134 1345 135 1355 136 1365 137ndash20

0

20

ndash4000

ndash3500

ndash3000

ndash2500

ndash2000

ndash1500

ndash1000

ndash500

0

500

The e

rror

of r

oll a

ngle

rate

(degs

)

10 150 5t (s)

Estimation error of pure Kalman filter estimationEstimation error with forgetting factor algorithmThe error solved by direct derivation

Figure 5 e error of the roll angle rate

5

0

ndash5

ndash10

ndash15

ndash25

ndash20

The e

rror

of r

oll a

ngels

(deg)

The error of roll angles

0

ndash10

ndash20

0

ndash1

ndash20 05

002

0

ndash002

5 52 54

1346 1348 135 1352 1354

151050t (s)

Estimation error of pure Kalman filterEstimation error with forgetting factor algorithmError of measurement

Figure 4 e error of roll angles

Table 1 e estimation error of the algorithm in the angular rate mutation stage

e stage of angularrate mutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)Estimated by pure

Kalman filterEstimated by the algorithm

with forgetting factorEstimated by pure

Kalman filterEstimated by the algorithm

with forgetting factor0 sndash055 s minus 23443 minus 02431 minus 13915 minus 7737215 sndash56 s minus 04660 minus 00080 minus 168456 minus 16650

6 Mathematical Problems in Engineering

the accuracy of the roll rotation rate is increased by morethan 4 times It can be seen from Figure 5 that in thestationary phase the accuracy of the roll rotation rate es-timated by the algorithm is 10 times higher than that ob-tained by direct derivation e validity of the algorithm isverified

42 Validation of Semiphysical Turntable Data After thealgorithm is verified it is verified according to the data of themagnetic measurement system With the three-axis mag-netometer HMC1053 produced by Honeywell company asthe only attitude sensing chip and the STM32 single chipmicrocomputer produced by ST Company as the controllerthe control circuit was designed to constitute the magneticattitude measurement system [21 22] and the magneticmeasurement system was fixed on the three-axis high-precision flight simulation turntable as shown in Figure 6e control table rotates around the X-axis Y-axis and Z-axis respectively and the roll angle is calculated by using themagnetic measurement system and the roll angle and theroll rotation rate are optimized and estimated by taking themagnetic measurement as the observation data

In this experiment the gyroscope is saturated and it isimpossible to further estimate the carrier roll angle so thegyro information is only used to estimate the roll angle rateestimated by the algorithm in the later stage

To simulate the flight state of the projectile body underthe maneuvering condition the flight turntable wasaccelerated uniformly around the X-axis to 5 rs and thendecelerated uniformly to 0 rs after maintaining 21 s asshown in Figure 7 In the model parameter setting Q 22R 052 and s 103 and the simulation results are shownin the figures

e estimated roll angle and roll rotation rate are shownin Figures 7 and 8 It can be seen from Figure 8 that at theangular rate mutation of 1 s 3 s and 23 s the roll angle androll rate estimated by the pure Kalman filter have large errorHowever the error estimated by adding forgetting factoralgorithm is significantly smaller as shown in Table 2

As can be seen from Table 2 in the angular rate mu-tation stage compared with the estimation result of thepure Kalman filter the roll angle accuracy estimated by thealgorithm with the forgetting factor is improved by anorder of magnitude e accuracy of the estimated rollrotation rate is improved by more than 4 times It can alsobe seen from Figures 9 and 10 that in the stationary phasethe roll angle error estimated by the algorithm is within 2degwhich is more than twice as accurate as the roll error of 5degmeasured e error of the roll rotation rate estimated bythe algorithm is within 5 degs which improves the accuracyby an order of magnitude compared with the error of theroll rotation rate obtained by direct derivation of 50 degswhich verifies the feasibility of the algorithm and thesystem

43 Verification of Bomb Test Data e feasibility of thealgorithm is verified by theoretical analysis and turntablesemiphysical simulation test Now the sensor data

Figure 6 ree-axis high-precision flight simulation turntable

10 15 25200 5t (s)

0

50

100

150

200

250

300

350

400

Roll

ange

l (deg)

Roll angle

Estimated by pureKalman filterMeasured roll angle

Estimated by forgettingfactor algorithmThe turntable feedback

Figure 7 e roll angles of turntable feedback and algorithmestimation

2 3 4

2000

1500

1000

1850

1800

1750122 124 126 128

1800

1600

1400

1200

1000

800

23 235 24

Roll rotation rate

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodThe turntable feedback

20 25 30151050t (s)

ndash1000

ndash500

0

500

1000

1500

2000

2500

Roll

rota

tion

rate

(degs

)

Figure 8 e roll angles rate of turntable feedback and algorithmestimation

Mathematical Problems in Engineering 7

collected from the ballistic flight test of the range is used forfurther verification A magnetic measuring system and anaxial MEMS gyro are installed in the test projectile bodye roll rotation rate of the gyro output is taken as areference to verify the accuracy of the estimated roll

rotation and roll rotation rate calculated by using only themagnetometer data e test results are shown in thefigures

Roll angel error

15

10

5

0

ndash523 24 25

12 122 124

5

0

ndash52 3 4

10

0

ndash10

5 150 35302010 25t (s)

ndash20

ndash15

ndash10

ndash5

0

5

10

15

Roll

ange

l err

or (deg

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

Figure 9 e error of the roll angle

1 2 3

100

0

ndash100

100

0

ndash10011 115 12 125 13

150100

500

ndash50235 24 245

Error of roll rotation rate

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

5 15 20 302510t (s)

ndash500

ndash400

ndash300

ndash200

ndash100

0

100

200

Erro

r of r

oll r

otat

ion

rate

(degs

)

Figure 10 e error of the roll angle rate

Sensor data

1510 200 5t (s)

0

05

1

15

2

25

3

Volta

ge v

alue

(V)

X axis of magneticsensorY axis of magneticsensor

Z axis of magneticsensorThe axial gyro

Figure 11 Sensor data of magnetic and gyro

10 20 30 40 50 60 700t (s)

0

100

200

300

400

Roll

Ang

le (deg

)

(a)

10 20 30 40 50 60 700t (s)

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

(b)

Figure 12 e system calculates the roll angle and the angular rateof the gyro output (a) e role angle that system calculates (b)Rotation rate of gyro type

Table 2 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular ratemutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)

Estimated by pure Kalmanfilter

Estimated by thealgorithm with forgetting

factor

Estimated bypure

Kalman filter

Estimated by thealgorithm

with forgetting factor1 sndash2 s minus 82261 minus 05716 minus 745018 minus 412543 sndash4 s 66852 minus 06276 776076 15466023 sndash24 s 73149 minus 00166 784181 160643

8 Mathematical Problems in Engineering

e figure above shows the system output and algorithmestimation results of the bomb test It can be seen fromFigure 11that the effective flight time of the ballistic test is 20 s It can beseen from Figures 12 and 13 that the gyro is saturated duringflight and cannot normally calculate the roll rotation rateHowever the roll rotation rate estimated by the algorithmmakesup for this defect e roll angle and roll rotation rate estimatedby the algorithm are shown in Figures 13 and 14 Figure 14shows that the roll angle estimated by the algorithm is betterthan the linearity of the roll angle calculated directly by thesystem which indicates that the roll angle estimated by thealgorithm compensates some errors caused by the systemmeasurement Figure 13 shows that the roll rotation rate

estimated by the algorithm compensates for the error caused bygyro saturation in the first two seconds In the stationary phasethe accuracy of the roll rotation rate estimated by the algorithmis 6 times higher than that obtained by direct derivation Fig-ure 15 shows that in the angular rate mutation stage the rollangular rate estimated by forgetting factor reduces the errorcaused by pure Kalman filter estimation and the mean value ofthe estimated error caused by angular rate mutation is shown inthe table

It can be seen from Table 3 that the angular rate at 11 sdoes not change much so the effect of the algorithm with theforgetting factor is not obvious However in other abruptchanges of angular rates the accuracy of the algorithm with

Roll rotation rate

5000

0

ndash5000

ndash1000005 1 15 2

2000

1800

1600

165 17 175 18

5 10 15 200t (s)

ndash8000

ndash6000

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodGyro calculating

Figure 13 e roll angle rate that the system calculates and the gyro output

Roll angle

72 74 76 78 8

3020 35 4025105 150t (s)

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

Roll

angl

e (deg)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmThe system calculates

Figure 14 e roll angle that the system calculates and the gyro output

Mathematical Problems in Engineering 9

the forgetting factor is more than 4 times higher than that ofthe pure Kalman filter

5 Conclusions

In this paper we propose a method to estimate the roll angleand roll angle rate of a projectile by using only the magneticfield information provided by a triaxial magnetometer and areal-time estimation algorithm based on the Kalman filterwith appropriate forgetting factor is proposed is methodsolves the problem that the projectile roll angle and roll anglerate cannot be obtained due to MEMS gyro overload anddegradation under the flight condition of high spin and highoverload e Kalman filter estimation algorithm with theoblivion factor is able to significantly reduce the error causedby estimation delay under high dynamic conditions

rough the above analysis and semiphysical simulationtest it can be concluded that the algorithm can estimate theroll angle and roll angle rate of the carrier in real time andquickly e experimental results show that the algorithmwith the forgetting factor reduces the influence of magneticsensor measurement error on the accuracy of roll angle andimproves the accuracy of roll angle by one time e ex-perimental results show that the error of the roll rotation rateestimated by this algorithm is within 5 degs and the accuracyis 6 times higher than that obtained by direct derivation Inthe angular rate mutation phase compared with the pureKalman filter estimation algorithm the accuracy of the roll

angle estimated by the algorithm that the Kalman filter withthe forgetting factor is improved by an order of magnitudeand the accuracy of the roll angle rate is improved by at leastfour times which canmeet the requirements of the projectileroll angle and roll angle rate of the guidance and controlsystem of general rotating bombs

Data Availability

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

Disclosure

Lizhen Gao and Yingying Zhang are co-first authors of thisarticle

Conflicts of Interest

e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was carried out in accordance with the require-ments of the National Natural Science Foundation of China(61873247) funded project corresponding test experimentswere carried out in the State Key Laboratory of ElectronicTesting Technology of North China University Key

Error of roll rotation rate

0 05 1 15 2 25ndash5000

0

5000

115 12 125 13

ndash200

0

200

20 21ndash1000

0

1000

2000

1510 200 5t (s)

ndash6000

ndash4000

ndash2000

0

2000

4000

6000

Erro

r of r

oll r

otat

ion

rate

(degs

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmEstimated by the direct derivation method

Figure 15 e error of roll angle rate that the system calculates and the gyro output

Table 3 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular rate mutationMean of roll angle rate error (s)

Estimated by pure Kalman filter Estimated by the algorithm with forgetting factor0 sndash02 s 14505 minus 31891042 sndash25 s 5706310 1440300115 sndash12 s 813683 62837720 sndash204 s 8241634 2084635

10 Mathematical Problems in Engineering

Laboratory of Instrumental Science and Dynamic Testing ofNorth China University Huaihai Industry Group inChangzhi City Shanxi Province and Alashan ShootingRange During this period the authors also got instructionsfrom Professor Zhang Xiaoming and Teacher Li Xiuyuane authors would like to thank them for their help andsupport

References

[1] L An L Wang and D Zhao ldquoAttitude determinationmethod of spinning projectile based on geomagnetic azi-muthrdquo Journal of Chinese Inertial Technology vol 27 no 5pp 618ndash624 2019

[2] A Grosz E Paperno S Amrusi and B Zadov ldquoA three-axialsearch coil magnetometer optimized for small size low powerand low frequenciesrdquo IEEE Sensors Journal vol 11 no 4pp 1088ndash1094 2011

[3] H Liu H Dong J Ge B Bai Z Yuan and Z Zhao ldquoResearchon a secondary tuning algorithm based on SVD amp STFT forFID signalrdquo Measurment Science and Technology vol 27no 10 pp 0957ndash0233 Article ID 105006 2016

[4] J Shang D Zhihong M Fu and S Wang ldquoA high-spin ratemeasurement method for projectiles using a magnetoresistivesensor based on time-frequency domain analysisrdquo Sensorsvol 16 no 6 p 894 2016

[5] C Mateo and J A Talavera ldquoShort-time fourier transformwith the window size fixed in the frequency domainrdquo DigitalSignal Processing vol 77 no 6 pp 13ndash21 2018

[6] X Yan G Chen and X Tian ldquoTwo-step adaptive augmentedunscented Kalman filter for roll angles of spinning missilesbased on magnetometer measurementsrdquo Measurement andControl vol 51 no 3-4 pp 73ndash82 2018

[7] T Addabbo R Biondi S Cioncolini A Fort F Rossetti andV Vignoli ldquoA zero-crossing detection system based on FPGAto measure the angular vibrations of rotating shaftsrdquo IEEETransactions on Instrumentation and Measurement vol 63no 12 pp 3002ndash3010 2014

[8] Y Zhou X Zhang and W Xiao ldquoSpinning projectilersquos an-gular measurement using crest and trough data of a geo-magnetic sensorrdquo Measurment Science and Technologyvol 29 no 9 Article ID 095007 2018

[9] H Zhao Z Su F Liu C Li Q Li and N Liu ldquoExtraction andfilter algorithm of roll angular rate for high spinning pro-jectilesrdquo Mathematical Problems in Engineering vol 2019Article ID 3181727 15 pages 2019

[10] S Carletta and P Teofilatto ldquoDesign and numerical validationof an algorithm for the detumbling and angular rate deter-mination of a CubeSat using only three-axis magnetometerdatardquo International Journal of Aerospace Engineeringvol 2018 Article ID 9768475 12 pages 2018

[11] L-B Li M-X Li L-X Jiang D-Y Wang F Zhan andT Sheng ldquoAngular rate estimation and damping control ofsatellite with magnetometer datardquo Optik vol 180 no 11pp 1049ndash1055 2019

[12] H Ma and S Xu ldquoMagnetometer-only attitude and angularvelocity filtering estimation for attitude changing spacecraftrdquoActa Astronautica vol 102 no 5 pp 89ndash102 2014

[13] S Sabzevari M R Arvan A R Vali S M M Dehghan andM H Ferdowsi ldquoSymmetry preserving nonlinear observer forattitude estimation with magnetometer onlyrdquo ISA Transac-tions vol 102 no 3 pp 314ndash324 2020

[14] J M Maley ldquoEfficient attitude estimation for a spin-stabilizedprojectilerdquo Journal of Guidance Control and Dynamicsvol 39 no 2 pp 1ndash12 2016

[15] G Hu W Wang Y Zhong B Gao and C Gu ldquoA new directfiltering approach to INSGNSS integrationrdquo Aerospace Sci-ence and Technology vol 77 no 7 pp 755ndash764 2018

[16] M Yunjian X Changfan J Yixian W Yao and Z YildquoAngular velocity estimation of rollingmdashammunition basedon magnetometerrdquo Journal of Projectiles Rockets Missiles andGuidance vol 36 no 1 pp 69ndash72 2016

[17] G Hu L Ni B Gao X ZhuWWang and Y Zhong ldquoModelpredictive based unscented Kalman filter for hypersonic ve-hicle navigation with INSGNSS integrationrdquo IEEE Accessvol 8 no 1 pp 4814ndash4823 2016

[18] C Chunhang L Chunsheng J Wendou et al ldquoe projectileattitude measuring method based on geomagnetic sensorrdquoJournal of Detection amp Control vol 40 no 12 pp 4814ndash48232020

[19] X Chao X-z Bu and Y Bo ldquoree different attitudemeasurements of spinning projectile based on magneticsensorsrdquo Measurement vol 47 no 1 pp 331ndash340 2014

[20] W Yu ldquoHalf-experiential formulas for calculating decreasingangular velocity of projectile in trajectoryrdquo Journal of De-tection and Control vol 231 no 5 pp 866ndash876 2003

[21] J Yu X Bu C Xiang and B Yang ldquoSpinning projectilersquosattitude measurement using intersection ratio of magneticsensorsrdquo Proceedings of the Institution of Mechanical Engi-neers Part G-Journal of Aerospace Engineering vol 231 no 5pp 1ndash6 2016

[22] X Zhao X Zhang D Long Z Bai and Y Wang ldquoe designof roll angle magnetic measurement system used in spinningprojectilesrdquo Chinese Journal of Sensors and Actuators vol 26no 9 pp 1309ndash1313 2013

Mathematical Problems in Engineering 11

right level and coordinates measured three component

Hfx H

fy H

fz1113960 1113961

Tis the initial three-component magnetic

field e missile system belongs to the moving coordinatesystem which is used to represent the motion attitude of theprojectile at any moment of its flight relative to the initiallaunch time Its origin is located in the center of mass of theprojectile and it moves with the projectilersquos flight motione X-axis is along the direction of the projectile axis andthe Y-axis and Z-axis are located in the transverse section ofthe center of mass and rotate with the projectile around theX-axis e geomagneticfield threecomponents Hb

x Hby Hb

z1113960 1113961Tmeasured in the missile co-

ordinate system is the three-axis magnetic field value at eachposition and attitude during the flight of the projectile Bothcoordinate systems satisfy the right-hand rule as shown inFigure 1

e transformation matrix of the launching system tothe projectile system can be obtained according to theprojection relation and rotation mode and the rotationmatrix results are different with different rotation orders Inthis paper rotation is carried out according to the rotationmode of 231 that is the yaw angle rotation matrix Cψ is

obtained by first rotating about the Y-axis en rotateabout the Z-axis to get the pitch angle rotation matrix CθFinally rotate about the X-axis to get the roll rotation matrixCc e details are shown below

Cψ

cos ψ 0 minus sin ψ

0 1 0

sin ψ 0 cos ψ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Cθ

cos θ sin θ 0

minus sin θ cos θ 0

0 0 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Cc

1 0 0

0 cos c sin c

0 minus sin c cos c

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(1)

Finally according to the rotation order of Y-Z-Xmultiply the three rotation matrices to obtain the rotationmatrix Cb

f of the launching system to the projectile system asfollows

Cbf CxCzCy CcCθCψ

cos ψ cos θ sin θ minus sin ψ cos θ

sin ψ sin c minus cos ψ sin θ cos c cos θ cos c cos ψ sin c + sin ψ sin θ cos c

sin ψ cos c + cos ψ sin θ sin c minus cos θ sin c cos ψ cos c minus sin ψ sin θ sin c

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)

us the relationship between the three componentsHf

of the initial magnetic field in the emission system and thethree components Hb of the magnetic field in the carriercoordinate system can be obtained

Hb C

bnH

f

Hb Hb

x Hby Hb

z1113960 1113961T

Hf H

fx H

fy H

fz1113960 1113961

T

(3)

3 Estimation Algorithm

31 Magnetic Information to Solve the Roll From the abovesection it can be seen that the attitude angle of the projectilebody can be solved by using geomagnetic field informationWhen the three components of the projectile bodyrsquos mag-netic field and the three components of the projectile bodyrsquosmagnetic field are known one of the attitude angles must beknown to solve the other two attitude angles Due to the factthat the yaw angle of the guided munitions such as therotary bomb is small in the trajectory fire plane duringflight the yaw angle can be set as 0 and the followingformula can be obtained

Hbx

Hby

Hbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

cos θ sin θ 0

minus sin θ cos c cos θ cos c sin c

sin θ sin c minus cos θ sin c cos c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Hfx

Hfy

Hfz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (4)

us the formula of pitch angle and roll angle of theprojectile can be obtained as follows

θ arctanH

nx

Hnz

1113888 1113889 minus arcsinH

bx

Hnx( 1113857

2+ H

nz( 1113857

21113969

Hbx

Hnx( 1113857

2+ H

nz( 1113857

21113969

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

c arctanH

nx sin θ + H

nz cos θ( 1113857H

by minus H

nyH

bz

Hnx sin θ + H

nz cos θ( 1113857H

bz + H

nyH

by

⎡⎢⎢⎣ ⎤⎥⎥⎦

(5)

32 Estimation of Roll Angle and Roll Rate In the kinematicsof the projectilersquos external trajectory the attenuation law ofthe axial rotational speed of the projectilersquos external ballistictrajectory can be obtained according to the Roguery formula[20]

_c _c0 exp minus 0075kLD

3

At1113888 1113889 (6)

Mathematical Problems in Engineering 3

where L is the projectile length D is the projectile diameterA is the moment of inertia of the projectile pole and k is thecoefficient

us we can know that the axial rotation speed ω of theprojectile decreases exponentially with time t e rotationalaxial rotational speed of the projectile conforms to theflexible formula in the uncontrolled-free flight stage or in thesingle control period so the roll angular velocity of theprojectile can be regarded as the change of uniform de-celeration in a short time According to this characteristic aquadratic kinematics equation can be established as theequation of state

c(t) 12at2 + bt + c (7)

where a b and c are constantsSuppose in the flight process of the rotating projectile

the rotational angle rotational angle rate and rotationalangle acceleration at time T are c(t) ω(t) and a(t) re-spectively From the above formula it can be seen that theangular velocity in a short time is uniform in other wordsthe roll angle plus acceleration is a random variable and theroll angle acceleration a(t) is driven by white noise If thewhite noise is j(t) the mean value of the system noise is zeroand the covariance matrix is Q and the following kinematicrelationship can be obtained

_c(t) ω(t)

_ω(t) a(t)

_a(t) j(t)

(8)

where E[jt] 0 and E[jtjTk ] Qtδtk

en the state equation of the model can be written asfollows

_X(t) FX(t) + Gj(t) (9)

where

X(t)

c(t)

ω(t)

a(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F

0 1 0

0 0 1

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

After that the roll angle calculated by the magneticmeasurement system is taken as the observation value andthe measurement model can be obtained as follows

Zt HXt + Vt (11)

where H is the measurement matrix

H 1 0 01113858 1113859 (12)

Vt is the measurement noise the mean value is zero andthe covariance matrix is R e equivalent of E[Vt] 0 andE[VtV

Tk ] Rtδtk

When the sampling period is Ts the model is discretizedand the results are as follows

Xk ϕkkminus 1Xkminus 1 + 1113946k

kminus 1ϕ(k τ)Gj(τ)dτ (13)

where ϕkkminus 1 I + FTs + (T2S2)F2 + (T3

S3)F3 + middot middot middotWhen the sampling interval is small that is the sampling

frequency is large the higher order term can be omitted andthe one-step transfer matrix is

ϕttminus 1

1 Ts 0

0 1 Ts

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (14)

At the same time the system noise driving array can beobtained as follows

Γ 0T2s2

Ts1113890 1113891

T

(15)

en the final equation of state of the system is

Xk ϕkkminus 1Xkminus 1 + Γjkminus 1 (16)

where Xk ck ωk ak1113858 1113859T

In the process of rotating projectile launch there areabrupt changes in the projectile roll angular rate which donot conform to the model setting of short-time uniformacceleration and thus the model is not accurate enough-erefore for the estimation of roll angle and roll angle ratethe measurement noise and system noise of the modelshould be appropriately modified to improve the weight ofnew information and reduce the interference of the previousestimation results to the current time estimation

Assuming that in the angular rate mutation stage thevariance matrix of measurement noise V(k) and systemnoise W(k) are respectively

E VkVTj1113960 1113961 s

Nminus kRkδkj

E WkWTj1113960 1113961 s

Nminus k+1Qkδkj

(17)

4 Mathematical Problems in Engineering

where k and j are both a certain moment in time series N Itcan be seen that when s is 1 the noise variance matrix doesnot change When s is greater than 1 the noise variancematrix increases According to the case that the model es-timation error increases when the angular rate changes sshould be a real number slightly greater than 1 us a newone-step prediction mean square error matrix and gainmatrix are

Pkkminus 1 ϕkkminus 1Pkminus 1ϕTkkminus 1 + Γkminus 1s

Nminus kQkminus 1Γ

Tkminus 1

Kk Pkkminus 1HTk HkPkkminus 1H

Tk + s

Nminus kRk1113872 1113873

minus 1

(18)

By multiplying the one-step prediction mean squareerror of the above equation by sminus (Nminus k) we can obtain

sminus (Nminus k)

Pkkminus 1 ϕkkminus 1sminus (Nminus k)

Pkminus 1ϕTkkminus 1 + Γkminus 1Qkminus 1Γ

Tkminus 1

(19)

Assume that

Plowastkkminus 1 ϕkkminus 1 sP

lowastkminus 1( 1113857ϕT

kkminus 1 + Γkminus 1Qkminus 1ΓTkminus 1 (20)

Similarly multiply the left and right sides of the esti-mated mean square error by sminus (Nminus k) to simplify and arrangethe final model equation as follows

1113954Xlowastkkminus 1 ϕkkminus 1 lowast 1113954X

lowastkminus 1

Plowastkkminus 1 ϕkkminus 1 sPlowastkminus 1( 1113857ϕT

kkminus 1 + Γkminus 1Qkminus 1ΓTkminus 1

Klowastk Plowastkkminus 1H

Tk HkP

lowastkkminus 1H

Tk + Rk1113872 1113873

minus 1

1113954Xlowastk 1113954X

lowastkkminus 1 + K

lowastk Zk minus Hk

1113954Xlowastkkminus 11113872 1113873

Plowastk I minus KkHk( 1113857P

lowastkkminus 1

(21)

where 1113954Xlowastkminus 1 1113954X

lowastk 1113954Xlowastkkminus 1 and Klowastk respectively represent the

estimated value at time k minus 1 and time k as well as the stateprediction and gain at time k and s is the forgetting factor

It can be seen that the simplified filtering model can onlybe multiplied by a forgetting factor before the estimatedmean square error at the previous moment

4 Results and Discussion

41 Validation of Simulation Data According to the abovemodel the initial value is given for simulation verification

First a set of roll angle and roll angle rate data with angularrate mutation are generated by simulation and appro-priate measurement error is added and then the algo-rithm is used for estimation Q 22 R 0012 and s 12are set in this group of simulation data e estimatedresults and the original generated data are shown in thefigure below

In the above simulation results it can be concluded fromFigures 2 and 3 that the estimation algorithm can simul-taneously estimate the roll angle and roll angle rate of theprojectile From Figures 3ndash5 it can be seen that the angularrate mutation exists at 0 s and 5 s It can be seen fromFigures 4 and 5 that the error of the roll angle and roll anglerate estimated by the Kalman filter is large due to the

Roll rotation rate

0 02 040

1000

2000

5 52 54

ndash3600

ndash3500

ndash3400

ndash3300

ndash4000

ndash3000

ndash2000

ndash1000

0

1000

2000

3000

4000

Roll

rota

tion

rate

(degs

)

10 150 5t (s)

Generated bysimulationEstimated by pureKalman filter

Estimated by forgettingfactor algorithmEstimated by the directderivation method

Figure 3 e roll angle rate of the generated and the estimated

0 5 10 15t (s)

Roll angle

0

50

100

150

200

250

300

350

400

Roll

angl

e (deg)

The roll angle measuredEstimated by pure Kalman filterEstimated by forgetting factor algorithm

Figure 2 e roll angles of the generated and the estimated

Position at any timeprojectile body coordinate

system (b)

Initial position-launching coordinate system (b) The traget

Ballistic plane

yn

znxn

o

o

zb (yb)

yb (zb)

xb

Figure 1 e relationship between the two coordinates

Mathematical Problems in Engineering 5

constraint of the estimation model while the error of thealgorithm estimated by adding the forgetting factor will begreatly reduced e error of roll angle and roll angle rate atthe angular rate mutation is shown in the table below

It can be seen from Table 1 that in the angular ratemutation stage compared with the pure Kalman filter es-timation the estimation accuracy of the algorithm with theforgetting factor is improved by an order of magnitude and

The error of roll angle rate

0 02 04

ndash1500

ndash1000

ndash500

0

5 52 54ndash80ndash60ndash40ndash20

020

1335 134 1345 135 1355 136 1365 137ndash20

0

20

ndash4000

ndash3500

ndash3000

ndash2500

ndash2000

ndash1500

ndash1000

ndash500

0

500

The e

rror

of r

oll a

ngle

rate

(degs

)

10 150 5t (s)

Estimation error of pure Kalman filter estimationEstimation error with forgetting factor algorithmThe error solved by direct derivation

Figure 5 e error of the roll angle rate

5

0

ndash5

ndash10

ndash15

ndash25

ndash20

The e

rror

of r

oll a

ngels

(deg)

The error of roll angles

0

ndash10

ndash20

0

ndash1

ndash20 05

002

0

ndash002

5 52 54

1346 1348 135 1352 1354

151050t (s)

Estimation error of pure Kalman filterEstimation error with forgetting factor algorithmError of measurement

Figure 4 e error of roll angles

Table 1 e estimation error of the algorithm in the angular rate mutation stage

e stage of angularrate mutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)Estimated by pure

Kalman filterEstimated by the algorithm

with forgetting factorEstimated by pure

Kalman filterEstimated by the algorithm

with forgetting factor0 sndash055 s minus 23443 minus 02431 minus 13915 minus 7737215 sndash56 s minus 04660 minus 00080 minus 168456 minus 16650

6 Mathematical Problems in Engineering

the accuracy of the roll rotation rate is increased by morethan 4 times It can be seen from Figure 5 that in thestationary phase the accuracy of the roll rotation rate es-timated by the algorithm is 10 times higher than that ob-tained by direct derivation e validity of the algorithm isverified

42 Validation of Semiphysical Turntable Data After thealgorithm is verified it is verified according to the data of themagnetic measurement system With the three-axis mag-netometer HMC1053 produced by Honeywell company asthe only attitude sensing chip and the STM32 single chipmicrocomputer produced by ST Company as the controllerthe control circuit was designed to constitute the magneticattitude measurement system [21 22] and the magneticmeasurement system was fixed on the three-axis high-precision flight simulation turntable as shown in Figure 6e control table rotates around the X-axis Y-axis and Z-axis respectively and the roll angle is calculated by using themagnetic measurement system and the roll angle and theroll rotation rate are optimized and estimated by taking themagnetic measurement as the observation data

In this experiment the gyroscope is saturated and it isimpossible to further estimate the carrier roll angle so thegyro information is only used to estimate the roll angle rateestimated by the algorithm in the later stage

To simulate the flight state of the projectile body underthe maneuvering condition the flight turntable wasaccelerated uniformly around the X-axis to 5 rs and thendecelerated uniformly to 0 rs after maintaining 21 s asshown in Figure 7 In the model parameter setting Q 22R 052 and s 103 and the simulation results are shownin the figures

e estimated roll angle and roll rotation rate are shownin Figures 7 and 8 It can be seen from Figure 8 that at theangular rate mutation of 1 s 3 s and 23 s the roll angle androll rate estimated by the pure Kalman filter have large errorHowever the error estimated by adding forgetting factoralgorithm is significantly smaller as shown in Table 2

As can be seen from Table 2 in the angular rate mu-tation stage compared with the estimation result of thepure Kalman filter the roll angle accuracy estimated by thealgorithm with the forgetting factor is improved by anorder of magnitude e accuracy of the estimated rollrotation rate is improved by more than 4 times It can alsobe seen from Figures 9 and 10 that in the stationary phasethe roll angle error estimated by the algorithm is within 2degwhich is more than twice as accurate as the roll error of 5degmeasured e error of the roll rotation rate estimated bythe algorithm is within 5 degs which improves the accuracyby an order of magnitude compared with the error of theroll rotation rate obtained by direct derivation of 50 degswhich verifies the feasibility of the algorithm and thesystem

43 Verification of Bomb Test Data e feasibility of thealgorithm is verified by theoretical analysis and turntablesemiphysical simulation test Now the sensor data

Figure 6 ree-axis high-precision flight simulation turntable

10 15 25200 5t (s)

0

50

100

150

200

250

300

350

400

Roll

ange

l (deg)

Roll angle

Estimated by pureKalman filterMeasured roll angle

Estimated by forgettingfactor algorithmThe turntable feedback

Figure 7 e roll angles of turntable feedback and algorithmestimation

2 3 4

2000

1500

1000

1850

1800

1750122 124 126 128

1800

1600

1400

1200

1000

800

23 235 24

Roll rotation rate

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodThe turntable feedback

20 25 30151050t (s)

ndash1000

ndash500

0

500

1000

1500

2000

2500

Roll

rota

tion

rate

(degs

)

Figure 8 e roll angles rate of turntable feedback and algorithmestimation

Mathematical Problems in Engineering 7

collected from the ballistic flight test of the range is used forfurther verification A magnetic measuring system and anaxial MEMS gyro are installed in the test projectile bodye roll rotation rate of the gyro output is taken as areference to verify the accuracy of the estimated roll

rotation and roll rotation rate calculated by using only themagnetometer data e test results are shown in thefigures

Roll angel error

15

10

5

0

ndash523 24 25

12 122 124

5

0

ndash52 3 4

10

0

ndash10

5 150 35302010 25t (s)

ndash20

ndash15

ndash10

ndash5

0

5

10

15

Roll

ange

l err

or (deg

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

Figure 9 e error of the roll angle

1 2 3

100

0

ndash100

100

0

ndash10011 115 12 125 13

150100

500

ndash50235 24 245

Error of roll rotation rate

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

5 15 20 302510t (s)

ndash500

ndash400

ndash300

ndash200

ndash100

0

100

200

Erro

r of r

oll r

otat

ion

rate

(degs

)

Figure 10 e error of the roll angle rate

Sensor data

1510 200 5t (s)

0

05

1

15

2

25

3

Volta

ge v

alue

(V)

X axis of magneticsensorY axis of magneticsensor

Z axis of magneticsensorThe axial gyro

Figure 11 Sensor data of magnetic and gyro

10 20 30 40 50 60 700t (s)

0

100

200

300

400

Roll

Ang

le (deg

)

(a)

10 20 30 40 50 60 700t (s)

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

(b)

Figure 12 e system calculates the roll angle and the angular rateof the gyro output (a) e role angle that system calculates (b)Rotation rate of gyro type

Table 2 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular ratemutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)

Estimated by pure Kalmanfilter

Estimated by thealgorithm with forgetting

factor

Estimated bypure

Kalman filter

Estimated by thealgorithm

with forgetting factor1 sndash2 s minus 82261 minus 05716 minus 745018 minus 412543 sndash4 s 66852 minus 06276 776076 15466023 sndash24 s 73149 minus 00166 784181 160643

8 Mathematical Problems in Engineering

e figure above shows the system output and algorithmestimation results of the bomb test It can be seen fromFigure 11that the effective flight time of the ballistic test is 20 s It can beseen from Figures 12 and 13 that the gyro is saturated duringflight and cannot normally calculate the roll rotation rateHowever the roll rotation rate estimated by the algorithmmakesup for this defect e roll angle and roll rotation rate estimatedby the algorithm are shown in Figures 13 and 14 Figure 14shows that the roll angle estimated by the algorithm is betterthan the linearity of the roll angle calculated directly by thesystem which indicates that the roll angle estimated by thealgorithm compensates some errors caused by the systemmeasurement Figure 13 shows that the roll rotation rate

estimated by the algorithm compensates for the error caused bygyro saturation in the first two seconds In the stationary phasethe accuracy of the roll rotation rate estimated by the algorithmis 6 times higher than that obtained by direct derivation Fig-ure 15 shows that in the angular rate mutation stage the rollangular rate estimated by forgetting factor reduces the errorcaused by pure Kalman filter estimation and the mean value ofthe estimated error caused by angular rate mutation is shown inthe table

It can be seen from Table 3 that the angular rate at 11 sdoes not change much so the effect of the algorithm with theforgetting factor is not obvious However in other abruptchanges of angular rates the accuracy of the algorithm with

Roll rotation rate

5000

0

ndash5000

ndash1000005 1 15 2

2000

1800

1600

165 17 175 18

5 10 15 200t (s)

ndash8000

ndash6000

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodGyro calculating

Figure 13 e roll angle rate that the system calculates and the gyro output

Roll angle

72 74 76 78 8

3020 35 4025105 150t (s)

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

Roll

angl

e (deg)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmThe system calculates

Figure 14 e roll angle that the system calculates and the gyro output

Mathematical Problems in Engineering 9

the forgetting factor is more than 4 times higher than that ofthe pure Kalman filter

5 Conclusions

In this paper we propose a method to estimate the roll angleand roll angle rate of a projectile by using only the magneticfield information provided by a triaxial magnetometer and areal-time estimation algorithm based on the Kalman filterwith appropriate forgetting factor is proposed is methodsolves the problem that the projectile roll angle and roll anglerate cannot be obtained due to MEMS gyro overload anddegradation under the flight condition of high spin and highoverload e Kalman filter estimation algorithm with theoblivion factor is able to significantly reduce the error causedby estimation delay under high dynamic conditions

rough the above analysis and semiphysical simulationtest it can be concluded that the algorithm can estimate theroll angle and roll angle rate of the carrier in real time andquickly e experimental results show that the algorithmwith the forgetting factor reduces the influence of magneticsensor measurement error on the accuracy of roll angle andimproves the accuracy of roll angle by one time e ex-perimental results show that the error of the roll rotation rateestimated by this algorithm is within 5 degs and the accuracyis 6 times higher than that obtained by direct derivation Inthe angular rate mutation phase compared with the pureKalman filter estimation algorithm the accuracy of the roll

angle estimated by the algorithm that the Kalman filter withthe forgetting factor is improved by an order of magnitudeand the accuracy of the roll angle rate is improved by at leastfour times which canmeet the requirements of the projectileroll angle and roll angle rate of the guidance and controlsystem of general rotating bombs

Data Availability

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

Disclosure

Lizhen Gao and Yingying Zhang are co-first authors of thisarticle

Conflicts of Interest

e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was carried out in accordance with the require-ments of the National Natural Science Foundation of China(61873247) funded project corresponding test experimentswere carried out in the State Key Laboratory of ElectronicTesting Technology of North China University Key

Error of roll rotation rate

0 05 1 15 2 25ndash5000

0

5000

115 12 125 13

ndash200

0

200

20 21ndash1000

0

1000

2000

1510 200 5t (s)

ndash6000

ndash4000

ndash2000

0

2000

4000

6000

Erro

r of r

oll r

otat

ion

rate

(degs

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmEstimated by the direct derivation method

Figure 15 e error of roll angle rate that the system calculates and the gyro output

Table 3 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular rate mutationMean of roll angle rate error (s)

Estimated by pure Kalman filter Estimated by the algorithm with forgetting factor0 sndash02 s 14505 minus 31891042 sndash25 s 5706310 1440300115 sndash12 s 813683 62837720 sndash204 s 8241634 2084635

10 Mathematical Problems in Engineering

Laboratory of Instrumental Science and Dynamic Testing ofNorth China University Huaihai Industry Group inChangzhi City Shanxi Province and Alashan ShootingRange During this period the authors also got instructionsfrom Professor Zhang Xiaoming and Teacher Li Xiuyuane authors would like to thank them for their help andsupport

References

[1] L An L Wang and D Zhao ldquoAttitude determinationmethod of spinning projectile based on geomagnetic azi-muthrdquo Journal of Chinese Inertial Technology vol 27 no 5pp 618ndash624 2019

[2] A Grosz E Paperno S Amrusi and B Zadov ldquoA three-axialsearch coil magnetometer optimized for small size low powerand low frequenciesrdquo IEEE Sensors Journal vol 11 no 4pp 1088ndash1094 2011

[3] H Liu H Dong J Ge B Bai Z Yuan and Z Zhao ldquoResearchon a secondary tuning algorithm based on SVD amp STFT forFID signalrdquo Measurment Science and Technology vol 27no 10 pp 0957ndash0233 Article ID 105006 2016

[4] J Shang D Zhihong M Fu and S Wang ldquoA high-spin ratemeasurement method for projectiles using a magnetoresistivesensor based on time-frequency domain analysisrdquo Sensorsvol 16 no 6 p 894 2016

[5] C Mateo and J A Talavera ldquoShort-time fourier transformwith the window size fixed in the frequency domainrdquo DigitalSignal Processing vol 77 no 6 pp 13ndash21 2018

[6] X Yan G Chen and X Tian ldquoTwo-step adaptive augmentedunscented Kalman filter for roll angles of spinning missilesbased on magnetometer measurementsrdquo Measurement andControl vol 51 no 3-4 pp 73ndash82 2018

[7] T Addabbo R Biondi S Cioncolini A Fort F Rossetti andV Vignoli ldquoA zero-crossing detection system based on FPGAto measure the angular vibrations of rotating shaftsrdquo IEEETransactions on Instrumentation and Measurement vol 63no 12 pp 3002ndash3010 2014

[8] Y Zhou X Zhang and W Xiao ldquoSpinning projectilersquos an-gular measurement using crest and trough data of a geo-magnetic sensorrdquo Measurment Science and Technologyvol 29 no 9 Article ID 095007 2018

[9] H Zhao Z Su F Liu C Li Q Li and N Liu ldquoExtraction andfilter algorithm of roll angular rate for high spinning pro-jectilesrdquo Mathematical Problems in Engineering vol 2019Article ID 3181727 15 pages 2019

[10] S Carletta and P Teofilatto ldquoDesign and numerical validationof an algorithm for the detumbling and angular rate deter-mination of a CubeSat using only three-axis magnetometerdatardquo International Journal of Aerospace Engineeringvol 2018 Article ID 9768475 12 pages 2018

[11] L-B Li M-X Li L-X Jiang D-Y Wang F Zhan andT Sheng ldquoAngular rate estimation and damping control ofsatellite with magnetometer datardquo Optik vol 180 no 11pp 1049ndash1055 2019

[12] H Ma and S Xu ldquoMagnetometer-only attitude and angularvelocity filtering estimation for attitude changing spacecraftrdquoActa Astronautica vol 102 no 5 pp 89ndash102 2014

[13] S Sabzevari M R Arvan A R Vali S M M Dehghan andM H Ferdowsi ldquoSymmetry preserving nonlinear observer forattitude estimation with magnetometer onlyrdquo ISA Transac-tions vol 102 no 3 pp 314ndash324 2020

[14] J M Maley ldquoEfficient attitude estimation for a spin-stabilizedprojectilerdquo Journal of Guidance Control and Dynamicsvol 39 no 2 pp 1ndash12 2016

[15] G Hu W Wang Y Zhong B Gao and C Gu ldquoA new directfiltering approach to INSGNSS integrationrdquo Aerospace Sci-ence and Technology vol 77 no 7 pp 755ndash764 2018

[16] M Yunjian X Changfan J Yixian W Yao and Z YildquoAngular velocity estimation of rollingmdashammunition basedon magnetometerrdquo Journal of Projectiles Rockets Missiles andGuidance vol 36 no 1 pp 69ndash72 2016

[17] G Hu L Ni B Gao X ZhuWWang and Y Zhong ldquoModelpredictive based unscented Kalman filter for hypersonic ve-hicle navigation with INSGNSS integrationrdquo IEEE Accessvol 8 no 1 pp 4814ndash4823 2016

[18] C Chunhang L Chunsheng J Wendou et al ldquoe projectileattitude measuring method based on geomagnetic sensorrdquoJournal of Detection amp Control vol 40 no 12 pp 4814ndash48232020

[19] X Chao X-z Bu and Y Bo ldquoree different attitudemeasurements of spinning projectile based on magneticsensorsrdquo Measurement vol 47 no 1 pp 331ndash340 2014

[20] W Yu ldquoHalf-experiential formulas for calculating decreasingangular velocity of projectile in trajectoryrdquo Journal of De-tection and Control vol 231 no 5 pp 866ndash876 2003

[21] J Yu X Bu C Xiang and B Yang ldquoSpinning projectilersquosattitude measurement using intersection ratio of magneticsensorsrdquo Proceedings of the Institution of Mechanical Engi-neers Part G-Journal of Aerospace Engineering vol 231 no 5pp 1ndash6 2016

[22] X Zhao X Zhang D Long Z Bai and Y Wang ldquoe designof roll angle magnetic measurement system used in spinningprojectilesrdquo Chinese Journal of Sensors and Actuators vol 26no 9 pp 1309ndash1313 2013

Mathematical Problems in Engineering 11

where L is the projectile length D is the projectile diameterA is the moment of inertia of the projectile pole and k is thecoefficient

us we can know that the axial rotation speed ω of theprojectile decreases exponentially with time t e rotationalaxial rotational speed of the projectile conforms to theflexible formula in the uncontrolled-free flight stage or in thesingle control period so the roll angular velocity of theprojectile can be regarded as the change of uniform de-celeration in a short time According to this characteristic aquadratic kinematics equation can be established as theequation of state

c(t) 12at2 + bt + c (7)

where a b and c are constantsSuppose in the flight process of the rotating projectile

the rotational angle rotational angle rate and rotationalangle acceleration at time T are c(t) ω(t) and a(t) re-spectively From the above formula it can be seen that theangular velocity in a short time is uniform in other wordsthe roll angle plus acceleration is a random variable and theroll angle acceleration a(t) is driven by white noise If thewhite noise is j(t) the mean value of the system noise is zeroand the covariance matrix is Q and the following kinematicrelationship can be obtained

_c(t) ω(t)

_ω(t) a(t)

_a(t) j(t)

(8)

where E[jt] 0 and E[jtjTk ] Qtδtk

en the state equation of the model can be written asfollows

_X(t) FX(t) + Gj(t) (9)

where

X(t)

c(t)

ω(t)

a(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F

0 1 0

0 0 1

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

After that the roll angle calculated by the magneticmeasurement system is taken as the observation value andthe measurement model can be obtained as follows

Zt HXt + Vt (11)

where H is the measurement matrix

H 1 0 01113858 1113859 (12)

Vt is the measurement noise the mean value is zero andthe covariance matrix is R e equivalent of E[Vt] 0 andE[VtV

Tk ] Rtδtk

When the sampling period is Ts the model is discretizedand the results are as follows

Xk ϕkkminus 1Xkminus 1 + 1113946k

kminus 1ϕ(k τ)Gj(τ)dτ (13)

where ϕkkminus 1 I + FTs + (T2S2)F2 + (T3

S3)F3 + middot middot middotWhen the sampling interval is small that is the sampling

frequency is large the higher order term can be omitted andthe one-step transfer matrix is

ϕttminus 1

1 Ts 0

0 1 Ts

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (14)

At the same time the system noise driving array can beobtained as follows

Γ 0T2s2

Ts1113890 1113891

T

(15)

en the final equation of state of the system is

Xk ϕkkminus 1Xkminus 1 + Γjkminus 1 (16)

where Xk ck ωk ak1113858 1113859T

In the process of rotating projectile launch there areabrupt changes in the projectile roll angular rate which donot conform to the model setting of short-time uniformacceleration and thus the model is not accurate enough-erefore for the estimation of roll angle and roll angle ratethe measurement noise and system noise of the modelshould be appropriately modified to improve the weight ofnew information and reduce the interference of the previousestimation results to the current time estimation

Assuming that in the angular rate mutation stage thevariance matrix of measurement noise V(k) and systemnoise W(k) are respectively

E VkVTj1113960 1113961 s

Nminus kRkδkj

E WkWTj1113960 1113961 s

Nminus k+1Qkδkj

(17)

4 Mathematical Problems in Engineering

where k and j are both a certain moment in time series N Itcan be seen that when s is 1 the noise variance matrix doesnot change When s is greater than 1 the noise variancematrix increases According to the case that the model es-timation error increases when the angular rate changes sshould be a real number slightly greater than 1 us a newone-step prediction mean square error matrix and gainmatrix are

Pkkminus 1 ϕkkminus 1Pkminus 1ϕTkkminus 1 + Γkminus 1s

Nminus kQkminus 1Γ

Tkminus 1

Kk Pkkminus 1HTk HkPkkminus 1H

Tk + s

Nminus kRk1113872 1113873

minus 1

(18)

By multiplying the one-step prediction mean squareerror of the above equation by sminus (Nminus k) we can obtain

sminus (Nminus k)

Pkkminus 1 ϕkkminus 1sminus (Nminus k)

Pkminus 1ϕTkkminus 1 + Γkminus 1Qkminus 1Γ

Tkminus 1

(19)

Assume that

Plowastkkminus 1 ϕkkminus 1 sP

lowastkminus 1( 1113857ϕT

kkminus 1 + Γkminus 1Qkminus 1ΓTkminus 1 (20)

Similarly multiply the left and right sides of the esti-mated mean square error by sminus (Nminus k) to simplify and arrangethe final model equation as follows

1113954Xlowastkkminus 1 ϕkkminus 1 lowast 1113954X

lowastkminus 1

Plowastkkminus 1 ϕkkminus 1 sPlowastkminus 1( 1113857ϕT

kkminus 1 + Γkminus 1Qkminus 1ΓTkminus 1

Klowastk Plowastkkminus 1H

Tk HkP

lowastkkminus 1H

Tk + Rk1113872 1113873

minus 1

1113954Xlowastk 1113954X

lowastkkminus 1 + K

lowastk Zk minus Hk

1113954Xlowastkkminus 11113872 1113873

Plowastk I minus KkHk( 1113857P

lowastkkminus 1

(21)

where 1113954Xlowastkminus 1 1113954X

lowastk 1113954Xlowastkkminus 1 and Klowastk respectively represent the

estimated value at time k minus 1 and time k as well as the stateprediction and gain at time k and s is the forgetting factor

It can be seen that the simplified filtering model can onlybe multiplied by a forgetting factor before the estimatedmean square error at the previous moment

4 Results and Discussion

41 Validation of Simulation Data According to the abovemodel the initial value is given for simulation verification

First a set of roll angle and roll angle rate data with angularrate mutation are generated by simulation and appro-priate measurement error is added and then the algo-rithm is used for estimation Q 22 R 0012 and s 12are set in this group of simulation data e estimatedresults and the original generated data are shown in thefigure below

In the above simulation results it can be concluded fromFigures 2 and 3 that the estimation algorithm can simul-taneously estimate the roll angle and roll angle rate of theprojectile From Figures 3ndash5 it can be seen that the angularrate mutation exists at 0 s and 5 s It can be seen fromFigures 4 and 5 that the error of the roll angle and roll anglerate estimated by the Kalman filter is large due to the

Roll rotation rate

0 02 040

1000

2000

5 52 54

ndash3600

ndash3500

ndash3400

ndash3300

ndash4000

ndash3000

ndash2000

ndash1000

0

1000

2000

3000

4000

Roll

rota

tion

rate

(degs

)

10 150 5t (s)

Generated bysimulationEstimated by pureKalman filter

Estimated by forgettingfactor algorithmEstimated by the directderivation method

Figure 3 e roll angle rate of the generated and the estimated

0 5 10 15t (s)

Roll angle

0

50

100

150

200

250

300

350

400

Roll

angl

e (deg)

The roll angle measuredEstimated by pure Kalman filterEstimated by forgetting factor algorithm

Figure 2 e roll angles of the generated and the estimated

Position at any timeprojectile body coordinate

system (b)

Initial position-launching coordinate system (b) The traget

Ballistic plane

yn

znxn

o

o

zb (yb)

yb (zb)

xb

Figure 1 e relationship between the two coordinates

Mathematical Problems in Engineering 5

constraint of the estimation model while the error of thealgorithm estimated by adding the forgetting factor will begreatly reduced e error of roll angle and roll angle rate atthe angular rate mutation is shown in the table below

It can be seen from Table 1 that in the angular ratemutation stage compared with the pure Kalman filter es-timation the estimation accuracy of the algorithm with theforgetting factor is improved by an order of magnitude and

The error of roll angle rate

0 02 04

ndash1500

ndash1000

ndash500

0

5 52 54ndash80ndash60ndash40ndash20

020

1335 134 1345 135 1355 136 1365 137ndash20

0

20

ndash4000

ndash3500

ndash3000

ndash2500

ndash2000

ndash1500

ndash1000

ndash500

0

500

The e

rror

of r

oll a

ngle

rate

(degs

)

10 150 5t (s)

Estimation error of pure Kalman filter estimationEstimation error with forgetting factor algorithmThe error solved by direct derivation

Figure 5 e error of the roll angle rate

5

0

ndash5

ndash10

ndash15

ndash25

ndash20

The e

rror

of r

oll a

ngels

(deg)

The error of roll angles

0

ndash10

ndash20

0

ndash1

ndash20 05

002

0

ndash002

5 52 54

1346 1348 135 1352 1354

151050t (s)

Estimation error of pure Kalman filterEstimation error with forgetting factor algorithmError of measurement

Figure 4 e error of roll angles

Table 1 e estimation error of the algorithm in the angular rate mutation stage

e stage of angularrate mutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)Estimated by pure

Kalman filterEstimated by the algorithm

with forgetting factorEstimated by pure

Kalman filterEstimated by the algorithm

with forgetting factor0 sndash055 s minus 23443 minus 02431 minus 13915 minus 7737215 sndash56 s minus 04660 minus 00080 minus 168456 minus 16650

6 Mathematical Problems in Engineering

the accuracy of the roll rotation rate is increased by morethan 4 times It can be seen from Figure 5 that in thestationary phase the accuracy of the roll rotation rate es-timated by the algorithm is 10 times higher than that ob-tained by direct derivation e validity of the algorithm isverified

42 Validation of Semiphysical Turntable Data After thealgorithm is verified it is verified according to the data of themagnetic measurement system With the three-axis mag-netometer HMC1053 produced by Honeywell company asthe only attitude sensing chip and the STM32 single chipmicrocomputer produced by ST Company as the controllerthe control circuit was designed to constitute the magneticattitude measurement system [21 22] and the magneticmeasurement system was fixed on the three-axis high-precision flight simulation turntable as shown in Figure 6e control table rotates around the X-axis Y-axis and Z-axis respectively and the roll angle is calculated by using themagnetic measurement system and the roll angle and theroll rotation rate are optimized and estimated by taking themagnetic measurement as the observation data

In this experiment the gyroscope is saturated and it isimpossible to further estimate the carrier roll angle so thegyro information is only used to estimate the roll angle rateestimated by the algorithm in the later stage

To simulate the flight state of the projectile body underthe maneuvering condition the flight turntable wasaccelerated uniformly around the X-axis to 5 rs and thendecelerated uniformly to 0 rs after maintaining 21 s asshown in Figure 7 In the model parameter setting Q 22R 052 and s 103 and the simulation results are shownin the figures

e estimated roll angle and roll rotation rate are shownin Figures 7 and 8 It can be seen from Figure 8 that at theangular rate mutation of 1 s 3 s and 23 s the roll angle androll rate estimated by the pure Kalman filter have large errorHowever the error estimated by adding forgetting factoralgorithm is significantly smaller as shown in Table 2

As can be seen from Table 2 in the angular rate mu-tation stage compared with the estimation result of thepure Kalman filter the roll angle accuracy estimated by thealgorithm with the forgetting factor is improved by anorder of magnitude e accuracy of the estimated rollrotation rate is improved by more than 4 times It can alsobe seen from Figures 9 and 10 that in the stationary phasethe roll angle error estimated by the algorithm is within 2degwhich is more than twice as accurate as the roll error of 5degmeasured e error of the roll rotation rate estimated bythe algorithm is within 5 degs which improves the accuracyby an order of magnitude compared with the error of theroll rotation rate obtained by direct derivation of 50 degswhich verifies the feasibility of the algorithm and thesystem

43 Verification of Bomb Test Data e feasibility of thealgorithm is verified by theoretical analysis and turntablesemiphysical simulation test Now the sensor data

Figure 6 ree-axis high-precision flight simulation turntable

10 15 25200 5t (s)

0

50

100

150

200

250

300

350

400

Roll

ange

l (deg)

Roll angle

Estimated by pureKalman filterMeasured roll angle

Estimated by forgettingfactor algorithmThe turntable feedback

Figure 7 e roll angles of turntable feedback and algorithmestimation

2 3 4

2000

1500

1000

1850

1800

1750122 124 126 128

1800

1600

1400

1200

1000

800

23 235 24

Roll rotation rate

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodThe turntable feedback

20 25 30151050t (s)

ndash1000

ndash500

0

500

1000

1500

2000

2500

Roll

rota

tion

rate

(degs

)

Figure 8 e roll angles rate of turntable feedback and algorithmestimation

Mathematical Problems in Engineering 7

collected from the ballistic flight test of the range is used forfurther verification A magnetic measuring system and anaxial MEMS gyro are installed in the test projectile bodye roll rotation rate of the gyro output is taken as areference to verify the accuracy of the estimated roll

rotation and roll rotation rate calculated by using only themagnetometer data e test results are shown in thefigures

Roll angel error

15

10

5

0

ndash523 24 25

12 122 124

5

0

ndash52 3 4

10

0

ndash10

5 150 35302010 25t (s)

ndash20

ndash15

ndash10

ndash5

0

5

10

15

Roll

ange

l err

or (deg

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

Figure 9 e error of the roll angle

1 2 3

100

0

ndash100

100

0

ndash10011 115 12 125 13

150100

500

ndash50235 24 245

Error of roll rotation rate

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

5 15 20 302510t (s)

ndash500

ndash400

ndash300

ndash200

ndash100

0

100

200

Erro

r of r

oll r

otat

ion

rate

(degs

)

Figure 10 e error of the roll angle rate

Sensor data

1510 200 5t (s)

0

05

1

15

2

25

3

Volta

ge v

alue

(V)

X axis of magneticsensorY axis of magneticsensor

Z axis of magneticsensorThe axial gyro

Figure 11 Sensor data of magnetic and gyro

10 20 30 40 50 60 700t (s)

0

100

200

300

400

Roll

Ang

le (deg

)

(a)

10 20 30 40 50 60 700t (s)

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

(b)

Figure 12 e system calculates the roll angle and the angular rateof the gyro output (a) e role angle that system calculates (b)Rotation rate of gyro type

Table 2 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular ratemutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)

Estimated by pure Kalmanfilter

Estimated by thealgorithm with forgetting

factor

Estimated bypure

Kalman filter

Estimated by thealgorithm

with forgetting factor1 sndash2 s minus 82261 minus 05716 minus 745018 minus 412543 sndash4 s 66852 minus 06276 776076 15466023 sndash24 s 73149 minus 00166 784181 160643

8 Mathematical Problems in Engineering

e figure above shows the system output and algorithmestimation results of the bomb test It can be seen fromFigure 11that the effective flight time of the ballistic test is 20 s It can beseen from Figures 12 and 13 that the gyro is saturated duringflight and cannot normally calculate the roll rotation rateHowever the roll rotation rate estimated by the algorithmmakesup for this defect e roll angle and roll rotation rate estimatedby the algorithm are shown in Figures 13 and 14 Figure 14shows that the roll angle estimated by the algorithm is betterthan the linearity of the roll angle calculated directly by thesystem which indicates that the roll angle estimated by thealgorithm compensates some errors caused by the systemmeasurement Figure 13 shows that the roll rotation rate

estimated by the algorithm compensates for the error caused bygyro saturation in the first two seconds In the stationary phasethe accuracy of the roll rotation rate estimated by the algorithmis 6 times higher than that obtained by direct derivation Fig-ure 15 shows that in the angular rate mutation stage the rollangular rate estimated by forgetting factor reduces the errorcaused by pure Kalman filter estimation and the mean value ofthe estimated error caused by angular rate mutation is shown inthe table

It can be seen from Table 3 that the angular rate at 11 sdoes not change much so the effect of the algorithm with theforgetting factor is not obvious However in other abruptchanges of angular rates the accuracy of the algorithm with

Roll rotation rate

5000

0

ndash5000

ndash1000005 1 15 2

2000

1800

1600

165 17 175 18

5 10 15 200t (s)

ndash8000

ndash6000

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodGyro calculating

Figure 13 e roll angle rate that the system calculates and the gyro output

Roll angle

72 74 76 78 8

3020 35 4025105 150t (s)

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

Roll

angl

e (deg)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmThe system calculates

Figure 14 e roll angle that the system calculates and the gyro output

Mathematical Problems in Engineering 9

the forgetting factor is more than 4 times higher than that ofthe pure Kalman filter

5 Conclusions

In this paper we propose a method to estimate the roll angleand roll angle rate of a projectile by using only the magneticfield information provided by a triaxial magnetometer and areal-time estimation algorithm based on the Kalman filterwith appropriate forgetting factor is proposed is methodsolves the problem that the projectile roll angle and roll anglerate cannot be obtained due to MEMS gyro overload anddegradation under the flight condition of high spin and highoverload e Kalman filter estimation algorithm with theoblivion factor is able to significantly reduce the error causedby estimation delay under high dynamic conditions

rough the above analysis and semiphysical simulationtest it can be concluded that the algorithm can estimate theroll angle and roll angle rate of the carrier in real time andquickly e experimental results show that the algorithmwith the forgetting factor reduces the influence of magneticsensor measurement error on the accuracy of roll angle andimproves the accuracy of roll angle by one time e ex-perimental results show that the error of the roll rotation rateestimated by this algorithm is within 5 degs and the accuracyis 6 times higher than that obtained by direct derivation Inthe angular rate mutation phase compared with the pureKalman filter estimation algorithm the accuracy of the roll

angle estimated by the algorithm that the Kalman filter withthe forgetting factor is improved by an order of magnitudeand the accuracy of the roll angle rate is improved by at leastfour times which canmeet the requirements of the projectileroll angle and roll angle rate of the guidance and controlsystem of general rotating bombs

Data Availability

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

Disclosure

Lizhen Gao and Yingying Zhang are co-first authors of thisarticle

Conflicts of Interest

e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was carried out in accordance with the require-ments of the National Natural Science Foundation of China(61873247) funded project corresponding test experimentswere carried out in the State Key Laboratory of ElectronicTesting Technology of North China University Key

Error of roll rotation rate

0 05 1 15 2 25ndash5000

0

5000

115 12 125 13

ndash200

0

200

20 21ndash1000

0

1000

2000

1510 200 5t (s)

ndash6000

ndash4000

ndash2000

0

2000

4000

6000

Erro

r of r

oll r

otat

ion

rate

(degs

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmEstimated by the direct derivation method

Figure 15 e error of roll angle rate that the system calculates and the gyro output

Table 3 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular rate mutationMean of roll angle rate error (s)

Estimated by pure Kalman filter Estimated by the algorithm with forgetting factor0 sndash02 s 14505 minus 31891042 sndash25 s 5706310 1440300115 sndash12 s 813683 62837720 sndash204 s 8241634 2084635

10 Mathematical Problems in Engineering

Laboratory of Instrumental Science and Dynamic Testing ofNorth China University Huaihai Industry Group inChangzhi City Shanxi Province and Alashan ShootingRange During this period the authors also got instructionsfrom Professor Zhang Xiaoming and Teacher Li Xiuyuane authors would like to thank them for their help andsupport

References

[1] L An L Wang and D Zhao ldquoAttitude determinationmethod of spinning projectile based on geomagnetic azi-muthrdquo Journal of Chinese Inertial Technology vol 27 no 5pp 618ndash624 2019

[2] A Grosz E Paperno S Amrusi and B Zadov ldquoA three-axialsearch coil magnetometer optimized for small size low powerand low frequenciesrdquo IEEE Sensors Journal vol 11 no 4pp 1088ndash1094 2011

[3] H Liu H Dong J Ge B Bai Z Yuan and Z Zhao ldquoResearchon a secondary tuning algorithm based on SVD amp STFT forFID signalrdquo Measurment Science and Technology vol 27no 10 pp 0957ndash0233 Article ID 105006 2016

[4] J Shang D Zhihong M Fu and S Wang ldquoA high-spin ratemeasurement method for projectiles using a magnetoresistivesensor based on time-frequency domain analysisrdquo Sensorsvol 16 no 6 p 894 2016

[5] C Mateo and J A Talavera ldquoShort-time fourier transformwith the window size fixed in the frequency domainrdquo DigitalSignal Processing vol 77 no 6 pp 13ndash21 2018

[6] X Yan G Chen and X Tian ldquoTwo-step adaptive augmentedunscented Kalman filter for roll angles of spinning missilesbased on magnetometer measurementsrdquo Measurement andControl vol 51 no 3-4 pp 73ndash82 2018

[7] T Addabbo R Biondi S Cioncolini A Fort F Rossetti andV Vignoli ldquoA zero-crossing detection system based on FPGAto measure the angular vibrations of rotating shaftsrdquo IEEETransactions on Instrumentation and Measurement vol 63no 12 pp 3002ndash3010 2014

[8] Y Zhou X Zhang and W Xiao ldquoSpinning projectilersquos an-gular measurement using crest and trough data of a geo-magnetic sensorrdquo Measurment Science and Technologyvol 29 no 9 Article ID 095007 2018

[9] H Zhao Z Su F Liu C Li Q Li and N Liu ldquoExtraction andfilter algorithm of roll angular rate for high spinning pro-jectilesrdquo Mathematical Problems in Engineering vol 2019Article ID 3181727 15 pages 2019

[10] S Carletta and P Teofilatto ldquoDesign and numerical validationof an algorithm for the detumbling and angular rate deter-mination of a CubeSat using only three-axis magnetometerdatardquo International Journal of Aerospace Engineeringvol 2018 Article ID 9768475 12 pages 2018

[11] L-B Li M-X Li L-X Jiang D-Y Wang F Zhan andT Sheng ldquoAngular rate estimation and damping control ofsatellite with magnetometer datardquo Optik vol 180 no 11pp 1049ndash1055 2019

[12] H Ma and S Xu ldquoMagnetometer-only attitude and angularvelocity filtering estimation for attitude changing spacecraftrdquoActa Astronautica vol 102 no 5 pp 89ndash102 2014

[13] S Sabzevari M R Arvan A R Vali S M M Dehghan andM H Ferdowsi ldquoSymmetry preserving nonlinear observer forattitude estimation with magnetometer onlyrdquo ISA Transac-tions vol 102 no 3 pp 314ndash324 2020

[14] J M Maley ldquoEfficient attitude estimation for a spin-stabilizedprojectilerdquo Journal of Guidance Control and Dynamicsvol 39 no 2 pp 1ndash12 2016

[15] G Hu W Wang Y Zhong B Gao and C Gu ldquoA new directfiltering approach to INSGNSS integrationrdquo Aerospace Sci-ence and Technology vol 77 no 7 pp 755ndash764 2018

[16] M Yunjian X Changfan J Yixian W Yao and Z YildquoAngular velocity estimation of rollingmdashammunition basedon magnetometerrdquo Journal of Projectiles Rockets Missiles andGuidance vol 36 no 1 pp 69ndash72 2016

[17] G Hu L Ni B Gao X ZhuWWang and Y Zhong ldquoModelpredictive based unscented Kalman filter for hypersonic ve-hicle navigation with INSGNSS integrationrdquo IEEE Accessvol 8 no 1 pp 4814ndash4823 2016

[18] C Chunhang L Chunsheng J Wendou et al ldquoe projectileattitude measuring method based on geomagnetic sensorrdquoJournal of Detection amp Control vol 40 no 12 pp 4814ndash48232020

[19] X Chao X-z Bu and Y Bo ldquoree different attitudemeasurements of spinning projectile based on magneticsensorsrdquo Measurement vol 47 no 1 pp 331ndash340 2014

[20] W Yu ldquoHalf-experiential formulas for calculating decreasingangular velocity of projectile in trajectoryrdquo Journal of De-tection and Control vol 231 no 5 pp 866ndash876 2003

[21] J Yu X Bu C Xiang and B Yang ldquoSpinning projectilersquosattitude measurement using intersection ratio of magneticsensorsrdquo Proceedings of the Institution of Mechanical Engi-neers Part G-Journal of Aerospace Engineering vol 231 no 5pp 1ndash6 2016

[22] X Zhao X Zhang D Long Z Bai and Y Wang ldquoe designof roll angle magnetic measurement system used in spinningprojectilesrdquo Chinese Journal of Sensors and Actuators vol 26no 9 pp 1309ndash1313 2013

Mathematical Problems in Engineering 11

where k and j are both a certain moment in time series N Itcan be seen that when s is 1 the noise variance matrix doesnot change When s is greater than 1 the noise variancematrix increases According to the case that the model es-timation error increases when the angular rate changes sshould be a real number slightly greater than 1 us a newone-step prediction mean square error matrix and gainmatrix are

Pkkminus 1 ϕkkminus 1Pkminus 1ϕTkkminus 1 + Γkminus 1s

Nminus kQkminus 1Γ

Tkminus 1

Kk Pkkminus 1HTk HkPkkminus 1H

Tk + s

Nminus kRk1113872 1113873

minus 1

(18)

By multiplying the one-step prediction mean squareerror of the above equation by sminus (Nminus k) we can obtain

sminus (Nminus k)

Pkkminus 1 ϕkkminus 1sminus (Nminus k)

Pkminus 1ϕTkkminus 1 + Γkminus 1Qkminus 1Γ

Tkminus 1

(19)

Assume that

Plowastkkminus 1 ϕkkminus 1 sP

lowastkminus 1( 1113857ϕT

kkminus 1 + Γkminus 1Qkminus 1ΓTkminus 1 (20)

Similarly multiply the left and right sides of the esti-mated mean square error by sminus (Nminus k) to simplify and arrangethe final model equation as follows

1113954Xlowastkkminus 1 ϕkkminus 1 lowast 1113954X

lowastkminus 1

Plowastkkminus 1 ϕkkminus 1 sPlowastkminus 1( 1113857ϕT

kkminus 1 + Γkminus 1Qkminus 1ΓTkminus 1

Klowastk Plowastkkminus 1H

Tk HkP

lowastkkminus 1H

Tk + Rk1113872 1113873

minus 1

1113954Xlowastk 1113954X

lowastkkminus 1 + K

lowastk Zk minus Hk

1113954Xlowastkkminus 11113872 1113873

Plowastk I minus KkHk( 1113857P

lowastkkminus 1

(21)

where 1113954Xlowastkminus 1 1113954X

lowastk 1113954Xlowastkkminus 1 and Klowastk respectively represent the

estimated value at time k minus 1 and time k as well as the stateprediction and gain at time k and s is the forgetting factor

It can be seen that the simplified filtering model can onlybe multiplied by a forgetting factor before the estimatedmean square error at the previous moment

4 Results and Discussion

41 Validation of Simulation Data According to the abovemodel the initial value is given for simulation verification

First a set of roll angle and roll angle rate data with angularrate mutation are generated by simulation and appro-priate measurement error is added and then the algo-rithm is used for estimation Q 22 R 0012 and s 12are set in this group of simulation data e estimatedresults and the original generated data are shown in thefigure below

In the above simulation results it can be concluded fromFigures 2 and 3 that the estimation algorithm can simul-taneously estimate the roll angle and roll angle rate of theprojectile From Figures 3ndash5 it can be seen that the angularrate mutation exists at 0 s and 5 s It can be seen fromFigures 4 and 5 that the error of the roll angle and roll anglerate estimated by the Kalman filter is large due to the

Roll rotation rate

0 02 040

1000

2000

5 52 54

ndash3600

ndash3500

ndash3400

ndash3300

ndash4000

ndash3000

ndash2000

ndash1000

0

1000

2000

3000

4000

Roll

rota

tion

rate

(degs

)

10 150 5t (s)

Generated bysimulationEstimated by pureKalman filter

Estimated by forgettingfactor algorithmEstimated by the directderivation method

Figure 3 e roll angle rate of the generated and the estimated

0 5 10 15t (s)

Roll angle

0

50

100

150

200

250

300

350

400

Roll

angl

e (deg)

The roll angle measuredEstimated by pure Kalman filterEstimated by forgetting factor algorithm

Figure 2 e roll angles of the generated and the estimated

Position at any timeprojectile body coordinate

system (b)

Initial position-launching coordinate system (b) The traget

Ballistic plane

yn

znxn

o

o

zb (yb)

yb (zb)

xb

Figure 1 e relationship between the two coordinates

Mathematical Problems in Engineering 5

constraint of the estimation model while the error of thealgorithm estimated by adding the forgetting factor will begreatly reduced e error of roll angle and roll angle rate atthe angular rate mutation is shown in the table below

It can be seen from Table 1 that in the angular ratemutation stage compared with the pure Kalman filter es-timation the estimation accuracy of the algorithm with theforgetting factor is improved by an order of magnitude and

The error of roll angle rate

0 02 04

ndash1500

ndash1000

ndash500

0

5 52 54ndash80ndash60ndash40ndash20

020

1335 134 1345 135 1355 136 1365 137ndash20

0

20

ndash4000

ndash3500

ndash3000

ndash2500

ndash2000

ndash1500

ndash1000

ndash500

0

500

The e

rror

of r

oll a

ngle

rate

(degs

)

10 150 5t (s)

Estimation error of pure Kalman filter estimationEstimation error with forgetting factor algorithmThe error solved by direct derivation

Figure 5 e error of the roll angle rate

5

0

ndash5

ndash10

ndash15

ndash25

ndash20

The e

rror

of r

oll a

ngels

(deg)

The error of roll angles

0

ndash10

ndash20

0

ndash1

ndash20 05

002

0

ndash002

5 52 54

1346 1348 135 1352 1354

151050t (s)

Estimation error of pure Kalman filterEstimation error with forgetting factor algorithmError of measurement

Figure 4 e error of roll angles

Table 1 e estimation error of the algorithm in the angular rate mutation stage

e stage of angularrate mutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)Estimated by pure

Kalman filterEstimated by the algorithm

with forgetting factorEstimated by pure

Kalman filterEstimated by the algorithm

with forgetting factor0 sndash055 s minus 23443 minus 02431 minus 13915 minus 7737215 sndash56 s minus 04660 minus 00080 minus 168456 minus 16650

6 Mathematical Problems in Engineering

the accuracy of the roll rotation rate is increased by morethan 4 times It can be seen from Figure 5 that in thestationary phase the accuracy of the roll rotation rate es-timated by the algorithm is 10 times higher than that ob-tained by direct derivation e validity of the algorithm isverified

42 Validation of Semiphysical Turntable Data After thealgorithm is verified it is verified according to the data of themagnetic measurement system With the three-axis mag-netometer HMC1053 produced by Honeywell company asthe only attitude sensing chip and the STM32 single chipmicrocomputer produced by ST Company as the controllerthe control circuit was designed to constitute the magneticattitude measurement system [21 22] and the magneticmeasurement system was fixed on the three-axis high-precision flight simulation turntable as shown in Figure 6e control table rotates around the X-axis Y-axis and Z-axis respectively and the roll angle is calculated by using themagnetic measurement system and the roll angle and theroll rotation rate are optimized and estimated by taking themagnetic measurement as the observation data

In this experiment the gyroscope is saturated and it isimpossible to further estimate the carrier roll angle so thegyro information is only used to estimate the roll angle rateestimated by the algorithm in the later stage

To simulate the flight state of the projectile body underthe maneuvering condition the flight turntable wasaccelerated uniformly around the X-axis to 5 rs and thendecelerated uniformly to 0 rs after maintaining 21 s asshown in Figure 7 In the model parameter setting Q 22R 052 and s 103 and the simulation results are shownin the figures

e estimated roll angle and roll rotation rate are shownin Figures 7 and 8 It can be seen from Figure 8 that at theangular rate mutation of 1 s 3 s and 23 s the roll angle androll rate estimated by the pure Kalman filter have large errorHowever the error estimated by adding forgetting factoralgorithm is significantly smaller as shown in Table 2

As can be seen from Table 2 in the angular rate mu-tation stage compared with the estimation result of thepure Kalman filter the roll angle accuracy estimated by thealgorithm with the forgetting factor is improved by anorder of magnitude e accuracy of the estimated rollrotation rate is improved by more than 4 times It can alsobe seen from Figures 9 and 10 that in the stationary phasethe roll angle error estimated by the algorithm is within 2degwhich is more than twice as accurate as the roll error of 5degmeasured e error of the roll rotation rate estimated bythe algorithm is within 5 degs which improves the accuracyby an order of magnitude compared with the error of theroll rotation rate obtained by direct derivation of 50 degswhich verifies the feasibility of the algorithm and thesystem

43 Verification of Bomb Test Data e feasibility of thealgorithm is verified by theoretical analysis and turntablesemiphysical simulation test Now the sensor data

Figure 6 ree-axis high-precision flight simulation turntable

10 15 25200 5t (s)

0

50

100

150

200

250

300

350

400

Roll

ange

l (deg)

Roll angle

Estimated by pureKalman filterMeasured roll angle

Estimated by forgettingfactor algorithmThe turntable feedback

Figure 7 e roll angles of turntable feedback and algorithmestimation

2 3 4

2000

1500

1000

1850

1800

1750122 124 126 128

1800

1600

1400

1200

1000

800

23 235 24

Roll rotation rate

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodThe turntable feedback

20 25 30151050t (s)

ndash1000

ndash500

0

500

1000

1500

2000

2500

Roll

rota

tion

rate

(degs

)

Figure 8 e roll angles rate of turntable feedback and algorithmestimation

Mathematical Problems in Engineering 7

collected from the ballistic flight test of the range is used forfurther verification A magnetic measuring system and anaxial MEMS gyro are installed in the test projectile bodye roll rotation rate of the gyro output is taken as areference to verify the accuracy of the estimated roll

rotation and roll rotation rate calculated by using only themagnetometer data e test results are shown in thefigures

Roll angel error

15

10

5

0

ndash523 24 25

12 122 124

5

0

ndash52 3 4

10

0

ndash10

5 150 35302010 25t (s)

ndash20

ndash15

ndash10

ndash5

0

5

10

15

Roll

ange

l err

or (deg

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

Figure 9 e error of the roll angle

1 2 3

100

0

ndash100

100

0

ndash10011 115 12 125 13

150100

500

ndash50235 24 245

Error of roll rotation rate

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

5 15 20 302510t (s)

ndash500

ndash400

ndash300

ndash200

ndash100

0

100

200

Erro

r of r

oll r

otat

ion

rate

(degs

)

Figure 10 e error of the roll angle rate

Sensor data

1510 200 5t (s)

0

05

1

15

2

25

3

Volta

ge v

alue

(V)

X axis of magneticsensorY axis of magneticsensor

Z axis of magneticsensorThe axial gyro

Figure 11 Sensor data of magnetic and gyro

10 20 30 40 50 60 700t (s)

0

100

200

300

400

Roll

Ang

le (deg

)

(a)

10 20 30 40 50 60 700t (s)

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

(b)

Figure 12 e system calculates the roll angle and the angular rateof the gyro output (a) e role angle that system calculates (b)Rotation rate of gyro type

Table 2 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular ratemutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)

Estimated by pure Kalmanfilter

Estimated by thealgorithm with forgetting

factor

Estimated bypure

Kalman filter

Estimated by thealgorithm

with forgetting factor1 sndash2 s minus 82261 minus 05716 minus 745018 minus 412543 sndash4 s 66852 minus 06276 776076 15466023 sndash24 s 73149 minus 00166 784181 160643

8 Mathematical Problems in Engineering

e figure above shows the system output and algorithmestimation results of the bomb test It can be seen fromFigure 11that the effective flight time of the ballistic test is 20 s It can beseen from Figures 12 and 13 that the gyro is saturated duringflight and cannot normally calculate the roll rotation rateHowever the roll rotation rate estimated by the algorithmmakesup for this defect e roll angle and roll rotation rate estimatedby the algorithm are shown in Figures 13 and 14 Figure 14shows that the roll angle estimated by the algorithm is betterthan the linearity of the roll angle calculated directly by thesystem which indicates that the roll angle estimated by thealgorithm compensates some errors caused by the systemmeasurement Figure 13 shows that the roll rotation rate

estimated by the algorithm compensates for the error caused bygyro saturation in the first two seconds In the stationary phasethe accuracy of the roll rotation rate estimated by the algorithmis 6 times higher than that obtained by direct derivation Fig-ure 15 shows that in the angular rate mutation stage the rollangular rate estimated by forgetting factor reduces the errorcaused by pure Kalman filter estimation and the mean value ofthe estimated error caused by angular rate mutation is shown inthe table

It can be seen from Table 3 that the angular rate at 11 sdoes not change much so the effect of the algorithm with theforgetting factor is not obvious However in other abruptchanges of angular rates the accuracy of the algorithm with

Roll rotation rate

5000

0

ndash5000

ndash1000005 1 15 2

2000

1800

1600

165 17 175 18

5 10 15 200t (s)

ndash8000

ndash6000

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodGyro calculating

Figure 13 e roll angle rate that the system calculates and the gyro output

Roll angle

72 74 76 78 8

3020 35 4025105 150t (s)

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

Roll

angl

e (deg)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmThe system calculates

Figure 14 e roll angle that the system calculates and the gyro output

Mathematical Problems in Engineering 9

the forgetting factor is more than 4 times higher than that ofthe pure Kalman filter

5 Conclusions

In this paper we propose a method to estimate the roll angleand roll angle rate of a projectile by using only the magneticfield information provided by a triaxial magnetometer and areal-time estimation algorithm based on the Kalman filterwith appropriate forgetting factor is proposed is methodsolves the problem that the projectile roll angle and roll anglerate cannot be obtained due to MEMS gyro overload anddegradation under the flight condition of high spin and highoverload e Kalman filter estimation algorithm with theoblivion factor is able to significantly reduce the error causedby estimation delay under high dynamic conditions

rough the above analysis and semiphysical simulationtest it can be concluded that the algorithm can estimate theroll angle and roll angle rate of the carrier in real time andquickly e experimental results show that the algorithmwith the forgetting factor reduces the influence of magneticsensor measurement error on the accuracy of roll angle andimproves the accuracy of roll angle by one time e ex-perimental results show that the error of the roll rotation rateestimated by this algorithm is within 5 degs and the accuracyis 6 times higher than that obtained by direct derivation Inthe angular rate mutation phase compared with the pureKalman filter estimation algorithm the accuracy of the roll

angle estimated by the algorithm that the Kalman filter withthe forgetting factor is improved by an order of magnitudeand the accuracy of the roll angle rate is improved by at leastfour times which canmeet the requirements of the projectileroll angle and roll angle rate of the guidance and controlsystem of general rotating bombs

Data Availability

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

Disclosure

Lizhen Gao and Yingying Zhang are co-first authors of thisarticle

Conflicts of Interest

e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was carried out in accordance with the require-ments of the National Natural Science Foundation of China(61873247) funded project corresponding test experimentswere carried out in the State Key Laboratory of ElectronicTesting Technology of North China University Key

Error of roll rotation rate

0 05 1 15 2 25ndash5000

0

5000

115 12 125 13

ndash200

0

200

20 21ndash1000

0

1000

2000

1510 200 5t (s)

ndash6000

ndash4000

ndash2000

0

2000

4000

6000

Erro

r of r

oll r

otat

ion

rate

(degs

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmEstimated by the direct derivation method

Figure 15 e error of roll angle rate that the system calculates and the gyro output

Table 3 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular rate mutationMean of roll angle rate error (s)

Estimated by pure Kalman filter Estimated by the algorithm with forgetting factor0 sndash02 s 14505 minus 31891042 sndash25 s 5706310 1440300115 sndash12 s 813683 62837720 sndash204 s 8241634 2084635

10 Mathematical Problems in Engineering

Laboratory of Instrumental Science and Dynamic Testing ofNorth China University Huaihai Industry Group inChangzhi City Shanxi Province and Alashan ShootingRange During this period the authors also got instructionsfrom Professor Zhang Xiaoming and Teacher Li Xiuyuane authors would like to thank them for their help andsupport

References

[1] L An L Wang and D Zhao ldquoAttitude determinationmethod of spinning projectile based on geomagnetic azi-muthrdquo Journal of Chinese Inertial Technology vol 27 no 5pp 618ndash624 2019

[2] A Grosz E Paperno S Amrusi and B Zadov ldquoA three-axialsearch coil magnetometer optimized for small size low powerand low frequenciesrdquo IEEE Sensors Journal vol 11 no 4pp 1088ndash1094 2011

[3] H Liu H Dong J Ge B Bai Z Yuan and Z Zhao ldquoResearchon a secondary tuning algorithm based on SVD amp STFT forFID signalrdquo Measurment Science and Technology vol 27no 10 pp 0957ndash0233 Article ID 105006 2016

[4] J Shang D Zhihong M Fu and S Wang ldquoA high-spin ratemeasurement method for projectiles using a magnetoresistivesensor based on time-frequency domain analysisrdquo Sensorsvol 16 no 6 p 894 2016

[5] C Mateo and J A Talavera ldquoShort-time fourier transformwith the window size fixed in the frequency domainrdquo DigitalSignal Processing vol 77 no 6 pp 13ndash21 2018

[6] X Yan G Chen and X Tian ldquoTwo-step adaptive augmentedunscented Kalman filter for roll angles of spinning missilesbased on magnetometer measurementsrdquo Measurement andControl vol 51 no 3-4 pp 73ndash82 2018

[7] T Addabbo R Biondi S Cioncolini A Fort F Rossetti andV Vignoli ldquoA zero-crossing detection system based on FPGAto measure the angular vibrations of rotating shaftsrdquo IEEETransactions on Instrumentation and Measurement vol 63no 12 pp 3002ndash3010 2014

[8] Y Zhou X Zhang and W Xiao ldquoSpinning projectilersquos an-gular measurement using crest and trough data of a geo-magnetic sensorrdquo Measurment Science and Technologyvol 29 no 9 Article ID 095007 2018

[9] H Zhao Z Su F Liu C Li Q Li and N Liu ldquoExtraction andfilter algorithm of roll angular rate for high spinning pro-jectilesrdquo Mathematical Problems in Engineering vol 2019Article ID 3181727 15 pages 2019

[10] S Carletta and P Teofilatto ldquoDesign and numerical validationof an algorithm for the detumbling and angular rate deter-mination of a CubeSat using only three-axis magnetometerdatardquo International Journal of Aerospace Engineeringvol 2018 Article ID 9768475 12 pages 2018

[11] L-B Li M-X Li L-X Jiang D-Y Wang F Zhan andT Sheng ldquoAngular rate estimation and damping control ofsatellite with magnetometer datardquo Optik vol 180 no 11pp 1049ndash1055 2019

[12] H Ma and S Xu ldquoMagnetometer-only attitude and angularvelocity filtering estimation for attitude changing spacecraftrdquoActa Astronautica vol 102 no 5 pp 89ndash102 2014

[13] S Sabzevari M R Arvan A R Vali S M M Dehghan andM H Ferdowsi ldquoSymmetry preserving nonlinear observer forattitude estimation with magnetometer onlyrdquo ISA Transac-tions vol 102 no 3 pp 314ndash324 2020

[14] J M Maley ldquoEfficient attitude estimation for a spin-stabilizedprojectilerdquo Journal of Guidance Control and Dynamicsvol 39 no 2 pp 1ndash12 2016

[15] G Hu W Wang Y Zhong B Gao and C Gu ldquoA new directfiltering approach to INSGNSS integrationrdquo Aerospace Sci-ence and Technology vol 77 no 7 pp 755ndash764 2018

[16] M Yunjian X Changfan J Yixian W Yao and Z YildquoAngular velocity estimation of rollingmdashammunition basedon magnetometerrdquo Journal of Projectiles Rockets Missiles andGuidance vol 36 no 1 pp 69ndash72 2016

[17] G Hu L Ni B Gao X ZhuWWang and Y Zhong ldquoModelpredictive based unscented Kalman filter for hypersonic ve-hicle navigation with INSGNSS integrationrdquo IEEE Accessvol 8 no 1 pp 4814ndash4823 2016

[18] C Chunhang L Chunsheng J Wendou et al ldquoe projectileattitude measuring method based on geomagnetic sensorrdquoJournal of Detection amp Control vol 40 no 12 pp 4814ndash48232020

[19] X Chao X-z Bu and Y Bo ldquoree different attitudemeasurements of spinning projectile based on magneticsensorsrdquo Measurement vol 47 no 1 pp 331ndash340 2014

[20] W Yu ldquoHalf-experiential formulas for calculating decreasingangular velocity of projectile in trajectoryrdquo Journal of De-tection and Control vol 231 no 5 pp 866ndash876 2003

[21] J Yu X Bu C Xiang and B Yang ldquoSpinning projectilersquosattitude measurement using intersection ratio of magneticsensorsrdquo Proceedings of the Institution of Mechanical Engi-neers Part G-Journal of Aerospace Engineering vol 231 no 5pp 1ndash6 2016

[22] X Zhao X Zhang D Long Z Bai and Y Wang ldquoe designof roll angle magnetic measurement system used in spinningprojectilesrdquo Chinese Journal of Sensors and Actuators vol 26no 9 pp 1309ndash1313 2013

Mathematical Problems in Engineering 11

constraint of the estimation model while the error of thealgorithm estimated by adding the forgetting factor will begreatly reduced e error of roll angle and roll angle rate atthe angular rate mutation is shown in the table below

It can be seen from Table 1 that in the angular ratemutation stage compared with the pure Kalman filter es-timation the estimation accuracy of the algorithm with theforgetting factor is improved by an order of magnitude and

The error of roll angle rate

0 02 04

ndash1500

ndash1000

ndash500

0

5 52 54ndash80ndash60ndash40ndash20

020

1335 134 1345 135 1355 136 1365 137ndash20

0

20

ndash4000

ndash3500

ndash3000

ndash2500

ndash2000

ndash1500

ndash1000

ndash500

0

500

The e

rror

of r

oll a

ngle

rate

(degs

)

10 150 5t (s)

Estimation error of pure Kalman filter estimationEstimation error with forgetting factor algorithmThe error solved by direct derivation

Figure 5 e error of the roll angle rate

5

0

ndash5

ndash10

ndash15

ndash25

ndash20

The e

rror

of r

oll a

ngels

(deg)

The error of roll angles

0

ndash10

ndash20

0

ndash1

ndash20 05

002

0

ndash002

5 52 54

1346 1348 135 1352 1354

151050t (s)

Estimation error of pure Kalman filterEstimation error with forgetting factor algorithmError of measurement

Figure 4 e error of roll angles

Table 1 e estimation error of the algorithm in the angular rate mutation stage

e stage of angularrate mutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)Estimated by pure

Kalman filterEstimated by the algorithm

with forgetting factorEstimated by pure

Kalman filterEstimated by the algorithm

with forgetting factor0 sndash055 s minus 23443 minus 02431 minus 13915 minus 7737215 sndash56 s minus 04660 minus 00080 minus 168456 minus 16650

6 Mathematical Problems in Engineering

the accuracy of the roll rotation rate is increased by morethan 4 times It can be seen from Figure 5 that in thestationary phase the accuracy of the roll rotation rate es-timated by the algorithm is 10 times higher than that ob-tained by direct derivation e validity of the algorithm isverified

42 Validation of Semiphysical Turntable Data After thealgorithm is verified it is verified according to the data of themagnetic measurement system With the three-axis mag-netometer HMC1053 produced by Honeywell company asthe only attitude sensing chip and the STM32 single chipmicrocomputer produced by ST Company as the controllerthe control circuit was designed to constitute the magneticattitude measurement system [21 22] and the magneticmeasurement system was fixed on the three-axis high-precision flight simulation turntable as shown in Figure 6e control table rotates around the X-axis Y-axis and Z-axis respectively and the roll angle is calculated by using themagnetic measurement system and the roll angle and theroll rotation rate are optimized and estimated by taking themagnetic measurement as the observation data

In this experiment the gyroscope is saturated and it isimpossible to further estimate the carrier roll angle so thegyro information is only used to estimate the roll angle rateestimated by the algorithm in the later stage

To simulate the flight state of the projectile body underthe maneuvering condition the flight turntable wasaccelerated uniformly around the X-axis to 5 rs and thendecelerated uniformly to 0 rs after maintaining 21 s asshown in Figure 7 In the model parameter setting Q 22R 052 and s 103 and the simulation results are shownin the figures

e estimated roll angle and roll rotation rate are shownin Figures 7 and 8 It can be seen from Figure 8 that at theangular rate mutation of 1 s 3 s and 23 s the roll angle androll rate estimated by the pure Kalman filter have large errorHowever the error estimated by adding forgetting factoralgorithm is significantly smaller as shown in Table 2

As can be seen from Table 2 in the angular rate mu-tation stage compared with the estimation result of thepure Kalman filter the roll angle accuracy estimated by thealgorithm with the forgetting factor is improved by anorder of magnitude e accuracy of the estimated rollrotation rate is improved by more than 4 times It can alsobe seen from Figures 9 and 10 that in the stationary phasethe roll angle error estimated by the algorithm is within 2degwhich is more than twice as accurate as the roll error of 5degmeasured e error of the roll rotation rate estimated bythe algorithm is within 5 degs which improves the accuracyby an order of magnitude compared with the error of theroll rotation rate obtained by direct derivation of 50 degswhich verifies the feasibility of the algorithm and thesystem

43 Verification of Bomb Test Data e feasibility of thealgorithm is verified by theoretical analysis and turntablesemiphysical simulation test Now the sensor data

Figure 6 ree-axis high-precision flight simulation turntable

10 15 25200 5t (s)

0

50

100

150

200

250

300

350

400

Roll

ange

l (deg)

Roll angle

Estimated by pureKalman filterMeasured roll angle

Estimated by forgettingfactor algorithmThe turntable feedback

Figure 7 e roll angles of turntable feedback and algorithmestimation

2 3 4

2000

1500

1000

1850

1800

1750122 124 126 128

1800

1600

1400

1200

1000

800

23 235 24

Roll rotation rate

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodThe turntable feedback

20 25 30151050t (s)

ndash1000

ndash500

0

500

1000

1500

2000

2500

Roll

rota

tion

rate

(degs

)

Figure 8 e roll angles rate of turntable feedback and algorithmestimation

Mathematical Problems in Engineering 7

collected from the ballistic flight test of the range is used forfurther verification A magnetic measuring system and anaxial MEMS gyro are installed in the test projectile bodye roll rotation rate of the gyro output is taken as areference to verify the accuracy of the estimated roll

rotation and roll rotation rate calculated by using only themagnetometer data e test results are shown in thefigures

Roll angel error

15

10

5

0

ndash523 24 25

12 122 124

5

0

ndash52 3 4

10

0

ndash10

5 150 35302010 25t (s)

ndash20

ndash15

ndash10

ndash5

0

5

10

15

Roll

ange

l err

or (deg

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

Figure 9 e error of the roll angle

1 2 3

100

0

ndash100

100

0

ndash10011 115 12 125 13

150100

500

ndash50235 24 245

Error of roll rotation rate

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

5 15 20 302510t (s)

ndash500

ndash400

ndash300

ndash200

ndash100

0

100

200

Erro

r of r

oll r

otat

ion

rate

(degs

)

Figure 10 e error of the roll angle rate

Sensor data

1510 200 5t (s)

0

05

1

15

2

25

3

Volta

ge v

alue

(V)

X axis of magneticsensorY axis of magneticsensor

Z axis of magneticsensorThe axial gyro

Figure 11 Sensor data of magnetic and gyro

10 20 30 40 50 60 700t (s)

0

100

200

300

400

Roll

Ang

le (deg

)

(a)

10 20 30 40 50 60 700t (s)

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

(b)

Figure 12 e system calculates the roll angle and the angular rateof the gyro output (a) e role angle that system calculates (b)Rotation rate of gyro type

Table 2 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular ratemutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)

Estimated by pure Kalmanfilter

Estimated by thealgorithm with forgetting

factor

Estimated bypure

Kalman filter

Estimated by thealgorithm

with forgetting factor1 sndash2 s minus 82261 minus 05716 minus 745018 minus 412543 sndash4 s 66852 minus 06276 776076 15466023 sndash24 s 73149 minus 00166 784181 160643

8 Mathematical Problems in Engineering

e figure above shows the system output and algorithmestimation results of the bomb test It can be seen fromFigure 11that the effective flight time of the ballistic test is 20 s It can beseen from Figures 12 and 13 that the gyro is saturated duringflight and cannot normally calculate the roll rotation rateHowever the roll rotation rate estimated by the algorithmmakesup for this defect e roll angle and roll rotation rate estimatedby the algorithm are shown in Figures 13 and 14 Figure 14shows that the roll angle estimated by the algorithm is betterthan the linearity of the roll angle calculated directly by thesystem which indicates that the roll angle estimated by thealgorithm compensates some errors caused by the systemmeasurement Figure 13 shows that the roll rotation rate

estimated by the algorithm compensates for the error caused bygyro saturation in the first two seconds In the stationary phasethe accuracy of the roll rotation rate estimated by the algorithmis 6 times higher than that obtained by direct derivation Fig-ure 15 shows that in the angular rate mutation stage the rollangular rate estimated by forgetting factor reduces the errorcaused by pure Kalman filter estimation and the mean value ofthe estimated error caused by angular rate mutation is shown inthe table

It can be seen from Table 3 that the angular rate at 11 sdoes not change much so the effect of the algorithm with theforgetting factor is not obvious However in other abruptchanges of angular rates the accuracy of the algorithm with

Roll rotation rate

5000

0

ndash5000

ndash1000005 1 15 2

2000

1800

1600

165 17 175 18

5 10 15 200t (s)

ndash8000

ndash6000

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodGyro calculating

Figure 13 e roll angle rate that the system calculates and the gyro output

Roll angle

72 74 76 78 8

3020 35 4025105 150t (s)

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

Roll

angl

e (deg)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmThe system calculates

Figure 14 e roll angle that the system calculates and the gyro output

Mathematical Problems in Engineering 9

the forgetting factor is more than 4 times higher than that ofthe pure Kalman filter

5 Conclusions

In this paper we propose a method to estimate the roll angleand roll angle rate of a projectile by using only the magneticfield information provided by a triaxial magnetometer and areal-time estimation algorithm based on the Kalman filterwith appropriate forgetting factor is proposed is methodsolves the problem that the projectile roll angle and roll anglerate cannot be obtained due to MEMS gyro overload anddegradation under the flight condition of high spin and highoverload e Kalman filter estimation algorithm with theoblivion factor is able to significantly reduce the error causedby estimation delay under high dynamic conditions

rough the above analysis and semiphysical simulationtest it can be concluded that the algorithm can estimate theroll angle and roll angle rate of the carrier in real time andquickly e experimental results show that the algorithmwith the forgetting factor reduces the influence of magneticsensor measurement error on the accuracy of roll angle andimproves the accuracy of roll angle by one time e ex-perimental results show that the error of the roll rotation rateestimated by this algorithm is within 5 degs and the accuracyis 6 times higher than that obtained by direct derivation Inthe angular rate mutation phase compared with the pureKalman filter estimation algorithm the accuracy of the roll

angle estimated by the algorithm that the Kalman filter withthe forgetting factor is improved by an order of magnitudeand the accuracy of the roll angle rate is improved by at leastfour times which canmeet the requirements of the projectileroll angle and roll angle rate of the guidance and controlsystem of general rotating bombs

Data Availability

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

Disclosure

Lizhen Gao and Yingying Zhang are co-first authors of thisarticle

Conflicts of Interest

e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was carried out in accordance with the require-ments of the National Natural Science Foundation of China(61873247) funded project corresponding test experimentswere carried out in the State Key Laboratory of ElectronicTesting Technology of North China University Key

Error of roll rotation rate

0 05 1 15 2 25ndash5000

0

5000

115 12 125 13

ndash200

0

200

20 21ndash1000

0

1000

2000

1510 200 5t (s)

ndash6000

ndash4000

ndash2000

0

2000

4000

6000

Erro

r of r

oll r

otat

ion

rate

(degs

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmEstimated by the direct derivation method

Figure 15 e error of roll angle rate that the system calculates and the gyro output

Table 3 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular rate mutationMean of roll angle rate error (s)

Estimated by pure Kalman filter Estimated by the algorithm with forgetting factor0 sndash02 s 14505 minus 31891042 sndash25 s 5706310 1440300115 sndash12 s 813683 62837720 sndash204 s 8241634 2084635

10 Mathematical Problems in Engineering

Laboratory of Instrumental Science and Dynamic Testing ofNorth China University Huaihai Industry Group inChangzhi City Shanxi Province and Alashan ShootingRange During this period the authors also got instructionsfrom Professor Zhang Xiaoming and Teacher Li Xiuyuane authors would like to thank them for their help andsupport

References

[1] L An L Wang and D Zhao ldquoAttitude determinationmethod of spinning projectile based on geomagnetic azi-muthrdquo Journal of Chinese Inertial Technology vol 27 no 5pp 618ndash624 2019

[2] A Grosz E Paperno S Amrusi and B Zadov ldquoA three-axialsearch coil magnetometer optimized for small size low powerand low frequenciesrdquo IEEE Sensors Journal vol 11 no 4pp 1088ndash1094 2011

[3] H Liu H Dong J Ge B Bai Z Yuan and Z Zhao ldquoResearchon a secondary tuning algorithm based on SVD amp STFT forFID signalrdquo Measurment Science and Technology vol 27no 10 pp 0957ndash0233 Article ID 105006 2016

[4] J Shang D Zhihong M Fu and S Wang ldquoA high-spin ratemeasurement method for projectiles using a magnetoresistivesensor based on time-frequency domain analysisrdquo Sensorsvol 16 no 6 p 894 2016

[5] C Mateo and J A Talavera ldquoShort-time fourier transformwith the window size fixed in the frequency domainrdquo DigitalSignal Processing vol 77 no 6 pp 13ndash21 2018

[6] X Yan G Chen and X Tian ldquoTwo-step adaptive augmentedunscented Kalman filter for roll angles of spinning missilesbased on magnetometer measurementsrdquo Measurement andControl vol 51 no 3-4 pp 73ndash82 2018

[7] T Addabbo R Biondi S Cioncolini A Fort F Rossetti andV Vignoli ldquoA zero-crossing detection system based on FPGAto measure the angular vibrations of rotating shaftsrdquo IEEETransactions on Instrumentation and Measurement vol 63no 12 pp 3002ndash3010 2014

[8] Y Zhou X Zhang and W Xiao ldquoSpinning projectilersquos an-gular measurement using crest and trough data of a geo-magnetic sensorrdquo Measurment Science and Technologyvol 29 no 9 Article ID 095007 2018

[9] H Zhao Z Su F Liu C Li Q Li and N Liu ldquoExtraction andfilter algorithm of roll angular rate for high spinning pro-jectilesrdquo Mathematical Problems in Engineering vol 2019Article ID 3181727 15 pages 2019

[10] S Carletta and P Teofilatto ldquoDesign and numerical validationof an algorithm for the detumbling and angular rate deter-mination of a CubeSat using only three-axis magnetometerdatardquo International Journal of Aerospace Engineeringvol 2018 Article ID 9768475 12 pages 2018

[11] L-B Li M-X Li L-X Jiang D-Y Wang F Zhan andT Sheng ldquoAngular rate estimation and damping control ofsatellite with magnetometer datardquo Optik vol 180 no 11pp 1049ndash1055 2019

[12] H Ma and S Xu ldquoMagnetometer-only attitude and angularvelocity filtering estimation for attitude changing spacecraftrdquoActa Astronautica vol 102 no 5 pp 89ndash102 2014

[13] S Sabzevari M R Arvan A R Vali S M M Dehghan andM H Ferdowsi ldquoSymmetry preserving nonlinear observer forattitude estimation with magnetometer onlyrdquo ISA Transac-tions vol 102 no 3 pp 314ndash324 2020

[14] J M Maley ldquoEfficient attitude estimation for a spin-stabilizedprojectilerdquo Journal of Guidance Control and Dynamicsvol 39 no 2 pp 1ndash12 2016

[15] G Hu W Wang Y Zhong B Gao and C Gu ldquoA new directfiltering approach to INSGNSS integrationrdquo Aerospace Sci-ence and Technology vol 77 no 7 pp 755ndash764 2018

[16] M Yunjian X Changfan J Yixian W Yao and Z YildquoAngular velocity estimation of rollingmdashammunition basedon magnetometerrdquo Journal of Projectiles Rockets Missiles andGuidance vol 36 no 1 pp 69ndash72 2016

[17] G Hu L Ni B Gao X ZhuWWang and Y Zhong ldquoModelpredictive based unscented Kalman filter for hypersonic ve-hicle navigation with INSGNSS integrationrdquo IEEE Accessvol 8 no 1 pp 4814ndash4823 2016

[18] C Chunhang L Chunsheng J Wendou et al ldquoe projectileattitude measuring method based on geomagnetic sensorrdquoJournal of Detection amp Control vol 40 no 12 pp 4814ndash48232020

[19] X Chao X-z Bu and Y Bo ldquoree different attitudemeasurements of spinning projectile based on magneticsensorsrdquo Measurement vol 47 no 1 pp 331ndash340 2014

[20] W Yu ldquoHalf-experiential formulas for calculating decreasingangular velocity of projectile in trajectoryrdquo Journal of De-tection and Control vol 231 no 5 pp 866ndash876 2003

[21] J Yu X Bu C Xiang and B Yang ldquoSpinning projectilersquosattitude measurement using intersection ratio of magneticsensorsrdquo Proceedings of the Institution of Mechanical Engi-neers Part G-Journal of Aerospace Engineering vol 231 no 5pp 1ndash6 2016

[22] X Zhao X Zhang D Long Z Bai and Y Wang ldquoe designof roll angle magnetic measurement system used in spinningprojectilesrdquo Chinese Journal of Sensors and Actuators vol 26no 9 pp 1309ndash1313 2013

Mathematical Problems in Engineering 11

the accuracy of the roll rotation rate is increased by morethan 4 times It can be seen from Figure 5 that in thestationary phase the accuracy of the roll rotation rate es-timated by the algorithm is 10 times higher than that ob-tained by direct derivation e validity of the algorithm isverified

42 Validation of Semiphysical Turntable Data After thealgorithm is verified it is verified according to the data of themagnetic measurement system With the three-axis mag-netometer HMC1053 produced by Honeywell company asthe only attitude sensing chip and the STM32 single chipmicrocomputer produced by ST Company as the controllerthe control circuit was designed to constitute the magneticattitude measurement system [21 22] and the magneticmeasurement system was fixed on the three-axis high-precision flight simulation turntable as shown in Figure 6e control table rotates around the X-axis Y-axis and Z-axis respectively and the roll angle is calculated by using themagnetic measurement system and the roll angle and theroll rotation rate are optimized and estimated by taking themagnetic measurement as the observation data

In this experiment the gyroscope is saturated and it isimpossible to further estimate the carrier roll angle so thegyro information is only used to estimate the roll angle rateestimated by the algorithm in the later stage

To simulate the flight state of the projectile body underthe maneuvering condition the flight turntable wasaccelerated uniformly around the X-axis to 5 rs and thendecelerated uniformly to 0 rs after maintaining 21 s asshown in Figure 7 In the model parameter setting Q 22R 052 and s 103 and the simulation results are shownin the figures

e estimated roll angle and roll rotation rate are shownin Figures 7 and 8 It can be seen from Figure 8 that at theangular rate mutation of 1 s 3 s and 23 s the roll angle androll rate estimated by the pure Kalman filter have large errorHowever the error estimated by adding forgetting factoralgorithm is significantly smaller as shown in Table 2

As can be seen from Table 2 in the angular rate mu-tation stage compared with the estimation result of thepure Kalman filter the roll angle accuracy estimated by thealgorithm with the forgetting factor is improved by anorder of magnitude e accuracy of the estimated rollrotation rate is improved by more than 4 times It can alsobe seen from Figures 9 and 10 that in the stationary phasethe roll angle error estimated by the algorithm is within 2degwhich is more than twice as accurate as the roll error of 5degmeasured e error of the roll rotation rate estimated bythe algorithm is within 5 degs which improves the accuracyby an order of magnitude compared with the error of theroll rotation rate obtained by direct derivation of 50 degswhich verifies the feasibility of the algorithm and thesystem

43 Verification of Bomb Test Data e feasibility of thealgorithm is verified by theoretical analysis and turntablesemiphysical simulation test Now the sensor data

Figure 6 ree-axis high-precision flight simulation turntable

10 15 25200 5t (s)

0

50

100

150

200

250

300

350

400

Roll

ange

l (deg)

Roll angle

Estimated by pureKalman filterMeasured roll angle

Estimated by forgettingfactor algorithmThe turntable feedback

Figure 7 e roll angles of turntable feedback and algorithmestimation

2 3 4

2000

1500

1000

1850

1800

1750122 124 126 128

1800

1600

1400

1200

1000

800

23 235 24

Roll rotation rate

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodThe turntable feedback

20 25 30151050t (s)

ndash1000

ndash500

0

500

1000

1500

2000

2500

Roll

rota

tion

rate

(degs

)

Figure 8 e roll angles rate of turntable feedback and algorithmestimation

Mathematical Problems in Engineering 7

collected from the ballistic flight test of the range is used forfurther verification A magnetic measuring system and anaxial MEMS gyro are installed in the test projectile bodye roll rotation rate of the gyro output is taken as areference to verify the accuracy of the estimated roll

rotation and roll rotation rate calculated by using only themagnetometer data e test results are shown in thefigures

Roll angel error

15

10

5

0

ndash523 24 25

12 122 124

5

0

ndash52 3 4

10

0

ndash10

5 150 35302010 25t (s)

ndash20

ndash15

ndash10

ndash5

0

5

10

15

Roll

ange

l err

or (deg

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

Figure 9 e error of the roll angle

1 2 3

100

0

ndash100

100

0

ndash10011 115 12 125 13

150100

500

ndash50235 24 245

Error of roll rotation rate

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

5 15 20 302510t (s)

ndash500

ndash400

ndash300

ndash200

ndash100

0

100

200

Erro

r of r

oll r

otat

ion

rate

(degs

)

Figure 10 e error of the roll angle rate

Sensor data

1510 200 5t (s)

0

05

1

15

2

25

3

Volta

ge v

alue

(V)

X axis of magneticsensorY axis of magneticsensor

Z axis of magneticsensorThe axial gyro

Figure 11 Sensor data of magnetic and gyro

10 20 30 40 50 60 700t (s)

0

100

200

300

400

Roll

Ang

le (deg

)

(a)

10 20 30 40 50 60 700t (s)

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

(b)

Figure 12 e system calculates the roll angle and the angular rateof the gyro output (a) e role angle that system calculates (b)Rotation rate of gyro type

Table 2 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular ratemutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)

Estimated by pure Kalmanfilter

Estimated by thealgorithm with forgetting

factor

Estimated bypure

Kalman filter

Estimated by thealgorithm

with forgetting factor1 sndash2 s minus 82261 minus 05716 minus 745018 minus 412543 sndash4 s 66852 minus 06276 776076 15466023 sndash24 s 73149 minus 00166 784181 160643

8 Mathematical Problems in Engineering

e figure above shows the system output and algorithmestimation results of the bomb test It can be seen fromFigure 11that the effective flight time of the ballistic test is 20 s It can beseen from Figures 12 and 13 that the gyro is saturated duringflight and cannot normally calculate the roll rotation rateHowever the roll rotation rate estimated by the algorithmmakesup for this defect e roll angle and roll rotation rate estimatedby the algorithm are shown in Figures 13 and 14 Figure 14shows that the roll angle estimated by the algorithm is betterthan the linearity of the roll angle calculated directly by thesystem which indicates that the roll angle estimated by thealgorithm compensates some errors caused by the systemmeasurement Figure 13 shows that the roll rotation rate

estimated by the algorithm compensates for the error caused bygyro saturation in the first two seconds In the stationary phasethe accuracy of the roll rotation rate estimated by the algorithmis 6 times higher than that obtained by direct derivation Fig-ure 15 shows that in the angular rate mutation stage the rollangular rate estimated by forgetting factor reduces the errorcaused by pure Kalman filter estimation and the mean value ofthe estimated error caused by angular rate mutation is shown inthe table

It can be seen from Table 3 that the angular rate at 11 sdoes not change much so the effect of the algorithm with theforgetting factor is not obvious However in other abruptchanges of angular rates the accuracy of the algorithm with

Roll rotation rate

5000

0

ndash5000

ndash1000005 1 15 2

2000

1800

1600

165 17 175 18

5 10 15 200t (s)

ndash8000

ndash6000

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodGyro calculating

Figure 13 e roll angle rate that the system calculates and the gyro output

Roll angle

72 74 76 78 8

3020 35 4025105 150t (s)

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

Roll

angl

e (deg)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmThe system calculates

Figure 14 e roll angle that the system calculates and the gyro output

Mathematical Problems in Engineering 9

the forgetting factor is more than 4 times higher than that ofthe pure Kalman filter

5 Conclusions

In this paper we propose a method to estimate the roll angleand roll angle rate of a projectile by using only the magneticfield information provided by a triaxial magnetometer and areal-time estimation algorithm based on the Kalman filterwith appropriate forgetting factor is proposed is methodsolves the problem that the projectile roll angle and roll anglerate cannot be obtained due to MEMS gyro overload anddegradation under the flight condition of high spin and highoverload e Kalman filter estimation algorithm with theoblivion factor is able to significantly reduce the error causedby estimation delay under high dynamic conditions

rough the above analysis and semiphysical simulationtest it can be concluded that the algorithm can estimate theroll angle and roll angle rate of the carrier in real time andquickly e experimental results show that the algorithmwith the forgetting factor reduces the influence of magneticsensor measurement error on the accuracy of roll angle andimproves the accuracy of roll angle by one time e ex-perimental results show that the error of the roll rotation rateestimated by this algorithm is within 5 degs and the accuracyis 6 times higher than that obtained by direct derivation Inthe angular rate mutation phase compared with the pureKalman filter estimation algorithm the accuracy of the roll

angle estimated by the algorithm that the Kalman filter withthe forgetting factor is improved by an order of magnitudeand the accuracy of the roll angle rate is improved by at leastfour times which canmeet the requirements of the projectileroll angle and roll angle rate of the guidance and controlsystem of general rotating bombs

Data Availability

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

Disclosure

Lizhen Gao and Yingying Zhang are co-first authors of thisarticle

Conflicts of Interest

e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was carried out in accordance with the require-ments of the National Natural Science Foundation of China(61873247) funded project corresponding test experimentswere carried out in the State Key Laboratory of ElectronicTesting Technology of North China University Key

Error of roll rotation rate

0 05 1 15 2 25ndash5000

0

5000

115 12 125 13

ndash200

0

200

20 21ndash1000

0

1000

2000

1510 200 5t (s)

ndash6000

ndash4000

ndash2000

0

2000

4000

6000

Erro

r of r

oll r

otat

ion

rate

(degs

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmEstimated by the direct derivation method

Figure 15 e error of roll angle rate that the system calculates and the gyro output

Table 3 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular rate mutationMean of roll angle rate error (s)

Estimated by pure Kalman filter Estimated by the algorithm with forgetting factor0 sndash02 s 14505 minus 31891042 sndash25 s 5706310 1440300115 sndash12 s 813683 62837720 sndash204 s 8241634 2084635

10 Mathematical Problems in Engineering

Laboratory of Instrumental Science and Dynamic Testing ofNorth China University Huaihai Industry Group inChangzhi City Shanxi Province and Alashan ShootingRange During this period the authors also got instructionsfrom Professor Zhang Xiaoming and Teacher Li Xiuyuane authors would like to thank them for their help andsupport

References

[1] L An L Wang and D Zhao ldquoAttitude determinationmethod of spinning projectile based on geomagnetic azi-muthrdquo Journal of Chinese Inertial Technology vol 27 no 5pp 618ndash624 2019

[2] A Grosz E Paperno S Amrusi and B Zadov ldquoA three-axialsearch coil magnetometer optimized for small size low powerand low frequenciesrdquo IEEE Sensors Journal vol 11 no 4pp 1088ndash1094 2011

[3] H Liu H Dong J Ge B Bai Z Yuan and Z Zhao ldquoResearchon a secondary tuning algorithm based on SVD amp STFT forFID signalrdquo Measurment Science and Technology vol 27no 10 pp 0957ndash0233 Article ID 105006 2016

[4] J Shang D Zhihong M Fu and S Wang ldquoA high-spin ratemeasurement method for projectiles using a magnetoresistivesensor based on time-frequency domain analysisrdquo Sensorsvol 16 no 6 p 894 2016

[5] C Mateo and J A Talavera ldquoShort-time fourier transformwith the window size fixed in the frequency domainrdquo DigitalSignal Processing vol 77 no 6 pp 13ndash21 2018

[6] X Yan G Chen and X Tian ldquoTwo-step adaptive augmentedunscented Kalman filter for roll angles of spinning missilesbased on magnetometer measurementsrdquo Measurement andControl vol 51 no 3-4 pp 73ndash82 2018

[7] T Addabbo R Biondi S Cioncolini A Fort F Rossetti andV Vignoli ldquoA zero-crossing detection system based on FPGAto measure the angular vibrations of rotating shaftsrdquo IEEETransactions on Instrumentation and Measurement vol 63no 12 pp 3002ndash3010 2014

[8] Y Zhou X Zhang and W Xiao ldquoSpinning projectilersquos an-gular measurement using crest and trough data of a geo-magnetic sensorrdquo Measurment Science and Technologyvol 29 no 9 Article ID 095007 2018

[9] H Zhao Z Su F Liu C Li Q Li and N Liu ldquoExtraction andfilter algorithm of roll angular rate for high spinning pro-jectilesrdquo Mathematical Problems in Engineering vol 2019Article ID 3181727 15 pages 2019

[10] S Carletta and P Teofilatto ldquoDesign and numerical validationof an algorithm for the detumbling and angular rate deter-mination of a CubeSat using only three-axis magnetometerdatardquo International Journal of Aerospace Engineeringvol 2018 Article ID 9768475 12 pages 2018

[11] L-B Li M-X Li L-X Jiang D-Y Wang F Zhan andT Sheng ldquoAngular rate estimation and damping control ofsatellite with magnetometer datardquo Optik vol 180 no 11pp 1049ndash1055 2019

[12] H Ma and S Xu ldquoMagnetometer-only attitude and angularvelocity filtering estimation for attitude changing spacecraftrdquoActa Astronautica vol 102 no 5 pp 89ndash102 2014

[13] S Sabzevari M R Arvan A R Vali S M M Dehghan andM H Ferdowsi ldquoSymmetry preserving nonlinear observer forattitude estimation with magnetometer onlyrdquo ISA Transac-tions vol 102 no 3 pp 314ndash324 2020

[14] J M Maley ldquoEfficient attitude estimation for a spin-stabilizedprojectilerdquo Journal of Guidance Control and Dynamicsvol 39 no 2 pp 1ndash12 2016

[15] G Hu W Wang Y Zhong B Gao and C Gu ldquoA new directfiltering approach to INSGNSS integrationrdquo Aerospace Sci-ence and Technology vol 77 no 7 pp 755ndash764 2018

[16] M Yunjian X Changfan J Yixian W Yao and Z YildquoAngular velocity estimation of rollingmdashammunition basedon magnetometerrdquo Journal of Projectiles Rockets Missiles andGuidance vol 36 no 1 pp 69ndash72 2016

[17] G Hu L Ni B Gao X ZhuWWang and Y Zhong ldquoModelpredictive based unscented Kalman filter for hypersonic ve-hicle navigation with INSGNSS integrationrdquo IEEE Accessvol 8 no 1 pp 4814ndash4823 2016

[18] C Chunhang L Chunsheng J Wendou et al ldquoe projectileattitude measuring method based on geomagnetic sensorrdquoJournal of Detection amp Control vol 40 no 12 pp 4814ndash48232020

[19] X Chao X-z Bu and Y Bo ldquoree different attitudemeasurements of spinning projectile based on magneticsensorsrdquo Measurement vol 47 no 1 pp 331ndash340 2014

[20] W Yu ldquoHalf-experiential formulas for calculating decreasingangular velocity of projectile in trajectoryrdquo Journal of De-tection and Control vol 231 no 5 pp 866ndash876 2003

[21] J Yu X Bu C Xiang and B Yang ldquoSpinning projectilersquosattitude measurement using intersection ratio of magneticsensorsrdquo Proceedings of the Institution of Mechanical Engi-neers Part G-Journal of Aerospace Engineering vol 231 no 5pp 1ndash6 2016

[22] X Zhao X Zhang D Long Z Bai and Y Wang ldquoe designof roll angle magnetic measurement system used in spinningprojectilesrdquo Chinese Journal of Sensors and Actuators vol 26no 9 pp 1309ndash1313 2013

Mathematical Problems in Engineering 11

collected from the ballistic flight test of the range is used forfurther verification A magnetic measuring system and anaxial MEMS gyro are installed in the test projectile bodye roll rotation rate of the gyro output is taken as areference to verify the accuracy of the estimated roll

rotation and roll rotation rate calculated by using only themagnetometer data e test results are shown in thefigures

Roll angel error

15

10

5

0

ndash523 24 25

12 122 124

5

0

ndash52 3 4

10

0

ndash10

5 150 35302010 25t (s)

ndash20

ndash15

ndash10

ndash5

0

5

10

15

Roll

ange

l err

or (deg

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

Figure 9 e error of the roll angle

1 2 3

100

0

ndash100

100

0

ndash10011 115 12 125 13

150100

500

ndash50235 24 245

Error of roll rotation rate

Estimated by pure Kalman filterEstimated by forgetting factor algorithmError of roll angel measured

5 15 20 302510t (s)

ndash500

ndash400

ndash300

ndash200

ndash100

0

100

200

Erro

r of r

oll r

otat

ion

rate

(degs

)

Figure 10 e error of the roll angle rate

Sensor data

1510 200 5t (s)

0

05

1

15

2

25

3

Volta

ge v

alue

(V)

X axis of magneticsensorY axis of magneticsensor

Z axis of magneticsensorThe axial gyro

Figure 11 Sensor data of magnetic and gyro

10 20 30 40 50 60 700t (s)

0

100

200

300

400

Roll

Ang

le (deg

)

(a)

10 20 30 40 50 60 700t (s)

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

(b)

Figure 12 e system calculates the roll angle and the angular rateof the gyro output (a) e role angle that system calculates (b)Rotation rate of gyro type

Table 2 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular ratemutation

Mean of roll angle error (deg) Mean of roll angle rate error (degs)

Estimated by pure Kalmanfilter

Estimated by thealgorithm with forgetting

factor

Estimated bypure

Kalman filter

Estimated by thealgorithm

with forgetting factor1 sndash2 s minus 82261 minus 05716 minus 745018 minus 412543 sndash4 s 66852 minus 06276 776076 15466023 sndash24 s 73149 minus 00166 784181 160643

8 Mathematical Problems in Engineering

e figure above shows the system output and algorithmestimation results of the bomb test It can be seen fromFigure 11that the effective flight time of the ballistic test is 20 s It can beseen from Figures 12 and 13 that the gyro is saturated duringflight and cannot normally calculate the roll rotation rateHowever the roll rotation rate estimated by the algorithmmakesup for this defect e roll angle and roll rotation rate estimatedby the algorithm are shown in Figures 13 and 14 Figure 14shows that the roll angle estimated by the algorithm is betterthan the linearity of the roll angle calculated directly by thesystem which indicates that the roll angle estimated by thealgorithm compensates some errors caused by the systemmeasurement Figure 13 shows that the roll rotation rate

estimated by the algorithm compensates for the error caused bygyro saturation in the first two seconds In the stationary phasethe accuracy of the roll rotation rate estimated by the algorithmis 6 times higher than that obtained by direct derivation Fig-ure 15 shows that in the angular rate mutation stage the rollangular rate estimated by forgetting factor reduces the errorcaused by pure Kalman filter estimation and the mean value ofthe estimated error caused by angular rate mutation is shown inthe table

It can be seen from Table 3 that the angular rate at 11 sdoes not change much so the effect of the algorithm with theforgetting factor is not obvious However in other abruptchanges of angular rates the accuracy of the algorithm with

Roll rotation rate

5000

0

ndash5000

ndash1000005 1 15 2

2000

1800

1600

165 17 175 18

5 10 15 200t (s)

ndash8000

ndash6000

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodGyro calculating

Figure 13 e roll angle rate that the system calculates and the gyro output

Roll angle

72 74 76 78 8

3020 35 4025105 150t (s)

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

Roll

angl

e (deg)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmThe system calculates

Figure 14 e roll angle that the system calculates and the gyro output

Mathematical Problems in Engineering 9

the forgetting factor is more than 4 times higher than that ofthe pure Kalman filter

5 Conclusions

In this paper we propose a method to estimate the roll angleand roll angle rate of a projectile by using only the magneticfield information provided by a triaxial magnetometer and areal-time estimation algorithm based on the Kalman filterwith appropriate forgetting factor is proposed is methodsolves the problem that the projectile roll angle and roll anglerate cannot be obtained due to MEMS gyro overload anddegradation under the flight condition of high spin and highoverload e Kalman filter estimation algorithm with theoblivion factor is able to significantly reduce the error causedby estimation delay under high dynamic conditions

rough the above analysis and semiphysical simulationtest it can be concluded that the algorithm can estimate theroll angle and roll angle rate of the carrier in real time andquickly e experimental results show that the algorithmwith the forgetting factor reduces the influence of magneticsensor measurement error on the accuracy of roll angle andimproves the accuracy of roll angle by one time e ex-perimental results show that the error of the roll rotation rateestimated by this algorithm is within 5 degs and the accuracyis 6 times higher than that obtained by direct derivation Inthe angular rate mutation phase compared with the pureKalman filter estimation algorithm the accuracy of the roll

angle estimated by the algorithm that the Kalman filter withthe forgetting factor is improved by an order of magnitudeand the accuracy of the roll angle rate is improved by at leastfour times which canmeet the requirements of the projectileroll angle and roll angle rate of the guidance and controlsystem of general rotating bombs

Data Availability

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

Disclosure

Lizhen Gao and Yingying Zhang are co-first authors of thisarticle

Conflicts of Interest

e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was carried out in accordance with the require-ments of the National Natural Science Foundation of China(61873247) funded project corresponding test experimentswere carried out in the State Key Laboratory of ElectronicTesting Technology of North China University Key

Error of roll rotation rate

0 05 1 15 2 25ndash5000

0

5000

115 12 125 13

ndash200

0

200

20 21ndash1000

0

1000

2000

1510 200 5t (s)

ndash6000

ndash4000

ndash2000

0

2000

4000

6000

Erro

r of r

oll r

otat

ion

rate

(degs

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmEstimated by the direct derivation method

Figure 15 e error of roll angle rate that the system calculates and the gyro output

Table 3 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular rate mutationMean of roll angle rate error (s)

Estimated by pure Kalman filter Estimated by the algorithm with forgetting factor0 sndash02 s 14505 minus 31891042 sndash25 s 5706310 1440300115 sndash12 s 813683 62837720 sndash204 s 8241634 2084635

10 Mathematical Problems in Engineering

Laboratory of Instrumental Science and Dynamic Testing ofNorth China University Huaihai Industry Group inChangzhi City Shanxi Province and Alashan ShootingRange During this period the authors also got instructionsfrom Professor Zhang Xiaoming and Teacher Li Xiuyuane authors would like to thank them for their help andsupport

References

[1] L An L Wang and D Zhao ldquoAttitude determinationmethod of spinning projectile based on geomagnetic azi-muthrdquo Journal of Chinese Inertial Technology vol 27 no 5pp 618ndash624 2019

[2] A Grosz E Paperno S Amrusi and B Zadov ldquoA three-axialsearch coil magnetometer optimized for small size low powerand low frequenciesrdquo IEEE Sensors Journal vol 11 no 4pp 1088ndash1094 2011

[3] H Liu H Dong J Ge B Bai Z Yuan and Z Zhao ldquoResearchon a secondary tuning algorithm based on SVD amp STFT forFID signalrdquo Measurment Science and Technology vol 27no 10 pp 0957ndash0233 Article ID 105006 2016

[4] J Shang D Zhihong M Fu and S Wang ldquoA high-spin ratemeasurement method for projectiles using a magnetoresistivesensor based on time-frequency domain analysisrdquo Sensorsvol 16 no 6 p 894 2016

[5] C Mateo and J A Talavera ldquoShort-time fourier transformwith the window size fixed in the frequency domainrdquo DigitalSignal Processing vol 77 no 6 pp 13ndash21 2018

[6] X Yan G Chen and X Tian ldquoTwo-step adaptive augmentedunscented Kalman filter for roll angles of spinning missilesbased on magnetometer measurementsrdquo Measurement andControl vol 51 no 3-4 pp 73ndash82 2018

[7] T Addabbo R Biondi S Cioncolini A Fort F Rossetti andV Vignoli ldquoA zero-crossing detection system based on FPGAto measure the angular vibrations of rotating shaftsrdquo IEEETransactions on Instrumentation and Measurement vol 63no 12 pp 3002ndash3010 2014

[8] Y Zhou X Zhang and W Xiao ldquoSpinning projectilersquos an-gular measurement using crest and trough data of a geo-magnetic sensorrdquo Measurment Science and Technologyvol 29 no 9 Article ID 095007 2018

[9] H Zhao Z Su F Liu C Li Q Li and N Liu ldquoExtraction andfilter algorithm of roll angular rate for high spinning pro-jectilesrdquo Mathematical Problems in Engineering vol 2019Article ID 3181727 15 pages 2019

[10] S Carletta and P Teofilatto ldquoDesign and numerical validationof an algorithm for the detumbling and angular rate deter-mination of a CubeSat using only three-axis magnetometerdatardquo International Journal of Aerospace Engineeringvol 2018 Article ID 9768475 12 pages 2018

[11] L-B Li M-X Li L-X Jiang D-Y Wang F Zhan andT Sheng ldquoAngular rate estimation and damping control ofsatellite with magnetometer datardquo Optik vol 180 no 11pp 1049ndash1055 2019

[12] H Ma and S Xu ldquoMagnetometer-only attitude and angularvelocity filtering estimation for attitude changing spacecraftrdquoActa Astronautica vol 102 no 5 pp 89ndash102 2014

[13] S Sabzevari M R Arvan A R Vali S M M Dehghan andM H Ferdowsi ldquoSymmetry preserving nonlinear observer forattitude estimation with magnetometer onlyrdquo ISA Transac-tions vol 102 no 3 pp 314ndash324 2020

[14] J M Maley ldquoEfficient attitude estimation for a spin-stabilizedprojectilerdquo Journal of Guidance Control and Dynamicsvol 39 no 2 pp 1ndash12 2016

[15] G Hu W Wang Y Zhong B Gao and C Gu ldquoA new directfiltering approach to INSGNSS integrationrdquo Aerospace Sci-ence and Technology vol 77 no 7 pp 755ndash764 2018

[16] M Yunjian X Changfan J Yixian W Yao and Z YildquoAngular velocity estimation of rollingmdashammunition basedon magnetometerrdquo Journal of Projectiles Rockets Missiles andGuidance vol 36 no 1 pp 69ndash72 2016

[17] G Hu L Ni B Gao X ZhuWWang and Y Zhong ldquoModelpredictive based unscented Kalman filter for hypersonic ve-hicle navigation with INSGNSS integrationrdquo IEEE Accessvol 8 no 1 pp 4814ndash4823 2016

[18] C Chunhang L Chunsheng J Wendou et al ldquoe projectileattitude measuring method based on geomagnetic sensorrdquoJournal of Detection amp Control vol 40 no 12 pp 4814ndash48232020

[19] X Chao X-z Bu and Y Bo ldquoree different attitudemeasurements of spinning projectile based on magneticsensorsrdquo Measurement vol 47 no 1 pp 331ndash340 2014

[20] W Yu ldquoHalf-experiential formulas for calculating decreasingangular velocity of projectile in trajectoryrdquo Journal of De-tection and Control vol 231 no 5 pp 866ndash876 2003

[21] J Yu X Bu C Xiang and B Yang ldquoSpinning projectilersquosattitude measurement using intersection ratio of magneticsensorsrdquo Proceedings of the Institution of Mechanical Engi-neers Part G-Journal of Aerospace Engineering vol 231 no 5pp 1ndash6 2016

[22] X Zhao X Zhang D Long Z Bai and Y Wang ldquoe designof roll angle magnetic measurement system used in spinningprojectilesrdquo Chinese Journal of Sensors and Actuators vol 26no 9 pp 1309ndash1313 2013

Mathematical Problems in Engineering 11

e figure above shows the system output and algorithmestimation results of the bomb test It can be seen fromFigure 11that the effective flight time of the ballistic test is 20 s It can beseen from Figures 12 and 13 that the gyro is saturated duringflight and cannot normally calculate the roll rotation rateHowever the roll rotation rate estimated by the algorithmmakesup for this defect e roll angle and roll rotation rate estimatedby the algorithm are shown in Figures 13 and 14 Figure 14shows that the roll angle estimated by the algorithm is betterthan the linearity of the roll angle calculated directly by thesystem which indicates that the roll angle estimated by thealgorithm compensates some errors caused by the systemmeasurement Figure 13 shows that the roll rotation rate

estimated by the algorithm compensates for the error caused bygyro saturation in the first two seconds In the stationary phasethe accuracy of the roll rotation rate estimated by the algorithmis 6 times higher than that obtained by direct derivation Fig-ure 15 shows that in the angular rate mutation stage the rollangular rate estimated by forgetting factor reduces the errorcaused by pure Kalman filter estimation and the mean value ofthe estimated error caused by angular rate mutation is shown inthe table

It can be seen from Table 3 that the angular rate at 11 sdoes not change much so the effect of the algorithm with theforgetting factor is not obvious However in other abruptchanges of angular rates the accuracy of the algorithm with

Roll rotation rate

5000

0

ndash5000

ndash1000005 1 15 2

2000

1800

1600

165 17 175 18

5 10 15 200t (s)

ndash8000

ndash6000

ndash4000

ndash2000

0

2000

4000

Roll

rota

tion

rate

(degs

)

Estimated by pureKalman filterEstimated by forgettingfactor algorithm

Estimated by the directderivation methodGyro calculating

Figure 13 e roll angle rate that the system calculates and the gyro output

Roll angle

72 74 76 78 8

3020 35 4025105 150t (s)

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

Roll

angl

e (deg)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmThe system calculates

Figure 14 e roll angle that the system calculates and the gyro output

Mathematical Problems in Engineering 9

the forgetting factor is more than 4 times higher than that ofthe pure Kalman filter

5 Conclusions

In this paper we propose a method to estimate the roll angleand roll angle rate of a projectile by using only the magneticfield information provided by a triaxial magnetometer and areal-time estimation algorithm based on the Kalman filterwith appropriate forgetting factor is proposed is methodsolves the problem that the projectile roll angle and roll anglerate cannot be obtained due to MEMS gyro overload anddegradation under the flight condition of high spin and highoverload e Kalman filter estimation algorithm with theoblivion factor is able to significantly reduce the error causedby estimation delay under high dynamic conditions

rough the above analysis and semiphysical simulationtest it can be concluded that the algorithm can estimate theroll angle and roll angle rate of the carrier in real time andquickly e experimental results show that the algorithmwith the forgetting factor reduces the influence of magneticsensor measurement error on the accuracy of roll angle andimproves the accuracy of roll angle by one time e ex-perimental results show that the error of the roll rotation rateestimated by this algorithm is within 5 degs and the accuracyis 6 times higher than that obtained by direct derivation Inthe angular rate mutation phase compared with the pureKalman filter estimation algorithm the accuracy of the roll

angle estimated by the algorithm that the Kalman filter withthe forgetting factor is improved by an order of magnitudeand the accuracy of the roll angle rate is improved by at leastfour times which canmeet the requirements of the projectileroll angle and roll angle rate of the guidance and controlsystem of general rotating bombs

Data Availability

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

Disclosure

Lizhen Gao and Yingying Zhang are co-first authors of thisarticle

Conflicts of Interest

e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was carried out in accordance with the require-ments of the National Natural Science Foundation of China(61873247) funded project corresponding test experimentswere carried out in the State Key Laboratory of ElectronicTesting Technology of North China University Key

Error of roll rotation rate

0 05 1 15 2 25ndash5000

0

5000

115 12 125 13

ndash200

0

200

20 21ndash1000

0

1000

2000

1510 200 5t (s)

ndash6000

ndash4000

ndash2000

0

2000

4000

6000

Erro

r of r

oll r

otat

ion

rate

(degs

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmEstimated by the direct derivation method

Figure 15 e error of roll angle rate that the system calculates and the gyro output

Table 3 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular rate mutationMean of roll angle rate error (s)

Estimated by pure Kalman filter Estimated by the algorithm with forgetting factor0 sndash02 s 14505 minus 31891042 sndash25 s 5706310 1440300115 sndash12 s 813683 62837720 sndash204 s 8241634 2084635

10 Mathematical Problems in Engineering

Laboratory of Instrumental Science and Dynamic Testing ofNorth China University Huaihai Industry Group inChangzhi City Shanxi Province and Alashan ShootingRange During this period the authors also got instructionsfrom Professor Zhang Xiaoming and Teacher Li Xiuyuane authors would like to thank them for their help andsupport

References

[1] L An L Wang and D Zhao ldquoAttitude determinationmethod of spinning projectile based on geomagnetic azi-muthrdquo Journal of Chinese Inertial Technology vol 27 no 5pp 618ndash624 2019

[2] A Grosz E Paperno S Amrusi and B Zadov ldquoA three-axialsearch coil magnetometer optimized for small size low powerand low frequenciesrdquo IEEE Sensors Journal vol 11 no 4pp 1088ndash1094 2011

[3] H Liu H Dong J Ge B Bai Z Yuan and Z Zhao ldquoResearchon a secondary tuning algorithm based on SVD amp STFT forFID signalrdquo Measurment Science and Technology vol 27no 10 pp 0957ndash0233 Article ID 105006 2016

[4] J Shang D Zhihong M Fu and S Wang ldquoA high-spin ratemeasurement method for projectiles using a magnetoresistivesensor based on time-frequency domain analysisrdquo Sensorsvol 16 no 6 p 894 2016

[5] C Mateo and J A Talavera ldquoShort-time fourier transformwith the window size fixed in the frequency domainrdquo DigitalSignal Processing vol 77 no 6 pp 13ndash21 2018

[6] X Yan G Chen and X Tian ldquoTwo-step adaptive augmentedunscented Kalman filter for roll angles of spinning missilesbased on magnetometer measurementsrdquo Measurement andControl vol 51 no 3-4 pp 73ndash82 2018

[7] T Addabbo R Biondi S Cioncolini A Fort F Rossetti andV Vignoli ldquoA zero-crossing detection system based on FPGAto measure the angular vibrations of rotating shaftsrdquo IEEETransactions on Instrumentation and Measurement vol 63no 12 pp 3002ndash3010 2014

[8] Y Zhou X Zhang and W Xiao ldquoSpinning projectilersquos an-gular measurement using crest and trough data of a geo-magnetic sensorrdquo Measurment Science and Technologyvol 29 no 9 Article ID 095007 2018

[9] H Zhao Z Su F Liu C Li Q Li and N Liu ldquoExtraction andfilter algorithm of roll angular rate for high spinning pro-jectilesrdquo Mathematical Problems in Engineering vol 2019Article ID 3181727 15 pages 2019

[10] S Carletta and P Teofilatto ldquoDesign and numerical validationof an algorithm for the detumbling and angular rate deter-mination of a CubeSat using only three-axis magnetometerdatardquo International Journal of Aerospace Engineeringvol 2018 Article ID 9768475 12 pages 2018

[11] L-B Li M-X Li L-X Jiang D-Y Wang F Zhan andT Sheng ldquoAngular rate estimation and damping control ofsatellite with magnetometer datardquo Optik vol 180 no 11pp 1049ndash1055 2019

[12] H Ma and S Xu ldquoMagnetometer-only attitude and angularvelocity filtering estimation for attitude changing spacecraftrdquoActa Astronautica vol 102 no 5 pp 89ndash102 2014

[13] S Sabzevari M R Arvan A R Vali S M M Dehghan andM H Ferdowsi ldquoSymmetry preserving nonlinear observer forattitude estimation with magnetometer onlyrdquo ISA Transac-tions vol 102 no 3 pp 314ndash324 2020

[14] J M Maley ldquoEfficient attitude estimation for a spin-stabilizedprojectilerdquo Journal of Guidance Control and Dynamicsvol 39 no 2 pp 1ndash12 2016

[15] G Hu W Wang Y Zhong B Gao and C Gu ldquoA new directfiltering approach to INSGNSS integrationrdquo Aerospace Sci-ence and Technology vol 77 no 7 pp 755ndash764 2018

[16] M Yunjian X Changfan J Yixian W Yao and Z YildquoAngular velocity estimation of rollingmdashammunition basedon magnetometerrdquo Journal of Projectiles Rockets Missiles andGuidance vol 36 no 1 pp 69ndash72 2016

[17] G Hu L Ni B Gao X ZhuWWang and Y Zhong ldquoModelpredictive based unscented Kalman filter for hypersonic ve-hicle navigation with INSGNSS integrationrdquo IEEE Accessvol 8 no 1 pp 4814ndash4823 2016

[18] C Chunhang L Chunsheng J Wendou et al ldquoe projectileattitude measuring method based on geomagnetic sensorrdquoJournal of Detection amp Control vol 40 no 12 pp 4814ndash48232020

[19] X Chao X-z Bu and Y Bo ldquoree different attitudemeasurements of spinning projectile based on magneticsensorsrdquo Measurement vol 47 no 1 pp 331ndash340 2014

[20] W Yu ldquoHalf-experiential formulas for calculating decreasingangular velocity of projectile in trajectoryrdquo Journal of De-tection and Control vol 231 no 5 pp 866ndash876 2003

[21] J Yu X Bu C Xiang and B Yang ldquoSpinning projectilersquosattitude measurement using intersection ratio of magneticsensorsrdquo Proceedings of the Institution of Mechanical Engi-neers Part G-Journal of Aerospace Engineering vol 231 no 5pp 1ndash6 2016

[22] X Zhao X Zhang D Long Z Bai and Y Wang ldquoe designof roll angle magnetic measurement system used in spinningprojectilesrdquo Chinese Journal of Sensors and Actuators vol 26no 9 pp 1309ndash1313 2013

Mathematical Problems in Engineering 11

the forgetting factor is more than 4 times higher than that ofthe pure Kalman filter

5 Conclusions

In this paper we propose a method to estimate the roll angleand roll angle rate of a projectile by using only the magneticfield information provided by a triaxial magnetometer and areal-time estimation algorithm based on the Kalman filterwith appropriate forgetting factor is proposed is methodsolves the problem that the projectile roll angle and roll anglerate cannot be obtained due to MEMS gyro overload anddegradation under the flight condition of high spin and highoverload e Kalman filter estimation algorithm with theoblivion factor is able to significantly reduce the error causedby estimation delay under high dynamic conditions

rough the above analysis and semiphysical simulationtest it can be concluded that the algorithm can estimate theroll angle and roll angle rate of the carrier in real time andquickly e experimental results show that the algorithmwith the forgetting factor reduces the influence of magneticsensor measurement error on the accuracy of roll angle andimproves the accuracy of roll angle by one time e ex-perimental results show that the error of the roll rotation rateestimated by this algorithm is within 5 degs and the accuracyis 6 times higher than that obtained by direct derivation Inthe angular rate mutation phase compared with the pureKalman filter estimation algorithm the accuracy of the roll

angle estimated by the algorithm that the Kalman filter withthe forgetting factor is improved by an order of magnitudeand the accuracy of the roll angle rate is improved by at leastfour times which canmeet the requirements of the projectileroll angle and roll angle rate of the guidance and controlsystem of general rotating bombs

Data Availability

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

Disclosure

Lizhen Gao and Yingying Zhang are co-first authors of thisarticle

Conflicts of Interest

e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was carried out in accordance with the require-ments of the National Natural Science Foundation of China(61873247) funded project corresponding test experimentswere carried out in the State Key Laboratory of ElectronicTesting Technology of North China University Key

Error of roll rotation rate

0 05 1 15 2 25ndash5000

0

5000

115 12 125 13

ndash200

0

200

20 21ndash1000

0

1000

2000

1510 200 5t (s)

ndash6000

ndash4000

ndash2000

0

2000

4000

6000

Erro

r of r

oll r

otat

ion

rate

(degs

)

Estimated by pure Kalman filterEstimated by forgetting factor algorithmEstimated by the direct derivation method

Figure 15 e error of roll angle rate that the system calculates and the gyro output

Table 3 e estimation error of the algorithm in the angular rate mutation stage

e stage of angular rate mutationMean of roll angle rate error (s)

Estimated by pure Kalman filter Estimated by the algorithm with forgetting factor0 sndash02 s 14505 minus 31891042 sndash25 s 5706310 1440300115 sndash12 s 813683 62837720 sndash204 s 8241634 2084635

10 Mathematical Problems in Engineering

Laboratory of Instrumental Science and Dynamic Testing ofNorth China University Huaihai Industry Group inChangzhi City Shanxi Province and Alashan ShootingRange During this period the authors also got instructionsfrom Professor Zhang Xiaoming and Teacher Li Xiuyuane authors would like to thank them for their help andsupport

References

[1] L An L Wang and D Zhao ldquoAttitude determinationmethod of spinning projectile based on geomagnetic azi-muthrdquo Journal of Chinese Inertial Technology vol 27 no 5pp 618ndash624 2019

[2] A Grosz E Paperno S Amrusi and B Zadov ldquoA three-axialsearch coil magnetometer optimized for small size low powerand low frequenciesrdquo IEEE Sensors Journal vol 11 no 4pp 1088ndash1094 2011

[3] H Liu H Dong J Ge B Bai Z Yuan and Z Zhao ldquoResearchon a secondary tuning algorithm based on SVD amp STFT forFID signalrdquo Measurment Science and Technology vol 27no 10 pp 0957ndash0233 Article ID 105006 2016

[4] J Shang D Zhihong M Fu and S Wang ldquoA high-spin ratemeasurement method for projectiles using a magnetoresistivesensor based on time-frequency domain analysisrdquo Sensorsvol 16 no 6 p 894 2016

[5] C Mateo and J A Talavera ldquoShort-time fourier transformwith the window size fixed in the frequency domainrdquo DigitalSignal Processing vol 77 no 6 pp 13ndash21 2018

[6] X Yan G Chen and X Tian ldquoTwo-step adaptive augmentedunscented Kalman filter for roll angles of spinning missilesbased on magnetometer measurementsrdquo Measurement andControl vol 51 no 3-4 pp 73ndash82 2018

[7] T Addabbo R Biondi S Cioncolini A Fort F Rossetti andV Vignoli ldquoA zero-crossing detection system based on FPGAto measure the angular vibrations of rotating shaftsrdquo IEEETransactions on Instrumentation and Measurement vol 63no 12 pp 3002ndash3010 2014

[8] Y Zhou X Zhang and W Xiao ldquoSpinning projectilersquos an-gular measurement using crest and trough data of a geo-magnetic sensorrdquo Measurment Science and Technologyvol 29 no 9 Article ID 095007 2018

[9] H Zhao Z Su F Liu C Li Q Li and N Liu ldquoExtraction andfilter algorithm of roll angular rate for high spinning pro-jectilesrdquo Mathematical Problems in Engineering vol 2019Article ID 3181727 15 pages 2019

[10] S Carletta and P Teofilatto ldquoDesign and numerical validationof an algorithm for the detumbling and angular rate deter-mination of a CubeSat using only three-axis magnetometerdatardquo International Journal of Aerospace Engineeringvol 2018 Article ID 9768475 12 pages 2018

[11] L-B Li M-X Li L-X Jiang D-Y Wang F Zhan andT Sheng ldquoAngular rate estimation and damping control ofsatellite with magnetometer datardquo Optik vol 180 no 11pp 1049ndash1055 2019

[12] H Ma and S Xu ldquoMagnetometer-only attitude and angularvelocity filtering estimation for attitude changing spacecraftrdquoActa Astronautica vol 102 no 5 pp 89ndash102 2014

[13] S Sabzevari M R Arvan A R Vali S M M Dehghan andM H Ferdowsi ldquoSymmetry preserving nonlinear observer forattitude estimation with magnetometer onlyrdquo ISA Transac-tions vol 102 no 3 pp 314ndash324 2020

[14] J M Maley ldquoEfficient attitude estimation for a spin-stabilizedprojectilerdquo Journal of Guidance Control and Dynamicsvol 39 no 2 pp 1ndash12 2016

[15] G Hu W Wang Y Zhong B Gao and C Gu ldquoA new directfiltering approach to INSGNSS integrationrdquo Aerospace Sci-ence and Technology vol 77 no 7 pp 755ndash764 2018

[16] M Yunjian X Changfan J Yixian W Yao and Z YildquoAngular velocity estimation of rollingmdashammunition basedon magnetometerrdquo Journal of Projectiles Rockets Missiles andGuidance vol 36 no 1 pp 69ndash72 2016

[17] G Hu L Ni B Gao X ZhuWWang and Y Zhong ldquoModelpredictive based unscented Kalman filter for hypersonic ve-hicle navigation with INSGNSS integrationrdquo IEEE Accessvol 8 no 1 pp 4814ndash4823 2016

[18] C Chunhang L Chunsheng J Wendou et al ldquoe projectileattitude measuring method based on geomagnetic sensorrdquoJournal of Detection amp Control vol 40 no 12 pp 4814ndash48232020

[19] X Chao X-z Bu and Y Bo ldquoree different attitudemeasurements of spinning projectile based on magneticsensorsrdquo Measurement vol 47 no 1 pp 331ndash340 2014

[20] W Yu ldquoHalf-experiential formulas for calculating decreasingangular velocity of projectile in trajectoryrdquo Journal of De-tection and Control vol 231 no 5 pp 866ndash876 2003

[21] J Yu X Bu C Xiang and B Yang ldquoSpinning projectilersquosattitude measurement using intersection ratio of magneticsensorsrdquo Proceedings of the Institution of Mechanical Engi-neers Part G-Journal of Aerospace Engineering vol 231 no 5pp 1ndash6 2016

[22] X Zhao X Zhang D Long Z Bai and Y Wang ldquoe designof roll angle magnetic measurement system used in spinningprojectilesrdquo Chinese Journal of Sensors and Actuators vol 26no 9 pp 1309ndash1313 2013

Mathematical Problems in Engineering 11

Laboratory of Instrumental Science and Dynamic Testing ofNorth China University Huaihai Industry Group inChangzhi City Shanxi Province and Alashan ShootingRange During this period the authors also got instructionsfrom Professor Zhang Xiaoming and Teacher Li Xiuyuane authors would like to thank them for their help andsupport

References

[1] L An L Wang and D Zhao ldquoAttitude determinationmethod of spinning projectile based on geomagnetic azi-muthrdquo Journal of Chinese Inertial Technology vol 27 no 5pp 618ndash624 2019

[2] A Grosz E Paperno S Amrusi and B Zadov ldquoA three-axialsearch coil magnetometer optimized for small size low powerand low frequenciesrdquo IEEE Sensors Journal vol 11 no 4pp 1088ndash1094 2011

[3] H Liu H Dong J Ge B Bai Z Yuan and Z Zhao ldquoResearchon a secondary tuning algorithm based on SVD amp STFT forFID signalrdquo Measurment Science and Technology vol 27no 10 pp 0957ndash0233 Article ID 105006 2016

[4] J Shang D Zhihong M Fu and S Wang ldquoA high-spin ratemeasurement method for projectiles using a magnetoresistivesensor based on time-frequency domain analysisrdquo Sensorsvol 16 no 6 p 894 2016

[5] C Mateo and J A Talavera ldquoShort-time fourier transformwith the window size fixed in the frequency domainrdquo DigitalSignal Processing vol 77 no 6 pp 13ndash21 2018

[6] X Yan G Chen and X Tian ldquoTwo-step adaptive augmentedunscented Kalman filter for roll angles of spinning missilesbased on magnetometer measurementsrdquo Measurement andControl vol 51 no 3-4 pp 73ndash82 2018

[7] T Addabbo R Biondi S Cioncolini A Fort F Rossetti andV Vignoli ldquoA zero-crossing detection system based on FPGAto measure the angular vibrations of rotating shaftsrdquo IEEETransactions on Instrumentation and Measurement vol 63no 12 pp 3002ndash3010 2014

[8] Y Zhou X Zhang and W Xiao ldquoSpinning projectilersquos an-gular measurement using crest and trough data of a geo-magnetic sensorrdquo Measurment Science and Technologyvol 29 no 9 Article ID 095007 2018

[9] H Zhao Z Su F Liu C Li Q Li and N Liu ldquoExtraction andfilter algorithm of roll angular rate for high spinning pro-jectilesrdquo Mathematical Problems in Engineering vol 2019Article ID 3181727 15 pages 2019

[10] S Carletta and P Teofilatto ldquoDesign and numerical validationof an algorithm for the detumbling and angular rate deter-mination of a CubeSat using only three-axis magnetometerdatardquo International Journal of Aerospace Engineeringvol 2018 Article ID 9768475 12 pages 2018

[11] L-B Li M-X Li L-X Jiang D-Y Wang F Zhan andT Sheng ldquoAngular rate estimation and damping control ofsatellite with magnetometer datardquo Optik vol 180 no 11pp 1049ndash1055 2019

[12] H Ma and S Xu ldquoMagnetometer-only attitude and angularvelocity filtering estimation for attitude changing spacecraftrdquoActa Astronautica vol 102 no 5 pp 89ndash102 2014

[13] S Sabzevari M R Arvan A R Vali S M M Dehghan andM H Ferdowsi ldquoSymmetry preserving nonlinear observer forattitude estimation with magnetometer onlyrdquo ISA Transac-tions vol 102 no 3 pp 314ndash324 2020

[14] J M Maley ldquoEfficient attitude estimation for a spin-stabilizedprojectilerdquo Journal of Guidance Control and Dynamicsvol 39 no 2 pp 1ndash12 2016

[15] G Hu W Wang Y Zhong B Gao and C Gu ldquoA new directfiltering approach to INSGNSS integrationrdquo Aerospace Sci-ence and Technology vol 77 no 7 pp 755ndash764 2018

[16] M Yunjian X Changfan J Yixian W Yao and Z YildquoAngular velocity estimation of rollingmdashammunition basedon magnetometerrdquo Journal of Projectiles Rockets Missiles andGuidance vol 36 no 1 pp 69ndash72 2016

[17] G Hu L Ni B Gao X ZhuWWang and Y Zhong ldquoModelpredictive based unscented Kalman filter for hypersonic ve-hicle navigation with INSGNSS integrationrdquo IEEE Accessvol 8 no 1 pp 4814ndash4823 2016

[18] C Chunhang L Chunsheng J Wendou et al ldquoe projectileattitude measuring method based on geomagnetic sensorrdquoJournal of Detection amp Control vol 40 no 12 pp 4814ndash48232020

[19] X Chao X-z Bu and Y Bo ldquoree different attitudemeasurements of spinning projectile based on magneticsensorsrdquo Measurement vol 47 no 1 pp 331ndash340 2014

[20] W Yu ldquoHalf-experiential formulas for calculating decreasingangular velocity of projectile in trajectoryrdquo Journal of De-tection and Control vol 231 no 5 pp 866ndash876 2003

[21] J Yu X Bu C Xiang and B Yang ldquoSpinning projectilersquosattitude measurement using intersection ratio of magneticsensorsrdquo Proceedings of the Institution of Mechanical Engi-neers Part G-Journal of Aerospace Engineering vol 231 no 5pp 1ndash6 2016

[22] X Zhao X Zhang D Long Z Bai and Y Wang ldquoe designof roll angle magnetic measurement system used in spinningprojectilesrdquo Chinese Journal of Sensors and Actuators vol 26no 9 pp 1309ndash1313 2013

Mathematical Problems in Engineering 11