Ground target localization algorithm for semi-active laser terminal correction projectile

Xing-long LI*，Wen-jin YAO，Li-kun ZHU，Xio-ming WANG，Ji-yn YU

aSchool of Mechanical Engineering，Nanjing University of Science and Technology，Nanjing，China

bInstitute of Chemical Materials of CAEP，Mianyang，Sichuan 621900，China

Ground target localization algorithm for semi-active laser terminal correction projectile

Xing-long LIa，b，*，Wen-jin YAOa，Li-kun ZHUa，Xiao-ming WANGa，Ji-yan YUa

aSchool of Mechanical Engineering，Nanjing University of Science and Technology，Nanjing，China

bInstitute of Chemical Materials of CAEP，Mianyang，Sichuan 621900，China

A target localization algorithm，which uses the measurement information from onboard GPS and onboard laser detector to acquire the target position，is proposed to obtain the accurate position of ground target in real time in the trajectory correction process of semi-active laser terminal correction projectile.A target localization model is established according to projectile position，attitude and line-of-sight angle.The effects of measurement errors of projectile position，attitude and line-of-sight angle on localization accuracy at different quadrant elevation angles are analyzed through Monte-Carlo simulation.The simulation results show that the measurement error of line-of-sight angle has the largest inf l uence on the localization accuracy.The localization accuracy decreases with the increase in quadrant elevation angle.However，the maximum localization accuracy is less than 7 m.The proposed algorithm meets the accuracy and real-time requirements of target localization.

Semi-active laser guidance；Terminal correction projectile；Target localization；Localization accuracy

1.Introduction

Usingreconnaissancesystemtoobtaintheexactcoordinatesof ground target and providing the target location for weapon systems timely has become an eff i cient combat mode.A reconnaissance system is used to locate a target，and the target localizationinformationisloadedtoanonboardcomputerbefore launchingatrajectorycorrectionprojectile.Currently，UAV ［1-3］，airborneelectro-opticalplatform ［4］，andairborneradarorground radar ［5-7］have been used for ground target localization.The localization accuracy of all the localization methods is related to position accuracy of carrier itself and target detection accuracy.

On the battlef i eld the localization methods mentioned above have a certain target localization accuracy，but they have some security issues.For example，when an unmanned aerial vehicle（UAV）electro-optical detection platform （EODP）is used to locate a target，there are excessive measurement error factors，and it is diff i cult to analyze the localization accuracy；the localization methods by radar and GPS are susceptibly inf l uenced by electromagnetic interference，and the cost is relatively high；localization method by reconnaissance aircraft is easy to expose to the target.In order to reduce the diff i culties and risk of reconnaissance，an autonomous ground target localization method for semi-active laser terminal correction projectile is proposed in the present paper.In the proposed method，the relationship between projectile and target location is derived by combining the onboard GPS measurement data with the laser spot signal received by laser detector，and then the target location is calculated exactly.The inf l uence of measurement errors on the localization accuracy obtained by the localization algorithm at different launching angles is analyzed through Monte Carlo method.The results show that the localization accuracy is higher in the case of small launching angle，and the inf l uence of line-of-sight angle measurement error on the localization accuracy is very small.The method realizes self-localization for ground target without other reconnaissance system.

2.Target localization method

The operating principle of semi-active laser terminal correction projectile is that a laser designator is used to irradiate a target，and then a laser spot signal ref l ected from the target is received by a laser detector，as shown in Fig.1；a controlcommand is generated based on the ref l ected signal as input；and f i nally，according to the control strategy，the trajectory error is correctedbyacontrolforceproducedbyanactuatorinprojectile.

Fig.1.Working principle of semi-active laser terminal correction projectile.

Laser spot signal ref l ected from target can be measured directly by laser detector，and then the line-of-sight（LOS）angle can be obtained，namely the angle between the projectile axis and line-of-sight.In this study，the target coordinate is derived based on line-of-sight angle，projectile position and speed.

2.1.Measurement model of laser detector

A detector of strapdown seeker is completely f i xed to a projectile.An imaging point of target on the image plane is obtained after a laser spot signal is received by laser detector. The ground frame O-XYZ and the body frame O-X1Y1Z1are def i ned in ［8］，and the image frame O′-XgYgZgis transformed from the body frame，as shown in Fig.2.

In the ground frame，the projectile coordinate is noted as O（xo，yo，zo），and the target coordinate is noted as T（xt，yt，zt）.The target coordinate in body frame is

where φa，φ2and γ are the elevation angle，projectile axis azimuth angle，and roll angle of projectile ［8］，respectively.f is expressed as the focal length of lens，as shown in Fig.2.In the image frame，xg=f is a constant since the image plane is placed on the focal plane.According to the imaging geometric relationship of laser spot，the imaging point of target in the image frame is （yg，zg）

The imaging point of target in image frame can be obtained by using Eqs.（1）and （2）.In Fig.2，Point T is the target point，and Point T′is the laser spot on the detector image surface. Then the line-of-sight angle can be obtained by the following formula

The magnitude of line-of-sight angle ref l ects the deviation of projectile longitudinal axis from line-of-sight.

2.2.Estimation method of projectile attitude angles

In order to realize the target localization，the projectile attitude angles need to be obtained at f i rst.For f l ight projectile，the angle between projectile longitudinal axis and projectile velocity is small，the deviation of trajectory from shooting surface is also small，and therefore the elevation angle φaand azimuth angle φ2can be estimated as follows

where θaand ψ2are velocity elevation angle and velocity azimuth angle，respectively；and δ1and δ2are angle of attack and the sideslip angle，respectively.The velocity elevation angle and velocity azimuth angle ［8］can be obtained according to the def i nition

where vx，vy，and vzare the velocity components in the base frame ［8］，which can be measured by onboard GPS in real time. External ballistics theory ［9］shows that，for non-rolling and fi n-stabilized projectile，its angle of attack is large early in the ballistic f l ight path but then stabilizes further along the projectile f l ight path.Its angle of attack can be approximated to 0°in the f i nal stage of f l ight.Therefore，θaand ψ2can be approximated to φaand φ2，respectively，based on theassumption of small angle of attack ［10］，that is φa≈ θa，and φ2≈ ψ2.

Fig.2.Transformation between imaging frame and body frame.

The base frame OXNYNZNis translated to the projectile center of gravity from the ground frame，and it moves with the center of gravity together.For the convenience of deriving the location relationship between projectile and target，the pitch angle ? and yaw angle ψ ［11］should be calculated from φaand φ2.Fig.3 shows the relationship among several angles in the base frame，where Point O is the projectile center of gravity，and Oξ is the longitudinal axis of projectile.

The pitch angle ? and yaw angle ψ can be obtained from the angle relationships in Fig.3

Fig.3.Conversion relationship of projectile attitude angles.

The pitch angle θ and yaw angle ψ can be approximated using Eqs.（4）-（6），and ?rand ψrare denoted as real pitch angle and real yaw angle，respectively.On the assumption of small angle of attack，the calculated errors of pitch angle and yaw angle are δ? = ?r- ? and δψ = ψr- ψ，respectively.

In order to verify the assumption of small angle of attack，a f i n-stabilized 120 mm projectile is used as an example，and the six-degrees-of-freedom trajectory simulations at quadrant elevation angles of 45°，55°，65°and 75°are taken，respectively. The calculated results of pitch angle error and yaw angle error at 3 s before projectile landed are shown in Figs.4 and 5，respectively.

Obviously，the pitch and yaw angle errors decrease with projectile dropping，and the pitch angle error does not exceed 0.022°and the yaw angle error does not exceed 0.2°at 4 quadrant elevation angles.It proves that the calculation method of pitch angle and yaw angle based on the small angle of attack assumption is available.

2.3.Target localization principle

The optical axis of strapdown laser detector coincides with the projectile axis.Point F′is an intersection between opticalaxis and ground，as shown in Fig.6.F′can be obtained from the real-time projectile coordinates and attitude angles.

Fig.4.Pitch angle error δ?.

Fig.5.Yaw angle error δψ.

In Fig.6，the point O（xo，yo，zo）is expressed as the vertex of cone，and the line-of-sight angle ε is expressed as half cone angle.A conical surface is constructed with the projectile axis as cone axis，and the ellipse E1is obtained from an intersection between conical surface and ground plane.The target point must be on the ellipse E1because all the angles between the projectile axis and the line from projectile center to E1are ε.

Fig.6.Positional relationship between the optical axis of detector and the target.

Fig.7.The schematic of ground target localization by laser detector.

Therefore，a ground target localization method was proposed.In the proposed method，the position and velocity of projectile are measured by onboard GPS and the line-of-sight angle is measured by laser detector at time t1，and then an ellipse E1on the ground is obtained，where the target point is on the ellipse.Similarly，when the projectile f l ies to another position at time t2，an ellipse E2is obtained through real time measurement. The target point is the intersection of E1and E2，but two intersection points exist for two ellipses，which results in location ambiguity.Therefore，the third ellipse E3at time t3is required so that the target point is the intersection point of the three ellipses.

The three ellipses intersect at a point in the ideal case，but in reality the three ellipses cannot intersect at a point due to assumption of small angle of attack and measurement errors.If every two ellipses intersect，the target point will be in a region in which the intersection points concentrate.The region is near Points L1，M1and N1，as shown in Fig.7.

3.Solution of localization algorithm

According to the target localization principle in Section 2.3，a relevant solution is needed to get the target location since three ellipses cannot intersect at a point.The projectile-target frame OrXrYrZris obtained by moving the base frame from origin point O to projection point Orof projectile on the ground，and the frame with angle ψ is rotated about the axis of OrYN，as shown in Fig.6.The center location of ellipse in projectiletarget frame can be obtained by the geometrical relationship between optical axis of detector and target，denoted as point E1（xe，0，0）.

The expressions of major axis a and minor axis b of ellipse are

If the ellipse equation is transformed from the projectiletarget frame to the ground frame，then the standard equation of ellipse is obtained as follows

Eq. （9）is converted into a general formula

where

where x′ = x0+xecosψ，z′ = z0+xesinψ.The general formulas of the three ellipses are obtained using the above calculation method.The proposed target localization can be transformed to a nonlinear programming issue，which is to fi nd an optimal point in the region around the estimated impact point，to satisfy the condition as follows

where Ai，Bi，Ci，Di，and Eiare the parameters of i-th ellipse；min F（x，z）is the objective function.For the nonlinear programming issue，the design variables are x and z with the constraints of xp-300

4.Simulation and analysis of target localization algorithm

A f i n-stabilized 120 mm terminal correction projectile was taken for example to demonstrate the effectiveness of the proposed target localization algorithm.The projectile mass m is 13.45 kg，the muzzle velocity v is 340 m/s，and the maximum f i eld-of-view angle of detector is ±8°.In the terminal trajectory，a laser detector receives a laser signal when a target is in the f i eld-of-view of detector.The measuring frequencies of both GPS and laser detector are 10 Hz.Moreover GPS and the laser detector are used to measure the position and velocity of projectile and the line-of-sight angle at the same time，respectively. A set of data is collected every 0.1 s during f l ight.The targetlocation could be calculated from every three sets of data by the localization algorithm in Section 2.

Table 1Measurement errors.

A six degrees of freedom （6-DOF）simulation was performed under the standard weather conditions.In the simulation，the quadrant elevation angle θ was 45°，and the uncontrolled ballistic impact point was P （7328.1，-31.4）.The parameters of individual measurement errors，as shown in Table 1，were def i ned by taking into account the measurement errors of GPS and laser detector ［12，13］.

To research the localization results of targets at different ranges in different directions achieved by the proposed target localization method，a number of different targets are placed around an uncontrolled ballistic impact point P at an interval of 45°with radii of 20 m，40 m，60 m，and 80 m evenly，as shown in Fig.8.Taking into account the correction capability and range of f i eld-of-view，the maximum distance between supposed target and impact point of trajectory is taken to be 80 m.

The acquisition times are not the same for the different targets.So for the convenience of research，the starting time of target localization is def i ned as the time when the remaining trajectory height is 1 km.A group of measurement parameters is obtained every 0.1 s.The target position can be calculated by using three groups of measurement data in neighboring time.A total of 30 calculations were taken.The obtained localization points are shown in Fig.8.

Fig.8.Localization results for different supposed targets.

Fig.9.The average localization error for different supposed targets.

where N=30，（xt，zt）is the location of actual target point. Three-dimensional surface f i tting was made for average localization error of each supposed target，and the average localization error surface was obtained，as shown in Fig.9.It can be seen from Fig.9 that the localization error increases with the increase in the distance from target to impact point，and the amplitude of localization error in the direction of 45°right rear of impact point is larger than the localization error in other directions，but the maximum localization error is less than 4 m，which indicates this method has a high localization accuracy.

5.Analysis of inf l uencing factors

Localization accuracy is important to the assessment of localization method.The analysis of the factors which have the inf l uences on localization accuracy can help guide a more reasonable distribution of the system errors to improve localization accuracy.

The ellipse parameters were calculated using the proposed localization algorithm based on the assumption of small angle of attack.The localization accuracy is related to the assumption of small angle of attack.Since the angles of attack are different at the different quadrant elevation angles，the localization accuracy is related to the quadrant elevation angle.During projectile f l ight，the f l ight states and parameters of projectile cannot be accurately measured due to measurement error and sensor noise.Therefore，the localization accuracy is also affected by the position，velocity and angle measurement errors，which are measured by onboard GPS and laser detector，respectively. 5.1.The effect of quadrant elevation angle on localization accuracy

Fig.10.Localization errors at different quadrant elevation angles.

Table 2Measurement errors of projectile position.

In Fig.10，n is the serial number of localization results. Apparently，the localization error does not exceed 4 m with the quadrant elevation angles of 45°and 55°.The localization error increases signif i cantly with the increase in quadrant elevation angle，and the maximum error reaches 7 m for θ =75°.It can be also known from Fig.10 that the greater the quadrant elevation angle，the larger the localization error.Because the φaincreases with the increase in quadrant elevation angle and the line-ofsight angle decreases at the same remaining height of different trajectories，the relative error of line-of-sight is greater under the condition of same measurement error of laser detector，and the localization error is greater.However，the target localization error of UAV electro-optical detection platform is about 37.1 m［3］，and the localization error of onboard radar reconnaissance system is about 30-40 m ［5］.The proposed method has improved the target localization accuracy signif i cantly，and then the semi-active laser terminal correction projectiles can defeat small fortif i cation target，self-propelled gun，communication and command vehicle，infantry combat vehicle and other lightly armored vehicle more effectively.

5.2.The effect of projectile position accuracy on localization accuracy

According to the error theory，the measurement error is usually divided into system error，random error and gross error. The measurement error of projectile position consists of system error and random error with ignoring the gross error.The system error always exists and the mean of system errors is a constant during the f l ight of projectile，then the measurement error is a normally distributed N ～ （μ，σ），where μ is the system error，i.e.，the measurement error of the mean deviation，and σ is the standard deviation of the normal distribution.

Let the projectile position errors in three directions δx，δy，δz be the same and the other error be zero，the measurement errors［12］are listed in Table 2.

According to the measurement errors in Table 2，the random measurement errors were obtained using Monte Carlo simulation.The target location was calculated 30 times for each trajectory，and 50 sets of trajectories were calculated to obtain 1500 calculated target locations.The average localization errorwas obtained through mathematical statistics.The average localization error in the measurement errors of different projectile position was obtained by substituting the measurement errors in Table 2 into 4 different quadrant elevation angles sequentially，as shown in Fig.11.

It can be seen from Fig.11 that the average localization error decreases fi rst and then increases with the increase in projectile position measurement errors.This is because the localization accuracy is affected by the measurement error of projectile position and the assumption of small angle of attack at the same time，and the projectile position error offsets a part of error caused by the assumption of small attack angle.

5.3.The effect of projectile velocity accuracy on localization accuracy

Let the projectile velocity errors in three directions δvx，δvy，δvzbe the same and the other errors be zero，the velocity errors［12］are listed in Table 3.

Fig.11.Effect on localization results by projectile location measurement error.

50 groups of simulation were done at each quadrant elevation angle，and then 1500 calculated target locations were obtained.The statistical results of average localization error are shown in Fig.12.

It can be known from Fig.12 that the localization error caused by 5 kinds of velocity errors is very small，the maximum localization error does not exceed 4.5 m，and the magnitude of localization error changes irregularly with the increase in the measurement error of projectile velocity.This is because，in the localization algorithm，the ellipse parameters are calculated under the assumption of small attack angle，the increase in themeasurement error of projectile velocity may increase or offset a part of localization errors caused by the assumption of small attack angle.

Table 4Measurement errors of line-of-sight angle.

5.4.The effect of line-of-sight angle accuracy on localization accuracy

The angle measurement accuracy ［13］can be controlled within 0.1°in the f i eld-of-view of four-quadrant detector as a commonly used laser detector.5 groups of angle measurement errors are shown in Table 4.

50 groups of simulation were done at each quadrant elevation angle.The statistical average localization error is shown in Fig.13.

Obviously，the localization precision decreases with the increase in the measurement error of line-of-sight angle.The greater the quadrant elevation angle，the larger the localization error in the condition of the same angle measurement error. This is because the greater the quadrant elevation angle，the larger the estimated errors of pitch angle and yaw angle，i.e.，the attitude angle error of projectile axis is larger.Because the ellipse is determined by the axis of projectile and the line-ofsight angle，and the measurement errors of projectile axis attitude and line-of-sight angle superpose on each other，the measurement error of ellipse is increased，thereby increasing the localization error.

Fig.13.Effect on line-of-sight measurement angle error of localization results.

6.Conclusions

In this effort，the model of ground target localization for semi-active laser terminal correction projectile was established，and the sources of localization error were analyzed.The effects of quadrant elevation angle and 3 kinds of measurement errors on localization accuracy were studied through Monte Carlo method.The results show that the localization error increases with the increase in the quadrant elevation angle of projectile，but the maximum localization error does not exceed 7 m.In 3 kinds of measurement errors，the localization error increases with the increase in the measurement error of line-of-sight angle，and the effects of projectile position error and velocity error on localization error are not obvious.The proposed localization method is simple in theory and meets the requirement of real-time localization accuracy.It provides a new localization method for semi-active laser terminal correction projectile，and it has an important signif i cance for engineering applications.

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Received 28 August 2015；revised 18 January 2016；accepted 19 January 2016 Available online 12 February 2016

Peer review under responsibility of China Ordnance Society.

*Corresponding author.Tel.:086 139 5176 6031.

E-mail address:lixinglong.sj@outlook.com （X.L.LI）.

http://dx.doi.org/10.1016/j.dt.2016.01.004

2214-9147/? 2016 China Ordnance Society.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license （http:// creativecommons.org/licenses/by-nc-nd/4.0/）.

? 2016 China Ordnance Society.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license（http://creativecommons.org/licenses/by-nc-nd/4.0/）.