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Stability Analysis of Strapdown Seeker Scale Factor Error and LOS Rate Woo Hyun Kim, School of Electrical Eng. & Computer Science, Seoul National University Jang Gyu Lee, School of Electrical Eng. & Computer Science, Seoul National University Chan Gook Park, School of Mechanical & Aerospace Engineering, Seoul National University BIOGRAPHY Woo Hyun Kim received Control and Instrumentation Engineering from Seoul his B.S. degree in physics National University in 1993. In 1998, he worked with from the Republic of Korea Prof. Jason L. Speyer about formation flight at UCLA Air Force Academy in 2000. as a visiting scholar. He is currently in charge of IEEE He received his M.S. degree AES Korean Society. His current research topics from Seoul National include the development of Inertial Navigation University in electrical Systems, GPS/INS integration, MEMS-based IMU engineering & computer applications, indoor navigation, Positioning for science in 2006. He is currently in the Ph.D. course at Ubiquitous Sensor Network and advanced filtering the Seoul National University in electrical engineering techniques. (email: chanpark@snu.ac.kr) & computer science. His research interests include INS (inertial navigation system), GPS, integrated system, Loran-C, and eLoran. (email: whyun77@snu.ac.kr) ABSTRACT Jang Gyu Lee is a Professor An inertial measurement unit (IMU) consisting of three in the School of Electrical gyroscopes and three accelerometers is utilized to carry Engineering & Computer out guidance performance in a short range ground to Science at Seoul National ground missile system. To improve the accuracy of University. He received his trajectory, a seeker is applied and this seeker Ph.D. degree in Electrical measurement is provided to a PNG law. Engineering from University A Strapdown seeker has advantages compared to a of Pittsburgh, USA in 1977. gimbaled seeker in terms of size and FOV limit. If the He was the Director of strapdown seeker output is combined with the output of Automation and System an inertial sensor, the potential for instability exists if Research Institute, Seoul the scale factors and gain of the two sensors are not National University, 1996- equal. Because the strapdown seeker provides LOS 1998. He worked at the Analytic Sciences Corp. angles from body to target in a body frame, the LOS (TASC) & Charles Stark Draper Lab., Technical Staff, angles need to be changed to their rates in the inertial 1977–1982. He is currently in charge of the Editorial frame for PNG. Advisory Board Member, International Journal of This paper presents the method of a LOS rate Space Technology and the Associate Editor, derivation from image plane values of the output of the International Journal of Parallel and Distributed strapdown seeker and analyzes the effects of the Systems and Networks. He charged the General instability generated by the scale factor errors of the Chairman, Editor, 14th IFAC Symposium on Automatic strapdown seeker and inertial sensor. The proposed Control in Aerospace, 98.8.24-28. His current research derivation of LOS rate and analysis of the effect of topics include INS, GPS, scale factor error are verified by a simple but realistic eLoran, Guidance, Estimation. navigation and guidance control simulation. When we (email: jgl@snu.ac.kr) assume the threshold of the miss-distance is within 10 meters, results are satisfied within the threshold with a Chan Gook Park is a Professor 0.5% scale factor error. When we assume the threshold in the School of Mechanical of the miss-distance is within 2 meters, results are and Aerospace Engineering at satisfied within the threshold with a 0.45% scale factor Seoul National University. He error. received his Ph.D. degree in Through the simulation, we can find the effect of scale PNG, the LOS angles need to be changed to their rates factor error, and this paper suggests further work in the inertial frame. towards a solution to this instability. This paper presents a method of LOS angle and LOS rate derivation from the image plane values, which are INTRODUCTION the output of the strapdown seeker. This paper also analyzes the effects of the instability generated by the Instead of increasing the quantity of missile warhead, scale factor errors of the strapdown seeker and inertial the improvement of the accuracy in guidance sensor. In order to evaluate the derived LOS angle and equipment brings on the accuracy rate on target with rate and to analyze the scale factor error, a short range respect to the striking power. There are seekers which missile employing a PNG law is simulated using the can improve the accuracy of the missile. The seeker guidance control navigation package simulation. The provides the LOS angle from missile to target through simulation results show that the method of LOS angle the pitch and yaw angle. The seeker consists of a and LOS rate derivation is accurate. Moreover, when gimbaled seeker and a strapdown seeker. we assume the threshold of the miss-distance is within In conventional gimbaled structures, a seeker provides 2 meters, results are satisfied within the threshold with the LOS rate (Jacques Waldmann, 2002). Recent a 0.45% scale factor error. advancements in seeker technology have resulted in seeker designs with much larger FOV (fields-of-view) and seeker tracking characteristics which do not DERIVATION OF LOS ANGLE AND LOS RATE require the seeker centerline to point in the general IN STRAPDOWN SEEKER MISSILE vicinity of the target. Examples of such seekers include optical and radar corelators, holographic lenses used The PNG law needs the line of sight rate with respect with laser detectors, and phased array antennas. The to the inertial frame in order to perform guidance. In potential advantages of such seekers are numerous, this section, the strapdown seeker is assumed to stemming fundamentally from the fact that the seeker measure the LOS angle in the body frame. To calculate can now be rigidly fixed to weapon body. the PNG law, the LOS angle needs to be translated These body-fixed strapdown seekers have the merit of from the body to the inertial frame. We have derived eliminating the tracking rate limits and structural the LOS angle and LOS rates in the inertial frame. limitations pertinent to inertially stabilized gimbaled seekers. They also reduce the mechanical complexity A. Derivation of LOS angle of implementation and calibration. The elimination of mechanical moving parts would in turn eliminate The target position vector and Vehicle position vectors frictional cross-coupling between pitch and yaw r r r PTI and P I . The LOS vector LI with respect to tracking channels, yielding the advantage of increased are V reliability of electronic components over mechanical inertial frame is defined as follows. ones. Finally, there is potentially a significant cost r r r T savings associated with eliminating the gimbals (Paul LI = PTI − P I = ⎡ LIx V ⎣ LIy LIz ⎤ ⎦ (1) L. Vergez and James R. McClendon, 1982). The strapdown seeker technology has been developed ψ θ The LOS angles ( ILOS , ILOS ) are defined in the significantly over the last several years for air-to- surface and air-to-air applications (Raman K. Mehra inertial frame. Figure 1 shows the LOS angles in the and Ralph D. Ehrich, 1984). One such application is inertial frame the LCPK (Low Cost Precision Kill) and the APKWS (Advanced Precision Kill Weapon System) in the USA. The proportional navigation guidance law is frequently employed in short range missiles because it can easily be implemented and provides a near optimal guidance for constant velocity targets. The conceptual idea behind PN guidance is that the missile should keep a constant bearing to the target at all times, which will result in an eventual impact (Jing Xu, Kai-Yew Lum and Jian Xin Xu, 2007). Acceleration commands in missiles guided by proportional navigation require the measurement of a relative LOS rate between missile and target. If the strapdown seeker output is combined with the Figure 1. LOS angles in inertial frame output of an inertial sensor, the potential for instability exists if the scale factors and gain of the two sensors ⎛ LIy ⎞ are not equal. Because the strapdown seeker provides ψ ILOS = tan −1 ⎜ ⎜ I ⎟⎟ (2)-a LOS angles from body to target in a body frame, for ⎝ Lx ⎠ In the image plane, the image plane value [y, z] is ⎛ ⎞ derived as follows (Joongsup Yun, Chang-Kyung Ryoo −1 ⎜ LIz ⎟ and Taek-Lyul Song, 2008). θ ILOS = tan ⎜ ⎟ (2)-b ⎜ ⎝ (L ) + (L ) I 2 x I 2 y ⎟ ⎠ Z = l tan θbLOS (5)-a r The LOS vector LI in inertial frame can be changed Y = Z 2 + l 2 tanψ bLOS (5)-b r ub into LOS vector L in body frame through a coordinate b From the image plane value which is the measurement transformation matrix CI . The LOS angles of strapdown seeker, we can derive the LOS angles ψ θ ( bLOS , bLOS ) are defined in the body frame. with respect to the inertial frame through equations 1 - 5. The notation used in Figure 2 is summarized as The target angular position in the image plane is follows: measured by the sensor. Figure 2 shows the LOS angles in the body frame and target angular position in r ub the image plane. L : LOS vector in body frame l : lens focal length b b b x , y , z : x-axis, y-axis and z-axis in body frame [y, z] : Image plane value ψ bLOS : LOS yaw angle represented in body frame θbLOS : LOS pitch angle represented in body frame In order to examine the equations, we performed a simulation using two simple reference trajectories. The assumptions of the first simulation are as follows. Figure 2. Image plane The vehicle’s initial attitude is roll 0deg, pitch 20deg and yaw 0deg. The vehicle’s final attitude is roll 0deg, r ub r uI pitch -20deg, yaw 0deg. The total flight time is L = CIb L = ⎡ Lb ⎣ x Lby Lb ⎤ z⎦ (3) 15seconds and vehicle speed is 200m/s. The target position is fixed and the strapdown seeker FOV limit is ⎛ Lb ⎞ ±40deg. Normal lens focal length is 50[mm]. Figure θbLOS = tan −1 ⎜ b ⎟ z (4)-a 3(a) shows the reference trajectory and the green ⎝ Lx ⎠ square represents the initial position, and the red circles represent points at every 2 seconds. ⎛ ⎞ As shown in Figure 3(a), the vehicle flies to the north ⎜ Lby ⎟ ψ bLOS −1 = tan ⎜ ⎟ (4)-b direction, so the path of the image plane value is ⎜ ⎝ (L ) + (L ) b 2 x b 2 z ⎟ ⎠ headed north. Figure 3(b) shows the image plane value. From this value, we can derive the LOS angle in inertial frame and body frame which are showed in b where CI denotes the coordinate transformation θ Figure 3(c). The initial bLOS is -20deg because the matrix from inertial frame to body frame. vehicle’s initial attitude is pitch 20deg. The initial C x (φ ) θ ILOS denotes the coordinate transformation is 0deg because the vehicle’s position z value matrix representing the rotation of angle φ and target’s position z value are the same. At the final about the x-axis. ψ θ point, bLOS and bLOS go to 0deg because the vehicle φ , θ and ψ are the vehicle’s roll, pitch, and yaw attacks the target at the final time. attitudes. In the second simulation, the assumptions are the same CIb = Cx (φ ) C y (θ ) Cz (ψ ) as the first simulation except for the vehicle’s initial attitude yaw 30deg and final yaw 30deg. Figure 4(a) shows the reference trajectory and Figure 4(b) shows ⎡1 0 0 ⎤ the image plane value. ⎢0 Cφ Sφ ⎥ C x (φ ) = ⎢ ⎥ As shown in Figure 4(a), the vehicle flies without ⎢0 − Sφ Cφ ⎥ ⎣ ⎦ changing the heading, so the path of the image plane value is the north direction. Sφ = sin φ , Cφ = cos φ 3-D Reference trajectory Measurement of Image Plane value 0.03 history start 400 0.02 final traj. 300 every 2sec 0.01 initial Z axis [m] 200 Up 0 100 -0.01 0 3000 1 -0.02 2000 0.5 1000 0 -0.5 -0.03 North 0 -1 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 East Y axis [m] (a) (b) Measurement of Image Plane value LOS [in inertial frame] LOS [in body frame] 0.03 0 0 history start -5 -5 Thetab [deg] Thetai [deg] 0.02 final reference -10 -10 simulation res. -15 -15 0.01 -20 -20 Z axis [m] 0 -25 -25 0 5 10 15 0 5 10 15 -0.01 40 1 30 0.5 Psib [deg] -0.02 Psii [deg] 20 0 -0.03 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 10 -0.5 Y axis [m] 0 -1 (b) 0 5 10 time [sec] 15 0 5 10 time [sec] 15 LOS [in inertial frame] LOS [in body frame] (c) 0 0 -5 -5 reference Figure 4. Simulation #2 Thetab [deg] Thetai [deg] -10 simulation res. -10 -15 -15 From this value, we can also derive the LOS angle in -20 the inertial frame and body frame which are shown in -20 0 5 10 15 -25 0 5 10 15 Figure 4(c). The initial ψ is 30deg and the initial 1 1 ψ bLOSis 0deg because the vehicle’s initial attitude is 0.5 0.5 yaw 30deg and final is 30deg. In this simulation, Psib [deg] Psii [deg] 0 0 ψ ( ILOS , ILOS ) and ( bLOS , θ ψ θbLOS ) are derived from -0.5 -0.5 -1 -1 the image plane value [y, z]. 0 5 10 15 0 5 10 15 time [sec] time [sec] B. Derivation of LOS rate (c) There are five coordinate systems to derive the LOS Figure 3. Simulation #1 rate: the inertial frame, body frame, strapdown seeker 3-D Reference trajectory frame, pointing frame, and LOS frame. The strapdown seeker is fixed in the vehicle, so we can assume that the strapdown seeker frame and body frame are same. 400 Each coordinate transformation matrix is summarized 300 traj. every 2sec as follows (Won-Sang Ra and Ick-Ho Whang, 2002): initial 200 CIB = Cx (φ ) C y (θ ) Cz (ψ ) Up 100 0 CbP = C y ( e y ) Cz ( ez ) (6) 3000 2000 1000 1500 C LOS P = C x (φ L ) = C y ( −λv ) Cz ( λh ) 1000 500 LOS North 0 0 C I East (a) e e Notations y and z are LOS angle error. The relations among the coordinate systems are illustrated in Figures 1, 5, and 6. r uuuuu uuuuur r uuu ωILOS = ωPLOS + CP ωIP LOS LOS LOS P (10) & ⎡ S λv λh ⎤ ⎡ pp ⎤ & ⎡φL ⎤ r uuuuu r uuu ⎢ & ⎥ ⎢ ⎥ uuuuur ⎢ ⎥ (11) ω LOS ILOS −C LOS P ω = ⎢ −λv ⎥ − CP P IP LOS ⎢ q p ⎥ = ωPLOS = ⎢ 0 ⎥ LOS ⎢C λv λh ⎥ & ⎢ rp ⎥ ⎢0⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Rearranging the equations, we get & λv = −CφL q p − SφL rp (12)-a & Sφ L Cφ λh = − q p + L rp (12)-b C λv C λv Figure 5. Inertial frame and LOS frame where SφL and CφL can be derived by the following equations ⎡ 0 ⎤ ⎡1 0 0 ⎤ ⎡0⎤ ⎡ 0 ⎤ ⎡0⎤ (13) CP ⎢0 ⎥ = ⎢0 CφL LOS ⎢ ⎥ ⎢ SφL ⎥ ⎢0 ⎥ = ⎢ SφL ⎥ = CILOS CbI CP ⎢0 ⎥ ⎥⎢ ⎥ ⎢ ⎥ b ⎢ ⎥ ⎢ 1 ⎥ ⎢ 0 − Sφ L ⎣ ⎦ ⎣ CφL ⎥ ⎢1 ⎥ ⎢CφL ⎥ ⎦⎣ ⎦ ⎣ ⎦ ⎢1 ⎥ ⎣ ⎦ SIMULATION FOR LOS ANGLE AND RATE In this section, the performance of LOS angles and LOS rates are investigated in a simple homing missile Figure 6. Strapdown seeker frame and pointing frame system scenario with PNG law. The simulation flow chart is illustrated in Figure 7. The simulation package The inertial angular rates of body frame and pointing consists of a navigation division and a guidance control frame are respectively denoted by: division. In the navigation division, the ATM_Sensor part senses the body’s acceleration and angular rate and r uuu image plane values. The ATM_PureINS part derives ωIb = [ p q r ] b T (7)-a the vehicle’s position, attitude, LOS angle and LOS rate with respect to inertial frame. These data are r uuu T applied to PNG law in the guidance control division. ωIP = ⎡ p p q p P ⎣ rp ⎤ ⎦ (7)-b The assumptions are as follows: the missile’s initial In the above equations, the superscript denotes the attitude is roll 0deg, pitch 20deg, yaw 0deg, and coordinate system in which the quantity is represented. guidance law is PNG. The vehicle’s initial speed is By basic kinematics, the following equations hold. 100m/s and Target speed is 10m/s. FOV limit is ± 40deg and miss-distance threshold is within 10 meters. r uuu ⎡0⎤ uuu r uuu r ⎡0⎤ ⎢ 0 ⎥ + C e ⎢e ⎥ (8)-a y ( y )⎢ y⎥ The image plane value is the measurement of the ω −C ω =ω = C ⎢ ⎥ P IP P b b Ib P bP P & b strapdown seeker. Figure 8(a) shows the image plane ⎢eZ ⎥ ⎣& ⎦ ⎣ ⎥ ⎢0⎦ values’ path which starts at a starting point and is terminated at a final point. From that value, we have derived LOS angles and LOS rates. r uuu ⎡ p ⎤ ⎡Cey 0 − Sey ⎤ ⎡ 0 ⎤ In Figure 8(a), the value of starting point [y,z] is ⎢ ⎥⎢ ⎥ ωIP = CbP ⎢ q ⎥ + ⎢ 0 P ⎢ ⎥ 1 & 0 ⎥ ⎢ey ⎥ [negative, negative], which means that the target is ⎢ r + eZ ⎥ ⎢ Sey 0 Cey ⎥ ⎢ 0 ⎥ ⎣ & ⎦ ⎣ ⎦⎣ ⎦ (8)-b located in the down and right directions in body frame. At the initial point, the vehicle is launched at pitch ⎡ pCey Cez + qCe y Sez − ( r + ez ) Sey ⎤ ⎡ p p ⎤ & 20deg attitude, so the LOS pitch angle with respect to ⎢ ⎥ ⎢ ⎥ =⎢ − pSez + qCez + ey & ⎥ = ⎢ qp ⎥ body frame is -20deg and LOS pitch angle with respect ⎢ pSey Cez + qSey Sez + ( r + ez ) Cey ⎥ ⎢ rp ⎥ to inertial frame is 0deg, as shown in Figure 8(b). At ⎣ & ⎦ ⎣ ⎦ the initial time, the target is located at the north east direction, so the LOS yaw angles are +(plus) deg. At ⎡0⎤ & ⎡ 0 ⎤ ⎡ S λv λh ⎤ final time, the vehicle attacks the target within 10 r uuuuu ω LOS =C LOS ⎢ 0 ⎥ + C (−λ ) ⎢ −λ ⎥ = ⎢ −λ ⎥ (9) & & meters, so LOS pitch and yaw with respect to body ILOS I ⎢ ⎥ y h ⎢ v⎥ ⎢ v ⎥ frame is approximately zero. & ⎢λh ⎥ ⎣ ⎦ ⎢ 0 ⎥ ⎢C λv λh ⎥ ⎣ ⎦ ⎣ & ⎦ Figure 7. Simulation flow chart As time goes by, the LOS rate converges to zero (b) LOS angles and LOS rates according to PNG law. Figure 9(a) shows the top-down trajectory and Figure 9(b) shows the side trajectory of Figure 8. Image plane value & LOS angles and rates the vehicle. The flight time is 10.02seconds and miss-distance is 9.8575 meters. These simulation results mean that the derivation of LOS angles and LOS rates has significance which can be applied to PNG law. (a) (a) Image plane value [y,z] (b) Figure 9. Result trajectory STABILITY ANALYSIS OF STRAPDOWN ⎡ s ⎤⎡ ⎛ s ⎞⎛ s ⎞⎛ s ⎞ ⎤ (16) SEEKER SCALE FACTOR ERROR ⎛ Ny ⎞ ⎢ 20 + 1 ⎥ ⎢ 0.965 ⎝ 60 + 1⎠ ⎝ 35.83 + 1⎠ ⎝ −34.60 + 1⎠ ⎥ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟=⎢ ⎥⎢ ⎥ ⎜ Ny ⎟ ⎢⎛ s ⎞ ⎥⎢ 2 ⎡ 0.4093 ⎤ ⎡ 0.4481 ⎤ ⎛ s ⎞ ⎥ ⎝ c ⎠ ⎢ + 1⎟ ⎥ ⎢ ⎜ + 1⎟ ⎜ ⎢ 14.98 ⎥ ⎢ 73.27 ⎥ ⎝ 77.8 ⎠ ⎣ ⎦⎣ ⎦ ⎥ Two problems can be generated when guidance-control ⎣ ⎝ 100 ⎠ ⎦ ⎣ ⎦ loop is implemented using the LOS angle from the strapdown seeker. The first problem is the trend that linearity errors and noises are proportional to the F.O.V. In Equation 14, if the strapdown seeker scale factor The second problem is as follows: because the K s and body gyro scale factor K g are equal, two loops strapdown seeker is rigidly fixed on the body, the can be compensated as follows: output of strapdown seeker is not the input for the PNG law but rather the LOS angles represented in body s 4Vc ⎛ N y ⎞ frame. At this point, the output of the strapdown seeker G (s) = Ks ⎜ ⎟ is affected by the missile angular motion and this ⎡ ζ ⎤ 1845 ⎜ N yc ⎟ ⎝ ⎠ degrades the stability of the system. ⎢ω ⎥ If the strapdown seeker output is combined with the ⎣ D⎦ (17) output of inertial sensors, the potential for instability exists if the scale factors and gain of the two sensors In this case, we performed the simulation. The miss- are not equal. The whole guidance-control system distance is 0.2913 meters and this value is the stable using the PNG law is depicted in Figure 10. This value when we assumed that the general miss-distance system is equipped with the strapdown seeker. The is 10 meters. total transfer function of this system derived by However, in the case of K s > K g , the total transfer Mason’s rule is illustrated in Equation 14. The function became negative feedback; therefore, negative vehicle’s equation and flight control system’s feedback occurred in the body angular rate, and the equations are illustrated in Equations 15 and 16. system can be unstable. In Figure 11, the LOS angle represented in the navigation frame is λr = ε + ψ . s 4Vc ⎛ N y ⎞ Ks ⎜ ⎟ ⎡ ζ ⎤ 1845 ⎜ N yc ⎟ ⎝ ⎠ After the scale factors discordance that is occurring, ⎢ω ⎥ (14) the LOS angle is changed to λr = ε −ψ and system G (s) = Ny = ⎣ D⎦ λy s 4Vc ⎛ N y ⎞⎛ r ⎞ 1 becomes unstable (Figure 12(b)). 1+ ⎜ ⎟⎜ ⎟ {K s − K g } ⎡ ζ ⎤ 1845 ⎜ N yc ⎟ ⎜ N y ⎟ s ⎝ ⎠⎝ ⎠ ⎢ω ⎥ ⎣ D⎦ ⎡ ⎛ s ⎞ ⎤ 0.888 ⎜ + 1⎟ ⎥ ⎛ r ⎞ ⎛ r ⎞⎛ y& ⎞ ⎢ ⎝ 0.7835 ⎠ ⎥ (15) ⎜ ⎜N ⎟ = ⎜ ⎟⎜ ⎟ ⎝ y ⎠⎜ N ⎟=⎢ ⎟ ⎢⎛ s ⎞⎛ s ⎞ +1 ⎥ ⎝ y ⎠ & ⎝ y ⎠ +1 ⎢ ⎜ 35.83 ⎟ ⎜ −34.60 ⎟ ⎥ ⎣⎝ ⎠⎝ ⎠⎦ Figure 11. Strapdown seeker angle Figure 10. Guidance control system (a) Figure 12. Effects of scale factor In addition, in the case of K s < K g , the total transfer function became positive feedback, therefore, positive feedback occurred in the strapdown seeker and system can be unstable (Figure 13). In Figure 11, the LOS angle represented in the body frame is ε = λr −ψ . After the scale factors discordance occurring, the LOS angle is changed to ε = λr + ψ and system became unstable (Figure 13(b)). (b) Figure 14. Miss-distance [0.1% scale factor error] (a) Scale factor equal (b) scale factor discordance Figure 13. Effects on scale factor (a) In the case of discordance, the authors performed the 100 times Monte Carlo simulation to analyze the scale factor error. The assumption of IMU is 1mg accelerometer, 10deg/hr gyroscopes. When we add the 0.1% scale factor error to the YZ image plane, the mean of miss-distance is 0.2415 meters with a standard deviation of 0.0210 meters. Miss-distance of 100 times simulation is depicted in Figure 14(a) and ascending sort is depicted in Figure 14(b). When we add the 0.5% scale factor error to the YZ image plane, the mean of the miss-distance is 1.1631 meters with a standard deviation of 0.6453 meters. Miss-distance of 100 times simulation is depicted in (b) Figure 15(a) and ascending sort is depicted in Figure 15(b). Figure 15. Miss-distance [0.5% scale factor error] When we add the 1.0% scale factor error to the YZ image plane, the mean of miss-distance is 72.8280 meters with a standard deviation of 64.3471 meters. Miss-distance of 100 times simulation is depicted in Figure 16(a) and ascending sort is depicted in Figure 16(b). As the scale factor error increases, the miss- distance increases exceedingly. Figure 17. Miss-distance according to scale factor error When we assume that the requirement of attack is within 10 meters, in other words, the threshold of the miss-distance is within 10 meters, the mean value of results is satisfied by adding 0.7% scale factor error (Figure 18) and whole results of 100 times Monte (a) Carlo simulation are satisfied with 0.5% scale factor error. From 0.6% scale factor error, under 95 times of Monte Carlo simulation results are satisfied. When we assume that the requirement of attack is within 2 meters, in other words, the threshold of the miss-distance is within 2 meters, the mean value of results is satisfied by adding 0.55% scale factor error (Figure 19) and whole results of 100 times Monte Carlo simulation are satisfied with 0.45% scale factor error. From 0.47% scale factor error, less than 95 times of Monte Carlo simulation results yield satisfactory results. Through the simulation, we find the fact that scale factor error affects the performance and stability of the (b) guidance missile in a great measure. The further work towards a solution to this instability is the research of Figure 16. Miss-distance [1.0% scale factor error] the dither adaptive method which applies high frequency to pitch and yaw and the EKF method which The mean and standard deviation of miss-distance is estimates the state variables and scale factors. illustrated in Table 1 and depicted in Figure 17 according to each scale factor error. Table 1. Mean and standard deviation of miss-distance Scale factor Mean[meters] STD[meters] error[%] 0.10% 0.2415 0.021 0.20% 0.3522 0.1182 0.30% 0.6321 0.2388 0.40% 0.6473 0.1819 0.45% 0.8635 0.3087 0.47% 1.0137 0.4499 0.50% 1.1631 0.6453 0.53% 1.4255 0.8743 Figure 18. Miss-distance within 10 meters 0.55% 1.8864 2.2968 0.60% 2.8001 3.5564 0.70% 10.9895 17.2225 0.80% 34.9645 38.2126 0.90% 51.7326 45.792 1.00% 72.828 64.3471 ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support of the Agency for Defense Development [UD070069CD]. This research is supported by the Agency for Defense Development, Korea and the ASRI (Automation and Systems Research Institute) of Seoul National University, Korea. REFERENCES D Jing Xu and Kai-Yew Lum and Jian_Xin Xu (2007). “Analysis of PNG Laws with LOS Angular Rate Figure 19. Miss-distance within 2mters Delay” AIAA 2007-6788 AIAA GNC Conference and Exhibit 20-23 August 2007, Hilton Head, South Carolina CONCLUSION Jacques Waldmann, Member, IEEE (2002). “Line-of- In this paper, we have presented the method of the Sight Rate Estimation and Linearizing Control of an LOS angle and LOS rate derivation from image plane Imaging Seeker in a Tactical Missile Guided by values provided by strapdown seekers. Also we have Proportional Navigation” IEEE Transactions on analyzed the effects of strapdown seeker scale factor control systems technology, Vol.10, No.4, July 2002 error. Joongsup Yun, Chang-Kyung Ryoo and Taek-Lyul The image plane provides a [y, z] value which is Song (2008). “Strapdown Sensors and Seeker Based represented in the seeker frame. The seeker frame is Guidance Filter Design”, International Conference on the same as the body frame. To translate this value into Control, Automation and Systems 2008 Oct. 14-17, the inertial frame value, we deal with a strapdown 2008 in COEX, Seoul, Korea seeker structure and geometric relation between inertial and body frame. Simple but realistic simulations Paul L. Vergez and James R. McClendon (1982). evaluate the method of LOS angle derivation. In order “Optimal Control and Estimation for Strapdown Seeker to derive the LOS rate in inertial frame, we handle 5 Guidance of Tactical Missiles”, AIAA 82-4125 Vol.5, coordinate systems and calculate the translational No.3 May-June 1982 connection between each coordinate system by the kinematics. Also presented is an application to the Raman K. Mehra, and Ralph D. Ehrich (1984). ” Air- calculation of PNG law for missiles. PNG law needs To-Air Missile Guidance For Strapdown Seekers”, the LOS rate in the inertial frame. The simulation for Proceedings of 23rd Conference on Decision and PNG law consists of two divisions which are the Control Las Vegas, NV, December 1984 navigation division and guidance-control division. LOS angle and LOS rate are calculated in the navigation Won-Sang Ra and Ick-Ho Whang (2002). “A Robust division and inputted into the PNG law in the guidance Horizontal LOS Rate Estimator for 2-Axes Gimbaled control division. The reasonable simulation Seeker”, Proceedings of the 41st IEEE Conference on demonstrates an appropriate result trajectory and shows Decision and Control Las Vegas, Nevada USA, that the method of LOS angle and LOS rate derivation December 2002 is correct. http://www.globalsecurity.org/military/systems/muniti In order to analyze the effects of strapdown seeker ons/apkws.htm scale factor error, we add the scale factor error gradually from 0.10% to 1.00%. When we assume that http://en.wikipedia.org/wiki/Advanced_Precision_Kill_ the requirement of attack is within 10 meters, in other Weapon_System words, the threshold of the miss-distance is within 10 meters, whole results of 100 times Monte Carlo simulation are satisfied with a 0.5% scale factor error. When we assume within 2 meters, whole results of 100 times Monte Carlo simulation are satisfied with a 0.45% scale factor error. We find that scale factor error affects the performance and stability of the guidance missile in a great measure. The further work towards a solution to this instability is the research of the dither adaptive method and the EKF method.

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