采用弹体追踪导引律的旋转弹锥形运动稳定性

doi:10.12382/bgxb.2022.0081

摘要/Abstract

摘要：

针对旋转弹采用捷联导引头和弹体追踪导引律时可能导致锥形运动失稳的问题,推导了捷联导引头在非旋转弹体坐标系下的动力学模型,建立了复数形式的弹体追踪制导控制系统数学模型;在不同弹体转速及阻尼回路增益情况下,分别考虑导引头响应延迟和陀螺标度因数误差,分析了上述制导控制系统的稳定性,并使用数值方法求得使其稳定的特征参数取值范围。研究结果表明:导引头延迟角越大,使系统稳定的制导回路增益上限越小;陀螺标度因数误差系数大于1时,使系统稳定的制导回路增益上限会变大,当陀螺标度因数误差系数小于1时,使系统稳定的制导回路增益稳定上限会变小。

关键词: 旋转弹, 锥形运动, 弹体追踪导引律, 稳定性

Abstract:

To address the problem of coning motion instability of spinning missiles induced by the introduction of the strapdown seeker and attitude pursuit guidance law, the dynamic model of the strapdown seeker in the non-spinning missile coordinate system is derived, and the mathematical model of the attitude pursuit guidance and control system in complex form is established. The response delay of the seeker and the gyro scale-factor error are considered under different spinning rates and damping loop gains. The stability of the spinning missile’s control and guidance system is analyzed, and the range of the corresponding characteristic parameters are solved by numerical methods. The results show that the larger the delay angle of the seeker is, the smaller the upper limit of the guidance loop gain that can stabilize the system is. When the gyro scale-factor error coefficient is greater than 1, the upper limit of the guidance loop gain become larger; when the scale-factor error coefficient is less than 1, the upper limit becomes smaller.

Key words: spinning missile, coning motion, attitude pursuit guidance (APG) law, dynamic stability

宋金超, 赵良玉. 采用弹体追踪导引律的旋转弹锥形运动稳定性[J]. 兵工学报, 2023, 44(6): 1795-1808.

SONG Jinchao, ZHAO Liangyu. Attitude Pursuit GuidanceLaw for Coning Motion Stability of Spinning Missiles[J]. Acta Armamentarii, 2023, 44(6): 1795-1808.

图/表 28

图1 非旋转弹体坐标系与旋转弹体坐标系之间的转换

Fig.1 Conversion between non-spinning and spinning missile coordinate systems

图2 攻角侧滑角示意图

Fig.2 Angle of attack and angle of side-slip

图3 俯仰平面弹目视线关系图

Fig.3 Relationship of missile-to-target LOS in pitch plane

图4 捷联导引头弹体追踪系统结构图

Fig.4 APG system of the strapdown seeker

表1 旋转弹参数表

Table 1 Parameters of spinning missile

参数	数值	参数	数值
C_Lα	11.932	C_D	0.507
C_m_α	-0.7504	C_mδ	0.2583
C_mpa	-0.1848	C_mq	1.35
T_d/s	0.003	k_d	1
T_s/s	0.012	μ_s	0.5
τ₁/ s	0.016	k_s	1
I_x/I_t	0.0030	V/(m·s^-1)	1200
τ₂/s	0.007	p/(rad·s^-1)	8π
l/m	7.6	S/m²	0.0707
λ_t/(°)	44.9	a₁	0.2125
a₂	0.2090	b₁₁	-23.7524
b₂₁	-0.2706	b₃	6.5933

表2 无延迟角的制导回路增益下限

Table 2 Lower limit of the control and guidance loop gain without delay angle

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	0.9930	0.9959	1.0003
0.10	1.0430	1.0459	1.0503
0.20	1.1430	1.1459	1.1513

表2 无延迟角的制导回路增益下限

Table 2 Lower limit of the control and guidance loop gain without delay angle

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	0.9930	0.9959	1.0003
0.10	1.0430	1.0459	1.0503
0.20	1.1430	1.1459	1.1513

表3 λg=3°制导回路增益上限

Table 3 Upper limit of the loop gain with λg=3°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	341.6	335.0	312.6
0.10	477.5	474.4	459.5
0.20	761.6	761.3	753.6

表3 λg=3°制导回路增益上限

Table 3 Upper limit of the loop gain with λg=3°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	341.6	335.0	312.6
0.10	477.5	474.4	459.5
0.20	761.6	761.3	753.6

表4 λg=6°制导回路增益上限

Table 4 Upper limit of the loop gain with λg=6°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	161.9	159.0	148.6
0.10	226.5	225.3	218.7
0.20	361.2	361.8	358.7

表4 λg=6°制导回路增益上限

Table 4 Upper limit of the loop gain with λg=6°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	161.9	159.0	148.6
0.10	226.5	225.3	218.7
0.20	361.2	361.8	358.7

图5 不考虑导引头响应延迟的收敛锥形运动

Fig.5 Convergent coning motion without considering seeker delay response

图6 λg=3°时收敛的锥形运动

Fig.6 Convergent coning motion with λg=3°

图7 λg=3°时弹体临界稳定的锥形运动

Fig.7 Critical stable coning motion with λg=3°

图8 λg=3°时弹体发散的锥形运动

Fig.8 Divergent coning motion with λg=3°

图9 λg=6°时收敛的锥形运动

Fig.9 Convergent coning motion with λg=6°

图10 λg=6°时弹体临界稳定的锥形运动

Fig.10 Critical stable coning motion with λg=6°

图11 λg=6°时弹体发散的锥形运动

Fig.11 Divergent coning motion with λg=6°

表5 kf=1.004无延迟角的制导回路增益上限

Table 5 Upper limit of the loop gain at kf=1.004

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	0.9970	0.9999	1.0043
0.10	1.0470	1.0499	1.0543
0.20	1.1470	1.1499	1.1553

表5 kf=1.004无延迟角的制导回路增益上限

Table 5 Upper limit of the loop gain at kf=1.004

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	0.9970	0.9999	1.0043
0.10	1.0470	1.0499	1.0543
0.20	1.1470	1.1499	1.1553

表6 kf=0.996无延迟角的制导回路增益上限

Table 6 Upper limit of the loop gain at kf=0.996

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	0.9890	0.9919	1.0963
0.10	1.0390	1.0419	1.0463
0.20	1.1390	1.1419	1.1473

表6 kf=0.996无延迟角的制导回路增益上限

Table 6 Upper limit of the loop gain at kf=0.996

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	0.9890	0.9919	1.0963
0.10	1.0390	1.0419	1.0463
0.20	1.1390	1.1419	1.1473

表7 kf=1.004,λg=3°制导回路增益上限

Table 7 Upper limit of the loop gain with, kf=1.004,λg=3°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	340.9	336.3	316.0
0.10	476.5	475.6	463.1
0.20	759.0	762.1	757.0

表7 kf=1.004,λg=3°制导回路增益上限

Table 7 Upper limit of the loop gain with, kf=1.004,λg=3°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	340.9	336.3	316.0
0.10	476.5	475.6	463.1
0.20	759.0	762.1	757.0

表8 kf=0.996、λg=3°制导回路增益上限

Table 8 Upper limit of the loop gain with kf=0.996,λg=3°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	339.3	330.8	306.7
0.10	474.6	469.2	452.1
0.20	757.1	754.7	744.2

表8 kf=0.996、λg=3°制导回路增益上限

Table 8 Upper limit of the loop gain with kf=0.996,λg=3°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	339.3	330.8	306.7
0.10	474.6	469.2	452.1
0.20	757.1	754.7	744.2

表9 kf=1.004、λg=6°制导回路增益上限

Table 9 Upper limit of the loop gain with kf=1.004,λg=6°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	162.3	159.5	149.0
0.10	227.0	226.0	219.1
0.20	361.8	362.3	358.9

表9 kf=1.004、λg=6°制导回路增益上限

Table 9 Upper limit of the loop gain with kf=1.004,λg=6°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	162.3	159.5	149.0
0.10	227.0	226.0	219.1
0.20	361.8	362.3	358.9

表10 kf=0.996、λg=6°制导回路增益上限

Table 10 Upper limit of the loop gain with kf=0.996,λg=6°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	161.5	158.6	147.9
0.10	226.0	224.9	218.1
0.20	360.8	361.5	358.0

表10 kf=0.996、λg=6°制导回路增益上限

Table 10 Upper limit of the loop gain with kf=0.996,λg=6°

k_ω	$γ ·$ /(π rad·s^-1)
k_ω	4	8	12
0.05	161.5	158.6	147.9
0.10	226.0	224.9	218.1
0.20	360.8	361.5	358.0

图12 仅考虑陀螺标度因数误差的收敛锥形运动

Fig.12 Convergent coning motion considering the gyro scale-factor error

图13 λg=3°、kf=1.004时收敛的锥形运动

Fig.13 Convergent coning motion with λg=3°、kf=1.004

图14 λg=3°、kf=1.004时临界稳定的锥形运动

Fig.14 Critical stable coning motion with λg=3°、kf=1.004

图15 λg=3°、kf=1.004时发散的锥形运动

Fig.15 Divergent coning motion with λg=3°、kf=1.004

图16 λg=6°、kf=0.996时收敛的锥形运动(kc=190)

Fig.16 Convergent coning motion with λg=6°、kf=0.996

图17 λg=6°、kf=0.996时临界稳定的锥形运动

Fig.17 Critical stable coning motion with λg=6°、kf=0.996

图18 λg=6°、kf=0.996时弹体发散的锥形运动

Fig.18 Divergent coning motion with λg=6°、kf=0.996

参考文献 24

[1]	杨树兴. 陆军多管火箭武器的发展与思考[J]. 兵工学报, 2016, 37(7):1299-1305. doi: 10.3969/j.issn.1000-1093.2016.07.019
	YANG S X. Progress and key points for guidance of multiple launch rocket systems[J]. Acta Armamentarii, 2016, 37(7):1299-1305. (in Chinese)
[2]	陈成, 赵良玉, 谢浩怡, 等. 单通道控制旋转弹角运动的复分析方法[J]. 兵工学报, 2021, 42(2):308-319.
	CHEN C, ZHAO L Y, XIE H Y, et al. Complex analysis for angular motion of a spinning projectile with one pair of canards[J]. Acta Armamentarii, 2021, 42(2):308-319. (in Chinese) doi: 10.3969/j.issn.1000-1093.2021.02.009
[3]	丁天宝, 何朝, 王良明, 等. 高速旋转炮弹宽海拔弹道解算方法[J]. 兵工学报, 2021, 42(1):209-213.
	DING T B, HE Z, WANG L M, et al. Calculation method of firing trajectory of high spinning projectile adapted to wide altitude[J]. Acta Armamentarii, 2021, 42(1):209-213. (in Chinese) doi: 10.3969/j.issn.1000-1093.2021.01.024
[4]	杨树兴, 赵良玉, 闫晓勇. 旋转弹动态稳定性理论[M]. 北京: 国防工业出版社, 2014.
	YANG S X, ZHAO L Y, YAN X Y. Dynamic stability of spinning missiles[M]. Beijing: National Defense Industry Press, 2014. (in Chinese)
[5]	YAN X Y, YANG S X, XIONG F F. Stability limits of spinning missiles with attitude autopilot[J]. Journal of Guidance, Control, and Dynamics, 2011, 34(1):278-283. doi: 10.2514/1.51627 URL
[6]	MURPHY C H. Free flight motion of symmetric missiles: BRL-1216[R]. Aberdeen Proving Ground, MD,US: U. S. Army Ballistic Research Laboratory, 1963.
[7]	陈亮, 刘荣忠, 郭锐, 等. 旋转尾翼弹箭极限圆锥运动稳定判据[J]. 兵工学报, 2019, 40(7): 1329-1339. doi: 10.3969/j.issn.1000-1093.2019.07.001
	CHEN L, LIU R Z, GUO R, et al. Stability criteria for limiting conical motion of rotary fin-stablized projectiles[J]. Acta Armamentarii, 2019, 40(7): 1329-1339. (in Chinese) doi: 10.3969/j.issn.1000-1093.2019.07.001
[8]	舒敬荣, 李红星, 李宏玲. 非线性力矩作用下气动偏心弹丸强迫圆锥运动稳定性条件[J]. 兵工学报, 2018, 39(5): 875-882. doi: 10.3969/j.issn.1000-1093.2018.05.006
	SHU J R, LI H X, LI H L. Forced conical motion stability conditions for pneumatic eccentric projectile under the action of nonlinear moment[J]. Acta Armamentarii, 2018, 39(5): 875-882. (in Chinese)
[9]	吴映锋, 钟扬威, 王良明. 旋转稳定二维弹道修正弹在固定舵作用下的角运动特性研究[J]. 兵工学报, 2017, 38(7): 1263-1272. doi: 10.3969/j.issn.1000-1093.2017.07.003
	WU Y F, ZHONG Y W, WANG L M. Study on angular motion characteristics of spin-stabilized 2D trajectory correction projectile under the effect of fixed canards[J]. Acta Armamentarii, 2017, 38(7): 1263-1272. (in Chinese)
[10]	LIU Y S, LI S, LI J L. Extraction and filter algorithm of guidance information for full-strapdown seeker on rotation missile[C]//Proceedings of MATEC Web of Conferences. EDP Sciences. Moscow, Moscow, Russia: Moscow State University of Civil Engineering (National Research University), 2018, 214(10):03008.
[11]	DU X, XIA Q L. Research on strap-down seeker guidance information for rolling interceptor[J]. Optik-International Journal for Light and Electron Optics, 2017, 129:183-199. doi: 10.1016/j.ijleo.2016.10.042 URL
[12]	ZIPFEL P H. Modeling and simulation of aerospace vehicle dynamics,[M]. 3rd ed. Reston,VA,US: AIAA, 2014.
[13]	ZARCHAN P. Tactical and strategic missile guidance[M]. 7th ed. Reston,VA, US:AIAA, 2019.
[14]	PARK B G. Impact angle control guidance law considering the seeker’s field-of-view limits[J]. Proceedings of Institution of Mechanical Engineers Part G Journal of Aerospace Engineering, 2013, 227(8):1347-1364. doi: 10.1177/0954410012452367 URL
[15]	LI W, WEN Q Q, YANG Y. Stability analysis of spinning missiles induced by seeker disturbance rejection rate parasitical loop[J]. Aerospace Science and Technology, 2019, 90:194-208. doi: 10.1016/j.ast.2019.04.013 URL
[16]	ZHENG D, LIN D F, XU X H, et al. Dynamic stability of rolling missile with proportional navigation&PI autopilot considering parasitic radome loop[J]. Aerospace Science and Technology, 2017, 67:41-48. doi: 10.1016/j.ast.2017.03.036 URL
[17]	HE S M, LIN D F, WANG J. Coning motion stability of spinning missiles with strapdown seekers[J]. The Aeronautical Journal, 2016, 120(1232): 1566-1577. doi: 10.1017/aer.2016.75 URL
[18]	HU X, YANG S X, XIONG F F, et al. Stability of spinning missile with homing proportional guidance law[J]. Aerospace Science and Technology, 2017, 71:546-555. doi: 10.1016/j.ast.2017.10.007 URL
[19]	FRANK E. On the zeros of polynomials with complex coefficients[J]. Bulletin of the American Mathematical Society, 1946, 52(2):144-157. doi: 10.1090/bull/2015-52-02 URL
[20]	李克勇. 旋转导弹制导控制与稳定性问题研究[D]. 北京: 北京理工大学, 2014.
	LI K Y. Guidance, control and stability of spinning missiles[D]. Beijing: Beijing Institute of Technology, 2014. (in Chinese)
[21]	胡啸. 制导旋转弹稳定性研究[D]. 北京: 北京理工大学, 2019.
	HU X. Dynamic stability of guided spinning missiles[D]. Beijing: Beijing Institute of Technology, 2019. (in Chinese)
[22]	YANG L, TANG S J, SHI J, et al. Attitude head pursuit transition guidance law[J]. Chinese Journal of Aeronautics, 2010, 23(3):359-363. doi: 10.1016/S1000-9361(09)60227-2 URL
[23]	沈浩, 祁载康. 提高弹体追踪制导律精度的研究[J]. 弹箭与制导学报, 2008, 28(4):7-10.
	SHEN H, QI Z K. Study of precision improvement of body-traced guidance law[J]. Journal of Projectiles, Rockets, Missiles and Guidance, 2008, 28(4):7-10. (in Chinese)
[24]	徐一航, 陈少松, 魏恺, 等. 正,余弦指令控制方式旋转导弹气动特性分析[J]. 哈尔滨工业大学学报, 2022, 54(1):114-122.
	XU Y H, CHEN S S, WEI K, et al. Analysis of aerodynamic characteristics of a rotating missile with sine and cosine command control[J]. Journal of Harbin Institute of Technology, 2022, 54(1):114-122. (in Chinese)