Attitude Pursuit GuidanceLaw for Coning Motion Stability of Spinning Missiles

doi:10.12382/bgxb.2022.0081

Abstract

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

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

Figures/Tables 28

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

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

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

Fig.4 APG system of the strapdown seeker

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

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

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

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

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

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

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

Fig.5 Convergent coning motion without considering seeker delay response

Fig.6 Convergent coning motion with λg=3°

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

Fig.8 Divergent coning motion with λg=3°

Fig.9 Convergent coning motion with λg=6°

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

Fig.11 Divergent coning motion with λg=6°

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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