﻿ 基于落点预测的高旋火箭弹弹道修正算法
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1. 西北工业大学 航天学院, 西安 710072;
2. 上海航天技术研究院, 上海 201109

Ballistic trajectory correction algorithms of high-spin rocket based on impact point prediction
YANG Sizhi1, GONG Chunlin1, HAO Bo2, WU Weinan1, GU Liangxian1
1. College of Astronautics, Northwestern Polytechnical University, Xi'an 710072, China;
2. Shanghai Academy of Spaceflight Technology, Shanghai 201109, China
Abstract: The trajectory corrective control of high-spin rocket is a two-dimensional correction control based on equivalent force. Due to gyro precession and the Magnus effect of high-spin rocket, the equivalent force is formed by the control force, and the force caused by the additional angle of attack under the action of the control force, the magnitude and direction of the equivalent force is constantly changing during the control. Therefore, the direction of control force can't be determined simply by the proportional relation of deviation (correction quantity). Based on this, the relationship between control force and equivalent force is analyzed in this paper. A corrective algorithm based on trajectory impact point prediction is proposed. At first, the deviation between the rocket's landing point and its target position is predicted in real-time using the impact point prediction model. Then the paper established corrective sensitivity coefficient matrix based on the small perturbation method, and the control quantity in both longitudinal and transverse directions were obtained through the deviation and sensitivity coefficient matrix. The combined vector and azimuth of the control quantity were obtained by using the relationship of velocity vector before and after the correction control, and the control period was calculated by using the control quantity and the equivalent force. In the control period, the direction of the control force was adjusted according to the azimuth of the equivalent force angle, the accurate control of the high-spin rocket was obtained, and the problem of non-linear coupling was solved. The simulation results show that the algorithm has high control accuracy, providing theoretical basis for engineering application.
Keywords: high-spin rocket    ballistic correction    impact point prediction    corrected sensitivity coefficient matrix    nonlinear coupled

1 动力学模型 1.1 控制力及其产生的等效力分析

 图 1 制导火箭弹的控制力 Fig. 1 Force of the guided rocket projectile
 ${{\bar F}_{\rm{R}}} = {F_{{\rm{R}}{\eta _1}}} + {\rm{i}}{F_{{\rm{R}}{\zeta _1}}} = {F_{\rm{R}}}{{\rm{e}}^{{\rm{i}}{\varphi _{\rm{f}}}}}$ （1）

 $\left\{ \begin{array}{l} {b_x} = \frac{{\rho S}}{{2m}}{C_x},{b_y} = \frac{{\rho S}}{{2m}}{{C'}_y},{b_z} = \frac{{\rho Sd}}{{2m}}C_z^{\prime \prime }\\ {k_z} = \frac{{\rho SL}}{{2{J_z}}}m_z^\prime ,{k_{zz}} = \frac{{\rho S{L^2}}}{{2{J_z}}}m_{zz}^\prime ,{k_y} = \frac{{\rho SLd}}{{2{J_z}}}m_y^{\prime \prime } \end{array} \right.$ （2）

 $\begin{array}{l} {\Delta ^{\prime \prime }} + \left( {H - {\rm{i}}P} \right){\Delta ^\prime } - (M + {\rm{i}}PT)\Delta = \\ \;\;\; - \frac{{\ddot \theta }}{{{v^2}}} - \left( {{k_{zz}} - {\rm{i}}P} \right)\frac{{\dot \theta }}{v} + \left( {\frac{{{L_{{\rm{CG}}}}}}{{{J_z}{v^2}}} - \frac{{{k_{zz}} - {\rm{i}}P}}{{m{v^2}}}} \right){F_{\rm{R}}}{{\rm{e}}^{{\rm{i}}{\varphi _{\rm{f}}}}} \end{array}$ （3）

 $\left\{ {\begin{array}{*{20}{l}} {{\delta _{2{\rm{p}}}} = - \frac{P}{{Mv}}\dot \theta - \left( {\frac{{PT}}{{{M^2}{v^2}}} - \frac{{2{P^3}T}}{{{M^3}{v^2}}}} \right)\ddot \theta }\\ {{\delta _{1{\rm{p}}}} = \left( {\frac{1}{{M{v^2}}} - \frac{{{P^2}}}{{{M^2}{v^2}}} + \frac{{{P^2}{T^2}}}{{{M^4}{v^2}}}} \right)\ddot \theta - \frac{{{P^2}T}}{{{M^2}v}}\dot \theta } \end{array}} \right.$ （4）

 ${\Delta _{\rm{G}}} = {\delta _{1{\rm{p}}}} + {\rm{i}}{\delta _{2{\rm{p}}}}$ （5）

 $\begin{array}{l} {{\Delta ''}_{{F_{\rm{R}}}}} + (H - {\rm{i}}P)\Delta _{{F_{\rm{R}}}}^\prime - (M + {\rm{i}}PT){\Delta _{{F_{\rm{R}}}}} = \\ \;\;\;\;\;\left[ {{L_{{\rm{CF}}}}/\left( {{J_z}{v^2}} \right) - \left( {{k_{zz}} - iP} \right)/\left( {m{v^2}} \right)} \right]{F_{\rm{R}}}{{\rm{e}}^{{\rm{i}}{\varphi _{\rm{f}}}}} \end{array}$ （6）

 ${\Delta _{{F_{\rm{R}}}}} = \Delta \delta {{\rm{e}}^{{\rm{i}}{\varphi _{\rm{n}}}}} = \frac{{{J_z}{k_{zz}} - m{L_{{\rm{CF}}}} - {\rm{i}}P{J_z}}}{{{J_z}m{v^2}(M + {\rm{i}}PT)}}{F_{\rm{R}}}{{\rm{e}}^{{\rm{i}}{\varphi _{\rm{f}}}}}$ （7）

 $\Delta \delta = \frac{{\sqrt {{{\left( {{J_z}{k_{zz}} - m{L_{{\rm{CF}}}}} \right)}^2} + {{\left( {P{J_z}} \right)}^2}} }}{{{J_z}m{v^2}\sqrt {{M^2} + {{(PT)}^2}} }}{F_{\rm{R}}}$ （8）
 $\begin{array}{l} {\varphi _{\rm{n}}} = {\varphi _{\rm{f}}} + \arctan \left( { - P{J_z}/\left( {{J_z}{k_{zz}} - m{L_{{\rm{CF}}}}} \right)} \right) - \\ \;\;\;\;\;\;\;\arctan (PT/M) \end{array}$ （9）

 ${F_{{\rm{L}}\Delta {F_{\rm{R}}}}} = \frac{{C_y^\prime \left( {{J_z}{k_{zz}} - m{L_{{\rm{CF}}}} - {\rm{i}}P{J_z}} \right)}}{{mLm_z^\prime (1 + {\rm{i}}PT/M)}}{F_{\rm{R}}}{{\rm{e}}^{{\rm{i}}{\varphi _{\rm{f}}}}}$ （10）
 ${F_{{\rm{M}}\Delta {F_{\rm{R}}}}} = \frac{{dC_z^{\prime \prime }\dot \gamma \left( { - m{L_{{\rm{CF}}}} - {\rm{i}}P{J_z}} \right)}}{{mLm_z^\prime }}{F_{\rm{R}}}{{\rm{e}}^{{\rm{i}}\left( {{\varphi _{\rm{f}}} + {\rm{ \mathsf{ π} }}/2} \right)}}$ （11）

 $\Delta \delta = \frac{{\sqrt {{{\left( {m{L_{{\rm{CG}}}}} \right)}^2} + {{\left( {P{J_z}} \right)}^2}} }}{{{J_z}m{v^2}M}}{F_{\rm{R}}}$ （12）
 ${\varphi _{\rm{n}}} = {\varphi _{\rm{f}}} + {\rm{ \mathsf{ π} }} + \arctan \left( {P{J_z}/\left( {m{L_{{\rm{CF}}}}} \right)} \right)$ （13）

 ${F_{{\rm{L}}\Delta {F_{\rm{R}}}}} = \frac{{C_y^\prime \left( { - m{L_{{\rm{CF}}}} - {\rm{i}}P{J_z}} \right)}}{{mLm_z^\prime }}{F_{\rm{R}}}{{\rm{e}}^{{\rm{i}}{\varphi _{\rm{f}}}}}$ （14）
 ${F_{{\rm{M}}\Delta {F_{\rm{R}}}}} = \frac{{dC_z^{\prime \prime }\dot \gamma \left( { - m{L_{{\rm{CF}}}} - {\rm{i}}P{J_z}} \right)}}{{mLm_z^\prime }}{F_{\rm{R}}}{{\rm{e}}^{{\rm{i}}\left( {{\varphi _{\rm{f}}} + {\rm{ \mathsf{ π} }}/2} \right)}}$ （15）
1.2 控制力对火箭弹运动的影响

1) 控制力对火箭弹的直接作用。

2) 控制力矩引起的附加攻角产生的升力和马格努斯力对火箭弹的作用。

 ${F_{\rm{c}}} = {F_{{\rm{L}}\Delta {F_{\rm{R}}}}} + {F_{{\rm{M}}\Delta {F_{\rm{R}}}}} + {F_{\rm{R}}}{{\rm{e}}^{{\rm{i}}{\varphi _{\rm{f}}}}}$ （16）

 ${F_{\rm{c}}} = \left( {\frac{{C_y^\prime \left( { - m{L_{{\rm{CF}}}} - {\rm{i}}P{J_z}} \right)}}{{mLm_z^\prime }} + 1} \right){F_{\rm{R}}}{{\rm{e}}^{{\rm{i}}{\varphi _{\rm{f}}}}}$ （17）

1.3 弹道模型建立

 $\left\{ \begin{array}{l} \dot v = \left( {{F_{{\rm{p}}x}} + {F_{{\rm{a}}x}} + {F_{{\rm{c}}x}}} \right)/m\\ \dot \theta = \left( {{F_{{\rm{p}}y}} + {F_{{\rm{a}}y}} + {F_{{\rm{c}}y}}} \right)/\left( {mv\cos \psi } \right)\\ \dot \psi = \left( {{F_{{\rm{p}}z}} + {F_{{\rm{a}}z}} + {F_{{\rm{c}}z}}} \right)/\left( {mv} \right)\\ {{\dot \omega }_{{\rm{f}}x}} = \left( {{M_{{\rm{f}}x}} + {M_{{\rm{af}}x}}} \right)/{J_{{\rm{f}}x}}\\ {{\dot \omega }_{{\rm{a}}x}} = \left( {{M_{{\rm{a}}x}} + {M_{{\rm{fa}}x}}} \right)/{J_{{\rm{ax}}}}\\ {{\dot \omega }_y} = \left( {{M_{{\rm{f}}y}} + {M_{{\rm{a}}y}} - {H^ * }{\omega _y} + {J_y}\omega _z^2\tan {\varphi _2}} \right)/{J_z}\\ {{\dot \omega }_z} = \left( {{M_{{\rm{f}}z}} + {M_{{\rm{a}}z}} - {H^ * }{\omega _z} - {J_y}{\omega _y}{\omega _z}\tan {\varphi _2}} \right)/{J_z}\\ {{\dot \gamma }_{\rm{f}}} = {\omega _{{\rm{f}}x}} - {\omega _z}\tan {\varphi _2},\dot x = v\cos \psi \cos \theta \\ {{\dot \gamma }_{\rm{a}}} = {\omega _{{\rm{a}}x}} - {\omega _z}\tan {\varphi _2},\dot y = v\cos \psi \sin \theta \\ {{\dot \varphi }_2} = - {\omega _y},{{\dot \varphi }_{\rm{a}}} = {\omega _z}/\cos {\varphi _2},\dot z = v\sin \psi \end{array} \right.$ （18）

2 制导控制系统设计 2.1 弹道落点预测模型设计

 $\left\{ \begin{array}{l} \frac{{{\rm{d}}v}}{{{\rm{d}}t}} = \frac{{{F_{{\rm{a}}x}}}}{m},\frac{{{\rm{d}}\theta }}{{{\rm{d}}t}} = \frac{{{F_{{\rm{a}}y}}}}{{mv\cos \psi }},\frac{{{\rm{d}}{\psi _2}}}{{{\rm{d}}t}} = \frac{{{F_{{\rm{a}}z}}}}{{mv}}\\ \frac{{{\rm{d}}x}}{{{\rm{d}}t}} = v\cos \psi \cos \theta ,\frac{{{\rm{d}}y}}{{{\rm{d}}t}} = v\cos \psi \sin \theta \\ \frac{{{\rm{d}}z}}{{{\rm{d}}t}} = v\sin \psi ,\frac{{{\rm{d}}\dot \gamma }}{{{\rm{d}}t}} = \frac{{{M_\xi }}}{{{J_x}}} \end{array} \right.$ （19）

 图 2 弹道预测与7自由度模型仿真结果对比 Fig. 2 Comparison between prediction of trajectory and simulation results of 7-DOF model

2.2 制导算法设计

 $\left\{ \begin{array}{l} X = {J_x}(v(t),\theta (t),\psi (t),x(t),y(t),z(t), \cdots )\\ Z = {J_z}(v(t),\theta (t),\psi (t),x(t),y(t),z(t), \cdots ) \end{array} \right.$ （20）

 $\left\{ \begin{array}{l} \Delta X = {J_x}(V(t),\theta (t) + \Delta \theta ,\psi (t) + \\ \;\;\;\;\;\;\Delta \psi ,x(t),y(t),z(t), \cdots ) - \\ \;\;\;\;\;\;{J_x}(V(t),\theta (t),\psi (t),x(t),y(t),z(t), \cdots )\\ \Delta Z = {J_z}(V(t),\theta (t) + \Delta \theta ,\psi (t) + \Delta \psi ,x(t),\\ \;\;\;\;\;\;y(t),z(t), \cdots ) - {J_z}(V(t),\theta (t),\psi (t),x(t)\\ \;\;\;\;\;\;y(t),z(t), \cdots ) \end{array} \right.$ （21）

 $\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial {J_x}}}{{\partial \theta }}\Delta \theta + \frac{{\partial {J_x}}}{{\partial \psi }}\Delta \psi = \Delta X}\\ {\frac{{\partial {J_z}}}{{\partial \theta }}\Delta \theta + \frac{{\partial {J_z}}}{{\partial \psi }}\Delta \psi = \Delta Z} \end{array}} \right.$ （22）

 $\left[ {\begin{array}{*{20}{c}} {\Delta \theta }\\ {\Delta \psi } \end{array}} \right] = {\left[ {\begin{array}{*{20}{c}} {\partial {J_x}/\partial \theta }&{\partial {J_x}/\partial \psi }\\ {\partial {J_z}/\partial \theta }&{\partial {J_z}/\partial \psi } \end{array}} \right]^{ - 1}}\left[ {\begin{array}{*{20}{l}} {\Delta x}\\ {\Delta z} \end{array}} \right]$ （23）

 ${{\bar \Delta }_\Sigma } = \Delta \theta + {\rm{i}}\Delta \psi$ （24）
2.3 控制算法设计

 图 3 修正前后速度矢量的位置关系 Fig. 3 Position relation of velocity vector before and after correction

 $\left\{ {\begin{array}{*{20}{l}} {{\Delta _\Sigma } = \arccos (\cos \Delta \theta \cos \Delta \psi )}\\ {{\varphi _{\rm{n}}} = \arcsin \left( {\sin \Delta \psi /\sin {\Delta _\Sigma }} \right)} \end{array}} \right.$ （25）

 ${\varphi _{\rm{n}}} = \left\{ \begin{array}{l} {\varphi _{\rm{n}}}\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{ \mathsf{ δ} }}\theta \ge 0,{\rm{ \mathsf{ δ} }}\psi \ge 0\\ {\varphi _{\rm{n}}} + 2{\rm{ \mathsf{ π} }}\;\;\;\;\;\;{\rm{ \mathsf{ δ} }}\theta \ge 0,{\rm{ \mathsf{ δ} }}\psi < 0\\ - {\varphi _{\rm{n}}} + {\rm{ \mathsf{ π} }}\;\;\;\;\;\;{\rm{ \mathsf{ δ} }}\theta \le 0,{\rm{ \mathsf{ δ} }}\psi \ge 0\\ {\varphi _{\rm{n}}} + {\rm{ \mathsf{ π} }}/2\;\;\;\;{\rm{ \mathsf{ δ} }}\theta \le 0,{\rm{ \mathsf{ δ} }}\psi \ge 0 \end{array} \right.$ （26）

 ${{\dot \Delta }_\Sigma } = {F_{\rm{c}}}/\left( {mv} \right)$ （27）

 ${T_{\rm{p}}} = {\Delta _\Sigma }/{{\dot \Delta }_\Sigma } = {\Delta _\Sigma }mv/{F_{\rm{c}}}$ （28）

 图 4 不同固定鸭舵偏角的修正能力 Fig. 4 Correction ability of different rudder deviation angles

3 仿真分析

 参数 数值 m/kg 16.1 L/m 0.967 LCG/m 0.420 d/m 0.107 Jz/(kg·m2) 1.065 Jx/(kg·m2) 0.034 Cx 0.214 C′y 0.059 C″z -2.77×10-3 M′z 0.024 M′zz -0.007 ky -1.01×10-2

 图 5 控制力的控制角度 Fig. 5 Control angle of control force
 图 6 修正控制和偏差量控制纵向和横向的修正量 Fig. 6 Longitudinal and transverse correction quantity of corrective controlled and deviation controlled
 图 7 修正控制和无控弹道的姿态角及姿态角速度 Fig. 7 Attitude angle and attitude angle velocity of corrective controlled and uncontrolled rocket
 图 8 修正控制和无控弹道倾角和弹道偏角曲线对比 Fig. 8 Comparison of trajectory inclination angle and deflection angle curves of corrective controlled and uncontrolled rocket
 图 9 修正控制和无控弹道曲线 Fig. 9 Ballistic curves of corrective controlled and uncontrolled rocket

4 结论

1) 在考虑陀螺和马格努斯效应的基础上分析了控制力对弹体的作用及角运动的影响，得到了修正等效控制力，建立了控制力与修正等效控制力之间的对应关系。

2) 建立了修正火箭弹的弹道落点预测模型，实时精确预测火箭弹的落点与目标的偏差量。利用小扰动法构造偏差量对控制量的敏感系数矩阵，根据偏差量解算出修正控制量，通过修正前后的坐标关系建立修正控制量的合矢量、方位角及控制周期。在控制周期内利用等效力与控制力之间的关系计算出控制力的方位角，实现修正火箭弹的修正控制系统闭环设计。

3) 该算法以修正终点为目标，解决了高旋火箭弹的非线性强耦合导致的实际控制力与需要的控制力不一致的问题，并通过对某型高旋修正控制的仿真分析，并与偏差量控制算法在纵向和横向的修正能力方面进行对比，验证该算法具有收敛速度快、控制精度高等特点，能够实现高旋火箭弹精确控制，具有一定的工程应用价值。

 [1] 许诺, 于剑桥, 王亚飞. 基于周期平均的固定翼双旋弹弹道修正方法[J]. 航空学报, 2015, 36(9): 2892-2899. XU N, YU J Q, WANG Y F. Trajectory correcting method of fixed-canard dual-spin projectiles based on period average[J]. Acta Aeronautica et Asrtonautica Sinica, 2015, 36(9): 2892-2899. (in Chinese) Cited By in Cnki (4) | Click to display the text [2] GAGNON E, LAUZON M. Course correction fuze concept analysis for in-service 155mm spin-stabilized gunnery projectiles[C]//AIAA Guidance, Navigation and Control Conference and Exhibit. Reston, VA: AIAA, 2008: 1-20. [3] THEODOULIS S, GASSMANN V, WERNERT P. Guidance and control design for a class of spin-stabilized fin-controlled projectiles[J]. Journal of Guidance, Control and Dynamics, 2013, 36(2): 517-531. Click to display the text [4] WERNERT P, LEOPOLD F, BIDINO D. Wind tunnel tests and open-loop trajectory simulations for a 155 mm canards guided spin stabilized projectile[C]//AIAA Atmospheric Flight Mechanics and Exhibit. Reston, VA: AIAA, 2008: 2-4. [5] COSTELLO M. Modeling and simulation of a differential roll projectile[C]//Modeling and Simulation Technologies Conference. Reston, VA: AIAA, 1998: 490-499. [6] COSTELLO M, PETERSON A. Linear theory of dual-spin projectile in atmospheric flight[J]. Journal of Guidance, Control and Dynamics, 2000, 23(5): 789-797. Click to display the text [7] 王毅, 宋卫东, 佟德飞. 固定鸭舵式弹道修正弹二体系统建模[J]. 弹道学报, 2014, 26(4): 36-41. WANG Y, SONG W D, TONG D F. Modeling of two-rigid-body system for trajectory correction projectile with fixed-canard[J]. Journal of Ballistics, 2014, 26(4): 36-41. (in Chinese) Cited By in Cnki (8) | Click to display the text [8] 许诺, 于剑桥, 王亚飞, 等. 固定翼双旋弹动力学特性分析[J]. 兵工学报, 2015, 36(4): 602-609. XU N, YU J Q, WANG Y F, et al. Analysis of dynamic characteristics of fixed-wing dual-spin projectiles[J]. Acta Armamenta Rii, 2015, 36(4): 602-609. (in Chinese) Cited By in Cnki (6) | Click to display the text [9] 王钰, 王晓鸣, 程杰, 等. 基于等效力方法的双旋弹侧向控制力落点响应分析[J]. 兵工学报, 2016, 37(8): 1379-1386. WANG Y, WANG X M, CHENG J, et al. Analysis on impact point response of a dual-spin projectile with lateral force based on equivalent force method[J]. Acta Armamenta Rii, 2016, 37(8): 1379-1386. (in Chinese) Cited By in Cnki (4) | Click to display the text [10] 张鑫, 姚晓先, 杨忠, 等. 周期平均控制下固定翼双旋弹角运动特性分析[J]. 航空学报, 2019, 40(4): 322452. ZHANG X, YAO X X, YANG Z, et al. Analysis of angular motion of dual-spin projectile with fixed-canards under period average control[J]. Acta Aeronautica et Astronautica Sinica, 2019, 40(4): 322452. (in Chinese) Cited By in Cnki | Click to display the text [11] 王毅, 宋卫东, 郭庆伟, 等. 固定鸭舵式二维弹道修正弹稳定性分析[J]. 军械工程学院学报, 2015, 27(3): 16-23. WANG Y, SONG W D, GUO Q W, et al. Stability analysis of two-dimension trajectory correction mortar with fixed-canard[J]. Journal of Ordnance Engineering College, 2015, 27(3): 16-23. (in Chinese) Cited By in Cnki (2) | Click to display the text [12] WERNERT P. Stability analysis for canard guided dual-spin stabilized projectiles[C]//AIAA Atmospheric Flight Mechanics Conference and Exhibit. Reston, VA: AIAA, 2009: 2-4. [13] WERNERT P, THEODOULIS S. Modeling and stability analysis for a class of 155mm spin-stabilized projectiles with course correction fuse(CCF)[C]//Proceedings of the 2011 AIAA Atmospheric flight Mechanics Conference and Exhibit. Reston, VA: AIAA, 2011: 1-13. [14] 王毅, 宋卫东, 宋谢恩, 等. 固定鸭舵式二维弹道修正榴弹偏流特性分析[J]. 系统工程与电子技术, 2016, 38(6): 1367-1371. WANG Y, SONG W D, SONG X E, et al. Ballistic drift analysis of two-dimensional trajectory correction projectiles with fixed-canards[J]. Systems Engineering and Electronics, 2016, 38(6): 1367-1371. (in Chinese) Cited By in Cnki (2) | Click to display the text [15] 朱大林.双旋弹飞行特性与制导控制方法研究[D].北京: 北京理工大学, 2015: 98-112. ZHU D L. Research on flight characteristics, guidance, and control for a dual-spin projectile[D]. Beijing: Beijing Institute of Technology, 2015: 98-112(in Chinese). Cited By in Cnki (1) | Click to display the text [16] 黎海清, 杨凯, 栗金平, 等. 落点预测制导律的旋转稳定弹制导控制[J]. 西安工业大学学报, 2015, 35(10): 855-861. LI H Q, YANG K, LI J P, et al. Course correction for spin-stabilized projectile based on impact point predication guidance law[J]. Journal of Xi'an Technological University, 2015, 35(10): 855-861. (in Chinese) Cited By in Cnki | Click to display the text [17] 郭致远, 姚晓先, 张鑫. 基于周期平均的固定舵双旋火箭弹控制方法[J]. 航空学报, 2017, 38(12): 321307. GUO Z Y, YAO X X, ZHANG X. Control method for a class of fixed-canard dual-spin rockets based on period average[J]. Acta Aeronautica et Astronautica Sinica, 2017, 38(12): 321307. (in Chinese) Cited By in Cnki | Click to display the text [18] 张衍儒, 肖练刚, 周华, 等. 固定翼鸭舵式双旋弹的制导控制算法研究[J]. 火炮发射与控制学报, 2016, 37(3): 20-24. ZHANG Y R, XIAO L G, ZHOU H, et al. Research on guidance control algorithm of dual-spin projectile with fixed canards[J]. Journal of Gun Launch & Control, 2016, 37(3): 20-24. (in Chinese) Cited By in Cnki | Click to display the text [19] 张衍儒, 肖练刚, 张继生, 等. 旋转控制固定鸭舵的导航初始化与控制算法研究[J]. 航天控制, 2014, 32(6): 34-39. ZHANG Y R, XIAO L G, ZHANG J S, et al. The navigation initialization and control algorithm of roll control fixed canards[J]. Aerospace Control, 2014, 32(6): 34-39. (in Chinese) Cited By in Cnki (4) | Click to display the text [20] COSTELLO M, PETERSON A. Linear theory of a dual-spin projectile in atmospheric flight[J]. Journal of Guidance, Control, and Dynamics, 2000, 23(5): 789-797. Click to display the text [21] WERNERT P. Stability analysis for canard guided dual-spin stabilized projectiles[C]//Atmospheric Flight Mechanics Conference and Exhibit. Reston, VA: AIAA, 2009: 1-24. [22] CHARLES H M. Symmetric missile dynamic instabilities[J]. Journal of Guidance and Control, 1981, 4(5): 464-472. Click to display the text [23] 韩子鹏. 弹箭外弹道学[M]. 北京: 北京理工大学出版社, 2014: 180-194. HAN Z P. Rocket exterior ballistic[M]. Beijing: Beijing Institute of Technology Press, 2014: 180-194. (in Chinese)
http://dx.doi.org/10.7527/S1000-6893.2019.23421

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#### 文章信息

YANG Sizhi, GONG Chunlin, HAO Bo, WU Weinan, GU Liangxian

Ballistic trajectory correction algorithms of high-spin rocket based on impact point prediction

Acta Aeronautica et Astronautica Sinica, 2020, 41(2): 323421.
http://dx.doi.org/10.7527/S1000-6893.2019.23421