# Research on Angular Motion Characteristics and Stability of Trajectory-Corrected Rocket Projectile with Isolated Rotating Tail Rudder

Authors: Chen Wang; Libo Ding; Yang Feng; Tan Lu
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

A trajectory-corrected rocket projectile with an isolated rotating tail rudder provides a new concept for a low-cost trajectory correction and guidance due to its simplified guidance principle. To study the angular motion characteristics of a trajectory-corrected rocket projectile with an isolated rotating tail rudder under the action of a periodic control force, the angular motion differential equation described in the complex plane is derived based on the rigid body trajectory equation. Furthermore, the effects of the initial conditions, gravity, and an asymmetric tail on the flight stability of the projectile in an uncorrected trajectory state are studied. The critical rotational speed of the tail rudder, which makes the projectile unstable due to the Magnus moment, and the rotational speed of the tail rudder, which causes projectile resonance, are derived. The transient and steady-state solutions of the angular motion of the projectile under trajectory correction and the variation law of the velocity direction for the projectile centroid are obtained. Finally, an example is given to verify the conclusions of this study. This research has important guiding significance for trajectory analysis, guidance control, and guidance law designs for isolated rotating rudder trajectory-corrected rocket projectiles.


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## Body

## 1. Introduction

Modern warfare has placed increasing demands on the cost-effectiveness of weapon systems, both in terms of increasing their accuracy and reducing costs as much as possible. The effective range of an unguided, man-portable, antiarmor weapon such as rocket-propelled grenades is generally within 500 m. Although these devices are inexpensive to build, they have a small effective range with accuracies that decrease significantly over distance. In comparison, man-portable antitank missiles, such as the “Javelin” missile of the United States, can strike fixed or moving targets within 2000 m with high precision using infrared imaging guides and four movable rudder blades [1]. However, because of its high cost, this type of weapon is only suitable to attack high-value targets.One effective scheme to increase the range and accuracy of unguided, man-portable, antiarmor weapons is to add a simple guidance module. A fixed canard system [2–6] is a low-cost, miniaturized, two-dimensional, trajectory-corrected steering system. Such a system was first used in the U.S. Army’s Precision Guidance Kit (PKG) [7–11] for 155 and 105 mm grenades. It can also be ported to 120 mm mortar rounds [12], and there have been reports in recent years of its application to long-range rockets. The periodic average control force method [13–18] has been adopted in actuator systems to realize two-dimensional trajectory correction using a single channel. This has the advantages of a simple structure, low cost, and high reliability.The fixed canard system is usually installed on the projectile head. Under noncorrected trajectories, the steering gear rotates at a uniform speed relative to the inertial system. In a trajectory-corrected state, the direction and magnitude of the periodic average aerodynamic force are changed by controlling the rotation speed of the steering gear to realize trajectory correction. Such trajectory correction projectiles are usually called dual-spin projectiles. To date, many works have modelled the external trajectory of dual-spin projectiles and studied their flight stability and control characteristics. Zhu et al. [16] established a seven degree-of-freedom dynamic motion equation for dual-spin projectiles in a fixed plane coordinate system. The differential equation for the complex angle of attack was derived using projectile linear theory without the assumption of a flat trajectory. A revised stability criterion was established from the Hurwitz stability criterion, and analytic solutions to the stability boundaries for trim angles were developed. To investigate the influence of yawing forces on angular motion, Wang et al. [19] derived a theoretical solution for the total yaw angle function with cyclic yawing forces using the designed seven degree-of-freedom model. Furthermore, a detailed simulation was performed to determine the influence rules of the yawing force on the angular motion. The calculated results illustrate that when the rotational speed of the forward part is close to the initial turning rate, the total yaw angle increases, and the flight range decreases sharply.Chang et al. [20] studied the swerve response of spin-stabilized projectiles on the canard control problem. A complex deviation angle was used to describe the motion of the projectile velocity vector over the controlled trajectory. The effect of distance between the center of pressure of the canard surface and the center of mass of the projectile on the swerving motion was explained from the perspective of the projectile velocity vector response. The results demonstrate that gravity plays a vital role in predicting the swerve response. Li et al. [21] established the exterior ballistic linearized equations to consider the control force by introducing the concept of the angular compensation matrix. The instability boundaries of the control force magnitude were derived. Numerical simulations demonstrate that if the control force magnitude is in the unstable scope, the projectile loses its stability. Furthermore, the effect of the projectile pitch, velocity, and roll rate on the flight stability during correction is investigated using the proposed instability boundaries. Guan and Yi [15] analyzed the ballistic characteristics of the projectile with a seven-degree-of-freedom projectile trajectory model. Their numerical simulations indicated that the dual-spinning projectile is different from the traditional spinning projectile mostly in that a degree of freedom is added in the direction of the projectile axis. The forebody of the projectile also spins at a low speed or even holds still to improve its control precision, while the afterbody spins at a high speed to maintain gyroscopic stability. Zheng and Zhou [22] established a mathematical model for a dual-spin projectile, and the angular motion equation was obtained using some linearized assumptions. The sufficient and necessary conditions of the coning motion stability for the dual-spin projectile with angular rate loops are analytically derived and further verified via numerical simulations.The above literature shows that the installation of PGK will affect the stability conditions of the original projectile. Before researching trajectory-corrected control algorithms, it is necessary to perform basic research on the flight dynamics and stability of projectiles. A fixed-wing canard system is applied to a grenade with a caliber of 105–155 mm. As the high-speed rotation of the projectile can produce a strong stability torque, installing the steering gear system on the projectile head has little impact on its stability. However, for individual antiarmor rockets, the tail wing is generally used to provide stability torque to the missile body. Installing the steering gear system at the head reduces the flight stability of the projectile [23–25], which readily causes the projectile to lose stability and overturn. Therefore, arranging the steering gear system at the tail improves the projectile stability. If the PGK system is transplanted to the tail of a subsonic single soldier rocket to give it the ability of two-dimensional trajectory correction, the angular motion and stability law of the projectile body will be different completely from that of a supersonic high rotation grenade. Research on the angular motion law and stability of projectiles is the basic concept for trajectory analysis, guidance control, and guidance law design. However, there have been no relevant reports on this type of trajectory-corrected projectile in recent years.This study analyzes the angular motion and stability of an isolated rotating rudder trajectory-corrected rocket projectile. The characteristic for this type of trajectory-corrected rocket projectile is that the actuator to generate the correction force is at the tail of the projectile and can rotate along its axis. The principle of the actuator is similar to the PGK, but it can provide corrective and stability torques. First, the guidance and control principle of the isolated rotating tail rudder trajectory correction rocket projectile is clarified. Second, the angular motion of the projectile on the complex plane is defined, and the angular motion differential equation is derived based on the rigid body trajectory equation. The angular motion and stability characteristics of the projectile under different tail speeds are studied for the uncorrected and corrected flight states. Finally, the conclusions are verified through an example.


## 2. Periodic Control Force Principle of Isolated Rotating Tail Rudder Rocket

Research into the development of isolated rotating tail rudder rockets helps compensate for the shortcomings of unguided, man-portable, antiarmor weapons due to their low precision and short range while keeping costs low. The caliber of the considered rocket is approximately 80 mm and is used primarily to strike fixed or low-speed moving targets within 2000 m. The overall layout of the rocket is shown in Figure1(a) [26]. The device is composed of a seeker, projectile body, and tail segment. The tail segment is driven by a motor and can rotate relative to the projectile body around its longitudinal axis. Four wings are fixed on the tail segment, as shown in Figure 1(b), and one pair of wings has a zero-degree deflection angle relative to the projectile body. These are the stabilizing rudders and provide a stable moment while the rocket is flying. The other pair of wings has a deflection angle of 5° relative to the projectile body and is the correction rudders. In Figure 1(b), the ω represents the speed of the tail segment, γ represents the angular displacement of the tail segment, and Fw represents the direction of the correction force generated by the correction rudders.Figure 1
Schematic diagram of the overall layout and periodic control force of the isolated rotating tail rudder rocket. (a) Overall layout of the isolated rotating tail rudder rocket; (b) principle of the periodic control force.
(a)(b)The correction rudders provide the radial aerodynamic force to the projectile body by rotating around its longitudinal axis with the tail segment. If the tail segment is fixed at a specific phase, the correction rudders provide continuous aerodynamic force in a specific direction to control the projectile body and rotate around its center of mass. This kind of rudder uses period-averaged forces to control the rocket’s position and attitude. When the rudder rotates at a constant speed relative to the ground, the average radial aerodynamic force generated by the entire wing is zero over a rotation period. This simplified assumption is generally only applicable to the case of small angles of attack and rotation speeds. For high angles of attack and rotation speeds, the rudder flow field is more complex, and this law is no longer applicable.


## 3. Dynamic Modelling of Rocket Projectile with Isolated Rotating Tail Rudder

The coordinate systems defined in this study are shown in Figures2(a) and 2(b). The coordinate system oExEyEzE is for the ground and can be regarded as the inertial system. The position of the origin oE is at the launch port, the oExE axis points to the firing direction along the horizon, the oEyE axis points upward along the vertical direction, and the oEzE axis is determined based on the right-hand rule. The ground coordinate system oExEyEzE is used to determine the spatial position of the projectile centroid. The coordinate system oxNyNzN is the reference coordinate system. The origin is at the projectile mass center and is a dynamic coordinate system that moves with the projectile. Each coordinate axis is obtained by translating the ground coordinate system oExEyEzE. The reference coordinate system oxNyNzN is used to determine the centroid velocity and azimuth of the projectile axis. The coordinate system ox1y1z1 is the trajectory coordinate system with its origin located at the projectile mass center. The ox1 axis points along the velocity direction of the mass center, the oy1 axis is vertical to the ox1 axis, and the oz1 axis is determined based on the right-hand rule. The trajectory coordinate system ox1y1z1 is determined by rotating the reference coordinate system oxNyNzN twice. First, the θa angle is rotated in a positive direction around the ozN axis to form the coordinate system ox2′y2zN. Second, the θb angle is rotated in a negative direction around the oy1 axis to form the coordinate system ox1y1z1. The θa is the height angle, and θb is the deflection angle of the trajectory. The ox2y2z2 is the projectile axis coordinate system with an origin located at the projectile mass center. The ox2 axis points along the forward direction of the projectile axis, the oy2 axis is vertical to the ox2 axis, and the oz2 axis is determined based on the right-hand rule. The projectile axis coordinate system ox2y2z2 can be determined by rotating the reference coordinate system oxNyNzN twice. First, the φa angle is rotated in a positive direction around the ozN axis to form the coordinate system ox2′y2zN. Second, the φb angle is rotated in a negative direction around the oy2 axis to form the coordinate system ox2y2z2. The φa is the height angle, and φb is the deflection angle of the projectile axis.Figure 2
Definition and rotation relationship for each coordinate system. (a) The ground coordinate system, reference coordinate system, and trajectory coordinate system; (b) the trajectory coordinate system and projectile axis coordinate system.
(a)(b)The translational differential equations for the center of mass of the rocket projectile [14, 27] are as follows:(1)v˙=Fxm,θ˙a=Fymvcosθb,θ˙b=Fzmv,(2)x˙=vcosθbcosθa,y˙=vcosθbsinθa,z˙=vsinθb,where v, θa, and θb represent the projectile centroid velocity, velocity elevation angle, and velocity deflection angle, respectively; x, y, and z represent the centroid displacements; m=m1+m2 represents the total projectile mass with m1 and m2 being the mass of the body and tail, respectively; and Fx, Fy, and Fz represent the force components of the projectile in the three axes of the trajectory coordinate system.The differential equations for the rocket projectile rotation around the center of mass [28–30] are as follows:(3)ω˙ξ1=Mξ1+Mξ21C1,ω˙ξ2=Mξ2+Mξ12C2,ω˙η=MηA−C1ωξ1+C2ωξ2ωζA+ωζ2tanφb,ω˙ζ=MζA+C1ωξ1+C2ωξ2ωηA−ωηωζtanφb,(4)γ˙1=ωξ1−ωζtanφb,γ˙2=ωξ2−ωζtanφb,φ˙a=ωζcosφb,φ˙b=−ωη,where γ1, γ2, φa, and φb represent the axial angular displacement of the projectile body, axial angular displacement of the tail, height angle of the projectile axis, and deflection angle of the projectile axis, respectively; ωξ1, ωξ2, ωη, and ωζ represent the components of the axial angular velocity of the projectile body, axial angular velocity of the tail, angular velocity of the projectile axis height angle, and angular velocity of the deflection angle of the projectile axis in the projectile axis coordinate system, respectively; Mξ1, Mξ2, Mξ21, and Mξ12 represent the components of the aerodynamic moment for the projectile body, aerodynamic moment of the tail, moment of the tail acting on the projectile body, and moment of the projectile body acting on the tail in the ox2 direction of the projectile axis coordinate system. The Mη and Mζ are the components of the aerodynamic moment acting on the projectile body in the oy2 and ozN directions of the projectile axis coordinate system, respectively.


## 4. Angular Motion Modeling of Projectile Body

### 4.1. Geometric Description of Angular Motion of Projectile Body

It is necessary to analyze and solve for the angular motion of the projectile to study the attitude motion of the projectile in flight and analyze its flight stability. The flight stability of the projectile is usually represented by variations in the angle of attackδ of the projectile with time t. Under normal conditions, the trajectory of the projectile varies around the ideal trajectory. Taking the velocity line of the ideal trajectory as the benchmark, the pitching and yaw motions of the projectile around this benchmark are the angular motion of the projectile.The unit sphere description of the projectile angular motion is shown in Figure3. The spherical surface is a unit sphere with the center of mass c. The unit vector r⟶0 has the same direction as the ideal trajectory velocity, and its intersection with the unit sphere is o. The plane cox is the horizontal plane of the trajectory coordinate system, and coy is the vertical plane. The unit vector r⟶1 has the same velocity direction as the projectile, and its intersection with the unit sphere is T. The unit vector r⟶2 has the same direction as the projectile axis, and the intersection with the unit sphere is B. The small range sphere over which the coordinate system xoy is located can be expanded into the plane as shown in Figure 4. If the x-axis is regarded as the imaginary axis and the y-axis as the real axis, the vectors oT⟶, oB⟶, and TB⟶ can be represented by the complex numbers ΨΨ=θ1+iθ2, ΦΦ=φ1+iφ2, and ΔΔ=δ1+iδ2, respectively, which satisfies the relationship Δ=Φ−Ψ. The δ1 represents the height angle of attack, and δ2 represents the lateral angle of attack.Figure 3
Unit sphere description for the angular motion.Figure 4
Complex plane description for the angular motion.


### 4.2. Differential Equation of Projectile Body Angular Motion

The following symbols are introduced to simplify the expressions of the forces and moments:(5)bx=ρScx2m,bz=ρSdcz″2m,by=ρScy′2m,bz=ρSdcz″2m,kz=ρSlmz′2A,kzz=ρSl2mzz′2A,kxz1=ρSldmxz1′2C1,kxz2=ρSldmxz2′2C2,ky=ρSldmy″2A,where cx, cy, cz, mz, mzz, mxz1, mxz2, and my represent the drag coefficient, lift coefficient, Magnus force coefficient, static moment coefficient, equatorial damping moment coefficient, projectile body pole damping moment coefficient, tail pole damping moment coefficient, and tail Magnus moment coefficient, respectively, and ρ, S, l, and d represent the air density, characteristic area, reference length, and projectile diameter, respectively. For small caliber rocket projectiles, the order of magnitude for each parameter at small angles of attack and subsonic velocities is shown in Table 1 [26].Table 1
Order of magnitudes for each parameter.
ParametersOrder of magnitudebx10−4by10−3bz10−7kz10−2kzz10−2kxz110−4kxz210−4ky10−3The forces and moments on the projectile are as follows:(6)Fx=Fpcosδ2cosδ1−mv2bx−mgsinθacosθb,Fy=Fpcosδ2sinδ1+mv2byδrcosδ2sinδ1sinδr−δNcosγ2+mvbzγ˙2δrsinδ2sinδr−mgcosθa,Fz=Fpsinδ2+mv2byδrsinδ2sinδr−δNsinγ2−mvbzγ˙2δrcosδ2sinδ1sinδr+mgsinθasinθb,Mξ1=−C1kxz1vωξ1,Mξ2=−C1kxz2vωξ2,Mξ12=−Mξ21=Mq,Mη=−Akzv2δrsinδ2cosδ1sinδr+Akzzvφ˙b+Akyvγ˙2δ2sinδ1sinδr+Akzv2δMsinγ2,Mζ=Akzv2δrsinδ1sinδr−Akzzvφ˙acosφb+Akyvγ˙2δ1sinδ2cosδ1sinδr−Akzv2δMcosγ2,where Fp, Mq, δ1, δ2, δr, δN, and δM represent the rocket engine thrust, tail motor torque, height angle of attack, lateral angle of attack, relative angle of attack, aerodynamic eccentricity angle of additional force, and aerodynamic eccentricity angle of additional moment, respectively. To facilitate the study of the angle of attack equation of the projectile rocket, the following ideal trajectory equation is introduced:(7)v˙i=ap−vi2bx−gsinθ,θ˙=−gcosθvi,x˙i=vicosθ,y˙i=visinθ,where vi, θ, xi, yi, and ap represent the center-of-mass velocity in the ideal trajectory, velocity angle, x-direction displacement, y-direction displacement, and acceleration generated by the rocket thrust, respectively. The complex plane description of the angular motion defined in Figure 4 indicates that the following equation holds: θa=θ+θ1, θb=θ2, φa=θ+φ1, and φb=φ2. According to the second and third equations in equation (1) and the second equation in equation (7), and considering that θ2, θ1, φ2, φ1, δ1, and δ2 are small quantities, the equation that describes changes in the velocity direction can be written as follows:(8)θ˙+θ˙1=apδ1v+byvδ1−byvδNcosγ2+bzγ˙2δ2−gcosθav,θ˙2=apδ2/v+byvδ2−byvδNsinγ2−bzγ˙2δ1+gθ2sinθa/v.Multiply the second equation in equation (8) by the imaginary unit i, and add the two equations in equation (8) to obtain:(9)Ψ˙=apv+byv−ibzγ˙2Δ+gsinθvΨ−byvδNeiγ2,where Ψ=θ1+iθ2 is the velocity complex deflection angle, and Δ=δ1+iδ2 is the complex angle of attack.Changes in the projectile axis direction are described by the third and fourth equations in equation (3). Using the relations ωζ≈φ˙a=θ˙+φ˙1, ωη=−φb=−φ2, ωξ1≈γ˙1, and ωξ1≈γ˙1 and omitting high-order small quantities allow simplification as follows:(10)−φ¨2+C1γ˙1+C2γ˙2θ˙+φ˙1A=−kzv2δ2+kzzvφ˙2+kyvγ˙2δ1+kzv2δMsinγ2,φ¨1+θ¨+C1γ˙1+C2γ˙2φ˙2A=kzv2δ1−kzzvθ˙+φ˙1+kyvγ˙2δ2−kzv2δMcosγ2.Multiply the first equation in equation (10) by −i and add the two equations to obtain:(11)Φ¨+kzzv−iC1γ˙1+C2γ˙2AΦ˙−kzv2−ikyvγ˙2Δ=iC1γ˙1+C2γ˙2Aθ˙−θ¨−kzzvθ˙−kzv2δMeiγ2,where Φ=φ1+iφ2 is the complex deflection angle of the spring axis. To eliminate the factor v before the state variable Ψ and Φ of equations (9) and (11), the independent variable is transformed from time t to arc length s, the symbol “′” is used to represent the derivative of the arc length, and a small amount of gsinθ/vΨ is omitted. Equations (9) and (11) are transformed into the following form:(12)Ψ′=apv2+by−ibzγ˙2vΔ−byδNeiγ2,(13)Φ″+v′v+kzz−iC1γ˙1+C2γ˙2AvΦ′−kz−ikyγ˙2vΔ=iC1γ˙1+C2γ˙2Av2−kzzvθ˙−θ¨v2−kzδMeiγ2.To derive the differential equation for the angle of attackΔ, the relation Φ′=Ψ′+Δ′ and using the first equation of (7) transforms Φ′ into:(14)Φ′=Δ′+apv2+by−ibzγ˙2vΔ−byδNeiγ2,=vΔ′v+bx+by+gsinθv2−ibzγ˙2vΔ−byδNeiγ2.Furthermore, taking the coefficient beforeΔ in equation (14) as a constant, Φ″ can be obtained as follows:(15)Φ″=vΔ″v−vΔ′v2v′+bx+by+gsinθv2−ibzγ˙2vvΔ′v−v′vΔ−ibyδNγ˙2veiγ2,=vΔ″v+bx+by+gsinθv2−ibzγ˙2v−v′vvΔ′v−bx+by+gsinθv2−ibzγ˙2vv′vΔ−ibyδNγ˙2veiγ2.Substituting equations (14) and (15) into equation (13) and replacing the state variable Δ with W=vΔ, the following differential equation for the attack angle can be derived:(16)W″+H−iPW′−M+iPTW=E,where the symbol “′” represents the derivative of the arc length s, and the other symbols are as follows:(17)H=bx+by+kzz+gsinθv2,P=C1γ˙1+C2γ˙2Av,M=kz,T=bx+by+gsinθv2−AkyC,E=EG+EF,EG=iP−kzzθ˙−θ¨v,EF=kzz+v˙/v2+iγ˙2v−iPbyvδNeiγ2−kzvδMeiγ2,where E at the right end of the complex angle of attack in equation (16) is the disturbance term, which can be divided into the gravity disturbance EG and wing aerodynamic disturbance EF. The EG mostly contains the θ term, which is caused by changes in gravity, and EF primarily contains the tail-spin angle term eiγ2.


## 4.1. Geometric Description of Angular Motion of Projectile Body

It is necessary to analyze and solve for the angular motion of the projectile to study the attitude motion of the projectile in flight and analyze its flight stability. The flight stability of the projectile is usually represented by variations in the angle of attackδ of the projectile with time t. Under normal conditions, the trajectory of the projectile varies around the ideal trajectory. Taking the velocity line of the ideal trajectory as the benchmark, the pitching and yaw motions of the projectile around this benchmark are the angular motion of the projectile.The unit sphere description of the projectile angular motion is shown in Figure3. The spherical surface is a unit sphere with the center of mass c. The unit vector r⟶0 has the same direction as the ideal trajectory velocity, and its intersection with the unit sphere is o. The plane cox is the horizontal plane of the trajectory coordinate system, and coy is the vertical plane. The unit vector r⟶1 has the same velocity direction as the projectile, and its intersection with the unit sphere is T. The unit vector r⟶2 has the same direction as the projectile axis, and the intersection with the unit sphere is B. The small range sphere over which the coordinate system xoy is located can be expanded into the plane as shown in Figure 4. If the x-axis is regarded as the imaginary axis and the y-axis as the real axis, the vectors oT⟶, oB⟶, and TB⟶ can be represented by the complex numbers ΨΨ=θ1+iθ2, ΦΦ=φ1+iφ2, and ΔΔ=δ1+iδ2, respectively, which satisfies the relationship Δ=Φ−Ψ. The δ1 represents the height angle of attack, and δ2 represents the lateral angle of attack.Figure 3
Unit sphere description for the angular motion.Figure 4
Complex plane description for the angular motion.


## 4.2. Differential Equation of Projectile Body Angular Motion

The following symbols are introduced to simplify the expressions of the forces and moments:(5)bx=ρScx2m,bz=ρSdcz″2m,by=ρScy′2m,bz=ρSdcz″2m,kz=ρSlmz′2A,kzz=ρSl2mzz′2A,kxz1=ρSldmxz1′2C1,kxz2=ρSldmxz2′2C2,ky=ρSldmy″2A,where cx, cy, cz, mz, mzz, mxz1, mxz2, and my represent the drag coefficient, lift coefficient, Magnus force coefficient, static moment coefficient, equatorial damping moment coefficient, projectile body pole damping moment coefficient, tail pole damping moment coefficient, and tail Magnus moment coefficient, respectively, and ρ, S, l, and d represent the air density, characteristic area, reference length, and projectile diameter, respectively. For small caliber rocket projectiles, the order of magnitude for each parameter at small angles of attack and subsonic velocities is shown in Table 1 [26].Table 1
Order of magnitudes for each parameter.
ParametersOrder of magnitudebx10−4by10−3bz10−7kz10−2kzz10−2kxz110−4kxz210−4ky10−3The forces and moments on the projectile are as follows:(6)Fx=Fpcosδ2cosδ1−mv2bx−mgsinθacosθb,Fy=Fpcosδ2sinδ1+mv2byδrcosδ2sinδ1sinδr−δNcosγ2+mvbzγ˙2δrsinδ2sinδr−mgcosθa,Fz=Fpsinδ2+mv2byδrsinδ2sinδr−δNsinγ2−mvbzγ˙2δrcosδ2sinδ1sinδr+mgsinθasinθb,Mξ1=−C1kxz1vωξ1,Mξ2=−C1kxz2vωξ2,Mξ12=−Mξ21=Mq,Mη=−Akzv2δrsinδ2cosδ1sinδr+Akzzvφ˙b+Akyvγ˙2δ2sinδ1sinδr+Akzv2δMsinγ2,Mζ=Akzv2δrsinδ1sinδr−Akzzvφ˙acosφb+Akyvγ˙2δ1sinδ2cosδ1sinδr−Akzv2δMcosγ2,where Fp, Mq, δ1, δ2, δr, δN, and δM represent the rocket engine thrust, tail motor torque, height angle of attack, lateral angle of attack, relative angle of attack, aerodynamic eccentricity angle of additional force, and aerodynamic eccentricity angle of additional moment, respectively. To facilitate the study of the angle of attack equation of the projectile rocket, the following ideal trajectory equation is introduced:(7)v˙i=ap−vi2bx−gsinθ,θ˙=−gcosθvi,x˙i=vicosθ,y˙i=visinθ,where vi, θ, xi, yi, and ap represent the center-of-mass velocity in the ideal trajectory, velocity angle, x-direction displacement, y-direction displacement, and acceleration generated by the rocket thrust, respectively. The complex plane description of the angular motion defined in Figure 4 indicates that the following equation holds: θa=θ+θ1, θb=θ2, φa=θ+φ1, and φb=φ2. According to the second and third equations in equation (1) and the second equation in equation (7), and considering that θ2, θ1, φ2, φ1, δ1, and δ2 are small quantities, the equation that describes changes in the velocity direction can be written as follows:(8)θ˙+θ˙1=apδ1v+byvδ1−byvδNcosγ2+bzγ˙2δ2−gcosθav,θ˙2=apδ2/v+byvδ2−byvδNsinγ2−bzγ˙2δ1+gθ2sinθa/v.Multiply the second equation in equation (8) by the imaginary unit i, and add the two equations in equation (8) to obtain:(9)Ψ˙=apv+byv−ibzγ˙2Δ+gsinθvΨ−byvδNeiγ2,where Ψ=θ1+iθ2 is the velocity complex deflection angle, and Δ=δ1+iδ2 is the complex angle of attack.Changes in the projectile axis direction are described by the third and fourth equations in equation (3). Using the relations ωζ≈φ˙a=θ˙+φ˙1, ωη=−φb=−φ2, ωξ1≈γ˙1, and ωξ1≈γ˙1 and omitting high-order small quantities allow simplification as follows:(10)−φ¨2+C1γ˙1+C2γ˙2θ˙+φ˙1A=−kzv2δ2+kzzvφ˙2+kyvγ˙2δ1+kzv2δMsinγ2,φ¨1+θ¨+C1γ˙1+C2γ˙2φ˙2A=kzv2δ1−kzzvθ˙+φ˙1+kyvγ˙2δ2−kzv2δMcosγ2.Multiply the first equation in equation (10) by −i and add the two equations to obtain:(11)Φ¨+kzzv−iC1γ˙1+C2γ˙2AΦ˙−kzv2−ikyvγ˙2Δ=iC1γ˙1+C2γ˙2Aθ˙−θ¨−kzzvθ˙−kzv2δMeiγ2,where Φ=φ1+iφ2 is the complex deflection angle of the spring axis. To eliminate the factor v before the state variable Ψ and Φ of equations (9) and (11), the independent variable is transformed from time t to arc length s, the symbol “′” is used to represent the derivative of the arc length, and a small amount of gsinθ/vΨ is omitted. Equations (9) and (11) are transformed into the following form:(12)Ψ′=apv2+by−ibzγ˙2vΔ−byδNeiγ2,(13)Φ″+v′v+kzz−iC1γ˙1+C2γ˙2AvΦ′−kz−ikyγ˙2vΔ=iC1γ˙1+C2γ˙2Av2−kzzvθ˙−θ¨v2−kzδMeiγ2.To derive the differential equation for the angle of attackΔ, the relation Φ′=Ψ′+Δ′ and using the first equation of (7) transforms Φ′ into:(14)Φ′=Δ′+apv2+by−ibzγ˙2vΔ−byδNeiγ2,=vΔ′v+bx+by+gsinθv2−ibzγ˙2vΔ−byδNeiγ2.Furthermore, taking the coefficient beforeΔ in equation (14) as a constant, Φ″ can be obtained as follows:(15)Φ″=vΔ″v−vΔ′v2v′+bx+by+gsinθv2−ibzγ˙2vvΔ′v−v′vΔ−ibyδNγ˙2veiγ2,=vΔ″v+bx+by+gsinθv2−ibzγ˙2v−v′vvΔ′v−bx+by+gsinθv2−ibzγ˙2vv′vΔ−ibyδNγ˙2veiγ2.Substituting equations (14) and (15) into equation (13) and replacing the state variable Δ with W=vΔ, the following differential equation for the attack angle can be derived:(16)W″+H−iPW′−M+iPTW=E,where the symbol “′” represents the derivative of the arc length s, and the other symbols are as follows:(17)H=bx+by+kzz+gsinθv2,P=C1γ˙1+C2γ˙2Av,M=kz,T=bx+by+gsinθv2−AkyC,E=EG+EF,EG=iP−kzzθ˙−θ¨v,EF=kzz+v˙/v2+iγ˙2v−iPbyvδNeiγ2−kzvδMeiγ2,where E at the right end of the complex angle of attack in equation (16) is the disturbance term, which can be divided into the gravity disturbance EG and wing aerodynamic disturbance EF. The EG mostly contains the θ term, which is caused by changes in gravity, and EF primarily contains the tail-spin angle term eiγ2.


## 5. Angular Motion Characteristics of Projectile under Uncorrected Trajectory

The lateral aerodynamic force generated from the fixed wings in the uncorrected trajectory state should be zero in a rotation cycle. Thus, the rotation speed of the tail wing around the projectile axis relative to the ground coordinate system should be a nonzero fixed value. The angular motion characteristics of the projectile in this state can be analyzed from the solution of equation (16), which contains the angular motion characteristics of the projectile. These are affected by three factors: angular motion caused by initial conditions, angular motion caused by gravity, and angular motion caused by asymmetric factors. Therefore, its solution can be decomposed into(18)W=W0+WG+WF,where W0 is the homogeneous solution of equation (16) and represents the angular motion caused by the initial conditions. The form of the solution is W0=C1el1s+C2el2s, where C1 and C2 are undetermined coefficients determined by the initial conditions. The WG is the angular motion due to the nonhomogeneous term EG, and WF is the angular motion due to the nonhomogeneous term EF.


### 5.1. Angular Motion Characteristics Caused by Initial Conditions

Both the initial angular displacement and initial angular velocity of the projectile can affect its angular motion response. Without considering the disturbanceE, the angular motion caused by the initial conditions is W1. Based on the coefficient freezing method, let l1=α1+iβ1 and l2=α2+iβ2 be the two characteristic roots of the following homogeneous equation:(19)W″+H−iPW′−M+iPTW=0.Then,W1 can be written as follows:(20)W1=K1eiβ10+β1s+K2eiβ20+β2s,where K1 and K2 are as follows:(21)K1=K10eα1s,K2=K20eα2s.TheK10, K20, β10, and β20 are determined by the initial conditions W10=v0δ0eiσ0 and W10′=v˙0/v0δ0+δ˙0eiσ0+δ0σ˙0eiσ0+0.5π. Then,(22)K10eiβ10=v˙0v0δ0+δ˙0+iδ0σ˙0−v0δ0l2eiσ0l1−l2,K20eiβ20=v˙0v0δ0+δ˙0+iδ0σ˙0−v0δ0l1eiσ0l2−l1.The angle of attackΔ1 is the angular motion caused by the initial disturbance and is expressed as follows:(23)Δ1=K1veiβ10+β1s+K2veiβ20+β2s,where β1 and β2 are the angular frequencies for one and two circular motions, and α1 and α2 in K1 and K2 represent changes in the circular motion amplitudes. If the circular motion amplitude is gradually reduced, the derivative of K1,2/v must be negative. That is,(24)α1,2−v˙v2<0,where α1,2 is the real part of the characteristic root, which is obtained from the complex square formula:(25)α1,2=−H2±24P2−4M−H22+4H−2T2P2+4M+H2−P2.Equation (24) is the full trajectory dynamic stability condition that the tail rotation speed should satisfy. For the active phase of the rocket, v˙/v2>0, equation (24) is more easily satisfied; therefore, it is sufficient to consider only the stability of the passive segment. For the passive phase of the rocket, v˙/v2≈−ρScx/2m and α1,2 monotonically increase with P; therefore, the left-hand side of equation (24) takes the maximum value at the minimum velocity. In summary, the stability of the entire trajectory under the initial disturbance can be guaranteed only by satisfying equation (26) at the minimum velocity v. Substituting equation (25) into equation (24), the tail rotation speed γ˙2 should satisfy(26)γ˙22<γ˙02,where γ˙0 is the critical speed. For the convenience of calculations, omitting the small passive section of v˙/v2 allows γ˙02 to satisfy the following condition:(27)γ˙02≈−MAvH/C22TT−H.


### 5.2. Angular Motion Characteristics Caused by Gravity

As the normal component of gravity acting on the projectile causes the velocity of the mass center to rotate downward, the angular displacement of the velocity vector for the mass center will cause changes in the angle of attack, which impacts the angular motion characteristics of the projectile. The angular motion equation caused by gravity is as follows:(28)W″+H−iPW′−M+iPTW=EG.The form of the solution is the superposition of the general solution for the homogeneous equation and the special solution of the nonhomogeneous equation. Namely,(29)WG′=W2+WG,where W2 is the homogeneous solution of equation (28) that considers the EG case, which has the same form as W0 and W1, differing only in the coefficients. The constant variation method can be used to solve the special solution WG of the nonhomogeneous equation as follows:(30)WG=1M+iPTθ¨v−θ˙iP−kzz−H−iPM+iPT2θ¨viP−kzz.Then,(31)ΔG=WGv,=1M+iPTθ¨v2−θ˙viP−kzz−H−iPM+iPT2θ¨v2iP−kzz.


### 5.3. Angular Motion Characteristics Caused by Asymmetric Tail

The tail wing provides the control force and torque to change the projectile motion. The radial periodic force is generated during the tail wing rotation due to the same direction deflection angle of a pair of wings. This can cause the projectile to have the angular motion characteristics of dynamic imbalance. If the angular frequency of the dynamic imbalance is close to the angular frequency of one or two circle motions, the projectile may resonate, which makes it unstable. The angular motion equation caused by the asymmetric tail is as follows:(32)W″+H−iPW′−M+iPTW=EF.If the additional force eccentricity angleδN is equal to the additional moment eccentricity angle δM, the aerodynamic disturbance term EF of the wing can be expressed as follows:(33)EF=Beiβ3s,where B=kzz+v˙/v2+iγ˙2/v−iPbyvδN−kzvδM, and β3=γ˙2/v. According to the coefficient freezing method, the solution of equation (32) can be expressed as follows:(34)WF′=C1el1s+C2el2s+C3eiβ3s,where(35)C1=C3l2−iβ3l1−l2,C2=C3l1−iβ3l2−l1,C3=Bl1−iβ3l2−iβ3.Then, the angle of attack generated by the asymmetric tail is as follows:(36)ΔF′=C3v1l1−l2l2−iω3el1s−l1−iω3el2s+eiβ3s.There are three circular motions. Namely, circular motion with the angular frequencies ofβ1 and β2 generated by the initial disturbance and the forced circular motion with an angular frequency of β3. If the damping factors α1 and α2 of the passive section satisfy the stability conditions, the amplitudes of the first two circular motions tend to 0 with an increased arc length s, and only the forced circular motion with an angular frequency of β3 remains. The amplitude of the forced circle motion is as follows:(37)Am=C3v=B/vβ3−β12+α12β3−β22+α22.


## 5.1. Angular Motion Characteristics Caused by Initial Conditions

Both the initial angular displacement and initial angular velocity of the projectile can affect its angular motion response. Without considering the disturbanceE, the angular motion caused by the initial conditions is W1. Based on the coefficient freezing method, let l1=α1+iβ1 and l2=α2+iβ2 be the two characteristic roots of the following homogeneous equation:(19)W″+H−iPW′−M+iPTW=0.Then,W1 can be written as follows:(20)W1=K1eiβ10+β1s+K2eiβ20+β2s,where K1 and K2 are as follows:(21)K1=K10eα1s,K2=K20eα2s.TheK10, K20, β10, and β20 are determined by the initial conditions W10=v0δ0eiσ0 and W10′=v˙0/v0δ0+δ˙0eiσ0+δ0σ˙0eiσ0+0.5π. Then,(22)K10eiβ10=v˙0v0δ0+δ˙0+iδ0σ˙0−v0δ0l2eiσ0l1−l2,K20eiβ20=v˙0v0δ0+δ˙0+iδ0σ˙0−v0δ0l1eiσ0l2−l1.The angle of attackΔ1 is the angular motion caused by the initial disturbance and is expressed as follows:(23)Δ1=K1veiβ10+β1s+K2veiβ20+β2s,where β1 and β2 are the angular frequencies for one and two circular motions, and α1 and α2 in K1 and K2 represent changes in the circular motion amplitudes. If the circular motion amplitude is gradually reduced, the derivative of K1,2/v must be negative. That is,(24)α1,2−v˙v2<0,where α1,2 is the real part of the characteristic root, which is obtained from the complex square formula:(25)α1,2=−H2±24P2−4M−H22+4H−2T2P2+4M+H2−P2.Equation (24) is the full trajectory dynamic stability condition that the tail rotation speed should satisfy. For the active phase of the rocket, v˙/v2>0, equation (24) is more easily satisfied; therefore, it is sufficient to consider only the stability of the passive segment. For the passive phase of the rocket, v˙/v2≈−ρScx/2m and α1,2 monotonically increase with P; therefore, the left-hand side of equation (24) takes the maximum value at the minimum velocity. In summary, the stability of the entire trajectory under the initial disturbance can be guaranteed only by satisfying equation (26) at the minimum velocity v. Substituting equation (25) into equation (24), the tail rotation speed γ˙2 should satisfy(26)γ˙22<γ˙02,where γ˙0 is the critical speed. For the convenience of calculations, omitting the small passive section of v˙/v2 allows γ˙02 to satisfy the following condition:(27)γ˙02≈−MAvH/C22TT−H.


## 5.2. Angular Motion Characteristics Caused by Gravity

As the normal component of gravity acting on the projectile causes the velocity of the mass center to rotate downward, the angular displacement of the velocity vector for the mass center will cause changes in the angle of attack, which impacts the angular motion characteristics of the projectile. The angular motion equation caused by gravity is as follows:(28)W″+H−iPW′−M+iPTW=EG.The form of the solution is the superposition of the general solution for the homogeneous equation and the special solution of the nonhomogeneous equation. Namely,(29)WG′=W2+WG,where W2 is the homogeneous solution of equation (28) that considers the EG case, which has the same form as W0 and W1, differing only in the coefficients. The constant variation method can be used to solve the special solution WG of the nonhomogeneous equation as follows:(30)WG=1M+iPTθ¨v−θ˙iP−kzz−H−iPM+iPT2θ¨viP−kzz.Then,(31)ΔG=WGv,=1M+iPTθ¨v2−θ˙viP−kzz−H−iPM+iPT2θ¨v2iP−kzz.


## 5.3. Angular Motion Characteristics Caused by Asymmetric Tail

The tail wing provides the control force and torque to change the projectile motion. The radial periodic force is generated during the tail wing rotation due to the same direction deflection angle of a pair of wings. This can cause the projectile to have the angular motion characteristics of dynamic imbalance. If the angular frequency of the dynamic imbalance is close to the angular frequency of one or two circle motions, the projectile may resonate, which makes it unstable. The angular motion equation caused by the asymmetric tail is as follows:(32)W″+H−iPW′−M+iPTW=EF.If the additional force eccentricity angleδN is equal to the additional moment eccentricity angle δM, the aerodynamic disturbance term EF of the wing can be expressed as follows:(33)EF=Beiβ3s,where B=kzz+v˙/v2+iγ˙2/v−iPbyvδN−kzvδM, and β3=γ˙2/v. According to the coefficient freezing method, the solution of equation (32) can be expressed as follows:(34)WF′=C1el1s+C2el2s+C3eiβ3s,where(35)C1=C3l2−iβ3l1−l2,C2=C3l1−iβ3l2−l1,C3=Bl1−iβ3l2−iβ3.Then, the angle of attack generated by the asymmetric tail is as follows:(36)ΔF′=C3v1l1−l2l2−iω3el1s−l1−iω3el2s+eiβ3s.There are three circular motions. Namely, circular motion with the angular frequencies ofβ1 and β2 generated by the initial disturbance and the forced circular motion with an angular frequency of β3. If the damping factors α1 and α2 of the passive section satisfy the stability conditions, the amplitudes of the first two circular motions tend to 0 with an increased arc length s, and only the forced circular motion with an angular frequency of β3 remains. The amplitude of the forced circle motion is as follows:(37)Am=C3v=B/vβ3−β12+α12β3−β22+α22.


## 6. Angular Motion Characteristics of Projectile in Trajectory Correction State

When the guidance system generates a deviation signal to correct the projectile attitude, the tail motor outputs a specific rotation speed to make the tail axially stable relative to the ground coordinate system. The direction of the additional torque generated by the aerodynamic eccentricity of the tail is fixed at the required phase, which changes the attitude of the projectile. Then, the tail rotation speed changes from constant toγ˙2=0. For convenience, the transition process of the tail rotation speed is omitted, and the response of the angle of attack under the step excitation of the tail rotation speed is studied.


### 6.1. Steady and Transient Solutions for Angle of Attack under Trajectory Correction

As the tail rotation speed isγ˙2=0 and the projectile body rotation speed is γ˙1≈0, the small quantity P in equation (16) can be omitted, and the projectile angle of attack satisfies the following equation:(38)W″+HW′−MW=B1eiγ2,where(39)B1=kzz+v˙v2byvδN−kzvδM.The rotation speed step excitation response of the system is equivalent to the solution of equation (38) when γ2 is equal to the fixed value γ20 under the initial conditions Wt0=W0 and W′t0=W0′. The initial conditions are determined from the uncontrolled flight terminal state. According to the coefficient freezing method, the form of the homogeneous solution of equation (38) is the same as that of equation (20), and the nonhomogeneous solution W∗ is as follows:(40)W∗=−B1Meiγ20.Then, the solution of equation (38) is as follows:(41)WC=C1∗el1∗s+C2∗el2∗s+W∗,where l1∗=−0.5H+i−M−0.5H2, l2∗=−0.5H−i−M−0.5H2, C1∗=W0′+W∗−W0l2/l1−l2, and C2∗=W0′+W∗−W0l1/l2−l1. If the condition M<0 is satisfied, the system will be stable, and the first two terms of equation (41) will decrease to 0 with an increased s. Then, W∗ is the steady-state solution of the system, and the corresponding steady-state angle of attack Δ∗ is as follows:(42)Δ∗=−B1Mveiγ20.


### 6.2. Effect of Trajectory Correction on Velocity Deflection Angle of Projectile

In the trajectory correction state, the tail wing of the projectile generates a control force along a fixed direction, which changes the projectile angle of attack. This generates a lift in the angle of attack plane and changes the direction of the velocity for the projectile mass center. The steady-state angle of attackΔ∗ is substituted into equation (9), and the independent variable is changed from time t to arc length s. Considering the trajectory passive section ap=0, the complex deflection angle differential equation for the projectile centroid velocity under the steady-state angle of attack is obtained as follows:(43)Ψ∗′=gsinθv2Ψ∗−byB1Mv+δNeiγ20.From equation (38), the following can be obtained:(44)B1Mv=kzz+v˙/v2bykzδN−δM≈−δM.Substituting equation (44) into equation (43) and ignoring a small amount of gsinθ/v2, the following is obtained:(45)Ψ∗′=δM−δNbyeiγ20.After linearization, the complex deflection angle of the projectile velocity at the steady angle of attack is as follows:(46)Ψ∗=δM−δNbyeiγ20s.


## 6.1. Steady and Transient Solutions for Angle of Attack under Trajectory Correction

As the tail rotation speed isγ˙2=0 and the projectile body rotation speed is γ˙1≈0, the small quantity P in equation (16) can be omitted, and the projectile angle of attack satisfies the following equation:(38)W″+HW′−MW=B1eiγ2,where(39)B1=kzz+v˙v2byvδN−kzvδM.The rotation speed step excitation response of the system is equivalent to the solution of equation (38) when γ2 is equal to the fixed value γ20 under the initial conditions Wt0=W0 and W′t0=W0′. The initial conditions are determined from the uncontrolled flight terminal state. According to the coefficient freezing method, the form of the homogeneous solution of equation (38) is the same as that of equation (20), and the nonhomogeneous solution W∗ is as follows:(40)W∗=−B1Meiγ20.Then, the solution of equation (38) is as follows:(41)WC=C1∗el1∗s+C2∗el2∗s+W∗,where l1∗=−0.5H+i−M−0.5H2, l2∗=−0.5H−i−M−0.5H2, C1∗=W0′+W∗−W0l2/l1−l2, and C2∗=W0′+W∗−W0l1/l2−l1. If the condition M<0 is satisfied, the system will be stable, and the first two terms of equation (41) will decrease to 0 with an increased s. Then, W∗ is the steady-state solution of the system, and the corresponding steady-state angle of attack Δ∗ is as follows:(42)Δ∗=−B1Mveiγ20.


## 6.2. Effect of Trajectory Correction on Velocity Deflection Angle of Projectile

In the trajectory correction state, the tail wing of the projectile generates a control force along a fixed direction, which changes the projectile angle of attack. This generates a lift in the angle of attack plane and changes the direction of the velocity for the projectile mass center. The steady-state angle of attackΔ∗ is substituted into equation (9), and the independent variable is changed from time t to arc length s. Considering the trajectory passive section ap=0, the complex deflection angle differential equation for the projectile centroid velocity under the steady-state angle of attack is obtained as follows:(43)Ψ∗′=gsinθv2Ψ∗−byB1Mv+δNeiγ20.From equation (38), the following can be obtained:(44)B1Mv=kzz+v˙/v2bykzδN−δM≈−δM.Substituting equation (44) into equation (43) and ignoring a small amount of gsinθ/v2, the following is obtained:(45)Ψ∗′=δM−δNbyeiγ20.After linearization, the complex deflection angle of the projectile velocity at the steady angle of attack is as follows:(46)Ψ∗=δM−δNbyeiγ20s.


## 7. Example and Stability Analysis

Numerical simulations are performed based on the known projectile body and aerodynamic parameters to verify the correctness of the angular motion characteristic analysis of the trajectory correction rocket projectile with an isolated rotating tail rudder as well as the stability analysis of a sample projectile. The relevant projectile body and aerodynamic parameters are shown in Table2 [26].Table 2
Projectile body and aerodynamic parameters.
ParametersValuesm (kg)6C1 (kg·m2)5.01e − 3bx (m−1)4.28e − 4bz (1)3.5e − 7kxz2 (m−1)2.5e − 4kz (m−2)3.47e − 2δN (rad)4.01e − 2A (kg·m2)2.48e − 1C2 (kg·m2)5.04e − 4by (m−1)8.94e − 3kxz1 (m−1)1.25e − 4ky (m−1)−9.01e − 3kzz (m−1)2.76e − 2δM (rad)8.11e − 2


### 7.1. Stability Analysis of Projectile under Magnus Moment

In the trajectory-corrected phase, the Magnus moment acting on the projectile body is generated due to the rotation of the tail wing relative to the inertial system as driven by the motor. The Magnus moment is positively correlated with the projectile velocity, tail rotation speed, and angle of attack. When other conditions are certain, a tail rotation speed that is too high will increase the Magnus moment to a degree that cannot be ignored. As the Magnus moment direction is roughly perpendicular to the projectile axis in the angle of attack plane, the moment will make the projectile move in a spiral around the speed line. If the damping moment received by the projectile cannot dissipate the oscillation energy of the projectile, the projectile axis oscillation will increase and result in projectile instability. The stability of the projectile under different tail wing speeds is shown in Figure5.Figure 5
Stability of the projectile under different tail speed conditions: (a)γ˙2<γ˙0; (b) γ˙2=γ˙0; (c) γ˙2>γ˙0.
(a)(b)(c)According to the characteristic equation of equation (19), the two characteristic frequencies β1 and β2 of the projectile are β1≈0.1856 and β2≈−0.1852, showing β1≈−β2 can be obtained. Therefore, the shape of the two characteristic frequencies is oval, as shown in Figure 4. According to equation (27), the critical rotation speed of the tail is γ˙0≈153.3rad/s. Figure 5 shows the movement of the projectile angle of attack under the initial conditions δ10=0.1rad, δ20=0, v0=280m/s, and θa0=0.07πrad. Without considering gravity and the correction force, the variable coefficient nonhomogeneous nonlinear equation (16) describing the complex angle of attack degenerates into the approximate second-order homogeneous linear expression of equation (19), which makes the analytic solution of equation (23) for the angle of attack very close to the numerical solution with a correlation coefficient of γ>0.98. Therefore, the analytic solution of equation (23) is considered accurate. Due to the influence of the Magnus moment, although the initial value of the directional angle of attack δ2 is 0, an increased arc length s causes the projectile axis to produce the phenomenon of a lateral reciprocating swing. As shown in Figure 5(a), when γ˙2<γ˙0, the Magnus moment is small, and the swing amplitude of the projectile body gradually decreases to 0. As shown in Figure 5(b), when γ˙2=γ˙0, the swing amplitude of the projectile eventually stabilizes at a certain angle. As shown in Figure 5(c), when γ˙2>γ˙0, the Magnus moment is large, and the swing amplitude of the projectile body first decreases before increasing, which eventually leads to instability of the projectile body.


### 7.2. Stability of Projectile Caused by Gravity

As the projectile is under the action of gravity, the height deflection angles of the projectile velocity bend downward, which results in a changed attack angle. Equation (29) describes this change, and the results are shown in Figures 6 and 7, and the initial conditions are δ10=δ20=0, v0=280m/s, and θa0=0.07πrad.Figure 6
Comparison of numerical and analytical solutions for the angular motion under the influence of gravity.Figure 7
Variations in the projectile height and lateral angle of attack under different tail rotation speeds. (a) Change of height angle of attackδ1 in steady state; (b) change of lateral angle of attack δ2 in steady state; (c) change of angle of attack δ1 and δ2 in unstable state.
(a)(b)(c)As shown in Figure6, at the initial stage of the trajectory, the projectile angle of attack swings around the velocity line. With an increased arc length s, if the projectile is stable, the swing amplitude of the angle of attack gradually decreases. The motion form in the swing stage of the angle of attack is derived from the homogeneous solution W2 in equation (28), and the motion form in the steady stage of the angle of attack is derived from the nonhomogeneous solution WG in equation (28). The movement trend for the angle of attack obtained by the numerical solution is the same as that obtained from the analytic solution, and only the homogeneous solution has slight errors with the numeric solution because of the approximate linearized results. Therefore, it is acceptable to use the analytic solution to analyze the angular motion of the projectile. As the homogeneous solution W2 in equation (28) is only different from W1 in its coefficients, its stability is the same as that without considering the influence of gravity. The effect of gravity is equivalent to applying the initial angular velocity to the height angle of attack, indicating the angle of attack produces a swing motion in the initial trajectory stage. Its motion mechanism is the same as that in Section 6.1. When the tail rotation speed is greater than the critical rotation speed, i.e., γ˙2>γ˙0, the angle of attack diverges, which results in projectile instability, as shown in Figure 7(c). Figure 7 shows changes in the projectile height angle of attack and lateral angle of attack under different tail rotation speeds when the projectile is stable. Changes in the amplitude of the height angle of attack δ1 are less affected by the tail rotation speed γ˙2, but changes in the lateral angle of attack δ2 amplitude increase with the tail rotation speed γ˙2, which is due to the influence of the increased lateral Magnus moment. It is noted that when the projectile is stable, the angular motion amplitude caused by gravity is small, which can be ignored when considering the effects of the correction force.


### 7.3. Resonance Stability of Projectile

When the projectile is in the uncorrected trajectory state, the periodic rotation correction force causes the projectile to exhibit forced vibrations. When the rotation frequency of the correction force is close to the natural frequency of the projectile swing, the projectile angle of attack amplitude response produces a large positive gain. If the projectile satisfies the stability condition in Subsection6.1, i.e., γ˙2<γ˙0, the projectile angle of attack amplitude can be calculated from equation (37), and the initial conditions are δ10=δ20=0, v0=280m/s, and θa0=0.07πrad as shown in Figure 8.Figure 8
Variations in the angle of attack amplitude with the tail rotation speed.When considering the rotation correction force that acts on the projectile, the projectile angular motion can be regarded as a damped forced vibration. According to equation (37), the amplitude of the angle of attack is determined by the damping factors α2 and the tail aerodynamic force B. As shown in Figure 8, when the tail rotation angle frequency β3 approaches the characteristic frequencies β1 and β2 of the projectile body, the projectile body angle of attack amplitude can exceed 0.4 rad. Although the angle of attack amplitude does not diverge, the aerodynamic force generated by its amplitude is in the nonlinear regime, which adversely impacts the projectile body control.The first method to avoid excessive angular amplitudeAm is to reduce the tail correction rudder deflection angle so that the tail aerodynamic force B is reduced. However, this reduces the correction capability of the projectile. The second method is to avoid the rotation speed of the tail to operate near the characteristic frequencies β1 and β2 of the projectile body. As the projectile needs to change between zero and nonzero in the corrected trajectory stage, if the rotation speed of the tail is too high in the uncorrected trajectory stage, the rotation speed of the tail must pass through the characteristic frequency of the projectile body when switching to the corrected state. Considering the critical rotation speed γ˙0>vβ1,2 of the tail obtained in Section 6.1, the optimal rotation speed range of the tail wing in the uncorrected phase of the projectile trajectory is 0.8vβ2<γ˙2<0.8vβ1.


### 7.4. Angular Motion of Projectile in Corrected Trajectory State

When the projectile is in the corrected trajectory state, the tail rotation speed isγ˙2=0. At this time, the Magnus force and moment can be ignored. Thus, the damping factor α1,2<0 can be achieved only if the static stability moment coefficient satisfies mz<0. At this time, the projectile is in the stable state. The analytic solution of the angular motion of the projectile in the corrected trajectory state is given by equation (41). Therein, the steering gear control system can stabilize the phase angle of the correction rudder at the γ20 position. Figure 9 shows the angular motion curves for the projectile body in the process of changing the corrected trajectory state after flying for 3 s under the uncorrected trajectory. Different colors represent the angular motion of the projectile when the correction rudder is stable at different phase angles γ20. The initial conditions for the simulations are v0=280m/s, δ10=δ20=0, and θa0=0.07πrad. The tail rotation speed in the uncorrected trajectory state is γ˙2=20rad/s.Figure 9
Angular motion of projectile when changing from the uncorrected trajectory state to the corrected trajectory state under different phase anglesγ20.Figure9 shows that in the first three seconds, the projectile axis processes around the velocity direction with a precession angular velocity that is at the tail rotation speed of γ˙2=20rad/s under the uncorrected trajectory state. As the tail rotation speed is within the stable speed range, the attack angle amplitude for the projectile body continues to decrease. When the control action is applied, the attack angle of the projectile body quickly stabilizes to a certain angle in the complex plane (direction of the midpoint line in Figure 9). At this time, the rotation speed of the projectile body is relatively small, and the influence of the Magnus moment can be ignored. Therefore, the stable attack angle direction for the projectile body is approximately equal to the phase angle γ20 of the tail wing in the corrected trajectory state. The deflection angle of the projectile centroid velocity in the complex plane in the trajectory correction state is given by equation (46). The deflection angle superimposed with the ideal trajectory velocity height angle θ is the velocity height angle θa and velocity lateral angle θb of the projectile in the reference coordinate system, as shown in Figure 10.Figure 10
Velocity height angleθa and velocity lateral angle θb of the projectile in the reference coordinate system.In Figure10, the different colors represent the projectile velocity height angle θa and velocity lateral angle θb when the correction rudder is stabilized at different phase angles γ20. In the uncorrected trajectory stage, the velocity direction of the projectile centroid swings slightly, and the angular velocity of the swing is the tail rotation speed of γ˙2=20rad/s. Herein, the projectile centroid average deflection angle deviates slightly to the positive direction of θb due to the influence of the Magnus moment. The superposition of the ideal trajectory velocity height angle θ causes the coordinate points of the projectile centroid velocity directions to form a circle on the complex plane (the dotted ellipse shown in Figure 10 is caused by different scale intervals of the horizontal and vertical coordinates). The center of the circle is below the origin, which represents the direction range in which the projectile centroid velocity can be controlled.


## 7.1. Stability Analysis of Projectile under Magnus Moment

In the trajectory-corrected phase, the Magnus moment acting on the projectile body is generated due to the rotation of the tail wing relative to the inertial system as driven by the motor. The Magnus moment is positively correlated with the projectile velocity, tail rotation speed, and angle of attack. When other conditions are certain, a tail rotation speed that is too high will increase the Magnus moment to a degree that cannot be ignored. As the Magnus moment direction is roughly perpendicular to the projectile axis in the angle of attack plane, the moment will make the projectile move in a spiral around the speed line. If the damping moment received by the projectile cannot dissipate the oscillation energy of the projectile, the projectile axis oscillation will increase and result in projectile instability. The stability of the projectile under different tail wing speeds is shown in Figure5.Figure 5
Stability of the projectile under different tail speed conditions: (a)γ˙2<γ˙0; (b) γ˙2=γ˙0; (c) γ˙2>γ˙0.
(a)(b)(c)According to the characteristic equation of equation (19), the two characteristic frequencies β1 and β2 of the projectile are β1≈0.1856 and β2≈−0.1852, showing β1≈−β2 can be obtained. Therefore, the shape of the two characteristic frequencies is oval, as shown in Figure 4. According to equation (27), the critical rotation speed of the tail is γ˙0≈153.3rad/s. Figure 5 shows the movement of the projectile angle of attack under the initial conditions δ10=0.1rad, δ20=0, v0=280m/s, and θa0=0.07πrad. Without considering gravity and the correction force, the variable coefficient nonhomogeneous nonlinear equation (16) describing the complex angle of attack degenerates into the approximate second-order homogeneous linear expression of equation (19), which makes the analytic solution of equation (23) for the angle of attack very close to the numerical solution with a correlation coefficient of γ>0.98. Therefore, the analytic solution of equation (23) is considered accurate. Due to the influence of the Magnus moment, although the initial value of the directional angle of attack δ2 is 0, an increased arc length s causes the projectile axis to produce the phenomenon of a lateral reciprocating swing. As shown in Figure 5(a), when γ˙2<γ˙0, the Magnus moment is small, and the swing amplitude of the projectile body gradually decreases to 0. As shown in Figure 5(b), when γ˙2=γ˙0, the swing amplitude of the projectile eventually stabilizes at a certain angle. As shown in Figure 5(c), when γ˙2>γ˙0, the Magnus moment is large, and the swing amplitude of the projectile body first decreases before increasing, which eventually leads to instability of the projectile body.


## 7.2. Stability of Projectile Caused by Gravity

As the projectile is under the action of gravity, the height deflection angles of the projectile velocity bend downward, which results in a changed attack angle. Equation (29) describes this change, and the results are shown in Figures 6 and 7, and the initial conditions are δ10=δ20=0, v0=280m/s, and θa0=0.07πrad.Figure 6
Comparison of numerical and analytical solutions for the angular motion under the influence of gravity.Figure 7
Variations in the projectile height and lateral angle of attack under different tail rotation speeds. (a) Change of height angle of attackδ1 in steady state; (b) change of lateral angle of attack δ2 in steady state; (c) change of angle of attack δ1 and δ2 in unstable state.
(a)(b)(c)As shown in Figure6, at the initial stage of the trajectory, the projectile angle of attack swings around the velocity line. With an increased arc length s, if the projectile is stable, the swing amplitude of the angle of attack gradually decreases. The motion form in the swing stage of the angle of attack is derived from the homogeneous solution W2 in equation (28), and the motion form in the steady stage of the angle of attack is derived from the nonhomogeneous solution WG in equation (28). The movement trend for the angle of attack obtained by the numerical solution is the same as that obtained from the analytic solution, and only the homogeneous solution has slight errors with the numeric solution because of the approximate linearized results. Therefore, it is acceptable to use the analytic solution to analyze the angular motion of the projectile. As the homogeneous solution W2 in equation (28) is only different from W1 in its coefficients, its stability is the same as that without considering the influence of gravity. The effect of gravity is equivalent to applying the initial angular velocity to the height angle of attack, indicating the angle of attack produces a swing motion in the initial trajectory stage. Its motion mechanism is the same as that in Section 6.1. When the tail rotation speed is greater than the critical rotation speed, i.e., γ˙2>γ˙0, the angle of attack diverges, which results in projectile instability, as shown in Figure 7(c). Figure 7 shows changes in the projectile height angle of attack and lateral angle of attack under different tail rotation speeds when the projectile is stable. Changes in the amplitude of the height angle of attack δ1 are less affected by the tail rotation speed γ˙2, but changes in the lateral angle of attack δ2 amplitude increase with the tail rotation speed γ˙2, which is due to the influence of the increased lateral Magnus moment. It is noted that when the projectile is stable, the angular motion amplitude caused by gravity is small, which can be ignored when considering the effects of the correction force.


## 7.3. Resonance Stability of Projectile

When the projectile is in the uncorrected trajectory state, the periodic rotation correction force causes the projectile to exhibit forced vibrations. When the rotation frequency of the correction force is close to the natural frequency of the projectile swing, the projectile angle of attack amplitude response produces a large positive gain. If the projectile satisfies the stability condition in Subsection6.1, i.e., γ˙2<γ˙0, the projectile angle of attack amplitude can be calculated from equation (37), and the initial conditions are δ10=δ20=0, v0=280m/s, and θa0=0.07πrad as shown in Figure 8.Figure 8
Variations in the angle of attack amplitude with the tail rotation speed.When considering the rotation correction force that acts on the projectile, the projectile angular motion can be regarded as a damped forced vibration. According to equation (37), the amplitude of the angle of attack is determined by the damping factors α2 and the tail aerodynamic force B. As shown in Figure 8, when the tail rotation angle frequency β3 approaches the characteristic frequencies β1 and β2 of the projectile body, the projectile body angle of attack amplitude can exceed 0.4 rad. Although the angle of attack amplitude does not diverge, the aerodynamic force generated by its amplitude is in the nonlinear regime, which adversely impacts the projectile body control.The first method to avoid excessive angular amplitudeAm is to reduce the tail correction rudder deflection angle so that the tail aerodynamic force B is reduced. However, this reduces the correction capability of the projectile. The second method is to avoid the rotation speed of the tail to operate near the characteristic frequencies β1 and β2 of the projectile body. As the projectile needs to change between zero and nonzero in the corrected trajectory stage, if the rotation speed of the tail is too high in the uncorrected trajectory stage, the rotation speed of the tail must pass through the characteristic frequency of the projectile body when switching to the corrected state. Considering the critical rotation speed γ˙0>vβ1,2 of the tail obtained in Section 6.1, the optimal rotation speed range of the tail wing in the uncorrected phase of the projectile trajectory is 0.8vβ2<γ˙2<0.8vβ1.


## 7.4. Angular Motion of Projectile in Corrected Trajectory State

When the projectile is in the corrected trajectory state, the tail rotation speed isγ˙2=0. At this time, the Magnus force and moment can be ignored. Thus, the damping factor α1,2<0 can be achieved only if the static stability moment coefficient satisfies mz<0. At this time, the projectile is in the stable state. The analytic solution of the angular motion of the projectile in the corrected trajectory state is given by equation (41). Therein, the steering gear control system can stabilize the phase angle of the correction rudder at the γ20 position. Figure 9 shows the angular motion curves for the projectile body in the process of changing the corrected trajectory state after flying for 3 s under the uncorrected trajectory. Different colors represent the angular motion of the projectile when the correction rudder is stable at different phase angles γ20. The initial conditions for the simulations are v0=280m/s, δ10=δ20=0, and θa0=0.07πrad. The tail rotation speed in the uncorrected trajectory state is γ˙2=20rad/s.Figure 9
Angular motion of projectile when changing from the uncorrected trajectory state to the corrected trajectory state under different phase anglesγ20.Figure9 shows that in the first three seconds, the projectile axis processes around the velocity direction with a precession angular velocity that is at the tail rotation speed of γ˙2=20rad/s under the uncorrected trajectory state. As the tail rotation speed is within the stable speed range, the attack angle amplitude for the projectile body continues to decrease. When the control action is applied, the attack angle of the projectile body quickly stabilizes to a certain angle in the complex plane (direction of the midpoint line in Figure 9). At this time, the rotation speed of the projectile body is relatively small, and the influence of the Magnus moment can be ignored. Therefore, the stable attack angle direction for the projectile body is approximately equal to the phase angle γ20 of the tail wing in the corrected trajectory state. The deflection angle of the projectile centroid velocity in the complex plane in the trajectory correction state is given by equation (46). The deflection angle superimposed with the ideal trajectory velocity height angle θ is the velocity height angle θa and velocity lateral angle θb of the projectile in the reference coordinate system, as shown in Figure 10.Figure 10
Velocity height angleθa and velocity lateral angle θb of the projectile in the reference coordinate system.In Figure10, the different colors represent the projectile velocity height angle θa and velocity lateral angle θb when the correction rudder is stabilized at different phase angles γ20. In the uncorrected trajectory stage, the velocity direction of the projectile centroid swings slightly, and the angular velocity of the swing is the tail rotation speed of γ˙2=20rad/s. Herein, the projectile centroid average deflection angle deviates slightly to the positive direction of θb due to the influence of the Magnus moment. The superposition of the ideal trajectory velocity height angle θ causes the coordinate points of the projectile centroid velocity directions to form a circle on the complex plane (the dotted ellipse shown in Figure 10 is caused by different scale intervals of the horizontal and vertical coordinates). The center of the circle is below the origin, which represents the direction range in which the projectile centroid velocity can be controlled.


## 8. Conclusions

(1)
The trajectory correction principle of a rocket projectile with an isolated rotating tail rudder is proposed, and the differential equation for the angular motion of the projectile body under a complex angle of attack plane is derived based on the rigid body trajectory equation. The analytic solution of the angular motion is approximately equal to the numeric solution with a correlation coefficient ofγ>0.98, which verifies the accuracy of the analytic solution for the angular motion.(2)
Angular motion of the projectile as caused by the initial conditions, gravity, and asymmetric rotating tail is studied. The critical rotation speedγ˙0 and optimal rotation speed range 0.8vβ2<γ˙2<0.8vβ1 of the tail are derived, which are key factors to ensure the stability of the projectile in the uncorrected trajectory state.(3)
The transient and steady-state solutions of the projectile in the corrected trajectory state are derived. The angular motion of the projectile in the process from uncorrected to corrected trajectories is explained through numerical simulations. In the state of the trajectory correction, an approximately equal relationship between the angle of attack and phase angle of the tail wing is visualized, and the variation range of the projectile centroid direction under controlled action is noted.



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*Source: 5880706-2023-02-15.xml*