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

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Introduction
Modern warfare has placed increasing demands on the costefectiveness of weapon systems, both in terms of increasing their accuracy and reducing costs as much as possible.Te efective 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 efective range with accuracies that decrease signifcantly over distance.In comparison, manportable antitank missiles, such as the "Javelin" missile of the United States, can strike fxed 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 highvalue targets.
One efective scheme to increase the range and accuracy of unguided, man-portable, antiarmor weapons is to add a simple guidance module.A fxed canard system [2][3][4][5][6] is a low-cost, miniaturized, two-dimensional, trajectory-corrected steering system.Such a system was frst used in the U.S. Army's Precision Guidance Kit (PKG) [7][8][9][10][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.Te periodic average control force method [13][14][15][16][17][18] has been adopted in actuator systems to realize two-dimensional trajectory correction using a single channel.Tis has the advantages of a simple structure, low cost, and high reliability.
Te fxed 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 fight stability and control characteristics.Zhu et al. [16] established a seven degree-of-freedom dynamic motion equation for dual-spin projectiles in a fxed plane coordinate system.Te diferential equation for the complex angle of attack was derived using projectile linear theory without the assumption of a fat 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 infuence 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-offreedom model.Furthermore, a detailed simulation was performed to determine the infuence rules of the yawing force on the angular motion.Te 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 fight range decreases sharply.
Chang et al. [20] studied the swerve response of spinstabilized 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.Te efect 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.Te 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.Te 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 efect of the projectile pitch, velocity, and roll rate on the fight 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-offreedom projectile trajectory model.Teir 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.Te 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.Te sufcient and necessary conditions of the coning motion stability for the dual-spin projectile with angular rate loops are analytically derived and further verifed via numerical simulations.
Te above literature shows that the installation of PGK will afect the stability conditions of the original projectile.Before researching trajectory-corrected control algorithms, it is necessary to perform basic research on the fight dynamics and stability of projectiles.A fxed-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 fight stability of the projectile [23][24][25], which readily causes the projectile to lose stability and overturn.Terefore, 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 twodimensional trajectory correction, the angular motion and stability law of the projectile body will be diferent 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.
Tis study analyzes the angular motion and stability of an isolated rotating rudder trajectory-corrected rocket projectile.Te characteristic for this type of trajectorycorrected rocket projectile is that the actuator to generate the correction force is at the tail of the projectile and can rotate along its axis.Te 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 clarifed.Second, the angular motion of the projectile on the complex plane is defned, and the angular motion diferential equation is derived based on the rigid body trajectory equation.Te angular motion and stability characteristics of the projectile under diferent tail speeds are studied for the uncorrected and corrected fight states.Finally, the conclusions are verifed through an example.

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.Te caliber of the considered rocket is approximately 80 mm and is used primarily to strike fxed or low-speed moving targets within 2000 m.Te overall layout of the rocket is shown in Figure 1(a) [26].Te device is composed of a seeker, projectile body, and tail segment.Te tail segment is driven by a motor and can rotate relative to the projectile body around its longitudinal axis.Four wings are fxed on the tail segment, as shown in Figure 1 Te 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 fxed at a specifc phase, the correction rudders provide continuous aerodynamic force in a specifc direction to control the projectile body and rotate around its center of mass.Tis 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.Tis simplifed 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 fow feld is more complex, and this law is no longer applicable.

Dynamic Modelling of Rocket Projectile with Isolated Rotating Tail Rudder
Te coordinate systems defned in this study are shown in Figures 2(a) and 2(b).Te coordinate system o E x E y E z E is for the ground and can be regarded as the inertial system.Te position of the origin o E is at the launch port, the o E x E axis points to the fring direction along the horizon, the o E y E axis points upward along the vertical direction, and the o E z E axis is determined based on the right-hand rule.Te ground coordinate system o E x E y E z E is used to determine the spatial position of the projectile centroid.Te coordinate system ox N y N z N is the reference coordinate system.Te 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 o E x E y E z E .Te reference coordinate system ox N y N z N is used to determine the centroid velocity and azimuth of the projectile axis.Te coordinate system ox 1 y 1 z 1 is the trajectory coordinate system with its origin located at the projectile mass center.Te ox 1 axis points along the velocity direction of the mass center, the oy 1 axis is vertical to the ox 1 axis, and the oz 1 axis is determined based on the right-hand rule.Te trajectory coordinate system ox 1 y 1 z 1 is determined by rotating the reference coordinate system ox N y N z N twice.First, the θ a angle is rotated in a positive direction around the oz N axis to form the coordinate system ox 2 ′ y 2 z N .Second, the θ b angle is rotated in a negative direction around the oy 1 axis to form the coordinate system ox 1 y 1 z 1 .Te θ a is the height angle, and θ b is the defection angle of the trajectory.Te ox 2 y 2 z 2 is the projectile axis coordinate system with an origin located at the projectile mass center.Te ox 2 axis points along the forward direction of the projectile axis, the oy 2 axis is vertical to the ox 2 axis, and the oz 2 axis is determined based on the right-hand rule.Te projectile axis coordinate system ox 2 y 2 z 2 can be determined by rotating the reference coordinate system ox N y N z N twice.First, the φ a angle is rotated in a positive direction around the oz N axis to form the coordinate system ox 2 ′ y 2 z N .Second, the φ b angle is rotated in a negative direction around the oy 2 axis to form the coordinate system ox 2 y 2 z 2 .Te φ a is the height angle, and φ b is the defection angle of the projectile axis.Te translational diferential equations for the center of mass of the rocket projectile [14,27] are as follows: where v, θ a , and θ b represent the projectile centroid velocity, velocity elevation angle, and velocity defection angle, respectively; x, y, and z represent the centroid displacements; m � m 1 + m 2 represents the total projectile mass with m 1 and m 2 being the mass of the body and tail, respectively; and F x , F y , and F z represent the force components of the projectile in the three axes of the trajectory coordinate system.
Te diferential equations for the rocket projectile rotation around the center of mass [28][29][30] are as follows:

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 fight and analyze its fight stability.Te fight stability of the projectile is usually represented by variations in the angle of attack δ 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.
Figure 2: Defnition and rotation relationship for each coordinate system.(a) Te ground coordinate system, reference coordinate system, and trajectory coordinate system; (b) the trajectory coordinate system and projectile axis coordinate system.
Te unit sphere description of the projectile angular motion is shown in Figure 3. Te spherical surface is a unit sphere with the center of mass c.Te unit vector r → 0 has the same direction as the ideal trajectory velocity, and its intersection with the unit sphere is o.Te plane cox is the horizontal plane of the trajectory coordinate system, and coy is the vertical plane.Te unit vector r → 1 has the same velocity direction as the projectile, and its intersection with the unit sphere is T. Te unit vector r → 2 has the same direction as the projectile axis, and the intersection with the unit sphere is B. Te 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 , respectively, which satisfes the relationship Δ � Φ − Ψ. Te δ 1 represents the height angle of attack, and δ 2 represents the lateral angle of attack.

Diferential Equation of Projectile Body Angular Motion.
Te following symbols are introduced to simplify the expressions of the forces and moments: where c x , c y , c z , m z , m zz , m xz 1 , m xz 2 , and m y represent the drag coefcient, lift coefcient, Magnus force coefcient, static moment coefcient, equatorial damping moment coefcient, projectile body pole damping moment coefcient, tail pole damping moment coefcient, and tail Magnus moment coefcient, 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].
Te forces and moments on the projectile are as follows: where F p , M q , δ 1 , δ 2 , δ r , δ N , and δ M 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: where v i , θ, x i , y i , and a p 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.Te complex plane description of the angular motion defned in Figure 4 indicates that the following equation holds: According to the second and third equations in equation ( 1) and the second equation in equation (7), and considering that θ 2 , θ 1 , φ 2 , φ 1 , δ 1 , and δ 2 are small quantities, the equation that describes changes in the velocity direction can be written as follows: Multiply the second equation in equation ( 8) by the imaginary unit i, and add the two equations in equation ( 8) to obtain: where Ψ � θ 1 + iθ 2 is the velocity complex defection angle, and Δ � δ 1 + iδ 2 is the complex angle of attack.
and ω ξ 1 ≈ _ c 1 and omitting high-order small quantities allow simplifcation as follows: Multiply the frst equation in equation (10) by − i and add the two equations to obtain: where Φ � φ 1 + iφ 2 is the complex defection 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 (g sin θ/v)Ψ is omitted.Equations ( 9) and ( 11) are transformed into the following form: To derive the diferential equation for the angle of attack Δ, the relation Φ ′ � Ψ ′ + Δ ′ and using the frst equation of (7) transforms Φ ′ into: Furthermore, taking the coefcient before Δ in equation ( 14) as a constant, Φ ″ can be obtained as follows: Mathematical Problems in Engineering Substituting equations ( 14) and ( 15) into equation ( 13) and replacing the state variable Δ with W � vΔ, the following diferential equation for the attack angle can be derived: where the symbol "′" represents the derivative of the arc length s, and the other symbols are as follows: 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 E G and wing aerodynamic disturbance E F .Te E G mostly contains the θ term, which is caused by changes in gravity, and E F primarily contains the tail-spin angle term e ic 2 .

Angular Motion Characteristics of Projectile under Uncorrected Trajectory
Te lateral aerodynamic force generated from the fxed wings in the uncorrected trajectory state should be zero in a rotation cycle.Tus, the rotation speed of the tail wing around the projectile axis relative to the ground coordinate system should be a nonzero fxed value.Te 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.Tese are afected by three factors: angular motion caused by initial conditions, angular motion caused by gravity, and angular motion caused by asymmetric factors.Terefore, its solution can be decomposed into where W 0 is the homogeneous solution of equation ( 16) and represents the angular motion caused by the initial conditions.Te form of the solution is W 0 � C 1 e l 1 s + C 2 e l 2 s , where C 1 and C 2 are undetermined coefcients determined by the initial conditions.Te W G is the angular motion due to the nonhomogeneous term E G , and W F is the angular motion due to the nonhomogeneous term E F .

Angular Motion Characteristics Caused by Initial
Conditions.Both the initial angular displacement and initial angular velocity of the projectile can afect its angular motion response.Without considering the disturbance E, the angular motion caused by the initial conditions is W 1 .
Based on the coefcient freezing method, let l 1 � α 1 + iβ 1 and l 2 � α 2 + iβ 2 be the two characteristic roots of the following homogeneous equation: Ten, W 1 can be written as follows: where K 1 and K 2 are as follows: Te K 10 , K 20 , β 10 , and β 20 are determined by the initial conditions W 10 � v 0 δ 0 e iσ 0 and W 10 ′ � (( _ v 0 /v 0 )δ 0 + _ δ 0 )e iσ 0 + δ 0 _ σ 0 e i(σ 0 +0.5π) .Ten, Te angle of attack Δ 1 is the angular motion caused by the initial disturbance and is expressed as follows: where β 1 and β 2 are the angular frequencies for one and two circular motions, and α 1 and α 2 in K 1 and K 2 represent changes in the circular motion amplitudes.If the circular motion amplitude is gradually reduced, the derivative of (K 1,2 /v) must be negative.Tat is, where α 1,2 is the real part of the characteristic root, which is obtained from the complex square formula: Equation ( 24) is the full trajectory dynamic stability condition that the tail rotation speed should satisfy.For the active phase of the rocket, _ v/v 2 > 0, equation ( 24) is more easily satisfed; therefore, it is sufcient to consider only the stability of the passive segment.For the passive phase of the rocket, _ v/v 2 ≈ − ρSc x /(2m) and α 1,2 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 _ c 2 should satisfy where _ c 0 is the critical speed.For the convenience of calculations, omitting the small passive section of _ v/v 2 allows _ c 2 0 to satisfy the following condition:

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.Te angular motion equation caused by gravity is as follows: Te form of the solution is the superposition of the general solution for the homogeneous equation and the special solution of the nonhomogeneous equation.Namely, where W 2 is the homogeneous solution of equation ( 28) that considers the E G case, which has the same form as W 0 and W 1 , difering only in the coefcients.Te constant variation method can be used to solve the special solution W G of the nonhomogeneous equation as follows: Ten, (31)

Angular Motion Characteristics Caused by Asymmetric
Tail.Te tail wing provides the control force and torque to change the projectile motion.Te radial periodic force is generated during the tail wing rotation due to the same direction defection angle of a pair of wings.Tis 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.Te angular motion equation caused by the asymmetric tail is as follows: If the additional force eccentricity angle δ N is equal to the additional moment eccentricity angle δ M , the aerodynamic disturbance term E F of the wing can be expressed as follows: Mathematical Problems in Engineering where B � ( According to the coefcient freezing method, the solution of equation ( 32) can be expressed as follows: where Ten, the angle of attack generated by the asymmetric tail is as follows: Tere are three circular motions.Namely, circular motion with the angular frequencies of β 1 and β 2 generated by the initial disturbance and the forced circular motion with an angular frequency of β 3 .If the damping factors α 1 and α 2 of the passive section satisfy the stability conditions, the amplitudes of the frst two circular motions tend to 0 with an increased arc length s, and only the forced circular motion with an angular frequency of β 3 remains.Te amplitude of the forced circle motion is as follows:

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 specifc rotation speed to make the tail axially stable relative to the ground coordinate system.Te direction of the additional torque generated by the aerodynamic eccentricity of the tail is fxed at the required phase, which changes the attitude of the projectile.Ten, the tail rotation speed changes from constant to _ c 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.

Steady and Transient Solutions for Angle of Attack under Trajectory Correction.
As the tail rotation speed is _ c 2 � 0 and the projectile body rotation speed is _ c 1 ≈ 0, the small quantity P in equation ( 16) can be omitted, and the projectile angle of attack satisfes the following equation: where Te rotation speed step excitation response of the system is equivalent to the solution of equation ( 38) when c 2 is equal to the fxed value c 20 under the initial conditions W(t 0 ) � W 0 and W ′ (t 0 ) � W 0 ′ .Te initial conditions are determined from the uncontrolled fight terminal state.According to the coefcient 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: Ten, the solution of equation ( 38) is as follows: where If the condition M < 0 is satisfed, the system will be stable, and the frst two terms of equation (41) will decrease to 0 with an increased s.Ten, W * is the steady-state solution of the system, and the corresponding steady-state angle of attack Δ * is as follows: (42)

Efect of Trajectory Correction on Velocity Defection Angle of Projectile.
In the trajectory correction state, the tail wing of the projectile generates a control force along a fxed direction, which changes the projectile angle of attack.Tis generates a lift in the angle of attack plane and changes the direction of the velocity for the projectile mass center.Te steady-state angle of attack Δ * is substituted into equation ( 9), and the independent variable is changed from time t to arc length s.Considering the trajectory passive section a p � 0, the complex defection angle diferential equation for the projectile centroid velocity under the steady-state angle of attack is obtained as follows: From equation (38), the following can be obtained: Substituting equation (44) into equation ( 43) and ignoring a small amount of g sin θ/v 2 , the following is obtained: (45) After linearization, the complex defection angle of the projectile velocity at the steady angle of attack is as follows: (46)

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.Te relevant projectile body and aerodynamic parameters are shown in Table 2 [26].

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.Te 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.Te stability of the projectile under diferent tail wing speeds is shown in Figure 5.
According to the characteristic equation of equation ( 19), the two characteristic frequencies β 1 and β 2 of the projectile are β 1 ≈ 0.1856 and β 2 ≈ − 0.1852, showing β 1 ≈ − β 2 can be obtained.Terefore, 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 _ c 0 ≈ 153.3rad/s.Figure 5 shows the movement of the projectile angle of attack under the initial conditions δ 10 � 0.1rad, δ 20 � 0, v 0 � 280m/s, and θ a0 � 0.07πrad.Without considering gravity and the correction force, the variable coefcient 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 coefcient of c > 0.98.Terefore, the analytic solution of equation ( 23) is considered accurate.Due to the infuence of the Magnus moment, although the initial value of the directional angle of attack δ 2 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 _ c 2 < _ c 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 _ c 2 � _ c 0 , the swing amplitude of the projectile eventually stabilizes at a certain angle.As shown in Figure 5(c), when _ c 2 > _ c 0 , the Magnus moment is large, and the swing amplitude of the projectile body frst decreases before increasing, which eventually leads to instability of the projectile body.

Stability of Projectile Caused by Gravity.
As the projectile is under the action of gravity, the height defection 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 δ 10 � δ 20 � 0, v 0 � 280m/s, and θ a0 � 0.07πrad.
As shown in Figure 6, 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.Te motion form in the swing stage of the angle of attack is derived from the homogeneous solution W 2 in equation (28), and the motion form in the steady stage of the angle of attack is derived from the nonhomogeneous solution W G in equation (28).Te 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.Terefore, it is acceptable to use the analytic solution to analyze the angular motion of the projectile.As the homogeneous solution W 2 in equation ( 28) is only diferent from W 1 in its coefcients, its stability is the same as that without considering the infuence of gravity.Te efect 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 2.76e − 2 δ M (rad) 8.11e − 2 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., _ c 2 > _ c 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 diferent tail rotation speeds when the projectile is stable.Changes in the amplitude of the height angle of attack δ 1 are less afected by the tail rotation speed _ c 2 , but changes in the lateral angle of attack δ 2 amplitude increase with the tail rotation speed _ c 2 , which is due to the infuence 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 efects of the correction force.

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 satisfes the stability condition in Subsection 6.1, i.e., _ c 2 < _ c 0 , the projectile angle of attack amplitude can be calculated from equation (37), and the initial conditions are δ 10 � δ 20 � 0, v 0 � 280m/s, and θ a0 � 0.07πrad as shown in Figure 8.
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 α 2 and the tail aerodynamic force B. As shown in Figure 8, when the tail rotation angle frequency β 3 approaches the characteristic frequencies β 1 and β 2 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.12 Mathematical Problems in Engineering Te frst method to avoid excessive angular amplitude A m is to reduce the tail correction rudder defection angle so that the tail aerodynamic force B is reduced.However, this reduces the correction capability of the projectile.Te second method is to avoid the rotation speed of the tail to operate near the characteristic frequencies β 1 and β 2 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 | _ c 0 | > v|β 1,2 | of the tail obtained in Section 6.1, the optimal rotation speed range of tail wing in the uncorrected phase of the projectile trajectory is 0.8vβ 2 < _ c 2 < 0.8vβ 1 .

Angular Motion of Projectile in Corrected Trajectory State.
When the projectile is in the corrected trajectory state, the tail rotation speed is _ c 2 � 0. At this time, the Magnus force and moment can be ignored.Tus, the damping factor α 1,2 < 0 can be achieved only if the static stability moment  coefcient satisfes m z < 0. At this time, the projectile is in the stable state.Te analytic solution of the angular motion of the projectile in the corrected trajectory state is given by equation (41).Terein, the steering gear control system can stabilize the phase angle of the correction rudder at the c 20 position.Figure 9 shows the angular motion curves for the projectile body the process of changing the corrected trajectory state after fying for 3 s under the uncorrected trajectory.Diferent colors represent the angular motion of the projectile when the correction rudder is stable at different phase angles c 20 .Te initial conditions for the simulations are v 0 � 280m/s, δ 10 � δ 20 � 0, and θ a0 � 0.07πrad.Te tail rotation speed in the uncorrected trajectory state is _ c 2 � 20rad/s.
Figure 9 shows that in the frst three seconds, the projectile axis processes around the velocity direction with a precession angular velocity that is at the tail rotation speed of _ c 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 infuence of the Magnus moment can be ignored.Terefore, the stable attack angle direction for the projectile body is approximately equal to the phase angle c 20 of the tail wing in the corrected trajectory state.Te defection angle of the projectile centroid velocity in the complex plane in the trajectory correction state is given by equation ( 46).Te defection angle superimposed with the ideal trajectory velocity height angle θ is the velocity height angle θ a and velocity lateral angle θ b of the projectile in the reference coordinate system, as shown in Figure 10.
In Figure 10, the diferent colors represent the projectile velocity height angle θ a and velocity lateral angle θ b when the correction rudder is stabilized at diferent phase angles c 20 .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 _ c 2 � 20rad/s.Herein, the projectile centroid average defection angle deviates slightly to the positive direction of θ b due to the infuence of the Magnus moment.Te 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 diferent scale intervals of the horizontal and vertical coordinates).Te center of the circle is below the origin, which represents the direction range in which the projectile centroid velocity can be controlled.Te 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 and phase angle of the tail wing is visualized, and the variation range of the projectile centroid direction under controlled action is noted.

Conclusions
(b), and one pair of wings has a zero-degree defection angle relative to the projectile body.Tese are the stabilizing rudders and provide a stable 2 Mathematical Problems in Engineering moment while the rocket is fying.Te other pair of wings has a defection 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, c represents the angular displacement of the tail segment, and F w represents the direction of the correction force generated by the correction rudders.

Figure 1 :
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.

4 )
where c 1 , c 2 , φ a , and φ b represent the axial angular displacement of the projectile body, axial angular displacement of the tail, height angle of the projectile axis, and defection angle of the projectile axis, respectively; ω ξ 1 , ω ξ 2 , ω η , and ω ζ 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 defection angle of the projectile axis in the projectile axis coordinate system, respectively; M ξ 1 , M ξ 2 , M ξ 21 , and M ξ 12 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 ox 2 direction of the projectile axis coordinate system.Te M η and M ζ are the components of the aerodynamic moment acting on the projectile body in the oy 2 and oz N directions of the projectile axis coordinate system, respectively.

Figure 6 :
Figure 6: Comparison of numerical and analytical solutions for the angular motion under the infuence of gravity.

Figure 7 : 2 Figure 8 :
Figure 7: Variations in the projectile height and lateral angle of attack under diferent tail rotation speeds.(a) Change of height angle of attack δ 1 in steady state; (b) change of lateral angle of attack δ 2 in steady state; (c) change of angle of attack δ 1 and δ 2 in unstable state.

( 1 ) 20
Te trajectory correction principle of a rocket projectile with an isolated rotating tail rudder is proposed, and the diferential 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.Te analytic solution of the angular motion is approximately equal to the numeric solution with a correlation coefcient of c > 0.98, which verifes 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.Te critical rotation speed _ c 0 and optimal rotation speed range 0.8vβ 2 < _ c 2 < 0.8vβ 1 of γ

Figure 9 :
Figure 9: Angular motion of projectile when changing from the uncorrected trajectory state to the corrected trajectory state under diferent phase angles c 20 .

Figure 10 :
Figure 10: Velocity height angle θ a and velocity lateral angle θ b of the projectile in the reference coordinate system.

Table 1 :
(3)er of magnitudes for each parameter.Changes in the projectile axis direction are described by the third and fourth equations in equation(3).Using the relations ω

Table 2 :
Projectile body and aerodynamic parameters.