# Flight Stability of Canard-Guided Dual-Spin Projectiles with Angular Rate Loops

Authors: Qiushi Zheng; Zhiming Zhou
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

Generally, as a precision-guided weapon, the missile has many disadvantage such as high price, difficult maintenance, and low yield. Modern war requires more and more new guided ammunition with high precision, low cost, and low collateral damage. Therefore, as a simple guided conventional ammunition technology, the dual-spin projectile has attracted the attention of ammunition experts recently. This paper proposes a dual-spin projectile scheme based on the rotation control method. Firstly, the concept of the dual-spin projectile is introduced. Secondly, the mathematical model of the dual-spin projectile is established, and the angular motion equation is finally obtained by using some linearized assumptions. Finally, the sufficient and necessary conditions of coning motion stability for dual-spin projectile with angular rate loops are analytically derived and further verified by numerical simulations. It is noticed that the upper bound of the control gain is affected by the delay angle of the control system and the spinning rate of the projectile.


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## Body

## 1. Introduction

The trajectory correction projectile is modified from the conventional ammunition, equipped with the ballistic detection device and the correction module. The ballistic detection device measures the trajectory elements and calculates the trajectory deviation in real time during the whole flight. Then, the projectile is guided by canards to reduce the impact miss distance and improve the damage probability. For the two-dimensional rajectory correction, the spin rate of the projectile is a major constraint to the correction effect. Conventional ammunitions always keep a very high spinning rate after launching to maintain the fly gyroscope stability. Thus, it is difficult to use canards to guide the projectile because general actuators do not have such a large operating frequency. In order to solve this problem, a dual-spin configuration principle has been proposed.The concept of low-cost trajectory correction ammunition first proposed by the United States in the 1970s adopted the dual-spin structure in [1]. The so-called dual-spin structure means that the projectile can be divided into two decoupled parts along the axial direction, which are connected by roller bearings. In the following years, many scholars have studied the dual-spin projectile. Costello and Peterson [2] developed the equations of motion for a dual-spin projectile in atmospheric flight. Guan and Yi [3] established a seven-degree-of-freedom projectile trajectory model for the dual-spin projectile and obtained a more accurate and reliable measurement data by filtering the trajectory of the dual-spin projectile using the UKF algorithm. Burchett et al. [4] obtained the closed form expressions for swerving motion under the action of lateral pulse jets for a dual-spin projectile in atmospheric flight by using the linear theory. Chang et al. [5, 6] studied the spin-rate characteristic of the dual-spin projectile and the dynamic response of the dual-spin projectile under the canard control effect. Liu et al. [7] deduced the analytical expressions without attitude information, which can predict the increment orientation for trajectory angular rate and swerving motion induced by canard control force. Wernert et al. [8] carried out the wind tunnel test of a canards guided 155 mm artillery and got the result of open-loop trajectory simulations.There has been a considerable literature on the flight stability of conventional spin-stabilized projectiles, such as [9–12]. But for this kind of dual-spin projectiles, there are some differences. Costello and Peterson [2] analysed the stability characteristic of the dual-spin projectile using modified projectile linear theory and obtained the expression of the gyroscopic stability factor and the dynamic stability factor which take on the same form as the rigid projectile. Wernert [13] addressed the problem of stability of such canard-guided dual-spin-stabilized projectiles and gave the stability criterion considering the influence of canards. He pointed out that the stability for this kind of projectiles is reduced due to the following two factors: the addition of canards and the reduced angular momentum of the forward part. Theodoulis and Wernert [14] present the quasilinear parameter-varying model of the dual-spin-stabilized projectile and analysed the stability of the projectile on different characteristic points of the whole flight envelope. Zhu [15] took account of the control effects of canards and then established a revised stability criterion according to the Hurwitz stability criterion which can be reduced to that of conventional spin-stabilized projectiles.However, few of the existing literatures have considered the coning motion of dual-spin projectile with the control loop. Thus, this paper focuses on the stability of coning motion for a dual-spin projectile with angular rate loops. The mathematical model of the dual-spin projectile is established. The sufficient and necessary condition of coning motion stability for dual-spin projectile with angular rate loops is analytically derived and further verified by numerical simulations. The stability boundary of control gains is obtained, and moreover, the influence of the total delay angle for the control system and spinning rate of the projectile on stability is analysed.


## 2. System Configuration

The so-called dual-spin projectile in this paper is improved from a conventional 155 mm spin-stabilized projectile by replacing the original fuse for a novel course correction fuse (CCF), which is illustrated in Figure1. The projectile can be divided into two parts: the forward part contains the necessary guidance and control hardware and software and the aft part contains the payload. These two parts are connected by rotating bearings and are completely roll decoupled from each other.Figure 1
Concept of the dual-spin projectile.As can be seen from Figure1, the forward part is equipped with two pairs of rotating fins which are used to generate normal force to guide the projectile. After launching, the aft part keeps a high spinning rate of several hundreds of radians per second in order to maintain the fly gyroscopic stability, while the spinning rate of the forward part can be reduced to keep a constant low value in a few seconds under the control of an axial motor. Therefore, this dual-spin structure design can not only solve the technical problems faced by the actuator when the projectile rotates at a high speed but also facilitate the design and normal work of the missile-loading measuring device and can also maintain the gyroscopic stability of the projectile.


## 3. Mathematical Model

According to the previous studies [2], the 7-DOF equations of motion of the dual-spin projectile are generally established in the body fixed plane (BFP) reference frame, as shown in Figure 2.Figure 2
The body fixed plane reference frame.The translational and attitude kinematics in the BFP reference frame can be derived as(1)ẋẏż=cosθcosψ−sinψsinθcosψcosθsinψcosψsinθsinψ−sinθ0cosθuvw,(2)ϕ̇fϕ̇aθ̇ψ̇=100tanθ010tanθ00100001/cosθpfpaqr.The translational and attitude airframe dynamics can be derived in the BFP reference frame as(3)u̇v̇ẇ=1mFxFyFz−0−rqr0rtanθ−q−rtanθ0uvw,(4)IxfṗfIxaṗaItq̇Itṙ=LfLaMN−00Ixapa+Ixfpfr+Itr2tanθ−Ixapa+Ixfpfq−Itqrtanθ,where u, v, and w are the linear velocities, x, y, and z are the linear positions, ϕf the roll angle of the forward part, ϕa the roll angle of the aft part, θ the pitch angle, ψ the yaw angle, pf the roll angular rate of the forward part, pa the roll angular rate of the aft part, and q and r the pitch and yaw angular rate. Fx, Fy, and Fz are the external forces containing aerodynamic and gravity components and can be derived as
(5)FxFyFz=−QSCD−mgsinθ−QSCNαβ−QSLVpaCypαα+QSCNδδy−QSCNαα+QSLVpaCypαβ−QSCNδδz+mgcosθ.Lf, LaM, and N are external moments containing aerodynamic and friction components:
(6)LfLaMN=QSLClδl−QSLClpfLVpf+Lf−a−QSLClpaLVpa−Lf−aQSLCmαα−QSLCmqLVq−QSLLVCnpαfpf+Cnpαapaβ+QSLCmδδz−QSLCmαβ−QSLCmqLVr−QSLLVCnpαfpf+Cnpαapaα+QSLCmδδy.The termCD is the drag coefficient, CNα the lift force coefficient, Clδ the roll-induced moment coefficient, Clpf the roll-damping moment coefficient of the forward part, Clpa the roll-damping moment coefficient of the aft part, Cmα the static moment coefficient, Cmq the lateral damping moment coefficient, Cnpα the Magnus moment coefficient, and Cmδ the canard control moment coefficient. δz and δy are the equivalent variables of the canard in the pitch and yaw channels.The definitions ofα and β are illustrated in Figure 1 and can be expressed as
(7)α=arctanwu,β=arctanvV.Other variables are the mass of the projectilem, the dynamic pressure Q=1/2ρV2 (where ρ is the air density), the airframe velocity V, the reference area S and length L, the moments of inertia Ixf, Ixa, and It, and the static and viscous friction coefficients CS and CV.It can be seen obviously from Equations (2) and (4) that the longitudinal and lateral dynamics of the dual-spin projectile contain two main couplings. On the one hand, the inertial coupling term Ixa/Itpa greatly depends on the high spin rate of the aft part. On the other hand, the aerodynamic coupling caused by Magnus moments depends on the angle of attack α, angle of sideslip β, and the high spinning rate pa. Furthermore, the aerodynamic coefficients appearing in Equations (5) and (6) are heavily nonlinear.


## 4. Angular Motion Equations of the Dual-Spin Projectile

Even though the mathematical model described in Equations (3) and (4) is more accurate and close to the real case, due to the highly nonlinear equations of motion, it is difficult to get the analytical solution and the obvious relationship between the flight characteristics of the missile and control parameters. To facilitate theoretical analysis, the general method is to apply the linearization theory of projectile. This theory has been regarded as an effective tool to analyse the flight stability of projectiles and applied [8–13]. Therefore, in order to linearize these two equations, the following assumptions are introduced firstly:
(1)
Relative to the longitudinal velocityu and the angular velocity pa, variables q, r, v, and w can be regarded as small quantities, so we have V=u2+v2+w2≈u(2)
During a short flight, the varieties of the projectile parameters are neglected, and the velocity and the spinning rates are considered to be constant(3)
The Magnus force and canard control force are negligible quantities compared to aerodynamic forces and the gravity effect is negligible(4)
Angle of attackα and angle of sideslip β are small, so we have α=arctanw/u≈w/V and β=arcsinv/V≈v/V.Under these assumptions, the lateral equations for the dual-spin projectile can be formulated as(8)v̇=−QSCNαβm−ruẇ=−QSCNααm+quq̇=QSLItCmα+Cmδα+CmqLVq−LVCnpαfpf+Cnpαapaβ+Cmδδz−Ixapa+IxfpfrItṙ=QSLIt−Cmα+Cmδβ+CmqLVr−LVCnpαfpf+Cnpαapaα+Cmδδy+Ixapa+IxfpfqIt.According to assumptions (1) and (4), we can get
(9)α̇≈ẇ−αu̇u=QS−CNα+CDmVα+q,β̇≈v̇−βu̇u=QS−CNα+CDmVβ−r.Thus, we can obtain the lateral equations of the dual-spin projectile in a simplified form as(10)α̇=−a1α+qβ̇=−a1β−rq̇=b11α+b12β+b21q−b22r+b3δzṙ=b12α−b11β+b22q+b21r+b3δy,where
(11)a1=QSCNα−QSCDmV,b11=QSLCmα+CmδIt,b12=−QSL2Cnpαpa+Cnpαpf/VIt,b21=QSL2Cmq/VIt,b22=Ixapa+IxfpfIt,b3=QSLCmδIt.By defining the complex variablesξ=β+iα, μ=q+ir, and δ=−δz+iδy, we can further obtain the angular motion equation of the dual-spin projectile in the complex form as
(12)ξ¨+a1−b21−ib22ξ̇−b11+a1b21+ib12+a1b22ξ=ib3δ.


## 5. Stability of the Moving Mass Spinning Missile with the Angular Rate Loop

### 5.1. Dual-Spin Projectile Control System with Angular Rate Loops

The control system with angular rate loops is shown in Figure3, in which ny and nz are control commands, θ̇ and ψ̇ are feedback signals which can be measured by IMU mounted on the forward part, kω is the control gain, and δz and δy are equivalent canard deflections.Figure 3
Simulation results forkω=0.31. (a) Curves of angle of attack and sideslip. (b) The critical coning motion.
(a)(b)When the control commands for the pitch and yaw channels areδz and δy, the ideal equivalent output of the actuator system should also be δz and δy. However, due to the signal delay, when the actuator system receives the control command, the angular position of the forward part has shifted by Δγ1. Furthermore, due to the dynamic response delay of the actuator, there will be a certain phase lag in the response output of the actuator system, which causes the actual equivalent output to generate an angular position deviation of Δγ2. Therefore, the total delay angle of the control system are γd=Δγ1+Δγ2.The actuator system is assumed to be a second-order system, only considering the steady-state output coupling, we can get(13)δzδy=krcosγd−sinγdsinγdcosγdδczδcy,(14)kr=ks1−Ts2pf22+2μsTspf2,(15)γd=pfτ+arccos1−Ts2pf21−Ts2pf22+2μsTspf2,where Ts is the time constant of the system, μs is the damping ratio, ks is the gain of the servo system, and τ is the command transmission delay time. Obviously, the control action in the pitch channel will cause the movement in the yaw channel, and the control action in the yaw channel will also affect the movement in the pitch channel.It can be seen from Figure3 that the input commands to the actuators can be described as
(16)nczncy=−kω00−kω−θ̇ψ̇.According to the definition of coordinate system and angle, positive equivalent canard deflections will generate positive angle of attack and negative angle of sideslip. And the negative angle of attack will generate negative pitching acceleration, and the positive angle of sideslip will generate positive yaw acceleration. Therefore, the displacement instruction of the canard is obtained as(17)δczδcy=1001nczncy.Meanwhile, based on the assumption that the missile is in horizontal flight, there exists an approximation relationship:α̇≈θ̇, β̇≈−ψ̇. Thus, Equation (17) can be expressed as
(18)δczδcy=−kω00−kωδczδcy.Finally, substituting Equation (18) into Equation (13), we can get the actual equivalent canard outputs with all the delays as
(19)δzδy=krcosγd−sinγdsinγdcosγdδczδcy.Converting Equation (19) into the complex form, one has
(20)δ=krkω−sinγd+icosγdξ̇.


### 5.2. Stability Conditions for the Dual-Spin Projectile Control System

Substituting Equation (20) into the angular motion Equation (12), we can get
(21)ξ¨+a1−b21+b3krkωcosγd−ib22−b3krkωsinγdξ̇−b11+a1b21+ib12+a1b22ξ=0.Rewrite the above equation into the simplified form as(22)ξ¨+HC−iPCξ̇−M+iQ=0,where
(23)HC=a1−b21+b3krkωcosγd,PC=b22−b3krkωsinγd.Apparently, the characteristic equation of Equation (22) has the form
(24)λ2+HC−iPCλ−M+iQ=0.Then, we can obtain the characteristic roots as(25)λ1,2=−HC−iPC±HC−iPC2+4M+iQ2.Assuming thatRc=Rcr+iRci=HC−iPC2+4M+iQ, one has
(26)Rcr=HC2−PC2+4M,Rci=−2HCPC+4Q.The real part ofλ1,2 is
(27)λR1,2=−HC±R+Rcr/22,where R=Rcr2+Rci2; it is obvious that the control system is stable only if the real parts of the characteristic roots are negative, that is
(28)−HC±R+Rcr2<0.The stability condition (28) can be formulated as
(29)−HC<0HC2>R+Rcr2.By substituting the expressions of these variables, the necessary and sufficient conditions for the stability of the dual-spin projectile control system can be obtained as(30)−L/VCmq+Cmδkrkωcosγd>0−Cmα−L/VCmq+Cmδkrkωcosγd2−ItLCnpα2pa2/QSV2+L/VCnpαpa−L/VCmq+CmδkrkωcosγdIxapa/QSL−Cmδkrkωsinγd>0.


### 5.3. Stability Analysis

Firstly, by analysing the first inequality in the stability condition (30), which is the pitching damping moment coefficient Cmq, it can be seen obviously that when the delay angle γd<90∘, the first inequality is satisfied. When γd>90∘, the control parameter kω has a upper limit as follows:
(31)kω<LCmqVCmδkrcosγd.Then, analysing the second inequality in the stability condition (30), since the dual-spin projectile is static unstable, we have Cmα>0. The first and the second terms of the second inequality are negative; thus, whether the second inequality is true depends on the third term. The magnitude and sign of the third term are mainly affected by the spinning rate of the aft part pa, the delay angle γd, and the control gain kω.The second inequality can be rewritten as a polynomial in terms ofkω:
(32)fkω=akω2+bkω+c>0,where
(33)a=kr2Cmδ2cosγd−Cmα−LVCnpαpasinγd,b=Cmδkr2LVCmαCmqcosγd+LV2CmqCnpαpasinγd+IxaQSVpaCmqCnpαpacosγd,c=−LCnpαpa2QSV2ItCnpα+Ixa.Whenγd<90∘, one gets a<0, b>0, and c<0. The two zero points of fkω are obtained as
(34)0<kω11=−b+b2−4ac2a<kω12=−b−b2−4ac2a.Only whenkω11<kω<kω12 one gets fkω>0. The sufficient and necessary condition for the coning motion stability can be derived as
(35)−b+b2−4ac2a<kω<−b−b2−4ac2a.Whenγd>90∘, one gets a>0, b<0, and c<0; the two zero points of fkω can be obtained as
(36)kω21=−b−b2−4ac2a<0<kω22=−b+b2−4ac2a.Only whenkω<kω21 or kω>kω22 one gets fkω<0. The sufficient and necessary condition for the coning motion stability can be derived as
(37)kω<LCmqVCmδkrcosγd∩kω<−b−b2−4ac2a∪kω>−b+b2−4ac2a.


## 5.1. Dual-Spin Projectile Control System with Angular Rate Loops

The control system with angular rate loops is shown in Figure3, in which ny and nz are control commands, θ̇ and ψ̇ are feedback signals which can be measured by IMU mounted on the forward part, kω is the control gain, and δz and δy are equivalent canard deflections.Figure 3
Simulation results forkω=0.31. (a) Curves of angle of attack and sideslip. (b) The critical coning motion.
(a)(b)When the control commands for the pitch and yaw channels areδz and δy, the ideal equivalent output of the actuator system should also be δz and δy. However, due to the signal delay, when the actuator system receives the control command, the angular position of the forward part has shifted by Δγ1. Furthermore, due to the dynamic response delay of the actuator, there will be a certain phase lag in the response output of the actuator system, which causes the actual equivalent output to generate an angular position deviation of Δγ2. Therefore, the total delay angle of the control system are γd=Δγ1+Δγ2.The actuator system is assumed to be a second-order system, only considering the steady-state output coupling, we can get(13)δzδy=krcosγd−sinγdsinγdcosγdδczδcy,(14)kr=ks1−Ts2pf22+2μsTspf2,(15)γd=pfτ+arccos1−Ts2pf21−Ts2pf22+2μsTspf2,where Ts is the time constant of the system, μs is the damping ratio, ks is the gain of the servo system, and τ is the command transmission delay time. Obviously, the control action in the pitch channel will cause the movement in the yaw channel, and the control action in the yaw channel will also affect the movement in the pitch channel.It can be seen from Figure3 that the input commands to the actuators can be described as
(16)nczncy=−kω00−kω−θ̇ψ̇.According to the definition of coordinate system and angle, positive equivalent canard deflections will generate positive angle of attack and negative angle of sideslip. And the negative angle of attack will generate negative pitching acceleration, and the positive angle of sideslip will generate positive yaw acceleration. Therefore, the displacement instruction of the canard is obtained as(17)δczδcy=1001nczncy.Meanwhile, based on the assumption that the missile is in horizontal flight, there exists an approximation relationship:α̇≈θ̇, β̇≈−ψ̇. Thus, Equation (17) can be expressed as
(18)δczδcy=−kω00−kωδczδcy.Finally, substituting Equation (18) into Equation (13), we can get the actual equivalent canard outputs with all the delays as
(19)δzδy=krcosγd−sinγdsinγdcosγdδczδcy.Converting Equation (19) into the complex form, one has
(20)δ=krkω−sinγd+icosγdξ̇.


## 5.2. Stability Conditions for the Dual-Spin Projectile Control System

Substituting Equation (20) into the angular motion Equation (12), we can get
(21)ξ¨+a1−b21+b3krkωcosγd−ib22−b3krkωsinγdξ̇−b11+a1b21+ib12+a1b22ξ=0.Rewrite the above equation into the simplified form as(22)ξ¨+HC−iPCξ̇−M+iQ=0,where
(23)HC=a1−b21+b3krkωcosγd,PC=b22−b3krkωsinγd.Apparently, the characteristic equation of Equation (22) has the form
(24)λ2+HC−iPCλ−M+iQ=0.Then, we can obtain the characteristic roots as(25)λ1,2=−HC−iPC±HC−iPC2+4M+iQ2.Assuming thatRc=Rcr+iRci=HC−iPC2+4M+iQ, one has
(26)Rcr=HC2−PC2+4M,Rci=−2HCPC+4Q.The real part ofλ1,2 is
(27)λR1,2=−HC±R+Rcr/22,where R=Rcr2+Rci2; it is obvious that the control system is stable only if the real parts of the characteristic roots are negative, that is
(28)−HC±R+Rcr2<0.The stability condition (28) can be formulated as
(29)−HC<0HC2>R+Rcr2.By substituting the expressions of these variables, the necessary and sufficient conditions for the stability of the dual-spin projectile control system can be obtained as(30)−L/VCmq+Cmδkrkωcosγd>0−Cmα−L/VCmq+Cmδkrkωcosγd2−ItLCnpα2pa2/QSV2+L/VCnpαpa−L/VCmq+CmδkrkωcosγdIxapa/QSL−Cmδkrkωsinγd>0.


## 5.3. Stability Analysis

Firstly, by analysing the first inequality in the stability condition (30), which is the pitching damping moment coefficient Cmq, it can be seen obviously that when the delay angle γd<90∘, the first inequality is satisfied. When γd>90∘, the control parameter kω has a upper limit as follows:
(31)kω<LCmqVCmδkrcosγd.Then, analysing the second inequality in the stability condition (30), since the dual-spin projectile is static unstable, we have Cmα>0. The first and the second terms of the second inequality are negative; thus, whether the second inequality is true depends on the third term. The magnitude and sign of the third term are mainly affected by the spinning rate of the aft part pa, the delay angle γd, and the control gain kω.The second inequality can be rewritten as a polynomial in terms ofkω:
(32)fkω=akω2+bkω+c>0,where
(33)a=kr2Cmδ2cosγd−Cmα−LVCnpαpasinγd,b=Cmδkr2LVCmαCmqcosγd+LV2CmqCnpαpasinγd+IxaQSVpaCmqCnpαpacosγd,c=−LCnpαpa2QSV2ItCnpα+Ixa.Whenγd<90∘, one gets a<0, b>0, and c<0. The two zero points of fkω are obtained as
(34)0<kω11=−b+b2−4ac2a<kω12=−b−b2−4ac2a.Only whenkω11<kω<kω12 one gets fkω>0. The sufficient and necessary condition for the coning motion stability can be derived as
(35)−b+b2−4ac2a<kω<−b−b2−4ac2a.Whenγd>90∘, one gets a>0, b<0, and c<0; the two zero points of fkω can be obtained as
(36)kω21=−b−b2−4ac2a<0<kω22=−b+b2−4ac2a.Only whenkω<kω21 or kω>kω22 one gets fkω<0. The sufficient and necessary condition for the coning motion stability can be derived as
(37)kω<LCmqVCmδkrcosγd∩kω<−b−b2−4ac2a∪kω>−b+b2−4ac2a.


## 6. Numerical Simulations

To demonstrate the proposed stability condition in the last section, numerical simulations are run for a 155 mm dual-spin projectile. All simulation results are based on a fixed-step (0.001 s), fourth-order Runge–Kutta numerical program by using the MATLAB simulation software. The parameters of the dual-spin projectile are listed in Table1.Table 1
Parameters of the 155 mm dual-spin projectile.
ParameterValueParametersValuems42.8It1.893L0.155S0.01887V450pa1450Cmα3.6Cmq13.2Cmδ3.6Cnpα0.7The delay angle of the control system is set as32.5∘; then, the calculated upper bound of the control loop gain can be obtained as 0.31 according to Equation (37). When the control loop gain kω=0.23, which satisfies the stability condition, the simulation results are obtained as shown in Figure 4. It can be seen obviously that the coning motion of the projectile converges to zero quickly. When kω=0.31, which is the critical value, the simulation results are illustrated as shown in Figure 3. It is observed that the coning motion of the missile neither converges nor diverges but presents a critical stable state. When kω=0.52, the simulation results are obtained as shown in Figure 5. It can be seen that the coning motion is divergent.Figure 4
Simulation results forkω=0.23. (a) Curves of angle of attack and sideslip. (b) Stable coning motion.
(a)(b)Figure 5
Simulation results forkω=0.52. (a) Curves of angle of attack and sideslip. (b) Unstable coning motion.
(a)(b)Furthermore, the influence of the total delay angle of the control system and the spinning ratepa of the aft part to the stability criterion is demonstrated. The stable regions corresponding to different total delay angles γd for the dual-spin projectile control system are illustrated in Figure 6. It can be seen that the stable region of the control system shrinks as the delay angle increases. Therefore, in the design of the control system for the dual-spin projectile, the delay angle should be reduced as far as possible to ensure that the control system is stable.Figure 6
The stable region for different delay angles.The stable regions corresponding to different spinning ratespa of the aft part are illustrated in Figure 7. It is noticed that the stable region of the control system shrinks as the spinning rate decreases. Furthermore, when the spinning rate of the aft part drops to a certain value, any control parameter kω cannot guarantee the stability of the dual-spin projectile system. In the actual flight, the spinning rate of the aft part gradually decreases with flight time, which will reduce the stability region of the projectile system. For this reason, the flight stability of the dual-spin projectile is more easily destroyed, which needs to be paid enough attention in the design process.Figure 7
The stable region for different spinning rate of the aft part.


## 7. Conclusions

In this paper, the mathematical equation of a dual-spin projectile is established. The sufficient and necessary condition of the coning motion stability for dual-spin projectiles with angular rate loops is analytically derived and further verified by numerical simulations. Simulation results show that there exists a stability boundary value for the control gain. If the control gain exceeds it, the coning motion of the dual-spin projectile will diverge, and the system will become unstable. It is also noticed that as the delay angle increases, the stability region of the system decreases, while, as the spinning rate of the aft part decreases, the stability region of the system decreases greatly. This paper is mainly based on the linearization theory of projectiles, so the stability condition obtained in this paper is applicable to the linearized missile model. In the future, we will focus on the stability analysis of nonlinear model of the dual-spin projectile.



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