# Stability of Moving Mass Control Spinning Missiles with Angular Rate Loops

Authors: Qiushi Zheng; Zhiming Zhou
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Moving mass control (MMC) is a new control method in control field. It is a potential way to solve the problem of aerodynamic rudder control insufficiency caused by the low density of upper atmosphere, to reduce the high speed missile aerodynamic thermal load, and to solve the problem of rudder surface ablation. However, the spinning of the airframe and the movement of internal moving mass induce the serious dynamic cross-coupling between pitch and yaw channels, which may lead to system instability in the form of a divergent coning motion. In this paper, the mathematical model of the MMC missile is established, and the angular motion equation is finally obtained by some linearized assumptions. Then, the sufficient and necessary conditions of coning motion stability for MMC missiles with angular rate loops under fast and slow spinning rates are analytically derived and further verified by numerical simulations. It is noticed that the upper bound of the control gain is affected by the location of the moving mass and the spinning rate of the missile.


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## Body

## 1. Introduction

The moving mass control (MMC) technology changes the position of center of mass of the system by the displacement of the internal moving mass to generate corresponding control torques, thereby changing the flight attitude of the missile [1–3]. Moving mass control missile has attracted much attention because of its special advantages. When the missile flies in the high altitude, the conventional aerodynamic control cannot provide the required lateral acceleration, as the density of air is too low. However, the moving mass control has the potential to solve this problem. Moreover, as the moving mass is arranged in the airframe, the missile has a better aerodynamic property, which reduces the aerodynamic heat of the warhead and avoids the problem of rudder ablation [2]. According to the number of moving masses of the actuator, the MMC missiles can be divided into three types: single MMC missiles [4], double MMC missiles [5], and triple MMC missiles.There is a heavy coupling between the pitch and yaw channel of the moving mass control spinning missile. On the one hand, it is due to Magnus and gyroscopic effects caused by the rotation of the missile. On the other hand, the motion of the moving mass causes the deviation of the center of mass of the system and the deviation of the principal axis of inertia, which aggravates the coupling between pitch and yaw channels. Many studies have been proposed focusing on the control for such a system with heavy coupling, nonlinear dynamics, and parametric uncertainties. Zhang et al. [6] divided the dynamics of the MMAV into the fast state part and the slow state part and designed an autopilot for a nonlinear 6-DoF mass moment aerospace vehicle based on fuzzy sliding mode control, using dynamic inversion techniques. Then, Zhang et al. [7] designed the flight control system for the MMAV via utilizing nonlinear predictive control approach.As for the stability for spinning missiles, many theoretical research studies have already been proposed. Murphy and Flatus [8–10] analyzed the factors that cause the coning motion of the missile, including the Magnus effect, inertial gyroscope effect, and aerodynamic asymmetry. Furthermore, for the stability of controlled spinning missiles, Yan et al. [11] studied the stability conditions of spinning missiles with rate loop, Li et al. [12] studied the stability of spinning missiles with an acceleration autopilot. In addition, Zhou et al. [13] studied the coning motion instability induced by hinge moment of the actuator.Previous research studies mainly focus on aerodynamic control missiles. For the study of the stability of spinning aircraft with internal moving masses, current research studies are mostly focused on the instability of coning motion induced by mass deviation. For example, Carrier and Miles [14] proposed that the center of mass of the rocket changes due to the internal fluid motion, which led to unstable coning motion of the rocket. El-Gohary [15, 16] studied the stability of the mass moment satellite by means of the Lyapunov equation and obtained the required force and torque of the servo system satisfying the stability conditions. However, few of the existing literatures have considered the coning motion of a moving mass control spinning missile with the control loop.Thus, this paper focuses on the stability of coning motion for a double moving masses control spinning missile with angular rate loops. The mathematical model of the missile system is established. The sufficient and necessary condition of coning motion stability for MMC spinning missiles 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 installation position of moving masses and spinning rate of the missile on stability is analyzed.


## 2. System Configuration

The moving mass control spinning missile proposed in this paper consists of a rigid bodyB and two radial internal moving masses m1 and m2 as shown in Figure 1. The moving mass m1 moves along the yb axis while the moving mass m2 moves along the zb axis. The mass of the body B is mb, and the two moving masses m1 and m2 have the same mass m; thus, the total mass of the missile system is ms=mb+2m. The mass ratio of the moving mass is μ=m/ms. l is the installation position of the moving mass, and δy and δz are the radial displacements of the two moving masses in the nonspinning coordinate system.Figure 1
System configuration of the moving mass control missile.


## 3. Mathematical Model

The missile system is composed of three parts: the projectile body B and two moving masses. According to the momentum theorem of particle system, the translational dynamics of the missile system can be described as(1)mBdVBdt+∑i=12midVidt=F+Lϑ,ψmsg,where VB is the velocity vector of the body B relative to the center of mass of the missile system S∗ and is given by(2)VB=V−γ×r1+r2,where V=uvwT is the velocity vector of the body B in the nonspinning coordinate system, γ=γ˙00T is the spinning velocity vector, and r1 and r2 are position vectors of the two moving masses in the nonspinning CS. The derivative of equation (2) can be derived as(3)dVBdt=V+ω4×V−γ×r1+r2+γ×r1+r2+ω4×γ×r1+r2,where ω4 is the angular rate of the nonspinning CS with respect to the inertial CS. The position vectors of the two masses in the nonspinning CS can be denoted as(4)r1=lδy40T,r2=l0δz4T,where δy4 and δz4 are projections of δy and δz on the nonspinning CS and are given by(5)δy4δz4=δycosγ−δzsinγδysinγ+δzcosγ.The velocity vector of each moving mass relative to the center of mass of the missile systemS∗ can be expressed as(6)Vi=V+dridt+ω4×ri.The derivative of equation (6) can be derived as(7)dVidt=V+ω4×V+ri+ω4×ri+2×ω4×ri+ω4×ω4×ri.Substituting equations (3) and (7) into equation (1) yields(8)∑i=12miri+ω4×ri+2×ω4×ri+ω4×ω4×ri−mBγ×r1+r2+γ×r1+r2+ω4×γ×r1+r2+msV+ω4×V=F+Lϑ,ψmsg,where F is the vector of aerodynamic force in the nonspinning CS and is given by(9)F=−XYZ=QS−CxCyαα−Cyαβ.According to the theorem of angular momentum, the rotational motion of the missile system can be described as(10)dHBdt+∑i=12dHidt=MS∗,where HB is the angular momentum of the body B, Hi is the angular momentum of the moving mass, and MS∗ is the external moments applied on the missile system, including aerodynamic moments and mass eccentricity moments. HB, Hi, and MS∗ are given by(11)HB=IBω1+μ1+μ22mB∑i=12ri×V1+V2,(12)Hi=1−∑i=12μi2miri×V1+V2,(13)MS∗=M−∑i=12μiri×Fnb,where ω1 is the angular rate of the body CS with respect to the inertial CS. Substituting equations (10)–(12) into equation (9) yields(14)IBω1+ω4×IBω1+mB∑i=12μiri×V1+V2+ω4×∑i=12μiri×V1+V2=M−∑i=12μiri×Fnb.The moments applied on the missile in the nonspinning CS are given by(15)M=MxMyMz=QSLmωx′L/Vωx4myαβ+mωy′L/Vωy4−mμγ˙αmyαα+mωy′L/Vωz4+mμγ˙β.By substituting equation (9) into equation (8) and equation (15) into equation (14), the dynamic equations of the missile system can be finally obtained as(16)u˙v˙w˙+ω4uvw+00μ100μ2δ¨y4δ¨z4+rω˙x4ω˙y4ω˙z4+r2ωx4−mBmsγ˙2ωy42ωz4+ω2−mBmsω4γμ1+μ2lδy4δz4=1ms−XYZ+Lϑ,ψ0g0,(17)Iω˙x4ω˙y4ω˙z4+mB−μ2δz4μ1δy40−μ1+μ2lμ1+μ2l0δ¨yδ¨z+Ixγ¨Ixγ˙ωz4−Ixγ˙ωy4+2mBrω40δ˙y4δ˙z4+ω2rμ1+μ2lδy4δz4=MxMyMz−r−XYZ,where(18)ω4=0−ωz4ωy4ωz40−ωx4−ωy4ωx40,γ˙=00000γ˙0γ˙0,ω2=ω4ω4,r1=00−δy400lδy4−l0,r2=0δz40−δz40l0−l0,r=μ1r1+μ2r2,I=Ix+mBμ1δy42+μ2δz42−mBμ1+μ2lδy4−mBμ1+μ2lδz4−mBμ1+μ2lδy4Iy+mBμ1+μ22l2+μ2δz42−mBμ1δy4δz4−mBμ1+μ2lδz4−mBμ2δy4δz4Iz+mBμ1δy42+μ1+μ22l2.


## 4. Angular Motion of the Moving Mass Control Spinning Missile

Even though the mathematical model described in equations (16) and (17) 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 analyze the flight stability of projectiles and applied in references [8–13]. Therefore, in order to linearize these two equations, the following assumptions are introduced:(1)
The mass ratio is small, soμ=μ1=μ2≪1, 1−μ≈1(2)
The spinning rate in the nonspinning CSωx4 keeps constant and is equal to zero, and γ¨ is small, so γ¨=0(3)
Variablesωy4, ωz4, v, w, α, and β are small(4)
The gravity effect is negligible(5)
l keeps constant, so l˙=l¨=0(6)
The missile is strictly axisymmetric, soIy=IzUnder these assumptions, the equations for lateral translational and rotational motion in equations (16) and (17) can be simplified to(19)v˙=Yms−ωz4u−mBmsγ˙δ˙z4,w˙=Zms+ωy4u+mBmsγ˙δ˙y4,(20)ω˙y4=MyI2+2μlZI2+μXδz4I2−I1I2ωz4γ˙+2μmBlδz¨I2,ω˙z4=MzI2−2μlYI2−μXδy4I2+I1I2ωy4γ˙−2μmBlδy¨I2.The angles of attackα and sideslip β are defined as(21)α=−arctanvu≅−vu,β=arcsinwV≅wV.By defining the complex angle of attackξ=−β+iα, the complex angular rate Ω=ωz4+iωy4, and the complex control instruction δ=δy4−iδz4, equation (19) can be reformulated as(22)Ω=−iξ˙+iQSCx−CyαmsVξ−imBγ˙msVδ˙.Equation (20) can be reformulated as(23)Ω˙=−iI1I2γ˙Ω−2μmBlI2δ¨+i2μlQSCyαI2ξ−iQSLmyαI2ξ−μQSCxI2δ+QSLmωy′I2LVΩ−QSLmμγ˙I2LVξ.Substituting equation (22) into equation (23), the angular motion equation of the moving mass spinning missile can be obtained as(24)ξ¨+Aξ˙+Bξ=C,where A=−mnω−k5+ik1, B=−mnα+k4+immα+k1k5, C=−imnδδ−−mnωk3+ik1k3δ˙−k3+ik2δ¨, k1=I1/I2γ˙, k2=2μmBl/I2, k3=mBγ˙/msV, k4=2μlQSCyα/I2, k5=QSCx−Cyα/mSV, mnω=QSL2Cωz′/I2, mnα=QSLmyα/I2, mmα=QSL2mμγ˙/I2V, and mnδ=QSCxμ/I2.According equation (24), the equilibrium point is determined by(25)ξe=CB=C1+C2B=ξe1+ξe2,where C1=−imnδδ and C2=−k1k3+ik˙3δ˙−k3+ik2δ¨.ξ
e
1 is the complex angle of attack generated by system centroid offset caused by the movement of the moving mass. Suppose that the spinning rate of the missile is zero and the position of the moving mass remains fixed, we get(26)ξe1=−imnδδ−mnα+k4.ξ
e
2 is the complex angle of attack generated by the offset of the principal axis of inertia and was estimated by Hodapp and Clark in [17] as(27)ξe2≅μmSlI2δmax.


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

The control system with angular rate loops is shown in Figure2, in which ny and nz are control commands, ϑ˙ and ψ˙ are feedback signals, and kω is the gain.Figure 2
Moving mass control missile system with angular rate loops.It can be seen from Figure2 that the input commands to the actuators can be described as(28)nynz=−kω00−kωϑ˙ψ˙.According to the definition of coordinate system and angle, negative angle of attack will generate positive pitching acceleration, while positive angle of sideslip will generate positive yaw acceleration. Therefore, the displacement instruction of the moving mass is obtained as(29)δycδzc=−1001nynz.Meanwhile, based on the assumption that the missile is in horizontal flight, there exists an approximation relationship:α˙=ϑ˙ and β˙=ψ˙. Thus, equation (29) can be expressed as(30)δycδzc=−kω00kωα˙β˙.Converting equation (30) into the complex form, one has(31)δ=ikωξ˙.Substituting equation (31) into equation (24) yields(32)−k2kω+ik3kω+1−k1k3kω−imnωk3kωξ¨+−kωmnδ−mnω−k5+ik1ξ˙+−mnα+k4+immα+k1k5ξ=0.


### 5.1. Slow Spinning Rate Case

For slowly spinning missiles, the main factor for the generation of angle of attack is the mass eccentric moment caused by the movement of moving masses. Therefore, when studying the stability of slowly spinning missiles, the first- and second-order derivatives of the position of moving masses can be ignored. Then, equation (32) can be simplified as(33)ξ¨+Hc+iPcξ˙−Mc+iQcξ=0,where Hc=−kωmnδ−mnω−k5, Pc=k1, Mc=mnα−k4, and Qc=−mmα+k1k5.The corresponding characteristic equation is(34)λ2+Hc+iPcλ−Mc+iQc=0.AssumingHc+iPc2+4Mc+iQc=Rc, where(35)Rc=Rcre+iRcim,one gets(36)Rcre=Hc2−Pc2+4Mc,Rcim=2HcPc+4Qc,.Then, the characteristic roots of equation (34) are given by(37)λ1,2=12−Hc±Rc+Rcre2+12−Pc±Rc−Rcre2i.According to Lyapunov stability theory, the sufficient and necessary condition for stability of the moving mass missile under low spinning rate with rate loops can be obtained as(38)−Hc±Rc+Rcre2<0.BecauseRc+Rcre/2>0, in order to ensure that equation (38) is true, the following inequality must be met:(39)−Hc<−Rc+Rcre2.SubstitutingHc, Rc, and Rcre into equation (39) yields(40)mnδ2kω2−mnα+k4+2mnω+k5−mnα+k4−k1mmα+k1k5mnδkω+mnω+k52−mnα+k4−mnω+k5k1mmα+k1k5−mmα+k1k52>0.To facilitate the analysis, a polynomialfkω is introduced:(41)fkω=akω2+bkω+c,where a=mnδ2−mnα+k4, b=2mnω+k5−mnα+k4−k1mmα+k1k5mnδ, and c=mnω+k52−mnα+k4−mnω+k5k1mmα+k1k5−mmα+k1k52.For slowly spinning missiles,k1 and mmα are small. The sign of a and b mainly depends on the sign of the first term on the right-hand side, so a and b have opposite signs. Two cases are discussed below:(1)
The first case is whena>0, one gets −mnα+k4>0, b<0, and c>0, and the curve of fkω is illustrated by Iin Figure 3. The intersections of fkω and the axis are given by(42)kω11=−b−b2−4ac2a,kω12=−b+b2−4ac2a.Thus, only whenkω<kω11 or kω>kω12, one gets fkω>0. The sufficient and necessary condition for the coning motion stability can be derived as(43)kω∈0,−mnω+k5mnδ∩0,kω11∩kω12,∞.(2)
The second case is whena<0, one gets −mnα+k4<0, b>0, and c could be positive or negative. Ignore the sign of c, and the intersections of fkω and the x axis are given by(44)kω21=−b+b2−4ac2a,kω22=−b−b2−4ac2a.Thus, only whenkω21<kω<kω22, one gets fkω>0. The sufficient and necessary condition for the coning motion stability can be derived as(45)kω∈0,−mnω+k5mnδ∩kω21,kω22.Figure 3
Simulation results forkω=0.1933. (a) The stable coning motion. (b) Curves of angle of attack and angle of sideslip.
(a)
(b)


### 5.2. Fast Spinning Rate Case

For fast spinning missiles, the main factor for generation of angle of attack is the deviation of the principal axis of inertia. For the convenience of analyzing, assume that moving masses are installed at the center of mass of the projectile body, that is,l=0. Then, one gets k2=0 and k4=0. By neglecting the effect of Magnus moment and considering k5 to be small, equation (32) can be simplified as(46)ξ…+a01+ia11ξ¨+a02+ia12ξ˙+ia13ξ=0,where a01=−mnω, a11=k1k3kω−1/k3kω, a02=k1/k3kω, a12=kωmnδ+mnω/k3kω, and a13=−mnα/k3kω.The characteristic equation of equation (46) is given by(47)λ3+a01+ia11λ2+a02+ia12λ+ia13=0.According to the theorem proved in [18], the sufficient and necessary condition for stability of the moving mass missile under fast spinning rate with rate loops can be expressed as(48)a01>0,a012a02+a01a11a12−a122>0,a012a02+a01a11a12−a122a01a12a13−a012a132>0.Becausea01=−mnω>0, equation (48) is rewritten as(49)c1=a012a02+a01a11a12−a122>0,c2=a01a12a13>0,c3=a012a02+a01a11a12−a122a01a12a13−a012a132>0.Substituting all the coefficients into equation (49) yields(50)c1=a012a02+a01a11a12−a122=−mnωmnδkω−mnωmnδk1k3+mnδ2kω2k32kω2>0,(51)c2=a01a12a13=−mnωmnαkωmnδ+mnωk32kω2>0,(52)c3=−p1kω2+p2kω+p3k34kω3>0,where(53)p1=mnω2mnδ2mnαk1k3+mnωmnδ3mnα,p2=2mnω2mnδ2mnα+mnω3mnδmnαk1k3+−mnω4mnα2k32,p3=mnω3mnδmnα.For the moving mass control missile under fast spinning rate, one hasmnωmnδk1k3>mnδ2; thus, equation (50) is always true. Because mnα>0, to make equation (51) true, one should have(54)kω<−mnωmnδ=kω21.To make equation (52) true, one should have(55)p1kω2+p2kω+p3<0.For fast spinning missile, one hasp1>0, p2<0, and p3<0. Thus, the true condition for equation (55) can be obtained as(56)0<kω<−p2+p22−4p1p32p1=kω22.Finally, the sufficient and necessary condition for stability of moving mass missile under fast spinning rate with rate loops can be expressed as(57)kω∈0,kω21∩0,kω22.


## 5.1. Slow Spinning Rate Case

For slowly spinning missiles, the main factor for the generation of angle of attack is the mass eccentric moment caused by the movement of moving masses. Therefore, when studying the stability of slowly spinning missiles, the first- and second-order derivatives of the position of moving masses can be ignored. Then, equation (32) can be simplified as(33)ξ¨+Hc+iPcξ˙−Mc+iQcξ=0,where Hc=−kωmnδ−mnω−k5, Pc=k1, Mc=mnα−k4, and Qc=−mmα+k1k5.The corresponding characteristic equation is(34)λ2+Hc+iPcλ−Mc+iQc=0.AssumingHc+iPc2+4Mc+iQc=Rc, where(35)Rc=Rcre+iRcim,one gets(36)Rcre=Hc2−Pc2+4Mc,Rcim=2HcPc+4Qc,.Then, the characteristic roots of equation (34) are given by(37)λ1,2=12−Hc±Rc+Rcre2+12−Pc±Rc−Rcre2i.According to Lyapunov stability theory, the sufficient and necessary condition for stability of the moving mass missile under low spinning rate with rate loops can be obtained as(38)−Hc±Rc+Rcre2<0.BecauseRc+Rcre/2>0, in order to ensure that equation (38) is true, the following inequality must be met:(39)−Hc<−Rc+Rcre2.SubstitutingHc, Rc, and Rcre into equation (39) yields(40)mnδ2kω2−mnα+k4+2mnω+k5−mnα+k4−k1mmα+k1k5mnδkω+mnω+k52−mnα+k4−mnω+k5k1mmα+k1k5−mmα+k1k52>0.To facilitate the analysis, a polynomialfkω is introduced:(41)fkω=akω2+bkω+c,where a=mnδ2−mnα+k4, b=2mnω+k5−mnα+k4−k1mmα+k1k5mnδ, and c=mnω+k52−mnα+k4−mnω+k5k1mmα+k1k5−mmα+k1k52.For slowly spinning missiles,k1 and mmα are small. The sign of a and b mainly depends on the sign of the first term on the right-hand side, so a and b have opposite signs. Two cases are discussed below:(1)
The first case is whena>0, one gets −mnα+k4>0, b<0, and c>0, and the curve of fkω is illustrated by Iin Figure 3. The intersections of fkω and the axis are given by(42)kω11=−b−b2−4ac2a,kω12=−b+b2−4ac2a.Thus, only whenkω<kω11 or kω>kω12, one gets fkω>0. The sufficient and necessary condition for the coning motion stability can be derived as(43)kω∈0,−mnω+k5mnδ∩0,kω11∩kω12,∞.(2)
The second case is whena<0, one gets −mnα+k4<0, b>0, and c could be positive or negative. Ignore the sign of c, and the intersections of fkω and the x axis are given by(44)kω21=−b+b2−4ac2a,kω22=−b−b2−4ac2a.Thus, only whenkω21<kω<kω22, one gets fkω>0. The sufficient and necessary condition for the coning motion stability can be derived as(45)kω∈0,−mnω+k5mnδ∩kω21,kω22.Figure 3
Simulation results forkω=0.1933. (a) The stable coning motion. (b) Curves of angle of attack and angle of sideslip.
(a)
(b)


## 5.2. Fast Spinning Rate Case

For fast spinning missiles, the main factor for generation of angle of attack is the deviation of the principal axis of inertia. For the convenience of analyzing, assume that moving masses are installed at the center of mass of the projectile body, that is,l=0. Then, one gets k2=0 and k4=0. By neglecting the effect of Magnus moment and considering k5 to be small, equation (32) can be simplified as(46)ξ…+a01+ia11ξ¨+a02+ia12ξ˙+ia13ξ=0,where a01=−mnω, a11=k1k3kω−1/k3kω, a02=k1/k3kω, a12=kωmnδ+mnω/k3kω, and a13=−mnα/k3kω.The characteristic equation of equation (46) is given by(47)λ3+a01+ia11λ2+a02+ia12λ+ia13=0.According to the theorem proved in [18], the sufficient and necessary condition for stability of the moving mass missile under fast spinning rate with rate loops can be expressed as(48)a01>0,a012a02+a01a11a12−a122>0,a012a02+a01a11a12−a122a01a12a13−a012a132>0.Becausea01=−mnω>0, equation (48) is rewritten as(49)c1=a012a02+a01a11a12−a122>0,c2=a01a12a13>0,c3=a012a02+a01a11a12−a122a01a12a13−a012a132>0.Substituting all the coefficients into equation (49) yields(50)c1=a012a02+a01a11a12−a122=−mnωmnδkω−mnωmnδk1k3+mnδ2kω2k32kω2>0,(51)c2=a01a12a13=−mnωmnαkωmnδ+mnωk32kω2>0,(52)c3=−p1kω2+p2kω+p3k34kω3>0,where(53)p1=mnω2mnδ2mnαk1k3+mnωmnδ3mnα,p2=2mnω2mnδ2mnα+mnω3mnδmnαk1k3+−mnω4mnα2k32,p3=mnω3mnδmnα.For the moving mass control missile under fast spinning rate, one hasmnωmnδk1k3>mnδ2; thus, equation (50) is always true. Because mnα>0, to make equation (51) true, one should have(54)kω<−mnωmnδ=kω21.To make equation (52) true, one should have(55)p1kω2+p2kω+p3<0.For fast spinning missile, one hasp1>0, p2<0, and p3<0. Thus, the true condition for equation (55) can be obtained as(56)0<kω<−p2+p22−4p1p32p1=kω22.Finally, the sufficient and necessary condition for stability of moving mass missile under fast spinning rate with rate loops can be expressed as(57)kω∈0,kω21∩0,kω22.


## 6. Numerical Simulation Results

To demonstrate the proposed stability condition above, numerical simulations are run for two sample moving mass missiles with different spinning rates.


### 6.1. Slow Spinning Rate Case

The parameters of a slowly spinning missile are listed in Table1.Table 1
Parameters of a slowly spinning missile.
Parameters
Value
m
s
kg
96.6
μ
0.04
L
m
1.5
S
m
2
0.2
γ
˙
rad
⋅
s
−
1
10
l
m
0.1
I
1
kg
⋅
m
2
5.4
I
2
kg
⋅
m
2
58.5
C
n
ω
z
−5.3
C
n
α
−0.1
C
m
α
−1.5
V
m
⋅
s
−
1
1140According to the formulae derived above, the calculated upper bound of the control loop gain is obtained as 0.3866. The simulation results for the control loop gainkω=0.1933, which satisfies the stability condition, are shown in Figure 3. It can be seen obviously that the coning motion of the missile converges to zero quickly.The simulation results for the critical control loop gainkω=0.3866 are shown in Figure 4. It is observed that the coning motion of the missile neither converges nor diverges but presents a critical stable state. The simulation results for kω=0.5798 are shown in Figure 5. It can be seen that the coning motion is divergent.Figure 4
Simulation results forkω=0.3866. (a) The critical coning motion. (b) Curves of angle of attack and angle of sideslip.
(a)
(b)Figure 5
Simulation results forkω=0.5798. (a) The unstable coning motion. (b) Curves of angle of attack and angle of sideslip.
(a)
(b)


### 6.2. Influence of the System Parameter

In this section, the influence of the location of the moving massl and the spinning rate of the missile γ˙ on the stability criterion is demonstrated. The relation between the installation position l of the moving mass and the upper bound of the control loop gain kω is shown in Table 2. It can be observed from the table that the upper bound of kω increases as the location of the moving mass moves towards the warhead. This is because with the increase of l, the static stability of the missile is continuously strengthened, which leads to the increase of the dynamic stability region and the increase of the upper bound of kω.Table 2
Upper bounds ofkω under different installation positions of moving masses.
Parameters
Value
l
m
−0.1
0
0.1
0.2
0.3
0.4
k
ω
0.31
0.35
0.385
0.416
0.44
0.46The relationship between the spinning rate and the upper bound of the design gain kw is shown in Table3.Table 3
Upper bounds ofkω under different spinning rates.
Parameters
Value
γ
˙
rad
/
s
1
2
3
4
5
6
7
8
9
10
k
ω
0.82
0.77
0.71
0.67
0.61
0.56
0.50
0.45
0.40
0.35As can be seen obviously, the increase of the spinning rate decreases the stable region of the control design gain. This is because the higher spinning rate leads to a more serious coupling between pitch and yaw channels.


### 6.3. Fast Spinning Rate Case

The parameters of a fast spinning missile are listed in Table4.Table 4
Parameters of a slowly spinning missile.
Parameters
Value
m
s
kg
96.6
μ
0.04
L
m
1.5
S
m
2
0.15
γ
˙
rad
⋅
s
−
1
1000
I
1
kg
⋅
m
2
5.4
I
2
kg
⋅
m
2
58.5
C
n
ω
z
−4.8
C
n
α
0.1
V
m
⋅
s
−
1
1140The upper bound ofkω in this case is 0.5522 according to the stability condition described in equation (57). The coning motions under kω=0.4418 and kω=0.6626 are shown in Figures 6 and 7, respectively, from which it can be seen that the coning motion is stable when kω=0.4418, while it is unstable when kω=0.6626.Figure 6
The stable coning motion.Furthermore, the upper bound ofkω under different spinning rates is shown in Table 5. It can be verified that the stable region of the control gain increases with the increase of the spinning rate. This is because the higher spinning rate causes a stronger inertia moment.Table 5
Upper bounds ofkω under different spinning rates.
Parameters
Value
γ
˙
rad
/
s
500
600
700
800
900
1000
1100
k
ω
0.14
0.25
0.33
0.41
0.49
0.55
0.61Figure 7
The unstable coning motion.


## 6.1. Slow Spinning Rate Case

The parameters of a slowly spinning missile are listed in Table1.Table 1
Parameters of a slowly spinning missile.
Parameters
Value
m
s
kg
96.6
μ
0.04
L
m
1.5
S
m
2
0.2
γ
˙
rad
⋅
s
−
1
10
l
m
0.1
I
1
kg
⋅
m
2
5.4
I
2
kg
⋅
m
2
58.5
C
n
ω
z
−5.3
C
n
α
−0.1
C
m
α
−1.5
V
m
⋅
s
−
1
1140According to the formulae derived above, the calculated upper bound of the control loop gain is obtained as 0.3866. The simulation results for the control loop gainkω=0.1933, which satisfies the stability condition, are shown in Figure 3. It can be seen obviously that the coning motion of the missile converges to zero quickly.The simulation results for the critical control loop gainkω=0.3866 are shown in Figure 4. It is observed that the coning motion of the missile neither converges nor diverges but presents a critical stable state. The simulation results for kω=0.5798 are shown in Figure 5. It can be seen that the coning motion is divergent.Figure 4
Simulation results forkω=0.3866. (a) The critical coning motion. (b) Curves of angle of attack and angle of sideslip.
(a)
(b)Figure 5
Simulation results forkω=0.5798. (a) The unstable coning motion. (b) Curves of angle of attack and angle of sideslip.
(a)
(b)


## 6.2. Influence of the System Parameter

In this section, the influence of the location of the moving massl and the spinning rate of the missile γ˙ on the stability criterion is demonstrated. The relation between the installation position l of the moving mass and the upper bound of the control loop gain kω is shown in Table 2. It can be observed from the table that the upper bound of kω increases as the location of the moving mass moves towards the warhead. This is because with the increase of l, the static stability of the missile is continuously strengthened, which leads to the increase of the dynamic stability region and the increase of the upper bound of kω.Table 2
Upper bounds ofkω under different installation positions of moving masses.
Parameters
Value
l
m
−0.1
0
0.1
0.2
0.3
0.4
k
ω
0.31
0.35
0.385
0.416
0.44
0.46The relationship between the spinning rate and the upper bound of the design gain kw is shown in Table3.Table 3
Upper bounds ofkω under different spinning rates.
Parameters
Value
γ
˙
rad
/
s
1
2
3
4
5
6
7
8
9
10
k
ω
0.82
0.77
0.71
0.67
0.61
0.56
0.50
0.45
0.40
0.35As can be seen obviously, the increase of the spinning rate decreases the stable region of the control design gain. This is because the higher spinning rate leads to a more serious coupling between pitch and yaw channels.


## 6.3. Fast Spinning Rate Case

The parameters of a fast spinning missile are listed in Table4.Table 4
Parameters of a slowly spinning missile.
Parameters
Value
m
s
kg
96.6
μ
0.04
L
m
1.5
S
m
2
0.15
γ
˙
rad
⋅
s
−
1
1000
I
1
kg
⋅
m
2
5.4
I
2
kg
⋅
m
2
58.5
C
n
ω
z
−4.8
C
n
α
0.1
V
m
⋅
s
−
1
1140The upper bound ofkω in this case is 0.5522 according to the stability condition described in equation (57). The coning motions under kω=0.4418 and kω=0.6626 are shown in Figures 6 and 7, respectively, from which it can be seen that the coning motion is stable when kω=0.4418, while it is unstable when kω=0.6626.Figure 6
The stable coning motion.Furthermore, the upper bound ofkω under different spinning rates is shown in Table 5. It can be verified that the stable region of the control gain increases with the increase of the spinning rate. This is because the higher spinning rate causes a stronger inertia moment.Table 5
Upper bounds ofkω under different spinning rates.
Parameters
Value
γ
˙
rad
/
s
500
600
700
800
900
1000
1100
k
ω
0.14
0.25
0.33
0.41
0.49
0.55
0.61Figure 7
The unstable coning motion.


## 7. Conclusion

In this paper, the mathematical equation of a moving mass spinning missile is established. The sufficient and necessary condition of the coning motion stability for moving mass missiles with angular rate loops is analytically derived under different spinning rates 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 missile will diverge and the system will become unstable. It is also noticed that for the slowly spinning missile, as the location of the moving mass increases, the stability region of the system increases, while the spinning rate of the missile increases and the stability region of the system decreases greatly. For the fast spinning missile, the system stability region increases with the increase of the spinning rate. 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 moving mass control missile.



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*Source: 7832602-2019-11-27.xml*