Stability of Moving Mass Control Spinning Missiles with Angular Rate Loops

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 speedmissile 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 MMCmissile is established, and the angular motion equation is finally obtained by some linearized assumptions. /en, the sufficient and necessary conditions of coning motion stability for MMCmissiles 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.


Introduction
e 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][2][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.
ere 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. en, 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][9][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.
us, this paper focuses on the stability of coning motion for a double moving masses control spinning missile with angular rate loops. e mathematical model of the missile system is established. e 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. e 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.

System Configuration
e moving mass control spinning missile proposed in this paper consists of a rigid body B and two radial internal moving masses m 1 and m 2 as shown in Figure 1. e moving mass m 1 moves along the y b axis while the moving mass m 2 moves along the z b axis. e mass of the body B is m b , and the two moving masses m 1 and m 2 have the same mass m; thus, the total mass of the missile system is m s � m b + 2m. e mass ratio of the moving mass is μ � m/m s . l is the installation position of the moving mass, and δ y and δ z are the radial displacements of the two moving masses in the nonspinning coordinate system.

Mathematical Model
e 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 where V B is the velocity vector of the body B relative to the center of mass of the missile system S * and is given by where V � u v w T is the velocity vector of the body B in the nonspinning coordinate system, γ � _ c 0 0 T is the spinning velocity vector, and r 1 and r 2 are position vectors of the two moving masses in the nonspinning CS. e derivative of equation (2) can be derived as where ω 4 is the angular rate of the nonspinning CS with respect to the inertial CS. e position vectors of the two masses in the nonspinning CS can be denoted as where δ y4 and δ z4 are projections of δ y and δ z on the nonspinning CS and are given by e velocity vector of each moving mass relative to the center of mass of the missile system S * can be expressed as e derivative of equation (6) can be derived as Substituting equations (3) and (7) into equation (1) yields where F is the vector of aerodynamic force in the nonspinning CS and is given by 2 Mathematical Problems in Engineering According to the theorem of angular momentum, the rotational motion of the missile system can be described as where H B is the angular momentum of the body B, H i is the angular momentum of the moving mass, and M S * is the external moments applied on the missile system, including aerodynamic moments and mass eccentricity moments. H B , H i , and M S * are given by where ω 1 is the angular rate of the body CS with respect to the inertial CS. Substituting equations (10)-(12) into equation (9) yields e moments applied on the missile in the nonspinning CS are given by 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 where Mathematical Problems in Engineering

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. is theory has been regarded as an effective tool to analyze the flight stability of projectiles and applied in references [8][9][10][11][12][13]. erefore, in order to linearize these two equations, the following assumptions are introduced: (1) e mass ratio is small, so μ � μ 1 � μ 2 ≪ 1, 1 − μ ≈ 1 (2) e spinning rate in the nonspinning CS ω x4 keeps constant and is equal to zero, and € c is small, so € c � 0 (3) Variables ω y4 , ω z4 , v, w, α, and β are small (4) e gravity effect is negligible (5) l keeps constant, so _ l � € l � 0 (6) e missile is strictly axisymmetric, so I y � I z Under these assumptions, the equations for lateral translational and rotational motion in equations (16) and (17) can be simplified to e angles of attack α and sideslip β are defined as By defining the complex angle of attack ξ � − β + iα, the complex angular rate Ω � ω z4 + iω y4 , and the complex control instruction δ � δ y4 − iδ z4 , equation (19) can be reformulated as Equation (20) can be reformulated as Substituting equation (22) into equation (23), the angular motion equation of the moving mass spinning missile can be obtained as According equation (24), the equilibrium point is determined by where ξ e1 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 ξ e2 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

Stability of the Moving Mass Spinning
Missile with the Angular Rate Loop e control system with angular rate loops is shown in Figure 2, in which n y and n z are control commands, _ ϑ and _ ψ are feedback signals, and k ω is the gain.
It can be seen from Figure 2 that the input commands to the actuators can be described as 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. erefore, the displacement instruction of the moving mass is obtained as δ yc Meanwhile, based on the assumption that the missile is in horizontal flight, there exists an approximation relationship: _ α � _ ϑ and _ β � _ ψ. us, equation (29) can be expressed as Converting equation (30) into the complex form, one has

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. erefore, when studying the stability of slowly spinning missiles, the first-and second-order derivatives of the position of moving masses can be ignored. en, equation (32) can be simplified as where e corresponding characteristic equation is one gets en, the characteristic roots of equation (34) are given by 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 Because ������������ (|R c | + R cre )/2 > 0, in order to ensure that equation (38) is true, the following inequality must be met: Substituting H c , R c , and R cre into equation (39) yields  To facilitate the analysis, a polynomial f(k ω ) is introduced: For slowly spinning missiles, k 1 and m mα are small. e 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) e first case is when a > 0, one gets − m α n + k 4 > 0, b < 0, and c > 0, and the curve of f(k ω ) is illustrated by Iin Figure 3. e intersections of f(k ω ) and the axis are given by us, only when k ω < k ω11 or k ω > k ω12 , one gets f(k ω ) > 0. e sufficient and necessary condition for the coning motion stability can be derived as (2) e second case is when a < 0, one gets − m α n + k 4 < 0, b > 0, and c could be positive or negative. Ignore the sign of c, and the intersections of f(k ω ) and the x axis are given by us, only when k ω21 < k ω < k ω22 , one gets f(k ω ) > 0. e sufficient and necessary condition for the coning motion stability can be derived as

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. en, one gets k 2 � 0 and k 4 � 0. By neglecting the effect of Magnus moment and considering k 5 to be small, equation (32) can be simplified as ξ ...
Because a 01 � − m ω n > 0, equation (48) is rewritten as c 1 � a 2 01 a 02 + a 01 a 11 a 12 − a 2 12 > 0, c 2 � a 01 a 12 a 13 > 0, c 3 � a 2 01 a 02 + a 01 a 11 a 12 − a 2 12 a 01 a 12 a 13 − a 2 01 a 13 2 > 0. where For the moving mass control missile under fast spinning rate, one has |m ω n m δ n k 1 k 3 | > (m δ n ) 2 ; thus, equation (50) is always true. Because m α n > 0, to make equation (51) true, one should have To make equation (52) true, one should have For fast spinning missile, one has p 1 > 0, p 2 < 0, and p 3 < 0. us, the true condition for equation (55) can be obtained as Finally, the sufficient and necessary condition for stability of moving mass missile under fast spinning rate with rate loops can be expressed as

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

Slow Spinning Rate
Case. e parameters of a slowly spinning missile are listed in Table 1.
According to the formulae derived above, the calculated upper bound of the control loop gain is obtained as 0.3866. e simulation results for the control loop gain k ω � 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. e simulation results for the critical control loop gain k ω � 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. e simulation results for k ω � 0.5798 are shown in Figure 5. It can be seen that the coning motion is divergent.    Table 2. It can be observed from the table that the upper bound of k ω increases as the location of the moving mass moves towards the warhead. is 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 k ω . e relationship between the spinning rate and the upper bound of the design gain kw is shown in Table 3.
As can be seen obviously, the increase of the spinning rate decreases the stable region of the control design gain.
is is because the higher spinning rate leads to a more serious coupling between pitch and yaw channels.

Fast Spinning Rate Case.
e parameters of a fast spinning missile are listed in Table 4. e upper bound of k ω in this case is 0.5522 according to the stability condition described in equation (57). e coning motions under k ω � 0.4418 and k ω � 0.6626 are shown in Figures 6 and 7, respectively, from which it can be seen that the coning motion is stable when k ω � 0.4418, while it is unstable when k ω � 0.6626. Furthermore, the upper bound of k ω 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. is is because the higher spinning rate causes a stronger inertia moment.     1000

Conclusion
In this paper, the mathematical equation of a moving mass spinning missile is established. e 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. is 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.

Nomenclature
Cx: Drag coefficient C α y : Lift coefficient F: Force vector, kg·m/s 2 H: Angular momentum vector, kg·m 2 /s I: Inertial moment, kg·m 2 k ω : Gain of angular rate feedback l: Installation position of the moving mass L: Airframe diameter, m M: Force moment vector, kg·m 2 /s 2 m: Mass m ωx ′ : Coefficient of roll damping moment m α y : Coefficient of static moment m ωy ′ : Coefficient of damping moment m μ : Coefficient of Magnus moment n y , n z : Input command Q: Dynamic pressure, N·m 2 r 1 , r 2 : Position vector of the moving mass S: Reference area, m 2 V: Velocity vector α: Angle of attack β: Angle of sideslip ϑ, ψ, c: Pitch, yaw, and roll angle, rad δ y , δ z : Radial displacement of the moving mass δ yc , δ zc : Radial displacement command of the moving mass μ: Mass ratio ω: Angular rate vector Subscripts B: Missile body S: Missile system.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.   Mathematical Problems in Engineering 9