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.
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 body B and two radial internal moving masses and as shown in Figure 1. The moving mass moves along the axis while the moving mass moves along the axis. The mass of the body B is , and the two moving masses and have the same mass m; thus, the total mass of the missile system is . The mass ratio of the moving mass is . l is the installation position of the moving mass, and and are the radial displacements of the two moving masses in the nonspinning coordinate system.
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 aswhere is the velocity vector of the body B relative to the center of mass of the missile system and is given bywhere is the velocity vector of the body B in the nonspinning coordinate system, is the spinning velocity vector, and and are position vectors of the two moving masses in the nonspinning CS. The derivative of equation (2) can be derived aswhere 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 aswhere and are projections of and on the nonspinning CS and are given by
The velocity vector of each moving mass relative to the center of mass of the missile system can be expressed as
The derivative of equation (6) can be derived as
Substituting equations (3) and (7) into equation (1) yieldswhere is the vector of aerodynamic force in the nonspinning CS and is given by
According to the theorem of angular momentum, the rotational motion of the missile system can be described aswhere is the angular momentum of the body B, is the angular momentum of the moving mass, and is the external moments applied on the missile system, including aerodynamic moments and mass eccentricity moments. , , and are given bywhere is the angular rate of the body CS with respect to the inertial CS. Substituting equations (10)–(12) into equation (9) yields
The 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 aswhere
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, so , (2)The spinning rate in the nonspinning CS keeps constant and is equal to zero, and is small, so (3)Variables , , , , α, and β are small(4)The gravity effect is negligible(5)l keeps constant, so (6)The missile is strictly axisymmetric, so
Under these assumptions, the equations for lateral translational and rotational motion in equations (16) and (17) can be simplified to
The angles of attack α and sideslip β are defined as
By defining the complex angle of attack , the complex angular rate , and the complex control instruction , 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 aswhere , , , , , , , , , , , and .
According equation (24), the equilibrium point is determined bywhere and .
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
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
5. Stability of the Moving Mass Spinning Missile with the Angular Rate Loop
The control system with angular rate loops is shown in Figure 2, in which and are control commands, and are feedback signals, and 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. Therefore, the displacement instruction of the moving mass is obtained as
Meanwhile, based on the assumption that the missile is in horizontal flight, there exists an approximation relationship: and . Thus, equation (29) can be expressed as
Converting equation (30) into the complex form, one has
Substituting equation (31) into equation (24) yields
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 aswhere , , , and .
The corresponding characteristic equation is
Assuming , whereone gets
Then, 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 , in order to ensure that equation (38) is true, the following inequality must be met:
Substituting , , and into equation (39) yields
To facilitate the analysis, a polynomial is introduced:where , , and .
For slowly spinning missiles, and 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 when , one gets , , and , and the curve of is illustrated by Iin Figure 3. The intersections of and the axis are given by Thus, only when or , one gets . The sufficient and necessary condition for the coning motion stability can be derived as(2)The second case is when , one gets , , and c could be positive or negative. Ignore the sign of c, and the intersections of and the x axis are given by Thus, only when , one gets . The sufficient and necessary condition for the coning motion stability can be derived as
(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, . Then, one gets and . By neglecting the effect of Magnus moment and considering to be small, equation (32) can be simplified aswhere , , , , and .
The characteristic equation of equation (46) is given by
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
Because , equation (48) is rewritten as
Substituting all the coefficients into equation (49) yieldswhere
For the moving mass control missile under fast spinning rate, one has ; thus, equation (50) is always true. Because , to make equation (51) true, one should have
To make equation (52) true, one should have
For fast spinning missile, one has , , and . Thus, 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
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 Table 1.
According 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 gain , 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 gain 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 are shown in Figure 5. It can be seen that the coning motion is divergent.
(a)
(b)
(a)
(b)
6.2. Influence of the System Parameter
In this section, the influence of the location of the moving mass l and the spinning rate of the missile 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 is shown in Table 2. It can be observed from the table that the upper bound of 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 .
The 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. 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 Table 4.
The upper bound of in this case is 0.5522 according to the stability condition described in equation (57). The coning motions under and are shown in Figures 6 and 7, respectively, from which it can be seen that the coning motion is stable when , while it is unstable when .
Furthermore, the upper bound of 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.
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.
Nomenclature
| : | Drag coefficient |
| : | Lift coefficient |
| : | Force vector, kg·m/s2 |
| : | Angular momentum vector, kg·m2/s |
| : | Inertial moment, kg·m2 |
| : | Gain of angular rate feedback |
| l: | Installation position of the moving mass |
| L: | Airframe diameter, m |
| M: | Force moment vector, kg·m2/s2 |
| m: | Mass |
| : | Coefficient of roll damping moment |
| : | Coefficient of static moment |
| : | Coefficient of damping moment |
| : | Coefficient of Magnus moment |
| , : | Input command |
| Q: | Dynamic pressure, N·m2 |
| , : | Position vector of the moving mass |
| S: | Reference area, m2 |
| : | Velocity vector |
| α: | Angle of attack |
| β: | Angle of sideslip |
| ϑ, ψ, γ: | Pitch, yaw, and roll angle, rad |
| , : | Radial displacement of the moving mass |
| , : | Radial displacement command of the moving mass |
| μ: | Mass ratio |
| ω: | Angular rate vector |
| B: | Missile body |
| S: | Missile system. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.