# Research on Instability Boundaries of Control Force for Trajectory Correction Projectiles

Authors: Rupeng Li; Dongguang Li; Jieru Fan
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The balance of stability and maneuverability is the foundation of the trajectory correction projectile. For the terminal correction projectile without an attitude feedback loop, a larger control force is expected which may cause an instability. This paper proposes a novel method to derive instability boundaries for the control force magnitude. No additional coordinate system is needed in this method. By introducing the concept of angular compensation matrix, the exterior ballistic linearized equations considering control force are established. The necessary prerequisite for a stable flight under control is given by the Routh stability criterion. The instability boundaries for the control force magnitude are derived. The results of example flights are 13.5% more accurate compared with that in relevant research. Numerical simulations demonstrate that if the control force magnitude lies in the unstable scope derived in this paper, the projectile loses its stability. Furthermore, the effects of the projectile pitch, velocity, and roll rate on flight stability during correction are investigated using the proposed instability boundaries.


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## Body

## 1. Introduction

Trajectory correction projectiles/fuzes can meet the requirement of low collateral damage and higher delivery accuracy in modern warfare. Moreover, existing stockpiles can be retrofitted and upgraded by just replacing a trajectory correction fuze with a low cost [1–3]. So it is no doubt that trajectory correction technologies have received much attention from researchers. Numerous correction models such as impact point prediction [4–7] and model trajectory prediction [8–10] can be found in the literature. For the pursuit of further improvements in accuracy, the guidance law based on target imager feedback is proposed and becomes a tendency in the future [11, 12]. Unlike the scheme mentioned above, this correction strategy can be used only during the terminal phase with a limited time-to-go. Therefore, a larger control force is expected. However, that may cause an instability for the projectile under control. For a successful terminal correction, the balance between the maneuverability and stability for the projectile under control is studied in this paper.Many efforts related to stability for ballistic flight have been explored in the literature. McCoy, R. L. [13], and Murphy [14–16] proposed the stability criteria for flight without control. Costello M [17] extended the gyro stability factor and dynamic stability factor to dual-spin projectile by linearization. In the area of flight stability with control, Wernert [18, 19] found that the control force induced by canards exerted an influence on steady angle of attack compared with conventional projectile without control. Cooper G and Fresconi F [20] regarded the projectile with activating canards asymmetric and investigated its flight stability under control. They established dynamic equations in the body reference frame. The research illustrated that flight instability would be caused when the control frequencies of actuator are close to fast epicyclic motion frequencies or slow epicyclic motion frequencies. Murphy [21] derived the stability boundaries for the maximum trim angles induced by control force. D Zhu [22] extended Murphy’s work through Hurwitz stability criterion and derived an analytic solutions of stability boundaries.The corresponding research made a significant contribution in the field of flight stability for controlled projectile. However, they did not reveal the relationship between control force magnitude and flight stability. Unlike missiles, the trajectory correction projectiles are not provided with a feedback loop for flight attitude. The investigation of force magnitude influence on the stability is more urgent for such projectiles and is beneficial to the design of actuator (such as the canard deflection or reference area). Lloyd K.H [23] made some efforts in this aspect. By proposing the non-spinning coordinate originally, the research investigated the influence of force magnitude on the flight stability and derived the analytic damping rate expression of the projectile under control by an analogy to non-controlled projectile. However, such an analogy can only get a rough scope of the control force for flight stability.This paper continues to bridge the gap in the area of controlled projectile stability and extend Lloyd K.H’s research without utilizing the non-spinning coordinate. Exterior ballistic differential equations for controlled projectile are expressed in fixed-plane coordinate and rewritten as the form of coefficient matrix. The items of control force are retained in these equations by a novel angular compensation matrix. Routh stability criterion is applied to the coefficient matrix. The magnitude boundaries of the control force that result in flight instability are derived and analyzed.In Section2, dynamical equations for projectile under control are established by the proposed compensation matrix. In Section 3, the instability boundaries of the control force are presented on the basis of the Routh stability criterion. In Section 4, an example controlled flight is calculated and 6 DOF numerical simulations are implemented. The results are favorable. The influence of flight parameters on the stability is analyzed by the instability boundaries.


## 2. Projectile Flight Dynamic Model

This section introduces the concept of the trajectory correction projectile, establishes the equations of motion in fixed plane coordinate, and demonstrates the invalidity of the classical linearization in stability research. The cause is analyzed in deep and a novel linearized method based on compensation matrix that is proposed.


### 2.1. Concept of the Trajectory Correction Projectile

The trajectory correction projectile discussed in this paper is used to improve the operational capability of the artillery when attacking the ground targets. The correction actuator is integrated into the projectile fuze in our design. Its appearance is shown in Figure1. The fuze consists of two parts defined as the forward part and aft part, respectively. The aft part is shown in green. It is fixedly connected with the projectile body by threaded connection. The projectile body is not illustrated here to highlight the fuze appearance. The forward part is shown in purple. It can rotate relative to the aft part of the fuze and the projectile body. The white nose represents the imager and is used for target detection. The blue part is the canard of a waffle style, in which each inner grid has a fixed deflection. It is strapdown with the forward fuze. When the projectile is in its ballistic flight, the canard is attached to the fuze surface. Once the correction is needed, the canard should be unfolded. Because of the oncoming flow and the deflection angle, the canard can produce an aerodynamic force which is used for projectile control. The direction of this control force is determined by roll angel of the forward fuze. Figure 2 illustrates the internal structure of the fuze. The red part represents a pair of bearings, which are used in providing a prerequisite for the relative rotation between the forward and aft part. The yellow part represents the motor shaft and gears, which are used to transfer the motor torque for the rotation.Figure 1
Appearance of the projectile fuze.Figure 2
Internal structure of the projectile fuze.When the projectile enters its descent trajectory, the detector starts to seek the target. The deviation between the target and the optical axis of detector is presented in detector imaging plane. The target orientation can be directly used to determine the roll angle of the forward fuze and as the input of the control system. The unfolding time of the canard depends on the magnitude of the deviation and the trajectory response under control. It can be seen that the correction actuator can complete the two-dimensional correction by a single motor.In general, conventional projectiles for artillery can be equipped with the proposed fuze and subsequently obtain a correction capability. Such trajectory correction projectiles work during the terminal trajectory because of the limited detection distance. So the left flight time for correction is constrained. Therefore, a sufficient control force is more necessary here for an effective correction. That is why the canard has a waffle configuration for more frontal area. For stability that is no less important than maneuverability for such projectiles, the balance between them is investigated in this paper.


### 2.2. Flight Model of the Projectile

The projectile flight model is expressed by differential equations in fixed plane coordinate which is defined as follows: the origin of coordinate system is located at the projectile centroid.X axis is aligned with the longitudinal axis of projectile, positive direction points to the projectile nose.Y axis is perpendicular to the longitudinal axis and lies in horizontal plane, positive direction points to right.Z axis is perpendicular toXY plane, positive direction points down. Projectile translational motion is written as follows:(1)u˙v˙w˙=FxmFymFzm+g-sin⁡θ0cos⁡θ+FxcmFycmFzcm+0r-q-r0-rtan⁡θqrtan⁡θ0uvwProjectile angular motion is written as (2).Fyc andFzc represent the horizontal and vertical components of the control force.MC andNC represent the corresponding control moments. While the canard is producing the control force, a drag force is also generated due to the canard deflection [18]. The induced drag force is denoted asFxc. Generally, a small deflection angle is considered for trajectory correction projectiles with canard. In this situation, the induced drag force is far less than the induced control force. So it is always ignored. For the drag force induced by canard that is along the longitudinal axis and through projectile centroid, the corresponding moment is zero.When the projectile is in a ballistic trajectory, the terms of control force and control moment are set to zero. And when the projectile needs a trajectory correction, these terms would have specific values according to operational requirements.(2)p˙q˙r˙=I-1LMN+0McNc-0-rqr0rtan⁡θ-q-rtan⁡θ0IpqrAngle of attackα and sideslip β are introduced and treated as small quantities. The expressions are shown as follows:(3)α=arctan-1⁡wV≈wV(4)β=arctan-1⁡uV≈uVCombining the latter two formulas in (1) and (2) with (3) and (4), a new equation set is completed. It is shown as follows:(5)β˙=Fymv+Fycmv-r-rα˙tan⁡θα˙=Fzmv+Fzcmv+gvcos⁡θ+q+rβ˙tan⁡θq˙=M+McIY-prIXIY-r2tan⁡θr˙=N+NcIY+pqIXIY+qrtan⁡θThe projectile body forces in (1) are composed of drag, lift, and Magnus force (always treated as small quantity and ignored). The projectile body moments in (2) are composed of static moment, pitch/yaw damping moment, and Magnus moment. The specific expressions for projectile body forces and moments are shown as follows:(6)Fy=-12ρSV2Clβ-12ρSV2CDβ-12ρSV2CypαplVαFz=-12ρSV2Clα-12ρSV2CDα+12ρSV2CypαplVαM=12ρSlV2CMαα+12ρSlV2CMpαplVβ+12ρSlV2CMqlVqN=-12ρSlV2CMαβ+12ρSlV2CMpαplVα+12ρSlV2CMqlVrBy comparing with the total projectile velocityV and the projectile roll ratep, the magnitude of projectile yaw rater and pitch tangent tan⁡θ are several orders inferior. The product ofr and tan⁡θ is a smaller value in (5). Therefore, the neglect of rtan⁡θ is a reasonable simplification. In addition, the total velocity and the roll rate change slowly compared with other parameters and are regarded as constant here. With the simplifications above, the four equations of (5) are largely linearized and uncoupled. Such simplifications and linearized methods are frequently used and helpful in exterior ballistics research. More details can be found in [17, 24].Based on the linearization, we rewrote (5) in the following form: X˙=KX+Q, in wihch the specific expression is shown as (7).K is the coefficient matrix, which is used to depict the projectile state. Because of the linearization, (7) becomes a set of differential equations with constant coefficients. And each unit in matrixK is invariable, which makes sure that the coefficients of corresponding eigenvalue equation are constant and provide a prerequisite for the future analysis of stability. (7)β˙α˙q˙r˙=AB0-1-BA10DCE-pIXIY-CDpIXIYEβαqr+FycmVmgcos⁡θ+FzcmVMCIYNCCIXwhere(8)A=-ρSVClα2m-ρSVCD2mB=-ρSCypαpl2mC=ρSlV2CMα2IYD=ρSl2VCMPαp2IYE=ρSl2VCMq2IYIt is remarkable that terms related to control force in (7) consist in matrix Q and its mathematical implications mean that the control force will cause a steady-state trim angle. This is in line with Wernert’s research [18]. However, these terms would not appear in the coefficient matrixK. The phenomenon is caused by the linearization according to the definition of the fixed plane coordinate. The fixed plane coordinate should have a roll rate-rtan⁡θ relative to the inertial coordinate and it is the prerequisite to ensure that they axis is located in the horizontal plane all the time. For ballistic trajectory without control force, the neglect of rtan⁡θ is a good assumption for linearization. But when control force is considered, the situation is different. Because once the target orientation relative to the projectile is determined, the control force direction would not vary. However, ignoring the item -rtan⁡θ forces the fixed plane coordinate rotate around its longitudinal axis at the roll rate rtan⁡θ. This implies the direction of the control force and is changing along with the fixed plane coordinate constantly. So a contradiction is caused. When it is reflected by the coefficient matrix, there would be no difference inK between the ballistic and controlled trajectory, and the control force would not appear in matrixK. Therefore, for further study of controlled trajectory, the effect of the neglect of -rtan⁡θ should be compensated before linearization.


### 2.3. Compensation Matrix for Control Force

Based on the analysis in Section2.1, the paper introduces an angular compensation matrix for control force and gravity to solve the problem above. We define the control force roll angle induced by linearization as Φ and express it as (9). The expression for roll angle induced by gravity has the same expression as follows:(9)Φ˙=rtan⁡θThe compensation matrix for control force and gravity is written as follows:(10)Tm=1000cos⁡Φsin⁡Φ0-sin⁡Φcos⁡ΦWith the compensation matrix, the projectile translational motion is rewritten as follows:(11)u˙v˙w˙=FxmFymFzm+g1000cos⁡Φsin⁡Φ0-sin⁡Φcos⁡Φ-sin⁡θ0cos⁡θ+1000cos⁡Φsin⁡Φ0-sin⁡Φcos⁡ΦFxcmFycmFzcm+0r-q-r0-rtan⁡θqrtan⁡θ0uvwThe projectile angular motion is rewritten as follows:(12)p˙q˙r˙=I-1LMN+1000cos⁡Φsin⁡Φ0-sin⁡Φcos⁡ΦLcMcNc-0-rqr0rtan⁡θ-q-rtan⁡θ0IpqrThen the latter two formulas in (11) and (12) can be linearized with the control force retained after the compensation. The projectile velocityV and roll ratep are regarded as constant in a sufficiently short time interval regarding the velocity componentu equal toV. The pitch angleθ is substituted byθ0+θd.θ0 is the initial pitch angle.θd is a small departure fromθ0. It should be noted that this method is applicable within a few seconds of the selected feature point. Fortunately, the time-to-go for terminal correction is limited to several seconds. The linear ballistic differential equations are eventually expressed as (13)-(16).(13)β˙=FymV+FzcmV+gVcos⁡θ0Φ-r+FycmV(14)α˙=FzmV+FzcmV+gVcos⁡θ0+q-FycmVΦ-gVsin⁡θ0θd(15)q˙=MIY-prIXIY+McIY+NcIYΦ(16)r˙=NIY+pqIXIY+NcIY-McIYΦCombining the four equations above with (9) and supplemental equation (17), the set of equations are completed.(17)θ˙d=q


## 2.1. Concept of the Trajectory Correction Projectile

The trajectory correction projectile discussed in this paper is used to improve the operational capability of the artillery when attacking the ground targets. The correction actuator is integrated into the projectile fuze in our design. Its appearance is shown in Figure1. The fuze consists of two parts defined as the forward part and aft part, respectively. The aft part is shown in green. It is fixedly connected with the projectile body by threaded connection. The projectile body is not illustrated here to highlight the fuze appearance. The forward part is shown in purple. It can rotate relative to the aft part of the fuze and the projectile body. The white nose represents the imager and is used for target detection. The blue part is the canard of a waffle style, in which each inner grid has a fixed deflection. It is strapdown with the forward fuze. When the projectile is in its ballistic flight, the canard is attached to the fuze surface. Once the correction is needed, the canard should be unfolded. Because of the oncoming flow and the deflection angle, the canard can produce an aerodynamic force which is used for projectile control. The direction of this control force is determined by roll angel of the forward fuze. Figure 2 illustrates the internal structure of the fuze. The red part represents a pair of bearings, which are used in providing a prerequisite for the relative rotation between the forward and aft part. The yellow part represents the motor shaft and gears, which are used to transfer the motor torque for the rotation.Figure 1
Appearance of the projectile fuze.Figure 2
Internal structure of the projectile fuze.When the projectile enters its descent trajectory, the detector starts to seek the target. The deviation between the target and the optical axis of detector is presented in detector imaging plane. The target orientation can be directly used to determine the roll angle of the forward fuze and as the input of the control system. The unfolding time of the canard depends on the magnitude of the deviation and the trajectory response under control. It can be seen that the correction actuator can complete the two-dimensional correction by a single motor.In general, conventional projectiles for artillery can be equipped with the proposed fuze and subsequently obtain a correction capability. Such trajectory correction projectiles work during the terminal trajectory because of the limited detection distance. So the left flight time for correction is constrained. Therefore, a sufficient control force is more necessary here for an effective correction. That is why the canard has a waffle configuration for more frontal area. For stability that is no less important than maneuverability for such projectiles, the balance between them is investigated in this paper.


## 2.2. Flight Model of the Projectile

The projectile flight model is expressed by differential equations in fixed plane coordinate which is defined as follows: the origin of coordinate system is located at the projectile centroid.X axis is aligned with the longitudinal axis of projectile, positive direction points to the projectile nose.Y axis is perpendicular to the longitudinal axis and lies in horizontal plane, positive direction points to right.Z axis is perpendicular toXY plane, positive direction points down. Projectile translational motion is written as follows:(1)u˙v˙w˙=FxmFymFzm+g-sin⁡θ0cos⁡θ+FxcmFycmFzcm+0r-q-r0-rtan⁡θqrtan⁡θ0uvwProjectile angular motion is written as (2).Fyc andFzc represent the horizontal and vertical components of the control force.MC andNC represent the corresponding control moments. While the canard is producing the control force, a drag force is also generated due to the canard deflection [18]. The induced drag force is denoted asFxc. Generally, a small deflection angle is considered for trajectory correction projectiles with canard. In this situation, the induced drag force is far less than the induced control force. So it is always ignored. For the drag force induced by canard that is along the longitudinal axis and through projectile centroid, the corresponding moment is zero.When the projectile is in a ballistic trajectory, the terms of control force and control moment are set to zero. And when the projectile needs a trajectory correction, these terms would have specific values according to operational requirements.(2)p˙q˙r˙=I-1LMN+0McNc-0-rqr0rtan⁡θ-q-rtan⁡θ0IpqrAngle of attackα and sideslip β are introduced and treated as small quantities. The expressions are shown as follows:(3)α=arctan-1⁡wV≈wV(4)β=arctan-1⁡uV≈uVCombining the latter two formulas in (1) and (2) with (3) and (4), a new equation set is completed. It is shown as follows:(5)β˙=Fymv+Fycmv-r-rα˙tan⁡θα˙=Fzmv+Fzcmv+gvcos⁡θ+q+rβ˙tan⁡θq˙=M+McIY-prIXIY-r2tan⁡θr˙=N+NcIY+pqIXIY+qrtan⁡θThe projectile body forces in (1) are composed of drag, lift, and Magnus force (always treated as small quantity and ignored). The projectile body moments in (2) are composed of static moment, pitch/yaw damping moment, and Magnus moment. The specific expressions for projectile body forces and moments are shown as follows:(6)Fy=-12ρSV2Clβ-12ρSV2CDβ-12ρSV2CypαplVαFz=-12ρSV2Clα-12ρSV2CDα+12ρSV2CypαplVαM=12ρSlV2CMαα+12ρSlV2CMpαplVβ+12ρSlV2CMqlVqN=-12ρSlV2CMαβ+12ρSlV2CMpαplVα+12ρSlV2CMqlVrBy comparing with the total projectile velocityV and the projectile roll ratep, the magnitude of projectile yaw rater and pitch tangent tan⁡θ are several orders inferior. The product ofr and tan⁡θ is a smaller value in (5). Therefore, the neglect of rtan⁡θ is a reasonable simplification. In addition, the total velocity and the roll rate change slowly compared with other parameters and are regarded as constant here. With the simplifications above, the four equations of (5) are largely linearized and uncoupled. Such simplifications and linearized methods are frequently used and helpful in exterior ballistics research. More details can be found in [17, 24].Based on the linearization, we rewrote (5) in the following form: X˙=KX+Q, in wihch the specific expression is shown as (7).K is the coefficient matrix, which is used to depict the projectile state. Because of the linearization, (7) becomes a set of differential equations with constant coefficients. And each unit in matrixK is invariable, which makes sure that the coefficients of corresponding eigenvalue equation are constant and provide a prerequisite for the future analysis of stability. (7)β˙α˙q˙r˙=AB0-1-BA10DCE-pIXIY-CDpIXIYEβαqr+FycmVmgcos⁡θ+FzcmVMCIYNCCIXwhere(8)A=-ρSVClα2m-ρSVCD2mB=-ρSCypαpl2mC=ρSlV2CMα2IYD=ρSl2VCMPαp2IYE=ρSl2VCMq2IYIt is remarkable that terms related to control force in (7) consist in matrix Q and its mathematical implications mean that the control force will cause a steady-state trim angle. This is in line with Wernert’s research [18]. However, these terms would not appear in the coefficient matrixK. The phenomenon is caused by the linearization according to the definition of the fixed plane coordinate. The fixed plane coordinate should have a roll rate-rtan⁡θ relative to the inertial coordinate and it is the prerequisite to ensure that they axis is located in the horizontal plane all the time. For ballistic trajectory without control force, the neglect of rtan⁡θ is a good assumption for linearization. But when control force is considered, the situation is different. Because once the target orientation relative to the projectile is determined, the control force direction would not vary. However, ignoring the item -rtan⁡θ forces the fixed plane coordinate rotate around its longitudinal axis at the roll rate rtan⁡θ. This implies the direction of the control force and is changing along with the fixed plane coordinate constantly. So a contradiction is caused. When it is reflected by the coefficient matrix, there would be no difference inK between the ballistic and controlled trajectory, and the control force would not appear in matrixK. Therefore, for further study of controlled trajectory, the effect of the neglect of -rtan⁡θ should be compensated before linearization.


## 2.3. Compensation Matrix for Control Force

Based on the analysis in Section2.1, the paper introduces an angular compensation matrix for control force and gravity to solve the problem above. We define the control force roll angle induced by linearization as Φ and express it as (9). The expression for roll angle induced by gravity has the same expression as follows:(9)Φ˙=rtan⁡θThe compensation matrix for control force and gravity is written as follows:(10)Tm=1000cos⁡Φsin⁡Φ0-sin⁡Φcos⁡ΦWith the compensation matrix, the projectile translational motion is rewritten as follows:(11)u˙v˙w˙=FxmFymFzm+g1000cos⁡Φsin⁡Φ0-sin⁡Φcos⁡Φ-sin⁡θ0cos⁡θ+1000cos⁡Φsin⁡Φ0-sin⁡Φcos⁡ΦFxcmFycmFzcm+0r-q-r0-rtan⁡θqrtan⁡θ0uvwThe projectile angular motion is rewritten as follows:(12)p˙q˙r˙=I-1LMN+1000cos⁡Φsin⁡Φ0-sin⁡Φcos⁡ΦLcMcNc-0-rqr0rtan⁡θ-q-rtan⁡θ0IpqrThen the latter two formulas in (11) and (12) can be linearized with the control force retained after the compensation. The projectile velocityV and roll ratep are regarded as constant in a sufficiently short time interval regarding the velocity componentu equal toV. The pitch angleθ is substituted byθ0+θd.θ0 is the initial pitch angle.θd is a small departure fromθ0. It should be noted that this method is applicable within a few seconds of the selected feature point. Fortunately, the time-to-go for terminal correction is limited to several seconds. The linear ballistic differential equations are eventually expressed as (13)-(16).(13)β˙=FymV+FzcmV+gVcos⁡θ0Φ-r+FycmV(14)α˙=FzmV+FzcmV+gVcos⁡θ0+q-FycmVΦ-gVsin⁡θ0θd(15)q˙=MIY-prIXIY+McIY+NcIYΦ(16)r˙=NIY+pqIXIY+NcIY-McIYΦCombining the four equations above with (9) and supplemental equation (17), the set of equations are completed.(17)θ˙d=q


## 3. The Instability Boundary for Control Force

The purpose of this section is to derive the boundary for control force that causes an instability and makes some efforts to build the balance between the maneuverability and stability for the projectile under control.We rewrite the set of equations (13)-(17) as matrix form: X˙=KX+Q:(18)β˙α˙q˙r˙Φ˙θ˙d=AB0-1F0BA10GHDCE-pIXIYR0-CDpIXIYES0000tan⁡θ000001000βαqrΦθd+FycmVFzcmV+gVcos⁡θ0MCIYNCCIX00where the new notations are as follows:(19)F=FZCmV+gVcos⁡θ0G=-FYCmVH=-gVsin⁡θ0R=NCIYS=-MCIYT=pIXIYTerms related to control force are now involved in the coefficient matrixK with the compensation matrix. The eigenvalue equation for the coefficient matrix in (18) is established as follows:(20)det⁡λE-A=a6λ6+a5λ5+a4λ4+a3λ3+a2λ2+a1λ+a0wherea0-a6 are consisted of the terms A, B, C, etc. These terms are the combination of physical parameters, state parameters, and aerodynamic coefficients of the projectile. The physical parameters represent the inherent properties of a projectile. So they are constant. State parameters and aerodynamic coefficients could be regarded as constant as well in a short time interval. When a long-duration flight is considered, they should be updated periodically. Each constituent is complex and brings difficulties to calculation and analysis. However, the dominant terms are limited relatively. For simplicity, secondary factors are ignored. The feasibility of this simplification will be verified later. The expressions fora0-a6 after simplification are written here directly.(21)a6=1a5=-2A-2Ea4=T2-2C-Stan⁡θ0a3=-2AT2+2AC+2CE+2DT-RTtan⁡θ0a2=C2+CStan⁡θ0-DRtan⁡θ0+2ARTtan⁡θ0a1=C2H-C2Ftan⁡θ0-ACStan⁡θ0+CHStan⁡θ0a0=-ACHStan⁡θ0-C2FHtan⁡θ0The Routh matrix corresponding to the eigenvalue equation (20) is established.(22)TROUTH=a6a4a2a0a5a3a1b1=-1a5a6a3-a5a4b2=-1a5a6a1-a2a5b3=a0c1=-1b1a5b2-a3b1c2=-1b1a5b3-a1b1d1=-1c11b1c2-b2c1d2=a0e1=-1d1c1d2-c2d1f1=a0The necessary prerequisite for a stable flight under control is given according to the Routh stability criterion: The coefficientsa0-a6 in (20) are positive and the first column elements of the Routh matrix are positive. Furthermore, if the control force magnitude cannot meet (23), the projectile will lose its stability. Then the control force is defined as an unstable force. The set of all unstable forces is called unstable scope. The boundaries of the scope are denoted as the instability boundary. A conclusion is derived evidently that if a control force magnitude belongs to the unstable scope, the flight is definitely unstable.(23)a5>0,a4>0,a3>0,a2>0,a1>0,a0>0b1>0,c1>0,d1>0,e1>0


## 4. Verification and Results

This section demonstrates the feasibility and liability of the theory derived in Section3. Generally, the flight instability can be clearly shown by numerical simulation. According to the research of Lloyd C.H [23], projectiles under control are the most sensitive to horizontal forces. Therefore, the simulations in this section pay emphasis on the effect of horizontal control forceFyc.The example projectile massm is 15kg, moment of inertia for longitudinal axisIx is 0.023kg·m2, moment of inertia for projectile transverse axisIy is 0.22kg·m2, the reference areaS is 0.0087m2, and the projectile diameterd is 0.105m. Initial states and meteorological conditions are shown in Table 1. The aerodynamic coefficients of the example projectile are shown in Table 2Table 1
Initial states and meteorological conditions.
Initial states Meteorological conditions Velocity 250m/s Ground pressure 1000hPa Elevation -43° Virtual temperature 288.9 Direction 0° Longitudinal wind 0m/s Spin-rate 1050rad/s Lateral wind 0m/sTable 2
Aerodynamic coefficients of the example projectile.
Aerodynamic coefficients lift force coefficient 1.7 drag force coefficient 0.13 Magnus force coefficient derivative -1.5 static moment coefficient derivative 3.8 Magnus moment coefficient derivative 0.2 damping moment coefficient derivative -8.0We take the example projectile to calculate the instability boundaries for the flight under control. The result of this unstable scope is-∞,-35.48∪58.33,+∞. According to the theory in Section 3, when control force magnitude belongs to the unstable scope (Fyc∈-∞,-35.48∪58.33,+∞), the projectile will lose its stability.Before the verification by simulation, the simplification in Section3 is recalled. It has been mentioned above that the result is derived from simplifieda0-a6. To demonstrate the practicability of this simplification, the instability boundaries are calculated again without any simplification. The result is Fyc∈-∞,-35.63∪58.00,+∞. It can be seen that the result varies a little compared with that of simplified model.The projectile angular motion without control force is presented as Figure3. The angle of attack motion and sideslip motion along with time are presented in Figure 4.Figure 3
Angular motion for the flight without control force.Figure 4
Angle of attack and sideslip for the flight without control force.These two figures illustrate that the angular motions for the flight without control force are converging gradually along with time and represent a stable flight.The projectile angular motion with a -55N force to left is presented as Figure5. In the same case, the angle of attack motion and sideslip motion along with time are presented in Figure 6.Figure 5
Angular motion for the flight with -55N control force.Figure 6
Angle of attack and sideslip for the flight with -55N control force.In Figures5 and 6, the fast epicyclic motion of the projectile is stable. However, the slow epicyclic motion is diverging along with time obviously. The results have illustrated that the slow epicyclic motion of the projectile is unstable when a -55N control force is exerted to left. The simulation results for Fyc=-55∈-∞,-35.63∪58.00,+∞ are in conformity to the negative part of the analytic instability boundaries.Figures7 and 8 present the numerical simulation results for the flight under an 80N control force. They illustrate that the slow epicyclic motion of the projectile is stable. However, the angle of attack and sideslip are diverging along with time obviously, and the fast epicyclic motion is unstable. For the control force 80N belonging to the unstable scope, the simulation results demonstrate the validity of the positive part of instability boundaries.Figure 7
Angular motion for the flight with 80N control force.Figure 8
Angle of attack and sideslip for the flight with 80N control force.Additionally, the instability boundaries for the flight under control derived in this paper are compared with Lloyd’s research. With the same example projectile, initial and meteorological conditions and the instability boundaries under horizontal control force by Lloyd are written asFyc∈-∞,-44.61∪66.91,+∞. As is shown in Figure 9, the blue slash represents the scope of horizontal control force that causes an instability derived by Lloyd, and the red slash represents the same scope derived in this paper.Figure 9
The comparison of instability boundaries.The figure indicates that the instability boundaries in this paper extend the scope by 13.5% compared with Lloyd’s research. To further demonstrate the reliability of this method, another two simulations are implemented. The control force magnitude is chosen from the extended scope by comparison.Figures10 and 11 exhibit the angular motion, angle of attack, and sideslip for the flight with a -40N control force (the value -40 is in the negative extended scope by comparison). The numerical simulation result illustrates clearly that the flight is unstable (specifically, the fast epicyclic motion is stable, while the slow epicyclic motion is unstable).Figure 10
Angular motion for the flight with -40N control force.Figure 11
Angle of attack and sideslip for the flight with -40N control force.Similarly, Figures12 and 13 exhibit the angular motion, angle of attack, and sideslip for the flight with a 60N control force (the value 60 is in the positive extended scope by comparison). The numerical simulation result indicates that the flight is unstable (specifically, the slow epicyclic motion is stable, while the fast epicyclic motion is unstable).Figure 12
Angular motion for the flight with 60N control force.Figure 13
Angle of attack and sideslip for the flight with 60N control force.The two simulation results give a further demonstration for the instability boundaries of the flight under control derived in this paper. In addition, it is 13.5% more accurate than Lloyd’s research.


## 5. The Effect of Projectile Parameters on Instability Boundaries

The effects of the projectile parameters during flight on stability are analyzed and discussed, respectively, utilizing the instability boundaries when a control force is exerted.


### 5.1. The Initial Pitchθ0

Figure14 dictates the effect of initial pitch on the instability boundaries. The line of red triangle represents the lower instability boundary variation of the positive control force (denoted as lower boundary for short). The line of blue diamond represents the upper instability boundary variation of the negative control force (denoted as upper boundary for short). As presented in Figure 14, the lower boundary increases progressively as the initial pitch angle decreases in descending flight, while the upper boundary decreases.Figure 14
The effect of initial pitch angle on the instability boundaries.The instability boundary variation in this case dictates a decreasing control force unstable scope. That is to say, the increase of the pitch angle in descending flight improves the projectile stability. The reason is that the pitch angle represents the curvature of the trajectory which will induce an aerodynamic trim angle. During the descending flight, the small pitch angle implies a severe trajectory curvature and a large trim angle. Therefore, the flight stability is weakened with the extra aerodynamic drag and lift forces induced by large trim angle.


### 5.2. The Projectile VelocityV

Figure15 dictates the effect of projectile velocity on the instability boundaries. The lines of red triangle and blue diamond represent the lower boundary and upper boundary variation, respectively, along with projectile velocity. As illustrated in Figure 15, the increase of projectile velocity causes an increase in lower boundary and a decrease in upper boundary. As discussed in Figure 14, the result illustrates that the flight stability increases as the projectile velocity increases. The conclusion of the analysis is in line with the literature [25].Figure 15
The effect of projectile velocity on the instability boundaries.


### 5.3. The Projectile Roll Ratep

The effect of projectile roll rate on the instability boundaries is illustrated in Figure16. The expression is almost identical to Figure 15 and to avoid repetition; it is not described here. The increase of projectile velocity causes a lower boundary increase and an upper boundary decrease. That is to say, the flight stability is positively associated with the projectile roll rate. The result is in line with the classical ballistic theory: the increase of projectile roll rate will directly improve the gyroscopic stability factor of projectiles.Figure 16
The effect of projectile velocity on the instability boundaries.This section shows the effects of projectile parameters on instability boundaries. The results illustrate that the proposed instability boundaries can be used to predict the stability variation under various flight parameters. The prediction results are in line with the relevant research or theory. Therefore, the validity of instability boundaries is demonstrated once again in this respect.


## 5.1. The Initial Pitchθ0

Figure14 dictates the effect of initial pitch on the instability boundaries. The line of red triangle represents the lower instability boundary variation of the positive control force (denoted as lower boundary for short). The line of blue diamond represents the upper instability boundary variation of the negative control force (denoted as upper boundary for short). As presented in Figure 14, the lower boundary increases progressively as the initial pitch angle decreases in descending flight, while the upper boundary decreases.Figure 14
The effect of initial pitch angle on the instability boundaries.The instability boundary variation in this case dictates a decreasing control force unstable scope. That is to say, the increase of the pitch angle in descending flight improves the projectile stability. The reason is that the pitch angle represents the curvature of the trajectory which will induce an aerodynamic trim angle. During the descending flight, the small pitch angle implies a severe trajectory curvature and a large trim angle. Therefore, the flight stability is weakened with the extra aerodynamic drag and lift forces induced by large trim angle.


## 5.2. The Projectile VelocityV

Figure15 dictates the effect of projectile velocity on the instability boundaries. The lines of red triangle and blue diamond represent the lower boundary and upper boundary variation, respectively, along with projectile velocity. As illustrated in Figure 15, the increase of projectile velocity causes an increase in lower boundary and a decrease in upper boundary. As discussed in Figure 14, the result illustrates that the flight stability increases as the projectile velocity increases. The conclusion of the analysis is in line with the literature [25].Figure 15
The effect of projectile velocity on the instability boundaries.


## 5.3. The Projectile Roll Ratep

The effect of projectile roll rate on the instability boundaries is illustrated in Figure16. The expression is almost identical to Figure 15 and to avoid repetition; it is not described here. The increase of projectile velocity causes a lower boundary increase and an upper boundary decrease. That is to say, the flight stability is positively associated with the projectile roll rate. The result is in line with the classical ballistic theory: the increase of projectile roll rate will directly improve the gyroscopic stability factor of projectiles.Figure 16
The effect of projectile velocity on the instability boundaries.This section shows the effects of projectile parameters on instability boundaries. The results illustrate that the proposed instability boundaries can be used to predict the stability variation under various flight parameters. The prediction results are in line with the relevant research or theory. Therefore, the validity of instability boundaries is demonstrated once again in this respect.


## 6. Conclusion

This work mainly focuses on the balance between the maneuverability and stability for the projectile under control. With a deep investigation, this paper indicates the invalidity of classical linearization in stability analysis and reveals the reason. Based on that, the exterior ballistic equations are established with the unique proposed compensation matrix and no additional coordinate system is needed. The necessary prerequisite for a stable flight under control is given by Routh stability criterion and the instability boundaries for the control force are derived. Numerical simulations demonstrate the validity of the analytic relationship between the flight instability boundaries and the control force magnitude. Additionally, the instability boundaries derived in this paper are compared with Lloyd’s research. The comparison results dictate that the former is more accurate and extends the scope by 13.5%. The results are also verified by simulations. The effects of the projectile pitch, velocity, and roll rate on the instability boundaries are analyzed, respectively, when a control force is exerted. The results illustrate that the unstable scope is negatively associated with the all the three projectile parameters during the descending flight, while the projectile stability is positively associated.



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*Source: 6362835-2019-03-11.xml*