# Research on Speed Scheme for Precise Attack of Miniature Loitering Munition

Authors: Zhidong Zhang; Jie Li; Yachao Yang; Chengwei Yang; Ruizhi Mao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The loitering munitions are advanced new ammunitions, which reflect both the characteristics of UAV and missile and have their own novel characteristics. This paper analysed the characteristics of the attack process of a small loitering munition and proposed a speed scheme suitable for precise strike of loitering munitions based on the concept of traditional ammunition’s speed-thrust schemes. Then, the boundary conditions and the existence of the solution have been discussed. Finally, flight test results showed that the scheme was effective.


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## Body

## 1. Introduction

A loitering munition (LM), as a new member of the precision-guided munitions [1–5], fits in the niche between cruise missiles and unmanned combat aerial vehicles (UCAVs) sharing characteristics with both. In the cruise section, the LM mainly conducts flight and mission target search in the form of a UAV, but in the launch section and the final guidance-attacking section, the LM works as a missile [5–7]. After launch, the original folded wings are unfolded, and the aircraft is converted from projectile form to drone form, and the configuration of the loitering munitions does not change again after the conversion of the machine finished [1, 8–10]. Therefore, it is a challenge for a LM to adapt to the control requirements of different mission phases under the same aerodynamic shape. The typical working process for a loitering munition includes pneumatic launch, loitering and gathering intelligence, locking on target, aborting mission, or precise attack. Some mature products of small LM are shown in Figure 1, and the working process diagram for a LM can be seen in Figure 2 [11].Figure 1
Some mature products of small LM: (a) HAROP drones, (b) Predator AX-1, and (c) Hero-900.
(a)(b)(c)Figure 2
Working process diagram for a LM.Traditional precision-guided munitions include various types of guided missile, guided bomb, guided projectile, and cruise missile [2, 12–14]. The design of the propulsion system and guidance control system occupies important positions in the system design, but they are kept independent as usual. The main goal of propulsion design is to achieve effective range and reach the final flight speed to meet the requirements of the attack mission [15]. For example, the antitank missile uses a rocket engine as the propulsion to fly at high speed, and its main speed schemes include the following three types: boost + glide flight, boost + acceleration cruising + glide flight, and acceleration boost + constant velocity (boost + cruising flight). A propulsion map and speed of typical high-speed kinetic energy antitank missile are shown in Figure 3, and in general, while designing the guidance algorithms ,it will be assumed that the flight speed is constant, especially at the terminal guidance phase [16–18], or the flight speed is time varying passively affected by gravity, propulsion, and aerodynamic forces [12, 19–21]. At the same time, the research topic of speed is mainly focused on the cruise flight process [22, 23] or is considered as a terminal constraint at some special flight mission, for example, a rendezvous mission to a tanker [24].Figure 3
Propulsion map and speed of typical high-speed kinetic energy antitank missile.
(a)(b)Because the pneumatic layout and aerodynamic characteristics of LMs are closer to UAV than to missiles [25, 26], it is unreasonable to assume that the speed is constant or ignore the dynamic adjustable characteristics of speed just like a traditional precision-guided munition in designing the guidance and control algorithms in the terminal guidance phase. Therefore, it is of great significance to carry out research on the speed scheme at the terminal attack stage for a LM.This paper is concerned with the problem of designing speed-trajectory characteristics of the LM within the attack phase. First of all, the speed requirement of guidance and control design in the LM’s precision attacking process is discussed. Then, a speed scheme is proposed to improve the accuracy of a LM strike. In the following sections, the boundary condition for the establishment of the scheme is given, and the existence conditions of the system solution is analysed. Finally, a flight test has been implemented to verify the theoretical rationality and practical value of the proposed scheme, where a method for parameter acquisition was given at the same time.It should be noted that the main topic of this paper is to analyse the dynamic characteristics of thrust and velocity, mainly from the theoretical analysis to solve the problems in practical applications, so the specific implementation of propulsion is not discussed.


## 2. Speed Scheme for Miniature LM’s Precision Attack

### 2.1. Equations of Aircraft Motion

As a novel precision-guided munition, LM in the form of small UAV could use the low-cost strap-down seeker [12, 27, 28] to achieve target capture and tracking during autonomous attack mission, so we discuss problems based on strap-down seeker against a stationary target.The flight model was expressed by differential equations in body-fixed coordinate which was defined as follows: the origin of coordinate system is located at the LM centroid.X axis is aligned with the longitudinal axis of LM; positive direction points to the LM’s nose. Y axis is perpendicular to the longitudinal axis and lies in horizontal plane; positive direction points to right. Z axis is perpendicular to XY plane; positive direction points down. This body-fixed coordinate system is illustrated in Figure 4.Figure 4
Body-fixed coordinate.In order to introduce the speed scheme clearly, only the longitudinal motion of a LM has been discussed. The force analysis of a LM is shown in Figure5, and the homing guidance geometry of missile-target engagement of the flight process is shown in Figure 6.Figure 5
Force analysis of LM.Figure 6
Two-dimensional homing guidance geometry of missile-target engagement.L is the lift force (aerodynamic force along z direction), D is the drag force (aerodynamic force along x direction), T is the thrust of engine, m is the mass of LM, g is the gravitational acceleration, and θ is the pitch angle (Euler angle along y axis) of LM.R is the instantaneous separation between LM and target, V is the speed of the loitering munition in x direction in body-coordinate, VΓ is the speed of the loitering munition in z direction in body-coordinate, η is the angle made by the line of sight with respect to the fixed reference, and q is the pitch rate (angular rate along y axis).Assume thatVΓ is small and θ is close to η, and the thrust T is assumed to act through the LM centroid and along the x axis. Then, from the engagement geometry and the force analysis of the aircraft, we can see the following.The relative motion relationship between the LM and the target can be described as the following equations:(1)R˙=−Vcosθ−η,η˙=−1RVsinθ−η.The dynamic equations of the LM can be written as follows:(2)V˙=1mT−D−mgsinθ,V˙Γ=1mmVq+mgcosθ−L,θ˙=q,q˙=MIy,where Iy is the LM’s moment of inertia in y direction.The equations of aerodynamic force and moment can be written as follows:(3)D=CDQS,L=CLQS,M=CmQScA,whereM is the aerodynamic moment component in y direction, CD is the drag coefficient, CL is the lift coefficient, Cm is the pitching moment coefficient, Q is the dynamic pressure, S is the reference area, and cA is the mean aerodynamic chord.In the above expressions, the pitching moment coefficient can be expressed as(4)Cm=CmLL+Cmδeδe+CmqqcA2V,where CmL is the moment derivative with respect to lift, Cmδe is the moment derivative with respect to control-surface deflection, δe is the control-surface deflection, and Cmq is the moment derivative with respect to pitch rate.The dynamic pressure can be expressed as(5)Q=12ρV2,where ρ is the density of air.Then, the equations describing the LM’s motion for the attacking stage can be expressed as(6)R˙=−Vcosθ−η,η˙=−1RVsinθ−η,V˙=1mT−12ρV2CDS−mgsinθ,V˙Γ=1mmVq+mgcosθ−L,θ˙=q,q˙=CmL1/2ρV2CLS+Cmδeδe+CmqqcA/2V1/2ρV2ScAIy.The state variables of the system during attacking can be selected as(7)X=R,η,V,VΓ,θ,q.The control variables are(8)U=T,δe.Equation (6) can be organized as(9)R˙=−Vcosθ−η,η˙=−1RVsinθ−η,V˙=−ρCDS2mV2−gsinθ+1mT,V˙Γ=−ρCLS2mV2+Vq+gcosθ,θ˙=q,q˙=CmLCL1/4ρ2S2cAIyV4+Cmq1/4ρScA2IyVq+Cmδe1/2ρScAIyV2δe.This is the system equation that describes the motion of the aircraft at the attacking phase.


### 2.2. A Speed Scheme Suitable for a LM with Strap-Down Seeker for the Attacking Stage

For a LM with strap-down seeker, it is important to keep flight stabilized to get an ideal target tracking feedback, which means there should be no excessive normal shifting motion during attacking. The desired normal acceleration should be set:(10)V˙Γ=0.According to the 4th expression of equation (9), this condition can be described as(11)−ρCLS2mV2+Vq+gcosθ=0.It can be solved from the above equation that the expected speed during the attacking process is(12)Vtc=mq+m2q2+2ρCLSmgcosθρCLS,where the subscript “tc” indicates that the value of the variable is an expected value which depends on the attacking process.When the LM is flying at this expected speed, it means(13)V˙=0.According to the 3rd expression of equation (9), this condition can be described as(14)−ρCDS2mV2−gsinθ+1mT=0.By substituting equation (12) into equation (14), the equilibrium thrust required should be(15)Ttc=12ρVtc2CDS+mgsinθ.A speed scheme that satisfies strap-down LM precision attack requirements could be defined by equations (14) and (15), and it can be formed as(16)Vtc=mq+m2q2+2ρCLSmgcosθρCLS,Ttc=12ρVtc2CDS+mgsinθ.


### 2.3. Analysis of the Speed Scheme’s Characteristic

#### 2.3.1. Boundary Conditions for the Solution

From equation (16), we can see that the parameters affecting the equations can be divided into two parts. We named them the static parameters and the dynamic parameters. The static parameters mainly include the gravitational acceleration, LM’s weight, the density of air, and the wing reference area. The dynamic parameters mainly include lift coefficient, drag coefficient, pitch rate, and pitch angle. In this paper, the static parameters were mainly regarded as known conditions and only the dynamic parameters have been analysed.(1) Pitch Angle Limited Value Ensures That the Thrust Has a Solution. The solution of the speed scheme requires(17)Vtc>0,Ttc∈0,Tmax,where Tmax is the maximum thrust of the engine.It can be seen from equation (16) that Vtc>0 is obvious; for Ttc, there is(18)0≤12ρVtc2CDS+mgsinθ≤Tmax.The design of the propulsion system should naturally meet the upper boundary requirements, so this paper studied the lower boundary of the pitch angle only.According to (18), when given a certain drag coefficient, there is(19)θ≥arcsin−1/2ρVtc2CDSmg,θ∈−π2,0.That means(20)θlb=arcsin−1/2ρVtc2CDSmg,where the subscript “lb” means this pitch angle is the lower boundary for speed scheme to have a positive thrust.Therefore, in order to meet the requirements of this scheme, the relationship between the limit value of the pitch angle and the flight speed during the attacking must obey the expression above, and it can be represented by a curve, as shown in Figure7:Figure 7
Lower boundary map of pitch angle according to the speed scheme. Different speeds give different lower boundaries for the pitch angle.According to equation (20), for different LMs or just for different flight status, if the drag coefficient CD is changing, the above curve in Figure 7 should be turned to a surface, which satisfies the relationship between the pitch angle value and the drag coefficient CD, as shown in Figure 8.Figure 8
Lower boundary map of pitch angle according to different CD.It can be seen from Figure8 that while the speed is constant, the greater the drag coefficient the steeper the pitch angle, and while the drag coefficient is constant, the greater the speed the steeper the pitch angle.(2) Balance between Flight Speed and Pitch Angle. While the LM’s lift coefficient is constant, it can be seen from the scheme equation (16) that the larger the pitch angle (absolute value) is, the smaller the flight speed that satisfies the normal equilibrium condition. According to the 1st expression of equation (16), a balance pitch can be calculated by equation (21), and the balance map of flight speed and pitch angle is shown in Figure 9.(21)θbalance=acrcosVtc2ρCLS−2Vtcmq2mg,where the subscript “balance” means this value is specific for the variable (pitch angle) to meet the speed scheme’s expected speed Vtc.Figure 9
Balance map of flight speed and pitch angle.In addition, when CL changes within a certain range with flight conditions, the relationship between balance pitch angle, CL, and speed is shown in Figure10:Figure 10
Balance map of pitch angle, flight speed, and CL.As can be seen from the above figure, in order to obtain a large attacking speed and obtain a steep attacking angle, the LM should have a small lift coefficient.(3)Valid Pitch Angles according to Prerequisites of Lower Boundary and the Balance Relationship. For a given set of aerodynamic characteristics [CD, CL], it can be known from the speed scheme that the pitch angle during attacking must satisfy both the lower boundary and the balance relationship we have given. We named these pitch angles “valid pitch angles.” This requirement is described in Figure 11.Figure 11
Valid pitch angle map according to both lower boundary and the balance relationship.When the aerodynamic characteristics [CD, CL] change within a certain range, valid pitch angles that satisfy the solution of the speed scheme under different conditions can be obtained, as shown in Figure12.Figure 12
Valid pitch angle map with different aerodynamic conditions: condition 1: CL = 0.1, CD = 0.07; condition 2: CL = 0.08, CD = 0.05; condition 3: CL = 0.06, CD = 0.03.Therefore, the valid pitch angles during attacking phase with respect to certain aerodynamic parameters can be expressed as(22)θvalid∈θ≥θlbθ=θblance,where the subscript “valid” means the pitch angle satisfies both the lower boundary and the balance relationship.According to equation (22), if the lift coefficient and the drag coefficient of a LM during attacking are known, we can calculate the boundary condition and the balance points of the pitch angle and then set the attack conditions according to a valid pitch angle. At the same time, the thrust needed can be calculated according to the speed scheme.


#### 2.3.2. The Existence and Uniqueness of Solution to the Motion of the LM

Having determined the boundary conditions of the solution, we only know that the scheme has the application basis with the corresponding conditions. In this section, the LM’s motion reliability during the attacking flight was confirmed. We attributed this problem to a uniqueness problem of the system solution [29].When the design of the control system is completed, the ideal attitude control loop can be equivalent to an inertia unit with a time constant ofTθ. Assume that the flight process is stable and the guidance process is implemented by the tracking guidance law, that is, the pitch rate is 0 (q≈0). Combine the proposed speed scheme equation (16) with the LM motion equation (9), and a new description of the system can be given:(23)R˙=−Vcosθ−η,η˙=−1RVsinθ−η,V˙=ρCDS2m2ρCLSmgcosθρCLS2−V2V˙Γ=−ρCLS2mV2+gcosθ,θ˙=η−θTθ.,System (23) can be linearized at any working point and expressed as follows:(24)x˙t=Ax.Based on Taylor’s expansion, the system characteristic matrix can be given as follows:(25)A=0−Vesinθe−ηe−cosθe−ηe0Vesinθe−ηeVeRe2sinθe−ηeVeRecosθe−ηe−1Resinθe−ηe0−VeRecosθe−ηe00−ρCDSVem0−CLCDgsinθe00−ρCLSVem0−gsinθe01Tθ00−1Tθ.The index of the variable “e” identifies the value at the working point, for example, flying according to the speed scheme is a special case.It can be seen from the system characteristic matrix that the elements in the fourth column are all zero. That means the normal velocityVΓ does not affect the condition of the system. So, in order to simplify the analysis of the system, the influence of the normal velocity can be ignored, and a simplified system characteristic matrix can be given:(26)A∗=0−Vesinθe−ηe−cosθe−ηeVesinθe−ηeVeRe2sinθe−ηeVeRecosθe−ηe−1Resinθe−ηe−VeRecosθe−ηe00−ρCDSVem−CLCDgsinθe01Tθ0−1Tθ.So, we get(27)A∗=CLVeCDgTθRe2sinθesinθe−ηecosθe−ηe.According to the compression mapping principle [30], when the system parameters meet the condition(28)A∗<1,the system should have a unique solution.


## 2.1. Equations of Aircraft Motion

As a novel precision-guided munition, LM in the form of small UAV could use the low-cost strap-down seeker [12, 27, 28] to achieve target capture and tracking during autonomous attack mission, so we discuss problems based on strap-down seeker against a stationary target.The flight model was expressed by differential equations in body-fixed coordinate which was defined as follows: the origin of coordinate system is located at the LM centroid.X axis is aligned with the longitudinal axis of LM; positive direction points to the LM’s nose. Y axis is perpendicular to the longitudinal axis and lies in horizontal plane; positive direction points to right. Z axis is perpendicular to XY plane; positive direction points down. This body-fixed coordinate system is illustrated in Figure 4.Figure 4
Body-fixed coordinate.In order to introduce the speed scheme clearly, only the longitudinal motion of a LM has been discussed. The force analysis of a LM is shown in Figure5, and the homing guidance geometry of missile-target engagement of the flight process is shown in Figure 6.Figure 5
Force analysis of LM.Figure 6
Two-dimensional homing guidance geometry of missile-target engagement.L is the lift force (aerodynamic force along z direction), D is the drag force (aerodynamic force along x direction), T is the thrust of engine, m is the mass of LM, g is the gravitational acceleration, and θ is the pitch angle (Euler angle along y axis) of LM.R is the instantaneous separation between LM and target, V is the speed of the loitering munition in x direction in body-coordinate, VΓ is the speed of the loitering munition in z direction in body-coordinate, η is the angle made by the line of sight with respect to the fixed reference, and q is the pitch rate (angular rate along y axis).Assume thatVΓ is small and θ is close to η, and the thrust T is assumed to act through the LM centroid and along the x axis. Then, from the engagement geometry and the force analysis of the aircraft, we can see the following.The relative motion relationship between the LM and the target can be described as the following equations:(1)R˙=−Vcosθ−η,η˙=−1RVsinθ−η.The dynamic equations of the LM can be written as follows:(2)V˙=1mT−D−mgsinθ,V˙Γ=1mmVq+mgcosθ−L,θ˙=q,q˙=MIy,where Iy is the LM’s moment of inertia in y direction.The equations of aerodynamic force and moment can be written as follows:(3)D=CDQS,L=CLQS,M=CmQScA,whereM is the aerodynamic moment component in y direction, CD is the drag coefficient, CL is the lift coefficient, Cm is the pitching moment coefficient, Q is the dynamic pressure, S is the reference area, and cA is the mean aerodynamic chord.In the above expressions, the pitching moment coefficient can be expressed as(4)Cm=CmLL+Cmδeδe+CmqqcA2V,where CmL is the moment derivative with respect to lift, Cmδe is the moment derivative with respect to control-surface deflection, δe is the control-surface deflection, and Cmq is the moment derivative with respect to pitch rate.The dynamic pressure can be expressed as(5)Q=12ρV2,where ρ is the density of air.Then, the equations describing the LM’s motion for the attacking stage can be expressed as(6)R˙=−Vcosθ−η,η˙=−1RVsinθ−η,V˙=1mT−12ρV2CDS−mgsinθ,V˙Γ=1mmVq+mgcosθ−L,θ˙=q,q˙=CmL1/2ρV2CLS+Cmδeδe+CmqqcA/2V1/2ρV2ScAIy.The state variables of the system during attacking can be selected as(7)X=R,η,V,VΓ,θ,q.The control variables are(8)U=T,δe.Equation (6) can be organized as(9)R˙=−Vcosθ−η,η˙=−1RVsinθ−η,V˙=−ρCDS2mV2−gsinθ+1mT,V˙Γ=−ρCLS2mV2+Vq+gcosθ,θ˙=q,q˙=CmLCL1/4ρ2S2cAIyV4+Cmq1/4ρScA2IyVq+Cmδe1/2ρScAIyV2δe.This is the system equation that describes the motion of the aircraft at the attacking phase.


## 2.2. A Speed Scheme Suitable for a LM with Strap-Down Seeker for the Attacking Stage

For a LM with strap-down seeker, it is important to keep flight stabilized to get an ideal target tracking feedback, which means there should be no excessive normal shifting motion during attacking. The desired normal acceleration should be set:(10)V˙Γ=0.According to the 4th expression of equation (9), this condition can be described as(11)−ρCLS2mV2+Vq+gcosθ=0.It can be solved from the above equation that the expected speed during the attacking process is(12)Vtc=mq+m2q2+2ρCLSmgcosθρCLS,where the subscript “tc” indicates that the value of the variable is an expected value which depends on the attacking process.When the LM is flying at this expected speed, it means(13)V˙=0.According to the 3rd expression of equation (9), this condition can be described as(14)−ρCDS2mV2−gsinθ+1mT=0.By substituting equation (12) into equation (14), the equilibrium thrust required should be(15)Ttc=12ρVtc2CDS+mgsinθ.A speed scheme that satisfies strap-down LM precision attack requirements could be defined by equations (14) and (15), and it can be formed as(16)Vtc=mq+m2q2+2ρCLSmgcosθρCLS,Ttc=12ρVtc2CDS+mgsinθ.


## 2.3. Analysis of the Speed Scheme’s Characteristic

### 2.3.1. Boundary Conditions for the Solution

From equation (16), we can see that the parameters affecting the equations can be divided into two parts. We named them the static parameters and the dynamic parameters. The static parameters mainly include the gravitational acceleration, LM’s weight, the density of air, and the wing reference area. The dynamic parameters mainly include lift coefficient, drag coefficient, pitch rate, and pitch angle. In this paper, the static parameters were mainly regarded as known conditions and only the dynamic parameters have been analysed.(1) Pitch Angle Limited Value Ensures That the Thrust Has a Solution. The solution of the speed scheme requires(17)Vtc>0,Ttc∈0,Tmax,where Tmax is the maximum thrust of the engine.It can be seen from equation (16) that Vtc>0 is obvious; for Ttc, there is(18)0≤12ρVtc2CDS+mgsinθ≤Tmax.The design of the propulsion system should naturally meet the upper boundary requirements, so this paper studied the lower boundary of the pitch angle only.According to (18), when given a certain drag coefficient, there is(19)θ≥arcsin−1/2ρVtc2CDSmg,θ∈−π2,0.That means(20)θlb=arcsin−1/2ρVtc2CDSmg,where the subscript “lb” means this pitch angle is the lower boundary for speed scheme to have a positive thrust.Therefore, in order to meet the requirements of this scheme, the relationship between the limit value of the pitch angle and the flight speed during the attacking must obey the expression above, and it can be represented by a curve, as shown in Figure7:Figure 7
Lower boundary map of pitch angle according to the speed scheme. Different speeds give different lower boundaries for the pitch angle.According to equation (20), for different LMs or just for different flight status, if the drag coefficient CD is changing, the above curve in Figure 7 should be turned to a surface, which satisfies the relationship between the pitch angle value and the drag coefficient CD, as shown in Figure 8.Figure 8
Lower boundary map of pitch angle according to different CD.It can be seen from Figure8 that while the speed is constant, the greater the drag coefficient the steeper the pitch angle, and while the drag coefficient is constant, the greater the speed the steeper the pitch angle.(2) Balance between Flight Speed and Pitch Angle. While the LM’s lift coefficient is constant, it can be seen from the scheme equation (16) that the larger the pitch angle (absolute value) is, the smaller the flight speed that satisfies the normal equilibrium condition. According to the 1st expression of equation (16), a balance pitch can be calculated by equation (21), and the balance map of flight speed and pitch angle is shown in Figure 9.(21)θbalance=acrcosVtc2ρCLS−2Vtcmq2mg,where the subscript “balance” means this value is specific for the variable (pitch angle) to meet the speed scheme’s expected speed Vtc.Figure 9
Balance map of flight speed and pitch angle.In addition, when CL changes within a certain range with flight conditions, the relationship between balance pitch angle, CL, and speed is shown in Figure10:Figure 10
Balance map of pitch angle, flight speed, and CL.As can be seen from the above figure, in order to obtain a large attacking speed and obtain a steep attacking angle, the LM should have a small lift coefficient.(3)Valid Pitch Angles according to Prerequisites of Lower Boundary and the Balance Relationship. For a given set of aerodynamic characteristics [CD, CL], it can be known from the speed scheme that the pitch angle during attacking must satisfy both the lower boundary and the balance relationship we have given. We named these pitch angles “valid pitch angles.” This requirement is described in Figure 11.Figure 11
Valid pitch angle map according to both lower boundary and the balance relationship.When the aerodynamic characteristics [CD, CL] change within a certain range, valid pitch angles that satisfy the solution of the speed scheme under different conditions can be obtained, as shown in Figure12.Figure 12
Valid pitch angle map with different aerodynamic conditions: condition 1: CL = 0.1, CD = 0.07; condition 2: CL = 0.08, CD = 0.05; condition 3: CL = 0.06, CD = 0.03.Therefore, the valid pitch angles during attacking phase with respect to certain aerodynamic parameters can be expressed as(22)θvalid∈θ≥θlbθ=θblance,where the subscript “valid” means the pitch angle satisfies both the lower boundary and the balance relationship.According to equation (22), if the lift coefficient and the drag coefficient of a LM during attacking are known, we can calculate the boundary condition and the balance points of the pitch angle and then set the attack conditions according to a valid pitch angle. At the same time, the thrust needed can be calculated according to the speed scheme.


### 2.3.2. The Existence and Uniqueness of Solution to the Motion of the LM

Having determined the boundary conditions of the solution, we only know that the scheme has the application basis with the corresponding conditions. In this section, the LM’s motion reliability during the attacking flight was confirmed. We attributed this problem to a uniqueness problem of the system solution [29].When the design of the control system is completed, the ideal attitude control loop can be equivalent to an inertia unit with a time constant ofTθ. Assume that the flight process is stable and the guidance process is implemented by the tracking guidance law, that is, the pitch rate is 0 (q≈0). Combine the proposed speed scheme equation (16) with the LM motion equation (9), and a new description of the system can be given:(23)R˙=−Vcosθ−η,η˙=−1RVsinθ−η,V˙=ρCDS2m2ρCLSmgcosθρCLS2−V2V˙Γ=−ρCLS2mV2+gcosθ,θ˙=η−θTθ.,System (23) can be linearized at any working point and expressed as follows:(24)x˙t=Ax.Based on Taylor’s expansion, the system characteristic matrix can be given as follows:(25)A=0−Vesinθe−ηe−cosθe−ηe0Vesinθe−ηeVeRe2sinθe−ηeVeRecosθe−ηe−1Resinθe−ηe0−VeRecosθe−ηe00−ρCDSVem0−CLCDgsinθe00−ρCLSVem0−gsinθe01Tθ00−1Tθ.The index of the variable “e” identifies the value at the working point, for example, flying according to the speed scheme is a special case.It can be seen from the system characteristic matrix that the elements in the fourth column are all zero. That means the normal velocityVΓ does not affect the condition of the system. So, in order to simplify the analysis of the system, the influence of the normal velocity can be ignored, and a simplified system characteristic matrix can be given:(26)A∗=0−Vesinθe−ηe−cosθe−ηeVesinθe−ηeVeRe2sinθe−ηeVeRecosθe−ηe−1Resinθe−ηe−VeRecosθe−ηe00−ρCDSVem−CLCDgsinθe01Tθ0−1Tθ.So, we get(27)A∗=CLVeCDgTθRe2sinθesinθe−ηecosθe−ηe.According to the compression mapping principle [30], when the system parameters meet the condition(28)A∗<1,the system should have a unique solution.


## 2.3.1. Boundary Conditions for the Solution

From equation (16), we can see that the parameters affecting the equations can be divided into two parts. We named them the static parameters and the dynamic parameters. The static parameters mainly include the gravitational acceleration, LM’s weight, the density of air, and the wing reference area. The dynamic parameters mainly include lift coefficient, drag coefficient, pitch rate, and pitch angle. In this paper, the static parameters were mainly regarded as known conditions and only the dynamic parameters have been analysed.(1) Pitch Angle Limited Value Ensures That the Thrust Has a Solution. The solution of the speed scheme requires(17)Vtc>0,Ttc∈0,Tmax,where Tmax is the maximum thrust of the engine.It can be seen from equation (16) that Vtc>0 is obvious; for Ttc, there is(18)0≤12ρVtc2CDS+mgsinθ≤Tmax.The design of the propulsion system should naturally meet the upper boundary requirements, so this paper studied the lower boundary of the pitch angle only.According to (18), when given a certain drag coefficient, there is(19)θ≥arcsin−1/2ρVtc2CDSmg,θ∈−π2,0.That means(20)θlb=arcsin−1/2ρVtc2CDSmg,where the subscript “lb” means this pitch angle is the lower boundary for speed scheme to have a positive thrust.Therefore, in order to meet the requirements of this scheme, the relationship between the limit value of the pitch angle and the flight speed during the attacking must obey the expression above, and it can be represented by a curve, as shown in Figure7:Figure 7
Lower boundary map of pitch angle according to the speed scheme. Different speeds give different lower boundaries for the pitch angle.According to equation (20), for different LMs or just for different flight status, if the drag coefficient CD is changing, the above curve in Figure 7 should be turned to a surface, which satisfies the relationship between the pitch angle value and the drag coefficient CD, as shown in Figure 8.Figure 8
Lower boundary map of pitch angle according to different CD.It can be seen from Figure8 that while the speed is constant, the greater the drag coefficient the steeper the pitch angle, and while the drag coefficient is constant, the greater the speed the steeper the pitch angle.(2) Balance between Flight Speed and Pitch Angle. While the LM’s lift coefficient is constant, it can be seen from the scheme equation (16) that the larger the pitch angle (absolute value) is, the smaller the flight speed that satisfies the normal equilibrium condition. According to the 1st expression of equation (16), a balance pitch can be calculated by equation (21), and the balance map of flight speed and pitch angle is shown in Figure 9.(21)θbalance=acrcosVtc2ρCLS−2Vtcmq2mg,where the subscript “balance” means this value is specific for the variable (pitch angle) to meet the speed scheme’s expected speed Vtc.Figure 9
Balance map of flight speed and pitch angle.In addition, when CL changes within a certain range with flight conditions, the relationship between balance pitch angle, CL, and speed is shown in Figure10:Figure 10
Balance map of pitch angle, flight speed, and CL.As can be seen from the above figure, in order to obtain a large attacking speed and obtain a steep attacking angle, the LM should have a small lift coefficient.(3)Valid Pitch Angles according to Prerequisites of Lower Boundary and the Balance Relationship. For a given set of aerodynamic characteristics [CD, CL], it can be known from the speed scheme that the pitch angle during attacking must satisfy both the lower boundary and the balance relationship we have given. We named these pitch angles “valid pitch angles.” This requirement is described in Figure 11.Figure 11
Valid pitch angle map according to both lower boundary and the balance relationship.When the aerodynamic characteristics [CD, CL] change within a certain range, valid pitch angles that satisfy the solution of the speed scheme under different conditions can be obtained, as shown in Figure12.Figure 12
Valid pitch angle map with different aerodynamic conditions: condition 1: CL = 0.1, CD = 0.07; condition 2: CL = 0.08, CD = 0.05; condition 3: CL = 0.06, CD = 0.03.Therefore, the valid pitch angles during attacking phase with respect to certain aerodynamic parameters can be expressed as(22)θvalid∈θ≥θlbθ=θblance,where the subscript “valid” means the pitch angle satisfies both the lower boundary and the balance relationship.According to equation (22), if the lift coefficient and the drag coefficient of a LM during attacking are known, we can calculate the boundary condition and the balance points of the pitch angle and then set the attack conditions according to a valid pitch angle. At the same time, the thrust needed can be calculated according to the speed scheme.


## 2.3.2. The Existence and Uniqueness of Solution to the Motion of the LM

Having determined the boundary conditions of the solution, we only know that the scheme has the application basis with the corresponding conditions. In this section, the LM’s motion reliability during the attacking flight was confirmed. We attributed this problem to a uniqueness problem of the system solution [29].When the design of the control system is completed, the ideal attitude control loop can be equivalent to an inertia unit with a time constant ofTθ. Assume that the flight process is stable and the guidance process is implemented by the tracking guidance law, that is, the pitch rate is 0 (q≈0). Combine the proposed speed scheme equation (16) with the LM motion equation (9), and a new description of the system can be given:(23)R˙=−Vcosθ−η,η˙=−1RVsinθ−η,V˙=ρCDS2m2ρCLSmgcosθρCLS2−V2V˙Γ=−ρCLS2mV2+gcosθ,θ˙=η−θTθ.,System (23) can be linearized at any working point and expressed as follows:(24)x˙t=Ax.Based on Taylor’s expansion, the system characteristic matrix can be given as follows:(25)A=0−Vesinθe−ηe−cosθe−ηe0Vesinθe−ηeVeRe2sinθe−ηeVeRecosθe−ηe−1Resinθe−ηe0−VeRecosθe−ηe00−ρCDSVem0−CLCDgsinθe00−ρCLSVem0−gsinθe01Tθ00−1Tθ.The index of the variable “e” identifies the value at the working point, for example, flying according to the speed scheme is a special case.It can be seen from the system characteristic matrix that the elements in the fourth column are all zero. That means the normal velocityVΓ does not affect the condition of the system. So, in order to simplify the analysis of the system, the influence of the normal velocity can be ignored, and a simplified system characteristic matrix can be given:(26)A∗=0−Vesinθe−ηe−cosθe−ηeVesinθe−ηeVeRe2sinθe−ηeVeRecosθe−ηe−1Resinθe−ηe−VeRecosθe−ηe00−ρCDSVem−CLCDgsinθe01Tθ0−1Tθ.So, we get(27)A∗=CLVeCDgTθRe2sinθesinθe−ηecosθe−ηe.According to the compression mapping principle [30], when the system parameters meet the condition(28)A∗<1,the system should have a unique solution.


## 3. Scheme Verification

After analysing the boundary conditions and the uniqueness of the solution, requirements of the program application have been grasped basically. So, we carry out the verification work next.First of all, the steps of the test were introduced. In this work, we found that there were many different methods to obtain the parameters the test needs, and different methods are different in difficulty and accuracy. Therefore, a method to get the lift coefficient and drag coefficient was introduced first, and then the experimental flow of scheme verification was given.


### 3.1. Acquisition of Parameters and the Scheme Testing Process

The thrust was calculated according to the equation (16), wherein the LM weight m, the density of air, the wing reference area, and the gravitational acceleration can be regarded as known parameters. The pitch angle and the pitch rate could be measured with sensors during the flight. Therefore, the lift coefficient and the drag coefficient were the key parameters for the realization of the speed and thrust scheme, which needs to be acquired for the verification of flight test. The following is a further explanation of the acquisition of the lift coefficient and the drag coefficient before the verification process.For small-sized aircraft, obtaining aerodynamics data through wind tunnel test has the disadvantages of complicated experiment and high cost. Especially for occasions requiring rapid prototyping and special control performance verification, traditional wind tunnel test cannot meet low cost and rapid evaluation in engineering application.The commonly used methods also include the aerodynamic parameter identification method based on flight test and the means of simulation calculation [31–34]. The main disadvantage of the simulation calculation is that the accuracy cannot be reliably guaranteed, and the result needs to be evaluated and revised using a series of experimental data.In order to reduce the cost of experiments and quickly obtain parameters closer to the actual situation than simulation calculations and in order to ensure the safety of the test, this paper adopted the aerodynamic parameter identification method of the aircraft balance point based on an object understanding calculation method.According to equation (9), the formulas we introduced to calculate the lift coefficient and the drag coefficient are(29)CL=Vfly_dataqfly_data−V˙Γfly_data+gcosθfly_data1/2ρVfly_data2S/m,CD=Tfly_data−V˙fly_data−mgsinθfly_data1/2ρVfly_data2S,where the subscript “fly_data” indicates that the specific values should be obtained from a fly test.In order to avoid the situation that the singular matrix cannot be solved and in order to obtain a more accurate solution using experimental data with measurement errors, the above equations were extended to overdetermined equations.Rewrite the above expressions as(30)ACLXCL=bCL,(31)ACDXCD=bCD,where(32)ACL=ρS2mVfly_data21ρS2mVfly_data22⋯ρS2mVfly_data2i,(33)XCL=CL,(34)bCL=Vfly_data1qfly_data1−V˙Γfly_data1+gcosθfly_data1Vfly_data2qfly_data2−V˙Γfly_data2+gcosθfly_data2⋯Vfly_dataiqfly_datai−V˙Γfly_datai+gcosθfly_datai,(35)ACD=ρS2Vfly_data21ρS2Vfly_data22⋯ρS2Vfly_data2i,(36)XCD=CD,(37)bCD=Tfly_data1−V˙fly_data1−mgsinθfly_data1Tfly_data2−V˙fly_data2−mgsinθfly_data2⋯Tfly_datai−V˙fly_datai−mgsinθfly_datai.In the above expressions, “i” is the number of equations in the overdetermined equations. Because solving the overdetermined equation is not the focus of this article, it will not be described in detail [35, 36].The parameter calculation and scheme test flow are given in Figure13:Figure 13
Flowchart of scheme test.The following is a detailed description of the verification flight test process.


### 3.2. Prophase Flight Test for Data Collection

The prophase flight test was designed to collect data for parameter calculations. The UAV used for flight test is shown in Figure14, and the parameters of the UAV are shown in Table 1.Figure 14
The UAV used for flight test.Table 1
Parameters of the UAV.
Wing reference area (m2)Wing span (m)Weight (kg)Length (m)0.461.23.51A simple data acquisition and flight control module was designed. The hardware system is shown in Figures15 and 16, and the device list is shown in Table 2.Figure 15
Hardware design structure diagram.Figure 16
The experimental equipment.Table 2
Main device list.
ModuleMain deviceUse of this moduleMaster processorSTM32F407Airspeed calculation and flight controlAirspeed sensorMS5525Dynamic pressure measurementRemote control receiverFutaba-R7008SBAccepting operator’s remote commandGPS moduleU-blox M8QMeasuring aircraft positionIMUMPU-9250Measuring the inertia of the aircraftMemory unitSanDisk-32MStoring experimental data in flightMinor processorRK3399Image compressionCameraFPV3.6Video captureAccording to the equations (32), (34), (35), and (37), the data needed to be collected include speed, pitch angle, thrust of the engine (in the experiment, the output percentage of the engine controller was collected and then converted into the actual thrust), pitch rate, and accelerations in x direction and z direction in body-coordinate frame. The data storage frequency was 50 Hz and the duration was 2 s.The obtained experimental data are shown in the following figures: Figure17 shows the output percentage of the engine controller, Figure 18 shows the acceleration in z direction in body-coordinate, Figure 19 shows the acceleration in x direction in body-coordinate, Figure 20 shows the pitch rate, Figure 21 shows the pitch angle, and Figure 22 shows the flight speed.Figure 17
Output percentage of the engine controller.Figure 18
Acceleration inz direction in body-coordinate.Figure 19
Acceleration inx direction in body-coordinate.Figure 20
Pitch rate.Figure 21
Pitch angle.Figure 22
Speed.


### 3.3. Calculation of Lift and Drag Coefficients

The lift coefficient and the drag coefficient were calculated according to equations (30) and (31) after the experimental data were obtained, and the calculation results are shown in Table 3.Table 3
Calculation result of the lift coefficient and the drag coefficient.
ParameterValueCL0.21CD0.1


### 3.4. Boundary Condition Calculation

According to equations (20)–(22) and the results of the previous section (see Table 3), the valid pitch angles could be calculated. The valid pitch angle map and the critical point of the valid pitch angle can be seen in Figure 23.Figure 23
Valid pitch angle map established by flight test data.It can be seen that when the pitch angle is in the range ofθvalid∈−25∘,0, there will be a valid solution of the system according to the scheme proposed in this paper, and the attacking mission could be performed.According to the previous flight test experience, the target attack range was 100–150 m after the target was clear in the field of view limited by the camera performance (the relationship between flight altitude, camera performance, and pitch angle is complex and is not the focus of this article, so it will not be described in detail). So, the reasonable range of attack altitude should be controlled within the following range:(38)hattack∈hminLattacktanminθvalid<h<maxLattacktanminθvalid,where the subscript “attack” means the value is at the time the attack started.That is,(39)hattack∈h46.63<h<69.94.


### 3.5. Judgement of the Existence and Uniqueness of Solutions to the Motion

According to the experimental data analysis, the valid pitch angle was in the range ofθvalid∈−25∘,0, the flight speed was in the range of V∈0,23, and the pitch angle response time was Tθ=0.2s. According to equation (27),(40)A∗=CLVeCDgTθRe2sinθesinθe−ηecosθe−ηe<CLmaxV2CDgTθRe2sinminθvalidsin2θe−ηe<CLmaxV2CDgTθRe2sinminθvalid.So, the system will have a unique solution when(41)CLmaxV2CDgTθRe2sinminθvalid<1.That is,(42)Re>2.2819.It can be seen from the above calculation result that there would be a unique state trajectory in the attacking process before the distance to the target is less than 2.2819 meters. This limit distance satisfied the conditions of the test; in other words, the speed scheme proposed in this paper was feasible.


### 3.6. Scheme Flight Test

#### 3.6.1. Control System Block Build according to the Speed Scheme

The logic of the flight test: use the scheme to calculate an expected thrust during the attacking phase and cooperate with the attitude control module to complete the impact task against a nonmoving target.The control system constructed in this paper consisted of an attitude control module and a speed control module. The attitude control module was realized by a classical PID controller. The speed control module was formed according to the speed scheme. The system block diagram is shown in Figure24.Figure 24
Control system block diagram.The speed controller parameters were calculated according to equation (16) and are shown in Table 4.Table 4
Speed controller parameters corresponding to the scheme.
ParameterValueK10.03K234.30K38.01K48.57


#### 3.6.2. Test Implementation Method

The UAV was the same one used in data acquisition (see Figure14), and the flight process can be seen in Figure 25 and can be described as follows:(1)
Take Off. The UAV takes off by remote controller and flies to the vicinity of the target.(2)
Regulate. Switch the control mode to the automatic driving by the remote controller and the UAV would adjust the route and attitude automatically.(3)
Attack. When the target enters the field of view, it can be selected on the video; at the same time, the controller designed in this paper starts working automatically until the UAV hits the target.Figure 25
Flight phase diagram.


#### 3.6.3. Flight Test Result

In this section, the result of the flight test is presented. The final 7 s flight path was selected for presentation purposes during attacking stages of the flight test, and the data are shown in the following figures: Figure26 shows the thrust, Figure 27 shows the speed, Figure 28 shows the pitch angle, Figure 29 shows the acceleration in z direction in body-coordinate, Figure 30 shows the flight path, Figure 31 shows the first perspective video screenshot, and Figure 32 shows the third perspective video screenshot.Figure 26
Thrust in flight test.Figure 27
Speed in flight test.Figure 28
Acceleration inz direction in body-coordinate.Figure 29
Pitch angle in flight test.Figure 30
Flight path in flight test.Figure 31
First perspective video screenshot in flight test.Figure 32
Third perspective video screenshot in flight test.Based on the experimental results, following conclusions can be given:First of all, Figures26 and 27 show that the thrust and speed were dynamic curves, which were very different from the traditional ammunition speed throttle scheme (as can been seen in Figure 3), and their changes were all within reasonable ranges.Secondly, as can been seen in Table5, the mean of acceleration in z direction from the scheme flight test was −0.6351 and that from remote control test was −9.323. The standard deviation of acceleration in z direction from the scheme flight test was 1.0596 and that from remote control test was 1.727. Compared with data from the remote control test (which has been shown in Figure 18), acceleration in z direction in body frame has been suppressed clearly (as can been seen in Figure 28).Table 5
Mean and standard deviation of flight tests.
Data sourceMeanStandard deviationRemote control test−9.3231.727Scheme flight test−0.63511.0596At the same time, the pitch angle and the flight trajectory curve is smooth (as shown in Figures29 and 30), which strongly verifies that the speed scheme proposed in this paper was helpful to improve the attitude and trajectory stability of the UAV during attacking.Finally, as can been seen in the screenshots of the flight videos (Figures31 and 32), the UAV successfully hit the target, which further illustrated the effectiveness of the scheme.


#### 3.6.4. Defect Analysis

However, the airspeed during the flight did not follow the expected trajectory perfectly, as shown in Figure33. Further analysis is given below.Figure 33
Tracking effect of airspeed command in automatic attack. The expected one was calculated according to equation (16).(1) Dynamic Response Characteristics of the Throttle. Since the airspeed did not change much during the whole flight process, it can be assumed that the thrust depended only on the working state of the propulsion system and was independent of the atmospheric conditions. According to equation (9), using the thrust as input, the transfer function of the speed output can be expressed as(43)ΔVΔT=kTss2−2ξωTs+ωT2.Assuming that the throttle thrust input signal is step-signalΔT1t, based on final value theorem of Laplace transform, the speed response is(44)ΔVt⟶∞=lims⟶0s•kTss2−2ξωTs+ωT2•ΔTs=0.This shows that the change of the throttle can only adjust the speed during the transition process but cannot fully achieve the purpose of controlling the speed according to the ideal trajectory.(2) Stability Analysis of Speed. From equation (24) and the system characteristic matrix equation (25), the differential expression describing the airspeed motion process at any operating point after linearization can be given by(45)V˙=−ρCDSVemV−CLCDgsinθeθ.When the system is aiming at the target, there is(46)V˙+ρCDSVemV+CLCDgsinθeθe=0.Its characteristic roots are(47)ζ1,2=−ρCDSVe/m±ρCDSVe/m2−4CL/CDgsinθeθe2,where(48)ρCDSVem>0,CLCDgsinθeθe>0.This means the two eigenvalues of the differential equation have negative real parts, so the change of the airspeed trajectory should satisfy the stability condition, and the test data also illustrated this feature: the speed curve rises following the expected one (Figure33).However, as shown in Figure23 (the control system block diagram), only an open-loop controller was used to control the speed to achieve the desired value Vtc. Obviously, an open-loop controller’s performance cannot meet the requirement of a complex testing environment, and the experimental results also verified this conclusion: the speed curve rises but slowly.It would be nice to design a better controller to improve the response characteristics of the speed, but how to improve the control law requires further analysis, and it is out of range of this research, so it is one of the tasks we will continue to expand in the future.


## 3.1. Acquisition of Parameters and the Scheme Testing Process

The thrust was calculated according to the equation (16), wherein the LM weight m, the density of air, the wing reference area, and the gravitational acceleration can be regarded as known parameters. The pitch angle and the pitch rate could be measured with sensors during the flight. Therefore, the lift coefficient and the drag coefficient were the key parameters for the realization of the speed and thrust scheme, which needs to be acquired for the verification of flight test. The following is a further explanation of the acquisition of the lift coefficient and the drag coefficient before the verification process.For small-sized aircraft, obtaining aerodynamics data through wind tunnel test has the disadvantages of complicated experiment and high cost. Especially for occasions requiring rapid prototyping and special control performance verification, traditional wind tunnel test cannot meet low cost and rapid evaluation in engineering application.The commonly used methods also include the aerodynamic parameter identification method based on flight test and the means of simulation calculation [31–34]. The main disadvantage of the simulation calculation is that the accuracy cannot be reliably guaranteed, and the result needs to be evaluated and revised using a series of experimental data.In order to reduce the cost of experiments and quickly obtain parameters closer to the actual situation than simulation calculations and in order to ensure the safety of the test, this paper adopted the aerodynamic parameter identification method of the aircraft balance point based on an object understanding calculation method.According to equation (9), the formulas we introduced to calculate the lift coefficient and the drag coefficient are(29)CL=Vfly_dataqfly_data−V˙Γfly_data+gcosθfly_data1/2ρVfly_data2S/m,CD=Tfly_data−V˙fly_data−mgsinθfly_data1/2ρVfly_data2S,where the subscript “fly_data” indicates that the specific values should be obtained from a fly test.In order to avoid the situation that the singular matrix cannot be solved and in order to obtain a more accurate solution using experimental data with measurement errors, the above equations were extended to overdetermined equations.Rewrite the above expressions as(30)ACLXCL=bCL,(31)ACDXCD=bCD,where(32)ACL=ρS2mVfly_data21ρS2mVfly_data22⋯ρS2mVfly_data2i,(33)XCL=CL,(34)bCL=Vfly_data1qfly_data1−V˙Γfly_data1+gcosθfly_data1Vfly_data2qfly_data2−V˙Γfly_data2+gcosθfly_data2⋯Vfly_dataiqfly_datai−V˙Γfly_datai+gcosθfly_datai,(35)ACD=ρS2Vfly_data21ρS2Vfly_data22⋯ρS2Vfly_data2i,(36)XCD=CD,(37)bCD=Tfly_data1−V˙fly_data1−mgsinθfly_data1Tfly_data2−V˙fly_data2−mgsinθfly_data2⋯Tfly_datai−V˙fly_datai−mgsinθfly_datai.In the above expressions, “i” is the number of equations in the overdetermined equations. Because solving the overdetermined equation is not the focus of this article, it will not be described in detail [35, 36].The parameter calculation and scheme test flow are given in Figure13:Figure 13
Flowchart of scheme test.The following is a detailed description of the verification flight test process.


## 3.2. Prophase Flight Test for Data Collection

The prophase flight test was designed to collect data for parameter calculations. The UAV used for flight test is shown in Figure14, and the parameters of the UAV are shown in Table 1.Figure 14
The UAV used for flight test.Table 1
Parameters of the UAV.
Wing reference area (m2)Wing span (m)Weight (kg)Length (m)0.461.23.51A simple data acquisition and flight control module was designed. The hardware system is shown in Figures15 and 16, and the device list is shown in Table 2.Figure 15
Hardware design structure diagram.Figure 16
The experimental equipment.Table 2
Main device list.
ModuleMain deviceUse of this moduleMaster processorSTM32F407Airspeed calculation and flight controlAirspeed sensorMS5525Dynamic pressure measurementRemote control receiverFutaba-R7008SBAccepting operator’s remote commandGPS moduleU-blox M8QMeasuring aircraft positionIMUMPU-9250Measuring the inertia of the aircraftMemory unitSanDisk-32MStoring experimental data in flightMinor processorRK3399Image compressionCameraFPV3.6Video captureAccording to the equations (32), (34), (35), and (37), the data needed to be collected include speed, pitch angle, thrust of the engine (in the experiment, the output percentage of the engine controller was collected and then converted into the actual thrust), pitch rate, and accelerations in x direction and z direction in body-coordinate frame. The data storage frequency was 50 Hz and the duration was 2 s.The obtained experimental data are shown in the following figures: Figure17 shows the output percentage of the engine controller, Figure 18 shows the acceleration in z direction in body-coordinate, Figure 19 shows the acceleration in x direction in body-coordinate, Figure 20 shows the pitch rate, Figure 21 shows the pitch angle, and Figure 22 shows the flight speed.Figure 17
Output percentage of the engine controller.Figure 18
Acceleration inz direction in body-coordinate.Figure 19
Acceleration inx direction in body-coordinate.Figure 20
Pitch rate.Figure 21
Pitch angle.Figure 22
Speed.


## 3.3. Calculation of Lift and Drag Coefficients

The lift coefficient and the drag coefficient were calculated according to equations (30) and (31) after the experimental data were obtained, and the calculation results are shown in Table 3.Table 3
Calculation result of the lift coefficient and the drag coefficient.
ParameterValueCL0.21CD0.1


## 3.4. Boundary Condition Calculation

According to equations (20)–(22) and the results of the previous section (see Table 3), the valid pitch angles could be calculated. The valid pitch angle map and the critical point of the valid pitch angle can be seen in Figure 23.Figure 23
Valid pitch angle map established by flight test data.It can be seen that when the pitch angle is in the range ofθvalid∈−25∘,0, there will be a valid solution of the system according to the scheme proposed in this paper, and the attacking mission could be performed.According to the previous flight test experience, the target attack range was 100–150 m after the target was clear in the field of view limited by the camera performance (the relationship between flight altitude, camera performance, and pitch angle is complex and is not the focus of this article, so it will not be described in detail). So, the reasonable range of attack altitude should be controlled within the following range:(38)hattack∈hminLattacktanminθvalid<h<maxLattacktanminθvalid,where the subscript “attack” means the value is at the time the attack started.That is,(39)hattack∈h46.63<h<69.94.


## 3.5. Judgement of the Existence and Uniqueness of Solutions to the Motion

According to the experimental data analysis, the valid pitch angle was in the range ofθvalid∈−25∘,0, the flight speed was in the range of V∈0,23, and the pitch angle response time was Tθ=0.2s. According to equation (27),(40)A∗=CLVeCDgTθRe2sinθesinθe−ηecosθe−ηe<CLmaxV2CDgTθRe2sinminθvalidsin2θe−ηe<CLmaxV2CDgTθRe2sinminθvalid.So, the system will have a unique solution when(41)CLmaxV2CDgTθRe2sinminθvalid<1.That is,(42)Re>2.2819.It can be seen from the above calculation result that there would be a unique state trajectory in the attacking process before the distance to the target is less than 2.2819 meters. This limit distance satisfied the conditions of the test; in other words, the speed scheme proposed in this paper was feasible.


## 3.6. Scheme Flight Test

### 3.6.1. Control System Block Build according to the Speed Scheme

The logic of the flight test: use the scheme to calculate an expected thrust during the attacking phase and cooperate with the attitude control module to complete the impact task against a nonmoving target.The control system constructed in this paper consisted of an attitude control module and a speed control module. The attitude control module was realized by a classical PID controller. The speed control module was formed according to the speed scheme. The system block diagram is shown in Figure24.Figure 24
Control system block diagram.The speed controller parameters were calculated according to equation (16) and are shown in Table 4.Table 4
Speed controller parameters corresponding to the scheme.
ParameterValueK10.03K234.30K38.01K48.57


### 3.6.2. Test Implementation Method

The UAV was the same one used in data acquisition (see Figure14), and the flight process can be seen in Figure 25 and can be described as follows:(1)
Take Off. The UAV takes off by remote controller and flies to the vicinity of the target.(2)
Regulate. Switch the control mode to the automatic driving by the remote controller and the UAV would adjust the route and attitude automatically.(3)
Attack. When the target enters the field of view, it can be selected on the video; at the same time, the controller designed in this paper starts working automatically until the UAV hits the target.Figure 25
Flight phase diagram.


### 3.6.3. Flight Test Result

In this section, the result of the flight test is presented. The final 7 s flight path was selected for presentation purposes during attacking stages of the flight test, and the data are shown in the following figures: Figure26 shows the thrust, Figure 27 shows the speed, Figure 28 shows the pitch angle, Figure 29 shows the acceleration in z direction in body-coordinate, Figure 30 shows the flight path, Figure 31 shows the first perspective video screenshot, and Figure 32 shows the third perspective video screenshot.Figure 26
Thrust in flight test.Figure 27
Speed in flight test.Figure 28
Acceleration inz direction in body-coordinate.Figure 29
Pitch angle in flight test.Figure 30
Flight path in flight test.Figure 31
First perspective video screenshot in flight test.Figure 32
Third perspective video screenshot in flight test.Based on the experimental results, following conclusions can be given:First of all, Figures26 and 27 show that the thrust and speed were dynamic curves, which were very different from the traditional ammunition speed throttle scheme (as can been seen in Figure 3), and their changes were all within reasonable ranges.Secondly, as can been seen in Table5, the mean of acceleration in z direction from the scheme flight test was −0.6351 and that from remote control test was −9.323. The standard deviation of acceleration in z direction from the scheme flight test was 1.0596 and that from remote control test was 1.727. Compared with data from the remote control test (which has been shown in Figure 18), acceleration in z direction in body frame has been suppressed clearly (as can been seen in Figure 28).Table 5
Mean and standard deviation of flight tests.
Data sourceMeanStandard deviationRemote control test−9.3231.727Scheme flight test−0.63511.0596At the same time, the pitch angle and the flight trajectory curve is smooth (as shown in Figures29 and 30), which strongly verifies that the speed scheme proposed in this paper was helpful to improve the attitude and trajectory stability of the UAV during attacking.Finally, as can been seen in the screenshots of the flight videos (Figures31 and 32), the UAV successfully hit the target, which further illustrated the effectiveness of the scheme.


### 3.6.4. Defect Analysis

However, the airspeed during the flight did not follow the expected trajectory perfectly, as shown in Figure33. Further analysis is given below.Figure 33
Tracking effect of airspeed command in automatic attack. The expected one was calculated according to equation (16).(1) Dynamic Response Characteristics of the Throttle. Since the airspeed did not change much during the whole flight process, it can be assumed that the thrust depended only on the working state of the propulsion system and was independent of the atmospheric conditions. According to equation (9), using the thrust as input, the transfer function of the speed output can be expressed as(43)ΔVΔT=kTss2−2ξωTs+ωT2.Assuming that the throttle thrust input signal is step-signalΔT1t, based on final value theorem of Laplace transform, the speed response is(44)ΔVt⟶∞=lims⟶0s•kTss2−2ξωTs+ωT2•ΔTs=0.This shows that the change of the throttle can only adjust the speed during the transition process but cannot fully achieve the purpose of controlling the speed according to the ideal trajectory.(2) Stability Analysis of Speed. From equation (24) and the system characteristic matrix equation (25), the differential expression describing the airspeed motion process at any operating point after linearization can be given by(45)V˙=−ρCDSVemV−CLCDgsinθeθ.When the system is aiming at the target, there is(46)V˙+ρCDSVemV+CLCDgsinθeθe=0.Its characteristic roots are(47)ζ1,2=−ρCDSVe/m±ρCDSVe/m2−4CL/CDgsinθeθe2,where(48)ρCDSVem>0,CLCDgsinθeθe>0.This means the two eigenvalues of the differential equation have negative real parts, so the change of the airspeed trajectory should satisfy the stability condition, and the test data also illustrated this feature: the speed curve rises following the expected one (Figure33).However, as shown in Figure23 (the control system block diagram), only an open-loop controller was used to control the speed to achieve the desired value Vtc. Obviously, an open-loop controller’s performance cannot meet the requirement of a complex testing environment, and the experimental results also verified this conclusion: the speed curve rises but slowly.It would be nice to design a better controller to improve the response characteristics of the speed, but how to improve the control law requires further analysis, and it is out of range of this research, so it is one of the tasks we will continue to expand in the future.


## 3.6.1. Control System Block Build according to the Speed Scheme

The logic of the flight test: use the scheme to calculate an expected thrust during the attacking phase and cooperate with the attitude control module to complete the impact task against a nonmoving target.The control system constructed in this paper consisted of an attitude control module and a speed control module. The attitude control module was realized by a classical PID controller. The speed control module was formed according to the speed scheme. The system block diagram is shown in Figure24.Figure 24
Control system block diagram.The speed controller parameters were calculated according to equation (16) and are shown in Table 4.Table 4
Speed controller parameters corresponding to the scheme.
ParameterValueK10.03K234.30K38.01K48.57


## 3.6.2. Test Implementation Method

The UAV was the same one used in data acquisition (see Figure14), and the flight process can be seen in Figure 25 and can be described as follows:(1)
Take Off. The UAV takes off by remote controller and flies to the vicinity of the target.(2)
Regulate. Switch the control mode to the automatic driving by the remote controller and the UAV would adjust the route and attitude automatically.(3)
Attack. When the target enters the field of view, it can be selected on the video; at the same time, the controller designed in this paper starts working automatically until the UAV hits the target.Figure 25
Flight phase diagram.


## 3.6.3. Flight Test Result

In this section, the result of the flight test is presented. The final 7 s flight path was selected for presentation purposes during attacking stages of the flight test, and the data are shown in the following figures: Figure26 shows the thrust, Figure 27 shows the speed, Figure 28 shows the pitch angle, Figure 29 shows the acceleration in z direction in body-coordinate, Figure 30 shows the flight path, Figure 31 shows the first perspective video screenshot, and Figure 32 shows the third perspective video screenshot.Figure 26
Thrust in flight test.Figure 27
Speed in flight test.Figure 28
Acceleration inz direction in body-coordinate.Figure 29
Pitch angle in flight test.Figure 30
Flight path in flight test.Figure 31
First perspective video screenshot in flight test.Figure 32
Third perspective video screenshot in flight test.Based on the experimental results, following conclusions can be given:First of all, Figures26 and 27 show that the thrust and speed were dynamic curves, which were very different from the traditional ammunition speed throttle scheme (as can been seen in Figure 3), and their changes were all within reasonable ranges.Secondly, as can been seen in Table5, the mean of acceleration in z direction from the scheme flight test was −0.6351 and that from remote control test was −9.323. The standard deviation of acceleration in z direction from the scheme flight test was 1.0596 and that from remote control test was 1.727. Compared with data from the remote control test (which has been shown in Figure 18), acceleration in z direction in body frame has been suppressed clearly (as can been seen in Figure 28).Table 5
Mean and standard deviation of flight tests.
Data sourceMeanStandard deviationRemote control test−9.3231.727Scheme flight test−0.63511.0596At the same time, the pitch angle and the flight trajectory curve is smooth (as shown in Figures29 and 30), which strongly verifies that the speed scheme proposed in this paper was helpful to improve the attitude and trajectory stability of the UAV during attacking.Finally, as can been seen in the screenshots of the flight videos (Figures31 and 32), the UAV successfully hit the target, which further illustrated the effectiveness of the scheme.


## 3.6.4. Defect Analysis

However, the airspeed during the flight did not follow the expected trajectory perfectly, as shown in Figure33. Further analysis is given below.Figure 33
Tracking effect of airspeed command in automatic attack. The expected one was calculated according to equation (16).(1) Dynamic Response Characteristics of the Throttle. Since the airspeed did not change much during the whole flight process, it can be assumed that the thrust depended only on the working state of the propulsion system and was independent of the atmospheric conditions. According to equation (9), using the thrust as input, the transfer function of the speed output can be expressed as(43)ΔVΔT=kTss2−2ξωTs+ωT2.Assuming that the throttle thrust input signal is step-signalΔT1t, based on final value theorem of Laplace transform, the speed response is(44)ΔVt⟶∞=lims⟶0s•kTss2−2ξωTs+ωT2•ΔTs=0.This shows that the change of the throttle can only adjust the speed during the transition process but cannot fully achieve the purpose of controlling the speed according to the ideal trajectory.(2) Stability Analysis of Speed. From equation (24) and the system characteristic matrix equation (25), the differential expression describing the airspeed motion process at any operating point after linearization can be given by(45)V˙=−ρCDSVemV−CLCDgsinθeθ.When the system is aiming at the target, there is(46)V˙+ρCDSVemV+CLCDgsinθeθe=0.Its characteristic roots are(47)ζ1,2=−ρCDSVe/m±ρCDSVe/m2−4CL/CDgsinθeθe2,where(48)ρCDSVem>0,CLCDgsinθeθe>0.This means the two eigenvalues of the differential equation have negative real parts, so the change of the airspeed trajectory should satisfy the stability condition, and the test data also illustrated this feature: the speed curve rises following the expected one (Figure33).However, as shown in Figure23 (the control system block diagram), only an open-loop controller was used to control the speed to achieve the desired value Vtc. Obviously, an open-loop controller’s performance cannot meet the requirement of a complex testing environment, and the experimental results also verified this conclusion: the speed curve rises but slowly.It would be nice to design a better controller to improve the response characteristics of the speed, but how to improve the control law requires further analysis, and it is out of range of this research, so it is one of the tasks we will continue to expand in the future.


## 4. Conclusions

In this paper, a speed scheme for small loitering munitions has been given for precision strike mission. The pitch angle boundary of the solution during attack was given, and the existence and unique conditions of the system solution were analysed. Then, a flight test was taken to verify the scheme, and the result showed that the scheme was effective. At last, based on the analysis of experimental data, this paper summarizes the shortcomings of the program and proposes a clear direction for further research.



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*Source: 4963738-2020-09-18.xml*