# Impact Angle Control Guidance to Intercept Moving Targets by Virtual Target Technique

Authors: Libing Hou; Jihong Zhu; Minchi Kuang; Heng Shi
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

To solve the problem regarding the impact angle of the missile, this paper proposes a novel guidance law, which can control the missile to hit the target at the desired angle. The key of the guidance law is selecting a moving point on the collision line as the virtual target, and the tactical requirements can be fulfilled by the missile directly pursuing the virtual target. The Lyapunov stable theory is used to prove the convergence of the proposed guidance law. The guidance command is generated by a PID controller to make the missile towards the virtual target. The proposed guidance law makes the lateral acceleration of the missile converge to zero, which leads the angle of attack to zero, and it theoretically guarantees the flight path angle equals the attitude angle. Numerical simulations demonstrate this impact angle control guidance law is very accurate and robust. Regardless of whether the initial heading error is large or small, the missile which employs the proposed guidance law can always hit the target from the preset direction and the guidance process is smooth.


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## Body

## 1. Introduction

Guidance laws aim to achieve desired impact angle, as well as zero miss distance, are pushed forward by a number of techniques. Literature [1–7] derived guidance laws with impact angle constraint based on proportional navigation guidance (PNG). Ratnoo and Ghose [2] explored the influence of navigation gain on impact angle and designed a two-stage PNG law to achieve the desired impact angle. But this method usually provides a long trajectory, which makes it difficult to be implemented on missiles with range limit in practice. A nonswitching guidance law for trajectory shaping control was proposed by Ratnoo [3], where the line-of-sight (LOS) angle is added into the PNG law. Though this impact angle control guidance (IACG) law is capable of approaching stationary targets at all angles, it fails to achieve the angle control when the target is moving. Vairavan and Ratnoo [1] made the improvement by using a variable navigation gain, which is able to deal with the moving target, yet it is not robust enough when the target’s velocity is varying. These PNG-based impact angle control guidance techniques are simple in form and cost little in the calculation, but some constraints on the impact angle cannot strictly satisfied when the initial heading coincides with the LOS. The sliding mode control (SMC) technique is another commonly used technique in the field of IACG. Literatures [8–19] use SMC to achieve desired impact angle. Guidance laws based on a conventional SMC can only ensure asymptotic convergence. A guidance law based on terminal sliding mode control was proposed in [10], which satisfied the finite-time convergence. Hu et al. [16] combined SMC and virtual target strategy to control collision time and angle. Guidance laws based on SMC have strong robustness, which are even effective on maneuvering targets. However, the discrete terms in the guidance formula always cause inevitable oscillations in the guidance commands, which is not suitable for actual implementations. The SMC technique was introduced to design a guidance law that makes multiple missiles hit the target in a preset direction at the same time [18], and it utilized range-to-go and radial relative velocity rather than time-to-go to construct the sliding surface to avoid performance degradation. Khankalantary and Sheikholeslam combined SMC, backstepping, and super-twist extend state observer to propose a robust guidance law [19] and verified its performance by a dynamics model. In addition to PNG and SMC, Ryoo et al. [20] proposed the IACG method to calculate the remaining time, and it succeeded in interception stationary targets at all angles. But it also failed against moving target. A trajectory shaping method based on Archimedean spiral was proposed in [21] to deal with the terminal angle constraint. But sometimes the problem cannot be solved since the calculated Archimedean spiral may be a bad curve. Yan et al. [22] introduced a IACG law based on computational geometry. This trajectory formed by this approach is composed of a log-aesthetic space curve segment and a collision line segment, and the curvature can be adjusted to satisfy the impact angle constraint. Despite these geometry-based methods are sufficient to produce desired trajectories, the amount of real-time calculation is relatively large, which makes it difficult to be implemented on a standard missile’s on-board computer. Moreover, Hu et al., Chen et al., and Shi et al. [16, 23, 24] used the virtual target method in the design of guidance laws. Chen et al. [23] took a virtual target on the reference trajectory, enabling the unmanned aerial vehicle to move along the reference trajectory. The key of this strategy is calculation of the speed of the virtual target. But this method cannot ensure the convergence within a finite time. In literature [16], the virtual target was utilized to optimize the impact time, and angle constraint is finished by SMC. Shi et al. proposed a predictive guidance law [24], the virtual target is the predictive intercept point, which is based on the target’s velocity and the time to go, and the missile pursues the virtual target by PNG law. A suboptimal IACG law based on finite-time state-dependent Riccati equation was proposed for an interceptor to pursue an aerial vehicle in a 3D engagement [25]. Kim et al. proposed a IACG law [26] for nose-dive missile which can maximize its terminal velocity, but the assumption that the initial angle is close to vertical is unreasonable.To overcome the shortcomings of the aforementioned methods, this paper is aimed at proposing a novel IACG strategy based on the virtual target approach to support the practical application. To this end, a virtual target is designed firstly via geometry analysis considering the impact angle constraint. The interceptor is then controlled to follow the virtual target. The guidance command is generated based on the error of the missile’s flight path angle and the LOS (the missile and the virtual target) angle. The convergence of the proposed method is proved analytically via Lyapunov stability theory, and the geometric conditions for convergence are derived. Numerical simulations with comparisons confirm the effectiveness and feasibility of the proposed strategy.The contributions of this paper are as follows:(i)
This IACG law achieves the desired impact angle with less calculation(ii)
The guidance process is smooth with no chattering(iii)
The lateral acceleration of the missile converges to zero, which leads to the angle of attack to zero, and it theoretically guarantees the trajectory angle can represent the attitude angleThe rest of the paper is organized as follows. Section2 presents the formulation of the impact angle control engagement mathematical model, followed by the derivation of the impact angle control guidance law in Section 3. Numerical simulations are presented in Section 4.


## 2. Problem Formulation

Consider a planar homing engagement scenario of a missile against a moving target, as shown in Figure1. M and T denote the missile and the target, and their velocities are VM and VT, and γM and γT represent their flight path angles, respectively. r stands for the distance between the missile and the target. The angle of line-of-sight is denoted by λ, and aM refers to the lateral acceleration of the missile. θc is the desired impact angle. The engagement kinematics for this problem can be given as follows:
(1)ẋM=VMcosγM,ẏM=VMsinγM,γ̇M=aMVM,ẋT=VTcosγT,ẏT=VTsinγT,r=xT−xM2+yT−yM2,ṙ=VTcosγT−λ−VMcosγM−λ,λ̇=VTsinγT−λ−VMsinγM−λr.Figure 1
Missile-target engagement.Assumingtf is the stop time of this engagement, and the missions of the paper can be described as the following objectives:
(2)rtf=0,γMtf−γTtf=θcaMtf⟶0.,If the target is stationary, its flight path angle is considered equal to0. In the sequel, the line segment between A and B and its length are denoted by AB and ∣AB∣, respectively. To facilitate the analysis of this engagement, assumptions are made as follows:Assumption 1.
The velocities of the missile and the target are constant and available.Assumption 2.
The autopilot dynamics is much faster than the missile dynamics.


## 3. Virtual Target Method for Impact Angle Control

In this section, a virtual target method is proposed to control the impact angle and eliminate miss distance. The key of this method is selecting a moving point from the collision line as the virtual target, and it approaches the real target along the collision line. The missile pursues the virtual target by pure pursuit guidance law, and it will first reach the collision line and then hit the target at the desired angle.


### 3.1. Design of Guidance Law with Impact Angle Control

In this subsection, the virtual target method is designed to make the missile hit the target from the desired direction. As is shown in Figure2, the target is stationary. The line from I to T represents the collision line, which is denoted by IT, and its angle is θc (the desired impact angle). η is the distance from M to line IT. The virtual target is a moving point on the collision line, and it is denoted by P. ∣PT∣=ξr, where ξ∈0,1, so ∣γM−λ∣<90∘. φ is the angle between line PT and line MT. Assuming φ is less than 90∘ and VM always points to the virtual target. The missile will first reach the collision line and then move to the real target along the collision line at the desired angle. Strict proofs will be given in the next subsection.Figure 2
Stationary target.The proposed virtual target method is strictly applicable only whenφ<90∘; otherwise, r may be not converged. But the two-stage guidance method can release the restricted condition. The two-stage method is shown in Figure 3.Figure 3
Omnidirectional attack.The guidance procedure is divided into two stages by the pointT1, the guidance methods employed by the two stages are both the proposed methods above. θc and θc1 are the desired impact angle and the first-stage desired impact angle, and the missile first reaches T1 at an angle of θc1 and then hits T at the desired angle of θc. The key of the two-stage guidance method is solving the length of ∣TT1∣ which can achieve the desired impact angle as well as the zero miss distance in the second stage. The second stage can be simplified to the engagement shown in Figure 4; the desired impact angle is −90∘. The algorithm to address the length of ∣TT1∣ is placed in the next subsection.Figure 4
Second stage.If the target is moving as shown in Figure5, the above conclusion could be extended against a moving target. The essential of the extension is to estimate the stop time, which is denoted by tf, and the approximation is called tgo. It must be noted that if the target escapes at a speed of VT and VT>VM, the mission of the missile is likely to fail, so we assume VT<VM. As is shown in Figure 5, VT is the velocity of the target. Target will arrive at T′ after tgo, and ∣TT′∣=VTtgo. If tgo=tf, T′ will be a stationary point. Assuming ∣TT′∣ is small, and it cannot result in φ≫90∘. Because ∣TT′∣ is small, r′≃r, and their convergences are consistent. According to the conclusion for stationary targets, if r′ is convergent and T′ can be regarded as a stationary target, M will first reach the collision line IT′ and then arrive at T′ with an angle of θc. The next subsection will give the convergence conditions of r′ and the algorithm to calculate tgo.Figure 5
Moving target.


### 3.2. Proofs of the Guidance Method

In this subsection, strict proofs of the guidance method will be provided.


#### 3.2.1. Stationary Target

Proposition 3.
If the missile target engagement is shown in Figure2, the missile will reach the collision line first and then move to the real target along the collision line. The terminal lateral acceleration of the missile converges to 0.Proof.
Referring to Figure2,
(3)PT=ξr,ξ∈0,1,∠PMT<∠MPT.
InΔPMT, if ∠PMT≥90∘ and ∠PMT+∠MPT>180∘, it breaks the theorem that the sum of the internal angles of a triangle is equal to 180, so
(4)∠PMT=∣γM−λ∣<90∘.
Substituting Equation (4) to Equation (1),
(5)ṙ=−VMcos∣γM−λ∣<0.
Choose Lyapunov function as follows:(6)V=r.
Considering Equation (5),
(7)V̇=ṙ<0,VV̇<0.
According to Lyapunov stability theory,r will converge to a certain value at the stop time of the engagement. If this value is not 0, the process will continue until r=0. Similarly, ∣PT∣ will also converge to 0 at that time. It means that the virtual target and the missile will reach the real target simultaneously.
Now considering the derivatives ofr and η,
(8)η̇=−VMsinγM−λ+φ.
Combining Equation (5) and Equation (8),
(9)r∣ṙ∣=ηVMcos∣γM−λ∣sinφ>ηVMsin∣γM−λ∣cosφ+cos∣γM−λ∣sinφ=ηVMsin∣γM−λ∣+φ=η∣η̇∣.
Equation (9) shows that the missile always takes more time to reach the target than it takes to reach line IT. So, we can conclude that the missile will first reach line IT, then move along the collision line with no maneuvering, and finally arrive at the real target, so the terminal lateral acceleration of the missile always converges to 0.Considering the two-stage guidance method shown in Figure3, if the lateral acceleration of the missile is not limited, the missile will hit the target at the desired angle as long as ∣TT1∣>0, but in reality, the lateral acceleration of the missile cannot exceed its maximum, and there is a minimum turning radius Rmin, so is ∣TT1∣. Assuming ∣TT1∣=kRmin, it is easy to find an approximate minimum of ∣TT1∣ with the numerical methods. Taking ξ=0.6 as an example, the desired impact angle is −90∘. The result is shown in Table 1kmin>5. The time lag due to the missile autopilot and actuator dynamics cannot be neglected, and the value of k we choose is always greater than kmin, so making too much efforts to determine the accurate value of kmin is not worthy, and 8 or 10 may be a good choice.Table 1
∣TT1∣’s influence on interception result.
k123456Impact angle (deg)-20.167-49.102-64.691-77.950-91.4516-89.997Miss distance (m)59.02760.32346.40926.3095.3100.4985


#### 3.2.2. Moving Target

Proposition 4.
If missile target engagement as is shown in Figure5, ∣TT′∣ is small and tgo=tf. The convergence condition of r is as follows:
(10)ξ<sincos−1VTVM.Proof.
According to Lyapunov stability theory,r′ will converge to 0 at the stop time of the engagement as long as r′̇ is always less than 0. Because ∣TT′∣ is small, r′≃r.
(11)r=VTcosγT−λ−VMcosγM−λ≤VT−VMcosγM−λ≤VT−VMcossin−1ξ<0.
We can obtain the following:(12)ξ<sincos−1VTVM.
Ifξ satisfies Equation (12), r will be shorter no matter which direction VT points that means r will converge to 0 at the end of the engagement, so do r′. Because tgo=tf, T′ can be regarded as a stationary target; according to the conclusion for stationary targets, the missile will first reach line IT′ then move to T′ at an angle of θc with no maneuvering.As analyzed above,∣TT′∣ must be small and precise to ensure the accuracy of the result, so the method to estimate tf is critical. Here are two algorithms to estimate tf; they are simple enough to assure an efficient calculation.
(13)tgo1=r∣ṙ∣,(14)tgo2=rδv,δv=VTcosγT−VAcosγA2+VTsinγT−VAsinγA2.Equation (13) is an algorithm to calculate tgo used in many guidance laws, but it may be invalid in the proposed guidance method, especially when VT is not small, because ṙ may be close to 0 and it could result in an infinity of ∣TT′∣, and it breaks the assumption that ∣TT′∣ is small; consequently, r may be not converged. So, we choose Equation (14) to calculate tgo, since it not only avoids ∣TT′∣ tending to infinity but also meets the precision required for the guidance method.


### 3.3. Generation of Guidance Command

The above derivations are based on the assumption thatVM always points to the virtual target. In reality, this assumption is not always satisfied. Take it as an optimization goal, and the command of autopilot is gained. The actual engagement is shown in Figure 6, VM does not point to the virtual target, and σ is the angle of line MP. If the error between γM and σ is eliminated, the above guidance method based on virtual target is valid. To eliminate the error, the PID controller is a suitable choice.
(15)e=σ−γM,a=KPe+KI∫edt+KDė,aM=aMmax,ifa>aMmax,a,if−aMmax≤a≤aMmax,−aMmax,ifa<−aMmax,where KP, KI, and KD are proportional gain, integral gain, and derivative gain, respectively. aMmax is the maximum allowable lateral acceleration of the missile.Figure 6
Actual missile target engagement.


## 3.1. Design of Guidance Law with Impact Angle Control

In this subsection, the virtual target method is designed to make the missile hit the target from the desired direction. As is shown in Figure2, the target is stationary. The line from I to T represents the collision line, which is denoted by IT, and its angle is θc (the desired impact angle). η is the distance from M to line IT. The virtual target is a moving point on the collision line, and it is denoted by P. ∣PT∣=ξr, where ξ∈0,1, so ∣γM−λ∣<90∘. φ is the angle between line PT and line MT. Assuming φ is less than 90∘ and VM always points to the virtual target. The missile will first reach the collision line and then move to the real target along the collision line at the desired angle. Strict proofs will be given in the next subsection.Figure 2
Stationary target.The proposed virtual target method is strictly applicable only whenφ<90∘; otherwise, r may be not converged. But the two-stage guidance method can release the restricted condition. The two-stage method is shown in Figure 3.Figure 3
Omnidirectional attack.The guidance procedure is divided into two stages by the pointT1, the guidance methods employed by the two stages are both the proposed methods above. θc and θc1 are the desired impact angle and the first-stage desired impact angle, and the missile first reaches T1 at an angle of θc1 and then hits T at the desired angle of θc. The key of the two-stage guidance method is solving the length of ∣TT1∣ which can achieve the desired impact angle as well as the zero miss distance in the second stage. The second stage can be simplified to the engagement shown in Figure 4; the desired impact angle is −90∘. The algorithm to address the length of ∣TT1∣ is placed in the next subsection.Figure 4
Second stage.If the target is moving as shown in Figure5, the above conclusion could be extended against a moving target. The essential of the extension is to estimate the stop time, which is denoted by tf, and the approximation is called tgo. It must be noted that if the target escapes at a speed of VT and VT>VM, the mission of the missile is likely to fail, so we assume VT<VM. As is shown in Figure 5, VT is the velocity of the target. Target will arrive at T′ after tgo, and ∣TT′∣=VTtgo. If tgo=tf, T′ will be a stationary point. Assuming ∣TT′∣ is small, and it cannot result in φ≫90∘. Because ∣TT′∣ is small, r′≃r, and their convergences are consistent. According to the conclusion for stationary targets, if r′ is convergent and T′ can be regarded as a stationary target, M will first reach the collision line IT′ and then arrive at T′ with an angle of θc. The next subsection will give the convergence conditions of r′ and the algorithm to calculate tgo.Figure 5
Moving target.


## 3.2. Proofs of the Guidance Method

In this subsection, strict proofs of the guidance method will be provided.


### 3.2.1. Stationary Target

Proposition 3.
If the missile target engagement is shown in Figure2, the missile will reach the collision line first and then move to the real target along the collision line. The terminal lateral acceleration of the missile converges to 0.Proof.
Referring to Figure2,
(3)PT=ξr,ξ∈0,1,∠PMT<∠MPT.
InΔPMT, if ∠PMT≥90∘ and ∠PMT+∠MPT>180∘, it breaks the theorem that the sum of the internal angles of a triangle is equal to 180, so
(4)∠PMT=∣γM−λ∣<90∘.
Substituting Equation (4) to Equation (1),
(5)ṙ=−VMcos∣γM−λ∣<0.
Choose Lyapunov function as follows:(6)V=r.
Considering Equation (5),
(7)V̇=ṙ<0,VV̇<0.
According to Lyapunov stability theory,r will converge to a certain value at the stop time of the engagement. If this value is not 0, the process will continue until r=0. Similarly, ∣PT∣ will also converge to 0 at that time. It means that the virtual target and the missile will reach the real target simultaneously.
Now considering the derivatives ofr and η,
(8)η̇=−VMsinγM−λ+φ.
Combining Equation (5) and Equation (8),
(9)r∣ṙ∣=ηVMcos∣γM−λ∣sinφ>ηVMsin∣γM−λ∣cosφ+cos∣γM−λ∣sinφ=ηVMsin∣γM−λ∣+φ=η∣η̇∣.
Equation (9) shows that the missile always takes more time to reach the target than it takes to reach line IT. So, we can conclude that the missile will first reach line IT, then move along the collision line with no maneuvering, and finally arrive at the real target, so the terminal lateral acceleration of the missile always converges to 0.Considering the two-stage guidance method shown in Figure3, if the lateral acceleration of the missile is not limited, the missile will hit the target at the desired angle as long as ∣TT1∣>0, but in reality, the lateral acceleration of the missile cannot exceed its maximum, and there is a minimum turning radius Rmin, so is ∣TT1∣. Assuming ∣TT1∣=kRmin, it is easy to find an approximate minimum of ∣TT1∣ with the numerical methods. Taking ξ=0.6 as an example, the desired impact angle is −90∘. The result is shown in Table 1kmin>5. The time lag due to the missile autopilot and actuator dynamics cannot be neglected, and the value of k we choose is always greater than kmin, so making too much efforts to determine the accurate value of kmin is not worthy, and 8 or 10 may be a good choice.Table 1
∣TT1∣’s influence on interception result.
k123456Impact angle (deg)-20.167-49.102-64.691-77.950-91.4516-89.997Miss distance (m)59.02760.32346.40926.3095.3100.4985


### 3.2.2. Moving Target

Proposition 4.
If missile target engagement as is shown in Figure5, ∣TT′∣ is small and tgo=tf. The convergence condition of r is as follows:
(10)ξ<sincos−1VTVM.Proof.
According to Lyapunov stability theory,r′ will converge to 0 at the stop time of the engagement as long as r′̇ is always less than 0. Because ∣TT′∣ is small, r′≃r.
(11)r=VTcosγT−λ−VMcosγM−λ≤VT−VMcosγM−λ≤VT−VMcossin−1ξ<0.
We can obtain the following:(12)ξ<sincos−1VTVM.
Ifξ satisfies Equation (12), r will be shorter no matter which direction VT points that means r will converge to 0 at the end of the engagement, so do r′. Because tgo=tf, T′ can be regarded as a stationary target; according to the conclusion for stationary targets, the missile will first reach line IT′ then move to T′ at an angle of θc with no maneuvering.As analyzed above,∣TT′∣ must be small and precise to ensure the accuracy of the result, so the method to estimate tf is critical. Here are two algorithms to estimate tf; they are simple enough to assure an efficient calculation.
(13)tgo1=r∣ṙ∣,(14)tgo2=rδv,δv=VTcosγT−VAcosγA2+VTsinγT−VAsinγA2.Equation (13) is an algorithm to calculate tgo used in many guidance laws, but it may be invalid in the proposed guidance method, especially when VT is not small, because ṙ may be close to 0 and it could result in an infinity of ∣TT′∣, and it breaks the assumption that ∣TT′∣ is small; consequently, r may be not converged. So, we choose Equation (14) to calculate tgo, since it not only avoids ∣TT′∣ tending to infinity but also meets the precision required for the guidance method.


## 3.2.1. Stationary Target

Proposition 3.
If the missile target engagement is shown in Figure2, the missile will reach the collision line first and then move to the real target along the collision line. The terminal lateral acceleration of the missile converges to 0.Proof.
Referring to Figure2,
(3)PT=ξr,ξ∈0,1,∠PMT<∠MPT.
InΔPMT, if ∠PMT≥90∘ and ∠PMT+∠MPT>180∘, it breaks the theorem that the sum of the internal angles of a triangle is equal to 180, so
(4)∠PMT=∣γM−λ∣<90∘.
Substituting Equation (4) to Equation (1),
(5)ṙ=−VMcos∣γM−λ∣<0.
Choose Lyapunov function as follows:(6)V=r.
Considering Equation (5),
(7)V̇=ṙ<0,VV̇<0.
According to Lyapunov stability theory,r will converge to a certain value at the stop time of the engagement. If this value is not 0, the process will continue until r=0. Similarly, ∣PT∣ will also converge to 0 at that time. It means that the virtual target and the missile will reach the real target simultaneously.
Now considering the derivatives ofr and η,
(8)η̇=−VMsinγM−λ+φ.
Combining Equation (5) and Equation (8),
(9)r∣ṙ∣=ηVMcos∣γM−λ∣sinφ>ηVMsin∣γM−λ∣cosφ+cos∣γM−λ∣sinφ=ηVMsin∣γM−λ∣+φ=η∣η̇∣.
Equation (9) shows that the missile always takes more time to reach the target than it takes to reach line IT. So, we can conclude that the missile will first reach line IT, then move along the collision line with no maneuvering, and finally arrive at the real target, so the terminal lateral acceleration of the missile always converges to 0.Considering the two-stage guidance method shown in Figure3, if the lateral acceleration of the missile is not limited, the missile will hit the target at the desired angle as long as ∣TT1∣>0, but in reality, the lateral acceleration of the missile cannot exceed its maximum, and there is a minimum turning radius Rmin, so is ∣TT1∣. Assuming ∣TT1∣=kRmin, it is easy to find an approximate minimum of ∣TT1∣ with the numerical methods. Taking ξ=0.6 as an example, the desired impact angle is −90∘. The result is shown in Table 1kmin>5. The time lag due to the missile autopilot and actuator dynamics cannot be neglected, and the value of k we choose is always greater than kmin, so making too much efforts to determine the accurate value of kmin is not worthy, and 8 or 10 may be a good choice.Table 1
∣TT1∣’s influence on interception result.
k123456Impact angle (deg)-20.167-49.102-64.691-77.950-91.4516-89.997Miss distance (m)59.02760.32346.40926.3095.3100.4985


## 3.2.2. Moving Target

Proposition 4.
If missile target engagement as is shown in Figure5, ∣TT′∣ is small and tgo=tf. The convergence condition of r is as follows:
(10)ξ<sincos−1VTVM.Proof.
According to Lyapunov stability theory,r′ will converge to 0 at the stop time of the engagement as long as r′̇ is always less than 0. Because ∣TT′∣ is small, r′≃r.
(11)r=VTcosγT−λ−VMcosγM−λ≤VT−VMcosγM−λ≤VT−VMcossin−1ξ<0.
We can obtain the following:(12)ξ<sincos−1VTVM.
Ifξ satisfies Equation (12), r will be shorter no matter which direction VT points that means r will converge to 0 at the end of the engagement, so do r′. Because tgo=tf, T′ can be regarded as a stationary target; according to the conclusion for stationary targets, the missile will first reach line IT′ then move to T′ at an angle of θc with no maneuvering.As analyzed above,∣TT′∣ must be small and precise to ensure the accuracy of the result, so the method to estimate tf is critical. Here are two algorithms to estimate tf; they are simple enough to assure an efficient calculation.
(13)tgo1=r∣ṙ∣,(14)tgo2=rδv,δv=VTcosγT−VAcosγA2+VTsinγT−VAsinγA2.Equation (13) is an algorithm to calculate tgo used in many guidance laws, but it may be invalid in the proposed guidance method, especially when VT is not small, because ṙ may be close to 0 and it could result in an infinity of ∣TT′∣, and it breaks the assumption that ∣TT′∣ is small; consequently, r may be not converged. So, we choose Equation (14) to calculate tgo, since it not only avoids ∣TT′∣ tending to infinity but also meets the precision required for the guidance method.


## 3.3. Generation of Guidance Command

The above derivations are based on the assumption thatVM always points to the virtual target. In reality, this assumption is not always satisfied. Take it as an optimization goal, and the command of autopilot is gained. The actual engagement is shown in Figure 6, VM does not point to the virtual target, and σ is the angle of line MP. If the error between γM and σ is eliminated, the above guidance method based on virtual target is valid. To eliminate the error, the PID controller is a suitable choice.
(15)e=σ−γM,a=KPe+KI∫edt+KDė,aM=aMmax,ifa>aMmax,a,if−aMmax≤a≤aMmax,−aMmax,ifa<−aMmax,where KP, KI, and KD are proportional gain, integral gain, and derivative gain, respectively. aMmax is the maximum allowable lateral acceleration of the missile.Figure 6
Actual missile target engagement.


## 4. Numerical Simulation

In this section, a series of numerical simulations verify the effectiveness and advantages of the guidance method proposed by this paper, and some of which include comparisons with existing angle control guidance law.


### 4.1. Case I: All-Aspect Attack

In this subsection, the capacity of the all-aspect attack of the proposed guidance method will be verified by numerical simulations. Simulation conditions are shown in Table2. The missile travels with a constant velocity, and aMmax is the maximum lateral acceleration of the missile. tτ is the time constant of missile’s autopilot, and the target is stationary.Table 2
Simulation conditions for Case I.
rT (m)VT0(m/s)γT0(deg)dVT(m/s2)rM(m)VM(m/s)γM0(deg)(10000,0)000(0,0)2500aMmax(m/s2)ξktτ(s)2000.6100.2The simulation result is presented in Figure7 and Table 3. The maximum miss distance is 0.2460 m; it can be considered that all missiles hit the target. All impact angle errors are less than 0.553∘, especially the errors of nonswitch methods are within 0.0097∘; this can already be applied to most combat scenarios. It shows that the proposed guidance method can achieve a precise omnidirectional attack.Figure 7
Case I: all-aspect attack.Table 3
Simulation result for Case I.
θc (deg)-180-90-60-300Impact angle (deg)-180.0553-90.0097-60.0028-30.00100Error of impact angle (deg)0.05530.00970.00280.00100Miss distance (m)0.07300.04360.19990.00450.2460θc (deg)306090180Impact angle (deg)30.001060.002790.0097180.0553Error of impact angle (deg)-0.001-0.0027-0.0097-0.0553Miss distance (m)0.00450.19990.04360.0730


### 4.2. Compare with Other Impact Angle Control Guidance Methods

In this subsection, guidance methods proposed by [1, 2] are used as comparison methods.


#### 4.2.1. Case II: Stationary Target

Three missiles employ the proposed method, literature [1] and literature [2] against a stationary target, respectively. Simulation conditions are shown in Table 4.Table 4
Simulation conditions for Case II.
rT (m)VT0 (m/s)γT0 (deg)dVT (m/s2)rM (m)VM (m/s)γM0 (deg)(5000,0)000(0,0)25030 or 5aMmax(m/s2)ξtτ(s)θc (deg)2000.60.2-90The simulation result is shown in Figure8 and Table 5. When the target is stationary and the initial heading error is large, the three guidance methods have good guidance effects. Trajectories of the proposed method and the method in [1] are smooth, but the trajectory of the method in [2] has obvious signs of switching, which is also reflected in acceleration. The method proposed in this paper has the best impact angle control effect. If the initial heading error is small, performances of the guidance laws proposed by [2, 11] are significantly worse. This could be attributed to PNG that the methods in [1, 2] rely on. The formula of PNG is a=Nṙλ̇; when the initial heading error is small, λ̇ is small, too. The lateral acceleration is too small to bend the trajectory enough, and the missile will not have a sufficiently large turning angle rate at the end of the engagement due to the limit of the lateral acceleration, so their missions failed. Figures 8(b) and 8(d) show that acceleration of the proposed method converges to near 0. The initial surge in acceleration is due to the initial velocity of the missile does not point to the virtual target, which shows that the initial heading error is essential to reduce the energy loss, although it did not affect the results here.Figure 8
Case II: stationary target.
(a)
Trajectories (γM0=30∘)(b)
Acceleration (γM0=30∘)(c)
TrajectoryγM0=5∘(d)
AccelerationγM0=5∘Table 5
Simulation result for Case II.
(a)γM0=30∘ProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0205-89.5548-90.6141Error of impact angle (deg)0.0205-0.44520.6141Miss distance (m)0.21310.16980.1853(b)γM0=5∘ProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0283-75.4084-117.1595Error of impact angle (deg)0.0283-14.591627.1595Miss distance (m)0.060435.8176151.3314


#### 4.2.2. Case III: Constant Velocity Target

Replace the stationary target with a constant speed target, comparing the performances of the three guidance methods. Simulation conditions are shown in Table6.Table 6
Simulation conditions for Case III.
rT(m)VT0 (m/s)γT0 (deg)dVT (m/s2)rM (m)VM (m/s)γM0 (deg)(5000,0)50 or 10000(0,0)25030aMmax(m/s2)ξtτ(s)θc(deg)2000.60.2-90The simulation result is presented in Figure9 and Table 7; three missiles all hit the target no matter the speed of the target is: 50m/s or 100m/s. The angle errors of the three guidance methods are no more than 0.4618∘ when VT=50 m/s; it can be considered that all the three guidance methods are successful to control the impact angle. But when VT=100m/s, the angle error of the method proposed by [2] is 2.9326∘; it failed to control the impact angle. Terminal acceleration of the method proposed in [1] is far more than 0; it is not conducive to building advantage. In the two numerical simulations, the proposed method based on virtual target has the least angle error, and its terminal acceleration converges to near 0; it has the best interception effect.Figure 9
Case III: constant velocity target.
(a)
Trajectories (VT=50m/s)(b)
Acceleration (VT=50m/s)(c)
Trajectories (VT=100m/s)(d)
Acceleration (VT=100m/s)Table 7
Simulation result for Case III.
(a)VT=50 m/sProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0148-90.4346-90.4618Error of impact angle (deg)0.01480.4340.4618Miss distance (m)0.14130.12920.1692(b)VT=100m/sProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0025-92.9326-90.2880Error of impact angle (deg)0.00252.93260.2880Miss distance (m)0.24880.25640.0408


#### 4.2.3. Case IV: Variable Velocity Target

In this subsection, variable velocity targets are considered. There are two kinds of variable speed targets, one with an initial speed of0m/s and an acceleration of 0.5m/s2, and the other with an initial speed of 100m/s and an acceleration of −0.5m/s2. Simulation conditions are presented in Table 8.Table 8
Simulation conditions for Case IV.
rT (m)VT0 (m/s)γT0 (deg)dVT (m/s2)rM (m)VM (m/s)γM0 (deg)(5000,0)0 or 10000.5 or -0.5(0,0)25030aMmax(m/s2)ξtτ(s)θc(deg)2000.60.2-90The simulation result is presented in Figure10 and Table 9. Methods proposed by [1, 2] failed to control their impact angles. The missile employs that the proposed guidance method based on virtual target hits the variable speed target at the desired impact angle, no matter the target is accelerated or decelerated. It shows that this method has the best robustness.Figure 10
Case IV: variable velocity target.
(a)
Trajectories (target’s speed up)(b)
Acceleration (target’s speed up)(c)
Trajectories (target’s speed down)(d)
Acceleration (target’s speed down)Table 9
Simulation result for Case IV.
(a)Speed upProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0287-84.8451-85.8370Error of impact angle (deg)0.0287-5.1549-4.163Miss distance (m)0.18140.06710.2607(b)Speed downProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0035-103.0161-99.1056Error of impact angle (deg)0.003513.01619.1056Miss distance (m)0.05550.15140.1097


## 4.1. Case I: All-Aspect Attack

In this subsection, the capacity of the all-aspect attack of the proposed guidance method will be verified by numerical simulations. Simulation conditions are shown in Table2. The missile travels with a constant velocity, and aMmax is the maximum lateral acceleration of the missile. tτ is the time constant of missile’s autopilot, and the target is stationary.Table 2
Simulation conditions for Case I.
rT (m)VT0(m/s)γT0(deg)dVT(m/s2)rM(m)VM(m/s)γM0(deg)(10000,0)000(0,0)2500aMmax(m/s2)ξktτ(s)2000.6100.2The simulation result is presented in Figure7 and Table 3. The maximum miss distance is 0.2460 m; it can be considered that all missiles hit the target. All impact angle errors are less than 0.553∘, especially the errors of nonswitch methods are within 0.0097∘; this can already be applied to most combat scenarios. It shows that the proposed guidance method can achieve a precise omnidirectional attack.Figure 7
Case I: all-aspect attack.Table 3
Simulation result for Case I.
θc (deg)-180-90-60-300Impact angle (deg)-180.0553-90.0097-60.0028-30.00100Error of impact angle (deg)0.05530.00970.00280.00100Miss distance (m)0.07300.04360.19990.00450.2460θc (deg)306090180Impact angle (deg)30.001060.002790.0097180.0553Error of impact angle (deg)-0.001-0.0027-0.0097-0.0553Miss distance (m)0.00450.19990.04360.0730


## 4.2. Compare with Other Impact Angle Control Guidance Methods

In this subsection, guidance methods proposed by [1, 2] are used as comparison methods.


### 4.2.1. Case II: Stationary Target

Three missiles employ the proposed method, literature [1] and literature [2] against a stationary target, respectively. Simulation conditions are shown in Table 4.Table 4
Simulation conditions for Case II.
rT (m)VT0 (m/s)γT0 (deg)dVT (m/s2)rM (m)VM (m/s)γM0 (deg)(5000,0)000(0,0)25030 or 5aMmax(m/s2)ξtτ(s)θc (deg)2000.60.2-90The simulation result is shown in Figure8 and Table 5. When the target is stationary and the initial heading error is large, the three guidance methods have good guidance effects. Trajectories of the proposed method and the method in [1] are smooth, but the trajectory of the method in [2] has obvious signs of switching, which is also reflected in acceleration. The method proposed in this paper has the best impact angle control effect. If the initial heading error is small, performances of the guidance laws proposed by [2, 11] are significantly worse. This could be attributed to PNG that the methods in [1, 2] rely on. The formula of PNG is a=Nṙλ̇; when the initial heading error is small, λ̇ is small, too. The lateral acceleration is too small to bend the trajectory enough, and the missile will not have a sufficiently large turning angle rate at the end of the engagement due to the limit of the lateral acceleration, so their missions failed. Figures 8(b) and 8(d) show that acceleration of the proposed method converges to near 0. The initial surge in acceleration is due to the initial velocity of the missile does not point to the virtual target, which shows that the initial heading error is essential to reduce the energy loss, although it did not affect the results here.Figure 8
Case II: stationary target.
(a)
Trajectories (γM0=30∘)(b)
Acceleration (γM0=30∘)(c)
TrajectoryγM0=5∘(d)
AccelerationγM0=5∘Table 5
Simulation result for Case II.
(a)γM0=30∘ProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0205-89.5548-90.6141Error of impact angle (deg)0.0205-0.44520.6141Miss distance (m)0.21310.16980.1853(b)γM0=5∘ProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0283-75.4084-117.1595Error of impact angle (deg)0.0283-14.591627.1595Miss distance (m)0.060435.8176151.3314


### 4.2.2. Case III: Constant Velocity Target

Replace the stationary target with a constant speed target, comparing the performances of the three guidance methods. Simulation conditions are shown in Table6.Table 6
Simulation conditions for Case III.
rT(m)VT0 (m/s)γT0 (deg)dVT (m/s2)rM (m)VM (m/s)γM0 (deg)(5000,0)50 or 10000(0,0)25030aMmax(m/s2)ξtτ(s)θc(deg)2000.60.2-90The simulation result is presented in Figure9 and Table 7; three missiles all hit the target no matter the speed of the target is: 50m/s or 100m/s. The angle errors of the three guidance methods are no more than 0.4618∘ when VT=50 m/s; it can be considered that all the three guidance methods are successful to control the impact angle. But when VT=100m/s, the angle error of the method proposed by [2] is 2.9326∘; it failed to control the impact angle. Terminal acceleration of the method proposed in [1] is far more than 0; it is not conducive to building advantage. In the two numerical simulations, the proposed method based on virtual target has the least angle error, and its terminal acceleration converges to near 0; it has the best interception effect.Figure 9
Case III: constant velocity target.
(a)
Trajectories (VT=50m/s)(b)
Acceleration (VT=50m/s)(c)
Trajectories (VT=100m/s)(d)
Acceleration (VT=100m/s)Table 7
Simulation result for Case III.
(a)VT=50 m/sProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0148-90.4346-90.4618Error of impact angle (deg)0.01480.4340.4618Miss distance (m)0.14130.12920.1692(b)VT=100m/sProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0025-92.9326-90.2880Error of impact angle (deg)0.00252.93260.2880Miss distance (m)0.24880.25640.0408


### 4.2.3. Case IV: Variable Velocity Target

In this subsection, variable velocity targets are considered. There are two kinds of variable speed targets, one with an initial speed of0m/s and an acceleration of 0.5m/s2, and the other with an initial speed of 100m/s and an acceleration of −0.5m/s2. Simulation conditions are presented in Table 8.Table 8
Simulation conditions for Case IV.
rT (m)VT0 (m/s)γT0 (deg)dVT (m/s2)rM (m)VM (m/s)γM0 (deg)(5000,0)0 or 10000.5 or -0.5(0,0)25030aMmax(m/s2)ξtτ(s)θc(deg)2000.60.2-90The simulation result is presented in Figure10 and Table 9. Methods proposed by [1, 2] failed to control their impact angles. The missile employs that the proposed guidance method based on virtual target hits the variable speed target at the desired impact angle, no matter the target is accelerated or decelerated. It shows that this method has the best robustness.Figure 10
Case IV: variable velocity target.
(a)
Trajectories (target’s speed up)(b)
Acceleration (target’s speed up)(c)
Trajectories (target’s speed down)(d)
Acceleration (target’s speed down)Table 9
Simulation result for Case IV.
(a)Speed upProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0287-84.8451-85.8370Error of impact angle (deg)0.0287-5.1549-4.163Miss distance (m)0.18140.06710.2607(b)Speed downProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0035-103.0161-99.1056Error of impact angle (deg)0.003513.01619.1056Miss distance (m)0.05550.15140.1097


## 4.2.1. Case II: Stationary Target

Three missiles employ the proposed method, literature [1] and literature [2] against a stationary target, respectively. Simulation conditions are shown in Table 4.Table 4
Simulation conditions for Case II.
rT (m)VT0 (m/s)γT0 (deg)dVT (m/s2)rM (m)VM (m/s)γM0 (deg)(5000,0)000(0,0)25030 or 5aMmax(m/s2)ξtτ(s)θc (deg)2000.60.2-90The simulation result is shown in Figure8 and Table 5. When the target is stationary and the initial heading error is large, the three guidance methods have good guidance effects. Trajectories of the proposed method and the method in [1] are smooth, but the trajectory of the method in [2] has obvious signs of switching, which is also reflected in acceleration. The method proposed in this paper has the best impact angle control effect. If the initial heading error is small, performances of the guidance laws proposed by [2, 11] are significantly worse. This could be attributed to PNG that the methods in [1, 2] rely on. The formula of PNG is a=Nṙλ̇; when the initial heading error is small, λ̇ is small, too. The lateral acceleration is too small to bend the trajectory enough, and the missile will not have a sufficiently large turning angle rate at the end of the engagement due to the limit of the lateral acceleration, so their missions failed. Figures 8(b) and 8(d) show that acceleration of the proposed method converges to near 0. The initial surge in acceleration is due to the initial velocity of the missile does not point to the virtual target, which shows that the initial heading error is essential to reduce the energy loss, although it did not affect the results here.Figure 8
Case II: stationary target.
(a)
Trajectories (γM0=30∘)(b)
Acceleration (γM0=30∘)(c)
TrajectoryγM0=5∘(d)
AccelerationγM0=5∘Table 5
Simulation result for Case II.
(a)γM0=30∘ProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0205-89.5548-90.6141Error of impact angle (deg)0.0205-0.44520.6141Miss distance (m)0.21310.16980.1853(b)γM0=5∘ProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0283-75.4084-117.1595Error of impact angle (deg)0.0283-14.591627.1595Miss distance (m)0.060435.8176151.3314


## 4.2.2. Case III: Constant Velocity Target

Replace the stationary target with a constant speed target, comparing the performances of the three guidance methods. Simulation conditions are shown in Table6.Table 6
Simulation conditions for Case III.
rT(m)VT0 (m/s)γT0 (deg)dVT (m/s2)rM (m)VM (m/s)γM0 (deg)(5000,0)50 or 10000(0,0)25030aMmax(m/s2)ξtτ(s)θc(deg)2000.60.2-90The simulation result is presented in Figure9 and Table 7; three missiles all hit the target no matter the speed of the target is: 50m/s or 100m/s. The angle errors of the three guidance methods are no more than 0.4618∘ when VT=50 m/s; it can be considered that all the three guidance methods are successful to control the impact angle. But when VT=100m/s, the angle error of the method proposed by [2] is 2.9326∘; it failed to control the impact angle. Terminal acceleration of the method proposed in [1] is far more than 0; it is not conducive to building advantage. In the two numerical simulations, the proposed method based on virtual target has the least angle error, and its terminal acceleration converges to near 0; it has the best interception effect.Figure 9
Case III: constant velocity target.
(a)
Trajectories (VT=50m/s)(b)
Acceleration (VT=50m/s)(c)
Trajectories (VT=100m/s)(d)
Acceleration (VT=100m/s)Table 7
Simulation result for Case III.
(a)VT=50 m/sProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0148-90.4346-90.4618Error of impact angle (deg)0.01480.4340.4618Miss distance (m)0.14130.12920.1692(b)VT=100m/sProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0025-92.9326-90.2880Error of impact angle (deg)0.00252.93260.2880Miss distance (m)0.24880.25640.0408


## 4.2.3. Case IV: Variable Velocity Target

In this subsection, variable velocity targets are considered. There are two kinds of variable speed targets, one with an initial speed of0m/s and an acceleration of 0.5m/s2, and the other with an initial speed of 100m/s and an acceleration of −0.5m/s2. Simulation conditions are presented in Table 8.Table 8
Simulation conditions for Case IV.
rT (m)VT0 (m/s)γT0 (deg)dVT (m/s2)rM (m)VM (m/s)γM0 (deg)(5000,0)0 or 10000.5 or -0.5(0,0)25030aMmax(m/s2)ξtτ(s)θc(deg)2000.60.2-90The simulation result is presented in Figure10 and Table 9. Methods proposed by [1, 2] failed to control their impact angles. The missile employs that the proposed guidance method based on virtual target hits the variable speed target at the desired impact angle, no matter the target is accelerated or decelerated. It shows that this method has the best robustness.Figure 10
Case IV: variable velocity target.
(a)
Trajectories (target’s speed up)(b)
Acceleration (target’s speed up)(c)
Trajectories (target’s speed down)(d)
Acceleration (target’s speed down)Table 9
Simulation result for Case IV.
(a)Speed upProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0287-84.8451-85.8370Error of impact angle (deg)0.0287-5.1549-4.163Miss distance (m)0.18140.06710.2607(b)Speed downProposedRef [2] methodRef [1] methodImpact angle (deg)-90.0035-103.0161-99.1056Error of impact angle (deg)0.003513.01619.1056Miss distance (m)0.05550.15140.1097


## 5. Conclusion

A new guidance method based on the virtual target technique that allows a missile to impose the desired impact angle on the interception of a moving target has been presented in this paper. The position of the virtual target can be obtained by simple calculation of the states based on the missile and target, and thereby, the missile chases the virtual target by the pure pursuit guidance law to achieve better intercept performance. Simple geometric principles and Lyapunov stability theory are utilized to prove the strict convergence of the virtual target method. By the approach,r and aM both converge to zero in the end, and the impact angle converges to θc. These features guarantee the perfect interception on the target at the expected angle without saturation of terminal acceleration. Regarding the problem of omnidirectional attack, this paper proposes a two-stage guidance method, which is able to achieve an all-aspect precision strike. Numerical simulation proves that the proposed method has good performance against stationary or moving targets with constant or variable velocity, regardless of large or small initial heading error, and the guidance process is smooth with no chattering. It shows the proposed method is an excellent impact angle control method.It must be pointed out that the value ofξ can adjust the length of trajectory, so controlling the impact time by calculating the value of ξ online is feasible. We will explore the algorithm of the value of ξ to design an impact time and angle control guidance law.



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