1. Introduction
As a cost-effective and efficient combat method, the salvo attack has received extensive attention in recent years. There are two primary guidance methods to perform a salvo attack: impact time control guidance (ITCG) and cooperative guidance. With ITCG, multiple missiles should be assigned the same impact time. The missiles fly independently according to the specified ITCG law, and there is no communication between them. With cooperative guidance, there is communication between the missiles, which coordinate with each other to identify a common impact time. Both methods have certain advantages, and this paper primarily focuses on the ITCG method [
1,
2,
3,
4,
5,
6].
The design of the ITCG law should ensure the constraints of impact time and terminal miss distance. In addition, the field-of-view (FOV) constraint should also be considered. To adjust the flight time of the missile, the flight trajectory may become highly curved, which may lead to target locking failure of the seeker, causing the combat mission to fail. Many pieces of literature investigate the design of ITCG laws while considering the FOV constraint. In [
7], a biased proportion navigation guidance (BPNG) law is designed. The biased term contains the impact time error, which is used to determine the impact time constraint. In addition, the cosine term of the lead angle is introduced to ensure the FOV constraint. In [
8], a new switching surface that does not require time-to-go estimation is first designed. Then, a terminal sliding mode guidance law is designed to reach the switching surface. Finally, this ITCG law is modified to ensure the FOV constraint. In [
9], the desired look angle that satisfies the impact angle and FOV constraints is determined. The desired look angle is a function of one adjustable gain, and the impact time constraint is achieved by adjusting the gain. Then, the ITCG law is derived to stabilize the real look angle to the desired look angle. In [
10], the guidance process is divided into two phases. In phase 1, deviated pure pursuit (DPP) is used to keep the lead angle constant; in phase 2, the pure proportional navigation guidance (PPNG) law is used to ensure zero miss distance. The impact time is controlled by adjusting the switching point between the two phases. In addition, the lead angle does not increase in these two phases; thus, the FOV constraint is ensured. In [
11], the guidance process is divided into two phases, and DPP is also used in phase 1. Different from [
10], a new guidance law is used in phase 2 rather than the PPNG law. With the new guidance law, the lead angle constraint can be guaranteed, and the flight time can be solved analytically. In [
12], a time-varying switching surface is first designed to ensure the impact time and FOV constraints. Then, a sliding mode guidance law is derived and modified to reach the switching surface.
These ITCG laws [
7,
8,
9,
10,
11,
12] can achieve the impact time and FOV constraints but have the same problem: they are derived based on the assumption of constant velocity. For missiles and guided projectiles flying in the atmosphere, thrust, aerodynamic forces and gravity can affect the velocity; thus, it is impractical to treat the velocity as constant in most cases. It is thus of practical importance to design ITCG laws that consider velocity variations, and some scholars have performed related research. In [
13], an ITCG law that considers time-varying velocity is derived based on integral sliding mode control (ISMC). Using this method requires the expected range of missile velocities. In [
14], the authors extend the existing time-to-go estimation algorithm under two-dimensional PPNG law to three-dimensional and design a new BPNG law, in which the biased term is used to eliminate the impact time error. Note that this method ignores the effect of gravity on velocity. In [
15], a geometry-based impact time and angle control guidance (ITACG) law is designed for variable-speed unmanned aerial vehicles (UAVs). First, the flight trajectory is designed using Bezier curves, and then, the guidance command is obtained using inverse dynamics to track the designed trajectory. In [
16], the effect of gravity on velocity is ignored, and the time-to-go estimation formula under the PNG law is derived. Then, the dynamics of the impact time error are analyzed, and the method in [
1] is extended to consider velocity variations. In [
17], assuming that the rate of change of velocity is a quadratic function of velocity, the time-to-go estimation formula under PNG law is derived, and then, the BPNG law is designed to achieve impact time control.
Although the methods in [
13,
14,
15,
16,
17] consider velocity variations, none consider the effect of gravity on velocity. In [
18], the ITCG law is derived based on the path-planning approach. The authors divide the flight trajectory into four segments, and this method can be extended to consider velocity variations using pre-flight analysis. However, this method does not consider the FOV constraint. In [
19], the authors design the ITCG law through the look-angle shaping method. By predicting the future mean velocity, the method can also be extended to the case of velocity variation. However, this method is only applicable to two-dimensional engagement. In [
20], the ITCG law is designed based on the data-driven method and PNG law. Using this method, a large offline database, which corresponds to a specific flight environment, must be established. When the real flight environment deviates from the flight environment of the database, the accuracy of the offline database will likely decrease. Therefore, the adaptability of this method to the flight environment is insufficient. In [
21,
22], the authors first derive the numerical estimation algorithm of time-to-go under the PNG law and then design a biased term for the PNG law to control the impact time. Note that the biased terms in [
21,
22] are given directly based on the feedback of impact time error, and there is no theoretical analysis or proof process. The biased term actually affects the length of the flight trajectory and the induced drag force, both of which can affect the impact time. In addition, the estimation algorithm in [
21] uses a recursive algorithm, which may lead to a high computational burden on the onboard computer, while the estimation algorithm in [
22] is only applicable to two-dimensional engagement.
Based on this review, the problem of three-dimensional impact time control guidance that considers field-of-view constraints and time-varying velocity is addressed in this study. Considering the unpowered flight of a guided projectile in the terminal guidance phase, a novel numerical estimation algorithm of time-to-go under the three-dimensional PNG law is derived. The effect of the aerodynamic forces and gravity on projectile velocity is fully considered in this algorithm. Then, the biased term is designed based on the PNG law. The effect of the biased term on flight time is analyzed, and the analysis results are given in the form of a theorem. Finally, numerical simulations are performed to verify the effectiveness of the proposed time-to-go estimation algorithm and ITCG law.
The primary advantages of the method proposed in this paper can be summarized as follows. First, compared with [
7,
8,
9,
10,
11,
12], the proposed method satisfies the impact time and FOV constraints and can consider velocity variations. Second, compared with [
13,
14,
15,
16,
17], the proposed method fully considers the effect of aerodynamic forces and gravity on velocity. Third, the proposed method analyzes the effect of the biased term on flight time and yields a sufficient condition for the design of the biased term, while the theoretical analysis is not performed in [
21,
22]. In addition, the method in [
22] is only applicable to two-dimensional engagement while the proposed method can be applied to three-dimensional engagement.
The remainder of this paper is organized as follows. In
Section 2, the guidance and dynamic models of the guided projectile are introduced, and the impact time control problem is described. In
Section 3, a simplified numerical estimation algorithm of time-to-go under the three-dimensional PNG law is given. In
Section 4, the effect of the biased term on flight time is analyzed, and the ITCG law considering FOV constraint and velocity variation is designed. Numerical simulation and comparisons are performed in
Section 5, and the conclusions are given in
Section 6.
3. Three-Dimensional Time-to-Go Estimation under Time-Varying Velocity
In this section, a novel time-to-go estimation algorithm that considers time-varying velocities under PNG law (11) is presented. Considering the real dynamics of the guided projectile, the flight time under PNG law (11) cannot be solved analytically; thus, a numerical solution is required. Limited by the computing power of the onboard computer, it is not advisable to solve the complete guidance and dynamic equations. Therefore, it is necessary to design a faster and high-precision time-to-go estimation algorithm for guided projectiles.
In practical applications, the terminal guidance phase generally follows a precise guidance handover from midcourse guidance; thus, the initial value of the lead angle
σm is usually small [
26]. Therefore, we can make the following approximation:
and (7) can be simplified to:
The PNG law (11) can also be simplified to:
Substituting (14) into (13), we can obtain:
By integrating (15), the functions of
θm and
φm concerning
R can be described as follows:
where
θm0,
φm0, and
R0 are the initial values of
θm,
φm, and
R, respectively. The velocity lead angle
σm can also be obtained according to (7) and (16) as:
Substituting (16) into (13), the function of
θL concerning
R can be described by:
where
θL0 is the initial value of
θL.
According to (10) and (12), we obtain
γ as a small angle; thus, (9) can be approximated as follows:
According to (20), the PNG law (11) can be approximated to the traditional PPNG law under the small lead angle assumption, and we can obtain:
By integrating (21) and combining (18), we can obtain:
where
θv0 is the initial value of
θv. The air density
ρ can be expressed as a function of altitude
yp as follows [
27]:
Note that the flight altitude
yp of the guided projectile can be obtained from (2) and (18) as:
Thus, the air density ρ can also be expressed as a function of R.
In this analysis, we can obtain the functions of
θm,
φm,
σm,
θL,
θv,
yp, and
ρ concerning
R under guidance law (11). According to (3) and (7), the final differential equations used for
tgo estimation are as follows:
Using (25) and (26), we can obtain the estimated flight time Te under PNG law (11). In addition, we can also obtain the estimated time-to-go tgoe and the estimation error ec at moment t.
Remark 2. The numerical solution of differential Equations (25) and (26) requires the aerodynamic coefficients CD0, CLα, Ccβ, kb, and gravitational acceleration g. For the terminal guidance process of the guided projectile, the flight time is typically short; thus, the changes in velocity and gravitational acceleration are small. Therefore, the aerodynamic coefficients and gravitational acceleration are treated as constants in this paper. It is also feasible to consider the effect of changes in these parameters on the tgo estimation. We thus update the parameter values based on the current Vp and R of each step in the integration.
Remark 3. The proposed estimation algorithm has two advantages in numerical calculation: (1) the number of differential equations to be solved is reduced, and only two differential equations must be solved; and (2) the relative distance R is used as the integration variable. To increase computational efficiency and estimation accuracy, the integration step size can be selected as a large value, which will be verified in the next numerical simulation. Therefore, the computational burden can be markedly reduced.