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Abstract

A novel adaptive sliding mode guidance law is proposed in this article. The target is assumed to have an arbitrarily but upper bounded maneuvering acceleration which is considered as the system disturbances and uncertainties. The guidance law is consisted of three terms. The first one is a proportional navigation–type term. The second one is a term used for compensating the target maneuvering acceleration. And the last one is a term for controlling the convergence time of the line-of-sight angular rate. In this guidance law, the upper bound of the target acceleration is estimated by an adaptive estimator with a tunable updating law. Hence, the prior knowledge of the upper bound of the target acceleration is not essential for this guidance law. The novel adaptive sliding mode guidance law can guarantee the asymptotical convergence of the line-of-sight rate to zero or its neighborhood, or even the finite time convergence of the line-of-sight rate conditionally. Finally, the new theoretical findings are demonstrated by numerical simulations.

Keywords

Sliding mode control arbitrarily maneuvering target asymptotic convergence finite time convergence adaptive tunable estimator

Introduction

Proportional navigation (PN) is widely used in the guidance community for its efficiency, robustness, and ease of implementation.^1–3 Although PN shows its advantages against nonmaneuvering targets or agile targets with low maneuverability, however, if the maneuverability of the target is close to that of the missile, the performance of PN will be largely degraded.

Robust guidance scheme is an effective methodology to tackle with the maneuvering target problem. A great number of existing robust guidance laws are obtained by using Lyapunov theories on asymptotic or exponential stability, such as H_∞ guidance law,⁴L₂ gain guidance law,⁵ Lyapunov-based nonlinear guidance laws,^6,7 and first-order sliding mode guidance laws.^8–12 For these guidance laws, only asymptotic or exponential stability is guaranteed, which indicates that the line-of-sight (LOS) rate or the designed sliding surface will converge to zero or its small neighborhood only as time approaches infinity.

The studies on guidance laws with finite time convergence (FTC) have become active and popular in the recent decade. Compared with asymptotic or exponential convergence, it is widely accepted that FTC has a better performance of handling the uncertainties and disturbances of a nonlinear system. Some FTC guidance laws were designed based on high-order (usually, second-order) sliding mode control theories.^13–24 Shetessel and colleagues^13–16 developed a smooth second-order sliding mode (SSOSM) control–based guidance law with FTC, and then applied it to the design of an integrated guidance and flight control system for endo-atmospheric interceptors. However, the proposed guidance algorithm could only handle sufficiently smooth uncertainties and disturbances, which were not quite possible in real interception scenarios. Based on the backstepping approach, Harl and Balakrishnan^17,18 presented a novel second-order sliding mode control approach which guaranteed that the sliding surface and its derivative would go to zero in finite time while ensuring that the sliding surface would not cross zero until the final time, and then they applied this method to the design of a terminal guidance law for an unpowered lifting reentry vehicle during the approach and landing phase and also applied this method to the design of an impact time and angle guidance law for airborne missiles. Based on the similar approach, Zhao et al.¹⁹ designed the terminal guidance law for an unpowered lifting reentry vehicle with impact angle constraints. For the above-mentioned high-order sliding mode control methods, the sliding surface is a linear function of the tracking error and its derivatives, and then the structures of deduced guidance laws become rather complex.

Another way to achieve FTC is the terminal sliding mode control (TSMC),^25,26 of which the sliding surface is a nonlinear function of the tracking error and its derivatives. However, TSMC methods usually involve singularity of control input. Therefore, nonsingular terminal sliding mode control (NTSMC) is proposed,^27–35 which avoids the singularity problem. NTSMC methods can be mainly divided into two types. The first one is the direct type, which is to construct a new nonsingular terminal sliding surface;^27–30 the other one is the indirect type, that is, to design a switching controller.^31–35 Kumar et al.^20,21 used TSMC and NTSMC to design FTC guidance laws for interceptors against maneuvering targets with all-aspect impact angle constraints, respectively. Sun et al.²² applied NTSMC to constructing guidance law for airborne missiles with a first-order dynamic lag, which could guarantee the FTC of LOS rate. However, for these guidance laws,^20–22 the prior knowledge of the upper bound of the system disturbances and uncertainties was required, and the estimation means of the upper bound were not given.

Some researchers tried to find new ways to achieve the FTC feature with the concept of adaptive control, and hence the prior knowledge of the upper bound of system disturbances and uncertainties does not need to be introduced. Wang and Wang^23,24 proposed an adaptive nonsingular terminal sliding mode control (ANTSMC) method which could guarantee FTC of the sliding surface in both reaching and sliding phases and applied it to the integrated guidance and control design of an airborne missile. It was claimed that the prior knowledge of the upper bound of system disturbances and uncertainties (i.e. target acceleration) is not needed for the FTC feature of the control law; however, in the proof the initial requirement on the estimation of the upper bound must be large enough to make the absolute value of the variable structure term larger than the actual upper bound.³⁶ Therefore, if the upper bound of the system uncertainties and disturbances is not previously known, the settling time of the sliding surface cannot be found and it cannot be strictly said that ANTSMC has the feature of FTC. Later, the authors made a modification of the updating law to guarantee the FTC of the ANTSMC.³⁷ Wang and Wang³⁸ also proposed a novel sliding mode guidance law with the saturation function and the same estimator of the upper bound of the system disturbances and uncertainties as used in Wang and Wang.²³ Zhou and Yang³⁹ presented the smooth adaptive nonsingular terminal sliding mode control (SANTSMC) method for a second-order nonlinear system with uncertainties, which can drive the system trajectory into a small neighborhood of the designed sliding surface in finite time without using the saturation function to replace the sign function, and then applied it to the outer loop design of the partial integrated guidance and control scheme of missile. It was also claimed that the upper bound of system disturbances and uncertainties is not essential for SANTSMC. However, a control parameter of SANTSMC is decided on the upper bound of system disturbances and uncertainties. Therefore, this claim cannot be taken as totally right. In fact, according to the authors’ knowledge, the FTC of a control law still has not been strictly and perfectly proven without the employment of the upper bound of system disturbances and uncertainties thus far.

Most of the above-mentioned high-order sliding mode guidance laws with FTC are designed for airborne missiles, where the commanded acceleration of the guidance law is usually assumed to be perpendicular to the missile velocity, mainly due to the aerodynamic force. Besides, the influence of dynamic lags on the guidance and control system is much more severe than that of the exoatmospheric interceptor. For exoatmospheric interception, based on the Lyapunov scalar differential inequality, Zhou et al.⁴⁰ proposed a novel finite time sliding mode guidance (FTSMG) law with a simple formation. The FTSMG was an extension of their previously proposed first-order sliding mode guidance law¹⁰ and can guarantee the FTC and stability of the LOS rate in both of the planar and three-dimensional (3D) environments. The commanded acceleration of FTSMG is perpendicular to LOS, and the control purpose is to drive the LOS rate to zero or its small neighborhood. Like most sliding mode guidance laws, the prior knowledge of the upper bound of the target maneuvering acceleration needs to be initially known for this guidance law. Some researchers used observers to achieve the FTC features of the designed guidance laws, for example, extended state observer (ESO)⁴¹ and finite time convergence disturbance observer (FTDOB).⁴² Adaptive control methods were also used to design guidance laws for missile with dynamic lags⁴³ and impact angle constraints.⁴⁴

In this article, we propose a novel adaptive sliding mode guidance law (NASMG) against the maneuvering target which has arbitrary acceleration projections along and perpendicular to LOS. Our purpose is to improve the performance of FTSMG in Zhou et al.,⁴⁰ without employing the prior knowledge of the upper bound of the target acceleration. Under this guidance law, the asymptotic convergence of the LOS rate from any initial value to zero can be guaranteed. And the FTC feature of the LOS rate to zero can also be guaranteed conditionally. Besides, the distribution of the commanded acceleration of NASMG will become more even than FTSMG, and the performance will become better, especially under the case that the target uses a small acceleration to maneuver while it actually has a large maneuverability. The work in this article can be taken as an improvement of the previously proposed FTSMG.

The structure of this article is organized as follows. In the “ Planar novel adaptive sliding mode guidance law” section, the 2D engagement kinematics will be introduced, and the design and analysis of the planar NASMG will be given. Then, this guidance law is extended to 3D space in the “Three-dimensional extension of NASMG” section. After that, the effectiveness of this guidance law is demonstrated by numerical simulations in the “Numerical simulation” section, and finally conclusions are drawn in the “Conclusion” section.

Planar novel adaptive sliding mode guidance law

Kinematics of planar engagement

The engagement geometry of planar interception is shown in Figure 1.

Figure 1.

2D engagement geometry.

The relative position vector $r$ is defined as

r = r_{t} - r_{m}

(1)

where $r_{t}$ and $r_{m}$ are the position vectors of the target and missile, respectively. The LOS vector $e_{r}$ is defined as the unit vector from the missile to the target, that is

\begin{matrix} e_{r} = \frac{r}{r}, & r = ‖ r ‖ \end{matrix}

(2)

And $e_{⊥}$ is defined as the unit vector vertical to $e_{r}$ and always on the left side of $e_{r}$ . Then, $e_{r}$ and $e_{⊥}$ form a polar coordinate system in 2D space.

The equations of motion are given by

V_{r} = \overset{\cdot}{r} = V_{t} \cos (λ - φ_{t}) - V_{m} \cos (λ - φ_{m})

(3)

V_{λ} = r \overset{\cdot}{λ} = - V_{t} \sin (λ - φ_{t}) + V_{m} \sin (λ - φ_{m})

(4)

where $V_{t}$ and $V_{m}$ are the speeds of the target and missile, respectively; $λ$ is the LOS angle, and $\overset{\cdot}{λ}$ the LOS angular rate; and $φ_{t}$ and $φ_{m}$ are the flight-path angles of the target and missile, respectively.

Taking the derivatives of the above two equations yield

\overset{\cdot\cdot}{r} - r {\overset{\cdot}{λ}}^{2} = a_{tr} - a_{mr}

(5)

r \overset{\cdot\cdot}{λ} + 2 \overset{\cdot}{r} \overset{\cdot}{λ} = a_{t ⊥} - a_{m ⊥}

(6)

where $a_{tr}$ , $a_{mr}$ , $a_{t ⊥}$ , and $a_{m ⊥}$ are the projections of the target and missile accelerations along $e_{r}$ and $e_{⊥}$ , respectively. The upper bounds of the target acceleration projections along and perpendicular to LOS are assumed to meet the following condition.

\begin{matrix} | a_{tr} (t) | < g, & | a_{t ⊥} (t) | < f \end{matrix}

(7)

where g and f are unknown constants.

It is commonly accepted that, when the initial closing speed is much smaller than zero, nullifying LOS rate will lead to an interception. Then a guidance law can be designed as follows.

Assume that $x = \overset{\cdot}{λ}$ , according to equation (6), we have

\overset{\cdot}{x} (t) = - \frac{2 \overset{\cdot}{r} (t)}{r (t)} x - \frac{u (t)}{r (t)} + \frac{w (t)}{r (t)}

(8)

where $u (t) = a_{m ⊥} (t)$ is the control input that is used to achieve the convergence of $\overset{\cdot}{λ}$ , and $w (t) = a_{t ⊥} (t)$ is taken as the system uncertainties and disturbances; then, it must have $| w (t) | \leq f$ . The starting time of the guidance process is taken to be zero (i.e., $t = 0$ ). The initial state variables in the guidance system are denoted as $r (0)$ , $\overset{\cdot}{r} (0)$ , and $x (0)$ .

Finite time stability of nonlinear systems

The finite time stability of a time-varying nonlinear system was proposed in Zhou et al.⁴⁰ and is employed in this article for the performance analysis of NASMG.

Definition 1

Consider a nonlinear system in the form of

\begin{matrix} \overset{\cdot}{x} (t) = f (x, t), & f (0, t) = 0, & x \in R^{n} \end{matrix}

(9)

where $f$ : $U_{0} \times R \to R^{n}$ is continuous on $U_{0} \times R$ , and $U_{0}$ is an open neighborhood of the origin $x = 0$ . The state of the system is said to converge to its local equilibrium $x = 0$ in finite time if, for any given initial time $t_{0}$ and initial states $x (t_{0}) = x_{0} \in U$ , there exists a settling time $T \geq 0$ , which is dependent on $x_{0}$ , such that every solution of the system (9), $x (t) = υ (t; t_{0}, x_{0}) \in U / {0}$ , satisfies

{\begin{matrix} \begin{matrix} lim_{t \to T (x_{0})} υ (t; t_{0}, x_{0}) = 0, & t \in [t_{0}, T (x_{0})] \end{matrix} \\ \begin{matrix} υ (t; t_{0}, x_{0}) = 0, & t \geq T (x_{0}) \end{matrix} \end{matrix}

(10)

Moreover, if the system (local) equilibrium $x = 0$ is Lyapunov stable with FTC in a neighborhood of the origin $U \in U_{0}$ , then the system equilibrium is called finite time stable. If $U \in R^{n}$ , then the origin is a global finite time stable equilibrium.

The following lemma⁴⁰ is popular in the guidance law design with FTC, for example, in Sun et al.²² It will prove useful in the design of NASMG, and hence is employed here.

Lemma 1

Consider the nonlinear system described by equation (9). Suppose that there is a $C^{1}$ (continuously differentiable) function $V (x, t)$ defined in a neighborhood of $\hat{U} \in R^{n}$ of the origin, and there are real numbers $α > 0$ and $0 < γ < 1$ , such that $V (x, t)$ is positive-definite on $\hat{U}$ and that $\overset{\cdot}{V} (x, t) + α V^{γ} (x, t) \leq 0$ . Then, the zero solution of system (9) is finite time stable.

The settling time is given by

T_{r} \leq \frac{V {(x_{0}, t_{0})}^{1 - γ}}{α (1 - γ)}

(11)

Remark 1

Note that if $\hat{U} \in R^{n}$ and $V (x, t)$ are radially unbounded, then the origin is globally finite time stable.

Novel adaptive sliding mode guidance law

For the time-varying nonlinear guidance system of equation (8), the guidance purpose is to nullify the LOS rate (i.e. nullify $x$ from any initial state $x (0)$ ). The NASMG is obtained in the following theorem.

Theorem 1

For the guidance system of equation (8), the NASMG is designed as

u = - N \overset{\cdot}{r} x + k \hat{f} sgn (x) + β {| x |}^{η} sgn (x)

(12)

where k satisfies

k > {\begin{matrix} \begin{matrix} 1, & r \geq 1 \end{matrix} \\ \begin{matrix} \frac{1}{r}, & r < 1 \end{matrix} \end{matrix}

(13)

and $β = const . > 0$ , $0 \leq η = const . < 1$ . The $\hat{f}$ is an estimation of the unknown constant $f$ , whose updating law is

\begin{matrix} \overset{\cdot}{\hat{f}} = k x^{\frac{p}{q}} sgn (x) \geq 0, & \hat{f} (0) > 0 \end{matrix}

(14)

where $p$ and $q$ are odd numbers. Then, the LOS angular rate x converges to zero as t→∞.

Proof

First, we consider the situation that is satisfied during the guidance process

\begin{matrix} k \hat{f} (t_{1}) \geq f, & \forall t_{1} \geq 0 \end{matrix}

(15)

The following continuously differentiable positive-definite Lyapunov function in the neighborhood of the origin $\hat{U} \in R^{n}$ is adopted

V_{1} = (\frac{q}{p + q}) x^{\frac{p}{q} + 1} = (\frac{q}{p + q}) {[x^{(\frac{p + q}{2 q})}]}^{2} \geq 0

(16)

In this article it is predefined that, for the n-order root of a negative real scalar, only the real root is valid. The complex roots are not considered here. Taking the derivative of $V_{1}$ yields

{\overset{\cdot}{V}}_{1} = x^{\frac{p}{q}} \overset{\cdot}{x} = x^{\frac{p}{q}} [- \frac{2 \overset{\cdot}{r}}{r} x - \frac{u}{r} + \frac{w}{r}]

(17)

Substituting equation (12) into equation (17) yields

\begin{matrix} {\overset{\cdot}{V}}_{1} = x^{\frac{p}{q}} [\frac{(N - 2) \overset{\cdot}{r}}{r} x + \frac{f - k \hat{f} sgn (x)}{r} - \frac{β {| x |}^{η} sgn (x)}{r}] \\ \leq \frac{(N - 2) \overset{\cdot}{r}}{r} x^{\frac{p + q}{q}} + \frac{(f - k \hat{f})}{r} x^{\frac{p}{q}} sgn (x) - \frac{β {| x |}^{η} x^{\frac{p}{q}} sgn (x)}{r} \\ \overset{i f N > 2}{\leq} \frac{(f - k \hat{f})}{r} x^{\frac{p}{q}} sgn (x) - \frac{β {| x |}^{η} x^{\frac{p}{q}} sgn (x)}{r} \end{matrix}

(18)

Substituting equation (15) into equation (18) and considering $\overset{\cdot}{\hat{f}} > 0 (x \neq 0)$ yields

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - \frac{β {| x |}^{η} x^{\frac{p}{q}} sgn (x)}{r} = - \frac{β {| x |}^{η + \frac{p}{q}}}{r}, & t \geq t_{1} \end{matrix}

(19)

During the guidance process, the following inequality must be valid⁴⁰

\begin{matrix} \overset{\cdot}{r} (t) < 0, & 0 < r (t) < r (0), & \forall t > 0 \end{matrix}

(20)

Therefore

{\overset{\cdot}{V}}_{1} \leq - \frac{β}{r (0)} {| x |}^{η + \frac{p}{q}} = - \frac{β}{r (0)} {(\frac{p + q}{q})}^{\frac{p + η q}{p + q}} V^{\frac{p + η q}{p + q}}

(21)

Let

\begin{matrix} α = \frac{β}{r (0)} {(\frac{p + q}{q})}^{\frac{p + η q}{p + q}}, & γ = \frac{p + η q}{p + q} \end{matrix}

(22)

We have

\begin{matrix} {\overset{\cdot}{V}}_{1} + α V_{1}^{γ} \leq 0, & t \geq t_{1} \end{matrix}

(23)

Then, according to lemma 1

\begin{matrix} V_{1}^{1 - γ} (x, t) \leq V_{1}^{1 - γ} (x (t_{1}), t_{1}) - α (1 - γ) t, & t_{1} \leq t \leq t_{1} + t_{r} \end{matrix}

(24)

and

\begin{matrix} V_{1}^{1 - γ} (x, t) = 0, & t > t_{1} + t_{r} \end{matrix}

(25)

where

t_{r} = \frac{V_{1}^{1 - γ}}{α (1 - γ)} = \frac{{| x (0) |}^{1 - η} r (0)}{β (1 - η)}

(26)

Hence, the LOS angular rate x converges to zero before t→∞.

Second, we consider the situation that equation (15) cannot happen during the whole guidance process, that is

\begin{matrix} k \hat{f} (t) < f, & \forall t \geq 0 \end{matrix}

(27)

Let $\tilde{f} = f - \hat{f}$ . The following continuously differentiable positive-definite Lyapunov function is adopted

V_{2} = (\frac{q}{p + q}) x^{\frac{p}{q} + 1} + \frac{1}{2} {\tilde{f}}^{2} = (\frac{q}{p + q}) {[x^{(\frac{p + q}{2 q})}]}^{2} + \frac{1}{2} {\tilde{f}}^{2}

(28)

Taking the derivative of V₂ yields

{\overset{\cdot}{V}}_{2} = x^{\frac{p}{q}} \overset{\cdot}{x} + \tilde{f} \overset{\cdot}{\tilde{f}} = x^{\frac{p}{q}} [- \frac{2 \overset{\cdot}{r}}{r} x - \frac{u}{r} + \frac{w}{r}] - \tilde{f} \overset{\cdot}{\hat{f}}

(29)

Substituting equations (12) and (14) into equation (29), we have

\begin{matrix} {\overset{\cdot}{V}}_{2} = x^{\frac{p}{q}} [- \frac{(N - 2) \overset{\cdot}{r}}{r} x + \frac{w - k \hat{f} sgn (x)}{r} - \frac{β {| x |}^{η} sgn (x)}{r}] \\ - (f - \hat{f}) k x^{\frac{p}{q}} sgn (x) \\ \overset{N > 2}{\leq} [\frac{(f - k \hat{f})}{r} - k (f - \hat{f})] x^{\frac{p}{q}} sgn (x) - \frac{β {| x |}^{η} x^{\frac{p}{q}} sgn (x)}{r} \\ = [f - k (\hat{f} + r (f - \hat{f}))] \frac{x^{\frac{p}{q}} sgn (x)}{r} - \frac{β {| x |}^{η} x^{\frac{p}{q}} sgn (x)}{r} \end{matrix}

(30)

According to the above inequality, recalling equations (13) and (27), it can be deduced that $\hat{f} (t) < f$ . Then, if r≥ 1

\begin{matrix} f - k (\hat{f} + r (f - \hat{f})) < f - \hat{f} - r (f - \hat{f}) \\ = (f - \hat{f}) (1 - r) \leq 0 \end{matrix}

(31)

Else, considering equation (13), we have

\begin{matrix} f - k (\hat{f} + r (f - \hat{f})) = f - k \hat{f} - rk (f - \hat{f}) \\ \leq f - \hat{f} - rk (f - \hat{f}) \\ = (f - \hat{f}) (1 - rk) < 0 \end{matrix}

(32)

According to equations (20) and (30)–(32)

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - \frac{β {| x |}^{η + \frac{p}{q}}}{r (0)} < 0 (x \neq 0), & \forall r (t) > 0 \end{matrix}

(33)

Let $K = β / r (0)$ , then ${\overset{\cdot}{V}}_{2} \leq - K {| x |}^{η + \frac{p}{q}} \leq 0$ holds when $x \neq 0$ , which means $x \in L_{\infty}$ and $\tilde{f} \in L_{\infty}$ . Integrating both sides of ${\overset{\cdot}{V}}_{2} \leq - K {| x |}^{η + \frac{p}{q}}$ from zero to infinity yields

\int_{0}^{\infty} {| x (t) |}^{η + \frac{p}{q}} dt \leq \frac{V_{2} (0) - V_{2} (\infty)}{K} \leq \frac{V_{2} (0)}{K}

(34)

Then, combined with Barbalat’s Lemma,⁴⁵ ${| x (t) |}^{η + p / q}$ converges to zero as t→∞, which means x asymptotically converges to zero. According to the above analysis, no matter $k \hat{f} (t) \geq f$ is satisfied or not during the guidance process, the NASMG of equations (12)–(14) will drive the LOS rate to converge to zero asymptotically. And if $k \hat{f} (t) \geq f$ is satisfied during the guidance process, NASMG will drive the LOS rate to converge to zero in finite time.

Remark 2

NASMG is a non-smooth controller which involves a signum function. This indicates that the control variable sometimes switch. In practical systems, the switching cannot be completely instantaneous, and the delay of switching will induce the chattering effect. To remove the chattering, the signum function needs to be smoothed. This is usually done by replacing the signum function with a saturation function $sa t_{δ} (x)$ , which is expressed as

sa t_{δ} (x) = {\begin{matrix} \begin{matrix} 1, & x > δ \end{matrix} \\ \begin{matrix} x / δ, & | x | \leq δ \end{matrix} \\ \begin{matrix} - 1, & x < - δ \end{matrix} \end{matrix}

(35)

where δ is a small positive constant. The updating law of $\hat{f} (t)$ is further modified into

\begin{matrix} \overset{\cdot}{\hat{f}} = {\begin{matrix} k x^{\frac{p}{q}} sgn (x) \geq 0, & | x | > δ \\ 0, & | x | \leq δ \end{matrix}, & \hat{f} (0) > 0 \end{matrix}

(36)

since there is no need to increase $\hat{f} (t)$ after the LOS rate converges into the boundary layer $| x | \leq δ$ . Then, NASMG of equations (12)–(14) with the saturation function of equation (35) and modified updating law of equation (36) guides the LOS rate to converge into a boundary layer $| x | \leq δ$ asymptotically.

Remark 3

According to the expression of NASMG of equations (12)–(14), this guidance law is constructed with three terms: the first one is a PN term, which functions as the traditional PN law; the second one is the adaptive variable structure term, which is used to compensate for the target acceleration; and the last one is the convergence time control term, which reduces the convergence time of the control law. Therefore, NASMG can also be taken as an augmented PN guidance law, which is with similar measurement requirements but has a better guidance performance compared to PN.

Three-dimensional extension of NASMG

The construction of FTSMG in 3D space has been discussed in Shin et al.⁴⁶ In this section, the NASMG will be extended in 3D environment by using the same method.

Extension of NASMG in 3D environment

Traditionally, the 3D pursuit was dealt with by constructing two independent guidance laws in the pitch and yaw planes of the missile respectively and considering the cross-coupling effect.^47–49 However, in this way the description of the relative motion was quite complex due to the cross-coupling and too many variables were involved.

Using the kinematic equations established in the rotating coordinate system of LOS can simplify the description of the 3D relative motion so that it can be divided into two decoupled sub-motions:^50–54 (1) the relative motion in the engagement plane spanned by the relative position and velocity vectors and (2) the rotation of this plane. In this section, we will realize the NASMG in 3D space by directly constructing a planar NASMG in the relative engagement plane.

Figure 2 depicts the 3D pursuit geometry, where $O_{A} - X_{A} Y_{A} Z_{A}$ denotes the inertial launch frame.

Figure 2.

3D engagement geometry.

The LOS kinematic equation set is

{\begin{matrix} {\overset{\cdot}{e}}_{r} = ω_{s} e_{θ} \\ {\overset{\cdot}{e}}_{θ} = - ω_{s} e_{r} + Ω_{s} e_{ω} \\ {\overset{\cdot}{e}}_{ω} = - Ω_{s} e_{θ} \end{matrix}

(37)

where $e_{r}$ is the unit vector along LOS, $e_{ω}$ is the unit vector along the LOS angular velocity, and $e_{θ} = e_{ω} \times e_{r}$ is the unit normal vector of LOS; $e_{r}$ , $e_{θ}$ , and $e_{ω}$ form the bases of the rotating LOS coordinate system; $e_{r}$ and $e_{θ}$ constitute the engagement plane; $ω_{s} \geq 0$ is the instantaneous LOS rate; the engagement plane may rotate around $e_{r}$ in 3D space, and $Ω_{s}$ is the rotation rate of the engagement plane.

The relative kinematic equation set between the missile and target is

{\begin{matrix} \overset{\cdot\cdot}{r} - r {ω_{s}}^{2} = a_{tr} - a_{mr} \\ r {\overset{\cdot}{ω}}_{s} + 2 \overset{\cdot}{r} ω_{s} = a_{t θ} - a_{m θ} \\ r ω_{s} Ω_{s} = a_{t ω} - a_{m ω} \end{matrix}

(38)

where $r$ is the relative range; $a$ represents the acceleration; subscripts “ $t$ and $m$ ” represent that the quantities belong to the target and interceptor, respectively; and subscripts “ $r$ , $θ$ , and $ω$ ” represent projections along $e_{r}$ , $e_{θ}$ , and $e_{ω}$ , respectively. The first two equations of equation (38) which describe the relative motion between missile and target in the engagement plane are decoupled with the third one which describes the rotational principle of the engagement plane. For more detailed derivations, the reader is referred to previous works.^3,50–54

The 3D LOS rate $ω_{s}$ is mainly decided by the first two equations of equation (38). We can see that equation (8) can also be deduced from the second equation of equation (38), when assuming $x = ω_{s}$ , $u = a_{m θ}$ , and $w = a_{t θ}$ . Then, 3D NASMG can be obtained by directly constructing a 2D NASMG in the engagement plane, that is

a_{mc} = (- N \overset{\cdot}{r} ω_{s} + k \hat{f} + β ω_{s}^{η}) e_{θ}

(39)

\begin{matrix} \overset{\cdot}{\hat{f}} = k ω_{s}^{\frac{p}{q}} \geq 0, & \hat{f} (0) \geq 0 \end{matrix}

(40)

where k is determined by equation (13). Note that there is no sign function in equations (39) and (40). The reason is that $ω_{s} \geq 0$ during the guidance process according to its definition. However, the commanded acceleration of 3D NASMG may reverse its direction sometimes when $ω_{s} = 0$ happens. This phenomenon will be discussed and handled in the next subsection.

Realization of 3D NASMG

The missile seeker usually measures the LOS elevation angle $q_{ε}$ and azimuth angle $q_{β}$ in 3D space, whose derivatives ${\overset{\cdot}{q}}_{ε}$ and ${\overset{\cdot}{q}}_{β}$ can be obtained via filtering. Besides, it is assumed in this article that the relative range $r$ and its derivative $\overset{\cdot}{r}$ can also be measured.

The inertial coordinate system $o_{A} - x_{A} y_{A} z_{A}$ and its geometric relationships with the traditional LOS coordinate system and the rotating LOS coordinate system are shown in Figure 3.

Figure 3.

Geometric relationship among three coordinate reference systems.

In Figure 3, $o_{A} - x_{A} y_{A} z_{A}$ firstly rotates around $y_{A}$ with angle $q_{β}$ and secondly rotates around $z_{s}$ with angle $q_{ε}$ , and the traditional LOS coordinate system $o_{A} - x_{s} y_{s} z_{s}$ is gained. The three axes of LOS coordinates can be expressed as

{\begin{matrix} x_{s} = \cos q_{ε} \cos q_{β} x_{A} + \sin q_{ε} y_{A} - \cos q_{ε} \sin q_{β} z_{A} \\ y_{s} = - \sin q_{ε} \cos q_{β} x_{A} + \cos q_{ε} y_{A} + \sin q_{ε} \sin q_{β} z_{A} \\ z_{s} = \sin q_{β} x_{A} + \cos q_{β} z_{A} \end{matrix}

(41)

The angular velocity of LOS coordinates is $ω_{LOScoor}$

\begin{matrix} ω_{LOScoor} = {\overset{\cdot}{q}}_{β} y_{A} + {\overset{\cdot}{q}}_{ε} z_{s} \\ = {\overset{\cdot}{q}}_{β} \sin q_{ε} x_{s} + {\overset{\cdot}{q}}_{β} \cos q_{ε} y_{s} + {\overset{\cdot}{q}}_{ε} z_{s} \end{matrix}

(42)

Since $e_{r}$ is equal to $x_{s}$ , then

\begin{matrix} {\overset{\cdot}{e}}_{r} = {\overset{\cdot}{x}}_{s} = ω_{LOScoor} \times x_{s} \\ = ({\overset{\cdot}{q}}_{β} \sin q_{ε} x_{s} + {\overset{\cdot}{q}}_{β} \cos q_{ε} y_{s} + {\overset{\cdot}{q}}_{ε} z_{s}) \times x_{s} \\ = - {\overset{\cdot}{q}}_{β} \cos q_{ε} z_{s} + {\overset{\cdot}{q}}_{ε} y_{s} \end{matrix}

(43)

Comparing equation (43) with the first equation of equation (37), the result is

e_{θ} = \frac{{\overset{\cdot}{q}}_{ε} y_{s} - {\overset{\cdot}{q}}_{β} \cos q_{ε} z_{s}}{\sqrt{{({\overset{\cdot}{q}}_{β} \cos q_{ε})}^{2} + {\overset{\cdot}{q}}_{ε}^{2}},} ω_{s} = \sqrt{{({\overset{\cdot}{q}}_{β} \cos q_{ε})}^{2} + {\overset{\cdot}{q}}_{ε}^{2}}

(44)

Recalling $e_{ω} = e_{r} \times e_{θ}$ , we have

e_{ω} = \frac{{\overset{\cdot}{q}}_{β} \cos q_{ε} y_{s} + {\overset{\cdot}{q}}_{ε} z_{s}}{\sqrt{{({\overset{\cdot}{q}}_{β} \cos q_{ε})}^{2} + {\overset{\cdot}{q}}_{ε}^{2}}}, ω_{s} = {\overset{\cdot}{q}}_{β} \cos q_{ε} y_{s} + {\overset{\cdot}{q}}_{ε} z_{s}

(45)

Then, the measurements of $\overset{\cdot}{r}$ , $ω_{s}$ , and $e_{θ}$ can be provided. Therefore, 3D NASMG can be realized.

Remark 4

During the guidance process, usually the LOS angles of $q_{ε}$ and $q_{β}$ do not change their signs, but the LOS rates of ${\overset{\cdot}{q}}_{ε}$ and ${\overset{\cdot}{q}}_{β}$ sometimes change signs. According to equation (44), when ${\overset{\cdot}{q}}_{ε}$ and ${\overset{\cdot}{q}}_{β}$ change signs together (i.e. $ω_{s} = 0$ happens), the direction of $e_{θ}$ reverses which causes the commanded acceleration to reverse. However, in practical systems, the reverse cannot be completely instantaneous, and the delay of reverse will induce chattering effect. Therefore, we use a saturation function $sa t_{δ} (ω_{s})$ to smooth the guidance command and weaken the chattering effect, as follows

a_{mc} = [- N \overset{\cdot}{r} ω_{s} + (k \hat{f} + β ω_{s}^{η}) sa t_{δ} (ω_{s})] e_{θ}

(46)

sa t_{δ} (ω_{s}) = {\begin{matrix} 1, & ω_{s} > δ \\ ω_{s} / δ, & ω_{s} \leq δ \end{matrix}

(47)

where $δ$ is a small positive constant. Recalling equation (36), the updating law of $\hat{f} (t)$ is also modified into

\overset{\cdot}{\hat{f}} = {\begin{matrix} k \cdot ω_{s}^{\frac{p}{q}}, & ω_{s} > δ \\ 0, & ω_{s} \leq δ \end{matrix}

(48)

Therefore, the 3D NASMG with the saturation function guides the LOS rate to converge into a boundary layer $ω_{s} \leq δ$ asymptotically.

Numerical simulation

A space interception problem is simulated in this section. An inertial reference system with origin initially fixed at the center of the launch site at the instant of launch is used. The X axis is in the horizontal plane and is along the launch direction. The positive Y axis is vertical to the horizontal plane. And the Z axis forms a right-hand coordinate system with the other two axes together. The initial states of the missile and target are shown in Table 1.

Table 1.

Initial states of missile and target.

Missile	X	Y	Z
Position (m)	0	0	0
Velocity (m/s)	711.2	2883.0	−427.3
Target	X	Y	Z
Position (m)	75,358	31,671	−43,508
Velocity (m/s)	−6062.2	0	3500.0

The maximum translation accelerations of the missile in the elevation and azimuth directions are assumed to be equal to 30 m/s². The sample period of the interceptor seeker is 10 ms. The 3D NASMG of equations (46)–(48) is used. For comparison, FTSMG in Zhou et al.⁴⁰ is also considered which can be directly constructed in the engagement plane and whose formation becomes⁴⁶

a_{FTSMG} = [- N \overset{\cdot}{r} ω_{s} + (f_{θ} + β ω_{s}^{η}) sa t_{δ} (ω_{s})] e_{θ}

(49)

where $f_{θ} \geq a_{t θ}$ is the prior knowledge of the upper bound of the target acceleration.

As we know, for PN guidance laws, the navigation gain N is usually 3–5. In this section, we set N = 3 for both NASMG and FTSMG. The influences on the guidance effect caused by parameters of $β$ and $η$ of the convergence time term had been well-investigated in Zhou et al.⁴⁰ Therefore, in this article, we mainly explore the influences on the guidance effect caused by guidance parameters of $k$ , $\hat{f} (0)$ , $p$ , and $q$ for NASMG. In the following cases, $β = 0.5$ and $η = 0.1$ are selected for both NASMG and FTSMG. To reduce chattering, $δ = 0.002 \deg / s$ is chosen for both guidance laws.

Case 1

Assume that the target escapes with constant acceleration component $a_{t θ} = 13 m / s^{2}$ and it cannot be estimated. For FTSMG, we assume $f_{θ} = 15 m / s^{2}$ . Apply NASMG with $k = 2.1$ , $\hat{f} (0) = 2$ , $p = 1$ , and $q = 3, 5, 7, 9$ , respectively. The miss distances of FTSMG and NASMGs with different q are less than 0.01 m. The simulation results are shown in the following figures.

From Figure 4, we can see that the 3D LOS rate of FTSMG can converge to the boundary layer of $ω_{s} \leq 0.002 \deg / s$ before the final time of interception, and the convergence rate of $ω_{s}$ of FTSMG is the largest. For NASMG, when q = 3, the 3D LOS rate $ω_{s}$ cannot converge to the boundary layer. It first increases and then decreases after a certain time, which shows the asymptotic convergence feature but not FTC feature. The reason is that the absolute value of the variable structure term $k \hat{f}$ is always smaller than the upper bound of the target acceleration f = 13 m/s², as shown in Figure 8. As q increases, the convergence rate of $ω_{s}$ of NASMG is increased, and $ω_{s}$ can converge to the boundary layer when $q \geq 5$ , since $k \hat{f}$ can be increased to be larger than f in the final stage of the guidance process by using the updating law, as shown in Figure 8.

Figure 4.

3D LOS rate in case 1.

The LOS azimuth and elevation rates ${\overset{\cdot}{q}}_{ε}$ and ${\overset{\cdot}{q}}_{β}$ of both guidance laws are shown in Figure 5, which are with the similar tendencies as $ω_{s}$ in Figure 4.

Figure 5.

LOS elevation and azimuth angular rates in case 1.

According to Figure 6, the total commanded acceleration $| a_{m} |$ of FTSMG is the largest in the beginning of the guidance process and decreasing in the first half of the guidance process, and then it gets to a constant stable value after $ω_{s}$ of FTSMG converges into the boundary layer. For NASMG, when q = 3, $| a_{m} |$ is the largest one in the latter half of the guidance process which cannot converge to a stable value before the end of the guidance. However, as q increases, $| a_{m} |$ of NASMG can converge to a stable value and the convergence rate is increased.

Figure 6.

Total missile commanded acceleration in case 1.

The lateral missile commanded accelerations $a_{my}$ and $a_{mz}$ along the Y axis and Z axis of the LOS coordinate system are shown in Figure 7, respectively, whose tendencies are similar as those of $| a_{m} |$ .

Figure 7.

Lateral missile commanded accelerations in case 1.

The estimation of the upper bound of the target acceleration $\hat{f}$ and the absolute value of the variable structure term of NASMG $k \hat{f}$ are shown in Figure 8. It can be seen that as q increases, the increase rate of $\hat{f}$ is increased, and so is that of $k \hat{f}$ .

Figure 8.

Estimation of the upper bound of the target acceleration and the absolute value of the variable structure term of NASMG in case 1.

The zero effort misses (ZEMs) of both guidance laws are shown in Figure 9. We can see that ZEM of FTSMG converge to zero with the most rapid speed. For NASMG, as q increases, the convergence rate of ZEM is increased.

Figure 9.

Zero effort miss in case 1.

It is noticed that there are abrupt increases and vibrations of $ω_{s}$ and $| a_{m} |$ for both guidance laws in the end of the engagement. The reason is that, when the relative range r reduces to zero, for intercepting maneuvering targets, $ω_{s}$ increases fiercely in a very short time; at this moment, the reaction of the guidance system is not quick enough to follow the abrupt change of $ω_{s}$ . This phenomenon is common for most interception scenarios. Since the guidance module usually cuts off the guidance command several hundred meters before the final intercept, the abrupt increase and vibration of the commanded acceleration will not influence the performance of the guidance system.

From the updating law of equation (48) we can see that the influence of p on the performance of NASMG works as the adverse effect of q. Hence, for the length of this article, we do not demonstrate the effect of p here. According to the simulation results in this case, it can be concluded that p and q must be chosen properly, since a large “p/q” may result into a non-convergent LOS rate and a small “p/q” may cause large $\hat{f}$ and $k \hat{f}$ . Therefore, a medium “p/q” is suggested for NASMG.

Case 2

Assume that the target escapes with constant acceleration component a_tθ = 6.5 (1 +sin(2πt+π/4)) m/s². The guidance parameters of FTSMG and NASMG are the same as those in case 1. The miss distances of both guidance laws are less than 0.01 m. The simulation results are shown in the following figures.

According to Figures 10 –15, we can see that the performance of FTSMG and that of NASMG under the sinusoidal target maneuvering acceleration are quite similar as those under the constant target maneuvering acceleration. Since a large prior estimation of the upper bound of the target acceleration is used for FTSMG, the convergence rate of 3D LOS rate is the fastest. However, a large initial commanded acceleration is also needed for FTSMG. For NASMG, the convergence rate of 3D LOS rate is influenced by the guidance parameters of k, p, and q. The influence effects of these guidance parameters are similar as those in Case 1. Hence, in the following discussion, only constant target maneuvering acceleration is considered.

Figure 10.

3D LOS rate in case 2.

Figure 11.

LOS elevation and azimuth angular rates in case 2.

Figure 12.

Total missile commanded acceleration in case 2.

Figure 13.

Lateral missile commanded accelerations in case 2.

Figure 14.

Estimation of the upper bound of the target acceleration and the absolute value of the variable structure term of NASMG in case 2.

Figure 15.

Zero effort miss in case 2.

Case 3

The target acceleration and FTSMG are the same as those in case 1. Apply the NASMG with $k = 2.1, 2.5, 3, 4$ ; $\hat{f} (0) = 2$ ; $p = 1$ ; and $q = 7$ , respectively. The miss distances of FTSMG and NASMGs with different k are less than 0.01 m. The simulation results are shown in the following figures.

According to Figure 16 we can see that, in this case, as k increases, the convergence rate of $ω_{s}$ of NASMG is increased, which is even larger than that of FTSMG when k is large enough. The curves of ${\overset{\cdot}{q}}_{ε}$ and ${\overset{\cdot}{q}}_{β}$ of NASMG are shown in Figure 17, which are with similar tendencies as those of $ω_{s}$ of NASMG.

Figure 16.

3D LOS rate in case 3.

Figure 17.

LOS elevation and azimuth angular rates in case 3.

The commanded accelerations of both guidance laws in this case are shown in Figure 18. It can be seen that as k increases, the initial and peak values of $| a_{m} |$ of NASMG are increased, and so is the convergence rate of $| a_{m} |$ .

Figure 18.

Total missile commanded acceleration in case 3.

The curves of $a_{my}$ and $a_{mz}$ in this case are shown in Figure 19, whose tendencies are similar as those of $| a_{m} |$ .

Figure 19.

Lateral missile commanded accelerations in case 3.

The curves of $\hat{f}$ and $k \hat{f}$ in this case are shown in Figure 20. From the figure it can be seen that as k increases, the increase rate of $\hat{f}$ is increased, but the peak value of $\hat{f}$ is decreased. However, for $k \hat{f}$ , as k increases, the increase rate and peak value are both increased.

Figure 20.

Estimation of the upper bound of the target acceleration and the absolute value of the variable structure term of NASMG in case 3.

The ZEMs of both guidance laws in this case are shown in Figure 21. It can be seen that as k increases, the convergence rate of ZEM of NASMG to the neighborhood of zero is increased, and ZEMs of NASMG with k = 3 and 4 converge to the neighborhood of zero faster than that of FTSMG.

Figure 21.

Zero effort miss in case 3.

According to the simulation results in this case, k should not be chosen too large for NASMG, since a large k usually results in a large variable structure term $k \hat{f}$ and a large commanded acceleration $| a_{m} |$ .

Case 4

The target acceleration and FTSMG are the same as those in case 1. Apply the NASMG with $k = 2.1$ ; $\hat{f} (0) = 2, 4, 6, 8$ ; $p = 1$ ; and $q = 7$ , respectively. The miss distances of FTSMG and NASMGs with different $\hat{f} (0)$ are less than 0.01 m. The simulation results are shown in the following figures.

From Figure 22 it can be seen that as $\hat{f} (0)$ increases, the convergence rate of $ω_{s}$ of NASMG is increased, and sometimes the convergence rate is even larger than that of FTSMG when $\hat{f} (0)$ is large enough. The curves of ${\overset{\cdot}{q}}_{ε}$ and ${\overset{\cdot}{q}}_{β}$ of NASMG in this case are shown in Figure 23, which are with similar tendencies as those of $ω_{s}$ of NASMG.

Figure 22.

3D LOS rate in case 4.

Figure 23.

LOS elevation and azimuth angular rates in case 4.

According to Figure 24, as $\hat{f} (0)$ increases, the initial commanded acceleration of NASMG is increased, and so is its convergence rate. Sometimes, the initial commanded acceleration of NASMG is even larger than that of FTSMG when $\hat{f} (0)$ is large enough.

Figure 24.

Total missile commanded acceleration in case 4.

The curves of $a_{my}$ and $a_{mz}$ of both guidance laws in this case are shown in Figure 25, whose tendencies are similar as those of $| a_{m} |$ .

Figure 25.

Lateral missile commanded accelerations in case 4.

The curves of $\hat{f}$ and $k \hat{f}$ in this case are shown in Figure 26. We can see that as $\hat{f} (0)$ increases, the increase rates of $\hat{f}$ and $k \hat{f}$ are not seriously influenced; but the time that they get to the stable, constant values are advanced.

Figure 26.

Estimation of the upper bound of the target acceleration and the absolute value of the variable structure term of NASMG in case 4.

The ZEMs of both guidance laws in this case are shown in Figure 27. It can be seen that as $\hat{f} (0)$ increases, the convergence rate of ZEM of NASMG to the neighborhood of zero is increased, and ZEMs of NASMG with $\hat{f} (0)$ = 6 and 8 converge to the neighborhood of zero faster than that of FTSMG.

Figure 27.

Zero effort miss in case 4.

According to the simulation results in this case, a small $\hat{f} (0)$ is better for the NASMG, since a large $\hat{f} (0)$ causes a large initial commanded acceleration which may exceed the physical limitation of the missile.

In the above four cases, the variable structure term of FTSMG is close to the magnitude of the target acceleration, which provides FTSMG with a good performance. However, in practical interception scenarios, sometimes the target may use a small acceleration while it has a large maneuverability. In the following case, this situation will be discussed.

Case 5

The target acceleration is the same as those in case 1. For FTSMG, we assume $f_{θ} = 25 m / s^{2}$ . Apply the NASMG with $k = 2.1$ , $\hat{f} (0) = 2$ , $p = 1$ , and $q = 7$ , respectively. The miss distances of FTSMG and NASMG are both less than 0.01 m. The simulation results are shown in the following figures.

According to Figure 28 it can be seen that, for FTSMG with a large $f_{θ}$ , the LOS rate $ω_{s}$ converges to the boundary layer much faster than that of NASMG. The ${\overset{\cdot}{q}}_{ε}$ and ${\overset{\cdot}{q}}_{β}$ are shown in Figure 29, which show the similar tendencies as the curves of $ω_{s}$ in Figure 28.

Figure 28.

3D LOS rate in case 5.

Figure 29.

LOS elevation and azimuth angular rates in case 5.

From Figure 30 it can be seen that the initial commanded acceleration of FTSMG is quite large and it converges to a constant value after $ω_{s}$ gets into the boundary layer. The commanded acceleration of NASMG is initially small. It first goes up and then decreases and becomes constant after the LOS rate gets into the boundary layer. The peak value of commanded acceleration of NASMG is much smaller than that of FTSMG in this case. The $a_{my}$ and $a_{mz}$ of both guidance laws in this case are shown in Figure 31, whose tendencies are similar as those of $| a_{m} |$ .

Figure 30.

Total missile commanded acceleration in case 5.

Figure 31.

Lateral missile commanded accelerations in case 5.

The estimation of the upper bound of the target acceleration of NASMG is shown in Figure 32. We can see that the variable structure term finally gets bigger than the upper bound of the target acceleration, which provides NASMG with the FTC feature.

Figure 32.

Estimation of the upper bound of the target acceleration and the absolute value of the variable structure term of NASMG in case 5.

The ZEMs of both guidance laws are shown in Figure 33. It can be seen that both ZEMs converge to the neighborhood of zero in finite time, and the convergence rate of ZEM of FTSMG is faster than that of NASMG.

Figure 33.

Zero effort miss in case 5.

It can be concluded from the above simulation results and discussions that the proposed NASMG can guarantee the asymptotic convergence of the LOS rate of the guidance system. If the guidance parameters are properly chosen, even the FTC feature of the guidance system can be guaranteed by NASMG. The performance of NASMG can be comparable to or even better than that of the previously proposed FTSMG, especially when the target adopts a small maneuvering acceleration while it actually has a large maneuverability. Besides, the prior knowledge of the target acceleration is no longer needed.

Through the above simulation and discussion, the advantage of NASMG over FTSMG has been demonstrated. In the following case, the performance of NASMG will be compared with that of the traditional PN guidance law when the maneuverability of the target is comparable to that of the missile.

Case 6

Assume that the target escapes with constant acceleration component $a_{t θ} = 20 m / s^{2}$ and it cannot be estimated. For PN, it has $a_{m θ} = - N \overset{\cdot}{r} ω_{s}$ and N = 3, 4, 5 is used. For NASMG, it is selected that k = 3, $\hat{f} (0) = 2$ , p = 1, and q = 5, and other parameters are the same as used in Case 1. The miss distances of PN with N = 3–5 are all larger than 1 m, while those of PN with N = 6 and NASMG is less than 0.01 m. The simulation results are shown in the following figures.

From Figure 34 it can be seen that, when N = 3–5, the LOS rate will go to infinite in the end of the guidance process. This causes the commanded acceleration of PN to go up beyond the maximum value of the acceleration that can be provided by the interceptor, as shown in Figures 35 and 36. PN with N = 6 can limit the LOS rate in a certain range; however, the LOS rate is generally increasing during the guidance process, and hence the commanded acceleration is also generally increasing, as shown in Figures 35 and 36. NASMG with properly chosen guidance parameters can guarantee the FTC of the LOS rate, and hence the commanded acceleration also converges to certain value during the guidance process. The curves of ${\overset{\cdot}{q}}_{ε}$ and ${\overset{\cdot}{q}}_{β}$ are shown in Figure 37, which show the similar tendencies as the curves of $ω_{s}$ in Figure 34.

Figure 34.

3D LOS rate in case 6.

Figure 35.

Total missile commanded acceleration in case 6.

Figure 36.

Lateral missile commanded accelerations in case 6.

Figure 37.

LOS elevation and azimuth angular rates in case 6.

The estimation of the upper bound of the target acceleration of NASMG is shown in Figure 38. We can see that the variable structure term gets bigger than the upper bound of the target acceleration around about 7 s, which provides NASMG with the FTC feature.

Figure 38.

Estimation of the upper bound of the target acceleration and the absolute value of the variable structure term of NASMG in case 6.

The ZEMs of both guidance laws are shown in Figure 39. It can be seen that PN with N = 3–5 cannot guarantee the convergence of ZEM to zero or its small neighborhood. PN with N = 6 can guarantee the convergence of ZEM; however, its convergence speed is slower than that of NASMG. The miss distances of both guidance laws are given in Table 2.

Figure 39.

Zero effort miss in case 6.

Tab. 2.

Miss distances of PN and NASMG.

Guidance law	Miss distance (m)
PN with N = 3	154.0610
PN with N = 4	58.8303
PN with N = 5	1.5577
PN with N = 6	<0.01
NASMG	<0.01

PN: proportional navigation; NASMG: novel adaptive sliding mode guidance law.

Conclusion

The NASMG proposed in this article is based on a novel tunable adaptive estimator of the upper bound of the target acceleration. The NASMG can be taken as an improvement of the previously proposed FTC sliding mode guidance law (FTSMG). NASMG can guarantee the asymptotic convergence of the LOS rate of the guidance system, while the prior knowledge of the upper bound of the target acceleration is no longer needed. The updating law of the estimator of the upper bound of the target acceleration is tunable, which makes NASMG more robust to uncertainties and disturbances of the guidance system. Second, the NASMG is extended to the 3D space with the help of the relative kinematic equations in the rotating LOS coordinate system. The realization method of the 3D NASMG in the traditional LOS coordinate system is also provided. Finally, a comparison between NASMG and FTSMG is conducted by using numerical simulations which indicates that the performance of NASMG is comparable or even better than that of FTSMG. The comparison of NASMG and PN with different navigation gains is also conducted, when the maneuverability of the target is comparable to that of the missile.

Footnotes

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was co-supported by the National Natural Science Foundation of China (Grant Nos. 61690210(2017-2021) and 61690213(2017-2021)) and the Hunan Provincial Natural Science Foundation of China (2019JJ50736).

ORCID iD

Ke-Bo Li

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Novel adaptive sliding mode guidance law against arbitrarily maneuvering target

Abstract

Keywords

Introduction

Planar novel adaptive sliding mode guidance law

Kinematics of planar engagement

Finite time stability of nonlinear systems

Definition 1

Lemma 1

Remark 1

Novel adaptive sliding mode guidance law

Theorem 1

Proof

Remark 2

Remark 3

Three-dimensional extension of NASMG

Extension of NASMG in 3D environment

Realization of 3D NASMG

Remark 4

Numerical simulation

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References