# Multiple Constraints-Based Adaptive Three-Dimensional Back-Stepping Sliding Mode Guidance Law against a Maneuvering Target

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Aiming at the maneuvering targets, a complex three-dimensional guidance model with highly coupling pitch plane and yaw plane dynamics is constructed while considering the motion information of targets as external disturbances. It is highlighted that the proposed method is more practical than previous studies on non-maneuvering targets and two-dimensional engagement cases.
- (2)
- An adaptive back-stepping sliding mode controller (BSMC) is proposed in order to compensate for the interference term caused by the maneuvering target. The design of an adaptive law to estimate and compensate for external disturbances effectively improves the universality for varieties of target motion. The proposed controller is nonlinear without a small angle hypothesis, which is more accurate than that with linearization.
- (3)
- Using the Lyapunov stability analysis method, we have integrated FOV limitation and the LOS constraint (represents TIA constraint) into the sliding mode control method. The FOV limitation represented by the overall leading angle is decoupled into two partial leading angles in pitch and yaw directions. Accordingly, Lyapunov functions consisting of partial leading angle and LOS angle error terms are separately derived while maintaining the stability of both subsystems, while the partial leading angle constraints are guaranteed by a specific transformation method.

## 2. Problem Formulation and Guidance Models Construction

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Remark**

**1.**

## 3. Multiple Constraints Based Adaptive BMSC Guidance Law

**D**denotes the external disturbances of the dynamics system, including motion information of the target and modeling uncertainty ($\Delta {D}_{1}$, $\Delta {D}_{2}$). In this paper, the first and second derivatives of the expected LOS angle is set to 0.

**Assumption**

**4.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

#### 3.1. Guidance Law Design in Pitch Plane

**Remark**

**6.**

**Remark**

**7.**

**Remark**

**8.**

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

**Theorem 1**, we have ${e}_{1}\to 0$, ${e}_{2}\to 0$, ${\sigma}_{1}\to 0$, and ${e}_{3}\to 0$. Therefore, it is concluded as follows:

**Theorem 1**. □

#### 3.2. Guidance Law Design in Yaw Plane

**Theorem**

**2.**

**Proof**

**of Theorem 2.**

**Theorem 2**, we have that ${e}_{4}\to 0$, ${e}_{5}\to 0$, ${\sigma}_{2}\to 0$, and ${e}_{6}\to 0$. Therefore, it is concluded as follows:

**Theorem 2**. □

## 4. Numerical Simulation Analysis

#### 4.1. Simulations for Constant Maneuvering Targets

- (1)
- Case 1: ${\theta}_{LF}=-{60}^{\circ},\phantom{\rule{4pt}{0ex}}{\varphi}_{LF}={45}^{\circ}$;
- (2)
- Case 2: ${\theta}_{LF}=-{60}^{\circ},\phantom{\rule{4pt}{0ex}}{\varphi}_{LF}={60}^{\circ}$;
- (3)
- Case 3: ${\theta}_{LF}=-{75}^{\circ},\phantom{\rule{4pt}{0ex}}{\varphi}_{LF}={90}^{\circ}$.

#### 4.2. Simulations for Variable Maneuvering Targets

- (1)
- Case 1: ${\theta}_{LF}=-{60}^{\circ},\phantom{\rule{4pt}{0ex}}{\varphi}_{LF}={45}^{\circ}$
- (2)
- Case 2: ${\theta}_{LF}=-{60}^{\circ},\phantom{\rule{4pt}{0ex}}{\varphi}_{LF}={90}^{\circ}$
- (3)
- Case 3: ${\theta}_{LF}=-{75}^{\circ},\phantom{\rule{4pt}{0ex}}{\varphi}_{LF}={90}^{\circ}$

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Variations of LOS angles against constant maneuvering targets: (

**a**) LOS angles in pitch plane; (

**b**) LOS angles in yaw plane.

**Figure 3.**Variations of LOS rates against constant maneuvering targets: (

**a**) LOS rates in pitch plane; (

**b**) LOS rates in yaw plane.

**Figure 4.**Variations of acceleration commands against constant maneuvering targets: (

**a**) acceleration commands in pitch plane; (

**b**) acceleration commands in yaw plane.

**Figure 5.**Variations of estimation errors against constant maneuvering targets: (

**a**) estimation errors in pitch plane; (

**b**) estimation errors in yaw plane.

**Figure 6.**Variations of partial leading angles against constant maneuvering targets: (

**a**) partial leading angles in pitch plane; (

**b**) partial leading angles in yaw plane.

**Figure 9.**Variations of LOS angles against variable maneuvering targets: (

**a**) LOS angles in pitch plane; (

**b**) LOS angles in yaw plane.

**Figure 10.**Variations of LOS rates against variable maneuvering targets: (

**a**) LOS rates in pitch plane; (

**b**) LOS rates in yaw plane.

**Figure 11.**Variations of acceleration commands against variable maneuvering targets: (

**a**) acceleration commands in pitch plane; (

**b**) acceleration commands in yaw plane.

**Figure 12.**Variations of estimation errors against variable maneuvering targets: (

**a**) estimation errors in pitch plane; (

**b**) estimation errors in yaw plane.

**Figure 13.**Variations of partial leading angles against variable maneuvering targets: (

**a**) partial leading angles in pitch plane; (

**b**) partial leading angles in yaw plane.

Case | Pitch LOS Angle Error $\Delta {\mathit{\theta}}_{\mathit{L}}$/° | Yaw LOS Angle Error $\Delta {\mathit{\varphi}}_{\mathit{L}}$/° | Maximum Leading Angle ${\mathit{\sigma}}_{\mathbf{Mmax}}$/° |
---|---|---|---|

Case 1 | 4.83 × 10${}^{-4}$ | 2.61 × 10${}^{-4}$ | 73.79 |

Case 2 | 1.24 × 10${}^{-3}$ | −1.04 × 10${}^{-4}$ | 73.96 |

Case 3 | 4.80 × 10${}^{-4}$ | 5.36 × 10${}^{-3}$ | 70.94 |

Case | Pitch LOS Angle Error $\Delta {\mathit{\theta}}_{\mathit{L}}$/° | Yaw LOS Angle Error $\Delta {\mathit{\varphi}}_{\mathit{L}}$/° | Maximum Leading Angle ${\mathit{\sigma}}_{\mathbf{Mmax}}$/° |
---|---|---|---|

Case 1 | 2.69 × 10${}^{-3}$ | 1.30 × 10${}^{-4}$ | 72.61 |

Case 2 | 7.84 × 10${}^{-3}$ | 1.43 × 10${}^{-4}$ | 71.15 |

Case 3 | 2.07 × 10${}^{-2}$ | −6.06 × 10${}^{-3}$ | 70.42 |

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**MDPI and ACS Style**

Shi, Q.; Wang, H.; Cheng, H.
Multiple Constraints-Based Adaptive Three-Dimensional Back-Stepping Sliding Mode Guidance Law against a Maneuvering Target. *Aerospace* **2022**, *9*, 796.
https://doi.org/10.3390/aerospace9120796

**AMA Style**

Shi Q, Wang H, Cheng H.
Multiple Constraints-Based Adaptive Three-Dimensional Back-Stepping Sliding Mode Guidance Law against a Maneuvering Target. *Aerospace*. 2022; 9(12):796.
https://doi.org/10.3390/aerospace9120796

**Chicago/Turabian Style**

Shi, Qingli, Hua Wang, and Hao Cheng.
2022. "Multiple Constraints-Based Adaptive Three-Dimensional Back-Stepping Sliding Mode Guidance Law against a Maneuvering Target" *Aerospace* 9, no. 12: 796.
https://doi.org/10.3390/aerospace9120796