# Cooperative Guidance Law for High-Speed and High-Maneuverability Air Targets

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## Abstract

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## 1. Introduction

- To the best of the authors’ knowledge, none of the studies in the literature have considered both predictive and cooperative guidance laws for multiple interceptors together.
- The reachable set was constructed utilizing the likelihood of the predicted target state, and the likelihood values were obtained for a finite prediction horizon, not just one-step-ahead. Therefore, all positions in the reachable set had some likelihood value at each prediction step.
- The predicted target states were obtained statistically for a finite prediction horizon, and the cooperative guidance law generated the control input for the interceptors within the prediction horizon.
- The launch time of the interceptors was considered in the proposed guidance law.
- The nonlinear engagement geometry and equations of motion were used, avoiding errors caused by the linearization of the engagement kinematics and small heading angle assumptions.
- The proposed algorithms does not need target acceleration information.

## 2. Problem Statement

## 3. Calculation of Predicted Target States

#### 3.1. State Estimation of Target

#### 3.2. Methods for Calculation of Predicted Target States

#### 3.2.1. Case 1: Target’s Maneuverability Limits Are Unknown

#### 3.2.2. Case 2: Target’s Maneuverability Limits Are Known

## 4. Cooperative Guidance Algorithm

## 5. Simulation Study and Results

- The target exhibits a random maneuver twice with a maximum maneuvering acceleration in opposite directions, which is one of the most-effective maneuver types to survive.
- The maneuvering time instants are random variables with a uniform distribution.
- The target speed is 500 m/s and remains constant throughout the flight. The initial range is about 12 km. The magnitude and duration of acceleration are 15 g (where g is 9.81 m/s${}^{2}$) and 1 s.
- The locations of Interceptor 1 $\left(M1\right)$, Interceptor 2 $\left(M2\right)$, and Interceptor 3 $\left(M3\right)$ are (10,000, 500), (10,000, −500), and (11,200, 0), respectively.
- The time constant of the interceptor and target is 0.2. The maximum acceleration of the interceptors is 10 g, and their speed is 400 m/s, which remains constant throughout the flight. ${T}_{s}$, ${\tau}_{thr}$, and ${X}_{thr}$ are 0.1, 10, and 10, respectively.
- The IMM filter consist of 1 constant velocity model and 8 coordinated turn models with varying turn rates. The turn rates for the coordinated turn models were selected as ±0.15 rad/s, ±0.30 rad/s, ±0.45 rad/s, and ±0.60 rad/s.
- The standard deviation of the process noise and measurement noise are 0.1 m/s${}^{2}$ and 10 m for all motion models, respectively.
- The diagonal element of the transition probability matrix is 0.98, and the transition probability between models is 0.0025. The initial mode probabilities are the same for all modes and equal to $1/9$.
- The single-point track initiation algorithm (SP), explained in [40], was used to calculate the initial values of ${\widehat{x}}_{k|k}$ and $Pk|k$. This ensured the rapid convergence of the IMM filter. The SP algorithm was designed for the CV model and, thus, only provides the initial position and velocity estimations. It also requires a maximum target speed as prior information, which was assumed to be 1000 m/s.

#### 5.1. Case 1: Target Position ID Is I

#### 5.2. Case 2: Target Position ID Is 2

#### 5.3. Case 3: Target Position ID Is 3

#### 5.4. Case 4: Target Position ID Is 4

#### 5.5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Motion Models

#### Appendix A.1. Nearly Constant Velocity Model

#### Appendix A.2. Coordinated Turn with Known Turn Rate

## Appendix B. Single Step of IMM Filter

- (1)
- Mixing: Let $\Pi =[{\pi}_{ji}\cong P({r}_{k}=j|{r}_{k-1}=i)]$ be the transition probability between different mode states ${r}_{k}$.
- (a)
- Calculate the mixing probabilities ${\left\{{\mu}_{k-1|k-1}^{ji}\right\}}_{i,j=1}^{Nr}$ as:$${\mu}_{k-1|k-1}^{ji}={\displaystyle \frac{{\pi}_{ji}{\mu}_{k-1}^{j}}{{\sum}_{l=1}^{Nr}{\pi}_{li}{\mu}_{k-1}^{l}}}.$$
- (b)
- Calculate the mixed estimates ${\left\{{\widehat{x}}_{k-1|k-1}^{0i}\right\}}_{i=1}^{Nr}$ and covariances ${\left\{{P}_{k-1|k-1}^{0i}\right\}}_{i=1}^{Nr}$ as:$${\widehat{x}}_{k-1|k-1}^{0i}=\sum _{j=1}^{Nr}{\mu}_{k-1|k-1}^{ji}{\widehat{x}}_{k-1|k-1}^{j},$$$${P}_{k-1|k-1}^{0i}=\sum _{j=1}^{Nr}{\mu}_{k-1|k-1}^{ji}\left[{P}_{k-1|k-1}^{j}\right.+\left({\widehat{x}}_{k-1|k-1}^{j}-{\widehat{x}}_{k-1|k-1}^{0i}\right)\left.{\left({\widehat{x}}_{k-1|k-1}^{j}-{\widehat{x}}_{k-1|k-1}^{0i}\right)}^{T}\right].$$

- (2)
- Mode-matched prediction update: For the i-th model, $i=1,...,{N}_{r}$, calculate the predicted estimate ${\widehat{x}}_{k|k-1}^{i}$ and covariance ${P}_{k|k-1}^{i}$ from the mixed estimate ${\widehat{x}}_{k-1|k-1}^{0i}$ and covariance ${P}_{k-1|k-1}^{0i}$ as:$${\widehat{x}}_{k|k-1}^{i}=A\left(i\right){\widehat{x}}_{k-1|k-1}^{0i},$$$${P}_{k|k-1}^{i}=A\left(i\right){P}_{k-1|k-1}^{0i}{A}^{T}\left(i\right)+Q\left(i\right).$$
- (3)
- Mode-matched measurement update: For the i-th model, $i=1,...,{N}_{r}$:
- (a)
- Calculate the updated estimate ${\widehat{x}}_{k|k}^{i}$ and covariance ${P}_{k|k}^{i}$ from the predicted estimate ${\widehat{x}}_{k|k-1}^{i}$ and covariance ${P}_{k|k-1}^{i}$ as:$${\widehat{z}}_{k|k-1}^{i}=C\left(i\right){\widehat{x}}_{k|k-1}^{i},$$$${\tilde{y}}_{k|k-1}^{i}={z}_{k}-{\widehat{z}}_{k|k-1}^{i},$$$${S}_{k|k-1}^{i}=C\left(i\right){P}_{k|k-1}^{i}{C}^{T}\left(i\right)+R\left(i\right),$$$${K}_{k}^{i}={P}_{k|k-1}^{i}{C}^{T}\left(i\right){\left({S}_{k|k-1}^{i}\right)}^{-1},$$$${\widehat{x}}_{k|k}^{i}={\widehat{x}}_{k|k-1}^{i}+{K}_{k}^{i}{\tilde{y}}_{k|k-1}^{i},$$$${P}_{k|k}^{i}={P}_{k|k-1}^{i}-{K}_{k}^{i}{S}_{k|k-1}^{i}{{K}_{k}^{i}}^{T}.$$
- (b)
- Calculate the likelihood ${\Lambda}_{k}^{i}$ and the updated mode probability ${\mu}_{k}^{i}$ as:$${\Lambda}_{k}^{i}=\mathcal{N}\left({z}_{k};{\widehat{z}}_{k|k-1}^{i},{S}_{k|k-1}^{i}\right),$$$${\mu}_{k}^{i}={\displaystyle \frac{{\Lambda}_{k}^{i}{\sum}_{j=1}^{Nr}{\pi}_{ji}{\mu}_{k-1}^{j}}{{\sum}_{l=1}^{Nr}{\Lambda}_{k}^{l}{\sum}_{j=1}^{Nr}{\pi}_{jl}{\mu}_{k-1}^{j}}}.$$

- (4)
- Output estimate calculation: Calculate the overall estimate ${\widehat{x}}_{k|k}$ and covariance ${P}_{k|k}$ as:$${\widehat{x}}_{k|k}=\sum _{i=1}^{Nr}{\mu}_{k}^{i}{\widehat{x}}_{k|k}^{i},$$$${P}_{k|k}=\sum _{i=1}^{Nr}{\mu}_{k}^{i}\left[{P}_{k|k}^{i}\right.\left.+\left({\widehat{x}}_{k|k}^{i}-{\widehat{x}}_{k|k}\right){\left({\widehat{x}}_{k|k}^{i}-{\widehat{x}}_{k|k}\right)}^{T}\right].$$

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**Figure 2.**Illustration of reachable sets, corresponding likelihood, and pdf of predicted target state when time instants (

**a**) $k+{t}_{1}$, (

**b**) $k+{t}_{2}$, and (

**c**) $k+{t}_{3}$.

**Figure 8.**An example of the likelihood of reachable sets where ${t}_{\alpha}=7$, MC Run No. 71. (

**a**) Method 1, (

**b**) Method 2.

**Figure 9.**An example of trajectories where ${t}_{\alpha}=7$, ${v}_{1}={v}_{2}=400$ m/s, MC Run No. 71. (

**a**) Method 1, (

**b**) Method 2.

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**MDPI and ACS Style**

Cevher, F.Y.; Leblebicioğlu, M.K.
Cooperative Guidance Law for High-Speed and High-Maneuverability Air Targets. *Aerospace* **2023**, *10*, 155.
https://doi.org/10.3390/aerospace10020155

**AMA Style**

Cevher FY, Leblebicioğlu MK.
Cooperative Guidance Law for High-Speed and High-Maneuverability Air Targets. *Aerospace*. 2023; 10(2):155.
https://doi.org/10.3390/aerospace10020155

**Chicago/Turabian Style**

Cevher, Fırat Yılmaz, and Mehmet Kemal Leblebicioğlu.
2023. "Cooperative Guidance Law for High-Speed and High-Maneuverability Air Targets" *Aerospace* 10, no. 2: 155.
https://doi.org/10.3390/aerospace10020155