# Modeling the Solution of the Pursuit–Evasion Problem Based on the Intelligent–Geometric Control Theory

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Related Works

- Integration of both optimal and heuristic algorithms within a single concept to improve decision efficiency under conditions of uncertainty;
- Creation of a comprehensive approach, including the development of behavioral strategies, mathematical models of players, and simulation under disturbances;
- Performing adequate linearization of differential equations describing the dynamics of UAVs and introducing other simplifications that do not lead to a significant loss of accuracy of the resulting solutions.

#### 1.3. Main Contributions

- The article describes the elements of the theory of intelligent–geometric control in relation to pursuit–evasion problems;
- Some game strategies have been developed for both the pursuer and the evader based on the solution of optimal problems and the application of heuristic rules;
- A model for predicting the movement of the evader is proposed, which expands the capabilities of the pursuer;
- Simulation of some game scenarios in a disturbed environment was carried out with the developed approaches.
- Subsequent sections are organized as follows:
- Section 2 contains problem statements for both the pursuer and the evader.
- Section 3 presents the architecture of intelligent–geometric control theory (Section 3.1), solutions to the optimization problem of the closest approach (Section 3.2 and Section 3.3), and heuristic strategies for players (Section 3.4). The problem of predicting the trajectory of the evader is considered in Section 3.5. The problem of moving along the required route given by waypoints is briefly described in Section 3.6.
- Section 4 is devoted to testing the proposed solutions. We first considered the planar case of the pursuit–evasion game in Section 4.1. Section 4.2 then discusses the dynamic model of an aircraft-type UAV and its linearization. Section 4.3 describes schemes for modeling the game in a disturbed environment for the spatial case. Simulation using the developed approach is carried out in Section 4.4.
- The final section, Section 5, contains the main conclusions and prospects for further research.

## 2. Problem Statement

#### 2.1. General Pursuit and Evasion Problems

**Evasion**

**problem.**

_{g}] under perturbations and constraints (2), (4) is to construct a control function ${u}_{e}\left(t\right)={(v}_{e}\left(t\right),{\theta}_{e}\left(t\right),{\psi}_{e}\left(t\right))$ such that:

**Pursuit**

**problem.**

#### 2.2. Optimal Convergence Problem

**Optimal Convergence**

**Problem.**

## 3. Materials and Methods

#### 3.1. Intelligent–Geometric Control Architecture

#### 3.2. Calculation of the Intercept Point

#### 3.3. Solution of the Optimal Convergence Problem

#### 3.4. Pursuit and Evasion Strategies

#### 3.4.1. Evader’s Strategy

- If there are rays that do not intersect any of the Apollonius spheres, then choose the closest ray ${F}_{esc}$ to the direction ${F}_{g}$;
- If ${F}_{esc}$ exists, then the evader should move in this direction;
- If the ray ${F}_{esc}$ cannot be calculated, then we suggest that the evader should move in the direction determined by vector ${F}_{res}$ (22).

#### 3.4.2. Pursuer’s Strategy

- If the evader is located in a closed region bounded by the spheres of Apollonius, it fails to escape;
- If it is possible to construct a half-line that does not intersect any of the spheres, then the evader manages to escape;
- In more complex cases—for instance, when the evader has a target point—to solve the problem, it is necessary to perform an accurate simulation of the game.

- If the evader’s trajectory intersects with the corresponding sphere, then the pursuer must move to the point of convergence, which is calculated by Formulas (11) and (12);
- If the evader moves towards the Apollonius sphere but the exact intersection point cannot be calculated, it is necessary to move in the direction of the maximum approach, which is determined by Formulas (19) and (20);
- If the evader is moving away from the Apollonius sphere, then the pursuer must fly parallel to the evader by setting the appropriate direction. An alternative option is to build a forecast of the evader’s movement $k$ steps ahead, after which the pursuer ${p}_{i}$ begins to move to the calculated point $\left({x}_{e}\left({t}_{n+k}\right),{y}_{e}\left({t}_{n+k}\right),{z}_{e}\left({t}_{n+k}\right)\right)$, where ${t}_{n}$ is the current point in time and $n$ is the number of observed waypoints.

#### 3.5. Neural Network Model for Predicting Evader’s Trajectory

- for $x>0$:

- for $x<0$:

#### 3.5.1. Algorithm for Training a Two-Layer ANN with the s-Parabola Activation Function

- Initialization of initial parameters and weights.
- Calculation of a neural network in the forward direction:
- Calculation of the first-layer signals:$${s}_{j}^{1}={w}_{0j}^{1}+\sum _{i=1}^{n}{x}_{i}{w}_{ij}^{1},\hspace{1em}{y}_{j}^{1}=f({s}_{j}^{1})=\left\{\begin{array}{c}\beta +\sqrt{2p{s}_{j}^{1}},\hspace{1em}\hspace{1em}if{s}_{j}^{1}0\\ \beta -\sqrt{-2p{s}_{j}^{1}},\hspace{1em}\hspace{1em}if{s}_{j}^{1}0\end{array}\right.$$
- Calculation of signals of the second (output) layer:$${s}_{1}^{2}={w}_{01}^{2}+\sum _{i=1}^{m}{y}_{i}^{1}{w}_{i1}^{2},\hspace{1em}{y}_{1}^{2}=f({s}_{1}^{2})=\left\{\begin{array}{c}\beta +\sqrt{2p{s}_{1}^{2}},\hspace{1em}\hspace{1em}if{s}_{1}^{2}0\\ \beta -\sqrt{-2p{s}_{1}^{2}},\hspace{1em}\hspace{1em}if{s}_{1}^{2}0\end{array}\right.$$

- Calculation of errors in inputs and outputs of neurons in the backward direction.
- Errors at the output and input of the last layer neuron.Output error: ${\delta}_{out}^{2}={y}_{g}^{2}-{y}_{out}$, where ${y}_{out}$ is the current value of the neuron output. The required value ${y}_{g}^{2}$ is set by the user. Error at neuron input: ${\delta}_{in}^{2}={s}_{g}^{2}-{s}^{2}$, where ${s}_{g}^{2}$ is the given input value of the second-layer neuron:$${s}_{g}^{2}=\frac{({y}_{g}^{2}-\beta {)}^{2}}{2p},if{s}_{1}^{2}0,{s}_{g}^{2}=-\frac{({y}_{g}^{2}-\beta {)}^{2}}{2p},if{s}_{1}^{2}0$$
- Errors at the output and input of the $i$th neuron of the first layer containing $m$ neurons.Errors in neuron outputs: ${\delta}_{iout}^{1}={\delta}_{in}^{2}{w}_{i1}^{2}/{K}^{2},$ where $i=1,\dots ,m,{K}^{2}=\left|{w}_{01}^{2}\right|+{\sum}_{i=1}^{m}\left|{w}_{i1}^{2}\right|$ is the normalizing coefficient. On the other hand, ${\delta}_{iout}^{1}={y}_{ig}^{1}-{y}_{i}^{1}$, where ${y}_{ig}^{1}$, ${y}_{i}^{1}$ are the given and current output values of the $i$th neuron of the first layer.Calculation of input signals for the activation function:$${s}_{ig}^{1}=\frac{({y}_{ig}^{1}-\beta {)}^{2}}{2p},if{s}_{1}^{1}0,{s}_{ig}^{1}=-\frac{({y}_{ig}^{1}-\beta {)}^{2}}{2p},if{s}_{1}^{1}0$$Error at the input of the $i$th neuron of the first layer: ${\delta}_{i,in}^{1}={s}_{ig}^{1}-{s}_{i}^{1}$.

- Correction of the neural network weights, which is carried out as follows:$${w}_{ij}^{1}\left(k+1\right)={w}_{ij}^{1}\left(k\right)+\frac{\eta {\delta}_{j,in}^{1}{x}_{i}}{{K}_{j}^{1}},{w}_{0j}^{1}\left(k+1\right)={w}_{0j}^{1}\left(k\right)+\frac{\eta {\delta}_{j,in}^{1}}{{K}_{j}^{1}},\phantom{\rule{0ex}{0ex}}{w}_{ij}^{2}\left(k+1\right)={w}_{ij}^{2}\left(k\right)+\frac{\eta {\delta}_{in}^{2}{y}_{i}^{1}}{{K}^{2}},{w}_{01}^{2}\left(k+1\right)={w}_{01}^{2}\left(k\right)+\frac{\eta {\delta}_{in}^{2}}{{K}^{2}},$$
- If the error at the output ${\delta}_{out}^{2}=\left|{y}_{g}^{2}-{y}_{out}\right|$ is less than the predetermined value, then a stop is performed. Otherwise, the learning rate is reduced by a certain amount, $\eta :=\eta -\Delta $, and the transition to step 2 is carried out.

#### 3.5.2. An Example of Predicting the Planar Trajectory of an Evader

#### 3.6. Trajectory-Tracking Problem

**Strategy**

**1.**

**Strategy**

**2.**

## 4. Results

#### 4.1. Planar Case

#### 4.2. UAV Dynamics Model

- a, b, and c with various subscripts are the parameters of longitudinal dynamics depending on the UAV type and flight mode;
- k, l, and n with various subscripts are the parameters of lateral dynamics depending on the UAV type and flight mode;
- $\theta $, $\psi $, and $\gamma $ are the pitch, yaw, and roll angles;
- ${\delta}_{e}$ is the elevator angle;
- ${\delta}_{a}$, ${\delta}_{r}$ are deflection angles of the ailerons and rudder;
- ${w}_{x},{w}_{y}$, and ${w}_{z}$ are the projections of wind velocities on the ${X}_{B},{Y}_{B}$, and ${Z}_{B}$ axes of the base coordinate system;
- ${V}_{b}$ is the UAV speed relative to air in the unperturbed mode;
- ${V}_{gx},{V}_{gy}$ are the projections of the UAV ground speed on the axes ${X}_{v}$ and ${Y}_{v}$;
- ${x}_{b},{y}_{b}$, and ${z}_{b}$ are the coordinates of the center of mass on the ${X}_{B},{Y}_{B}$, and ${Z}_{B}$ axes;
- ${\omega}_{gx},{\omega}_{gy}$, and ${\omega}_{gz}$ are the projections of the angular speed relative to air on the ${X}_{v},{Y}_{v},and{Z}_{v}$ axes;
- $k=\frac{1}{57.3}$ is the correcting coefficient;
- $s$ is the Laplace transform parameter.

- The lateral and longitudinal motions are independent;
- The wind speed is significantly less than the speed of the UAV;
- The horizontal rectilinear flight under no wind conditions without roll and sliding is taken as the unperturbed mode.

#### 4.3. Matlab Simulation Scheme

#### 4.4. Spatial Case

## 5. Discussion

- For both the pursuers and the evader, we propose simple but effective heuristic rules that imitate the reasonable actions of a human operator in accordance with precise geometric calculations and the modified potential field approach.
- The problem of predicting the movement of an evader using a two-layer, fully connected feed-forward network is considered. A distinctive feature of the model is the use of a special activation function, which reduces the calculation time in conditions of limited computing power and the need to take prompt action.
- The solution of the trajectory tracking problem based on the principles of functioning of intelligent dynamic systems is discussed.
- A scheme for simulating pursuit–evasion games is proposed and studied that takes into account the dynamic models of participants and wind disturbances. A series of simulations conducted in MATLAB/Simulink environment demonstrates that the proposed strategies determine the natural behavior of dynamic objects under uncertainty.

- Pursuers can be considered as intelligent agents, endowed with the ability to exchange information and make collective decisions. This approach determines the need to conduct research at the strategic level, related to issues of goal setting and distribution of tasks between participants.
- The problem statement can be expanded, for example, by considering scenarios with the low-speed agile evader or where the pursuers know the evader’s target point.
- Further improvement of the intelligent component is seen in the introduction of a knowledge representation model for describing the workspace in the form of semantic networks and the creation of appropriate decision-making procedures, increasing the intelligence of the participants themselves through the development of goal-setting approaches. This, of course, will require the creation of new strategies and algorithms.
- Finally, natural experiments are needed to demonstrate in practice the effectiveness of the developed intelligent–geometric control algorithms.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol Name | Definition |

${X}_{B},{Y}_{B},{Z}_{B}$ | Axes of the base coordinate system |

$X,Y,Z$ | Axes of the coordinate system fixed to the vehicle, and parallel to axes ${X}_{B},{Y}_{B}$, and ${Z}_{B}$. |

${X}_{v},{Y}_{v},{Z}_{v}$ | Axes of the coordinate system fixed to the vehicle (the axis ${X}_{v}$ is directed along the longitudinal axis, the axis ${Y}_{v}$ is directed upward, and the axis ${Z}_{v}$ is directed rightward). |

$e$ | Evader. |

${p}_{i}$ | $i$th pursuer in group $P$. |

${(x}_{e},{y}_{e},{z}_{e})$ | Coordinates of the evader. |

${(x}_{p,i},{y}_{p,i},{z}_{p,i})$ | Coordinates of the pursuer ${p}_{i}$. |

${v}_{e}$ | Speed of the evader. |

${v}_{p,i}$ | Speed of the pursuer ${p}_{i}$. |

${\theta}_{e},{\psi}_{e}$ | Pitch and yaw angles of the evader. |

${\theta}_{p,i},{\psi}_{p,i}$ | Pitch and yaw angles of the pursuer ${p}_{i}$. |

$g({x}_{g},{y}_{g},{z}_{g})$ | Target point of the evader. |

${T}_{g}$ | Time moment when the evader reaches its target. |

${T}_{c}$ | Time moment when at least one of the pursuers captures the evader. |

${\epsilon}_{min}$ | Required distance between the evader and its target point. |

$R$ | Radius of the geometric model of a UAV. |

$\theta ,\psi ,\gamma $ | Pitch, yaw, and roll angles. |

${Q}_{p,i}$ | State of pursuer ${p}_{i}$ (coordinates and velocity). |

${Q}_{e}$ | State of the evader (coordinates and velocity). |

$d({o}_{1},{o}_{2})$ | Distance between objects ${o}_{1},{o}_{2}$. |

${F}_{i}$ | Unit vector that starts from evader’s location point $e$ and coincides with the direction from ${p}_{i}$ to $e$. |

${F}_{g}$ | Unit vector that starts from evader’s location point $e$ and coincides with the direction from $e$ to $g$. |

${F}_{esc}$ | Ray that does not intersect any of the Apollonius spheres. |

$a,b$,$c$ | Parameters of longitudinal dynamics. |

$k,l,n$ | Parameters of lateral dynamics. |

${\delta}_{e}$ | Elevator angle. |

${\delta}_{a}$, ${\delta}_{r}$ | Deflection angles of the ailerons and rudder. |

${x}_{b},{y}_{b},{z}_{b}$ | Coordinates of the center of mass on the ${X}_{B},{Y}_{B}$,${Z}_{B}$ axes. |

$V$ | UAV airspeed. |

${V}_{b}$ | UAV speed relative to air in the unperturbed mode. |

${V}_{gx},{V}_{gy}$ | Projections of the UAV ground speed on the axis $X,Y$. |

${\omega}_{gx},{\omega}_{gy},{\omega}_{gz}$ | Projections of the angular speed on the $X,Y,Z$ axes. |

$s$ | Laplace transform parameter. |

${W}_{{\omega}_{gz}/\Delta {\delta}_{E}}$ | Transfer function from the control action of the elevator $\Delta {\delta}_{e}$ to the pitch angular velocity ${\omega}_{gz}$. |

${W}_{{\omega}_{gz}/{w}_{x}},{W}_{{\omega}_{gz}/{w}_{y}}$ | Transfer functions from the wind components ${w}_{x}$ and ${w}_{y}$ to the pitch angular velocity ${\omega}_{gz}$. |

${W}_{{\omega}_{gy}/{\Delta \delta}_{r}}$ | Transfer function from the control action of the rudder $\Delta {\delta}_{r}$ to the yaw angular velocity ${\omega}_{gy}$. |

${W}_{{\omega}_{gy}/{w}_{z}}$ | Transfer function from the wind component ${w}_{z}$ to the yaw angular velocity ${\omega}_{gy}$. |

${t}_{j}^{\left(e\right)}$ | Reference time moment corresponding to the passage of the point ${E}_{j}$ by the UAV. |

${t}_{j}$ | Moment of closest approach of the UAV to the waypoint ${E}_{j}$. |

${E}_{j}$ | Intermediate reference waypoint of the route. |

$e\left({t}_{j}\right)$ | Coordinates of the UAV at time ${t}_{j}$. |

$c$ | Pseudo-target that simulates an ideal flight path. |

β | Displacement of the parabola along the $OY$ axis. |

${s}_{j}^{1},{s}_{j}^{2}$ | Sum of weighted inputs on the $j$th neuron of the first/second layer. |

${y}_{j}^{1},{y}_{j}^{2}$ | Output signal of the $j$th neuron of the first/second layer. |

${w}_{ij}^{1}$ | Weight of the connection between the $i$th input value and the $j$th neuron of the first layer. |

${y}_{out}$ | Neural network output. |

${\delta}_{out}^{2},{\delta}_{in}^{2}$ | Error at the output/input of the second layer. |

${\delta}_{j,out}^{1},{\delta}_{j,in}^{1}$ | Error at the output/input of the $j$th neuron of the first layer. |

$g$ | Subscript $g$ indicates the target values of the corresponding parameters $s,y$. |

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**Figure 2.**Pursuit of an evader by a group of dynamic objects: (

**a**) The evader’s movement is goal-directed; (

**b**) There is no target point.

**Figure 3.**Examples of s-shaped curves based on parabola branches for: (

**a**) $p=\frac{1}{4}$; (

**b**) $p=\frac{1}{20}$.

**Figure 4.**The result of predicting the movement of the evader using a neural network with an activation function of the “s-parabola” type.

**Figure 5.**Scenario in which an evader manages to escape but does not reach the target point: (

**a**) The initial stage; (

**b**) The final stage.

**Figure 6.**Scenario in which an evader manages both to escape and reach the target point: (

**a**) The initial stage; (

**b**) The final stage.

**Figure 7.**Scenario in which an evader is surrounded by all pursuers: (

**a**) The initial stage; (

**b**) The final stage.

**Figure 8.**General intelligent–geometric schemes for controlling the motion of dynamic objects: (

**a**) For an evader; (

**b**) For a pursuer.

**Figure 10.**Motion trajectories of the participants (for one evader and five pursuers): (

**a**) At the initial stage; (

**b**) At the final stage.

**Figure 11.**Motion trajectories of the participants (for one evader and six pursuers): (

**a**) At the initial stage; (

**b**) At the final stage.

No | Evader | Pursuer 1 | Pursuer 2 | Pursuer 3 | Pursuer 4 | Pursuer 5 | Target | Evader Escapes | Evader Reaches the Target |
---|---|---|---|---|---|---|---|---|---|

1 | (0, 0), 5 | (20, 30), 4 | (−20, 20), 4 | (−20, −20), 4 | (100, 20), 4 | (−20, −90), 4 | (−5, −100) | Yes/Yes | No |

2 | (0, 0), 5 | (30, 20), 4 | (−30, −50), 4 | (−70, 40), 3 | (100, −20), 4 | (100, −90), 4 | (100, 150) | Yes/Yes | Yes |

3 | (0, 0), 5 | (30, 30), 4 | (−30, 30), 4 | (−30, −30), 4 | (200, −20), 4 | (50, −190), 4 | - | No/Yes | - |

4 | (0, 0), 5 | (20, 20), 4 | (−20, 20), 4 | (−20, −20), 4 | (100, −20), 4 | (100, −90), 4 | (−50, −50) | Yes/Yes | No |

5 | (0, 0), 5 | (20, 20), 4 | (−20, 20), 4 | (−20, −20), 4 | (100, −20), 4 | (100, −90), 4 | (−150, −150) | Yes/Yes | Yes |

6 | (0, 0), 5 | (30, 20), 4 | (−30, 20), 4 | (−30, −20), 3 | (100, −20), 4 | (100, −90), 4 | (−100, −150) | Yes/Yes | Yes |

7 | (0, 0), 5 | (30, 30), 3 | (−30, 30), 3 | (−30, −30), 3 | (200, −20), 4 | (50, −190), 4 | - | Yes/Yes | - |

8 | (0, 0), 5 | (20, 20), 4 | (−20, 20), 4 | (−20, −20), 4 | (100, 0), 4 | (0, −90), 4 | - | No/Yes | - |

9 | (0, 0), 5 | (20, 20), 3 | (−20, 20), 3 | (−20, −20), 4 | (100, −10), 4 | (−40, −100), 4 | - | Yes/Yes | - |

10 | (0, 0), 5 | (20, 20), 4 | (−20, 20), 4 | (−20, −20), 4 | (50, 0), 4 | (0, −50), 4 | - | No/No | - |

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**MDPI and ACS Style**

Khachumov, M.; Khachumov, V.
Modeling the Solution of the Pursuit–Evasion Problem Based on the Intelligent–Geometric Control Theory. *Mathematics* **2023**, *11*, 4869.
https://doi.org/10.3390/math11234869

**AMA Style**

Khachumov M, Khachumov V.
Modeling the Solution of the Pursuit–Evasion Problem Based on the Intelligent–Geometric Control Theory. *Mathematics*. 2023; 11(23):4869.
https://doi.org/10.3390/math11234869

**Chicago/Turabian Style**

Khachumov, Mikhail, and Vyacheslav Khachumov.
2023. "Modeling the Solution of the Pursuit–Evasion Problem Based on the Intelligent–Geometric Control Theory" *Mathematics* 11, no. 23: 4869.
https://doi.org/10.3390/math11234869