A new mathematical model and algorithm for the optimal path to intercept a moving target

Path planning for intercepting a moving target has been a significant challenge, demanding efficient strategies for path planning to ensure successful interception. This has applications in various domains such as robotics, aerospace, and agricultural applications [1]-[9]. The motivation behind the study of pursuer-target interception lies in real-world scenarios where one object needs to intersect with another. Researchers are exploring advanced algorithms, mathematical models, and artificial intelligence methods to enhance the accuracy and speed of interception. Our study focus deeply into the essential realm of pursuer-target interception path planning. By exploring innovative mathematical models and employing advanced computational techniques, our aim is to contribute significantly in this area. We propose efficient ways to minimize the pursuer’s path when intercepting a moving target. Our research endeavors not only towards theoretical advancements but also practical solutions that can be universally applied, addressing the evolving challenges associated with intercepting a moving target in various domains.

Dubins [10] introduced the shortest path for vehicles moving only forward and turning in unidirectional turns (either left or right) from specific starting to ending positions in the 2D plane. The author described paths as combinations of curves ($C$) and straight ($S$) lines, where the vehicle starts and ends with a turn, and in the middle, it could be straight or curved. There are six possible combinations of these paths, represented by $D=\{LSL,RSL,RSR,LSR,LRL,RLR\}$, where L and R mean left and right turns, respectively. Reeds and Shepp [11] later expanded on Dubins work, allowing backward adaptability in these paths. Dubins results have since been further developed, incorporating geometric principles [12] and leveraging advanced theories in nonlinear optimal control [13]-[20]. Specifically, Pontryagin’s Maximum Principle (PMP) [21] has been applied, enhancing the understanding and applications of Dubins paths. These developments have expanded the scope of Dubins work, making it a pivotal foundation in the study of optimal vehicle paths.

Numerous researchers have also been made an effort to develop a model to figure out the shortest distance for intercepting a moving target. Boissonnat and Bui’s study [14] on the Relaxed Dubins Problem (RDP) focuses on finding the shortest Dubins path from a given configuration to a stationary target. Dubin’s path problem from a fixed initial configuration to intercept a target have been extensively researched in several fields [22, 23]. When intercepting a moving target, researchers have also explored Dubins and relaxed Dubins optimal pathways [24]-[27]. Looker [28] proposes a search algorithm to determine the shortest $CS$ type path for interception, assuming the pursuer behaves like a Dubins vehicle and the target moves at a constant speed. Meyer et al. [29] analyze problems related to minimum-time interception and interception at a predetermined time for a target moving in a straight line. Additionally, Meyer et al. [30] address finding the shortest path for a Dubins vehicle without a terminal angle constraint, referred to as the ‘relaxed Dubins’ problem. In our research, we explore similar scenarios and attempt to reformulate various mathematical models for intercepting a moving target.

Kaya [31] studied the reformulation of the Dubins path based on the optimality principle, aiming to find a time-optimal, collision-free trajectory for the Dubins vehicle. The author extended this approach to compute Dubins interpolating curves, which are the shortest curvature-constrained paths through a given sequence of points [32]. Zheng [33] introduced a model focusing on the Minimum-Time Intercept Problem (MTIP). This problem involves guiding a Dubins vehicle from a specific location with a specified initial heading angle to intercept a moving target in the shortest time possible. Later, the author discussed the same scenario with prescribed impact angle constraints [34]. Recently, Forkan et al. [35] introduced an innovative mathematical model to find a minimum path length or minimum time for touring a finite number of targets by unmanned aerial vehicle (UAV), both the UAV and targets are at the same altitude. The authors used the arc parameterization technique to determine the optimal path. We employ the same methodologies to develop the mathematical model for finding the optimal path of the pursuer’s pathway to intercept a moving target that travels through a straight line at a constant speed. In recent times, Hou [36] suggests a geometry-based impact time and angle control guiding rules without calculating the time-to-go that guarantees target captureability and zero terminal guidance command.

This paper proposes a mathematical model formulated as an optimal control problem, leveraging Dubin’s path with a combination of circular and straight-line segments. The model’s geometric interpretation is illustrated in Section 3, and an efficient algorithm is developed to approximate the optimal path of the pursuer for intercepting a moving target. Extensive numerical experiments demonstrate the model’s efficiency, high convergence rate, and advantages over existing methods. The proposed model is also suitable for minimizing the time it takes while a moving target is intercepted by a pursuer travelling at a constant speed. The model is expected to have the following useful features:

1. (i)

   When faced with a complex problem that involves numerous state and control variables, it’s crucial for the method to provide approximations of optimal solutions. This is especially important in real-life applications where the problem-solving process can get increasingly intricate. The biggest hurdle in such scenarios is to develop an algorithm that can effectively and efficiently solve the problem which is the main motivation of this paper.

2. (ii)

   The method should construct an algorithm to generate solutions efficiently at a reasonable computational time. Computational time management in our analysis is important because solving a large problem can be very costly.

3. (iii)

   The method should optimize the heading angles to determine the pursuer’s direction to generate the shortest path of interception. This is an essential feature since it guarantees that, given those heading angles, one may optimize the pursuer’s path or time without considering each of the feasible paths separately. Therefore, our goal is to propose a mathematical model and increase its capacity for approximating the optimum trajectory in terms of length and time through the solution of the mathematical model.

We are therefore interested in studying more about how the suggested model might construct the optimum path with minimal computational effort. The strengths and advantages of the proposed method with features (i)-(iii) are illustrated through the examples. The proposed model is compared with an existing model presented in [28], and the findings are demonstrated in Figures (3-8).

The rest of this paper comprises the following sections. In Section 2, we illustrate reformulation of Dubins problem as a time-optimal control problem and Pontryagin’s Maximum Principle. The geometrical interpretation of optimal path planning of a pursuer for intercepting a moving target is described in Section 3. In Section 4, the model for finding an optimal path of the pursuer to intercept a moving target is proposed. In Section 5, we introduce an algorithm to implement the proposed model. Numerical experiments are conducted and we provide the results and discussions based on numerical experiments in Section 6. The last section presents the conclusion of the paper.

Let us consider the optimum path, denoted as $z(t):[t_{0},t_{f}]\to\mathbb{R}^{2}$, starting from the initial position to the interception point. This path $z(t)$ measure the entire travel path for both the pursuer and the target until they meet each other. We assume that the target is moving in a straight line with a constant speed and the pursuer does not face any obstacles during the motion. We also assume that the pursuer can not reverse its direction to intercept a target. The pursuer also has complete knowledge of the target’s future trajectory and speed.

We consider $(x(t),y(t))\in\mathbb{R}^{2}$ along $z(t)$ with $\dot{x}(t)=|\mathbf{V}|\cos\theta(t)$ and $\dot{y}(t)=|\mathbf{V}|\sin\theta(t)$, where $\theta(t)$ is the heading angle, which is measured from the $x$-axis through counterclockwise and $\theta(t)\in[0,2\pi]$, $|\mathbf{V}|$ is the speed, $R$ is the minimum turning radius of the pursuer.

Let $t\in[t_{0},t_{f}],$ where $t_{f}$ be the terminal time which is unknown and needs to be determined, and $t_{0}$ be the initial time. Therefore, the minimum path of the pursuer for intercepting a moving target is determined by the arc length functional

where, $||\dot{z}(t)||=|\mathbf{V}|,$ $\displaystyle\dot{z}(t)=\frac{dz}{dt}$, and $||.||$ is the Euclidean norm.

Furthermore, the unit tangent vector is defined by $\displaystyle T(t)=\frac{\dot{z}(t)}{||\dot{z}(t)||}=(\cos\theta(t),\sin\theta(t))$, and the curvature at each point of the path is calculated as $\displaystyle a=\frac{||\dot{T}(t)||}{||\dot{z}(t)||}=\frac{|\dot{\theta}(t)|}{|\mathbf{V}|}$.

The quantity $\displaystyle\frac{\dot{\theta}(t)}{|\mathbf{V}|}$ can be positive or negative, and referred to as the signed curvature. If $\dot{\theta}(t)>0$ then the pursuer moves in the counter-clockwise direction (left turns ($L$)) and if $\dot{\theta}(t)<0$ then the pursuer moves in the clockwise direction (right turns ($R$)), and also $\dot{\theta}(t)=0$, if the pursuer moves in the straight line ($S$). Let $u(t)=\displaystyle\frac{\dot{\theta}(t)}{|\mathbf{V}|}$ be a control variable.

Let us consider that heading angle of pursuer at the starting position $P(x_{P_{0}},y_{P_{0}})$ is $\theta_{P_{0}}$, and the constant velocities of pursuer and target are $\bf{V_{P}}$ and $\bf{V_{T}}$ respectively.

Hence the optimal control problem for the pursuer can be formulated as follows.

where ($x_{P}(t)$, $y_{P}(t)$ $\theta_{P}(t)$) is represent the position along the path of pursuer.

Let’s consider the target position is ($x_{T}(t)$, $y_{T}(t)$ $\theta_{T}$) moving with constant speed $|\mathbf{V_{T}}|>0$ without making any turns or maneuvers. The direction of the target’s movement is indicated by $\theta_{T}$, which belongs to $[0,2\pi]$. Since the target does not change its direction, the value of $\theta_{T}$ remains fixed throughout the engagement. Assuming the target’s initial position is $(x_{T_{0}},y_{T_{0}})$, its equations of motion can be expressed as follows:

Let the pursuer and the target initiate their journey from the points $P(x_{P_{0}},y_{P_{0}},\theta_{P_{0}})$ and $T(x_{T_{0}},y_{T_{0}},\theta_{T_{0}})$, respectively and after certain period of time the pursuer intersect the target at a arbitrary unknown position. Following Dubin’s path the pursuer adheres to a minimum turning radius, allowing it the flexibility to make left or right turns to optimize its path. According to Figure 1, the pursuer can follow either path $PX_{1}$ or $PX_{2}$ to travel across a circular trajectory. Subsequently, based on its choice of the left or right turn, the pursuer follows a straight-line path, either $X_{1}X_{3}$ or $X_{2}X_{4}$, with the aim of intercepting the moving target, while maintaining a constant speed of $|\bf{V_{P}}|$. Simultaneously, the target is in motion at a constant speed of $|\bf{V_{T}}|$, moving towards its path. It is assumed that after a certain time interval $t$, both the pursuer and the target will intersect at a point denoted as $X_{3}$ or $X_{4}$. To ensure the success of these intercepts, we make the assumption that the pursuer’s speed, $|\bf{V_{P}}|$, exceeds that of the target, $|\bf{V_{T}}|$. This creates two possible feasible paths of the type $CS+S_{T}$ which are as follows:

Let us assume an initial time $t_{0}=0$ and a terminal time $t_{f}=t_{3}$ (at $X_{3}$ or $X_{4}$) for the both pursuer and target, so that the length of each sub-path can be defined as $\xi_{i}=|\mathbf{V_{P}}|(t_{i}-t_{i-1})$, where $i=1,2,3$. The notation $\xi_{i}=|\mathbf{V_{T}}|(t_{i-1}-t_{i})$, where $i=4$ represents the distance traveled by the target until it is intercepted by the pursuer. Now we solve the ordinary differential equations given in (2) for $x_{P}(t)$, $y_{P}(t)$, $\theta_{P}(t)$ during the time interval $t_{i-1}\leq t\leq t_{i}$, $i=1,2,3$ yields,

From (14), we obtain the position $(x_{P}(t_{i}),y_{P}(t_{i}))$, $i=1,2$ along turning curve ($C$) yields

The position $(x_{P}(t_{i}),y_{P}(t_{i})),$ with $i=3$ at which the pursuer intercepts the moving target along a straight line ($S$) can be described as

Since the speed of the pursuer ($|\mathbf{V_{P}}|$) is constant over the time duration $[t_{i}-t_{i-1}]$, $~{}~{}i=3$, so the length of straight path for pursuer can be written as

Now using (18), we rewrite (17) as below:

and heading angles of pursuer along the both curve ($C$) and straight line ($S$) can be obtained as

where $u(t)=(\dot{\theta}_{P}(t)/|\mathbf{V_{P}}|)=a$, for left-turn circular arc, $u(t)=(\dot{\theta}_{P}(t)/|\mathbf{V_{P}}|)=-a$, for right-turn circular arc and $u(t)=(\dot{\theta}_{P}(t)/|\mathbf{V_{P}}|)=0$, for straight line.

Using (15), (16), and (19), the optimum path $CS$ of the pursuer for the time duration from $t_{0}$ to $t_{3}$ is obtained below.

Now, we calculate the target path for the time duration from $t_{0}$ to $t_{3}$ as below,

By simplification the above relations and using $\displaystyle\xi_{4}=\displaystyle|\mathbf{V_{T}}|(t_{3}-t_{0})$, we obtain the path for moving target as

It is note that when a moving target is intercepted by the pursuer, their positions coincide. In other words, $(x_{P}(t_{3}),y_{P}(t_{3}))=(x_{T}(t_{3}),y_{T}(t_{3}))$ at the interception moment. As a result combining $\eqref{path1}$-$\eqref{path4}$ and thus we obtain,

and

Since the pursuer and target move with separate constant velocities and intercept to each other at the same moment, we must include an additional constraint to the optimization problem as

Considering equations (20) and (25)-(27), we can formulate the following mathematical model for target interception.

where

The possible pathways that described in (28) intersect a moving target satisfy the minimum turn radius criteria established in the following Proposition (5.1).

In Step (3) of the algorithm, we utilize the output obtained in Step (2) to determine the positions of both pursuer and target trajectories at time $t\in[t_{o},t_{f}]$. Initially, in Step (3)(a), we locate the center of the turning radius circle of the pursuer, a necessary step for visualizing the turning curves. Subsequently, in Step (3)(b)(i) and (ii), we compute the positions $(x,y)$ of both the target and pursuer from their starting points up until the intercept occurs.

In Example 1, we tested our proposed Algorithm (5.1) with various combinations of pursuer and target positions, velocities, and turning radii. The pursuer’s objective is to intercept the target in the shortest possible distance or time. As the turning radius is influenced by both angular and linear speeds, Example 2 tested the model, calculating the turning radius using the parameters such as pursuer and target positions, and velocities. We explored different optimal path combinations by adjusting angular speeds, both increasing and decreasing them, to evaluate their impact on interception strategies. In experiments conducted in Examples 1 and 2, Algorithm (5.1) demonstrated remarkable efficiency in computing optimal paths, as evidenced by Figures 3–7 and the detailed results presented in Tables 1-5.

In the experiments, we have implemented Algorithm (5.1) writing codes in AMPL [37], utilizing various solvers such as Ipopt [38] and KNITRO [39], all with default settings, to solve the problems. Almost both solvers exhibited a high convergence rate in solving the proposed model. The obtained solutions are simulated by using MATLAB for the numerical experiments. In our analysis, the computations are conducted on an HP Evo laptop with 16 GB RAM and a Core i7 processor running at 4.6 GHz.

Example 1

Let an initial position for the pursuer, denoted as  $P(0,0,2\pi/3)$ and a target point $T(-5,0,\pi/2)$. The objective is for the pursuer $P$, moving at a constant speed $\mathbf{|V_{P}|}$, to intercept the target $T$, which is also moving at a constant speed $\mathbf{|V_{T}|}$. This interception is accomplished by finding the shortest path $f$, as determined by Algorithm (5.1). It is important to note that we assume a minimum turning radius of $R=1$ for the pursuer’s turning curve. The pursuer takes time $t$ to intercept the target.

The results of our analysis reveal that the $LS+S_{T}$ path proves to be the optimal choice to successfully achieve the task, as illustrated in Figure 3(a). Another feasible path is the $RS+S_{T}$ path, showcased in Figure 3(b). Table 1 provides a detailed breakdown of the lengths and time periods associated with each segment of these paths. Specifically, the total length of the path obtained by our algorithm $f=6.26$, and the corresponding time is $t=1.04$. In particular, the non-optimal path requires two times more length ($f=14.16$) and time ($t=2.36$), when compared to the optimal path, as illustrated in Table 1 and demonstrated in Figure 3(a) and (b). The computational time taken by our system to derive the optimum path is around $0.078$ seconds.

Now, we fixed the target point to the right side of the pursuer, specifically at coordinate $T(8,-2,2\pi/3)$, while increasing the target’s speed to $\mathbf{|V_{T}|}=2$. Additionally, we adjusted the initial angle of the pursuer to $\pi/3$. The pursuer’s turning curve meets a minimum radius of $R=1$.

Our analysis concluded that the optimal choice to successfully achieve this task is the $RS+S_{T}$ path, as shown in Figure 4(a). Other feasible path is $LS+S_{T}$, displayed in Figure 4(b). Table 2 provides a comprehensive breakdown of the lengths and time intervals associated with each segment of these paths. Specifically, our algorithm yielded an optimal path length of $f=9.55$ and a corresponding time of $t=1.36$. Almost the previous analysis, the non-optimal path requires twice the length and time compared to the optimal path, as depicted in Figure 4(a) and 4(b). The computational time needed for our system to determine the optimal path is around $0.093$ seconds.

We further evaluated the adaptability of Algorithm (5.1) by conducting tests in various directions, including scenarios where the target travels horizontally relative to the pursuer. The analysis for these scenarios is presented in Figure 5(a)(b) and data summarized in Table 3, considering an initial position at $P(0,0,\pi/3)$ and $T(8,3,\pi)$.