Motion control of multiple autonomous ships to approach a target without being detected

Abstract

We introduce a motion control which is a stealth strategy of a team of autonomous ships to approach a target without being detected. The target is equipped with a radar to detect a nearby ship. In the team, one ship, the leader, has a stealth ability not to be detected by a radar as it approaches the target. We steer the leader so that it moves to the target while appearing to be at the same bearing line from the target to the leader. Assuming that the target can readily observe optical flow, but only poorly sense looming, this type of motion by the leader is difficult to be detected by the target. We steer every ship, except for the leader, so that a pulse signal emitted from the radar on the target cannot reach any ship, since the signal is blocked by the leader. In this way, even in the case where multiple ships approach the target, the target cannot detect any ship. In our control law, a ship, except for the leader, does not have to access the position of the target in real time. However, a ship must access the bearing line from the target to the leader in real time. Our motion control is inspired by line-of-sight guidance control laws and is developed in discrete-time systems. We present MATLAB simulations to verify the performance of our motion control law.

Introduction

Nowadays, stealth ships are built, employing stealth technology in an effort to ensure that it is harder to be detected by a radar. The main idea of stealth technology is to absorb a pulse signal emitted from a radar or to reflect the pulse in such a way that the signal cannot return to the radar which emitted it.

Let us consider a scenario of deploying multiple autonomous ships to approach and attack a target without being detected. Since weapon range of a ship is bounded, they approach a target until they are sufficiently close to the target. As they are sufficiently close to the target, they attack the target simultaneously to maximize the attack efficiency. Usually, a stealth ship is expensive due to its stealth technology. Thus, deploying multiple stealth ships may not be feasible.

This article introduces a motion control which is a stealth strategy of a team of autonomous ships to approach a target without being detected. The target is equipped with a radar to detect a nearby ship. In the team, one ship, the leader, has a stealth ability not to be detected by a radar as it approaches the target.

We steer the leader so that it moves to the target while appearing to be at the same bearing line from the target to the leader. This type of motion by the leader is called the motion camouflage with respect to a fixed point at infinity. ¹ This motion camouflage is employed by various visual insects and animals to achieve prey capture, mating, or territorial combat. ^{1 –4} This motion is a time-optimal solution to capture a prey moving along a straight line. ² Also, Ghose et al. ² argued that this motion minimizes time-to-capture of an unpredictably moving prey. Assuming that the prey can readily observe optical flow, but only poorly sense looming, this type of motion by the predator is difficult to be detected by the prey. ¹

Justh and Krishnaprasad ¹ developed a feedback control law to achieve this kind of motion camouflage. As far as we know, Justh and Krishnaprasad ¹ are unique in proving that under their controller, the predator converges to the motion camouflage state in finite time. We use the controller of Justh and Krishnaprasad ¹ to achieve the motion camouflage of the leader.

We steer every ship, except for the leader, so that a pulse signal emitted from the radar on the target cannot reach any ship, since the signal is blocked by the leader. In this way, even in the case where multiple ships approach the target, the target cannot detect any ship. There are many papers on control of multiple autonomous vehicles. ^{5 –16} Ji and Egerstedt ⁹ introduced distributed coordination control of multi-agent systems while preserving connectedness. Zhang and Leonard ¹² presented formation control approach to improve source seeking using distributed sensors. Cao et al. ¹⁴ discussed the overview of literature on multi-agent systems. As far as we know, stealth motion control for multiple robots has not been considered in the literature on formation control.

In our article, we design the control law for a ship except for the leader. We develop the control law in discrete-time systems. In our control law, a ship, except for the leader, does not have to access the position of the target in real time. However, a ship must access the bearing line from the target to the leader in real time.

The marine community has been using various guidance algorithms to make a marine vehicle follow a predefined path. ^{17 –19} Line-of-sight (LOS) guidance control has been widely used to make the vehicle follow the path robustly. ^17,19 LOS guidance control demonstrated a high level of robustness to even a discontinuous path. ²⁰ LOS guidance control is designed to steer the vehicle in such a way that it stays on the path, or to lead it toward the path if the cross-track error (the shortest distance to the path) is not zero. LOS guidance control makes the cross-track error converge to zero as time goes on. ¹⁷

Our motion control is inspired by the LOS guidance control law in the study by Lekkas and Fossen. ¹⁷ In our article, each ship is controlled so that it moves toward the bearing line from the target to the leader while getting closer to the target. The cross-track error for a ship is the shortest distance between the ship and the bearing line. We use a conventional LOS guidance control, which is based on a constant look-ahead distance. We present MATLAB simulations to verify the performance of our motion control law.

The article is organized as follows: The section “Assumptions and definitions” introduces assumptions and definitions in our article. The section “The motion control” presents the motion control law. The section “MATLAB simulations” introduces MATLAB simulations to verify the effectiveness of our control law. The last section provides conclusions.

Assumptions and definitions

We introduce several assumptions and definitions before presenting our main results. $\angle (v_1,v_2)$ denotes the angle formed by two vectors $v_1$ and $v_2$ . Mathematically, $\angle (v_1,v_2)=\text{arccos}(\frac{v_1\cdot v_2}{\parallel v_1\parallel \parallel v_2\parallel })$ . Here, $\angle (v_1,v_2)$ exists between 0 and π.

We consider the team of pursuers consist of one leader and N followers. There are N + 1 pursuers in total. v^l denotes the speed of the leader and v^f denotes the speed of each follower. Note that every follower moves with the same speed.

We say that a target is captured in the case where it is within the weapon range of every pursuer. Note that we can capture the target only if it is within the weapon range of all pursuers. The problem tackled in our article is as follows: develop a controller for every follower to capture the target as fast as possible without being detected.

To capture the target in finite time, we assume that the speed of a target, v^t , is strictly less than v^l or v^f . We simplify all pursuers and the target as circles in two dimensions. Moreover, collision avoidance among circular pursuers is not within the scope of this article. To avoid collision among pursuers, we can use a decentralized reactive collision avoidance algorithm in the study by Lalish and Morgansen. ²¹ The algorithm in the study by Lalish and Morgansen ²¹ can work in conjunction with our motion control law to guarantee that all pursuers remain free of collisions.

We assume that every follower localizes itself and accesses L in real time. Here, L denotes the infinite line crossing both the target and the leader. We argue that this assumption is feasible in practice. The leader and followers can use Inertial Measurement Units or Global Positioning Systems to localize themselves, respectively. The leader can use sensors, such as onboard radars, to access the target position in real time. Ristic et al. ²² presented various methods to estimate the target position using sensor measurements. Since the leader can localize itself as well as the target in global coordinate systems, the leader can access L in real time. Then, a follower can communicate with the leader to access L in real time.

In this article, we consider discrete-time systems. The subscript k is used to represent the time step k. In other words, for an arbitrary notation A, A_k represents A at time step k. Let Fⁱ denote the i th follower, where i ≤ N. $f_k^i$ denotes the center of Fⁱ at time step k. $t_k$ denotes the center of the target at time step k and $l_k$ denotes the center of the leader at time step k. L _k+1 is the infinite line crossing both $t_{k+1}$ and $l_{k+1}$ . Let R^l denote the radius of the leader. Also, let R^f denote the radius of a follower.

Let $c_k^i$ denote the point on L _k+1, which is closest to $f_k^i$ . $h_k^i$ denotes the point on L _k+1, which is v^fT distance from $c_k^i$ and is located between $c_k^i$ and $t_{k+1}$ . Here, T denotes the sampling interval of our control law. $d_k^i$ denotes $\parallel f_k^i-c_k^i\parallel$ . See Figure 1 for an illustration of these concepts.

Figure 1.

An illustration of concepts.

Let us consider Fⁱ at time step k. A follower’s reachable area within one time step forms the circle, called the reachable circle, whose radii has length v^fT. $H_k^i$ denotes the point on the reachable circle, which is closest to $h_k^i$ . θ ₀ denotes $\angle (f_{k+1}^i-f_k^i,c_k^i-f_k^i)$ . See Figure 2 for an illustration. $H_k^i$ is marked as the red circle in this figure.

Figure 2.

The reachable circle of Fⁱ within one time step.

Conditions for stealth state

In the team, one ship, the leader, has a stealth ability not to be detected by a radar as it approaches the target. We steer every ship, except for the leader, so that a pulse signal emitted from the radar on the target cannot reach any ship, since the signal is blocked by the leader. In this way, even in the case where multiple ships approach the target, the target cannot detect any ship.

We assume that a radar emitting a pulse signal is located at the center of the target. Considering the size of each pursuer, a pulse signal cannot reach Fⁱ in the case where

$\angle (f^i-t,l-t)+\text{arcsin}\frac{R^f}{\parallel t-f^i\parallel }<\text{arcsin}\frac{R^l}{\parallel t-l\parallel }$

1

The stealth task of Fⁱ is to achieve equation (1). In the case where Fⁱ achieves the stealth task at time step k, we say that Fⁱ is in the stealth state S_s at time step k. This implies that the target cannot detect Fⁱ even in the case where Fⁱ approaches the target.

See Figure 3 for an illustration of the case where the stealth task of Fⁱ is achieved. A pulse signal emitted from the target cannot reach Fⁱ , blocked by the leader. Thus, Fⁱ cannot be detected by the target’s radar. To make the definition of this stealth task feasible, both $\frac{R^f}{\parallel t-f^i\parallel }$ and $\frac{R^l}{\parallel t-l\parallel }$ must be less than one. Otherwise, the arcsin function does not exist at all.

Figure 3.

Fⁱ is in the stealth state. However, F^j is not.

Figure 3 also shows the case where F^j is not in the stealth state. A pulse signal emitted from the target can reach F^j . Thus, F^j is detected by the target.

In the case where $\text{arcsin}\frac{R^f}{\parallel t-f^i\parallel }>\text{arcsin}\frac{R^l}{\parallel t-l\parallel }$ , the stealth task of Fⁱ is not achievable. On the other hand, the difficulty of stealth task of Fⁱ is alleviated in the case where $R^l\gg R^f$ or $\parallel t-f^i\parallel \gg \parallel t-l\parallel$ .

To make Fⁱ stay in the stealth state, it is desirable to make Fⁱ converge to L. Suppose that $\parallel f_k^i-c_k^i\parallel >v^{f}T$ and that Fⁱ is on L_k . In this case, Fⁱ cannot reach L _k+1 within one time step, since the reachable circle does not meet L _k+1. To make Fⁱ converge to L at each time step, it is necessary that $\parallel f_k^i-c_k^i\parallel \le v^{f}T$ at every time step k. We say that strict convergence condition is satisfied at time step k in the case where $\parallel f_k^i-c_k^i\parallel \le v^{f}T$ .

The motion control

In this section, we introduce our motion control law. At each time step k, our control law is given as follows: Fⁱ moves to $H_k^i$ within one time step. Since $h_k^i$ exists on L _k+1 and is located between $c_k^i$ and $t_{k+1}$ , Fⁱ moves toward L _k+1 while getting closer to the target.

Stability analysis

We analyze the stability of our control law. Let us consider the special case where L_k does not change with respect to k. In this special case, Fⁱ converges to L_k as k increases, while heading toward the target. The proof for this convergence is provided in the study by Lekkas and Fossen. ¹⁷

To make our problem more interesting, we consider the general case where L_k changes with respect to k. We have the following theorem.

Theorem 1

Under our control law, we achieve the stealth task of Fⁱ at time step k + 1 in the case where $\text{arcsin}((1-\frac{v^{f}T}{\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}})\frac{d_k^i}{\parallel f_{k+1}^i-t_{k+1}\parallel })+\text{arcsin}\frac{R^f}{\parallel t_{k+1}-f_{k+1}^i\parallel }<\text{arcsin}\frac{R^l}{\parallel t_{k+1}-l_{k+1}\parallel }$ .

Proof

The stealth task of Fⁱ is to achieve equation (1). Suppose that $\text{sin}(\angle (f_{k+1}^i-t_{k+1},l_{k+1}-t_{k+1}))=(1-\frac{v^{f}T}{\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}})\frac{d_k^i}{\parallel f_{k+1}^i-t_{k+1}\parallel }$ . Using equation (1), the stealth task of Fⁱ is achieved at time step k + 1 in the case where $\text{arcsin}((1-\frac{v^{f}T}{\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}})\frac{d_k^i}{\parallel f_{k+1}^i-t_{k+1}\parallel })+\text{arcsin}\frac{R^f}{\parallel t_{k+1}-f_{k+1}^i\parallel }<\text{arcsin}\frac{R^l}{\parallel t_{k+1}-l_{k+1}\parallel }$ , which proves this theorem.

We prove that under our control law, $\text{sin}(\angle (f_{k+1}^i-t_{k+1},l_{k+1}-t_{k+1}))=(1-\frac{v^{f}T}{\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}})\frac{d_k^i}{\parallel f_{k+1}^i-t_{k+1}\parallel }$ . Consider the triangle formed by the following three vertices: $f_k^i$ , $h_k^i$ , and $c_k^i$ . Using this triangle, we get

$\text{sin}(\pi /2-{\theta }_0)=\frac{d_k^i}{\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}}$

2

since $\angle (h_k^i-c_k^i,f_k^i-c_k^i)$ is 90°. See Figure 4 for an illustration.

Figure 4.

An illustration.

We have

$f_{k+1}^i=f_k^i+\frac{h_k^i-f_k^i}{\parallel h_k^i-f_k^i\parallel }{v}^{f}T$

3

Using equation (3), we derive

$\parallel f_{k+1}^i-h_k^i\parallel =\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}-v^{f}T$

4

Consider the triangle formed by the following three vertices: $f_{k+1}^i$ , $h_k^i$ , and $t_{k+1}$ . Using both equation (4) and the Law of Sines on the triangle, we have

$\frac{\parallel f_{k+1}^i-t_{k+1}\parallel }{\text{sin}(\pi /2+{\theta }_0)}=\frac{\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}-v^{f}T}{\text{sin}({\theta }_e)}$

5

where θ_e denotes $\angle (f_{k+1}^i-t_{k+1},l_{k+1}-t_{k+1})$ . In addition, using equation (2), we derive

$\frac{\parallel f_{k+1}^i-t_{k+1}\parallel \sqrt{d_k^i{ }^2+{(v^{f}T)}^2}}{d_k^i}=\frac{\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}-v^{f}T}{\text{sin}({\theta }_e)}$

6

where $\text{sin}(\pi /2+{\theta }_0)=\text{sin}(\pi /2-{\theta }_0)$ is used.

Using equation (6), we get

$\text{sin}({\theta }_e)=\frac{d_k^i(\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}-v^{f}T)}{\parallel f_{k+1}^i-t_{k+1}\parallel \sqrt{d_k^i{ }^2+{(v^{f}T)}^2}}$

7

Since ${\theta }_e=\angle (f_{k+1}^i-t_{k+1},l_{k+1}-t_{k+1})$ , we get $\text{sin}(\angle (f_{k+1}^i-t_{k+1},l_{k+1}-t_{k+1}))=(1-\frac{v^{f}T}{\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}})\frac{d_k^i}{\parallel f_{k+1}^i-t_{k+1}\parallel }$ .

□

Let us assume that the relative distance between a follower and the target is bigger than or equal to that between the leader and the target. In other words, $\parallel t-f^i\parallel \ge \parallel t-l\parallel$ . This is a reasonable assumption, since we design the control law for a follower so that it “hides” behind the leader while chasing the target. In this case, we derive Lemma 1 using both Theorem 1 and Corollary 1. Lemma 1 shows that in the case where $2R^f\le R^l$ , $2\parallel f_k^i-c_k^i\parallel \le R^l$ is sufficient to achieve the stealth task of Fⁱ at time step k + 1.

Corollary 1

Let us consider two constants a and b within the interval $[-\pi /2,\pi /2]$ . $\text{arcsin}a<\text{arcsin}b$ if and only if a < b.

Lemma 1

Assume that $\parallel t_{k+1}-f_{k+1}^i\parallel \ge \parallel t_{k+1}-l_{k+1}\parallel$ . Under our control law, a sufficient condition to achieve the stealth task of Fⁱ at time step k + 1 is $2\text{max}(d_k^i,R^f)\le R^l$ .

Proof

$(1-\frac{v^{f}T}{\sqrt{d_k^i{ }^2+{(v^{f}T)}^2}})\frac{d_k^i}{\parallel f_{k+1}^i-t_{k+1}\parallel }$ in Theorem 1 is less than $\frac{d_k^i}{\parallel f_{k+1}^i-t_{k+1}\parallel }$ . Using both Theorem 1 and Corollary 1, we have the following statement:

Under our control law, a sufficient condition to achieve the stealth task of Fⁱ at time step k + 1 is

$\text{arcsin}\frac{d_k^i}{\parallel f_{k+1}^i-t_{k+1}\parallel }+\text{arcsin}\frac{R^f}{\parallel t_{k+1}-f_{k+1}^i\parallel }\le \text{arcsin}\frac{R^l}{\parallel t_{k+1}-l_{k+1}\parallel }$

8

Using Corollary 1 and the assumption that $\parallel t_{k+1}-f_{k+1}^i\parallel \ge \parallel t_{k+1}-l_{k+1}\parallel$ , we derive that $\text{arcsin}\frac{d_k^i}{\parallel f_{k+1}^i-t_{k+1}\parallel }+\text{arcsin}\frac{R^f}{\parallel t_{k+1}-f_{k+1}^i\parallel }$ in equation (8) is less than or equal to $\text{arcsin}\frac{d_k^i}{\parallel t_{k+1}-l_{k+1}\parallel }+\text{arcsin}\frac{R^f}{\parallel t_{k+1}-l_{k+1}\parallel }$ . Hence, equation (8) is satisfied in the case where

$\text{arcsin}\frac{d_k^i}{\parallel t_{k+1}-l_{k+1}\parallel }+\text{arcsin}\frac{R^f}{\parallel t_{k+1}-l_{k+1}\parallel }\le \text{arcsin}\frac{R^l}{\parallel t_{k+1}-l_{k+1}\parallel }$

9

Moreover, since $\text{arcsin}\frac{d_k^i}{\parallel t_{k+1}-l_{k+1}\parallel }+\text{arcsin}\frac{R^f}{\parallel t_{k+1}-l_{k+1}\parallel }$ in equation (9) is less than or equal to $2\text{arcsin}\frac{\text{max}(d_k^i,R^f)}{\parallel t_{k+1}-l_{k+1}\parallel }$ , equation (9) is satisfied in the case where

$2\text{arcsin}\frac{\text{max}(d_k^i,R^f)}{\parallel t_{k+1}-l_{k+1}\parallel }\le \text{arcsin}\frac{R^l}{\parallel t_{k+1}-l_{k+1}\parallel }$

10

Using the fact that $2\text{arcsin}a=\text{arcsin}2a\sqrt{1-a^2}$ , equation (10) leads to

$\text{arcsin}\frac{2m}{\parallel t_{k+1}-l_{k+1}\parallel }\sqrt{1-\frac{m^2}{\parallel t_{k+1}-l_{k+1}{\parallel }^2}}\le \text{arcsin}\frac{R^l}{\parallel t_{k+1}-l_{k+1}\parallel }$

11

Here, $m=\text{max}(d_k^i,R^f)$ for convenience. Using Corollary 1, equation (11) leads to

$2m\sqrt{1-\frac{m^2}{\parallel t_{k+1}-l_{k+1}{\parallel }^2}}\le R^l$

12

Since $2m\sqrt{1-\frac{m^2}{\parallel t_{k+1}-l_{k+1}{\parallel }^2}}$ in equation (12) is less than or equal to 2m, we have the following statement:

Under our control law, a sufficient condition to achieve the stealth task of Fⁱ at time step k + 1 is $2\text{max}(d_k^i,R^f)\le R^l$ .

□

Theorem 2 proves convergence of our control law under the following three assumptions. The first assumption is that $\parallel t_k-f_k^i\parallel \ge \parallel t_k-l_k\parallel$ at every time step k ≥ k ^*. The second assumption is that the size of a follower is smaller than half the size of the leader. The third assumption is that $2\parallel f_k^i-c_k^i\parallel \le R^l$ at every time step k ≥ k ^*. This implies that the relative distance between Fⁱ and L is smaller than half the size of the leader.

Theorem 2

Fⁱ moves under our control law. Assume that $\parallel t_k-f_k^i\parallel \ge \parallel t_k-l_k\parallel$ at every time step k ≥ k ^*. Also, we assume that $2R^f\le R^l$ . We further assume that $2\parallel f_k^i-c_k^i\parallel \le R^l$ at every time step k ≥ k ^*. In this case, Fⁱ is in the stealth state at every time step k ≥ k ^*.

Proof

We assume that $2R^f\le R^l$ and $2\parallel f_k^i-c_k^i\parallel \le R^l$ at every time step k ≥ k ^*. This leads to $2\text{max}(d_k^i,R^f)\le R^l$ at every time step k ≥ k ^*.

Moreover, $\parallel t_k-f_k^i\parallel \ge \parallel t_k-l_k\parallel$ at every time step k ≥ k ^*. Thus, using Lemma 1, Fⁱ is in the stealth state at every time step k ≥ k ^*.

□

Assume that $\parallel t_k-f_k^i\parallel \ge \parallel t_k-l_k\parallel$ at every time step k. Also, assume that the size of a follower is smaller than half the size of the leader. In the case where our control law yields $\parallel f_{k+1}^i-c_{k+1}^i\parallel <\parallel f_k^i-c_k^i\parallel$ at every time step k, Fⁱ enters the stealth state in finite time and never leaves the state using Theorem 2.

We provide the conditions to satisfy that $\parallel f_{k+1}^i-c_{k+1}^i\parallel <\parallel f_k^i-c_k^i\parallel$ . We assume that L _k+1 and L _k+2 are parallel to each other, since we steer the leader so that it moves to the target while appearing to be at the same bearing line from the target to the leader.

We need to introduce new notations before presenting the conditions. Let $P_2$ denote the point on L _k+2, such that the line segment connecting $f_k^i$ and $P_2$ is normal to L _k+2. We introduce one notation l ₂, which is the signed distance between $P_2$ and $c_k^i$ . In the case where $(P_2-c_k^i)\cdot (f_k^i-c_k^i)$ is positive, l ₂ is positive. In this case, the line segment connecting $f_k^i$ and $P_2$ does not meet L _k+1. In the case where $(P_2-c_k^i)\cdot (f_k^i-c_k^i)$ is negative, l ₂ is negative. In this case, the line segment connecting $f_k^i$ and $P_2$ meets L _k+1. Note that $-v^{t}T\le l_2\le v^{t}T$ . See Figure 5 for an illustration. l ₂ is negative in this figure.

Figure 5.

L _k+1 and L _k+2 are parallel to each other. l ₂ is negative.

Lemma 2

Consider the case where L _k+1 and L _k+2 are parallel to each other. $d_{k+1}^i-d_k^i<0$ if and only if $|d_k^i-v^{f}T\text{cos}({\theta }_0)-l_2|<d_k^i$ .

Proof

Figure 5 shows the case where L _k+1 and L _k+2 are parallel to each other.

We draw both x-axis and y-axis on the plane satisfying the following conditions:

x-axis and y-axis are normal to each other.

$c_k^i$ is set as the origin. In other words, x-axis and y-axis intersect at $c_k^i$ .

The xy coordinate of $f_k^i$ is $(0,d_k^i)$ .

Using the geometry in Figure 5, the y-coordinate of $c_{k+1}^i$ is l ₂. Also, the y-coordinate of $f_{k+1}^i$ is $d_k^i-v^{f}T\text{cos}({\theta }_0)$ . $d_{k+1}^i$ is the difference between the y-coordinate of $c_{k+1}^i$ and that of $f_{k+1}^i$ . Thus, we derive

$d_{k+1}^i=|d_k^i-v^{f}T\text{cos}({\theta }_0)-l_2|$

13

This lemma is proved.

Corollary 2 proves convergence of our control law under the following three assumptions. The first assumption is that $\parallel t_k-f_k^i\parallel \ge \parallel t_k-l_k\parallel$ at every time step k ≥ k ^*. The second assumption is that the size of the leader is sufficiently large to satisfy both $2v^{f}T\le R^l$ and $2R^f\le R^l$ . The third assumption is that the strict convergence condition is satisfied at every time step k ≥ k ^*. In this case, we derive the following corollary using Theorem 2.

Corollary 2

Fⁱ moves under our control law. Assume that $\parallel t_k-f_k^i\parallel \ge \parallel t_k-l_k\parallel$ at every time step k ≥ k ^*. Also, we assume that $2R^f\le R^l$ . We further assume that $2v^{f}T\le R^l$ . Also, the strict convergence condition is satisfied at every time step k ≥ k ^*. In this case, Fⁱ is in the stealth state at every time step k ≥ k ^*.

Proof

Since $2v^{f}T\le R^l$ , $2\parallel f_k^i-c_k^i\parallel \le R^l$ at every time step k ≥ k ^*. Since the strict convergence condition is satisfied at every time step k ≥ k ^*, $\parallel f_k^i-c_k^i\parallel \le v^{f}T$ at every time step k ≥ k ^*. Using Theorem 2, Fⁱ is in the stealth state at every time step k ≥ k ^*.

MATLAB simulations

In this section, we verify the effectiveness of our motion control law using MATLAB simulations. In all scenarios, T = 1 s. Also, the leader and five followers move with unit speed, and the target moves with the speed 0.7. The initial position of the target is (100, 100), and that of the leader is (0, 0). The initial positions of all followers are as follows

$(-10,-10),(0,-5),(0,30),(-10,20),(-10,-20)$

R^l = 2 and R^f = 1, and we check whether the stealth task of Fⁱ is achieved by measuring $S_s(k)=\text{arcsin}\frac{R^l}{\parallel t_k-l_k\parallel }-\text{arcsin}\frac{R^f}{\parallel t_k-f_k^i\parallel }-\angle (f_k^i-t_k,l_k-t_k)$ . $S_s(k)$ is derived using equation (1). In the case where $S_s(k)$ is positive, then the stealth task of F_i is achieved at time step k.

We control the motion of the leader using the motion camouflage control in the study by Justh and Krishnaprasad. ¹ The motion camouflage control is designed so that the bearing of the leader with respect to the target does not change as time goes on. However, as we use the motion camouflage control in the study by Justh and Krishnaprasad, ¹ the bearing of the leader with respect to the target changes slowly as time goes on. We will show that the stealth task of each follower is achieved despite this slow bearing change.

Justh and Krishnaprasad ¹ introduced Frenet–Serret frames ²³ to model the movement of a target

$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\dot{t}=v^t{x}^t\\ {\dot{x}}^t=v^t{u}^t{y}^t\\ \hspace{0.17em}{\dot{y}}^t=-v^t{u}^t{x}^t\end{array}$

14

where u^t is the steering (i.e. curvature) control of the target and v^t is the speed of the target, which is 0.7. $t$ is the position of the target. Moreover, $x^t$ is the unit tangent vector to the trajectory of the target and $y^t$ is the corresponding unit normal vector. We use equation (14) to simulate the motion of the target.

Recall we assumed that every follower localizes itself and accesses L in real time. Here, L denotes the infinite line crossing both the target and the leader. In practice, there is an error as a follower measures L in real time. Thus, in simulations, we generate a noisy target position $t_{n,k+1}$ by adding zero-mean Gaussian noise with standard deviation std to each element in the true target position $t_{k+1}$ . Then, each follower uses L _k+1, which intersects $t_{n,k+1}$ instead of $t_{k+1}$ , to generate its control law. In this way, we simulate noisy environments.

Each MATLAB simulation runs for finite time, at the end of which the distance between the leader and the target is equal to or less than 5 distance units.

Figure 6 shows the behavior of the system in the case where u^t in equation (14) is $-0.1\text{sin}(e/10)$ . Here, e denotes the elapsed time in seconds. We simulate noisy environments using std = 0.5. Figure 6 further shows both S_s and the bearing of each pursuer from the point of view of the target. The bearing of each follower almost converges to that of the leader. See that S_s of every follower becomes positive as time goes on. This implies that the stealth task of every follower is achieved as time goes on.

Figure 6.

The target and pursuer positions ( $u^t=-0.1\text{sin}(e/10)$ ).

Figure 7 shows the behavior of the system in the case where u^t in equation (14) is 0.1. We simulate noisy environments using std = 0.5. Figure 7 further shows both S_s and the bearing of each pursuer from the point of view of the target. The bearing of each follower almost converges to that of the leader. See that the stealth task of each follower is achieved as time goes on.

Figure 7.

The target and pursuer positions (u^t = 0.1).

Figure 8 shows the behavior of the system in the case where u^t in equation (14) is 0.45 during the time interval from 50 to 60 s. u^t is 0 outside this interval. In this way, we simulate abrupt maneuvers of the target. We also simulate noisy environments using std = 0.5. Figure 8 shows that as the target maneuvers abruptly, every pursuer maneuvers significantly under our control law. Figure 8 further shows both S_s and the bearing of each pursuer from the point of view of the target. The bearing of each follower almost converges to that of the leader. See that the stealth task of each follower is achieved as time goes on.

Figure 8.

The target and pursuer positions (u^t changes abruptly).

Conclusions

We introduce a motion control which is a strategy of a team of pursuers to approach the target without being detected. Our control law is to steer followers so that they remain stealthy under the cover of the leader with stealth abilities. We present MATLAB simulations to verify the performance of our motion control law.

Our motion control can be applied in various military scenarios. For instance, assume that the target is equipped with LOS weapons, such as cannons. The leader is equipped with defense systems to protect itself against the LOS weapons. On the other hand, followers are vulnerable to the LOS weapons. In this case, our motion control can be applied to attack the target while protecting all pursuers.