Collision-Free Shepherding Control of a Single Target within a Swarm^††thanks: This work was supported in part by JSPS KAKENHI Grant Number JP21H01352.

It has long been observed that various biological individuals tend to exhibit swarming behavior. Examples of swarms include flocks of birds, schools of fish, herds of animals, and colonies of bacteria [1, 2], and engineering applications include the formation control of multi-robot teams and autonomous aircraft. Many studies [3, 4, 5] have investigated the relevant properties of swarm motion based on the attraction-repulsion swarm model and its applications in areas such as UAV swarms and multi-robot systems.

As research on swarm intelligence advances, there has been increasing attention to the use of an external special agent (called shepherd) to control the movement of swarm agents (called sheep). This process is referred to as shepherding [6]. The shepherding model specifically refers to the situation in which “sheep” agents avoid a “shepherd” agent while also interacting with other sheep, according to a swarm model. In this context, we can find various methods [7, 8, 9] to guide a whole swarm of sheep agents.

There is emerging interest in guiding a part of the swarm (e.g., a target sheep) to the goal because of its potential engineering applications in the context of micro-/nano-robotic swarms [10, 11]. For example, Deptula et al. [12] proposed a control algorithm for a shepherd to guide a targeted sheep agent towards its destination. Licitra et al. [13] designed a controller for herding one sheep within a swarm to its destination without considering swarm interaction forces in the dynamics model. Using the framework of reinforcement learning, Sebastián and Montijano [14] developed a method for guiding a few agents within a swarm with heterogeneous agent dynamics. Le et al. [15] proposed a method to use a shepherd to regulate a sheep within a small swarm. Zhang et al. [16] designed an algorithm for capturing and regulating a portion of a swarm using multiple shepherds.

However, the shepherd control methods mentioned above [12, 13, 14, 15, 16] do not consider collisions within the swarm; their algorithms assume that no pairs of agents in the swarm collide. Because the occurrence of a collision can disrupt an entire system and lead to failed tasks [17, 18, 19], it is practically important to develop a shepherding control method with guaranteed collision-free properties. Although several studies have proposed distributed [20, 21] and leader-based [22] control methods for swarm systems with collision-free guarantees, these studies have not been directly extended to the context of shepherding control. Therefore, to design a stable shepherd controller to achieve certain control objectives, the first consideration must be to carefully analyze the collision properties of shepherding swarms.

In this study, the control objective is to use a shepherd agent to guide a sheep agent (target) in a swarm to the origin (goal). First, the repulsion-bounded swarm model is described. We then carefully analyze and rigorously prove that under relatively mild conditions, sheep in the swarm do not collide, even under the influence of a shepherd. Based on this analysis, we use the Lyapunov stability principle to design a motion controller for the shepherd to guide the target to the endpoint (i.e., the origin). The effectiveness of the proposed method is illustrated by numerical simulations, in which we compare it with a baseline strategy based on the farthest agent-targeting strategy presented in [23].

This paper is organized as follows: Section II describes the shepherding swarm model and the problem for controlling one target to the origin; in Section III we analyze the collision-free property of the shepherding swarm model; in Section IV we design a shepherd motion controller to guide a specific target to the destination. Section V provides several numerical simulations that highlight the shepherding behavior.

We consider a swarm of $N>2$ individuals (sheep) and one herder (shepherd) in $\mathbb{R}^{2}$. The positions of sheep $i$ and shepherd at time $t\geq 0$ are denoted by $x_{i}(t)\in\mathbb{R}^{2}$ and $y(t)\in\mathbb{R}^{2}$, respectively. Let $\mathcal{N}=\{1,2,...,N\}$.

We assume that, for all $i\in\mathcal{N}$, the velocity $\dot{x}_{i}$ of the $i$th sheep is specified by a linear combination of three components:

where $f_{ai}$ and $f_{bi}$ represent the attraction and repulsion functions from the other sheep to the $i$th sheep, respectively, and $f_{yi}$ represents the function from the shepherd to the $i$th sheep.

Below, we describe how $f_{ai}$, $f_{bi}$, and $f_{yi}$ are constructed. First, the attraction function of the $i$th sheep is given by:

where the function $\phi_{a}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ is defined as

for a positive constant $m_{a}$. Then, the bounded repulsion function of the $i$th sheep is given by

where the function $\phi_{b}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ is defined as

for positive constants $\ell_{b}$, $m_{b}$, and $M_{b}$ satisfying

Finally, we define $f_{yi}$ as

where the function $g\colon[0,\infty)\to[0,\infty)$ is defined by:

and the parameters $\gamma_{1}$, $\gamma_{2}$, and $\ell_{y}$ are positive constants that satisfy

We can now state the problem studied in this paper.

In this section, we present a sufficient condition that guarantees that no pairs of sheep will collide within a swarm. We begin by presenting the main results of this section. Within the theorem, the maximum shepherd force for one sheep is defined as follows:

which plays an important role in this process.

In the remainder of this section, we present a proof of Theorem 1 using the following notation. For all $i,j\in\mathcal{N}$, let

The first collision time of the system is defined as follows:

Set $\mathcal{C}$ is defined as

We then define the complement set as

We prove Theorem 1 by contradiction. First, the proof is outlined. Let us introduce the nonnegative function

If a finite-time collision occurs, that is, if the conclusion of Theorem 1 is violated, then $t_{1}$ is finite; therefore, we have $\mathcal{X}_{2}(t_{1})=0$. Conversely, as will be shown later, a careful analysis of the derivative of $\mathcal{X}_{2}(t)$ allows us to show that function $\mathcal{X}_{2}(t)$ increases on the interval $[t_{1}-\tau,t_{1})$ for a sufficiently small $\tau>0$ under inequality (12). This observation leads to $\mathcal{X}_{2}(t)$ being negative on the interval, which contradicts the intrinsic nonnegativity of function $\mathcal{X}_{2}$.

To proceed further, we first evaluate the derivative of $\mathcal{X}_{2}$. A straightforward calculation shows that the derivative can be decomposed as follows:

where

Because the symmetry [24] of these expressions with respect to indices $i$ and $j$ allows us to show $\mathcal{I}_{2}=\mathcal{I}_{3}$ and $\mathcal{I}_{4}=\mathcal{I}_{5}$, we obtain

To further evaluate the derivative $\dot{\mathcal{X}}_{2}$, we provide lower bounds for $\mathcal{I}_{2}$ and $\mathcal{I}_{4}$. A lower bound for $\mathcal{I}_{2}$ is presented in the following proposition:

Then, we derive a lower bound for $\mathcal{I}_{4}$. We begin by presenting the following technical lemma. The proof of the lemma 3 is omitted due to space limitations.

Using Lemma 3, we can prove the following lower bound for $\mathcal{I}_{4}$. The proof of the proposition 4 is omitted due to space limitations.

We are now ready to prove Theorem 1.

From Theorem 1, we conclude that no collision occurs, the dynamics of the system are continuous, and we do not need to consider collisions when using the model (1) to design a controller for regulating the target to the origin. In this section, we describe the design of the controller for the shepherd to ensure that the target sheep is controlled to the origin. We specifically use the Lyapunov stability theorem to ensure that the shepherd has ability of locally regulating the target to the origin.

Let us denote the sum of the interaction forces of the target as

We remark that this quantity is supposed to be available by the shepherd. Then, using (7), we can rewrite the velocity of the target as

The system (40) is controllable when $f_{y}^{\max}\geq w_{T}$ for $t>0$. The controller design should base on the controllability of the (40). Thus we select the gains that satisfy

To realize our control objective, we set the desired position of the shepherd, denoted by $y_{d}(t)\in\mathbb{R}^{2}$, as

where $K_{1}$ is a positive constant. This choice is intuitive because if the shepherd is always at the desired position specified above, the target sheep will be regulated to the origin by the repulsive force of the shepherd. Then, we quantify the mismatch between the actual position $y$ and desired position of the shepherd as

In this study, we consider the following first-order dynamics for the shepherd:

The following theorem shows that we can construct a feedback controller to locally achieve the control objective stated in Problem 1.

In this section, we demonstrate the effectiveness of the proposed feedback controller in regulating one target sheep to the origin in a large-scale swarm. The initial conditions for the first experiment were set to $x_{T}(0)=[13,14]^{\top}$ and $y(0)=[17,17]^{\top}$. We randomly generated $N=200$ non-overlapping sheep in the square region of $[0,17]^{2}$. We selected the constants as $m_{a}=10$, $m_{b}=5$, $M_{b}=10$, $\ell_{y}=0.5$, $\gamma_{1}=1$, $\gamma_{2}=e^{2}$, $l=0.5$, and $K_{1}=1.5$. These constants were selected to satisfy inequality (12) in Theorem 1; therefore, the swarm guarantees the collision-free property and the controller can safely regulate the target to the origin.

To further illustrate the proposed method’s effectiveness, we compare its performance with a baseline strategy from the farthest-agent algorithm [23]. The farthest-agent algorithm utilizes the sum of the following two vectors as the velocity input for the shepherd, where one vector drives the shepherd to approach the target, and the other vector avoids the shepherd exceeding the target.

Figure 1(a) depicts two sets of comparative experiments, illustrating the change of the swarm trajectories by the proposed and baseline methods. As shown in Figure 1(a), we conducted comparative experiments to show the swarm trajectories change from $t=0$ until either $\lVert x_{T}\rVert<0.1$ or a designated time limit of $t=200$. Figures 1(a) and 1(b) show the overall trajectories of the proposed and baseline methods on a $N=10$ swarm with the same initial placements, respectively, while Figures 1(c) and 1(d) show those on a $N=50$ swarm with the same initial placements. We observe that the baseline method guides the target around the origin but fails in making the target’s position converge to the origin. On the other hand, the proposed method is guaranteed to make the tracking error converge to zero and, therefore, succeeds in guiding the target to the origin asymptotically.

We further illustrate the effectiveness of the proposed method in different swarm sizes (from $N=10$ to $N=200$) as shown in Figure 2. The dashed line represents the baseline method, and the solid line represents the proposed method. Each plot is obtained by averaging 100 different simulations with different initial placements. From Figure 2, we observe the proposed method regulates the target to the origin before $t=200$ for all different swarm sizes and performs better than the baseline. This further corroborates the conclusion of the comparative analysis of the two methods mentioned above.

In this study, we investigated the problem of using a shepherd to accurately guide a target in a large swarm to its destination. We first analyzed the properties of the swarm and rigorously proved through mathematical analysis that under certain inequality constraints, no sheep in the swarm will collide. Subsequently, we used the Lyapunov stability principle to design a shepherd control law that precisely guides the target to the origin. We also compared our model with a baseline shepherding control algorithm. Our experimental results demonstrate that the shepherd can precisely guide the target to the goal in both small and large swarms. Our future work includes finding necessary and sufficient conditions of the collision-free property of shepherding swarm.