Self-Organized Patchy Target Searching and Collecting with Heterogeneous Swarm Robots Based on Density Interactions

Abstract

The issue of searching and collecting targets with patchy distribution in an unknown environment is a challenging task for multiple or swarm robots because the targets are unevenly dispersed in space, which makes the traditional solutions based on the idea of path planning and full spatial coverage very inefficient and time consuming. In this paper, by employing a novel framework of spatial-density-field-based interactions, a collective searching and collecting algorithm for heterogeneous swarm robots is proposed to solve the challenging issue in a self-organized manner. In our robotic system, two types of swarm robots, i.e., the searching robots and the collecting robots, are included. To start with, the searching robots conduct an environment exploration by means of formation movement with Levy flights; when the targets are detected by the searching robots, they spontaneously form a ring-shaped envelope to estimate the spatial distribution of targets. Then, a single robot is selected from the group to enter the patch and locates at the patch’s center to act as a guiding beacon. Subsequently, the collecting robots are recruited by the guiding beacon to gather the patch targets; they first form a ring-shaped envelope around the target patch and then push the scattered targets inward by using a spiral shrinking strategy; in this way, all targets eventually are stacked near the center of the target patch. With the cooperation of the searching robots and the collecting robots, our heterogeneous robotic system can operate autonomously as a coordinated group to complete the task of collecting targets in an unknown environment. Numerical simulations and real swarm robot experiments (up to 20 robots are used) show that the proposed algorithm is feasible and effective, and it can be extended to search and collect different types of targets with patchy distribution.

1. Introduction

The task of target searching and collecting (TSC) refers to the issue of searching, finding, and gathering scattered targets in an unknown environment in some way to a specified location. In the field of robotic studies, the TSC task is related to a wide range of potential applications such as the collection of raw materials in the industry, harvesting of field crops in agriculture, garbage collection in space or in the ocean, search and rescue in earthquake or fire disaster, etc. Meanwhile, the research of TSC task also has huge potential application value in intelligent production and Industry 5.0 scenarios [1,2,3,4,5]. Swarm robots can autonomously deliver the collected raw materials to the production line without the need for manual material addition. There is no doubt that the TSC task is very challenging for autonomous robots, which is partly due to the fact that the space to be searched is often much larger than the perceived range of robots, and the environmental landscape is unknown, time changing, and uncertain; other reasons come from the spatial–temporal randomness of the distribution of the target and the time and energy constraints of robots in handling the targets.

If we try to solve the TSC task with a single robot, it is undoubtedly very difficult; however, if we use a swarm of robots for searching and collecting the scattered targets, it is expected to be very efficient because a group of robots can work in parallel at different locations in a coordinated manner. Such swarm robots, which mimic the social animals in nature, are a rapidly emerging domain in recent years. For the biological swarming systems, their group-level collective behaviors are complex and robust, and they are often subjected to excellent characteristics such as self-organization, self-adaptation, and scalability, which emerged from the relatively simple inter-mate interactions and/or certain coordination mechanisms. Correspondingly, the swarm robots enable a group of robots to coordinate their actions and cooperate with each other without relying on a central controller or a global communication network. This feature makes swarm robots suitable for various applications that require scalability, robustness, and flexibility, such as target search [6,7,8], target handling [9,10,11], multiple subgroups collecting [12], multi-target trapping [13], group chase [14,15], invasion and defense [16,17], etc. The great advantages of swarm robots provide us with a new way to solve the TSC task.

In nature, many group-living animals, such as ants, termites, bees, fish schools, and bird flocks, can perform the target (i.e., food, nest, or mate) searching and/or collecting task very efficiently, which is known as the group foraging behavior or homing behavior in biology and ecology [18]. The acquisition of this excellent capability for animal groups is the product of evolution over billions of years, and it can be treated as a prototype to inspire us to design some simple and efficient robotic solutions for the TSC task. In this paper, inspired by the mechanisms of the division of labor and recruitment in animal groups, such as scout ants/bees for searching and patrolling [19,20], and working ants/bees for carrying food [21], we propose a novel self-organized strategy to solve the TSC task by using heterogeneous swarm robots. Two kinds of swarm robots are introduced in our robotic system: the searching robots are responsible for finding, evaluating, and marking the targets; the collecting robots are equipped with a pair of the brush to collect and stack the targets in situ. Numerical simulations and real swarm robot experiments (up to 20 robots are used) show that the proposed methodology is feasible and effective for searching and collecting different types of targets with patchy distribution in an unknown environment.

1.1. Related Work

Target searching in an unknown environment is one of challenging issues in swarm robotics. Many of the current works featuring target search can be viewed as optimization search algorithms performed in physical space. In [22], the authors proposed a particle swarm optimization (PSO) technique for swarm robots to perform single- and multi-target searches. The algorithm improves search efficiency and success rates by dynamically modifying the exploration range and speed restriction, and a collaboration factor is introduced to determine whether to share information or request help from other robots. Another work in [23] outlined the basic characteristics, variations, difficulties, and trends of PSO for swarm robots to perform target search missions. The authors pointed out that PSO faces challenges in target searching such as dealing with complex constraints and improving computational and communication efficiency while enhancing robustness and fault tolerance. In [24], a multi-target search approach based on learning automation (LA) is reported by dividing the search space into sub-regions and assigning an LA to each to learn the probability of the target’s existence. A more systematical review of the advantages, drawbacks, and applicability of various tactics for target-searching problems can be found in [25]; the authors emphasized the need to take into account the complex and dynamic environments, to improve robot–human interaction, and to increase the resilience and scalability of swarm robots. Although the optimization-based target search approaches are effective, they still suffer from a general limitation in that they require robots with global sensing and memory capabilities to obtain environmental gradient or optimal position information.

When targets are found, the following task is to collect them, which aims to gather and stack the dispersed targets in some way to a certain location. Recently, researchers have become interested in the multi-robot collaborative collection strategy, the main motivation of which is to collect targets in the region through full area coverage. For example, in [26], a coverage path-planning method based on area partitioning was developed to tackle the multi-robot cooperative coverage problem by creating a unique route for each robot. A bio-inspired neural network is designed in [27] to model the workspace and guide a swarm of robots for area coverage missions. Each robot regards other robots as moving obstacles and autonomously generates a path from the neural activity landscape and previous robot position. In [28], the authors present a quad-tree data structured method for complete area coverage path planning in a known environment with arbitrary shape obstacles by applying the concepts of contour map and spanning tree.

In fact, the spatial distribution of targets is usually not uniform; it tends to aggregate into some dense patches with varying sizes and densities. For example, the distribution of minerals, water pollutants, and marine debris all display a dispersed pattern, with a patchy distribution. This feature leads to the fact that while the target collection method of swarm robots based on regional full coverage is efficient in collecting homogeneously distributed targets, it will lose its efficiency dramatically when encountering targets with patchy distribution. This is due to the presence of large-scale sparse target regions and/or non-target regions. Additionally, most existing methods focus on only one part of the TSC task, i.e., either target search or target collection, and they cannot perform a full task flow from target search to the collection using the same modeling framework. The last point is that the existing target search algorithms based on the gradient or evolutionary algorithms require robots to access some global information as the primary input to support their decision making, but this global information is difficult to obtain for swarm robots with only local perception capabilities.

1.2. Motivation and Contributions

In nature, food and many subsistence-related resources show patchy distribution. Examples include patches of blooming nectar sources, a fruiting tree, a plankton community [29], and an aggregation of gray slugs in the cultivated field [30]. For such a patchy and sparse distribution of targets, biological foragers have developed very efficient searching and foraging strategies, i.e., divisions of labor and recruitment. These mechanisms have developed within many biological groups, such as scouts and workers in social ant/termite/bee colonies, to improve the efficiency of group‘s searching and foraging performance with specialized behaviors.

On the other hand, animals can draw on spatial density or concentration signals as the mediator to drive behavioral adjustments or make decisions. For example, ants change their colony structure based on pheromone concentration [31] and optimize nest site selection according to the density of food sources [32]; ants, as a team, deposit soil wastes in constructing nests also according to the density of wastes already present [33]. As the primary driving factor of these behaviors, the density information characterizes the spatial distribution of objects, and it plays a crucial role in animals’ decision making.

As for the control of swarm robots, most of the current work employs the motion information of neighbors such as positions, headings, and velocities to coordinate the motion between them. There is still an insufficient investigation and utilization of density information to reflect the group-level configurations. In our previous work [12,13,34], we use a novel density-based framework to design the control policy of swarm robots; these studies showed that density-based interactions can endow the swarm robots with various unique capabilities, such as spontaneously forming multiple-/single-ring configurations and symmetric shrinking and expanding [34]. These backgrounds motivate us to address the TSC task by using the density-based control framework for swarm robots.

In this paper, inspired by the mechanisms of the division of labor and recruitment in biological groups, and drawing on the density-driven control framework of swarm robots, we propose a bio-inspired methodology to solve the TSC task by employing heterogeneous swarm robots in a self-organizing manner. Our methodology first uses the Gaussian kernel function [35] to establish the density field of the robot’s neighborhood environment based on which robot–robot and robot–target interactions are formed. Two kinds of robots are introduced in our swarm robotic system: one is the searching robots, which are responsible for searching the unknown environment, finding the patchy targets, and evaluating the distribution of targets; another is the collecting robots, which are equipped with a brush to collect and stack the targets by edge surrounding and spiral shrinking behaviors. Uniquely, we treat the maximum density point of patchy targets as the in situ accumulation location. This strategy greatly saves time and energy costs for target transportation. In this way, two groups of robots work cooperatively to search for and collect the targets with patchy distribution. Below, we summarize the main contributions of this paper:

(1) A heterogeneous swarm-robotic system is developed to solve the TSC task, in which the searching robots and collecting robots are specialized to perform different sub-tasks; and their behaviors are coordinated by a recruitment mechanism.

(2) We establish a density field to characterize the environment and model the distribution of patchy targets by using the Gaussian kernel function; the spatial density information is employed as the primary guiding factor in the control of swarm robots.

(3) The spatial distribution of patchy targets is evaluated distributively by the edge-surrounding behavior of swarm robots.

1.3. Structure of the Paper

The remainder of this paper is organized as follows. In Section 2, we provide a detailed description of the TSC problem and introduce relevant prior knowledge that will aid in understanding our proposed methodology. In Section 3, we explain our methodology in detail, including the behavioral design principles and the implementation policy of the searching robots and the collecting robots. Numerical simulations and real robotic experiments are presented to verify the effectiveness of our approach in Section 4. Finally, Section 5 concludes this paper by summarizing our results and discussing some potential avenues for future work.

2. Problem Formulation

Consider an unknown space where there are M separate, fragmented target units, and they need to be collected together. These target units are unevenly distributed but clustered to form one or more relatively high-density patchy distribution patterns, just like the fallen leaves under the tree and the patches of flower fields. We assume that (1) the target unit is small-size and light-weight enough to be pushed by a single robot; (2) each target patch consists of a large number of separate units which can be detected and recognized by robot’s sensor; and (3) the distribution range and location of the target patches are unknown.

The swarm-robotic system is heterogeneous, in which two subgroups of robots with different physical capabilities are consisted, which are referred to as the searching robots (i.e., searcher) and the collecting robots (i.e., collector), respectively; see the illustration scenario shown in Figure 1. The $N_{\mathrm{s}}$ searching robots move quickly and have a strong sense of perception but cannot collect targets. The $N_{\mathrm{c}}$ collecting robots are equipped with a pair of brush modules and a high reduction ratio motor, enabling them to clean and push heavy targets. Note that each searcher (collector) has a limited sensing range to detect its local surrounding, including con-specific robots, heterogenic robots and targets. The sensing range of the searcher and collector are denoted as $R_{\mathrm{s}}$ and $R_{\mathrm{c}}$, respectively. We assume that (1) the collecting robot moves noticeably slower than the searching robot; (2) the searching robot’s sensing range is more larger than that of the collecting robot; and (3) the explicit information is not allowed to be exchanged and shared between robots in all task processes except for supporting the recruitment.

Based on the above description, the TSC mission can be described as follows:

Design a distributed control policy for heterogeneous swarm robots (i.e., searchers and collectors) to drive the team of searchers to autonomously search and locate the target patch and also drive the team of collectors to collect and stack the patchy targets into a pile with the following two constraints:

(1) Neither searchers nor collectors know any global information about the center and distribution range of targets in advance.

(2) Once targets are found by searchers, collectors can be recruited with certain minimalist long-range signaling or communication.

3. Preliminary

In this section, some fundamental concepts and methods to be employed later are provided, including the density estimation method based on kernel function, the motion control model of swarm robots, and the density-field-driven control framework for swarm robots.

3.1. Spatial Density Estimation

For a swarm of robots or a cluster of targets, all referred to as objects in this subsection, spatial distribution density is a very intuitive and natural description for such objects that often occurs as a group or a cluster in their operating space. Intuitively, the smaller the spacing between these objects, the higher the spatial density of their distribution. We use spatial density as the primary measurement to characterize the distribution of robots (targets) and also utilized it to design swarm robot behaviors.

To estimate the spatial density field of objects (i.e., the robots and targets) by our swarm robots, we adopt the kernel-function-based definition of spatial density from our previous works [13,34]. For an object $i$ with position $\mathrm{X}$, a locally spatial density field is attached to its neighborhood, and the intensity (i.e., density) of this spatial-relevant field decays monotonically with increasing distance from the object. Formally, the density corresponding to any position $\mathrm{Y}$ located near the object $i$ is defined as:

$${\rho }_{\mathrm{Y}}={\displaystyle \sum _{}^{}W\left(\mathrm{X},\mathrm{Y}\right)}={\displaystyle \sum _{}^{}\exp \left(-\frac{{\Vert \mathrm{X}-\mathrm{Y}\Vert }^2}{2{\sigma }^2}\right)}$$

(1)

where $W(*,*)$ is a kernel function used to calculate the spatial density; and $\sigma >0$ is a scalar parameter to adjust the supporting range of density field generated by each object, as shown in Figure 2a. If there is more than one object around the position $\mathrm{Y}$, their density fields need two be stacked, i.e., the summation operation shown in Equation (1).

In practical tasks, we can choose various types of kernel functions and set different parameters for them to estimate the spatial density field. In our work, the Gaussian-like function, see the right part of Equation (1), is used as the kernel function to estimate the spatial density of searchers, collectors, and targets in the robot’s sensing range. In Figure 2b, a case of spatial density field is shown which is formed from two target patches, where the density is mapped with different colors: the darker the color, the higher the density value of the region, i.e., the more targets are distributed there. Clearly, the Gaussian-like kernel function can well characterize the spatial distribution of targets with patch distribution.

3.2. Model of Swarm Robots

The swarm robots (i.e., the searching robot and the collecting robot) move over flat ground with a constant transitional speed $v_0^{}$. The position and direction (unit vector) of swarm robot $i$ at time $t$ are denoted as $x_i^{}(t)$ and $v_i^{}(t)$, respectively. The motion state of swarm robot $i$ at time $t+\Delta t$ satisfies:

$$\begin{array}{c}{x}_i^{}(t+\Delta t)=x_i^{}(t)+v_i^{}(t+\Delta t)v_0^{}\Delta t\\ v_i^{}(t+\Delta t)=v_i^{}(t)+{\omega }_i^{}(t+\Delta t)\Delta t\\ {\omega }_i^{}(t+\Delta t)=\min \left(\frac{\angle (v_i^{}(t),u_i^{}(t+\Delta t))}{\Delta t},{\omega }_{\max }^{}\right)\end{array}$$

(2)

where $u_i^{}(t+\Delta t)$ is the swarm robot’s control input and denotes the desired direction of motion at next control cycle $t+\Delta t$; ${\omega }_i^{}(t+\Delta t)$ is the angular speed of robot at time $t+\Delta t$, ${\omega }_{\max }^{}$ is its maximum angular speed, and $\angle (*,*)$ stands for calculating the angle between two (unit) vectors.

The control inputs of the searching robot ($u_i^{\mathrm{s}}$) and the collection robot ($u_i^{\mathrm{c}}$) can be given as following:

$$\begin{array}{l}{u}_i^{\mathrm{s}}(t+\Delta t)=u_i^{\mathrm{self}}(t+\Delta t)+u_i^{\mathrm{rep}}(t+\Delta t){+}^{\mathrm{s}}{u}_i^{\mathrm{task}}(t+\Delta t)\\ u_i^{\mathrm{c}}(t+\Delta t)=u_i^{\mathrm{self}}(t+\Delta t)+u_i^{\mathrm{rep}}(t+\Delta t){+}^{\mathrm{c}}{u}_i^{\mathrm{task}}(t+\Delta t)\end{array}$$

(3)

where $u_i^{\mathrm{self}}$ and $u_i^{\mathrm{rep}}$ are two instinctive driven forces for robot. The self-driven term $u_i^{\mathrm{self}}$ is used to guide the robot moving with its current direction:

$$u_i^{\mathrm{self}}(t+\Delta t)={\lambda }^{\mathrm{self}}\frac{x_i(t)-x_i(t-\Delta t)}{\Vert x_i(t)-x_i(t-\Delta t)\Vert }$$

(4)

where ${\lambda }^{\mathrm{self}}>0$ is the self-driven weight. Collision avoidance term $u_i^{\mathrm{rep}}$ is used to achieve collision-free motion between robots, which is achieved by repulsive forces as:

$$u_i^{\mathrm{rep}}(t+\Delta t)=-{\lambda }^{\mathrm{rep}}{\displaystyle \sum _{j\in N_i}\left(\frac{x_j(t)-x_i(t)}{{\Vert x_j(t)-x_i(t)\Vert }^2}\right)}$$

(5)

where ${\lambda }^{\mathrm{rep}}>0$ is the collision avoidance weight. $N_i$ is the set of other individual robots within the robot’s sensing range.

The remaining terms, i.e., ${}^{\mathrm{s}}{u}_i^{\mathrm{task}}$ and ${}^{\mathrm{c}}{u}_i^{\mathrm{task}}$, are task-related control inputs for searchers and collectors, respectively. They are the core of this work, and we will describe them in detail in the Methodology section.

3.3. Density-Driven Control Framework for Swarm Robots

In our earlier studies [13,34], based on the concept of the spatial density field, we proposed a novel density-based control framework for swarm robotics. In [13], we used the density-based control framework to achieve swarm robots’ trapping of multiple targets, and the properties of the density-based control framework are analyzed in detail in [14]. Here, we take the searching robots as an example to briefly review the framework.

As shown in Figure 3, if a robot’s sensing range contains both other robots and targets, the density value at the location of this robot is the superposition of the density fields generated by both. Based on the kernel function-based density estimation method, we can give the definition of the real-time density of the searching robot $i$ as:

$$\begin{array}{l}{\rho }_i^{\mathrm{s}}(t)={\displaystyle \sum _{j\in N_i^{\mathrm{s}}}W\left(x_i^{\mathrm{s}}(t),x_j^{\mathrm{s}}(t)\right)}+{\displaystyle \sum _{k\in G_i^{\mathrm{s}}}W\left(x_i^{\mathrm{s}}(t),g_k(t)\right)}\\ ={\displaystyle \sum _{j\in N_i^{\mathrm{s}}}\exp \left(-\frac{1}{2{\sigma }^2}{\left(\frac{\Vert x_i^{\mathrm{s}}(t)-x_j^{\mathrm{s}}(t)\Vert }{R_{\mathrm{s}}}\right)}^2\right)}+{\displaystyle \sum _{k\in G_i^{\mathrm{s}}}\exp \left(-\frac{1}{2{\sigma }^2}{\left(\frac{\Vert x_i^{\mathrm{s}}(t)-g_k(t)\Vert }{R_{\mathrm{s}}}\right)}^2\right)}\end{array}$$

(6)

where $g_k(t)$ is the target position, and $x_i^{\mathrm{s}}(t)$, $x_j^{\mathrm{s}}(t)$ are the positions of the searchers $i$ and $j$. $N_i^{\mathrm{s}}$ and $G_i^{\mathrm{s}}$ are the set of neighboring robots and targets that can be perceived.

$$N_i^{\mathrm{s}}=\left\{j|d_{ij}<R_{\mathrm{s}},j=\{1,\cdots ,N_{\mathrm{s}}\}\backslash i\right\}$$

(7)

$$G_i^{\mathrm{s}}=\left\{k|d_{ik}<R_{\mathrm{s}},k=1,\cdots ,M,i=1,\cdots ,N_{\mathrm{s}}\right\}$$

(8)

where $d_{ij}$ denotes the distance between two robots, and $d_{ik}$ denotes the distance between the robot $i$ and target $k$. For the collecting robot, $N_i^{\mathrm{c}}$ and $G_i^{\mathrm{c}}$ have the same meanings.

Then, the control policy of the robot based on the density information can be presented as follows:

$$u_i^{\mathrm{des}}(t+\Delta t)=\frac{1}{{\rho }_i^{\mathrm{s}}(t)}\left({\left(\frac{{\rho }_i^{\mathrm{s}}(t)}{{\rho }_0^{\mathrm{s}}(t)}\right)}^7-1\right)[{\lambda }^r{\displaystyle \sum _{j\in N_i^{\mathrm{s}}}{\nabla }_{i}W\left(x_i^{\mathrm{s}}(t),x_j^{\mathrm{s}}(t)\right)}+{\lambda }^{\mathrm{g}}{\displaystyle \sum _{k\in G_i^{\mathrm{s}}}{\nabla }_{i}W\left(x_i^{\mathrm{s}}(t),g_k(t)\right)}]$$

(9)

where ${\lambda }^{\mathrm{r}}>0$ is the strength of robot–robot interaction, ${\lambda }^{\mathrm{g}}>0$ is the strength of robot–target interaction, and ${\rho }_0^{\mathrm{s}}(t)>0$ is the reference density for the searching robot. ${\nabla }_i$ is the gradient operator used to calculate the gradient of the density field at the position of robot $i$.

Based on the density-driven control framework mentioned above, the swarm robot can perform specific abilities such as group splitting, group merging, multi-target trapping, and multi-ring encirclement with remarkable robustness and scalability [34]. This paper focuses on the TSC mission, we employ the density-driven control framework to design the distributed control policy for the searchers and the collectors to search, evaluate, and collect the targets with patch distribution.

4. Methodology

In this section, we first provide a brief overview of the control policy. Then, in Section 4.2 and Section 4.3, the detailed implementation steps of our policy are presented.

4.1. Overview of the Control Policy

To solve the TSC problem with swarm robots, we tend to deploy heterogeneous robots as briefly described in the Problem Formulation section. This setting is inspired by the mechanisms of the division of labor and recruitment in social insets, such as scout ants/bees for searching and patrolling [19,20], and working ants/bees for carrying food [21]. Corresponding to the searcher–collector setting, we divided the TSC mission into two sub-task phases: the search-evaluation phase and the recruitment-collection phase.

During the search-evaluation phase, the searching robots in formation first explore the environment by random wandering behavior. Once a searcher finds a target, the team of searchers try to evaluate the density of the target patch by encircling it. This evaluation provides information about the density at the periphery of the patch. Then, a searcher is randomly selected to enter the interior of the patch to evaluation its internal density and stop at the place with the highest density; i.e., it become a robotic beacon for recruiting the collecting robots.

During the recruitment-collection phase, based on the spatial density information from both inside and outside of the target patch as well as guidance from the robotic beacon, the collecting robots are recruited and move toward the patchy targets. Then, they form a ring-shaped envelope around the patch’s periphery using the edge encirclement behavior. From then on, the collectors continuously shrink their ring formation, pushing the targets inwards to gather.

Eventually, thanks to the cooperation of the two-type robots in these two sub-task phases, the swarm robots will complete the target search-collection task in an unknown environment.

4.2. Search-Evaluation Phase

The task of searching robots consists of three main steps: (1) random wandering in formation to search targets; (2) evaluating the patchy targets by encircling behavior; and (3) establishing a robotic guiding beacon. Figure 4 shows a detailed diagram of these steps.

4.2.1. Random Exploration

The searching robots explore the space when they have not detected any targets (i.e., $G_i^{\mathrm{s}}=Ø$). As shown in Figure 4a, the control input of the searching robot consists of two parts and can be written as:

$${}^{\mathrm{s}}{u}_i^{\mathrm{task}}(t+\Delta t)=u_i^{\mathrm{des}}(t+\Delta t)+u_i^{\mathrm{ran}}(t+\Delta t)$$

(10)

where $u_i^{\mathrm{ran}}$ is a random walk term, which drives the robot to perform a random wandering motion, and $u_i^{\mathrm{des}}$ is a robot formation term, which controls a group of robots to move in formation.

There are various random walk models that can be used for designing the term $u_i^{\mathrm{ran}}$, such as the Markov chain [36], random walk [37], and Lévy flight [38]. We choose Lévy flight as the random walk model because it is frequently used to describe the foraging behaviors of animals [39,40]. The control term $u_i^{\mathrm{ran}}$ can be expressed as follows:

$$\begin{array}{l}{u}_i^{\mathrm{ran}}(t+\Delta t)=\frac{(x_i^{\mathrm{s}}(t)+\psi S)-x_i^{\mathrm{s}}(t)}{\Vert (x_i^{\mathrm{s}}(t)+\psi S)-x_i^{\mathrm{s}}(t)\Vert },\\ S=\frac{u}{{\left|v\right|}^{\frac{1}{\beta }}},\gamma ={\left\{\frac{\Gamma \left(1+\beta \right)\sin \left(\frac{\pi \beta }{2}\right)}{\beta \Gamma \left(\frac{1+\beta }{2}\right)2^{\frac{\beta -1}{2}}}\right\}}^{\frac{1}{\beta }}\end{array}$$

(11)

where $x_i^{\mathrm{s}}(t)$ is the position of the searching robot and $\psi$ is the step control parameter. Here, we set $\psi =1,\beta =1.5$, $u\sim N\left(0,{\gamma }^2\right),v\sim N\left(0,1\right)$. Based on this control setting, the searching robot can perform the random walk during the search phase with a Lévy flight trajectory.

For formation control of the searching robots, we employ the density-based control framework. Specifically, we choose the weights of the term $u_i^{\mathrm{des}}$ with ${\lambda }^{\mathrm{r}}\gg {\lambda }^{\mathrm{g}}$ in Equation (9). This control setting can achieve a circular formation behavior for the searching robots. Meanwhile, we can set different values of ${\rho }_0^{\mathrm{s}}(t)$ to adjust the radius of robotic formation to suit the needs of different scenarios.

As shown in Figure 4a, combining these two behaviors, the searching robots can move in formation with the Lévy flight trajectory, achieving the goal of exploring the unknown environment for searching targets in the way of random wandering. Note that this swarming search behavior of robots compensates for the limitation that a single robot cannot perform target evaluation and also provides support for the subsequent steps, i.e., target evaluation.

4.2.2. Target Evaluation

Once one searching robot detects a target (i.e., $G_i^{\mathrm{s}}\ne Ø$), the searching robots immediately stop their exploring behavior and immediately start the target evaluation behavior. As shown in Figure 4b, the core of this process is that robots need to form a ring-shaped envelope around the target patch, and it can also be achieved by adjusting the parameter of $u_i^{\mathrm{des}}$, i.e.,

$${}^{\mathrm{s}}{u}_i^{\mathrm{task}}(t+\Delta t)=u_i^{\mathrm{des}}(t+\Delta t)$$

(12)

with the weights ${\lambda }^{\mathrm{g}}\gg {\lambda }^{\mathrm{r}}$. This setting makes the searching robots more attracted by the targets rather than other conspecifics, and it leads to the robot swarm moving along the edge of the target patch and eventually forming a single-layer ring structure around these targets.

Note that the searching robots will remain in the encirclement configuration unless a switching condition is satisfied to trigger the next process, i.e., establishing a guiding beacon. To this end, we first define the uniformity ${\eta }_{\mathrm{s}}$ as follows:

$${\eta }_{\mathrm{s}}(t)=\frac{1}{N_{\mathrm{s}}}{\displaystyle \sum _{i=1}^{N_{\mathrm{s}}}{\left(l_i(t)-\frac{1}{N_{\mathrm{s}}}{\displaystyle \sum _{i=1}^{N_{\mathrm{s}}}{l}_i}(t)\right)}^2}$$

(13)

where $l_i(t)=\left(1/N_i\right){\displaystyle {\sum }_{j\in N_i^{\mathrm{s}}}\Vert x_i^{\mathrm{s}}(t)-x_j^{\mathrm{s}}(t)\Vert }$ is the average distance between robot $i$ and other searching robots within its sensing range. Then, as shown in Figure 4b, once a uniformly spaced ring structure is formed, i.e., once ${\eta }_{\mathrm{s}}<{\delta }_{\mathrm{s}}$, the next process starts. Here, ${\delta }_{\mathrm{s}}>0$ is a threshold used to assess the uniformity of the ring structure.

4.2.3. Guiding Beacon Establishment

In this process, a searching robot is selected randomly from the group of searchers to enter the evaluation envelope; then, it will act as a robotic beacon to recruit and guide the collecting robots, as shown in Figure 4c. The robotic guiding beacon is controlled based on the density gradient search behavior evolved from $u_i^{\mathrm{des}}$ by setting ${\rho }_0^{\mathrm{s}}(t)\gg {\rho }_i^{\mathrm{s}}(t)$ and does not require any additional force. The principle of this behavior is that the robot density converges continuously to the desired reference density ${\rho }_0^{\mathrm{s}}(t)$.

It is worth noting that although this control method for the robotic guiding beacon seems simple, the beacon robot will eventually stay at the target patch’s density center, i.e., the position with the highest density in the target patch, which is determined by the nature of the density-driven control method. If we take this position as the stacking point of targets, the collection time and energy consumption for swarm robots can be saved.

4.3. Recruitment-Collection Phase

The task of collecting robots consists of two main steps: (1) moving toward and surrounding the patchy targets and (2) collecting the target patch based on the shrinkage strategy.

4.3.1. Edge-Surrounding Behavior

Once the recruiting signal is broadcast by the robotic guiding beacon, the team of collectors immediately sets the location, i.e., the density center of the target patch, as the goal and moves toward it. This goal-oriented collective behavior can be implemented as:

$${}^{\mathrm{c}}{u}_i^{\mathrm{task}}(t+\Delta t)=\frac{x_b^{\mathrm{s}}(t)-x_i^{\mathrm{c}}(t)}{\Vert x_b^{\mathrm{s}}(t)-x_i^{\mathrm{c}}(t)\Vert }$$

(14)

where $x_b^{\mathrm{s}}(t)$ and $x_i^{\mathrm{c}}(t)$ are the current position of the guide beacon and the collecting robot $i$, respectively.

In the process of moving, once the target (i.e., $G_i^{\mathrm{c}}\ne Ø$) is detected by one collecting robot. They immediately perform an edge-surrounding behavior to trap the patch of targets. Essentially, this edge-surrounding behavior for collectors is the same as the target evaluation behavior of searchers. The corresponding control input can be described as:

$${}^{\mathrm{c}}{u}_i^{\mathrm{task}}(t+\Delta t)=u_i^{\mathrm{des}}(t+\Delta t).$$

By this control method, a group of collectors can form a ring configuration with uniform spacing around the target patch, and they will remain in the encirclement configuration unless a certain switching condition is satisfied to trigger the next target collecting process. Here, we use the same trigger condition as those given in Section 4.2.2, which will not be discussed here.

4.3.2. Collecting Behavior

In this process, the encirclement configuration formed by collectors shrinks inward to push the peripheral targets moving toward the center of the target patch. To completely cover the target distribution area, a vortex-like shrinkage strategy is designed to drive the motion of collectors. As shown in Figure 5, the control input of the collecting robot consists of two parts:

$${}^{\mathrm{c}}{u}_i^{\mathrm{task}}(t+\Delta t)=u_i^{\mathrm{des}}(t+\Delta t)+u_i^{\mathrm{scr}}(t+\Delta t)$$

(15)

where $u_i^{\mathrm{scr}}$ is a tangential force and $u_i^{\mathrm{des}}$ is a contraction force. These two forces are jointly driven by the collecting robots moving around the patch and gradually shrinking inward to form a vortex-like trajectory. Here, the term $u_i^{\mathrm{scr}}$ is constructed based on the robotic guide beacon:

$$u_i^{\mathrm{scr}}(t+\Delta t)=A\frac{x_b^{\mathrm{s}}(t)-x_i^{\mathrm{c}}(t)}{\Vert x_b^{\mathrm{s}}(t)-x_i^{\mathrm{c}}(t)\Vert }$$

(16)

where $x_b^{\mathrm{s}}(t)$ is the location of the robotic beacon and $A\in {ℝ}^{1\times 2}$ represents the rotation matrix.

The contraction force can also be achieved via our density-based control framework: specifically, by increasing the reference density ${\rho }_0^{\mathrm{c}}$ gradually in $u_i^{\mathrm{des}}$. If the shrinking condition is satisfied at time $t_1$, we increase ${\rho }_0^{\mathrm{c}}$ adaptively with the following rule:

$${\rho }_0^{\mathrm{c}}(t)={\rho }_0^{\mathrm{c}}(t-1)+\frac{{\rho }_{\mathrm{b}}^{\mathrm{s}}(t)-{\rho }_0^{\mathrm{c}}(t)}{M}$$

(17)

where ${\rho }_{\mathrm{b}}^{\mathrm{s}}(t)$ is the density at the position of the guiding beacon, and M is the number of target units. In this way, ${\rho }_0^{\mathrm{c}}(t)$ grows incrementally based on ${\rho }_0^{\mathrm{c}}(t-1)$. As a result, this strategy can accelerate the contraction rate at the start of the collecting process and obtain a precise shrinkage at the collecting end stage, which can create a balance between the mission’s speed and efficiency.

5. Numerical Simulations

In this section, we conducted simulation experiments to verify the effectiveness of our proposed control policy for swarm robots to solve the TSC mission in an unknown environment. All simulation experiments in this section are based on MATLAB implementation.

5.1. Simulation Set-Up

To simulate the movement of targets being pushed by robots, we define $D_i$ as the set of robot $i$ that is in contact with target $k$. If $D_i=Ø$, the target is stationary in place; if $D_i\ne Ø$, for simplicity, the target moves at the same speed as the robot. The motion equation of the target is updated as:

$$\begin{array}{l}{g}_k(t+\Delta t)=g_k(t)+v_k^{\mathrm{g}}(t+\Delta t)v_0^{\mathrm{c}}\Delta t\\ v_k^{\mathrm{g}}(t+\Delta t)=-{\displaystyle \sum _{i\in D_i}\frac{x_{{}_i}^{\mathrm{c}}(t)-g_k(t)}{\Vert x_{{}_i}^{\mathrm{c}}(t)-g_k(t)\Vert }}\end{array}$$

(18)

where $g_k(t)$ and $v_k^{\mathrm{g}}(t)$ denote the position and velocity of target $k$ at time $t$, respectively. In addition, the robot–target contact is defined as the distance between robots and the target being less than 60 mm.

To quantitatively characterize the task process and evaluate the effectiveness of target collection, three metrics are introduced: (1) the average density of the searching robots,${\overline{\rho }}^{\mathrm{s}}$; (2) the average density of the collecting robots, ${\overline{\rho }}^{\mathrm{c}}$; (3) the area of the convex packet formed by the collecting robots, ${\mathrm{S}}^{\mathrm{r}}$. Formally, ${\overline{\rho }}^{\mathrm{s}}$ and ${\overline{\rho }}^{\mathrm{c}}$ are defined as:

$$\begin{array}{l}{\overline{\rho }}^{\mathrm{s}}(t)=\left({\displaystyle \sum _{i=1}^{N_{\mathrm{s}}}{\rho }_i^{\mathrm{s}}(t)}\right)/N_{\mathrm{s}}\\ {\overline{\rho }}^{\mathrm{c}}(t)=\left({\displaystyle \sum _{i=1}^{N_{\mathrm{c}}}{\rho }_i^{\mathrm{c}}(t)}\right)/N_{\mathrm{c}}\end{array}$$

(19)

where $N_{\mathrm{s}}$ and $N_{\mathrm{c}}$ are the number of searchers and collectors, and ${\rho }_i^{\mathrm{s}}(t)$ and ${\rho }_i^{\mathrm{c}}(t)$ are the corresponding density of them. The calculation of the convex package can be found in [41]. The simulation parameters are set as: $v_0^{\mathrm{c}}=10\mathrm{mm}/\mathrm{s}$, $v_0^{\mathrm{s}}=20\mathrm{mm}/\mathrm{s}$, ${\omega }_{\max }=28.65\mathrm{deg}/\mathrm{s}$, $R_{\mathrm{s}}=0.7\mathrm{m}$, $R_{\mathrm{c}}=0.5\mathrm{m}$, ${\lambda }^{\mathrm{self}}=1$, ${\lambda }^{\mathrm{rep}}=20$, ${\lambda }^{\mathrm{r}}=100$, ${\lambda }^{\mathrm{g}}=1000$, ${\delta }_{\mathrm{s}}=0.5$, ${\delta }_{\mathrm{s}}=0.1$, ${\rho }_0^{\mathrm{s}}=1.5$, $\Delta t=0.1 \mathrm{s}$.

5.2. Case 1: Circle-like Target Patch

In this simulation, there are 5 searchers and 20 collectors employed to search and collect a target patch with 26 units. Figure 6a shows the trajectory of the searching robots in the search-evaluation phase. Figure 6b shows the trajectory of the collecting robots in the recruitment-collection phase. The evolving curves of three metrics, i.e., ${\rho }^{\mathrm{s}}$, ${\rho }^{\mathrm{c}}$ and ${\mathrm{S}}^{\mathrm{r}}$, are shown in Figure 7.

Initially, target patch and swarm robots are deployed far away from each other. Subsequently, the team of searching robots begins to explore space in the form of Levy flight, and the random searching phase takes about 62 s ($\mathrm{T}=0\sim 62\mathrm{s}$). Because there is no target detected by the searching robots at this period, the value of ${\overline{\rho }}^{\mathrm{s}}=0$. Soon afterward, one target is detected by searching robots at $\mathrm{T}=63\mathrm{s}$, target evaluation is started, and the group’s average density remains constant at this stage. Then, at time $\mathrm{T}=174\mathrm{s}$, the switching condition is satisfied; the No. 5 searcher is selected to enter the target patch, moving along the direction of density increase, and finally standing at the center of the target patch with maximum density ${\overline{\rho }}_5^{\mathrm{s}}=14.6$. Meanwhile, the other searching robots reset their task state and continue to explore the space. At that point, the search-evaluation phase ends.

The recruitment-collection phase begins at time $\mathrm{T}=190\mathrm{s}$. The collecting robots first move toward the target patch, recruiting and guided by the robotic beacon. At $\mathrm{T}=210\mathrm{s}$, one of the collecting robots detects targets, and then, their edge-surrounding behavior of triggered. In this process ($\mathrm{T}=211\sim 300\mathrm{s}$), the collecting robots move along the contour of the target’s density field in sequence until the target patch is surrounded by them. Starting from $\mathrm{T}=301\mathrm{s}$, the behavior switching condition is satisfied, and collectors start to shrink inward to the center of the target patch along with a vortex-like trajectory. The entire collecting process takes about 400 s ($\mathrm{T}=301\sim 700\mathrm{s}$). Correspondingly, the average density ${\overline{\rho }}^{\mathrm{c}}$ increases from 5.2 to 13.9, and the surrounding area by collecting robots ${\mathrm{S}}^{\mathrm{r}}$ decreases from $3.62{\mathrm{m}}^2$ to $0.56{\mathrm{m}}^2$, indicating that the scattered target units have been successfully collected by collecting robots.

5.3. Case 2: Irregular Target Patch

To further verify the adaptability of our methodology when meeting different shapes of target patches, we extend the simulation experiments by considering several irregular patches with different shapes and sizes. Figure 8 shows one example of an irregular target patch, where a star-like target patch is distributed. Clearly, the simulation results show that our control policy can control the swarm robot to evaluate and collect effectively. The searching and collecting process is similar to that the of previous case, and it is not repeated here.

5.4. Comparative Simulation Experiment

To verify the superiority of the algorithm proposed in this paper, the algorithm in this paper and the partitioned plowing algorithm are used to collect the target with the patch distribution shown in Figure 9a. Among them, the blue line is the trajectory of the target being carried, which can be used to approximate the extra energy consumption of the robot. We have already analyzed the specific performance of the algorithm in this paper in Section 5.1 in detail, and we will not discuss it here. As shown in Figure 9c, the partitioned plowing algorithm needs to cover the entire area, which leads to a huge time cost in areas without targets. Moreover, in areas with dense targets, the partitioned plowing algorithm poses a great challenge to the collection ability of a single robot.

The experimental results show that the task completion time of the algorithm in this paper is reduced by 37% compared with the partitioned plowing algorithm, which can effectively reduce the time to complete the task and improve the target collection efficiency. The blue line trajectory of the algorithm in this paper is obviously shorter than that of the partitioned plowing algorithm, which means that the extra energy consumption of the algorithm in this paper when collecting targets is less.

6. Real Robotic Experiments

This section further experiments with real swarm robots to validate our proposed methodology.

6.1. Experiment Set-Up

Figure 10 shows the SwarmBang robot, which is a two-wheel differential driven mobile robot. The robot’s diameter is approximately 80 mm and can push targets weighing ~100 g. The collecting robot’s head is equipped with a pair of brushes for collecting the targets. The brush’s length $l=45\mathrm{mm}$, which is a little bigger than the radius of the robot. The targets to be collected are irregular light-weight foam blocks. In our experiments, the robot’s position, speed, and heading are captured in real time by the NOKOV moving capture system. The real-time position of target units is collected by a top-view camera. Note that although this information is collected globally, the control of each robot only uses the local information within its sensing range. A more detailed description about the SwarmBang robot system can be found in our previous work [12,13,34]. The control parameters of the searching robots are $v_0^{\mathrm{s}}=20\mathrm{mm}/\mathrm{s}$, $R_{\mathrm{s}}=2\mathrm{m}$, ${\delta }_{\mathrm{s}}=0.5$; the collecting robots are set as $v_0^{\mathrm{c}}=10\mathrm{mm}/\mathrm{s}$, $R_{\mathrm{c}}=1\mathrm{m}$, ${\delta }_{\mathrm{c}}=0.1$. The remaining parameters are set as: $\Delta t=0.1\mathrm{s}$, ${\lambda }^{\mathrm{self}}=1$, ${\lambda }^{\mathrm{rep}}=20$, ${\rho }_0^{\mathrm{s}}=1.5$, ${\lambda }^{\mathrm{r}}=100$, ${\lambda }^{\mathrm{g}}=1000$.

6.2. Experimental Result

In our robotic experiments, we employ 15 robots as collectors and 5 robots as searchers to collect a target patch with 26 units. The initial configurations of the target patch and the robot’s swarm, the experimental snapshots, and the robot’s motion trajectory are shown in Figure 11, and the corresponding evolving curves of three metrics, i.e., ${\overline{\rho }}_{}^{\mathrm{s}}$, ${\overline{\rho }}_{}^{\mathrm{c}}$ and ${\mathrm{S}}^{\mathrm{r}}$, are given in Figure 12.

At the initial moment, the swarm robots and the target patch are distributed in the experimental site without overlapping each other. Then, the searching robots begin to explore the space in formation. The stage of random search continues for about 20 s. When a searching robot detects the target at $\mathrm{T}=21\mathrm{s}$, it immediately halts its random search and begins to evaluate the target patch. In the following target evaluation process ($\mathrm{T}=21-70\mathrm{s}$), the average density of searching robots ${\overline{\rho }}_{}^{\mathrm{s}}$ increases and fluctuates around 4.2, which is due to the searching robots adjusting their heading frequently. At time $\mathrm{T}=61\mathrm{s}$, the guiding beacon establishment begins, and the No. 2 searcher enters the target patch and locates at the density center of the target patch, which has the maximum density (${\rho }_2^{\mathrm{s}}=15.24$). Meanwhile, the collecting robots are recruited by the robotic beacon to perform the collecting task, while other searching robots will return to a designated area.

Then, the recruitment-collection phase starts at $\mathrm{T}=76\mathrm{s}$. The collecting robots keep approaching the guiding beacon until some targets are detected. The edge-surrounding behavior begins at $\mathrm{T}=80\mathrm{s}$ and ends at $\mathrm{T}=254\mathrm{s}$. After $\mathrm{T}=264\mathrm{s}$, the collecting robots begin to shrink inward via the vortex shrinkage strategy, and this collecting process is ended at $\mathrm{T}=585\mathrm{s}$, with the collector’s encirclement area decreasing from $2.95{\mathrm{m}}^2$ to $0.32{\mathrm{m}}^2$, and the target patch area decreasing from $0.83{\mathrm{m}}^2$ to $0.29{\mathrm{m}}^2$. The physical robotic experiment shows that our proposed control policy for heterogeneous swarm robots can effectively solve the TSC problem (The experimental video of this experiment can be found in supplementary materials).

7. Conclusions

We consider a new target collection scenario in which the targets are not evenly distributed but clustered into some denser patches, and a team of robots needs to find these target patches without prior knowledge while collecting them together into stacks. This feature of locally high-density patchy distribution leads to the poor performance of traditional collection algorithms based on full spatial coverage, which results in the challenge of designing target-collecting algorithms. We propose a heterogeneous swarm-robotic solution for the mission of target search and collection, in which a team of searching robots is responsible for searching and locating the target patch, while a team of collecting robots serves to gather and stack the patchy targets. Our methodology is inspired by the division of labor and density-driven behaviors observed in social animals; each robot uses only local information to coordinate its motion, and the remote communication happens only when the searchers try to recruit the collectors. Therefore, our proposal is distributed, while the searching and collecting behaviors of heterogeneous robots are self-organizing. Numerical simulations and real robotic experiments were conducted to validate the effectiveness of the proposed algorithm in an unknown environment with patchy targets. However, most real TSC tasks occur in complex environments, such as: the ground may be uneven or there are a lot of obstacles. The simulation and experiment of this paper only consider the TSC task in a 2D simple scenario, which cannot be directly applied to the real target search and collection task. At the same time, in the collection task, people usually tend to pile up debris in the corners, which requires that the swarm robots not only need obstacle avoidance capabilities but also some ability to use environmental information.

In future work, we intend to consider the TSC missions in complex and obstacle environments. Another interesting issue that we will investigate is to consider the time and energy consumption minimization of swarm robots in the TSC missions. In addition, the algorithm can be extended to three-dimensional space and complete the target search and collection tasks in complex scenarios based on AUV or UAV swarm. We will further explore the applicability of the algorithm in different scenarios.