Swarm intelligence principles observed in nature have been widely studied and applied to a number of collective tasks where a group of autonomous robots are used to replace a single intelligent robot for the advantage of robustness, flexibility and scalability [
4,
5]. These collective tasks include aggregation [
6,
7], pattern formation and self-assembly [
8‐
11], pursuit and evasion [
12,
13], defense and intrusion [
14,
15], coordinately navigation [
16], collective construction [
17], task allocation [
18], etc. Research works have been also carried out to the emergence of simple cognitive abilities, such as leader election [
19], autonomous task sequencing [
20] and collective decision-making [
21,
22].
Multi-target trapping is one of challenging research areas in swarm robotics. It takes advantage of cooperation of simple robots in large numbers to entrap multiple moving targets that cannot be stopped by a single one. Such phenomenon is very common in animal world and has been extensively studied in prey–predator interactions. For predator, the hunting behavior usually relies on circle-like formation so that a group of weaker predators are capable of catching a stronger prey, as reported in ant colony predation [
23] and wolf pack predation [
24]. For prey, the defense behavior (anti-attack behavior) also shows entrapment dynamics. In [
25], the authors reported that spotted hyenas, which always lives in packs, display anti-attack behavior when the pack is under attack by lions. This defense behavior places the individual survival at great risk but protects the puppies or the younger members. In swarm robotic systems, the multi-target trapping has been successfully applied in applications such as search and rescue [
26], collective transportation and construction [
27], convoy/escorting missions [
28,
29] and perimeter defense tasks [
30].
Over the past decade, the artificial potential field method has been widely used in multi-target trapping, where the robot moves along the gradient of several potential field functions. Generally speaking, the robot swarm is mainly driven by the attractive–repulsive force, in which the generation of trapping shape is achieved by attractive force; the collision avoidance and the obstacle avoidance are achieved by repulsive force. In [
31], the attractive force between hunters and prey, the repulsive force between a pair of hunters were only introduced to achieve entrapment task. There is no need to introduce an encirclement function in this method, but the robot swarm is only able to entrap a single target with a circle shape. Similarly, in [
32] the authors considered the entrapment task based on the virtual structure. The robot swarm moves towards the virtual center of a circle, and due to the repulsive force among inter-robot interaction, regular polygon formations are realized. In [
33], the inter-robot interactions were assumed to be globally repulsive and selectively attractive, so that trapping shapes can be generated by choosing different interactive topologies. However, the expected trapping shapes are limited to a few of formations, such as line, circle and ring.
By introducing an encirclement function, the diversity of trapping shape has been significantly improved. In [
34], the authors introduced the elliptical encirclement function into the control policy. By combing normal and sigmoid limited functions, they can organize the robot swarm into elliptical trapping shapes. However, it is limited to entrap a single target or multiple targets with elliptical shapes. To generate an arbitrary shape [
35], used radial basis implicit function to describe an arbitrary pattern and the robot swarm could be driven to those predefined shapes. Although the flexibility of shape formation gets great improvement, the authors only focused on pattern formation control of swarm robots, thus no further work considering the transformation of trapping shapes was conducted. This limitation is also seen in [
36] where the authors replaced the radial basis implicit function with non-uniform rational B-spline function to describe the expected pattern. The gene regular network-based controller was adopted to drive the robot swarm into the predefined patterns with no transformation and multi-target trapping abilities. These limit the applications of multi-target trapping methods in real-world problem. In our previous work [
10,
11], we enhanced the flexibility of radial basis implicit function in describing the trapping shapes. The modified method is capable of driving the robot swarm into a dynamically changing shape according to the specific distribution of the moving targets. Meanwhile, the abilities of splitting and merging of robot swarm are also fulfilled. In addition, influence of disturbances, modeling errors and various uncertainties in the real systems limit the applications of many multi-target trapping methods, fault diagnosis methods and reinforcement learning methods could be good potential solutions [
37‐
39].
In general, the previous works show the applicability of proposed methods for multi-target trapping tasks that can be accomplished by pairwise interactions. The pairwise interaction means one robot interacts with another robot in a moment with such as attractive–repulsive interaction or obtains the weighted sum of these interactions linearly. In contrast to pairwise interaction, the density-based or concentration-based interaction uses the property of group’s spatial distribution to influence the actions of individuals. For example, odor concentration is largely behind the individual’s decision-making of several species, such as larvae of cockroaches prefer to follow their own strain odor rather than that of others [
40], and carpenter ants were reported to be capable of aversive learning and can confirm previous findings about the different resources solicited by differential odor-heat conditions [
41]. The huddling behavior of emperor penguins was reported to be a thermoregulatory behavior which reduces exposure and preserves body heat. This temperature field guided behavior enables all breeders to get a regular and equal access to the warmth of the huddles during the Antarctic winter [
42]. Light traps of various species especially for flying moths were observed as light-strength-based interaction, in which the moths method the light source according to the light intensity distribution [
43]. Another important example is embryogenesis where tissue differentiation and embryonic development are guided by morphogenesis which is adjusted by concentration of gene and its products [
44]. Although such phenomena have been extensively studied by biologists, few robotics researchers have taken them as inspiration for control policy design in swarm robotic tasks.
Contribution
In this paper, we consider the case of a swarm of homogenous robots entrapping multiple moving targets with appropriate encirclements. However, contrary to what is usually used in the rest of the literature [
31‐
33], our controller design does not depend on the pairwise attractive–repulsive interactions, but we rather leverage the density-based interactions in both inter-robot interaction and robot–target interaction, which allow the robots to consider the local spatial density as the main clue for the individual-level decision-making.
With respect to the solutions in the rest literature [
10,
11,
34‐
36], our method does not involve any use of
ad hoc formation control strategies to force the robots surround the targets. The proposal allows the swarm robots to take either the single-ring formation, multi-ring formation or multi-subgroup formation to entrap the targets in a distributed way. These enclosed patterns cannot be obtained by a single framework in previous work, thus significantly improving the flexibility and adaptability of the proposed trapping method. Generally speaking, it is density-based interactions that drive the swarm robots like a fluid to encircle the targets, thus the enclosed formations are obtained by the property of fluid rather than the explicit formation functions. The enclosed configurations emerge from the individual-level interactions rather than being explicitly designed; therefore, the proposal is also self-organized. The effectiveness, robustness and scalability of the proposal are evaluated in numerical simulations. The validity of the proposal is also confirmed in real robotic experiments.