A self-adaptive landmark-based aggregation method for robot swarms

Abstract

Aggregation, a widely observed behavior in social insects, is the gathering of individuals on any location or on a cue. The former being called the self-organized aggregation, and the latter being called the cue-based aggregation. One of the fascinating examples of cue-based aggregation is the thermotactic behavior of young honeybees. Young honeybees aggregate on optimal temperature zones in the hive using a simple set of behaviors. The state-of-the-art cue-based aggregation method BEECLUST was derived based on these behaviors. The BEECLUST method is a very simple, yet a very capable method that has favorable characteristics such as robustness to noise and simplicity to apply. However, the BEECLUST method does not perform well in low robot densities. In this article, inspired by the navigation techniques used by ants and bees, a self-adaptive landmark-based aggregation method is proposed. In this method, robots use landmarks in the environment to locate the cue once they “learn” the relative position of the cue with respect to the landmark. With the introduction of an error threshold parameter, the method also becomes adaptive to changes in the environment. Through systematic experiments in kinematic and realistic simulators with different parameters, robot densities, and cue sizes, it was observed that using the information of the environment makes the proposed method to show better performance than the BEECLUST in all the settings, including low robot density, high noise, and dynamic conditions.

Keywords

Bio-inspired swarm robotics cue-based aggregation landmark-based navigation self-adaptive

1. Introduction

Environment plays a significant role in different aspects for animals. They interact with the environment by all means to survive. One aspect of the environment is its critical role in the navigation of some animals. For example, ants form pheromone trails between a foraging site and their nest then use this trail as a landmark for navigation (Hölldobler et al., 1990). Honeybees are one another species that use environmental cues for navigation (Srinivasan, 2010). They use visual (Collett, 1996) or olfactory cues (Reinhard, Srinivasan, Guez, & Zhang, 2004; Reinhard, Srinivasan, & Zhang, 2004) to find their way in foraging. Both ants and bees are able to memorize the distance and direction they traveled with respect to a landmark; then, they can integrate this information to return back to their nest (Collett & Collett, 2002). Another aspect of the environment is its protection for animals. Species usually aggregate and hide for protection on the habitat they live in (Grünbaum & Okubo, 1994). In aggregation behavior, animals such as amoeba (Rappel et al., 1999) or cockroaches (Camazine et al., 2003) gather as a single group that is beneficial for their survival. There are two types of aggregation behavior observed in nature: self-organized and cue-based. In self-organized aggregation, species aggregate without the need of any external cue (Halloy et al., 2007). In contrast, in cue-based aggregation, species aggregate on an external cue that is a zone with a distinctive physical feature such as temperature or light intensity in the environment (Frank et al., 2015). Young honeybees are one such species exhibiting this behavior to aggregate on locations that have optimal temperature range favorable for their growth (Heran, 1952) called the thermotactic behavior.

A behavior-based approach is used to model thermotatic behavior of young honeybees (Schmickl & Hamann, 2011). The behavior of bees is modeled as follows: a bee moves randomly in the hive until it encounters another bee. It waits for a particular amount of time based on the local temperature, then starts to move randomly again. From this model, the novel cue-based aggregation method, BEECLUST, is derived based on a simple state machine: (1) move randomly, (2) avoid obstacles, and (3) wait for a certain time proportional to the local light intensity. When the waiting time is over, the robot goes back to state (1) (Kernbach et al., 2009).

The BEECLUST method has been implemented in different studies for over a decade. In Schmickl et al. (2009), the BEECLUST method was studied in both static and dynamic environments. In the static environment, there was only one light source, while, in the dynamic environment, two light sources were located in the arena, and the intensity of the light sources was varied during the experiments. It was shown that the BEECLUST method was robust in dynamic environments. In Arvin et al. (2011), parameters of the BEECLUST method were studied systematically, and velocity and waiting time parameters were modified and tested systematically in a dynamic environment. The results showed that both the aggregation time and aggregation performance improved. In Arvin et al. (2014), a fuzzy logic–based aggregation method was proposed. Through systematic experiments in single- and multiple-cue arenas with static and dynamic settings, the proposed method performed better than the BEECLUST method. The BEECLUST method was modified in Wahby et al. (2016) such that robots were able to calibrate the waiting time based on the intensity of the cue which increased the aggregation performance. In a follow-up study (Wahby et al., 2019), two additions were made to the BEECLUST method so that robots were able to measure local robot density and light intensity and share this information with their neighbors. Through systematic experiments, it was shown that robots were able to adapt to dynamic lighting conditions and, as a result, the performance of aggregation was improved in the low-density robot population. The BEECLUST method was modified in Vardy (2016) such that the “wait” state was changed with the “seek” state. When a robot encountered another robot, instead of waiting, it sought the location of its last encounter with another robot for a predefined amount of time. Systematic experiments were conducted, and the proposed method performed better than the BEECLUST method in larger arenas with a large swarm size. While all the aforementioned works considered behavioral homogeneity, in Bodi et al. (2012), a swarm with two different behavioral groups, one with the tendency to aggregate in the area with a high illuminance and the other with the tendency to aggregate in the area with a low illuminance, were created using the BEECLUST method. Through systematic experiments, it was shown that the performance of aggregation depends on the density of robot population. A performance increase was observed particularly for low densities. Similarly, in Kengyel et al. (2015), four different behavioral groups (goal finder, wall follower, random walker, and immobile agent) were created and the BEECLUST method was applied. Evolutionary experiments showed that a certain combination of behavioral groups (in the descending order of ratios: wall follower, immobile agent, random walker, and goal finder) gives the best aggregation performance. A decentralized collaboration strategy to achieve aggregation is proposed in Güzel and Kayakökü (2017). In their work, agents attended a group and allocated a leader for each group. While the leader keeps moving toward a goal, other robots stay coordinated with the leader and the center of gravity of the group. In a follow-up work (Güzel et al., 2019), leaders used a vision-based goal detection algorithm to find the goal in a cluttered environment. Furthermore, to enable collaborative and efficient exploration of goals, an exploration algorithm was proposed.

Different landmark-based navigation techniques have been adopted in swarm robotics. Using pheromones is one such technique applied by different means in different studies (Arvin et al., 2014; Garnier et al., 2013; Mayet et al., 2010; Payton et al., 2001). In Arvin et al. (2018), an artificial pheromone system (Arvin et al., 2015; Na et al., 2020) with an LCD and a USB camera was used. Through systematic experiments, the effects of different parameters such as evaporation and diffusion rates were studied. It was shown that pheromone-based landmark navigation improved the aggregation performance. In Lemmens and Tuyls (2009), an adaptive foraging method using landmark navigation was proposed. The method was based on path integration capabilities of bees. Radio-frequency identification (RFID) tags and other agents were used as the landmarks, and it was shown that the proposed method improved the foraging performance. In Alers et al. (2013), different ways of realizing landmarks including the QR-codes in a real environment were evaluated in different settings, and it was concluded that feasibility of using a particular landmark realization depends on its complexity and the computational capabilities of the robot.

In this article, inspired from the navigation techniques used by the ants and bees (Collett & Collett, 2002), a self-adaptive landmark-based aggregation (LBA) method is proposed based on the BEECLUST method. The main motivation of the proposed method is to increase the poor performance of the BEECLUST method in low densities (Arvin et al., 2016). The performance of the method is tested and compared with the BEECLUST method in static and dynamic environments with different robot densities, cue sizes, and noise.

The contribution of this article is twofold. First, to the best of our knowledge, this is the first implementation of landmark-based navigation in a cue-based aggregation setting. Second, through systematic studies, it was shown that the performance of the proposed method outperforms the BEECLUST method both in low- and high-density populations. In addition, the proposed method is compared to another state-of-the-art aggregation method, ODOCLUST (Vardy, 2016). Results illustrate that the proposed method was able to outperform the ODOCLUST method as well.

The rest of the article is organized as follows: In the next section, the BEECLUST and LBA methods are introduced. Following that, the experimental setup is discussed, and details on the two simulation platforms are provided. Then, the results of the experiments are represented and discussed. Finally, in the last section, the conclusions and the vision for the future works are given.

2. Aggregation method

In this article, three aggregation methods are implemented. Besides the BEECLUST and ODOCLUST methods, the proposed LBA method is discussed in this section.

2.1. BEECLUST method

In the BEECLUST method, robots aggregate on a cue present in the environment. In this method, shown in Algorithm 1, when a robot detects another robot, it stops and waits for a predetermined time. The waiting time depends on the intensity of the cue at the location where the robot is detected. It is calculated as

w_{s} = w_{\max} \frac{I_{c}^{2}}{I_{c}^{2} + 25, 000}

(1)

where $w_{\max}$ represents the maximum waiting time which is set to be 120 s and $I_{c}$ is the intensity of the cue at the location of the encounter. When the waiting time is over, robots turn $θ$ degrees, where $θ$ is a random variable having uniform distribution within the range [90°, 270°], and then they move forward.

Algorithm 1: BEECLUST method
1 Procedure BEECLUST aggregation
2 move forward
3 if $obstacle detected$ then
4 turn $θ °$
5 else if $robot detected$ then
6 measure cue intensity, $I_{c}$
7 if $I_{c} > 0$ then
8 calculate $w_{s}$
9 wait for $w_{s}$
10 turn $θ °$

2.2. ODOCLUST method

The main idea behind the ODOCLUST method was to increase the aggregation performance using odometry. In this method, robots explore the arena, and save the position of the last two collision points by using odometry. They go in between these two collision points for a specific amount of time called the timeout. In the original version of the ODOCLUST method (Vardy, 2016), the timeout was decided randomly. However, since our article is focused on cue-based aggregation, the ODOCLUST method was modified and implemented such that the timeout duration was decided based on the cue intensity. Figure 1 shows the state machine of the modified ODOCLUST method.

Figure 1.

State machine of the ODOCLUST method.

2.3. The LBA method

The LBA method, shown in Algorithm 2, is developed based on the BEECLUST method. It is assumed that there are landmarks and a cue present in the environment. Landmarks do not contain any a priori information about the location of the cue. They are just distinct identifiers detectable by the robots.

Algorithm 2: Landmark-based aggregation
1 Procedure Landmark-based aggregation
2 go forward
3 if $obstacle detected$ then
4 turn $θ °$
5 else if $landmark k detected$ then
6 if ${\vec{S}}_{total}^{k} exists$ then
7 go toward the cue using ${\vec{S}}_{total}^{k}$
8 if $cue not found$ then
9 increase error variable one step, $e$ ++
10 if $e \geq τ_{e}$ then
11 reset ${\vec{S}}_{total}^{k}$ and $e$
12 else
13 start calculating displacement vectors, ${\vec{S}}_{i}^{k}$
14 else if $robot detected$ then
15 measure cue intensity, $I_{c}$
16 if $I_{c} > 0$ then
17 if ${\vec{S}}_{total}^{k}$ $does not exist$ then
18 calculate and store ${\vec{S}}_{total}^{k}$
19 calculate $w_{s}$
20 wait for $w_{s}$
21 turn $θ °$

Initially, a focal robot, as in the BEECLUST method, starts to explore the environment randomly. When it detects one of the landmarks (ID, relative position, and orientation of the landmark are extracted) for the first time, it starts to integrate all displacement vectors and continues to explore the environment randomly until the focal robot encounters another robot. Then, it checks the intensity at the location of the encounter. If there is a cue, then it stores the total displacement vector, which is determined by adding the individual displacement vectors, as detailed in Figure 2. The resulting vector conveys (noisy) information about the relative position of the cue from the landmark. So, when the robot detects the same landmark for the next time, it uses the total displacement vector for this particular landmark to go directly toward the cue without the need for further exploration, unlike the BEECLUST method. The magnitude and angle of the displacement vectors are calculated using odometry. Odometry is implemented by counting the pulses from the optical encoder on each wheel. By doing so, it is possible for a robot to calculate the traveled distance as well as the angle that it rotated. The distance that a robot traveled from a given point can be derived by using the incremental travel distance for the left and right wheels $(Δ U_{L}$ and $Δ U_{R})$ using the total wheel revolution as in Gutiérrez et al. (2009)

\begin{matrix} Δ U_{L} = c_{m} n_{l} \\ Δ U_{R} = c_{m} n_{r} \end{matrix}

(2)

where $n_{r}$ and $n_{l}$ are the pulse counts of the right and left wheels of a robot, respectively. $c_{m}$ is the conversion factor from encoder pulses to linear wheel displacement and it is calculated as

c_{m} = \frac{π D_{n}}{α C_{e}}

(3)

where $D_{n}$ , $C_{e}$ , and $α$ are the nominal wheel diameter, the encoder resolution, and the speed reduction ratio, respectively. The linear displacement, $s$ , of a robot can be determined by

s = \frac{Δ U_{L} + Δ U_{R}}{2}

(4)

Figure 2.

Demonstration of how ${\vec{S}}_{total}^{2}$ (red vector) of Robot 1 is calculated by integrating its displacement vectors ${\vec{S}}_{i}^{2}$ (green vectors). Robot 1 integrates all the displacement vectors from the time it detected the Landmark 2 $(t = t_{1})$ until it encountered Robot 2 on the cue $(t = t_{2})$ . The blue arc shown in front of Robot 1 at $t = t_{1}$ shows the field of view of its camera when detecting the landmarks.

Similarly, when a robot turns, the angular displacement, $ϕ$ , can be calculated as

ϕ = \frac{Δ U_{L} - Δ U_{R}}{b}

(5)

where $b$ is the distance between two contact points between wheels and the floor.

It is assumed that robots can detect the relative position and orientation of the landmarks. So, when a robot detects a landmark, it calculates its position and orientation with respect to the coordinate system of the landmark as a reference and thereafter, keeps track of its orientation using odometry.

The ith displacement vector, ${\vec{S}}_{i}^{k}$ , is represented as

{\vec{S}}_{i}^{k} = s_{i}^{k} ∠ β_{i}^{k}

(6)

where $k$ denotes the ID number of the landmark that the displacement vectors correspond to. $S_{i}^{k}$ and $β_{i}^{k}$ are the magnitude and angle of displacement vector in the landmark coordinate system, respectively. When a robot reaches the cue after detecting the kth landmark, the total displacement vector ${\vec{S}}_{total}^{k}$ is calculated by integrating all the displacement vectors that were accumulated as

{\vec{S}}_{total}^{k} = \sum_{i = 0}^{M} {\vec{S}}_{i}^{k}

(7)

where $M$ is the number of displacement vectors that a robot has traveled to reach the cue after detecting the landmark. Each displacement vector is calculated between two consecutive turns of a robot. Figure 2 shows how ${\vec{S}}_{total}^{k}$ is calculated using three consecutive displacement vectors. The dark disk is the cue, and six rectangles represent the areas where the corresponding landmark is detectable by a robot. Green arrows represent the displacement vectors, ${\vec{S}}_{i}^{k}$ , that a robot traveled from the last detected landmark until it detects a robot on the cue. The red vector represents the total displacement vector, ${\vec{S}}_{total}^{k}$ , which represents an approximate shortest path from the landmark to the cue. The robot stores this vector and uses it to go directly to the cue, for the next time when it detects the same landmark. It should be noted that the vector ${\vec{S}}_{0}^{k}$ , which is the initial relative position of the robot with respect to the landmark, is also used to find the total displacement vector. This way, the base point of the resulting vector ${\vec{S}}_{total}^{k}$ is the landmark, rather than the robot, which makes the vector independent of the initial position of the robot.

Each robot saves a new displacement vector when it detects a new landmark, and assigns this vector to the ID of the detected landmark. Hence, a robot is able to store as many ${\vec{S}}_{total}^{k}$ as the number of landmarks, $L$ , in the arena: ${\vec{S}}_{total}^{1}$ , ${\vec{S}}_{total}^{2}$ , …, ${\vec{S}}_{total}^{L}$ .

To add the effect of noise due to detection and odometry, noise is added to the actual direction, $β_{i}^{k}$ , and the noisy direction, ${\tilde{β}}_{i}^{k}$ , is calculated as

{\tilde{β}}_{i}^{k} = β_{i}^{k} + σ η_{i}^{k}

(8)

In this equation, $σ$ is the strength of the noise, and $η$ is a uniformly distributed dimensionless random variable in the range $[- 1, 1]$ .

To make the LBA method adaptive, an error variable, $e^{k}$ , and an error threshold, $τ_{e}$ , are defined. Each $e^{k}$ is associated with the corresponding ${\vec{S}}_{total}^{k}$ for each robot. Every time a robot moves based on ${\vec{S}}_{total}^{k}$ but does not reach the cue, the error variable $e^{k}$ is incremented by one. $τ_{e}$ denotes the maximum acceptable error for each ${\vec{S}}_{total}^{k}$ . When the variable $e^{k}$ becomes greater than the threshold $τ_{e}$ $(e^{k} > τ_{e})$ , the robot does not use ${\vec{S}}_{total}^{k}$ anymore and the vector resets. When $k th$ landmark is encountered next time, ${\vec{S}}_{total}^{k}$ is recalculated.

An increment in error variable $e^{k}$ may occur due to three reasons: (1) cue change, which happens when the location of the cue is changed in a dynamic environment (to be explained in the next section), and the robot goes to the previous location of the cue; (2) noise in the measurements of environmental variables, like the relative orientation and position of landmarks; and (3) inaccuracies in the odometry calculation, which can be the result of wheel slippage, or other external factors. When a robot travels relatively long distances, errors in the odometry calculations accumulate, and the deviation of the calculated vector and the real displacement vector that a robot has traveled becomes significant. In all these cases, a robot may not reach the cue when using a particular displacement vector, hence resets the corresponding vector.

3. Experimental setup

In the experimental analysis, two different simulation platforms were used. The first was the kinematic simulator and the other was the realistic simulator. For the sake of comparison, in all the simulations, both the LBA method and the BEECLUST method were implemented.

3.1. Kinematic simulations

In kinematic simulations, shown in Figure 3(a), robots were modeled as two-wheeled agents that have a certain size, linear speed, and rotational speed but no dynamics. The specifications of the robots were taken to be the same as the Kobot robot (Turgut et al., 2007), as shown in Figure 4.

Figure 3.

Simulation platforms. (a) The kinematic simulator. The light blue inner and outer circles represent a robot and sensing range of its sensors, respectively. The red line represents the direction of a robot. Six black semicircles represent the location of landmarks. The detectable region for the landmarks is represented by black rectangles. The cue is shown by a gray-black disk. (b) The realistic simulator. Robots and QR-codes are simulated realistically.

Figure 4.

A picture of the Kobot robot CAD model. The Kobot robot has a diameter of 12 cm and a height of 11 cm.

A $5 \times 10 m^{2}$ rectangular arena was used for the simulations, as shown in Figure 2. A disk was placed on the arena as the cue. The intensity of the cue decreased gradually toward its perimeter. The distribution of the intensity of the cue along the radial direction was considered to be a two-dimensional (2D) Gaussian centered at the cue. The cue radius was set to be $0.7 m$ in such a way that it was large enough to accommodate all robots on the cue (Arvin et al., 2016). The location of the cue was changed from A $[2.5 m, 2.5 m]$ to B $[7.5 m, 2.5 m]$ at a predefined time during the simulations that were being the dynamic experiments. In cases where the location of the cue is not changed during the simulation, like the first part of the above-mentioned setup, the experiment is considered as static. Three distinct landmarks were placed evenly on each long side of the arena making a total of six landmarks as shown in Figure 3(a). Small rectangles in the figure indicate the detectable area $(0.77 \times 1.55 m^{2})$ of the landmarks by the robots. Since noise was taken into account, each experiment was repeated 20 times. The initial position and orientation of each robot were chosen randomly for each experiment. Unless otherwise noted, the standard values shown in Table 1 were used in all experiments.

Table 1.

Standard values used in the simulations.

Parameter	Description	Value/range
$w_{a}$	Arena width	$10 m$
$h_{a}$	Arena height	$5 m$
$r_{c}$	Cue radius	$0.7 m$
$I_{c}$	Measured cue intensity	$[0, 255]$
$w_{\max}$	Maximum waiting time	$120 s$
$r_{\det}$	Detection range for robots and obstacles	$0.06 m$
$N$	Number of robots	$20$
$L$	Number of landmarks	$6$
$τ_{e}$	Error threshold	$4$
$σ$	Angular noise	$15 °$

To illustrate various aspects of the proposed method, different experiments were implemented. All these experiments were repeated for a given number of Monte Carlo trials, to decrease the effect of unwanted factors on the evaluation, such as initial conditions and noise. More specifically, initial positions of robots within the arena followed a random distribution.

Evaluation of the non-adaptive LBA method: In these experiments, to study the LBA method without its adaptation capability, the error threshold was set to a very large number, such that error variables never reset, so that the LBA method became non-adaptive. The duration of the experiments was 80,000 s, and the cue was changed at 40,000 s. The static part of these simulations was taken longer than the other experiments to make sure that robots utilize all the landmarks during the first part of the experiments. Consequently, all the calculated total displacement vectors pointed to the wrong cue at “A,” as shown in Figure 2, in the second part of the simulation. This assumption was intentionally made to highlight the consequences of lack of adaptability for the LBA method.

Error threshold experiments: In these experiments, the effect of error threshold on the performance of the adaptive LBA method was investigated using $τ_{e} = {1, 2, 3, 4, 6, 8, 10, 12}$ . The duration of the experiments was 80,000 s and the cue was changed at 20,000 s. To emphasize the effect of the error threshold, the angular noise value was taken larger than the standard value as $σ = 30 °$ .

Noise experiments: In these experiments, the effect of noise on the performance of the adaptive LBA method was investigated using $σ = {0 °, 15 °, 30 °, \dots, 180 °}$ . Only static experiments were performed with a duration of 20,000 s.

Population size experiments: In these experiments, the effect of population size was studied as in Arvin et al. (2016) using $N = {10, 15, 20, 25, 30}$ robots with a fixed arena size as shown in Table 1. The duration of the experiments was 20,000 s, and only static experiments were performed.

Cue size experiments: In these experiments, the effect of cue size was investigated using $r_{c} = {0.25, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0, 5.0}$ . The arena size was fixed and used as in Table 1 in the experiments.

3.2. Realistic simulations

In realistic simulations, shown in Figure 3(b), all the actuators and sensors of Kobots were modeled using the Webots simulator (Michel, 2004). No robot had access to any global information of either itself or other robots, and all the processing was done on board. Static experiments were performed using 10 simulated Kobot robots with a duration of $10, 000 s$ . The arena size was scaled from the standard size to $3.55 \times 7.1 m^{2}$ and, similarly, the cue radius was taken as $0.5 m$ . The intensity distribution of the cue along the radial direction was also Gaussian as in the kinematic simulations. The error threshold was 4 and the experiments were repeated 5 times with random initial positions and orientations.

During the experiments, fiducial markers, in particular, QR-codes were used as the landmarks. The advantage of using QR-codes is that they can easily be detected by readily available computer vision libraries using a simple complementary metal–oxide–semiconductor (CMOS) camera in a fairly robust way even when they are partially visible (Liu et al., 2008). Relative distance and orientation information can be extracted from QR-codes using Perspective-n-Point (PnP) algorithms with low computational requirements, and all these can easily be implemented in a real-world scenario. Therefore, during realistic simulations, 2D images of QR-codes with a size of $20 \times 20 c m^{2}$ were attached to the walls of the arena to represent the landmarks as shown in Figure 5. Three QR-codes were placed evenly on each long edge of the rectangular arena making a total of six QR-codes the same as the kinematic simulations. A QR-code’s unique ID, its relative position, and relative orientation are estimated when detected by a robot.

Figure 5.

A snapshot of a QR-code in Webots. QR-codes are implemented as 2D images attached to the boundaries of the arena. The QR-code is detected using the OpenCV library implemented in Webots. The relative position and orientation of the QR-code are determined using the PnP algorithm.

During the experiments, images were acquired by a camera installed on each robot. If a QR-code was present in the image, its ID, relative position, and orientation were obtained using the QRCodeDetector() function of the OpenCV library (Baggio et al., 2012). The vector ${\vec{S}}_{0}^{k}$ was estimated using the relative position and orientation of the detected QR-code using the PnP algorithm (Lepetit et al., 2009). The range of detection was realistically simulated based on the specifications of the camera used in the robots as illustrated in Figure 6. No additional noise was added other than the one present in the camera and the DC motors in the experiments.

Figure 6.

The error caused by image processing of a QR-code for different points of the arena. The QR-code (shown by the black half disk) was not detectable from the white points. The black dashed lines show the approximate bounds of the detectable area of the QR-code. (a) Angular error of the measured relative orientation compared to the actual orientation. (b) Distance error of the measured relative distance compared to the actual distance.

3.3. Metrics

As of the performance metric, the normalized aggregation size (NAS), which is the number of robots aggregated on the cue divided by the population size, was used. For some experiments, the time evolution of the metric was shown to illustrate the transient behavior of the methods. For the others, the steady-state value of the metric was taken into account by averaging the last 100 samples of the metric. In this regard, the steady state was assumed to be reached when the metric value reached and settled within the 5% range of its mean value of the last 500 samples.

4. Results and discussions

4.1. Kinematic simulations

4.1.1. Evaluation of the non-adaptive LBA method

The results of these experiments are depicted in Figure 7. In the first part of the experiment, with the LBA method, NAS reaches almost 0.8, whereas it is 0.3 with the BEECLUST method. In the second part of the experiment in which the position of the cue changed from “A” to “B” as shown in Figure 2, NAS decreases dramatically reaching almost 0 for both methods. The BEECLUST method recovers faster than the LBA method, both attaining an NAS of 0.3. As already mentioned, to give enough time for all the robots to utilize all the landmarks to learn the location of the cue, the cue changed at $40, 000 s$ , so that all the landmarks point to the wrong location, “A” in the second half of the experiment.

Figure 7.

Non-adaptive LBA method experiments. The normalized aggregation size for the non-adaptive LBA and BEECLUST methods. The duration of the experiment is $80, 000 s$ , and the location of the cue changed at $40, 000 s$ indicated by the yellow arrow. The dashed blue and the solid red lines are the mean performance of the non-adaptive LBA and BEECLUST methods for 20 repetitions, respectively. The shades show the first and third quartiles of the results. All the other variables are taken as in Table 1.

With the LBA method, robots are able to use the landmarks in the environment making them “learn” the location of the cue and aggregate rapidly on the cue around $10, 000 s$ . When the position of the cue is changed at $40, 000 s$ , the method cannot adapt to the change and robots continue to move toward to the “memorized” location of the cue, “A” where it does not exist anymore. This decreased the performance of the method dramatically. Since the BEECLUST method is based solely on random encounters of the robots, robots explore the environment without any preference, which makes the method to have a mediocre performance, but it can adapt to the changes rapidly.

To show the underlying differences between the LBA method and the BEECLUST method, the heat maps of the static part $(t \in [0, 40, 000])$ of one of the simulations are depicted in Figure 8. The heat of the cue (shown as a red circle) is higher in the LBA method than in the BEECLUST method. Furthermore, the off-cue regions are colder and less dense for the proposed method; that is, robots wandered outside the cue less than those in the BEECLUST method, since they are able to use the landmarks. This is also evident from the distribution of the temperature around the left-most top and bottom landmarks showing that robots are able to move directly toward the cue after they learn and detect the landmark. The evenly distributed temperature on the off-cue regions of the BEECLUST method is due to the stochastic nature of the method making the robots explore the environment randomly. A small detail to mention is the low presence of robots in the detectable area of landmarks. This means that when robots learn the displacement vector from a particular landmark, once the same landmark is detected, they go directly along the corresponding displacement vector without any further exploration. These facts show that the LBA method is able to use the information available in the environment, hence explores less than the BEECLUST method boosting its performance once robots learn the displacement vectors from the landmarks to the cue.

Figure 8.

Non-adaptive LBA experiments. Heat maps of a randomly selected experiment of the non-adaptive LBA method (top) and the BEECLUST method (bottom). Positions of the robots are accumulated during $t \in [0, 40, 000]$ s and results are color coded. The red circles represent the cue, and the black rectangles in the top figure represent the detectable regions of the corresponding landmarks.

4.1.2. Error threshold experiments

In these experiments, the effect of error threshold on the adaptability of the LBA method is investigated in a dynamic environment. Time evolution of the NAS values is depicted in Figure 9(a) for the LBA $(τ_{e} = {2, 10})$ , ODOCLUST, and the BEECLUST methods. The steady-state NAS values for various error thresholds just before the change of the location of the cue $(20, 000 s)$ , after the change $(30, 000 s)$ , and at the end of the experiment $(80, 000 s)$ are shown in Figure 9(b), (c), and (d), respectively. In the first part of the experiments, before the change of cue position in Figure 9(a), due to the use of the landmarks, the NAS values of the LBA method rapidly reach to 0.7 and then suddenly decrease to almost 0 at $20, 000$ due to the change of the position of the cue. After the change, robots with the error threshold equal to 2 have a faster transient response than the robots having an error threshold of 10 and adapt to the change of the cue much faster, both reaching an NAS value of 0.8 at the end of the experiment as shown in Figure 9(d). The BEECLUST method rapidly reaches an NAS value of 0.2 that also decreases rapidly during the change of the position of the cue. After the change, an NAS value of 0.3 is reached. The ODOCLUST method reaches an NAS value of 0.5 that decreases to almost 0 during the change of the cue. After the change, NAS reaches to its original value.

Figure 9.

Error threshold experiments. The normalized mean aggregation size for the adaptive LBA method for (a) $τ_{e} = 2$ (red dashed line), $τ_{e} = 10$ (blue solid line), the BEECLUST method (green dashed line), and the ODOCLUST method (purple dashed line). The steady-state normalized mean aggregation size for the adaptive LBA method for $τ_{e} = {1, 2, 3, 4, 6, 8, 10, 12}$ , (b) before the cue change (20,000), (c) short time after the cue change (30,000), and (d) at the end of the experiment (80,000). The duration of the experiment is 80,000, and the position of the cue is changed at 20,000 indicated by the yellow arrow. The noise is taken as $σ = 30$ .

The error threshold changes the behavior of the LBA method drastically. In the static part of the experiment $(t \in [0, 20, 000] s)$ , the prominent factor is the noise (taken as $σ = 30)$ . Higher error threshold compensates the effect of errors due to noise, increasing the performance as shown in Figure 9(b) and (d). However, the performance saturates for threshold values greater than 3, since the noise is not high enough to necessitate higher threshold values. In other words, higher error threshold increases the robustness of the method to noise. The situation is just the opposite when the adaptability of the method is considered. Higher error threshold values make robots stubborn and they cannot adapt to the changes in the environment rapidly as shown in Figure 9(c).

4.1.3. Noise experiments

In these experiments, the effect of noise on the performance of the LBA method is analyzed in a static environment and the steady-state results are depicted for $τ_{e} = 4$ in Figure 10. For the LBA method, the NAS values decrease as noise increases from $0 °$ to $180 °$ . Noise was not added to the BEECLUST method since it does not have any effect on its performance, for the robots explore the environment in a fully random manner in the BEECLUST method. The NAS values of the BEECLUST method are shown for comparison purposes and they are around 0.2.

Figure 10.

Noise experiments. The steady-state normalized mean aggregation size for the adaptive LBA (red solid line) and BEECLUST (blue dashed line) methods. For the adaptive LBA method, the noise is taken as $σ = {0 °, 15 °, 30 °, \dots, 180 °}$ .

As discussed in the previous section, noise has a negative effect on the performance of the LBA method. Although the overall performance decreases, it can still be compensated by the error threshold. Increasing the error threshold as shown in Figure 9(b) increases the performance when the noise is constant. The advantage of the BEECLUST method is that it is totally robust against noise albeit its low performance when compared to the LBA method. It can be deduced that when $σ = 180$ , the performance of the LBA method is almost the same as the BEECLUST method, since noise suppresses all the information gained from the environment. As it can be observed in Figure 10 around $180 °$ , there is still a performance difference between the two methods. This is due to the fact that the detectable area of landmarks makes the robots not enter these regions when using the LBA method as seen in Figure 8, which, statistically speaking, increases the probability of finding the cue, hence the performance a bit.

4.1.4. Population size experiments

In population size experiments, the size of the arena is kept fixed and the number of robots is increased from 5 to 45, and the steady-state NAS values are depicted in Figure 11 for the LBA and BEECLUST methods. For the LBA method, the NAS values increase reaching 0.7 and then decrease to 0.6 as the number of robots increases. For the BEECLUST method, the trend is somehow different; the NAS value increases from a very small value, 0.1–0.4, where it is saturated as the number of robots increases.

Figure 11.

Population size experiments. The steady-state normalized mean aggregation size for the adaptive LBA (red solid line) and BEECLUST (blue dashed line) methods. The population size is taken as $N = {10, 15, 20, 25, 30}$ robots.

When the LBA method is considered, increasing the population size (or the density of robots) increases its performance (up to 20 robots) and then the performance saturates and starts to decrease slowly as the size increases. The performance increase is expected since a robot needs another robot to calculate the displacement vector (so that robot learns its way from the landmark to the cue) and to wait on the cue (so that aggregation happens) else the robot does not make intensity measurement and misses the cue as shown in the last part of Algorithm 2. So, when the number of robots increases, there is a higher chance for a robot to meet another robot increasing the learning rate and performance of the method. The reason behind the decrease in performance is more subtle; when the population size is larger than a certain value, the probability of encounter with another robot off-the-cue increases. And if a robot is moving toward the cue using the total displacement vector, it would stop, measure the cue intensity (that would be 0 since it is an off-the-cue encounter), turn randomly, and move toward this random direction instead of going straight to the cue, decreasing the performance of the LBA method. However, when the BEECLUST method is considered, increasing the population size increases the performance reaching an NAS value of 0.3. The reason for the increase in performance is the increased probability of encounters of the robots on the cue. The performance saturates around 0.3 due to the overcrowding of robots in the arena. The performance difference between the two methods is more prominent in low densities (or low population size). High robot density hinders the performance of the LBA method due to overcrowding, preventing to use environmental information.

4.1.5. Cue size experiments

In the cue size experiments, the size of the arena is fixed, yet the size of the cue is increased from 0.25 to 10, and the steady-state NAS values are depicted in Figure 12 both for the LBA and BEECLUST methods. When the cue size increases, the performance of the LBA method increases sharply, saturating at $1.0$ at a cue radius of $5 m$ . The same trend is also observed for the BEECLUST method; increase in the cue radius increases performance of the method that gets saturated at a cue radius greater than $5 m$ .

Figure 12.

Cue size experiments. The steady-state normalized mean aggregation size for the adaptive LBA (red solid line) and BEECLUST (blue dashed line) methods. The cue radius is taken as $r_{c} = {0.25, 0.5, 0.75, 1.0, 1.5, 2.0, \dots, 5.0}$ in meters.

When the LBA method is considered, the increase in the cue size increases the probability of finding the cue. This increases both the learning rate of total displacement vectors and the probability of finding another robot on the cue causing a rapid increase in the performance as seen in Figure 12. Since the performance of the BEECLUST method solely depends on the random encounters of robots on the cue, increasing the size of the cue increases the number of robots on the cue, hence increasing the performance of the BEECLUST method. The rate of performance increase is higher for the LBA method since it also utilizes environmental information. The last point to note is that the performance difference between the two methods decreases, similar to the population size experiments, as the cue size increases. In the limiting case, performance of the BEECLUST method catches that of the LBA method. That is an expected result, since when the cue radius is $5 m$ , almost $75 %$ of the arena is covered by the cue and robots do not need landmarks to find the cue anymore.

4.2. Realistic simulation

These simulations are performed to illustrate the applicability of the LBA method in a realistic setting using the simulated models of all actuators and sensors, including the camera, as discussed in the previous section. Due to heavy computational load, a systematic investigation of all parameters has not been performed using realistic simulations; instead, a representative set of values is investigated, and the results for the LBA and BEECLUST methods are depicted in Figure 13. Both methods show a similar trend, though the LBA method performs better than the BEECLUST method. The kinematic and realistic simulation results are very similar, showing that the kinematic simulations are reliable.

Figure 13.

Realistic experiments. The time evolution of the normalized mean aggregation size for the adaptive LBA method with the realistic simulation (red solid line) and with the kinematic simulation (green dotted line) and the BEECLUST method with the realistic simulation (blue dashed line). The population size is $10$ robots. Only static experiments with a duration of $10, 000 s$ are performed. The experiments are repeated for 5 times for each setting. For the kinematic simulations, the noise is taken as $σ = 15$ . The error threshold is taken as 4.

One of the main concerns in realistic simulations is the noise. Since there is noise inherently in actuation and sensing of robots in realistic simulations, noise was not added artificially during the simulations. The noise in detecting the QR-codes affects both the estimation of the relative orientation of the QR-codes $β_{0}$ and the magnitude of the initial relative position vector $s_{0}$ . A snapshot of a QR-code and an overlay of three principal axes after detection are shown in Figure 5. To investigate the noise in QR-code detection further, additional systematic experiments were performed. In these experiments, a robot was located in different locations on the arena, its camera being directed toward the QR-code indicated by a black semi-circle on the x-axis. Then, the gathered image by the camera was processed based on the OpenCV library, and the $β_{0}$ and $s_{0}$ were estimated. The error of estimation was then calculated, and the results for the angular and distance errors were depicted as a heat map in Figure 6(a) and (b), respectively.

In both plots, there are regions shown in white indicating that the positions where the robot was not able to detect the QR-code. When the angular error is considered, a symmetric pattern of positive and negative relative angles is observed. The heat map is mostly blue indicating that the angular error is relatively small, almost below 2. Moreover, the detectable region has a bow shape, and the range is from $- 45$ to $+ 45$ . The distance error plot is half blue and half light blue, indicating that the error is mostly below $0.05$ . The mean and standard deviation of the angular and distance errors are calculated and the results are shown in Table 2.

Table 2.

Realistic simulation errors.

Variable	Mean	Standard deviation
$\| β_{0} - {\hat{β}}_{0} \|$	$0.62 °$	$1.21 °$
$\| s_{0} - {\hat{s}}_{0} \|$	$0.02 m$	$0.03 m$

In both the plots, the maximum error is observed around the central axis of the detectable region that is furthest from the QR-code. Since the angle is calculated using the slope of the vertical lines of the rectangle on the periphery of the QR-code, a few-pixel error causes a big change in the slope for very small relative angles resulting in larger errors than the moderate relative angles. Finally, the undetectable points (white regions) can be categorized into three groups: (1) those which are too close to the landmark, where the camera could not capture the whole QR-code; (2) the points which are too far from the QR-code, and due to the limited resolution of the image, the detection is not achieved; and (3) the regions having a larger relative angle with respect to the QR-code.

5. Conclusion and future work

In this article, a new aggregation method was proposed based on the BEECLUST algorithm. The proposed method added the capability of using landmarks for navigation. Triggered by landmarks, robots recall their memories to move toward the cue. The experiments supported the idea that compared to the BEECLUST, the LBA method makes robots to wander less off-the-cue, and to move toward the cue as soon as they meet a landmark. It was proven that exploiting the data of the landmarks significantly increases the aggregation performance, when the environment is static. In a dynamic environment, however, the situation was more challenging. The dynamic feature of such environments makes the old information not valid anymore; hence, relying on outdated data is detrimental to the aggregation performance. The concept of the adaptive algorithm is introduced to cope with this challenge by using an error variable and a corresponding error tolerance threshold. By so doing, the proposed method reacts to the changes in the environment, makes the robots able to erase the wrong perceptions of the environment, and subsequently updates them. By evaluating the performance of the LBA method in dynamic environments, it was revealed that the proposed method was more adaptive to the environment changes than the BEECLUST method. Thus, it was concluded that the proposed method not only outperformed the original BEECLUST method in stationary environments but also was able to adapt faster to the changing environment. Hitherto, we draw the conclusion that for a dynamic environment, non-adaptive LBA is not able to aggregate the swarm, thus considering the adaptive feature of the method is inevitable.

The study on different error threshold parameters, $τ_{e}$ , demonstrated that in a noisy environment, increasing the threshold makes the system more resistant to the noise. On the contrary, the effect of the threshold on the adaptation rate is the opposite, that is, decreasing $τ_{e}$ increases the agility of the method to adapt to new conditions of the environment. It is essential to mention that the effect of noise, as a demonstration of uncertainties in the environment, on the aggregation performance is destructive. The degradation of the performance versus noise is such that after a critical value of noise strength, using the information of landmarks did not make a change. Consequently, it can be concluded that for the environments with a high degree of uncertainties, it is not justifiable to employ the LBA method and it is not reasonable to afford the cost.

Similar to another study by Arvin et al. (2016), on the effect of the population density, the results bore the conclusion that the BEECLUST method lacks a proper performance in low-density swarms. More than that, the difference between the LBA method and the BEECLUST method widened for the swarms with low-density populations. Thus, an important conclusion to be made here is the fact that in such cases, adopting the LBA method significantly increases the performance. In the sample experiment, the amount of increase was so much that even a 4 times denser swarm of robots with the BEECLUST method could not outperform the LBA method. As one of the most main conclusions of this article, compared to increasing the number of robots with the BEECLUST method, equipping robots with the LBA method costs less, yet performs better.

The effect of cue size on the aggregation performance was also studied in the last investigation which reflected the fact that in an environment with low density of cue, exploiting the environmental information plays a vital role in the aggregation performance of robots. Therefore, in an environment with low probability of finding the cue, storing data and referring to them can considerably boost the aggregation performance.

As a future work, the implementation of the LBA method on real robots will be investigated. More than that, the advantage of using an artificial neural network–based algorithm will help the method to estimate the center of the cue to increase the probability of collision. Furthermore, the possibility of sharing information via a pheromone-based communication system might help the swarm to increase the performance of aggregation. Another important aspect is to analyze the number, distribution, and position of landmarks and their effects on the aggregation performance as a future study. Besides, the presence of static or dynamic obstacles was not considered in this work. In future works, stationary and moving objects will be added to the scenario, and the performance of the proposed method will be evaluated under these new circumstances. Last but not the least, for all the aforementioned studies, meta-heuristic algorithms such as genetic algorithms might be used to automate the design of the controllers.

Footnotes

Handling Editor: Hiroyuki Iizuka, Hokkaido University, Japan

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Arash Sadeghi Amjadi

Mohsen Raoufi

Supplemental material

Supplemental material for this article is available online.

About the Authors

Arash Sadeghi Amjadi is an MSc student at the Kovan Research Laboratory at Middle East Technical University, Turkey. Currently, he studies in the mechanical engineering department under the supervision of Prof. Ali Emre Turgut. He received a BSc in electrical engineering from the University of Tabriz, Iran, in 2019. His research interest includes swarm robotics, deep learning, machine learning, control and systems engineering, and autonomous robotic systems.

Mohsen Raoufi received a BSc in aerospace engineering and a master’s degree in dynamics and control from Sharif University of Technology, Iran, with a study on the state estimation of non-linear systems using a swarm-based optimization algorithm. In 2020, he worked as a researcher at the Kovan Research Laboratory at Middle East Technical University, Turkey, and currently is a doctoral researcher at the Cluster Science of Intelligence, Germany. His research interests are mainly focused on swarm intelligence, robotics, collective decision-making, and system dynamics.

Ali Emre Turgut has received a BSc in mechanical engineering from Middle East Technical University, Turkey, in 1996; an MSc in mechanical engineering from Middle East Technical University, Turkey, in 2000; and PhD in mechanical engineering from Middle East Technical University at Kovan Research Laboratory, Turkey, in 2008. He worked as a post-doctoral researcher at Universite Libre de Bruxelles, IRIDIA, Belgium, and as a research associate at the department of biology at KU Leuven, Belgium during 2008–2012. In 2013, he worked as an assistant professor in the Department of Mechatronics Engineering at the University of Aeronautical Association of Turkey. He worked as a research associate in the Laboratory of Socioecology and Social Evolution, KU Leuven. He started working in the mechanical engineering department at Middle East Technical University as an assistant professor in 2015.

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