Positioning multiple mobile robots for geometric pattern formation: An ...

Positioning Multiple Mobile Robots for Geometric Pattern Formation: An Empirical Analysis Avinash Gautam

Apoorv Umang Saxena

Department of Computer Science Birla Institute of Technology and Science Pilani, India [email protected]

Department of Computer Science Birla Institute of Technology and Science Pilani, India [email protected]

Prerak Mall

Sudeept Mohan

Department of Computer Science Birla Institute of Technology and Science Pilani, India [email protected]

Department of Computer Science Birla Institute of Technology & Science Pilani, India [email protected]

Abstract—This paper presents an experimental setup for absolute positioning of multiple mobile robots in an indoor environment using a low cost camera. Localization or positioning of mobile robot in its environment is crucial for deciding its future course of action. In this paper we have proposed to use an overhead camera for positioning multiple mobile robots which are required to act as a team. Also we have tested the efficacy of two existing distributed algorithms for circle formation using a team of five e-puck robots. The first algorithm is mathematically proven with many assumptions about the sensing and motion capabilities of mobile robots which are not feasible in the real world. In the second algorithm the authors have considered explicit inter robot communication and have utilized the distributed solution of a well known algorithm often discussed in distributed computing - the Dinning Philosophers Problem for the robots to synchronize during their activation cycle. The contribution of this paper is twofold i.e., first, a practical low cost, multi-robot positioning system is proposed and second, experimental evaluation of two distributed algorithms for circle formation by a team of mobile robots have been carried out. It is seen that the second algorithm outperforms the first. Keywords— Distributed Algorithms, Multi-Robot Systems, Mobile Robot Positioning, Image Processing, Pattern Formation, e-puck

I. INTRODUCTION The use of multiple mobile robots for solving complex problems has gained considerable attention in the past two decades. Deploying multi-robot systems to carry out complex missions and/or operations can be advocated with two school of thoughts: (a) some tasks are inherently redundant and take a long time for a single robot to complete (b) a single monolithic robot equipped with complex sensors cannot complete the task in a robust manner. The design complexity of individual robots in multi robot systems is assumed to be fairly simple i.e., they are equipped with cheap sensors and low onboard processing and memory. These robots are neither omniscient nor This work is being funded by CAIR-DRDO, Ministry of Defense (Government of India) via its Grant# CAIR/ROB/CARS-2010/2. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding agency.

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omnipotent. It makes it clear that the use of multi-robot systems require individual robots to interact, coordinate, and negotiate with each other to decompose a bigger task into subtasks and dispatch these subtasks to individual robots. Today researchers worldwide are studying the feasibility of using multi-robots systems in several applications i.e., target tracking and enclosing [1], surveillance [1, 2], data sharing [3], landmine detection and removal [4], area exploration [5] and, sensing grids [6, 7]. Pattern formation problem by multi-robot systems is central to all the aforementioned tasks. Rather it can be considered as first step towards coordination. In this paper we have reviewed several pattern formation algorithms by multiple mobile robots [3, 18, 19, 22-28]. In particular we have studied the circle formation problem. It has been observed that many of these approaches are theoretical or computer simulations and have taken impractical assumptions about the sensing and navigation capabilities of mobile robots. We have established an experimental test bench for control and monitoring of multiple mobile robots. Two distributed algorithm for circle formation [22, 23, 29] by a team of mobile robots have been implemented, wherein we have dropped some of their impractical assumptions. We have compared the efficiency of these two algorithms for circle formation using five e-puck robots and determined that the algorithm suggested in [29] is more efficient than the one which is presented in [22, 23]. The paper is divided into six sections. Section-II presents the literature review and brief analysis of several research works on pattern formation problem. Section-III presents a discussion on the two distributed algorithms [22, 23, 29]. A detailed description of the experimental setup is included in Section-IV. Experimental results and their analysis are reported in Section-V. Conclusion and future scope is presented in Section-VI. II. RELATED WORK Pattern formation is one of the fundamental activity which has also been observed in several living organisms [8] wherein

the individuals try to remain in some orientation and distance from each other so as to maintain certain shape like an arrow or a chain. They do so in search of new feeding and breeding grounds (ant forming chains), to prevent attacks from the predators (fish schools), migration to favorable places for survival (bird flocks) etc. Pattern formation in multi robot systems has emerged as a new discipline which provides convenient coordination capabilities with approaches ranging from centralized coordination which are "leader-follower" based [9, 10, 11] to "virtual leader" [12, 13, 14] and completely decentralized algorithms which are leader-less [15, 16, 17]. While centralization has its own advantages i.e., motion plan can be generated for all the individuals simultaneously with a centralized planner [11], these approaches don't scale with the team size. With "virtual leader" based approaches the robots coordinate to jointly synthesize a lone fictitious leader whose trajectory is then followed by the robot team. This approach cancels out disturbances inherent in the leader-follower based algorithms at the expense of high volume of communication and computations. To synthesize virtual leader it is vital to timely announce/ communicate its latest position coordinates to other robots to support their real-time control. The assumption on the communication topology is one thing which is common in both "leader-follower" and "virtual-leader" based approaches. There is a whole body of work on decentralized approaches on pattern formation by multiple mobile robots [17-21]. An almost impractical assumption has been reported in all these research works i.e., the availability of global information wherein small robots are considered to be equipped with sensors with unlimited visibility which can tell them the instantaneous positions of all other robots situated in their environment. Such an advanced sensing system for global localization is not recommended for swarm robots or multi robot systems which are deliberately designed to be low cost machines for obvious reasons. Agreement problem as a pattern formation problem has been addressed in [20] which considers homogenous robot team in two dimensional space. In [21] the authors have considered nonholonomic team of robots but the utility of the work is only investigated in numerical solution. Again in [15, 16, 17] authors have taken impractical assumptions that each robot is equipped with a global position sensor for the purpose of localization and with reliable communication system. Recently the pattern formation problem by multiple mobile robots has been discussed from computational perspective. In the process of generalization of this problem several approaches [3, 18, 19, 22-28] have treated the robots as a point in Euclidean plane R2. Common assumptions which have been taken in all these works are, all robots are Anonymous- cannot be distinguished from each other i.e., there are no identification marks on them. Visibility- this assumption is directly associated with the sensing capability of the robot i.e., robots perception of the world, how it can localize itself, its peers and other objects in the universe. Some of the research works have considered limited visibility, while there are other works which consider unlimited visibility. Autonomous- the robots act completely independent of each other and expects no intervention. Oblivious- robots have no memory i.e., they execute algorithm based on their current state only. Synchronization- there are various levels of

synchronization i.e., synchronous, semi-synchronous, and asynchronous. The robots exhibit various levels of synchronization in their actions, computations and movements. Orientation- the robots have no common sense of orientation/ heading. Local Coordinate System- the robots have no common knowledge or understanding of the reference frame i.e., all robots operate in their own local reference frame. No Communication- the robots cannot explicitly communicate with each other, the only means of communication for the robots is to observe each other. Mobile- all robots can freely move in their environment. The feasibility of these approaches in other applications like rescue missions, area exploration etc. are not investigated. Also none of these works have given any quantitative measures on the performance of the robot team i.e., we cannot say anything about convergence time, the distance travelled by the robots, computational complexity in terms of amount of decision making. In the next section we present a detailed discussion of two different algorithms presented in [22, 23, 29]. III. DISCUSSION ON ALGORITHMS This section presents a brief discussion on two existing distributed algorithms on circle formation by multiple mobile robots. The first algorithm [22] is a theoretical work but it has not given any quantifiable measures how good or bad the robots have performed as a team. Subsequently one more work [23] has appeared where the authors have simulated a part of the algorithm presented in [22] and have given quantifiable measure in terms of the total number activation steps for robot team in achieving convergence. A more recent algorithm which is suggested in [29] stands out to be a practically feasible approach. In all of these works the robots are considered as mobile processors with a sensor to detect instantaneous position of all robots. Table-I lists down a set of assumptions about the sensing, motion and processing capabilities of mobile robots which differentiates these approaches. TABLE I. Reference

ASSUMPTIONS ABOUT THE ROBOTS [22]

[23]

[29]

Mobile







Autonomy





Anonymous





 X





Visibility

Unlimited

Unlimited

Synchronization

Asynchronous

Asynchronous

Orientation

Not Applicable

Not Applicable

Unlimited SemiSynchronous Not Applicable

Local

Local

Local

X

X



Oblivious

Coordinate System Communication



Further in [22] the authors have considered some assumptions which are difficult to implement in practice, for example, robots have infinite memory. At the same time they have said that the robots are oblivious because they are unable to remember their past states and actions. It is also considered that the robots are infinitesimally small, they never collide, and

that two or more robots can occupy the same position. However the execution of the algorithm never allow individual robots to intrude in the spatial territories of other robots. One assumption which is rather difficult to realize in the practical implementations is that the robots can observe, compute, and move with infinite decimal precision. Taking all the assumptions into consideration the main problem of circle formation is carefully decomposed in two different problems. The first problem requires all the robots to be arranged on the circumference of a non degenerated circle. After sensing the immediate positions of other robots, all the robots calculate the smallest enclosing circle in their own local coordinate systems and develop common consensus i.e., the center/origin of the smallest enclosing circle calculated by each robot coincides and thus the smallest enclosing circle is used as a common reference. The smallest enclosing circle remains the invariant, such that, each robot complies by the following self imposed restrictions on their movement (a) no robot ever moves beyond the boundary of the smallest circle enclosing all robots (b) All robots located on the boundary of the smallest enclosing circle remain on that boundary (c) Robots located on the circumference of the smallest enclosing circle do not move unless there are at least three such robots with distinct positions (d) One more self imposed restriction on the movement of robots is suggested on the Voronoi partitions calculated by each robot which states that "A robot always moves toward a point that is inside its Voronoi cell". In order to prevent collisions and conflicts of interests about occupying the same position by two or more robots, all the robots are required to find Voronoi partitions. This restriction has been imposed in order to make robots movement restricted in their own Voronoi partitions and hence let them operate in mutually exclusive zones which is the region of dominance of individual robots. Followed by this process all robots infinitely gets activated and execute the Uniform Circle Formation algorithm. It is also assumed that obtaining the information about the system (positions of the direct neighbors clockwise and anticlockwise), computing the new position (target position), and moving towards the target position are all instantaneous operations and are atomic. This assumption is taken to avoid the situation of livelock which is explained in section 2.2 of [22]. The algorithm converges towards uniform configuration but does not terminate deterministically. In [23] the convergence of the Uniform Circle Formation algorithm which is proposed in [22] is quantitatively measured using computer simulation. This work starts with an understanding that the robots are already situated on the circumference of the smallest enclosing circle. All robots are activated in a round robin manner. The activation schedule of the robots is probabilistic, unpredictable and unknown to the robots. This ensures that each robot becomes active at infinitely many instances and minimum one robot is active during each time instant. Authors allowed only a subset of robots to be active in other robot's round. This is achieved by assigning to robots a probability of activation p. For number of robots (n) equal to 16 and p=0 the robots execute in a strict round robin fashion and algorithm converges much slower i.e., it requires more than 100 activation steps, whereas if p=1 all the robots are activated together and runs in fully synchronized manner and the algorithm converges in 55 activation steps. For p=0.5 it is

shown that algorithm converges after approximately 80 activation steps. Thus it is shown that the algorithm converges much faster for higher probabilities. Raising the probability from 0 to 1 increases the level of synchronization. Uniform Circle Formation algorithm proposed in [22] is implemented in [29] under the mode of explicit communication. The basic assumptions considered here are specified in Table-1. Here the robots have been assigned unique identifications (IDs) and thus they are not anonymous, they can explicitly send messages to other robots and are more involved with each other. Unlike [24] where the activation schedule of the robots is probabilistic, in this approach the authors have used the distributed solution of dinning philosopher problem [30]. All robots run the same deterministic algorithm. This approach ensures that if a robot is active and is executing the algorithm its neighbors stay silent. This approach is a semi-synchronized approach and it guarantees that the robots converge to a uniform state deterministically. In the subsequent sections of this paper we have empirically demonstrated that the approach for Uniform Circle Formation suggested in [29] converges faster than the approach presented in [23]. The experimental results their analysis and the test bench are described in Section-IV and Section-V respectively. IV. EXPERIMENTAL RESULTS AND ANALYSIS In this section we present the results and the analysis of our experiment with five e-puck robots for 20 runs. The two distributed algorithms suggested in [22, 23, 29] are implemented and their performance is compared with respect to each other. The first algorithm [22, 23] decides the activation of the robots in round robin fashion based on probabilities. From here we will refer to this algorithm as RR. In the second algorithm [29], the activation schedule of the robots is governed by the distributed solution of dining philosopher problem. From here we will refer to this algorithm as DP. Both the algorithms (RR and DP) for uniform circle formation start with an understanding that all the robots are situated on the circumference of the smallest enclosing circle. The activation cycle of the robot comprises of four different states i.e., idle- initially all robots are idle and do nothing, sense- each robot senses the position of its direct neighbors (clockwise and anticlockwise), compute- each robot computes its next target position and move- each robot moves to the position it has just computed. These four states make one activation cycle of the robot. Both RR and DP assume that the activation cycle of the robot is atomic. Three different parameters are considered for comparing RR and DP. They are: (a) number of activation cycles (b) number of rounds and (c) total distance travelled. It is important to understand that the spatial distribution of the robots has significant impact on the three parameters suggested above. We call this spatial distribution as configuration of the robot team. To compare the two algorithms we need to execute them by keeping the same configuration for the robot team. Here we have considered 20 different configurations and noted down the results which are presented below.

highlights the execution of the two algorithms we have tested on our test-bench.

Fig. 1. Comparison of RR and DP based on number of activation steps.

RR and DP are compared based on the number of activation steps they take to achieve convergence for 20 different configurations, see Fig. 1. RR is tested for three probabilities of activation i.e., p=0, 0.5 and 1. The results thus obtained substantiated the authors claim in [23] that the number of activation steps reduce with the increase in probability of activation. It is interesting to observe that the number of activations steps for DP reduce substantially when the robot team is subjected to the synchronization rules defined for dining philosopher in [30]. Refer to Fig. 2 for analysis of average number of activations. In [22] the probability (p) values which are considered tell how many other robots can be active in one robot's round. We have considered rounds as a parameter for measuring the performance of RR against DP. It is found that compared with RR, DP achieves convergence in significantly less rounds, see Fig. 3 as compared to RR. Refer to Fig. 2 for average analysis of rounds. When it comes to total distance travelled by the robot team, RR outperforms DP as it requires the robot team to travel less distance, see Fig. 4. Although DP requires robots to travel more it is not a very noticeable distance under majority of the configurations.

Fig. 3. Comparison of RR and DP based on number of Rounds

Fig. 4. Comparison of RR and DP based on total distance travelled

A. System's Configuration The basic configuration of the system comprises of four main components, namely:  Overhead Camera: The overhead camera is a USB 2.0 RGB camera that delivers 640 x 480 resolution colored images at 30 frames per second.  Platform: The platform is a 3 m x 3 m white board on which the robots move. It has been placed under fixed diffused lights in order to maintain constant lighting for accurate color segmentation.  Robots: The robots used are e-puck [33] which are small differential wheeled mobile robots. Each e-puck is powered by a dsPIC processor and is equipped with Bluetooth for communication with a central server. A total of five e-pucks have been used for conducting the experiment.

Fig. 2. Averaging Activation Steps and Rounds

V.

THE EXPERIMENTAL TEST BENCH

This section gives a brief description of the experimental test-bed and is divided into three sub-sections. In the section-A the systems configuration is explained followed by section-B, which explains the process that is used for finding the absolute positions/ localization of multiple mobile robots and Section-C,

 A Laptop or a Personal Computer: This is used as a server to which the camera is connected. Various image processing algorithms which are used to determine the absolute positions of multiple robots execute on this computer. B. Localization/ Positioning Multiple Robots Each robot is equipped with a marker so that its position and direction can be determined using the overhead camera.

The marker is a 10 cm x 10 cm square of solid color along with a black rectangle placed on it at a distance of 4 cm from its centre, as shown in Fig. 5.A. Every robot has a marker of different color so that it can be identified uniquely.

Fig. 5. (A) Marker used for determining robot's stance and (B) Direction vector obtained from blob centroids

1) Identification of markers a) Color Image Segmentation: The first step involves segmenting the color images into seven colors i.e., five colors corresponds to the robot, black color is used for the direction strip and the white color is for the background. The color thresholding is done in YUV color space using the algorithm by Bruce et al. [31]. RGB look-up tables for each color are calculated offline so as to reduce run-time computation. b) Blob Detection: For each of the five robot color a binary image is created by using the look-up table for its corresponding color. A blob detection algorithm is applied on this image [32] and the centroid of the largest blob is taken to be the robot’s position. Within the blob of each robot, the blob detection algorithm is applied again to identify black blobs. The vector joining the largest black blob and the robot centroid is used to give the direction of the robot, as shown in Fig. 5.B. C. Executing the Two Algorithms The main program runs on the central server and sends motion commands to the robots depending on the algorithm being executed [23] or [29]. Before running the main program, thresholds for the seven colors are calibrated and RGB look-up tables are generated. The program consists of two main threads: (A) Camera Control Thread: This thread acquires the image from the camera and performs all the image processing steps for identifying the markers. It stores the robot stance (position and orientation), visibility (robot visible/ not visible) and data for all five robots in a shared data structure BeliefState. This thread runs at around 30 frames per second, which is also the maximum frame rate of the camera, so as to provide the most up-to-date data possible to the Algorithm Control Thread. (B) Algorithm Control Thread: This thread communicates with the robots and runs the circle formation algorithms. It receives robot stance information via the BeliefState shared data structure. The algorithm runs in several steps as the program moves through four states, which is the activation cycle of the robot [29]. The robots configuration at different stages of algorithm execution is shown in Fig. 6.

Fig. 6. (A) Initial random configuration of robots (B) Finding Smallest Enclosing Circle (C) Robots moved to circumference of the smallest encolsing circle (D) Uniform Circle Formation by Multiple Real Mobile Robots

VI.

CONCLUSION AND FUTURE WORK

An experimental test bed has been established for control and monitoring of multiple mobile robots. An overhead camera has been used for absolute positioning of multiple low cost mobile robots which are unable to localize themselves. Successful validation of two distributed algorithms for circle formation using five e-puck robots has been achieved. It has been observed that the algorithm suggested in [29] outperforms the algorithm suggested in [22, 23]. Two exceptions which exist because of the initial spatial distributions of the robot team are (a) if the initial configuration itself makes a circle which is mostly balanced, then there is no noticeable difference between the two algorithms (b) When the distance which is required to be travelled by the robots is large i.e., if they have to make a bigger circle in that case [29] always outperforms [22, 23]. The empirical results obtained prove that explicit communication and the increase in level of synchronization improves the performance of the robot team. Explicit communication, however suffers from several kinds of noise in the environment which may result in data loss or worse there is a possibility of complete communication failure. The multi robot system should be designed to be fault tolerant i.e., it is highly desirable to have a solution which performs well both in the presence of communication and no communication. In future we will study and design fault tolerant pattern formation algorithms in the presence of obstacles. REFERENCES [1] Sugihara, K.; Suzuki, I., "Distributed motion coordination of multiple

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[5] [6] [7]

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[14]

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