LARS 2009: LATIN AMERICAN ROBOTICS SYMPOSIUM

Abstract—The task of formation maintenance is of great importance in high level applications being executed by multi-robot systems, such as exploration, surveillance and cleaning, among others. This article presents a strategy for formation maintenance in a group of mobile, cooperative, homogenous robots, which work together with a software agent known as “virtual robot”. This virtual agent acts as reference for the spatial localization of each robot, and facilitates the navigation tasks of the group. This approach combines the characteristics of other well known formation reference strategies: leader- and unit-center referenced formations. The proccess of validating this strategy was obtained from both simulated and real multi-robot systems.

Index Terms—Behavior-based control, Cooperative robotics, Formation control, Virtual robot.

I. INTRODUCTION

OOPERATIVE robotics is a topic of great interest, due to the advantages a group of robots have in comparison

to single specialized robots, such as distributed sensing, distributed action, fault tolerance, among others [1]. But at the same time many dificulties arise when working with multi-robot systems. There is an inherent complexity for the coordination of groups of autonomous agents. One of the main complexities to deal with is the coordination of movement. It is desirable to minimize interferences between robots when they traverse the environment, without sacrifying coherency. In order to fulfill this objective it has been proposed that the navigation could be achieved by maintaining regular geometric patterns known as formations. Thus, formation maintenance arises as an important task in multirobot systems. A coherent spacial organization within the group of robots serves for optimization of high level tasks, such as cleaning, surveillance and rescue applications, among others. Formation maintenance is also bioinspired, as it can be found in many animal behaviors. Several social

C

Manuscript received September 11, 2009. Jonathan A. Hernandez. is a M.E student at the School of Electrical and

Electronics Engineering of the University of Valle, Cali, Colombia (e-mail: jonathanalejandro@gmail.com).

Alejandro M. Pustowka is a M.E student at the School of Electrical and Electronics Engineering of the University of Valle, Cali, Colombia (e-mail: alejandro.pustowka@gmail.com).

Eduardo F. Caicedo is with the School of Electrical and Electronics Engineering of the University of Valle, Cali, Colombia (e-mail: ecaicedo@univalle.edu.co).

Eval B. Bacca is with the School of Electrical and Electronics Engineering of the University of Valle, Cali, Colombia (e-mail: evbacca@univalle.edu.co).

organisms demonstrate the importance of group organization. This organization offers many benefits that involves energy optimization, greater protection from predators and facilitates the search for food. Due to those benefits, social animal organization as seen in herds, flocks and shoals could have been a key element to the success of that organisms. For these reasons, social behaviors can be considered an inspiration for the control of multirobot systems.

In this paper we present an approach for formation maintenance of a group of robots using a virtual robot as the reference for the formation. The next section describes some previous work done in this area. Section III and IV present the characterization of formations and the proposed solution. Section V describes the experimental setup being used for running the tests and obtain the results, which are included in section VI. Finally, conclusions are presented in section VII followed by some future work mentioned in section VIII.

II.BACKGROUND

Several approaches have been used for the task of formation maintenance, but mainly three of them are the most common: Leader-follower controllers, behavior-based controllers and virtual structures [4]. Some of the strategies are based in centralized algorithms where the navigation tasks are planned on a central unit, such as a robot of the group with better processing capabilities or another dedicated processing device. Other solutions are focused on a descentralized approach, where the navigation tasks are done individually by each one of the robots.

The leader-follower approach defines the path planning for each robot by using the kinematic information of the leader [4], [8], [12]. The control action applied to the system uses both the leader and the follower position information to define the route to the planned path. A slightly different leader-follower approach uses a grid map and an A* search algorithm in order to plan a route for the leader robot [9]. This route is then used by the trajectory generator, which defines a continuous path. The follower robots calculate their own trajectories based on the position of the leader.

Behavior-based controllers have also been successfully used for formation maintenance purposes. It is posible to achieve good performance in high level behaviors (like flocking) using only local information and simple behaviors, such as aggregation, avoidance and dispersion [13]. Balsh et al. [3] present also a behavior based control, and define three alternatives for the definition of formation reference: center-unit reference, leader reference and neighbour reference.

Formation control of cooperative robots with limited sensing using a virtual robot as

reference

Jonathan A. Hernandez*, Alejandro Pustowka*, Eduardo F. Caicedo, Eval B. Bacca.

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LARS 2009: LATIN AMERICAN ROBOTICS SYMPOSIUM

Negotiation can be used to define the robot's position in the formation [15]. The number of robots in the formation can therefore be increased or decreased, using negotiation. An approach inspired by the way molecules connect themselves when forming crystal structures is presented in [2]. Each robot counts with union points or virtual attachments, which serve as attractors for the other robots. Thus certain configurations of the union points around the robot can serve to define rigid formation patterns. Fredslund and Mataric [5] present a behavior based control where each robot uses only local information to locate its friend (a neighboring robot) by using a camera and a laser range finder. Because of the distributed nature of the approach, formation maintenance is an emergent behavior.

Another approach for formation maintenance uses a concept known as “virtual structure” [11]. Each robot is modeled as a particle that belongs to a rigid geometric pattern. The navigation is implemented using potential fields, so that each point belonging to the virtual structure is an attractor for the robots according to their ID. Thus it is only necessary to control the virtual structure's movement across the environment.

A self-organized approach implementing negotiation is presented in [10]. Negotiation is used to define the robot's ID according to its spatial location. The authors divide the solution in three stages: pattern generation, flocking and change of pattern depending on the enviromental restrictions. They compare the algorithms for a leader and a neighbour reference and propose an hybrid approach, which chooses the reference that best fits for the actual pattern of formation.

Many approaches have been proposed for dealing with formation maintenance. Each one of them presents inherent advantages and limitations. Based on this previous work we present our solution, specifying a behavior based control for formation maintenance and using a virtual robot as the reference.

III. FORMATION DEFINITION

Formations can be defined as “a geometric disposition of individuals in the environment”. It is essential to define a common reference in order to specify the position of each one of the agents in the formation. There are several techniques proposed, being the most common the following ones, explained in detail in [3]:

1) leader-reference: The position of one of the robots is taken as the reference for the others, in order to calculate the desired position for each one.

2) neighbour-reference: the robots assume the position of the closest partner as the reference to calculate its own position.

3) unit-center reference: the unit-center location of the group is calculated by each one of the robots and is taken as the reference to calculate its desired position.

Fig. 1 illustrates those techniques. The first two have the advantage that they can be implemented using only local information, detecting the leader or neighbour position with its local sensors, while the last one requires the use of global information, as it implies that each one of the robot in the

group should know a priori the position of the others in the environment.

(a) (b) (c)

Fig. 1. Reference types.(a) is unit-center reference, (b) is neighbour reference and (c) is leader reference

Formation Characterization

The model used to define different formation patterns makes the following assumptions: Each robot has its own ID, which will be used to define its position in the formation. This characteristic does not affect the scalability of the model, but limits the position of the robot in the formation. This can be avoided by using negotiation of each robot's ID.

Each formation pattern is characterized using a composite function. This function defines the desired geometric figure contour, see equation (1). An additional parameter m is used in order to define the width of the pattern. The reference of the formation is found in the origin of the coordinate system, which should equals the unit-center of the pattern.

F form x ,m={F1x ,m

F2 x ,m

F3x ,m

...} (1)

The position of each robot in the formation is defined using a vector referenced to the unit-center of the pattern, and it can be easily specified using polar coordinates as in (2), where PRK is the position of the k-nth robot. The calculation of the angle θk depends on the number of robots in the group, where (3) is used if the number of robots is odd, or (4) if the number of robots is even. Variable n represents the cuantity of robots, and k represents the robot's ID.

P Rk=r k ,k

(2)

k odd =2n

k (3)

k even =n2k1 (4)

The calculation of the rk distance is done by intersecting he composite function (1) and the line defined by angle θk of the robot and the reference point FR, as seen in (5). The point of intersection represents the cartesian coordinates for the position of the k-nth robot, (xk, yk). Thus rk is found by converting the cartesian coordinates to the equivalent polar coordinates (6).

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F form x ,m−F Rx =0 (5)

r k= xk2 yk

2 (6)

In that way it is possible to obtain the position of the k-th robot referenced to the center of the formation. In order to include the orientation of the formation pattern, the model has to be corrected, adding the θref parameter, which deals with the reference orientation (7).

P Rk=r k ,kref (7)

Fig. 2 illustrates an example modelling a square formation pattern. Line 1 to 4 conform the composite function (1). With four robots, equation (4) should be used. For example, the angle is calculated for the robot with ID = 0, resulting in an angle of п/4. r0 is finally calculated using equation (7).

Fig. 2. Example of a formation pattern

By following this method it is posible to model different formation patterns which share a common characteristic: all the agents will be located in the contour of the geometric pattern. Four patterns were defined, and are shown on Fig. 3: square, row, column and arrow.

(a) (b) (c) (d)

Fig. 3. Implemented formation patterns. a) Square pattern b) arrow pattern c) row pattern d) column pattern

IV. PROPOSED SOLUTION

Now that a model for the patterns has been defined, a reference for them has to be chosen. The approach taken for this work uses a software entity for the reference. This entity is known as “virtual robot”, and its position is communicated to the other robots for navigation purposes.

A behavior based approach is used in our solution. The control of the navigation is implemented using potential fields. The behaviors are defined by following the schema-

model proposed by Arkin [1], using perception and motor schemas. A cooperative arbiter is also used rather than a subsumption one. The robots are modeled as particles, that means its position will be controlled but not its orientation. Altough this is a minor issue, in most cases the desired orientation is reached indirectly when the formation is moving. We used the unicycle-like model with differential drive to model the robots. The relationship between the control variables and the outputs (kinematic model) can be found in (8).

[x 'y ' ' ]=[

cos 0sin 0

0 1] [ vw ] (8)

A. The virtual robot

Based on the previous work reviewed in the background section, mainly three reference types were defined: leader, neighbour and center-unit reference [3]. The reference selection for the solution involves some constraints to be evaluated. Communication dependency is not desirable, due to inherent wireless limitations, such as distance, signal power, among others. It is also desirable to select a method that leads to a more fault-tolerant solution. Thus a descentralized approach is convenient for the fulfill of this requirement.

We propose the use of a virtual robot as the pattern reference. This is mainly a leader reference strategy, but shares some of the characteristics of the unit-center approach. Being a software entity, it can be executed in a centralized as in a distributed system, improving fault tolerance. In leader referenced approaches the fault tolerance can be improved by using negotiation of a new leader, in case the actual one fails. With a virtual robot the negotiation can be easier, as all references depends on a virtual agent rather than a real one. There is no need to re-negotiate position references for each robot. The virtual robot is located in the unit-center of the pattern. This leads to a better formation stability when moving, as stated in [3].

The strategy implemented with the virtual robot follows these steps. First, attractors for each robot are defined around the virtual robot, according to each robot's ID and the formation pattern. The robots use local control, based on behaviors, to reach those points. The virtual robot can then be controlled, and the robots will follow the virtual robot maintaining the desired formation pattern. Fig. 4 illustrates the strategy.

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Fig. 4. Attractor to the desired position in the pattern

B. Virtual Robot Control

The behavior stack of the virtual robot is shown on Fig. 5. The following perception and motor shemas were defined for the control of the virtual robot:

1) Detect-Obstacles: Perception-schema that detects obstacles in the environment using a global sensor. It is important to note that the virtual robot is unaware of obstacle configurations that does not imply a risk to the whole group. In such cases, obstacle avoidance is solved by the local control of each robot in the group. In cases where the configuration of obstacles can be seen as a risk for the whole group (such as a corridor), the virtual robot can change the formation pattern to improve the obstacle avoidance.

2) Detect-Centroid: This schema reads the position of each robot and calculates the unit-center of the formation. The virtual robot should not get too far away from the real robots. Real robots could have been left behind when the environment impose many restrictions. This is corrected in our approach by defining an attractor to the unit-center of the formation. This is the only schema that requires global communication.

3) Avoid-Obstacles: Motor-schema used to avoid detected obstacles by the perception-schema Detect-Obstacles.

4) Goto-Point: Motor-schema that defines an attractor to the goal point.

5) Goto-Centroid: Motor-schema that calculates an attractor to the unit-center of formation, localized by the perception-schema detect-centroid.

Fig. 5. Behavior's stack of the virtual robot

C. Control of the real robots

The behavior stack of the real robots is shown on Fig. 6. The following perception and motor shemas were defined for the control of real robots:

1) Detect-obstacles: Perception-schema that calculates the position of obstacles in the environment, using information from the proximity sensors.

2) Detect-formation: Perception-schema used to define the desired position in the pattern by using the virtual robot's position and its own ID (See Fig. 4).

3) Avoid-obstacles: Motor-schema used to avoid obstacles detected by the perception-schema detect-obtacles.

4) Maintain-formation: Motor Schema that generates an attractor to the position calculated by detect-formation schema.

Fig. 6: Behavior's stack of real robots

V.EXPERIMENTAL SETUP

The experiments done for the validation of the proposed solution were executed in a simulated platform as in real robots.

A. Simulator

The Stage simulator from the open source project Player/Stage was selected for this tests. Simulation in a 2D-environment can be done, and for computational-efficiency purposes Stage uses simplified models of robots and devices. This allows simulation of large multi-robot systems. Player [6][7], the most important part of this project, is a server used for robot control. It permits an abstraction of different robot devices, which simplifies the design of the controller. Player is a network server, that means it uses TCP/IP as the base protocol for the communication. One of the main advantages of using the Player/Stage framework is that the clients being programmed, tuned and tested in simulation with the Stage simulator will work fine with little or any modification in real robots [6].

B. The group of Robots

The experiments with real robots were done using the multi-robot platform of the University of Valle, NX-Bot. This platform was developed focused on cooperative-robotics experimentation. Each robot runs a Linux distribution for embedded systems. The Player server is running as a process of the OS. The NX-Bots are differential drive robots with proximity sensors and Wifi for its communication. Table I illustrates the main characteristics of the platform.

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TABLE I. ROBOT SPECIFICATIONS

Variable Va.lue

Dimension Radius: 12 cmHeight: 15 cm

Kinematic model Differential drive

Max. Velocity 15 cm/s

Encoders 360 tics/rev

Wheel radius 3,325 cm

Sensors Six proximity sensorsFour Bumpers

In order to improve the positioning system of the robots, a localization system with a global camera was used. The robots acquire its own position by contacting a network server.

VI. TESTS AND RESULTS

Two tests were selected for the evaluation and validation of the proposed solution. The first one evaluates the stability of the formation (defining the term “stability” as in [5]), by observing the robots maintaining a formation pattern while traversing across an obstacle-free environment. The second test evaluates the robustness of the model to changes of the pattern. A manual change of formation will be applied, in order to see how the group of robots adapts itself to the new pattern.

The estimated average position error eavrg (i.e. the average of the individual error positions of each robot, see (9)) will be used to perform a quantitative evaluation of the proposed solution. The position error is calculated by suming the distance between the actual position of each robot (Pi in (9)) and the desired position in the formation pattern (Di in (9)), and averaging it by the number of robots in the pattern (n in (9)). For satisfactory formation stability, we assume that the average position error should be less than the size radius of the robots. That means, the expected errors for our simulation tests have to be less than 25 cm (radius of simulated robots), and less than 12 cm (radius of NX-Bots) for the real scenario. The four formation patterns defined in section III were tested in the experiments.

eavrg=∑i=0

n Pi−Di

n(9)

A. Formation stability

Fig. 7 illustrates the simulated group of robots, when establishing a square pattern, and moving across the environment while keeping in formation. The figure also shows the average position error calculated every instance of the experiment. After establishing the formation, the average and max. errors are lower than 25 cm (the radius of simulated robots). The average error is around 1,19 cm (see table II).

Fig. 7. Average position error in a simulated square formation pattern

Fig. 8 illustrates the simulation results for the group of robots while maintaining a row pattern. The average and max. errors are lower than 25 cm (radius of the simulated robot): 1,67 cm as stated in table II. Results for the other patterns are considered in table II.

Fig. 8. Average position error in a simulated row formation pattern

The tests evaluated in real robots also showed a satisfactory behavior. Fig. 9 illustrates a series of images of the group of NX-Bots while driving across the real environment, maintaining a square pattern. Fig. 10 shows the behavior of the average error during the traverse. It remains lower than 12 cm (the radius of real robots).

Fig. 9. Maintaining Square pattern with real robots

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Fig. 10. Average position error in real tests maintaining square pattern

Fig. 11 shows the group of real robots maintaining a row-pattern formation, and Fig. 12 shows the calculated average error of this test. As with the previous pattern, the average error is below the limit defined as the robot radius.

Fig. 11. Maintaining row pattern with real robots

Fig. 12. Average position error in real tests maintaining row pattern

A summary of the average errors for each formation pattern can be found in Table II, including both simulated and real results. Absolute and relative errors are included in the table. Relative errors are calculated using the radius of the robot. That means, relative errors should be bellow 100% in order to be satisfying. As it can be seen, all errors are below the corresponding limits, being those the radius of the robot (for absolute errors), and the 100% limit (for relative errors).

TABLE IIRELATIVE AND ABSOLUTE AVERAGE ERROR EAVRG (CM)

Formation Pattern

Simulation (r = 25 cm)

Real(r = 12 cm)

Square 1,19 4,76% 1,83 15,25%

Arrow 0,88 3,52% 1,08 9,00%

Row 1,67 6,68% 2,79 23,25%

Column 1,78 7,12% 3,83 31,92%

B. Change of formation pattern

A pattern change from row to column can be seen on Fig. 13. This is a manual change order given by the user trough the command interface of the application.

Initially, the robots drive across the environment maintaining a row pattern. In an instance the pattern is changed to column formation. Thus, the average error increases (see Fig. 14), because the robots are not in the desired position. The reactive control then acts in order to let the robots reach the desired position in the pattern. Some constraints, like the velocity of the robots and their kinematic model limit the response time the robots need in order to traverse again maintaining the desired pattern.

Fig. 13. Manual change of pattern formation

Fig. 14: Average Error for a manual change of pattern (simulation)

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Fig. 15 shows the behavior of the average position error for a manual change of pattern using real robots. The behavior is very similar to the previous simulated results.

Fig. 15: Average Error for a manual change of pattern (real robots)

With a position error lower than the defined limit for both simulated and real results, it can be stated that the formations remain stable while the robots drive across the environment. The tests also showed that a change of pattern is posible with some time constraints imposed by the maximum velocity of the robots, its kinematic model and the geometric nature of the new pattern. The obtained results are satisfactory and validate the approach taken in this work.

VII. CONCLUSIONS

A successfull approach for maintaining four defined formations was implemented. The strategy is based on a virtual robot as the pattern reference and a group of four robots acting reactively in a simulated as in a real environment. The proposed tecnique is scalable, referring to the number of robots, and it allows to use any pattern that can be mathematically modeled by the described process of section III.

The virtual-robot reference is mainly a leader reference, requiring more bandwidth but remaining lower than a unit-center reference approach.

The use of a virtual robot facilitates some high level tasks. Teleoperation of the whole group of robots can be easily implemented by just controlling the virtual robot. Sensorial fusion can be useful to model the environment using information of each one of the robots. There is also the possibility to define behaviors using sensor information from real robots.

VIII. FUTURE WORK

The abstraction of the virtual robot serves for sensorial fusion purposes. Strategies for automatic change of pattern can be implemented in situations where a certain configuration of the environment inhibits the normal traverse of the group when maintaining a specific pattern. This is the case when the robots have to drive across a corridor whose width is not enough for the actual pattern. The notion of virtual robot could also serve for fault tolerance strategies.

Being a software entity, it is more feasible to recover from a software crash than from a hardware one.

IX. ACKNOWLEDGMENT

The authors thank the University of Valle and Colciencias (Departamento Administrativo de Ciencia, Tecnología e Innovación), for the research scholarships awarded through the “Young Researchers” program to Jonathan A. Hernandez under grant p-2008-0210 and to Alejandro M. Pustowka under grant p-2008-0930.

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