Adaptable Robot Formations Using Fast Marching Square Working Under Uncertainty Conditions Javier V. G´ omez Robotics Lab. Universidad Carlos III de Madrid Legan´ es, 28914, Spain Email: jvgomez@pa.uc3m.es Santiago Garrido Robotics Lab. Universidad Carlos III de Madrid Legan´ es, 28914, Spain Email: sgarrido@ing.uc3m.es Luis Moreno Robotics Lab. Universidad Carlos III de Madrid Legan´ es, 28914, Spain Email: moreno@ing.uc3m.es Abstract—Robot formations are getting important since they can develop tasks that only one robot could not do or could take too much time. Also, they can perform some tasks better than humans.This paper provides a new algorithm to control robot formations working under uncertainty conditions such as errors in robot positions, errors when sensing obstacles or walls, etc. The proposed approach provides a solution which is based on leader- followers architecture (real or virtual leaders) with a prescribed geometry of the formation and it adapts dynamically to the environment, avoiding obstacles or changing itself when required by the environment (i. e. narrow corridors). The algorithm applies the Fast Marching Square (FM 2 ) method to the path planning of mobile robot formations, which have been proved to work fast and efficently. The FM 2 method is a potential based path planning method with no local minima which provides smooth and safe trajectories. The algorithm described here allows to easily set different behaviours to the formation during its motion depending on the objectives, being possible to set the flexibility. The results presented here show that using this method allows to the formation reacting to either static and dynamic obstacles with an easily changeable behaviour. I. I NTRODUCTION Due to the multiple applications of robot formations (surveillance, cooperative mapping, etc) the interest on this field has increased enormously during the last years, being actually one of the main topics on robotics research. Right now, a single robot is able to perform very complex tasks on its own but some of these tasks can be perfomed in a more efficent way using a group of robots. A good enough algorithm to control the motion of a robots formation can influence in a lot of very different fields. For example, using a group of small robots allows to carry out a complex task which, if it has to be done with only one robot this have to be a very complex robot, more expensive and with more limited applications (i. e. transportation of big pieces), thus it is possible to save time and money. When developing exploration or surveillance tasks a robot formation can give better results since the exploration can be done faster or can patrol a wider area, being more effective when developing their objectives. Another important social impact of using robot formations is in search and rescue objectives. In environmental disasters, such as earthquakes or tsunamis, it is very important to explore all the area as fast as possible, and there are lot of cases where humans are not able to go (i. e. destroyed buildings) so using more than one robot is required. If the robots used are able to keep a formation they can explore more efficently the disaster area finding faster the survivors, creating a map of the destroyed zone (which allows to plan the action to do) or even move dribs without need a very complex robot. Also, in evacuation tasks (i. e. fires) robot formations can be very interesting since they can guide people to the exit using an optimal path, recalculating faster the path when the desired one is impossible. Also, it is important that the robots do not feel, do not doubt and do not fear, allowing them to develop this kind of tasks in a better way than humans. Therefore applying robot formations to risky or complex tasks can improve their development and can save time, money, increase safety in some kind of jobs and dangerous situations or even save lives. The algorithm proposed here is focused in leader-follower architecture, where the formation is deformable depending on the position of the robots in function of a virtual potential field. The Fast Marching Square (FM 2 ) method [1] is introduced in robot formations method including new advantages. The paper is divided in four sections (apart from this one): in Section II the Fast Marching Square algorithm is summarized. How it is applied to robot formations is explained in Section III, where a base algorithm is proposed and some modifi- cations that can be included (avoiding obstacles, changing flexibility, etc.). The final conclusions are shown in section IV. mds January 11, 2007 II. FAST MARCHING AND FAST MARCHING SQUARE A. Introduction to Fast Marching and Level Sets In 1996, J. Sethian proposed the Fast Marching algorithm to approximate the viscosity solution of the Eikonal equation |∇(D(x))| = P (x) (1) The level set {x/D(x)= t} of the solution represents the wave front advancing with a media velocity P (x), which is the inverse of the media refraction index R(x). Therefore, the Eikonal equation can be written as |∇(D(x))| =1/R(x). The resulting function D is a distance function, and if the