Convex Shape Generation by Robotic Swarm

Transcription

1 2016 International Conference on Autonomous Robot Systems and Competitions Convex Shape Generation by Robotic Swarm Irina Vatamaniuk 1, Gaiane Panina 1, Anton Saveliev 1 and Andrey Ronzhin 1 1 Laboratory of Autonomous Robotic Systems St. Petersburg Institute for Informatics and Automation of the Russian Academy of Sciences St. Petersburg, Russia Corresponding author: Irina Vatamaniuk, vatamaniuk@iias.spb.su Abstract Forming a predetermined spatial shape is one of the most important problems of modern swarm robotics. The work presented here focuses on the motion planning for the swarm of identical robots in the task of spatial reconfiguration. The main requirement is that the reconfiguration must be collision free, that is, at each moment the pairwise distances between robots should not be smaller than some fixed minimal admissible distance. A collision-free navigation algorithm that generates a prescribed convex surface is presented. It includes trajectory computation combined with delay assignment. The computer simulation results demonstrate that the proposed algorithm works efficiently even for big swarms. Keywords-a swarm of robots; surface imitation; group control; robotic systems; motion planning I. INTRODUCTION In contrast to the single robot, systems of autonomous mobile robots are able to fulfill a larger class of civil and military missions, including firefighting, rescue operations, agricultural and other tasks [1]. The main problem of swarm navigation is to bring robots from initial positions to the target positions in a collision-free way. In respect that the robots in a swarm are identical, they do not have individual assignments and their motions are predicted by the current swarm location and energy as well as time savings. A traditional approach makes use of communications between robots, that is, it works with a robotic network. In addition, traditional algorithms are gradient descent ones with asymptotic convergence (originally with continuous time, but for implementation - with discrete time adapted). In many applications, a group of robots is required to interact in a certain way as a single unit to accomplish some complex tasks. While performing a group task, each robot obtains its individual trajectory to take corresponding place in the group [2]. Let us sketch briefly some existing approaches. In [3], the multi-robot motion-planning problem is considered. The authors discuss the motion planning of m unlabeled unit-disc robots in a simple polygon with n vertices. Solving this problem for a single robot and introducing the application of the multi-component system, they propose the collision-free algorithm with complexity O (nlogn+mn+m 2 ). Sometimes there are several tasks that need to be solved at the same time. To execute the assigned tasks a swarm of autonomous mobile robots must divide itself into several balanced groups. An approach to multi-robot coalition formation in real time scenarios based on the multi-robot task allocation is described in [4]. The partitioning problem is also discussed in [5]. The authors propose algorithms for controlling position and orientation of robot groups while forming a certain structure. Synchronization and orientation of particular robots are the two main aspects of the proposed algorithms. The partitioning problem is considered for the asynchronous full-compass and semi-synchronous halfcompass models, and a randomized algorithm is given for the semi-synchronous no-compass model. A leader follower structure in the presence of obstacles is discussed in [6]. The authors use the Asexual Reproduction Optimization (an evolutionary-based algorithm) that mathematically models the budding mechanism of asexual reproduction. The robots plan their paths based on the potential field method. The leader moves according to the desired trajectory, which can help the whole system, and the other robots follow the leader by the relative knowledge achieved from their neighbors and the leader. The main problem in the task of cooperative control of multi-robot systems is positioning accuracy. A region-based shape controller for multi-robot systems is presented in [7]. The authors propose a Lyapunov-like function for stability analysis of the multi-robot systems and show that the robots are able to move as a group inside the desired region maintaining minimum distance from each other. The other difficulty of shape formation by a swarm of robots is the complexity of the prescribed shape. In [8], two kinds of the formation control problem are discussed: the efficient initial formation and formation control while avoiding obstacles. The authors propose a behavior-based control design approach for the formation control of swarm robot systems while navigating in an unknown environment with obstacles. A multi-group coordination control methodology for robot swarms is presented in [9] to establish a complex formation. The authors propose to decompose the robot swarms to multiple groups. The inter-group formation and the intra-group formation are considered, which mean coordination of groups in the swarm and the formation of individual robots in the groups respectively. Additionally, the authors use a complex leader hierarchy in the multigroup formation control architecture, which allows creating various complicated shapes. Another decentralized control strategy to realize formations of a robot swarm is presented in [10]. The authors describe designing artificial potential fields to obtain robot formation in the shape of a desired polygon. To attain /16 $ IEEE DOI /ICARSC

2 different objectives (movement to the target position, avoiding collisions and obstacles, coherent movement, etc.) different potential fields can be designed. As in [7], the computations and the formal proofs are based on the Lyapunov function. In [11], a new topology method based on a multilevel structure is employed to present different shapes of swarm robots in the desired region. To control the robots while forming the shape and to reduce the possibility of robots getting stuck or colliding, the authors propose a potential function-based controller. II. COLLISION-FREE RECONFIGURATION ALGORITHM Suppose that there is a swarm of n identical robots and a convex surface S needed to be formed by these robots. The robots have to cover the surface with the given density ρ. It is assumed that there are enough robots to form the surface. The density ρ means the desired number of robots on the square 10 x 10. We assume that the surface is determined by its vertices υ i = (x i, y i, z i ) and by the set of the triangular faces (i, j, k) numbered from 1 to M. We assume that each robot can move with the speed not exceeding V. The minimal distance between the centers of every pair of robots is limited by the value MinDist. If this distance is smaller than MinDist during the reconfiguration, we say that the point robots collide. The meaning of the constant MinDist is clear: the real robots are not points; they have some physical size, and it is unsafe for them to come close to each other. The task is to reconfigure a part of the swarm in a collision-free way to form the surface S. The energy and time costs of the reconfiguration should be possibly minimized. The output of the proposed quadratic time algorithm is: The target coordinates of the point robots. The assigned trajectories of all robots taking part in the reconfiguration. The assigned relative delays of all robots taking part in the reconfiguration. Estimated time of the swarm reconfiguration. Estimated number of resolved collisions. The proposed algorithm includes the following stages [12, 13]: 1) Analysis of the surface and target points assignment. The first stage includes calculating the target and portal points coordinates according to the geometry of the shape and the required density ρ. The target points fall into three types: peak points (vertices of the surface S), edge points and face points, according to their location on the surface S. The third type target points are assigned at the affine image of the nodes of the standard lattice associated to the triangular face. Alternatively, one can choose honeycomb lattice, which gives a more economic way of packing robots on the surface. The second and the third type points are calculated based on the distance δ between target points: 10. 2) Portal points assignment. Their informal meaning is the point where a robot can cross the surface avoiding collisions. Put a portal point at the barycenter of each of the faces. If the number of the faces is relatively small (with respect to the number of robots), it is recommended to further refine the triangular faces of the surface. It is important that the mutual distances between the portal and target points are greater than MinDist. 3) Equipment of the points. Each of the target and portal points is equipped with two equidistant normal points. We equip each of the points (both target and portal ones) with two normal points, one inside and another outside the surface (Fig.1). Each normal point is positioned at the distance MinDist from the target or portal point by the normal vector to the surface. These points are used at stage 5 of the algorithm. 4) Matching the robots and target points. We match the robots and the target points in a way which minimizes the total distance. This has another advantage: the number of collisions to be resolved is also minimized. The number of target points may be smaller than the number of robots in the swarm. At this step we decide, which of the robots will participate in the reconfiguration ( active robots ) and which ones will not ( passive robots ). Each target point is matched with the closest uninvolved robot; hence the list of the active robots is formed in ascending order of the distance to the proper target point. In the case when n > N, the most distant unmatched robots are considered passive and do not take part in the swarm reconfiguration. We proved that with this matching passive robots never collide with the active ones. 5) Trajectory assignment. We make use of the following principle: a robot approaches the surface along the directon of the normal vector. It is explained by the fact that if the robot approaches the target point by a trajectory tangent to the surface, there may occur multiple collisions with the robots that have already reached their target points. For each of the active robots we prescribe the trajectory from its starting point to the assigned target point by the following rule. Take the segment connecting the starting X(k) and the outside normal point Y1(k) of target point Y(k): [X(k), Y1(k)]. If the segment does not intersect the surface S, the trajectory is X(k) Y1(k) Y(k), where Y1(k) is the normal point of the target point Y(k) lying on the same side of the surface S as the point X(k). Otherwise the trajectory crosses the surface S at a portal point i: X(k) Y1(i) Y2(i) Y3(k) Y(k). Here the portal point i belongs to the face where the segment [X(k), Y(k)] intersects the surface S; Y1(i) and Y2(i) are the normal points of i; Y3(k) is the normal point of the target point Y(k). Fig. 1 shows the example of possible robot trajectories. Note that the smaller MinDist is, the closer is the assigned trajectory to a straight line segment

3 Begin all T d := 0 Robot trajectories for i:=2 to n for k:=1 to i-1 Figure 1. The robots X move to their target points Y through the normal points Y1, Y2 and Y3. The target and the portal points are marked black and white respectively. No Is there any collision between robots i and k? 6) Delays assignment. To completely avoid collisions, we assign the starting time delays to some of the robots. Simple examples show that if all the robots start moving simultaneously, the collisions still exist. However, all of them can be resolved by assigning delays to some of the robots. Assume that at the beginning all delays are zero, that is, robots start moving at one and the same time: T d (k)=0. We run the collision test for each pair of the robots. The test computes the minimal distance between the moving robots under assumption that the robots move along the prescribed trajectories. The distance is an easily computable quadratic function in time, so its minimum comes from a simple polynomial expression. In case of collision we increase the time delay T d (k) for the robot with the greater index: T k T k, where τ is a time slice: d MinDist. V After that the collision test is performed again (considering the assigned delay T d (k)) until all the collisions are resolved. The flowchart describing delays assignment is presented in Fig. 2. Experiments show that although we need iterations of the collision resolving, the actual number of cycles is small, see next section for details. d T d (i) := T d (i)+τ k:=1 End Yes Figure 2. A flowchart of the algorithm of the delays assignment. III. EXPERIMENTS Computer experiments certify that the algorithm works efficiently for the number of robots up to 1500 (we did not test beyond this number; however, we anticipate an efficient work of the algorithm for bigger swarms). To evaluate the proposed method a regular octahedral surface was simulated. The size of the shape was determined by the necessary number of robots at the maximum areal density ρ max. The starting coordinates of the robots were set randomly in a predetermined domain. Two options were considered: 1) all robots are located in the neighborhood of the center of the octahedron limited by 3l, where l is the length of the octahedron s edge ( around the center ); 2) all robots are located to the center of the octahedron not closer than 3l and not farther than 6l ( outside ). Further, we considered such parameters as: 1) computation time (Fig. 3a); 2) the amount of collisions that were resolved at ρ max (Fig. 3b); 3) maximum and average delays assigned to the robots at stage 6 of the algorithm (Table 1)

4 Figure 3. (a) Estimated computation time. (b) Estimated number of collisions. We can conclude the following: as is shown in Fig. 3b, the number of collisions under described conditions does not exceed 12% of the number of robots. Fig. 3a shows the average computation time of the robot trajectories. It increases in proportion to the number of robots and actually does not depend on the starting disposition of the robots. Let us give an example. Take 102 robots with a radius R=1 positioned randomly in the cube 120x120x120. Take MinDist=3R. For the octahedral surface S with vertices at the points (20, 0, 0) ( 20, 0, 0) (0, 20, 0) v, (0, 20, 0) (0, 0, 20) (0, 0, 20) we put one portal point at the center of each face i of the surface S. Although this was not an economical way (remind that for big number of robots a further refinement of the mesh is recommended), the algorithm worked well. The number of prevented collisions is 4. The maximal time delay in the given example was 6τ. The maximum delay that appeared in the experiments and the average delay depend on the number of robots (Table 1). TABLE I. THE MAXIMUM AND AVERAGE DELAY Number Maximum Delay, τ Average Delay, τ of Robots Inside Outside Inside Outside IV. CONCLUSION The aim of this paper was to introduce an algorithm for collision free reconfiguration of a swarm of robots in the convex surface formation. The advantages of our approach are: Quadratic computational time; Almost rectilinear trajectories; Each robot changes the direction of movement at most three times, which is also economic; Relatively small delays imply economy in reconfiguration time. Relatively small number of collisions to be resolved. The developed software model demonstrates the movement of a large number of small robotic systems in the formation of spatial figures. Experimental simulation of swarm movement was carried out for different quantities of robots (from 10 to 1500 units) in order to verify the scalability of the software and the computational costs. The future work will be related to the application of the algorithm in the real engineering robotic systems, to research and complication of formed surfaces, as well as (for the sake of robustness) to using mutual communications between robots. ACKNOWLEDGMENT This work is supported by state research and the Russian Foundation for Basic Research (grant ofi_m). REFERENCES [1] T. Sousselier, J. Dreo, and M. Sevaux, Line Formation Algorithm in a Swarm of Reactive Robots Constrained by Underwater Environment, Expert Systems with Applications, vol.42, issue 12, 2015, pp [2] L. Shuang, S. Dong, and Z. Changan, Coordinated Motion Planning for Multiple Mobile Robots along Designed Paths with Formation Requirement, IEEE/ASME Transactions on Mechatronics, vol. 16, 2011, pp [3] A. Adler, M. de Berg, D. Halperin, K. Solovey, Efficient Multi- Robot Motion Planning for Unlabeled Discs in Simple Polygons, IEEE Transactions on Automation Science and Engineering, vol. 12 (4), 2015, pp

5 [4] J. Guerrero, Gabriel Oliver, Multi-Robot Coalition Formation in Real-Time Scenarios, Robotics and Autonomous Systems, vol. 60, 2012, pp [5] A. Efrima, D. Peleg, Distributed algorithms For Partitioning a Swarm of Autonomous Mobile Robots, Theoretical Computer Science, vol. 410, 2009, pp [6] A. Asl, M. Menhaj, and A. Sajedin, Control of Leader Follower Formation and Path Planning of Mobile Robots Using Asexual Reproduction Optimization (ARO), Applied Soft Computing, vol. 14, 2014, pp [7] C.C. Cheah, S.P. Hou, and J.J. Slotine, Region-Based Shape Control for a Swarm of Robots, Automatica, vol. 45(10), 2009, pp [8] D. Xu, X. Zhang, Z. Zhu, C. Chen, P. Yang, Behavior-Based Formation Control of Swarm Robots, Mathematical Problems in Engineering, vol. 2014, Article ID , 13 pages, [9] L. Sabattini, C. Secchi, and C. Fantuzzi, Potential Based Control Strategy for Arbitrary Shape Formations of Mobile Robots, Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, 2009, pp [10] R. Haghighi, C. Cheah, Multi-Group Coordination Control for Robot Swarms, Automatica, vol. 48, 2012, pp [11] X. Yan, J. Chen, and D. Sun1, Multilevel-Based Topology Design and Shape Control of Robot Swarm, Automatica, vol. 48, 2012, pp [12] I.V. Vatamaniuk, G.Yu. Panina and A.L. Ronzhin, Modeling of Robotic Systems' Trajectories in Spatial Reconfiguration of Swarm, Robotics and Technical Cybernetics, vol. 3(8), 2015, pp (In Russ.). [13] I.V. Vatamaniuk, G.Yu. Panina, and A.L. Ronzhin, Reconfiguration of Space Formation of Robot Swarm, Large-scale Systems Control, issue 58, Moscow: Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 2015, pp (In Russ.)