Formation Control for Mobile Robots with Limited Sensor Information

Transcription

1 Formation Control for Mobile Robots with imited Sensor Information Tove Gustavi and Xiaoming Hu Optimization and Systems Theory Royal Institute of Technology SE 1 44 Stockholm, Sweden gustavi@math.kth.se Abstract In this paper mobile multi-agent systems with limited sensor information are studied. Some control algorithms are proposed that do not require global information, and are easy to implement. First two basic controls for a single mobile agent tracking another moving object are derived. Then it is shown how these basic controls can be combined in order to achieve more complex formations. Index Terms Formation control, imited sensor information I. INTROUCTION For mobile robots two of the basic functionalities are to navigate and to follow. Naturally both of them are well studied for either single agent systems or multi-agent systems [3], [4], [5], [8], [9], [12], [16], [17]. Many methods in the literature, however, is focused on the problem of designing full state feedback controllers for goal reaching, target tracking or maintaining a preassigned formation. The issue of sensor and actuator limitations seems to be more or less overlooked, even though it has been a major issue in research areas such as SAM [14], [15]. In this paper we will focus on mobile multi-agent systems with limited information from range sensors. Our approach differs from the other approaches in the literature such as [4], [9], [16], [17] in that different sensing information and actuator constraints are assumed. Therefore a new control strategy is needed in order to have a robust formation control. We introduce first two basic controls for a single mobile agent tracking another moving object. Then we show how these basic controls can be combined in order to obtain more complex formations. First we need to explain what we mean by limited sensor information. Through out this paper we will assume that all sensors have a limited detection range. This means that every agent in a formation must keep within a certain specified distance of its neighbor/neighbors in order to be able to communicate. When designing a control algorithm for a system of this type, high priority must be given to control actions that aim at keeping distances small. Furthermore, some sensors may be directional and have a This work is in part sponsored by the European IST project RECSYS, the Swedish Research Council and the Centre of Autonomous Systems, KTH. limited angle of view. In those cases the agents must keep not only the distances, but also the relative rotations, more or less fixed. Finally, we will treat the case where data from the positioning sensors are so noisy or unreliable that it is necessary for two or more agents to share information on the surroundings in order to be able to operate. In this case, constraints on the control system are similar to the previous cases, but here the demands on robustness are extremely high since the system relies on the assumption that the agents can communicate with each other at all times. It is clear from this reasoning that limited sensor information leads to additional constraints on the behavior of the system, and that any limitations of data must therefore be taken into account already when designing the basic control algorithms. II. BASIC BEHAVIORS In the following sections we suggest two basic controllers for a mobile robot with sensor limitations following another mobile agent, which we from now on will refer to as the leader. The first controller (Section III) is designed such that the robot follows the leader s trajectory while trying to keep the distance to the leader constant. The second controller (Section IV) is designed to make the robot move side by side with the leader at some specified distance, while keeping the same orientation as the leader. In the first case it is possible to apply linearization, but in the second case, we will need to study a nonlinear control system. Combining the two basic controllers in various ways allows us to define a wide range of more complex controllers for multi-agent systems, as will be shown in Section V. Through out this paper we use the following nonholonomic model for a robot: where, and denote the speed, rotation and angular velocity of the robot with respect to some fixed coordinate system. (1)

2 III. VERTICA TRACKING - PATH FOOWING Our vertical-tracking control is developed from the pathfollowing control presented in [3]. The control in [3] was developed to steer a mobile robot along a pre-defined trajectory. To obtain this, a virtual vehicle approach was used. A reference point was defined at a distance from the center of the robot and the control was designed to steer this point along the given path. The look-ahead distance,, had to be given due to the necessity to avoid singularities. Otherwise, as the distance to target decreases, the angle to target ( in Figure 1), measured from the robot s axis of orientation, will eventually become undefined. With this approach it is easy to see the similarities with tracking. Instead of letting the reference point follow a path, one may easily modify the algorithm so that the point instead follows a moving target. In [3], the direction relative the robot at which the look-ahead distance should be kept was optional. In this application we want the mobile robot at all times to be oriented towards its leader and therefore we shall here choose the reference point to be on the robot s axis of orientation. With this particular choice of reference point we obtain not only a point that can be driven arbitrarily close to the leader without causing the angle to target to be undefined. We also obtain control of the robot s orientation, since controlling towards the center of the target (or in this case the leader),, means simultaneously controlling the angle of orientation,, so that the relative angle to target,, approaches zero. Now let the point be defined as above, i.e., located y Fig. 1. ( p(s), q(s) ) φ ρ (x, y ) Revised Virtual vehicle. on the robot s axis of orientation at a distance from the center of the robot. If the robot is centered in, then (2) erivation of equation (2) in combination with the unicycle model (1) gives the connection between the velocity and angular velocity of the robot, and, and the motion of x the point : If and (3) are chosen as!"#! $ % (4) for some constant, including both a proportional and a derivative part, and are driven towards and. The corresponding controls for the robot, i.e. for and, are obtained by inserting the expressions for " in equation (3). Rewriting the expression yields & ' $ ( *) +-,./1 &) 2 (5) where is the relative angle to the leader measured by the robot, ' is the distance to the leader and and ) are the speed and orientation of the leader. In case the leader is virtual, both the speed at which the leader should move and the speed at which the robot should follow the leader can be designed to gain robustness. One example of this is the path following problem considered in [3]. The path was parameterized and the control algorithm was designed to make the robot track a point *3 14!" :987 : on the curve. The parameter, 7, was then defined in order to adjust the speed of the point so that the robot could keep up with the leader. When following another agent in a formation it may not be possible to implement such a feedback control, for instance because of limitations in bandwidth that put restrictions on how much information that can be transmitted between the agents. Since it is not necessary for stability, and since we want to define a control that is as general as possible, we do not assume here that the leader can adjust its speed to the tracking robot. Then we must, however, impose a restriction on the speed of the leader with respect to the maximum speed of the followers. With a communication system that allows leader and follower to communicate with each other and exchange information about speed, acceleration and orientation, (5) can be implemented directly. As mentioned before, this may not always be the case. A common scenario is that a robot only have access to data that are collected with its own positioning sensors. Based on these data estimations can be made of, for instance, speed and acceleration of the leader. If the target moves smoothly, the problem can be avoided by simply ignoring the target velocity in (5). If ; >?;A@CB one wants to follow the target with a directional angle (Figure 2), then (5) can be rewritten as > E F' E ' E! : > G2 (6)

3 IV. HORIZONTA TRACKING The objective of the horizontal tracking control is to align the robot with another moving object so that they move in parallel with the same orientation (Figure 3). Ideally, the angle from the leader s axis of orientation to the following robot should be B. It is possible to extend this control to multi-agent systems by letting a third robot follow the original follower and so on. Eventually we get a chain of robots, all moving in parallel in the same direction. This type of control can be used in applications where large areas should be explored or covered by a number of mobile units. A more concrete example of this is the searching for land-mines with a team of mobile agents. Since our eventual control objective is to align the middle points of two robots, the linearization technique used in the previous ; >A; section can not be applied here (we would have B ). One might argue that two off-the-axis points with small distance to the axes can be considered instead [4], so feedback linearization can be applied. However, in this case the control action could be very big and ill conditioned. What we propose here is instead a nonlinear control strategy. As we will show, the control is guaranteed to stay well bounded for some very mild assumptions on the motion of the leader. To simplify the equations we shall here assume that we know the speed and angular velocity of the object we want to follow, either from estimation based on data from the positioning sensors on board, or from information transfered to the robot via some communication system. Note that if the robot should always be positioned at a 9-degrees-angle from the leaders axis of orientation then (xt,yt) Fig. 3. Fig. 2. d β Tracking with an angle. beta v Horizontal tracking and formation keeping. v v the speed of the robot will not necessarily be equal to the speed of the leader. Suppose that the measured distance between the robot and the leader is while is the desired distance. Then the control we propose for the robot is ( " (7) where, for some constants $ (8) Here. and are defined as the rotation angle of the leader and the robot and is the measured relative B angle to the leader which should be approximately. Theorem 1: If the control for the follower is given by (7), then the equilibrium B is locally asymptotically stable when is set to a positive constant and " is set to zero. Thus, when varies slowly and " is small, the distance and relative angle errors are bounded. With these conditions fulfilled, it is clear from (8) that the control action will also be bounded. Remark: It is well known that for such nonholonomic systems, stabilization to an equilibrium point by any state feedback control is not possible. Thus when both and are set to zero, it is not possible to stabilize the system. Proof: enote B, and B, where is the relative angle to the following robot measured by the leader, then ( " After plugging in (7) and setting, we have 8 # E : (9) E et, then ( # is a center manifold, where are some positive constants. One can easily show that the flow on the center manifold is asymptotically stable. Then by center manifold theory, (9) is locally asymptotically stable. When is set to zero, may not tend to zero.

4 2 V. COMBINING BASIC FORMATIONS In this section we show that the two control-algorithms for horizontal and vertical tracking described in previous sections can be used as building-blocks when designing control algorithms for more complex multi-agent formations. More specific, we will use (7) and a simplified version of (5) to design a control-algorithm for a triangular formation consisting of two mobile agents following a third agent (see Figure 4). In [11], some mathematical properties of a formation defined by a rigid graph are discussed. In particular we know that it is possible to use algebraic constraints in the form of distance to define a rigid formation. Although in this paper we only consider the case of a triangular formation, control for many other rigid formations can be easily obtained by putting together other constellations of our basic controllers. 1 #. # - # $#%%#&! / # $*+%*,!. / $'(%')! - "! Fig. 4. Two robots tracking a target B. Multi-agent coordination 1 A. Problem statement - limited sensor information The problem we will focus on in this section is to design a control for a system of two robots with limited sensor input/output tracking a leader. We here assume that the two tracking robots can communicate with each other if they are sufficiently close and within the working range of each others communication receivers/transmitters. We also assume that the reliability of their positioning sensors decreases with distance to target. To obtain the sufficient information of the position and speed of the lead-robot, the two tracking robots must share information. In general, sharing information between agents increases the robustness of a formation. In this case, however, sharing distance and angle information between the tracking robots is not only important for robustness but sometimes even necessary. Therefore it is important that the robots always stay close to each other in order to maintain contact. When tracking a target/following a leader with a team of robots, one way to keep the robots sufficiently close to each other is to use an independent lead-follower control of the type described by (6) for each of the tracking robots. This approach is simple and it works fine in many applications where the relative position of the agents is not critical. The drawback of this control is that the relative distances and angles between the agents are maintained only indirectly. When only small deviations from the desired relative positions can be tolerated, a more robust strategy, where the controls of the agents are coupled, must be used. With this consideration in mind, we shall use the following subsection to derive a coupled control for a triangular formation. Remark: For the application of tracking it is interesting to note that only the relative distances and angles between target and tracking robots are of interest and that the control is therefore independent of the coordinate system. In this subsection we shall use the two controls (3) and (7) to derive a coupled control for two robots tracking a leader in a triangular formation. As mentioned before, our method is based on the assumption that the two robots have limited sensor range. For a proximity sensor, the ability to detect distances radially, i.e., distances between the sensor and features in the surrounding environment, rapidly decreases outside the sensors working range. The angular measurements are less dependent on distance and can therefore be used to calculate an estimated position of a distant feature if data from several sensors is available, so called triangulation. The estimation of distance to target can be made as long as some necessary conditions are fulfilled. The target must not be located on the same line as the two tracking robots and the robots must be separated by a distance that is sufficiently large relative the distance to the target. Furthermore, the robots must always be within communication range of each other in order to be able to exchange information. The motion of the two robots must therefore be coordinated in order to keep the distance and relative angle between them, as well as the detection angle to the target, within a given range. Here we shall assume that the communication sensors on the two tracking robots are positioned on the sides, so that the robots must always move parallel to each others with relative angles and (defined in Figure 4 and the table below) sufficiently close to 3)4 and 3)4% respectively. enote the two tracking robots Robot 1 and Robot 2, in accordance with Figure 4, and let us use the following table to define some variables we will use. Note that the angles are defined so that they are positive if measured counter-clockwise. As a consequence, the angle will be negative (approximately B ) since the relative angle to Robot 1 will be measured clockwise by Robot 2.

5 G Position coordinates x-coordinate for Target y-coordinate for Target x-coordinate for Robot no. i, i1,2 y-coordinate for Robot no. i, i1,2 rotation angle of Robot no. i, i1,2 Sensor readings distance to target for Robot no. i, i1,2 angle to target measured by Robot no. i, i1,2 distance to the other robot measured by Robot no. i, i1,2 angle to the other robot measured by Robot no. i, i1,2 Parameters distance robot-reference point desired distance between robots constant We immediately note that we can solve the problem of communication between the two tracking robots by letting one of them, say Robot 1, track the other robot using the control for horizontal tracking presented in the previous section. et and be the speed and velocity of Robot 1, and be the speed and velocity of Robot 2 and let be the desired distance between the tracking robots. Then, in accordance with equations (7) and (8), we choose with for some constants and, (1) $ (11) We could now easily define a tracking angle to the leader of the formation and let the motion of Robot 2 be decided by (6). Instead, based on the consideration in the previous subsection, we will use another approach and define a coupled control for Robot 2 that depend not only on the leader, but also on Robot 1. As in Section III we define a reference point at a distance from each of the two tracking robots, in the direction of their respective orientation. We denote the points and, with the indices referring to the number of the corresponding robots. Our control strategy will now be to steer the point,defined as to the target/leader. (12) The connection between the motion of the point and the steering parameters of Robot 1 and Robot 2 can be calculated using (2): 4 F (13) et the motion of the point simple controller: be decided by a (14) Combining (13) and (14) now gives an equation connecting the control parameters for Robot 1 and Robot 2: 4 4( (15) 4 The above equation coordinates the motion of the two robots in such a way that the point is steered towards the target. By inserting (7) in (15), it is possible to solve for and, but first it is necessary to express all the variables in (15) and (7) in angles and distances that are known, or that can be measured by the sensors of the robots. Using that 3, it is possible to rewrite the right-hand-side of (15). A geometric study of the angles and distances shows that the left-hand-side of (15) is equivalent to 4 (16) with and defined as "! % #! %$& (')$ *+ /, +.-, / (17) -1 and can be expressed in terms of known and measurable parameters as &) 3 *) 3 ) 5 (18) BA C. BA C 6$& ('.$ 87 9;:>?@): BA, C) :>?E: BA, C >?F 9;:HJIK@K:, :HJIK#E:, BA C HJIKBF, M " efine N to be N (19) It is then possible to solve (15) for and as long as PO N is invertible. We note that the matrix is invertible unless RQ 3, i.e., when the two robots face opposite directions. Thus, the control for Robot 2 is * O, N S 4( T N Since, the control for Robot 2 must be bounded as long as is bounded and VU WQ 3. If the initial positions for the robots are chosen not too close to such an extreme

6 and the motion of the target is smooth, it is reasonable to assume that the difference in orientation between the robots will be quite small and considerably less than 3 at all times. From theorem 1 we know that is bounded when varies slowly and is small. VI. SIMUATIONS In this section we show some results from a simulation in Matlab of two robots tracking (following) a target (leader). In Figure 5 the trajectories of the target and the two agents are plotted. In Figure 6 the distance and angle errors are plotted. The initial positions and orientations for the robots were chosen to produce a small initial error in both alignment and orientation. The effect of these errors can be seen in the figures. Initially the errors are big and fluctuating but eventually they decrease and stabilize close to zero. In particular, it is interesting to study the second (d1 d)/d beta1 pi/2 phi1 phi Fig. 5. Two robots tracking a target. istance error between robots, (d1 d)/d vs time Error in detection angle, t (beta1 pi/2) vs time ifference in rotation between t robots, (phi1 phi2) vs time t Fig. 6. istance and relative orientation errors. plot in Figure 6. The maximum and minimum values of the curve give us a hint on how big range the sensors that keep track of the other robots must have to allow the robots to communicate with each other at all times. Here the maximal deviation from the desired angle is.28 radians, or 16 degrees, which means that with this control, the sensors should at least have a detection range of 4 degrees for stable detection. VII. CONCUSIONS Formation control of multi-agent systems is well studied in the literature. In this paper we have focused on formations where sensor information is limited. We have proposed some new control algorithms that are only based on local information and that are easy to implement, and we have shown how these algorithms can be combined to build up control systems for more complex formations. It remains to be investigated how switching between the different algorithms affects the stability of the system. A better understanding of the switching behavior is desirable. For example it could be made use of in the development of an efficient obstacle avoidance algorithm for formations. REFERENCES [1] R.C. Arkin. Behavior-Based Robotics. The MIT Press, Cambridge, Massachusetts, [2] T. Balch and R. Arkin. Behavior-based formation control for multi-robot teams. IEEE Transaction on Robotics and Automation, 14(6): , [3] M. Egerstedt, X. Hu and A. Stotsky, Control of Mobile Platforms Using a Virtual Vehicle Approach, IEEE Trans. Aut. Control vol. 46 no. 11, 21. [4] A. K. as, R. Fierro, V. Kumar, J. P. Ostrowski, J. Spletzer, and C. J. Taylor, A vision based formation control framework, IEEE Transactions on Robotics and Automation, vol. 18, pp , 22. [5] J. awton, R. Beard and B. Young, A ecentralized Approach to Formation Manuervers, IEEE Transactions on Robotics and Automation, vol 19, pp , 23. [6] P. Ögren, M. Egerstedt and X. Hu, A Control yapunov Function Approach to Multi-Agent Coordination, IEEE trans. Robotics and Automation vol. 18, 22. [7] Manuel Mazo Jr., Alberto Speranzon, Karl H. Johansson and Xiaoming Hu, Multi-Robot Tracking of a Moving Object Using irectional Sensors, to appear in Proc. of ICRA 24. [8] M. Egerstedt and X. Hu, Formation constrained multi-agent control, IEEE Transactions on Robotics and Automation, vol. 17, pp , 21. [9] W. Kang, N. Xi, Y. Zhao, J. Tan, and Y. Wang, Formation control of multiple autonomous vehicles: Theory and experimentation, IFAC 15th Triennial World Congress, 22. [1] X. Hu,. Fuentes, and T. Gustavi, Sensor-based navigation coordination for mobile robots, in IEEE Conference on ecision and Control, 23. [11] Resa Olfati-Saber and R. M. Murray, Graph Rigidity and istributed Formation Stabilization of Multi-Vehicle Systems, in IEEE Conference on ecision and Control, 22. [12] A. Jadbabaie, J. in and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control 48, pp , 23. [13] Herbert G. Tanner, George J. Pappas, and Vijay Kumar. eader-toformation stability. IEEE Transactions on Robotics and Automation, 24. [14] M. issanayake, P. Newman, S. Clark, H. urrant-whyte and M. Csorba, A solution to the simultaneous localization and map building (SAM) problem, IEEE Transactions on Robotics and Automation, Vol: 17 (3), Jun 21. [15] Chieh-Chih Wang, Charles Thorpe, and Sebastian Thrun, Online Simultaneous ocalization and Mapping with etection and Tracking of Moving Objects: Theory and Results from a Ground Vehicle in Crowded Urban Areas, Proceedings of ICRA 23, pp , 23. [16] Rene Vidal, Omid Shakernia, and Shankar Sastry, Formation Control of Nonholonomic Mobile Robots with Omnidirectional Visual Servoing and Motion Segmentation, Proceedings of ICRA 23, pp , 23. [17] Estela Bicho and Sergio Monteiro, Formation control for multiple mobile robots: a non-linear attractor dynamics approach, Proceedings of IROS 23, pp , 23.