A Pursuit-Rendezvous Approach for Robotic Tracking 471 4.5 The time-to-go The time-to-go is the time it takes the robot to reach the moving goal. The time-to-go is very important for any comparison between control strategies. The time-to-go can be estimated by the following equation: r r t goto  −= − (30) In general, it is difficult to estimate the time-to-go since it depends on many factors that are time-varying. The most important factors are the velocity ratio, and the target manoeuvers. The time-to-go may be used to determine the appropriate value of b to adjust c(t). The only case where it is possible to find the time-to-go analytically is when the goal moves in a straight line, ( const G = θ ), v R and v G are constant, and the robot is applying a pure rendezvous approach. In this case, the time-to-go is given by () () ληθ cvv r t RGG goto coscos 0 −− −= − (31) It is obvious that the time-to-go is proportional to the initial range. 5. In the presence of obstacles It is clear that the problems of navigation and reaching a moving object in the presence of obstacles are among the most difficult problems in robotic navigation. They combine local path planning for collision avoidance with global path planning for reaching the goal. In our formulation, the robot moves in two modes, the navigation mode and the obstacle avoidance mode. Clearly, the obstacle avoidance mode has the priority. The collision avoidance is accomplished by building a polar histogram of the environment. The polar histogram is based on the angular information obtained from the sensory system. Only obstacles that appear within a given region called the active region are considered. The polar histogram allows determining free directions and directions corresponding to the obstacles. A snapshot of the local environment from a given position of the robot is a characterization of the visible obstacles and the angles they make with the robot. The first stage in the polar histogram is to represent the robot’s surrounding environment using angular information provided by the robot’s onboard sensors. The angles nj i1 and nj i2 are the limit angles characterizing obstacle B i as shown in figure 7. The polar diagram denoted by D is obtained as follows: ¦ = = k i i dD 1 (32) where k denotes the number of obstacles in the active region, and d i is given as follows: d i =1 if [] ηληληθ −−∈− 21 , iiR (33) d i =0 otherwise Note that the polar histogram is constructed based on the angle η θ β −= R , therefore the pure pursuit corresponds to 0= β , and the pure rendezvous corresponds to λ β = . The obstacle avoidance mode is activated when at least one obstacle appears in the active region, and the robot navigates by using the polar histogram. It is also easy to represent the goal’s 472 Mobile Robots, Perception & Navigation orientation angle in the polar histogram. The robot deviates from its nominal path only if an obstacle appears in its path. The algorithm for collision avoidance is the following: Procedure Deviation 1. Choose an intermediary point M such that ηη − M has the same sign as η θ − G . M η is the visibility angle between the robot and point M. 2. Navigate towards this point using the pure pursuit. A heading regulation procedure is used to keep the smoothness of the path. The equation for the heading regulation is similar to (21). B 1 B Ȝ i2 Ȝ i1 1 B i B 2 Ȝ i2 M Ș M G B 1 B Ȝ i2 Ȝ i1 1 B i B 2 Ȝ i2 M Ș M G Fig. 6. Collision avoidance. Collision avoidance algorithm: 1. If obstacle detected within the active region, then the collision avoidance mode is activated. 2. If the robot is in a collision course with obstacle B i , then call procedure deviation 3. After obstacle passed go back to the pursuit-rendezvous mode. Since ηη − M and η θ − G have the same sign, the robot orientation angle and the goal orientation angle are on the same side of the visibility line. d 2 d i d 1 d 2 d i d 1 Fig. 7. Polar histogram for the environment in figure 6. 6. Pursuit-rendezvous for target dynamic coverage Dynamic target coverage by a wheeled mobile robot or a group of mobile robots has been considered in the literature recently. This problem is important in various applications, such A Pursuit-Rendezvous Approach for Robotic Tracking 473 as cleaning, security and patrolling, and sensor network deployment. Dynamic target coverage aims to generate a trajectory and the corresponding linear velocities. In the previous section, we designed a control law that allows the robot to reach the moving goal from an arbitrary initial state. In this section our goal is to design a second control law to keep the moving object within a given distance from the robot so that the goal stays in the robot’s field of view. That is, () desdes rtrr 21 ≤≤ (34) with desdesdes rrr 21 ≤≤ . r des is the desired value of the coverage range, des r 1 and des r 2 are the range limits for r des . The coverage range is represented by a circle as shown in figure 8. Dynamic coverage is necessary in various surveillance and tracking applications. For example, in many situations it is important to keep the goal in the field of view of the robot’s sensory system. To accomplish this task, it is necessary to design a control law for the robot’s linear velocity. Note that a constant range between the robot and the moving object corresponds to 0=r  ; that is, () () ληθ cvv RGG coscos =− (35) In order to combine the navigation mode with the tracking at a constant distance mode, we use the method which is known as feedback linearization (Drakunov et. 1991) in combination with backstepping or block control (Drakunov et. 1991) which gives ( ) des r rrKr −−=  (36) where K r is a real positive number. Equation (36) allows to drive the relative range smoothly to its desired coverage range. By replacing r  by its value, we obtain () () ( ) des rRGG rrKcvv −−=−− ληθ coscos (37) From which the relative velocity of the robot can be obtained as follows: ( ) () () λ ηθ c vrrK v GG des r R cos cos −+− = (38) The system converges to a steady state that satisfies equation (35). We have the following remarks concerning equation (38): 1. The term K r ( r - r des ) goes to zero with time. 2. If the goal applies a pure escape strategy, then lj R = Lj and v R = v G . This is true for both the pure pursuit and the pure rendezvous. 3. In general, the required value of v R is smaller in the case of the pure pursuit. In the case of the pure pursuit, the dynamic coverage of a target is characterized by an important property, which can be states as follows: Proposition Under the pure pursuit, the dynamic coverage is characterized by η θ → R and GR vv → . This means that the robot’s orientation angle will track the target’s orientation angle, and the robot’s linear velocity will track the target’s linear velocity. 474 Mobile Robots, Perception & Navigation Proof The kinematics model under the pure pursuit is written as () ()() 1 sin cos − −= −−= rv vvr GG RGG ηθη η θ   (39) The equilibrium position for the second equation is given by T θη = * . By using the classical linearization, it turns out that this equilibrium position is asymptotically stable. Therefore, G θ η → , since R θ η = under the pure pursuit, which gives GR θ θ → . From equation (38) under the pure pursuit (c = 0), we have GR vv → as GR θθ → . r 2 des r des r 1 des G r 2 des r des r 1 des G Fig. 8. An illustration of the dynamic coverage ranges. 7. Simulation Here we consider several simulation examples to illustrate the suggested approach. Example 1: A comparison between the pure pursuit (PP) and the pure rendezvous (PR) Three scnearios are shown here. The first scneario shown in figure 9 corrresponds to a goal moving in a straight line. The second scenario shown in figure 12 corresponds to a goal moving in a circle. The third scenario is shown in figure 13, the goal moves in a sinusoidal motion, which is among the most difficult paths to reach. Note that the path of the goal is not a-priori known to the robot. For the scenario of figure 9, the visibility angle is shown in figure 10, and the robot orientation angle in figure 11. From figure 10, the visibility angle is constant under the PR. From figure 11, it is clear that more corrections and manoeuvers are required under the PP. Figure 14 shows the robot path for different values of c. In all cases the robot reaches the goal successfully. A Pursuit-Rendezvous Approach for Robotic Tracking 475 0 20406080 0 20 40 60 80 x y PP PR Goal R 0 0 20406080 0 20 40 60 80 x y PP PR Goal R 0 Fig. 9. Reaching a goal moving in a line. 0 20406080 -1.5 -1.0 -0.5 0.0 PR PP Ș time 0 20406080 -1.5 -1.0 -0.5 0.0 PR PP Ș time Fig. 10. Evolution of the visibility angle for the scenario of figure 9. 0 20406080 -1.5 -1.0 -0.5 0.0 PP time PR ș R 0 20406080 -1.5 -1.0 -0.5 0.0 PP time PR ș R Fig. 11. Evolution of the robot’s orientation angle for the scenario of figure 9. 476 Mobile Robots, Perception & Navigation -50 -25 0 0 25 50 75 x y Goal PP PR -50 -25 0 0 25 50 75 x y Goal PP PR Fig. 12. Reaching a goal moving in circle. 0 20406080 0 30 60 90 120 PP PR Robot Goal G 0 R 0 x y 0 20406080 0 30 60 90 120 PP PR 0 20406080 0 30 60 90 120 PP PR Robot Goal G 0 R 0 x y Fig. 13. Reaching a goal moving in a sinusoidal motion. 0 30 60 90 0 5 10 15 20 25 c=0.8 c=0.5 PR PP G 0 R 0 0 30 60 90 0 5 10 15 20 25 c=0.8 c=0.5 PR PP G 0 R 0 Fig. 14. Robot’ path for different Values of c. Example 2: in the presence of obstacles: Two scenarios are shown in figures 14 and 15 to illustrate the navigation towards a moving goal in the presence of obstacles. The paths of the robot under the PP and the PR are different as shown in the figures. The robot accomplish the navigation and obstacle avoidance tasks successfully. A Pursuit-Rendezvous Approach for Robotic Tracking 477 0 1530456075 0 15 30 45 60 3 Pure rendezvous Pure pursuit 0 1530456075 0 15 30 45 60 3 Pure rendezvous Pure pursuit 020406080 0 20 40 60 PR PP Target 020406080 0 20 40 60 PR PP Target Fig. 15. Tracking and navigation in the presence of obstcales, goal moving in a line presence of obstacles. 8. Conclusion We presented a method for robotic navigation and tracking of an unpredictably moving object. Our method is kinematics-based, and combines the pursuit law with the rendezvous law. First a kinematics model is derived. This kinematics model gives the motion of the goal with respect to the robot. The first equation gives the range rate between the robot and its goal. The second equation gives the turning rate of the goal with respect to the robot. The control law is then derived based on this kinematics model. This law is controlled by a real variable, which may be constant or time-varying. The most important properties of the control law are discussed. The dynamic coverage of the target is also discussed, where a second law for the robot’s linear velocities is derived. 9. References Adams, M.D. (1999) High speed target pursuit and asymptotic stability in mobile robotics. IEEE Transactions on Robotics and Automation, Vol. 15, No. 2, pp. 230-236. Chaumette, F.; Rives P. and Espiau, B. (1991) Positioning of a robot with respect to an object, tracking it and estimating its velocity by visual servoing, Proceedings of the IEEE International Conference on Robotics and Automation, pp. 2248-2253 California. Davis, L; Philomin, V. and Duraiswami, R. (2000). Tracking humans from a moving platform, Proceedings of IEEE International Conference on Pattern recognition, pp. 171- 178. Drakunov, S.; Izosimov, D.; Lukjanov, A.; Utkin, V.I. and Utkin, V. Block control principle,. Automation and Remote Control, Vol. 51, No. 6, pp. 737.746, 1991. Drakunov, S.; Izosimov, D.; Lukjanov, A.; Utkin, V.I. and Utkin, V. Block control principle,. Automation and Remote Control, Vol. 51, No. 5, pp. 601.608, 1991. Feyrer, S. and Zell, A. (1999). Detection, tracking, and pursuit of humans with an autonomous mobile robot. Proceedings of the IEEE International Conference on Intelligent Robots and Systems, pp. 864-869. Feyrer, S. and Zell, (1999). Tracking and pursuing persons with a mobile robot Proceedings of the International Workshop on Recognition, Analysis and Tracking Faces and Gestures in Real time Systems, pp. 83-88, California. 478 Mobile Robots, Perception & Navigation Kim, B.H.; Roh, D.K; Lee, J.M; Lee, M.H; Son, K; Lee, M.C; Choi, J.W; and Han, S.H. (2001) Localization of a mobile robot using images of a moving target, Proceedings of the IEEE International Conference on Robotics and Automation, pp. 253-258 Seoul. Lee, S.O.; Cho, Y.J.; Hwang-Bo, M.; You, B.J. and Oh, S.R.(2000) A stable target- trackingcontrol for unicycle mobile robots, Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1822-1827. Murrieta, R.; Sarmiento, A. and Hutchinson, S. (2003) On the existence of a strategy to maintain a moving target within the sensing range of an observer reacting with delay, Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 1184-1191. Oh, P.Y. and Allen, P.K. (2001). Visual servoing by partitioning degrees of freedom. IEEE Transactions on Robotics and Automation, Vol. 17; No. 1, 1-17. Thuilot, B; Martinet, P; Cordesses, L. and Gallice, J. (2002) Position based visual servoing: Keeping the object in the field of vision, Proceedings of the IEEE International Conference on Robotics and Automation, pp 1624-1629 Washington DC. Tsai, M ; Chen, K; Cheng, M; and Lin, K. (2003). Implementation of a real-time moving object tracking system using visual servoing. Robotica, Vol. 21, No. 6, 615-625. Parker, L.; Birch, B.; Reardon, C. (2003) Indoor target intercept using an acoustic sensor network and dual wavefront path planning, Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, 278-283. Spletzer, J.and Taylor,C. (2003) Dynamic Sensor Planning and Control for Optimally Tracking Targets, The International Journal of Robotics Research, Vol. 22, No. 1, 7-20. Yamaguchi, H. (1999) A cooperative hunting behavior by mobile-robot troops, The International Journal of Robotics Research, Vol. 18, No. 9, 931-940. Belkhouche F. and Belkhouche B. (2005) A method for robot navigation toward a moving goal with unknown maneuvers, Robotica, Vol. 23, No 6, 709-720. 22 Sensor-based Global Planning for Mobile Manipulators Navigation using Voronoi Diagram and Fast Marching S. Garrido, D. Blanco, M.L. Munoz *, L. Moreno and M. Abderrahim University Carlos III of Madrid, Spain * Polytechnic University of Madrid, Boadilla del Monte, Spain. 1. Introduction Mobile Manipulators are special cases of mobile robots that need a careful consideration when planning their motion. Due to their higher and changing centre of gravity caused by the position of the manipulator and its configuration, abrupt motion or change of direction can have dangerous consequences and the toppling of the whole mobile manipulator. Therefore, the motion planning process has to produce smooth and safe trajectories. Since the 1980’s mobile robot motion planning problems have been an important research topic that has attracted the attention of many researchers who worked extensively to obtain efficient methods to solve these problems. Such problems have been approached in two general ways: one approach has concentrated on solving motion planning problems by considering a previously known global environment or obstacles information and robot characteristics; the second approach has focused on planning motions by using local sensor information and robot characteristics. The first approach has been used extensively at a global planning level in robotics to plan trajectories from one environment location to another. This plan functions under the assumption of a perfectly controlled and modelled environment that in most situations does not exist. Unexpected obstacles, persons or moving elements make this approach difficult to utilize except in robustly controlled environments, such as in industrial manipulation environments (manufacturing). The second approach became evident in mobile robotics from the very beginning. A mobile robot’s environment presents unexpected objects which makes it impossible to use the first approach exclusively. The second approach produces local plans by using local sensor information to avoid obstacles. Obviously a local plan is a solution for a local scale problem and needs to be integrated with a global planner or with global information to guarantee the existence of a solution to the global problem. This method received different names such as collision avoidance methods , local planners, and navigation methods. To navigate in complex environments, an autonomous mobile robot needs to reach a compromise between the need for reacting to unexpected events and the need for efficient and optimized trajectories. Usually, path planning methods work in two steps: in the first 480 Mobile Robots, Perception & Navigation step, a global path over an existing map is calculated, and in the second step, the robot reads the sensor data and modifies the local trajectory in a reactive way. This provides the path adaptation mechanism to cope with unexpected obstacles or dynamic environments. These methods calculate the global trajectory off-line in a priori known map, and while the robot is moving local modifications are made continuously based on the sensor data. The reason for using a two level planning approach is due to high computational cost that is required in most motion planning techniques to achieve an updated environment model (and to plan a smooth and dynamically adapted trajectory). The use of a two level planner strategy decreases computational cost by activating the global planner occasionally (it can be done off line) while the local planner which is much faster runs on line. This two level approach affects the control architecture of the mobile robot. The first mobile robots were based on a sense-model-plan scheme referred to in the literature as planned architectures (Meystel, 1986). These architectures present some difficulties providing fast responses to environmental changes. Posterior reactive architectures (Brooks, 1986) have the advantage of using fast response methods based on a sensor-decision-action scheme to react to environmental changes, but also show the difficulty of extending the reactivity to upper levels. Finally, hybrid deliberative/reactive architectures (Arkin, 1990), (Arkin, 1998), (Alami et al., 1998), (Chatila, 1995), (Bonasso et al., 1996) and (Low et al., 2002) have emerged as the result of recognizing the advantages provided with planning at high control levels and reactive architectures at lower control levels. The Voronoi Fast Marching method is based on a sensor-based global path planning paradigm. This is a planning approach based on a fast sense-model-plan scheme able to integrate sensor information in a simple grid based environment model and to calculate a globally consistent, smooth and safe trajectory, fast enough to be used as a reactive navigation method. This approach presents some advantages. One is the ability of global planning methodologies to guarantee a path between a given point and the goal point, if one exists. And the other is the smoothness and safety of the solution obtained. This solution eliminates the local minima trap problem and the oscillations in narrow places present in other methods, and also indirectly eliminates the use of a supervision system able to detect local minima failures (obstructed paths), in order to initiate the search for a new and feasible global path from the current position to the goal point. To calculate the trajectory, the proposed method combines the Voronoi Distance transform and the Fast Marching Method. The Voronoi approach to path planning has the advantage of providing the safest trajectories in terms of distance to obstacles, but because its nature is purely geometric it does not achieve enough smoothness. On the other hand, the Fast Marching Method has been applied to path planning (Sethian, 1996a), and their trajectories are of minimal distance, but they are not very safe because the path is too close to obstacles, and more importantly the path is not smooth enough. In order to improve the safety of the trajectories calculated by the Fast Marching Method, two solutions are possible. The first possibility, in order to avoid unrealistic trajectories, produced when the areas are narrower than the robot, the segments with distances to the obstacles and walls less than half the size of the robot need to be removed from the Voronoi Diagram before the Distance Transform. The second possibility, used in this paper, is to dilate the objects and the walls in a safely distance to ensure that the robot does not collide nor accepts passages narrower than the robot size. The last step is to calculate the trajectory in the image generated by the Voronoi Diagram using the Fast Marching Method, the path obtained verifies the smoothness and [...]... number of test navigation point) to 1 And then calculate and update the state of this new registered Cd_NP in next sampling interval 506 Mobile Robots, Perception & Navigation C D When the current test navigation point (T_NPn) is similar to the extra past confirmed navigation point (E_Cd_NP), then reset the current test navigation point and set n to 1 After that, register a new test navigation point... decreases 494 Mobile Robots, Perception & Navigation 8.3 Architectural effect Due to the fact that the VFM method integrates global motion planning and obstacle avoidance capabilities in an algorithm, which lets us simplify the mobile robot control architecture This technique has been originally designed for mobile manipulators, which requires a smoother and safer motion plan compared to mobile robots From... the collision avoidance problem is the potential field approach developed by Khatib (Khatib, 1986 and Khatib & Chatila, 1995) This approach is based on the creation of an artificial potential field in which the target is an attractive pole and the obstacles are repulsive surfaces The robot follows the gradient of 482 Mobile Robots, Perception & Navigation this potential toward its minimum The derived... 363–369, ISSN: 01628828 Low, K H ; K Leow, W & Ang, M H (2002) A Hybrid Mobile Robot Architecture with Integrated Planning and Control, Proceedings of the 1st International Joint Conference on: Autonomous Agents and Multiagent Systems, Nº 2, pp 219-226, Bologna-Italy, July 2002, ACM press NY 496 Mobile Robots, Perception & Navigation Maurer, C R., Qi, R & Raghavan, V (2003) A Linear Time Algorithm... interval If the current test navigation point is not similar to the current potential navigation point, then search the nearest confirmed navigation point (Cd_NP) in the list relative to the current potential navigation point If this resulting point is similar to the current potential point and their distance is smaller than “MinNavTravelDis”, then reset the current test navigation point and set n to... smooth changes in the trajectory (figures 8 and 9) Sensor-based Global Planning for Mobile Manipulators Navigation using Voronoi Diagram and Fast Marching 491 Fig 8 Laser scan data Fig 9 Extended Voronoi transform of the scanned data and trajectory obtained with the Fast Marching Method 492 Mobile Robots, Perception & Navigation In the second test, in order to show global plan capabilities, the method... neuralnetworks [9, 33-35, 38, 51] or combination of them [27] are designed by this reactive navigation 498 Mobile Robots, Perception & Navigation approach Since a totally reactive behavior uses only locally available environment information, without any memory of the previously encountered situations, the autonomous robots are found to suffer from local minima situations For that reason, a reactive approach... (1986) Planning in a Hierarchical Nested Autonomous Control System, Mobile Robots, SPIE Proc., Vol 727, pp 42–76 Minguez, J & Montano, L (2001) Global Nearness Diagram navigation, Proc IEEE Int Conf on Robotics and Automation, Vol 1, pp.33–39, ISSN: 1050-4729, Seoul- Korea, May 2001, IEEE NJ Minguez, J & Montano, L (2004) Nearness Diagram navigation: collision avoidance in troublesome scenarios, IEEE Trans... in the environment map avoiding complex modelling or information fusion 2 In the second step the objects are enlarged in the radius of the mobile robot, which will eliminate unfeasible paths, and avoid additional path verifications 484 Mobile Robots, Perception & Navigation 3 The updated environment model is then converted to an environment safety image by using the Euclidean distance transform in the... should be used, such as L_S’ and R_S’ sensor reading scaling factor σ S , F_S’ 502 Mobile Robots, Perception & Navigation sensor reading scaling factor σ F , change of side sensor reading (ΔL_S and ΔR_S) scaling factor σ ΔS and L_M and R_M velocity scaling factor σ V Δ Δ Σ Σ Σ Σ Σ Σ Σ Δ Δ Δ Δ Δ Δ Fig 2 The HFFS reactive navigation structure The singleton fuzzy set is used for the all output variables . 83-88, California. 478 Mobile Robots, Perception & Navigation Kim, B.H.; Roh, D.K; Lee, J.M; Lee, M.H; Son, K; Lee, M.C; Choi, J.W; and Han, S.H. (2001) Localization of a mobile robot using. obstacles are repulsive surfaces. The robot follows the gradient of 482 Mobile Robots, Perception & Navigation this potential toward its minimum. The derived force induces a collinear and. enlarged in the radius of the mobile robot, which will eliminate unfeasible paths, and avoid additional path verifications. 484 Mobile Robots, Perception & Navigation 3. The updated environment