Abstract— This article presents a decoupled approach for conflict free coordinated manipulation of multiple industrial robot agents operating in a typical industrial environment. Decoupled method proposed in this work plans the path for the robots in two steps. In the first step, for each participating robot agent a path collision free with respect to stationary obstacle is obtained. The coordination among agents is achieved in the second step of the approach by use of off-line strategy for conflict resolution. The key objective of the conflict resolution is dynamic assignment of priorities for path modification of the robot agents. Thus, the agents involved in a local conflict situation coordinate their movement by adopting the lowest movement cost motion paths. Results obtained from realistic simulation of multi-robot agent environment with three industrial robots demonstrate that the proposed approach assures the rapid, concurrent and conflict free coordinated motion of the robot agents.

I. INTRODUCTION

N this paper a decoupled approach for collision free and conflict free motion of multiple six degree of freedom

industrial robots operating in a typical industrial environment is introduced. The robot agent system considered for study is a decoupled multi-agent robot system, where each agent executes its task (i.e. pick and place, spray painting etc) independently and without support of any centralized control.

Fig. 1. Robot agents operating in typical industrial environment

Manuscript received February 15, 2009. Shital S. Chiddarwar is doing Ph.D. in Department of Mechanical

engineering , Indian Institute of Technology Madras, Chennai 600036 India. She. (email: s.chiddarwar@ gmail.com).

N. Ramesh Babu is a distinguished professor from Department of Mechanical Engineeering, Indian Institute of Technology Madras, Chennai 600036 India. (Corresponding author) (Phone: 0091-44-2257 4675 (office) 0091-9444048379 (Mobile)Fax:0091-44-2257 5705 / 2257 4652) (e-mail: nrbabu@iitm.ac.in).

Fig. 1 demonstrates a typical multi-agent scenario where each agent Ri is assigned a task, for example to move object from its starting position Si to a corresponding goal position Gi. To achieve this task, the agent is anticipated to find a collision free path using suitable technique and avoid stationary and dynamic obstacles by using the information shared by other agents. In this case, the mapping information of stationary obstacles is already known whereas the information about moving agents i.e. dynamic obstacles can be obtained through inter-agent communication.

There are several contributions pertaining to the collision and the conflict free motion planning of multiple industrial and mobile robot agent agents. The traditional approaches for multi-agent control are classified into two approaches – centralised and decentralised, or centralised and decoupled approaches [1], [2].

In the centralised approach, multiple agents operating in workspace are treated as a single multi-bodied agent operating in a composite and multiple degree of freedom configuration space [1]. Centralised approach acts as a planner and optimally plans the motion for multiple agents. On the contrary, the decentralised approach treats each agent as a single independent system [3]. Practical multi-agent path planning problems are addressed by a two phase decoupled approach. In first phase, a collision free path for each agent is computed by considering the stationary obstacles and ignoring the presence of other agents in the environment, whereas, second phase determines the coordination among the agents. These three approaches form a traditional foundation for motion planning of multi-agent system.

Motion planning of multiple agents is still a challenging field of research because of the technical difficulties in resolving conflicts between the agents. Centralised approach single handedly tries to plan the motion for all the agents in an interactive fashion [4]. Moreover, an essential assumption, the information from all agents can be transmitted to a superior system is also seems to be impracticable. Due to these aspects, centralised approaches face the problems like curse of dimensionality, computational complexity and NP hard problems [5].

The decentralised approach has its own inherent problem like mutual interference, local minima, parallel run phenomenon, and failure of negotiation. Moreover, this approach gives only sub-optimal solution because of the limitation of computational capacity of the architecture [6], [7]. Above all, the conflict among agents cannot be settled by mere arbitration.

The decoupled path planning is inherently incomplete and is not guaranteed to give a solution always but the loss of

Dynamic Priority Allocation for Conflict Free Coordinated Manipulation of Multiple Agents

Shital S. Chiddarwar and N. Ramesh Babu

I

5th Annual IEEE Conference on Automation Science and EngineeringBangalore, India, August 22-25, 2009

978-1-4244-4579-0/09/$25.00 ©2009 IEEE 549

completeness is relatively small and worth computational gain. This approach can be used for applications where interactions among the agents are moderately constrained [8].

Decoupled methods present a coordination phase separated from the path planning phase. Coordination among the robot agents can be achieved by adjusting the geometric paths [9], introducing time delay [10] or modifying the velocity profiles [11]. The adjustment in the geometric path was done by identifying the regions of the space swept by the robotss and then modifying the path planned a priori so that the robots do not occupy these regions simultaneously [9]. The problems existing with this method are overcome by delaying the motion of robot by a pre-computed value at the beginning of the execution of the movements of one robot so that it does not collide with other robot agents [10]. Another way of coordinating the motion of robot was achieved by modifying or tuning the velocity profile without changing the geometric path. By velocity tuning, i.e. by assigning different velocities to various links of robots, the collision among the robots, along the paths computed in the path planning phase, was resolved [11]. A variant of the basic decoupled approach is prioritised path planning in which the motion of each robot is determined consecutively in the order given by prioritisation scheme. These schemes made use of a variety of techniques for priority determination and assignment. The most straightforward way of priority assignment was to fix the priority of robot movement based on the size of robot. Highest priority was given to a large robot while planning its collision free path with the stationary obstacles in the workspace. Then, the path for the next lower priority robot was planned by velocity tuning to avoid both the stationary obstacles and the higher priority robot, treated as a moving obstacle [12]. This approach is mainly applicable for heterogeneous robot agents and cannot be used for applications involving the same type of robots. To avoid this problem, a simple heuristic was proposed for priority assignment [13]. In order to compute the priority of robot agent, the shortest distance between the start and goal configurations of each robot was used as a heuristic. Predefined prioratisation scheme [14] was developed for different applications. In this case, the operation sequence was utilized to define the priority of the robot. All these approaches are found to be suitable for certain specific situations. Moreover, the success of these methods mainly depends upon the assignment of priority to the robot.

The coordination of robots can be carried out before the task execution (offline) or during the movement of robot agents (online) [15]. Online coordinated task planning of multiple robots fundamentally demands for stopping of robot agents during the path planning and the computational requirements are quite extensive. Hence, this approach is mostly suited for situations where the tasks are simple and can be taught to robot agent by trial and error. In typical manufacturing processes like spray painting, spot welding or precision assembly, it is essential to generate the path in advance by offline methods so as to avoid the interruption of robot agents performing the tasks continuously. In this

context, decoupled path planning approaches are the most appropriate ones for multi-agent motion planning. The popularly used prioritised decoupled path planning has an inherent drawback of deadlock situation [1], which arises when higher priority robot blocks the path of lower priority robot. Under such circumstances, the planner fails to give the feasible path and do not offer any scope for backtracking. Ultimately, prioritised path planner has to quit and alternate planner needs to be invoked for further path planning.

From the above discussion, it is clear that there is a need to develop an effective approach that encompasses the propensity to attain the objective of concurrent movement of all the participating robots with optimum speed and minimum travelling time in a competent manner. Moreover, for efficient resolution of the conflicts arising during task execution by multiple robots, the devised methodology should provide a strategy to dynamically determine the sequence in which path of the robot is to be modified based on their current state in the workspace. Hence, in this paper a two-phase decoupled approach for collision free and conflict free motion of multiple six degree of freedom industrial robots is proposed. In the first phase of approach, a path for each robot that is collision free path with respect to stationary obstacles is determined. The coordination among robots is achieved in second phase by modifying geometric paths of the conflicting robots. A heuristic method based on length of path to be travelled by robot after path modification is developed to dynamically determine a path modification sequence (PMS). The decision regarding path modification of a particular robot is taken based on its position in the optimal PMS.

The proposed approach and its various aspects are explained in the next section. The results obtained by implementing proposed technique are discussed in subsequent section. Finally, conclusions made from the study and scope for future work is presented.

II. DECOUPLED AGENT BASED CONTROL STRATEGY

The architecture of proposed decoupled approach comprises of two major distinct phases, independent path planning and coordinated path planning. In order to attain robot’s goal directed motion planning through an integrated and rapid cycle of planning and motion execution, both phases needs to be synchronized in the suitable demeanor. The task of finding collision free path with respect to stationary obstacles for each robot is performed in independent path planning phase whereas, coordination among multiple robots is achieved by means of coordinated path planning phase of the proposed approach which adopts negotiation strategy to resolve the conflicts. Being in charge for determining coordinated path for multiple robots, the coordinated path planning phase is outfitted with inter-intelligent space, strategy for conflict determination and stratagem for dynamically deciding the path modification sequence (PMS).

For execution of first phase of decoupled approach, configuration space (Cspace) for each robot agent was

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determined by adopting swept sphere volume (SSV) technique [16] for modelling of robot and its environment. The algorithms used to determine the minimum safe distance between link of the robot agent and obstacle were developed based on SSV [17] and same are used in this work. All the robot agent joint angles were varied by one degree at a time and collision check was done with respect to the stationary obstacles for every such configuration. The collision free configurations were also subjected to singularity check in order to ensure it to be singularity free. The Cspace thus obtained consist of configurations which are collision free as well as singularity free and then transformed into Configuration space-time (CST) by adding one more dimension i.e. time to the robot Cspace. CST is the proper manifold for representing a dynamic environment and is thus suitable for multi-agent motion planning purposes. The time required for end effector of the robot to travel from current configuration to the requisite configuration was computed using linear velocity of the end-effector. If end-effector moves from pose1 (X1, Y1, Z1, Roll1, Pitch1,Yaw1) to pose2 (X2, Y2, Z2, Roll2, Pitch2,Yaw2) with the linear velocity of end-effector (Vx, Vy,Vz, x, y, z) then the time required for moving from pose1 to pose 2 is maximum of ( 2 1 2 1 2 1 2 1 2 1 2 1, , , , ,

x y z x y z

X X Y Y Z Z Roll Roll Pitch Pitch Yaw YawV V V

).

The CST thus obtained was searched using a grid based A* search algorithm for finding the optimal path for each robot agent, which is collision free with respect to stationary obstacles.

In the coordinated path planning phase, the path determined in the independent path planning phase is considered as the requisite path for all the robot agents. Since, robot agents are sharing common workspace, there is a chance of occupancy of certain configurations by two or more agents at the same time which will lead to a conflict condition. In order to identify such collision or conflict conditions, the concept of intelligent CST is introduced in this work. An intelligent CST is build from the selective information gathered from various participating robot agents. Hence, it consists of the information regarding the intended path of each robot agent i.e. path determined in independent path planning phase and position of each robot in the time domain. In order to determine the possibility of the conflict conditions arising due to the movement of the agents along the intended path in the common workspace, conflict determination strategy that makes use of information stored in the intelligent CST is formulated.

Each robot agent gets the intended path planned in first phase to move an object from initial position to goal position. During the movement along the given path, robot agent is capable of reaching to the environmental unforeseen movement of the other agents. The strategy adopted to determine the possible conflict condition is presented in Fig 2. The demonstration is given in the form of a network that consists of activities and events. The position of a robot agent is considered as an event and the action taken to make that event happen is considered as an activity. The event is represented by an octagonal geometry whereas the activity is represented by a straight line.

Fig. 2. Strategy for conflict detection Where, a = robot agent R1 moving autonomously from position Pi to

Pg through position P2b = another robot agent R2obstructs the path c = information sent to robot agent R1d = robot agent R1moves from its current position to Pw and waits e = robot agent R1negotiates with conflicting robot agent R2f = negotiation information sent to waiting robot agent R1g = robot agent R1moves from waiting position to previous position to resume the motion h = robot agent R1 moves along coordinated path i = conflict free coordinated movement of robot agent R1 towards goal

As shown in Fig. 2, the robot agent ‘R1’ has to move fromits initial ‘Pi’ to goal position ‘Pg’ via intermediate position ‘P2’. The robot agent ‘R1’ moves autonomously from position ‘Pi’ towards the next position ‘P2’ through an activity ‘a’. When the robot agent is moving from ‘P2’ to ‘Pg’, at certain location ‘Pc’ other robot agent ‘R2’ obstructs the immediate path of the robot agent ‘R1’ . The information regarding conflict is given to robot agent ‘R1’ placed at ‘P2’by activity ‘c’. Once moving robot agents get the information regarding conflict condition, robot agent ‘R1’moves from its current position ‘P2’ to dummy position ‘Pw’ and waits there. The dummy position ‘Pw’ and dummy activities ‘d’ and ‘g’ are represented by a dotted line. As this approach works in off-line mode, this dummy position is considered same as of the position ‘P2’. Conflict detection is performed in a “hand-shake” fashion between the potentially conflicting robot agents. This aspect of conflict resolution assures the win-win condition for all the participating robot agents. The negotiation between the robot agent waiting at position ‘Pw’ and the conflicting robot agent at position ‘Pc’ is carried out at the position ‘Pn’. The information regarding decision taken at negotiation is sent to the robot agent waiting at ‘Pw’ by the activity ‘f’. The robot agent waiting at position ‘Pw’ moves to its original position ‘P2 through activity ’g’ and moves along modified path through position Pe and by following the activity ‘h’. The final movement of the robot agent towards its goal position Pg is achieved through activity ‘i'. During the individual motion, the robot agent may also encounter a group of already conflicting

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robot agents. At this point, the robot agent is expected to stop and wait until conflict has been resolved (Pw). By use of this waiting state, an expansion of conflict situation can be avoided.

Using above methodology, the conflicting configuration was identified and removed from the CST of every participating agent with the assumption that other agents will not modify their path to resolve the conflict. The CST of all the participating agents thus modified was explored using incremental A* algorithm [19] in orderto determine optimal modified path. The advantage of this algorithm lies in the stratagem adopted for carrying out the search. It searches the discritised CST in incremental fashion rather than searching it from scratch when CST is modified. The first search is based on the A* algorithm and search information thus gathered is saved by the planner and used for further searches.

The modified paths thus obtained were used for determining the PMS and the cost associated with it based on the length of the modified path. The sets of PMS in the case of two or three robot agents can be obtained as shown in Fig. 4. Each PMS is given by a series of robot agent’s ids, e.g. R2R1R3 with R2 has the highest and R3 has the lowest priority in the path modification. As stated previously, path for the robot agent can be modified depending on each PMS, e.g. for R2R1R3 the following set {R2, R2R1, R2R1R3}represents the PMS for R2, R1, R3 respectively. For each PMS, cost value was defined which evaluates sequence based on length of the modified path to the goal. To evaluate this cost, the additional cost of the sequence without any priorities was calculated. Hence, the cost of PMS = cost of PMS with priority – cost of PMS without priority. For this computation, the cost of respective paths were taken into account as for example (R2R1R3) = [ (R2R1) + (R2R1R3)] - [ (R1) + (R3)]. The key task was to determine the minimum cost PMS, which automatically yields the optimal set of path modification sequence to resolve a conflict situation. For this purpose, overall sets of PMS based on all possible combinations of conflicting robot agents were collectively generated and the optimal set for the subsequent task of path modification was used. Based on the selected sequence, the decision regarding path modification was done. For example, if the sequence R2R1R3 has least cost then the agents R2 and R1 are to be moved along modified path whereas the agent R3 is to be moved along its intended path. The course of action taken to execute collision free and conflict free motion of the various agents is explained in the next section.

III. MULTI-AGENT MOTION PLANNING

Decoupled conflict resolution by modifying the geometric path of the robot agents during motion execution is based on the location of the robot agent in the PMS. Thus, the path modification is done dynamically based on the current situation of the robot agent and not in the fixed order. Proposed multi-agent motion planner can ask robot agent to either move along the Cartesian path planned in the first phase of the approach or ask it to wait until conflict is settled down. Depending upon position of agent in PMS,

two major cases are distinguished. First, path modification is not required for robot agent that is placed at the last position in the PMS. In fact, it can continue its motion only when obstruction formed due to conflict is resolved, e.g. robots change their path. Secondly, for robot agents whose path is to be modified, conflict resolution and path replanning can be generated only after repeatedly consulting their respective CST.

Fig. 3. Sets of path modification sequence for a) two robots b) three robots

Based on the proposed approach, the conflict resolution procedure adopted for various groups of the robot agents is explained in the following section. a. Two robot agent conflict: In case of two agent conflict,

two PMS can be considered, i.e. R1 R2 and R2 R1 (Fig. 3 (a)). As soon as conflict has been detected by strategy givrn in previous section, robot agents exchange this information to intelligent CST. Once robot agent has received the intended motion of the conflicting one, it starts planning a new path as if other one will not modify the path. For this purpose, it makes use of its CST in the form of grid and current information in the intelligent CST. Because of this decoupled planning, each robot agent computes a candidate motion path for the conflict situation and calculates the cost or length of underlying path modification sequence. The optimal PMS can be selected by comparing the costs of computed PMS. Subsequently, each robot agent executes its motion, which is either the originally intended path or the path modified through multi-robot agent motion planning.

b. Three robot agent conflict: When three robot agents come at the same configuration at the same time then in order to resolve the conflict, six PMS can be distinguished as shown in Fig. 3(b) and same procedure as explained previously can be adopted to determine the conflict free path for all the agents.

c. n-robot agent conflict: When ‘n’ robot agents are involved in a conflict situation, there are generally n! different PMS. An expansion of the proposed scheme for negotiation to n > 3 robot agents would result again in the optimum motion paths. However, the required planning efforts would also expand if the number of robot agents is increased. For realistic applications, problem complexity should be kept limited. Therefore, conflicts including more than 3 robot agents are to be decomposed into either 2 or 3 robot agent schemes. This means that the first two or three robot agents, which

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detects conflict elaborates an optimum solution. Other robot agents are automatically given lower precedence as compared to those actively participating in negotiation procedure.

IV. RESULTS AND DISCUSSION

The proposed approach is demonstrated with an example consisting of three robot agents performing coordinated task in a workcell environment. From Fig. 4, it can observed that, at time t1, agent R1, R2 and R3 start moving individually towards their respective goals. A conflict situation C1 arising for R1 and R2 can be seen at t2 and similar conflict situation C2 can be seen for R1, R2 and R3 at t4 (Fig. 5). In order to bring out the coordination among the robots which are sharing the common workspace and compare the performance of the proposed decoupled approach, the task of conflict resolution was also attempted by allotting fixed priority in the movement to robots, reduction in the joint velocity of the robots and introducing delay in startup time of the robots. The time required to carry out the task or to travel desired distance, length of path to be traversed and overall computation time are compared to evaluate effectiveness of various conflict resolution techniques.

Fig. 4. Three agent scenario

1) Path modification based on optimal PMS The participating robots in first conflict ‘C1’ are R1 and R2. This conflict was resolved by modifying the path of each robot by assuming that other agent will not modify its path. The length of modified path to be traversed by both robots was computed. After comparing the path lengths, it was found out that if path of R2 is modified then it has to move along longer path than R1. Hence, R2 was moved along its original path whereas R1 was moved along the modified path. Aftre resolving first conflict, it was found that second conflict still persists. Three robots participate in conflict at ‘C2’ and hence as explained earlier the conflicting configuration is removed from CST of respective robots and modified path is determined. Based on the length of modified paths, various possible PMS’s and the cost associated with them were determined and same is presented in Table I. From the data given in this table, it can be inferred that the cost of PMS R3R2R1 is least. Hence, path of R3 and R2 is to be modified where as R1 is to be moved along its original path.

Fig. 5. Conflict situation for three agent scenario

TABLE I COMPUTATION OF PMS FOR SECOND CONFLICT SITUATION

2) Fixed priority assignment In this method, motion priority was assigned to participating robots in the fixed order. The conflict condition was avoided by giving first priority in movement to robot R1, second to R2 and last to R3. This priority allotment was done randomly. From data given in Table II, it can be seen that, R2 needs to wait until R1 finishes the task and R3 has to wait until R1 and R2 complete the task. The total travelling time required for R2 and R3 included the waiting time incurred due to fixed priority assignment.

TABLE II TIME REQUIRED FOR COMPLETION OF TASK WITH FIXED PRIORITY AND

DYNAMIC PMS

3) Velocity reduction Conflict resolution can also be achieved by using the velocity reduction. Based on the position of robot in PMS, velocity of R1 is kept same as of original and the velocity of R2 is reduced by 1/3 and that of R3 by 1/4 of the original velocity. From the results presented in Table III, it is clear that reduction in the joint velocity introduces the delay in

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completion of the task and hence the robot agents R2 and R3need more time for completion of the task.

TABLE III TIME REQUIRED FOR COMPLETION OF TASK BY VELOCITY REDUCTION

AND INTRODUCING DELAY IN STARTUP TIME

4) Delay in startup time The problem of collision avoidance is also attempted by introducing a delay in startup time of the robotic agent with loss of flexibility of operating all agents simultaneously. As per PMS, startup of agent R2 is delayed by 10 s and that of R3 by 15 s in order to discard the probability of collision among the robotic agents.

From the results presented in Tables II and III, it can also be observed that, each agent travels along the shortest path when coordination is achieved using fixed priority, velocity reduction, and by introducing delay in startup time. When the conflicts are resolved by computing dynamic priorities for path modification, the agent R2 travels 0.218 m and R3covers 0.179 m more distance. Moreover, the proposed approach needs slightly more computation time as it computes possible priority sequences, cost associated with them and determines the modified path for each agent. Since the approach is for off-line purpose, the computational time is not of much importance. Moreover, the time taken by each agent to travel desired distance is of prime importance, which is very less with the proposed approach when compared with other approaches. This key advantage associated with proposed approach overshadows the trivial shortcomings like change in path length and a little more computational time. Moreover, it allows simultaneous motion of all agents which avoid occurrence of deadlock situation due to preoccupancy of space by higher priority agents. Also, participating agents need not wait until higher priority agents finish their job and hence the total travelling time required is less as compared to other methods. As there is no delay in startup, the movement of all agents can be started at same time which is very important for realization of various industrial operations. The velocity of the agents is not reduced which in turn minimizes the time taken to travel the desired path. Hence, from the above discussion, it is clear that the travelling time required for each agent using the proposed approach of path modification with dynamic priority sequence is less and the flexibility of starting movement of all agents at the same time makes it very effective and efficient for achieving coordination among multiple agents.

V. CONCLUSIONS

The proposed offline decoupled approach for coordinated manipulation of multiple agents computes dynamic priorities to modify the path and resolves the possible conflicts. It is a simple and effective method to realize conflict free coordinated manipulation of several industrial robotic agents operating in common workspace. Multi-agent motion planning is based on search of optimal path using incremental A * algorithm in the discritized configuration space-time. The use of intelligent CST for storing current position of agent in its CST and the path to be followed by each agent helps to transfer information to the motion planner effectively. The proposed approach of dynamic priority computation for modifying the path assures the concurrent movement of all the agents irrespective of the number of agents and nature of obstacles in the environment.

REFERENCES

[1] J. C. Latombe, “Robot motion planning,” Kluwer Academic Publishers, 1991.

[2] K. Fujimura, “Motion planning in dynamic environment,” Springer-Verlag, New York, 1991.

[3] Jun Ota, “Multiagent systems as distributed autonomous systems,” Advanced Engineering Informatics, vol. 20, pp. 59-70, 2006.

[4] J.F. Canny, “The complexity of Robot motion planning,” MIT Press, 1988.

[5] S. Akella, S. Hutchison, “Coordinating the motions of multiple robot with specified trajectories,” IEEE International Conference on Robotics and Automation, Washington, May 2002, pp. 624-631.

[6] J. Hasegawa, K. Kurihara, N. Nishiuchi, “Collision free path planning method for mobile robot t,” Proc of IEEE International Conference on Systems, Man and Sybernetics, vol.3, 2002.

[7] C. Mudgal, J. Vassileva, “An influence diagram model for multi-agent negotiation,” Proc of IEEE Int. Conf on Multiagent Systems, July 2000, pp. 451-452.

[8] G. Sanchez, J. C. Latombe, “Using a PRM planner to compare centralized and decoupled planning for multi-robot systems,” Int. Conf. on Robotics and Automation, vol.2, 2002, pp. 2112-2119.

[9] X. Cheng, “Online collision free path planning for service and assemblytask by a two arm robot agent,” Int. Conf. on Robotics and Automation, 1995, pp. 1523-1528.

[10] P. A. O'Donnell, T. Lozano-Perez , “Deadlock-free and collision-free coordination of two robot manipulators,” Proc. IEEE Int. Conf. Robotics and Automation,1989, pp. 484-489.

[11] K. G. Kant, S. W. Zucker, “Towards efficient trajectory planning: path velocity decomposition,” Int. Jrnl. of Robotics Research, vol. 5, 1986, pp. 72-89.

[12] J. P.van der Berg , M. H. Overmars, “Prioritized motion planning for multiple robots,” Proc. IEEE/RSJ Int. Conf. Robotics and Systems,2005, pp. 2217-2222.

[13] M. Bennewitz, W. Burgard, S. Thrun, “Finding and optimizing solvable priority schemes for decoupled path planning techniques for teams of mobile robot,” Robotics and Autonomous Systems, vol. 41, no.2, 2002, pp. 89- 99.

[14] Th. Fraichard, “Trajectory planning in a dynamic workspace state time space approach,” Advanced Robotics, vol. 13, no. 1, 1999, pp. 75-94.

[15] E. Todt, G. Raush, R. Suarez, “Analysis and classification of multiple robot coordination methods,” Proc. IEEE Int. Conf. Robotics and Automation, vol. 4, 2000, pp. 3152-3157.

[16] T. Harden, C. Kapoor, D. Tesar, “Obstacle avoidance influence coefficients for manipulator motion planning,” Proc ASME IDETC/CIE, Long beach, USA, Sep 24-28, 2005,pp.1-13.

[17] Shital Chiddarwar, N. Ramesh Babu, “Off-line decoupled path planning approach for effective coordination of multiple robots,” Robotica, 2009 (accepted for publication)

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