EDAR - mobile robot for parts moving based on a game-theoretic approach

EDAR (event-driven assembler robot) — a mobile robot capable of moving a collection of disk-shaped parts located on a two-dimensional workspace from an arbitrary initial configuration to a desired configuration while avoiding collisions in a purely reactive manner, is presented. Since EDAR uses a higher-level scheduler to switch among the subtasks of moving individual parts, it is viewed as mediating a noncooperative game played among the parts. Comments Copyright 2002 IEEE. Reprinted from IEEE Electronics Letters, Volume 38, Issue 3, January 2002, pages 147-148. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. NOTE: At the time of publication, Daniel Koditschek was affiliated with the University of Michigan. Currently, he is a faculty member of the School of Engineering at the University of Pennsylvania. This journal article is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/329 LAICHI, E, ABOULNASR, T., and STEENAART, W.: ‘Effect Of delay on the performance of the leaky LMS adaptive algorithm’, IEEE Trans. Signul Process., 1997, 45, pp. 811-813 HAYKIN, s.: ‘Adaptive filter theory’ (Prentice Hall, 1996), 3rd edn. TOBIAS, o.J., BERMUDEZ, J.C.M., and BERSHAD, N.J.: ‘Stochastic analysis of the delayed LMS algorithm for a new model’. Proc. Int. Conf. Acoust., Speech, Signal Process., 2000, pp. 404-407 EDAR mobile robot for parts moving based on a game-theoretic approach C.S. Karagoz, H.I. Bozma and D.E. Koditschek EDAR (event-driven assembler robot) a mobile robot capable of moving a collection of disk-shaped parts located on a two-dimensional workspace from an arbitrary initial configuration to a desired configuration while avoiding collisions in a purely reactive manner, is presented. Since EDAR uses a higher-level scheduler to switch among the subtasks of moving individual parts, it is viewed as mediating a noncooperative game played among the parts. Introduction: In this Letter we describe EDAR (event-driven assembler robot) a mobile robot designed and developed for moving a set of rigid disk-shaped parts from random initial placements to a final assembled configuration. The plethora of robots developed for this task have been based on feedfonvard approaches using a sequence of (i) plan generation, (ii) trajectory generation, and (iii) control stages [ I , 21. However, the sensor and actuator uncertainties as well as the changing environment have made it imperative that some level of reactivity should be integrated into such strategies [3]. In this Letter, we push this paradigm to the extreme and describe a robot that operates purely on event-driven principles [4]. Doing this represents small progress in developing a complete formalism of reactive systems, which needs to be done if we are to explore trade-offs along the spectrum from feedfonvard to feedback systems. Approach: Roughly two ideas are at work in EDAR: artificial potential functions and game-theoretic interpretation of the resulting system. First, the notion of artificial potential functions is employed to encode the subtask of positioning each part to its destination. A standard method of deriving feedback controllers from potential fields is then used to construct a closed loop within each subtask. The advantage of this approach is that it is known that if an artificial potential function has certain mathematical properties [ 5 ] , then an object moving in a gradient field generated by this artificial potential function will inevitably end at its prespecified goal position without collisions or getting stuck along the way. Since, as has been shown, no single closed loop can result in a completed parts moving task, a higher-level organising principle has to switch between the alternative closed loops. This idea autonomous scheduling of subtasks is then addressed using the concept of a game [6]. By interpreting the artificial potential functions determining the closed loop dynamics characterising each of the subtasks as pay-off functions, the higherlevel automaton can be seen to be refereeing a game played among the parts to be moved. As there is a set of pay-off functions one associated with each subtask the resulting problem leads to a noncooperative game interpretation. As the obstacles presented by the ungrasped parts present a different geometry depending on whether the robot is moving alone or which part it is coupled to, a single artificial potential function no longer suffices to solve the problem. Rather, a class of artificial potential functions needs to be introduced, each encoding one subtask characterising either one robot-to-part motion or one robot-coupled-to-part motion. In this case, the subtasks become conflicting and the higher-level organising principle is interpreted as governing a noncooperative game played among the subtasks: (i) Next-part: A switching mechanism chooses the next part to be moved by the robot. The robot achieves the subtask of moving this part via a sequence of mate-part and move-part states. Next-part state is re-invoked at the end of mate-part and move-part states. (ii) Mate-part: A set of feedback controllers, one for each different part, is used. The robot actuation is generated by the negative gradient vector field of the artificial potential function defined for the target part. The robot moves until a minimum point of the function is attained. If the robot stops at a position where its gripper could successfully grasp the part, then a transition occurs to move-part state. However, the robot motion could be blocked on the way before reaching the target part, i.e. a local minimum of this function is attained. In this case, the subtask of moving the part is terminated and a transition occurs back to next-part state to choose another part subtask. (iii) Move-part: After the robot grasps the part, the coupled object consisting of the robot and the part moves to the goal position of the part. Again, a set of feedback controllers are designed to accomplish this. The robot actuation is generated by the negative gradient vector field of the artificial potential function defined for the mated part. The minimum value of function is attained when the robot moves the mated part to its goal position. However, the motion of the coupled object could be blocked on the way before this goal position is achieved. In both cases, the subtask of moving part is terminated, the robot ungrasps the part and a transition occurs back to next-part state to choose another part subtask. The switching among the part subtasks are invoked repetitiously in a reactive manner until all the parts are moved to their goal positions by the robot via the following discrete dynamical system: b[n + I] =f(b[n]) where b denotes the augmented position vector of the parts andfis the transition map from one blocked part state to the next. This discrete dynamical system can be interpreted as a noncooperative game played by the parts and refereed by the robot. The fixed points of the dynamical system addresses the solutions of the game. Implementation: EDAR has been designed and constructed with the purpose of implementing this approach (Fig. 1). It is a 2DOF mobile robot with a 3DOF arm mechanism and a lDOF gripper mechanism. Its projection onto a two-dimensional workspace is a disk. Its gripper is able to hold disk-shaped objects, e.g. pipes, EDAR can sense its joint positions via optical encoders mounted on the joints. The positions and the sizes of the parts are obtained using a stationary camera mounted on the top of the workspace. Its translational velocity is about 6 cm/s and the time required for grasping/ungrasping is about 30 s/part. EDAR moves parts in purely event-driven manner; thus the control software of EDAR is based on the proposed gametheoretic approach. Fig. 1 Photograph of EDAR with the parts Experiments: EDAR has been tested extensively in experiments involving cylindrical pipes of height 1.5 m and of radii varying from 6 to 11 cm. Owing to physical space restrictions, we have been able to conduct experiments involving at most three parts. A typical run of an experiment with three parts is shown in Fig. 2. A set of random goal configurations of varying complexity, measured by the packed tightness of the goal locations, has been defined for the parts. Starting from random initial workspace configurations, the following measures of performance have been studied statistically: (i) the variation of the part path lengths against complexity; (ii) the robot path length against complexity; (iii) the positioning inaccuracy against complexity. Furthermore, experimental results are compared with the simulations of the corresponding tasks. ELECTRONICS LETTERS 31st January 2002 Vol. 38 No. 3 147 The following conclusions are obtained. The path lengths of the parts and the robot increase with increasing task complexity. This can be attributed to two factors: first, the closer the parts need to be packed together, the more careful and precise the robot has to be in its movements; secondly, as there is an increased possibility of collision with the parts, there is more dodging around. Interestingly, compared to simulations, the paths taken by the parts are only 10% longer on average (Fig. 3). As expected, path lengths in real experiments tend to vary more owing to the sensor inaccuracies and non-ideal motion capabilities of the robot. The positional inaccuracies in experiments range between 2.0-5.3 cm/part. In simulations we observe much lower inaccuracies which we attribute again to the EDAR sensor and actuator hardware limitations and not to our approach. -aJn +cIuIl -tun ’ 0 DDRR: G: min{/,&,} 0 .* 0