AGV Navigation in Flexible Manufacturing Systems using Formation Control

Transcription

1 AGV Nagation in Flexible Manufacturing Systems using Formation Control A. Sanchez E. Aranda-Bricaire E. G. Hernandez-Martinez J. Magallon J. Molina Unidad de Ingenieria Avanzada, Centro de Investigacion y Estudios Avanzados (Cinvestav), AP , Zaoan 45050, Mexico( arturo@gdl.cinvestav.mx, jmagallo@gdl.cinvestav.mx, jamolina@gdl.cinvestav.mx). Seccion de Mecatronica, Centro de Investigacion y Estudios Avanzados (Cinvestav), AP , Mexico D.F. 7000, Mexico ( earanda@cinvestav.mx, eghm2@yahoo.com.mx) Abstract: A formation control strategy is roosed for the nagation of non-holonomic (2,0) automated guided vehicles (AGVs) in flexible manufacturing systems (FMS). Each AGV is equied with a formation control, which is based on artificial vector fields to guarantee convergence to each of the location goals. Reulsive vector fields are emloyed to define the nagation layout and to avoid collisions with other AGVs. The AGV nagates reaching each of the location goals without the need of route lanning. The formation control was imlemented as art of an architecture and successfully tested in a rtual reality FSM with 4 AGVs. Keywords: AGV nagation, formation control. 1. INTRODUCTION Flexible manufacturing systems (FMS) are commonly emloyed in different industries as suitable alternatives to resond to the highly demanding markets (e.g. Desai et al. (2001); Cao et al. (1997)). The transortation of raw materials and finished roducts can be carried out different deces such as maniulator robots, conveyor belts or automated guided vehicles (AGV). Emloying AGVs for these tasks imlies many other oerational challenges such as collision-avoidance, on-line rescheduling, battery recharging, low-level manufacturing conflict resolution, etc. Therefore, the oerational comlexity of AGVnagation in FMS imoses an interdiscilinary aroach for constructing hierarchical control strategies. This aer rooses a nagation strategy of AGVs oerating on a delimited area of the factory-floor using formation control. From a control-theoretical oint of ew, the most common aroach to the nagation control of the AGVs is to design a control law enabling each AGV to follow a resecified trajectory within the FMS enronment (e.g. Arai et al. (2002); Balch and Arkin (1998)). However, the resulting controllers are difficult to imlement and maintain because any change in the roduct manufacturing rules of the FMS may lead to re-comuting the re-secified trajectories. Collision-avoiding is also difficult to achieve resulting into unaccetable hazards. Decentralized control strategies roosed for multi-agent robot systems (MARS) have shown to rode an interesting alternative for the coordination of grous of mobile robots using incomlete information and accomlishing a common task (e.g. Chen, E.H. acknowledges financial suort from Conacyt, Mexico in the form of scholarshi J.Mo. acknowledges financial suort from Prome, Mexico in the form of a PhD scholarshi. and Wang (2005); Leonard and Fiorelli (2001)). One classical roblem of the theory of MARS is formation control, in which a set of mobile robots each one subject to artial knowledge of the system is desired to reach a articular attern, avoiding ossible collisions. The aer considers non-holonomic wheeled mobile robots of the tye (2, 0), as defined in Gholiour and Yazdananah (2003). The roduction of a articular good is assigned to an AGV as a sequence of locations goals. Each location goal corresonds to a formation defined with resect to a fixed rtual leader as roosed Zhang and Hu (2007). This allows, simultaneously, to reach absolute desired ositions (instead of relative ones) and to secify a safety area which the AGVs must not tresass during the FMS oeration. Collisions are avoided defining suitable reulsive vector fields. The dynamic model of the AGV considered together with the formation control are discussed in section 2. In order to test the erformance of the roosed control, a rtual-reality FMS was built with four AGVs as shown in section 3. Each AGV is equied with a modular-hierarchical control architecture built ad-hoc to test the ability of the controller to reach desired formations whilst nagating on the FSM floor without colliding with other AGVs. Section 4 shows the results of simulations tests under three situations: convergence in finite time of the AGV to a sequence of desired location goals, collision-avoiding for achieng fixed formations with other AGVs and collision-avoiding while nagating freely to achieve a desired osition on the FSM-floor. Including the formation control law as art of the AGV-control hierarchy seems to be romissory. The simlicity of the mathematical exressions makes it easy to imlement and to maintain. Finally, section 5 discusses

2 some limitations of the current aroach and future work heading towards the design of a comlete architecture for controlling AGV oerations in FMS. 2. AGV KINEMATIC MODEL AND FORMATION CONTROL Attractive otential functions are emloyed to declare location goals. The negative gradient of these functions are used as control inuts for the AGVs. While this simle strategy allows to ensure convergence to the desired locations, ossible collisions can not be ruled out. In order to discard the event of a collision, the control law is modified using an artificial unstable focus-tye vector field centered at the osition of every other AGV. The same tye of unstable behaor is forced with resect to a rtual leader (Zhang and Hu (2007)). The rtual leader does nos exist hysically. However, it allows to ensure two key features of the closed-loo system: Firstly, absolute ositions are reached, as the osition of the rtual leader is referred to an inertial coordinate frame. Secondly, scaling aroriately a reulsive vector field centered at the rtual leader location, a safety area can be ensured. Denote N = {R 1,..., R 5 }, the set of the non-holonomic (2,0) AGVs mong on the factory-floor. AGV R 5 is considered as the rtual leader and the rest are follower AGVs. The kinematic model of each R i, as shown in Fig. 1, is described ẋi ẏ i θ i = R i (θ i ) [ w i ], R i (θ i ) = cos θi 0 sin θ i 0, (1) 0 1 i = 1,..., n where v i is the linear velocity of the midoint of the wheels axis and w i is its angular velocity. In the rest of the aer, the formation strategy is related to the coordinates α i = ( i, q i ) shown in Fig. 1 which are given i α i = = q i xi + l cos (θ i ), i = 1,..., n (2) y i + l sin (θ i ) (e.g. Desai et al. (2001)). The dynamics of (2) are given cos θi l sin θ α i = A i (θ i ), A w i (θ i ) = i, (3) i sin θ i l cos θ i i = 1,..., n The so-called decouling matrix A(θ i ) of each AGV is nonsingular. Then, it is ossible to design a control strategy for ositioning α i at a desired location using the control law = A w 1 (θ i ) α id, i = 1,..., n (4) i where α id is the desired dynamics of coordinates α i. To design a formation strategy, we define N i the subset of ositions of the AGV which are detectable for R i. In this aer, the subsets N i are defined N i = {z 5 }, i = 1,..., 4 (5) N 5 = Let c ji = [h ji, v ji ] T denote a vector which reresents the desired osition of R i with resect to R j in a articular formation. Thus, we define αi = f(n i), i = 1,..., n as the desired relative osition of every R i in the formation given α i = α 5 + c 5i, i = 1,..., 4 (6) α 5 = m (7) where m R 2 is an secific osition in the FMS of the rtual leader AGV. The ositions αi, as shown below, can be the ositions of workstations in the FMS. The formation strategy using R 5 as rtual leader is shown in Fig. 2. This formation can be considered as a star formation centered in the rtual leader. 1 2 Y l i = ( i, q i ) c 51 u i c 53 5 c 52 c 54 y i i i 3 4 Fig. 1. Kinematic model of unicycles x i The oint α i can be the center of mass of the AGV or the lace where a sensor or actuator is located. The idea of controlling coordinates α i instead of the center of the wheels axis is frequently found in the mobile robot literature in order to avoid singularities in the control law X Fig. 2. Formation Control Strategy using a rtual leader The goal of each AGV R i, i = 1,..., 4 is to be steered to a relative osition c 5i with resect to the rtual leader avoiding collisions, i.e. it is necessary to design a control law u i (t) = g i (N i (t)) for every robot R i, such that lim t (α i αi ) = 0, i = 1,..., n and α i(t) α j (t) > d, t 0, i j where d is the diameter of a circle centered in the coordinate α i that circumscribes each AGV.

3 In order to guarantee the convergence to the desired formation, local otential functions are defined γ i = α i α 5 2, i = 1,..., 4 (8) γ 5 = α 5 m 2 3. THE VIRTUAL FMS A diagram of the FMS considered in this aer is deicted in Fig. 3. Four AGVs are initially located in the arkingbattery-recharging area. The two large boxes reresent the feedstock and the finished goods warehouses. The functions γ i are ositive definite and reach their global minimum (γ i = 0) when α i α 5 = c 5i, i = 1,..., 4 and α 5 = m. A control law based on the negative gradient of functions γ i steering every agent to the minimum of these otential function but collisions can occur. For avoiding this, we roose reulsive vector fields (between air of agents) given 1 2 where (i β ij = δ ij V j ) (q i q j ) ij, j i (9) ( i j ) + (q i q j ) Feed stock Safety area Finish ed goods { 1, if αi α δ ij = j 2 d 2 0, if α i α j 2 > d 2 (10) ( V ij = 1 α i α j 2 1 d 2 and for the safety area an square reulsive area given ) 2 g = ( i 5 ) 4 + (q i q 5 ) 4 ( 1 V i5 = g 1 ) 2 l 2 (11) The reulsive vector field is a clockwise unstable focus centered at the osition of the another AGV. This vector field is scaled a function V. This function tends monotonously to infinity when the distance between agents tends to zero and V = 0 in the limit of the minimum allowed distance. Thus, the reulsive forces aear only in a danger of collision. For V i5, the reulsive field takes an almost-square shae of side length l defining the nontresassing zone in the FSM layout. Using the reous vector fields, we define a control law given ] = A w 1 (θ i ) 1 i 2 k γ i + η α i j i [ β ij, i = 1,...5(12) where k, η R and k, η > 0. The dynamics of the coordinates α i for the closed-loo system (3)-(12) is given α i = 1 2 k γ i + η β ij, i = 1,...5 (13) α i j i The control law (12) steers the coordinates α i to a desired osition. However, the angles θ i remain uncontrolled. These angles do not converge to any secific value. Thus, the control law is to be considered as a formation control without orientation. 4 Fig. 3. The FSM Virtual Enronment 3 The small boxes reresent the working stations, where the AGVs deliver and, after a while, ick u a articular material to follow u the manufacturing rocess. The behaor secifications of the AGVs are as follows: (1) Each AGV must ick u first feedstock from the feedstock-warehouse. (2) Each AGV must carry the material through a secified sequence of working stations. This sequence is given a roduction lanner not considered in this aer. (3) Each AGV must deliver the rocessed roduct to the finished roduct-warehouse. (4) Two or more AGVs must not occuy simultaneously the same osition. (5) The non-tresassing zone is delimited a square area identified as a Safety area in Fig. 3 which the AGVs must not invade. The formation control is imlemented as art of a hierarchical control architecture oerating each AGV of the FSM (Molina (2009)) as shown in Fig. 4. The strategic and tactical level tasks are carried out searately from the oeration of the AGV whilst the oerational asects are resolved on-line nagation control module of each AGV. For the sake of testing the formation control, a simle scheduling strategy was established in which each roduct to be manufactured is assigned to an AGV. An uer coordination module at the tactical level deals with construction of the art roduction rules (i.e. roduction recies) that are assigned to the AGV (Sanchez et al. (2009)). Production rules are locally translated the recie interreter block of the nagation module into a sequence of location goals that must be executed the hysical dece layer (not

4 Strategical (Suerior) Production Planning 2 3 Tactical (Uer) Oerational (Lower) Coordination Control Module Actity or Recie Translator Nagation control Module Actity or Recie Interreter Behaor Control Rules or Laws Alied to AGV s Formation Control Sequential Tasks Communication Between AGV S Fig. 4. Hybrid control architecture including the formation control module as art of the nagation control module shown in the figure). The formation control block is emloyed within the nagation control module to drive the nagation on the FSM floor. Molina (2009) deals also with on-blocking conditions and other oerational issues (such as battery-recharging, failure mode oeration, etc.) resolved the behaor control block. Magallon (2008) imlemented the control architecture in Matlab-Simulink and the FMS reously described was deloyed in a rtual enronment as shown in Fig. 3. The rtual leader was laced at the centre of the layout with quadratic reulsive fields to create the nagation area Safety area Fig. 5. Imlementation of subgoal locations SIMULATION RESULTS The formation control module was tested in the rtual FMS described reously for the following cases: i) nagation of one AGV and convergence on finite time, ii) collision-avoiding of 2 AGVs while nagating., iii) nagation of multile AGV to achieve a fixed formation. Location subgoals were defined outside each warehouse and workstation in order to resolve situations of two or more AGVs requesting access to these resources when is being used another AGV. Each subgoal reresents a queueing osition that an AGV can occuy in a fixed formation. Fig. 5 shows the queueing osition for each FSM resource. For the first case, a roduction recie dictates collecting first feedstock and then siting in sequence each of the four workstations, delivery of the finished roduct to the warehouse and return to the arking area. The nagation trajectories deloyed on a x-y axis are deicted in Fig.6. Convergence times on each axis are shown in Fig.7. Note that the AGV seed diminishes when aroximating the designated location goals reresented the inflexion oints of the uer line. Collision-avoiding maneuvering of two AGVs can be observed in trajectories shown in Figs. 8 and 9. A collision situation is created assigning to AGV1 location sequence ( 1, 2, 3, 4, 5, 6 ) and to AGV2 location sequence ( 6, 5, 4, 3 ). Collision is averted near working station 4 the action of unstable reulsive fields of both AGVs Fig. 6. Nagation of one AGV siting both warehouses and all working stations in sequence to fulfill a roduction recie. Nagation trajectories Trajectories achieng a desired formation (e.g. queuing at a feedstock-warehouse inlet) without colliding are resented in Fig. 10. Queue ositions were assigned arbitrarily to each AGV. In this case, the AGV identification number corresonds to its osition in the queue. All AGV initiated from the arking area simultaneously. AGV1 reached its location goal 11 with minor disturbances. AGV3 gave way to AGV2 that initiated its trajectory from a further location than AGV3. With a higher riority, AGV2 reached its location goal followed AGV3. 5. CONCLUSIONS AND FUTURE WORK The roosed formation control law seems to be a simler alternative for the nagation of AGVs comared to other aroaches based on calculating trajectories. Other imortant low-level functional issues remain to be exlored such as emergency rocedures, fault-recovery, collision avoidance of non-identified objects or the dynamic construction of orderly formations to resolve AGV queueing outside the rocessing facilities. The rtual-reality enronment has

5 Goal onvergence on x onvergence on y Nagation AGV1 Nagation AGV2 Nagation AGV3 Nagation AGV4 Location subgoal 1 Location subgoal 2 Location subgoal t(s) Fig. 7. Nagation of one AGV siting both warehouses and all working stations in sequence to fulfill a roduction recie. Convergence to location goals in finite time Fig. 8. Collision avoiding during nagation. Trajectory aths Recie AGV1 Recie AGV t(s) Fig. 9. Collision avoiding during nagation. Convergence to location goals in finite time roved to be a useful tool for raid rototying and testing of the roosed ideas in which an agent-based aroach will be incororated Fig. 10. Nagation of 4 AGVs to achieve a desired fixed formation T. Balch, R.C. Arkin. Behaor-based formation control for multirobot teams. IEEE Transactions on Robotics and Automation, vol. 14(3): ,1998. Y. U. Cao, A.S. Fukunaga, A.B. Kahng. Cooerative mobile robotics: Antecedents and directions. Autonomous Robots, vol. 4(1):34-46, Y. Q. Chen, Z. Wang. Formation control: A Reew and a New Consideration. International Conference on Intelligent Robots and Systems, ,2005. J.P. Desai, J.P Ostrowski, V. Kumar. Modeling and control of formations of nonholonomic mobile robots,. IEEE Transactions on Robotics and Automation, vol. 6(17): , A. Gholiour, M.J. Yazdananah. Dynamic tracking control of nonholonomic mobile robot with model reference adatation for uncertain arameters. Proc. ECC2003 Euroean Control Conference, Cambridge, UK, Se, N. E. Leonard, E. Fiorelli. Virtual Leaders, Artificial Potentials and Coordinated Control of Grous. IEEE Conference on Decision and Control, ,2001. J. Magallon. Design and Simulation of an Hybrid Controller for AGV Nagation on Manufacturing Systems. M. Sc. thesis (in sanish). Unidad de Ingenieria Avanzada, Cinvestav, J. Molina. AGV Oeration in Flexible Manufacturing Systems teams. PhD thesis. Unublished: In rearation. Unidad de Ingenieria Avanzada, Cinvestav, A. Sanchez, E. Aranda-Bricaire, F. Jaimes, E. Hernandez, A. Nava. Synthesis of roduct-driven coordination controllers for a class of discrete-event manufacturing systems. Acceted for ublication. Robotics and Comuter Integrated Manufacturing, W. Zhang and J. Hu. A Case Study of Formation Constrained Otimal Multi-Agent Coordination. Proc. 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, Dec , 2007 REFERENCES T. Arai, E. Pagello, L. E. Parker. Guest Editorial Advances in Multirobot systems. IEEE Transactions on Robotics and Automation, vol. 18(5): , 2002.