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University of Zagreb, Faculty of EE & C SWAN '06

University of Zagreb, Faculty of EE & C SWAN '06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 200 6.

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University of Zagreb, Faculty of EE & C SWAN '06

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  1. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 String Algebra-based Approach to Dynamic Routing in Multi-LGV Automated Warehouse Systems Presented by Zdenko Kovačić Faculty of Electrical Engineering and Computing University of Zagreb Unska 3, 10000 Zagreb, CROATIA http://flrcg.rasip.fer.hr Laboratory for Robotics and Intelligent Control Systems 1

  2. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Presentation outline I. Introduction II. A multi-LGV plant layout representation III. The shortest path determination IV. Implementation examples Laboratory for Robotics and Intelligent Control Systems 2

  3. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 I. Introduction • laser guided vehicles (LGVs) in flexible manufacturing systems: • => increased system throughput, • => reduced operational costs, • => consistent execution of predetermined tasks, • => … • requirement - the use of effective supervisory control strategies which • are able to solve conflict and deadlock problems in the system layout • on-line (NP - hard). • two approaches to routing are prevailing in the literature: • - static routing - concerned only with the spatial dimension of the routing • problem (determination of paths in the space domain), • - dynamic routing - treats routing as a time-space problem dealing with • determination of paths feasible both in space and time (in some cases a • time-space approach could be seen also as static routing). Laboratory for Robotics and Intelligent Control Systems 3

  4. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 The aim of the proposed dynamic routing method: Determination of the shortest path (in terms of a traveling time) by solving a shortest path problem with time windows (SPPTW) and by using a string matrix composition. Dynamic routing the number of active vehicles and the corresponding missions change in time  determined shortest path should be feasible  time windows elongation provides conflict-free and deadlock-free paths for all active vehicles. Laboratory for Robotics and Intelligent Control Systems 4

  5. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 A multi-LGV plant layout representation • layout of a multi-LGV plant can be represented by a graph whose • intersections and ends of paths are referred to as graph nodes, while • paths themselves are represented by weighted arcs. directed graph • a nominal arc weight minimal traverse time of arc aj for vehicle ri, because time windows are used for shortest path calculation where lj is the length of arc aj, and vij is the maximal admissible speed for vehicle ri on arc aj. Note: in other cases arc weights could correspond to the lengths of the arcs, to the energy needed to pass the arcs etc. Laboratory for Robotics and Intelligent Control Systems 5

  6. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 An example of a real multi-LGV plant layout Laboratory for Robotics and Intelligent Control Systems 6

  7. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 • a set of active vehicles (vehicles with assigned missions) • a set of active missions . • a path – set of arcs • the weight of the pathσ is equal to the release time of its destination arc di W(σ) = Laboratory for Robotics and Intelligent Control Systems 7

  8. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Representation of LGV mission both can be changed during mission execution • a mission the shortest path between oi and di mission priority • a set of all paths that connect origin arc oi and destination • arc di of mission mi is • we have to find whether is feasible Laboratory for Robotics and Intelligent Control Systems 8

  9. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Definition of mission priority • a missionpriority Pi(t) is calculated according to the relation: where tdi is a due time of mission mi, and Pi0 is an initial mission priority (the mission with the highest priority has the lowest value of Pi0). • the priority of the mission of the vehicle that is far from its destination is higher than the priority of the mission of the vehicle that is already close to its goal. • the mission whose due time is expiring has a higher priority than the mission with a larger time reserve. Laboratory for Robotics and Intelligent Control Systems 9

  10. More about mission priorities University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 • Each time the request for a new mission arrives or current paths become unviable, initially assigned missions’ priorities are recalculated by the dispatching system  the missions with low initial priorities do not wait infinitely in a queue the influence of livelock is thus significantly reduced. • A care should be taken as more than one mission might have priority -∞. In that case,the mission that requests the vehicle first gets a higher priority. • When mi is finished without new mission assignments, an idle positioning mission is activated and vehicle ri is driven to the nearest idle positioning arc (a dead-end arc which is treated only as oi or di). • Being on the idle positioning arc, the vehicle is waiting for a new mission assignment. • We assume that: • a vehicle can reside only on arcs • only one vehicle at the time is allowed to occupy one arc Laboratory for Robotics and Intelligent Control Systems 10

  11. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Definition of a time window • Assigned to mission mi, vehicle ri occupies particular arc aj during a time window defined as whereis a release time of arc aj from mission mi, andis a time of entry on arc aj by mission mi. vehicle r3 first occupied arc a during time window w3a etc. then vehicle r1 occupied arc a during time window w1a These are vectors withthe 1st component corresponding to the highest priority mission, nth component to the lowest priority mission and n=│Ma│, i.e., the dimension of all three vectors is equal to the number of active missions (variable) Laboratory for Robotics and Intelligent Control Systems 11

  12. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 More about characteristics of time windows • In case of moving on a circular path, a vehicle may visit an arc aj more than once, hence, more than one component of time vector wjwould correspond to the same mission, i.e. n≠│Ma│ index iυj corresponds to the υth time window of mission mi on arc aj. • The components of vector wj, related to active missions which do not use arc aj, are set to zero, while the related components of vectors and are set to ∞. • From defined time vectors we know which missions visit which arcs, but we cannot tell directly in which orderfor the purpose of time windows insertion, we must sort components of time vectors in a chronological order. • In this way, vector x= [xi] can be converted into a sortedvector= [xi], where Laboratory for Robotics and Intelligent Control Systems 12

  13. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 • In the example shown below, missions m1 and m2 have the highest and the lowest priority, respectively • Time vectors of a given arc aaare the following • Sorted time vectors for arc aa are written as (it should be noted that, although seven missions are active, only four of them are using arc aa). Laboratory for Robotics and Intelligent Control Systems 13

  14. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 The shortest path determination The method solves the Shortest Path Problem with Time Windows (SPPTW). The method finds the mission candidate paths by string matrix composition A string matrixS, associated with graph G = (N, A) and its arc adjacency matrix G, is an nxn matrix with string entry sij obtained as follows: For each gij which has entry 1, sij=Ai-Aj, where Ai is a word identifying arc ai and Aj is a word identifying arc aj. If gij = 0, sij= 0, that is, if there is no node between arcs, the entry is an empty string. Powers of string matrix S are calculated as follows: i.e. an entry of Sρ is found as: where n is a number of arcs in the corresponding graph. Laboratory for Robotics and Intelligent Control Systems 14

  15. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 The shortest path determination Standard multiplication in should be replaced with series string composition, while standard addition should be replaced with parallel string composition. Then, component comprises all paths from arc i to arc j that include ρ nodes; these paths are accordingly called ρth order paths (the components of string matrix S represent the 1st order paths). A setcontains all paths of the corresponding graph. When a new mission mm is requested at the moment tm, the supervisor is looking for an idle vehicle rm to assign it to that mission (with initial mission priority). As a goal of dynamic routing is to determine the shortest path for mission mm, only powers of vector (i.e., the row of matrix S that corresponds to the origin arc om) must be calculated by using a string composition Laboratory for Robotics and Intelligent Control Systems 15

  16. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 The shortest path determination Having vector, the weight of each path, represented by a string in, has to be determined and then vector is formed in the following way: if there exist ρth order paths that connect om and dm, then Furthermore, if W() < W(), then the string that stood foris replaced by the string representing . In the case W() ≥ W(), the path in vector is replaced by the null string, otherwise the path remains the component of the vector. Initially, when mission mm is requested and W() = ∞. Further step in algorithm is testing of the path feasibility Laboratory for Robotics and Intelligent Control Systems 16

  17. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 1. Initialization of time vectors Let us choose a candidate path. For each arc aj ,its time vectors are initialized as missions with lower priority than a new mission and missions which released the arc before new missionrequest unknown  intmj = tm Initialization of the origin arc om the average vehicle speed the remaining length of the arc at the moment of request Laboratory for Robotics and Intelligent Control Systems 17

  18. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 2. Insertion of time windows • Having time windows of all arcs that belong to the candidate path initialized, starting from the second arc on the path, we are looking on each arc for the first available time window that fulfils two requirements: • it is wide enough to accommodate vehicle rm for a predetermined period, • its entry time is set after the release time of the upstream arc, i → j. request for new mission Laboratory for Robotics and Intelligent Control Systems 18

  19. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 3. Time windows elongation and overlaps conflict request for new mission resolution of conflict by window elongation Laboratory for Robotics and Intelligent Control Systems 19

  20. Installation in Cavi Triveneta - Simulator University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Laboratory for Robotics and Intelligent Control Systems 20

  21. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Pre-installation testing in Euroimpianti Vicenza Laboratory for Robotics and Intelligent Control Systems 21

  22. Installation Cavi Triveneta, Italy University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Laboratory for Robotics and Intelligent Control Systems 22

  23. Installation Haribo Linz, Austria University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Laboratory for Robotics and Intelligent Control Systems 23

  24. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Q & A Laboratory for Robotics and Intelligent Control Systems 24

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