1. 1 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 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

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

3. 3 Laboratory for Robotics and Intelligent Control Systems 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). University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006

4. 4 Laboratory for Robotics and Intelligent Control Systems 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. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006

5. 5 Laboratory for Robotics and Intelligent Control Systems 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. a nominal arc weight  minimal traverse time of arc a j for vehicle r i, where l j is the length of arc a j, and v ij is the maximal admissible speed for vehicle r i on arc a j. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 directed graph Note: in other cases arc weights could correspond to the lengths of the arcs, to the energy needed to pass the arcs etc. because time windows are used for shortest path calculation

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

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

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

9. 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. 9 Laboratory for Robotics and Intelligent Control Systems Definition of mission priority a mission priority P i (t) is calculated according to the relation: where t di is a due time of mission m i, and P i0 is an initial mission priority (the mission with the highest priority has the lowest value of P i0 ). University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006

10. More about mission priorities 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 m i is finished without new mission assignments, an idle positioning mission is activated and vehicle r i is driven to the nearest idle positioning arc (a dead-end arc which is treated only as o i or d i ). 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 10 Laboratory for Robotics and Intelligent Control Systems University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006

11. 11 Laboratory for Robotics and Intelligent Control Systems Definition of a time window University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 vehicle r 3 first occupied arc a during time window w 3a then vehicle r 1 occupied arc a during time window w 1a etc. These are vectors with the 1 st component corresponding to the highest priority mission, n th component to the lowest priority mission and n=│M a │, i.e., the dimension of all three vectors is equal to the number of active missions (variable) Assigned to mission m i, vehicle r i occupies particular arc a j during a time window defined as where is a release time of arc a j from mission m i, and is a time of entry on arc a j by mission m i.

12. 12 Laboratory for Robotics and Intelligent Control Systems More about characteristics of time windows University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 In case of moving on a circular path, a vehicle may visit an arc a j more than once, hence, more than one component of time vector w j would correspond to the same mission, i.e. n≠│M a │  index iυj corresponds to the υ th time window of mission m i on arc a j. The components of vector w j, related to active missions which do not use arc a j, 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 = [x i ] can be converted into a sorted vector = [x i ], where

13. 13 Laboratory for Robotics and Intelligent Control Systems 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, m issions m 1 and m 2 have the highest and the lowest priority, respectively Time vectors of a given arc a a are the following Sorted time vectors for arc a a are written as (it should be noted that, although seven missions are active, only four of them are using arc a a ).

14. 14 Laboratory for Robotics and Intelligent Control Systems 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 matrix S, associated with graph G = (N, A) and its arc adjacency matrix G, is an nxn matrix with string entry s ij obtained as follows: For each g ij which has entry 1, s ij =A i -A j, where A i is a word identifying arc a i and A j is a word identifying arc a j. If g ij = 0, s ij = 0, that is, if there is no node between arcs, the entry is an empty string. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Powers of string matrix S are calculated as follows: i.e. an entry of S ρ is found as: where n is a number of arcs in the corresponding graph.

15. 15 Laboratory for Robotics and Intelligent Control Systems 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 ρ th order paths (the components of string matrix S represent the 1 st order paths). A set contains all paths of the corresponding graph. University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 When a new mission m m is requested at the moment t m, the supervisor is looking for an idle vehicle r m to assign it to that mission (with initial mission priority). As a goal of dynamic routing is to determine the shortest path for mission m m, only powers of vector (i.e., the row of matrix S that corresponds to the origin arc o m ) must be calculated by using a string composition

16. 16 Laboratory for Robotics and Intelligent Control Systems 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 ρ th order paths that connect o m and d m, then University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 Furthermore, if W( ) < W( ), then the string that stood for is 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 m m is requested and W( ) = ∞. Further step in algorithm is testing of the path feasibility

17. 17 Laboratory for Robotics and Intelligent Control Systems Let us choose a candidate path. For each arc a j, its time vectors are initialized as 1. Initialization of time vectors Initialization of the origin arc o m University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 missions with lower priority than a new mission and missions which released the arc before new mission request unknown  in t mj = t m the average vehicle speed the remaining length of the arc at the moment of request

18. 18 Laboratory for Robotics and Intelligent Control Systems 2. Insertion of time windows request for new mission University of Zagreb, Faculty of EE & C SWAN'06 Department of Control and Computer Engineering Arlington, USA, Decembe 7-9, 2006 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: a)it is wide enough to accommodate vehicle r m for a predetermined period, b)its entry time is set after the release time of the upstream arc, i → j.

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

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

21. 21 Laboratory for Robotics and Intelligent Control Systems 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

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

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

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