Hierarchical hybrid control for Multiple Mobile Robot Systems⁎

10th IFAC Symposium on Intelligent Vehicles 10th IFAC Symposium on Intelligent Autonomous Autonomous Vehicles 10th IFAC Symposium on Autonomous Vehicles 10th IFACPoland, Symposium on Intelligent Intelligent Autonomous Vehicles Gdansk, July 3-5, 2019 Available online at www.sciencedirect.com Gdansk, Poland, July 3-5, 2019 Gdansk, Poland, July 3-5, 2019 10th IFAC Symposium on Intelligent Autonomous Vehicles Gdansk, Poland, July 3-5, 2019 Gdansk, Poland, July 3-5, 2019

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IFAC PapersOnLine 52-8 (2019) 452–457

Hierarchical hybrid control for Multiple Hierarchical hybrid control for Hierarchical hybrid control for Multiple Multiple Mobile Robot Systems Mobile Robot Systems Mobile Robot Systems  Mobile ∗Robot Systems ∗ ∗

Elzbieta Roszkowska Roszkowska ∗∗ Piotr Piotr Dulewicz Dulewicz ∗∗ Lukasz Lukasz Janiec Janiec ∗∗ Elzbieta Elzbieta Elzbieta Roszkowska Roszkowska ∗∗∗ Piotr Piotr Dulewicz Dulewicz ∗∗∗ Lukasz Lukasz Janiec Janiec ∗∗∗ Elzbieta Roszkowska Piotr Dulewicz Lukasz Janiec ∗ ∗ Department Cybernetics and Robotics, Electronics Faculty, ∗ Department of of Cybernetics and Robotics, Electronics Faculty, ∗ of Cybernetics and Robotics, Electronics Faculty, ∗ Department Department of Cybernetics and Robotics, Electronics Faculty, Wroclaw University of Science and Technology, Poland ∗ Wroclaw University of Science and Technology, Poland Department of Cybernetics and Robotics, Electronics Faculty, Wroclaw (email: University of Science and Technology, Poland Wroclaw University of Science and Technology, Poland [email protected]). [email protected]). Wroclaw (email: University of Science and Technology, Poland (email: [email protected]). (email: [email protected]). (email: [email protected]). Abstract: The paper presents a (discrete-event/continuous-time) control concept for a Abstract: The presents aa hybrid hybrid (discrete-event/continuous-time) control Abstract: The paper paper presents hybrid (discrete-event/continuous-time) control concept concept for for aaa Abstract: The paper (discrete-event/continuous-time) control concept for system of mobile mobile robotspresents sharing a a hybrid common motion space. space. The The logics logics underlying underlying its construction system of robots sharing a common motion its construction Abstract: The paper presents aa hybrid (discrete-event/continuous-time) control concept for a system mobile robots common space. The logics its system of ofensures mobile collision-free robots sharing sharing a deadlock-free common motion motion space. The logics underlying underlying its construction construction formally and motion of the robots, and the modular structure formally and motion of the robots, and the modular structure system ofensures mobile collision-free robots sharing a deadlock-free common motion space. The logics underlying its construction formally ensures collision-free and deadlock-free motion of the robots, and the modular structure formally ensures collision-free and deadlock-free motion of the robots, and the modular structure of the controller allows for its modification and experimental optimization. of the controller allows for its modification and experimental optimization. formally ensures collision-free and deadlock-free motion of the robots, and the modular structure of the controller allows for its modification and experimental optimization. of the controller allows for its modification and experimental optimization. of allows forFederation its modification and experimental optimization. © the 2019,controller IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Multiple mobile robot system, formally correct behavior, hybrid control. Keywords: Keywords: Multiple Multiple mobile mobile robot robot system, system, formally formally correct correct behavior, behavior, hybrid hybrid control. control. Keywords: Multiple mobile robot system, formally correct behavior, hybrid control. Keywords: Multiple mobile robot system, formally correct behavior, hybrid control. 1. INTRODUCTION INTRODUCTION providing the the above above properties properties is is mainly mainly due due to to the the asas1. providing 1. INTRODUCTION INTRODUCTION providing the is due to asproviding the above above properties properties is mainly mainly due to the the as1. sumed representation of the robot system, whose opersumed representation of the robot system, whose oper1. INTRODUCTION providing the above properties is mainly due to the assumed representation of the robot system, whose opersumed representation of the robot system, whose operation is abstracted in continuous time. Devoid of these The use of a mobile robot team in place of one robot ation is abstracted in continuous Devoid of these The use of aa mobile robot team in place of one robot sumed representation of the robottime. system, whose operation is abstracted in continuous time. Devoid of these The use of mobile robot team in place of one robot ation is abstracted in continuous time. Devoid of these is the approach introduced in Roszkowska The use of a increases mobile robot team in place of onerobotic robot disadvantages substantially the performance of many disadvantages is introduced in substantially the performance of many is abstracted continuous time. Devoid of these disadvantages is the theinapproach approach introduced in Roszkowska Roszkowska The use of a increases mobile robot team in place of onerobotic robot ation substantially increases the performance ofarea many robotic disadvantages is the approach introduced in Roszkowska substantially increases the performance of many robotic (2005); Reveliotis and Roszkowska (2011); Roszkowska and applications, including those related to searching, (2005); Reveliotis and Roszkowska (2011); Roszkowska and applications, including those related to area searching, disadvantages is the approach introduced in Roszkowska (2005); Reveliotis and Roszkowska (2011); Roszkowska and substantially increases the performance of many robotic applications, including those related related to area area extraction searching, Reveliotis applications, including those to searching, (2005); Reveliotis and Roszkowska (2011); Roszkowska and (2013), where the logic of robot coordination is search and rescue, interplanetary exploration, Reveliotis (2013), where the logic of robot coordination is search and rescue, interplanetary exploration, extraction Reveliotis and Roszkowska (2011); Roszkowska and Reveliotis (2013), where the logic of robot coordination is applications, including those related to area extraction searching, (2005); search and rescue, interplanetary exploration, Reveliotis (2013), where the logic of robot coordination is developed using the DES (Discrete Event System) formalsearch and rescue, interplanetary exploration, extraction of minerals, agriculture, forestry, or transport. A key developed using the DES (Discrete Event System) formalof minerals, agriculture, forestry, or transport. A key Reveliotis (2013), where the logic of robot coordination is developed using the DES (Discrete Event System) formalsearch and rescue, interplanetary exploration, extraction of minerals, agriculture, forestry, or transport. A key developed using the DES (Discrete Event System) formalism. The proposed model ensures correct space sharing by of minerals, agriculture, forestry, or transport. A key issue in the design of such systems is to coordinate the ism. The proposed model ensures correct space sharing by issue in the design of such systems is to coordinate the developed using the DES (Discrete Event System) formalism. The proposed model ensures correct space sharing by of minerals, agriculture, forestry, or transport. A key issue in in the theofdesign design of such such systems is to to coordinate coordinate the ism. The of proposed ensures correct space constraints sharing by issue of systems is the a group group of robots model through imposing certain constraints movement aa number of robots operating in the same robots through imposing certain movement number of robots operating in the same The of proposed model ensures correct space constraints sharing by aon group robots through imposing certain issue in theof design of such systems is to coordinate the aism. movement of a number of robots operating in the same movement of a number of robots operating in the same a group of robots through imposing certain constraints their motion, and can be applied to mobile robots workspace. Regardless of their tasks, the robots must be on their motion, can be applied to mobile robots workspace. Regardless tasks, the robots must be aongroup of robotsand through imposing certain constraints their motion, and can be applied to mobile robots movement of a numberof oftheir robots operating in the same workspace. Regardless of their tasks, the robots must be on their motion, and can be applied to mobile robots accomplishing any arbitrarily assumed missions, and for workspace. Regardless of their tasks, the robots must be able to effectively share a common area in order to prevent accomplishing any arbitrarily assumed missions, and for able to effectively share a common area in order to prevent on their motion, and can be applied to mobile robots accomplishing any arbitrarily assumed missions, and for workspace. Regardless of their tasks, the robotstomust be accomplishing able to effectively share a common area in order prevent any arbitrarily assumed missions, and for both centralized and distributed supervisory control arable to effectively share a common area in order to prevent the mutual disruption of traffic and effectively pursue their both centralized and distributed supervisory control arthe mutual disruption of traffic and effectively pursue their accomplishing any arbitrarily assumed missions, and for both centralized and distributed supervisory control arable to effectively share a common area in order to prevent the mutual disruption of traffic and effectively pursue their both centralized and distributed supervisory control arthe mutual disruption of traffic and effectively pursue their chitecture. To To respect respect the the requirements requirements of of aa so-defined so-defined missions. chitecture. missions. both centralized and distributed supervisory control architecture. To respect the requirements of a so-defined the mutual disruption of traffic and effectively pursue their missions. chitecture. To respect the requirements ofrobots a so-defined missions. DES-based supervisory-control scheme, the need to DES-based supervisory-control scheme, the need to chitecture. To respect the requirements ofrobots a so-defined DES-based supervisory-control scheme, the robots need to missions. Representative results for this type of research can be DES-based supervisory-control scheme, the robots need to Representative results for this type of research can be modify their individual motion control, based on the CTS Representative results for this type of research can be modify their individual motion control, based on the CTS Representative results for this type of research can be DES-based supervisory-control scheme, the robots need to modify their individual motion control, based on the CTS found, among others, in the following papers. Tomlin modify their individual motion control, based on the CTS found, among others, in the papers. Tomlin Time System) abstraction. Representative results for this following type of research can be (Continuous found, among others, in the following papers. Tomlin (Continuous Time System) abstraction. modify their individual motion control, based on the CTS (Continuous Time System) abstraction. found, among others, in the following papers. Tomlin et al. (1998) and Lygeros et al. (1998) contribute with et al. (1998) Lygeros et al. (1998) contribute with Time System) abstraction. found, amongand others, in the following papers. Tomlin et al. al. (1998) (1998) and Lygeros et al. (1998) contribute with (Continuous (Continuous Time System) abstraction. The aim of this work is to present aa hybrid, hierarchical et and Lygeros et al. (1998) contribute with hybrid control solutions for aircraft and highway systems, The aim of this hybrid, hierarchical hybrid control solutions for aircraft and highway systems, The aimfor of multiple-mobile this work work is is to to present present hybrid, hierarchical et al. (1998) and Lygeros et al. (1998) contribute with The hybrid control solutions for aircraft and highway systems, aim of this work is to present aa hybrid, hierarchical control robot systems that ensures a a hybrid control solutions for aircraft and highway systems, respectively. Pecora et al. (2005)and Valle and Hutchinson control for multiple-mobile robot systems that ensures respectively. Pecora et al. (2005)and Valle and Hutchinson The aim of this work is to present a hybrid, hierarchical control for multiple-mobile robot systems that ensures a hybrid control solutions for aircraft and highway systems, respectively. Pecora et al. (2005)and Valle and Hutchinson control for multiple-mobile robot systems that ensures a respectively. Pecora et al. (2005)and Valle and Hutchinson correct and efficient co-operation of a group of robots (1998) focus on conflict resolution and optimal motion correct and efficient co-operation of a group of robots (1998) focus on conflict resolution and optimal motion control for multiple-mobile robot systems that of ensures a correct and efficient co-operation of a group robots respectively. Pecora et al. (2005)and Valle and Hutchinson (1998) focus on conflict resolution and optimal motion correct and efficient co-operation of a group of robots sharing a common 2D space. In Section 2, we present in (1998) focus on conflict resolution and optimal motion planning for multiple mobile robots. Inalhan et al. (2002) sharing aa common 2D space. In Section 2, we present in planning for multiple mobile robots. Inalhan et al. (2002) correct and efficient co-operation of a group of robots sharing common 2D space. In Section 2, we present in (1998) focus on conflict resolution and optimal motion planning for multiple mobile robots. Inalhan et al. (2002) a common 2D space. In Section 2, we present in more detail the objectives of this paper. In Sections 5–6, planning multiple mobile robots. Inalhan etfor al. solving (2002) sharing presents aafordecentralized optimization method more detail the of paper. In Sections 5–6, presents decentralized optimization method a common 2D space. In Section we present in more detail the objectives objectives of this this paper.of2, Inthe Sections 5–6, planning multiple mobile robots. Inalhan etfor al. solving (2002) sharing presents afordecentralized decentralized optimization method for solving more detail the objectives of this paper. In Sections 5–6, we systematically pursue the synthesis controller. presents a optimization method for solving the coordination problem of interconnected systems with we systematically pursue the synthesis of the controller. the coordination problem of interconnected systems with more detail the objectives of this paper. In Sections 5–6, we systematically pursue the synthesis of the controller. presents a decentralized optimization method for solving the coordination coordination problem of interconnected interconnected systems with we systematically pursue the synthesis of the controller. the problem of systems with Section 77 is devoted to the implementation of the concept multiple decision makers. Andreasson et al. (2015) adopts devoted to the implementation of the concept multiple decision makers. Andreasson et al. (2015) adopts we systematically pursue the synthesis of the controller. Section 77 is is devoted to the implementation of the concept the coordination problem of interconnected systems with Section multiple decision makers. Andreasson Andreasson etspace al. (2015) (2015) adopts Section is devoted to the implementation of the concept and simulation tests. The last section presents the conclumultiple decision makers. et al. adopts constraints on trajectories in time and that are imand simulation tests. The last section presents the concluconstraints on trajectories in time and space that are imSection 7 is devoted to the implementation of the concept and simulation tests. The last section presents the conclumultiple decision makers. Andreasson etspace al. (2015) adopts constraints on trajectories in time and that are imand simulation tests. The last section presents the conclusions. constraints on trajectories in time and space that are imposed dynamically on the system. Some works, as Basilico sions. posed dynamically on the system. Some works, as Basilico and simulation tests. The last section presents the conclusions. constraints on trajectories in timeSome and space that are im- sions. posed dynamically on the system. works, as Basilico posed dynamically on the system. Some works, as Basilico et al. (2009), apply Leader-Follower strategy for planning et al. (2009), apply Leader-Follower strategy for planning sions. posed dynamically on the system. Some works, as Basilico et al. al. (2009), (2009), apply Leader-Follower strategyoffor forprioritized planning 2. PROBLEM et apply Leader-Follower strategy planning mobile robot patrolling. Useful application 2. PROBLEM PROBLEM STATEMENT STATEMENT mobile robot patrolling. Useful application prioritized 2. et al. (2009), apply Leader-Follower strategyof planning mobile robot patrolling. Useful application offorprioritized prioritized 2. PROBLEM STATEMENT STATEMENT mobile robot patrolling. Useful application of control can be found in Cap et al. (2015). Decentralized control can be found in Cap et al. (2015). Decentralized 2. PROBLEM STATEMENT mobile robot patrolling. Useful application of prioritized We consider a Multiple Mobile control can be found in Cap et al. (2015). Decentralized Robot System System (MMRS) (MMRS) control canfor be collision found inavoidance Cap et al.are (2015). Decentralized algorithms proposed in Ferrera We consider consider aa Multiple Multiple Mobile Mobile Robot Robot algorithms for collision avoidance are proposed in Ferrera We System (MMRS) control canfor be collision found inavoidance Cap et al.are Decentralized algorithms for collision avoidance are(2015). proposed in Ferrera Ferrera We consider a Multiple Mobile Robot System (MMRS) viewed as a group of autonomous mobile robots sharing algorithms proposed in et al. (2013) and Pallottino et al. (2007). Difficulty of the viewed as a group of autonomous mobile robots sharing We consider a Multiple Mobile Robot System (MMRS) et al. (2013) Pallottino et al. (2007). Difficulty of the viewed as aa Each group of autonomous mobile robots sharing algorithms forand collision avoidance are proposed in Ferrera et al. (2013) and Pallottino et al. (2007). Difficulty of the viewed as group of autonomous mobile robots sharing a 2D space. robot performs a mission that requires et al. (2013) and Pallottino et al. (2007). Difficulty of the system control that increases with density of robots in a 2D space. Each robot performs a mission that requires viewed as a group of autonomous mobile robots sharing system control that increases with density of robots in space. Each performs mission requires et al. (2013) andthat Pallottino et al. (2007). Difficulty of the system control increases with density of robots robots in aait 2D space. along Each aarobot robot performs mission that requires it 2D to travel travel along specific path. aaThe The pathsthat are planned planned system control that increases with density of in shared workspace is studied in Ferrera et al. (2017). to specific path. paths are a 2D space. Each robot performs a mission that requires shared workspace is studied in Ferrera et al. (2017). it to travel along a specific path. The paths are planned system control that increases with density of robots in it shared workspace workspace is studied studied in Ferrera Ferrera et al. al. (2017). (2017). to travel along a specific path. paths arepositional planned independently, without taking into account any shared is in et independently, without taking intoThe account any positional it to travel along a specific path. The paths arepositional planned independently, without taking into account any shared workspace is studied in Ferrera et al.methodologies (2017). The main contribution of these works are independently, without taking into account any positional constraints introduced by the paths of other robots. The The main contribution of these works are methodologies constraints introduced by the into pathsaccount of other other robots. The The main contribution contribution of robot these movement. works are are methodologies methodologies independently, without taking any positional constraints introduced by the paths of robots. The The main of these works of collision-free control of The resulting constraints introduced by the paths of other robots. The robots operate asynchronously and are able to control their of collision-free control of robot movement. The resulting robots operate asynchronously and are able to control their The main contribution of robot these movement. works are methodologies of collision-free control of The resulting constraints introduced by the paths of other robots. The robots operate asynchronously and to control their of collision-free control of robot movement. The resulting algorithms have, however, two major drawbacks: a) due robots operate asynchronously and are are able able toallow control their algorithms have, however, two major drawbacks: a) due motion with path-following algorithms that allow each of motion with path-following algorithms that each of of collision-free control of robot movement. The resulting algorithms have, however, two major drawbacks: a) due robots operate asynchronously and are able to control their motion with path-following algorithms that allow each of algorithms have, however, two major drawbacks: a) due to high computational complexity, they are not scalable, motion with path-following algorithms thatwhen allowalone each on of them to correctly accomplish its mission to high computational complexity, they are not scalable, them to correctly accomplish its mission when alone on algorithms have, however, two major drawbacks: a) due to high computational complexity, they are not scalable, motion with path-following algorithms thatwhen allowalone each on of them to correctly accomplish its mission to high computational complexity, they are not scalable, and b) they do not guarantee the correct operation of them to correctly accomplish its mission when alone on the stage. When sharing the motion space, the robots must and b) they do not guarantee the correct operation of the stage. When sharing sharing the motion motion space, the the robots must to computational complexity, they are not scalable, andhigh b) they they doterms not guarantee guarantee thethe correct operation of the them to correctly accomplish its mission when alone on stage. When the space, robots must and b) do not the correct operation of the system in of ensuring completion of the the stage. When sharing the motion space, the robots must refine their motion strategies in order to avoid collisions. In the terms of ensuring the completion of the refine theirWhen motion strategies in order orderspace, to avoid avoid collisions. In and system b) theyin not guarantee operation of refine the system indoterms terms of ensuringthethe thecorrect completion of the the the stage. sharing the motion the collisions. robots must their motion strategies in In the system in of ensuring completion of robot missions. The ineffectiveness of these models in refine theirwe motion strategies in modifications order to to avoid collisions. In this work, we assume that such modifications only concern robot missions. The ineffectiveness of these models in this work, assume that such only concern the system in terms of ensuring the completion of the robot missions. missions. The The ineffectiveness ineffectiveness of of these these models models in in this refine their motion strategies in order to avoid collisions. In work, we assume that such modifications only concern robot  this work, we assume that such modifications only concern the robot velocity profile, whereas their planned paths This work work was supported supported by the the National National Science Science Center grant no. no.  the robot we velocity profile, whereas their planned planned paths robot missions. The ineffectiveness of these models in the This was by Center grant  this work, assume that such modifications only concern robot velocity profile, whereas their paths This work was supported by the National Science Center grant no.  This work was supported by the National Science Center grant no. the robot velocity profile, whereas their planned paths remain unchanged. 2016/23/B/ST7/01441. remain unchanged. 2016/23/B/ST7/01441.  the robot velocity profile, whereas their planned paths remain unchanged. 2016/23/B/ST7/01441. This work was supported by the National Science Center grant no. remain unchanged. 2016/23/B/ST7/01441. remain unchanged. 2016/23/B/ST7/01441. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Copyright 2019 Copyright © 2019 IFAC IFAC Peer review© under responsibility of International Federation of Automatic Control. Copyright © Copyright © 2019 2019 IFAC IFAC 10.1016/j.ifacol.2019.08.093 Copyright © 2019 IFAC

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The overall goal of this paper is a methodology of control synthesis for MMRS, that ensures provably correct and efficient co-operation of the robots. The notion of correctness is a qualitative criterion and requires that each robot be able to complete its mission without colliding with other robots. That is, depending on the state of other robots, a robot may have to slow down or even come to a stop and wait until the situation changes and it can safely resume further travel. Thus, the correct control must also ensure for each robot, the possibility to resume its travel after a break, i.e., eliminate from the MMRS behavior such phenomena as deadlocks and robot starvation. The induced modifications of the robot velocity inevitably cause an increase of their mission completion time, thus possibly changing the value of global indicators of MMRS efficiency, e.g., the time of all task completion C = Cmax = max∈1,...,n Ci . These measures can differ depending on the ’slow-down’ strategy, as well as the policy deciding in the case of conflict occurrence, which robots should slow down and which robots can progress without changing their motion profile. Consequently, there are two main questions driving the development of such a control. (1) How to coordinate the asynchronous motion of the robots so that the following requirements be satisfied: (a) all the robots will accomplish their missions in a finite time interval [0, T ], (b) at any moment t ∈ [0, T ] the areas occupied by any arbitrarily given pair of robots are disjoint. (2) How to induce, within the admissible (i.e. observing requirements (1.a) and (1.b)) robot concurrent operation, efficient MMRS behavior. As can be noticed, the control satisfying requirements (1.b) and (1.a) ensures, respectively, collision-free and deadlockfree concurrent motion of the robots. Requirement (2) implies the need of a flexible model of MMRS control, that leaves room for the research towards optimization of system efficiency, and of tools to carry it out. To achieve realization of these postulates, we propose a hierarchical, hybrid control system that consists of four control levels: • • • •

concurrency control (DES model), priority control (DES model, timed DES model), mode control (DES model), motion control (CTS model)

The operation of these subsystems is interelated and they are assumed to play the following roles. The concurrency control provides the logic of the admissible progress of each particular robot on its path, relating it to its own and the other robots’ state. The priority control provides a policy for conflict detection and resolution by deciding which robots should modify their velocity in the case when further progress of two or more robots in the initially assumed motion mode is inadmissible. The mode control defines a set of robot motion modes (e.g., the initially assumed control, acceleration, deceleration, standstill), and a state-based logic of their application, that gives each robot the possibility to modify its velocity in order to observe the progress-admissibility conditions set at the concurrency control level. Finally, the motion control is a set of CTS robot control algorithms that induce the motion of a robot in the modes defined by the mode control. 453

position sensor

ap,cp,rp

453

motion control

robot A1

ap,cp,rp

position sensor

motion control

robot An

motion mode

mode control

motion mode

mode control

next

next

e Δ(s) e {ap,cp,rp}

s'=f(s,e)

ξ-MMRS=(S,E,Γ,f,s0,sM)

feasible events safety

E'=Γ(s)

check

admissible events priority

Γα(s) Γ(s)

control

selected events

Δ(s)

concurrency control model synthesis

ξ-MMRS=G(P,ξ)

MMRS SUPERVISOR

parameters (P,ξ)

Fig. 1. Architecture of the considered controller. Whereas development of the logic that formally guarantees correct MMRS control is feasible and has been done in our earlier works, analytical optimization of MMRS efficiency is a hard problem due to the very high complexity of the system. An essential element of the latter process is elimination of the unnecessary robot waiting through maximization of the permissiveness level of the admissibility conditions set at the concurrency control level. However, the examination of the impact of other control solutions, in particular prioritization policies, on quantative measures of MMRS behavior can only be done through simulation experiments. The main contribution of this work is a comprehensive methodology of the construction of MMRS control and its experimental implementation in the form of the algorithms and code of the controller that can be coupled with both, a simulated and a real MMRS. The modular structure of the control concept, reflected in the structure of the controller, allows for its modification and experimental optimization. Figure 1 presents the architecture of the considered controller, and its component blocks are discussed in the subsequent sections. 3. CONCURRENCY CONTROL As mentioned in the previous section, the objective of the concurrency control is to provide the logic that observes Requirement (1). The concept underlying this control level assumes a breakdown of the robot paths into sectors and imposing certain state-dependent constraints on the robot transitions between these sectors (Roszkowska (2005)). More specifically, each robot Ai , is abstracted by a disc ai large enough to cover the robot in any of its configuration. Then, to ensure that such disks never overlap (and thus satisfy Requirement (1.b)), we establish a conflict relation between robot sectors, defined as follows.

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Definition 1. Consider a set of n robots and let Pi = {pki | k = 0..mi } bethe set of consecutive path sectors of robot n Ai , and P = i=1 Pi . The conflict relation is given by the set of pairs ξ ⊆ P × P such that ∀i, j = 1..n, ∀k = 0..mi , ∀l = 0..mj , (pki , plj ) ∈ ξ iff i = j and dmin (pki , plj ) < ρi +ρj . That is, sectors pki and plj are in conflict if and only if they belong to the paths of different robots and the minimal distance between them is less than the sum of the radiuses of their disks. Definition 1 directly implies the following property. Property 1. The disks of any two robots will never overlap if their conflict sectors are used in the mutually exclusive way, i.e., for each conflict pair (pki , plj ) ∈ ξ, it is ensured that only one of the sectors can be occupied at a time. In the considered MMRS, it is assumed that the first and the last sector of each robot’s path are private, that is, they are not in conflict with any sector pki ∈ P . Then, based on Property 1, we can define the following deterministic finite state automaton (DFSA) model of MMRS. Definition 2. Consider a set of n robots, their sector set P and conflict relation ξ. ξ-M M RS = (S, E, Γ, f, s0 , SM ) is a DFSA such that: (1) S = {s = (s1 , s2 , ..., sn ) | si ∈ 0..mi } is the set of states. The components si , i = 1..n, of each vector s are the numbers of the path sectors currently occupied by the respective robots Ai . (2) E = {ei : i = 1..n} is the set of events. Event ei represents the transition of robot Ai to the next path sector. i (3) Γ(s) = Γ(s1 , s2 , ..., sn ) = {ei : si = pm ∧ ∀j = i si +1 sj 1..n, (pi , pj ) ∈ ξ} is the feasible-event function that returns the subset of events that can occur in state s. Event ei is in Γ(s) if robot Ai can transit to its next path sector pisi +1 , i.e., psi i is not its last sector and pisi +1 is not in conflict with any of the sectors currently occupied by the other robots. (4) f (s, ei ) = f ((s1 , s2 , ..., sn ), ei ) = (s1 , ..., si−1 , si + 1, si+1 , ..., sn ) is the partial next-state function, only defined for events ei that are feasible in state s, i.e., for pairs (s, ei ) ∈ Γ(s). (5) s0 = (0, 0, . . . , 0) is the initial state, representing the situation when all the robots are in their initial sectors. (6) SM = {sm }, where sm = (m1 , m2 , . . . , mn ) is the marked state, representing the situation when all the robots are in their final sectors. The operation of ξ-M M RS satisfies Requirement (1.b), thus ensuring collision-free concurrent motion of the robots. However, Requirement (1.a) is not generally satisfied as the proposed model is not deadlock free, i.e., can include states from which the final state cannot be reached. In order to ensure the correct operation of the system, it is necessary to to substitute the feasible-event function Γ(s) with a more restrictive admissible-event function that constrains the behavior of ξ-M M RS so that for each admissible event sequence, there exists its admissible extension driving the system to the final state. The proposed solution employs the notion of an ordered state, defined as follows. 454

Definition 3. In system ξ-M M RS, state s = (s1 , . . . , sn ) is α-ordered iff there exists an order on the set of robots A, α : A → {1, 2, . . . , n} that satisfies the following condition: s ∀i, j s.t. α(Aj ) > α(Ai ), ∀k = si ..qi , (pki , pj j ) ∈ ξ, where qi is the smallest number s.t. qi ≥ si and sector pqi i is private. Property 2. In ξ-M M RS, the final state can be reached from any state s that is α-ordered. Proof. The condition defining the α-ordered state provides the robots Ai , i = 1..n, with the ability to progress one-byone, in the order given by α(Ai ), to their respective closest private sectors pqi i . Since in the state s = (q1 , . . . , qn ) no robot occupies a sector that is in conflict with any sector to be further travelled by any robot, the robots can, one by one, complete their missions. Definition 4. α-controlled ξ-M M RS is an extension of the automaton defined in Def. 2, obtained by: • substituting the feasible event function Γ(s) with admissible event function Γα (s) = {e : e ∈ Γ(s) ∧ s = f (s, e) is α-ordered}, and • substituting the transition function f with f α such that f α (s, e) = f (s, e), but it is only defined for admissible pairs (s, e), i.e., such that e ∈ Γα (s). The following theorem proves that the operation of αcontrolled ξ-M M RS satisfies Requirement (1.a). Theorem 1. In α-controlled ξ-M M RS, the final state sm is reachable from each state s reachable from the initial state s0 . Proof. Based on Definition 4, each state reachable in αcontrolled ξ-M M RS is α-ordered. Thus, by Property 2, the theorem holds. Testing the admissibility of a feasible event ei in state s can be done with a polynomial-complexity algorithm, which allows its application in the control of MMRS. 4. PRIORITY CONTROL As depicted in Figure 1, the transition of robots between the sectors is controlled by the MMRS supervisor through generating events next, granting the respective robots the permission to proceed to the next sector. To make such decisions, the supervisor must follow the state of the robots, updating it at the occurence of each event that is sent/received to/from any robot, check the admissibility conditions established in Section 3, and resolve conflicts among robots using some priority policy. Prioritization of the robots is necessary to decide which of them should be allowed to continue their motion, and which should be temporarily suspended in the case when concurrent operation of all of them is not admissible. It should be noted that to establish the admissibility conditions in Section 3, it was sufficient to view the change of a sector by a robot as a single event and, consequently, abstract the state of MMRS as a distribution of the robots in the path sectors. However, the supervision of the robot motion requires the recognition of also transient states, corresponding to the movement of a robot from some sector pij to the next sector pi,j+1 . A transient state of robot Ai starts with the occurrence of event next, a

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permission of the supervisor sent to robot Ai to let it enter sector pj+1 , and ends with the occurrence of event i rp, sent by the robot to the supervisor, when its disk stops overlapping sector pj+1 . As during this period, the disk i , it is assumed that both overlaps both sectors, pji and pj+1 i sectors are occupied by Ai , whose state is represented then by the pair of consecutive sector numbers si = (j, j + 1). In other words, when issuing event next, the supervisor allocates the next sector, pj+1 , to robot Ai , and having i received signal rp from Ai , the supervisor deallocates sector pji from the robot. According to the model established in the previous section, a transition of robot Ai to the next section pj+1 at state i s is only admissible if the corresponding event ei ∈ Γα (s), is occupied that is if no sector that is in conflict with pj+1 i and the potential next state s = f (s, ei ) is α − ordered (which ensures, respectively, no collisions and no deadlocks among the robots). As this rule is time-independent, the allocation of the next sector can take place at any time during the travel of robot Ai in the previous one. However, from this moment, sector pj+1 becomes considered as i occupied by Ai , which will block the access of other robots to the sectors that are in conflict with pj+1 . i Thus, it sounds reasonable to allocate a sector to a robot only then when the robot is approaching it. This is signaled with an event ap sent by the robot to the supervisor when a certain point at the robot’s path is reached. Consequently, the pool of robots that can receive the permission to continue their motion (by allocation to them their respective sectors) consists of all the robots Ai , whose transition to the next sector is admissible, i.e., ei ∈ Γα (s), and that are sufficiently close to the border of their current sector. As the set Γα (s) may contain events that are in mutual conflict, i.e., such that the occurrence of one of them disables the other, all of the events in the pool cannot be activated at the same time, and there is a need of some mechanism to decide about their activation order. To do that, we employ a policy that, based on some heuristic rule, selects a maximal subset of non-conflict admissible events, that is, a a set ∆(s) ⊆ Γα (s), such that: (i) if Ai is in a transient state then ei ∈ ∆(s) (ii) each pair of events ei , ek ∈ ∆(s), is non-conflict, that is, robots Ai and Ak can transit to their next sectors concurrently (iii) ∆(s) is a maximal subset of non-conflict admissible events, that is, for each event ei ∈ ∆(s), there exists event ek ∈ Γα (s) that is in conflict with ei .

The set ∆(s) is recalculated at each state change, and command next is sent to all the robots Ai that have passed their approaching point ap and ei ∈ ∆(s).

In this paper, we do not provide any exact algorithms to calculate ∆(s), but Section 7, briefly describes three rules applied for selecting events to be included in this set during the simulation experiments. 5. MODE CONTROL Implementation of the established logics raises the problem of how to prevent a robot from entering a new sector when it is has not been allocated to the robot. As mentioned 455

455

ρ

ap

cp

rp

Fig. 2. A path sector with its characteristic points. rp

regular to ap

ap

regular to cp next

regular to rp

cp

decel.

v=vmax next

accel.

v=0 next standstill

Fig. 3. Robot motion-mode control. in the previous section, the proposed solution assumes the employment of a central supervisor that interacts with the robots on a DES-based scheme. The supervisor generates only one event, next, and the robots inform the supervisor about reaching three characteristic points on their sectors: ap, cp, and rp (Fig. 2). Fig. 3 depicts an automaton modeling the robot motion-mode control during its travel in a sector and transition to the next one. We distinguish four motion modes: regular, deceleration, acceleration, and standstill, as well as three phases of the regular motion: regular-to-ap, regular-to-cp, and regularto-rp, corresponding to the travel of the robot to point ap, to point cp, and to point rp, respectively. In the initial or the final state, and while waiting for the permission to enter the next sector, the robot remains in the standstill state. Occurrence of event next allows it to start moving and results in the transition to state acceleration. The robot remains in this state until the velocity v = vmax , moving in the meantime to the next sector. As point rp is located at a distance ρ from the sector border, when reaching it, the robot occupies no more the previous sector, and communicates it to the supervisor. This brings the robot to state regular-to-ap. The event associated with reaching point ap brings the robot to state regular-to-cp and is a signal for the supervisor that the robot is approaching its critical point cp, where it has to start decelerating if not allowed to transit to the next sector. If the event next occurs before the critical point cp is reached, it changes the state to regular-to-rp, and the robot proceeds to the next sector. Otherwise, at point cp, the robot changes its motion mode to deceleration and starts awaiting signal next, which can occur after the robot has come to the standstill or still when decelerating. In both cases, having received the next signal, the robot starts accelerating and heads to the next sector.

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y0

A8

θ y

A3

A5

A1

x

x0

Fig. 4. Graphical representation of the monocycle. 6. MOTION CONTROL The correct operation of the considered MMRS requires the robots to follow their assumed paths in the three distinguished motion modes - regular motion, acceleration, deceleration, and temporary standstill, which can be enforced by using adequate programs provided for real robots or specialized simulation platforms. An alternative approach is a specific design of a multimode controller for a class of robots assumed for MMRS, or modification of an existing control algorithm so that it can be used in the considered three control modes. For example, Fig. 4 depicts a mobile robot of class (2, 0), a unicycle with nonholonomic wheel slipping constraints. The state of the unicycle is described by three state variables: position of the central point with respect to the x and y axes, and the angle θ, which describes the orientation of the platform in the global coordinate system: qm = (x, y, θ) (Fig. 4). Then two control variables u = (v, ω) are assumed, where v is the linear velocity and ω is the angular velocity of the platform. The kinematic model of such robot is described by the following equation:       x˙ cos θ 0      q˙m = y˙ = sin θ v + 0  ω 0 1 θ˙ A well known solution of the path following problem for a robot described with the above formulated kinematics is the Samson controller, described by the following formula Micaelli and Samson (1992).  vr = v(t) c(s) θ˜ ωr = −k2 lvr sin − k3 θ˜ + vr cosθ˜ 1+c(s)l θ˜ where: • k2 , k3 - amplification of the algorithm, determined experimentally, • θ˜ - robot orientation tracking error, defined in Frenet coordinates (Mazur (2004)) • l - distance between the robot center and the path, defined in Frenet coordinates, • c(s) - the curvature of curve c, derived from the robot path. The above given formula describes the motion of the robot in its regular mode, and further research is required on its modification towards the representation of the remaining 456

A2

A4

A6

A7

Fig. 5. Path layout of the robots used in the simulation experiment. three motion modes. It goes, however, beyond the scope of this paper. Formal establishment of such a control makes it possible to simulate more accurately the motion of each particular robot in time, thus to identify the moments of occurrence of paricular events, motion mode changes, and eventually the time behavior of the whole MMRS, necessary for the evaluation of its performance. 7. IMPLEMENTATION AND EXPERIMENTS The hierarchical control for MMRS proposed in this paper was implemented in the ROS/Stage environment using the kinematic modeling support offered by the system. The simulation parameters were similar to those characterizing the laboratory system of turtlebots, where the considered control concept is currently being implemented. The motion space was assumed to be 19 m long and 16 m wide, and the maximal velocity of the robots was 0.65 m s . The robots were assumed to do a specified number of laps on their respective circular paths, shown in Figure 5 together with their partitioning into sectors. The sectors depicted in the same color are in conflict one with another, whereas the remaining, not colored sectors, are non-conflict. The main objective of the simulation was to generate an animation illustrating the concurrent motion of the robots under the discussed control, and to demonstrate the possibility of influencing MMRS performance through different priority control policies. The assumed performance criterion was the makespan, i.e., the time required for the three concurrently driving robots to complete a specified number of laps. We implemented three priority rules, PR1, PR2, and PR3. PR1 selects the set ∆(s) with the maximal number of elements. PR2 applies the same rule, but additionally takes into account the distances to their nearest private stages of the accepted robots, i.e., robots Ai such that ei ∈ ∆(s). It is inspired by the intuitive assumption that the more time the robots spend

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in their private sectors, the longer is the time when they travel concurrently, and hence the shorter is the time when all the robots complete their tasks. The third rule, PR3, selects the set of events ∆(s) that maximizes the sum of the remaining distances to the end of the path of all accepted robots Ai . The simulation was performed six times for each of the described priority rule. The measured makespans and their mean values are presented in Table 1. As can be noticed, the makespan with the shortest time was obtained for PR3. A video demonstrating the concurrent robot motion under this priority rule can be found at https://www.dailymotion.com/video/x75lpsu or can be obtained from the authors at email request. The animation shows that no collisions or deadlocks occur among the robots, which could have been expected as these features are formally ensured by the model. Yet, it can also be observed that a further optimization of the concurrent robot motion towards shortening the makespan is still possible. Table 1. Simulation results. Criterium 1

Criterium 2

Criterium 3

Time [s]

210.06 202.02 215.91 208.62 193.12 194.44

200.56 192.92 173.54 173.73 185.84 202.07

175.10 159.90 154.19 175.88 164.56 178.67

Mean [s]

204.03

188.11

168.08

8. CONCLUSION In this paper we presented a modular, hybrid control system for mobile robots sharing a common 2D motion space. The concept underlying its construction assumes discretization of robot paths and control of robot transitions between path sectors. This involves two types of DES-based supervisors, concurrency control and mode control, issuing for the robots permissions to enter the next sector, and requests to change its motion mode, respectively. The supervisors operate in an asynchronous feedback loop. The control decisions of each supervisor are calculated based on the state of its DES model, which is updated as a result of events occurring in the real MMRS. The assumed control architecture consists of a central concurrency controller and multiple mode controllers, each associated with one particular robot. However, the developed logic allows the full spectrum of its implementation, from the system where mode-change decisions are made centrally, to the system where each robot has a copy of the concurrency control model and updates its state through communication with other robots. Further work assumes, among others, automatization of the process of path partitioning into sectors and calculation of the conflict relation, as well as the analysis of the relation between the sector structure and the system efficiency. Research into more advanced efficiency optimization, through adaptation of Operation Research methods to on-line scheduling, is also planned.

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REFERENCES Andreasson, A., Bouguerra, A., Cirillo, M., Dimitrov, D., Driankov, D., Karlsson, L., Lilienthal, A., Pecora, F., Saarinen, J., Sherikov, A., and Stoyanov, T. (2015). Autonomous transport vehicles: Where we are and what is missing. IEEE Robotics and Automation Magazine, 22(1), 64–75. Basilico, N., Gatti, N., and Amigoni, F. (2009). LeaderFollower Strategies for Robotic Patrolling in Environments with Arbitrary Topologies. (Aamas), 10–15. Cap, M., Novak, P., Kleiner, A., and Selecky, M. (2015). Prioritized Planning Algorithms for Trajectory Coordination of Multiple Mobile Robots. IEEE Transactions on Automation Science and Engineering. Ferrera, E., Capit´an, J., Casta˜ no, A., and Marr´ on, P. (2017). Decentralized safe conflict resolution for multiple robots in dense scenarios. Robotics and Autonomous Systems. Ferrera, E., Casta˜ no, A., Capit´an, J., Ollero, A., and Marr´on, P. (2013). Decentralized collision avoidance for large teams of robots. In 2013 16th International Conference on Advanced Robotics, ICAR 2013. Inalhan, G., Stipanovic, D.M., and Tomlin, C.J. (2002). Decentralized optimization, with application to multiple aircraft coordination. Proc. of CDC’02. IEEE, 1147– 1155. Lygeros, J., Godbole, D.N., and Sastry, S. (1998). Verified hybrid controllers for automated vehicles. IEEE Transactions on Automatic Control, 43, 522–539. Mazur, A. (2004). Hybrid adaptive control laws solving a path following problem for non-holonomic mobile manipulators. International Journal of Control, 77, 1297–1306. Micaelli, A. and Samson, C. (1992). Path following and time-varying feedback stabilization of a wheeled robot. In Proc. of ICARCV’92. Pallottino, L., Scordio, V.G., Bicchi, A., and Frazzoli, E. (2007). Decentralized cooperative policy for conflict resolution in multivehicle systems. IEEE Transactions on Robotics, 23, 1170–1183. Pecora, F., Cirillo, M., and Dimitrov, D. (2005). On mission-dependent coordination of multiple vehicles under spatial and temporal constraints. Proc. IEEE/RAS Int. Conf. on Intelligent Robots and Systems, 5262– 5269. Reveliotis, S. and Roszkowska, E. (2011). Conflict resolution in free-ranging multivehicle systems: A resource allocation paradigm. IEEE Transactions on Robotics, 27(2), 283–296. Roszkowska, E. (2005). Provably correct closed-loop control for multiple mobile robot systems. Proc. of ICRA’05. IEEE, 2810–2815. Roszkowska, E. and Reveliotis, S. (2013). A distributed protocol for motion coordination in free-range vehicular systems. Automatica, 49(6), 1639–1653. Tomlin, C., Pappas, G.J., and Sastry, S. (1998). Conflict resolution for air traffic management: a study in multiagent hybrid systems. IEEE Transactions on Automatic Control, 43, 509–521. Valle, S.M.L. and Hutchinson, S.A. (1998). Optimal motion planning for multiple robots having independent goals. IEEE Transactions on Robotics and Automation, 14, 912–925.