Real Time Man-Robot Control of a Group of Specialized Mobile Robots

Transcription

1 БЪЛГАРСКА АКАДЕМИЯ НА НАУКИТЕ BULGARIAN ACADEMY OF SCIENCES ПРОБЛЕМИ НА ТЕХНИЧЕСКАТА КИБЕРНЕТИКА И РОБОТИКАТА, 65 PROBLEMS OF ENGINEERING CYBERNETICS AND ROBOTICS, 65 София 2012 Sofia Real Time Man-Robot Control of a Group of Specialize Mobile Robots Vassil Sgurev, Stanislav Drangajov, Lyubka Doukovska Institute of Information an Communication Technologies, 1113 Sofia s: srangajov@gmail.com l.oukovska@mail.bg Abstract: A man-robot system for fire extinguishing is iscusse in the paper. Mobile robots have achieve wie application in ifferent areas of material prouction an social practice. Different tasks have arisen which have lea to wiening of the application area of mobile robots. Their application is of great importance in cases of natural isasters, an especially in inustrial accients, such as fires, environmental pollutions an other aggressive environment, in which the presence of a human operation near the outbreaks of the isaster is angerous an in many cases even impossible. In many cases the mobile robots act together on the same site, so that they must be controlle an manage in the proper way. No oubt, there must be a human operator, coorinating an supervising the robots actions. Keywors: Fire fighting robots, robots optimal path control, network flow algorithms. 1. Introuction Mobile robots have obtaine wie application in the last two ecaes in material prouction in various areas [4, 5, 6]. Different problems have appeare which have greatly enlarge the area of mobile robots application. An important place in them take some applications relate to operation in aggressive environment in situations of isasters, floos, fires where ue to the high risk it is ifficult for a human to penetrate. 3

2 The task of using mobile robots in high temperature zones is of consierable importance for carrying out fire extinguishing activities. In such cases it is often necessary to use not one, but a group of mobile robots. They carry out joint coorinate activity to perform the necessary fire fighting functions at ifferent places on the site where fire outbreaks are locate. The task arises of real time operative control of the fire fighting robots in a group which circulate between the sources of the fire extinguishing agent an the burning objects. The movement of the mobile robots is suppose to be carrie out along preefine sections which are a part of a path network of known configuration. Besies this, it is necessary mobile robots to be loae with fire extinguishing agents ifferent in consistency an composition. Depening on the sequence of transportation of this agent by the mobile robots it is necessary that the requirements of averaging the composition of the fire extinguishing agent to be observe. As the time for extinguishing the fire outbreaks is limite, it is necessary that the fire fighting robots move along separate paths so that their total run is minimize, which in its turn results in maximization of the supplie fire extinguishing agents. It follows from this that the combine control of the mobile robots group shoul be carrie out on the basis of simultaneous observation of the following two criteria: a) minimization of the total run of the mobile robots on the roa network, which results in maximization of the transporte fire extinguishing agent for the same perio of time; b) keeping the requirement that the composition of the fire extinguishing agent is not out of the amissible bounaries, i.e., effective averaging to be carrie out for the agent composition. It shoul be taken in min that the mobile robots movement is of efinitely stochastic character so that it cannot be performe by a provisional scheule which is strictly observe. The consierable stochastic character requires precise information about the location of each mobile robot an this is one best by using local or global positioning systems. With such layout each time a mobile robot reaches the en of a path section which bifurcates, the control system computes the new aresses on the path network for the mobile robots on the basis of existing information an control criteria. 2. The roa network an control methos In the present work approximate an exact methos are propose for solving the problem escribe, base on the network flow interpretation of the current state of the group of controlle mobile robots on the transportation network [2, 3]. The first one of these methos allows almost optimal control to be achieve by using the metho of the relative network flows. This metho is of polynomial computational complexity. The secon metho is exact an the result is optimal control, but through algorithms of Integer Linear Programming (ILP), which are of pseuo polynomial computational complexity an require more computational time a memory [2]. 4

3 Further on we will use the graph theory terminology of the network flows without explicitly efining it [1]. If the mobile robot moves along a path section from its initial noe x i to noe x j along arc x ij of length (time, istance) l ij, a new, artificial (fictitious) noe x k is introuce with a new arc x kj an length l kj which exactly correspons to the istance of this mobile unit to the en noe x j on the section of the network. Besies, l kj l ij. A general roa network is conitionally shown in Fig. 1, with three loaing points of fire extinguishing agent T = {x 1, x 2,..., x t }, two fire outbreaks S = {x t+1,..., x s }, five intermeiate points on the network R = {x s+1,..., x r } an five artificial noes on the network, each one corresponing to a mobile fire fighting robot D = {x r+1,..., x }. The noes of the sets T, S, R, D are enote by the following inices: I t = {1, 2,..., t}; I s ={t+1,..., s}; I r = {s+1,, r}; an I = {r+1,..., }. Numbering of {x i } in Fig. 1 is one in the same way. The lengths (times, values) of the corresponing arcs are encircle. T x 1 x 2 x x 8 x x 13 x x x 4 S x 3 x x 7 4 x 5 x 10 x 14 Fig. 1 If the Shortest Routes (SR) to each one of the fire outbreaks are apriori etermine an encircle in the figure, an some technological constraints are presente on new arcs between noes D an T, then the reuce network, shown in Fig. 2 will be obtaine. In orer to efine a network flow on it, an aitional noe x 0 with arcs x 4,0 an x 5,0 with null values of the lengths l 4,0 an l 5,0 is neee. x 15 x x x 11 3 y 3 y 4 x 14 y 8 y5 y 2 y 7 y 1 9 y 6 Fig. 2 x 4 y 9 y 10 x 5 S x 0 5

4 It is suppose that only one type of fire extinguishing agent is loae at each supply point that is ifferent from the other points. The following enotations will be introuce: X = {T S D D}, X'= {D S {x 0 }}; I = {I t I s Ι r Ι }, I'= {I I s {0}}; f ij = f (x i, x j ) an arc flow function on the arc x ij ; Г 1 an Г 1 irect an inverse mapping on the graph; M ij the set of the inices of all routes {μ} with an initial noe x i an a final noe x j ; L ij = min l (μ) the shortest istance (time, value) between noes x i an x j, μ M ij at that L ij = 0 if there oes not exist any route or arc from x i towars x j ; (1) U'= U 1 U 2, where U 1 = {x ij / L ij > 0, i I, j I s }; U 2 = {x ij i I s, j = {0}}; G' (x', U') a reuce network; I(i, j) the set of the pairs of inices (i, j) on all arcs {x ij }, belonging to the reuce network; k i (t) relative loaing of the mobile robots with fire extinguishing agent content from points with inex i I t ; (2) 0 k i (t) 1; k j (t)= 1, i I t ; j I t k i (s) planne relative content of the fire extinguishing agent at the fire outbreak with inex i I s (3) 0 k i (s) 1; k j (s)= 1, i I s ; j I s k i () relative content of the fire extinguishing agent on a loae mobile robot with inex i I, (4) k i () { k i (t)}, i I ; j I t k 1 i() relative content of the fire extinguishing agent in a loae fire fighting mobile robot i I with respect to other loae mobile robots, at that (5) k 1 k ( ) i()= i, i I ; j I k ( ) k 2 i() normalize relative content of the extinguishing agent in a loae mobile robot with inex i I with respect to other loae mobile robots, 1 (6) k 2 k i ( ) i()=, i I 1 ; max k ( ) j I j i 6

5 g ij maximal ensity of the flow between noes x i an x j, which accounts the istances (times, values) between these two noes an the normalize relative content of the extinguishing agent in a loae mobile robot with inex i I, 2 k i ( ) (7) g ij =, i I, j I s ; Lij g(i) maximal ensity of the flow from mobile robots with inex i I towars the closest one among fire outbreaks S, where (8) g (i)= max g ij, i I ; j I s h(i) the maximal possible relative flow of the fire fighting robots with inex i I towars the closest fire outbreak points in S, (9) h(i) = g( i), i I ; g( j) j I h ij the maximal possible relative flow of mobile robots with inex i I towars fire outbreak points with inex j I s g ij (10) h ij = h(i), i I, j I s ; g i Ι 1 I the set of the inices of the mobile robots alreay aresse to any fire outbreak points from S; q(j) the sum of the relative flows of the mobile robots with inices from Ι 1 alreay aresse to fire outbreak point x j S, (11) q (j) = max h i,j, j I s, i Ι \Ι' where in case Ι 1 = I, q (j) = 0; ξ (j) the ifference between the planne relative content of the fire extinguishing agent towars a fire outbreak with inex j I s an the relative flows towars the same fire point (12) ξ (j) = k j (s) q (j), j I s ; ξ (j*) the maximal value of the values ξ (j) (13) ξ (j*) = max ξ (j); j I s h (i*) a value inicating the relative flow at optimal aressing of the mobile robots with inex i* I towars fire outbreak points with inex j* I s, at that (14) h (i*) = max h i,j an j = j*. i Ι \Ι 1 The upper an lower bouns of the amissible fluctuation of the total content of the fire extinguishing agent at the fire outbreak places is efine by the following inequalities: 7

6 i I (15) k j (s) ε m I n I k ( ) f s i k m ij ( ) f m, n k j (s) + ε, j I s, where ε is the amissible eviation of the corresponing fire extinguishing agent from the planne content at x j S. Besies this, if z ij enotes the volume of the fire extinguishing agent content from i-th point irecte towars j-th fire outbreak point, then quite often aitional technological constraints arise zij (16) w ij w w ij + w, i I, j I s, m zmj I where {w ij } are coefficients epening on the current situation an w is a parameter inicating the probable fluctuations of the ifferent extinguishing agent consistencies at the fire outbreak points with respect to the planne average values. 3. Approximate metho The enotations introuce enable the proposal of the following approximate metho an proceure of the relative network flows for efficient aressing an control of the robotize transport processes, in which minimal transport istances are sought, simultaneously observing requirements (15) an (16) for average fire extinguishing agent contents at the fire sites. This iterative metho can be escribe by the following three steps: Step 1. Applying relations from (6) up to (10) the parameters {g ij }, {g(i)}, {h(i)}, {h ij } are efine an it is assume that I 1 =, where is the symbol of an empty set. Step 2. Using relations from (11) up to (14) the values {q(j)}, {ξ (j)}, {ξ (j*)} an h(i*) are compute an the mobile robot with inex i* I is aresse towars fire outbreak point with inex j* I s. With this it is irecte to the next section of the route, leaing to the respective final fire point. The following alterations are mae: k j* (s):= k j* (s) h i*j*, Ι 1 := Ι {i*}. Step 3. In case Ι = Ι 1 Stop, otherwise go to Step 2. All transportation units with inices in I are successively aresse with the help of this iterative proceure. Each time when an alteration occurs in the general situation, the complete iterative proceure is starte an successive new aressing of the mobile robots is realize. It is quite probable that the new aresses of some mobile robots may coincie with the preceing ones. 8

7 A numerical example emonstrating the operation of the metho of the relative network flows will be presente, using the reuce network in Fig. 2 an the ata from Tables 1, 2 an 3. Table 1 gives all the ata use to execute Step 1, incluing the values of the coefficients {k i (), {k 1 i()} an {k 2 i()}. Table 1 x ij L ij, g ij x 4 x 5 k i () k 1 i () k 2 i () g(i) h(i) L x ij 3 11 g ij x 12 L ij g ij x 13 L ij 9 11 g ij x 14 x 15 L ij 9 g ij L ij g ij Table 2 isplays ata for the parameters {g ij /g(i)}, {h ij }, as well as the final results after application of the iterative algorithm, inicating at what iteration a given mobile robot is irecte to any fire outbreak point. The solutions selecte are shaowe in the table. Table 2 x ij x 11 x 12 x 13 x 14 x 15 R S k jj (s) x 4 x g ij /g(i) 1 h ij 0.38 g ij /g(i) h ij g ij /g(i) h ij g ij /g(i) 1 h ij 0.12 g ij /g(i) h ij Aresses Iteration No x 4 1 x 5 2 x 4 3 x 5 5 x 4 4 Total Table 3 inicates the relative ensities {q(i)}, the planne relative extinguishing agent compositions {k i (s)} at every fire outbreak point, as well as the ifferences {ξ(j)} between the actual an the planne situation of the network. This table shows the ata from the first iteration of the algorithm propose when the mobile robot with an initial noe x 11 is irecte towars fire point x 4. In this case ξ(4) = 0.22, greater than ξ(5) = 0.21, i.e., a ecision must be mae for aressing towars a fire outbreak point with a bigger ifference, namely ξ(4). 9

8 10 Table 3 Parameter x 4 x 5 Sum along rows k j (s) q(j) ξ(j) As following from Table 2, the total run of all five transportation units is only 2% longer compare to the case when they move along the shortest routes of the network. This insignificant increase of the run is ue to the aitional requirements for averaging of the extinguishing agent composition in the liqui volume transporte. It follows from Tables 1 an 2 that accoring to the solution obtaine the average content of the extinguishing agent at the two fire outbreak points is 0.54 for fire outbreak point x 4, an 0.47 for point x 5 respectively. The values planne for the two fire points are 0.6 an 0.4 accoringly, i.e., the eviation is smaller than the amissible value ε = ± 0.1. The propose metho of relative network flows is insensible towars any probable altering in the number of the fire fighting mobile robots. This is ue to the fact that the comparisons between the planne an the actual inicators are accomplishe in relative units. The propose approximate iterative algorithm for fining almost optimal control is realize by a reasonable number of computing operations with polynomial complexity which is important for its computer realization in ispatchers systems. 4. Exact metho The optimal control of the mobile robots can be realize with the help of the suggeste exact metho that minimizes the total run of all mobile robots, keeping the requirements for averaging of the extinguishing agent composition in the transporte liqui volume. It uses the same enotations as the former metho of relative network flows. The metho is reuce to solving the following problem of integer network flow programming with aitional linear constraints referring to the reuce network presente in Fig. 3 [5]: (17) L = l min ij f ij ( i, j) I ( u) uner the constraints: for every i I 1if i I, (18) fij f ji = D if i= 0, 1 1 j Γi j Γi 0 in the remaining cases; (19) (k j (s) ε) km( ) fm, n ki( ) fij 0, j I s ; m I n Is i I 0, j I s ; (20) k ( ) f ( k ( s) + ε ) k ( ) f, i ij j m m n i I m I n Is

9 (21) f ij 0, (i, j) Ι (i, j), where functions {f ij } accept integer values only an D enotes the power of set D. From the equations of flow preservation it follows that (22) D = f,0. i Is The complementary linear constraints (19) an (20) are a irect sequence of requirements (15). 5. Solution results If the example given in Fig. 3 is consiere, using the output ata presente in Table 1, the problem presente by relations (17) up to (22) will take the form (23) L=3y y 2 + 9y y y y 6 + 9y 7 +14y 8 min uner the constraints: (24) y 9 y 1 y 2 y 3 y 4 = 0; (25) y 10 y 5 y 6 y 7 y 8 = 0; (26) y 1 = 1, y 2 + y 5 = 1, y 3 + y 6 = 1, y 7 = 1; (27) y 4 + y 8 = 1, y 9 + y 10 = 5; (28) 0.2 y y y y y y y y 8 0; (29) 0.06 y y y y y y y y 8 0; (30) 0.2 y y y 4 0.9; (31) 0.5 y y 7 0.8; (32) y i {0, 1}, i = 1, 2,, 8. Functions y 9 an y 10 accept integer values. The arc flow functions {y i i = 1, 2,, 10} corresponing to {f ij }, are presente in Fig. 3. The values of {l ij } are taken from the same figure. The inequalities (28) an (29) correspon to (19) an (20), the relations (30) an (31) to (16) an their coefficients {k j (s)} an {k i ()} are accepte from Table 1 an Table 3 respectively. It is assume that ε = ±0.1. Solving the example formulate in (22) up to (32), an optimal solution presente in Table 4 is obtaine. WebOptim software system is use for this problem solving [3]. Table 4 y i y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y 10 Fire point x x If in the pair (x i, x j ) the symbol x i enotes the number of the aresse mobile robot an x j the fire outburst point it is irecte to, then the optimal solution from Table 4 correspons to (x 11, x 4 ); (x 12, x 4 ); (x 13, x 4 ); (x 14, x 5 ) an (x 15, x 6 ). The solution with the help of the approximate algorithm gives the following aresses: (x 11, x 4 ); (x 12, x 5 ); (x 13, x 4 ); (x 14, x 4 ) an (x 15, x 5 ), i.e., the two solutions iffer in the aresses of the mobile units x 12 an x 15 only. i 11

10 If the istances values on the reuce network given in Fig. 2 are use, as well as the ata from Table 1, then for the exact algorithm the total run of the aresse mobile robots towars fire outbreak point x 4 is equal to 29 units, an towars x 5 to 23 units, i.e., 52 units of istance totally. In the solution obtaine by the approximate algorithm these inicators are 28 units towars x 4 an 24 units towars x 5, i.e., total run of 52 istance units as well. If the technological constraints are observe, in the exact algorithm the relative value of the transporte fire extinguishing agent towars fire outbreak x 4 is an towars x The approximate algorithm gives a solution of towars x 4 an towars x 5. The two solutions lea to equal ifference of ±0.667 with respect to the planne values that are 0.6 for x 4 an 0.4 for x 5 respectively. This is smaller than the amissible eviation of ±0.1. Hence, the two solutions obtaine by the exact an by the approximate algorithm, are optimal an equivalent. However, the approximate algorithm is of polynomial computing complexity, while the exact algorithm is of exponential computing complexity. This makes the approximate algorithm preferable in many cases. Of course, there might be situations, when the approximate solution oes not lea to optimal, but to almost optimal solutions, but even then the ifference woul be slight an negligible in practice. 6. Conclusion The present paper iscusses the problem of combine real-time control of mobile robots, specialize for operation in extremely high temperature environment, when it is necessary to put out fires. The criteria for such control are minimization of the total run of the mobile robots, as well as the necessity to average the extinguishing agent content in the liqui, transporte by the mobile robots. An exact network flow metho is suggeste for optimal aressing of the mobile robots observing the technological requirements towars the content of the extinguishing agent transporte by them. An approximate network flow metho is evelope that on the basis of relatively simple proceures of polynomial complexity obtains solutions, close to the optimal ones. A numerical example is presente illustrating the efficiency of the two methos propose for operative control of the transportation of fire extinguishing agent from the loaing points towars the fire outbreak points. R e f e r e n c e s 1. C h r i s t o f i e s, N. Graph Theory An Algorithmic Approach. N. Y. Acaemic Press. C. Crosson, M. Tonking, W. Moffat, Es Palabora System of Truck Control. Mining Magazine, February E v a n s, J., E. M i n i e k a. Optimization Algorithms for Networks an Graphs. N. Y., M. Dekker, Inc.,

11 3. Genova, K., L. Kirilov, V. Guliashki, B. Staykov, D. Vatov. A. Prototype of a Web- Base Decision Support System for Builing Moels an Solving Optimization an DecisionMaking Problems. In: Proceeings of XII International Conference on Computer Systems an Technologies CompSysTech 11. B. Rachev, A. Smrikarov Es. Wien, Austria, June 2011, ACM ISBN , ACM ICPS, Vol. 578, 2011, S i e g w a r t, R, I. N o u r b a k h s h, D. S c a r a m u z z a. Introuction to Autonomous Mobile Robots. 2n Eition. Intelligent Robotics an Autonomous Agents series. Massachusetts Institute of Technology Cambrige, Massachusetts The MIT Press, ISBN D u b e l, W., et al. An Autonomous Firefighting Robot. Floria International University, Miami, FL Department of Electrical an Computer Engineering Firefighting Robots to Help Fire Crews Battling Wilfires. Управление в реальном времени группой мобильных роботов при помощи системы человека робота Васил Сгурев, Станислав Дрангажов, Любка Дуковска Институт информационных и коммуникационных технологий, 1113 София s: srangajov@gmail.com l.oukovska@mail.bg (Р е з ю м е) В работе обсуждается система человек робот для погашения пожаров. Мобильные роботы применяются в разных областях материального производства и социальной практики. Появились новые задачи, которые расширили применения мобильных роботов. Их использование очень важно в случаях природных бедствий или промышленных аварий, такие как пожары, загрязнение среды, когда присуствие человека в агрессивной среде опасно и в многих случаях невозможно. В множестве таких случаев мобильные роботы действуют совместно на одной и той же локации, так что ими надо правильно управлять. При этом, присуствие человека безсомненно необходимо, чтобы координировать и наблюдать действия роботов. 13