Coverage Control with Information Decay in Dynamic Environments

Transcription

1 Coverage Control wth Informaton ecay n ynamc Envronments Nco Hübel S. Hrche A. Gusrald T. Hatanaka M. Fujta O. Sawodny Insttute for System ynamcs, Unverstät Stuttgart Fujta Laboratory, ept. of Mechancal and Control Engneerng, Tokyo Insttute of Technology Abstract: In ths paper a method for coverage control for a convex regon R 2 n a dynamc envronment s studed. An nformaton map s ntroduced n whch the nformaton about each pont s decayng wth respect to tme s.t. the robots must revst them perodcally. Also a tmevaryng densty functon s used for modelng movng ponts of nterest. The consdered gradent based control approach causes the cost functon to stay wthn the desred bounds. But due to the non-statonary problem setup caused by the nformaton decay t does not converge to a sngle pont but to a bounded set, such that the robots keep gatherng nformaton contnuously. Wth ths method t s possble to gather nformaton about several ponts of nterest wthn the regon wth only a few robots. In the end smulaton results are presented to outlne the effectveness of the proposed control law. 1. INTROUCTION Natural dsasters wth ther need for quck humantaran help as well as mltary and survellance operatons and tasks n hazardous envronments are major applcaton for robots. In these knd of applcatons we often have to deal wth addtonal dffcultes lke an adversaral envronment, constrants (e.g. tme restrctons, lmted communcaton capabltes, etc.) and changng msson objectves. But for the use of robots n these scenaros they also must be economcally reasonable. Ths s the motvaton for the approach of ths paper to use only a few robots, whch should gather nformaton n a specfed area. Recently a lot of research results came up for the area of coverage control. A very nce ntroductory overvew can be found n Martínes [2007]. Most of the results rely on Vorono tessellatons. For example n Cortés [2004] a gradent descent to reach the optmal Vorono confguraton s proposed and n Martínes [2004] coverage control algorthms for robot groups wth lmted-range nteractons are presented. Other results use some explct nformaton measures to express a gan of nformaton. For nstance Olfat-Saber [2007] proposes an algorthm whch causes moble agents wth a dynamc network topology to mprove ther estmaton of a movng target. In Martínez [2005] moton coordnaton algorthms whch maxmze the determnant of a Fsher Informaton Matrx are presented and n Basr [1995] a soluton to an actve sensng task s gven whch mnmzes the varance of the estmaton error and thus reduces the uncertanty of the target state. In addton there are also results lke Ahmadzadeh [2007] whch are based on recedng horzon control or MPC. In ths paper the problem of gatherng nformaton and montorng an area wth only a few robots s addressed. The am was to derve results, whch are applcable for nfnte tme and therefore not converge to a sngle pont but to a bounded set. In order to acheve that, an dea smlar to the effectve coverage functon n Hussen [2007] but wth the novel concept of an nformaton model wth nformaton decay s ntroduced. The paper s organzed as follows. In secton 2 the problem setup and the necessary defntons are presented and explaned. Then the control law wll be ntroduced and dscussed n secton 3. The results were valdated by extensve smulatons. In secton 4 a smulaton for coverng a square area wth a movng pont of nterest s presented before a concluson and the possble extensons of ths work are stated n secton PROBLEM SETUP In ths paper Q = R 2 denotes the confguraton space of the agents. Let be the area whch should be montored by the agents. must be a convex subset of Q. Throughout the paper R + = {a R a 0} s used. 2.1 Agent model Let A = {A S = {1,2,3,...,N}} be the set of agents consstng of the sngle agents A wth N beng the number of agents and S beng the ndex set of the fully connected network whch means that t contans the ndces of all agents. Let q Q denote the poston of agent A. All agents A satsfy the knematc equaton q = u, S (1) wth u R 2 as the control nput of agent A. It s assumed that the underlayng dynamcs of the agents are controlled by low level controllers whch use u as reference nput. 2.2 Measurement functon The measurement functon M (s ) : Q R + wth s = p q 2 (2) of the agent A s defned as a C 1 -contnuous map that descrbes the sensng performance of that agent. Sensng

2 performance n ths case means how much nformaton the agent can get about a fxed pont p dependng on hs own poston. Naturally sensng performance s decreasng when the dstance to the agent ncreases, whch leads to the assumpton of M beng a monotoncally decreasng functon of the dstance between a pont p and M the poston of the robot q : s s= p q 0 p 2 and q Q S. Addtonally the agent can only gather nformaton n a certan area around hs poston. Therefore the sensor area of the agent A s defned as W = { } p p q r 2 wth the sensor range r. Ths leads to the followng addtonal assumpton on the measurement functon: M (s ) = M s s= p q = 2 0 p \ W = { } p s r 2. Now let the measurement map M(s) = M (s ) S (3) be defned as the sum of all measurement functons and s be the set contanng all s. An example of such a measurement functon s C M (s ) = r 4 (s r 2 ) 2 f s r (4) 0 f s > r whch s also depcted n fg. 1. In ths method δ, I ref (p) and k are consdered as desgn parameters. I ref (p) s the reference nformaton map whch gves the level of nformaton the robots should gather for each pont of and k are the feedback gans. Whle accordng to (7) the feedback gans are drectly related to the velocty of the robots, the choce of δ and I ref (p) s more complcated. But accordng to (5) δ I ref (p) + M(0) 0 should hold for all p because otherwse I ref (p) never can be reached n all ponts p. An explct relaton between the speed of the robots and the desgn parameters wll be the subject of further research. 2.4 ensty functon Let φ(p, t) : R + R + be the tme-varant postve sem-defnte densty functon whch represents the regons of nterest n the area. The robots wll always spend partcular nterest on gatherng nformaton n regons where the densty functon has a hgh value and wll gather nformaton about adjacent regons only f they already have gathered enough nformaton about that regons of nterest. Movng ponts of nterest such as the targets of a survellance operaton can be represented by the movng peaks of a tme-varant densty functon. Ths wll cause the robots to follow these peaks. There are two assumptons on φ(p, t). Frst, t must be two tmes contnuously dfferentable wth respect to tme. Ths assumpton s not very restrctve due to the fact that real ponts of nterest also can not nstantly change ther poston. And possble newly detected ponts of nterest can be ncorporated by a slowly rsng peak. Second, φ (p, t) k and 2 φ (p, t) k 2 must hold whch physcally means that the speed and the acceleraton of the target has to be small compared to the robot. In ths paper the tme-varyng densty functon s assumed to be pre-known and the problem of onlne-estmaton and adapton s not consdered. But there are already exstng results and algorthms avalable from the feld of computer scence. Fg. 1. Meassurement functon (4) wth C = 2 and r = Informaton model Let the nformaton map be defned as I(p, t) : R + R +. The evoluton of the map s modeled wth the followng partal dfferental equaton: I(p, t) = δ I(p, t) + M(s) (5) wth the decay rate δ 0. The nformaton map ndcates how much nformaton the agents have gathered about the ponts p. It conssts of two parts. The frst part δ I(p, t) represents the nformaton decay whereas the nformaton map M(s) n the second part represents the nformaton gan. For δ < 0 the nformaton s decayng over the tme s.t. all the ponts have to be revsted frequently n order to keep the nformaton level hgh. For δ = 0 the results of Hussen [2007] are a specal case of the results presented n ths paper. 2.5 Cost functon Let J(t) = h(i ref (p) I(p, t)) φ(p, t) dp (6) be the cost functon whch should be mnmzed wth the penalty functon h(e I ) whch penalzes a lack of coverage whch n addton s weghted accordng to the mportance of the area denoted by the densty functon. There are some necessary assumptons on the penalty functon h(e I ) wth e I = I ref (p) I(p, t): A1 h(e I ) must be pecewse C 1 A2 h(e I ), h e I (e I ), 2 h(e I ) > 0, e I > 0 e 2 I A3 h(e I ), h e I (e I ), 2 h(e I ) = 0, e I 0 e 2 I A2 and A3 bascally mean that only a lack of nformaton s penalzed but not havng too much nformaton. For nstance the functon h(x) = (max(0, e I )) 2 wll be used for the smulaton.

3 3. MAIN RESULT Now the control law s ntroduced whch wll cause the objectve functon (6) to stay bounded and ts valdty s proven by a Lyapunov functon approach. In addton there wll be a short physcal nterpretaton of the problem and a relaton between the decay rate δ and the acheved performance wll be dscovered. Then the theorem wll be extended for partally or even non connected robot groups. The followng abbrevatons wll be used: h (e I ) = h e I h (e I ) = 2 h e 2 I ei =I ref (p) I(p,t) ei =I ref (p) I(p,t) M (s ) = M s s= p q 2 Fully connected robot group: As a startng pont a fully connected robot group s consdered, whch means that each agent can communcate wth all other agents. Thus the ndex set s S = {1,2,3,...,N}. Consder the followng control law u (q ) = k W h (e I ) M (s ) (p q ) φ(p, t) dp (7) wth the feedback gans k R +. Please note that the nput for each agent A only depends on hs own poston q explctly. The nformaton about the postons of the other robots are only used ndrectly through the use of the nformaton map I(p, t) n the penalty functon. Accordng to equaton (5) only for the evoluton of the nformaton map the nformaton about all robot postons s requred. Ths corresponds to the concept of a world model whch often can be found n computer scence lterature on artfcal ntellgence. Note further that t s not necessary to evaluate the ntegral over the whole area of. Because M (s ) = 0 p / W holds by defnton, t s suffcent to evaluate the ntegral over W. Whle ths reduces computatonal effort t causes the problem of local mnma whch wll be solved later. A closer look at the control law wll reveal that t sums up the weghted vectors from the robot to each pont p W. The use of the gradent of the error dstrbuton nsde W causes the robot to move n the drecton wth the maxmum error nsde ts sensor range. Through the followng remark ths can be seen easly. Remark 1. Here some propertes whch are often used n mechancs but also hold for more general problems wll be revewed. For a regon V R n and a generalzed mass densty functon ρ(p) wth p V, the generalzed mass and generalzed center of mass s gven as follows: M V = ρ(p) dp (8) V C V = 1 p ρ(p) dp. (9) M V By splttng up the ntegral and usng (8) and (9) wth V = W and ρ(p) = h (e I )M φ(p) we fnd out that the control law (7) can also be expressed as follows: u (q ) = k M W (C W q ) (10) Note that the control law can become zero f h (e I ) s equal to zero everywhere n W even f J(t) s not equal to zero. Ths happens f the followng condton holds: V I I ref p W A3 h = 0 p W (7) u (q ) = 0 (11) To move the robot away from condton (11) the followng smple lnear control law s used û (q ) = ˆk (q ˆp ) (12) wth the control gans ˆk R +. The only requrement on the pont ˆp s that t has to drve the robots away from condton (11). How to choose ˆp specfcally s not consdered n ths paper. But one possblty s to choose ˆp as the nearest pont for whch condton (11) does not hold. A more detaled dscusson of the second control law can be found n the remark after the proof of the followng theorem. Theorem 1. Under the gven assumptons the control law { u u (q (q ) = ) f h 0 for some p W û (q ) f h (13) = 0 p W wth suffcently large gans k, ˆk R + wll hold the objectve functon wthn the bounds C > J(t) 0 t > 0. Therefore t wll cause the robots to contnuously gather nformaton n. Proof. For the proof (6) wll be used as a common Lyapunov functon V (t) for the swtchng control law (13). ue to the non-statonary problem setup t s only possble to keep the cost functon wthn the stated bounds. In order to acheve that the control law s used to ensure V (t) < 0 whenever V (t) > 0 holds. As stated before t s suffcent to ntegrate the control law only over W but n order to unfy the ntegraton process t s also feasble to ntegrate over because M (s ) equals zero outsde of W. But frst the stated bounds should be derved. From (5) t s obvous that the lower bound I mn (p) = 0 p for I(p, t) exsts. Then the maxmum dfference, and because of A2 therefore also the maxmum value of the penalty functon h(e I ), s e I = I ref I mn = I ref. And because - accordng to A3 - dfferences n the other drecton (e I < 0) are not penalzed ths s the case n whch the penalty functon reaches ts maxmum. Hence the upper bound C for the cost functon s J(t) C = h(i ref)φ(p)dp. For N > 0 equalty can only be acheved for t = 0 and an ntal nformaton map I 0 (p,0) = 0 p but not for any t > 0 because as long as there s at least one robot n t wll cause the measurement map to be M(s) > 0 for some p and accordng to (5) ths wll cause the nformaton map to be I( p, t) > 0. Hence the value of cost functon wll be smaller. Snce ths s a very conservatve upper bound whch s also not related to the control law, more research wll be done on fndng a better upper bound C. Accordng to A3 the lower bound s J(t) 0 for e I 0. The lower bound s only reached f there are enough robots s.t. M(s) δ I ref p holds for all tme. Ths bascally means that there are enough robots to keep the measurement map hgh enough s.t. the nformaton map never drops below the reference nformaton map after t exceeds t for the frst tme. The common Lyapunov functon and ts dervatves wll be dscussed n the followng. The exclamaton marks should ndcate that these nequaltes are not necessarly fulflled but should be fulflled n the below stated cases.

4 V (t) = V (t) = + V (t) = + h (e I ) h(e I ) φ(p, t) dp 0 (14) ( δ I(p, t) M(s) ) φ(p, t) dp } {{ } φ <0 h(e I ) (p, t) dp! 0 (15) } {{}? h (e I ) ( δ I(p, t) M(s)) 2 φ(p, t) dp δ h (e I ) ( δ I(p, t) M(s))φ(p, t) dp M W (C W q )u S ! h φ (e I ) ( δ I(p, t) M(s)) (p, t) } {{}? 2 φ h(e I ) (p, t) dp! 0 2? (16) From A2 and A3 t s obvous that nequalty (14) s fulflled wth equalty ff e I = I ref (p) I(p, t) 0 p holds. But nequalty (15) does not necessarly hold. It can be reformulated as follows such that t shows us bounds on δ. (0 ) δ h (x)m(s)φ(p, t)dp + φ h(x) (p, t)dp h (x)i(p, t)φ(p, t)dp (17) If ths nequalty s volated the objectve functon wll start to rse. So the goal s to keep the cost functon wthn bounds around that crtcal pont. To acheve that, t has to be verfed that the cost functon wll decrease agan after a certan tme. Therefore nequalty (16) has to hold n that case. The ndcated sgns of the terms n (16) are also only necessarly true n that case. Please note that ( δi(p, t) M(p, t)) > 0 also does not necessary hold pontwse but should ndcate the sgn of the ntegraton result. The nput u must be chosen s.t. the nequalty holds. Ths can only be acheved wth a suffcently bg postve nput u as ndcated n (16). Snce we assumed φ and 2 φ to be small they were neglected n these thoughts. But t mght 2 also be consdered as a dsturbance and n the future one mght look nto that problem from the robust control pont of vew. But frst let S be defned as the arbtrary ndex set that contans those agents, whch use the control law u, and Ŝ as the arbtrary ndex set that contans those agents, whch use û. Of course S Ŝ = S and S Ŝ = have to hold. Wth the use of (10), (12), (13) and v = C W q the term of (16) whch contans the nput becomes: 2 S M W v u = 2 S k M 2 W v 2 2 j Ŝ ˆk j M Wj v j (q j ˆp j ) Please note that for all agents A j wth j Ŝ condton (11) holds and thus because of h (e I ) = 0 (8) M Wj = 0 p W j wth j Ŝ the last term of the equaton above s equal to zero and therefor we fnally obtan the nequalty: V = h (e I ) ( δ I(p, t) M(s)) 2 φ(p, t) dp + δ h (e I ) ( δ I(p, t) M(s)) φ(p, t) dp 2 k MW 2 (C q ) 2 S + 2 h (e I )(( δ I(p, t) M(s))) φ (p, t) + 2 h(x) 2 φ (p, t)dp 0 2 whch holds for suffcently bg k. Therefore the cost functon wll stay wthn the desred bounds. Remark 2. (scusson of control law û (q ):) As obvous from the proof, swtchng to the second control mode û (q ) does not change the stablty argument. But t s mmedately clear (by applyng smple lnear analyss) that ths control law wll drve the agents A j to the pont ˆp j n nfnte tme. Though, t s not necessary for the robot to reach that pont because as soon as there s a pont n hs sensory range for whch condton (11) does not hold (e.g. ˆp j enters W j ) t wll swtch back to the frst control law. Ths shows that (12) drves the agents A j to a state where (10) wll be used agan. We exclude cases wth nfnte swtchng here. Future work wll address ths possble problem. Partally connected robot group: Now the valdty of theorem 1 for partally or even non connected robot groups wll be shown. But frst a worst case scenaro s dscussed. Theorem 1 also holds for an ndex set S 1 = 1. Ths means that there s only one sngle agent A 1 avalable for coverng the area. Thus theorem 1 holds obvously for a group of dsconnected robots. In that case each robot tres to cover by hmself wthout recognzng the efforts of the others. Startng from ths pont the ndex sets Ŝ S wth = 1,...,n can be defned for n dsconnected subgroups wth the propertes n =1 Ŝ = S and Ŝ Ŝ j =. It s,j j assumed that the robots of each group can communcate wth all of ther group members but not wth robots of other groups. Now a group specfc objectve functon Ĵ for each group can be defned as Ĵ (t) = h(i ref (p) Î(p, t)) φ(p, t) dp (18) wth the group specfc nformaton map Î(p, t), whch also evolves accordng to equaton (5) but wth a group specfc measurement map ˆM (s ) = j Ŝ M j. Obvously the nequalty Ĵ(t) J(t) wth equalty ff Ŝ = S holds. The new common Lyapunov functon can be defned as ˆV (t) = Ĵ (t) (19) Ths wll lead to smlar expressons as we have obtaned n the proof for the fully connected robot group and therefore

5 the proof follows exactly the same strategy. Ths shows us that each robot group for tself tres to cover the ponts of nterest. 4. SIMULATION In ths secton we wll see 3 robots coverng a square area of 50x50 unts length n whch a pont of nterest s movng. The pont of nterest s modeled by a Gaussan whch s movng n a counter-clockwse crcular moton around the mddle pont of the map. The smulaton starts wth an ntal nformaton map I(p,0) = I 0 = 0 p. As reference nformaton map I ref (p) = 5 p s used. The measurement functon (4) wth the sensor range r = 2 and the peak sensng capacty C = 2 and the penalty functon h(x) = (max(0, x)) 2 are used. The decay rate and the control gans are set to δ = 0.03 and k = ˆk = 5 respectvely. For the ntegratons a smple trapezodal method s appled. In fg. 2 the evoluton of the cost functon over the tme s shown. It can be seen that even wth the tny sensor range of r = 2 the robots manage to keep the cost functon bounded. On the last page the smulaton results belongng to the cost functon depcted n 2 are shown n two columns. In the left column the tme-varant densty functon φ(p, t) s depcted at dfferent tme steps. In the rght column the correspondng evoluton of the nformaton map I(p, t) can be seen at the same tme steps. It can be seen that the robots create a hgh level of nformaton always near the peak of the densty functon and therefore they have to follow ts movng peak. There are always 2 robots mantanng the hgh level of nformaton near ths peak and the thrd robot tres to gather nformaton about other areas 1. If we would ncrease the sensor range, the robots could gather more nformaton and at a certan pont all robots would move away from the peak of the densty functon to gather nformaton somewhere else untl the level of nformaton at the peak has dropped suffcently. Then one or more would move back to the peak. 5. CONCLUSION In ths paper we have formulated a coverage control problem wth a large varety of applcatons. We have ntroduced the novel concept of an nformaton model wth nformaton decay n order to derve a control law that - under the gven assumptons - causes small groups of robots to gather nformaton about an area wthout convergng to fnal postons and thus beng capable of montorng an area for nfnte tme. Addtonally ths control law can be appled n dynamc envronments and a modelng possblty for movng ponts of nterest was descrbed. Then t was shown that wth the ntroduced control law the cost functon s bounded and t was ponted out that ths control law s also vald for partally connected robot groups wth only mnor changes. Fnally a numercal smulaton was presented. For the future there are several areas of work n addton to the ponts already mentoned n the paper. Especally collson avodance, more realstc models of the robots, dynamc network topologys and the use of ansotropc sensors mght be nterestng. In addton a better cooperaton of the robots mght be acheved usng predctve control methods lke MPC. ACKNOWLEGEMENTS Ths work was nspred by Hussen [2007] and the authors would lke to thank I. Hussen and.m. Stpanovć for ths nspraton and the grateful supply of manuscrpts. REFERENCES S. Martínes, J. Cortés and F. Bullo. Moton Coordnaton wth strbuted Informaton. IEEE Control Systems Magazne, pages 75-88, J. Cortés, S. Martnez, T. Karatas and F. Bullo. Coverage control for moble sensng networks. IEEE Transactons on Robotcs and Automaton, pages , S. Martínes, J. Cortés and F. Bullo Spatally-strbuted Coverage Optmzaton and Control wth Lmted- Range Interactons. ESAIM: Control, Optmzaton and Calculus of Varatons, pages , R. Olfat-Saber. strbuted Trackng for Moble Sensor Networks wth Informaton-rven Moblty. Proc. of the Amercan Control Conference, S. Martínez. On Optmal Sensor Placement and Moton Coordnaton for Target Trackng. Proc. of the IEEE Internatonal Conference on Robotcs and Automaton, pages , A. Ahmadzadeh, A. Jadbabae, V. Kumar and G. J. Pappas. Cooperatve Coverage usng Recedng Horzon Control. European Control Conference, I. I. Hussen and. M. Stpanovć. Effectve Coverage Control for Moble Sensor Networks Wth Guaranteed Collson Avodance. IEEE Transactons on Control Systems Technology, Vol. 15, No. 4, July O. A. Basr and H.C. Shen. Informatonal Maneuverng n Fg. 2. Cost functon ynamc Envronment. IEEE Internatonal Conference on Systems, Man and Cybernetcs, pages vol.2, There are vdeos of other smulatons avalable at http : //fujta.fl.ctrl.ttech.ac.jp/researches/2007/search/3robots dot1delta, ph = ones.mpg and http : //fujta.fl.ctrl.ttech.ac.jp/researches/2007/search/3rob15range ph mpg.

6 Fg. 3. Smulaton: In the left column you can see the tme-varant densty functon φ(p, t) whereas n the rght column the accordng evoluton of the nformaton map I(p, t) s depcted.