Adaptive Flocking Control for Dynamic Target Tracking in Mobile Sensor Networks

Transcription

1 The 29 IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems October -5, 29 St. Lous, USA Adaptve Flockng Control for Dynamc Target Trackng n Moble Sensor Networks Hung Manh La and Wehua Sheng Abstract Target trackng s an mportant task n sensor networks, especally n moble sensor networks. Flockng control s used to control a moble sensor network to track a target. However, there are some exstng problems n ths control method, such as network fragmentaton, loss of formaton and poor trackng performance. In order to handle these problems we propose a novel approach to flockng control of a moble sensor network to track a movng target wthn changng envronments. In our approach, each agent can cooperatvely learn the network s parameters to decde the sze of network n a decentralzed fashon so that the connectvty, formaton and trackng performance can be mproved when avodng obstacles. In addton, to demonstrate the beneft of our approach a comparson between ths approach and the exstng method s gven. Computer smulatons are performed to demonstrate the effectveness of the proposed approach. Keywords: Flockng, target trackng, moble sensor network. A. Motvaton I. INTRODUCTION Moble sensor networks [] have advantages over statonary sensor networks such as adaptaton to envronmental changes and reconfgurablty for better performance. A man ssue for multple moble sensors to track a movng target s that these sensors have to move together wthout collson among them durng trackng. Ths requres us to apply cooperatve control methods, and one of these methods s flockng control [2], [3], [4], [5]. However, these exstng works have some lmtatons when the envronment changes, for example when the moble sensor network has to pass through a narrow space among obstacles. These lmtatons nclude:. Connectvty s lost because of the fragmentaton phenomenon. 2. Formaton of the network s totally changed. 3. Low speed or gettng stuck causes poor trackng performance. Therefore to desgn an adaptve flockng control algorthm to deal wth these problems s a challengng task. In ths paper, we present a novel approach to flockng control of a moble sensor network to track a movng target n changng envronments. In ths approach, each agent cooperatvely learns the network s parameters n a decentralzed fashon Ths project s supported by the Vetnamese Government, MOET (Mnstry of Educaton and Tranng) program and the DoD ARO DURIP grant CS-RIP. Hung Manh La and Wehua Sheng are wth the school of Electrcal and Computer Engneerng, Oklahoma State Unversty, Stllwater, OK 7478, USA (hung.la@okstate.edu, wehua.sheng@okstate.edu). so that the connectvty, formaton and trackng performance can be mproved when avodng obstacles. The reason of mantanng the connectvty and smlar formaton s that when the network shrnks ts sze to deal wth changng envronments the neghborhood of each agent can be mantaned. Ths allows the network to keep the same topology, whch reduces the complexty of control durng the trackng process. Computer smulatons are conducted to prove our theoretcal results. B. Related work Flockng control has receved consderable attenton due to ts wde applcatons, such as space exploraton and survellance. In [2], the theoretcal framework for desgn and analyss of dstrbuted flockng algorthms was proposed. These algorthms solved the flockng n free space and n the presence of obstacles. The statc and dynamc vrtual leaders were used as a navgatonal feedback for all moble sensors. Ths establshed a background for flockng control desgn for a group of sensors. Adaptve flockng control, an extenson of flockng control, has also ganed attenton from researchers n recent years. Yang et al. [6] proposed an adaptve flockng control algorthm to avod collson among robots themselves and between robots and obstacles. However, ther algorthm dd not consder the problem of formaton, connectvty and trackng performance n complex envronments. In addton, ther algorthm only consdered a statc target or a rendezvous pont, whch leads all agents to get there. Lee and Chong [7] ntroduced a moton plannng framework for a large number of autonomous robots that enables the robots to confgure themselves adaptvely nto an area of arbtrary geometry. Ther proposed method allows the robots to converge to the unform dstrbuton by formng an equlateral trangle wth ther two neghbors. However, the problem of target trackng was not addressed n ther work. An extenson of ther work was developed n [8] by the same authors to allow the swarm of robots to go to predetermned rendezvous ponts. Ther approach was based on a decentralzed approach that enables a swarm of robots navgate autonomously n complex envronments populated by obstacles. The problem of splttng/mergng moble robots n the network accordng to the envronments s addressed n ther paper. Namely, when the swarm of robots detects obstacles, each robot splts from the network and determnes ts drecton toward the statc goal based on the wdth of space among obstacles. However, n realty t s dffcult for each robot to sense the whole envronment and compute the wdth of space among obstacles. Also, n ther work /9/$ IEEE 4843

2 the problem of controllng the sze of the network was not consdered, and the connectvty and formaton were not guaranteed n complex envronments. In summary, most of exstng work focused on the coordnaton, formaton and splttng/mergng problems n both fxed and swtchng topologes. The problem of how to control the sze of the network n a decentralzed and adaptve fashon n complex envronments whle mantanng connectvty, formaton and trackng performance s stll an open problem. The rest of ths paper s organzed as follows. In the next secton we present the background of flockng control. Secton III descrbes our adaptve flockng control algorthm to track a movng target whle avodng obstacles. Secton IV provdes the smulaton results. Fnally, secton V concludes ths paper. II. FLOCKING CONTROL BACKGROUND In ths secton we wll present the graph prelmnary and the flockng control background. We consder n sensors movng n an m (e.g.,m = 2,3) dmensonal Eucldean space. The dynamc equaton of each sensor s descrbed as follows: q = p () ṗ = u, =,2,...,n. To descrbe the topology of flocks or swarms we consder a dynamc graph G consstng of a vertex set ϑ =,2...,n} and an edge set E (, j) :, j ϑ, j }. In ths topology each vertex denotes one member of flocks, and each edge denotes the communcaton lnk between two members. Let q, p R m be the poston and velocty of node, respectvely. We know that durng the movement of sensors, the relatve dstance between them may change, hence the neghbors of each sensor also change. Therefore, we can defne a set of neghborhood of sensor as follows: N = j ϑ : q j q r, ϑ =,2,...,n}, j }, (2) here, r s an actve range (radus of neghborhood crcle n the case of two dmensons, m = 2, or radus of neghborhood sphere n the case of three dmensons, m = 3), and. s the Eucldean dstance. The geometry of flocks s modeled by an -lattce [2] that meets the followng condton: q j q = d, j N, (3) here d s a postve constant ndcatng the dstance between sensor and ts neghbor j. To construct a collectve potental that s dfferentable at sngular confguraton (q = q j ), the set of algebrac constrans s rewrtten n term of σ - norm as follows: q j q σ = d, j N, (4) here the constrant d = d σ wth d = r/k c, where k c s the scalng factor. The σ - norm [2],. σ, of a vector s a map R m = R + defned as z σ = /ε[ +ε z 2 ] wth ε >. Unlke the Eucldean norm z, whch s not dfferentable at z =, the σ - norm z σ, s dfferentable every where. Ths property allows to construct a smooth collectve potental functon for agents. The flockng control law n [2] controls all sensors to form an -lattce confguraton. Ths algorthm conssts of three components as follows: u = f + f β + f γ. (5) The frst component of (5) f, whch conssts of a gradent-based component and a consensus component (more detals about these components see [9], [], []), s used to regulate the potentals (mpulsve or attractve forces) and the velocty among sensors. f = c φ ( q j q σ )n j +c 2 a j (q)(p j p ), (6) j N j N where each term n (6) s computed as follows [2]: The acton functon φ (z) that vanshes for all z r wth r = r σ s defned as follows: φ (z) = ρ h (z/r )φ(z d ) (7) wth the uneven sgmodal functon φ(z) defned as φ(z) =.5[(a+b)σ (z+c)+(a b)], here σ (z) = z/ +z 2, and parameters < a b, c = a b / 4ab to guarantee φ() =. The bump functon ρ h (z) wth h (,), z [,h) ρ h (z) =.5[+cos(π( z h h ))], z [h,) (8), otherwse. Vector along the lne connectng q to q j s defned as n j = (q j q )/ +ε q j q 2. (9) The adjacency matrx a j (q) s defned as ρh ( q a j (q) = j q σ /r ), f j, f j =. () The second component of (5) f β s used to control the sensors to avod obstacles, f β = c β φ β ( ˆq,k q σ ) ˆn,k + c β 2 b,k (q)( ˆp,k p ), k N β k N β () where the set of β neghbors (vrtual neghbors, [2]) s } N β = j ϑ β : ˆq,k q r,ϑ β =,2,...,K}, (2) here K s the number of obstacles, r s an obstacle detectng range, and ˆq,k, ˆp,k are the poston and velocty of sensor projected on the obstacle k, respectvely (more detals please see [2]). Smlar to vector n j establshed n (9), vector ˆn,k s defned as ˆn,k = ( ˆq,k q )/ +ε ˆq,k q 2. (3) The heterogeneous adjacent matrx b,k (q) s defned as b,k (q) = ρ h ( ˆq,k q σ /d β ), (4) 4844

3 where d β = r σ. The repulsve acton functon of β neghbors s defned as φ β (z) = ρ h (z/d β )(σ (z d β ) ). (5) The thrd component of (5) f γ s a dstrbuted navgatonal feedback. f γ = c γ (q q γ ) c γ 2 (p p γ ), (6) where the γ - agent (q γ, p γ ) s the vrtual leader that leads the flock to follow ts trajectory, and t s defned as follows qγ = p γ (7) ṗ γ = f γ (q γ, p γ ). The constants of three components used n (5) are chosen as c < cγ < cβ, and cν 2 = 2 c ν. Here cν η are postve constants for η =,2 and ν =,β,γ. III. ADAPTIVE FLOCKING CONTROL FOR TRACKING A MOVING TARGET In ths paper, we consder the γ agent as a movng target. Hence, based on Olfat-Saber s flockng control [2] we desgn a control law wth a movng target as u = c φ ( q j q σ )n j + c 2 a j (q)(p j p ) j N j N +c β k N β φ β ( ˆq,k q σ ) ˆn,k + c β 2 b,k (q)( ˆp,k p ) k N β (q q mt ) c mt 2 (p p mt ), (8) here (q mt, p mt ) s the poston and velocty of the movng target, respectvely, and c mt are postve constants. In ths control law, we assume that each agent has ablty to sense the poston and velocty of the movng target. The problem here s how to cooperatvely control the sze of the network n an adaptve and decentralzed fashon n order to mantan the network s connectvty, smlar formaton and trackng performance n the presence of obstacles. One example of such flockng control s llustrated n Fgure. Fg.. Illustraton of the adaptve flockng control A. Adaptve flockng control To control the sze of the network, we need to control the set of algebrac constrants n Equaton (4), whch means that f we want the sze of the network to be smaller to pass the narrow space then d should be smaller. Ths rases the queston of how small the sze of network should be reduced and how to control the sze n a decentralzed and dynamc fashon. To control the constrant d one possble method s based on the knowledge of obstacle obtaned by any sensor n the network, whch wll broadcast a new d to all other sensors. However, t s dffcult for a sngle sensor to learn the sze of the obstacles due to ts lmted sensng range. To overcome ths problem we propose a method based on the repulsve force, β k N φ β ( ˆq,k q σ ), whch s generated by the β-agent (vrtual agent) projected on the obstacles. If any sensor n the network gets ths repulsve force t wll shrnk ts own d. If ths repulsve force s bg (sensor s close to obstacle(s)) d wll be further reduced. Then, n order to mantan the neghborhood (topology) the actve range of each sensor s re-desgned. To create the agreement on the relatve dstance and actve range among sensors n a decentralzed way, a consensus or a local average update law s proposed. Furthermore, to mantan the connectvty each sensor s desgned wth an adaptve weght of attractve force from the target and an adaptve weght of nteracton force from ts neghbors so that the network reduces or recovers the sze gradually. That s, f an sensor has weak connecton to the network t should have a bg weght of attracton force to the target and a small weght of nteracton force from ts neghbors. Frstly, we control the set of algebrac constrants as q j q σ = d, j N, (9) and let each agent have ts own d, whch s desgned as d, f d β k N φ β ( ˆq,k q σ ) = = c a β φ β ( ˆq,k q σ ) +, f k N β φ β ( ˆq,k q σ ), k N (2) here c a s the postve constant. From Equaton (2) we see that f the repulsve force generated from the obstacles β k N φ β ( ˆq,k q σ ) = or N β = / (empty set) then the agent wll keep ts orgnal d. When the agent senses the obstacles t reduces ts own d, and the value of d depends on the repulsve force that the agent gets from obstacles. In order to control the sze of network each sensor needs ts own r that relates to d as follows: r = k c d σ wth d σ = d (εd or d = +)2 ε. Explctly, r s computed as n Equaton (2). r, f β k N φ β ( ˆq,k q σ ) = r = ε [ k 2 (εd +)2 c ε + ], (2) f β k N φ β ( ˆq,k q σ ).

4 Smlar to computng r, r s computed as r, f β k N φ β ( ˆq,k q σ ) = r = ε [(εr + ) 2 ], f β k N φ β ( ˆq,k q σ ). (22) It should be ponted out that the actve range r s dfferent from the physcal communcaton (sensng) range. Namely, the actve range s the range that each agent decdes ts neghbors to talk wth, but the physcal communcaton range s the range defned by the RF module. Ths mples that even a robot can communcate wth many other robots n the network, t wll only talk (nteract) wth robots n ts actve range. That s why we want to control the actve range of each robot n order to reduce the communcaton and mantan the smlar formaton when the network shrnks. To acheve agreement on d, r and r among sensors n the connected network we use the followng update law based on local average for d, r and r : d = N + N + j= d j r = N + N + j= r j r = N + N + j= r j, (23) here N s the number of neghbors of agent. In addton, to better mantan the network connectvty each agent should have an adaptve weght of attractve force from the target and nteracton force from ts neghbors as dscussed before. Frstly, n the control protocol (8), the frst two terms are used to control the formaton (velocty matchng, collson avodance among robots). The thrd and fourth terms are used to allow robots to avod obstacles, and the last term s used for target trackng. If the last term s absent the control wll lead to network fragmentaton [2]. The coeffcents of the nteracton forces (c, c 2 ), (cβ, cβ 2 ) and attractve force (c mt ) whch delver desred swarmlke behavour are used to adjust the weght of nteracton forces and attractve force. The bgger (c mt ) the faster convergence to the target. However f (c mt ) s too bg the center of mass (CoM) as defned n Equaton (24) q = n n = q p = n n = p (24) oscllates around the target, and the formaton of network s not guaranteed. In addton, n order to guarantee that no agent ht obstacles, the par (c β, cβ 2 ) s selected to be bgger than the other two pars, (c, c 2 ) and (cmt ). Fnally we have the relatonshp among these pars as: (c,2 < cmt,2 < c β,2 ). From the above analyss we see that these adaptve weghts allow the network to reduce and recover the sze gradually. They also allow the network to mantan the connectvty durng the obstacle avodance. We let each sensor have ts own weght of the nteracton forces as n Equaton (25) and attractve force as n Equaton (26). In the -lattce confguraton f the sensor has less than 3 neghbors t s consdered as havng a weak connecton to the network. Ths means that ths sensor s on the border of network, or far from the target hence t should have bgger weght of attractve force from ts target and smaller weght of nteracton forces from ts neghbors to get closer to the target. Ths desgn also has the beneft of makng the whole network track the target faster. From ths analyss c,2 are desgned as follows: and cmt,2 of each agent c () = c, f N 3 c, f N < 3, (25) here c < c, c 2 () = 2 c (), and =,2,...,n. c mt () = c mt, f N 3 c mt, f N < 3, (26) here c mt > c mt () = 2 c mt (), and =,2,...,n. Now, the neghborhood of sensor (N ), the new adjacency matrx a j (q) and the new acton functon φ (z) are redefned as follows: N = j ϑ : q j q r, ϑ =,2,...,n}, j } ; (27) a j (q) = ρh ( q j q σ /r ), f j (28), f j = ; φ ( q j q σ )=ρ h ( q j q σ /r )φ( q j q σ d ). (29) Fnally, the adaptve flockng control law for dynamc target trackng s u = c () j N +c 2 () j N +c β k N β φ ( q j q σ )n j a j(q)(p j p ) φ β ( ˆq,k q σ ) ˆn,k + c β 2 b,k (q)( ˆp,k p ) k N β ()(q q mt ) c mt 2 ()(p p mt ). (3) B. Stablty Analyss By applyng the control protocol (3), the CoM (defned n Equaton (24)) of postons and veloctes of all moble sensors n the network wll exponentally converge to the target n both free space and obstacle space. In addton, the formaton (collson free and velocty matchng among moble sensors) wll mantan n the process of the target trackng. Let us consder adaptve flockng control n free space and obstacle space, respectvely. Case (Free space): In free space, β k N φ β ( ˆq,k q σ ) =, hence we can rewrte the control protocol (3) by gnorng constants c ν η (for η =,2 and ν =,β) as follows: u = q ψ ( q j q σ )+ a j (q)(p j p ) j N j N (q q mt ) c mt 2 (p p mt ) (3) 4846

5 where ψ (z) = z d φ (s)ds s the parwse attractve/repulsve potental functon. From (3), we can compute the average of control law u as follows: n u = = n =u n n = ( j N q ψ ( q j q σ ) + a j (q)(p j p )) j N (q q mt ) c mt 2 (p p mt ). (32) Obvously, we see that the par (ψ,a(q)) are symmetrc. Hence we can rewrte (32) as: u = (q q mt) c mt 2 (p p mt). (33) Equaton (33) mples that q = p ṗ = (q q mt) c mt 2 (p p mt). (34) The soluton of (34) ndcates that the CoM of postons and veloctes exponentally converge to those of the target. The formaton (collson-free and velocty matchng among moble sensors) s mantaned n the free space trackng because the gradent-based term and the consensus term are consdered n ths stuaton (more detals please see [2]). Case 2 (Obstacle space): Snce d s desgned to be reduced when each agent senses the obstacles. Therefore, when the sensor network have to pass through the narrow space between two obstacles ts sze wll be shrunk gradually, and when the network already passed ths narrow space t grows back to the orgnal sze gradually. Ths reduces the mpact of the obstacle on the network hence the speed of sensors can be mantaned or the CoM keeps trackng the target. Also, the connectvty and smlar formaton can be mantaned n ths scenaro. IV. SIMULATION RESULTS In ths secton we wll test our adaptve flockng control algorthm (3) and compare t wth the exstng flockng algorthm (8) n terms of the network connectvty, formaton and trackng performance. The parameters used n ths smulaton are specfed as follows: - Parameters of flockng: number of sensors = 5 (randomly dstrbuted n the box of x sze); a = b = 5; d = 7; the scalng factor k c =.2; the actve range r = k c d = 8.4; ε =. for the σ-norm; h =.2 for the bump functon (φ (z)); h =.9 for the bump functon (φ β (z)). - Parameters of target movement: The target moves n the lne trajectory: q mt = [+3t, t] T wth t 3.5, and p mt = (q mt (t) q mt (t ))/ t wth step sze t =.2. To analyze the connectvty of the network we defne a connectvty matrx c j (t) as follows:, f j N c j (t) = (t), j, f j N (t), j (35) and c =. Because the rank of Laplacan of a connected graph [2] c j (t) of order n s at most (n ) or rank(c j (t)) (n ), the relatve connectvty of a network at tme t s defned as C(t) = n rank(c j(t)). (36) If C(t) < the network s broken, and f C(t) = the network s connected. Based on ths metrc we can evaluate the network connectvty n our adaptve flockng control algorthm (3). Fgures 2 represents the results of movng target (red/dark lne) trackng n the lne trajectory usng the exstng flockng control algorthm (5). Fgures 3 represents the results of movng target trackng n the lne trajectory usng the adaptve flockng control algorthm (3). Fgure 4 shows the results of velocty matchng among sensors (a, a ), connectvty (b, b ) and error postons between the CoM (black/darker lne) and the target (trackng performance) (c, c ) of both flockng control algorthms (3) and (5), respectvely. To compare these algorthms we use the same ntal state (poston and velocty) of moble sensors. By comparng these fgures we see that by applyng the adaptve flockng control algorthm (3) the connectvty, smlar formaton and trackng performance are mantaned when the network passes through the narrow space between two obstacles (two red/dark crcles) whle the exstng flockng control algorthm (5) could not handle these problems. In Fgures 3 when the network enters the small gap between two obstacles ts sze s shrunk gradually n order to pass ths space, then the network sze grows back gradually when t passed. Therefore the connectvty and smlar formaton are mantaned. V. CONCLUSION Ths paper studed the approach to flockng control of a moble sensor network to track and observe a movng target n changng envronments. We desgned an adaptve flockng control algorthm that can cooperatvely learn the network s parameters n a decentralzed fashon to change the sze of the network n order to mantan connectvty, formaton and trackng performance when passng through obstacles. In addton, to see the beneft of the adaptve flockng algorthm we compared t wth the normal flockng control algorthm, and we found that the connectvty, smlar formaton and trackng performance n the adaptve flockng control algorthm are better than those n the exstng flockng control algorthm. The computer smulaton verfed our theoretcal results. REFERENCES [] S. Kamath, E. Mesner, and V. Isler. Trangulaton based mult target trackng wth moble sensor networks. IEEE Internatonal Conference on Robotcs and Automaton, pages , 27. [2] R. Olfat-Saber. Flockng for mult-agent dynamc systems: Algorthms and theory. IEEE Transactons on Automatc Control, 5(3):4 42, 26. [3] R. Olfat-Saber. Dstrbuted trackng for moble sensor networks wth nformaton drven moblty. Proceedngs of the 27 Amercan Control Conference,, pages ,

6 Fg. 2. Snapshots of the moble sensor network (a) when the moble sensors form a network, (b) when the moble sensors avod obstacles, (c) when the moble sensors get stuck n the narrow space between two obstacles. These results are obtaned by usng algorthm (5) Fg. 3. Snapshots of the moble sensor network (a) when the moble sensors form a network, (b) when the moble sensors avod obstacles, (c) when the moble sensors successfully passed through the narrow space between two obstacles, (d) when the moble sensors recover the orgnal sze. (a, b, c, d ) are closer look of (a, b, c, d), respectvely. These results are obtaned by usng algorthm (3) Fg. 4. Velocty matchng among sensors, connectvty, and error of postons between CoM and the movng target n (a, b, c) usng algorthm (3), (a, b, c ) usng algorthm (5), respectvely. [4] H. G. Tanner, A. Jadbaba, and G. J. Pappas. Stable flockng of moble agents, part : fxed topology. Proceedngs of the 42nd IEEE Conference on Decson and Control, pages 2 25, 23. [5] H. G. Tanner, A. Jadbaba, and G. J. Pappas. Stable flockng of moble agents, part : dynamc topology. Proceedngs of the 42nd IEEE Conference on Decson and Control, pages 26 22, 23. [6] Y. Yang, N. Xong, N. Y. Chong, and X. Defago. A decentralzed and adaptve flockng algorthm for autonomous moble robots. The 3rd Internatonal Conference on Grd and Pervasve Computng Workshops, IEEE computer socety, pages , 28. [7] G. Lee and N. Y. Chong. Adaptve self-confgurable robot swarms based on local nteractons. Proceedngs of the 27 IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems, pages , 27. [8] G. Lee and N. Y. Chong. Adaptve flockng of robot swarms: Algorthms and propertes. IEICE transactons on Communcatons, (9): , 28. [9] P. Ogren, E. Forell, and N. E. Leonard. Cooperatve control of moble sensor networks: Adaptve gradent clmbng n a dstrbuted envronment. IEEE Transactons on Automatc Control, 49(8):292 32, 26. [] R. Olfat-Saber and R. M. Murray. Consensus problems n networks of agents wth swtchng topology and tme delays. IEEE Transactons on Automatc Control, 49(9):52 533, 24. [] R. Olfat-Saber, J. Alex Fax, and R. M. Murray. Consensus and cooperatve n networked mult-agent systems. Proceedngs of the IEEE, 95():25 233,