Proceedings of the 2001 IEEE Internationa Conference on Robotics & Automation Seou, Korea May 21-26, 2001 Tracking Contro of Mutipe Mobie Robots A Case Study of Inter-Robot Coision-Free Probem Jurachart JONGUSUK, Tsutomu MITA Mita Laboratory, Contro and Systems Engineering Tokyo Institute of Technoogy, emai:jira@ctr.titech.ac.jp Abstract A virtua robot tracking contro approach with a consideration of cearance is proposed, in the same time, an agorithm which intends to avoid coision between robots is aso introduced using a transformation of Cartesian to distance coordinate. Each robot is indexed by different number, priority number to indicate that which robot is prior to another. A smainterva feedback contro is appied to a ow-priority robot to avoid coision with the higher one when it detects a possibiity of coision. At the end of sma-interva feedback contro, the coision-free condition is estabished. The virtua robot tracking contro is appied again to maintain the overa formation. As a resut, the distribution of robots in a system is guaranteed to be in imited communication range aong the motion. Numerica simuation highights the efficiency of our contro techniues. Keywords: mutipe mobie robots, feedback, coision, tracking contro. 1 Introduction Contro of mutipe mobie robots has been subject of a considerabe research effort over the ast few years. The rapid progress in this fied has been the interpay of systems, theories and probems[1. Three exampes are introduced as the recent research trends, namey foraging, box-pushing and traffic contro, foraging addresses the probem of earning and distributed agorithm; box-pushing focuses particuary on cooperative manipuation; and traffic contro addresses the probem occurring when mutipe robots move in a common environment that typicay attempt to avoid coisions. Despite the vast amount of papers pubished on the coision avoidance motivated by the case of traffic contro, the majority has concentrated on centraized path and motion pannings, which mosty utiize feed-forward contro in order to accompish a goa eading aso to a computationa time consumption, see for exampes Arkin[2 and Lavae[3, whie ess attention has been paid to contro of the mutipe mobie robots by a concept of feedback contro. Desai[5 has studied the probem of contro and formation where a notation of - contro is originay presented, however, the issue of sensor capabiity is absent. - contro gives a motivation of using reative dynamics in the system, which wi be further extended in our paper. In this paper, we focus on tasks in which the mutipe mobie robots are reuired to foow a given trajectory in eader-foower formation whie avoiding coision among themseves. In addition, because of the high-bandwidth of inter-robot communication, consideration of communication capabiity becomes necessary. Here, a imit range of communication is considered. The probem of contro and coordination for mutipe mobie robots is formuated using decentraized controers. We assume that there are no obstace in the work space, which simpifies the probem to avoiding inter-robot coision. We aso propose a simpe techniue that heps in decision process of coision avoidance based on geometric anaysis. This proposition corresponds to the issue of Latombe[4 stated that coision avoidance is an intrinsicay geometric probem in configuration space. 2 Probem Formuation A unicyce-type mobie robot, which kinematic mode is defined by i = B i u i = cos i 0 sin i 0 u i (1) 0 1 where i =[x i, y i, i T and u i =[v i, ω i T is considered. This mode captures the characteristic of a cass of restricted mobie robot which is what we need to investigate. The robot are assumed to satisfy pure roing and non-sipping conditions, which ead to nonhoonomic veocity constraints, non-sipping : ẋ i sin i ẏ i cos i =0, (2) pure roing : ẋ i cos i +ẏ i sin i = v i (3) 2.1 Assumptions We summarize here a the assumptions used in this paper. (i) System is homogeneous, i.e., a robots are of the same mode described in (1) and satisfy the veocity constraints, (2) and (3). (ii) The workspace (xy-pane) is fat and contains no obstace. (iii) The reference robot foows a smooth trajectory, and maintains positive veocity. (iv) Each foower robot is indexed by different priority number. (v) Each robot can extract any necessary information from its communication euipment. 0-7803-6475-9/01/$10.00 2001 IEEE 2885
2.2 Probem Statement To keepour investigation on the way, the probem is preiminariy stated as foows. Statement 1 Giving initia positions and orientations, i for the foower robot i, and the motion of reference robot, it is reuired that contro aw u i gives a motion satisfied assumptions (i)-(v) such that, as t, two feedback contro approaches, VR tracking contro and - contro that aso fufi the reuirement in item (3) of the probem, which are the keys of the main issue. 3.1 Detecting Coision 1. formation is estabished, 2. no coision among robot i and any robot j, and 3. a whoe motion satisfies the imitation of communication range. 3 Controer Design and Anaysis We arrive here with assumptions defined in the ast section. We wi first appy a VR tracking contro approach 1 to the system to compete the reuirement in item (1) of the probem whie adding an abiity of coision checking. The check is done in a sufficienty sma samping time. When a robot detects possibiity to coide with another, it checks if its priority number is higher (sma vaue) or not. If not it has to change the contro to - contro 2 in order to avoid the coision and compete the reuirement of item (2) of the probem. It is easy to understand the agorithm above by the fowchart shown in Figure 1. Figure 2 : Coision avoidance mode A simpe techniue used to detect the coision between any two robots can be reaized by modeing both robots to be covered with doube circes centering at the contro points, see Figure 2. The soid ones cover the entire robots with radius D whie the outer dotted circes with radius d + D is designed for the reuired cearances between robots. For convenience, we define the distance between two robots in the foowing definition. Definition 1 Let (x i,y i ) and (x j,y j ) denote the contro points of robot i and robot j, respectivey. Distance between robots is defined by ρ ij = (x i x j ) 2 +(y i y j ) 2 (4) Then, we define a function that can be used to check the coision between robots i and j as foows. f ij = ρ 2 ij 4(D + d) 2 f ij > 0 safety (5) 0 coision f ij mathematicay represents a condition for item (2) of the probem statement. Note that the choice of d shoud be designed with appropriate vaue. A arge d increases safety eve but decreases the smoothness of the robot because it detects the possibiity of coision more often, which yieds more switching of contro inputs. Figure 1 : Agorithm fowchart Referring to this agorithm, we need to design a simpe coision detecting techniue and introduce 1 See detais in section 3.2. 2 See detais in section 3.3. 3.2 Virtua Robot Tracking Contro This approach offers a tracking contro with arbitrary cearance between robots, which is, in fact, necessary for navigating the mutipe mobie robots whie maintaining the formation as we need. However, the coision that can occur during transient state of the contro is not considered. We give the detai of this approach in this section. To easiy separate many robot configuration symbos, we wi use the foowing definitions thereafter. r =[x r, y r, r T refers to a reference robot, i = [x i,y i, i T, i =1, 2,... refers to foower robot i and vi =[x vi, y vi, vi T is for VR of foower robot i. 2886
where B vi is the matrix defined in (7), v r denotes reference robot s transationa veocity and u i denotes foower robot s veocity vector, and z vi = [ xvi y vi, z r = [ xr y r, b r = [ cos r sin r After appying I/O inearization, we obtain u i = B vi 1 (b r v r λz ei ) (9) Figure 3 : VR tracking mode In order to separate foower from reference robot avoiding coision, a concept of VR is defined with additiona objective to force error parameters to zero when system approaches fina time, as foows. Definition 2 Virtua robot (VR) : a hypothetica robot whose orientation is identica to that of its corresponding foower robot, but position is paced apart from by the predefined r- cearances. The symbos, r denote ongitudina cearance and cearance aong rear whee axis, respectivey. The reation between VR and the foower robot is written as foows. x vi = x i r sin i + cos i y vi = y i + r cos i + sin i (6) vi = i Note that the above euation is satisfied when the foower is on VR s right-hand side. From (6) and assumptions (2), (3), we can derive the kinematic mode of VR as foows. vi = cos i r cos i sin i sin i r sin i + cos i u i = [ 0 1 B vi 0 1 u i (7) Figure 3 describes the parameters of VR on xy-pane. A standard techniue of I/O inearization is used to generate a contro aw for each foower[7. A property inherited in this controer is described by the foowing statement. Property 1 VR tracking contro gives soution that exponentiay converges in an interna shape x e and y e. Proof. An error mode is formuated as ż ei = z vi z r = B vi u i b r v r (8) where λ denotes positive-constant diagona matrix. The soution of this controed system, is z ei = e λt z ei (0) (10) which itsef tes that the soution exponentiay converges in an interna shape x e and y e. Moreover, soution is we-defined because parameter is designed not to be zero, hence, B vi 1 is non-singuar. Curve in Figure 3 iustrates this property. In addition, zero dynamics given in term of i must be proved to be stabe in order to guarantee the use of contro aw in (9). The differentia euation for i is given by i = ( λ 1x ei v r cos r ) sin i + ( λ 2y ei + v r sin r ) cos i (11) or can be written in the form i = κ 1 sin( i + κ 2 ) (12) where κ 1 = 1 (λ 1 x e v r cos r) 2 +( λ 2 y e + v r sin r) 2 κ 2 = arctan λ 2y e + v r sin r λ 1 x e v r cos r The trajectory, i e = κ 2 arcsin wr κ 1 is shown to be exponentiay stabe by inearization[8 of (12) as foows. i = κ 1 cos( i + κ 2 ) i=e i i (13) = κ 2 1 w2 r i (14) Note that cos(arcsin(a)) = 1 a 2 and denotes a deviation operator. Coroary 1 Property 1 ensures that appying (9) to foower robot wi automaticay avoid coision with reference robot if the condition beow is satisfied. ρ ri (t 0 ) > 2(d + D) (15) where ρ ri denotes distance between reference and foower robot i and t 0 is initia time. Proof. Reca that (10) ensures exponentia convergence of ρ ri. Aso, the predetermined r- cearance impies that ρ ri r 2 + 2. Obviousy, if robots do not start with coision or (15), the motions of foower robots are free of coision with reference robot. 2887
3.3 - Contro The aim of this contro is to maintain the desired engths, d 13 and d 23 of considered robot (Robot 3 in Figure 4) from its two eaders. Refer to the main agorithm, at any coision occurrence, the owest priority robot wi be switched from VR tracking contro to this contro. - contro is originay presented in Desai[5, however the convergence of states is not estabished within finite time and is aimed for maintaining formation. Here, we utiize - contro as coision avoidance controer, which reuires that formation must be estabished within finite time. Figure 4 : Notation for - contro mode The kinematic euations for the Robot 3 is given as foows. [ [ [ 13 cos γ1 D sin γ = 1 v3 23 cos γ 2 D sin γ 2 ω 3 [ v1 cos ψ 13 v 2 cos ψ 23 ψ = B u 3 v (16) and 3 = ω 3 where γ i = i + ψ i3 3,(i =1, 2). Remark B is singuar ony if γ 2 γ 1 = nπ, (n =..., 1, 0, 1,...) which determines the three robots ie on the same ine connecting them. For the purpose of coision avoidance, a property beow is reaized by I/O inearization. Property 2 - contro gives soution that monotonicay converges in an interna shape 13 and 23 within finite time. Proof. Reuired that the controed variabes 13 and 23 monotonicay converge to 13 d and 23, d respectivey, within finite time T r, we set [ [ [ 13 α1 0 ( = 13 d 13) p 23 0 α 2 (23 d 23) p (17) = α e (18) which is identica to a termina attractor mode where (p, ) can be, for exampe, set to (2, 3). See appendix ψ B for detais. By arbitrary setting reaching time T r, we obtain a formua for finding α 1, α 2 as foows. α 1 = sign( d 13 13(0)) d 13 13(0) 1 3 /( T r 3 ) α 2 = sign(23 d 23(0)) 23 d 23(0) 1 T r 3 /( 3 ) The sign( ) and are used to avoid compex soutions. We can generate a feedback contro aw as foows. u 3 = B 1 (v + α e ), 0 t T r (19) The soution of this controed system is ( i3 = (i3 d i3(0)) 1 α ) 3 i 3 3 t,i=1, 2 (20) which proves the properties. Note that the term in parenthesis is inear. Again, zero dynamics in term of 3 can be proved stabe, ceary when two eaders foows parae straight ines with the same veocities or v 1 = v 2 = v >0, ω 1 = ω 2 =0, 1 (0) = 2 (0) = 0. From (16), (19), we have 3 = v cos ψ 13 cos(ψ 23 + 0 3 ) D sin(ψ 13 ψ 23 ) v cos ψ 23 cos(ψ 13 + 0 3 ) D sin(ψ 13 ψ 23 ) = v D sin( 0 3 ) (21) Lyapunov candidate, V =1 cos( 0 3 ) guarantees the stabiity of (21). In case of more compicate motions of two eaders, the stabiity is guaranteed by observing that ρ 12 exponentiay decreases according to property 1. This means 13, 23 are aways determined or the ony singuarity γ 1 γ 2 = 0 is not passed. Remark Note that (20) aso gives an interesting convergent motion in an interna shape 13 and 23. It matches with our reuired motion when coision is detected, i.e., when Robot 3 gets cose to the two eaders, with the contro (19), it tends to monotonicay separate from the eaders avoiding coision. 3.4 Combination of VR and - Contro We can formuate a contro of the whoe system using the combination of three techniues as stated in section 3.1-3.3. The fow of contro process is formery presented in this section, however conditions in which the contro process can be performed propery are not anayzed. We summarize a imitations occurs from the combination as foows. 1) r 2 + 2 > 4(D + d) 2 (22) 2) i3 d > 2(D + d), i =1, 2 (23) 3) ρ 12 <13 d + 23 d (24) 2888
Condition (22) is extracted from the first reuirement in item (1) of the probem statement and the design of coision detection in section 3.1. Condition (23) comes from the restriction of the robots not to get coser than safety regions when - contro is performed. Again, in - contro phase, 13 d and d 23 must be designed in such a way that three robots can form a triange formation at the end of contro, which yieds (24). As stated in section 3.3, when three robots ie on the same ine connecting them, the contro aw u 3 in (19) is not determined. In addition, the reuirement in item (3) of the probem states that communication range is imited, namey R as maximum range. These two facts enabe us to define the accessibe area, shaded ones in Figure 5, where the ow priority robot can move on during coision avoidance. Hence, in - contro phase, the design of 13 d and 23 d (euivaent to the design of P x ) is simpy done by setting the vaues within accessibe area. by VR tracking contro aw; u i in (9) as a formation controer and - contro aw; u 3 in (19) as a coision controer, gives a motion of the form, ρ ri R, i =1, 2 (25) whie avoiding inter-robot coision and maintaining a predetermined formation in the steady state. R is positive scaar corresponding to maximum communication ength of reference robot. Proof. It can be simpy checked that the distribution of the three robots is guaranteed to be inside a circe of radius R centered at reference robot by property 1, property 2 and design process in section 3.4 aong the motion if the two foower robots are initiay in this circe. 4 Simuation Resuts The aim of these simuations is to vaidate our tracking contro method for a mutipe mobie robot (three-robot case). Our goa is to make the three robots form a eader-wing motion. Initia parameters are set as foows. Figure 5 : [Left Target TG 2 is outside accessibe area : a < b [Right Target TG 2 is inside accessibe area : a > b In fact, the design of P x can be cassified into two cases: desired target is outside accessibe area, Figure 5 [Left and inside the area, Figure 5 [Right. The two cases can be distinguished by the condition beow. (For the sake of simpicity, we assume that reference robot moves aong the positive y-direction.) if ese a < b Target is outside accessibe area Target is inside accessibe area where a, b [0, 2π) are measured in cockwise fashion. a is an ange measured from foower 2 to foower 1 centered at reference robot whie b is measured from foower 2 to its formation target TG 2. Remark In case of Figure 5 [Left, it is strongy recommended to define P x behind foower 1 to ensure that coision wi hardy or not repeat after switching to VR tracking contro. 3.5 Main Issue Finay, we summarize a resuts in this paper as a main issue in theorem beow. Theorem 1 A system of three mobie robots whose modes are defined by (1), operating in an environment described by assumptions (i)-(v) and controed Common : T r =1.5s, D =2.25, d=1, R =25 Reference : r (0) = [0, 0, π 2 T,u r =[5, 0 T Foower 1 : 1 (0) = [6, 6, π 2 T, (r, ) =(6, 6) Foower 2 : 2 (0) = [10, 2, 3π 4 T, (r, ) =( 6, 6) Assume aso that reference robot has the highest priority and Foower robot 2 has the owest one. From Figure 7, we see that motion of foower robot 2 is kept within the maximum communication range R with a desired formation estabished in the steady state. See snapshot in Figure 6 for an entire motion of system. According to tabe 1, four of coision occurrences between the two foowers are found. The number of coision can be decreased by modifying d 13, d 23 and T r in each occurrence, here we simpy use constant vaues. No.\Info. t 13 d 23 d case 1 0.3925 18.1325 10.0000 outside 2 2.2648 18.1325 10.0000 outside 3 4.1099 18.1325 10.0000 inside 4 5.9663 18.1325 10.0000 inside Tabe 1 : Coision information 2889
These topics are the subject of our future work. References Figure 6 : Snapshot of motion Figure 7 : Distance from Reference to Foower 2 in r- direction 5 Concusion and Future Work In this paper, we have studied strategies for controing mutipe mobie robots (three-robot case) focused on a coision-free movement. The foower robots track the gait of reference robot using virtua robot tracking contro. The extended contro techniue used to avoid the coision of the threetupe robots is appied during finite time interva in such a way that the owest priority robot can move away from the others. After the coision-free condition is estabished, the controer is switched back to the virtua robot tracking contro and maintain the desired formation and so on. The convergences and stabiities of both controers are throughy studied, eading to the main issue in theorem 1. We aso pan to enarge our framework to the probems of obstace avoidance and formation changes, which we beieve that successfu strategies are strongy based on the geometric cacuation as stated in Latombe[4. According to theorem 1, the extension of our work to the system of more than three robots is possibe particuary when we have a number of groups of three-tupe robots to be controed. [1 Y. U.Cao, A. S. Fukunaga, A. B. Kahng: Cooperative Mobie Robotics; Autonomous Robots, Vo. 4, pp. 7 27 (1997) [2 T. Bach, R. Arkin: Communication in Reactive Mutiagent Robotic Systems; Autonomous Robots, Vo. 1, pp. 27 52 (1994) [3 S. LaVae, S. Hutchinson: Optima Motion Panning For Mutipe Robots Having Independent Goas IEEE Transaction on Robotic and Automation Vo. 14, No. 6, pp. 925 958 (1998) [4 J. C. Latombe: Motion Panning: A Journey of Robots, Moecues, Digita Actors, and Other Artifacts; Internationa J. of Robotics Research, Vo. 18, No. 11, pp. 1119 1128 (1999) [5 J. P. Desai, J. Ostrowski, V. Kumar: Controing Formations of Mutipe Mobie Robots; IEEE Proceeding of Int. Conf. Robotics and Automation, pp. 2864 2869 (1998) [6 J. P. Desai, J. Ostrowski, V. Kumar: Contro Of Changes In Formations For A Team Of Mobie Robots; IEEE Proceeding of Int. Conf. Robotics and Automation, pp. 1556 1561 (1999) [7 J. E. Sotine, W. Li: Appied Noninear Contro, Prentice Ha, NJ (1991) [8 H. Kogo, T. Mita: Introduction to Contro System Theory (2 nd Ed., Japanese), Jikkyo Press, Japan (1997) A Appendix Termina Attractor To estabish the convergence of z ei in arbitrary finite time, T r, consider the foowing emma. Lemma 1 The origin of a differentia euation ż = λz p, λ > 0 (A-1) is a termina attractor with a finite reaching time p T r = z(0)1 λ(1 p ), p (0, 1) (A-2) where p, I +, I + = {positive integer} Proof. Direct integration of (A-1) yieds z(t) 1 p = z(0) 1 p (1 p )λt (A-3) Reuiring z(t r) = 0 impies formua (A-2). Note that T r can be chosen arbitrary from the choice of λ. We shoud take care of the choice p,, which is suggested in proposition beow. Proposition 1 The choice of p and in (A-1) shoud be seected such that (p, ) Ω. Ω={(p, ) p <, is odd} (A-4) According to (A-1) and (A-2), p and 1 p must be designed to be odd-root in order to avoid 2i-root of negative vaues. In addition with the assumption that p < 1, Ω is estabished. 2890