Cooperative Collision Avoidance between Multiple Mobile Robots

Transcription

1 Cooeratve Collson Avodance between Multle Moble Robots Atsush Fujmor,* Masato Teramoto Deartment of Mechancal Engneerng Shzuoka Unversty 3-5-1, Johoku, Hamamatsu , Jaan e-mal: Peter N. Nkforuk, Madan M. Guta Deartment of Mechancal Engneerng Unversty of Saskatchewan Saskatoon, Saskatchewan S7N 5A9, Canada Receved March 27, 1999; acceted February 10, 2000 Ths aer resents a new collson avodance technque, called cooeratve collson avodance, for multle moble robots. The detecton of the danger of collson between two moble robots s dscussed wth resect to the geometrc asects of ther aths as are cooeratve collson avodance behavors. The drecton control command and the velocty control command for the cooeratve collson avodance are then roosed. The avodance technque s extended to cases n whch the number of moble robots s more than two. Furthermore, the condtons for collson avodance are consdered wth resect to the navgaton arameters and gudelnes of desgnng the navgaton arameters are obtaned. The effectveness of the roosed technque s demonstrated by means of numercal smulaton and navgaton exerments usng two real moble robots named Poneer John Wley & Sons, Inc. I. INTRODUCTION Moble robots have been the subject of much research and develoment due to ther alcablty n a wde varety of tasks n ndustry, human su- * To whom all corresondence should be addressed. orted works, and elsewhere. If the task requested cannot be easly carred out by a sngle robot, multle robots should be cooeratvely used. 1 4 Snce the use of multle robots may lead to collson, ther navgaton and collson avodance have been dscussed from varous onts of vew. In Ref. 1, for examle, a arallel rocessng system was con- () ( ) Journal of Robotc Systems 17 7, by John Wley & Sons, Inc.

2 348 Journal of Robotc Systems 2000 structed for the cellular robotc system and was aled to the behavor decson of multle moble robots. Matarc 5,6 roosed avodance behavor rules when a collson between two moble robots was redcted: stong the robots for a fxed erod or changng ther drectons. Arkn 7 and Sughara et al. 8 roosed algorthms combnng these rules. Shan and Hasegawa 9 roosed behavorbased moton lannng of multle moble robots n a narrow assage. On the other hand, learnng technques were ncororated nto neural networks andor fuzzy logc controllers for moble robot navgaton. 10,11 Iteratng navgaton of moble robots n a gven envronment, navgaton arameters assocated wth robot s behavors were adjusted by the learnng technques. These aroaches dd not gve analytcal dscussons for selectng the navgaton arameters although the effectveness was shown n smulaton and navgaton exerments. Therefore, the ntal selecton of the navgaton arameters remans to be solved. Ths aer resents the navgaton of multle moble robots wth collson avodance n a two-dmensonal free-sace envronment and dscusses collson avodance condtons from the theoretcal onts of vew. Each robot has ts own goal or oston to be navgated. The objectve s to navgate the multle moble robots to ther goals wthout collsons between them. To do ths, ths aer rooses a new collson avodance technque called cooeratve collson avodance and gves the avodance condtons wth resect to the navgaton arameters. The navgaton scheme used n ths aer s based on the adatve navgaton technque 12 n whch the navgaton law s gven by a frst-order dfferental equaton, and the navgaton of the robot to the goal and obstacle avodance are done by swtchng the drecton angle command adatvely. In ths aer, the cooeratve collson avodance behavors of the multle moble robots are then ncororated nto the adatve navgaton technque. The remander of ths aer s organzed as follows. The statements of roblem are resented n Secton II. The cooeratve collson avodance s dscussed n Secton III. The detecton of collson between two moble robots s frst dscussed wth resect to asects under whch the two robots encounter each other. The drecton control command and the velocty control command for the cooeratve collson avodance are then roosed. The avodance technque s extended to cases n whch the number of moble robots s more than two. The condtons for collson avodance are consdered wth resect to the navgaton arameters n Secton IV. Accordng to the condtons, a desgn rocedure of the navgaton arameters s gven. The effectveness of the roosed technque s demonstrated by means of numercal smulaton examles n Secton V. In Secton VI, the roosed technque s aled to an exermental robot named Poneer-1. The concluson s resented n Secton VII. II. STATEMENT OF PROBLEM Ths secton descrbes a navgaton roblem of multle moble robots n a two-dmensonal free-sace envronment. Let n be the number of moble robots. r The th moble robot s denoted as R and ts oston s reresented by the cartesan coordnates Žx Ž t., y Ž t.. where t s tme. The drecton angle of the th robot s Ž t., Ž Ž t. rad., whch s measured from the x-axs. The velocty of the th robot s v Ž t.. The equatons of moton of the nr moble robots are then gven by x Ž t. v Ž t. cos Ž t. Ž 1,..., n.. Ž 1. r y Ž t. v Ž t. sn Ž t. The objectve s to navgate the nr moble robots from ther startng onts S Ž 1,..., n. r to ther own goals G Ž 1,..., n. r wthout any collsons between them. The followng assumton s gven on the robots. ( ) A1 All of moble robots have the same sze and the same erformance. The radus of the robots s r. R ( A2) Žx Ž t., y Ž t., Ž t., v Ž t.. Ž 1,..., n. r are known to each other by some means of communcaton. The drecton angle of the moble robots s gven by Ž t. Ž t. Ž t. Ž 1,..., n., Ž 2. r where s a ostve constant. Ths rocess can be used not only to navgate moble robots to ther goal, but also to avod collsons by swtchng the drecton angle command Ž t.. 12 Equaton Ž. 2 s called the drecton control.

3 Fujmor et al.: Collson Avodance between Robots 349 Smlarly, the velocty of the moble robots s gven by v Ž t. v Ž t. v Ž t. Ž 1,..., n., Ž 3. v r where 0v v t v. 4 mn max s a ostve constant, v Ž. v t s the velocty com- mand, vmn s the mnmum velocty, and vmax s the maxmum velocty. Equaton Ž. 3 s called the velocty control. Based on the adatve navgaton technque, 12 the drecton angle command Ž. t and the velocty command v Ž. t are gven accordng to the follow- ng three modes: Fgure 1. Two moble robots. Navgaton mode: Navgate a robot to ts goal f no collson s detected and the robot s not at goal. Lettng Ž. t be the angle from the oston of the th robot R to ts goal G, Ž t. s gven by Ž t. Ž t., whle v Ž t. s gven by v Ž. t v 0, where v0 s the navga- ton velocty. Cooeratve avodance mode: Make a robot avod collsons wth other robots f collsons are redcted. Ž. t and v Ž. t wll be gven n the later sectons. Fnal mode: Aroach a robot to ts goal when the robot s near the goal. Let lg be the dstance between the th robot R and ts goal G. Near means that l d, where d s a G f f range to the fnal mode. When navgatng robot R to goal G, let t f be the tme n whch l becomes l d frst. Then, Ž t. s fxed G G f as Ž t. Ž t. for tt. v Ž. f f t s decreased to zero whle aroachng to the goal. control commands for the avodance. The cooeratve collson avodance between two moble robots s then extended to cases n whch the number of moble robots s more than two. A. Collson Detecton and Mode Selecton Fgure 1 shows a stuaton where the oston, drecton angle, and velocty of two moble robots R 1 and R are Žx Ž. t, y Ž.. t, Ž. t, and v Ž.Ž t 1, 2. 2, resectvely. If the robots face n the drecton of the dashed arrows, a collson wll not haen. However, f the robots face n the drecton of the sold arrows, the two moble robots may collde wth each other at a ont P 12, whch s called the redcted crossng ont. Let lr j be the dstance between robots R and R j. In Fg. 1, lr and dlr dt are gven by The detals of the navgaton mode and the fnal mode are descrbed n Ref. 12. The rest of ths aer therefore deals wth the cooeratve avodance mode. III. COOPERATIVE COLLISION AVOIDANCE Ths secton resents the collson avodance between multle moble robots, that s, the cooeratve avodance mode. Frst, a concet of the cooeratve collson avodance s descrbed wth two moble robots, that s, the collson detecton, the cooeratve collson avodance behavors, and the 2 2 R l 'Ž x x. Ž y y. Ž 5. dlr 12 1 Ž v cos v cos.ž x x dt l R12 v sn v sn y y Usng them, the danger of collson s classfed as Table 1. Evaluatng the danger of collson, t s seen that an avodance behavor s requred for Case 4. Thus, lr and dlr dt can be used as a swtch to the cooeratve avodance mode. Usng l G, l R, and j dl dt, the mode selecton of robot R s gven as Rj

4 350 Journal of Robotc Systems 2000 Table 1. Danger of collson wth resect to l and R dl dt. R Case lr dlrdt Danger of collson 1 long 0 Danger s small and s gong to be decreased. 2 long 0 Danger s small but s gong to be ncreased. 3 short 0 Danger s great but s gong to be decreased. 4 short 0 Danger s great and s gong to be ncreased. follows: f lg d f then fnal mode s selected. else f lr d and dlr dtd j j v then cooeratve avodance mode s selected. else navgaton mode s selected. end f d Ž 0. and d Ž 0. v are, resectvely, the dstance- and the dervatve-swtch to the cooeratve avodance mode to be desgned. B. Cooeratve Collson Avodance Behavors As descrbed n Secton II, moble robots have two control strateges, the drecton angle control and the velocty control for navgaton and collson avodance. The urose of ths secton s to resent an avodance technque usng these strateges for the cooeratve collson avodance. Let and Ž,. be defned as the crossng angle and the relatve angle resectvely as shown n Fg. 1. Usng the crossng angle, asects under whch two moble robots may encounter each other n a two-dmensonal free-sace envronment are classfed as gven n Fg. 2. In Fg. 2Ž. a, a collson wll occur unless the robots change ther drectons. In ths case, the drecton control s requred to avod collson. The velocty control, however, s of no use. In Fg. 2Ž. c, on the other hand, a collson may be avoded by changng the drectons of the two robots, or changng ther veloctes so that they do not go through the redcted crossng ont at the same tme. From the vewont of safety and control, the velocty control s more effectve n avodng collson than the drecton control. When usng the velocty control, a decson has to be made as to whch robot should go through the redcted crossng ont frst. In ths aer, two rortes of Fgure 2. Asects under whch two moble robots encounter each other. two moble robots, hgh rorty and low rorty, are frst defned, and the behavors of the two moble robots are then decded accordng to these rortes. As an examle, n Fg. 2Ž. c, the robot wth the hgh rorty s accelerated to go through the redcted crossng ont frst, whle the other robot wth the low rorty s decelerated. In Fg. 2Ž. e, where v2v 1, both the drecton control and the velocty control should be used to avod a collson. For examle, the drecton of robot R2 could be changed and ts velocty ncreased, whle the drecton of robot R1 could be ket constant and ts velocty decreased. These resonses tend to reduce the danger of collson as quckly as ossble. Fgure 2Ž b. s an ntermedate asect between Fgs. 2Ž. a and 2Ž. c, whle Fg. 2Ž. d s between Fgs. 2Ž. c and2ž. e. For these asects, the drecton control and the velocty control are used nterolatvely. The detals wll be gven later. C. Defnton of Prortes of Two Moble Robots As descrbed n the revous secton, a redcted collson s avoded so that the robot wth the hgh

5 Fujmor et al.: Collson Avodance between Robots 351 rorty acts ror to the other robot wth the low rorty. The hgh rorty should be then assgned to a robot that wll arrve at the redcted crossng ont frst. That s, the rortes of two moble robots are defned as the nverse of ther arrval tme at the redcted crossng ont n ths aer. In Fgs. 2Ž b., 2Ž c., and 2Ž d., the redcted crossng ont P s calculated by usng Ž x, y,. Ž 1, Denotng the dstance between the oston of robot R and the redcted crossng ont as RP 12, the rortes w Ž 1, 2. are gven by v w Ž 1,2.. Ž 7. RP 12 If there s no redcted crossng ont because the robots are movng along a straght lne such as shown n Fgs. 2Ž. a and 2Ž. e, the rortes are gven by w v Ž 1,2.. Ž 8. That s, the robot whose velocty s greater s assgned to the hgh rorty. If the rortes are obtaned as w1w 2, robot R1 s assgned to the hgh rorty whle robot R2 s the low rorty n ths aer. where and ch cl are addtonal angles of hgh and low rorty. They are gven by 2 chk sgn 1 Ž 11. cl sgn satž,0,,k,0, 2 / Ž 12. where 1 x0 sgn x ½ Ž x0 satž x, a, b, c, d. c xa dc Ž xa. c axb Ž 14. ba d xb. k s the maxmum of, satž. c s the saturaton functon and s shown n Fg. 3, and and ch cl are as shown n Fg. 4. Smlarly, let v c be the velocty command for the cooeratve avodance mode. It s gven by v ch ww j v, j1 or2, j, Ž 15. c ½ vcl wwj D. Drecton Angle Command and Velocty Command Ths secton resents a desgn of the drecton angle command and the velocty command to realze the cooeratve collson avodance behavors descrbed n Secton III B. Let Ž. c t be the drecton angle command for the cooeratve avodance of robot R. It s gven by where ž / v chsat,0,,v 0, vmax 2 Ž 16. v clsatž,0,,v 0, v mn, 2 / Ž 17., 9 c c where c s an addtonal angle for the cooera- tve avodance mode. As mentoned n Secton III B, the cooeratve collson avodance behavors deend on the rorty and the crossng angle. Accordng to the rorty, s defned as c ½ ch ww j, j1 or2, j, Ž 10. c cl wwj Fgure 3. Functon sat x, a, b, c, d.

6 352 Journal of Robotc Systems 2000 Fgure 4. Ž t.. c Addtonal angle for cooeratve avodance and v and v ch cl are, resectvely, the velocty com- mands of hgh and low rorty and are shown n Fg. 5. At 0 Fg. 2Ž. a, the drecton control s fully used whle the velocty control s not used. At 2 rad Fg. 2Ž. c, on the other hand, only the velocty control s used accordng to the rorty. Durng 02, and v c c are lnearly nter- olated as shown n Fgs. 4 and 5. Ths corresonds to Fg. 2b. E. Extenson of Cooeratve Collson Avodance Ths secton resents an extenson of ths cooeratve collson avodance to stuatons where the number of moble robots s more than two. In stuatons such as ths, t s to be noted that there exst multle relatons between any two moble robots. A moble robot has nr 1 relatons wth the rest of robots. Therefore, the drecton angle command c and the velocty command v c are decded so as to ntegrate the multle relatons. Consder frst the stuaton shown n Fg. 6 where three moble robots are movng toward each other. By evaluatng l and dl dt Ž, j1, 2, 3, Rj Rj Fgure 6. Three moble robots. j., the danger of collson s detected at two onts, P and P, where P Ž P j j s the redcted crossng ont of robots R to R j. Let wj be the rorty of robot R relatve to robot R j. When the danger of collson of robot R wth robot R j s detected, the rorty w s gven by j v w, Ž 18. j RP j where w s set to w0. If Pj s not found, wjv. Moreover, f the danger of collson s not detected, w 0. Equaton Ž 18. j s an extended ex- resson of Eq. Ž. 7. Table 2 shows the rortes for Fg. 6. The avodance behavors of robots R1 and R3 are resectvely decded accordng to the sngle relaton wth resect to robot R 2. On the other hand, robot R2 has two relatons wth resect to robots R1and R 3. It s referable that the behavor of robot R2 s gven by evaluatng all the relatons n whch t s nvolved. Then, the avodance behavor of robot R s determned by takng the center of gravty whose weght s the rorty w. Then, j c Table 2. Fg. 6. Weghts between three moble robots of Fgure 5. v c. Velocty command for cooeratve avodance Oonent R R R R1 0 w12 0 R2 w21 0 w23 R3 0 w32 0

7 Fujmor et al.: Collson Avodance between Robots 353 and v are gven by c Ý n r j1 j cj n Ý r j1 wj w Ž 19. c w v v, Ž 20. c Ý n r j1 j cj n Ý r j1 wj where and v cj cj are the drecton angle command and the velocty command for the cooeratve collson avodance of robot R wth robot R j, resec- tvely. IV. COLLISION AVOIDANCE CONDITION AND NAVIGATION PARAMETERS As mentoned so far, there are several navgaton arameters assocated wth moble robots, avodance behavors, and mode selecton. They are classfed as follows: derve the avodance condtons: ( A3) Let t0 be the tme when the cooeratve avodance mode s selected. The veloctes of two moble robots are v t v t v ( ) A4 The navgaton velocty v s gven by 0 vmax vmn v. Ž () 1 0. Under assumtons Ž A1. Ž A3., a condton for collson avodance of 0 rad s gven as follows. Theorem 1: Consder an asect under whch two moble robots encounter each other wth the redcted crossng angle 0 rad. Then, a collson between the robots s avoded only f the followng relaton s held durng the cooeratve avodance mode: Robot arameter: r, v,, R 0 v Avodance arameter: k, v, v max mn Mode selecton arameter: d, dv The robot arameters are assocated wth moble robots themselves and are gven n advance. The avodance arameters are assocated wth the cooeratve collson avodance behavors and reman to be desgned. The mode selecton arameters are also to be desgned. Ths secton frst dscusses the condtons for collson avodance between two moble robots wth the navgaton arameters. The condtons are used for desgnng the avodance and the mode selecton arameters. For convenence n the followng sectons, d s relaced by a non-d- mensonal arameter l defned as where 1 cosž l 1. A 4 Ž 24. 1A k rr A. Ž 25. Proof: At 0, only the drecton control s used. Fgure 7 shows a stuaton n whch two moble robots aroach each other closest. The trajectores of the robots are symmetrc wth resect to the redcted crossng ont P 12. Durng the cooeratve avodance mode, s ket at 0 rad. From v 0 d l. Ž 21. 2r R A. Collson Avodance Condton Although collson avodance condtons derved n ths secton are for tycal avodance cases whose redcted crossng angle s 0 and 2 rad, they are used as gudelnes for desgnng the navgaton arameters. The followng assumtons are gven to Fgure 7. Cooeratve avodance for 0 rad.

8 354 Journal of Robotc Systems 2000 Eqs. Ž.Ž 9 12., the drecton angle command for the cooeratve collson avodance s gven by k 1,2. 26 c Ž. Ž. Substtutng Eq. 26 nto Eqs. 1 and 2 and ntegratng them, we have t k tt t v k x Ž t. sn Ž t. sn Ž t.4 x Ž t. v0 Ž 28. y Ž t. cos Ž t. cos Ž t.4 y Ž t. 0 0 k 1,2. Wthout loss of generalty, t0 s gven by t00. The startng onts of robots R1 and R2 can be coordnated as Ž. x t, y t, t 0,0,0, Ž. x t, y t, t 2r l,0, R If rr RP 1 12 R2P12 s satsfed durng the cooeratve avodance mode, a collson s avoded. Let t a be the tme when the dstance of robots R1 and R2 s the shortest durng the cooeratve avodance mode and let ta be the tme when the two moble robots collde wth each other wthout changng ther drectons; that s, x Ž t. Ž l 1. 1 a rr t. Ž 31. a v v 0 0 c velocty v0 s ncreased, the rotatonal acceleraton of the drecton control, that s, and k, must be ncreased to kee a secfed A. () 2 2. Under assumtons Ž A1. Ž A4., a condton for collson avodance of 2 rad s gven as follows. Theorem 2: Consder an asect under whch two moble robots encounter each other wth the redcted crossng angle 2 rad. Then, a collson between the robots s avoded only f the followng relaton s held durng the cooeratve avodance mode, where t b r R gž t,., Ž 34. b v v vmax vmn v Ž v t gž t,. t Ž 1e b.. Ž 36. b v b s the tme whch satsfes the followng equaton: v v ' v t 2 r l v t Ž 1e b. 0. Ž 37. R max b Proof: At 2, only the velocty control s used. Fgure 8 shows a stuaton n whch two moble robots R1 and R2 are encountered where s 2 rad. At tt 0, the startng onts of robots R1 and R can be resectvely coordnated as Ž x,0,. v 2 10 Durng the cooeratve avodance mode, the followng relatons are held: Ž. Ž. t t, y t y t, cos t cos t a a 1 a 1 a 1 a 1 a 32 Usng Eqs. Ž 27. and Ž 28., a suffcent condton for r RP y Ž t. cos Ž t. s gven by R a 1 a 5 y Ž t. 1 a v0 1 r 1. Ž 33. R cos Ž t. k ½ cos Ž t. 1 a 1 a Defnng A as Eq. Ž 25., Eq. Ž 33. s wrtten as Eq. Ž 24.. In Eq. Ž 25.,, k, and rr are roortonal to A, whle v s nversely roortonal. If the navgaton 0 Fgure 8. Cooeratve avodance for 2 rad.

9 Fujmor et al.: Collson Avodance between Robots 355 and Ž 0, y, 2. 20, where x 10, y200. The redcted crossng ont P12 s then at the orgn. If x y and v Ž t. v Ž t v 0, the relaton of the rorty of the two robots s gven by w1w 2. For tt Ž 0., the veloctes of the two robots are 0 v t v t v ve 38 1 max v t v t v ve mn The ostons of the robots are gven by H x Ž t. x v Ž. d t v Ž v t x v t 1e. Ž max H t v y Ž t. y v Ž. d v Ž v t y v t 1e.. Ž mn The worst case for collson avodance s that y 20 x 0 and x ' rrl. For ths case, lettng tb be the tme when robot R1 arrves at the redcted crossng ont P, t satsfes Eq. Ž b. If 2rR y Ž t. x Ž t. 2 b 1 b s satsfed durng the cooeratve avodance mode, the collson s avoded. Usng gt, Ž. defned by Eq. Ž 36., we have b v 2r y t x t 2vg t, y x. R 2 b 1 b b v A suffcent condton for Eq. Ž 42. s gven by Eq. Ž 34.. For a fxed r, gt, Ž. R b v should be large enough to satsfy Eq. Ž 34.. gt, Ž. b v s a monotonously ncreas- ng functon because g g 0, 0 for t, 0. Ž 43. b v t b v Moreover, from Eq. Ž 37., l s also monotonously ncreasng wth resect to t b; we have v l 0, for t 0. Ž 44. b t b Therefore, a large v and a large l are referred to hold Eq. Ž 34.. That s, the acceleraton of the velocty should be great and the dstance-swtch to the cooeratve avodance mode d Ž 2r l. should be long. R B. Desgn of Navgaton Parameters Ths secton resents a desgn of the navgaton arameters k, v max, vmn d, and dv by takng nto consderaton the collson avodance condtons: Theorem 1 and 2. An objectve of moble robots n ths aer s to arrve at ther own goals. Although collsons must be avoded, the trajectores of moble robots should be made short from ther startng onts to goals as much as ossble. Equatons Ž 24. and Ž 34. n Theorems 1 and 2 are suffcent condtons for collson avodance. To revent from overactng the avodance behavors, the navgaton arameters are therefore desgned by takng the values near the lower bound whch satsfy Eqs. Ž 24. and Ž 34. n Theorems 1 and 2. A rocedure of desgnng the navgaton arameters s gven as follows. Ste 1. Gve the values of the robot arameters r R,, and v from the erformance of moble robots used. Decde v max, v mn, and dv from the navgaton objectves. d f s desgned so as to satsfy Theorem 1 n Ref. 12. Ste 2. Draw functon gt, Ž. b v and choose the value of tb near the lower bound. Ste 3. Calculate l by usng Eq. Ž 37.. Obtan d usng Eq. Ž 21.. Ste 4. Usng l obtaned n Ste 3, draw func- ton fž A. defned as 1 fž A. cosž l 1. A 4. Ž 45. 1A A ostve fž A. means that Eq. Ž 24. s held. Choose the value of A near the lower bound. Ste 5. Calculate k by usng Eq. Ž 25.. In Theorems 1 and 2, the dervatve-swtch d v was not dscussed. Snce the mode should be swtched to the cooeratve avodance mode for Case 4 n Table 1, a canddate of dv s zero. It should be adjusted by evaluatng avodance erformance. V. NUMERICAL SIMULATION Ths secton resents numercal smulaton examles to demonstrate the effectveness of the ro-

10 356 Journal of Robotc Systems 2000 osed cooeratve avodance behavors and collson avodance condtons for multle moble robots. Frst, two moble robots cases are shown accordng to the desgn rocedure of the navgaton arameters. Usng the navgaton arameters desgned, general cases n whch the number of robots n s 3 and 5 are shown next. Assumtons Ž A1. r Ž A4. were suosed to be held n these smulaton examles. A. Two Moble Robots Ste 1. The robot arameters used n the numercal smulaton were gven as rr 0.3, v0 4.0, 8.488, v Ž 46. Moreover, v, v, and d were gven as max mn v v 8.0, v 0.0, d max mn v From Ref. 12, d f was gven by Ste 2. Fgure 9 shows the lot of functon gt, Ž. wth The lower bound of collb v v son avodance condton Eq. Ž 34. was rrv The lower bound was then gven as tb 0.22 n Fg. 9. Ste 3. Substtutng the obtaned tb nto Eq. Ž 37., the lower bound of l was calculated as 2.8. To hold Eq. Ž 34., the value of l was chosen as l 3.1 and d was then calculated as Fgure 10 shows the trajectores of two moble robots for 2 rad and l 3.1 to hold Eq. Ž 34.. The trajectores were dslayed for a secfed erod k, and the number wrtten n the fgures ndcates the value of k. Fgure 11 shows the tme resonses of the velocty v Ž sold lne. and the command v Ž dashed lne.. Accordng to the dec- son of the rortes, the rorty of robot R1 was greater than that of robot R 2. It s seen from these fgures that a collson was avoded by usng only the velocty control. Robot R1 wth hgh rorty was accelerated whle robot R2 wth low rorty was decelerated after the cooeratve avodance mode was selected. Ste 4. Fgure 12 shows the lot of functon fž A. where l was gven as l 3.1. A ostve fž A. means that the collson avodance condton s satsfed. Postve ranges of fž A. for A 3.7 are meanngless because the robots turn more than 2 Fgure 9. Functon gt, wth b v v

11 Fujmor et al.: Collson Avodance between Robots 357 Fgure 10. Trajectores of two moble robots for 2 rad and l 3.1. durng the cooeratve avodance mode. The value of A was chosen as A Ste 5. k was calculated as rad 40.5 deg. Fgure 13 shows the trajectores of two moble robots for 0 rad where k rad to hold Eq. Ž 24.. Fgure 14 shows the tme resonses of the drecton angle Ž sold lne. and the command Ž dashed lne.. A collson was avoded by usng only the drecton control. To valdate Theorems 1 and 2, the followng smulaton examles are shown. Fgure 15 shows the trajectores of two moble robots wth l 2.4 whch dd not satsfy Eq. Ž 34., whle Fg. 16 shows the trajectores wth l 3.1 and k 0.22 rad 12.6 deg whch dd not satsfy Eq. Ž 24.. In both cases, collsons occurred near the redcted crossng ont. ( ) B. n 2 Moble Robots r Usng the desgned navgaton arameters n the case of two moble robots, numercal smulaton examles of robots nr s 3 and 5 are shown. Fgure 17 shows the trajectores of three moble robots wth Fgure 11. Tme resonses of velocty v Ž sold lne. and command v Ž dashed lne..

12 358 Journal of Robotc Systems 2000 Fgure 12. Functon fž A. wth l 3.1. Fgure 13. Trajectores of two moble robots for 0 rad.

13 Fujmor et al.: Collson Avodance between Robots 359 Fgure 14. Tme resonses of drecton angle Ž sold lne. and command Ž dashed lne.. dfferent startng onts and goals. Fgure 18 shows the trajectores of fve moble robots. Although some robots had multle relatons wth resect to others, ther avodance behavors were decded by ntegratng the multle relatons and all the robots were navgated to ther own goals wthout collsons. Now, we gve comments on the collson avodance condtons, Eqs. Ž 24. and Ž 34., when nr 2. Snce a robot has multle relatons wth the rest of robots, the collson avodance condtons may be volated even f the navgaton arameters were chosen accordng to the rocedure gven n Secton IV B. For examle, f robot R2 avoded robot R1 accordng to the cooeratve avodance behavors but robot R3 was ostoned very close to the area n whch robot R2 was movng, the avodance condtons between robots R2 and R3 may not had held. In the numercal smulaton, ths henomenon occurred n roorton as the number of robots was ncreased. For three moble robots, collsons were avoded n almost all examles, whle collsons were sometmes observed for fve moble robots esecally where the robots are ostoned densely. To mrove ths, the avodance behavors are deter- Fgure 15. Trajectores of two moble robots for 2 rad and l 2.4.

14 360 Journal of Robotc Systems 2000 Fgure 16. Trajectores of two moble robots for 0 rad, l 3.1, and k 0.22 rad. Fgure 17. Trajectores of three moble robots.

15 Fujmor et al.: Collson Avodance between Robots 361 Fgure 18. Trajectores of fve moble robots. mned by takng nto account not only the crossng angle but also the robot densty n a regon. That s, when ncreasng the densty, t may be a way that robots wth lower rorty sto or go back to hold the avodance condtons. Ths ssue wll be dscussed n a future research. VI. NAVIGATION EXPERIMENT USING PIONEER-1 Ths secton shows a navgaton exerment usng two real moble robots named Poneer-1 to verfy the roosed cooeratve collson avodance. Fgure 19 shows a hoto of Poneer-1. The radus s 0.15 m and the weght s 8 kg. Poneer-1 s a small moble robot develoed by Artfcal Intellgence and Grnnell More of Real World Interface Inc. 13 It contans all of basc comonents for robotcs sensng and navgaton n a real-world envronment, ncludng battery ower, drve motors and wheels, oston encoders, and ultrasonc sonar transducers, all managed va an onboard MC68HC11-based mcrorocessor. Poneer-1 communcates the states of robot and the commands wth a clent comuter through a rado modem. 14 To erform the roosed cooeratve collson avodance technque exermentally, two vehcles of Poneer-1 were used as shown n Fg. 20. The states of the robots Žx Ž t., y Ž t., Ž t., v Ž t.. Ž 1, 2. were n common wth the robots through a memo-lnk. A navgaton rogram n whch the navgaton, cooeratve avodance, and fnal modes were ncluded was erformed by each PC comuter. The navgaton arameters used n the navgaton exerment were gven as r 0.15 m, v 1.6 msec, R 0 8 1sec, sec, v 3.2 msec, v 0 msec, max v mn d 1.2 m, d m, d 0.1 m, k 1. Ž 48. f Substtutng these values nto equatons n Theorem 1, we had fž A because of A Smlarly n Theorem 2, we had rrv , tb sec, and gt, Ž. b v Thus, the collson avodance condtons, Eqs. Ž 24. and Ž 34., were satsfed by these values. Fgure 21 shows three exermental results n whch the two robots encounter each other wth the redcted crossng angle 0, 4, and 2 rad. A sequence of crcles means a trajectory of robot and s dslayed by 4 sec. Accordng to the cooeratve avodance behavors, collsons were avoded and the robots were navgated to ther goals safely n the three cases. v

16 362 Journal of Robotc Systems 2000 Fgure 19. Photo of Poneer-1. VII. CONCLUDING REMARKS Ths aer has resented a new collson avodance technque, called cooeratve collson avodance, for multle moble robots. The detecton of the danger of collson between two moble robots was dscussed wth resect to the geometrc asects of ther aths. The drecton control command and the velocty control command for the cooeratve collson avodance were then roosed. The avodance technque was extended to cases n whch the number of moble robots s more than two. Furthermore, the condtons for collson avodance were consdered wth resect to the navgaton arameters and gudelnes of desgnng the navgaton arameters were obtaned. The effectveness of the roosed technque was demonstrated by means of numercal smulaton and navgaton exerments usng two real moble robots named Poneer-1. Fgure 21. Trajectores of two vehcles of Poneer-1. Fgure 20. Navgaton exermental system usng two vehcles of Poneer-1.

17 Fujmor et al.: Collson Avodance between Robots 363 Ths aer gave an nsght nto the navgaton arameters of multle moble robots from the theoretcal onts of vew. The obtaned results are also helful for the learnng technques to choose the ntal values of the arameters. As further subjects to research, the avodance behavors should be mroved to hold the avodance condtons when ncreasng the number of robots. In ths aer, t was suosed that there was no obstacle n the envronment. The roosed technque should be aled to more comlcated envronments n whch statc and movng obstacles are scattered. REFERENCES 1. T. Fukuda, T. Ueyama, A. Saka, A. Ca, T. Sugura, A. Saka, and K. Yamada, Dynamcally reconfgurable robotc systems Ž21st reort, behavor decsons of autonomous moble robots by multrocess., JSME Trans C 61 Ž 1995., S. Hrose, R. Kurazume, and S. Nagata, Cooeratve ostonng system wth multle robots, J Robot Soc Jaan 13 Ž 1995., J. Ohta, Y. Bue, T. Ara, H. Osum, and K. Suyama, Transferrng control by cooeraton of two moble robots, J Robot Soc Jaan 14 Ž 1996., L.E. Parker, ALLIANCE: an archtecture for fault tolerant multrobot cooeraton, IEEE Trans Robot Automat RA-14 Ž 1998., M.J. Matarc, Mnmzng comlexty n controllng a collecton of moble robots, Proc IEEE Int Conf Robotcs Automat, Nce, France, 1992, M.J. Matarc, Desgnng emergent behavors: from local ntegratons to collectve ntellgence, From Anmals to Anmats 2: Int Conf Smulaton of Adatve Behavor, MIT Press, Cambrdge, MA, 1993, R.C. Arkn, Cooeraton wthout communcaton: multagent schema-based robot navgaton, J Robot Syst 9 Ž 1992., K. Sughara and I. Suzuk, Dstrbuted algorthms for formaton of geometrc atterns wth many moble robots, J Robot Syst 13 Ž 996., L. Shan and T. Hasegawa, Sace reasonng from acton observaton for moton lannng of multlerobots: mutual collson avodance n a narrow assage, J Robot Soc Jaan 14 Ž 1996., C.G. Km and M.M. Trved, A neuro-fuzzy controller for moble robot navgaton and multrobot convoyng, IEEE Trans Sys Man Cybernet B SMC-28 Ž 1998., N. Kubota, T. Moroka, F. Kojma, and T. Fukuda, Adatve behavor of moble robot based on sensory network, JSME Trans C 65 Ž 1999., A. Fujmor, P.N. Nkforuk, and M.M. Guta, Adatve navgaton of moble robots wth obstacle avodance, IEEE Trans Robot Automat RA-13 Ž 1997., ActvMeda, Inc., Poneer-1 Oeraton Manual Verson 1, ActvMeda, Inc., Poneer-1 Software Manual Verson 4.1.2, 1996.