I see you, you see me: Cooperative Localization through Bearing-Only Mutually Observing Robots

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1 2012 IEEE/RSJ International Conference on Intelligent Robots an Systems October 7-12, Vilamoura, Algarve, Portugal I see you, you see me: Cooperative Localization through Bearing-Only Mutually Observing Robots Philippe Giguere 1 an Ioannis Rekleitis 2 an Maxime Latulippe 1 Abstract Cooperative localization is one of the funamental techniques in GPS-enie environments, such as unerwater, inoor, or on other planets, where teams of robots use each other to improve their pose estimation. In this paper, we present a novel schema for performing cooperative localization using bearing only measurements. These measurements correspon to the angles of pairs of lanmarks locate on each robot, extracte from camera images. Thus, the only exteroceptive measurements use are the camera images taken by each robot, uner the conition that both cameras are mutually visible. An analytical solution is erive, together with an analysis of uncertainty as a function to the relative pose of the robots. A theoretical comparison with a stanar stereo camera pose reconstruction is also provie. Finally, the feasibility an performance of the propose metho were valiate, through simulations an experiments with a mobile robot setup. I. INTRODUCTION The methoology of Cooperative Localization (CL) is a popular localization approach for teams of robots, in situations where external measurements of the environment are not reliable or even unavailable. The key concept is to utilize proprioceptive measurements together with measurements relative to other robots to get an accurate estimate of the pose of each robot. In this paraigm, no external positioning systems are available, such as GPS or external cameras setups that are frequently use in motion capture systems. Unerwater an unergroun environments, inoor areas, as well as, areas on other planets are all GPS enie areas where CL can improve the accuracy of state estimation. The most important contribution of CL is the ecoupling of uncertainty from the environment. When robots operate in an unknown environments, the statistical properties of the noise affecting their sensors is, at best, an eucate guess. For example, poor reflectance of walls can give skewe results in range finers an istorte images to cameras. Similarly, spills on the floor can affect the oometric measurements. By carefully engineering robot tracking sensors, usually a sensor on one robot an a target on the other, the uncertainty increase is boune by the known parameters of the system. For robots moving in 2D, the relative information between any two robots can be measure as a triplet of one istance an two angles z = [l, θ, φ]; where l is the istance between the two robots, θ is the bearing at which the observing robot sees the observe robot, an finally, φ is the perceive orientation of the observe robot. When robustness an minimal uncertainty is more value than efficiency [1], [2], a team of robots will always keep some robots stationary to act 1 Département informatique et e génie logiciel, Université Laval, firstname.lastname@ulaval.ca 2 School of Computer Science, McGill University, yiannis@cim.mcgill.ca Fig. 1. An irobot Create with a USB camera an a four marker target mounte on it. A Hokuyo laser range finer can be seen behin which was use to collect groun truth measurements. as fixe reference points, while others move. Consequently the uncertainty accumulation results only from the noise of the robot tracker sensor. When efficiency is important an all robots move at the same time [3], [4] the uncertainty reuction is proportional to the number of robots [5]. Most approaches up to now utilize irect measurements of the relative istance l between two robots. In this paper, we propose a novel scheme where only relative angles are use. As such, a vision base sensor mounte on each robot suffice to achieve localization; see Fig. 1 for an early prototype 1. Traitionally, ifferent probabilistic reasoning algorithms have been employe to combine the sensor ata into an accurate estimate of the collective pose of the robot team [6], [7]. In this paper, we present an analytical calculation of the robot s pose together with a stuy on the ifferent factors that affect the uncertainty accumulation, such as, istance between the robots an also relative orientation; probabilistically fusing oometry measurements with the robot tracker ata is beyon the scope of this paper. In the following section, an overview of relate work is provie. Section III escribes the CL problem we are aressing, as well as its approximate solution. Next in Section IV, we present an uncertainty stuy base on an omniirectional camera moel. This stuy inclues a comparison with a stereo camera localization system. Experimental results from a prototype system are presente. Finally, we present future irections for this work. II. RELATED WORK The first time cooperative localization was use was in the work of Kurazume et al. [1] terme cooperative positioning. The authors stuie the effect of CL to the uncertainty reuction by keeping at least a member of the robot team 1 In this setup, we place two sets of two ifferent lanmarks, in orer to fin the best type of LED markers /12/S IEEE 863

2 stationary. Early work presente results from simulation, later work [8] use a laser scanner an a cube-target to recover range an bearing between robots an erive optimal motion strategies. A vision base system with a helicalpattern was presente in [9] recovering range bearing an the orientation of the observe robot an was use in a followthe-leaer scenarion. The term cooperative localization was use in [10] using the helical pattern to facilitate mapping. The accuracy of the vision base system was severely limite by iscretization errors. Later work use a laser range finer an a three plane target proucing estimates on the orer of 0.01 m accuracy [11]. Alternative vision base etection [12] use colore cyliners. Using CL with a team of small robots Grabowski et al. [13] employe range only estimates from a single sonar transucer with centimeter accuracy. Ultrasoun transucers were use also in [14] with similar accuracy. Alternative approaches use ifferent sensors to provie: the bearing of the observe robot using omniirectional cameras [15], [16]; or both bearing an istance, with stereo vision an active lighting in [17] an combining vision an laser range finers in [18]. Relative bearing an/or range has been employe in a constraint optimization framework [19]. An analysis of the ifferent sensor moalities with solutions to range only [20] in 2D [21] an in 3D [22] was presente. The first ever analytical evaluation of estimating the uncertainty propagation uring CL was presente in [5]. The initial formulation was base on the algorithm escribe in [23]. The main ifference was that the robots instea of measuring their relative orientations, ha access to absolute orientation measurements. Further stuies of performance were presente in [24]. III. DESCRIPTION OF THE PROBLEM The problem of cooperative localization of two robots we are aressing consists of combining measurements relative to each other, in orer to recover the iniviual robot poses in a common frame of reference. For simplicity, let us consier two robots (A an B) that are moving on a plane, their state escribe as follows: Fig. 2. The three angles ϕ 1 ϕ 2 an ϕ 3 measure by a single camera. The angles ϕ 1 an ϕ 3 correspon to the angular positions, in the image, of the lanmarks. The angle ϕ 2 correspon to the angular position of the other camera. 6 angular measurements: ϕ A 1, ϕ A 2, ϕ A 3, ϕ B 1, ϕ B 2 an ϕ B 3. These angles ϕ correspon to the three visual markers on the other robot, as epicte in Fig. 2. Central to the propose solution is ientifying the orientation an location of the other camera, which will be use as the reference point for the robot location. The relative localization of B compare to A will be euce from: the angle α = ϕ B 1 ϕ B 3, corresponing to the angle between the outer markers of robot A, as seen by camera B; the angle β = ϕ A 2, corresponing to the bearing of the location of camera B within the frame of reference of A. These measurements, α an β, inuce the following constraints on the location of camera B relative to camera A: camera B must be locate on a circle of raius r = /(2 sin α) passing through markers D an E (inscribe an central angle theorem); camera B must be locate along a line of angle β, with respect to camera A. The intersection of these two constraints, illustrate in Fig. 3, uniquely locates camera B with respect to camera A. The relative position of A compare to B can also be compute, using ϕ A 1, ϕ A 3 an ϕ B 2 in a similar manner. Next, we are going to erive the analytical expression of the pose of robot B in the frame of reference of robot A. x i = [x i, y i, θ i ] T (1) where [x, y] escribes the position an θ the orientation. Each robot i is equippe with a number of ientifiable co-linear visual markers; in our implementation three lanmarks were use (LED lights) place at equal istance /2 from each other. In aition, each robot is equippe with a camera locate irectly below the central marker, as epicte in Fig. 2. Using only relative orientation measurements from the two cameras, the relative position an orientation of one robot with respect to the other can be erive. In general, the two robots move on the plane using their oometry, or other sensors to localize. When two robots meet, they orient themselves in such a way that each camera can see the markers on the other robot, an thus, the other robot s camera. From this configuration, each robot takes a picture an transmits it to the other robot. From the pair of images I A an I B, each robot extracts the following Fig. 3. Schema of the localization technique between two robots separate by an unknown istance l. The angle α between the two visual markers D an E space by is measure from the image taken by camera B, yieling a circular constraint. The camera A is use to measure the relative angle β. Camera B is at the intersection of the circle an the line. 864

3 A. Analytical Solution for the position of robot B relative to robot A Let us set the origin A = (0, 0) as the position of the camera on the A robot. The outer lanmarks D an E of robot A are locate at L A l = ( /2, 0) an L A r = (/2, 0), respectively. The position of the camera on the B robot is at B = (x b, y b ), with the center of the circumscribing circle containing the position of robot B an the position of the two markers D an E on robot A locate at C = (x c, y c ). All of these positions are epicte in Fig. 3. The values of α, β an will then be use to estimate the unknown position of robot B relative to A. Let us first calculate the center an raius r of the circumscribing circle. From the law of the inscribe angle theorem, the angle DCE between the two markers at the center of the circle C is 2α. With the origin (0,0) at A an the optical axis of the camera pointing along the y-axis, the circle must be locate irectly in front of camera A, because of the symmetric location of the markers D an E with respect to camera A. The center of the circle (0, y c ) an its raius r are y c = 2 tan α, r = 2 sin α. (2) The istance l = AB between the two robots can be calculate from the triangle ACB, where, AC = y c, CB = r an the BAC angle is β. From the construction of the circle r 2 = AB = l = yc 2 + AB 2 2y c AB cos β ( cos α cos β + 1 cos 2 α sin 2 β 2 sin α Given l, then the position of B is: [ ] [ ] xb l cos(90 B = = o β) y b l sin(90 o = β) where l is calculate from Eq. 3. [ l sin(β) l cos(β) ).(3) ], (4) B. Approximate Solution for the position of robot B relative to robot A By assuming that l, a number of approximations can be mae. Most importantly, we will assume that the camera A is on the circumscribing circle, as shown as point P in Fig. 4, as oppose to (0,0). This approximation is possible, since the istance between P an (0,0) tens to 0 as l/ grows. Fig. 4. Geometric relationships use in the approximate solution for the localization problem. The angle between the tangent at P an the line P B is equal to 90 o β. Using the fact that the angle forme by a chor an a tangent that intersect on a circle is half the measure of the intercepte arc, we have that the angle γ in Fig. 4 is equal to: γ = 2(90 o β). (5) The approximate position x b from the camera A is: x b r sin γ = r sin (2(90 o β)) = r sin 2β. (6) Combining Eq. 2 an 6, we have: x b (α, β, ) The approximate solution for y b is: sin 2β 2 sin α. (7) y b r(1 cos(γ)) = r(1 + cos(2β)). (8) Combining Eq. 2 an 8, we have: y b (α, β, ) (1 + cos(2β)). (9) 2 sin α IV. UNCERTAINTY STUDY: JACOBIAN OF x b AND y b The precision for the localization of robot B relative to A is a function of the Jacobian J of the measurement functions x b (α, β, ) an y b (α, β, ): ] J = [ xb α α (10) an of the measurement errors of the various angles ϕ j. In this stuy, we will assume that these errors are istribute normally, with a stanar eviation σ ϕ. Since the angle α is base on the ifference ϕ 1 ϕ 3 an assuming that the errors are uncorrelate 2, we have the following stanar eviation on α: σ α = 2σ ϕ. The angle β simply correspons to the secon angle β = ϕ 2, leaing to the following stanar eviation on β: σ β = σ ϕ. In orer to better see the impact of the istance l on the precision of localization, we can reformulate Eqs. 7 an 9 in terms of (l, β, ) instea of (α, β, ). This can be one using the epenency between α an β, through the law of sines applie to the triangle of Fig. 4 b): sin α sin(ω) l sin(β + 90o ) l = cos(β). (11) l Using this approximation an cos 2 (β) = 1 2 (1 + cos(2β)), we get α = 1 + cos(2β) l2 cos(α) 1 + cos(2β) = l2 cos(α). (12) 2 There is some amount of correlation, but the error ue to neglecting it is smaller than the actual noise. 865

4 Finally, for the conitions where l, we get α 1 an cos(α) 1: α l2 (13) The next partial erivative is approximate by using Eq.11 = sin(2β) sin(α) l sin(2β) cos(β) = 2l sin(β) (14) By making use of the ientity sin(2β) = 2 sin(β) cos(β). The last element for y b, using Eq. 11, gives: = (1 + cos(2β)) 2 sin α l(1 + cos(2β)) 2 cos(β) = l (cos(β)). (15) In like manner, we approximate the partial erivatives of x b : an α cos(α) sin 2β 2 sin 2 α cos 2β sin α l2 tan β, (16) = l cos(2β) cos(β). (17) l sin 2β 2 cos(β) = l (sin(β)). (18) Thus, the Jacobian J of the system is approximate as: [ ] l2 J tan β l cos(2β) l cos(β) (sin(β)) l2 l 2l sin(β) (cos(β)). (19) Fig. 5 shows the istributions of the estimates of the position of robot B relative to A, for ifferent angles β an l = 10 m, using a simulation in matlab. Robot A is locate at the origin (0,0), with its markers D an E locate at (-1,0) an (1,0), respectively. Analysis of these simulation results confirme the valiity of Eq. 19. Fig. 5. Distribution of pose estimates of camera B with a istance between the robots of l = 10 m at ifferent relative bearings β, in simulation. The samples are generate with Gaussian noise 2σ ϕ for α an σ ϕ for β, with σ ϕ =1.745e-3 ra. As can be seen from the istributions at location 1 an 2, the value of σ y is relatively inepenent of β. uality principle [25]. The relative heaing error, however, is much smaller for our metho. For this comparison, we moel a pair of stereo cameras with a baseline of ientical to the istance between the lanmarks in our configuration. Both cameras are oriente so as to see the marker L on robot B, as shown in Fig. 6 b). This baseline is the same as the istance between the lanmarks use in our propose metho, an is reprouce again in Fig. 6 a). B. Derivation of Precision for Stereo Camera V. COMPARISON WITH OTHER TECHNIQUES A. Theoretical Comparison with Stereo Camera Base Localization Distance an relative heaing can be estimate also if both cameras are locate on a single robot, forming a stereo camera pair. However, our metho presents a number of avantages compare to a stereo system: only one camera per robot is require, reucing the computing loa, weight an cost; perceive heaing φ between the robots can be measure with higher precision; an with more than two robots, all the robots nee to be equippe with stereo pairs, in orer to avoi the case when two robots with only targets meet, thus being unable to localize. In this section, we emonstrate that the localization errors σ x an σ y are, for all purposes, similar for both systems. The reason for this similarity is that cameras an view points can be interchange, as capture by the Carlsson-Weinshall Fig. 6. Comparison between lanmark (L) an camera (C) configurations for our metho a) an an equivalent stereo system b). The positions of the cameras an markers are interchange. For this equivalent stereo camera system, the variance in the location of the target in the space (x, y) is [26]: an σ 2 xstereo = (b2 + x 2 )y 2 2b 2 f 2 σ 2 p, (20) σ 2 ystereo = y4 2b 2 f 2 σ2 p, (21) 866

5 where σ p is the stanar eviation of the lanmark in the image plane, f is the focal length an b is the baseline of the system. Since our system operates in 2D, we have change the original notation so that z = y, to conform with our axes. The error functions for σ xstereo an σ ystereo in Eqs. 20 an 21 can be expresse in terms of β an σ β, to compare irectly with the errors in our system, base on angles α an β. First, we take the square root an use f = 1: σ xstereo = b2 + x 2 2bf yσ p = b2 + x 2 2b yσ p. (22) From the efinition of β, we replace x by x = y tan(β) an move the enominator b insie the square root: 1 + y2 b tan(β) 2 σ xstereo = 2 yσ p. (23) 2 From the efinition of β an a focal istance f = 1, we get that the position on the image plane is p = tan β. This converts the pixel noise σ p into angular noise σ β = σ ϕ : σ p = p σ ϕ = tan βσ ϕ = sec 2 (β)σ ϕ. (24) Replacing σ p in Eq. 23 by Eq. 24 an from the problem efinition y = l cos(β), we get that the error on position σ xstereo expresse as a function of β an σ ϕ is: 1 + ( l b σ xstereo = sin(β)) 2 lσ ϕ. (25) 2 cos(β) In a similar manner, we can show that error on position σ ystereo, expresse as a function of σ ϕ, is equal to: σ ystereo = C. Comparison with our Metho l2 2b σ ϕ. (26) We can estimate the error on the x position using partial erivatives: ( ) 2 ( ) 2 σx 2 xb = σ xb β + α σ α (27) since we assume that σ = 0. We have σ α = 2σ ϕ an σ β = σ ϕ. Taking the expressions from Eq. 19, we have: ( σx 2 = l cos(2β) ) 2 ( ) 2 cos(β) σ ϕ + l2 tan β 2σ ϕ. (28) Substituting = 2b for the baseline efinition in [26] an using stanar trigonometric ientities, we get: 1 + cos(4β) + ( l b σ x = sin(β)) 2 lσ ϕ. (29) 2 cos(β) For a similar level of angular noise σ ϕ, both systems perform nearly ientically. In fact, Eq 29 only iffers from Eq. 26 by the extra cos(4β) term. However, this term oes not contribute significantly beyon 10 o when l b, since it is being ominate by ( l b sin(β))2. Similarly for σ y an using the partial erivatives in Eq. 19, we have: σ 2 y = ( yb σ β ) 2 ( ) 2 ) yb + α σ α = (2 sin 2 (β) + l2 4b 2 l 2 σϕ. 2 (30) If we only consier the cases where the robots are within their fiel of views (β < 45 o ) an at a istance l b, then 2 sin 2 (β) l2 4b 2 an we can approximate Eq. 30: σ y l2 2b σ ϕ. (31) Thus, we have a epth evaluation better by a factor of 1/ 2, compare to a stereo system with one lanmark observe. If two lanmarks are observe with the stereo system an the estimate epth average, then the error σ ystereo shoul be the same as in Eq. 31. One must keep in min that the analysis above was one for the case where a stereo baseline istance 2b is equal to. However, most stereo systems on mobile robots use short baseline istances 2b, to facilitate image registration. Since our approach is not boun by such limitations, we can use an arbitrarily large 2b between our lanmarks. In practice, this allows our system to significantly outperform stanar stereo systems on mobile robots. D. Perceive Heaing φ Compare to a single stereo pair, we are able to recover the perceive heaing φ between the two cameras with very high precision. The perceive heaing estimate for our approach is equal to: φ = β 2 β 1, (32) with a noise equal to σ φ = 2σ ϕ. (33) Measuring this perceive heaing, using a single stereo pair, woul not be able to capture an estimate of φ with this level of precision. The reason is that fining φ involves the following computation: φ stereo cos 1 ( (ϕ3 ϕ 1 )l measure ). (34) With ϕ δ = ϕ 3 ϕ 1, the estimate noise on φ stereo is ( ) σφ 2 φstereo 2 ( ) 2 φstereo stereo 2σϕ + σ l ϕ δ l (35) σφ 2 1 ( stereo 2l 2 σϕ 2 + ϕ 2 ( lϕ δ )( + lϕ δ ) δσl 2 ) (36) The best possible case is when ϕ δ = 0 an the other robot is locate straight ahea, which means that: σ φstereo 2 l σ ϕ (37) 867

6 an is always greater than our perceive heaing estimate noise in Eq. 33, for l. E. Simulate Comparison against a one Camera an 3 Noncolinear Lanmarks It is possible to retrieve both heaing an istance information without exchanging images between robots, if the robots have at least 3 non-colinear markers on them. For comparison, we simulate the performance of such system with one camera, an 3 markers locate on a square of sie = 1, with a istance l = 10 between the robots an σϕ =1.745e-3 ra. For our mutual camera approach, information gathere from both sies of the square were optimally combine. Simulation results, presente in Fig. 7, show that the uncertainty on the location of the robot is much larger for the 3 markers case. 11 Fig. 8. Platform use for the ata collection (top), with one picture use for localization (bottom). Distance Y (m) True Location Robot B Localization 3 markers Our Localization Distance X (m) Fig. 7. Simulations results between a one camera an 3 non-colinear markers vs. our approach, for various locations of robot B at a istance l = 10. The markers on the robot A are locate at (, 0), (0, 0) an (0, ) (outsie the graph). VI. E XPERIMENTAL R ESULTS We evaluate the performance of our localization metho on two real ata sets. The experimental setup comprise one irobot Create robot with an on-boar computing moule, 2 Logitech C x1200 pixels webcams an a number of LED marker glowsticks. The outer markers were separate by a istance of = m on the robot, with a camera an another marker exactly in the mile, as shown in Fig. 8. The other camera was mounte on a fixe setup, with an LED marker above. Groun truth was establishe by using a Hokuyo URG-04LX-UG01 LIDAR mounte on the robot an placing 10 boxes in the environment as lanmarks. The robot move forwar by 5.25 cm between each step an along a nearly straight trajectory towars the fixe camera. Pictures were taken at each step forwar an use as the test sets. Fig. 9 shows the estimate robot position foun by our metho, using one α an one β per sample, compare with groun truth. The average absolute position error over the complete trajectory was 4.09 cm for experiment 1 (140 samples) an 4.40 cm for experiment 2 (110 samples). When combining with the range information obtaine by the other α, β pair, the average absolute measurement error for experiment 1 roppe to 2.82 cm, a reuction by a factor Fig. 9. Comparison between the estimate robot position using our metho an groun truth, for two trials. The fixe camera was locate at (0, 0). Fig. 10. Comparison between the absolute position error an the preicte error, as a function of the istance l. As can be seen, the preicte errors for an angular noise of σϕ = ra are consistent with the measure errors. With a focal istance of f = 1320 pixels for the webcam, the noise is equivalent to 0.4 pixel. of 2, as expecte when averaging two uncorrelate noisy estimates. The istribution of absolute errors, shown in Fig. 10, are consistent with the noise σy preicte by Eq. 31. Histograms of step measurements, for three brackets of istance l, are shown in Fig. 11. These correspon to 868

7 Fig. 11. Robot step (5.25 cm) measurement istributions with our metho (a-c) an LIDAR (). Fig. 12. Robot perceive heaing estimates, uring experiment 1. As a comparison, laser scan alignments performe with ICP resulte in noisier estimates. the estimate istance between the 5.25 cm steps of the robot. Thus, a perfect system woul measure this value, an noisy systems shoul have a istribution with a sprea relate to the measurement noise. The stanar eviation of these istributions are, again, in agreement with our noise moel. The perceive heaing φ for experiment 1 is plotte in Fig. 12, along with the perceive heaing erive from the Hokuyo laser ata. One can see that the heaing φ compute with our metho is much less noisy than the one foun with a LIDAR, which is the common metho for evaluating the heaing of a robot in its environment. VII. CONCLUSION AND FUTURE WORK In this paper we presente a novel approach to cooperative localization, base on mutual observation of lanmark bearing angles between two robots. We erive a mathematical solution along with a theoretical noise moel that compare favorably against an equivalent stereo camera system. During experiments with a mobile robot, our system emonstrate goo position estimation (average error aroun 4 cm over 250 samples), espite the use of off-the-shelf cameras an markers. This makes our solution particularly well suite for eployment on fleets of inexpensive robots. The biggest challenge with the current physical implementation is to establish mutual observations, without interfering with the normal operation of the vehicles. To eliminate this problem, omniirectional cameras can be employe. Our metho is completely compatible with such cameras. We plan on extening this methoology to vehicles that move freely in three imensions. Of particular interest are uneractuate square blimps that move at slow spees [27], since the require harware for our solution is very light. Applications to unerwater vehicles [28] are also consiere, since they are generally eploye in unstructure environments without GPS. Another current irection of research is to employ Unscente Kalman Filtering (UKF) to improve the state estimation, by exploiting oometry measurements. REFERENCES [1] R. Kurazume, S. Nagata, an S. Hirose, Cooperative positioning with multiple robots, in ICRA, vol. 2, 1994, pp [2] I. Rekleitis, G. Duek, an E. Milios, Multi-robot exploration of an unknown environment, efficiently reucing the oometry error, in IJCAI, vol. 2, 1997, pp [3] S. Roumeliotis an G. Bekey, Collective localization: A istribute kalman filter approach to localization of groups of mobile robots, in ICRA. IEEE, 2000, pp [4] D. Fox, W. Burgar, H. Kruppa, an S. 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