Cooperative, distributed localization in multi-robot systems: a minimum ...

Cooperative, distributed localization in multi-robot systems: a minimum-entropy approach Vincenzo Caglioti, Augusto Citterio, Andrea Fossati Department of Electronics and Information Politecnico di Milano Milano, Italy

Abstract— In this paper, we consider the problem of localization in a multi-robot system. We present a new approach focused on distribution, scalability, and minimum-uncertainty perception. An Extended Kalman Filter (EKF) is used to update an estimate of the robot poses in correspondence to each sensor measurement. An entropic criterion is used, in order to select optimal measurements that reduce the global uncertainty relative to the estimate of the robot poses. It is shown that, in addition to EKF, also the selection of the optimal measurement can be distributed among the robots, in a scalable fashion. The proposed approach has been validated by simulations and preliminary experimental results.

I. I NTRODUCTION The localization problem of mobile robots is a critical factor since most tasks require an accurate knowledge of position and orientation; localization is even more crucial in multi-robot systems, in fact for a tight cooperation each robot needs to know the position of the other members of the team. The solution we propose allows, starting from a known state and given the environment map, to compute the estimates of the poses (i. e. 2D coordinates and orientation) of all the robots belonging to the system; moreover it is also possible to characterize the uncertainty related to these estimates. The state of the whole system is kept updated following both robots movements (measured by odometric sensors) and exteroceptive measurements. A simple and frequent approach considers each agent as a stand-alone entity, we instead consider the team of robots as a single distributed system. Robots actively cooperate in order to accurately localize themselves efficiently fusing all the available information collected by proprioceptive and exteroceptive sensors. State information and computational load are moreover completely distributed among the agents; this is obtained using a distributed version of the Kalman filter which allows each robot to update the locally stored information in a O(n) time, being n the number of robots in the system. Also the memory space needed by each of the robots is O(n), while a centralized agent would require O(n2) time and O(n2 ) space. Another key aspect of the approach proposed in this paper is minimumuncertainty perception, i.e. the capability of selecting the best measurement from the point of view of information increment. For example, in our experiments, we consider the case of an orientable range finder that can be oriented with high accuracy: the selection of the optimal measurement consists in

choosing the range finder orientation, leading to the maximum reduction of the uncertainty of the a posteriori pose estimate. The presented approach is general enough to be extended to handle different types of sensors and measurements. The adopted uncertainty criterion is based on the expected entropy variation deriving from the measurement. Often, the cost associated to each measurement is not considered and localization systems focus on making more measures instead of performing high quality measures. Minimum uncertainty perception has been already used for localization of a single mobile robot ([1], [2]), but never applied to a multi-robot system. Moreover this process can be distributed so that the measurement selected by any robot of the system, according to the adopted entropic criterion, is optimal not only for the single robot itself, but also for the whole multi-robot system. From the analysis in [3], it can be derived that the robot poses can be updated according to a distributed, scalable algorithm. We show that also the selection of optimal measurements can be distributed among the robots according to a scalable algorithm. A distributed localization algorithm based on the determination and use of optimal measurements has been implemented. Both simulation results and experiments on real robots are presented, showing that the use of optimal measurements significantly increases the localization accuracy. Section 2 discusses some related work, Section 3 illustrates distributed localization. The entropy criterion adopted in the selection of optimal measurements is presented in Section 4, while the localization process is outlined in Section 5. Simulation results and experiments with real robots are discussed in Sections 6 and 7, while Section 8 concludes the paper. II. R ELATED W ORK Numerous applications of robotics involve a team of cooperating robots that have to complete some required task (e.g. [4], [5]). The localization problem is a key issue and, in these cases, multi-robot cooperation to update the pose estimates will produce a better localization by compensating for measurement errors. Roumeliotis and Rekleitis [6] use the relative position information to improve the localization accuracy in a robot team. They derived an analytical formula that expresses the upper bound of the uncertainty accumulation as a function of time and noise characteristics of the

robot sensors. The uncertainty growth has been shown to be inversely proportional to the number of robots. Collaborative multi-robot localization has been often considered within an estimation-theoretic framework. In [7] collaborative localization of indoor robots is performed using a sample-based version of the Monte Carlo Localization (MCL) algorithm. Some robots localize themselves in the same environment, whose map is supposed to be known, and probabilistic methods are employed to synchronize each robot’s belief whenever a robot detects another one. In other studies, the cooperative navigation problem is formulated in a Kalman filter (KF) framework. In [3] a Kalman filter is used to process the available positioning information from all the members of the team. A distributed localization algorithm is obtained by writing the equations for the centralized Kalman filter, resulting in a number of smaller communicating filters. Each of these filters processes sensor data collected by its host robot and communication between filters occurs when two robots detect each other and measure their relative pose. Madhavan et al. improve this approach, considering a nonlinear kinematic model of the robots and adopting an Extended Kalman Filter-based algorithm for the localization of a team of robots operating on outdoor terrain [8]. Also Martinelli et al. start from [3] and derive the equations for an Extended Kalman Filter based estimator for the most general relative observation between two robots [9]. Several previous approaches to multiple robots are based on a simple strategy: a team of robots is divided into two subteams and, at each instant, one subteam is in motion while the other subteam remains stationary and acts as a landmark. This approach was first proposed by Kurazume et al. and then improved in [10], [11] and [12]. Rekleitis et al. use the same strategy to reduce dead reckoning inaccuracies and to obtain a map of the environment, by exploiting the ability of the robots to see each other and to detect obstacles between them ([13], [14]); they also analyze the trade-off between accuracy and efficiency in multi-robot localization ([15]). A recent paper tries instead to find the best trajectores for a group of robot to follow, in order to obtain the best localization information possible [16]. None of the above approaches considers the issue of using the current information about the state in order to plan future sensor measurements that maximize the increase in the localization accuracy; moreover our objective is both to allow all the robots to move at the same time and not to impose a particular trajectory. In [17] the tradeoff between the value of information an exteroceptive measurement carries, and the cost of processing it are considered, in order to find an optimal measurement frequency to obtain the highest possible position accuracy. Stroupe et al. [18] propose an approach for merging Gaussian distributions to fuse measurements collected by different robots, in order to increase the localization accuracy. In the present paper, the sensor measurements are selected according to an uncertainty criterion. Uncertainty criteria based on entropy are among the most popular ones: in [1], [2] a single robot actively explores a known environment so as to

best localize itself, while in [19] minimum-entropy exploration is used to build a map of an unknown environment. To the best of our knowledge, no approaches have been presented yet, where multi-robot cooperative localization has been integrated with minimum uncertainty sensor planning. III. D ISTRIBUTED L OCALIZATION In the approach we propose, complete information about robot pose is maintained and updated through an efficient distributed process: this is made possible by adopting Gaussian distributions for the involved parameters. In this framework, there is no need for any centralized agent: each single mobile robot is equipped with one or more sensors (no central sensor exists). In addition, the localization is performed by updating a current estimate of the robot poses in correspondence of each sensor measurement: to update the current estimate, the robots communicate with each other and update locally stored information. Again, no central agent exists, where the whole pose information is stored and maintained. The map of the robot environment is supposed to be known, modulo moving obstacles. Each robot i performs sensor measurements, which can be aimed either at the environment or at another mobile robot j. It is shown that, under broad hypotheses, the problem of maintaining and updating pose information is scalable, i.e., the time and space needed for updating the estimate at each of the n mobile robots is of the order of 1/n of the time and space that would be needed for updating the estimate of the robots by a centralized agent. The following hypotheses are assumed: - the pose estimation errors and the measurement errors are supposed to be normally distributed; - the pose estimation errors are kept small enough (e.g., by dead reckoning and frequent measurements), such that the relationships between errors in the different parameters can be well approximated by linear relations; - the computational cost for broadcasting a datum from a single robot to the remaining ones is O(n). First, let us calculate the space complexity and the time complexity associated to updating the pose estimation of n robots by means of a centralized agent within the above described gaussian framework. To correctly update pose estimates, both the pose estimates and their covariance matrix must be maintained and updated. This requires O(n2) time using O(n2) storage, in correspondence to each sensor measurement. This process is described by the Kalman estimator. By using measures related to the relative positions between two robots (or between a robot and the environment), the pose estimate updating problem is shown to be scalable: the estimate update corresponding to each sensor measurement can be performed in O(n) time by distributing the process among the n robots. In addition, each robot only needs to store O(n) data in its memory. Now we consider a system consisting of n mobile robots, which navigate within a known environment. In order to maintain an estimate of the pose of the robots, each robot i can perform a sensor measurement aimed either at the environment or at another robot j. Once such a sensor measurement

has been performed, the measuring robot i updates its pose estimate and the correlation between its pose and the pose of the robot j. Then, it propagates the information towards the remaining robots, which in turn update both estimate and correlations. T Let xi = Xi Yi θi be the vector of the pose parameters of themobile robot i. T Let x = xT1 xT2 . . . xTn indicate the column vector formed by the pose vectors of the n mobile robots, and x ¯ its current estimate. Let Σii indicate the 3 × 3 covariance matrix of the pose parameters of robot i, and let Σij indicate the correlation matrix between the pose parameters of robot i and the pose parameters of robot j: the covariance matrix of the whole pose vector x is given by   Σ11 · · · Σ1N  ..  . .. Λ =  ... . .  ΣN 1

···

ΣN N

Suppose that a measurement z is taken, whose result depends on some of the poses xi : the linearization of the relationships between z and x can be expressed as z = Hx + w where w is a measurement error, which is supposed to be independent of x, and H is the Jacobian matrix of z with respect to x. If σ2 is the variance of the error w, the a posteriori Kalman estimate of the pose parameters, and its covariance matrix are given by: T

T

2 −1

x ˆ=x ¯ + ΛH (HΛH + σ )

(z − H x ¯)

Λ0 = Λ − ΛH T (HΛH T + σ2)−1 HΛ.

(1)

follows: each robot k will store and keep updated all the pose estimates x ¯l and the sub-matrices Σkl with l = 1..n. The O(1) sized matrixes hi and hj can be calculated by the measuring ¯j , the shape of robot i, knowing the estimated poses x ¯i and x the measured robot j or the shape of the environment. It can be noticed that each term needed to calculate expressions (3) and (4) is of constant size and is locally available or can be retrieved, through message exchange, at a constant cost. Therefore, it can be concluded that the problem of updating the pose estimate of n robots, and their covariance matrix by means of sensor measurements, each of which depends on the relative position between two robots, is scalable. This property can also be derived from [3]. IV. M INIMUM -E NTROPY L OCALIZATION This section presents a formalization of the problem of selecting sensor measurements characterized by minimum residual uncertainty about the state of the multi-robot system. In the sequel a criterion based on the concept of entropy is derived; since only continuous, normally distributed variables are present, the criterion purpose reduces to the minimization of the determinant of the covariance matrix Λ. As reported in [20], the criterion used to optimize a sensor measurement in a system described by continuous probability distributions is based on the minimization of the expected value of the variation of the entropy of the system itself, that is described by the following expression: J(x/z) = E[∆Hx/z] = E[∆Hz/x] = Z Z Z = f(z) ln f(z)dz − f(z/x) ln f(z/x)dz · f(x)dx.

(2)

(6)

Now consider the case where the measurement z only depends on the relative pose between robot i and robot j: from z = hi xi + hj xj + w, the expression of H becomes   H = 0 · · · 0 hi 0 · · · 0 hj 0 · · · 0

Since the probability distributions involved in the description of the state x are assumed to be gaussian, the entropybased optimization criterion can be calculated as follows:

and the expression of HΛH T becomes: HΛH T = hiΣii hTi + 2hiΣij hTj + hi Σjj hTj .

(3)

The a posteriori estimates of the robots poses are x ˆk = x ¯k + (Σki hTi + Σkj hTj )· · (hi Σii hTi + 2hi Σij hTj + hj Σjj hTj + σ2 )−1 ·

(4)

· (z − hi x ¯i − hj x ¯j ) while the a posteriori correlations are Σ0kk = Σkk − (Σki hTi + Σkj hTj )· · (hi Σii hTi + 2hiΣij hTj + hj Σjj hTj + σ2 )−1 ·.

(5)

· (hi Σik + hj Σjk) These expressions can be locally calculated by any single robot, using an O(n) storage, in O(n) time. In fact, the pose estimates and the covariance matrix Λ can be distributed as

J(z/x) = E[∆Sz/x] = Z 1 1 2 = · ln (2πσz ) − · ln (2π(HΛH T + σz2 )) · f(x)dx = 2 2 1 σz2 = · ln . 2 HΛH T + σz2 (7) All the terms involved in the criterion are available or can be locally calculated by any robot of the system before performing a range measure, whose error variance is σz . To choose the optimal measurement, it is sufficient to evaluate J for all the possible measurements (either with different types of sensors or with the same sensor), and select the one that minimizes it. In the sequel, we show that the adopted entropy criterion can be evaluated and minimized by means of a distributed algorithm. To do this, we start from the following result, proved in [20]:

Theorem 1: Suppose the state parameter vector x can be decomposed into two subvectors x1 and x2 such that f(z/x1 , x2) = f(z/x1 ). Then: E[∆H(x/z)] = E[∆H(x1/z) ]. Though intuitive, this property is not satisfied by a large set of criteria, as, e.g., the trace of the covariance matrix of the cartesian coordinates. This property allows to reduce the problem of evaluating (and hence, minimizing) the expected variation of the entropy of the whole system state to the problem of evaluating (or, minimizing) the entropy relative to the only state variables that are observable from the measure output z. In our framework, each sensor measurement z only involves (at most) two robots i and j: therefore, the only state variables that are observable from z, are the pose parameters of the involved robots i and j. Thus, the expected entropy variation associated to a measurement coincides with the expected entropy variation relative to the pose parameters of the robots involved in the measurement. This expected entropy variation can locally be evaluated by one of the involved robot. Let us denote the locally optimal measurement at robot i as the measurement, involving the robot i plus either an other robot j or the environment, characterized by the minimum expected entropy variation. Each robot i can determine the locally optimum measurement. The globally optimum measurement, i.e. the measurement which minimizes the expected variation of the entropy of the whole state vector, is one of the locally optimum measurements. Therefore, the best measurement a robot can perform in order to minimize uncertainty on its own pose, coincides with the best measurement it can perform in order to minimize the uncertainty of the whole multi-robot system: since information is shared among individuals, “ego-centric” information maximization meets “collective” information maximization. The globally optimum measurement can be selected among the locally optimum ones, by finding that measurement whose criterion value is minimum: in principle, distributed minimization can be executed in O(log n) time. Notice that, from Theorem 1, the local optimization at a given robot i, consisting in minimizing (7), reduces to finding the measurement which maximizes (3) (observe that hi and hj depend on the measurement). Therefore, also the problem of determining the optimal measurement is scalable. In fact, a centralized agent would have to evaluate expression (3), for all i. V. T HE

LOCALIZATION PROCESS

The localization process consists in the re-iteration of a sequence including: motion of the robots, determination of the optimal measurement, execution of the optimal measurement, update of the pose estimates. All the robots of the system know their own a priori pose estimate (i.e., the estimate before the movement), with its covariance matrix, and the environment map. In addition, each

robot i stores the pose estimates of all the robots j plus the i-th row of the global covariance matrix Λ (this row contains all the correlation matrices Σij ). The storage at each robot amounts to O(n), i.e., O(1/n) times the total storage needed for pose estimates and covariance matrix. When a robot i executes a movement, it broadcasts the data retrieved from the odometric sensors to all the other robots j, so that they can update the current pose estimate of robot i and their correlation matrix with robot i. The current error (i.e., the error after the movement) relative to the pose estimate of robot i depends both on the a priori error (i.e. the error before the movement) and on the error affecting the odometric measures. The covariance of the a priori error is Σii , while the covariance of the odometric error is Di . The current pose estimate is obtained from the a priori pose estimate according to: x ¯0i = x ¯i + [∆Xi ∆Yi ∆θi ]T

(8)

where ∆Xi , ∆Yi, ∆θi represent the estimated variation in the pose parameters, expressed in global coordinates, associated to the movement: they can correspond either to the odometric readings (if available) or simply to the prescribed motion. The covariance of the estimate error of the moving robot, after its movement, can be shown to be: Σ0ii = Ki · Σii · KiT + Li · Di · LTi

(9)

while the correlation matrix between a robot j and the moving robot i is given by: Σ0ij = Ki · Σij · KjT ,

(10)

since the odometric error of the moving robot i is independent of the pose estimate error of robot j. In the above formulas, K and L are propagation (jacobian) matrices:   1 0 −∆Yi Ki = 0 1 ∆Xi  0 0 1   cos θi − sin θi 0 Li =  sin θi cos θi 0 0 0 1 where θi is the orientation component of the pose estimate before the movement. We calibrated the covariance matrix D of the odometric error, both for straight and curvilinear translations and for rotations. The calibrated model of the odometric error associated to straight or curvilinear translations includes an error in the wheel orientation, plus a path length error proportional to the covered distance. The error involved in rotations is modeled as an additive error, proportional to the angle of rotation, affecting the three components of the pose estimate. Expressions (9), (10), (11) are used to compute the current pose estimates and covariance matrices. These expressions can be used to calculate the entropic criterion and to determine the optimal measurement, as shown in the previous section.

After the selected range measurement is executed, pose estimates and matrices are updated, by using equations (4), (5) and (6) so that each robot keeps updated information about all the robot poses. VI. S IMULATION

RESULTS

The distributed localization process described in the previous section has been implemented. The implemented system can work with a known environment, possibly including some unknown obstacles. Each robot is equipped with an odometric sensor and an orientable range finder. The range finder can be used to perform measurements aimed at the environment or at other robots. Both the range measurements and the odometric measurements are supposed to be affected by a Gaussian error with a known variance. The error in the orientation angle of the range finder is supposed to be negligible. During the selection of the optimal measurement, a candidate measurement is discarded, if the incidence angle on the measured surface (i.e., the angle between the measurement line and the surface normal) is higher than a certain threshold. Such a measurement, in fact, would be unreliable due to the insufficient signal reflected towards the sensor. During measurement execution, a measurement result is discarded, if its value is external to a 95 % confidence interval centered on its a priori estimate. This situation, in fact, can arise in the presence of an unknown obstacle crossed by the measurement line. Both simulations and experiments on real robots have been performed, to compare the accuracy obtained by using optimal measurements (according to the entropy criterion (7)) with the accuracy obtained by using measurements along random directions (i.e. choosing a random orientation for the range finder). A single scalar index has been adopted to describe the localization error: this scalar Q is given by the sum, over all the robots, of the mean distances between the estimated position of each robot vertex and its actual position. Several open-loop simulations have been performed, in which the current pose estimate does not affect the next robot movement. In all these simulations, the pose estimate error associated to the use of optimal measurements is significantly smaller than the pose estimate error associated to the use of random measurements. Also, several closed-loop simulations have been performed, where the next robot movement is based on the current pose estimate, trying to execute an assigned trajectory. The first simulation we present is a closed-loop simulation, in which a team of five robots tries to execute the trajectory illustrated in Figure 1, while performing a fixed number of measurements. The final situation of the system, after optimal measurements, is depicted in Figure 2. The true position of the robots is shown on the right, while the estimated position is shown on the left. Notice that the real trajectory is affected by the pose estimate errors, since the pose estimates are used to control the robot motion. Figure 3 shows the performance of trajectory control based on measurements along random directions.

Fig. 1.

Simulation - Robot trajectory.

Fig. 2. Simulation results - Optimal measurements. Left: Estimated final state - Right: True final state.

Fig. 3. Simulation results - Random measurements. Left: Estimated final state - Right: True final state.

We present a second simulation, where five robots follow a random trajectory: in turn, each robot randomly performs a simulation step, which can consist in either a forward movement, a rotation or a measurement. The total simulation consists of 130 steps for each robot.

Fig. 4.

Global Simulation - Localization error.

This simulation is performed both with measurements along optimal directions and along random directions. In Figure 4 the localization error Q defined above is plotted both for measurements along optimal directions and for measurements along random directions.

VII. P RELIMINARY EXPERIMENTAL RESULTS Preliminary experiments have been carried out using two mobile robots. Each robot is equipped with an odometric sensor and an orientable range finder: the robots can communicate through a wireless network adapter. Two environments have been used: the first environment is a standard indoor one. The second environment (see Figure 5) is an outdoor area, whose floor is slightly non-planar. The ground truth pose of the robots is obtained through an accurate triangulation based on the known environment map. In the first experiment, one robot executed a 10 meter linear trajectory while the second robot executed a 11 meter polygonal trajectory. During motion, each of the two robots performed a range measurement every 50 cm. In the second experiment, robots executed a 7.5 meter linear trajectory along opposite directions. During motion, each of the two robots performed a range measurement every 75 cm. Within each environment, this experiment has been repeated with locally optimal measuraments and with measurements along random directions.

Fig. 5.

The experimental set-up.

Within the indoor environment, the following mean localization errors have been obtained: Optimal measurements: 8.5 cm Random measurements: 12.1 cm Within the outdoor environment, the following mean localization errors have been obtained: Optimal measurements: 21.9 cm Random measurements: 32.7 cm This confirms that optimal measurements yield better localization results. Larger localization errors are obtained within the non-planar environment, since a planar model has been adopted. VIII. C ONCLUSIONS In this paper, distributed cooperative localization in multirobot systems has been addressed. Besides issues related to the scalable distribution of the pose estimate update among the robots, we focused on minimum uncertainty measurements. We showed that, using an entropy-based criterion, also the determination of optimal measurements allows a scalable distribution among the robots. Both simulations and experiments with real robots, equipped with orientable range sensors, show a significant gain in

accuracy with respect to approaches not using optimal measurements. Although the method has been implemented for range sensor, the approach can be applied to robots equipped with heterogeneous sensors. Further experimentations with real robots and extensions to heterogeneous sensors constitute object for ongoing activity. R EFERENCES [1] G. Borghi and V. Caglioti, “Uncertainty minimization in the selflocalization of mobile robots in curvilinear environments,” IEEE Transactions on Robotics and Automation, vol. 14, pp. 902–911, 1998. [2] W. Burgard, D. Fox, and S. Thrun, “Mobile robot localization by entropy minimization,” in Advanced Mobile Robots, 1997. Proceedings., Second EUROMICRO workshop on 22-24 Oct. 1997, 1997, pp. 155–162. [3] S. Roumeliotis and G. A. Bekey, “Distributed multirobot localization,” IEEE Transactions on Robotics and Automation, vol. 18, no. 5, p. 781, 2002. [4] M. J. Mataric, M. Nilsson, and K. T. Simsarin, “Cooperative multirobot box-pushing,” Intelligent Robots and Systems 95. ’Human Robot Interaction and Cooperative Robots’, Proceedings. 1995 IEEE/RSJ International Conference on, vol. 3, pp. 556 – 561, 1995. [5] G. Dudek, M. Jenkin, E. Milios, and D. Wilkies, “Experiments in sensing and communication for robot convoy navigation,” in Proceedings IEEE International Conference on Intelligent Robots and Systems, vol. 2, 1995, pp. 268–273. [6] S. Roumeliotis and I. Rekleitis, “Propagation of uncertainty in cooperative multirobot localization: Analysis and experimental results,” Autonomous Robots, vol. 17, pp. 41–54, 2004. [7] D. Fox, W. Burgard, H. Kruppa, and S. Thrun, “A probabilistic approach to collaborative multi-robot localization,” Autonomous Robots, vol. 8, p. 325344, 2000. [8] R. Madhavan, K. Fregenet, and L. E. Parker, “Distributed heterogeneous outdoor multi-robot localization,” in International Conference on Robotics and Automation, 2002. [9] A. Martinelli, F. Pont, and R. Siegwart, “Multi-robot localization using relative observations,” IEEE International Conference on Robotics and Automation, 2005. [10] R. Grabowski and P. Khosla, “Localization techniques for a team of small robots,” in EEE/RSJ International Conference on Intelligent Robots and Systems, vol. 3, 2001, pp. 1067–1072. [11] R. Kurazume and S. Hirose, “Study on cooperative positioning system: optimum moving strategies for cps-iii.” in IEEE International Conference in Robotics and Automation, vol. 16-20, 1998, pp. 2896–2903. [12] ——, “An experimental study of a cooperative positioning system,” Autonomous Robots, vol. 8 (1), pp. 43–52, 2000. [13] I. Rekleitis, G. Dudek, and E. Milios, “Multi-robot exploration of an unknown environment, efficiently reducing the odometry error.” in International Joint Conference on Artificial Intelligence, vol. 2, 1997. [14] ——, “Multi-robot collaboration for robust exploration,” in In Proceedings of International Conference in Robotics and Automation, 2000, pp. 3164–3169. [15] ——, “Multi-robot cooperative localization: A study of trade-offs between efficiency and accuracy,” in In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2002. [16] N. Trawny and T. Barfoot, “Optimized motion strategies for cooperative localization of mobile robots,” In Proceedings of the IEEE International Conference on Robotics and Automation, vol. 1, pp. 1027–1032, 2004. [17] F. Zhang, B. Grocholsky, and V. Kumar, “Formations for localization of robot networks,” In Proceedings of the IEEE International Conference on Robotics and Automation, 2004. [18] A. Stroupe, M. Martin, and T. Balch, “Merging probabilistic observations for mobile distributed sensing,” CMU technique report, 2000. [19] Y. Yu and K. Gupta, “C-space entropy: A measure for view planning and exploration for general robot-sensor systems in unknown environments,” International Journal of Robotics Research,, vol. December 2004., pp. pp. 1197–1223, 2004. [20] V. Caglioti, “An entropic criterion for minimum uncertainty sensing in recognition and localization tasks.” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 31, pp. 187–196, 2001.