Autonomous Robotic Exploration Using Occupancy Grid Maps and Graph SLAM Based on Shannon and Rényi Entropy

Transcription

1 Autonomous Robotic Explortion Using Occupncy Grid Mps nd Grph SLAM Bsed on Shnnon nd Rényi Entropy Henry Crrillo, Philip Dmes, Vijy Kumr, nd José A. Cstellnos Abstrct In this pper we exmine the problem of utonomously exploring nd mpping n environment using mobile robot. The robot uses grph-bsed SLAM system to perform mpping nd represents the mp s n occupncy grid. In this setting, the robot must trde-off between exploring new re to complete the tsk nd exploiting the existing informtion to mintin good locliztion. Selecting ctions tht decrese the mp uncertinty while not significntly incresing the robot s locliztion uncertinty is chllenging. We present novel informtion-theoretic utility function tht uses both Shnnon s nd Rényi s definitions of entropy to jointly consider the uncertinty of the robot nd the mp. This llows us to fuse both uncertinties without the use of mnul tuning. We present simultions nd experiments compring the proposed utility function to stte-of-the-rt utility functions, which only use Shnnon s entropy. We show tht by using the proposed utility function, the robot nd mp uncertinties re smller thn using other existing methods. I. INTRODUCTION Autonomous explortion is high level tsk tht is required for mny rel-world robotic pplictions. The primry gol is to cquire the most complete nd ccurte mp of n environment in finite time. Constructing high qulity mp requires the robot to mintin good locliztion, even while trversing unknown spce where by definition the uncertinty of the robot grows. Considering grid mp representtion of the environment, this utonomous explortion tsk cn be divided into three generl steps: 1) identifiction of possible loctions to explore in the current estimte of the mp 2) evlution of the utility of possible ctions 3) execution of the ction with the highest utility. In the first step, we would idelly evlute every possible ction in the robot nd mp spce. However, this proves to be computtionlly intrctble in rel pplictions [1]. In prctice, we select smll subset of loctions in the mp bsed on their locl informtion, using techniques such s frontier-bsed explortion [2] where destintions re the boundries between known nd unknown spce. HC nd JC grtefully cknowledge funding from MINECO-FEDER project DPI , reserch grnts BES nd EEBB , nd DGA Grupo(T04). PD nd VK grtefully cknowledge funding from AFOSR Grnt FA , ONR Grnts N , N , nd N , nd NSF Grnt IIS PD is the recipient of Ntionl Defense Science nd Engineering Grdute Fellowship. PD nd VK re with the GRASP Lbortory, University of Pennsylvni, Phildelphi, USA. {pdmes, kumr}@ses.upenn.edu. JC is with the Universidd de Zrgoz, Zrgoz, Spin. jcste@unizr.es. HC is with the Depto. de Ingenierí Electrónic, Pontifici Universidd Jverin, Bogotá, Colombi. h.crrillo@jverin.edu.co In the second step, the robot computes the utility of performing ech of the cndidte ctions from the first step. We would idelly use the full joint distribution of the mp m nd robot poses x before (P (x, m)) nd fter (P (x, m u, z)) tking the cndidte ction u nd receiving mesurements z. We compute some mesure of informtion gin for ech ction using these prior nd posterior distributions. One mjor problem is tht computing the forementioned joint probbility nlyticlly is, in generl, intrctble [3], [4]. In prctice, n pproximtion of the joint probbility is used, such s ssuming tht the mp nd robot uncertinties re independent [5] or conditionlly independent [3]. These pproches often rely on heuristic liner combintion of the robot nd mp uncertinties [6] [9]. A cvet of the bove is tht the scle of the numericl vlues of the two uncertinties is not comprble, i.e. the mp s uncertinty is often orders of mgnitude lrger thn the robot s uncertinty, requiring the user to mnully tune the weighting prmeters. In this pper, we ssume tht the mp is represented by n occupncy grid, one of the most common pproches, nd tht grph SLAM system is used, the stte-of-the-rt SLAM solution. We propose novel utility function tht drives the robot towrds res of high expected reduction of uncertinty within the mp, s mesured by Shnnon s entropy. We discount the informtion vlue of visiting region if we expect the robot to hve high uncertinty in its pose. We do this using the Rényi entropy, linking the prmeter of the Rényi entropy to the predicted uncertinty in the robot s locliztion. II. STATE-OF-THE-ART UTILITY FUNCTIONS FOR AUTONOMOUS ROBOTIC EXPLORATION The literture on utonomous robotic explortion is diverse, dting bck more thn 30 yers. We focus on recent pproches tht use n informtion-theoretic frmework, s these most closely relte to our proposed utility function. We discuss the underlying ssumptions, nd ssocited shortcomings, of these utility functions. A. Overview In this section, we focus on step 2 of the utonomous explortion tsk outlined in Sec. I. We focus on the uncertinty terms of the utility functions, leving out other uxiliry costs such s energy consumption or heuristic trvel costs. The seminl work in utonomous explortion is from Ymuchi [2]. However, this initil work ssumes perfect knowledge of the robot s pose, so the utility function depends only on the mp. Bourgult, et l. [6] pioneer the use

2 of entropy-bsed utility functions for utonomous robotic explortion, using convex combintion of the mp nd the robot s pose entropies. They compute the posterior uncertinty of the mp ssuming tht the robot hs no error when executing ctions, using (15) from [6]: H[P (m x, d)] P (m ij ) log(p (m ij )) i,j + (1 P (m ij )) log(1 P (m ij )), (1) where m ij is the Bernoulli rndom vrible ssocited with cell ij of the occupncy grid mp nd P (m ij ) is the probbility of tht cell being occupied. However, robot with high uncertinty in its pose my incorrectly cler lrge number of cells in the mp, yielding less ccurte mp while reducing the mp entropy ccording to (1). To ccount for this, Bourgult, et l. use sclr function of the robot s pose covrince mtrix given by the SLAM system proposed in [10]. One drwbck of this pproch is tht the scle of the mp s uncertinty is orders of mgnitude lrger thn the scle of the robot s uncertinty, so chnge in the robot s pose uncertinty hs negligible effect on the vlue of the utility function. To tke the bove into ccount, Bourgult, et l. propose utility function tht is liner weighted combintion of the robot s nd mp s uncertinty, but the weights re set in heuristic fshion. Crlone, et l [4] nd Stchniss [11, Ch. 7] show tht such entropy-bsed strtegies outperform the frontier-bsed method of Ymuchi [2]. Stchniss, et l. [3] propose utility function for Ro- Blckwellized prticle filter bsed SLAM systems. Bsed on severl conditionl independence ssumptions, the utility function is liner sum of the entropy of the robot s poses nd the expected entropy of the possible mps ssocited with ech prticle: #P H[P (x, m d)] H[P (x d)]+ w [i] H[P (m [i] x [i], d)] i=1 (2) where #P is the number of prticles, w [i] is the likelihood of prticle i, nd d = (u, z) is the history of dt, i.e. the control inputs nd received mesurements. Using the stndrd ssumption tht the robot s pose is represented using Gussin distribution, the first term in (2) is: H[P (x d)] = n 2 (1 + log(2π)) + 1 log det Σ (3) 2 where n is the dimension of the robot s pose nd Σ is the n n covrince mtrix. This pproch suffers from the sme issue in the reltive scles of the entropy vlues, s discussed in [8] nd [4]. Blnco, et l. [8] nd Crlone et l. [4] present utility functions tht mke the entropy computtions independent of the occupncy grid size nd only consider cells seen by the robot. However, these pproches re restricted to prticle filter bsed SLAM systems, which re known not to scle s well s grph-bsed pproches with the mp size. Our pproch most closely resembles tht of Crlone, et l. [4], discounting the informtion gin of n ction bsed on the probbility of hving good locliztion of the robot. Vlenci, et l. [5] nd Kim, et l. [9] use grph-bsed SLAM systems [12], ssuming tht the men of the mp nd the robot s poses is close to the mximum likelihood estimte [13, Ch. 11], [12]. Vlenci, et l. represent the mp s n occupncy grid nd Kim, et l. s set of fetures. Both ssume tht the uncertinty in the mp nd the robot s pose re independent, so the posterior entropy is the weighted sum of the individul entropies. These weights re chosen heuristiclly to trde-off between explortion nd exploittion. B. Remrks The utility function from [5] tkes the form: I G [, ẑ] = H[P (x, m d)] H[P (x, m d,, ẑ)] current entropy future/predicted entropy where = 1:T is cndidte ction with time horizon T nd ẑ = z 1:T re the mximum likelihood sensor mesurements received while executing ction. These mesurements re usully clled hllucinted mesurements. The set of future mesurements ẑ is commonly computed vi pproximte ry-csting techniques in conjunction with plusible sensor model, s in [1], [3]. This reduces the computtionl complexity by not considering ll possible combintions of mesurements for n ction. To select n ction, the typicl procedure is to greedily optimize (4) over the set of possible ctions: = rg mx H[P (x, m d)] current entropy H[P (x, m d,, ẑ)] future/predicted entropy Vlenci, et l. [5] noted tht the strting position of the ctions is the sme, so the first term in (5) is equl for ll ctions. This simplifies the objective to: = rg min rg min H[P (x, m d,, ẑ)] predicted joint entropy H[P (x d,, ẑ)] pose entropy + H[P (m x, d,, ẑ)] mp entropy The utility function (4) is usully computed using (1) nd (3), s in [3], [5], [14]. As mentioned previously, the scles of the mp nd pose entropy vlues re very different, i.e. H[P (x d,, ẑ)] H[P (m x, d,, ẑ)], effectively neglecting the effect of the robot s pose uncertinty. To illustrte this, consider simple exmple scenrio in which robot explores m environment with n occupncy grid resolution of 0.05 m. Let us ssume tht only 1% of the cells re unknown, i.e. the probbility of occupncy is 0.5, nd the remining cells re known perfectly, i.e. the probbility of occupncy is 0 or 1. Using (1), the entropy of the mp is 400 bits. Let the pose of the robot, which consists of 2D position nd orienttion, be represented by Gussin distribution with sphericl covrince mtrix. If the stndrd devition of the robot is equl to the environment size (10 m), then, using (3), the (4) (5) (6) (7)

3 entropy is only 16.1 bits. To hve entropy equl to the mp entropy requires the robot s pose to hve stndrd devition of Given this, ll the bove optimiztion problems from (5) to (7) re equivlent in prctice nd effectively neglect the uncertinty in the robot s locliztion. The heuristic weighting from [6], [7] cn overcome this, but requires creful mnul tuning. Given the forementioned scle problem, in prctice the utility (7) reduces to: rg min H[P (m x, d,, ẑ)] predicted mp entropy Note tht computing (8) requires full updte of the occupncy grid, time consuming opertion. Erly pproches to robotic explortion using informtion theory, such s [6], [7], turn this into the equivlent problem of plcing the sensor over regions of mximum entropy in the current mp. This voids the need to updte the mp using unknown future mesurements, nd the objective becomes: rg mx m m() H[P (m x, d)] current entropy where H[P (m x, d)] is the current entropy of cell m nd m() is the set of cells tht the robot my see by tking ction. It is lso possible to speed up the computtion of future mesurements by using plusible sensor model nd only considering the cells in m() s in [3]. III. PROPOSED UTILITY FUNCTION Entropy is mesure of the uncertinty of rndom vrible [15], [16]. The most commonly ccepted definition is from Shnnon [15]. Rényi generlized Shnnon s definition of entropy in [16], to be: H α [p(x)] = 1 1 α log 2 ( n i=1 P α i ) (8) (9) (10) where x is discrete vrible with possible outcomes x 1,..., x r nd P i is the probbility ssocited with the outcome x i ; in our frmework the rndom vrible is the occupncy of ech cell P (m ij ), which cn hve two possible outcomes: 0 (free spce) nd 1 (obstcles). The vrible α [0, 1) (1, ) is free prmeter, though in this work we restrict our ttention to the rnge (1, ). The α prmeter in (10) hs n intuitive, geometricl interprettion. Consider the simplex formed by ll possible discrete probbility distribution functions over set of rndom vribles. The Rényi entropy with prmeter α is relted to the α norm of point in tht vector spce, i.e. probbility distribution. See [17, Ch. 2] for more complete description. Shnnon entropy is specil cse of Rényi entropy, in the limit s α 1 [17]. A. A Shnnon nd Rényi Bsed Utility Function Our proposed utility function is: = rg mx H[p(m x, d)] m m() } {{ } Shnnon entropy H α() [P (m x, d)] Rényi entropy (11) Fig. 1. Vlue of the utility function for Bernoulli rndom vrible m. The figure is mirrored for P (m) [0.5, 1]. where α = α() depends on the uncertinty in the robot s pose fter tking ction. As in (9), the entropy is computed using only the cells tht the robot will see by crrying out the ction. The key difference between our utility function nd (9) is the Rényi entropy, which is used to discount the informtion vlue of the robot visiting cell depending on its locliztion uncertinty. Intuitively, the Shnnon entropy term in (11) is n optimistic mesure of the mp uncertinty, ssuming the robot will not experience locliztion errors when crrying out ction. For exmple, imgine scenrio where robot hs significnt drift in its odometry nd it is not possible to perform loop closure to correct for this. If the robot continues to explore, the SLAM system my yield poor locliztion nd mp estimtes. However, the mp uncertinty computed using (1) will likely decrese becuse the robot will continue to visit unexplored cells. Conversely, (11) combines the two uncertinties in unified mnner without ny heuristic tuning, linking the uncertinty in the mp nd the robot s pose. Given the bove interprettion, nd the reltionship [17]: H[P (x)] > H α [P (x)] H α [P (x)] 0, 1 < α α the proposed utility function (11): (12) 1) is non-negtive; 2) is bounded from bove by the Shnnon entropy nd from below by zero; 3) monotoniclly decreses with α. Note tht the first property complies with the Shnnonin belief tht informtion does not hurt, i.e. dding more mesurements does not increse entropy in expecttion. B. The Prmeter α We relte the prmeter α() in (11) to the predicted uncertinty of the robot fter tking ction. When the robot hs perfect locliztion, i.e. miniml uncertinty, then the informtion gin should be mximl nd when the robot is completely lost, i.e. mximl uncertinty, then the informtion gin should be zero. In other words, we wnt α 1 s the uncertinty becomes infinite since the two

4 entropies in (11) cncel out, nd α s the uncertinty pproches zero, since this minimizes the Rényi entropy. Fig. 1 shows the vlue of (11) for Bernoulli rndom vrible m representing the probbility of occupncy of single cell. In the cse tht P (m) 0, mening we hve (nerly) perfect informtion bout the cell, then single ction will hve little effect on the estimte, nd uncertinty of the robot should not mtter. This is reflected in the cost function, where the informtion gin is pproximtely zero for ll α. In the cse tht P (m) 0.5 we hve (nerly) no informtion bout the cell. Since we hve n uninformtive prior, ny mesurement my be considered good mesurement, nd the informtion gin gin does not depend upon the robot uncertinty. When we hve little bit of informtion bout cell, i.e P (m) 0.45, then the robot uncertinty is most importnt, s n incorrect mesurement (due to poor locliztion of the robot, not sensor noise) will contrdict prior informtion, incresing the uncertinty bout the mp. In other words, this cse should hve the lrgest dependence on the uncertinty of the robot, which we see is true from Fig. 1. We wnt the sclr prmeter α to hve monotone dependence on the uncertinty of the robot s locliztion. A simple cndidte reltionship is: α = (13) σ where σ is sclr relted to the predicted uncertinty of the robot s pose fter tking ction. More complex reltionships should be the focus of further study, but we will show in Sec. V tht (13) is sufficient to demonstrte the efficcy of our proposed utility function. The computtion of σ is left until Sec. IV s it hs n indirect connection to the proposed utility function. IV. AUTONOMOUS EXPLORATION FRAMEWORK In this section we detil frmework for robot-bsed utonomous explortion. Our gol is to cquire mp of n unknown environment in finite time, which implicitly requires the robot to mintin good estimte of its pose. We ssume tht the robot hs SLAM system running, simultneously estimting the robot s pose nd generting n occupncy grid of the environment from the collected sensor dt. Our SLAM front-end is n Itertive Closest Point (ICP)- bsed lser scn mtcher [18]. Our SLAM bck-end is the Incrementl Smoothing nd Mpping (isam) librry [19], which builds pose grph using the lser odometry to constrin consecutive pose nodes. In our frmework, ech node in the pose grph lso hs n ssocited set of fetures extrcted from the lser scn tken t tht loction. These fetures re computed using the Fst Lser Interest Region Trnsform (FLIRT) [20]. We compre the feture sets from different nodes using modified RANSAC lgorithm, dding loop closure constrints if the mtch is sufficiently strong. We lso ssume tht the robot hs nvigtion system cpble of driving it to ny ccessible point in the environment. Fig. 2. Cndidte pth through the environment. The occupncy grid mp is shown in the bckground, with white representing free spce, blck is occupied spce, nd grey is unknown. The robot is t the bottom nd the dshed line is pth through free spce to the frontier gol loction, denoted with the X t the top. Blck squres indicte existing isam nodes while the hollow circles indicte potentil new nodes long the pth. The dotted line is potentil loop closure between the cndidte pth nd previous node in the pose grph. We use the lgorithm proposed in [21], which tkes the most recent lser scn, infltes ll obstcles by specified mrgin, nd then drives the robot towrds the point in free spce tht is closest to the gol loction. While this method works well to void locl obstcles, the robot often gets stuck when the gol loction is fr wy in the mp, e.g. when driving down series of corridors with multiple turns. To void this issue, the robot plns pth in the current mp nd intermedite wypoints re pssed to the nvigtion routine, replnning if ny of the wypoints re found to be inccessible. Our frmework is divided in three high-level steps, s outlined in Sec. I. We describe our pproch to these tsks in the reminder of this section. A. Identifiction of Cndidte Destintions In order to generte gol loctions, the robot computes explortion frontiers [2] from the ltest occupncy grid mp. A cell of the occupncy grid is lbeled s being t frontier if it is unoccupied, djcent to n unknown cell, nd not djcent to n occupied cell. Such cells re clustered in the mp to generte frontier regions, with the gol loctions being the men positions of the cells within ech cluster. We crete n ction pln for ech of these gol loctions. An ction pln is set of wypoints, in free spce, tht led the robot from its current loction to gol: = {(x 0, y 0 ), (x 1, y 1 ),..., (x n, y n )}. We use the AD lgorithm from the SBPL librry [22] to crete the ction plns, using the current occupncy grid. B. Evlution of Action Utilities The robot then computes the utility of ech cndidte ction ccording to (11), in which every term cn be computed using stndrd ry-csting techniques such s the one described in [3]. The min chrcteristic of our utility function is the prmeter α, which depends on the future uncertinty of n ction pln. As discussed in Sec. III- B the prmeter α is relted to the predicted uncertinty of the robot during n ction through (13). This rises two questions: how to get good pproximtion of the robot s

5 E-optimlity (E-opt) minimizes the mximum eigenvlue of the covrince mtrix, Σ, mx(λ k ). (16) Fig. 3. The figure depicts n ction pln with three steps. The men estimted pose (x k ) of the robot, the footprint of lser sensor ttched to the robot (f k ), nd the covrince mtrix ellipsoid of the robot s pose (Σ k ) re shown t ech step. locliztion uncertinty during n ction nd then how to extrct meningful uncertinty sclr σ from it. 1) Uncertinty Prediction: To ddress the first question we utilize isam. An initil fctor is plced t the robot s current estimted loction, with the covrince mtrix tken from the most recent node in the grph. We interpolte the ction pln with some fixed step size nd dd pose nodes long the length of the pth, dding odometry constrints between them. If the ction pln tkes the robot ner other existing nodes in the grph, then the robot hs the potentil to close loop. To tke this into ccount, the robot dds dditionl fctors for ny nerby nodes in the pose grph tht hve sufficiently high number of FLIRT fetures, i.e. the res tht hve gret del of structure which the robot cn use to loclize itself. The position nd covrince mtrix of these fctors re tken from the isam grph of the full mp, nd we dd constrints bsed on the trnsformtion between the existing nd potentil nodes. We use isam to optimize the miniture pose grph creted from the ction pln. The computtionl overhed is miniml since the ctions consist of only 10 s of nodes. Fig. 2 illustrtes this process. 2) Uncertinty Sclr: We compute the uncertinty sclr σ using the covrince mtrix estimtes from the ction pose grph. From the Theory of Optiml Experimentl Design (TOED) [23] [25], there re severl stndrd optimlity criteri tht mp covrince mtrix to sclr while retining useful sttisticl properties. The three most widely used criteri re: A-optimlity (A-opt) minimizes the verge vrince, 1 n trce(σ) = 1 n n λ k (14) k=1 where n is the dimension of the covrince mtrix Σ nd λ k is its kth eigenvlue. D-optimlity (D-opt) minimizes the volume of the covrince mtrix ellipsoid, ( ) det(σ) 1/n 1 n = exp log(λ k ). (15) n k=1 These criteri cn be pplied to the covrince mtrix estimte from the isam solution of the ction pln grph, using either the full covrince mtrix or the mrginl covrince mtrices from ech node in the grph. If the full covrince mtrix is used, there is single α vlue long the entire pth. Then (11) is pplied to the subset of the mp visited by the robot when executing ction. The use of mrginl covrince mtrices is more subtle, s different nodes my hve different α vlues. For ech cell m m(), the robot finds the lst node j from which the cell ws visible nd uses tht α j to compute the informtion gin (11) in tht cell. In this wy, the informtion gin is discounted using the uncertinty of the robot when viewing ech individul cell. This is illustrted in Fig. 3. Note tht ctions my consist of vrible number of wypoints, depending on the distnce through the mp to the frontier. Longer pths llow the robot to explore more re t the expense of higher uncertinty, unless the robot is ble to close loop. The proposed pproch implicitly penlizes long pths due to the expected increse in the pose uncertinty, blncing this with the potentil to gin more informtion by viewing lrger re. C. Execution of Actions The robot crries out the ction with the mximum informtion gin, using the nvigtion lgorithm from [21]. It is possible to experience filure in the nvigtion or SLAM system while executing n ction. If the filure occurs within predetermined time of the beginning of the ction, the next best ction is executed, without recomputing the ction set. In this cse, smll neighborhood round the finl destintion of the fulty ction is blcklisted until it is cler tht the gol is ccessible to the robot. This prevents the robot from repetedly ttempting n impossible ction. V. SIMULATED EXPERIMENTS We perform series of experiments in the cve-like nd Autolb 2D environments in Fig. 5, both of which re vilble in the Rdish repository [26]. The code is written in C++ using ROS nd run on computer with n Intel i7 processor, 8GB of RAM, nd running 64-bit Ubuntu Let SH denote the stndrd utility function (4) bsed on Shnnon entropy. For our proposed utility function, we test two optimlity criteri for the σ computtion: A-opt (A) nd D-opt (D). Both use the full covrince mtrix for ech ction pln. A. Experimentl Procedure We strt the robot from five different loctions in ech environment nd give the robot 3 minute time budget to complete the explortion tsk. With more time the robot would explore the whole environment, but it is more interesting to compre the different utility functions under the sme

6 () Autolb (b) Cve (c) Autolb (d) Cve Fig. 4. CDF of the uncertinty in the robot pose t every time step of every simultion (-b) nd CDF of the running verge uncertinty of the robot pose over ech simultion (c-d). In this figure SH strtegy is colored in green, A strtegy in blue, nd D in red. Fig. 5. () Autolb (b) Cve Mps used in the simultion experiments. conditions. In the ccompnying multimedi mteril there re snpshots of the resulting occupncy grids for some of the simultion nd hrdwre experiments. The robot is differentil drive pltform with mximum velocity of 0.4 m/s. The robot is equipped with lser scnner with mximum lser rnge of 10 m, field of view of 270, nd 1080 bems per scn. We use our own kinemtic nd lser simultors. The odometry noise is Gussin with stndrd devition of 5 cm for every m in the (x, y) position nd 1 for every 45 in the orienttion. The lser noise is lso Gussin, with stndrd devition of 1 cm. The robot begins ech experiment with no informtion bout the environment. For ech tril, we record the uncertinties of the current pose nd the history of poses nd the percentge of re correctly explored t every time step. We mesure the uncertinty of covrince mtrix using D- opt, i.e. bsed on the determinnt of the covrince mtrix, s suggested in [24]. B. Presenttion of the Results One problem with evluting robot-bsed utonomous explortion experiments is tht trils my hve rdiclly different trjectories, mking pirwise comprisons difficult. Moreover, presenting just the men or the medin of the trils cn be misleding. Inspired by the solution of similr problem [18], we summrize the results of the experiments using the cumultive distribution function (CDF) of the metrics of interest. The CDF provides richer representtion of the vribility of the results thn the men or medin, while voiding misleding necdotl evidence due to noise. For ech prmeter of interest, e.g. uncertinty in the robot s pose, we compute the CDF from histogrm of the vilble dt t ech time step. The bin size is utomticlly set using the procedure described in [27]. To more clerly quntify the differences between CDFs, we extrct three point estimtes: the 50 th, 75 th nd 95 th percentiles. TABLE I 50 th, 75 th AND 95 th PERCENTILES OF THE CDFS IN FIG. 4. Robot pose Pose history Autolb Cve P50 P75 P95 P50 P75 P95 A D SH A D SH For metrics in which high vlue implies worse performnce, e.g. uncertinty or trnsltionl error, we would like CDF tht reches 1 s quickly s possible. C. Simultion Results 1) CDFs of Uncertinty: Fig. 4 shows the CDFs of the pose uncertinty in ech test environment. Fig. 4 nd Fig. 4b show the uncertinty in the robot s pose t ech individul node in the pose grph s computed by isam, mesuring the worst-cse performnce of the explortion strtegies. Fig. 4c nd Fig. 4d show the running verge uncertinty over the history of robot s poses, mesuring the verge performnce over tril. Tb. I shows the percentiles of the CDF. Overll, our proposed utility function hs lower uncertinty in both environments. The robot using our utility function with A-opt results in 49.70% less uncertinty in the pose t the 75 th percentile thn the robot using Shnnon entropy. This difference is still lrge (46.84%) t the 95 th percentile. 2) Percentge of Explored Are: Evluting the percentge of mp correctly identified by robot during explortion is inherently clssifiction problem: we wnt to determine if ech cell is correctly identified s being free or occupied. However, the number of free cells is typiclly much greter thn the number of occupied cells. So robot my return lrge number of mtched free cells even if there re very few mtches of the occupied cells. To void this bis towrds free spce, we mesure the mp ccurcy by independently estimting free nd occupied cells. The percentge of the mp correctly explored is computed s the blnced ccurcy (BAC), concept borrowed from the mchine lerning community [28]. The BAC eqully weights the ccurcy of the estimtes of the free nd occupied cells: BAC = 1 ( ) # correct free cells # correct occupied cells + 2 # totl free cells # totl occupied cells (17)

7 TABLE II MEAN AREA CORRECTLY EXPLORED BY EACH EXPLORATION STRATEGY Autolb Cve Free cells Occupied Cells BAC Free cells Occupied Cells BAC Accurcy ±σ Accurcy ±σ Accurcy Accurcy ±σ Accurcy ±σ Accurcy A % 6.09 % % 4.86 % % % 6.27 % % 1.96 % % D % 5.60 % % 5.43 % % % 5.50 % 9.41 % 1.01 % % SH % 7.95 % 8.77 % 5.41 % % % 5.50 % 2.61 % 0.61 % % it is more conservtive thn D-opt, which grees with the findings in [24]. The experiments lso revel the fct tht the utility function is not robust to filure in the SLAM or nvigtion system. In others words, the utility function is not fult-tolernt to filures of the lser-bsed loop closure system, or to diffuse reflections of the lser scn tht produce phntom explortion frontiers. This will be ddressed in future work s it is necessry for truly utonomous robots. Fig. 6. Scrb pltform used to perform the utonomous explortion tsk. Tb. II shows the percentge of ech mp correctly explored by robot using ech utility function, with the proposed utility function performing fvorbly, prticulrly with respect to occupied cells. VI. HARDWARE EXPERIMENTS We perform series of experiments using the Scrb pltform, shown in Fig. 6. The robot is equipped with Hokuyo UTM-30LX lser scnner (30 m rnge nd 270 field of view) nd n on-bord computer with n Intel i5 processor nd 4GB of RAM running Ubuntu nd ROS. The robot explores the office environment t the University of Pennsylvni shown in Fig. 7. A. Experimentl Procedure The robot begins the utonomous explortion tsk t different strting loctions within the environment nd runs until it hs explored the entire environment. The mximum velocity of the robot is set to 0.45 m/s. The robot runs the utonomous controller using its on-bord computer. The robot strts ech experiment with no informtion bout the environment. Since it ws not possible to obtin ground truth for the tril, we cnnot provide n insightful comprison ginst the Shnnon bsed utility function. Nevertheless, the hrdwre experiments llow us to compre different prmeteriztions of our frmework nd check how they behve with rel dt. B. Results We compre the results of robot using the A nd D methods to teleoperted explortion. Fig. 7 depicts exmple occupncy grids of the environment, fter 5 minutes nd fter the robot hs completed the explortion tsk. Overll, the hrdwre experiments show the conservtive behvior of our utility functions, with the robot retrversing known res of the mp in order to mintin good locliztion. The experiments with A-opt show tht VII. CONCLUSIONS In this pper we present novel informtion-theoretic utility function to select ctions in n utonomous explortion tsk. The proposed utility function uses both the Shnnon nd Rényi definitions of entropy to utomticlly blnce explortion nd exploittion by considering the uncertinties in both the robot pose nd the mp. Our utility function hs severl key properties: it is non-negtive, it is bounded from bove by the Shnnon entropy (i.e. the existing pproches), nd monotoniclly decreses with the uncertinty in the robot s locliztion. These properties stem from the mthemticl reltionship between the Shnnon nd Rényi definitions of entropy. Unlike previous ttempts to define utility functions using convex combintion of the mp nd robot pose uncertinty, our utility function does not require ny mnul per cse prmeter tuning. Insted we rely on Rényi s definition of entropy to discount the expected informtion gin nd directly relte the α prmeter in (11) to the predicted uncertinty. This reduces to stndrd pproches when the locliztion uncertinty is eliminted. Our simultion nd experimentl results show substntil reduction of the robot pose nd mp uncertinties when using our proposed utility function compred to the stte of the rt utility functions. This decrese in uncertinty is due to the exploittion of previous mp informtion, resulting in more loop closures. However, explortion is more conservtive nd the predicted pose uncertinty used to compute α ssumes tht loops cn be closed relibly. Clerly chrcterizing the relibility of the loop closure system, nd the subsequent effects on the informtion gin, is n importnt direction for future work. REFERENCES [1] W. Burgrd, M. Moors, C. Stchniss, nd F. Schneider, Coordinted Multi-robot Explortion, IEEE Trnsctions on Robotics (TRO), vol. 21, no. 3, pp , Jun [2] B. Ymuchi, Frontier-bsed Explortion Using Multiple Robots, in Proceedings of the Second Interntionl Conference on Autonomous Agents, ser. AGENTS 98. ACM, 1998, pp [3] C. Stchniss, G. Grisetti, nd W. Burgrd, Informtion Gin-bsed Explortion Using Ro-Blckwellized Prticle Filters, in Proceedings of Robotics: Science nd Systems Conference (RSS), Cmbridge, MA, USA, Jun

8 () Teleoperted 5min (b) A 5 min (c) D 5 min (d) Teleoperted 5 min 21 sec (e) A 30 min 57 sec (f) D 16 min 41 sec Fig. 7. Exmples of the resulting mps nd pose grphs from hrdwre experiments. The robot is either teleoperted, or utonomously explores using the A nd D methods from Sec. V. The blue edges indicte odometry constrints in the pose grph while red edges indicte loop closure events. ( c) show the occupncy grids nd pose grphs built by the robot fter 5 min. (d f) show the finl occupncy grids nd pose grphs. [4] L. Crlone, J. Du, M. Kouk, B. Bon, nd M. Indri, Active SLAM nd Explortion with Prticle Filters Using Kullbck-Leibler Divergence, Journl of Intelligent & Robotic Systems, vol. 75, no. 2, pp , Oct [5] R. Vlenci, J. Vllve, G. Dissnyke, nd J. Andrde-Cetto, Active Pose SLAM, in Proceedings of the IEEE/RSJ Interntionl Conference on Intelligent Robots nd Systems (IROS), Oct. 2012, pp [6] F. Bourgult, A. Mkrenko, S. Willims, B. Grocholsky, nd H. Durrnt-Whyte, Informtion Bsed Adptive Robotic Explortion, in Proceedings of the IEEE/RSJ Interntionl Conference on Intelligent Robots nd Systems (IROS), 2002, pp [7] A. Mkrenko, S. B. Willims, F. Bourgult, nd H. F. Durrnt-Whyte, An Experiment in Integrted Explortion, in Proceedings of the IEEE/RSJ Interntionl Conference on Intelligent Robots nd Systems (IROS), Sep. 2002, pp [8] J. Blnco, J. Fernndez-Mdrigl, nd J. Gonzlez, A Novel Mesure of Uncertinty for Mobile Robot SLAM with RoBlckwellized Prticle Filters, The Interntionl Journl of Robotics Reserch (IJRR), vol. 27, no. 1, pp , Jn [9] A. Kim nd R. M. Eustice, Perception-driven Nvigtion: Active visul SLAM for Robotic Are Coverge, in Proceedings of the IEEE Interntionl Conference on Robotics nd Automtion (ICRA), My 2013, pp [10] H. J. S. Feder, J. J. Leonrd, nd C. M. Smith, Adptive Mobile Robot Nvigtion nd Mpping, The Interntionl Journl of Robotics Reserch (IJRR), vol. 18, no. 7, pp , Jul [11] C. Stchniss, Robotic Mpping nd Explortion. Berlin, Germny: Springer, 2009, vol. 55. [12] G. Grisetti, R. Kuemmerle, C. Stchniss, nd W. Burgrd, A Tutoril on Grph-bsed SLAM, IEEE Intelligent Trnsporttion Systems Mgzine, vol. 2, no. 4, pp , Jn [13] S. Thrun, W. Burgrd, nd D. Fox, Probbilistic Robotics. Boston, MA, USA: MIT Press, [14] J. Du, L. Crlone, M. Kouk, B. Bon, nd M. Indri, A Comprtive Study on Active SLAM nd Autonomous Explortion with Prticle Filters, in Proceedings of IEEE/ASME Interntionl Conference on Advnced Intelligent Mechtronics, Jul. 2011, pp [15] C. Shnnon nd W. Wever, The Mthemticl Theory of Communiction, ser. Illinois Books. Chmpign, IL, USA: University of Illinois Press, [16] A. Rényi, On Mesures Of Entropy And Informtion, in Proceedings of the 4th Berkeley Symposium on Mthemtics, Sttistics nd Probbility, 1960, pp [17] J. Principe, Informtion Theoretic Lerning: Rényi s Entropy nd Kernel Perspectives, ser. Informtion Science nd Sttistics. Berlin, Germny: Springer, [18] F. Pomerleu, F. Cols, R. Siegwrt, nd S. Mgnent, Compring ICP Vrints on Rel-world Dt Sets, Autonomous Robots (AR), vol. 34, no. 3, pp , Apr [19] M. Kess, A. Rngnthn, nd F. Dellert, isam: Incrementl Smoothing nd Mpping, IEEE Trnsctions on Robotics (TRO), vol. 24, no. 6, pp , Dec [20] G. D. Tipldi nd K. O. Arrs, FLIRT-interest regions for 2D rnge dt, in Proceedings of the IEEE Interntionl Conference on Robotics nd Automtion (ICRA), My 2010, pp [21] J. Guzzi, A. Giusti, L. M. Gmbrdell, G. Therulz, nd G. A. Di Cro, Humn-friendly Robot Nvigtion in Dynmic Environments, in Proceedings of the IEEE Interntionl Conference on Robotics nd Automtion (ICRA), My 2013, pp [22] M. Likhchev, Serch-Bsed Plnning Librry, sbpl/sbpl, ccessed: [23] F. Pukelsheim, Optiml Design of Experiments, ser. Clssics in Applied Mthemtics. Phildelphi, PA, USA: Society for Industril nd Applied Mthemtics (SIAM), [24] H. Crrillo, I. Reid, nd J. A. Cstellnos, On the Comprison of Uncertinty Criteri for Active SLAM, in Proceedings of the IEEE Interntionl Conference on Robotics nd Automtion (ICRA), St. Pul, MN, USA, My 2012, pp [25] H. Crrillo, Y. Ltif, J. Neir, nd J. A. Cstellnos, Fst Minimum Uncertinty Serch on Grph Mp Representtion, in Proceedings of the IEEE/RSJ Interntionl Conference on Intelligent Robots nd Systems (IROS), Vilmour, Portugl, Oct. 2012, pp [26] A. Howrd nd N. Roy, Rdish: The Robotics Dt Set Repository, ccessed: [27] H. Shimzki nd S. Shinomoto, A Method for Selecting the Bin Size of Time Histogrm, Neurl Computtion, vol. 19, no. 6, pp , Jun [28] K. Brodersen, C. S. Ong, K. Stephn, nd J. Buhmnn, The Blnced Accurcy nd its Posterior Distribution, in Proceedings of the Interntionl Conference on Pttern Recognition (ICPR), Aug. 2010, pp