Adaptive Path Planning for Effective Information Collection

Transcription

1 Adaptive Path Planning for Effective Information Collection Ayan Dutta, Prithviraj Dagupta Abtract We conider the problem of information collection from an environment by a multi-robot ytem, where the location to ample information from are known with ome amount of uncertainty by the robot. Different robot are equipped with different type of enor and the quality of information collected can be improved if information i ampled from the environment with different enor, wherever poible. The problem facing the robot i to determine their path through information ampling location, o that the overlap between path of robot with identical enor i reduced, while the robot with complementary enor are llowed to have overlapping path. To addre thi problem, we decribe a ditributed path planning algorithm that comprie of two tep - path merging and path decoupling o that the robot are able to collect more information from the environment. We have hown analytically that our propoed algorithm ha polynomial time complexity and our experimental reult, in imulation, how that by dynamically adapting path, the robot are able to collect higher information than by following initially generated individual path. INTRODUCTION Dynamic path planning i a well-known and crucial apect of autonomou navigation of multiple mobile robot. Recently, reearcher have conidered a practical apect of the multi-robot path planning problem where each robot ha to dynamically adapt it path baed upon the information collected while ampling certain trategic point along their path [9]. In thee technique, each robot ha the ame et of enor and allocating multiple robot to collect information along the ame path i conidered ub-optimal. Alo, the path update mechanim of a robot incorporate only the information collected by a robot itelf while navigating. In thi paper, we conider uch a information collection problem with heterogeneou robot (enor-wie). We poit that the performance of multi-robot information collection with multiple enor can be improved, if a robot path i not only updated uing it elf-ampled information, but, the information collected by other robot a well a the enor capabilitie of thoe robot i incorporated dynamically into robot path plan. However, thi alo introduce additional overhead to avoid redundant information collection by robot and to direct robot with appropriate enor to improve the quality of collected information, and, finally to determine opportune location and time interval to periodically hare the information between robot. The problem i further complicated in realitic cenario, where robot have limited battery and might intermittently get out of communication range with each other. Computer Science Department, Univerity of Nebraka at Omaha {adutta, pdagupta}@unomaha.edu To addre thee challenge, we propoe a novel framework were robot initially generate path individually auming a probabilitic information ditribution in the environment. Thee path are then hared between robot and two technique - merging path between robot with different enor or decoupling path between robot with imilar enor, are employed, to increae the information gain from the collective path. Our propoed algorithm ha polynomial time complexity. We have teted the quality of our propoed path adaptation algorithm in imulation. Our reult how that with dynamic adaptation of path baed on robot enor capabilitie, multiple robot can collect information more effectively than without adaption of path. The run time of our propoed path adaptation algorithm are alo computationally feaible - maximum of 2 econd for adapting a path, coniting 5 cell. RELATED WORK Multi-robot path planning ha been a well reearched apect of robotic and everal technique for waypoint navigation and coverage with multiple robot have been propoed. Mutli-robot informative path planning (MIPP) involve an apect of the general multi-robot path planning problem where each robot ha to determine waypoint between given tart and end location in the environment o that the information gain of the reulting path i increaed. In one the earliet work on MIPP, Guetrin et al. [4] have modeled it a a enor placement problem and propoed a greedy algorithm that enure that the mutual information gain acro the enor i maximized. Singh et al. [9] have propoed a recurive, branch and bound algorithm to olve the MIPP problem that find the bet, budget-limited path through a graph of poible waypoint. Their algorithm i verified for a maritime information collection application and how improvement over greedy path election. Thi work i inpired from a recurive greedy bet walk earch algorithm, a decribed in [2]. The MIPP problem with periodic connectivity between robot ha been tudied in [5]. Here robot do not need to maintain continuou connectivity, but form a connected network at certain interval. In contrat to thee work, our work in thi paper conider robot with different enor and attempt to merge or eparate path to improve the information collection baed upon the imilarity of enor of robot allocated to thoe path. Path merging algorithm for a ingle robot path have been propoed in [6]. The robot firt generate multiple path between given tart and goal location uing any ampling baed path generation method. Then, a hybrid path that improve an objective function baed on the quality

2 of the waypoint i generated by electing ubet of waypoint from the different generated path. In [7], author have propoed a multi-robot path planning algorithm, where robot adjut their initially generated path depending on their repective prioritie, while in [1], a contrained path planning problem by multiple robot i propoed. Mot of thee algorithm are intended for waypoint navigation and are not directly applicable to information collection. In our propoed method, robot firt generate their path uing a greedy method, imilar to [4] and then the generated path are dynamically adjuted to achieve higher information gain. MODEL Let R = {r 1, r 2,.., r N } denote a et of N robot and S i denote the et of enor on robot r i. Each enor S i ha an aociated reliability value ρ that repreent the quality of information that i ampled by that enor from the environment. For the purpoe of navigation, each robot ue a map of the environment; the map i decompoed into a grid-like cell uing a cellular decompoition technique [3]. A robot enter a cell to collect information from the region within the cell. Let C denote the et of cell in the environment. Robot r i path, Π i i defined a an ordered equence of cell it viit, i.e., Π i = {c 1, c 2,...}. For information collection, each cell c j C i further ub-divided into a et of information point. Upon entering a cell, a robot dynamically elect a ubet of the information point in the cell to viit and take reading from uing it on-board enor. Following Shannon information entropy formula [8], the information gain from point p k,j P j ampled by robot r i in cell c j i given by I i,j = p k,j P j P (p k,j ) log b P (p k,j ) S i ρ, where P (p k ) denote the probability that point p k provide high quality information and ρ i the reliability value of enor S i. We aume that the information collected from information point in a cell follow the law of diminihing return. That i, the initially viited point in a cell provide new information, but a the robot collect more information from that cell, future point might add only repetitive or redundant information. To model diminihing return from information collection, we limit the maximum allowable information gain from a cell to a threhold, MAX INF O. Once the information gain from a cell exceed M AX IN F O, the robot doe not gain any new information but will only expend time and battery to viit more information point in that cell. The total information gain from robot r i path Π i i given by I(Π i ) = c j Π i I i,j. Robot incur cot in term of battery power pent for navigating in the environment and collecting information uing their enor. The cot of viiting cell c j Π i by robot r i can be written a, cot i,j = cot i,j ene + cot i,j travel. The ening cot i calculated from the enor power requirement while the travel cot i calculated a the ditance between the centroid of cell (c j 1, c j ) Π i. The utility of path Π i can then be written a U(Π i ) = I(Π i ) cot(π i ), where cot(π i ) = c j Π i cot i,j. Fig. 1. (left)the environment i divided into cell uing grid-baed decompoition. Each cell c j contain a et of information point that are ditributed uniformly acro the cell (black dot). A robot elect a ubet of thee information point to viit and ample data from (highlighted point). (middle) Robot path through the elected information point in a cell; (right)one robot path through the environment. For combining the utilitie of overlapping path followed by two different robot, we conider their enor type. If the enor are identical, the combined utility i ub-additive a the robot each expend the cot to navigate the path but get the ame information on their enor. On the other hand, if the enor are complementary, the utilitie are uperadditive a more information i gained by ampling the ame point uing different enor. The utility function i given in Equation 1. U(Π i ) + U(Π j ) if S U(Π ij = i S j { } U(Π i ) + U(Π j ) +( I i,k ρ ) c k Π i Π j S i S j if S i S j = { } (1) The objective of the informative path planning problem i to find a et of path {Π 1, Π 2,.., Π R },.t. max U(Π i ). c k Π i Π j I i,k i R PATH ADAPTATION ALGORITHMS FOR INFORMATION COLLECTION Initially all the robot will generate path for moving through the environment to collect information. Any path generation algorithm for information collection can be ued. In thi paper, we have ued path generation algorithm for information collection, imilar to a decribed in [4]. According to thi path generation algorithm, we add 2 new cell to the current path in one cycle, addition of which to the current path yield highet amount of utility. Let C denote the et of 2 cell which will be added to robot r i current path Π i. Let c lat be the lat cell added to Π i. We earch for 2 uch cell, {c 1, c 2 }, which being added to Π i, yield highet utility. c 1 i one-tep ditant cell from c lat, i.e., c 1 neighbor(c lat ), c 2 i 2-tep away from c lat. Once we are done adding bet 2 uch cell to C, we add C to Π i. We repeat thi procedure until current path etimated battery expenditure (EtB(Π i )) doe not cro the battery budget. Merging of robot path We have dicued earlier that if robot have different enor, then they would gain higher amount of information from viiting ame cell. The objective of path merging algorithm i to merge path of robot, having different

3 Algorithm 1: Path Merging of Robot 1 (b) 2 3 Fig. 2. (left)two robot generate two path; (right)left robot ha changed it path to merge with the right one path. (b)(left)two robot generate two path; (right)left robot ha changed it path to decouple from the other path (Dahed line indicate the older path) enor to increae the amount of information gain. For path merging, robot will exchange the information about generated path with the robot which are in communication range. Thi merging of path will be done equentially by ordered robot. Firt every robot will broadcat information about their generated path along with the etimated amount of utility they will achieve by following it path. Upon receiving thi path information from in-range robot, each robot will ort thee path, along with it own, according to the etimated utility of thee path. Kort denote the et of all path in a orted order. The equence of path-merging will be according to the order of thee orted path and it i denoted a Rort, where highet utility path generating robot will be the firt member in Rort and o on. Path merging i done by mean of replacing cell in one robot path with cell in another robot path. One example of path merging i hown in Figure 2(left), where two robot having different et of enor merge their initially generated path. Path merging procedure i hown in Algorithm 1. Merging of path will be done only between robot which do not have ame et of enor. Let aume ri Rort i the robot which i merging it path, Πi with path of other robot, which do not poe the ame et of enor a ri. For every cell ck Πi, ri will check whether ck i replaceable or not with any cell cl Πj and Si Sj! = { }, j 6= i. Whether ck and cl are replaceable or not, can be decided by replaceable() function. The function replaceable() return TRUE, when poible replacement of two candidate cell ck and cl earn higher utility than without the replacement (baed on the utility function decribed earlier); otherwie replaceable() return FALSE (line 12 15). Firt robot member, r1 Rort will merge it path firt and when it i done merging, r1 will end new path information Π1, along with the information of about which cell are merged with other path to all in-range robot and then r2 Rort will do merging. Thi procedure will go on until all robot are done merging their path with all other robot. Any robot will not change the portion of it path, which i already been merged with, by another robot path earlier. Note that path merging doe not necearily lead to the fact that, ci+1 neighbor(ci ), where, ci, ci+1 Πmerged. Although one robot ha to cover the cell between ci and ci+1, but it will not do ening (or, information collection) through the intermediate cell. We aume that low-level path-planning algorithm uch a D [] algorithm to go from ci to ci+1 a ci+1 6 neighbor(ci ) i already available pathmerge() Input: Kort : Sorted et of all path. Output: K : A et of new merged path. K { }. Rort : Sorted et of all robot, acc. to Kort. Each ri Rort will do the following: BESTU U (Πi ) for all (Πi, Πj )pair, Πj Kort, i 6= j, Si Sj! = { } do for each ck Πi do if ( cl Πj, cl 6 Πi and replaceable(ck, cl ) == TRUE) then Πi Πi {cl } \ {ck }. //replace ck with cl. K Πi. Send Kort and K to ri+1 Rort. replaceable(ck, cl ) Input: ck Πi and cl Πj. Output: TRUE or FALSE. Πi Πi {cl } \ {ck }. if P (U (Πi ) + U (Πj ) + Ii,l Si Sj ρ > BESTU + U (Πj )) then return TRUE; ele return FALSE; to the robot. Path Decoupling If two or more robot, having ame et of enor viit ame cell to collect information, then they are not gaining any new information. Becaue ame enor will collect ame type of information from ame information point. So, they will jut incur higher cot without gaining new information. Thu the total utility will be lowered. To avoid thi ituation, robot having ame et of enor hould not viit ame cell. To enable thi, they hould follow pathdecoupling() method, hown in Algorithm 2. We will introduce 2 new term in thi ubection. Firt one i imilar path. Path Πi i labeled a imilar to path Πj, if correponding robot, ri and rj have ame et of enor S and Πi Πj > p% Πi, where p i a contant, which i the imilarity threhold for determining whether the path i imilar or not. If very few number of cell are ame in two path, then the robot do not need to change that, a thi i a computationally cotly operation. Second new term introduced in Algorithm 2 i OK cell. An OK cell i a cell, adding of which to a path Πi, will not lead it to be imilar to any other path, Πk, k 6= i. Path are firt orted from highet cot to lowet cot order by every robot. The robot with highet cot path will firt get chance to decouple it path from other path, and then robot with lower cot path will execute pathdecoupling() algorithm (line 2 4). Next, for every cell cj, which i alo preent in other path, the robot will find the bet utility providing OK cell, cl (6= cj ) from the neighbor cell of cj predeceor in the path and will replace cj with cl. An example of path decoupling operation i hown in Figure 2.(b), where robot

4 in the left change it path to decouple from imilar path of the other robot. Once the decoupling i done, robot will hare their new path information with other in-range robot. Algorithm 2: Decoupling of imilar robot path. 1 pathdecoupling() Input: K: Set of all path. Output: Π i: Modified path of robot r i. 2 K i K: Sorted et of imilar path with robot r i path Π i. 3 R ort: Sorted et of robot according to K i. 4 Each robot r k R ort will do the following: 5 for each path Π k K i do 6 for each cell c j (k Π i) do 7 c l Highet utility-earning OK cell in neighbor(c j 1) \ c j. (Detail in text) 8 Π i Π i \ c j c l. 9 Update K i. return Π i. Path planning algorithm: Thi i the main algorithm (hown in Algorithm 3), from which all other procedure are being called. Initially after generating the information point, robot generate their path uing pathgeneration() algorithm. Robot do not have unlimited communication range; they can communicate only within a fixed radiu. It i computationally and battery power-wie very cotly to keep all the robot in communication range at all time. But we aume that initially, before tart of the information collection proce, all the robot are in each other communication range, i.e., ditance between any two robot i le than the communication range radiu. After all the robot have generated their path, they exchange their path information with all other robot and execute pathdecoupling() and pathmerge() algorithm (detail in previou ection). Now that the robot have finally decided which path they are taking, they tart following their repective path. We aume that the initial knowledge about the information point i uncertain. The environment can change over time. While actually exploring, robot can find ome new point to collect information from or the robot may perceive that ome a priori elected point are currently unavailable. Again, robot can pend more/le battery than etimated amount. Thu the etimated utility and battery expenditure of path Π i may change when the robot actually viit cell. Thi uncertainty i accommodated in the model by introducing noie to the number of elected point viited by the robot in each cell. If initially in cell c j robot r i wa uppoed to viit K number of cell, which wa the etimator of the utility earned, then while actually viiting c j, thi K i adulterated to K ± δ, where δ [, K]. Thi change effect robot information gain, incurred cot (thu utility) and actual amount of battery pent. A actual and initial etimated utility and battery expenditure of a path might be different, o each robot need to refine it earlier generated path periodically to accommodate the change. At every time interval T α, where T i a contant and α = {, 1, 2,..}, if the difference between actual and Algorithm 3: Multi-Robot Informative Path Planning under Uncertainty 1 Each robot r i will follow thee tep: 2 Generate path Π i, uing pathgeneration() algorithm. 3 Exchange path information with in-range robot. 4 Decouple Π i from imilar path, uing pathdecoupling() algorithm. 5 Merge Π i with appropriate robot path, uing pathmerge() algorithm. 6 Send new path information to the next robot in R ort. 7 Start following Π i. 8 Calculate actual utility earned and actual amount of battery pent. 9 if ( U(Π i) U act(π i) ) > T HRES U ( EtB(Π i) ActB(Π i) ) > T HRES B at time interval T α then Generate a new path Π i and follow the path. 11 ele 12 Follow path Π i. 13 if r i come in comm. range of r k for the firt time in time interval T α and T α+1 then 14 if S i S k! = { } then 15 Follow tep in line 4 and 6 END. 16 ele 17 Follow tep in line 4, 5 and 7 END. 18 ele 19 Follow path Π i. 2 Termination Condition: Generated path ha been completely viited OR pent budgeted amount of battery. initial etimated utilitie and battery expenditure cro repective threhold value T HRES U and T HRES B, the robot will generate new path (line 8 12). A we know that robot have limited communication range - they necearily do not need to tay within other robot communication range and at every time interval T α each robot generate a new path, therefore it would reult in better utility achievement, if the robot can merge and decouple path, when poible and appropriate after new path generation. That i why, whenever any robot r i come in communication range with r k (k i), for the firt time between two ucceive path generation interval, i.e., between time interval T α and T α+1, they firt exchange their path and then either decouple or merge depending on the pecific criteria dicued in previou ection (line 13 19). Thi algorithm run until the robot have completed viiting their generated path or they have pent their allocated battery budget (line 2). ANALYSIS Complexity of path merging algorithm: Let n and l denote the total number of robot and maximum number of cell in any robot path. Line 3 in Algorithm 1 would take O(nlogn) computation for orting. For the loop in line 6, there can be (n 1) of uch pair in wort cae and for each of the (n 1) pair, inner loop in line 7 will run maximum of l 2 time. replaceable() function take only contant time. Thu the final time complexity for path merging algorithm i O(nl 2 ) (a logn i much leer than l 2 ), wherea time

5 Path decoupling algorithm Total utility earned 12 Before decoupling After decoupling Fig. 3. Run time of decoupling algorithm. (b) Utility earned by 2 robot - before and after path decoupling. 25 Total utility earned The algorithm preented in thi paper are teted in imulated environment. An environment with cell, in a grid tructure ha been ued. Initial cell poition of robot are drawn from U[(, 3), (, 3)]. We aumed that initially all the robot are with everyone communication range. Robot firt generate path and then depending on enor et on each robot, pathdecoupling() or pathmerging() algorithm i executed. Each robot i only allowed to viit upto a certain number of cell in the environment (which i the repreentative of battery budget). that each robot can collect information from i varied through 5 to 5. Maximum number of information point in each cell in drawn from N (15, 3). Information collection i abtracted in the experiment and travel cot inide each cell i alo not taken into account. But ening cot i accounted for the cot calculation. We aume that each robot i equipped with one enor. Reliability value of the enor are drawn from U[, 1]. Senor cot, in term of electrical energy needed, i alo normalized to [, 1]. Each tet i run for 5 time, but the deviation i nominal; thu not included in the figure. Note that, the run time reported in thi paper are running time of the algorithm, not the time to viit cell and information collection by robot. We have teted the pathdecoupling(), with 2 robot having ame enor. Run time of thi decoupling algorithm i very nominal. For decoupling path with coniting 5 cell, algorithm took only 5 econd. Our main objective wa to how that by decoupling their path, robot with ame enor earn more utility than if they followed the ame path and the reult i hown in Figure 3(b). A can be een in thi figure that with increaing length of the path, with decoupling, robot earn more utility than following the ame path and collecting redundant information. For example, with 5 cell-coniting path, in total, robot earn an utility of 3.53, after executing pathdecoupling() algorithm, a oppoed to an utility of without decoupling. Run time for thi algorithm i alo very nominal; for 2 robot, 4.5 Run time (econd) E XPERIMENTAL E VALUATION 5 Run time (econd) complexity of comparable path hybridization algorithm [6] i O(n2 l2 ), which i clearly much wore than our propoed path merging algorithm. Complexity of path decoupling algorithm: Similar to pathmerging() algorithm, orting will take O(nlogn) computation, outer loop (line 5 in Algorithm 2) will run for maximum of (n 1) time and the inner loop will run for l time, which i the maximum number of cell in any path. But computation inide the inner loop will not be of contant time. Complexity of finding bet cell and checking for OK cell will be O(b) and O(n) repectively. Thu time complexity of path decoupling algorithm will be O(n2 bl). Any communication, with all other robot in wort cae, will be of O(n2 ) complexity. Note that, in real world cenario, a the robot explore different region, not all the robot will come into within every robot communication range very often (except for the initial tate). Thu actual communication complexity will be much le. Path merging algorithm Utility before merging 25 Utility after merging (b) Fig. 4. Run time of merging algorithm. (b) Utility earned by 2 robot - before and after path merging. thi algorithm only take 4.76 econd for decoupling 5 cell-coniting path. A cenario of 5 cell-coniting path ha been implemented with 2 robot, where they had ame enor and their initial path were alo exactly ame. Actual reultant decoupled path are hown in Figure 5[right]. Next we have teted the pathmerging() algorithm, with 2 robot having different enor. Run time for thi algorithm i lightly higher than the previou algorithm; for 2 robot, thi algorithm take.41 and 2.13 econd for decoupling and 5 cell-coniting path repectively (Figure 4). With path merging, total utility earned increae conitently from the utility earned from path before merging, a the path length increae (Figure 4(b)). For example, with path length, total utility before merging wa 22.9 and it increaed to 65.8 after merging and with path length 5, utility increaed to from In thi cae alo, we have implemented a cenario with 2 robot, where they have different enor and the robot have to generate 5 cellconiting path. Actual reultant merged path are hown in Figure 5(b)[right]. Notice that, actual path length of the reultant merged path i more than 5, but only 5 cell marked S are viited by the robot for information collection, while other cell are paed by the robot to reach different

6 Fig. 5. (b) Decoupling of 2 robot path. (b) Merging of 2 robot path. S marked cell. A our reult indicate, by dynamically adapting path, intead of following initial generated path, robot can earn higher information and thu utility. C ONCLUSIONS AND F UTURE W ORK In thi paper, we have propoed path adaptation algorithm for information collection from an environment by a et of mobile robot. Thee path adaptation algorithm take enor information into account and adjut the initially generated path to collect higher amount of information. Our propoed algorithm are fat and they aure higher information collection than information collected by initially generated path. In thi paper, we have ued already exiting algorithm for informative path generation. We are working toward developing more ophiticated path generation algorithm and we will alo tet our propoed path adaptation algorithm with more number of robot and will compare the reult with exiting comparable algorithm. R EFERENCES [1] Pramod Abichandani, Hande Y Benon, and Mohe Kam. Decentralized multi-vehicle path coordination under communication contraint. In Intelligent Robot and Sytem (IROS), 211 IEEE/RSJ International Conference on, page IEEE, 211. [2] Chandra Chekuri and Martin Pal. A recurive greedy algorithm for walk in directed graph. In Foundation of Computer Science, 25. FOCS th Annual IEEE Sympoium on, page IEEE, 25. [3] Howie Choet, Kevin M. Lynch, Seth Hutchinon, George A. Kantor, Wolfram Burgard, Lydia E. Kavraki, and Sebatian Thrun. Principle of Robot Motion: Theory, Algorithm, and Implementation. MIT Pre, 25. [4] Carlo Guetrin, Andrea Kraue, and Ajit Paul Singh. Near-optimal enor placement in gauian procee. In Proceeding of the 22nd international conference on Machine learning, page ACM, 25. [5] Geoffrey Hollinger and Sanjiv Singh. Multi-robot coordination with periodic connectivity. In Robotic and Automation (ICRA), 2 IEEE International Conference on, page IEEE, 2. [6] Barak Raveh, Angela Enoh, and Dan Halperin. A little more, a lot better: Improving path quality by a path-merging algorithm. Robotic, IEEE Tranaction on, 27(2): , 211. [7] Ralf Regele and Paul Levi. Cooperative multi-robot path planning by heuritic priority adjutment. In Intelligent Robot and Sytem, 26 IEEE/RSJ International Conference on, page IEEE, 26. [8] Claude E Shannon. Prediction and entropy of printed englih. Bell ytem technical journal, 3(1):5 64, [9] Amarjeet Singh, Andrea Kraue, Carlo Guetrin, and William J. Kaier. Efficient informative ening uing multiple robot. J. Artif. Intell. Re. (JAIR), 34:77 755, 29. [] Anthony Stentz. Optimal and efficient path planning for unknown and dynamic environment. Technical report, DTIC Document, 1993.