Stochastic Search and Graph Techniques for MCM Path Planning Christine D. Piatko, Christopher P. Diehl, Paul McNamee, Cheryl Resch and I-Jeng Wang

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1 Stochatic Search and Graph Technique for MCM Path Planning Chritine D. Piatko, Chritopher P. Diehl, Paul McNamee, Cheryl Rech and I-Jeng Wang The John Hopkin Univerity Applied Phyic Laboratory, Laurel, MD ABSTRACT We have been developing path planning technique to look for path that balance the utility and rik aociated with different route through a minefield. Such method will allow a battlegroup commander to evaluate alternative route option while earching for low rik path. A rik management framework can be ued to decribe the relative value of different factor uch a rik veru time to objective, giving the commander the capability to balance path afety againt other miion objective. We will decribe our recent invetigation of two related path planning problem in thi framework. We have developed a tochatic earch technique to identify low rik path that atify a contraint on the tranit time. The objective i to generate low rik path quickly o that the uer can interactively explore the time-rik tradeoff. We will compare thi with the related problem of finding the fatet bounded-rik path, and the potential ue of dynamic graph algorithm to quickly find new path a the rik bound i varied. Keyword: mine, path planning, rik management, optimization. OVERVIEW Mine and minefield data i probabilitic due to many factor uch a poition uncertainty, detection probabilitie, claification confidence level, and claification error. In the face of thi uncertainty, a commander need a meaningful way to meaure rik in order to determine the bet way to cro a potential minefield, or decide that the rik to cro i too great. Effective path planning technique can provide valuable information in a deciion aid for mine countermeaure (MCM). Path planning i a well-tudied problem in the etting where there i a ingle criterion to optimize. It i well known how to find the path of hortet ditance, and well known how to find the path of leat cot (rik). However, to optimize both criteria at once i a challenging problem. Intead of ingle path quality of length/time-to-objective, we want to include other optimization criteria that decribe the deirability of path uch a level of rik. In addition, it i deirable for command and control to have alternative route preented to help make a deciion. Therefore, beide a ingle optimal path we would like to find everal choice of path to diplay a alternative. The mot common path quality attribute i path length or time to goal, a a model of the time required to achieve a ucceful landing. Another important quality that we model i the probability of a ucceful landing, given the path rik from mine. In previou work (Piatko et al. 2) we decribed initial effort on thi problem uing an A* planner. In thi paper we preent our recent implementation experience uing Dijktra algorithm for thi problem. We alo invetigate the ue of dynamic graph algorithm to efficiently update the hortet path when imulating the removal of a ingle mine. We have developed method to addre a new problem of minimizing rik while meeting a hard contraint on time to goal.. Grid graph model for path planning We dicretize the minefield into a graph, with vertice evenly ditributed over the minefield. We aume a tart vertex and a goal vertex g. Rik value r i are aociated with each gridpoint vertex i according to it proximity to mine. Factor to conider for developing a rik model can be found in (Piatko et al. 2 and Wang et al. 2). Length lij repreent the etimated length/time to travere between two adjacent gridpoint i and j. Eight edge emerge from each vertex. The edge are directional, and are defined by the from vertex, the to vertex, and the weight, or cot of travering the edge. The weight of each edge i found uing the following equation: Chritine.Piatko@jhuapl.edu; phone 77-65; Chri.Diehl@jhuapl.edu; Paul.McNamee@jhuapl.edu; Cheryl.Rech@jhuapl.edu; and I-Jeng.Wang@jhuapl.edu. Detection and Remediation Technologie for Mine and Minelike Target VII, J. Thoma Broach, Ruell S. Harmon, Gerald J. Dobeck, Editor, Proceeding of SPIE Vol. 72 (22) 22 SPIE X/2/$5. 5

2 W ij = α r to + (-α) l ij, where r to i the rik value at the to vertex, and l ij i the ditance from the from vertex to the to vertex. Thi ditance i 2 for diagonal edge, and otherwie. The parameter α i et by the uer, and i a rik tolerance factor. Setting α to zero correpond to earching for the hortet ditance path; etting α to one correpond to earching for the lowet rik path without regard to ditance..2 Linear combination of time and rik One approach to making multicriteria path planning problem tractable i to reduce them to ingle metric problem. For example, we can ue a linear combination of the path quality metric of path length and rik, and find the bet path uing thi ingle metric. Given a tarting location and a deired goal g, thi correpond to finding the path p that minimize the value O p) W ij p ij ( =. Thi can be computed optimally in polynomial time uing Dijktra' algorithm for fixed α.. Fixed contraint on rik Another approach to making multicriteria path planning problem tractable i to fix ide criteria a contraint. The uer or application can then explore variou value for the ide contraint. For mine path planning, one uch contrained problem i to find the hortet length path with rik le than a fixed bound R. Thi correpond to finding the path p that minimize the value O ( p) = (in thi cae (-α) i zero), uch that each vertex along the path ha rik r i R. l ij p ij Thi can alo be computed optimally in polynomial time uing Dijktra' algorithm, by uing the ubgraph of the full grid graph that doe not conider any edge that violate the rik threhold contraint.. Fixed contraint on time/length An alternative way to fix a ide criterion for mine path planning i to find the hortet path with rik le than a fixed time contraint T. Auming contant velocity, fixed time correpond to a fixed length L. For thi verion, we wih to find the path p that minimize the value O ( p) = (in thi cae α i zero) uch that the total path length l ij p ij r ij p ij l ( p) = i le than L. Thi i harder than the fixed contraint on rik problem above, becaue the ide contraint i on a um rather than on the individual edge of the path. While thi verion of the problem i NP-hard in general, uing a retricted verion of the grid graph that ue only horizontal and vertical edge (and thu unit length) we can olve thi problem in polynomial time. We alo preent a fater tochatic earch algorithm that find locally reaonable path..5 Waypoint and multiple goal Each of the above path-planning problem can be generalized to include pecified intermediate waypoint w...wm along w, and o on, until the path. In thi cae, we plan the hortet path from to the intermediate goal w, from w to 2 reaching wm and then planning to final goal g. Similarly, we can generalize to plan path to multiple goalpoint. 2. DIJKSTRA S ALGORITHM FOR PATH-PLANNING We can ue Dijktra algorithm to find the hortet path between waypoint. In previou work, we decribed our effort uing a Java implementation of an A* planner for the linear combination and fixed contraint on rik problem. We have redone thi original planner to ue a tatic Dijktra implementation. In the next ection, we alo decribe our reult from coding a dynamic verion to update Dijktra hortet path a the graph i modified. To date, we have found A* and tatic Dijktra to be comparable in term of running time. Our particular implementation of Dijktra eem to produce more aethetic path than our A* implementation. Firt, the graph for the minefield mut be built and all the weight for the edge calculated for the particular choice of α. Dijktra algorithm then proceed a follow (ee alo Cormen, Leieron, and Rivet 997 and Dijktra 955). The 5 Proc. SPIE Vol. 72

3 vertex cloet to the tarting waypoint i firt conidered, and given a path cot of zero. Thi tarting vertex i put then in a et of vertice for which cot ha been calculated (thi et ha only one vertex in it at thi point). Next, all the vertice connected by an edge to that vertex are put in a et to be conidered. The vertex with the lowet cot, i.e., the lowet edge weight, i aigned a cot equal to the edge weight between the ource and that vertex. That vertex i then put in the et of vertice for which cot ha been calculated. Next all vertice connected to the vertex for which the cot wa jut calculated are put in the et to be conidered. Again, the vertex with the lowet cot from the ource i conidered next. The cot i the um of the edge weight from the ource to the vertex. Thi vertex i aigned that cot and put in the et of vertice for which the cot ha been calculated, and all vertice connected to the vertex are added to the et of vertice to be conidered, unle the vertex i already in that et. The algorithm end when each vertex ha been conidered, i.e., the cot from the ource to each vertex ha been calculated. Since thi algorithm compute the hortet path to all vertice, it can be ued to compute the hortet path to multiple goal point. The proce can alo be terminated if there i jut a ingle goal g when that particular vertex i reached. Figure how an intance of Dijktra algorithm. For each vertex, the um of the edge weight along the hortet path from the ource can be aved, a well a the previou vertex along the path from the ource to that vertex. Thi give an encoding of the hortet path tree from the tart vertex that can be ued to recontruct the vertice along the hortet path. The path from the tarting waypoint to the goal waypoint i found by locating the vertex cloet to the goal, then tepping through the previou vertice until the ource i reached. Figure 2 how a naphot of a path through a minefield calculated uing Dijktra algorithm. Dijktra algorithm can be implemented to have a running time of O((V + E) log V), where V i the number of vertice and E i the number of edge.. DYNAMIC GRAPH UPDATES TO SIMULATE MINE NEUTRALIZATION EFFECTS We can imulate the removal or neutralization of a ingle mine by recomputing the rik of affected edge in the neighborhood of a mine, and recomputing the hortet path from the tart to the goal. Thi enable interactive exploration of which mine or minefield are bet to neutralize. However, recomputing the hortet path uing Dijktra algorithm can be expenive for a large grid graph. Fortunately, Dijktra need to be run in full jut once for each et of waypoint and each value of α. Recent reult have hown how to efficiently maintain hortet path in a dynamic etting, where et of edge inertion, deletion or cot change occur. An algorithm by Ramalingam and Rep [Ramalingam and Rep 996] can be ued to update the hortet path efficiently. Only thoe vertice whoe hortet path cot change becaue of the change in the rik field due to the removed mine need to be updated. Such method can alo be ued to upport dynamic change on bound of acceptable rik for a path. They can alo upport other dynamic apect of path planning, including incorporating updated information about mine. When a mine i removed, the weight of each edge in that affected area of the rik field i recalculated. The to vertex of each edge whoe weight ha changed i put into a et of vertice to be updated. With each of thee vertice i aved the old cot of the hortet path from the ource to that vertex, and the new cot of the path when the cot of jut that one edge i changed. Thee vertice are conidered according to the lowet new path cot from the ource to that vertex, much the ame way a in Dijktra algorithm. A vertice are conidered, their lowet path cot i updated, and they are removed from the et of vertice to be conidered. After the vertex i conidered, the cot of the hortet path to all vertice connected by an edge to the conidered vertex i compared to the cot that would reult from a new path through the updated vertex. If the new cot to that vertex i lower than the previou aved cot, that vertex i added to the et of vertice to be updated. When the et of vertice to be updated i empty, the algorithm terminate. Thu, only vertice whoe hortet path change are updated, intead of all vertice. Figure how an intance of Ramalingam and Rep algorithm. Thi updating algorithm ha a running time of O(A log A), where A i the um of affected vertice and the edge into thee vertice. The cloer the removed mine i to the ource, the larger A i, and the longer the running time of the algorithm. We implemented Ramalingam and Rep algorithm to update the hortet path when a ingle mine i removed, and compared thi to the cot of redoing Dijktra algorithm for the entire graph each time. Figure and 5 how ome reult from thee experiment with our implementation. We randomly generated different minefield for each of 6 different grid reolution (25 by 25, by, and o on up to 5 by 5). Each minefield contained up to randomly generated mineline, each with 5 to 5 mine. The Proc. SPIE Vol

4 imulated minefield alo contained up to randomly placed noie point (with lower mean poterior probabilitie than the mine). We alo generated a random tart vertex to the left of the minefield and a random goal to the right. We ued the linear combination metric with fixed α of. for thee experiment. We computed the hortet path from the tart to goal uing Dijktra algorithm. We imulated two kind of mine deletion/neutralization, a mine near the tart vertex and one farther away. Thi capture the effect of different number of affected vertice on the running time. Figure how the 2 individual data point of thee experiment, howing the time to update the hortet path a a function of the ditance to the ource for variou graph ize. The trend i that removing a mine cloe to the ource take longer to update, ince more of the hortet path tree i affected. Figure 5 compare uing the full Dijktra algorithm veru the graph updating algorithm. It alo how more clearly the effect of increaing graph ize. The upper curve how the mean and tandard deviation of the running time to compute the initial hortet path uing Dijktra algorithm. We did not time uing the full Dijktra algorithm to compute the two mine deletion path, but thoe running time would be imilar. The lower curve how the mean and tandard deviation of the update running time (uing the ame 2 experiment a in Figure ). Note that there i a ubtantial time advantage to updating rather than computing the hortet path from cratch.. ALGORITHMS FOR FIXED TIME-TO-GOAL The previou technique can be ued to identify path that atify contraint on the acceptable amount of rik, or linear combination of time-to-goal and rik. We are alo intereted in the alternative contrained problem of finding low rik path that atify a contraint on the tranit time. Such method would identify low rik path through a minefield that allow a hip to arrive at it detination at a pecified time. (We aume the hip travere the path at a contant peed in order to implify thi planning problem.) That i, given a fixed bound L on the length of the path (fixed time-to-goal), we wih to find the lowet rik path.. Optimal algorithm for fixed time-to-goal In general, the problem i NP-hard. However, in the grid cae, with jut direction, L i dicretized to exactly unit increment. The lowet rik path of length L can then be computed by uing a modified verion of a Bellman-Ford algorithm (Cormen et al. 997). A decribed above, path rik i ued for the objective (ditance) function. Several modification are neceary. We need to make L pae over the graph. A oppoed to the uual aynchronou approach of Bellman-Ford (which overwrite ditance information in the middle of each pa), we need to maintain both the previou phae i- and current phae i ditance information to each vertex. The previou ditance information record the lowet rik path from the ource with i- or fewer link to each vertex. Initially all uch ditance are et to infinity, except the ource vertex ditance i to itelf i et to zero. At pa I, we initialize the i link ditance to each vertex v with the value of the previou (i- or fewer link) lowet rik path to v. Then, for each edge (u,v), we tet if uing thi edge reult in a lower rik path to v with i link; i.e., if the lowet rik path with i- or fewer link to u plu the cot of edge (u,v) i le than the value of the current lowet rik path with i or fewer link to v. If o, we record the value of thi new path of i or fewer link to v. After all edge have been conidered, we have found the hortet rik path with i or fewer link to each vertex. (We have no negative cot edge and thu do not need to worry about negative cot cycle). After L uch pae, if we have reached the goal vertex g, we have found the hortet rik path that take <= L tep. If we wih to recontruct the path, we alo need to keep an auxiliary ize L data tructure at each vertex to record the predeceor vertex for each link value i (ince predeceor are different for different value of i). Thi modified Bellman-Ford approach can be implemented in O(kE) time and O(kV) pace for k phae. For our mine problem, thi correpond to O(LE) and O(LV) pace. Thi approach become even more expenive when adding more direction to the grid vertice. For direction (adding the diagonal), a diagonal tep add 2 length, which are not equal to the vertical/horizontal tep. One option i to dicretize each edge into finer increment and ditribute the rik of the original edge proportionally on thee finer edge. The goal i make each "tep" mall enough to be a unit increment a above. Since thi i a bigger grid, thi computation will be even lower. Another option would be to dicretize the minefield into a hex grid. In thi cae, all tep would be of unit length. 56 Proc. SPIE Vol. 72

5 .2 Stochatic earch for fixed time-to-goal Since thi optimal algorithm can be o computationally expenive, we developed a fat, randomized method finding the minimum rik path where there i a contraint on time-to-goal. In thi cae our objective i to define a path planner that generate low rik path quickly o that the uer can interactively explore the time-rik tradeoff. We alo require the planner to produce alternative to the lowet rik path a well a intermediate olution when the planning proce cannot be completed due to time contraint. We ue a population-baed, multireolution tochatic earch to generate a number of low rik path through a minefield. The baic idea i to generate random path, locally improve them, and ave the bet one. Thi type of approach i alo being ued for more general type of motion planning (Hocaglu and A. C. Sanderon 99 and 2). When the planner i executed, an initial population of candidate path i created by randomly ampling the pace of path that deliver the hip to the detination at the appropriate time. The path are encoded a ymbol tring that indicate the equence of move through the grid overlaid on the minefield. A the planning proceed, the reolution of the grid i incrementally increaed. At each reolution level, the current population of path i firt randomly perturbed to explore the path pace. Then the path are determinitically perturbed into local minima. Thi proce i repeated for a number of iteration at each reolution level. Then, the bet path in the current population are aved if they yield improvement over any of the lowet rik path dicovered previouly. The advantage of thi approach i that it quickly generate path, including intermediate reult. The main diadvantage i that it provide no guarantee of optimality on the reulting path. Figure 6 how a path generated by thi planner coded in MATLAB. Initial reult have been encouraging; many reaonable path can be generated very quickly. It i alo poible to ue thi method to generate alternative choice of path. We alo ported thi planner to C++ to invetigate the effectivene of the proce on larger planning tak, and found that it caled well. 5. CONCLUSIONS We have developed efficient method for everal verion of mine path planning problem with multiple contraint. Thi work ha demontrated the effectivene of uch algorithm for mine path planning. In future work, we would like to implement the polynomial time algorithm for the fat time-bounded problem to compare how cloe the tochatic approach come to finding optimal path. We alo plan on uing the Ramalingam and Rep algorithm to create a rik lider that ue batche of update to allow a uer to vary the rik threhold interactively. We alo eek to develop method to addre the very intereting problem of deciding which multiple mine would be mot effective to remove. Finally, there are a number of very recent reult that ue different technique for contrained hortet path planning that could alo prove valuable to ue for MCM path planning application (Mehlhorn and Ziegelmann 2). 6. ACKNOWLEDGEMENTS Thi work i partially upported by Office of Naval Reearch Contract N2-9-D-2. The author thank Fernando Pineda of JHU/APL for hi help developing the rik model. The author alo thank Richard Chang for hi inight on modifying the Bellman-Ford algorithm. Proc. SPIE Vol

6 7. REFERENCES. T. H. Cormen, C. E. Leieron, and R. L. Rivet, Introduction to Algorithm, McGraw-Hill, New York, E. W. Dijktra, A note on two problem in connexion with graph. In Numeriche Mathematik, Vol., pp , C. Hocaglu and A. C. Sanderon, Evolutionary Path Planning uing Multireolution Path Repreentation. In Proceeding of the 99 IEEE International Conference on Robotic and Automation, pp. -2, May 99.. C. Hocaglu and A. C. Sanderon, Planning Multiple Path with Evolutionary Speciation. In IEEE Tranaction on Evolutionary Computation, Vol. 5, No., June K. Mehlhorn and M. Ziegelmann, CNOP A Package for Contrained Network Optimization. In ALENEX 2, Algorithm Engineering and Experimentation: Third International Workhop (ALENEX 2), Lecture Note in Computer Science, Volume 25, Spring Verlag, C. D. Piatko, C. Priebe, L. Cowen, I.-J. Wang, and P. McNamee, Path Planning for Mine Countermeaure Command and Control. In Proceeding of the SPIE Volume 9, Detection and Remediation Technologie for Mine and Minelike Target VI, Orlando, Florida, pp. 6-, G. Ramalingam and T. Rep, An Incremental Algorithm for a Generalization of the Shortet-Path Problem. In Journal of Algorithm, Vol. 2, No. 2, pp , I-J. Wang, C. D. Piatko, and F. Pineda, Rik Model for Mine Countermeaure, manucript, 2. 5 Proc. SPIE Vol. 72

7 Step one - et cot to ource to zero and put in et of vertice for which path i calculated Step two - add vertice connected to ource to et to be conidered Step three - et cot for vertex with lowet value 2 2 Step four - add for conideration vertice adjacent to the vertex jut calculated Step five - et cot for next lowet vertex Step ix - add for conideration vertice adjacent to the vertex jut calculated Step even - et cot for next lowet vertex Figure. An Intance of Dijktra Algorithm Proc. SPIE Vol

8 Figure 2. A naphot of the Dijktra mine path 59 Proc. SPIE Vol. 72

9 Step one original graph Step two - decreae value of an edge and place vertex in et to be conidered Step three - update vertex and place adjacent vertice in et to be conidered only if their cot decreae Step four - update vertex with lowet new value and add adjacent vertice to et, if neceary Repeat Step four until et to be conidered i empty Figure. An intance of Ramalingam and Rep Algorithm Proc. SPIE Vol

10 Time to Update Vertice 96 Vertice 22 Vertice 6 Vertice 2 Vertice 25 Vertice 6 Vertice 2 Ditance from Source 6 Figure. The effect of mine location on update time. Time to Solve 2 Mean Mean +- Standard Deviation Full Diktra Update 5 2 Number of Vertice 25 x Figure 5. Comparion of running time of Dijktra algorithm on a full graph and updating the hortet path after the removal of one mine. 592 Proc. SPIE Vol. 72

11 Figure 6. A naphot of the tochatic earch mine path planner. Proc. SPIE Vol