Path Planning for Mine Countermeasures

Transcription

1 ath lanning for Mine Countereaure Cherl Rech *a Chritine iatko a Fernando J. ineda b Jeica itole a I-Jeng Wang a a John Hokin Univerit Alied hic Laborator Laurel Marland b John Hokin Blooberg School of ublic Health Baltiore Marland ABSTRACT We have develoed ath-lanning technique and tool to look for ath through inefield. Our technique eek to balance the length and rik aociated with different route through a inefield. Our ethod are intended to rovide battlegrou coander owerful new tool to evaluate alternative route while earching for low rik ath. It i well known how to find the ath of hortet ditance and well known how to find the ath of leat rik. However to otiize both criteria at once i a challenging roble. We have develoed and coared two ethod for ulticriteria ath lanning. We have found that the better ethod ue a linear cobination of the criteria with a uer elected rik tolerance araeter. We alo decribe the novel ue of dnaic grah algorith to quickl find new ath a the rik i changed due to the neutralization deletion of ine and the dicover inertion of ine. We will alo decribe an algorith ued to convert ath to a et of traight-line ath for hi navigation. Our tool allow a coander to find a ath of accetable length and rik to elore the effect of eliinating ine and to obtain a et of waoint for inefield navigation.. INTRODUCTION Minefield data are robabilitic due to an factor uch a oition uncertaint detection robabilitie claification confidence level and claification error. Nonethele in the face of thi uncertaint a coander need a eaningful wa to eaure rik in order to ae the bet wa to cro a otential inefield or decide that the rik to cro i too great. ath lanning i a well-tudied roble in the etting where there i a ingle criterion to otiize. In thi aer we decribe two ethod to find otial ath baed on ultile criteria. Intead of a ingle ath qualit of length we want to include other otiization criteria that decribe the deirabilit of ath uch a level of rik. In addition it i deirable for coand and control to have alternative route reented to hel ake a deciion. Therefore beide a ingle otial ath we would like to find everal choice of ath to dila a alternative. The tool we decribe in thi aer allow the uer to view everal ath.. RISK MODL Here we decribe the odel for calculating the rik of detonating a ine at an location. The aroiate location of ine are obtained fro enor lu the detection and claification algorith. The rik at a given location contributed b a ingle ine deend on the aroiate location of the ine and the odel for the uncertaint that the aroiate location i correct. Let be the unknown location of a ine and be the eaureent outut fro the enor lu the detection and claification algorith. Aue that i norall ditributed with the following denit: π det e T Σ i the covariance atri of the difference between and. * cherl.rech@hual.edu Detection and Reediation Technoie for Mine and Minelike Target VIII Ruell S. Haron John H. Hollowa Jr. J. T. Broach ditor roceeding of SI Vol SI X/03/$

2 The rik alo deend on the triggering odel for the ine; thi i the odel for the area around the ine that will caue a ine to detonate. Let be the oition of the obect atteting to detect or avoid ine. Let the rando event denote the event that the ine i triggered. Aue the following Gauian-like triggering odel for : e c T c i the confidence that the ine i there fro enor lu detection and claification algorith. Σ i the covariance atri of the triggering odel or the ditance between and required to detonate the ine. We are intereted in characterizing the oterior robabilit that the ine i triggered given the hi oition and the enor eaureent. Thi can be derived via alication of Bae rule: d d Aue that ine location i uniforl ditributed o that i a contant and that the oition of the hi i indeendent of the ine location. Then we obtain the following ereion: d Thu the oterior robabilit of interet i the convolution of the two ditribution. Thi reult in the following ereion for the oterior robabilit of triggering a ine given a ine location and a hi location: e det det c T + + π π We define the oint rik at contributed b a ingle ine uing the following equation: r Net we obtain the oint rik contributed b ultile ine located at 3. with aociated enor eaureent 3.. Aue that the ine are indeendent of one another and the enor eaureent are indeendent given the ine location. The robabilit that an of the ine are triggered i the following:... The oint rik at r i defined b the following ereion: r r... The oint rik reulting fro ultile ine i eactl the u of the oint rik contributed b each individual ine. 80 roc. of SI Vol. 5089

3 Net we will eand our oint rik odel to calculate the rik along a ath. Conider a dicrete ath a defined b the oint on the ath that i: a 3 n The robabilit of triggering an ine while traveling over the ath i: a Y n i i Y where we ue Y to denote all the enor eaureent of ine location. We define the rik aociated with the ath a b: r a a Y n i Y i n Y i i i n r Thu the rik of triggering a ine over an entire ath i il the u of the oint rik value along the ath. 3. GRID GRAH MODL FOR ATH LANNING We dicretize the inefield into a grah with vertice evenl ditributed over the inefield. Rik value ri are aociated with the verte at i according to the collection of ine and the rik odel. The collection of ine i defined b a location and confidence c for each ine. Location i converted to the cloet gridoint location i. The triggering ditance i converted into the nuber of gridoint in each direction i and. For ever verte the rik contributed b ever ine in the collection that i within i and i calculated and ued. The rik due to the ine in the collection at location i i thu calculated uing the following equation: r ine i i+ i + a i i b c a b π det π det + e + * i a i + b c ab i the confidence for the ine at location a b. Thi value i zero if there i no ine there. At ever verte there i alo a oibilit of an undetected ine. The confidence that there i an undetected ine at an verte i calculated a c undetected nuber of undetected ine/nuber of vertice where the nuber of undetected ine i found uing the following ereion. N undetected N ine D N ine D i the robabilit of detection and N ine i the nuber of detected ine. A background rik i calculated for ever verte. It i the u of the rik at ever verte in the triggering area urrounding that verte. The following ereion decribe the calculation of the background rik: i cundetected π det rbackground e + * a + b π det + a i b Thu ri r background +r ine i. Vertice are connected to each other b edge. The edge are directional and are defined b the fro verte the to verte and the weight or cot of travering the edge. ight edge eerge fro each verte. The weight of each edge i calculated a follow: W edge -α r to + α l edge roc. of SI Vol

4 r to i the rik value at the to verte and l edge i the ditance fro the fro verte to the to verte. Thi ditance i for diagonal edge and otherwie. The araeter α i et b the uer and i a rik tolerance factor. Setting α between zero and one correond to otiizing over a linear cobination over the two criteria of length and rik. Thi aroach ake the ulticriteria ath lanning roble tractable b reducing it to a ingle etric roble. Setting α to one correond to earching for the hortet ditance ath. Cobining thi with a aiu bound on the rik reult in a ath that iniize the length of the ath for which each verte ha a rik value below the bound. Thi i a econd aroach to aking the ulticriteria ath lanning roble tractable. We will elore both aroache below. Given a tarting location and a deired goal g we find the ath that iniize the following araeter: O W edge edge For fied α thi i couted otiall in olnoial tie uing Diktra algorith. Figure how a ale inefield with a ath fro tart to goal calculated. 4. DYNAMIC GRAH UDATS We have ileented a dnaic grah udate algorith to recalculate the hortet ath when onl art of the inefield i changed. When a ine i added or reoved the weight of each edge in the affected area of the rik field i recalculated. The to verte of each edge whoe weight ha changed i ut into Figure - A Minefield with the Otiu ath fro to g Deterined a et of vertice to be udated. With each of thee vertice i aved the old cot of the hortet ath fro the ource to that verte. Figure A A and A3 in the aendi how the algorith adated fro Raalinga and Re 3 ued to udate the cot to all affected vertice. Thi udating algorith ha a running tie of OA A where A i the u of affected vertice and the edge into thee vertice. The cloer the udated ine i to the ource the larger A i and the longer running tie of the algorith 4. We erfored a tiing tud to coare the tie to find the hortet ath uing Diktra algorith the tie to udate the hortet ath when a ine i reoved and the tie to udate the hortet ath when a ine i added. We randol generated 0 different inefield for each of 6 different grid reolution 50 b b 300 and o on u to 500 b 500. ach inefield contained u to 0 randol generated ineline each with 5 to 5 ine. The iulated inefield alo contained u to 00 randol laced noie oint with lower ean oterior robabilitie than the ine. For each inefield we alternatel reoved and added ine; four ine were reoved and four were added. We varied the ditance to the 30 ource of the reoved and added ine. Thi Dnaic Udate for Deleting Mine cature the effect of different nuber of affected 5 Full Diktra vertice on the running tie. Figure coare 0 uing the full Diktra algorith veru the grah Dnaic Udate for Adding Mine udating algorith ued to add a ine or reove a 5 ine. The olid line are the ean and the dahed line above and below the olid line indicate one 0 tandard deviation above and below the ean. Running Tie 5 Figure how that the dnaic udate are fater than erforing Diktra algorith on the entire grah. Adding a ine take ore tie than deleting a ine. Thi i becaue the vertice ut be added to the Nuber of Vertice Figure - Tiing Coarion for Diktra Algorith for ntire Grah veru Udating for Adding or Deleting a Mine 8 roc. of SI Vol. 5089

5 hea udated and then added again a hown in Figure A-A3. The tandard deviation of the tie required for dnaic udate i higher than that for running Diktra algorith. Thi i becaue the running tie for dnaic udate deend on the nuber of affected vertice which varie according to how cloe the added or deleted ine i to the ource. The tandard deviation of the tie required to add a ine i higher than that for deleting a ine. If a ine i added near the ource the tie to olve the dnaic udate algorith aroache the tie required to erfor Diktra algorith for the entire grah. 5. OTIMIZING FOR LNGTH AND RISK A dicued above we are atteting to olve a ulti-criteria otiization roble. We deire a hort ath and a ath that caue iniu rik. Thee two criteria work againt one another. We have ileented two ethod to attet to balance ath length and ath rik. Firt we weight the edge according to a linear cobination of length and rik uing the α araeter dicued above. When α i toward zero length i acrificed for low rik and when the value i toward one rik i acrificed for a hort ath. Figure 3 how the variou otial ath roduced when α i varied fro 0.0 to. A α tend toward a value of the ath becoe traighter. Figure 3 - Snahot of ath lanner with Varing Rik Tolerance Second we ileented a ethod where the aiu allowable rik at a ingle oint i et. The hortet ath where all oint have a rik below the aiu allowable ingle oint rik i found. Figure 4 how different ath found where the aiu allowable rik i varied. In thi figure a the aiu allowable rik i increaed the ath becoe traighter. Curved ath indicate a lower aiu allowable rik. The aiu allowable rik for the inefield hown in Figure 4 could onl be lowered to a value of 0.5. If the aiu allowable rik i lower there are no ath between the waoint. Net we eaine which ethod give ore deirable ath. We coare the aiu oint rik along the ath to the length of a ath. We alo coare the total rik along a ath to the length of a ath. The darker curve in Figure 5 how the coarion for the eerient where the aiu allowable rik along the ath i varied and α ha a contant value of. The olid dark curve how that the total rik along the ath decreae a the ath length increae. The dahed dark curve how that the actual aiu rik along the ath decreae eonentiall a the length increae. The lighter curve in Figure 5 how the coarion for the eerient where the aiu allowable rik along the ath i held contant at the larget value in the inefield and the alha araeter i varied fro zero to one. The olid lighter curve how that the rik total decreae eonentiall a length increae. The dahed lighter curve how that the aiu rik along the ath i high for everal ath length where the alha araeter i high then dro down Figure 4 - Snahot of ath lanner with Varing Maiu oint Rik Value ignificantl and decreae linearl a length increae. Coaring the two et of curve the eerient where α i varied roduce ignificantl lower total ath rik value than the eerient where aiu rik along the ath i varied. At the cot of higher oint rik along the ath the overall ath rik i ignificantl lower. Since the goal i to travere the entire ath without detonating a ine the overall ath rik hould be the figure of erit. Varing the alha araeter roduce ath that better reduce the total ath rik. Although choicall a coander a not want to travere an area with a high rik doing o will reduce the overall rik. erha it would be bet to autoaticall find inflection oint uch a that indicated b the circle in Figure 5. Thi aear to be an otial cobination of low aiu rik along the ath low total rik and hort length. roc. of SI Vol

6 .0 50 var a allowable rik a rik rik total 0.8 Rik Total var alha a rik rik total Ma Rik Along ath Length Figure 5 - Coarion of Rik Total and Maiu Rik Along ath When Varing Alha or Maiu Allowable Rik Figure 6 - Snahot of ath lanner with Silified ath 6. ATH SIMLIFICATION The grah hown in Figure i an inforative look at a ath through a inefield. In the end though a hi or a arine or a tank will need ecific direction on where to go. Thee direction will include ecific location to head for and direction to travel in. Thu the ath hown in Figure ut be reduced to a reaonable nuber of traight-line egent. We ue the algorith given b Dougla and euker 5 to ilif the ath into a et of oint between which traight-line egent adequatel decribe the ath. The algorith draw a traight line between the tart and the end of the ath. Net the ditance between ever verte on the line i calculated. The verte with the larget ditance to the line i arked and coared to eilon a tolerance value for how cloe the ilified line ut be to the original line. If the ditance i larger than eilon the algorith i called recurivel on the line egent between the tart and the verte with the larget ditance and on the line egent between the verte and the end. If the ditance i aller than eilon tart and end are added to the lit of oint ued to decribe the line. Figure 6 how a nahot of the ath lanning oftware howing a are ath. 7. CONCLUSIONS We have develoed and ileented efficient ethod for ine ath lanning roble with ultile contraint. Thi work ha deontrated the effectivene of uch algorith for ine ath lanning. In future work we will ue the Raalinga and Re algorith to create a rik lider that ue batche of udate to allow a uer to var the rik threhold interactivel. We alo eek to develo ethod to addre the ver intereting roble of deciding which ine would be ot effective to reove. We would alo like to etend thi work to include navigabilit of terrain when finding the otial ath. 84 roc. of SI Vol. 5089

7 RFRNCS. Coren T. H. Leieron C.. and Rivet R. L. Introduction to Algorith McGraw-Hill New York Diktra. W. A note on two roble in conneion with grah Nuerihe Matheatik Raalinga G. and Re T. An Increental Algorith for a Generalization of the Shortet-ath roble Journal of Algorith iatko C. D. Diehl C.. McNaee J.. Rech C. L. and Wang I. Stochatic Search and Grah Technique for MCM ath lanning roceeding of the SI Dougla D. H. and eucker T. K Algorith for the Reduction of the Nuber of oint Required to Rereent a Line or it Caricature The Canadian Cartograher roc. of SI Vol

8 ANDIX For each edge in area where ine added or deleted Change edge weight Calculate new cot for ath to to verte with that edge value changed. Udate to verte and lace on in-hea According to new cot if edge weight decreaed over-conitent According to old cot if edge weight increaed under-conitent roce Hea Figure A - Dnaic Grah Udate Algorith While hea i not et o If verte i over-conitent new cot < old cot Fro all the edge with thi verte a it to verte find the edge that roduce new cot the edge that the hortet ath coe through If found Set cot aociated with verte equal to new cot Set reviou verte aociated with verte to edge fro verte Mark the verte a roceed If not found Find the edge that roduce the lowet cot If the fro verte for thi edge ha been roceed Set cot to the lowet cot Set reviou verte to the edge fro verte Mark the verte a roceed If the fro verte for thi edge hant been roceed Set new cot equal to the lowet cot ut verte back on hea according to new cot a over-conitent Udate ucceor If verte i under-conitent new cot > old cot Fro all the edge with thi verte a it to verte find the edge that roduce the lowet cot to the verte Set new cot to equal to the lowet cot ut the verte back on the hea according to new cot a over-conitent Udate ucceor Figure A - rocedure "roce hea" in Dnaic Grah Udate Algorith For all edge with fro verte equal to verte ut roceed Get to verte for edge Get old cot current value aociated with verte Get new cot cot to fro verte ut roceed lu edge weight If new cot < old cot ut to verte on hea according to new cot a over-conient If new cot > old cot Find lowet cot to verte If old cot <lowet cot ut to verte on hea according to old cot a under-conitent If old cot lowet cot&hortet ath to to went through fro verte ut to verte on hea according to old cot a under-conitent Figure A3 - rocedure "udate ucceor" in Dnaic ath Udate Algorith 86 roc. of SI Vol. 5089