Toward Computing an Optimal Trajectory for an Environment-Oriented Unmanned Aerial Vehicle (UAV) under Uncertainty

Size: px

Start display at page:

Download "Toward Computing an Optimal Trajectory for an Environment-Oriented Unmanned Aerial Vehicle (UAV) under Uncertainty"

Rudolph Rich
5 years ago
Views:

Jounal of Uncetain Systems Vol9, No2, pp84-94, 2015 Online at: wwwjusoguk Towad Computing an Optimal Tajectoy fo an Envionment-Oiented Unmanned Aeial Vehicle (UAV) unde Uncetainty Jeald Bady, Octavio

1 Jounal of Uncetain Systems Vol9, No2, pp84-94, 2015 Online at: wwwjusoguk Towad Computing an Optimal Tajectoy fo an Envionment-Oiented Unmanned Aeial Vehicle (UAV) unde Uncetainty Jeald Bady, Octavio Lema, Vladik Keinovich, Caig Tweedie Cybe-ShARE Cente, Univesity of Texas at El Paso, El Paso, TX 79968, USA Received 1 Januay 2014; Revised 11 June 2014 Abstact Ove the past decade a few but inceasing numbe of eseaches have begun using Unmanned Aeial Vehicles (UAVs) to expand and impove upon existing emote sensing capabilities in the Actic Due to the limited flight time, it is impotant to make sue that the UAV follows an optimal tajectoy in which it cove all the points fom a given aea within the smallest possible tajectoy length Unde the usual assumptions that we cove a ectangula aea and that each on-boad senso coves all the points with a given adius, we descibe the optimal tajectoy A moe complex optimal tajectoy is also developed fo the situations in which we need to get a moe spatially detailed pictue of some sub-egions of inteest (in which we should have a smalle value ) and it is sufficient to get a less detailed pictue (with lage ) in othe sub-egions We also descibe the best ways to cove the tajectoy in situations in which an UAV missed a spot due to excess wind o to an inexact contol c 2015 Wold Academic Pess, UK All ights eseved Keywods: Unmanned Aeial Vehicles (UAV), envionmental studies, optimal tajectoy 1 Intoduction Need fo envionment-oiented UAVs Actic obseving systems need to be enhanced with impoved emote sensing technologies and capabilities paticulaly mid-altitude emote sensing using ai-bone platfoms; see, eg, [9] Ove the past decade a few but inceasing numbe of eseaches have begun using Unmanned Aeial Vehicles (UAVs) to expand and impove upon existing emote sensing capabilities in the Actic Need fo customizable UAVs Typically UAVs tend to be designed fo a specific task o aea of opeation and so Unmanned Aicaft Systems (UASs) ae usually not easily customizable It is desiable to develop UASs that allow fo customizable senso packages, eliable communications between gound and aicaft, tools to optimize flight contol, eal time data pocessing, the ability to visually ascetaining the quantity of data while the UAV is ai-bone, and the ability to launch and land safely in these emote egions Ou system We have developed a pototype softwae system that allows fo the customization of UAVs This softwae has enhanced communication between gound and the UAV, can synthesize nea eal time data acquied fom sensos on-boad, can log opeation data duing flights, can visually demonstate the amount/quality of data fo a sampling aea The softwae has been designed to benefit an existing Actic Obseving Netwok poject sponsoed by the US National Science Foundation (NSF), a poject that will focus on the emote sensing of landscape-scale vegetation stuctue and function Ou UAS includes a paaglide UAV that has a suite of sensos suitable fo chaacteizing hypespectal eflectance and othe suface popeties This paaglide UAV allows low and slow flying, has a limited Coesponding autho s: jealdbady@gmailcom (J Bady), lolema@episdog (O Lema), vladik@utepedu (V Keinovich), ctweedie@utepedu (C Tweedie)

Jounal of Uncetain Systems, Vol9, No2, pp84-94, 2015 85 ange but a elatively lage (ca 13 kg) payload Sensos on-boad elay opeational flight data (aispeed, gound speed, latitude, longitude, pitch, yaw,

2 Jounal of Uncetain Systems, Vol9, No2, pp84-94, ange but a elatively lage (ca 13 kg) payload Sensos on-boad elay opeational flight data (aispeed, gound speed, latitude, longitude, pitch, yaw, oll, and video) as well a seies of customizable senso packages Additional sensos can be added to an on-boad laptop o a CR1000 data logge; see Fig 1 Figue 1: Ou UAV in flight Need fo coveage The pupose of the UAV measuements is to descibe the values of the envionmentelated physical quantities such as tempeatue, humidity, etc, at all possible locations within the ectangula obsevation aea Of couse, this all cannot be undestood liteally: the obsevation aea has infinitely many points, and it is not possible to measue the value of the quantity is all these points Fom the pactical viewpoint, it is not necessay to take the measuements in all infinitely many points: usually, we know that the values at neaby points ae pactically indistinguishable Specifically, a use usually povides us with a theshold 0 such that the values of the desied quantity at points P and P of distance d(p, P ) 0 ae indistinguishable In this sense, to make sue that we know the value at each point within the obsevation aea, we have to make measuements in such a way that evey point fom the ectangle is at a distance 0 fom some point at which a measuement was made ie, fom one of the points on the UAV s tajectoy

3 86 J Bady et al: Towad Computing an Optimal Tajectoy Need to take uncetainty into account In pactice, it is not possible to maintain the exact tajectoy of a UAV, we can only maintain the desied tajectoy with a cetain accuacy 1 In view of this uncetainty, if we simply make sue that evey point P in the aea is at a distance d(p, P ) = 0 fom some point P on the desied tajectoy, the actual tajectoy point P may be at a distance d(p, P ) = 1 fom P and thus, at a distance d(p, P ) = > 0 fom P Thus, to make sue that even with this uncetainty, we have the desied coveage (with a distance theshold 0 ), we need to guaantee that evey point P fom the obsevation aea is at such a distance d(p, P ) fom some point P fom the tajectoy that d(p, P ) d(p, P ) + d(p, P ) 0 even when the distance d(p, P ) attains the lagest possible value 1 In othe wods, we need to make sue that d(p, P ) Fo this inequality to be satisfied, we must make sue that d(p, P ) def = 0 1 Thus, to povide the desied coveage unde this uncetainty, we need to make sue that evey point P fom the obsevation aea is at a distance fom some tajectoy point Need fo tajectoy optimization Due to the limited flight time, it is impotant to make sue that the UAV follows an optimal tajectoy in which it cove all the points fom a given aea within the smallest possible flight time ie, at the smallest possible tajectoy length Such a tajectoy is descibed in this pape Comment Most of ou esults wee fist announced in [3, 4] 2 Towads an Optimal Tajectoy The poblem: eminde We opeate unde the usual assumptions that we cove a ectangula aea and that each on-boad senso coves all the points with a given adius (see discussion above) We ae looking fo tajectoies that povide the desied coveage of the aea ie, fo which evey point fom the aea is located at a distance fom some point on this tajectoy Ou objective is to come up with the tajectoy that is optimal in the sense that it is the shotest among the tajectoies that povide the desied coveage Analysis of the poblem Each tajectoy piece of length L i coves the aea A i 2 L i ; see Fig 2 L i Figue 2: Coveage So, a tajectoy of length L = i L i coves the aea A i A i = i (2 L i ) = 2 L i = 2 L i Thus, to cove a egion of aea A 0, we need a tajectoy of length L A 0 /2 Asymptotically optimal tajectoy The following natual tajectoy is theefoe asymptotically optimal, see Fig 3

4 Jounal of Uncetain Systems, Vol9, No2, pp84-94, L 2 L 1 Figue 3: An almost optimal tajectoy Indeed, in the egion of aea A 0 = L 1 L 2, we have L 1 /2 pieces of length L 2 each The total length is L L 2 (L 1 /2) = (L 1 L 2 )/2 = A 0 /2, ie, this tajectoy is (almost) optimal The above asymptotically optimal tajectoy does not cove all the points The mino poblem with this tajectoy is that the cone points (maked bold on Fig 4) ae not coveed, because the distance fom the tajectoy to each cone point is = 2 > Figue 4: Fo the asymptotically optimal tajectoy, cone points ae not coveed Pecise fomulation of ou optimization poblem In this pape, we will conside tajectoies which consist of two linea segments in each cone aea, ie, tajectoies in which the oiginal linea tajectoy coming to the point P 1 is followed by two segments P 1 P 2 and P 2 P 3 that go into the next coido of width 2; see Fig 5

5 88 J Bady et al: Towad Computing an Optimal Tajectoy C P P 3 P 2 P 1 Figue 5: An optimal tajectoy Unde this assumption, the question is how to select the points P 2 and P 3 in such a way that the total path l = d(p 1, P 2 ) + d(p 2, P 3 ) is the shotest among all the paths that cove the whole cone aea (ie, fo which evey point fom this aea is at a distance of fom some point on a tajectoy) Pactical comment The estiction to two-segment tajectoies comes fom pactice: it is much easie to contol the UAV along a linea tajectoy, so adding too many segments would make the contol difficult to execute in pactice Mathematical comment Fom the puely mathematical viewpoint, it may be inteesting to analyze which tajectoies ae optimal among all possible tajectoies not necessaily the two-segment ones This poblem is simila to the known Kolmogoov s definition of an ε-entopy of a set S (given oiginally in [6, 7]) as the smallest numbe of points fo which evey point in the set S is at a distance ε fom one of the selected points What we ae looking fo can be viewed as a 1-D analogue of this notion: find the smallest length of a connected cuve fo which evey point in the set S is at a distance ε fom one of the points on this cuve Instead of consideing all possible cuves, we can also take into account the fact that shap tuns of a UAV ae sometimes difficult to execute, and thee is usually a bound on the cuve s cuvatue Fom this viewpoint, it may be inteesting to conside cuves whose cuvatue is bounded by a given value B An optimal tajectoy: desciption To cove the cone points like C, we popose a fin-like modification of the above tajectoy; see Fig 5 Specifically, afte following the oiginal tajectoy up to the point P 1 which is located 2 units below the uppe bounday, we then go staight to the point P 2 on the line CP at a distance fom the cone point C Then, we follow anothe staight line to the point P 3, etc Let us pove that this tajectoy indeed coves all the points, and that, among tajectoies that cove all the points, this fin-line tajectoy is optimal (we will descibe in what sense this tajectoy is optimal) To illustate this poof, we will use Fig 6 Poof that the new tajectoy coves all the points Let us fist show that this tajectoy indeed coves all the points Indeed, in the given coido of width 2, evey point P below the line P 1 P 0 is coveed by the tajectoy point which lies on the same hoizontal line as P Whethe this point P is to the left o to the ight of the tajectoy, the distance is always A simila agument can be made about all the othe points within this coido, except fo the points within the squae P 0 P 1 P P 3 To pove the coveage fo points fom this squae, we daw two lines though the point P 0 :

6 Jounal of Uncetain Systems, Vol9, No2, pp84-94, C C 0 P 2 P 3 P P 3 P 1 P 0 P 1 P Figue 6: Poof of coveage and optimality of the new tajectoy the line P 0 P 1 paallel to the segment P 3 P 2 of the tajectoy, and the line P 0 P 3 paallel to the segment P 1 P 2 of the tajectoy Both lines ae at distance fom the coesponding tajectoy segments Thus: all the points fom the squae P 0 P 1 P P 3 which ae above the new line P 0 P 1 ae coveed, because they ae at a distance fom the segment P 3 P 2 ; all the points fom the squae P 0 P 1 P P 3 which ae to the left of the new line P 0 P 3 ae coveed, because they ae at a distance fom the segment P 1 P 2 Thus, all the points fom the squae P 0 P 1 P P 3 ae indeed coveed This completes the poof of coveage Poof that the new tajectoy is indeed optimal Fist, let us take into account that we need to cove points ight above the point P 0 (which is 2 units below the uppe bounday) If P 3 is less than units away fom the uppe bounday, then these points ae not coveed So, P 3 must be at least units away fom the uppe bounday Similaly, to cove the point C 0 on the uppe bounday, we need to make sue that the point P 3 is at a distance of at most fom the uppe bounday By combining these two conclusions, we thus deduce that the point P 3 must be at exactly points fom the uppe bounday ie, at the same location as fo the oiginal tajectoy The only emaining question is whee to place the tuning point P 2 Fo fixed points P 1 and P 3, and fo a fixed total length l = d(p 1, P 2 ) + d(p 2, P 3 ), the set of all the coesponding points P 2 foms an ellipse, with P 1 and P 3 as foci; see, eg, [2, 5] At least one of these points of this two-segment piece of the tajectoy must cove the cone point C, ie, it must be at a distance fom C Thus, at least one point fom this tajectoy must be eithe within o at the bode of the cicle of adius centeed at the cone point C Thus, the cicle and the ellipse must intesect If they ae not tangent to each othe at the intesection point, then we can decease the value l, and get a smalle ellipse which will still be intesecting Thus, fo the smallest value, the cicle and the ellipse must be tangent to each othe at the intesection point One can show that unde this condition, the point P 2 should be on the line P C at a distance fom the cone point C, ie, exactly whee it is in ou aangement The optimality is poven

7 90 J Bady et al: Towad Computing an Optimal Tajectoy Comment It is woth mentioning that the situation is symmetic with espect to eflection ove the line CP P 0 : unde this eflection, P 1 tuns into P 3 and vice vesa Pactical comment: how to actually contol the UAV Once the optimal tajectoy has been detemined, the next question is how to contol the UAV so that it follows this tajectoy Fo manual contol, it is impotant to povide a good visualization of the past tajectoy and of what it has aleady coveed; such a visualization is descibed, eg, in [8] Fo automatic contol, contol algoithms ae descibed, eg, in [1, 10] 3 What if We Want Diffeent Coveage in Diffeent Sub-Regions Fomulation of the poblem In some situations, we need to get a moe spatially detailed pictue of some sub-egions of inteest (in which we should have a smalle value ), and it is sufficient to get a less detailed pictue (with lage ) in othe sub-egions Solution: main idea In this case, it is easonable to use optimal (o asymptotically optimal) aangement in each sub-egion Example: case of fou sub-egions Fo example, if we need fou diffeent values i in fou diffeent quate-egions, then we should combine the coesponding optimal tajectoies in fou subegions as on Fig 7 Optimal tajectoy Optimal tajectoy fo 2 fo 3 Optimal tajectoy fo 1 Optimal tajectoy fo 4 Figue 7: Case when we need diffeent coveage in diffeent subegions In paticula, if we use asymptotically optimal tajectoy in each subegion, we get the following tajectoy; see Fig 8 Geneal case The coesponding sub-division can be iteated if within each quate-egion, we have subegions with diffeent desied coveage 4 Tailwind Poblem Idealized case In the above text, we assumed that a UAV follows the desied tajectoy In this case, we get a full coveage of the desied egion Tailwind: a poblem In pactice, an UAV can deviate fom the planned tajectoy As a esult, we may not cove some points in the egion

8 Jounal of Uncetain Systems, Vol9, No2, pp84-94, Figue 8: Poof of coveage and optimality of the new tajectoy: example One eason why this may happen is tailwind In the pesence of a stong tailwind, the UAV flies too fast, thee is not enough time fo sensing A solution to the tailwind poblem In the case of a tailwind, a natual solution is to change the diection of the tajectoy, so that the wind would no longe be a tailwind This solution is illustated below In Fig 9, we show the oiginal plan In Fig 10, we show how this plan is disupted by tailwind In Fig 11, we show how the change in diection changed the oiginal tajectoy Figue 9: Oiginal plan

9 92 J Bady et al: Towad Computing an Optimal Tajectoy Figue 10: Tailwind poblem: plan disupted by tailwind Figue 11: Solution to the tailwind poblem: change diection 5 Missed Spot Poblem Missing spot: fomulation of the poblem In the ideal case, we should get a pefect coveage of the aea; see Fig 9 In pactice, howeve, a senso may malfunction when the UAV is flying ove a cetain aea In this case, while the tajectoy is still coveing the whole aea, the measuement coveage misses a spot; see Fig 12 Additional poblem The additional poblem is that by the time we lean about the disuption, the plane has aleady moved along the planned tajectoy; see Fig 12 A seemingly natual idea In this case, if we left a missing spot, a natual idea is: to come back, to cove this spot, and then to continue along the oiginal tajectoy

10 Jounal of Uncetain Systems, Vol9, No2, pp84-94, Figue 12: Missed spot poblem This idea is illustated on Fig 13 C B A Figue 13: Missed spot poblem, seemingly natual idea: come back, then continue Limitation of this seemingly natual solution The main disadvantage of the above (seemingly natual) solution is that we waste time by coveing the same segment AB (see Fig 13) thee times: when we followed the oiginal path, when we go back, fom the point A at which we ealized that we missed the point, to the point B that we missed; and when we go back, fom the point B, to the point A, to esume the oiginal tajectoy A bette idea: epai the spot on the next iteation A bette idea an idea that avoids the abovementioned waste is to continue and to e-visit the missed spot on the next iteation; see Fig 14

11 94 J Bady et al: Towad Computing an Optimal Tajectoy Acknowledgments Figue 14: New idea: epai the spot on the next iteation This wok was suppoted in pat by gants fom the US National Science Foundation (NSF): Cybe-ShARE Cente of Excellence (HRD ) and Computing Alliance of Hispanic-Seving Institutions CAHSI (CNS ), and by Gant 1 T36 GM fom the US National Institutes of Health (NIH) The authos ae thankful to the anonymous efeees fo valuable suggestions Refeences [1] Allaie, FCJ, Tabouchi, M, Labonté, G, and G Fusina, FPGA implementation of genetic algoithm fo UAV eal-time path planning, Jounal of Intelligent and Robotic Systems, vol54, pp , 2009 [2] Bege, M, Geomety II, Spinge, Belin, Heidelbeg, New Yok, 1987 [3] Bady, J, Lema, O, Keinovich, V, and C Tweedie, Towad computing an optimal tajectoy fo an envionment-oiented Unmanned Aeial Vehicle (UAV), Abstacts of the 7th Joint UTEP/NMSU Wokshop on Mathematics, Compute Science, and Computational Sciences, 2010 [4] Bady, J, Lema, O, Keinovich, V, Tweedie, C, and R Romeo, Towad computing an optimal tajectoy fo an envionment-oiented Unmanned Aeial Vehicle (UAV), Extended Abstacts of the NSF Joint Annual Meeting of the Centes of Reseach Excellence in Science and Technology (CREST) Histoically Black Colleges and Univesities Reseach Infastuctue fo Science and Engineeing (HBCU-RISE) CREST/HBCU-RISE-JAM 2010, pp69 70, 2010 [5] Coxete, HSM, Intoduction to Geomety, Wiley, New Yok, 1989 [6] Kolmogoov, AN, Theoy of tansmission of infomation, Ameican Mathematical Society Tanslation, Se2, vol33, pp , 1963 [7] Kolmogoov, AN, and VM Tikhomiov, ε-entopy and ε-capacity of sets in function space, Ameican Mathematical Society Tanslation, Se2, vol17, pp , 1961 [8] Sikanth, MB, Mathias, PC, Nataajan, V, Naidu, P, and T Poston, Visibility volumes fo inteactive path optimization, The Visual Compute, vol24, pp , 2008 [9] Stephens, GL, Ellingson, RJ, Vitko J, J, Bolton, W, Tooman, TP, Valeo, FPJ, Minnis, P, Pilewskie, P, Phipps, GS, Sekelsky, S, Caswell, JR, Mille, SD, Benedetti, A, McCoy, RB, McCoy J, RF, Ledebuh, A, and R Bambhaj, The depatment of enegy s atmospheic adiation measuement (ARM) unmanned aeospace vehicle (UAV) pogam, Bulletin of the Ameican Meteoological Society, vol81, pp , 2000 [10] Yang, K, Gan, SK, and S Sukkaeih, An efficient path planning and contol algoithm fo RUAV s in unknown and clutteed envionment, Jounal of Intelligent and Robotic Systems, vol57, pp , 2010

Positioning of a robot based on binocular vision for hand / foot fusion Long Han

Positioning of a robot based on binocular vision for hand / foot fusion Long Han 2nd Intenational Confeence on Advances in Mechanical Engineeing and Industial Infomatics (AMEII 26) Positioning of a obot based on binocula vision fo hand / foot fusion Long Han Compute Science and Technology,