Dubins Path Planning of Multiple UAVs for Tracking Contaminant Cloud

Transcription

1 Proeedings of the 17th World Congress The International Federation of Automati Control Dubins Path Planning of Multiple UAVs for Traking Contaminant Cloud S. Subhan, B.A. White, A. Tsourdos M. Shanmugavel, R. Żbikowski Department of Aerospae, Power & Sensors, Cranfield University, Shrivenham, Swindon SN6 8LA, United Kingdom. Abstrat: : This paper presents ooperative path planning approah for multi UAVs to detet, model and trak the shape of a ontaminant loud boundary. The objetive of this researh study is to manage the resoures (airborne sensors suh as to determine the boundary of a loud and trak its motion with minimum information. The dispersing model of the ontaminant loud boundary is based on SCIPUFF. The UAVs are assumed to just have a sensor pakage whih an sense nulear, biologial and hemial (NBC ontaminants. Therefore as a UAVs fly through a ontaminant loud the NBC sensors will reognise the entry and exit points of the UAVs from the ontaminant loud boundary and give these two points as measurements. Based on the measurements the splinegon approahuses to predit the ontaminant loud boundary and produes a segment for the next UAVs path. 1. INTRODUCTION The advanes of sensor tehnology and data fusion have broadenedaninterestinthe use unmannedaerialvehiles (UAVs espeially with more information is available for deision making. This benefits of UAVs have been used either in ivilian or military appliations. Civilian appliations involve grid searhing for mining, oil exploration, surveillane and reonnaissane for traffi ontrol, resue missions, fire extintion, identifiations of hazardous materials, oeanographi/geologialsurveys, marine and border inspetion, see e.g. Swaroop (1999, Burns (2000 and Girard (2003. Military appliations entail similar tasks suh as targetreognition, forward-deployedoffensive missions, information gathering and ommuniations, see e.g. Jeyaraman (2005. In seurity and defene unmanned aerial systems an provide signifiant redutions in manpower and risk to humans. Otherpotentialbenefits entailosteffiieny, apabilityto minimize the risk to human life, ability to perform in hostile, hazardous and geometrially omplex environments where diret human intervention is undesirable, enhaned detetion apabilities based on novel sensor tehnologies and its ability to arry out long-time monotonous missions. This paper fouses on the study of ooperative path planning for UAVs to trak, detet and model the shape of a ontaminant loud using Dubins path planning. The threat of pollution from hemials that are poisonous, odour less and opaque gases or due to biologial agents or the threat of radiation due to industrial aidents or seurity hallenges is high, the apability to monitor and trak one suh a loud has been released is of utmost importane. This is beause the rapid detetion and geoloation of possible ontaminant distribution is of paramount importane to prevent injury to the population. 2. PROBLEM DESCRIPTION The main objetive of this paper is that unmanned aerial system is able to sense and trak a ontaminant loud. The senario would be to use UAVs as a monitoring system launhed upon suspiion or onfirmation of the existene of a ontaminant loud. One the UAVs have reahed the ontaminant loud, they will need to hek that it exists. If the UAVs sensor swarms follow a defined searh pattern, when a UAVs detets the ontaminant loud it an send out a signal that it has deteted the ontaminant loud. The UAVs an then entre on that loation and begin to trak. At this stage the UAVs needs to organise itself to be in positions that enable to sense and trak it. Thus swarm ontrol and guidane is a subset of the whole logistial problem of putting a suite of sensors into the right plae at the right time to be able to provide situational awareness of ontaminant loud releases. In this paper we propose an algorithm to trak, estimate and reonstrut the boundary of the ontaminant loud. For this senario we assume that there is a UAVs sensor swarm to take measurements of the air borne ontaminant louds. More over sine we assume that the air borne ontaminants are opaque gases there are no vision sensors to sense the ontaminant loud. The UAVs are assumed to just have a sensor pakage whih an sense nulear, biologial and hemial (NBC ontaminants. Therefore as a UAVs flies through a ontaminant loud the NBC sensors will reognise the entry and exit points of the UAVs from the ontaminant loud boundary and give these two points as measurements. The definition of the loud boundary will depend on the loud omposition. For a radiologial loud, this would be to sense the presene or absene of radioativity. For a biologial or hemialloud, the onentrationlevelatwhihthe loud is harmful would need to be determined. Similarly the UAVs in the sensor swarm will take measurements. But it should be noted that all the measurements will be /08/$ IFAC / KR

2 . t s n s r s a s a Fig. 1. Dubins Ar Geometry taken asynhronously. Thus the task is to approximate a shape of the ontaminant loud from an emerging set of disrete loation points obtained from the asynhronous measurements. The shape of the ontaminant loud will deform and will hange shapes ontinuously and rapidly. This hange in shape should not only be traked, but if possible they should be predited as well so that this preditionofthe ontaminatedregionanbe sheduledto be visited by the UAVs in a systemati fashion i.e. these points will be given as way points to the path planning algorithm of the UAVs sensor swarms. a f 3. DUBINS PATH PLANNING A Dubins path is the shortest path onneting two onfigurations in a plane under the onstraint of a bound on urvature, see e.g. Shanmugavel (2007, Dubins (1957. In the plane, the line is the shortest distane between two points and a irular ar is the shortest turn of onstant urvature. Combining these two provides the shortest path. The Dubins path is formed either by onatenation of two irular ars with their ommon tangents or by three onseutive tangential irular ars. For a two dimensionalmanoeuvre, the initialandfinaltangentvetors are oplanar, hene the initial and final turning irles and the onneting tangent lie in a plane. A 2D Dubins path is shown in Figure 1. The sign of the initial and final manoeuvre an be determined by designating either a left or right turn. Viewed from eah position, a positive or negative rotation will define the sign of the urvature for eah manoeuvre. Also, from the figure, we have: 0 r s = e s, e s = [ t s n s ] (1 where is the urvature of the initial manoeuvre and: r f = e f 0, e f = [ t f n f ] (2 where is the urvature of the final manoeuvre. The initial and final manoeuvre vetors t s and t f are related by: t f = R(θt s (3 where R(θ is the rotation matrix required to hange the axis set from initial to final axes, see also (13 below. Hene, we have: os(θ = t f t s (4 r f n f t f. The onneting vetors a s, a f and a form an orthogonal setofvetors. Inordertodetermine the vetors, firstdefine the onneting vetor a as: t = R(θ s t s (5 where t is the basis vetor defining the onneting vetor. If the position of the final point p f relative to the start position p s is measured in start axes e s, we have: pt p f p s = e s p, p = (6 p n Hene, the vetor sum for the position vetor in start axes is given by: p = r s a s + a + a f r f (7 The left hand side of this equation represents the vetor onneting the entres of the turn irles. Hene: t = a s + a + a f (8 where is the length of the entre vetor. The remaining onneting vetors a s, a f and a an be written in terms of the start basis vetors, as: a s = R(θ s ( 0, a f = R(θ s a = R(θ s ( a 0 0 The entre vetor equation equation (8, now beomes: a t = R(θ s (10 This is a rotation equation, hene the right hand vetor must have the same magnitude as the left, to give: ( a ( 2 1 = (11 This an be used to test for a feasible solution, by: 1 1 ( 2 2 > 0 (12 In order to ompute the rotation angle θ s, the equation an be written in the form: 2 2 t = R(θ s ( ( os(θs sin(θ R(θ s = s (13 sin(θ s os(θ s (9 Solving for θ s gives: os(θs = R(, κ sin(θ s s, t (14 where: R(,, = ( 2 ( 2 (15 The final angle θ f an then be determined using: θ f = θ θ s (

3 Thus the urvature onstraint is met by satisfying the equation P si (x si, y si, θ si ri(t P fi (x fi, y fi, θ fi, κ i (t < κ i,max (17 The path length is alulated by summation of the ar lengths and the onneting tangent length. L = L ars + L tangent = θ s + a + θ f (18 v f n f f r f t f a f v I 4. CLOUD MODELLING USING SPLINEGONS t I n I a s The ontaminant loud behaviour is omplex. There are several ways to model its behaviour in a omplex environment. Physial modelling of the loud an be done using Gaussiandispersionmodels suh as SCIPUFF, whihpredits loud behaviour using a statistial dispersion tehnique. Suh tehniques will have limited use for traking as the loud s behaviour must be expressed in a manner that allows the guidane of a group of UAVs equipped with suitable sensors to detet and trak suh a loud. The requirement is for a loud model that an be expressed in a ompat format, thus enabling the exhange of a defining loud dataset amongst the UAVs group with minimal ommuniation overhead and with maximum utility in guidane algorithms. The UAVs are required to fly feasible paths that maximise the overage of the loud whilst avoiding loal obstales. The flight paths of the UAVs will be ditated by Dubins or related paths. These are defined using irular ars and straight line segments. This suggests that the appropriate strategy should be to fly through the loud and note the entry and exit points rather than transition the loud boundary. This is further supported by the fat that the loud density will not have a physial boundary, but will have a density profile that asymptotially approahes zero. Hene the sensor will produe a threshold transition rather than a physial boundary. There will also be density variations aross the loud that an be deteted by transition rather than by boundary following. Finally, transitioning the loud will enable the UAVs to perform searh operations outside the detet loud boundary to ensure that maximum use of the area overage is maintained. 4.1 Splinegon Constrution A splinegon with onstant urvature line segments an be defined with C 2 ontat at the verties. This implies that the line segments share both a ommon vertex and that the tangents at the verties are also the same. In order to ensure C 2 ontat between verties, the line segments must meet both position and tangent end point onstraints. A single ar segment between verties only has one degree of freedom: the ar urvature. This is not enough to be able to math the tangentonstraintatboth end verties, as at least two degrees of freedom are neessary. Extra degrees of freedom are thus required to ensure the C2 onstraints at both line segment end verties an be met. One solution to inreasing the degrees of freedom is to introdue an intermediate vertex suh that the line segment is replaed by two ar segments of different urvature, as shown in Figure 2. The entry and exit verties identified by the UAV transitions are shown by the symbol x and the Fig. 2. Ar segment with C 2 ontat intermediate vertex p f b f d s Fig. 3. Ar segment with C 2 ontat intermediate vertex intermediate vertex is shown by the symbol o in Figure 6. Hene two ars of differing urvature will onnet the UAV verties via the intermediate vertex. In order to developthe defining equations for suh a solution, onsider the intersetion of two onstant urvature ars at a point with C 2 ontat. Consider all tangent vetors in Figure 3, only the intermediate tangent vetor t I is unknown. Hene [ t t I t s t f ] v f f t I s v I t f r s + b f b f n s p I t = 0 Ts = 0 (19 Hene the solution vetor lengths lie in the right null spae of TN = 0 (20 Thus a family of solution exist for a variety of t I vetor. In order to explore the bounds for feasible solutions, onsider again Figure 3. Using the sin rule on the two triangles [v s, p s, p I ] and [v f, p f, p I ] yields: sinφ s d s = sinθ s sinφ f = sinθ f (21 d f b f Now the vetor p f p s, along whih the tangent t I lies, must have length + b f for a feasible solution and hene v s d s s v s t s t s p s 5720

4 d s + d f = + b f (22 Substituting for d s and d f from equation (21 yields: Now sinφ s sinθ f + b f = + b f sinθ I sinθ I sinθ I = sinφ s + b f sinφ f bs + b f = 1 bs = + b f bf = b f (23 + b f bf = 1, 0 < < 1 (24 Hene a range solutions exist from = 0 to = 1. Thus the solution range is: sinθ I = sinφ s + (1 sinφ f = sinφ s, = 1 = sinφ f, = 0 (25 These parameters will give the requisite information to solve the ar parameters in Figure 2. Hene a s = t s + t I a s = a s sinθ s = t s t I osθ s = t st I = 2 sin(θ s/2 a s, a s 0 = 0, a s = 0 a f = b f t f + b f t I a f = a f sinθ f = t I t f osθ f = t It f 4.2 Solution Seletion = 2 sin(θ f/2 a f, a f 0 = 0, a f = 0 v I = v s + a s (26 There are many different seletion mehanisms for the hoie of a solution over the range of solution given by equation 25. The initial hoie of solution for the loud traking ontainment is determined by onsidering the differene in absolute urvature κ d, where: κ d = min( (27 If this is minimised, the two ars will be balaned and the intermediate vertex will lie lose to the middle of the two UAV verties. Extreme solutions will result in a high urvature short segment and a low urvature long segment whih will plae the intermediate vertex lose to the vertex onneted to the high urvature segment. Choosing a ontour with minimum hanges in urvature between segments will produe splinegons with less abrupt diretion hanges. Using this approah, the ontaminant loud an be adequately modelled by a splinegon. 5. MISSION SCENARIO AND SIMULATION Consider a mission in whih two UAVs team are traking a ontaminant loud. The UAVs are assumed to be homogenous in their physial apabilities and flying at onstant speed at onstant altitude. The task is to sense and trak a ontaminant loud while oordinating the data measurement to approximate the shape of the loud. The representative ontaminant distribution is based on the SCIPUFF, see Sykes (1996. This shows ontours of onstant density at disrete levels set by SCIPUFF. Assuming the UAVs sensor is set to the lowest level, the desription must desribe the outer sensor threshold ontour in suffiient detail to ontain it and trak it. In this paper an approximate shape of the ontaminant loud is formed based on a set of point measurements using the splinegons. In order to trak the loud, the simplest approah would be just to form a shape of the loud using splinegons every time a measurement arrives. This would involve sending the UAVs sensor swarms in random searh paths in order to verify the presene or absene of the ontaminantpartiles. The desriptionwill take into aount the fat that the UAVs will fly through the ontaminant loud and hene will detet an entry and exit point on eah transition. This suggests that the mosteffiientmodelingapproahshouldbe to define these points as verties and form a polygon with line segments. This raises an issue as to how to represent the urved nature of the density ontour. One suh approah is to use a generalisation of polygons to produe a set of verties that are onneted by line segments of onstant urvature. This is a subset of a lass of objet named as splinegons, see e.g. Dobkin (1988, Dobkin ( Algorithm and implementation The algorithm detail for traking ontaminant loud boundary is shown in Figure 4. The detail implementation of the algorithm an be given as follows. First, a waypoint must be defined for the UAVs sensor swarms to trak the loud. In Figure 5, the first UAVs (UAV-A passes the waypoint (t 0 t 1, magenta and takes a measurement for the entry and exit points. One the first two sets of measurements have been taken by the UAVs, the shape of the loud at the first two measurement instanes are approximately formed using the splinegons. The tangent and normal veloity for eah of the verties in the splinegon are then omputed. The entre point of the splinegons is alsoomputed, whihis thenusedto ompute the veloity at the entre of the splinegon. Based on the highest order of the urvature the new four verties are introdued for the next waypoint. UAV-A broadasts the first two verties to the seond UAVs (UAV-B. While UAV-B passes the loud (t 1 t 2, blak and broadasts the path length, UAV-A goes to the next segment using Dubins path (blue-redmagenta and extended path (yan to arrive at t 2 seond. This time onstraint is introdued to avoid ollision during 5721

5 4 2 1 Start Define a waypoint for UAV-A 3 Trak the loud Yes Measure the entry and exit points No 11 Define a new waypoint 7 No 5 Boundary Approximation using Splinegon & produes some new verties 6 Dubin path planning for UAV-A with the same path length of UAV-B path 9 Collision avoidane Yes Broadast the new verties 10 8 UAV-B passes the loud Measure the entry and exit points Fig. 5. First transition. UAV B 12 Tol < E No 13 Yes Finish Fig. 4. Algorithm for traking ontaminant loud. a mission and to make sure only one UAVs passes the ontaminant loud. Figure 6 shows UAV-B passes the loud (t 1 -t 2, magenta and takes a measurement for the entry and exit points. Again the boundary approximation is performed by splinegons with two sets more of measurements. The two new verties for the next waypoint are introdued for UAV-B instead of given to UAV-A. UAV-A broadasts the path length (t 2 t 3, blue while UAV-B passes the Dubins path (blue-red-magenta to the next segment. Figure 7 shows UAV-A passes the loud (t 2 -t 3, magenta and takes a measurementfor the entry and exit points. The splinegons is used to approximate the loud boundary. The two new verties for the next waypoint are introdued. At (t 3 t 4 UAV-B passes the ontaminant loud (blue while UAV-A is passing the Dubins path (blue-red-magenta and extended path (yan. Similarly, Figure 8-10 show the transition of UAVs sensor swarms path for traking ontaminant loud. In Figure 11 the Dubins path will ollide with the obstale, therefore a new waypoint is introdued to avoid ollision to the building, see Figure 12 for the orreted path. In this implementation the ontaminant loud boundary approximated by 16 verties. It an be seen in Figure 12 that the approximated shape is good enough to ompare with the SCIPUFF result. 6. CONCLUSION In this paper, some omputational results of traking ontaminant loud boundary using UAVs path planning have been presented. The approah based on the ombination Fig. 6. Seond transition Fig. 7. Third transition of Dubins path planning and traking the evolving ontaminant boundary using splinegons. The path planning solutions ompose of segments whih are produed by splinegons and the Dubins path to join both segments. Splinegons are a generalisation of polygons whih produe a set of verties that are onneted by line segments of onstant urvature. Splinegons are used beause the requirement is to model the ontaminant loud using a ompat format there by enabling exhange of loud datasets amongst the UAVs sensor swarms with minimal ommuniation overhead and with maximum utility in guidane algorithms. 5722

6 UAV B 44 Fig. 8. Fourth transition Fig. 9. Fifth transition UAV B Fig. 10. Sixth transition Fig. 11. Seventh transition, where the Dubins path must be orreted before the UAVs ollides to the obstale Fig. 12. Seventh transition with a new waypoint to avoid ollision The design of Dubins path is shown in both by priniple of Eulideananddifferentialgeometries. Itis shownthatthe existene and length of the Dubins path are funtion of urvature of turning irle. The mission senario is that only one UAVs passes the ontaminant loud to avoid ollision. Therefore the Dubins path length onstraints is proposed to guaranteed this mission fulfill. REFERENCES D. Swaroop and J. K. Hedrik. Constant spaing strategies for platooning in automated highway systems. ASME Journal of Dynamial Systems, Measurement and Control, 129: pages , R. Burns and C. A. MLaughlin and J. Leitner and M. Martin. Teh 21: Formation design, ontrol and simulation. Proeedings of the IEEE Aerospae Conferene, pages 19-25, A. R. Girard and J. B. Sousa and J. K. Hedrik. An overview of emerging results in networkedmulti-vehile systems. Proeedings of the th IEEE Conferene on Deision and Control, pages , S. Jeyaraman and A. Tsourdos and R. Żbikowski and B. A. White. Kripke modelling of multiple robots with de-entralised o-operation speified with temporal logi. Journal of Systems and Control Engineering, Proeedings of the IMehE: Part I, page 15-31, R. I. Sykes and D. S. Henn and S. F. Parker and R. S. Gabruk. SCIPUFF - A generalized hazard dispersion model. Ninth Joint Conferene on the Appliations of Air Pollution Meteorology with A&WMA, L. E. Dubins On urves of minimal length with a onstraint on average urvature, and with presribed initial andterminal positions andtangents. AmerianJournal of Mathematis, 79, pages , M. Shanmugavel and A. Tsourdos and B. A. White and R. Żbikowski. Differential geometri path planning of multiple UAVs. Journal of Dynami sytems, Measurement and Control, 129, pages 620-6, D. P. Dobkin and D. L. Souvaine and C. J. Van Wyk. Deomposition and Intersetion of Simple Splinegons. Algorithmia, 3, , 1988 D. P. Dobkin and D. L. Souvaine. Computational Geometry in a Curved World. Algorithmia, 5(3, 1-457,