NOISE ABATEMENT PATH PLANNING USING VIRTUAL FORCES

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1 Harm M.J. Kujpers*, Hendrkus G. Vsser* * Faculty of Aerospace Engneerng, Delft Unversty of Technology h.g.vsser@tudelft.nl Keywords: nose abatement, trajectory optmzaton, vrtual forces, communty mpact Abstract Ths paper presents a study on the development of a real-tme path plannng algorthm for the synthess of nose abatement routngs based on a so-called vrtual force method. More specfcally, a snake method s proposed to calculate a near-optmal nose abatement departure route (ground track) n real tme, assumng a gven (optmal) vertcal flght profle. The accuracy of the developed vrtual force snake algorthm s demonstrated n numercal examples, n whch smulated results are compared wth known exact trajectory optmzaton solutons that were generated offlne. It s shown that usng the snake method, a ground track can be produced for a realstc departure scenaro wth an accuracy margn of less than 4% n terms of the defned nose performance crteron, relatve to the exact optmal soluton. 1 Introducton The nose resultng from flght operatons at major arports s a contnung source of annoyance n nearby resdental communtes. One opton to reduce the nose mpact s to reshape the arrval and departure trajectores nto and out of an arport. For ths purpose, noseabatement routes and procedures have been desgned and mplemented. Most current nose abatement procedures are local adaptatons of generc procedures, amed at optmzng arcraft nose footprnts [1]. It s readly clear though that a trajectory that s optmzed wth respect to a generc crteron, such as the nose footprnt sze, provdes a poor ndcaton of the true nose mpact, whch evdently depends on the actual populaton dstrbuton n the communtes surroundng an arport [2]. To enable the calculaton of ndvdually customzed envronmentally frendly approach and departure trajectores, a number of trajectory optmzaton frameworks have been developed, ncludng a tool called NOISHHH [3-5]. Ths nnovatve tool combnes a nose model, a dose-response relatonshp, en emsson nventory model, a geographc nformaton system, and a dynamc trajectory optmzaton algorthm. The NOISHHH tool generates routngs and flght-paths that mnmze the envronmental mpact n the resdental communtes surroundng the arport, whle satsfyng all mposed operatonal and safety constrants. As the NOISHHH tool s based on optmal control theory, t provdes optmal control solutons n open-loop form usng an teratve numercal procedure. Solvng a trajectory optmzaton problem usng an optmal control approach typcally requres a huge computatonal effort due to the large number of trajectory ntegratons necessary to obtan a sngle optmal soluton. It s readly clear that an optmal control theory based tool such as NOISHHH does not lend tself to realtme on-board applcatons. In ths study we propose the development of a nose abatement path plannng algorthm based on so-called vrtual force technques that provdes a near-optmal soluton n real tme, sutable for on-board applcaton n a trajectorybased operatons envronment. The employed force feld method orgnates from the feld of robotcs where t s wdely used for autonomous moble robot path plannng, prmarly owng to ts elegant mathematcal analyss and smplcty 1

2 KUIJPERS, VISSER [6]. More specfcally, we seek to develop a socalled snake method to calculate a nearoptmal nose abatement departure route (ground track), assumng a gven procedure (.e., speed/alttude/thrust profles). 2 Snake Concept Descrpton One of the most elegant vrtual force technques for real-tme suboptmal path plannng s the snake method [6]. In a mechancal analogy, the snake can be thought of as an elastc band, represented by a chan of pont masses connected to one another by sprngs and dampers (see Fgure 1). One end of the chan s attached to the ntal pont of the departure route, whle the other end s attached to the termnal (ext) pont of the route. Wthout any addtonal external forces actng on the masses, the shape of the snake n steady state s defned by the shortest path between the ntal and termnal ponts,.e., a straght lne track. By havng nose exposed communtes exert a vrtual external repulsve force on the snake control ponts (the masses), the snake wll expand and start to move away from these nose exposed stes. In order to mantan the shortest possble path, whlst crcumventng the communtes, the nternal forces wll then try to contract the snake, causng an oscllatory moton untl an equlbrum between external and nternal forces s obtaned. Once a new potental energy mnmum (or equlbrum) s reached a new curved optmal path s obtaned. By alterng the sprng stffness or dampng rato the curvature of the snake as well as the settlng tme of the nterconnected sprngmass-damper system can be nfluenced. Fg. 1. Path plannng usng snake method [6]. The governng set of equatons of moton for the basc snake model s gven by: N x mq m F F Fˆ Fˆ Fˆ, 1 1 k, y k 1 (1) where F represents the sprng force exerted on control pont and s gven by Eq. (2): F F k q q q q 1 1 1, (2) s where k s s the sprng constant. Note that q represents the poston coordnates of control pont (mass m ). In Eq.(1), F ˆ represents the damper force exerted on control pont and s gven by Eq. (3):, (3) Fˆ Fˆ c q q q q where c s the damper constant. In Eq.(1), F ˆk, s the vrtual repellng force exerted on control pont by the nose -exposed communty (obstacle) k. The repellng force exerted by a noseexposed communty should obvously depend on the sze of the communty, the populaton densty and the level of the exposure. The four man parameters that affect the sound exposure level n affected communtes are the arcraft s thrust settng, ts true arspeed, ts slant range and elevaton angle wth respect to a gven communty. The nfluence of thrust level and slant range are evdent. When the arspeed ncreases, the tme durng whch a communty s exposed to the nose s reduced and, consequently, so s the sound exposure level SEL (the nose metrc employed n ths study). The elevaton angle of the arcraft relatve to the observer locaton n a communty has sgnfcant nfluence on the attenuaton of sound due to over-ground propagaton [1]. To assess the true nose mpact n a communty t s useful to have a set of relatonshps that show whch annoyance level s assocated wth a gven nose exposure level. In essence, such so-called dose-response 2

3 relatonshps combne the physcal measurement of sound wth a scentfc assessment of the communty percepton of sound. Wth the mplementaton of a dose-response relatonshp, the nose mpact n nose-senstve areas can be drectly assessed. In the present study use s made of a sleep dsturbance relatonshp as proposed by the FICAN [1]. Ths dose-response relatonshp, whch s derved from laboratory experments and feld studes, gves an upper bound on the number of expected awakenngs due to a sngle nght-tme flyover. The resultng curve can be found n Fgure 2. In order to calculate the number of awakenngs at a partcular geographc locaton, the percentage of awakenngs as predcted by the FICAN dose-response relatonshp s coupled to a Geographc Informaton System contanng populaton densty data. To facltate acoustc calculatons the well-known Integrated Nose Model INM has been employed n ths study [1]. Fg. 2. FICAN dose-response relatonshp [1]. When modelng the vrtual force exerted by an exposed communty, the number of people or rather, the populaton densty n a certan communty matters as much as the communty sze. For ths reason a novel repulsve force model has been conceved that captures these essental characterstcs. The proposed repulsve model s gven by the followng expresson: ˆ 1 1 q pk Fk, kkrk, 2 q p q p f q p k k k 0, (4) where k k s the repulsve coeffcent, p k the locaton and r k the radus of communty k. In the model descrbed by Eq.(4), the repulsve force of the snake s nversely proportonal to the dstance from the consdered control pont to the nose-affected communty. For the repulsve coeffcent, the followng relatonshp has been emprcally establshed: k k 4 k tunng k 10, (5) where k tunng s the tunng parameter and ρ k s the populaton densty per square klometer. The best value for the tunng parameter has been establshed expermentally on the bass of a varety of departure scenaros, for whch exact optmal solutons were obtaned usng the NOISHHH tool. It turns out that a parameter value k k = 0.25 offers a good overall performance. For ths reason, ths partcular value has been adopted n the numercal examples presented n ths study. When employng the repulsve force model specfed by Eq. (4), the repulsve force wll, n prncple, always act on a control pont, even at a dstance at whch no nose polluton s dscernable. To avod ths form occurrng, an artfcal barrer has been created around a communty that lmts the nfluence range of the vrtual force. On control ponts resdng outsde the barrer, no vrtual force s exerted by the consdered communty. Based on numercal expermentaton, a fxed barrer sze (for any communty) of 2000 m was decded upon n ths study. 3 Intal Results for the Snake Path Planner The trajectory of an arcraft can be decomposed nto two components, the horzontal trajectory,.e. the ground track (route) and the vertcal trajectory,.e. the profle, whch descrbes the alttude and correspondng thrust settng. In ths study, the prmary goal s to develop a vrtual 3

4 KUIJPERS, VISSER force model to shape the ground track, for a gven vertcal profle. The vertcal profle consdered n the smulaton of the snake path planner s extracted from the optmal 3D trajectory, calculated usng the NOISHHH tool for a gven scenaro. Ths has the advantage that the valdaton of the 3D trajectory produced by the snake method remans essentally restrcted to the geometry of the ground track. The accuracy of the ground track produced by the snake path planner s demonstrated n a numercal example related to an exstng departure procedure currently n use at Schphol arport n the Netherlands, vz. the Spjkerboor departure [7]. The accuracy wll be assessed by comparng the performance of the snake soluton to the correspondng NOISHHH soluton n terms of the performance crteron, vz., the expected number of awakenngs due to a sngle flyover event. It needs to be noted that the Spjkerboor departure was not used n the tunng process of the snake model parameters. In the numercal example, performance data (ncludng fuel flow characterstcs) pertanng to a Boeng arcraft s used. The data set comprses separate drag polars for each flap settng and for the confguratons wth the undercarrage extended or retracted. All calculatons are performed usng the assumpton of standard atmospherc condtons, and no wnd present. Fgure 3 shows the nose-optmzed soluton n a 3D space for a Spjkerboor departure (overlayng a map showng populaton denstes), as produced by NOISHHH. Although the expected number of awakenngs has been the prmary optmzaton crteron used to establsh ths partcular nose-optmzed soluton, a small fuel consumpton penalty has been ncluded as a secondary crteron such as to avod excessve path extensons to crcumnavgate nose exposed communtes. In Fgure 4, the correspondng SEL level contours (a) and local awakenngs (b) are shown. The expected number of awakenngs for ths noseoptmzed NOISHHH soluton s 2,805. The NOISHHH soluton s based on a populaton densty grd comprsng 6400 (80x80) cells of 1x1 km 2. Ths populaton densty map now needs to be converted nto somethng the snake algorthm can use. Although each cell could n prncple be used as a sngle communty, t makes sense to cluster cells nto larger communtes and select only the relevant ones. Fgure 5 presents a nose obstacle map that has been nferred from the populaton densty map n the vcnty of the Spjkerboor departure route, usng essentally an ad-hoc approach. It s readly clear that the seven modelled communtes are wdely varyng n terms of sze as well as populaton densty. a) b) Populaton densty (10 4 persons/km 2) x 10 4 Fg. 3. NOISHHH generated nose-optmzed trajectory for the Spjkerboor departure. Fg. 4. SEL and local awakenngs for the Spjkerboor departure usng NOISHHH

5 Populaton densty (10 4 persons/km 2) n Fgure 4 reveals that the two solutons appear to be n close agreement. Nevertheless, the number of awakenngs has ncreased n the snake soluton to 2,912, whch s about 4% n relatve terms. Populaton densty (10 4 persons/km 2) Fg. 5. Nose obstacle map for snake applcaton to the Spjkerboor departure scenaro. In [8] a more structured approach to clusterng a populaton grd to generate a set of vrtual nose obstacles s presented. In ths approach, the populaton data s pre-processed usng a K-means clusterng algorthm to aggregate the dstrbuted populaton cells nto a gven number of populaton clusters. In future research, ths partcular approach wll be lkely adopted n buldng nose abatement scenaros for the snake algorthm. In the smulaton of the snake, the followng (sprng-mass-damper) system parameters have been adopted: mass m = 1; sprng constant k s = 0.5; dampng rato ζ = 1. In Fgure 6, the resultng ground track for the Spjkerboor departure obtaned by usng the modfed snake method, s compared to the one obtaned usng NOISHHH. An nspecton of the results reveals a very close correspondence for the two ground tracks. Only towards the end of the trajectory, near the communty of Stede Broec, a slght dfference between the two ground tracks can be observed. As the populaton densty n ths regon s rather low, t s expected that ths wll result n only a mnor ncrease n the total number of expected awakenngs. In Fgure 7, the SEL contours and geographc awakenngs dstrbuton assocated to the snake soluton are llustrated. A comparson of the snake results wth the correspondng NOISHHH results shown earler Fg. 6. Comparson between NOISHHH and Snake ground tracks for Spjkerboor departure. In addton to the Spjkerboor departure presented heren, several more Standard Instrument Departure scenaros pertanng to Schphol arport have been examned [9]. It turns out that wth the proposed snake technque, usng exactly the same system model and settngs, even better results can be obtaned than n the Spjkerboor scenaro, frequently resultng n an accuracy level of better than 1%. 4 Conclusons Ths paper has presented a study on the development of real-tme path plannng algorthms for the synthess of nose abatement routngs based on vrtual force methods. In partcular, concept descrptons and ntal results related to the snake methods were presented. To render the snake method - whch orgnates from the feld of robotcs applcable to nose abatement route plannng, a novel repulsve force model has been developed whch captures the essental characterstcs of the nose-exposed communtes. 5

6 KUIJPERS, VISSER for a realstc departure scenaro wth an accuracy margn of less than 4% n terms of the expected number of awakenngs relatve to the correspondng NOISHHH result. Future research wll focus on mproved technques for clusterng populaton nto nose obstacles, and nto mproved repulsve force models. In addton, the ntroducton of torson sprngs wll be consdered to mprove trajectory shapng n tght turns. a) b) Fg. 7. SEL and local awakenngs for the Spjkerboor departure usng the snake approach. The accuracy of the developed vrtual force snake algorthm s demonstrated n numercal examples, n whch smulated results are compared wth known exact trajectory optmzaton solutons that were generated offlne. The snake performance has been assessed based on realstc nght-tmes scenaro s pertanng to Standard Instrument Departures currently n place at Schphol arport n the Netherlands, usng expected number of awakenngs due to a sngle flyover as the prmary optmzaton crteron. The research bears out that the snake method holds out great promse for nose abatement path plannng applcatons. Usng the snake method, a ground track can be produced References [1] Vsser H.G., Hebly S. and Wjnen R.A.A. Management of the Envronmental Impact of Arport Operatons. Nova Scence Publshers, Inc., New York, U.S.A., 2009 [2] Vsser H.G. and Wjnen R.A.A. Optmsaton of Nose Abatement Arrval Trajectores. Aeronautcal Journal, Vol.107, No. 1076, pp , [3] Vsser H.G. Envronmentally Optmzed Resolutons of In-Tral Separaton Conflcts for Arrval Flghts. Journal of Arcraft, Vol. 45, No.4, pp , [4] Heblj S.J. and Vsser H.G. Advanced Nose Abatement Departure Procedures: Custom optmzed departure Profles, Proc AIAA Gudance, Navgaton, and Control Conference, Honolulu, Hawa, U.S.A., [5] Hogenhus, R.H., Heblj, S.J. and Vsser H.G. Optmzaton of RNAV Nose Abatement Approach Trajectores. Proceedngs of the Insttuton of Mechancal Engneers Part G - Journal of Aerospace Engneerng, Vol. 225, Issue 5, pp , [6] Qu Y., Pan Q and Yan J. Flght path plannng of UAV based on heurstcally search and genetc algorthms, Proc IEEE Industral Electroncs Socety Conference, Ralegh, North Carolna, U.S.A., [7] Hartjes S, Vsser H.G and Hebly S. Optmsaton of RNAV Nose and Emsson Abatement Standard Instrument Departures. Aeronautcal Journal, Vol.114, No. 1162, pp , [8] De Olvera R.F. and Büskens C. On-Board Trajectory Optmzaton of RNAV Departure and Arrval Procedures Concernng Emssons and Populaton Annoyance, Proc SAE 2011 AeroTech Congress & Exhbton, Toulouse, France, SAE Techncal Paper , [9] Kujpers H.M.J. Nose Abatement Path Plannng Usng Vrtual Forces. MSc thess, Delft Unversty of Technology, Delft, the Netherlands,

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