Composite Hermite Curves for Time-Based Aircraft Spacing at Meter Fix

Transcription

1 AIAA Guiance, Navigation an Control Conference an Exhibit 0-3 August 007, Hilton Hea, South Carolina AIAA Composite Hermite Curves for Time-Base Aircraft Spacing at Meter Fix Thierry Miquel * Direction es Services e la Navigation Aérienne, Toulouse, FRANCE Félix Mora-Camino Ecole Nationale e l Aviation Civile an LAAS u CNRS, Toulouse, France an Jean-Marc Loscos Direction es Services e la Navigation Aérienne, Toulouse, FRANCE The elegation to the flight cre of some tasks currently performe by air traffic controllers provies ne perspectives potentially to increase air traffic control efficiency. More specifically, the task of establishing properly space laning sequences is very emaning in heavy traffic conitions for the air traffic controllers in charge of the terminal maneuvering area. Automatic merging operations coul relieve air traffic controllers of proviing time-consuming raar-vectoring instructions. The outcomes of this paper are tofol: firstly composite Hermite curves to generate a path ith length an enpoint constraints are presente. This reference trajectory is use to stretch the path of an airliner in orer to elay its arrival on a meter fix chosen by air traffic control. Seconly, the esign of a nonlinear control la base on feeback linearization to track the reference trajectory is presente. This communication treats level flight, but in is consiere. The esign is folloe by to illustrative examples hich sho the effectiveness of the propose approach. I. Introuction he overall goal of the R&D community has alays been to improve Air Traffic Management (ATM operations T through a greater involvement of pilots in cooperation ith air traffic controllers. The technology calle ADS-B (Automatic Depenent Surveillance-Broacast is no available for civil aircraft 1 to enhance flight cre air traffic situational aareness. In aition, the Airborne Separation Assistance System (ASAS concept as propose to ICAO in March 1995 to take avantage of the airborne capabilities in particular for communication, navigation an surveillance. The range of ASAS applications is very ie 3 : those applications are esigne to be fully integrate into ATM operations, an most of them involve the active cooperation of air traffic controllers an pilots. By ICAO efinition 4 ASAS is an airborne system that allos the flight cre to maintain separation beteen their aircraft an one or more other aircraft, an provies information concerning the surrouning traffic. ASAS on boar an aircraft interacts ith the existing systems (particularly the Flight Management System - FMS or even ith the installe equipment (particularly the isplay screens; this may also require ne equipment epening on the architecture neee. Not only the system, but also its utilization as efine by ICAO: an ASAS application is a set of operational proceures for air traffic controllers an flight cres, hich makes use of the capabilities of airborne separation assistance systems to meet a clearly-efine operational objective. This paper consiers Airborne Spacing applications. In this category of ASAS applications for maintaining spacing (ithout transfer of responsibility the flight cres must acquire an maintain a given istance or time ith another esignate aircraft, after a ne Air Traffic Control (ATC instruction. Even if the flight cres inherit ne * PhD, Civil Aviation Design an Operation Engineer, Separation an Alerting Systems Unit, miquel@cena.fr Senior Researcher at LAAS u CNRS, Professor of Automatic Control an Avionics at Ecole Nationale e l Aviation civile, Felix.MORA@enac.fr Civil Aviation Design an Operation Engineer, Hea of Separation an Alerting Systems Unit, loscos@cena.fr Copyright 007 by DSNA. Publishe by the American Institute of Aeronautics an Astronautics, Inc., ith permission. 1 American Institute of Aeronautics an Astronautics Copyright 007 by DSNA. Publishe by the American Institute of Aeronautics an Astronautics, Inc., ith permission.

2 tasks, the separation beteen aircraft is still the controller's responsibility. This ASAS application can thus be regare as a ne instruction base on the aircraft's ne capabilities, on the controller's initiative. Amongst the various applications consiere, Sequencing & Merging appears to be promising for sequencing aircraft that arrive at large terminal maneuvering areas (TMA. The task of establishing properly space laning sequences is very emaning in heavy traffic conitions for the air traffic controllers in charge of the terminal maneuvering area. With the sustaine increase of air traffic leaing to airspace saturation, ne flight maneuvering capabilities are expecte for commercial aircraft. Among these, automatic path stretching an tracking onboar airliners to achieve time-base separation at given meter fix appears as a promising solution to ease air traffic controllers orkloa. Inee, the envisione path stretching maneuver coul relieve air traffic controllers of the nee to provie time-consuming raar-vectoring instructions. It is realize onboar the aircraft using their sensing an communication facilities, in a ecentralize ay. This paper focuses on path stretching an tracking for time-base aircraft spacing at meter fix. The path stretching problem consists in fining curves that satisfy constraints on the positions an tangents at the to enpoints. Fining a smooth path ith continuous curvature that can match arbitrary enpoint constraints an length constraint is nontrivial. Several techniques can be use, incluing geometric approaches 5, 6, 7, the use of optimal control 8, 9 as ell as evolutionary algorithms 10. In that paper, a geometric approach using composite Hermite curves is folloe. The paper is organize as follos: in the next section, the path stretching planner base on composite Hermite curves to generate a path ith length an enpoint constraints is escribe. This reference trajectory is use to stretch the path of an airliner in orer to elay its arrival on a meter fix chosen by air traffic control. Then, the esign of a nonlinear control la base on feeback linearization to track the reference trajectory is presente. The esign is folloe by to illustrative examples hich treats level flight but here in is consiere. Finally, conclusions on the effectiveness of the propose approach are ran. II. PATH STRETCHING A. Preliminaries In the folloing, a fully coorinate aircraft (i.e. sie-slip angle is assume to be zero flying over a flat an nonrotating earth ithin stanar atmosphere is assume. Furthermore, an as in many moern jets, it is assume in the folloing that a Flight Management System (FMS an an autopilot function ealing ith heaing control is available onboar the trailing aircraft. x W V χ f χ i Gs P(0: initial position an orientation P(T: final position an orientation Fig. 1: Spee triangle As a consequence, the kinematics equations of motion of the aircraft are the folloing (see Fig. 1: x& = V cos( W cos( = Gs cos( χ (1 y& = V sin( W sin( = Gs sin( χ In the preceing equations is the irection from here the in is bloing, W is the spee of the in, the heaing of the aircraft an V its airspee, Gs the grounspee an χ the track angle. In orer to be inepenent from in, the path stretching planner generates an airborne open loop trajectory for the folloing system, here (x a (0, y a (0, a (0 an (x a (T, y a (T, a (T are given constraints at the beginning an the en of the trajectory: x& a = V cos( ( a y& a = V sin( a y American Institute of Aeronautics an Astronautics

3 B. Composite Hermite curves The objective for the path planning is to stretch the trajectory hile maintaining a smooth motion. The problem consists of generating a smooth trajectory curve by joining to istinct configurations, the oriente start an en points, ith a constraint on the time at hich en point is reache. In this paper composite Hermite curve are use 11. To pieceise C -continuous segments are use. Each segment is a cubic polynomial Hermite curve Q(s, s [s 0, s 1 ] here s 0, s 1 R an s 0 < s 1 satisfy the folloing enpoint location an tangent vector conitions: Q( s0 = P0, Q' ( s0 = a0v0 (3 Q( s1 = P1, Q' ( s1 = a1v1 Points P 0 an P 1 are given D (or 3D points, an V 0 an V 1 are given tangent vectors at P 0 an P 1 respectively. Q(s can be expresse as follos: Q( s = ( τ + 1( τ 1 P0 + ( τ + 3 τ P (4 1 + ( 1 τ τ ( s1 s0 a0v0 + ( τ 1 τ ( s1 s0 a1v1 Where s s0 t (5 s = V t, s0 = 0, s1 = V T, τ = = s1 s0 T A negative value of a 0 or a 1 means that the tangent vector of the Hermite curve is in the opposite irection of V 0 or V 1. C. Open loop trajectory The open loop trajectory is obtaine through a composite Hermite curve compose by segment P(0Pi an segment PiP(T as epicte in Figure. (P(0,χ 0 represents the starting position an orientation of the aircraft hereas (P(T,χ T represents the final position an orientation of the aircraft: Pi χ i x χ 0 τ υ P(0: initial position an orientation y χ T P(T: final position an orientation Fig. : Parameterization of composite Hermite curves Intermeiate point Pi is positione to the line perpenicular to the segment P(0P(T, here P(0 esignates the initial position of the aircraft, an P(T its esire position after T time of flight. Require time T is suppose to be given by air traffic control through the help of an Arrival Manager (AMAN Denoting by r υ the vector perpenicular to segment P(0P(T, vector OPi is efine by: r r r P( T (6 OPi = τ + υ here τ = Orientation χ i of intermeiate point Pi is taken to be the angle of the vector bisecting segment P(0Pi an segment PiP(T. Let L be the esire length of the Composite Hermite curve. The esire length L = V T is achieve by setting appropriately the length of vector r υ perpenicular to segment P(0P(T. III. Path Tracking The purpose of path tracking is to make the aircraft able to follo the reference trajectory establishe by the path stretching planner. 3 American Institute of Aeronautics an Astronautics

4 The control of the lateral motion of the trailing aircraft to follo the reference trajectory is achieve by proviing heaing cues to the autopilot. It is base on feeback linearization 1 hich allos the transformation of a nonlinear system into an equivalent linear system. Desire position an orientation erive from the reference trajectory x x χ c Gs χ c χ τ Trailing aircraft current position y Fig. 3: Cross track istance ν an the along track istance τ beteen current an esire position From Figure 3, the cross track istance ν an the along track istance τ from the current position (x, y of the aircraft to the axis oriente by the esire track χ an passing through the esire position (x, y are given by: τ cos ( χ sin( χ = ( ( x x (7 υ sin χ cos χ y y In aition, the ynamics of the cross track istance ν is given by: (8 υ = Gs sin ( χc χ t No, let us impose a first orer ynamic for the cross track istance: υ = λ υ (9 t Combining the to previous equations leas to the expression for the commane track angle χ c : λ υ (10 λ υ = Gs sin ( χc χ χ c = χ arcsin Gs Previous equation shos that the ratio λ ν/gs shall lie beteen 1 an +1. In aition, the ifference beteen the esire track χ an the commane track χ c shall be limite beteen π an + π. The previous equation associate ith (1 leas to the expression of the commane heaing angle c : λ υ (11 Gs sin χ ( arcsin + W sin Gs sin( χ ( c + W sin Gs ( ( c = arctan = arctan Gs cos χc + W cos λ υ ( Gs cos χ arcsin + W cos Gs Gs = V + W V W cos( In the folloing, the ratio beteen the airspee V an parameter λ has been set to the raius of turn R of the aircraft: V V g tan( ϕ (1 max = R = λ = λ g tan( ϕmax V Assuming a maximum bank angle φ max of 30 egrees an an airspee V of 149 m/s, this leas to a value of parameter λ at sec -1. y χ υ axis 4 American Institute of Aeronautics an Astronautics

5 IV. Illustrative examples A. Scenarios In this section, to situations are presente in orer to evaluate the properties of the path stretching an tracking control previously escribe. The aircraft moels hich have been use is a fictitious commercial aircraft moel (calle RCAM hich as evelope by GARTEUR - Group for Aeronautical Research an Technology in Europe (Magni, et al., The aircraft mass has been set to kg. The airspee (TAS of the aircraft has been chosen to be 89 knots (149 m/s hich correspons to a conventional airspee (CAS of 50 knots at flight level 100. This airspee is compliant ith current proceure an observations on raar ata. The scenario is epicte in Fig. 4, an is built aroun real life meter fixes of an air navigation proceure in France. The leaing aircraft starts its trajectory 40 NM before meter fix SOKMU, hereas the RCAM aircraft starts its trajectory vertical to meter fix DPE (Dieppe. It firstly converges toars meter fix SOKMU an then moves toars meter fix MERUE. DPE RCAM aircraft 37 NM DVL leaing aircraft 4 NM 17 NM 44 NM SOKMU : Meter fix 1 MERUE : Meter fix Fig. 4: Case stuy scenario In the first situation, no in is assume. When no path stretching is applie, the leaing aircraft passes meter fix SOKMU at t = 497 sec hereas the RCAM aircraft passes meter fix SOKMU before the leaing aircraft at t = 456 sec. In the secon situation, a in of 97 knots (50 m/s bloing from East is assume. When no path stretching is applie, the leaing aircraft passes meter fix SOKMU at t = 748 sec hereas the RCAM aircraft passes meter fix SOKMU before the leaing aircraft at t = 538 sec. Fig. 5: Trajectory ithout path stretching The objective of the simulations is to elay the arrival time of the RCAM aircraft to SOKMU by 90 sec. In the folloing, parameters a 0 an a 1 in (4 have been set to American Institute of Aeronautics an Astronautics

6 B. Results hen no in is consiere The estimate time at hich the leaing aircraft is at closest istance of SOKMU is obtaine by iviing the istance beteen the aircraft an SOKMU by its groun spee, hich gives 497 sec (that is 40 NM ivie by 149 m/s. Then the esire elay of 90 sec is ae. As a consequence, the estimate time T use in (5 is = 587 sec. This leas to the esire length L = V T of the Composite Hermite curve, here V is the airspee of the RCAM aircraft, hich is 149 m/s. To achieve that length, the norm of vector r υ in (6 shall be set to 1.76 NM. Fig. 6: Trajectory ith path stretching an no in The simulation shos that the actual time at hich the leaing aircraft is at closest istance of SOKMU is 497 sec, hereas the RCAM aircraft is at closest istance of SOKMU at 589 sec: the time ifference is 9 sec, that is + sec of error compare ith the esire elay of 90 sec. C. Results hen in is consiere In that situation, a in of 97 knots (50 m/s bloing from East is assume, but the same methoology than is the previous section is applie. The estimate time at hich the leaing aircraft is at closest istance of SOKMU is obtaine by iviing the istance beteen the aircraft an SOKMU by its groun spee, hich gives 748 sec (that is 40 NM ivie by 99 m/s. Then the esire elay of 90 sec is ae. As a consequence, the estimate time T use in (5 is = 838 sec. This leas to the esire length L = V T of the Composite Hermite curve, here V is the airspee of the RCAM aircraft, hich is 149 m/s. To achieve that length, the norm of vector r υ in (6 shall be set to.16 NM. Fig. 6: Trajectory ith path stretching an in of 50 m/s 6 American Institute of Aeronautics an Astronautics

7 The simulation shos that the actual time at hich the leaing aircraft is at closest istance of SOKMU is 748 sec, hereas the RCAM aircraft is at closest istance of SOKMU at 837 sec: the time ifference is 89 sec, that is 1 sec of error compare ith the esire elay of 90 sec. V. Conclusion In this paper, the esign of a ne FMS function eicate to path stretching has been consiere. This ne function aims at achieving a specifie elay beteen commercial aircraft at a specifie meter fix. This envisione ne capability onboar commercial aircraft takes avantage of the Automatic Depenent Surveillance-Broacast (ADS-B an provies ne perspectives to potentially increase air traffic control efficiency. It coul be seen as the airborne counterpart of the groun base arrival manager (AMAN. In this paper, the path stretching is base on composite Hermite curves to generate a path ith length an enpoint constraints, hereas the tracking of the reference trajectory is mae through heaing control hich is base on feeback linearization. This allos the trailing aircraft to accurately follo the reference trajectory. Airspee is maintaine at a constant value in orer to avoi throttle eman. Simulation results illustrate the efficiency of the propose esign. Nevertheless, aitional stuies in terms of operational scenarios, hich take into account the vertical motion of the aircraft, are neee in orer to refine an valiate the propose esign. Future ork ill focus on the assessment of the propose esign using a set of numerous simulate encounters for the purpose of statistical evaluation. In aition aaptive path stretching ill be aresse in orer to take into account possible airspee variations. Acknolegments The authors ish to thank Philippe Louyot for helpful inputs an comments References The Eleventh ICAO Air Navigation Conference, Automatic epenent surveillance broacast (Concept of Use, WP6 Item 1. in the list of ocumentation AN-Conf11 (003 ( Air Traffic Control Quarterly, Volume 13 Number Special Issue on ASAS, Francis Casaux Eitor (005 Loscos J.M., ASAS: toars ne cooperation base on airborne separation, Revue technique e la DTI no., pp (005 The Eleventh ICAO Air Navigation Conference, Circular on airborne separation assistance system (ASAS, IP5 Item 1. in the list of ocumentation AN-Conf11 (003 ( Yamamo M., Iamura M., Mohri A., Quasic-Time-Optimal Motion Planning of Mobile Platforms in the Presence of Obstacles, International Conference on Robotics an Automation, pp (1999 Fraichar Th., Ahuactzin J. M., Smooth Path Planning for Cars, IEEE International Conference On Robotics an Automation May, pp. 1-6 (001 Lamiraux F., Kavraki L. E., Planning paths for elastic objects uner manipulation constraints, International Journal of Robotics Research, vol. 0, no. 3, pp (001 Hagelauer, Mora-Camino, Evaluation of Practical Solutions for On Boar Aircraft Fourth Dimensional Guiance, Journal of Guiance, Control an Dynamics, Vol 0(5, pp , 1997 Bertsekas, Dynamic Programming an Optimal Control, Vol.1, Athena Scientific, Belmont, Massachusetts, 1995 Gianazza D., Duran N., Assessment of the 3D-separation of Air Traffic Flos, 6th USA/Europe Seminar on Air Traffic Management Research an Development, Baltimore, June 005 Yong, J.H., Cheng F. (004. Geometric Hermite curves ith minimum strain energy, Computer Aie Geometric Design, 1, Khalil H.K., Nonlinear Systems, Thir Eition, Prentice Hall, 00 Magni J.F., Bennani S., Terlou J., Robust Flight Control : A Design Challenge, Springer-Verlag Publications ( American Institute of Aeronautics an Astronautics