Prediction of Top of Descent Location for Idle-thrust Descents

Transcription

1 Prediction of Top of Descent Location for Idle-thrust Descents Laurel Stell NASA Ames Research Center

2 Background 10,000 30,000 ft Vertical profile from cruise to meter fix. Top of Descent (TOD) idle thrust meter fix nmi Idle-thrust descent reduces fuel consumption and emissions. 2

3 Example Meter Fix Time Errors for Operational Data er of ts Top of Descent Path Distance Number of Flights Meter Fix Crossing Time Airbus 320 B757 B B Error (nmi) Error (sec) More than 90% of the meter fix crossing time predictions have absolute error less than 30 sec. 3

4 Example Prediction Errors for Operational Data Number of Flights Top of Descent Path Distance Meter Fix Crossing Time Airbus 320 B757 B B Error (nmi) Error (sec) Less than 50% of the TOD location predictions have absolute error less than 5 nmi. 4

5 Outline of Presentation Mathematical analysis and approximation of TOD location based on NASA trajectory predictor Comparison of results with FMS test bench data and with operational data Possible applications 5

6 Mathematical Analysis 6

7 Treatment of Wind For this mathematical analysis, wind is zero Incorporating wind into the results is explained in paper Analysis of operational data included wind 7

8 Kinematic Equations ds dt = V t dh = γ a V t dt t : time s : the ground path distance relative to the meter fix h : the altitude V t : the true airspeed γ a : is the flight path angle with respect to the air mass 8

9 Kinetic Equation m dv t dt = T D mgγ a T : thrust D : drag m : aircraft mass g : gravitational acceleration 9

10 All Equations for Vertical Profile 1 dv t g dt = T D mg ds dt = V t dh = γ a V t dt γ a Specify 2 out of 3: flight path angle γ a, airspeed V t, thrust T 10

11 Descent Procedure. Top of Descent (TOD) constant Mach cruise constant calibrated airspeed (CAS) Idle thrust throughout descent Known horizontal trajectory deceleration meter fix 11

12 Constant Mach and CAS Segments V t (h) specified: 1 dv t g dt ds = T D mg = V t dt dh = γ a V t dt ds dh = 1 ( = γ a 2g γ a dvt 2 ) mg dh T D 12

13 Top of Descent Location hfix ( S TOD = h crz 2g 1 g Vfix ) dvt 2 mg dh T D dh V c V t mg T D dv t S TOD : TOD location as path distance relative to meter fix h crz : cruise altitude h fix : meter fix altitude V c : descent CAS V fix : meter fix CAS 13

14 Top of Descent Location hfix ( S TOD = h crz 2g 1 g Vfix ) dvt 2 mg dh T D dh V c V t mg T D dv t Goal Polynomial approximation in terms of h crz, V c, h fix, V fix, W = mg 13

15 Approach Analyzed trajectories from NASA predictor to develop the polynomial approximations Test matrix with 9000 entries varying cruise altitude h crz cruise Mach descent speed V c meter fix altitude h fix meter fix speed V fix aircraft weight W B and B

16 Approximation for Constant Mach and Constant CAS hfix h crz ( g dvt 2 ) W dh P(V c, W ) T D dh hfix h crz ( g ( h)v(v c )P(V c, W ) dvt 2 ) dh dh ( h) [linear function of V c, W ] linear function of V c, W, h Each of the last three lines gives a polynomial approximation, with decreasing accuracy but increasing simplicity. 15

17 Accuracy of Approximations for TOD Location B cumulative distribution function 1 approx approx linear residual (nmi) 16

18 Accuracy of Approximations for TOD Location B cumulative distribution function 1 approx approx linear residual (nmi) 16

19 Validation Against Real-World Data 17

20 FMS Test Bench Experimental Setup Commercial FMS connected to flight simulator Same aircraft types as in mathematical analysis Test matrix is (3 values of descent CAS V c ) (3 values of weight W ) All other inputs the same in all runs Recorded TOD locations computed by FMS 18

21 FMS Test Bench Data Results Least squares fit with a model linear in V c and W gave absolute errors less than 2 nmi for both aircraft types Much more accurate than expected from previous results Approximations above partly explain this Test matrix does not adequately exercise predictor 19

22 Operational Data Collection Procedures Collected data for flights arriving at Denver International Airport September 8 23, 2009 United Air Lines and Continental Airlines participated Humans in controller facility wrote down: Flights that participated and whether descent interrupted Speed profiles Pilots wrote down aircraft weight Extracted cruise altitude and horizontal trajectory from radar data 20

23 Operational Data Results Analyzed about 70 descents each for Airbus 319/320 and B from one airline Least squares fit with a model linear in h, V c, and W gave absolute errors less than 4 nmi for all except: Four (6%) of Airbus descents Six (8%) of B757 descents 21

24 Applications 22

25 Possible Uses of These Approximations Design of FMS test bench experiments Simplified sensitivity analysis Kinematic predictor of TOD location Dynamic machine learning of coefficients to handle de-icing settings, for example Continuously updated error models Only 5-10 coefficients; NASA predictor has 2000 entries in its table for B777 thrust 23

26 More About Kinematic Predictor For what-if tool with data link Might only consider changing V c and horizontal trajectory Only need to assume S TOD is linear in V c Only need to estimate one coefficient To fill in predicted trajectory between S TOD and meter fix Kinematic approach: straight line might be better than current predictions with 10 nmi error in S TOD Kinetic approach: modify W /(T D) so that integrator gives S TOD close to desired value Replace with constant Multiply by constant Add constant... 24

27 Conclusions Need to improve prediction of vertical profile to enable fuel-efficient descents in congested airspace Largest source of error is difference in W /(T D) between FMS and ground predictor TOD location can be approximated well by a simple function of h crz, V c, h fix, V fix, W These approximations might guide research to improve the predictor 25