A Study on Trajectory Optimization for the Terminal Area Keywords : - PowerPoint PPT Presentation

A Study on Trajectory Optimization for the Terminal Area Keywords : BADA, conflict resolution, terminal area 2014/05/30 ICRAT2014 Doctoral session Yokohama National University O Daichi Toratani Seiya Ueno

Table of contents 1, Introduction - Background and target of this study 2, Problem formulation (without conflict) - Model of the trajectory 3, Simulation results (without conflict) 4, Problem formulation (Conflict resolution) - Introducing conflict resolution 5, Simulation results (Conflict resolution) 6, Conclusion and future plan 1/25

1, Introduction 1/6 Continuous descent operation (CDO) CDO • Descending constant rate CDO is able to improve; • Constant thrust Step-by-step • Fuel consumption • Noise pollution Conventional etc. descent operation Climb Descent Tokyo international airport (Haneda airport), HND Proposed for the CDO Crossing 2/25

1, Introduction 2/6 Air traffic management in the terminal area • • Conventional descent operation Continuous descent operation (Step-by-step) Fixed! Conflict Climb Air port Air port Descent Continuous climb operation (CCO) CCO The purpose of this study is … the optimal conflict-free trajectory Stepped climb for the CCO. 3/25

1, Introduction 3/6 Optimal control theory and trajectory optimization • 𝑧 State equation (Dubins car) 𝜄 𝑣 𝑊 = 1 (const.) 𝑒 sin 𝜄 cos 𝜄 𝑦 = Input: 𝑣 𝑒𝑢 𝑧 sin 𝜄 e.g.) Cruising aircraft • 𝜄 Criterion (Minimum input) 𝑢 𝑔 1 2 𝑣 2 𝑒𝑢 cos 𝜄 𝑦 Minimizing 𝐾 = 𝑢 0 • 𝑧 0 [deg] 1 1 𝑌 𝑔 = Boundary conditions 𝑌 0 = 90 [deg] 0 0 0 [deg] 1 1 𝑌 𝑔 = Optimal trajectory where 𝑌 𝑈 = 𝜄 → Arc 𝑦 𝑧 𝑌 0 = 90 [deg] 0 0 0: Initial 𝑦 f : Terminal 4/13

1, Introduction 4/6 Related previous studies (Optimal control approach) • A. Andreeva-Mori et al., “Scheduling of Arrival Aircraft Based on Minimum Fuel Burn Descents” • Fuel burn model • Optimal trajectory → CDO • J. Hu et al., “Optimal Coordinated Maneuvers for Three-Dimensional Aircraft Conflict Resolution ” • Constraint for conflict resolution • Multiple aircraft conflict resolution • Constant velocity 5/13

1, Introduction 5/6 Problem of the trajectory optimization In the practice of the air traffic control, … Vectoring (Spatial control) Changing velocity (Temporal control) ? Slowdown Spatial Temporal Spatial and Temporal conflict resolution conflict resolution conflict resolution ! It is difficult to treat spatial and temporal conflict resolutions simultaneously. 6/16

1, Introduction 6/6 Space-time coordinate system (STCS) In the STCS,; • The vertical axis means time. y y x x • It is able to treat the time t = t 1 along with the position. t = t 2 • It is also able to calculate t t t = t 3 the altitude. 4D trajectory The target of this study is ... To develop the optimization method in the STCS. - Conflict resolution, minimum fuel, minimum time, etc. Which conflict resolutions (spatial or temporal) are optimal to resolve conflict? 7/25

2, Problem formulation (w/o conflict) 1/5 Base of aircraft data (BADA) • ℎ 𝑞 Total-energy model (TEM) 𝑒ℎ 𝑒𝑊 𝑈𝐵𝑇 𝑀 𝑈ℎ𝑠 − 𝐸 𝑊 𝑈𝐵𝑇 = 𝑛𝑕 0 𝑒𝑢 + 𝑛𝑊 𝑈𝐵𝑇 𝑈ℎ𝑠 𝑒𝑢 ↔ 𝑒𝑊 𝑈𝐵𝑇 = 1 γ 𝑛 𝑈ℎ𝑠 − 𝐸 − 𝑛𝑕 sin 𝛿 𝑒𝑢 𝑦 • 𝐸 Azimuth angle 𝑒𝜔 𝑒𝑢 = 𝑕 0 𝑛𝑕 tan 𝜚 𝑊 𝑈𝐵𝑇 𝐼 𝑞 • Fuel flow 𝑊 𝑈𝐵𝑇 𝑔1 1 + 𝑊 𝑈𝐵𝑇 𝐺𝐺 = 𝐷 𝑈ℎ𝑠 𝐷 𝑧 𝑔2 • Maximum climb thrust 𝛿 𝑈ℎ𝑠 = 𝐷 𝑈𝑑,1 1 − ℎ 𝑞 𝜔 2 + 𝐷 𝑈𝑑,3 ℎ 𝑞 𝐷 𝑈𝑑,2 𝑦 8/25

2, Problem formulation (w/o conflict) 2/5 Base of aircraft data (BADA) ℎ 𝑞 • State equations 𝑀 𝑕 𝑈ℎ𝑠 tan 𝜚 𝜔 γ 𝑊 𝑈𝐵𝑇 1 𝑦 𝑊 𝑈𝐵𝑇 𝑛 𝑈ℎ𝑠 − 𝐸 − 𝑛𝑕 sin 𝛿 𝑒 𝐸 = 𝛿 𝑞 𝑒𝑢 𝑛𝑕 𝑦 𝑊 𝑈𝐵𝑇 cos 𝛿 cos 𝜔 𝑧 𝑊 𝑈𝐵𝑇 cos 𝛿 sin 𝜔 𝐼 𝑞 𝑊 𝑈𝐵𝑇 ℎ 𝑞 𝑊 𝑈𝐵𝑇 sin 𝛿 • Fuel flow 𝑧 𝑔1 1 + 𝑊 𝑈𝐵𝑇 𝐺𝐺 = 𝐷 𝑈ℎ𝑠 𝐷 𝛿 𝑔2 𝜔 𝑞 : Rate of flight path angle 𝑦 9/25

2, Problem formulation (w/o conflict) 3/5 Space-time coordinate system (STCS) dy 𝜔 𝑡 y 𝜔 𝑢 • State equation dx 𝜆 𝑡 dl 𝜔 𝑡 x 𝜆 𝑢 𝜔 𝑢 𝑒 cos 𝜔 𝑡 cos 𝜔 𝑢 = Subscript 𝑦 𝑒𝑡 sin 𝜔 𝑡 cos 𝜔 𝑢 ds 𝑡 : Spatial 𝑧 t sin 𝜔 𝑢 𝑢 : Temporal 𝑢 Independent variable 𝜆 : Curvature Length of the trajectory 𝑡 • Velocity and acceleration 𝜔 𝑢 𝑊 = 𝑒𝑚 1 dl 𝑒𝑢 = tan 90° − 𝜔 𝑤 = tan 𝜔 𝑢 𝜆 𝑢 ↔ 𝜆 𝑢 = −𝑏sin 3 𝜔 𝑢 𝑏 = − dt ds sin 3 𝜔 𝑢 10/25

2, Problem formulation (w/o conflict) 4/5 • BADA (Independent variable: time) • STCS 𝑕 𝑒𝜔 𝑢 tan 𝜚 𝑒𝑡 = −𝑏sin 3 𝜔 𝑢 𝜔 𝑡 𝑊 𝑈𝐵𝑇 𝑊 𝑈𝐵𝑇 1 𝑒𝑢 𝑛 𝑈ℎ𝑠 − 𝐸 − 𝑛𝑕 sin 𝛿 𝑒 𝛿 + 𝑒𝑡 = sin 𝜔 𝑢 = 𝑦 𝑞 𝑒𝑢 𝑧 1 𝑊 𝑈𝐵𝑇 cos 𝛿 cos 𝜔 𝑡 𝑊 𝑈𝐵𝑇 = ℎ 𝑞 𝑊 𝑈𝐵𝑇 cos 𝛿 sin 𝜔 𝑡 tan 𝜔 𝑢 𝑊 𝑈𝐵𝑇 sin 𝛿 • BADA (STCS) 𝑕 sin 𝜔 𝑢 tan 𝜔 𝑢 tan 𝜚 𝜔 𝑡 1 −sin 3 𝜔 𝑢 𝜔 𝑢 𝑛 𝑈ℎ𝑠 − 𝐸 − 𝑛𝑕 sin 𝛿 𝛿 𝑒 sin 𝜔 𝑢 𝑞 𝑦 = cos 𝜔 𝑡 cos 𝜔 𝑢 cos 𝛿 𝑒𝑡 𝑧 sin 𝜔 𝑡 cos 𝜔 𝑢 cos 𝛿 ℎ 𝑞 cos 𝜔 𝑢 sin 𝛿 𝑢 sin 𝜔 𝑢 11/25

2, Problem formulation (w/o conflict) 5/5 Optimal control problem and calculation method • Optimal control problem Constraint equation Criterion Boundary conditions 𝑡 𝐺 𝒀 𝑡 0 = 𝒀 𝟏 𝑒𝒀 𝑒𝑡 = 𝑮 𝐾 = 𝑀 𝑒𝑡 Minimizing 𝒀 𝑡 𝑔 = 𝒀 𝒈 0 State equation Fuel flow Initial and terminal state Optimal control theory • Two-point boundary value problem (TPBVP) Simultaneous non-linear differential equations Linear approximation • Simultaneous non-linear equations Simultaneous non-linear equations solver 12/25

3, Simulation results (w/o conflict) 1/3 Simulation conditions Terminal condition 𝜔 𝑡𝑔 𝑊 𝑈𝐵𝑇𝑔 𝛿 𝑔 𝑦 𝑔 𝑧 𝑔 ℎ 𝑞𝑔 𝑢 𝑔 = 0 250.0 𝐺 𝐺 0 10000 𝐺 485.0 [kt] 32808 [ft] 32808 [ft] 54.00 [nm] 107.99 [nm] Initial condition 𝜔 𝑡0 𝑊 𝑈𝐵𝑇0 𝛿 0 𝑦 0 𝑧 0 ℎ 𝑞0 𝑢 0 = 0 150.0 5 0 0 3000 0 Data of aircraft 291.6 [kt] 9843 [ft] Optimal climbing trajectory in the 3D space Boeing 777-200 Units: 𝜔 𝑡 𝑊 𝑈𝐵𝑇 𝛿 𝑦 𝑧 ℎ 𝑞 𝑢 deg m/s deg m m m s = 𝐺 : Terminal free 13/25

3, Simulation results (w/o conflict) 2/3 Trajectories in the 3D space and the STCS 32808 [ft] 3D space Space-time coordinate system 14/25

3, Simulation results (w/o conflict) 3/3 TAS, altitude, fuel flow, and fuel consumption 583.2 [kt] 39370 [ft] 700.3 [s] 11.02 [lb/s] 6614 [lb] 2509 [kg] • The optimal trajectory in the STCS is derived. 15/25

4, Problem formulation (Conflict resolution) 1/2 Spatial conflict resolution y 5 [nm] x time Spatial interval Vectoring (Spatial control) Temporal conflict resolution 250 [m/s] y 5 [nm] x 37.0 [s] = 9260 [m] time Changing velocity (Temporal control) Temporal interval in the STCS 16/25

4, Problem formulation (Conflict resolution) 2/2 Interior-point constraint If you wont to add a new constraint, … Waypoint 17/25

4, Problem formulation (Conflict resolution) 2/2 Interior-point constraint Split! Terminal1 Terminal2 Initial2 Initial1 Trajectory1 Trajectory2 Position1 = Position2 (Specified) Angle1 = Angle2 (Free) ⋮ 17/25

4, Problem formulation (Conflict resolution) 2/2 Interior-point constraint • The original trajectory optimization is transformed to the trajectory If you wont to add a new constraint, … optimization problem with a new constraint. • To see clearly which conflict resolutions are optimal, conflict Split! Waypoint resolution with specified point is shown. Terminal1 Terminal2 Initial2 Initial1 Trajectory1 Trajectory2 Position1 = Position2 (Specified) Angle1 = Angle2 (Free) ⋮ Trajectory with constraint 17/25

5, Simulation results (Conflict resolution) 1/6 Simulation conditions (Interior point constraint) Conflict point 𝜔 𝑡 𝑊 𝑈𝐵𝑇 𝛿 𝑦 𝑧 ℎ 𝑞 𝑢 = 0 225.9 2.410 73842 0 7553 384.4 439.0 [kt] 39.87 [nm] 24781 [ft] 32808 [ft] Descending aircraft Fixed 54.00 [nm] 107.99 [nm] -54.00 [nm] 18/25

5, Simulation results (Conflict resolution) 2/6 Simulation conditions (Spatial conflict resolution) Interior point conditions 𝜔 𝑡𝑗𝑜𝑢 𝑊 𝑈𝐵𝑇𝑗𝑜𝑢 𝛿 𝑗𝑜𝑢 𝑦 𝑗𝑜𝑢 𝑧 𝑗𝑜𝑢 ℎ 𝑞𝑗𝑜𝑢 𝑢 𝑗𝑜𝑢 Spatial w/o conflict = 0 𝐺 𝐺 73842 9260 𝐺 𝐺 32808 [ft] 39.87 [nm] 5 [nm] 54.00 [nm] 107.99 [nm] -54.00 [nm] 19/25