Panagiotis Tsiotras. Dynamics and Control Systems Laboratory Daniel Guggenheim School of Aerospace Engineering Georgia Institute of Technology

Transcription

1 Panagiotis Tsiotras Dynamics and Control Systems Laboratory Daniel Guggenheim School of Aerospace Engineering Georgia Institute of Technology ICRAT 12 Tutotial on Methods for Optimal Trajectory Design Berkeley, California, May 22, 2012

2 Contents Some motivation for on board a/c optimal trajectory generation Brief overview of the trajectory optimization problem Numerical optimal control via direct transcription Density function based mesh refinement Optimal path tracking Time optimal path tracking operation Extension for fuel optimal path tracking operation Initial guess generation Examples Conclusions 2

3 Motivation: Some Statistics Over 70% of fatal aviation accidents are in take off/landing phases 3

4 Swissair Flight 111 5

5 US Airways Flight

6 Obstacle Avoidance LinZhi airport, China. 2,949 meters above sea level, surrounded by over 4000 m high mountains. Required Navigation Performance (RNP) procedure, Naverus, GE Aviation 7

7 Aircraft Emergency Landing Time is the MOST critical factor Swissair flight 111: 14min US Airways flight 1549: 3min Fuel may be a limiting factor too Pilots may benefit from an automation aid for: Choosing landing sites/airport, runway, Planning landing trajectory Time or fuel optimal Real Time requirement Convergence guarantees Challenge 8

8 Current Approach Optimal Control Problem Optimal trajectory and controls A D Numerical Optimal Control Software SOCS GPOPS DIDO DENMRA TOMLAB B C NLP Solver SNOPT NPSOL MINOS Fmincon CVX SQP Interior point Trust region Convex Opt A: Cost function, differential equations, state/constraints, path constraints, etc. B: Decision variables, Cost function, algebraic constraints, bounds, Jacobian, Sparsity, etc. C: Optimal solution to the NLP problem; D: Optimal solution to the optimal control problem. 9

9 Pros/Cons PROS: Feasible (and optimal) trajectory generation Direct incorporation of vehicle dynamics State constraints Control constraints Cook book process CONS: Computationally intensive (not RT) Not easy to code, in general No convergence guarantees Solution susceptible to initial guess Solution accuracy depends on discretization, integration schemes Very large NLP problem 10

10 An Alternative Use a hierarchical approach Geometric planner State constraints, obstacles Path generator Motion planner Time parameterization Trajectory generator Divide and Conquer strategy 11

11 Pros/Cons PROS: Computationally very efficient (RT possible) Physical intuition guides trajectory design Converge guarantees CONS: No exact optimality guarantees No feasibility (gap between the two layers) 12

12 Some Questions How do we reduce the dimensionality of the OCP (~ no of constraints)? How do we increase convergence robustness? How do we use good initial guesses for the OCP? How do we choose a good geometric path in the hierarchical scheme? How do we modify the path if original path is not dynamically feasible? How do we achieve real time execution on board the aircraft (e.g., during emergency)? GOAL: Real Time robust (sub)optimal trajectory generation 14

13 Landing Traj. Optimization Schematic Landing Task Geometric Path Planner Time Parameterization Feasible? No Path Smoothing Yes Numerical Optimal Control Back-up Trajectory Optimal Trajectory 15

14 OC Problem Formulation

15 Discretization The state and control variables in an optimal control problem are discretized on a mesh in the time domain The mesh is defined by Size determining the number of decision variables, hence the size of the Nonlinear Programming problem. Distribution affecting the accuracy of the solution and convergence of the solver With an appropriate distribution, a small mesh may provide faster and more accurate solution 17

16 A Density Function Based on Curvature 18

17 Example 19

18 A Density Function Based on Curvature 20

19 A Density Function Based on Curvature 21

20 Min Energy Double Integrator 22

21 Optimal Landing with Limited Thrust

22 Landing Traj. Optimization Schematic Landing Task Geometric Path Planner Time Parameterization Feasible? No Path Smoothing Yes Numerical Optimal Control Back-up Trajectory Optimal Trajectory 24

23 Equations of motion Aircraft Dynamics ẋ = v cos γ cos ψ, ẏ = v cos γ sin ψ, ż = v sin γ, v = 1 m [T F D(C L,v,z) mg sin γ], 1 γ = mv [F L(C L,v,z)cosφ mg cos γ], ψ = F L(C L,v,z)sinφ, mv cos γ 25

24 Aircraft Dynamics Aircraft dynamics in the path coordinate domain: 26

25 Kinematics 27

26 Problem Description Given a geometric path Find optimal speed profile using aircraft dynamics, then obtain the time optimal trajectory, determine the optimal control inputs such that the aircraft travel along the path in minimum time. State and Control constraints: 28

27 Admissible Velocity Set s-e Feasibility check Feasibility check & Speed Optimization 29

28 Lift Coefficient Constraint 30

29 Bank Angle Constraint 31

30 Differential Constraint 32

31 Optimal Control Formulation The main difficulty is the identification of the switching structure 33

32 Time optimal Parameterization Search Integrate and Check pattern 34

33 Control Inputs 35

34 Numerical Example Landing path time optimal parameterization for B747 aircraft 36

35 Numerical Example Optimal speed profile (path coordinate) and optimal thrust profile (time domain) 37

36 Numerical Example History of other controls (lift coefficient and bank angle) 38

37 Numerical Example Optimality validation using the landing path generated by DENMRA 39

38 Numerical Example 40

39 Minimum Fuel Landing 41

40 Minimum Energy OCP 42

41 Optimal Switching Structure 43

42 Singular Surface 44

43 Switching Structure 45

44 Optimal Control 46

45 Numerical Example 47

46 Numerical Example 48

47 Numerical Example 49

48 Landing Traj. Optimization Schematic Landing Task Geometric Path Planner Time Parameterization Feasible? No Path Smoothing Yes Numerical Optimal Control Back-up Trajectory Optimal Trajectory 50

49 Aircraft Kinematic Model To generate a geometric landing path we consider the following simplified kinematic model 51

50 Families of Dubins Paths Dubins paths are concatenations of line segments and circular arcs 1. Two circular arcs interconnected by a line segment 2. Three circular arcs 52

51 Path Planning Problem in Vertical Plane Assume γ max =0 (continuous descent) 53

52 Vertical Path Planning Problem Because γ min is large the last part/s of the path correspond to level flight Because γ min is too large, we choose γ<γ min for the last part of the path Because γ min is not large enough, full loops need to be added at the last arc 54

53 Simulation Results Increased convergence rate from 49.3% to 99.2% 500 cases Blue colored paths are the solutions of DENMRA and the red colored paths are the initial guesses 55

54 Landing Traj. Optimization Schematic Landing Task Geometric Path Planner Time Parameterization Feasible? No Path Smoothing Yes Numerical Optimal Control Back-up Trajectory Optimal Trajectory 56

55 Problem Statement Generating a smooth and collision free path : Bending rigidity Curvature 57

56 Path Discretization 58

57 Cost Functional Discretization Trapezoidal discretization of the cost functional: Positive definite 59

58 Linear Constraints Collision avoidance, tangency constraints: 60

59 Quadratic Programming Formulation Discretized path smoothing problem in a Quadratic Programming (QP) formulation length curvature collision avoidance boundary condition 61

60 Example 62

61 Some References 64

62 Acknowledgements NASA NRA NNX08AB94A (Aviation Safety Program) Advanced Methods for Intelligent Flight Guidance and Planning in Support of Pilot Decision Making NASA Ames Research Center Kalmanje KrishnaKumar Stephen Jacklin Corey Ippolito John Kaneshige Students Yiming Zhao Efstathios Bakolas