Analysis and Control of Large Scale Aerospace Systems

Transcription

1 Analysis and Control of Large Scale Aerospace Systems from Planetary Landing to Drone Traffic Management Abhishek Halder Department of Mechanical and Aerospace Engineering University of California, Irvine Irvine, CA

2 Motivation: Drone Traffic Management Controlling A Drone Controlling Swarm of Drones

3 Motivation: Drone Traffic Management Controlling A Drone Controlling Swarm of Drones Large number of agents Population density

4 Motivation: Mars Entry-Descent-Landing Navigational uncertainty Heating uncertainty Landing footprint uncertainty Chute deployment uncertainty Image credit: NASA JPL

5 Motivation: Mars Entry-Descent-Landing Navigational uncertainty Heating uncertainty Landing footprint uncertainty Chute deployment uncertainty Image credit: NASA JPL Large number of uncertain scenarios Probability density

6 Motivation: Mars Entry-Descent-Landing higher Mach numbers result in increased aerothermal heating of parachute structure, which can reduce material strength; and (3) at Mach numbers above Mach 1.5, DGB parachutes exhibit an instability, known as areal oscillations, which result in multiple partial collapses and violent re-inflations. The chief concern with high Mach number deployments, for parachute deployments in regions where the heating is not a driving factor, is therefore, the increased exposure to areal oscillations. Supersonic parachute Figure 2. Mars Science Laboratory DGB parachute undergoing full-scale wind-tunnel testing. The Viking parachute system was qualified to deploy between Mach 1.4 and 2.1, and a dynamic pressure between 250 and 700 P a [1]. However, Mach 2.1 is not a hard limit for successfully operating DBG parachutes at Mars and there is very little flight test data above Mach 2.1 with which to quantify the amount of increased EDL system risk. Figure 3 shows the relevant flight tests and flight experience in the region of the planned MSL1.parachute deploy. While parachute Table MSL Candidate Landing Sites experts agree that higher Mach numbers result in a higher probability of failure, they have different opinions on where the limit Lat. Gillis [5]Lon. Elevation should besite placed. For example, has proposed an upper bound of Mach 2 for parachute Name (deg) aerodynamic (deg) decelerators (km) at Mars. However, Cruz [3] places the upper o o Mach number Mawrth Valis between E range somewhere two N and three. deliver such a large and capable rover safely to a scientifically compelling site, which is rich in minerals likely to trap and preserve biomarkers, presents a myriad of engineering challenges. Not only is the payload mass significantly larger Gale Crater 4.49o S o E than all previous Mars missions, the delivery accuracy and This presents a challenge foroedl system odesigners, who terrain requirements are also more stringent. In August of Eberswalde Crater S E must then weigh the system performance gains and risks as2012, MSL will enter the Martian atmosphere with the largest o o Holden Crater S sociated with deploying the parachute earlier,eat E) both-1.94 higher aeroshell ever flown to Mars, fly the first guided lifting entry Gale Crater (4.49S, altitudes and Mach numbers, against a very real, but not well at Mars, generate a higher hypersonic lift-to-drag ratio than quantified, probability of parachute failure. It is clear that any previous Mars mission, and decelerate behind the largest deploying a DGB at Mach 2.5 or 3.0 represents a significant supersonic parachute ever deployed at Mars. The MSL EDL pose of proposing andanselecting landing sites. itwhile increase in risk over inflation possible at Mach 2.0. However, is system will also, for the first time ever, softly land Curiosity many siteshow were initially proposed the first ofby these worknot clear much additional risk isatencumbered deploydirectly on her wheels, ready to explore the planet s surface. shops, theparachute outcomeatof2.25 the 4th Landing SiteThis Workshop in 2008 ing the instead of is especially was listmars of four sites, listed large in Table 1. Of the trueafor EDLcandidate in light of the extremely uncertainties Parachute Decelerators for Mars

7 What to Analyze and Control Signal Sets Densities

8 Outlook

9 Outline of Today s Talk Part I: An Application Propagating Density in Planetary EDL Part II: A Theory Controlling Density Part III: Ongoing and Future Research Unmanned Aerial Systems Traffic Management

10 Part I. An Application Propagating Density in Planetary EDL Forecasting, Estimation, Validation, Verification Joint work with R. Bhattacharya (Texas A&M), J. Balaram (JPL)

11 State-of-the-art Nonlinear Dynamics with Monte Carlo on Samples Linear Dynamics with Gaussian Uncertainty Dynamics Dynamics

12 State-of-the-art Nonlinear Dynamics with Monte Carlo on Samples Linear Dynamics with Gaussian Uncertainty Dynamics Dynamics too expensive for EDL simulation too ideal for EDL simulation

13 How Bad is Gaussian Fit Source: Golombek et. al., J. Geophys. Research Credit: NASA JPL, Univ. Washington, St. Louis, JHU APL, Univ. Arizona.

14 Propagating Joint Density Function Trajectory dynamics ẋ(t) = f (x(t), p), x R n s, p R n p ; x(0), p random [ ] x x e (t) = f e (x e (t)), x e := R p n s+n p, x e (0) ρ 0 (x e ) Density dynamics Liouville PDE for joint density ρ (x e (t), t) ρ t + (ρf) = 0 Method of characteristics (MOC) x e (t) = f e (x e (t)), ρ(t) = ρ f, [ ] [ ] xe (0) xe (0) = ρ(0) ρ 0 (x e (0))

15 PDE BVP MOC ODE IVP MC simulation Liouville MOC Offline post-processing Histogram approximation Grid based ns ODEs per sample Online Exact arithmetic Meshless ns + 1 ODEs per sample

16 Application to Mars EDL Landing Footprint Uncertainty

17 Application to Mars EDL Chute Deployment Uncertainty A.H., R. Bhattacharya, Dispersion Analysis in Hypersonic Flight During Planetary Entry Using Stochastic Liouville Equation, Journal of Guidance, Control, and Dynamics, A.H., R. Bhattacharya, Beyond Monte Carlo: A Computational Framework for Uncertainty Propagation in Planetary Entry, Descent and Landing, AIAA GNC, 2010.

18 Extension for Process Noise Trajectory St Dynamics ochast ic is evolution Stochastic Differential equat ion Equation Process Noise noise Nonparametric Non-parametric KL expansion KL Parametric am r Forward Kolmogorov operator Perron -Frob en iu s (PF) operator Fokker-Planck FPK PDE (2nd order) Liouville Liouville PDE (1st PDE order) Function approximation Funct ion approximation MOC MOC ρ ρ P. Dutta, A.H., R. Bhattacharya, Uncertainty Quantification for Stochastic Nonlinear Systems using Perron-Frobenius Operator and Karhunen-Loève Expansion, MSC, 2012.

19 Application to Nonlinear Filtering Mean and Std. Dev. for W Dist time (s) P. Dutta, A.H., R. Bhattacharya, Nonlinear Estimation with Perron-Frobenius Operator and Karhunen-Loève Expansion, IEEE Transactions on Aerospace and Electronic Systems, P. Dutta, A.H., R. Bhattacharya, Nonlinear Filtering with Transfer Operator, ACC, 2013.

20 Model and Controller V&V K. Lee, A.H., R. Bhattacharya, Performance and Robustness Analysis of Stochastic Jump Linear Systems using Wasserstein Metric, Automatica, A.H., R. Bhattacharya, Probabilistic Model Validation for Uncertain Nonlinear Systems, Automatica, A.H., L. Lee, R. Bhattacharya, A Dynamical System Pair with Identical First Two Moments But Different Probability Densities, CDC, K. Lee, A.H., R. Bhattacharya, Probabilistic Robustness Analysis of Stochastic Jump Linear Systems, ACC, A.H., R. Bhattacharya, Frequency Domain Model Validation in Wasserstein Metric, ACC, A.H., R. Bhattacharya, Further Results on Probabilistic Model Validation in Wasserstein Metric, CDC, A.H., R. Bhattacharya, Model Validation: A Probabilistic Formulation, CDC, 2011.

21 F-16 Flight Controller Verification LQR vs. gslqr Results: (MC)

22 F-16 Flight Controller Verification LQR vs. gslqr Results: (MOC) A.H., K. Lee, R. Bhattacharya, Optimal Transport Approach for Probabilistic Robustness Analysis of F-16 Controllers, Journal of Guidance, Control, and Dynamics, A.H., K. Lee, R. Bhattacharya, Probabilistic Robustness Analysis of F-16 Controller Performance: An Optimal Transport Approach, ACC, 2013.

23 Model Refinement A.H., R. Bhattacharya, Geodesic Density Tracking with Applications to Data Driven Modeling, ACC, 2014.

24 Part II. A Theory Controlling Density Finite Horizon LQG Density Regulator Joint work with E.D.B. Wendel (Draper Laboratory)

25 How to Go from One Density to Another

26 or Close to Another

27 LQG State Regulator [ t1 ] min φ (x 1, x d ) + E x (x Qx + u Ru) dt u U 0 dx(t) = Ax(t) dt + Bu(t) dt + F dw(t), x(0) = x 0 given, x d given, t 1 fixed, Typical terminal cost: MSE φ (x 1, x d ) = E x1 [ (x1 x d ) M(x 1 x d ) ]

28 LQG Density Regulator min u U ϕ (ρ 1, ρ d ) + E x [ t1 ] (x Qx + u Ru) dt dx(t) = Ax(t) dt + Bu(t) dt + F dw(t), x(0) ρ 0 given, x d ρ d given, t 1 fixed, Proposed terminal cost: MMSE ϕ (x 1, x d ) = where y := (x 1, x d ) 0 inf E [ y (x1 x d ) M(x 1 x d ) ], y ρ P 2 (ρ 1,ρ d )

29 Formulation: LQG Density Regulator min u U ϕ(ρ 1, ρ d ) inf E [ y (x1 x d ) M(x 1 x d ) ] y ρ P 2 (ρ 1,ρ d ) [ t1 ] + E x (x Qx + u Ru) dt 0 dx(t) = Ax(t) dt + Bu(t) dt + F dw(t), x(0) ρ 0 = N (µ 0, S 0 ), x d ρ d = N (µ d, S d ), t 1 fixed, U = {u : u(x, t) = K(t)x + v(t)}

30 dim. TPBVP ( n 2 + 3n ) dim. TPBVP ( ) µ(t) ż(t) ( A BR = 1 B Q A ) ( µ(t) z(t) ), Ṡ(t) = (A + BK o )S(t) + S(t)(A + BK o ) + FF, Ṗ(t) = A P(t) P(t)A P(t)BR 1 B P(t) + Q, Boundary conditions: µ(0) = µ 0, z(t 1 ) = M(µ d µ 1 ), S(0) = S 0, P(t 1 ) = ( ) 1 ) 2S 2 S 1 d (S 1 2 d S S 2 2 d d I n M

31 Controlled State Covariance x 2 ρ d = N (µ d, S d) ρ 1 = N (µ 1, S 1) x 1

32 Expected Optimal Control E[u o (t)] t A.H., E.D.B. Wendel, Finite Horizon Linear Quadratic Gaussian Density Regulator with Wasserstein Terminal Cost, ACC, 2016.

33 Part III. Ongoing and Future Research UTM Unmanned Aerial Systems Traffic Management

34 Vision for UAS Traffic Management (UTM) Class G airspace extends up to 1200 ft AGL 500 ft AGL 200 ft AGL Weight no more than 55 lbs Requires: Automated V2V separation management Yield manned traffic Avoid obstacles (buildings, towers etc.)

35 AIRSPACE OPERATIONS & MANAGEMENT Technical Challenges ~500 ft. and below Geographical needs and applications Rules of the airspace: performance-based Geofences: dynamic and static Dynamic Geofencing Control over LTE Image credit: NASA Ames Research Center Wind Uncertainty 5 Provable Safety

36 Protocols Laws of the Sky Offline Protocol Motion Protocol How FAA approves a flight path request? What does an individual drone do in real time? Communication Protocol What and how should a drone in flight talk? Database Protocol Which other drones to talk with and when?

37 Offline Protocol How FAA approves a flight path request?

38 Offline Protocol Path Planning and Deconfliction UAS 1 requested flight paths Path planner 1 Server

39 Offline Protocol Path Planning and Deconfliction UAS 1 {Wind, geofencing, obstacle} database forecasts requested flight paths Server Path planner 1 suggested flight paths

40 Offline Protocol Path Planning and Deconfliction Approved flight path history update database update history {Wind, geofencing, obstacle} database forecasts accepted paths, drone license IDs requested flight paths Server Path planner 1 suggested flight paths UAS 1 Path manager 1 Drone team 1 commanded paths

41 Offline Protocol Path Planning and Deconfliction Approved flight path history update database update history {Wind, geofencing, obstacle} database forecasts accepted paths, drone license IDs requested flight paths Server Path planner 1 suggested flight paths UAS 1 Path manager 1 Drone team 1 suggested flight paths commanded paths UAS 2 requested flight paths accepted paths, drone license IDs Path planner 2 Path manager 2 Drone team 2 commanded paths

42 Motion Protocol What does an individual drone do in real time?

43 Input: Approved Flight Path

44 Reach Set Evolution due to Wind Uncertainty

45 Discrete Decision Making Instances

46 4D Flight Tubes F [tj,t j+1 )

47 4D Flight + Landing Tubes {F [tj,t j+1 ), L [tj+1,t j+2 )}

48 Motion Protocol: t = t0

49 Motion Protocol: t [t 0, t 1 )

50 Motion Protocol: t = t1

51 Motion Protocol: t = t1

52 Motion Protocol: t = t1

53 Algorithms for Motion Protocol Compute minimum volume outer ellipsoids: SDP

54 Proposed Architecture: Performance Throughput w 2 (t) w 1 (t) w 1 (t) and w 2 (t) are different wind trajectories Too few take-offs Too many aborts Number of offline approvals

55

56 Thank You

57 Backup Slides for Part I

58 Backup Slides for Part II

59 ϕ (N (µ 1, S 1 ), N (µ d, S d )) equals (µ 1 µ d ) M (µ 1 µ d ) + min C R n n tr ((S 1 + S d 2C)M) s.t. [ ] S1 C 0 C S d

60 ϕ (N (µ 1, S 1 ), N (µ d, S d )) equals (µ 1 µ d ) M (µ 1 µ d ) + min C R n n tr ((S 1 + S d 2C)M) s.t. [ ] S1 C C S d 0 max tr (CM) s.t. S C R n n 1 CS 1 d C 0 C = S 1 S 1 2 d (S 1 2 d S 1 S 1 2 d ) 1 2 S 1 2d

61 This gives ϕ (N (µ 1, S 1 ), N (µ d, S d )) = (µ 1 µ d ) M (µ 1 µ d ) ( [ ]) ( Sd ) ( ) 1 +tr MS 1 + MS d 2 MS 1 Sd Sd S 2 1 Sd Applying maximum principle: K o (t) = R 1 B P(t), v o (t) = R 1 B (z(t) P(t)µ(t))

62 Matrix Geometric Mean The minimal geodesic γ : [0, 1] S n + connecting γ(0) = S d and γ(1) = S 1 1, associated with the Riemannian metric g A (S d, S 1 1 ) = tr ( ) A 1 S d A 1 S 1 1, is γ (t) = S d # t S 1 1 = S 1 2 d ( S 2 1 d S 1 1 S 1 2 d ) t S 1 2 d ( ) 1 t = S 1 1 # 1 t S d = S S S d S S Geometric ( ) Mean: 1 γ = S d # 1 S = S 1 1 # 1 S d 2

63 Example ( ) dx1 = dx 2 [ ] ( ) x1 dt + x 2 [ ] 0 u dt + 1 [ ] 0.01 dw 0.01 ρ 0 = N ( (1, 1), I 2 ), ρd = N ( (0, 0), 0.1 I 2 ), Q = 100 I 2, R = 1, M = I 2, t 1 = 2

64 Backup Slides for Part III

65 Input-Output for Motion Protocol {Drone ID k, F [tjk,tj k +1), L [tjk,tj k +2)} {Drone ID l, F [tjl,tj l +1), L [tjl +1,tj l +2)} Incoming Req Outgoing Req Motion Incoming AD {A, A,?, A,?, A, D, A,?} Protocol for Drone ID l {A} Outgoing AD Communication Protocol Communication Protocol