Robot Dynamics Fixed-wing UAVs: Dynamic Modeling and Control

Transcription

1 Robot Dynamics Fixed-wing UAVs: Dynamic Modeling and Control V Marco Hutter, Roland Siegwart, and Thomas Stastny

2 Contents Fixed-wing UAVs 1. ntroduction 2. Aerodynamic Basics 3. Aircraft Dynamic Modeling 4. Fixed-wing Control

3 ntroduction

4 ntroduction Fixed-wing aircraft Definition: fixed-wing aircraft are capable of flight using wings that generate lift caused by the vehicle s forward airspeed and the shape (geometry) of the wings. flightsimhandbook/43-1.jpg f the wings are traveling faster than the fuselage, it's probably a helicopter...and therefore, unsafe. Advantages: Long range / endurance (cover larger areas faster, and stay airborne longer - efficient!) Mechanically simple Disadvantages: Typically requires some infrastructure (e.g. catapult for take-off, flat open landing area or net for landing) Unless hybrid no hovering

5 ntroduction Small fixed-wing UAVs

6 ntroduction Small fixed-wing UAVs Wide variety of configurations for different purposes / applications No community standard requires modeling and specific tuning per platform!

7 Aerodynamic Basics

8 Aerodynamic Basics Basic Principles Analysis on differential volumes: With viscosity: Navier-Stokes Equation Without viscosity: Euler Equation ncompressible along streamline: Bernoulli Equation v 2 2 gh p const t_tube.htm as on 29th July

9 Aerodynamic Basics Basic Principles LFT! V=FASTER P=LOWER V=SLOWER P=HGHER

10 Aerodynamic Basics Basic Principles But Bernoulli isn t the whole story! Watch out for common misconceptions of lift E.g. the distance traveled argument for speed difference What is really going on? streamline curvature induced pressure gradients. fluid particle p out v centripetal force: p out > p in p in Nice lecture by Dr. Holger Babinsky, University of Cambridge streamline

11 Aerodynamic Basics Basic Principles p upper < p atm p lower > p atm p atm Therefore: p upper < p lower LFT! p upper p lower p atm

12 Aerodynamic Basics Airfoils 2-Dimensional Flow Analysis Flow field (pressure distribution, laminar/turbulent) highly dependant on angle of attack, Reynolds number and Mach number Suction Laminar boundary layer Transition point Turbulent boundary layer Separated boundary layer Overpressure Stagnation point

13 Aerodynamic Basics Airfoils Leading edge dl c Angle of attack a v dm dd Trailing edge Chord 25 % Chord Thickness Pressure distribution can be reduced to two forces and one moment per unit length: Lift force Drag force Moment dl Cl c dy V 2 dd Cd c dy V 2 V 2 2 dm Cm c dy : Density of fluid (air) [kg/m 3 ] c V : Chord length [m] : Flight speed (w.r.t. air) [m/s] C l : Airfoil lift coefficient [-] C d : Airfoil drag coefficient [-] C m : Airfoil moment coefficient [-]

14 Aerodynamic Basics Angle of attack / stall C l Polars of Airfoil we Re C d a

15 Airfoil Lift, Drag and Moment Methods to determine airfoil lift, drag and moment coefficients: Theoretically using 2D-CFD software Javafoil Xfoil Experimentally in a wind tunnel Extruded airfoil mounted on a measurement system Laminar flow produced by fans Experimentally from flight data System identification on logged sensor measurements of static and dynamic maneuvers Javafoil

16 Control surfaces For small airplanes, the standard control surfaces are: Ailerons (rolling) Elevator (pitching) Rudder (yawing) For larger airplanes, they can be more complex Ailerons: 2. Low-Speed Aileron 3. High-Speed Aileron Lift increasing flaps and slats: 4. Flap track fairing 5. Krüger flaps 6. Slats 7. Three slotted inner flaps 8. Three slotted outer flaps Spoilers: 9. Spoilers 10. Spoilers-Air brakes

17 Propulsion Group Placement n the front n the back On the wings

18 Aircraft Dynamic Modeling

19 Why Model the Dynamics of an Airplane? System analysis: model allows evaluating flight characteristics Stability Controllability Power required fuel needs Controllability in the case of actuator failure Autopilot design and simulation: model allows comparing different control techniques and autopilot parameter tuning Gain of time and money Higher performance of the autopilot No risk of damage compared to real tests Pilot training (in Simulator) Allows simulating and training especially emergency situations

20 Aircraft Dynamic Modeling ntro Dynamics of an airplane... Are very different from an acrobatic aircraft to a jet-liner airplane... but the principles remain the same for all Wings, stabilizers control surfaces (ailerons, rudder, elevator, flaps, spoilers, ) propulsion group (motor-gearbox-propeller, turbine, rocket, ) n this lecture, we will model a typical fixed-wing UAV 1. Assumptions/Simplifications 2. Kinematics (frames of reference + state definitions) 3. Dynamics (forces + moments) 4. Equations of motion (Newton)

21 Aircraft Dynamic Modeling ntro Actuator Dynamics Propulsion Body Dynamics Aerodynamics

22 Aircraft Dynamic Modeling Assumptions and simplifications Definitions Origin of body-fixed coordinate frame set into center of gravity Assumptions and simplifications Rigid and symmetric structure: constant, (almost) diagonal inertia matrix Constant mass Motor without dynamics and without gyroscopic effects (can be adapted) Aerodynamics (list not complete): We don t enter stall (operation in the linear lift domain) Neglect fuselage lift/sideslip force

23 Aircraft Dynamic Modeling On rigidity

24 Aircraft Kinematics

25 Aircraft Kinematics Reference axes nertial (local) reference frame North, East, Down (or NED) Flat Earth assumption Where to define the origin? E.g. UTM (Universal Transverse Mercator) coordinates Body-fixed frame x out the nose y out the right wing z (with right hand rule) down Origin typically located at center of gravity

26 Aircraft Kinematics Reference axes Body-axis

27 Aircraft Kinematics Reference axes Body-axis Body velocity: Air-mass relative speed (airspeed): V is always positive

28 Aircraft Kinematics Reference axes Body-axis Body velocity: Air-mass relative speed (airspeed): V is always positive Body rates:

29 Aircraft Kinematics Reference axes Body-axis Body velocity: Air-mass relative speed (airspeed): Wind-axis Airflow angles: Angle of attack Sideslip angle Wind frame is opposite to free-stream velocity, or wind (i.e. the air-mass) V is always positive Body rates:

30 Aircraft Kinematics Reference axes *Possible point-of-confusion: wind-axis does NOT consider wind in the inertial sense. Both body and wind frames only consider air-mass relative motion Body-axis Body velocity: Air-mass relative speed (airspeed): Wind-axis Airflow angles: Angle of attack Sideslip angle Wind frame is opposite to free-stream velocity, or wind (i.e. the air-mass) V is always positive Body rates:

31 Aircraft Kinematics Coordinate transformation Euler angles, roll, pitch, and yaw, are used to transform between inertial and body axes

32 Earth fixed frame (regarded as inertial): e xe,e ye,e ze Body fixed frame: e xb,e yb,e zb Aircraft Kinematics Coordinate transformation Rotation Matrix ( to ) is parametrized with 3 successive rotations using the ZYX Tait-Brian Angles (specific kind of Euler Angles): 1 Yaw: around - Frame 1 2 Pitch: around - Frame 2 3 Roll: around - Frame B

33 Aircraft Kinematics A note on angular rates Angular Rates: Time variation of Tait-Bryan angles Singularity: for «Gimbal Lock» Body angular rates p q r J r 2 J r p, q, 0 r cos sin sin sin cos cos cos (J r becomes singular)

34 Aircraft Kinematics Polar coordinates Longitudinal nertial Frame (*side view) : Flight path angle, defined from horizon to : Pitch angle, from horizon to x-body axis : Roll angle, rotation about x- body axis (note: pointing out of slide) (N, E) (D) (N, E) (N, E)

35 Aircraft Kinematics Polar coordinates (N) Lateral-directional nertial Frame (*top view) : Heading angle, defined from North to : Yaw angle, from North to x- body axis Note: this STLL does not consider wind. (E)

36 Aircraft Kinematics Polar coordinates (N) (E) Lateral-directional nertial Frame (*top view) : Heading angle, defined from North to : Yaw angle, from North to x- body axis Adding a constant, horizontal wind: : Course angle, defined from North to : Ground-based inertial velocity (or ground speed ) : Wind velocity Note: the constant wind assumption effects only position dynamics

37 Aircraft Kinematics Polar coordinates

38 Aircraft Dynamics

39 Aircraft Dynamics Forces & Moments Forces and moments acting on the airplane Weight at the center of gravity Thrust of propeller: complex task will not be presented here Sum aerodynamic forces and moments from each part of the airplane: L Wing D Tail Fuselage Y F T L m Note: Thrust force can be offset from x-body axis. Typically denoted by angle ε. M m mg N m

40 Aircraft Dynamics Forces & Moments Forces and moments acting on the airplane Lift Drag Thrust Gravity D L Lift and Drag always perpendicular and parallel, respectively, to free-stream. F T For geometry definitions (i.e. S, b, c) ҧ see Wing Geometry in backup slides Y L m Moments Rolling moment: Pitching moment: Yawing moment: M m mg N m Free-stream velocity

41 Aircraft Dynamics Component build-up The aerodynamic forces and moments discussed in the previous lecture are build up of both static and dynamic components summed from each part of the aircraft. E.g. 1 st order Taylor Expansion (only linear terms) 2 nd order Taylor Expansion (some coupled terms) Other model structures could be used, but the build-up approach typically takes a polynomial form in nominal flight regimes

42 Represented in body frame, at the CoG: Aircraft Dynamics Forces and moments a a a a a a a a cos cos cos sin sin cos sin sin sin cos cos 0 0 sin 0 cos cos sin sin cos mg L D F mg Y mg L D F g m F F L D Y L D T T B T T F tot C T T T m m m m m m tot N M L N M L M

43 Application of Newton s Second Law Aircraft Dynamic Modeling Equations of motion Euler Derivatives xz xx yy zz xz xz zz xx yy xz yy zz xz xx zz xz yy xz xx zz xz yy xz xx qr pq r p p r pr q qp qr r p r q p r q p r q p v F B tot m dt d qu pv w pw ru v rv qw u m w v u r q p w v u m r q p dt d dt d zz xz yy xz xx B tot M Typically small

44 Aircraft Dynamic Modeling Equations of motion Translation 1 u rv qw FT cos D cosa m 1 v pw ru Y g sin cos m 1 w qu pv T m F sin D sina Lcosa g cos cos Lsina g sin x y z C B u v w w

45 Rotation (simplified with xz 0): Aircraft Dynamic Modeling Equations of motion xx yy m m zz zz xx m m yy yy zz m m xx pq N N r pr M M q qr L L p T T T cos cos cos sin sin cos cos tan sin tan 1 r q r q r q p r q p J r

46 Fixed-wing Control

47 Fixed-wing Control ntroduction Control of airplanes is not easy: nherently non-linear (especially in longitudinal axis) Low control authority Actuator saturation double integrator characteristics MMO: 4 inputs, 6 DoF, thus underactuated

48 Fixed-wing Control Control Concepts Many control techniques: Cascaded PD loops Optimal control LQR Robust control H-infinity H-2 loop-shaping Adaptive control Model Predictive Control Linear/Nonlinear Chose according to: Computational Power Type of flight (e.g. aerobatics vs. level flight) Availability / fidelity of model

49 Fixed-wing Control The plant Elevator Aileron Rudder Throttle Propulsion, Mechanics, Aerodynamics Forces Moments Nonlinear Aircraft Dynamics Velocities (Body Fr.): Turn rates (Body Fr.): Position (ntertial Fr.): Tait-Bryan angles: nput vector: State vector: Output: e.g. Some remarks about the conventions used in this lecture: 1,1, 1,1, 1,1, 0,1 nput limits: Ailerons: e a may be mixed to symmetric or differential deflections We define positive actuator inputs as those that produce positive moments r T

50 Fixed-wing Control Control & Guidance A popular concept: cascaded control loops Control = low level part Stabilize attitude (and sometimes speed) Guidance = high level part Follow paths or trajectory (position control) Effect: Reject constant low frequency perturbation (constant wind) Guidance nner loop High-level Low-level UAV Desired path/waypoint

51 Fixed-wing Control Simple cascaded control thr Trajectory Generation and Guidance d P 1 P 1 J r p d D 2 q d r d D 2 D 2 ail elev rudd y Attitude Controller Body Rate Controller Airplane Dynamics Constrain to coordinated turn: 1 P with anti-reset wind-up 2 Gain scaled with 1/V T 2 Bandwidths of inner Loops must be sufficiently larger!

52 Fixed-wing Control Steady level turning flight Turning (not straight) <- steady <- level <- turning Assuming NO sideslip, i.e. F T D Demand for coordinated turn: Y=0 L increases with V min increases with L (D) Recall Y is composed of only aerodynamic forces, which must be zero, thus the lateral force here only comes from centripetal acceleration. mg (N, E)

53 Fixed-wing Control Steady level turning flight Heading-rate can be found for a given roll angle with a force balance (assuming ) Note this assumes we have thrust force only acting in the same axis as drag. n reality, thrust force likely will add a small vertical component to the lift (e.g. if we fly at any pitch/angle of attack). Force balance:

54 Fixed-wing Control Guidance Following a Trajectory on the Horizontal Plane Theroy and Graphics from: S. Park, J. Deyst, and J. P. How, A New Nonlinear Guidance Logic for Trajectory Tracking, Proceedings of the AAA Guidance, Navigation and Control Conference, Aug AAA

55 Fixed-wing Control TECS (Total Energy Control System) Control Altitude and Airspeed

56 Fixed-wing Control TECS (Total Energy Control System)

57 Fixed-wing Control Note the model abstraction n lower-level loops, dynamics are modeled from actuators attitude/airspeed Note that aside from computaitonal costs, these dynamics are challenging to globally identify in a nonlinear, high fidelity form. Thus linearizations are often made. Higher-level loops often model the aircraft in a threedegrees-of-freedom (3DoF) sense, mapping attitude/airspeed position Modeling of high-level dynamics does not require identification, as typically only kinematics are used

58 See you tomorrow for the exercise! Next week Fixed-wing case studies! Hybrid VTOL fixed-wings Solar-powered UAVs

59

60 Backup Slides

61 References B.W. McCormick. Aerodynamics, Aeronautics, and Flight Mechanics. Wiley, SBN: B. Etkin. Dynamics of Atmospheric Flight. Wiley, SBN: G.J.J. Ducard. Fault-Tolerant Flight Control and Guidance Systems: Practical Methods for Small Unmanned Aerial Vehicles. Advances in ndustrial Control. Springer, SBN: R.W. Beard and T.W. McLain. Small Unmanned Aircraft: Theory and Practice. Princeton University Press, SBN: R.F. Stengel. Flight Dynamics. Princeton University Press, SBN:

62 c c 0 Basics of Aerodynamics Wing Geometry Wing Geometry x Mean geometric chord y b: Wingspan c: Chord cҧ c 0 : Root Chord c t S c t : Tip Chord S: Reference Area b AR: Aspect Ratio AR b S

63 Modeling for Control Linearization Decouple plant into lateral-directional and longitudinal states Linearize about trim airspeed Typically straight and level steady flight

64 Modeling for Control The (linearized) plant Subsystem e T Longitudinal Plant Δu, Δw Δq Δ a r Lateral Plant Corresponding Poles (Aerobatic Model Airplane) -2 im 2 re -2 Short Period Mode: ω = 5 rad/s Phugoid Mode: ω = 0.6 rad/s -4 im 4 re -4 Roll Subsidence Mode Spiral Mode Dutch Roll Mode ω = 5 rad/s

65 Modeling for Control The (linearized) plant Phugoid mode: exchange between kinetic and potential energy. SLOW Longitudinal Modes Short Period Mode: oscillation of angle of attack. FAST Spiral Divergence: often slightly unstable, but controllable. Lateraldirectional Modes Dutch Roll Mode: combined yawroll oscillation Grafics adapted from: and