A Blade Element Approach to Modeling Aerodynamic Flight of an Insect-scale Robot

Transcription

1 A Blade Element Approach to Modeling Aerodynamic Flight of an Insect-scale Robot Taylor S. Clawson, Sawyer B. Fuller Robert J. Wood, Silvia Ferrari American Control Conference Seattle, WA May 25,

2 Introduction 2

3 RoboBee Background Wing stroke angle ϕ w controlled independently for each wing Thrust and body torques controlled by modulating stroke angle commands Pitch Roll 100 mg 14 mm 120 Hz Image Credit: [Ma K.Y., 12], [Ma K.Y., 13] Video of RoboBee test flight courtesy of the Harvard Microrobotics 3 Lab

4 Introduction and Motivation Applications Navigation in cluttered environments, requiring precise reference tracking Robust stabilization, subject to large disturbances such as winds and gusts Research Goals Control design, implementation, and guarantees Develop high-fidelity simulation tools Previous work Simplified RoboBee Flight Model [Fuller, S.B. 14], [Chirarattananon, P. 16] 6 DOF body motion, no wing modeling Linearized, uncoupled, stroke-averaged aerodynamic forces Controlled with hierarchical PID and iterative learning RoboBee Wing Aerodynamics [Whitney, J.P. 10], [Jafferis, N.T. 16] Model wing aerodynamics with blade-element theory Omit body dynamics (constant body position and orientation) 4

5 Blade-Element Overview Wing is divided span-wise into rigid 2D differential elements Differential forces are computed for each element, and then integrated along wingspan for total force on wing Tuned to provide close approximation of actual forces in an expression that is: Closed-form Computationally-efficient Provides insight into dominant underlying physics 5

6 Model Description 6

7 Modeling Setup 3 Rigid bodies Main body + two wings 8 DOF model Main body: 6 DOF Wings: 1 DOF each (pitch angle ψ w ) Stroke angle ϕ w treated as an input Fixed frame to body frame F i, Ƹ j, Ƹ k B x, y, z No stroke-plane deviation θ w Wing Euler Angles Stroke Angle Stroke-Plane Deviation Wing Pitch Similar for: B to left wing L x l, y l, z l B to right wing R x r, y r, z r 7

8 ሷ ሶ States and Inputs ሶ x = Θ T r T ΘT r T Θ l u = φ 0 φ p φ T r Θ T rሶ T Θሶ T r Θሶ T T l Stroke angle trajectory ϕ w modeled as a function of input u following linear second-order system: φ w t + 2ζω n φ w t + ω 2 n φ w t = A w sin ω f t + തφ w For the right wing, for example, A w = φ 0 φ r 2, തφ w = φ p x State φ W Wing stroke angle u Control Input φ 0 Nominal stroke amplitude Θ Body orientation φ p Pitch input r Body position φ r Roll input Θ r Right wing orientation A w Wing stroke amplitude ω f Flapping frequency തφ w Mean stroke angle 8

9 ሶ ሶ ሶ ሶ ሶ Rigid Body Dynamics Angular momentum balance about body CG: M G = Hሶ G M L G + M R G = H B G + ሶ H G L + Blade-element theory used to calculate aerodynamic forces and moments Hሶ R G Aerodynamic forces act at instantaneous centers of pressure CP L, CP R M L G = M L rd L + r CPL /G F aero + r L/G m L g Angular momentum about G calculated as a sum of contributions from each frame H B G = I B ω B + ω B I B ω B H L G = I L ω L + ω L I L ω L + r L/G m L a L 9

10 ሶ ሶ Wing Rigid Body Dynamics Single DOF: wing pitch ψ w Angular momentum balance in span-wise direction y r M A = y r ሶ H A Negligible M A = M R rd + r CPR /A FR R aero + r R/G m R g + M k H A = I R ω R + ω R I R ω R + r R/A m R a R r CPR /A = y CP y r + z CP α z r Center of Pressure location constant in span-wise direction Non-Negligible a R = a G + a A/G + a R/A a R/A = ωሶ R r R/A + ω R (ω R r R/A ) Negligible wing mass, but very high angular rate/acceleration 10

11 Rigid Body Dynamics B Body frame CP L Center of pressure of L R Right wing frame G Center of gravity of B L Left wing frame R Center of gravity of R CP R Center of pressure of R L Center of gravity of L M rd r A/B F aero m g Hሶ A G I A ω A a R Rotational damping moment Position of A w.r.t. B Total aerodynamic force Mass Gravity vector Angular momentum of frame A about G Inertia tensor of frame A Angular rate of frame A Acceleration of point R 11

12 Blade-Element Aerodynamics Wing is divided spanwise into rectangular, 2D, rigid differential elements Differential force df aero a function of force coefficient C F, local airspeed V δw, dynamic pressure q, reference area ds df aero = C F α qds q = 1 2 ρv δw V δw ds = c r dr V δw = V G + V A/G + V δw/a 12

13 Blade-Element Aerodynamics Integrate along wingspan to obtain total force F aero df aero = C F α qds F aero = 1 R 2 C F α ρ න V δw V δw c r dr 0 Angle of attack α approximately constant along wingspan, because velocity V δw is dominated by angular rate ω R small α t = tan 1 V δw x w V δw z w V δw = V G + V A/G + V δw/a V δw/a = ω R r δw/a Integral can be decomposed so that it does not have to be evaluated at each step of simulation 13

14 Blade-Element Aerodynamics q Dynamic Pressure ρ Ambient air pressure V δw Velocity of differential element V G Velocity of robot body CG V A/G Velocity of hinge point relative to robot body CG V δw/a Velocity of differential element relative to hinge point α Angle of attack C F (α) Force coefficient ds Differential reference area R Wingspan r Wingspan coordinate c(r) Chord length 14

15 Controller Modeling 15

16 Controller Modeling Motivation Open-loop flight deviates quickly from hovering To validate model against hovering flight requires duplicating flight test controller for closed-loop simulations Video Credit: [Ma K.Y., 13] 16

17 Controller Overview Flight test controller detailed in [Ma, K.Y. 13] Control design replicated in simulation for purpose of validation Altitude: Lateral: Attitude: Signal: (PID) Desired lift force (PID) Desired body orientation (PID) Desired torque Generate signal for piezoelectric actuators 17

18 Altitude Controller t f L,des = k pa e k ia න e dτ k da eሶ 0 e = z des z Compute desired lift f L,des from the error in altitude f L,des Desired lift force k pa e k ia k da z des z Proportional gain Error Integral gain Derivative gain Desired altitude Current altitude 18

19 Lateral Controller z des = k pl r r d k dl rሶ rሶ d Compute desired body orientation from the position error and velocity error z des k pl r r d k dl Desired body vector Proportional gain Position of robot Desired position of robot Derivative gain 19

20 Attitude Controller τ des = k p z des k d Lχ where the body Euler angles Θ are used to compute ω = LΘሶ χ = s s + λ Θ τ des k p z des k d L ω Desired body torque Proportional gain Desired body vector Derivative gain Nonorthogonal transformation matrix Body angular rate 20

21 Model Validation 21

22 Forced Response High frequency forced response of simulation closely matches experimental data 22

23 Open-loop Flight Simulation (bottom) shows similar instability to experiment (top) in open-loop flight 23

24 Closed-loop Flight Qualitative comparison of simulated (left) experimental (right) closed-loop trajectories 24

25 Simulation Closed-loop simulation of hovering flight 25

26 Flight Comparisons Video Credit: (top left and top right) [Ma K.Y., 13] 26

27 Future Work and Conclusion Proportional Integral Filter (PIF) Compensator Submitted to CDC 17 Based on linearization of full equations of motion about hovering Intelligent control Preliminary work: [Clawson, T.S. 16] Use adaptive control architecture to learn on-line Detailed dynamics analysis Analyze periodic maneuvers and find set points Determine stability of various set points This model combines accurate aerodynamic force calculations with dynamic modeling to create an integrative flight model 27

28 A Blade Element Approach to Modeling Aerodynamic Flight of an Insect-scale Robot Taylor S. Clawson, Sawyer B. Fuller 1, Robert J. Wood 2, Silvia Ferrari 1 Mechanical Engineering, University of Washington, Seattle, WA 2 Engineering and Applied Sciences + Wyss Institute, Harvard University, Cambridge, MA Further questions: Taylor Clawson tsc83@cornell.edu Related Work Clawson, Taylor S., et al. "Spiking neural network (SNN) control of a flapping insect-scale robot." Decision and Control (CDC), 2016 IEEE 55th Conference on. IEEE,