Nonlinear Wind Estimator Based on Lyapunov

Transcription

1 Nonlinear Based on Lyapunov Techniques Pedro Serra ISR/DSOR July 7, 2010 Pedro Serra Nonlinear 1/22

2 Outline 1 Motivation Problem 2 Aircraft Dynamics Guidance Control and Navigation structure Guidance Dynamics 3 Geometrical Properties Error dynamics Control Design 4 Pedro Serra Nonlinear 2/22

3 Motivation Motivation Problem Difficulty to accurately measure the vehicle s position regarding the local environment GPS provides positioning in the (ECEF) without considering local topography measurement rate is not sufficient for some applications quality of the height s measure is poor signals are subjected to shortages some environments Pedro Serra Nonlinear 3/22

4 Motivation Motivation Problem Using cameras as primary sensors for relative position allows to cast the problem into an Image-Based Visual Servo Control possibility to perform autonomous tasks in low-structured environments with no external assistance Avisionsystemcanbecheap,lightandadaptable can be used to provide robust relative pose information Pedro Serra Nonlinear 4/22

5 Problem Motivation Problem Alignment Glide-Slope Flare Taxiing 4 o Observed Lines Runway Tracking parallel linear visual features though IVBS control Estimate the wind velocity in the orthogonal direction to the linear visual features Pedro Serra Nonlinear 5/22

6 Aircraft Dynamics Aircraft Dynamics Guidance Control and Navigation structure Guidance Dynamics v = v a + v w ξ = R(v a + v w ) m v = Ω mv + F g + F aero + F engine Ṙ = RΩ I Ω = Ω IΩ+Γ aero F aero = QS [ C X (α, β) C Yβ β C Zα (α α 0 ) ] T Pedro Serra Nonlinear 6/22

7 Aircraft Dynamics Guidance Control and Navigation structure Guidance Dynamics Guidance, Control and Navigation structure regulating the norm of the airspeed V a = v a on a desired velocity V d a ; regulating the sideslip angle β to zero stabilizing the orientation dynamics through a fast inner-loop controller such that assignments in (φ, α, β) are correctly stabilized stabilizing the translational dynamics by considering β =0 and V a constant using (φ, α) as control inputs. Pedro Serra Nonlinear 7/22

8 Aircraft Dynamics Guidance Control and Navigation structure Guidance Dynamics Guidance, Control and Navigation structure q IBVS Algorithm h i (α, β, φ, Ω) (α c,β c,φ c ) v d a Navigation System (Camera + IMU) Orientation Controller Airspeed Controller (Ω,v a) δ i T Airplane Aircraft Pedro Serra Nonlinear 8/22

9 Guidance Dynamics Aircraft Dynamics Guidance Control and Navigation structure Guidance Dynamics ξ = R(v a + v w ) (1) v w = Ω v w (2) v a = Ω v a + π va u a (α, φ) (3) e y b B e z b eb x v a v w I e x e y e z π va = I d v av T a Va 2 u a (α, φ) = mg cos θ sin φe y + QSC Zα (α α 0 )e α Pedro Serra Nonlinear 9/22

10 Geometrical Properties Geometrical Properties Error dynamics Control Design Linear features representation e b x e b y e b z P P h 1 2 h 1 2 U 2 U 1 e x e y e z Centroid h i = P i U i P i U i q = h 1 + h 2 Desired centroid Error vector q = R T b δ = q q Pedro Serra Nonlinear 10/22

11 Error Dynamics Geometrical Properties Error dynamics Control Design Error dynamics δ = Ω δ Q[(v a + v w ) U] Let ˆv w be an estimative of v w U ṽ w = v w U ˆv w Then δ = Ω δ + Q[sk(U)v a ˆv w ṽ w )] Pedro Serra Nonlinear 11/22

12 Geometrical Properties Error dynamics Control Design Need to ensure ˆv w <εv a,then ˆv w = ε y V a 1+ y 2 Choosing the dynamics for y as an external pertubation constant in the inertial frame ẏ = Ω y + π U u w, y(0) = 0 where u w acts as the innovation of the wind estimator. Pedro Serra Nonlinear 12/22

13 Geometrical Properties Error dynamics Control Design After some tedious computations Pˆvw = ε V a ˆv w = Ω v w + Pˆvw π U u w 1 ˆv w 2 ε 2 V 2 a ( I ˆv T wvˆ w ε 2 V 2 a ) ( ) 1+ε, ε 2, 1. The estimation remains in the plane orthogonal to U. Pedro Serra Nonlinear 13/22

14 Control Design Geometrical Properties Error dynamics Control Design Let va d be the desired airspeed v d a = sk(u)(k 1 δ ˆv w )+ V 2 a k 1 δ ˆv w 2 U Define a second error term δ 2 = v a v d a and let S 1 and S 2 be two storage functions S 1 = δ ṽ k wδ T k1 2 ṽ w 2 S 2 = δ Pedro Serra Nonlinear 14/22

15 Control Design Geometrical Properties Error dynamics Control Design Theorem Specify the inovation term of the wind estimate and the control input as u w = k 2 δ u a = 2k 3V a v a + v d a 2 π v a v d a then there exists positive gains (k 1,k 2,k 3,K) such that the function L = S 1 + KS 2 is a Lyapunov function for the guidance dunamics that guarantees that the error signals (δ, δ 2, ṽ w ) converge exponentially to zero. Pedro Serra Nonlinear 15/22

16 Control Design - proof Geometrical Properties Error dynamics Control Design Proof. Computing the time derivative of S 1 and S 2 and applying the control input an the innovation term one can show that ( ) T ( ) L δ + ṽw M 1 (k i,k,q,pˆvw k ) δ + ṽw 1 k 1 g 1 (k 2,k 1,K)δ T Pˆvw δ k 3 K δ 2 2 Kδ T 2 M 2(U, Q, k i,v d a )δ 2 where it is possible to choose positive gains (k 1,k 2,k 3,K) such that M 1 (k i,k,q,pˆvw ) and M 2 (U, Q, k i,va d ) are positive defenite matrices and g 1 (k 2,k 1,K) is a positive scalar. Pedro Serra Nonlinear 16/22

17 Control Design - proof Geometrical Properties Error dynamics Control Design Proof - cont. Lyapunov direct method ensures that (Pˆvw δ, δ + k 1 1 ṽw,δ 2 ) converge towords zero. Note that is Pˆvw positive as soon as ṽ w <ε V a and due to the restrictions imposed for the gain k 1, δ + k1 1 ṽw =0implies that v w <ε V a Thus Pˆvw is a positive definite matrix at the equilibrium, implying that (δ, ṽ w,δ 2 ) converge to zero exponentially. Pedro Serra Nonlinear 17/22

18 v w (1) [m/s] v w est U v w U v w (2) [m/s] v w (3) [m/s] times [s] Pedro Serra Nonlinear 18/22

19 Estimation Error 8 estimation error velocity [m/s] time [s] Pedro Serra Nonlinear 19/22

20 Landing Maneuver Pedro Serra Nonlinear 20/22

21 Short Bibliography Bruno Herisse, Tarek Hamel, Robert Mahony, and Francois-Xavier Russotto. Anonlinearterrain-followingcontrollerforavtolunmannedaerialvehicleusing translational optical flow. In Robotics and Automation, ICRA 09. IEEE International Conference on, pages , May F. Le Bras, T. Hamel, C. Barat, and R. Mahony. Nonlinear image-based visual servo controller for automatic landing guidance of a fixed-wing aircraft. In European Control Conference, pages ,Aug R. Mahony and T. Hamel. Image-based visual servo control of aerial robotic systems using linear image features. Robotics, IEEE Transactions on, 21(2): ,April2005. Pedro Serra Nonlinear 21/22

22 Nonlinear Based on Lyapunov Techniques Pedro Serra ISR/DSOR July 7, 2010 Pedro Serra Nonlinear 22/22