Nonlinear trajectory tracking control based on adaptive backstepping - PowerPoint PPT Presentation

Introduction Trajectory tracking controller Simulation results DSOR Conclusions dynamical systems and ocean robotics LAB Nonlinear trajectory tracking control based on adaptive backstepping David Cabecinhas Dynamic Stochastic Filtering, Prediction, and Smoothing Instituto Superior T´ ecnico July 7th, 2010 David Cabecinhas Trajectory tracking control of a quadrotor 1/19

Introduction Trajectory tracking controller Simulation results DSOR Conclusions dynamical systems and ocean robotics LAB Outline Introduction 1 Desired trajectory tracking controller Quadrotor model Trajectory tracking controller 2 Problem statement Controller design Simulation results 3 Conclusions 4 David Cabecinhas Trajectory tracking control of a quadrotor 2/19

Introduction Trajectory tracking controller Desired trajectory tracking controller Simulation results Quadrotor model DSOR Conclusions dynamical systems and ocean robotics LAB Trajectory tracking for quadrotors Quadrotor – underactuated vehicle, nonlinear dynamics Desired controller properties Guarantee asymptotic stability of the tracking error at the origin with actuation bounded in tracking error in the presence of constant force disturbances Design methodology Dynamic model handled in its natural space SE(3) = R 3 × SO(3) Nonlinear Lyapunov-based techniques Adaptive backstepping Dynamic augmentation of the actuation David Cabecinhas Trajectory tracking control of a quadrotor 3/19

Introduction Trajectory tracking controller Desired trajectory tracking controller Simulation results Quadrotor model DSOR Conclusions dynamical systems and ocean robotics LAB Trajectory tracking for quadrotors Quadrotor – underactuated vehicle, nonlinear dynamics Desired controller properties Guarantee asymptotic stability of the tracking error at the origin with actuation bounded in tracking error in the presence of constant force disturbances Design methodology Dynamic model handled in its natural space SE(3) = R 3 × SO(3) Nonlinear Lyapunov-based techniques Adaptive backstepping Dynamic augmentation of the actuation David Cabecinhas Trajectory tracking control of a quadrotor 3/19

Introduction Trajectory tracking controller Desired trajectory tracking controller Simulation results Quadrotor model DSOR Conclusions dynamical systems and ocean robotics LAB Quadrotor model – Actuation 2 pairs of counter-rotating rotors ( F 1 , F 3 ) and ( F 2 , F 4 ) Input transformation T = � 4 i =1 F i   α 1 ( F 1 − F 3 )   n = α 2 ( F 2 − F 4 ) T F 4 α 3 ( F 1 − F 2 + F 3 − F 4 ) n n 2 3 F 1 F 3 Underactuated vehicle n 1 One-directional thrust T Full torque control n = [ n 1 n 2 n 3 ] T F 2 1 force + 3 torques vs. 6 DoF David Cabecinhas Trajectory tracking control of a quadrotor 4/19

Introduction Trajectory tracking controller Desired trajectory tracking controller Simulation results Quadrotor model DSOR Conclusions dynamical systems and ocean robotics LAB Quadrotor Model – Equations of motion Dynamics Kinematics − S ( ω ) v + 1 p ˙ = R v ˙ = v m f ˙ − J − 1 S ( ω ) J ω + J − 1 n R = − S ( ω ) R ˙ = ω External force includes thrust and gravitational contributions f = − T u 3 + mgR T u 3 + R T b , u 3 = [0 0 1] T External force disturbance: b ∈ R 3 Using input transformation n = J τ + S ( ω ) J ω , Euler equations are reduced to integrator form ω = τ = [ τ 1 τ 2 τ 3 ] T ˙ David Cabecinhas Trajectory tracking control of a quadrotor 5/19

Introduction Trajectory tracking controller Problem statement Simulation results Controller design DSOR Conclusions dynamical systems and ocean robotics LAB Trajectory tracking – Sketch of problem Premise Vehicle is underactuated – cannot describe arbitrary trajectories in SE(3) Solution Enforce position tracking only (define desired position p d ( t ) ) Orientation automatically constrained to direct T (desired rotation matrix R d = [ r 1 d r 2 d r 3 d ] satisfies r 3 d T d = mg u 3 + b − m ¨ p d ) T T     0 x ( t )     p d ( t ) = 0 p d ( t ) = 0 r 1 r 1 r 3 0 0 r 3 f g f g David Cabecinhas Trajectory tracking control of a quadrotor 6/19

Introduction Trajectory tracking controller Problem statement Simulation results Controller design DSOR Conclusions dynamical systems and ocean robotics LAB Trajectory tracking – Problem statement Assumption Desired trajectory p d ( t ) is a class C 4 function Control objective Design a control law for T and τ such that p → p d with the largest possible basin of attraction. David Cabecinhas Trajectory tracking control of a quadrotor 7/19

Introduction Trajectory tracking controller Problem statement Simulation results Controller design DSOR Conclusions dynamical systems and ocean robotics LAB Controller design Position error e 1 = p − p d Initial candidate Lyapunov function V 1 = 1 2 e T 1 e 1 Go through backstepping procedure until actuation is available ˙ V 1 = e T 1 ˙ e 1 ¾ ( x ) ˙ 1 ( σ ( e 1 ) + 1 p max V 1 = − k 1 e T 1 σ ( e 1 ) + k 1 e T k 1 ˙ e 1 ) 1 ( σ ( e 1 ) + 1 x ˙ V 1 = − W 1 ( e 1 ) + k 1 e T ( R v − ˙ p d )) ¡ p max k 1 � �� � e 2 David Cabecinhas Trajectory tracking control of a quadrotor 8/19

Introduction Trajectory tracking controller Problem statement Simulation results Controller design DSOR Conclusions dynamical systems and ocean robotics LAB Controller design Position error e 1 = p − p d Initial candidate Lyapunov function V 1 = 1 2 e T 1 e 1 Go through backstepping procedure until actuation is available ˙ V 1 = e T 1 ˙ e 1 ¾ ( x ) ˙ 1 ( σ ( e 1 ) + 1 p max V 1 = − k 1 e T 1 σ ( e 1 ) + k 1 e T k 1 ˙ e 1 ) 1 ( σ ( e 1 ) + 1 x ˙ V 1 = − W 1 ( e 1 ) + k 1 e T ( R v − ˙ p d )) ¡ p max k 1 � �� � e 2 David Cabecinhas Trajectory tracking control of a quadrotor 8/19

Introduction Trajectory tracking controller Problem statement Simulation results Controller design DSOR Conclusions dynamical systems and ocean robotics LAB Controller design New Lyapunov function candidate V 2 = V 1 + 1 2 e T 2 e 2 ˙ 1 k 2 ( 1 1 V 2 = − W 2 ( e 1 , e 2 ) + k 1 k 2 e T m R f − ¨ 2 ( e 2 + p d )) k 2 With full force control in f , we could obtain ˙ V 2 = − W 2 ( e 1 , e 2 ) < 0 However f = − T u 3 + mgR T u 3 + R T b and b is unknown! Define new error as 1 k 2 ( 1 1 m ( − RT u 3 + mg u 3 + � e 3 � e 2 + b ) − ¨ p d ) k 2 ˙ 1 2 � V 2 = − W 2 ( e 1 , e 2 ) + k 1 k 2 e T 2 e 3 + mk 1 e T b and iterate backstepping process twice more David Cabecinhas Trajectory tracking control of a quadrotor 9/19

Introduction Trajectory tracking controller Problem statement Simulation results Controller design DSOR Conclusions dynamical systems and ocean robotics LAB Controller design New Lyapunov function candidate V 2 = V 1 + 1 2 e T 2 e 2 ˙ 1 k 2 ( 1 1 V 2 = − W 2 ( e 1 , e 2 ) + k 1 k 2 e T m R f − ¨ 2 ( e 2 + p d )) k 2 With full force control in f , we could obtain ˙ V 2 = − W 2 ( e 1 , e 2 ) < 0 However f = − T u 3 + mgR T u 3 + R T b and b is unknown! Define new error as 1 k 2 ( 1 1 m ( − RT u 3 + mg u 3 + � e 3 � e 2 + b ) − ¨ p d ) k 2 ˙ 1 2 � V 2 = − W 2 ( e 1 , e 2 ) + k 1 k 2 e T 2 e 3 + mk 1 e T b and iterate backstepping process twice more David Cabecinhas Trajectory tracking control of a quadrotor 9/19

Introduction Trajectory tracking controller Problem statement Simulation results Controller design DSOR Conclusions dynamical systems and ocean robotics LAB Controller design New Lyapunov function candidate V 2 = V 1 + 1 2 e T 2 e 2 ˙ 1 k 2 ( 1 1 V 2 = − W 2 ( e 1 , e 2 ) + k 1 k 2 e T m R f − ¨ 2 ( e 2 + p d )) k 2 With full force control in f , we could obtain ˙ V 2 = − W 2 ( e 1 , e 2 ) < 0 However f = − T u 3 + mgR T u 3 + R T b and b is unknown! Define new error as 1 k 2 ( 1 1 m ( − RT u 3 + mg u 3 + � e 3 � e 2 + b ) − ¨ p d ) k 2 ˙ 1 2 � V 2 = − W 2 ( e 1 , e 2 ) + k 1 k 2 e T 2 e 3 + mk 1 e T b and iterate backstepping process twice more David Cabecinhas Trajectory tracking control of a quadrotor 9/19

Introduction Trajectory tracking controller Problem statement Simulation results Controller design DSOR Conclusions dynamical systems and ocean robotics LAB Controller design – Adaptive backstepping 4 ] T ∈ R 12 V 4 = 1 2 e T e e = [ e T 1 e T 2 e T 3 e T � � ˙ T, R, ω , p (4) ˙ h ( e , T, ˙ � 1 V 4 = − W 4 ( e ) + e T d ) + RM ( T )¯ u + b 4 k 2 1 k 2 ˙ b T ( e 2 + e 3 + e 4 )) 1 b + 1 � k 1 � + 1 k 2 e T 3 k 2 � 0 0 − T � � ¨ � T , M ( T ) = c 1 u = ¯ , nonsingular if T � = 0 T τ 1 τ 2 0 T 0 m 1 0 0 Add estimation error to the Lyapunov function V 4 = 1 1 4 ] T ∈ R 12 � b T Γ − 1 � 2 e T e + e = [ e T 1 e T 2 e T 3 e T b 2 k 1 � � ˙ T, R, ω , p (4) ˙ h ( e , T, ˙ 1 � V 4 = − W 4 ( e ) + e T d ) + RM ( T )¯ u + b 4 k 2 1 k 2 b T � � ˙ − Γ − 1 ˙ � k 1 � � 1 b + 1 + 1 k 2 e T b + ( e 2 + e 3 + e 4 ) 3 k 2 David Cabecinhas Trajectory tracking control of a quadrotor 10/19