Outline New Developments in Point-Stabilization and Path-Following � First Part: Point Stabilization Switched seesaw system � A. Pedro Aguiar Seesaw control systems design � pedro@isr.ist.utl.pt Stabilization of underactuated vehicles � � The Extended Nonholonomic double Integrator (ENDI) ISR/IST Institute for Systems and Robotics Instituto Superior Técnico � The underactuated autonomous underwater vehicle (AUV) Lisbon, Portugal � Second Part: Path-following In collaboration with: António M. Pascoal (ISR/IST), João P. Hespanha (UC Santa Barbara), and Petar V. Kokotovi ć (UC Santa Barbara) � Limits of performance CDC’06 Workshop � Geometric path-following New Developments in Point-Stabilization, Trajectory-Tracking, Path- � Speed assigned path-following Following, and Formation Control of Autonomous Vehicles December 12, 2006 • San Diego, CA, USA A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles 2 1
Input-to-state practically stable Input-to-state practically stable Nonlinear system Nonlinear system � � domain that Set of measurable domain that Set of measurable Measuring function Measuring function � � contains x=0 essentially bounded signals contains x=0 essentially bounded signals Continuous nonnegative function Continuous nonnegative function Input-to-state practically stability (ISpS) on w.r.t. ω Input-to-state practically stability (ISpS) on w.r.t. ω � � continuous, strictly increasing, and γ (0) = 0 continuous, for each fixed t ∈ R , the function β ( · , t ) is of class K , and for each fixed r ≥ 0 the function β ( r , t ) decreases with respect to t and β ( r , t ) → 0 as t → ∞ . is the (essential) supremum norm of a signal u : I → R n on an interval I ⊂ [0, ∞ ) A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles 3 4 2
Unstable/stable switched system Unstable/stable switched system Instability ( σ = 1) Unstable/stable switched system � � disturbance Piecewise constant switching signal Stability ( σ = 2) � ISpS on X w.r.t. ω . . . . . . A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles 5 6 3
Unstable/stable switched system Switched seesaw system The unstable/stable switched system on w.r.t. ω is ISpS at t = t k if � Switched System � satisfies Two measuring functions � If the inequality is independent of r, � Temporal representation � and degenerate into Sets a lower bound on the periods of time over which the switching system is required to be stable ! If the differences between consecutive switching times ∆ i are uniformly � bounded, then the system is ISpS. A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles 7 8 4
Switched seesaw system Seesaw control systems design Nonlinear system � If the gains satisfy the small-gain theorem � Step 1. (Detectability property) � w ISpS � Find two measuring functions (outputs) that are IOSS Step 2. (Switched seesaw system) � ISpS � Design two feedback laws such that the nonlinear system w together with the switching controller is a switched seesaw system w.r.t. Then, if σ is chosen such that the stability conditions hold, the closed-loop Then, the seesaw switched system is ISpS on w.r.t. system is ISpS w.r.t. A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles 9 10 5
The extended nonholonomic double integrator (ENDI) The extended nonholonomic double integrator (ENDI) Model with input disturbances and state measurement noise Model � � It captures the kinematics and dynamics of a wheeled mobile robot The ENDI falls into the class of control affine nonlinear systems with drift and cannot be stabilizable via a time-invariant continuously differentiable feedback law! A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles 11 12 6
The extended nonholonomic double integrator (ENDI) The extended nonholonomic double integrator (ENDI) Time-derivative of the measuring functions: � Model with input disturbances and state measurement noise � Feedback laws: � Measuring functions: � (witn v = n = 0) (witn v = n = 0) � � Under a suitable choice of the controller gains, the closed-loop system verifies the It can be viewed as conditions of a switched seesaw system on positive semi-definite w.r.t. Lyapunov functions Satisfies the detectability condition A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles 13 14 7
The underactuated autonomous underwater vehicle The extended nonholonomic double integrator (ENDI) Vehicle modeling (horizontal plane) 20 300 � 200 ω su ( t ) x 1 ( t ) ω su x 1 10 Kinematics � 100 0 0 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 35 40 45 50 time time 20 300 200 ω us ( t ) x 2 ( t ) ω us x 2 10 100 Objective: Dynamics � Derive a feedback control law for 0 0 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 35 40 45 50 τ u and τ r to stabilize the time time underactuated AUV to a small 10 3 σ ( t ) x 3 ( t ) neighborhood of a desired 2 x 3 0 σ position and orientation. 1 -10 0 There is no side thruster! 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 35 40 45 50 time time Mass and hydrodynamic added mass: Hydrodynamic damping terms: Measurement noise: Zero-mean uniform random noise with amplitude 0.1 Dwell-time constants: Input disturbances: A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles 15 16 8
The underactuated autonomous underwater vehicle The underactuated autonomous underwater vehicle Coordinate Transformation (state and control) Coordinate Transformation (state and control) � � Transformed system Transformed system � � It is ISS with x viewed as input ! A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles 17 18 9
Conclusions (part I) The underactuated autonomous underwater vehicle Position and orientation Linear and angular velocities � A new class of switched systems was introduced and mathematical 5 2 x ( t ) tools were developed to analyze their stability and disturbance/noise u ( t ) 1 u [m/s] x [m] 0 0 attenuation properties. -1 -5 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 � A so-called seesaw control design methodology was also proposed. time [s] time [s] -3 x 10 5 5 � To illustrate the potential of this control design methodology, v ( t ) y ( t ) v [m/s] y [m] 0 0 applications were made to the stabilization of the ENDI and to the -5 dynamic model of an underactuated AUV in the presence of input -5 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 time [s] time [s] disturbances and measurement noise. 1 0.2 r ( t ) r [rad/s] ψ ( t ) ψ [rad] 0 0.5 -0.2 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 time [s] time [s] Dwell-time constants: Initial condition: A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles 19 20 10
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