Outline New Developments in Point-Stabilization and Path-Following - - PowerPoint PPT Presentation

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• A. Pedro Aguiar

pedro@isr.ist.utl.pt

New Developments in Point-Stabilization and Path-Following

CDC’06 Workshop

New Developments in Point-Stabilization, Trajectory-Tracking, Path- Following, and Formation Control of Autonomous Vehicles

December 12, 2006 • San Diego, CA, USA

ISR/IST Institute for Systems and Robotics Instituto Superior Técnico Lisbon, Portugal

In collaboration with: António M. Pascoal (ISR/IST), João P. Hespanha (UC Santa Barbara), and Petar V. Kokotović (UC Santa Barbara) 2

• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Outline

First Part: Point Stabilization

• Switched seesaw system
• Seesaw control systems design
• Stabilization of underactuated vehicles

The Extended Nonholonomic double Integrator (ENDI) The underactuated autonomous underwater vehicle (AUV)

Second Part: Path-following

Limits of performance

Geometric path-following Speed assigned path-following

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Input-to-state practically stable

• Nonlinear system
• Measuring function
• Input-to-state practically stability (ISpS) on w.r.t. ω

domain that contains x=0 Set of measurable essentially bounded signals 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 → ∞. Continuous nonnegative function is the (essential) supremum norm of a signal u: I → Rn on an interval I ⊂ [0, ∞) 4

• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Input-to-state practically stable

• Nonlinear system
• Measuring function
• Input-to-state practically stability (ISpS) on w.r.t. ω

domain that contains x=0 Set of measurable essentially bounded signals Continuous nonnegative function

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Unstable/stable switched system

• Unstable/stable switched system

disturbance Piecewise constant switching signal

. . .

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Unstable/stable switched system

• Instability (σ = 1)
• Stability (σ = 2)

. . .

ISpS on X w.r.t. ω

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Unstable/stable switched system

• The unstable/stable switched system on w.r.t. ω is ISpS at t = tk if

satisfies

• If the inequality is independent of r,

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.

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Switched seesaw system

• Temporal representation
• Switched System
• Two measuring functions

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Switched seesaw system

Then, the seesaw switched system is ISpS on w.r.t.

• If the gains satisfy the small-gain theorem

ISpS ISpS

w w

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Seesaw control systems design

• Nonlinear system
• Step 1. (Detectability property)

Find two measuring functions (outputs)

that are IOSS

• Step 2. (Switched seesaw system)

Design two feedback laws such that the nonlinear system

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 system is ISpS w.r.t.

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

The extended nonholonomic double integrator (ENDI)

• 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!

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

The extended nonholonomic double integrator (ENDI)

• Model with input disturbances and state measurement noise

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

The extended nonholonomic double integrator (ENDI)

• Model with input disturbances and state measurement noise
• Measuring functions:

It can be viewed as positive semi-definite Lyapunov functions Satisfies the detectability condition

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

The extended nonholonomic double integrator (ENDI)

• Time-derivative of the measuring functions:
• Feedback laws:
• (witn v = n = 0)
• (witn v = n = 0)

Under a suitable choice of the controller gains, the closed-loop system verifies the conditions of a switched seesaw system on w.r.t.

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

The extended nonholonomic double integrator (ENDI)

5 10 15 20 25 30 35 40 45 50 100 200 300

time ω su

5 10 15 20 25 30 35 40 45 50 100 200 300

time ω us

5 10 15 20 25 30 35 40 45 50 1 2 3

time σ

10 20 30 40 50 60 70 80 90 100 10 20

time x 1

10 20 30 40 50 60 70 80 90 100 10 20

time x 2

10 20 30 40 50 60 70 80 90 100

• 10

10

time x 3

x1(t) x2(t) x3(t) ωsu(t) ωus(t) σ (t)

Measurement noise: Zero-mean uniform random noise with amplitude 0.1 Input disturbances: Dwell-time constants:

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

The underactuated autonomous underwater vehicle

• Vehicle modeling (horizontal plane)
• Kinematics
• Dynamics

Objective: Derive a feedback control law for τu and τr to stabilize the underactuated AUV to a small neighborhood

• f

a desired position and orientation.

There is no side thruster!

Mass and hydrodynamic added mass: Hydrodynamic damping terms:

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

The underactuated autonomous underwater vehicle

• Coordinate Transformation (state and control)
• Transformed system

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

The underactuated autonomous underwater vehicle

• Coordinate Transformation (state and control)
• Transformed system

It is ISS with x viewed as input !

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

The underactuated autonomous underwater vehicle

20 40 60 80 100 120 140 160 180 200

• 1

1 2

time [s] u [m/s]

20 40 60 80 100 120 140 160 180 200

• 5

5 x 10

• 3

time [s] v [m/s]

20 40 60 80 100 120 140 160 180 200

• 0.2

0.2

time [s] r [rad/s]

20 40 60 80 100 120 140 160 180 200

• 5

5

time [s] x [m]

20 40 60 80 100 120 140 160 180 200

• 5

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time [s] y [m]

20 40 60 80 100 120 140 160 180 200 0.5 1

time [s] ψ [rad]

Dwell-time constants: Position and orientation Linear and angular velocities

x(t) y(t) ψ(t) u(t) v(t) r(t)

Initial condition:

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Conclusions (part I)

A new class of switched systems was introduced and mathematical tools were developed to analyze their stability and disturbance/noise attenuation properties. A so-called seesaw control design methodology was also proposed. To illustrate the potential of this control design methodology, applications were made to the stabilization of the ENDI and to the dynamic model of an underactuated AUV in the presence of input disturbances and measurement noise.

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Reference-tracking versus path-following

• Nonlinear regulator problem:
• Tracking error:
• In reference-tracking, the control objective is to force the output y(t) to converge to

a reference signal r(t).

exosystem Plant 22

• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Reference-tracking versus path-following

• Desired path
• Path-following error
• In path-following, the control objective is to force the output y(t) to follow a

geometric path yd(θ) without a timing law assigned to it.

Additional design of freedom Path-following is motivated by applications in which spatial errors are more critical than temporal errors

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Path-following problems

• Geometric path-following

For a desired path yd(θ), design a controller that achieves

• boundedness - the state x(t) is uniformly bounded for all t ≥ 0 and every

initial condition x(t0),w(θ0) in some neighborhood of (0,0).

• error convergence - the path error eP(t) converges to zero as t → ∞, and
• forward motion - θ(t) > c, c > 0, for all t ≥ 0 .
• Speed-assigned path-following

In addition to the geometric path-following task it is required that either

• θ(t) → vd, vd > 0 as t → ∞, or θ(t) = vd for all t ≥ T and some T ≥ 0.

. . .

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Problem statement

• Reference-tracking :

Given an arbitrarily small δ > 0, is it possible or not to find a tracking controller that make ?

Kwakernaak and Sivan, 1972 Middleton, Freudenberg,1991 Qui and Davison, 1993 Su, Qiu, and Chen, 2003 Seron, Braslavsky, and Kokotovic, 1999

• Path-following :

Both for the geometric and the speed-assigned path-following, given an arbitrarily small δ > 0, is it possible or not to find a controller that make ?

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Reference-tracking (linear systems)

• If (A, B, C, D) is a non-minimum phase system, a fundamental limitation

exists in the achievable transient performance JT

• Constant references r(t) = η

zeros of (A, B, C, D) contained in the open right complex half-plane

• Sinusoidal references r(t) = η1 sin(ω t) + η2 cos (ω t), η = col(η1 , η2)

Qiu and Davison, Automatica 93 26

• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Geometric path-following (linear systems)

• If (A,B) is stabilizable, then for the geometric path-following problem there exist

constant matrices K and L, and a timing law θ(t) such that the feedback law achieves for any given positive constant δ.

satisfies some timing law to be specified

• Desired path

Aguiar, Hespanha, Kokotovic, IEEE TAC 05

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Speed-assigned path-following (linear systems)

• Let vd be specified and assume that (A,B) is stabilizable and that there exist

matrices Π and Γ satisfying Then, for the speed-assigned path-following problem there exist controllers of the form that achieve for any given positive constant δ.

satisfies some timing law to be specified piecewise-constant matrices The Sylvester equations are solvable if system (A, B, C, D) is right-invertible and its zeros do not coincide with the eigenvalues of vdS. Aguiar, Hespanha, Kokotovic, IEEE TAC 05 28

• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Limits of performance (nonlinear systems)

• Nonlinear systems that are locally diffeomorphic to systems in strict-feedback form
• Reference-tracking

Coordinate transformation error system

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Limits of performance (nonlinear systems)

• Cheap control problem:

Find the optimal feedback law that minimizes the cost functional

• Best-attainable cheap control performance for reference-tracking

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Limits of performance (nonlinear systems)

• Cheap control problem:

Find the optimal feedback law that minimizes the cost functional

• Best-attainable cheap control performance for reference-tracking
• Minimum-energy problem:

For the system Find the optimal feedback law that minimizes the cost functional

viewed as the input

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Limits of performance (nonlinear systems)

• Reference-tracking

The best attainable value of JT is the lowest control energy needed to stabilize the zero-dynamics of the error system !

Seron, Braslavsky, Kokotovic, Mayne, 1999 Aguiar, Hespanha, Kokotovic, 2005 32

• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Limits of performance (nonlinear systems)

• Reference-tracking

The best attainable value of JT is the lowest control energy needed to stabilize the zero-dynamics of the error system !

Seron, Braslavsky, Kokotovic, Mayne, 1999 Aguiar, Hespanha, Kokotovic, 2005

This fundamental performance limitation does not exist in path-following !

• Path-following

In particular this is true even for the speed-assigned path-following !

Aguiar, Hespanha, Kokotovic, 2005

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Geometric path-following

Assign the timing law reference-tracking problem with r(t) generated by

to be selected

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Illustrative example

• A vehicle with mass M moving in the plane, on top of which lies a mass m

position of the vehicle position of the mass applied force the top of the vehicle is not flat and gravity drives the mass away from the equilibrium position z=y with this force

Goal: Force the vehicle to follow a circular path with radius R centered at the origin

with some desired velocity (in steady-state) vd

One approach to solve this problem is to recast it as a trajectory tracking problem by creating the reference signal

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• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Simulation results

Reference-tracking Path-following

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• 1
• 0.5

0.5

time [s] e1

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• 0.2

0.2 0.4

time [s] e2

20 40 60 80 100 120 140 160 180 200 220 0.2 0.4 0.6 0.8

time [s] J

• 1.5
• 1
• 0.5

0.5 1 1.5

• 1.5
• 1
• 0.5

0.5 1 1.5

y1 [m] y2[m]

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• 1
• 0.5

0.5

time [s] e1

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• 0.1

0.1 0.2 0.3

time [s] e

2 20 40 60 80 100 120 140 160 180 200 220 0.05 0.1

time [s] J

• 1.5
• 1
• 0.5

0.5 1 1.5

• 1.5
• 1
• 0.5

0.5 1 1.5

y1 [m] y2 [m]

JT = 0.41 JP = 0.09 N = 18 36

• A. Pedro Aguiar - CDC’ 06 Workshop New Developments in Point-Stabilization, Trajectory-Tracking, Path-Following, and Formation Control of Autonomous Vehicles

Conclusions (part II)

We highlighted an essential difference between reference-tracking and path-following. The reference-tracking problem is subjected to the limitations imposed by the unstable zero-dynamics. This limitation is due to the need to stabilize the zero-dynamics by the tracking error, which therefore prevents the output y(t) from achieving perfect tracking. The path-following problem is not subjected to the limitations of reference-tracking. This conceptual result may be of practical significance, because the path-following formulation is convenient for many applications. The freedom to design a timing law is a major advantage of path- following over reference tracking.