Guidance non-linear.pdf

8/11/2019 Guidance non-linear.pdf

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2002 - Pedro Lima, M. Isabel RibeiroRobtica Mvel

NONNON--LINEAR CONTROL OFLINEAR CONTROL OFWHEELED MOBILE ROBOTSWHEELED MOBILE ROBOTS

Maria Isabel Ribeiro

Pedro Lima

[email protected] [email protected]

Instituto Superior Tcnico (IST)

Instituto de Sistemas e Robtica (ISR)

Av.Rovisco Pais, 1

1049-001 LisboaPORTUGAL

May.2002

all rights reserved


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Course Outline

tarefasPLANEAMENTO DE

TAREFAS

PLANEAMENTO DE

MOVIMENTO

AUTO-LOCALIZAO(long baseline, short baseline,feature matching)

MAPA DOMUNDO

FUSOSENSORIAL

CONTROLO

CONDUO

ACTUADORESSENSORES

VECULO

informao processada apartir dos dados dos sensores

actualizaodo mapa

mapaglobal

Informao enviadaaos actuadores

Deteco deobstculos

Posio e/ouvelocidadesdesejadas

(seguir trajectria

/evitar obstculos)

Motores, rodas,hlices, superfciiesde defleco

mapalocal

Trajectria desejada(pos+vel+ace)

Caractersticas domeio envolvente

Trajectria estimada(pos+vel+ace)

Sonar, laser, viso,encoders, giroscpios


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Guidance

sensor

measurements

operati

onpoint

posture

estimate

joint

state

feedback

PathPlanner

GuidanceJoint

ControllerVehicle

Localization

obstacles

target

pat

h

or

tra

jectory

jo

int

set

po

ints

(e.g.,

whee

l

ve

loc

it

ies

)

jo

in

t

torques

(e.g.,m

otor

inputs

)

GUIDANCEtake the robot from the current posture to the desired

posture, possibly following a pre-determined path ortrajectory, while avoiding obstacles

GUIDANCEGUIDANCEtake the robot from the current posture to the desired

posture, possibly following a pre-determined path ortrajectory, while avoiding obstacles


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Guidance Methodologies

State(posture)-feedback methods:

posture stabilization (initial and final postures given; no

path or trajectory pre-determined; obstacles notconsidered; may lead to large unexpected paths)

trajectory tracking (requires pre-planned path) virtual vehicle tracking (requires pre-plannedtrajectory)

Potential-Field like methods

potential fields (holonomic vehicles) generalized potential f ields (non-holonomic vehicles) modified potential fields (non-holonomic vehicles)

Vector Field Histogram (VHF) like methods

nearness diagram navigation (holonomic vehicles) freezone (non-holonomic vehicles)

Some Guidance methodologiesSome Guidance methodologies

NON-LINEAR CONTROL DESIGNFOR MOBILE ROBOTS


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Control of Mobile Robots

Three distintic problems: Trajectory Tracking or Posture Tracking

Path Following

Point Stabilization


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Trajectory Tracking

Vehicle of unicycle type

The trajectory tracking problem for a WMR of the unicycletype is usually formulated with the introduction of a

vir tual reference vehicle to be tracked.

=

==

!

!

!

sinvy

cosvx

),y,x(z =

v

position and orientation withrespect to a fixed frame

linear velocity

angular velocity

kinematic model

Is a simplified model, but Captures the nonholonomy property which

characterizes most WMR and is the core ofthe difficulties involved in the control of thesevehicles

reference vehicle

ref

x

y

refy

refx

{ }B


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Trajectory Tracking

refref

refrefref

refrefref

sinvy

cosvx

=

=

=

!

!

!

Kinematic model of the Reference Vehicle

)t(vref )t(ref Bounded Bounded derivatives Do not tend to zero as t tends to infinity

),y,x(z refrefrefref =

Control Objective

Drive the errors refrefref -,yy,xx to zero

Express the errors in the {B} frame

=

=

ref

ref

ref

3

2

1

yy

xx

100

0cossin

0sincos

e

e

e

e

DifferentiatingIntroducing the change of inputs

=+=

ref2

3ref1

u

ecosvvu


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Trajectory Tracking

+

+

=2

1

ref3u

u

10

00

01

v

0

esin

0

e

100

00

00

e!

non-linear dynamic systemcontrol variables

Question? Is it possible to design a feedback law u=f(e) such that the error

converges to zero? Is this law linear or non-linear ?

Two different solutions:

Linear feedback control

Nonlinear feedback control


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Trajectory Tracking

LINEAR FEEDBACK CONTROL

+

+

=2

1

ref3u

u

10

00

01

v

0

esin

0

e

100

00

00

e!

0u

0e

==linearize about the

equilibrium point

+

=2

1

ref

ref

u

u

10

00

01

e

100

00)t(

0)t(0

e!

linear time-varying dynamic system

refref

refref

v)t(v

)t(

==

Assuming

linear time invariant dynamic system

Is it possible to design a linear feedback law u=f(e) such thatthe error converges to zero?

Is the dynamic system controllable ?


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Trajectory Tracking

=

000010

00v00v0001

refref

refref

2

ref

c

0v refref ==If the SLIT is non-controllable

the reference robot at rest

the error cannot be taken tozero in finite time

otherwise

Keu=

=

3

2

1

232221

131211

2

1

e

e

e

KKK

KKK

u

u

ijK Chosen by pole placement

closed loop poles 0)aas2s)(a2s( 22 =+++

undetermined

system

ku1 =

332ref22

111

eke)vsgn(ku

eku

==

a2k

|v|

ak

a2k

3

ref

2r

2

2

1

=

=

=

If vr tends to zero, k2 increases withouth boun


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Trajectory Tracking

closed loop poles 0)aas2s)(a2s( 22 =+++

332r22

111

eke)vsgn(ku

eku

==

2ref

2ref3

ref2

2ref

2ref1

bvw2k

|v|bk

bvw2k

+=

=

+=

To avoid the previous problem:

The closed-loop poles depend on the values of vrand wr

2ref

2ref bvwa +=


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Trajectory Tracking

NONLINEAR FEEDBACK CONTROL

+

+

=2

1

ref3u

u

10

00

01

v

0

esin

0

e

100

00

00

e!

3refref32

3

3ref42

1refref11

e)w,v(kee

esinvku

e)w,v(ku

=

=

k4 positive constant

k1 continuos function strictly positive in RxR-(0,0)k3 continuos function strictly positive in RxR-(0,0)

See the analogy with the linear contol

PropertyProperty: This control globally asymptoticallystabilizies the origin e=0

demonstration using Lyapunov stability theory

bk

bvw2)w,v(k)w,v(k

4

2ref

2refrefref3refref1

=

+==


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Path Following

Objective:Objective:

Steer the vehicle at a constant forward speedalong a predefined geometric path that is givenin a time-free parametrization.

x

y{ }B

"

Path

d

Approach:Approach: The controller should compute

The distance of the vehicle to the path

The orientation error between the vehicles main axis andthe tangent to the path

The controller should act on the angular velocity

to drive both to zero


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Path Following

x

y{ }B

"

Path

d

M

M is the orthogonal projection of the robots position P on the pathM exists and is uniquely defined if the path satisfies some conditions

),,P( NT Serret-Frenet frame moving along the path. The vector is the tangent vector to the path in the

closest point to the vehicle The vector is the normal

N

N

T

T

is the distance between P and M

s is the signed curvilinear distance along the path, from some initial path tothe point M

d(s) is the angle betwen the vehicles x-axis and the tange to the path at thepoint M.

c(s) is the paths curvature at the point M, assumed uniformly bounded anddifferentiable

is the orientation error

"

d~ =


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Path Following

A new set of state coordinates for the mobile robot

The error dynamics can be derived writing thevehicle kinematic model in the Serret-Frenet frame:

)~,,s( "

They coincide with x,y, in the particular caswhere the path coincides with the x-axis

"

!

"!

"

!

)s(c1

)s(c~

cosvw

~

~sinv

)s(c1

~cosv

s

=

=

=

PROBLEM FORMULATION

Given a path in the x-y plane and the mobile robot translational velocity,v(t), (assumed to be bounded) together with its time-derivative dv(t)/dt,

the path following problem consists of finding a (smooth) feedbackcontrol law

such that

))t(v,~

,,s(k = "

0)t(~

lim

0)t(lim

t

t

=

=

"


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Path Following

LINEAR FEEDBACK CONTROL

"

!

"!

"

!

)s(c1

)s(c~

cosvw

~

~sinv

)s(c1

~cosv

s

=

=

=

u~

~sinv

)s(c1

~cosv

s

=

=

=

!

"!

"

!

")s(c1

)s(c~

cosvwu

=

Two different solutions:

Linear feedback control

Nonlinear feedback control

Linearize the dynamics around the equilibrium point )0~,0( =="

)t(u)t(~

)t(~

)t(v)t(

=

=!

"!

Linearization of thelast two equations


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Path Following

The linear system isCONTROLLABLE

ASYMPTOTICALY

STABILIZABLE BY LINEARSTATE FEEDBACK

Linear stabil izing feedback

= ~|v|kvku 32 " 0k,0k 32 >>

0vk|v|k 223 =++ ""!"!! Closed-loop differential equation

0kk 2'

3'' =++ """

0aas2s 22 =++

=

=

t

0

d|v|

' "

"transformation

Distance gone by point M along the p

Desired closed-loopcharaceristic equation

If

v(t)=v=cte 0


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Path Following

NONLINEAR FEEDBACK CONTROL

u~

~sinv

)s(c1

~cosv

s

=

=

=

!

"!

"

!

Nonlinear control law

= ~)v(k~

~sin

vku 2 "

with

0k2>

).(k continuous function strictly positive

In order to have the two (linear and nonlinear) controllersbehave similarly near chose0~,0 =="

|v|k)v(k 3=

with

22

3

ak

a2k

=

=


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Path Following

Property

Under the assumption

0)t(vlimt

the non-linear control

=~

)v(k~

~sin

vku 2 "asymptotically stabilizes )0

~,0( =="

provided that the vehicles initial configuration is such that

2

2

2

2

)s(csup(lim

1)0(

~

k

1)0(


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Point Stabilization

Given

An arbitrary posture

Find

A control law

which stabilizes asymptotically z-zd about zero,whatever the initial robots posture z(0)

)t,zz(kv

d=

),y,x(zd =

COROLLARY

There is no smooth control law k(z) that can solve thepoint stabilization problem for the considered class ofsystems.

3 ALTERNATIVES

Smooth (differentiable) time-varying nonlinear feedback k(z,t) Piecewise continuous control laws k(z) Time-varying piecewise continuous control laws k(z,t)


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References

C. Canudas de Wit, H. Khennouf, C. Samson, O.Sordalen, Nonlinear Control Design for Mobile Robots inRecent Developments in Mobile Robots, World Scientific,1993. Reading assignment

Carlos Canudas de Wit, Bruno Siciliano and Georges

Bastin (Eds), "Theory of Robot Control", 1996.

Documents

Guidance non-linear.pdf