NON-LINEAR CONTROL OF WHEELED MOBILE ROBOTS

12002 - © Pedro Lima, M. Isabel RibeiroRobótica Móvel

NONNON--LINEAR CONTROL OF LINEAR CONTROL OF WHEELED MOBILE ROBOTSWHEELED MOBILE ROBOTS

Maria Isabel RibeiroPedro Lima

[email protected] [email protected] Superior Técnico (IST)

Instituto de Sistemas e Robótica (ISR)Av.Rovisco Pais, 11049-001 Lisboa

PORTUGAL

May.2002

all rights reserved


Course Outline

tarefasPLANEAMENTO DE

TAREFAS

PLANEAMENTO DEMOVIMENTO

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

MAPA DOMUNDO

FUSÃOSENSORIAL

CONTROLO

CONDUÇÃO

ACTUADORESSENSORES

VEÍCULO

informação processada a partir dos dados dos sensores

actualização do mapa

mapa global

Informação enviada aos actuadores

Detecção de obstáculos Posição e/ou

velocidades desejadas

(seguir trajectória /evitar obstáculos)

Motores, rodas, hélices, superf´ciies de deflecção

mapa local

Trajectória desejada (pos+vel+ace)

Características do meio envolvente

Trajectória estimada (pos+vel+ace)

Sonar, laser, visão, encoders, giroscópios


Guidance

sensor measurements

operation point

posture estimate

joint state feedback

Path Planner Guidance Joint

Controller Vehicle

Localization

obstacles

target path or trajectory

joint set points

(e.g., wheel velocities)

joint torques

(e.g., motor inputs)

GUIDANCE–take the robot from the current posture to the desired posture, possibly following a pre-determined path or trajectory, while avoiding obstacles

GUIDANCEGUIDANCE–take the robot from the current posture to the desired posture, possibly following a pre-determined path or trajectory, while avoiding obstacles


Guidance Methodologies

• State(posture)-feedback methods:• posture stabilization (initial and final postures given; no path or trajectory pre-determined; obstacles not considered; may lead to large unexpected paths)• trajectory tracking (requires pre-planned path)• virtual vehicle tracking (requires pre-planned trajectory)

• Potential-Field like methods• potential fields (holonomic vehicles)• generalized potential fields (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 DESIGN FOR MOBILE ROBOTS


Control of Mobile Robots

• Three distintic problems:– Trajectory Tracking or Posture Tracking– Path Following– Point Stabilization


Trajectory Tracking

• Vehicle of unicycle type

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

virtual reference vehicle to be tracked.

ω=θθ=θ=

!

!

!

sinvycosvx

),y,x(z θ=

v

ω

position and orientation with respect 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 of the difficulties involved in the control of these vehicles

reference vehicle

θ

refθ

x

y

refy

refx

{ }B


Trajectory Tracking

refref

refrefref

refrefref

sinvycosvx

ω=θ

θ=θ=

!

!

!

Kinematic model of the Reference Vehicle

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

),y,x(z refrefrefref θ=

Control Objective

Drive the errors refrefref - ,yy ,xx θθ−− to zero

Express the errors in the {B} frame

θ−θ

−

−

θθ−

θθ

=

=

ref

ref

ref

3

2

1

yy

xx

100

0cossin

0sincos

e

e

e

e

DifferentiatingIntroducing the change of inputs

ω−ω=+−=

ref2

3ref1

uecosvvu


Trajectory Tracking

+

+

ω−

ω

=2

1ref3 u

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


Trajectory Tracking

LINEAR FEEDBACK CONTROL

+

+

ω−

ω

=2

1ref3 u

u

10

00

01

v

0

esin

0

e

100

00

00

e!

0u0e

==linearize about the

equilibrium point

+

ω−

ω

=2

1ref

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(=

ω=ωAssuming

linear time invariant dynamic system

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

Is the dynamic system controllable ?


Trajectory Tracking

ω−

ωω−

=Γ

000010

00v00

v0001

refref

refref2ref

c

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

the reference robot at rest

the error cannot be taken to zero in finite time

otherwiseKeu =

=

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 =+ξ+ξ+

undetermined system

ku1 −=

332ref22

111

eke)vsgn(kueku

−−=−=

a2k|v|

ak

a2k

3

ref

2r

2

2

1

ξ=

ω−=

ξ=

If vr tends to zero, k2 increases withouth bound


Trajectory Tracking

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

332r22

111

eke)vsgn(kueku

−−=−=

2ref

2ref3

ref2

2ref

2ref1

bvw2k

|v|bk

bvw2k

+ξ=

=

+ξ=

To avoid the previous problem:The closed-loop poles depend on the values of vr and wr

2ref

2ref bvwa +=


Trajectory Tracking

NONLINEAR FEEDBACK CONTROL

+

+

ω−

ω

=2

1ref3 u

u

10

00

01

v

0

esin

0

e

100

00

00

e!

3refref323

3ref42

1refref11

e)w,v(kee

esinvku

e)w,v(ku

−−=

−=

k4 positive constantk1 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 asymptotically stabilizies the origin e=0

demonstration using Lyapunov stability theory

bkbvw2)w,v(k)w,v(k

4

2ref

2refrefref3refref1

=+ξ==


Path Following

•• Objective:Objective:– Steer the vehicle at a constant forward speed

along a predefined geometric path that is given in a time-free parametrization.

θ

x

y { }B

"

Path

dθ

•• Approach:Approach:– The controller should compute

– The distance of the vehicle to the path– The orientation error between the vehicle’s main axis and

the tangent to the path

– The controller should act on the angular velocity to drive both to zero


Path Following

θ

x

y { }B

"

Path

dθM

• M is the orthogonal projection of the robot’s position P on the path– M exists and is uniquely defined if the path satisfies some conditions

),,P( NT ηη • 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

Nη

Nη

Tη

Tη

• is the distance between P and M• s is the signed curvilinear distance along the path, from some initial path to

the point M• θθθθd(s) is the angle betwen the vehicle’s x-axis and the tange to the path at the

point M. • c(s) is the path’s curvature at the point M, assumed uniformly bounded and

differentiable• is the orientation error

"

d~ θ−θ=θ


Path Following

• A new set of state coordinates for the mobile robot

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

)~,,s( θ"

They coincide with x,y,θ in the particular case where the path coincides with the x-axis

"

!

"!

"!

)s(c1)s(c~cosvw~

~sinv)s(c1

~cosvs

−θ−=θ

θ=

−θ=

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) feedback control law

such that

))t(v,~,,s(k θ=ω "

0)t(~lim

0)t(lim

t

t

=θ

=

∞→

∞→"


Path Following

LINEAR FEEDBACK CONTROL

"

!

"!

"!

)s(c1)s(c~cosvw~

~sinv)s(c1

~cosvs

−θ−=θ

θ=

−θ=

u~

~sinv)s(c1

~cosvs

=θ

θ=

−θ=

!

"!

"!

")s(c1)s(c~cosvwu

−θ−=

Two different solutions:

Linear feedback control

Nonlinear feedback control

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

)t(u)t(~)t(~)t(v)t(

=θ

θ=!

"!Linearization of the last two equations


Path Following

≠ The linear system is

CONTROLLABLE

ASYMPTOTICALY STABILIZABLE BY LINEAR

STATE FEEDBACK

Linear stabilizing feedback

θ−−= ~|v|kvku 32 " 0k ,0k 32 >>

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

0kk 2'

3'' =++ """

0aas2s 22 =+ξ+

∫ τ=γ

γ∂∂=

t

0

d|v|

' ""transformation

Distance gone by point M along the path

Desired closed-loop characeristic equation

Ifv(t)=v=cte 0


Path Following

NONLINEAR FEEDBACK CONTROL

u~

~sinv)s(c1

~cosvs

=θ

θ=

−θ=

!

"!

"!

Nonlinear control law

θ−θ

θ−= ~)v(k~~sinvku 2 "

with0k2 >

) . (k continuous function strictly positive

In order to have the two (linear and nonlinear) controllers behave similarly near chose 0~ ,0 =θ="

|v|k)v(k 3=

with2

2

3

aka2k

=

ξ=


Path Following

• Property

Under the assumption0)t(vlim

t≠

∞→the non-linear control

θ−θ

θ−= ~)v(k~~sinvku 2 "

asymptotically stabilizes )0~ ,0( =θ="

provided that the vehicle’s initial configuration is such that

22

2

2

)s(csup( lim1)0(~

k1)0( <θ+"

Condition to guarantee that 1-c(s)l remains positive

The vehicle’s location along the path is characterized by the value of s (the distance gone along the path)

Depends on v(t)

This degree of freedom can be used to stabilize s about a prescribed value sd


Point Stabilization

• Given– An arbitrary posture

• Find– A control law

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

)t,zz(kv

d−=

ω

),y,x(zd θ=

COROLLARY

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

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)


References

• C. Canudas de Wit, H. Khennouf, C. Samson, O. Sordalen, “Nonlinear Control Design for Mobile Robots” in Recent Developments in Mobile Robots, World Scientific, 1993.

– Reading assignment

• Carlos Canudas de Wit, Bruno Siciliano and GeorgesBastin (Eds), "Theory of Robot Control", 1996.

Documents

NON-LINEAR CONTROL OF WHEELED MOBILE ROBOTS