Seminar at Monash University, Sunway Campus, 14 Dec 2009 Mobile Robot Navigation – some issues in...

Seminar at Monash University, Sunway Campus, 14 Dec 2009

Mobile Robot Navigation – some issues in controller design and

implementation

L. Huang

School of Engineering and Advanced Technology Massey University


Outlines

1. Introduction2. Target tracking control schemes

based on Lyapunov method Potential field method

3. Speed control 4. Conclusion


Introduction

A wheeled mobile robot (WMR) can be driven by wheels in various formations:

Differential Omni Directional Steering


Differential Wheel Robot

Omni Wheel Robot


A

B

Two basic issues:

1. How to move a robot from posture A to posture B stand alone ?

2. How to determine postures A and B for a robot when a group of robots performing a task (such as soccer playing) ?


StrategyPlanning

PathGeneration

TrajectoryGeneration

MotionControl

SpeedControl

,A B ,i ix y ( ), ( )x t y t ( ), ( )d dv t ω t ( ), ( )v t ω t

Line Line+Time Desired Velocity Actual VelocityPoint


Robot’s posture (Cartesian coordinates) cannot be stablized by time-invariant feedback control or smooth state feedback control (Brockett R. W. etc.).

Stabilization problem was solved by discontinuous or time varying control in Cartesian space (Campion G. B., Samson C. etc.)

Asymptotic stabilization through smooth state feedback was achieved by Lyapunov design in Polar coordinates – the system is singular in origin, thus avoids the Brockett’s condition (Aicardi M. etc ).

Trajectory tracking control is easier to achieve and is more significant in practice (desired velocity nonzero) (Caudaus De Wit, De Luca A etc.).

Differential wheel driven robot (no-holonomic):


It is fully linearisable for the controller design (D’Andrea-Novel etc.)

Dynamic optimal control was implemented ( Kalmar-Nagy etc.)

Robot modeled as a point-mass

Application of Lyapunov-based and potential field based methods in the development of target tracking control

scheme

Issues to be addressed

Omni-wheel driven robot

Potential field method was used for robot path planning (Y.Koren and J. Borenstein)


General control approaches

0cossin yx Differential Wheel

Robot

• Nonholonomic Constraint (rolling contact without slipping)

vy

x

1

0

0

0

sin

cos

• Kinematic Model

Nonhonolonic (No-integrable) and under actuated (2-input~3-output)cannot be stabilized by time-invariant or smooth feedback control


Trajectory tracking (Cartesian coordinates based)

Given dddd yxyx and, ,

andvfind

to make dd yyxx ,


)](sin)([cos)cos( 1 yyxxkvv dddd

)()](cos)()[sinsgn( 32 ddddd kyyxxvk

Desired angular velocity

Desired direction

Desired linear velocity (along the trajectory)

Note: 0dv

22ddd yxv

22dd

ddddd yx

yxxy

The trajectory needed to be specified in prior; the controller fails when

It can be proved (due to Lyapunov and Barbalat), the following control can meet the objective :

From

the p

lan

ned

traje

ctory

kxyATAN ddd ),(2

ddd vbkbvkk 222

31 ,2


)](sin)([cos)cos( 1 yyxxkvv dddd

))(,()](cos)([sin)sin(

32

dddddd

ddd vkyyxxvk

bk 2

with nonlinear modifications to adjust angular motion:

where,


Control task: move the robot from its original posture: to the target posture

( , , )p p px y θ ( , , )g g gx y θ

Goal / target tracking (Polar coordinates based)

). :0( arkingparallel pg


22 )()( yyxx gg

The system model described in polar coordinates:

)(tan 1

xx

yy

g

g

The model is singular at 0

sin

,sin

,cos vvv


cos1kv 12 2 2 3

2

sin 2= + ( + )

2

k γω k a γ a γ a δ

a γLet

0,0,0

It can be proved that ( due to Lyapunov and Barbalat)

0>a ,a ,a ,a2

1a

2

1

2

1321

23

22

21 aV

0cos 2222

2211 akkaV

with the Lyapunov function candidate

• large control effort or fluctuation when the angle tracking error is near zero or the linear tracking error is big

• the target is assumed to be stationary


Potential field approach (point mass model)

Attractive and repulsive fields:

Robot move along the negative gradientof the combined field:

• The law only specifies the direction of the robot velocity• target is assumed to be stationary• local minima

else 0

if,)(2

12

1

021

01

2

1

rep

rtTrtatt

U

ppU

else ,

if ,)(

)()(

1

021

01

21

rt

prt

reppattp

p

p

pUpUv


Lyapunov based target tracking controller with limited control efforts

System model (extended from the conventional one by including the velocity of the target):

0,sinsin

sinsin

coscos

tt

t

t

vv

vv

vv

t ,


Controller 1: Extension of the general control approach

Note: • target motions directly affects the control efforts• sinusoidal functions of the systems states attenuate the magnitude of control • tracking errors appear as the denominators in the terms of the

controller• linear tracking and angular tracking errors are treated equally – too demanding ?

It can be proved with Lyapunov method, that under the controller,

(Lyapunov function candidate: )

)(2

2sin)

sincos

2

2sin(

cos)cos(

vtt

vt

v

vv

0 and0,0

)(2

1 222 V


Controller 2: Improvement from Controller 1

Prioritise and change the control objectives:

and reflect them in the definition of the Lyapunov function:

New controller:

which can also achieve the convergence of the tracking errors, but withless control efforts

bounded)(or 0 bounded),(or 0,0

222 )(2

1

2

1

2

1 V

tvt

vt

v

vv

2)

2

2sin)sincos

2

2sin((

2

cos)cos(


Comparison: control efforts of Controllers 1 and Controller 2

)(2

2sin)

sincos

2

2sin(1

vttv

or

tvtv

2

)2

2sin)sincos

2

2sin((

22

22

11

k

k

kk

v

t

vt

2

1,,

2

2sin)sincos

2

2sin(

21


• By observation, the magnitude of controller 2 is less than that controller 1

• Analysing the factors ( ) affecting the controller magnitude, it is obvious that, except for the region near that affecting Controller 1 is larger in magnitude than that affecting Controller 2.

____1

1

1

k

2

k


Simulation Results (tracking a target Moving along a circle)

Linear tracking Angular tracking

2.1),08.0sin(1547),08.0cos(153 ttt vtytx

15.0,075.0 v


Linear velocity Angular velocity


Experiments

Robot trajectory under Controller 1

Robot trajectory under Controller 2


Under Controller 1:

Under Controller 2:

Tracking errors

Tracking errors

Velocities

Velocities


Demonstrations

Controller 1 Controller 2


Conclusions:

It is feasible to reduce the control efforts through prioritization of control objectives defining of Lyapunov function to reflect that priority attenuation of controller outputs with some special functions

of the system states (like sinusoidal functions etc.) while achieving the same or better control results in

comparison with the conventional controllers The performance of the controller is affected by the

noises of the sensors for state feedback (esp. velocity).


Potential field based control approach for robot’s target tracking

System model:

Potential fields:

sinsin

coscos

][

vvy

vvx

yxp

tartartar

tartarrt

Trtrtrt

else 0

if,)(2

12

1

021

01

2

1

rep

rtTrtatt

repatt

U

ppU

UUU


Case 1: Moving target free of obstaclesMinimization of the angle between the gradient of the field and the

direction of robot motion relative to the target.


• Direction

2),

)sin(arcsin(

v

v tartar

0))((

ofon Minimisati

3

rtrt

rtrt

rt

rt

attrt

rt

attrt

att

rt

rtattpatt

xy

yx

xy

Uy

x

Up

U

p

pUU

rt

Robot direction is adjusted around the directional line pointing to the target


It leads to:

0)))(sin()cos((

)cos)cos((

2

1222

1

11

tartartartarrt

tartarrtrtTrtatt

vvvp

vvpppU

2

122

112 ))cos(2( rttarrttartar ppvvv

0)0( 12 tattatt eUU

0rtp

Intuitively

• Speed

It is chosen to decrease , or

)sin( tartarvv

attU

One of the choices is:

The speed determined by the relative linear distance, the target velocity and there directional relationship.


The robot does not need to be always faster than the target (e.g. . when )

Comparison of the robot and target speeds:

tar

rttar

tar v

p

v

v ,))cos(21( 2

122

11

)( tar

1.01

4.01

8.01

2)(

tar


The approach can be extended to solve the path/speed planning of the robot surrounded by multiple obstacles.

Case 2: Moving target with moving obstacles

)(,

coscos

sinsinarctan

)(sin))cos()cos(

)sin(arcsin

10

1122

1

1

1

1

2221

iroiii

rt

roiii

n

iroii

n

iroii

n

itartarrtroiobsiobsiitartar

tartar

pp

p

vpvvv

v

v


Simulation Results:

1,0.1,

cos0.2,sin0.3

1 tartar

tt

vt

tytx

Trajectories Relative Distance

Solid line: targetDashed line : robot under the proposed controllerDotted line :robot under the conventional potential field controller


Speed Angle

Solid line: targetDashed line : robot under the proposed controllerDotted line :robot under the conventional potential field controller


Performance of the conventional field method with a high gain

Trajectories Speed


Conclusion:• the speed as well as the direction of the robot motion are determined with potential field method• the velocity of the moving target is taken into consideration• the proposed approach maintains or improves tracking accuracy and reduce control efforts, in comparison to the traditional approaches• further study on the determination of the optimum speed of the robot can be done by specifying additional performance requirements.


Speed control considering dynamic coupling between the actuators

• Synchronisation of the wheels’ motion affects the robot’s trajectory• Coupling between the actuators needs to be considered


Dynamic model:

Model based adaptive control

)(CM

w

w

IImbb

rImb

b

r

Imbb

rIImb

b

r

M)(

4)(

4

)(4

)(4

22

22

2

2

22

22

2

2

0

0)(

rl

lrC

wc

c mmmb

drm2,

4 2

3

mcwc IIbmdmI 22 22

Tlr ][

parameters geometric theare,,

wheels theandrobot theof parameters inertia theare ,,,

rdb

IImm cmwc


Introducing new variables

Dynamic model is transformed to a more compact form :

lrlr

lrlr

TT

21

21

2121

,

,

,

CM 2

then

11

11

2

1,, TTT

01

10,

0

0

2

11 CMTTM

ww Ib

IrI

mr

2

2

2

2

1 2,

2


Based on the transformed dynamic model, the adaptive speed controllers are derived:

Modified to reduce the amplitudes of the control outputs:

222111

21122222111

22212121

22212121

,

)(ˆ,ˆ,ˆ

ˆ)ˆˆ(2

1)(

2)(

2

ˆ)ˆˆ(2

1)(

2)(

2

dd

dddd

rddldrrdlldl

lddldlldrrdr

ee

ee

kkkk

kkkk

)(

)(ˆ

ˆ)ˆˆ(2

1)(

2)(

2

ˆ)ˆˆ(2

1)(

2)(

2

21122

2112

2212121

2212121

eek

eek

kkkkk

kkkkk

k

rdldrrdlldl

ldldlldrrdr


(sec)t

(sec)t

rad/slr

Simulation results

Nml


r

+ +

+

-

-

+

rd

ld +

ru

lu l

r

-

+

- +

)(sK l )(sGl

)(sK r )(sGr

+

+ -

)(sK a

l

Model free PID control

A loop for the coupling of the wheels’ speeds is added.


When

)()()()()(

)()()()()(

ssGssGs

ssGssGs

lsynrdindr

rsynldindl

,)()())(2)()(()(

)()(

,)()())(2)()(()(

))(2)()(()()(

sKsKsKsKsKsG

sKSG

sKsKsKsKsKsG

sKsKsKsGSG

aa

sind

aa

sind

Transfer functions :

)()()(),()()( sKsKsKsGsGsG rlrl

• First order motor model is adopted: • PID controller is used for the speed control• Implemented with one PIC18F252 microcontroller

2/,1

)( tamm

m KJRs

ksG


Modeling (Kinematics)

b

r

rX

rY

rO

1

2 3

1v

2v

3v

v

12

3

Y

X

O

Omni Wheel Robot

Speed Control of an Omni-wheel robots


Inverse kinematic model:

)3,2,1( ivvbr iri

T

T

T

v

v

v

]3

sin3

[cos

]3

sin3

[cos

]01[

3

2

1

)3

sin3

cos(

)3

sin3

cos(

)(

13

12

11

ryrx

ryrx

rx

vvbr

vvbr

vbr

cossin

sincos

][

yxry

yxrx

Tryrxr

vvv

vvv

vvv


• Chooped fed motors with drivers to drive the wheels

• PID controller implemented with one one 80296 microcontrollers (three PWM outputs)

• Encoder resolution 512 ppr• Sampling time 1 ms• Control loop completed within 0.5ms

This is achieved through:

• codes written in an assembly language without using floating point libraries (too slow)

• fixed point notation and a look up table of whole numbers to represent a floating point number with reasonable accuracy

• only the simple operations like addition, substration, multiplication and bits-shifting are used.


Implementation


Demonstrations


Conclusion

Lyapunov and potential field based target tracking controllers, and speed controller for dynamically coupled wheels for mobile robots were presented

Both position and velocity of the target were considered in the target tracking

controller design Functions of the system states, especially those of the target, are are designed to moderate the magnitude or fluctuation of the control effort The states of the system were assumed to be available; sensor noises

affect the performance of the controller. To get a good system states estimation and prediction from the sensor

data is another big issue to be addressed together with the controller design (Kalman filtering, Bayesian method etc.) Further study can be undertaken on integrating open-loop optimal control, closed-loop control and system states estimation and prediction

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Seminar at Monash University, Sunway Campus, 14 Dec 2009 Mobile Robot Navigation – some issues in...

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