seminar at monash university, sunway campus, 14 dec 2009 mobile robot navigation – some issues in...
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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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
Outlines
1. Introduction2. Target tracking control schemes
based on Lyapunov method Potential field method
3. Speed control 4. Conclusion
Seminar at Monash University, Sunway Campus, 14 Dec 2009
Introduction
A wheeled mobile robot (WMR) can be driven by wheels in various formations:
Differential Omni Directional Steering
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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) ?
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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):
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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)
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
Trajectory tracking (Cartesian coordinates based)
Given dddd yxyx and, ,
andvfind
to make dd yyxx ,
Seminar at Monash University, Sunway Campus, 14 Dec 2009
)](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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
)](sin)([cos)cos( 1 yyxxkvv dddd
))(,()](cos)([sin)sin(
32
dddddd
ddd vkyyxxvk
bk 2
with nonlinear modifications to adjust angular motion:
where,
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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 ,
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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(
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
• 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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
Experiments
Robot trajectory under Controller 1
Robot trajectory under Controller 2
Seminar at Monash University, Sunway Campus, 14 Dec 2009
Under Controller 1:
Under Controller 2:
Tracking errors
Tracking errors
Velocities
Velocities
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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).
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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.
Seminar at Monash University, Sunway Campus, 14 Dec 2009
• 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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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.
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
Speed Angle
Solid line: targetDashed line : robot under the proposed controllerDotted line :robot under the conventional potential field controller
Seminar at Monash University, Sunway Campus, 14 Dec 2009
Performance of the conventional field method with a high gain
Trajectories Speed
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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.
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
(sec)t
(sec)t
rad/slr
Simulation results
Nml
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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.
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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
Seminar at Monash University, Sunway Campus, 14 Dec 2009
• 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.
Seminar at Monash University, Sunway Campus, 14 Dec 2009
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