pp robot.pdf
(
499 KB
)
Pobierz
Adaptive pole placement for a mimo quasi-linear parameter varying mobile robot - Control, Automatiom, Robotics and Vision, 2002. ICARCV 2002. 7th International Conference on
Seventh International Conference
on
Control, Automation,
Robotics
And
Vision (lCARCV’O2),Dec
2002,
Singapore
Adaptive Pole Placement for a MIMO Quasi-Linear
Parameter Varying Mobile Robot
Antonios Tsourdos, John
T.
Economou,
Brian
A. White
Department of Aerospace, Power and Sensors
Royal Military College of Science
Cranfield University
Shrivenham,SN6
8LA
England,UK
Email:
A.Tsourdos@rmcs.cranfield.ac.uk
,
J.T.Economou@rmcs.cradeld.ac.uk,
B.A.
White@rmcs.cranfield.ac.uk
Abstract
In this paper, an indirect adaptive control scheme
IS
proposed for
a
multiple-input multiple-output differen-
tially steered mobile robot. The resulted closed-loop
system has an effective control for both the vehicle
longitudinal velocity and the vehicle yaw rate. The
closed-loop system stabilityis examined via
LMI
quad-
tratic stability test. The proposed scheme shows effec-
tiveness for
a
variaty of realistic simulation scenarios.
become adaptive and quasi-linear, to provide perfor-
mance over
a
greater range, in the face of changing
operating conditions.
The tracking performance of
a
mobile robot is
also
dependent on the location within the steering envelope
and varies with factors such
as
yaw rate and acceler-
ation. Several approaches, including adaptive control
[l],
[2],
nonlinear control
[3],
and gain scheduling [4]
have been used to alleviate these tracking problems.
One
of
the most popular methods for applying
linear time-invariant
(LTI)
control theory to time-
varying and/or quasi-linear systems is gain schedul-
ing
[5].
This strategy involves obtaining Taylor
lin-
earised models for the plant
at
fhitely many equilibria
(“set points”), designing an LTI control law (“point
design”) to satisfy local performance objectives for
each point, and then adjusting (“scheduling”)the con-
troller gains in
real
time
as
the operating conditions
vary.
This
approach
has
been applied successfullyfor
many years, particularly
for
aircraft and process con-
trol problems. Relatively recent examples (some of
which involve modern control design methods) include
active suspensions [6], high-speed drives
[7].
Despite past success of gain scheduling in practice,
until recently little has been known about it theoret-
ically
as
a
time-varying and/or quasi-linear control
technique. Also, determining the actual scheduling
routine is more of an
art
than
a
science. While
ad
hoc
approaches such
as
linear interpolation and curve
fit-
ting may be sufficient for simple static-gain controllers,
doing
the
same
for dynamic multivariable controllers
can be
a
rather tedious process.
An early theoretical investigation into the perfor-
mance
of
parameter-varying systems can be found in
[8].
During the
1980’s,
Rugh and
his
colleagues de-
veloped
an
analytical framework for gain scheduling
using extended linearisation
[5].
Also, Shamma and
Athans
[9]
introduced linear parameter-varying (LPV)
systems as a tool for quantifying such heuristic design
rules
as
“the resulting parameter must vary slowly”
and “the scheduling parameter must capture the non-
linearities of the plant”. Shahruz and Behtash
[lo]
suggested using
LPV
systems for synthesising gain-
scheduled controllers.
NOMENCLATURE
mobile robot longitudinal velocity
mobile robot yaw rate
motor armature voltage
common-mode artuator voltage
differential actuator volt
mobile robot mass
mobile robot Inertia
resistive moment of inertia
effective tyre force
Motor gearbox ratio
PMDC
motor torque constant
Motor armature resistance
Wheel radius
C.G. to left track
C.G. to right track
C.G
to front track
C.G
to rear track
1
Introduction
The performance of vehicle and robotics systems
is
hghly dependent on the capabilities of the guidance,
navigation and control systems.
To
achieve improved
performance
in
such robotic system systems, it is im-
portant that more sophisticated control systems be de-
veloped and implemented. In particular,
as
the perfor-
mance envelope is expanded, the control schemes must
Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:46 from IEEE Xplore. Restrictions apply.
Attention has since turned to performance and de-
sign of parameter-dependent controllers for
LPV
sys-
tems. Various design methods which have been pro-
posed share several common features, e.g., the cur-
rent methods are based on extended state-space ap-
proaches to
B,
optimal control for LTI systems
[12],
[U],
and LTV systems
[13].
Performance
is
usually
measured in terms of the induced Lz-norm, and con-
trollers are designed for certain classes of parameter
variations, e.g., real or complex values, arbitrarily
fast
or
bounded rates of variation, shape of the parame-
ter envelope etc. The resulting parameter-dependent
controllers are scheduled automatically,
so
that the of-
ten arduous task of scheduling
a
complex multivariable
controller
a
posteriori
is
avoided.
In this paper adaptive pole-placement control de-
sign technique
is
applied to the motion control for the
model. The mobile robot motion is modelled to be
quasi-linear with varying parameters. Based on the
quasi-linear model, we adopt for design procedure the
adaptive pole-placement method. In this scheme, un-
known parameters are estimated and based on these
estimates, control parameters are updated. Computer
simulations show that this approach is very promising
to apply the motion control design for mobile robots,
which are highly quasi-linear in dynamics.
2
Mobile Robot model
1
(4)
The quasi linear parameter varying model inputs
&e
transformed using the common-mode
6.2
and differ-
ential
6
-
demands
as
shown in 6
using
the transfor-
mation%
5.
Assume that the left and right vehicle side actuators
are operating
at
the same armature voltages respec-
tively. The derived QLPV model structure is shown
ri
i
3
Figure
1:
Vehicle wheels force diagram
The
given-in
7
where
x
is the state miable vector to be
determined in
terms
of
x
and the reference signal.
in
1.
U,
=
-K(p)Tx
(7)
It
should be noted that
K(p)
is determined recur-
sively but its structure and
in
particular the values
of the lognitudinal and lateral controllers
Kl(p)
and
Kz(p)
are obtained using the pole-placement tech-
nique.
Substituting the control law
in
the state equation
yields:
591
Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:46 from IEEE Xplore. Restrictions apply.
The state space now is described by:
x
=
A'x
(8)
with the augmented matrix
A'
to be given by
A'
=
[ ]
=
[
-rc-
-;
A(P)
0
] [
]
+
[ ]
[
:zi 3
+
[
f
]
[
r:
]
(
12)
Ab)
-
BK(p)*.
The characteristic equation of the augmented sys-
The compensated system therefore becomes:
tem can now be determined:
The coefficients
Ai,j
are parameter dependant.
4
Closed-Loop
Responses
The adaptive controller effectiveness is shown by
means
of
three reallistic scenarios.
Equating the above mathematical expression of the
characteristic polynomial of the augmented system
with the one of the desired (obtained using the de-
4.1
Scenario
1:
Rectilinear motion
els are easily obtained.
But this controller would result only to
a
desired
transient of
all-
local models by placing-the.poles
of
all the local systems within
a
specified area. However
since our
aim
is good tracking for the mobile robot
we should include to the design specification except
peak overshoot and settling time, zero steady-state er-
ror. Ths can be achieved with an integral term in the
forward path
as
it
is
shown
at
the block diagram
2.
In this case the recilinear system mode is excited and
therefore
thd
=
1.5ms-1
and
I'd
=
0.
Figure 3(a)
clearly shows the velocity convergence to the demand.
In
figure 4(b) the Jacobian system coefficients are
shown which are non-zero for the longitudinal model
J11
#
0,
Jzz
#
0.
However coeflicients
512
=
521
=
0
which clearly identify
a
decoupled sytem.
For
this
robotic vehicle the centre of gravity is positioned
so
that
wl
=
wr
and hence the yaw rate is zero. For cer-
tain applications in which the opposite holds
wl
#
wT
even
at
tbd
f
0,rd
=
0
demands the mobile robot
would tend to drift towards the direction that the cen-
tre of gravity is shifted.
,
... .
.
. . .
,
.. . .
so
.
.,
TUEINs
MH,
Figure
2:
Adaptive
MIMO
Pole Placement Structure
Figure 3: Longitudinal Velocity and Jacobian Matrix
coeflicients
The new augmented model would contain for this
system two more state variables to account
for
this
integral term.
These new state variables are defined
as:
4.2
Scenario
2:
Pivot Turn motion
Another simulation scenario is for
ud
=
O,rd/neqO
in
which the yaw rate mode is excited. Figure
4(a)
shows the yaw rate convergence for
a
step demand
of
Td
=
2.5~~-'.
In
figure 4(b) similarly to the
JXO-
bian coefficients of the rectilinear motion scenario the
system is decoupled due to the parameters properties.
xi
=
1:
edt
=
1;
([
::
]
-
[
:
I)
dt
(10)
Therefore,
(11)
592
Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:46 from IEEE Xplore. Restrictions apply.
-
sired performance characteristics) the coefficients of
the pole-placement controller for each
of
the local mod-
...
..
.., .....
/
.
..
...........
.
_.
..............................
.
.
*
.
.
.
.
.
.
...
-....
211:
....
=
................................
- -
:
'I
THINS
Figure 4: Yaw rate and Jacobian Matrix Coefficients
4.3
Scenario
3:
Combined motion
During
this
scenario both modes are instanteneously
excited
and
therefore the coupled system generates
non zero Jacobian elements
as
shown in figure 4(d).
The demanded inputs are
tbd
=
1.5ms-'
and
Td
=
0.5~~-1
s
8.
respectively. The resulting step responses con-
vergence are shown in figure 4(a)(b). The associated
state variable erros are shown in figure
4(c).
Figure
6:
Controller Gains Space and Combined
Mo-
tion Gain Adaptation
".I-,
...............................
the variations in the parameters of the system. In this
section we consider a system defined by
.................... .........
.........
............
.....................
.....
.....
.......
................
.
=
x
=
A(p)z
(14)
where the state matrix
Ab)
is a function of a real pa-
rameter vector
p
=
@I,.
..
,pk)
E
Rk.
Let
X
=
R*
be the state space of
this
system. One
can
think of
this (autonomous) system
as
a feedback interconnec-
tion of a plantsand a control system. We will analyze
the stability of the equilibrium point
x*
=
0 of this
system. More precisely, we analyze
to
what extent
the equilibrium point
x*
=
0
is asymptotically stable
when
p
varies in
a
prescribed set, say
P,
of uncertain
or
varying parameters.
There are two particular cases
of
this stability prob-
lem that are of special interest.
1.
the parameter vector
p
is a
fixed but
known
ele-
ment of a parameter set
P
C
Rk.
2. the parameter vector
p
is
a
time
wrying
p
:
R
+
Rk
which belongs to some set
P
of functions in
(Rk)R.
The differential equation
(14)
is
then to
be interpreted
as
$(t)
=
A(p(t))z(t).
The first case typically appears in models in which
the physical parameters
are
fixed but only approxi-
mately known up to some accuracy. Note that for
these parameters (14) defines a time-invariant system.
The second case involves time-varying parameters.For
this case one can in addition distinguish between the
situation where
P
consists of one element only (known
time varying perturbations) and the situation where
P
is a higher dimensional set of time functions (arbitrary
time varying perturbations). Stability against time-
varying parameters is generally
a
more demanding re-
quirement for the system than stability against time-
TIUEINs
Figure
5:
Longitudinal Velocity, Yaw Rate and
Error
Responses and Jacobian Matrix Coefficients
4.3.1
Adaptive
Gains
The adaptive gain space for the pole placement con-
troller
are
shown
in
figures
6(a)(b)(c) and (d).
For the
combined motion the actual adaptive
gain
trajectories
are also superimposed on the gain space for the
30s
test run.
5
Stability analysis
An important issue in the design of control systems
involves the question to what extent the stability and
performance of the controlled system is robust against
593
Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:46 from IEEE Xplore. Restrictions apply.
....
.,
invariant unknown parameter.This is because second
case is obviously
a
special case of first one.
Since
A
is
a
continuous function of parameter
p
E
P,
and
P
is compact, it is clear that the condition (16)
implies that the left hand side is uniformly negative
definite. That is, there exists a scalar
6
>
0,
such
that
for all
p
E
P,
AT@)P
+
PA@)
5
41n.
The different necessary and sufficient conditions for
quadratic stability are given in following theorem.
Theorem.[l5]Given a compact set
P
C
RS,
and
an integer
m
>
0,
then the following conditions are
equivalent:
1. function
A(p)
is quadratically stable over
P,
i.e.
there exists a matrix
P
E
S;xn,
such that for all
PEP
5.1
Quadratic Stability
A
sufficient condition for
x*
=
0 to be an asymptoti-
cally stable equilibrium point of
(14)
is the efistence
of a quadratic Lyapunov function
V(s)
=
ZTPZ
with
P
=
PT
>
0 such that
AT(p)P
+
PA@)
<
0
2.
for any
C
E
Co(R6,PXm),
along state trajectories
x
of
(14)
that originate in
a
neighbourhood of the equilibrium
x*
=
0. The
system
(14) is said to be quadratically stable for perturbations
P
if
there exists a matrix
P
=
PT
>
0
such that
A@(t))TP
+
PA(p(t))
<
0
for
all
perturbations
p
E
P.
If
the system
(14)
is
quadratically stable for perturbations
P
then
V(x)
=
xTPx
is a quadratic Lyapunov function for
(14)
for
all
p
E
P.
The existence of a quadratic Lyapunov
function implies that the equilibrium point
x*
=
0
is
asymptotically stable. Quadratic stability for pertur-
bations
A
is therefore equivalent to the existence
of
a
quadratic Lyapunov function
V(x)
=
xTPx,P
>
0
such
that
there exists
a
matrix
X
E
STxn,
such that for all
p
E
P
AT(p)X
+
XA(p)
+
XC(p)CT(p)X
<
0
there exists a matrix
Y
E
S:xn,
such that
for
all
p
E
P
AT(p)Y
+
YA@)
+
YB@)BT(p)Y
<
0
Definition
[15]:
For LPV systems (15), if
A@(t))
is quadratically stable over
P,
then
(15) is
a quadrat-
ically stable LPV system.
The test requires that A@(t))be modeled
as
a
con-
vex polytope of matrices, from the possible set of
d-
ues
of
A@(t)).
To
construct the polytopic model of
A@(t)),
we evaluate
A@@))
over the extreme values
of the parameters and their derivatives.
If
there are
total
1
parameters and their derivatives, where
1
2
k;
then
ACp(t))
may be considered
as
being contained
within a convex set with
2'
vertices.
for
all
p
E
P.
Note that in general quadratic stability of the sys-
tem
for
an
uncertainty class
P
places
an
infinite num-
ber of constraints on the symmetric matrix
P.
A(&))
E
4{Ai,Az,..-,Az}
The conservative assumption of the quadratic Lya-
punov stability test is that the system can change
in-
hitely fast. For example, the values of the matrix
A(p(t))
could instantly change from those of the ma-
trix
A1
to those of another matrix, say
Al.
As
im-
plemented by
[l6],
the test amounts to searching for
the matrix solution,
P,
to the following linear matrix
inequality over the vertices of the convex set
5.2
Quadratic Stability
forQLPV
Sys-
Recently developed stability tests may be employed to
assess the stability of the parameter varying systems
such
as
(15)
[14].
x=
ACp(t))z
+
B@(t))u
Y
=
C@(t))x
+aP(t)).
p(t)
E
P,W
2
0
(15)
The quadratic Lyapunov test is a general, although
conservative test applicable
to
the systems like the one
encountered in this paper.
DefinitiomGiven
a
compact set
P
C
RS,
and
a
function
A
E
C"(R",
Rnxn
2'
a<o
(17)
If a solution exists, the Lyapunov function
V(x(t),p(t))
demonstrates stability by satisfying
,,
the function
A
is quadrati-
cally stable over
P
if there exists
a
matrix
P
E
STxn,
such that for all
p
E
P
A~@)P+PA~)
(16)
594
Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:46 from IEEE Xplore. Restrictions apply.
3.
for any
B
E
CO(Rs,
Rmxm),
tems
ATP+PAi
<
2d
P
1
I
for
i=l--.
<o
Plik z chomika:
Oferus
Inne pliki z tego folderu:
web robot mechanism excample.pdf
(89 KB)
TRAJECTORY CONTROL OF ROBOT MNIPULATORS.pdf
(368 KB)
system identification and self tuning PP.pdf
(474 KB)
Stabilizing robots with uncertain parameters actuated by DC motors with flexible coupling shafts.pdf
(490 KB)
robust pole placement.pdf
(786 KB)
Inne foldery tego chomika:
Akwizycje
DSP
Ekspertowe
Inne
KNEKSA
Zgłoś jeśli
naruszono regulamin