14th World Congress ofIFAC
NONTIME REFERENCED TRACKING CONTROL WITH APPLICAT...
E2cl11
Copyright cg 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
NONTIME REFERENCED TRACKING CONTROL WITH APPLICATION IN UNMANNED VEHICLE Wei Kang * Ning Xi **,1
* Mathematics Department, Naval Postgmduote School, Monterey, CA 93943_, USA ** Department of Electrical Engineering, Michigan State University, East Lansing, MI 488241226 USA J
Abstract: A new design method for nontiTIle based tracking controller is presented. The key step is the introduction of a motion reference variable based on the desired and measurable system output. It enables the construction of control systems V\l'ith integrated planning capabilitY1 in which planning becomes realtime closed~ loop process. The control system is capable of coping with unexpected events and
uncertainties. The new design method converts a controller designed by traditiona.l timebased approach to a new controller with respect to a nontime based action reference. The design method is exemplified by a unmanned vehicle tracking control problem. Copyright © 1999 IFAC
KeY"lords: Feedback control, Autonomous vehicles
1. INTRODUCTION
In the eventbased planning and .control scheme,
the function of the Action Reference block in The objective of tracking control is to ensure the output of a system to track a given reference input. A common feature of many path tracking feedback design is a state trajectory following approach. The reference input or desired path is usually described as a function of time. Therefore, most of systems are modeled as a timebased dynamic system. However, nontime based controllers are often preferred in applications of robotics. Developing nontime based controller
has attracted the attention of researchers in different fields. In (Xi, 1993a), the eventbased con
Figure 1, (b) is to compute the action reference p~rameter s online, based on sensory measure . ments. The planner generates the desired value
to the system, according to the online computed action reference parameter s. In other words, the
action plan is adjusted on line which enables the planner to handle unexpected events or uncertain
ties. For example, the eventbased planning and control scheme has been successfully applied to deal with unexpected obstacle in a robot motion (Xi, 1~93a.). If the robot is blocked by the obstacle, the action reference also stops evolving. As a result the desired input will be constant and the error will remain unchanged. In this case, if the conventional timebased motion plan were imple
troller design was first introduced. Since then, it has been successfully applied to robot motion control (Tarn, 1993), multirobot coordination (Xi, 93b)} force and impact control (Wu, 1995), robotic teleoperation (Tarn, 1995) and manufacturing au
mented, the system error would keep increasing as the planner generates the desired input according
tomation (Xi, 1997).
to the original fixed plan. As a result, the system \~lould
1
Rezsearch 'supported in part by NSF IRI 9796287 and
NSF IRl 9796300.
become unstable.
Nontime based controller has also been applied to mobile robot tracking controL DeSantis (1994)
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NONTIME REFERENCED TRACKING CONTROL WITH APPLICAT...
and Sampei (1991) provide controllers to track lines and circles~ In (Kaminer, 1998), tracking of
14th World Congress of IFAC
the desired trajectory
Xd(t)~
There are many de
a trimming trajectory based on the distance to
sign methods in the literature for tracking controL Since the design is based on the model driven by
the trajectory and the arc length of the trajectory
t, the controller is timebased. Ho,vever, in the
is derived and implemented for unmanned air vehicles.
control of autolloIIIOUS vehicles, the desired path can be easily described as a function of nontime
In this paper, we adopt a viev.~ point which is different from all the nontime based approaches mentioned above. We focus on the problem of finding controllers for the tracking of a desired path described as a function of nontime based action reference. Instead of developing new design algorithms based on the nontime based action reference, we use the existing design method for timebased controller. Then, the time variable in the feedback is substituted by a special transformation so called statetareference projection.
The resulting feedback becomes nontime based. The new approach has the following four 39vantages. First, the model of the system is a general nonlinear dynamical control system. The method is applicable to a wide range of tracking control problems. Second; a transformation is provided to transform a time dependent controller into a non
time based controller. It bridges the traditional state trajectory tracking with the nontime based design. The results enable us to design nontime based feedback using the existing, well known methods of feedback design in time domain such
as LQR and H 00 controI~ The third advantage of our design algorithm is the flexibility in choosing action references. The arc length of the desired path is often used as the motion reference. However, there are many curves for which the formula of computing arc length is complicated.
based action reference s. In this section, we prove the result which transforms a timebased feedback controller into a nontime based controller for a desired path driven by motion reference s.
A system is defined by the equation
(1) where u E lR m is the control input, and x E .IR7'J is
the state of the system. The desired path is usually
=
given in a output space. Suppose that y h(x) represents the output of the system. The desired path is given by a nontime based motion plan Y
action
ref~rence
is not necessarily the orthogonal
Yd (s), where s is
the motion reference4
The first step in the controller design is to obtain a corresponding timebased motion plan based on the given velocity.. If s is the nontime based motion reference, it requires that s is an strictly
increasing function of time t, Le. s
= vet).
How
to pick the function v(t) depends on the desired speed.. For example, if the problem require that the system is operated so that s is increasing at a constant speed Vc, then vet) ·vot.
=
The second step is to find a timebased feedback, u(x, t), to track the path Yd(V(t). There are many . well known design algorithms can be found in the literature of linear and nonlinear control theory. The feedback satisfies
Parameters other than arc length can be used as action references in the design to simplify the controller. The fourth advantage of our method is that the transformation from state space to the
=
lim (h(x(t))  yi( v(t))) == O.
t~oo
Furthermore, if the initial position is on the desired path, then the trajectory of the controlled
system follows the path. More specifically, there
projection. For many curves, orthogonal projec
exists an initial condition of the system
tion requires online numerical solution of minimization. Our method allows the designer to use
that· the trajectory satisfies h(x(t)) = Yd(t). Denote this path by Xd(S) or X'd(V(t).
any mapping satisfying a projection condition. A
good choice for the projection mapping can dramatically reduce the online computational time.
XQ
such
The third step is to find a suitable transformation,
s = 1"(x), from the state space to the reference
8 ..
The transformation satisfies
(2) 2. MOTION REFERENCE PROJECTION
METHOD The dynamics of mechanical or electrical systems is usually modeled by differential equations in which the free variable is the time variable t. A desired trajectory is often described as a function of time, denoted by X'd(t). X represents the state of
the system. In a typical tracking control problem,
For example, given any state 2:0, let Xd(SO) be the orthogonal projection from Xo to Xd(S). If we define l(xo) == So, then it satisfies (2). The transformation satisfying (2) is called a statetoreference projection. HO"lever, orthogonal projection is not the only way to define ,. In section 3, the projection to xaxis is adopted as " which is a simpler formula than orthogonal projection.
a controller can be designed so that the trajectory
The last step is to construct the nontime based
of the plan system x(t) asymptotically approaches
feedback. Let
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= U(X,
u(x)
VI
Theorem 2.1. There exists a neighborhood W(o) of the desired trajectory x d (s) such that the trajectory x(t) of (7) satisfies
(3)
(,(X)))
where U(X, t) is the feedback found in the second step, "Y(x) is a statetareference projection. The closedloop system is
x := f(x, u(x»).
limt+OCld( x(t), v 1 (,(x(t))) )
t
=
I d(x,t)
The trajectory of (5) with initial time := XQ is denoted by ~(t, to, xo). Similarly, a trajectory of (7) is denoted by ~(t,to,xo). By the definition of V(X,t), it is easy to check that the derivative of V(x, t) along ~(t, to, xo), a trajectory of (5), satisfies Proof.
to and initial state x(io)
A § neighborhood of Xd is a subset of JRn defined by W(c5) = {x E JRn I d(x, Xd) < £}. In this paper, we assume that the vector field F(x, u), the feedback u(x, t), and their derivatives are bounded. Assume that the derivatives of v(t) and vI (s) are bounded. The closed loop system with feedback fi( x, t) is
x ==
j(x, u(x, t))
d
dt V(x(t), t)
= u(x, v
1
(J'(x)))
:=
~d(x(t), t)~ Therefore,
8V(x,t)/(
ox
= d(x, t).
( t)) x, u x,
+
aV(x,t) at
(11)
The desired trajectory Xd( v(t)) is a solution to the equation (5). Therefore, Xd(V(t)) == f(Xd, U(Xd' t)). Since s i(Zd(S)), we have t = v 1(,(xa(v(t)). The derivative of this equation with respect to t yields
(5)
=
The nontime based feedback is
u(x)
= O.
provided x(O) is in W(&). This implies that the trajectory of the closedloop system with a timeinvariant feedback asymptotically approaches the desired path.
(4)
Given any state x and any time t the distance from x to the point Xd(t) in the desired path is denoted by d(x, t). So, d(x, t) Ilx  Xd(t)1l is a function from JRn X lR to lFl. The open set U(x a , r) consists of (x, t) E JRn x JFl such that d(x, t) is less than T, Le.
U(Xd:r):= {(x,t)EJR n xlR
14th World Congress ofIFAC
(6) (12)
The closedloop system with nontime based feedback is
x = f(x, u(x)).
Now, let's consider the closedloop system (7) with the feedback (6). The derivative of V(x, VI (")'(x))) in the direction of (7) is
(7)
Let ~(t, to, :co) be the trajectory of (5) with the initial time to and initial condition x(to) = xo. We assume that there exists a neighborhood U(Xd~ r) so that (xa, to) E U(Xd' r) implies that ~(t, to, xo) approaches Xd exponentially. Furthermore, the trajectory satisfies
V(x, ,v 1 (y(x))) =
~~ I(x, u(x, vIb(x»)))
(av
1 av 1 8, +m(x,v (/(x»)) + a; ox! 
)
1
aV at·
From (11), we have
mea1(tto)d(xo, to) ~ d(~(t, to, xo), t) (8) :5 1\1 e 0:2 (t  to) d ( x 0, to)
> 0, M > 0, 0'1 > 0 and 0'2 > o. The following function is used as our Lyapunov function. It is defined by for some m
00
V(xo,t o)
= Id(~(t,to,xo),t)dt.
(9)
By the definition of V(x, t), it is easy to check that V (x, t) > 0 for all x in a neighborhood of, Xd(t) and V(Xd(V(t)}t) O. Therefore, V(x,t) ' has a minimum value at Xd(t). It can be· proved that
=
to
V(x, t)
We assume that all derivatives of V(x, t) of order less than or equal to three are bounded in a neighborhood U(Xd' r). From (8), it is easy to show that
md(x, t) :::; V(x, t) :::; M d(x 7 t) for some m
>
0 and M
>
(IQ)
= (x 
Xcl(t))TQ(x, t)(x  Xd(t))
for some positive definite matrix Q(x, t). There8V fore, at (x, t) == P(x, t)(x  Xd(t)) for some row vector P(x, t). In a neighborhood U(Xd, r), P(x, t) is bounded. So,
(14)
O.
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NONTIME REFERENCED TRACKING CONTROL WITH APPLICAT...
in a neighborhood U(Xd' r) for some M 1
>
O.
Meanvllhile, (12) implies
8v 1 8
a; 8x1 f (x, u(x)) 
1= 0
if X is a point on xa(s). Therefore, OVl VI I' Ia; 8x_!(x,u(x))  1 <
1 2M
(15)
1
in aneighborhood vll(o). From (13), (14) and (15), we have
14th World Congress of IFAC
I is the length between the front and the rear wheels. In the following, we will first derive a timebased feedback for an arbitrary path given by y = f(x). The design does not depend on its motion reference. A motion reference, then is d~fined for a given path (a sine curve is adopted as our example), and the statetoreference projection will be found to convert the timebased feedback to a nontime based feedback. Simulations are carried out to show the stability of the tracking control.
\'ie defined. the following change of coordinates to
V(x"v1(y(x))) < _ d(x,V:(J(x))) ,
achieve linearization of the last three equations in
(1),
From (10)1
.
. 1 1 V(x,) v1(,(x))) <  2~M V(x" v (,(x)))
= Y,
Zl
in a neighborhoad of Xd. By Gronwall's inequality, V (x ~ ~ vI ("y (x) )) approaches zero exponentially along a trajectory 1>(t, to, xo) if ;Co is in a neighborhood of Xd. From (10), md(x, t) ~ V(x, t).
(}'l v
Z2
u1
= sin 8,
1
Z3
•
==
2
1 cos (J tan
=  [2 SIn 9(tan~) ~ {31
==
0'1
COS (J = I cos 2 rj;
( 2)
+ {3i U2
The last three equations in (1) ate transformed into
Therefore, d(x, v 1 (/(x))) approaches zero expo
nentially along a. trajectory ,p(t, to, xo) if Xo is in a neighborhood, W(6), of Xd(S).
Suppose that the desired path is the following function of time
x
ba\k can be obtained to stabilize the tracking
control. Furthermore all design methods for traditional timebased controller can still be used. This significantly simplifies the design procedure for a nontime based controller.
= xd(i), Y = Yd(t).
t
Let's first find a feedback for
==
Ut
Ut
Since
~t (x 
Xd)
cos 8  Xd, define Ul
= _l_(xd  (x  :td)).
(4)
CDS()
3. PATH TRACKING FOR A UNMANNED Under (4), the first equation of (1) becomes
VEHICLE
d dt (x
In this section, tracking control of a unmanned vehicle using state feedback is discussed~ Consider a kinematic model of the vehicle with front and rear wheels, as shown Figure 2. The rear wheels are aligned with the vehicle while the front wheels

Zd)
= (x 
Xd).
Its solution asymptotically approaches zero as t + feedback for U2 comes from (3). Define
00. ~he
are allowed to spin about the vertical axes. The constraints on the system arise by allowing the wheels to roll and spin, but not slip. Let '(x 7 y,
where
Ul
e3
,
=
Z3 
1 d
Yd
(5)
()
Ul
dt
Ul
Then,
(1)
(6)
corresponds to the forward velocity of
The right side of the last equation, in (6) satisfies
the rear lvheels of the car and
U2
corresponds
to the velocity 'of the steering wheel, the angle of the car body with respect to the horizontal is e, the steering angle with respect to the car body is
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NONTIME REFERENCED TRACKING CONTROL WITH APPLICAT...
The derivatives of Ul can be computed from (4). Notice that Ul contains the term 4>1 which depends on U2. Therefore, the expression(7) has the form (k2(X,
y, f),
rPl t) + j62 (x
Y, (),
t
rP, t)U2
(8)
To stabilize (6), we define v
== 02(X, y, 9,4>, t) + f32(X, y, f), cP, t)U2 (at el + a2 e 2 + a3 e 3)
+Ul
(9)
Then, (6) is equivalent to =Ule2, e2 = UleS ea = U1 (atel + a2 e 2 + a3 e g)
el
Select the values of eigenvalues of
ai 1
(10)
i = 1, 2, 3, so that all the
are on the left half plane. Assume that 1T
11"
2
2
 < O(t) < .
(11)
The solution of(lO) is e(t) == eo exp(A J~ul(7)dr). From (4), Ul(T)dT increases as Xd increases. Since A is stable, we know that e(t) approaches zero exponentially as Xd 'increases. Therefore, yYd(t) 7 0 as Xd increases. To find a formula for U2, we solve for 'U2 from (9) and the last equation in (2). The feedback is
J;
02 
U2
To
= j3
1 
0'1
Ih
Ut
+ f31 
f3. (alel 2
14th World Congress of IFAC
the curve y = sin x and the lines y == ax + b satisfy the condition. However, a circle does not satisfy (11). Feedbacks to track circles can be found in the literature (see~ for instance, (DeSantis, 1994)).
,(x, y) == x.
A timebased feedback is computed using the formula (13) to track the curve
+ a2 e 2 + a3 e 3)4
summarize~ define
(14)
x
= t,
Y = sin t.
For the reason of simplicity, the function s = v(t) is vet) 1~ This implies that the speed of x on
=
the desired curve is 1. However, it is not necessary to assume constant velocity. Our design method is applicable even if the speed is not a constant.
The feedback is calculated by using (13). The time variable t in the feedback is substituted by the statetareference projection ,(x, y) x to achieve a timeinvariant state feedback.
=
The feedback is Ut 'U2
In the simulation, the length of the vehicle is I == 1.
= cas _l_(xd 19
(x  Xd» = _ 02(X, y, (J,.p, t)  at (9, tP) f32(X, y, 0) 4>, t)  131 ({J, tjJ)
+Ul(alel
The coefficients in (13) are al = 5, (13)
+ a2 e 2 + a3 e 3).
Th~orem 3~ 1. Under the feedback (13), (x(t), y(t») of (1) exponentially approaches a desired path (Xd(t), Yd(t)) as Xd increases, provided that (11) is satisfied.
Remark. The graph of any differentiable function y == /(:1:) satisfies the condition (11). For instance,
aa
= 5.
a2
== 9 and
The initial position of the vehicles are (0,2) and (3, 1.8589). The simulation results are shown in Figure 3 and Figure 44 The dotted curves in Figure 3a and 4a are the paths of the vehicle in xyplane. Figure 3b and 4b shows the value YYd, the difference between the actual path of the vehicle and the desired path. Although the stablity of the system is local, the simulation sho\vs that the system is stable for large variations of the initial conditions x(O) and y(O) under the nontime based feedback. Actually, the initial condition x(O) can be anywhere.
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ISBN: 0 08 043248 4
14th World Congress ofIFAC
NONTIME REFERENCED TRACKING CONTROL WITH APPLICAT...
. 4. CONCLUSION A
ne~
BejczYf A.K~ (1993b) . EventBased Planning and Control for MuItiRobot Coordination. Proceedings of the 1993 IEEE International Conference on Robotics and Automation, Atlanta, Georgia, pp. 251258.
xl, N., Tarn, T.J. and
design method for nontime based track
ing c6ntroller has been developed. It can convert a controller designed by tra(litional timebased approach to a new nontime based controller. It significantly simplifies the design procedure. The method has been applied to design a tracking controller of a unmanned vehicle. The simulation results ha.ve clearly demonstrated the advantage of nontime' based controller. More importantly, the results have provided a efficient and systematic approach to design nontime based tracking
(a) traditional control
controller for a general nonlinear dynamic system.
It is an important step in the advancement of developing a theoretical foundation for sensorreferenced intelligent control.
REFERENCES DeSantis, R. M. (1994). Pathtracking for a tractortrailerlike robot. The International Journal of Robotics Research, Vol. 13, No. 6, 533544. Kaminer, I., Pascoal, A., Hallberg, E. and Silvestre, C. (1998). Trajectory tracking f9r autonomous vehicles: an integrated approach to guidance and controL J. of Guidance, Control and Dynamics, Vol. 21, pp. 2938. Murry~R. M. a.nd Sastry, S. S. (1993). Nonholonomic IDotion planning: steering using sinusoids. IEEE Trans. Automat. Contr., Vol. 38, pp. 700716. . Sampei, M., Tamura, T .• Itoh, T. and Nakamichi~
M. (1991). Path tracking control of trailerlike mobile robot. IEEE/RSJ International Workshop on Intelligent R.obots and Systems IROS'91, pp. 193198, Osaka, Japan. Tarn, T.J., Xi, N., Guo, C. and Bejczy, A.K. (1995). Fusion of Human and Machine intelligence for Telero~otic Systems..The Proceedings' of the 1995 IEEE International Conference on Robotics and Automation. Nagoya, Japan.
(b) nontime based control
Figure 1; Comparison of nontime based and time based control .
y
• x
Figure 2: The configuration of the car is determined
by the location of the back wheels (x ~ Y), and the
q;.
angles (j and
F~3b
flQure3a
..l' ....
2 \
2 .~.\
:
\.
4
\ ..
>0'"
1':\, lo.5.'~"''''''''''''''''''''''
1
.
0··· .. · .... .~ ..
_1l..~~~'
Tarn, T. J., Bejczy, A.K. and Xi~ N. (1993). Intelligent Planning and Control for Robot Arms. Proceedings of the IFAC 1993 World Congress;
'.
o
5
10
5
. t(aeoond~
lC(me&w)
Figure 3: Tracking a sine curve
Sydney, Australia, 1823. Wu, V .., Tarn, T.J. and Xi, N. (1995). Force and Transition Control with Environmental Uncertainties1 The Proceedings of the 1995 IEEE International Conference on Robotics and Automation Nagoya, Japan. Xi, N. and Tarn, T.J. (1997). Integrated Task
Figur84b
0 . 5 . . . .      ..... ..
: , :. ....,, \,
.........:
, .
l
Scheduling and Action Planning/Control for Robotic Systems Based on a MaxPlus Algebra. ModeL Proceedings of IEEE/RSJ lnternational Conference on Intelligent Robotics and Systems, Grenoble, France. Xi, N. (1993a). EventBased Planning and Control for Robotic Systems. Doctoral Dissertation, Washington University.
10
10
ti ll(meter)
Figure 4: Tracking a sine curve
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Copyright 1999 IFAC
ISBN: 0 08 043248 4