Non-time referenced tracking control with application in unmanned vehicle

Non-time referenced tracking control with application in unmanned vehicle

14th World Congress ofIFAC NON-TIME REFERENCED TRACKING CONTROL WITH APPLICAT... E-2c-l1-1 Copyright cg 1999 IFAC 14th Triennial World Congress, Be...

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14th World Congress ofIFAC

NON-TIME REFERENCED TRACKING CONTROL WITH APPLICAT...

E-2c-l1-1

Copyright cg 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China

NON-TIME 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 48824-1226 USA J

Abstract: A new design method for non-tiTIle 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 real-time 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 time-based approach to a new controller with respect to a non-time 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 event-based 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, non-time based controllers are often preferred in applications of robotics. Developing non-time based controller

has attracted the attention of researchers in different fields. In (Xi, 1993a), the event-based con-

Figure 1, (b) is to compute the action reference p~rameter s on-line, based on sensory measure- . ments. The planner generates the desired value

to the system, according to the on-line 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 event-based 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 time-based motion plan were imple-

troller design was first introduced. Since then, it has been successfully applied to robot motion control (Tarn, 1993), multi-robot 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.

Non-time based controller has also been applied to mobile robot tracking controL DeSantis (1994)

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and Sampei (1991) provide controllers to track lines and circles~ In (Kaminer, 1998), tracking of

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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 time-based. 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 non-time

In this paper, we adopt a viev.~ point which is different from all the non-time based approaches mentioned above. We focus on the problem of finding controllers for the tracking of a desired path described as a function of non-time based action reference. Instead of developing new design algorithms based on the non-time based action reference, we use the existing design method for time-based controller. Then, the time variable in the feedback is substituted by a special transformation so called state-ta-reference projection.

The resulting feedback becomes non-time 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 non-time based design. The results enable us to design non-time 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 choos-ing 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 time-based feedback controller into a non-time 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 non-time 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 time-based motion plan based on the given velocity.. If s is the non-time 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 time-based 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 on-line 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 on-line 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 state-toreference projection. HO"lever, orthogonal projection is not the only way to define ,. In section 3, the projection to x-axis 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 non-time based

of the plan system x(t) asymptotically approaches

feedback. Let

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= U(X,

u(x)

V-I

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 state-ta-reference projection. The closed-loop 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 v-I (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 non-time based feedback is

u(x)

= O.

provided x(O) is in W(&). This implies that the trajectory of the closed-loop 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

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(6) (12)

The closed-loop system with non-time based feedback is

x = f(x, u(x)).

Now, let's consider the closed-loop system (7) with the feedback (6). The derivative of V(x, V-I (")'(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, v-Ib(x»)))

(av-

1 av -1 8, +-m(x,v (/(x»)) + ---a;- ox! -

)

1

aV at·

From (11), we have

me-a1(t-to)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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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, OV-l VI I' I----a;- 8x_!(x,u(x)) - 1 <

1 2M

(15)

1

in aneighborhood vll(o). From (13), (14) and (15), we have

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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 state-to-reference projection will be found to convert the time-based feedback to a non-time based feedback. Simulations are carried out to show the stability of the tracking control.

\'ie defined. the following change of coordinates to

V(x"v-1(-y(x))) < _ d(x,V-:(J(x))) ,

achieve linearization of the last three equations in

(1),

From (10)1

.

. 1 1 V(x,) v-1(,(x))) < - 2~M V(x" v- (,(x)))

= Y,

Zl

in a neighborhoad of Xd. By Gronwall's inequality, V (x ~ ~ v-I ("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 time-based controller can still be used. This significantly simplifies the design procedure for a non-time 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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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

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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 time-based 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 state-ta-reference projection ,(x, y) x to achieve a time-invariant 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 xy-plane. Figure 3b and 4b shows the value Y-Yd, 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 non-time based feedback. Actually, the initial condition x(O) can be anywhere.

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. 4. CONCLUSION A

ne~

BejczYf A.K~ (1993b) . Event-Based Planning and Control for MuItiRobot Coordination. Proceedings of the 1993 IEEE International Conference on Robotics and Automation, Atlanta, Georgia, pp. 251-258.

xl, N., Tarn, T.J. and

design method for non-time based track-

ing c6ntroller has been developed. It can convert a controller designed by tra(litional time-based approach to a new non-time 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 non-time' based controller. More importantly, the results have provided a efficient and systematic approach to design non-time 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). Path-tracking for a tractor-trailer-like 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. 29-38. Murry~R. M. a.nd Sastry, S. S. (1993). Nonholonomic IDotion planning: steering using sinusoids. IEEE Trans. Automat. Contr., Vol. 38, pp. 700-716. . 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. 193-198, 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) non-time based control

Figure 1; Comparison of non-time 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, 18-23. 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 Max-Plus Algebra. ModeL Proceedings of IEEE/RSJ lnternational Conference on Intelligent Robotics and Systems, Grenoble, France. Xi, N. (1993a). Event-Based 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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