Path Tracking Control of Wheeled Mobile Robots

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

PATH TRACKING CONTROL OF WHEELED MOBILE ROBOTS S. Patarinski*,l, H. Van Brussel** and H. Thielemans** *Department of Mechatronics and Precision Engineering, Tohoku University, Sendai 980, Japan **Division PMA, Department ofMechanical Engineering, KU Leuven, 3001 Leuven, Belgium

Abstract. Kinematic modeling of WMR is considered regarding the choice of the generalized coordinates and the control variables, so that the model is independent of the particular design, structure, number of wheels, etc. Three methods for kinematic control are proposed, based on: (i) transformation of the model as seen from the moving coordinate frame associated with the desired motion and Taylor series linearization; (ii) model order reduction. and (iii) direct Lyapunov synthesis. The results are evaluated via numerical simulation of a WMR with two driven wheels and a passive castor. Key Words. Vehicles; robots; tracking systems: modeling; linearization techniques: model rednction; Lyapunov methods

1. INTRODUCTION During the last decade WMR attracted substantial research efforts motivated by the variety of potentia.! practical applications and the challenges nonholonomic mechanical systems still present to control and robotics communities. The stateof-the-art in the field was reviewed by Patarinski (1991), but no details are recalled here due to the lack of space. In summary, \VMR modeling, both kinematic and dynamic, is solved to a satisfactory extent, while there are still open problems regarding their control.

~ 2. KINEMATIC t-.l0DELING OF

.ICR

W~IR

In this section kinematic modeling of WMR is briefly considered. Further on by • the respective desired quantities are denoted, by II - the deviations from tlwm, and "pose" stays for both "position and orientation". WMR motion is described relative to an inertial coordinate frame O"y. Refering to Fig. 1. two moving coordinate frames arc also introduced: C""y' - the WMR fixed coordinate frame, where y' is alligned with its longitudinal axis, and C' "'y' - the coordinate frame associated with the desired motion.

Figure 1: Nomenclature

Denoting by x = [.T Y ef and u = [v wjT the state HlHlllw control vectors respectively, the kinematic model of "'\Ill can be presentl"d as: sin

The pose of \VMR is naturally and conveniently descrilwd by the Cartesian coordinates :1', y and the angle e. Actual control inputs are the rates of rotation of thl" dri"ing, and the steering angles of tlK stl"ering whl"els, but they arl" not suitable for developing universal kinematic models of W~IR. The huge kinematic redundancy, that seems to exist in the case of more than three V\,heels, is completely eliminated by 1.11(' rolling compatibility conditions (Alexander and t-.Iaddocks, 1990). Therefore, v and w, the linear and thl" angular velocities of \VMR, are commonly considered as control inputs. .I

eo]

x = Ax + B(x)u = [ co;e

~

u

11)

or in a scalar forlll:

where A

i

sin e.l'

f;

cose.v

iJ

w

= 0 3•3 is the null 3 x

(:?)

3 matrix.

The kinemat.ic model (1) - (3) is general and applies t.o 'lilY \VMR, providing a unified means for motion plannilll'. illId control. To describe a particular vehicle, the equatiolls 1('-

On leave fromt.he Depart.ment. of Rohot.ics and Mechat.ronics, In-

stItute of ?dechalllcs and BlOlllechanks. Dulgariall Academy of Sd. f'nces, Sofia.

893

lating v and w to the rates of rotation of the driving and the steering wheels (Muir and Neuman, 1987; Alexander and Maddocks, 1990) should be added.

where:

A(t) =

[

O~ ~ ~"]; o

0

3. CONTROL OF MOTION In this section three approaches are developed and evaluated through numerical simulation. 3.1. Taylor series linearization approach

and ~u = u - u". Obviously, the system (5) is controllable, unless v" = O. The control ~u should have the form of a state feedback:

This approach has already been considered in (Song et al., 1989), expanding the model (1) in Taylor series along the desired motion and transforming the linearized system as seen from the moving coordinate frame C' X'y'. Here the two steps are performed in a reversed order - firstly, eq. (1) is transformed as seen from C' X'y' and secondly, the resulting model is linearized in the neighbourhood of C". As shown below, the formal exchange of the order of the two steps leads to a non-trivial result, because the model of the system becomes quite simple ("almost linear") right after the first step.

~u= K~x

where K is a 2 x 3 feedback gain matrix. It is straightforward seeing, that whatever the K matrix, the poles of the closed loop system depend upon v". In spite of its very simple structure, the model (5) is timevariant and still difficult for control synthesis. There are two alternatives: either to synthesize a time-varying control law (eg with a quadratic performance index), or to approximate v"(t) by a piece-Wise constant function. Both methods have found practical application, but certain preference should be given to the second one due to both its simplicity and because WMR are typically required to move with a constant velocity most of the time.

Consider the following coordinate transformation: (3) where: the subscripts (omitted further on for brevity) denote the reference frame - i for the inetrtial, m for the moving one, and COS B' sin B" 0] T(x") = -sinB" cosB" 0 [ o 0 1

A basic advantage of this approach is, that it provides a formal way to cope with the nonholonomic constraints - the deviation of \VMR from the desired motion is automatically split into positional and orientational errors, that are properly combined and compensated through the state feedback. Its main drawbacks are the time-variance of the linearized system (5) and the numerical difficulties appearing at low velocity (v" ~ 0).

Differentiating (3) and substituting (1) yields: ~x = A~x

+ Eu + Gu"

(4)

where:

3.2. Model order reduction approach Here, the main idea is to present WMR kinematics in a symmetric (wrt position/orientation) form, that makes the genesis of the pose error transparent.

and u", u are the feed forward (defined in the course of motion planning) and the feedback (synthesised below) control components, respectiYely.

In feedback linearization, as applied to WtvlR, preference is commonly given to the positional variables .or and y (Campion et al., 1991; d'Andrea-Novel et al., 1991; Pomet et al., 1992). The differential order of the system (1) is 2 and understanding that vehicle's orientation is at least as important as its position (in fact - more important regarding the accumulated pose error), it is desirable to keep up the third equation in (3), while combining the first two ones. A physically sound way of combining 1: and y is through the path travelled by WMR:

The following is noteworthy: 1. Eq. (4) demonstrates very clearly the nonholonomic nature of WMR: the motion along the x' axis is not controllable at all through u·, and (because of the small values of tiB in a normal mode of operation) is, roughly speeking, "less controllable" with respect to (wrt) u (or in other words requires much higher control effort) than the one along y". 2. Again due to the small values of tiB, the E matrix can be simplified, substituting SlllllB ~ llB and cosllB ~ 1, that yields:

5=

10' 1I(t)
and v =

=0

)1: + i;2 2

(6)

(7)

Relations (6), (7) suggest to square and sum the first two equations in (3), that yields:

3. A general conclusion streaming from eq. (4) about the control is that llx can be only compensated through llB, ie - besides llB, w should also feedback 111'. The linearized system is: ~x = A~x+B~u

5(0)

s

v, 5(0) = 0

iJ

w, B(O) = B(O)

(8)

Given some feasible desired motion x"(t), the control should have a feedforward-plus-state-feedback form:

(5)

894

I'

w

I" ...J'

+ k.'::'8 + ko'::'O

ity). The third t.enll in \. is chosen to prO\'ide compensation of 6.1' through '::'(1, as suggested abOH' .

(9)

where: il.s = 5 - s'; ~(I = (I - 0": s' is ddined through (6), (7): 1'" = s"; ...J" = 0'; 1>.. and k o are feedback gains, that should be chosen to pn)\'ide system stabilit.y and proper dynamic beha\·ior.

Differentiating (10) wrt t IH' time yie-lds: W

+ k~r ).::,.1' + k..,.'::'(I) siu:::"O + PO + k..r'::'·I)";

(( (1 +

( 11)

il.y cos .::,(1) I'

If t.he control is chosen as:

It is clear now, t.hat if .1"(0) = .r'(O). y(O) = .11"(0), and 0(0) = (1'(0). then .r(t) == .r'(t), y(t) == y"(t). and (I(t) == O·(t). Because always 5(0) = sOlO) = 0, t.he princial source of errors is t.he initial orientat.ion (I(O) of Wi\·m, but of comse, bot.h 5 and (I measurement. inaccuracies cont.ribllt,· to t.he t.rajectiry following errors during w;.,m motion.

l'

-k,.( (( 1 + k~,. )'::'.1

w

-ldil.(I+k.. ,.'::'.r)

where k" and

k~

+ k..,.'::'O) sin'::'O + '::'11 ('()s '::'0) (12)

ar<> positiw fe"dhack gains. the fuuction

W in (11) is negative, unless the thre-e :::".1' = O. .::,.'/ = 0 and 6,0 = 0, that guarantees the stahility of th .. dos,'d loop

The following is worth noting:

system (-1)-(12).

1. At first sight this approach seems too simplified. It should be well understood, howe\'('r, that following the- fundame-ntal controllability, accessibility and feedback linearizability results (Bloch, 1989; Campion et aI., 1991), nothing more can be obtained using a smooth, tinle-im'ariant state feedback.

Taking into account that .::,(1 is small. the cOlHrol (12) redu('('s to:

+ k:u)'::'.r + k_,'::'(I)'::'O + '::'Ij)

-k,,(((1

I'

-k..(o'::'O

w

+ '::'.1')

Besides its relatiwly simple form. a maiu adY;llltage of tIll' control law (12) is the guaranteed stability of till' closed luop system - ie both "!Tors in \\':--m position ami oril'ntation \·anish. Certain difficulties in its I1pplication COIII','rn till' choice of the fe-edback gains k,. and k... and Ih.. \H'ightiug factor kwr .

2. The model (8) is quite realistic because the path traq'lled is the most conmlonl~' and in som(' cases .. the single directly measured variable for \\I1vm feedback cont.rol. 3. The require-ment for accur
3.4. Numerical simulation J'('sults

4. The transformation (6), (7) is not a bij"ction: it defines a unique 't' (and 5), given .i', .Ii together with th,' respf'ctiw initial conditions, but tlw inVl'rse t.ransformat.ion is not unique, that is a consequence of tht' nonholonomic natmf' of \VMR - the actual coordinatt's .r, .If can lle estimated integrating the kinemat.ic model (3).

Here, the three methods are applied to the I'xalllpk of a H'hicle with two driving wheels and a passiH' ca,;tor 13,'sid,'s tlIP fact that only a particular design is considelw!. till' n'sldts are generally representatiw, that follows from the COIICl'pt of kinematically equivalent WMR (Patarinski. 1991). The desiredmutioIl, described by:

Going back to the gcm'ral conclusion about t.he cont.rol, df'ri\'ed in the precf'eding sllbsedion. t hen' is st.ill room t.o improve system's abilit.y for compcnsat.inl?; posit.ional errors, including them in the orientational feedback, eg assuming w (9) in the form:

.I'"(t) = 21 1 (3 - 2/)

= /1 = t 4 (3 -

y'(t)

2t)1

.i·'(t) , 2) (I"(t) = - arctl1n -.= arctan-lI-(3 - . 1 y'(t)

is obviously feasible. The system is subject to certain initial errors: SI(O) = 0.1[111). '::'y(O) = 0.1[111]. and '::'0(0) = 0.1 [m<4-

where kOr and koy are t.he respcctin' fp('dback gains. This however de\'iates from the original idea of the met hod and will not be considered in mol'l' detail h",re.

The feedback gains are chosen as fullo\\'s: In summary, t he model urder rcd uct ion approach giws preference to the orient at ion of Wi\m, while bot h positional eoordinat.es are combilH'd to ~'ield the second order system (8). A main ad\'antage is its extreme simplicit.y'· the system is linear, decoupled a.nd synmH'tric wrt. linear/angular coordinates. The practical application howe\'('r requires periodic updating of the current. pose of WMI1 relat iw to t.he inertial coordinate frame (that is, by till' way. quite typica.l for the field).

• The linearization approach -

K_[k 0

Fig. 2:

o

11

1

Im,:':F'=7==C-==========-::C::;'=1

I·odl·

..

..

,

;

;

,

'.'

6,2....

••1/

.

,

........

i

,..........

,

,

, ,

.

..

'.' ./="'- .. ,.........

3.3. Direct LyapunO\' approach

;

, .

.

~

,

.... ,

,

,

,

.

. .

i.

~

'1

In this subsectiun a cOlltrol law is s~'nthesised for the system (4), based on thl' fullowillg LYl1puuo\' function candidate:

.

1

\ ='2('::'1'

2')

.,

+D..(r+(({'::'{I+k~,.'::'.I)-)

..·.. ,

fi~·;· ..·

·.. A·',·..

,

, " ',

,

'

~S"-·.·.··

.·· .

' . 'i

·

---..... 'J

(10)

where I.:~r is a weight ing factor and ({ > 0 is a scaling coefficient (further on it.s 111l1llf'ricaJ value is assumed 1 for simplie-

Figure 2: The linearization approach

895

,

I

where: k ll = -aa/v' if Iv'l> V, k ll = -aa/v in the opposite case, and ii = 0.001 for the results shown; ai, i E Q,2 assign the desired spectrum {-I; -2: -3} to the closed loop system.

some for control synthesis and real-time implementation.

2. As a consequence of the understanding that WMR position and orientation are equally important, the dimension of the configuration space is reduced using a proper projection - the reduced order state vector comprises the lenght of the path travelled by WMR and its orientation, and the reduced order system is symmetric wrt the linear and the angular coordinates. This model gives an adequate presentation of the nonholonomic nature of WMR, but its practical application requires periodical updating of WMR pose relative to the inertial frame in which the desired motion has been specified.

Fig. 3: k. = k e =

• The model order reduction approach -15.

..

, , (m) "'F=-;::=::;:===:::C==:;="'T'===:;=="'l (,.d).. A

,! ,;

A~ ,

:·:\I\~

,..

i

;

i

"

+

........\\""',.".. ", "+ , +."." + .'"."..".",."

::'~J~"'. ""::i

,.".",.".." !

"

,

;"

!"..,." ;.."..,

;." "

i

j "

,•....,.."..j

3. Following the conclusion that, besides for /1(}, the angular feedback should also contribute for the compensation of -"i.:1', proper Lyapunov function candidate is suggested. The control law derived provides closed loop system's stability, ie guarantees precise tracking of the desired motion, being in the same time relatively simple for real-time implementation.

I

! "

··1

;, "".."." "".""'",,··,,·,,·1

+·""·····j··,·······,········j·,,·····j·,··,,···t·······,, ,

"

\

The methods are comparatively evaluated by numerical simulation of a WMR with two driving wheels and a passive castor.

Figure 3: The model order reduction approach

• The direct Lyapunov approach -

Fig. 4: k" = kw = -15.

5. ACI":NOWLEDGEMENTS

'm"::-r"=:-;==T=r::==;===::::r==:;=:::;:=:::=:::::::::::7l

I"d' >j .. ,

::

:.

;

~..!

+ _..+ +.. ···+·.. ··..+··.. ·····,·········1

;

;~~ :." !P" ,."" .~" .~, +.f',

::~\~

This research was supported by the Commission of the European Communities under contract FI2T-0006-C-(A) as part of the TM8 project of the TELEMAN program. The discussions with P. Willems and G. Cmnpion of Universite Catholique de Louvain within the frames of the Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming, is acknowleged. SP gratefully acknowledges the facilities made available during his visiting professorship at the Department of Mechanical Engineering, Katholieke Universiteit Leuven, and especially the support of the Head of Division PMA and the collaboration of its faculty, as well as the partial support by the Bulgarian National Science Research Foundation under grant NIMM 44/91 "Control of Intelligent Machines". The numerical simulation was carried out using the simulation package DYNAST (Mann, 1992).

·, ..···"···i·······,,,·,,· ·,············,[·········,··········;·········+·· .. ···l

:.,:. .:' '-' '.'

•.. . '";'.;: , ., '••••.••••.:.·F; .. -/ :;;.·..-r··[/c

r::..:.:.:.:.!,:.:.:.:::.:.:.:;:

,v ,

,': :. " ..:

+."" , . " ,[

1 1

1

Figure 4: The direct Lyapunov approach

As it should have been expected, if the initial conditions are precise and no disturbances are involved, the three methods provide motions of compatible quality. In the general case both the linearization and the Lyapunov approaches guarantee satisfactory behavior of WMR.

6. REFERENCES Alexander, J .• and J. Maddocks (1990). On the kinematics of robots. In: Autonomous Robot Vehicles, (J. Alexander, and J. Maddocks, Eds.), 5-24. Springer-Verlag, New York. Bloch, A.M. (1989). Stabilizability of nonholonomic control systems. Proc. IFAC Symp. Nonlinear Control Systems Design, Capri,I73-178. Campion, G., B. d'Andrea-Novel, and G. Bastin (1991). Modelling and state feedback control of nonholonomic mechanical systems. Proc. 30th IEEE Con/. Decision and Control, Brighton, 1184-1189. d'Andrea-Novel, B., G. Bastin, and G. Campion (1991). Modelling and control of non holonomic wheeled mobile robots. Proc. IEEE Con/. Robotics and Automation, Sacramento, 1130-1135. Mann, H. (1992). DYNAST User's Manual. Division PMA, K.U.L., Leuven. Muir, P., and C. Neuman (1987). Kinematic modeling of wheeled mobile robots. J. Robotic Systems, VolA, No.2, 281-340. Patarinski, S. (1991). SMT 1 Mobile Platform: Modeling, Motion Planning, Kinematics Evaluation, Control and Simulation. Technical Report, Division PMA, K.U.L., Leuven. Pomet, J.B., B. Thuilot, G. Bastin, and G. Campion (1992). A hybrid strategy for the feedback stabilization of nonholonomic mobile robots. Proc. IEEE Con/. Robotics and Automation, Nice, 129-134. Song, K.-T., J. De Schutter, and H. Van Brussel (1989). Design and implementation of a path-following controller for an autonomous mobile robot. In: Proc. Int. Con/. Intelligent Autonomous Systems (T. Kanade, F.C.A. Groen, and L.O. Hertzberger, Eds.), 253-263, Krips Repro, Meppel.

A more precise analysis shows that certain preference should be given to the Lyapunov method that both is relatively easy to implement in real time (cl the discontinuity of the linearization method when v' ~ 0) and provides closed loop system's stability; its performance can be further improved by a proper choice of the feedback gains. The model order reduction method is also simple enough to implement, but fails to compensate for the initial pose errors.

4. COr-;CLUSIONS In the paper kinematics and control of WMR are considered. Three methods for kinematic control are proposed: 1. The non-linear model is transformed as seen from the moving coordinate frame, assosiated with the desired motion, that makes the nonholonomic nature of \VMR transparent and leads to some important conclusions regarding the control. After t,hat, the system is linearized in the neighbourhood of the origin. Although linear and of a simple structure. the model is time-variant and still a bit cumber-

896