Proceedings of the 33rd
Conference on Decision and Control
Lake Buena Vista, FL

December 1994
FA12
10110
Stabilization
of
NonHolonomic Mobile Robots using Lyapunov
Functions
for
Navigation and Sliding Mode Control
.Jiirgon Giildiier
Vadim I. Ct,kin
DLR, Inst. for Robotics urd Sgsteni Dgnumics
Postfuch
11 16,
082230 Wessling, Germany
Abstract
Mohile robots with noiiholonomic kinematics have three degrees of freedom for planar motion, but only there are
two control inputs available. The stabilization problem for such robots is known not to be solvable via smooth
timeinvariant feedback. We propose to utilize a Lyapunov function to prescribe a set of desired trajectories
to navigate the robot to a specified configuration. Ideal tracking of the prescribed traject.ories is achieved by
exploiting the invariance property ancl the order reduction property of sliding mode control. The mobile robot
is shown to
be
exponentially stahilizable for a class of quadratic Lyapunov functions.
1
Introduction
The problem of steering a mobile robot with non
lioloiioniic kinematics to a specified configuration
(.ro,
yo,
do)
has recently enjoyed quit,e some at,tent,ion
in the robot,ics community. The challenge of t.he prob
leni is reflcct,c!d in t.he fact that,
a
mobile robot in the
plane possesses t,liree degrees of freedom of motion,
wliicli have be cont,rolled by only two control inputs
and under nonholonomic motion constraii1t.s. Several
researchers have shown, based on Brocketk's Theo
rem
[l],
that, such
a
system is openloop controllable,
but, not, stabilizable by pure smooth t,imeinvariant
feedback, see for example
[241.
To overcome the above difficulties, several di
rections of research have evolved. Among the pro
posed solutions are timevarying feedback
[5],
result
ing in oscillatory trajectories
[6];
nonlinear continuous
state feedback depending on an exogeneous time vari
able
[7]
and yielding asymptotic stabilizat,ion about
(soly0,@o)
at. a rate of
l/t;
and piecewise continuous
control resulting in exponential st,ability
[SI.
Further
more, there have been a number of openloop strate
gies seeking a bounded sequence of control inputs, t,he
existence of such sequences having been established
in
[9].
We propose to utilize sliding mode control t,o solve
the stabilization problem, i.e. to employ a discon
tinuous control strategy. The first step is to design
a Lyapunov navigat,ion function such that its gradi
ent prescribes a suitable set of trajectories leading
to the goal configuration in the robot workspace in
the desired manner. Exact tracking of the t.rajecto
ries is guarant.eed via sliding mode cont,rol and hence
the robot. is successfully navigated to t,he goal con
figuration. In a case study with a class of quadrat,ic
078031 9680/94$4.0001994 IEEE
Lyapunov functions, exponential stabi1it.y is achieved
along parabolic trajectories. Simulations are utilized
to illustrate the performance of the proposed control
algorithm.