Proceedings of the American Control Conference
Chicago, Illinois • June 2000
Full S t a t e T r a c k i n g and I n t e r n a l D y n a m i c s of
Nonholonomic Wheeled Mobile Robots
Danwei Wang
Gllangyan Xu
S c h o o l o f Elect.rical ~ n d E l e c t r o n i c E n g i n e e r i n g
Nanyang Technological University
Singapore 639798, [email protected]
y~
Abstract
jtt
Output tracking control and internal stability are analyzed for restricted mobile robots. ~'acking error dynamics offers insights into the properties and stability of
tile input-output subsystem as well as the internal dynamics. Sufficient conditions for the flill state tracldng
are developed. A type (1,1) mobile robot is studied in
details and simulation results are presented to confirm
the theory.
Xv
0
X
F i g u r e 1: A mobile robot with steerable wheels
1 Introduction
In the last decade, feedback control tbr tile trajectory
tracking problem of nonholonomic wheeled mobile robots has been extensively studied based on the inputoutput dynamics [1, 2, 3, zl]. Especially, (static or dynamic) input-output feedback linearization is considexed as a standard technique. However, few efforts were
spent to analyze the nonlinear internal dynamic behaviors of those control schemes. One interesting observation was made in [5] on the internal s:.,~bility of a 2-wheel
differentially steered mobile robot. [ts internal dynamics exhibits unstable properties when tile mobile robot
tracks a trajectory for backwards motion.
In this paper, we study tile tracking error internal
dynamics for general configl~rations of nonholonomic
wheeled mobile robots. Our results provide the sufficient condition for the fifil state tracking stability by
using the tracking control schemes based on the inputoutput dynamics. A special car-like robot is studied in
details to enhance and visualize the results. The analysis to the car-like robot shows that the proposed sufficient condition is practical applicable by adjusting tile
parameters in the output fimction depending on the behavior of the desired trajectory.
2 Dynamics
x
and Form,,lation
We consider wheeled mobile robots moving on a horizontal plane, as shown in Figure 1. Tile robots are
3274
classified according to tile mobility 1 < rn _< 3 and the
steeribility 0 < s < 2 as type (m, s) mobile robot [3]. If
the robot is equipped with fixed a n d / o r steering wheels,
its mobility is restricted (m ~ 2) and the system is nonholonomic. Suppose there is no skidding between tile
wheels and the ground. Tile robot dynamics can be
described as follows (an extension fi'om [3]).
=
c (q)/z
(1)
with
q --
~ =
C(q) =
~Y((7)
w[lere~ ~ z [x y]T represent tile coordinates of a reference
point, P on the robot ill the inertial frame X O Y . 0 is the
heading angle as defined in Figure 1. 3' = 171 "'" 7~] T
represents the steering coordinates of independent steering wheels. Both vectors v E R m and w E R '~ are homogeneous to velocities. Both vectors u~. E R ~' and
us E R" are control inputs homogeneous to torques.
R(O) is the standard rotation matrix. The matrix Q(7)
and vector b(7) for each type of nonholonomic wheeled
mobile robots are listed in Table 1. For tile restricted
mobile robot, tile steering coordinate vector 7 can be
further partitioned as
7 .... 171
72]T
V ~ I E R ~+" 2 72E R2
Table
(2,0)
T y p e (rn, s)
1:
(1,2)
cosffl cos T2 ]
sin (71 + 7 2 ) / 2 /
(1,1)
(2,1)
Q (~) e R 2×m
b~' (~) e n m
71 E [5~rn,
ts-2
72 E R 2-m
1-1
[0
ll/a
null
mill
sin 7/a
null
7
1]
null
IoL
d (~) ~ n2
Reglflar
conditions
i¢0
171 <
1¢0
Oh(q)
o
0
IE(7 )
(4)
1,.+,~_2
0
is nonsingular if the regular conditions in Table 1 are
satisfied. Note that, parameters I and p in (3) have
explicit physical meanings and (:an be selected at will.
Moreover, the steering angles ~l and y2 are always restricted by the robot mechanism such that their maximums are smaller than 90 degree. Therefore, the regular conditions in Table 1 can always be satisfied for real
restricted mobile robots.
The control task is to track a feasible desired trajectory
(qd(t), #a(t)), which is pre-specified by an open-loop motion planner such t h a t the dynamics (1)(2) are satisfied
for a uniformly bounded input ud(t) and corresponding
uniformly bounded velocity #d(t) , i.e.,
0~
=
c(o,~, ~d)t',:
(5)
t~d
--
~d
(6)
7r
We should note that the output fimction h(q) in (3) is
an epimorphism, So, the stable full state tracking implies the stable output tracking. However, the reverse
might not he true. [t is understood that, for a given
trajectory za(t), the generalized states qd(t) might not
be unique, e.g., a straight line motion of za(t) may be
caused by forward or backward motion of a mobile robot and corresponds to different solutions of generalized
states q~t(t) and Iza(t). [n the extreme case, the output
tracking of zd(t) may require the solution of qd(t) getting out of its admissible range. For instance, steering
angle 7 is required to have a vahm out of its physical
limitation, such that the output tracking control design
is not implementable. Next, we shall show that controllers, which stabilize the input-output dynamics of ~,
may achieve the stable full state tracking under certain
condition.
Clearly, the desired trajectory can also be expressed in
the form of output (3) as
h(q,,)
l¢0
~
Since the same structure between system (1)(2) and the
trajectory (5)(6), one may establish the full order dynamics in terms of the state tracking error/~ = q--qd and
fi = t ~ - >a. We say that the system (1)(2) achieves stable full state tracking to trajectory (5)(6) if a control law
u makes the full order dynamics of (q, ,5) uniformly stable. Similarly, since the system (1)-(a) is input-output
decoupled, one may establish the i n p u t - o u t p u t dynamics in terms of the output tracking error ~ = z - zaWe say that the system (1)-(3) achieves stable output
tracking to trajectory (7) if a control law u makes the
input-output dynamics of ~ uniformly stable.
with
zd=
lp¢ o
171 < 7max <
(3)
[/i~T(0)
;~q c ( q ) =
l sin (P'Y)
717
where, vector @, 72 and d(7 ) for each type of robot
are given by Table 1. Note that, the first two entries
of z represents a virtual reference point in the XOY
plane. This output fimction is such defined that it may
be decoupled to input u, i.e., the following decoupling
matrix
E(q)-
Yl
72
a + /cos(T2)
Isin (Z~)
la+lcos(PT) l
_
Since 0 < m + s - 2 <~ 1 amt 0 < 2 - m
< 1 must be
valid, both 3A and ~2 are either a scalar or a null. Then,
we define the output function as follows.
v~
sm (71 - 7 2 ) / 2 .
(r)
3275
3
Full
state tracking
Since tile decoupling matrix E(q) is nonsingular, one
may check that, by defining a function
~, = k(q) = I0
~21r
(s)
tile following maps
j
L~(q)j
(9)
F-h(')]F
and
e,1
Lk(q)
1 Consider" the tracking problem, of robot
(I)(2) to a moving desired trqjectory (qd, #d) satisfying
(5)(6) with 7d(t) #d(t), and ud(t) uniformly bounded.
Theorem
l
(lO)
L (q) J
are b o t h diffeomorphisms. Clearly, ~(-) and ~(.) also
m a p the desired t r a j e c t o r y (1)(2) to the one in the new
coordinates (~d, ~2d, ~d).
Define tracldng errors g = ~i - ~id (i = 1: 2), ~ = r~ -- r~d,
and # - z - Zd. One m a y find a feedback u sud~ t h a t the
d y n a m i c s of tracking errors can be o b t a i n e d as follows
~1
:
g2
(i1)
~
=
g(i,,&)
(12)
=
it
(141)
(a) Suppose that control input u is specified such that,
by using th.e tran.sformation (I0), the closed-loop
t-subsystem (11)(12) is stable.
(b) Furthermore, by denoting
A(q/d,~d) = D.fo(~, Td#,d)8~
~=(,
(19)
suppose the linear system
= A(~d, .e)~
(20)
for o,,ery f,'o~n (~'e(t), #e(t)) = (Ire, ~e) is exponentially stable in a neighborhood of :/2 = O.
where,
P(~I, ~2, ~, t) = F(~ 1 + ~ld(t), ~ + 'ld(t))(~
--F(~ld(t), ~d(t) )~2d(t)
Then, the robot system. (1)(2) locally achieves stable full
state tracking to desired trajectory (5)(6').
+ @d(t))
(15)
and
r(.)-
a/c(q)
~Jq c(q) E-1 (q)q=4, '
(16)
(.)
This set of tracking error d y n a m i c equations (1 1)-(141)
consists of two parts. T h e first p a r t is the & s u b s y s t e m
(1 1)(12), wlfich characterizes tire i n p u t - o u t p u t d y n a m ics. T h e second p a r t is the rl-subsystem (la), which
is not controllable and characterizes the internal dynamics. In the case t h a t the i n p u t - o u t p u t d y n a m i c s
& s u b s y s t e m is stabilized, the stability of internal dynamics r/-subsystem d e t e r m i n e s w h e t h e r the stable flfll
state tracking and even the stable o u t p u t tracking can
be achieved. In particular, the zero dynamics, equation
(13)(15) when the s y s t e m o u t p u t is set to zero (~ = 0),
is given as
;~ = ~,,(~, t)
(it)
- IF(Girl(t), ~ + ~l~t(t)) - F(~,~(t), rld(t))l ~Ud(t)
and its stability is critical to the internal stability. To
gain more insights to the tracking error zero dynamics,
t,y using (16) and (4), (17) can be expressed in terms of
the original mobile r o b o t generalized coordinates as
~i
J',, (~, 7d, #d)
(18)
O u t l i n e o f proof: Because of the diffeomorphism (10),
the proof can be c o m p l e t e d by showing t h a t suppositions (a)(b) iml)ly tire uniform stability of the tracking
error dynanfics (11)-(13). T h e proof consists of flowing
two steps.
(i) fo in (18) and hence ~)fo/9~ is Lipschitz in ~ in a
n e i g h b o r h o o d of ~ = 0, uniformly in t. So, (20) is
the linear a p p r o x i m a t i o n of (18). lqlrthermore, since
~yd(t) = wd(t) E ~d and lid(t) = ud(t) are uniformly
bounded, linear s y s t e m (20) is a slowly time varying
system by L e m m a 2.414 in [6]. Therefore, supposition
(b) locally implies the uniform a s y m p t o t i c stability of
the tracking error zero d y n a m i c s (18) and hence (17).
(it) Noticing the s t r u c t u r e of matrices E(q), G(q) and
Table l, F(.) is Lipschitz, uniformly in t. lqu'thermore, ~2d = E (qd)#d is mfiformly bounded. Then,
~#((1, ~2, ~, t) is Lipschitz in (~1, ~2,/j), uniformly in t.
Finally the claims follow Corollary in page 445 of 17]
and the result (i).
[]
In the case t h a t o u t p u t tracking control law is used,
T h e o r e m 1 offers sufficient conditions for the stable lull
state tracking and thus the stable o u t p u t t r a c k i n g in a
n e i g h b o r h o o d of ~ = 0.
4 Tracking stability o f a car-like r o b o t
(1
I,,+, ~
0
I2-,-~,
ttd
Here, we m a y give a main result by the following theo-
reIll.
3276
T h e stability analysis m e t h o d p r o p o s e d in T h e o r e m 1 is
generally suitable for analyzing the t r a j e c t o r y tracking
stability of any wheeled mobile r o b o t u n d e r an o u t p u t
tracking control scheme. W i t h o u t loss of generality, we
i n v e s t i g a t e t h e c a r - l i k e r o b o t in F i g u r e 2, w h o s e d y n a m ics is g i v e n by (1)(2) w i t h e l e m e n t s of t y p e (1,1) in T a b l e
1. In l h i s s p e c i a l c a s e , . , is t h e l o n g i t u d i n a l v e l o c i t y ; co
is t h e s t e e r i n g r a t e ; a is a p o s i t i v e c o n s t a n t of t h e w h e e l base. P a r a m e t e r s (1, p) defines a v i r t u a l r e f e r e n c e p o i n t
P~.
iy
t
,
!
/
•,
~
X
/
>,
/
l
i
its eigenvalues at every fiozen (~d, 04) are
7)d
A1 --
"04
A2 -
a
and b o t h of them are negative if vd, 1 > 0 and 1741 < 7r/2.
A p p l y i n g T h e o r e m l, we conclude t h a t the stable lull s t a t e
trackhag of a car-like r o b o t to a feasible t r a j e c t o r y (17d] <
7r/2, Vd, Wd, 'Ud are uniformly b o u n d e d ) t h a t moves fbrwards
(vd > 0) can be achieved by choosing o u t p u t function such
t h a t the virtual reference point P, is in fi'ont of t h e fi'ont
wheel axle (l > 0) in the steering direction (p - l).
\i.,'"
: '! ."~C
; .k%.¢-'p r
va = -15 m/s
%= .20~20/s
" , , . ~.'¢.""
~0= ~/3
90
1
/
2
\
~
0
I
/
\
",<",,:.:¢.%%/o>,
0
,/-"./",../1~ ~xg
.i" 0
(22)
1 cos 5'a
X
~
240 ~ _ , _ _ q - ~ @ o
F i g u r e 2: A ear-like robot configuration
270
(a) High speed
After some derivations, the linear approximation (20) of
t h e e a r - l i k e r o b o t is o b t a i n e d as
;~ = Vall(Td'vd'cod)
Lam(~d,
al2(Td, Ud, cod)]
a22(Td,,,d:Wd)
Vd, cod)
(21)
vd = -1 m/s
90
yo.=z/3
z/3
with
all
',~4
4
412
plcva sin (274)
a (cos (pV4 - 2Z~) + cos ~4)
2vapl sin 2 7d + 2vda COS (PTd) + p21awa sin (27d)
24o "----_Lq. ~2o0
o.; (c,,s (p74 - 2~4) + c,,s w )
27O
2r'd la cos (p~'d -- 27d)
(I,21
=:
a (cos (P~d
2~) +
-
Figure
2a + / ( 2 + p) cos (PTd)
- , , 4 pla (cos (m'4 -- 2~4) + cos ~4)
~dPasin (p~d) + p a s i n (pTd -- 27d) -- pl sin (274)
a (c,)s (pT~
-
2~)
~ cos 7d)
This linear a p p r o x i m a t i o n has 5 p a r a m e t e r s (Td, "d, ~d, l, p).
It is so complex t h a t its eigenvalues analytic studies are, in
general, difficult, if not impossible. However, in some special
cases, eigenvalues analysis can offer m o r e insights. O n e such
case is given by setting p = l as
=
vd
[0
_
1
l cos "7,./
+~
3: B a c k w a r d m o v i n g s t a b l e s e t s
cos,d)
a cos (Pfd -- 27d) -- 21 sin 2 7d
"'~ a~ (cos (p~4 - 2~'~) + cns w )
A(74, Vd)
(b) Low speed
-- l 2 sin 2 7d -- a2 -- la cos (P74)
pZa2 (~,,s (p~'4 - 2~4) + c , , s ~ 4 )
a s i n (pya) + a sin (p~d - 2%~) - I sin (274)
~4
a22
1 8 ~ 0 0
2l s i n 2 74 - a cos (P74 - 2 7 4 ) + a cos (PTd)
a~ (cos (PT~ - 27~) + ~os~'~)
1]
1 ~
l cos "Yd
T h e above analyses offer applicable sufficient conditions for
the full s t a t e tracking of the car-like robot: (a) setting 1 > 0
and p = 1 when Vd > 0; (b) setting I < - a and Pv. < P < 0
with p , d e t e r m i n e d by the behavior of the desired trajectory,
i
a
sin 74 ('va t a n 7d + awd)
To flarther investigate ttle stable thll s t a t e tracking p r o b l e m
of a car-like r o b o t to a feasible t r a j e c t o r y t h a t moves backwards (va < 0), eigenvalues analysis faces limitation. Here,
we use numerical search to explore the sets of five design par a m e t e r s (Vd, wa, 7d, l, p) t h a t ensure the full s t a t e tracking
stability'. In Figure 3, the shaded areas are the sets of locations of the virtual reference points (l < - a and p < 0) t h a t
are able to ensure the stable tull s t a t e t r a c k i n g at different
setting of (~d,&d, '/)d)- Figm'e 3 shows t h a t lower velocities
or higher steering rates come with smaller p a r a m e t e r values of p. Note t h a t such sets of virtual referem:e points are
open sets and the boumtaries do not g u a r a n t e e the lull state
tracking stability.
_11
3277
t
i.e., w i t h desired s t e e r i n g r a t e i n c r e a s i n g a n d desired velocity
decreasing, Pu t e n d s to zero.
--
I
1
=-I..~
v=.Z
p = -o.2
I
5 Simulation
results
In t h e s i m u l a t i o n study, t h e desired t r a j e c t o r y consists of
t h r e e s t r a i g h t lines a n d two curves. T h e first c u r v e is designed w i t h a m a x i m u m c u r v a t u r e k , , , , = 0.5238 a n d a
m a x i n m m c u r v a t u r e c h a n g e r a t e ]kma.x = 0.6864. T h e s e c o n d
c u r v e h a s a h i g h e r m a x i m u m c u r v a t u r e k~.a,~ = 0.8479 a n d
a m u c h h i g h e r c u r v a t u r e c h a n g e r a t e k . . . . = 1.7599. [n t h e
fl)llowing, we use a s i m p l e i n p u t - o u t p u t feedback linearizat i o n c o n t r o l law to veri(y t h e a n a l y s e s in t h e last section.
0
2
4
6
8
10
12
14
16
(a) p = - o . 2
][Y
~ = .1.2a
v=. 2
8
P = -0,1
In t h e first case, t h e car-like r o b o t is e x p e c t e d to m o v e forw a r d w i t h t h e desired velocity of va = 2 m / s e c . T h e v i r t u a l
reference point is c h o s e n at P0 = (l,p) = (0.5a, 1). B a s e d
on T h e o r e m 1, t h e l i n e a r a p p r o x i m a t i o n (21) of t h e t r a c k i n g
e r r o r zero d y n a m i c s is a s y m p t o t i c a l l y s t a b l e a n d so is t h e
r o b o t tracking. T h e s i m u l a t i o n result, as s h o w n in F i g u r e
4, c o n f i r m s t h a t t h e vehicle follows t h e desired t r a j e c t o r y
t h r o u g h out.
•
s
s
i0 i2 14 1C
(b) p = - 0 . 1
I
l I
10 - -
Figure
l = O..~a
v=2
5: L 0 o k - b e h i n d
tracking
- ~_~
8 --
offer a general a p p r o a c h for analysis u s i n g l i n e a r a p p r o x i m a tions. T h e d e t a i l i n v e s t i g a t i o n of a ear-like m o b i l e r o b o t
i n d i c a t e s t h a t sufficient c o n d i t i o n for s t a b l e t r a c k i n g c a n b e
i m p l e m e n t e d by a d j u s t i n g p a r a m e t e r s in t h e o u t p u t function.
i
IF
T6 ,
Figure
8
1'0
1'2
4: Look-ahead
14
1'6
1'8
tracking
In t h e s e c o n d case, t h e car-like r o b o t is t u r n e d a r o u n d
a n d e x p e c t e d to m o v e b a c k w a r d s to t r a c k t h e s a m e t r a j e c t o r y w i t h t h e desired velocity of Vd = - - 2 m / s e e . T h e
v i r t u a l reference p o i n t is c h o s e n at, two different l o c a t i o n s
/'1 = ( l , p ) = ( - 1 . 2 < - 0 . 2 )
a n d P~ = (1,p) = ( - 1 . 2 a , - 0 . 1 ) .
B a s e d on Figure 3, [a2 is ch)ser to t h e vet!icle s y m m e t r i c axis
line a n d a b l e to h a n d l e m o r e difficult m a n e u v e r s , such ms
t h e se(:ond c m v e , w h i c h h a s a h i g h e r m a x i m u m c u r v a t u r e
c h a n g e rate. T h e s i m u l a t i o n results, as s h o w n in F i g u r e 5,
show t h a t , w i t h p = - 0 . 2 , t h e r o b o t fails to t r a c k t h e desired
t r a j e c t o r y at t h e s e c o n d curve, a n d t h a t , w i t h p = - 0 . 1 , t h e
r o b o t succeeds in t r a c k i n g t h e c o m p l e t e desired t r a j e c t o r y .
Ore" i n t u i t i o n a n d e x p e r i e n c e verify" t h a t d r i v i n g a car b a c k w a r d s w i t h h i g h e r s t e e r i n g r a t e s will cause difficulties a n d
even instability.
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6 Conclusion
T h i s p a p e r a n a l y z e t h e s t a b l e full s t a t e t r a c k i n g p r o b l e m
of n o n h o l o n o m i c w h e e l e d m o b i l e r o b o t s u n d e r control laws
b a s e d o n t h e i n p u t - o u t p u t d y n a m i c s . It is s h o w n t h a t t h e
t r a c k i n g e r r o r i n t e r n a l d y n a m i c s a n d zero d y n a m i c s play a
critical role of t h e full s t a t e t r a c k i n g s t a b i l i t y of s u c h m o b i l e
robots. Sufficient c o n d i t i o n s for t h e s t a b l e flfll s t a t e t r a c k i n g
3278
[6]
K. S. Tsakalis a n d P. A. l o a n n o u , Linear TimeVcwging Systems Cont~vl and Adaptation. P r e n t i c e Hall,
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