In this paper, we address the problem of ε-tracking and ε-stabilization for a class of SE(2) vehicles with second-order nonholonomic constraints. We introduce a class of transformations called near-identity diffeomorphism that allow dynamic partial feedback linearization of the translational dynamics of this class of SE(2) vehicles. This allows us to achieve global exponential ε-stabilization and ε-tracking (in position) for the aforementioned classes of autonomous vehicles using a coordinate-independent dynamic state feedback. This feedback is only discontinuous w.r.t. the augmented state. We apply our results to ε-stabilization and ε-tracking for an underactuated surface vessel

Olfati-Saber, Reza

English

Caltech Authors - Main

Proceedings of the American Control Conference 
Anchorage, AK May 8-10, 2002 
Near-Identity Diffeomorphisms and Exponential 
E-Tracking and 6-Stabilization of First-Order 
Nonholonomic SE( 2) Vehicles 
Reza Olfati-Saber 
Control and Dynamical Systems 
Division of Engineering and Applied Science, 107-81 
California Institute of Technology 
Pasadena, CA 91125 
olfati@cds.caltech.edu 
Abstract 
In this paper, we address etracking and €-stabilization 
for a class of SE(2)  autonomous vehicles with first- 
order nonholonomic constrains. We introduce a class 
of transformations called near-identity diffeomorphisms 
that allow dynamic partial feedback linearization of the 
translational dynamics of this planar vehicle. This al- 
lows us to achieve global exponential €-stabilization and 
€-tracking (in position) for the aforementioned class 
of planar vehicles using a coordinate-independent dy- 
namic state feedback. This feedback law is only discon- 
tinuous w.r.t. the augmented state. We apply our re- 
sults to €-stabilization/tracking of a nonholonomic mo- 
bile robot. 
Keywords: nonholonomic systems, nonlinear con- 
trol, autonomous vehicles, €-stabilization, etracking, 
mobile robots, dynamic partial feedback linearization, 
dynamic state feedback 
1 Introduction 
Control of autonomous vehicles is currently an impor- 
tant field of research. Many vehicles of interest includ- 
ing mobile robots [l, 2, 31, surface vessels [4], VTOL 
and CTOL aircraft [5, 61, hovercraft, CalTech ducted 
fan [7, 81, helicopters, aircraft, and underwater vehicles 
are systems moving in R2 or R3. The configuration 
space of these systems (in the simplest form) is SE(2)  
or SE(3) ,  respectively (where SE(n)  denotes the Spe- 
cial Euclidean group of rigid motions in Rn). All these 
vehicles are control systems with first-order or second- 
order nonholonomic constraints. 
The main purpose of this paper is to address stabi- 
lization and tracking problems for the dynamic model 
of a class of SE(2) vehicles with first-order nonholo- 
nomic constraints. A similar treatment for tracking 
and stabilization of SE(2) vehicles with second-order 
nonholonomic constraints is presented in [9]. ' 
Stabilization of the kinematic model of special exam- 
ples of SE(2) vehicles with first-order nonholonomic 
constraints including a two-wheeled mobile robot and a 
rolling disk have been addressed by several researchers 
in the past. These systems were used as benchmark 
examples for nonlinear control of nonholonomic sys- 
tems. Among all methods are motion planning using 
sinusoids [lo], applying time-varying change of coor- 
dinates to driftless nonholonomic systems [ll], local 
exponential stabilization of homogeneous systems us- 
ing periodic inputs [la], €-tracking for a nonholonomic 
integrator [13], and finally the use of discontinuous 
change of coordinates [I] and quasi-smooth dynamic 
state feedback for stabilization of chained form systems 
[14]. Many of the aforementioned results are local, re- 
quire the use of small inputs, and suffer from undesired 
singularities and/or lack of stability in the sense of Lya- 
punov [lo, 12, 11. In contrast, our results are global, the 
proposed controllers are coordinate-independent, the 
stabilization/tracking is rather aggressive (i.e. expo- 
nential), the control design is directly performed on the 
dynamic model, and our approach is readily applicable 
to SE(2) vehicles with second-order nonholonomic con- 
straints [9] and SE(3)  vehicles. 
The key tool in our approach is a class of trans- 
formations called near-identity diffeomorphisms (NID) 
which allow dynamic partial feedback linearization of 
the translational dynamics of SE(2) vehicles. This in 
turn results in a dynamic state feedback that achieves 
global exponential €-stabilization and etracking in p~ 
sition for the aforementioned class of SE(2) vehi- 
cles. As an example, we apply our results to E- 
0-7803-7298-0/02/$17.00 0 2002 AACC 4690 
stabilization/tracking of a wheeled mobile robot. 
The outline of the paper is as follows. In section 2, 
near-identity diffeomorphisms are introduced and the 
notions of estabilization and €-tracking under dynamic 
state feedback are defined. In section 3, the dynam- 
ics of a planar vehicle with nonholonomic velocity con- 
straint is given. Our main results are presented in sec- 
tion 4. In section 5, the application of these results to 
estabilization/tracking of a mobile robot is provided. 
Finally, in section 6 concluding remarks are made. 
2 Near-Identity Diffeomorphisms 
Near-identity diffeomorphisms and their applications 
were first discussed in [15]. Here, we define what 
we mean by near-identity diffeomorphisms and provide 
some examples. 
Definition 1. (near-identity diffeomorphism) Let 
$(.,A) : R" x RP -+ R" be a smooth function and 
z = $(.,A) be a global diffeomorphism in x for  all 
X E RP, i.e. there exists a smooth function d(z,X) : 
R" x RP + R" such that 
4($k X I '  4 = 2, $(4(z, 4 7 4  = z 
' for all x, z ,  A. 
diffeomorphism iff for all x E R" 
W e  say z = $(x, A) is a near-identity 
+(z ,X)=x e X = O  
Remark 1. By uniform continuity of $(x, A) at X = 0 
w.r.t. x that belongs to a compact domain K, it follows 
that 
V€ > 0,36 > 0 : lpll I 6 + Il$(x, A) - 211 5 e 
for all x E IC. In other words, for e << 1, $(x,X) is in 
an t-neighborhood of x-thus the name near-identity. 
Two simple examples of a near-identity diffeomorphism 
are as the following: 
i) $(x, A) = x + AS, 
ii) $(x,X) = x + AX, x E R",X E RP,p < 
x, 3 E R", X E R, JJSJJ = 1. 
n, A has full column rank. 
According to remark 1, in case i) 6 ( ~ )  = E and in case 
ii) 6(€) = €/omaZ(A).  In this paper, we use NID's that 
are similar to the one in i). 
Now, consider a nonlinear control system with a 
(state,input) pair (x ,u)  E R" x R" augmented with 
another nonlinear system with a (input,state) pair 
(.,A) E I[$" x RP as the following 
(1) 
x =  f (2, U )  
= g ( x , X )  
and let z = $(x ,  A) be a near-identity diffeomorphism. 
In new coordinates, we obtain the following (2, A)- 
system 
(2) 
i = f ( z , X , u )  
X = i j (z ,X)  
with obvious definitions of f, i j .  
Definition 2. (€-stabilization) For a fix E > 0, let X f  = 
6 ( ~ )  satisfy the property in remark 1. Let xo be a desired 
equilibrium point of x = f ( x , u )  with U = 0. W e  say 
the dynamic state feedback 
(3) 
U = k ( x , X )  
i = g(x ,X) ,  X(0) = Xo 
achieves globally asymptotic estabilization of xo for  
the x-subsystem in (1) iff for  the closed-loop (.,A)- 
system 
i = f(z,X,k(z,X)) 
x = i j ( z ,X )  
with k(z,X) = k (b ( z ,A ) ,X) ,  ( x0 ,X f )  is  a globally 
asymptotically stable equilibrium in the sense of Lya- 
punov. 
The notion of e-tracking given a dynamic state feedback 
is defined in a rather simpliar way (see [15, p. 2121). 
Definition 3. (€-tracking) For a jix E > 0, let X f  = 
& ( E )  satisfy the property in remark 1. Consider the 
following nonlinear system with an (input, output) pair 
( U ,  Y )  
(4) 
a : =  f ( x ,  U )  
Y = &-) 
after applying a near-identity diffeomorphism z. = 
$ ( x , X )  and augmenting the system in (4) with X = 
g(x,X),  the dynamics of the augmented system in new 
coordinates takes the form 
i = j ( z , X , u )  
A =  a27 4 (5) 
Y = h(d(t.,N) 
Let U = k(z ,X ,yd ,yd , .  . . , y r ' )  be a control law that 
achieves asymptotic tracking of a desired trajectory 
yd(.) for  (5) such that X ---f Xf as t + m. Then, we 
say 
= k ( x ,  Y d ,  Y d r .  . . 7 Y r ' )  := &($(x,  A), Y d ,  y d r . .  . 7 Y t ) )  
achieves asymptotic e-tracking of the desired trajectory 
Y d ( ' )  f o r  (4)- 
4691 
3 Kinematic SE(2) Vehicles 
A two-wheeled mobile robot depicted in Fig. 1 is an ex- 
ample of an SE(2) vehicle. A coordinate-independent 
representation of the dynamics of this mobile robot is 
in the form 
x =  ( R e l ) v l  
R = R L j  (6) 
i , =  7 
where (2, R) E R2 x SO(2) = SE(2), R is a rotation 
matrix in W2 satisfying RTR = I2 with det(R) = 1. 
Also, w = ( 2 1 1 , ~ ) ~  and 3 is a skew-symmetric matrix 
given by 
0 -w 
5 = [ w  0 1  
By a SE(2) wehicle, we mean a robot that its config- 
uration space is SE(2). Consider the following near- 
identity diffeomorphism 
z = $(z, A) := 2 + X(Re1) (7) 
where X E R, e, is the i th standard basis in W", and 
R is the rotation matrix. The dynamics of a kinematic 
SE(2) vehicle in z-coordinates can be expressed as 
i. = RAW + (Rxel)X 
i , =  7 
RA = Rx3(X,A) (8) 
where v = ( w ~ , w ) ~  and R x , G ( X , i )  are given by 
r o -xW 1 
where [CllC21.. . ICm] denotes an n x m matrix with 
columns C1, C2,. . . , C,,, E B". The properties of RA in 
the following lemma is stated for the future use. 
Lemma 1. RA satisfies the following properties: 
det(Rx) = A, (Rx)-l = RT A-' (9) 
The proof of Lemma 1 is very elementary. According 
to Lemma 1, for X # 0, RA is invertible and for X # 1, 
Remark 2. In coordinates, RA takes the following form 
RA !2 SO(2). 
1 cos(8) - A  sin(0) R x =  [ sin(0) X cos(@) 
We make use of the following Lemma later. 
Lemma 2. Assume (,i,X) + 0 as t + 00 and X(t) # 
0,Vt .  Then, w(t )  + 0 as t + 00 as well. 
Proof: Based on property i) in Lemma 1 and the fact 
that X(t) # 0 for all t ,  Rqt )  is invertible for all t and 
w = R,'(i. - ( R e 1 ) i )  
By assumption, the right hand side of the last equation 
vanishes and thus w ( t )  = ( ~ ( t ) ,  w( t ) )  4 0 as t -+ 0. 0 
4 Main Results 
In this section, we present our main €-stabilization and 
€-tracking results for the class of kinematic SE(2) ve- 
hicles. 
Proposition 1. Consider the kinematic SE(2) vehicle 
in (8) augmented with 
X = -cx(X - E ) ,  X(0) > € > 0, CA > 0 
Then, applying the change of coordinates (q, p )  = (z, i )  
as the following 
(10) 
q = z + X ( R e l )  
= R ~ v + X ( R ~ ~ ~ )  
transforms the dynamics of the system into 
i =  P 
@ = 
( j =  72 
RAT + Rx3(X, X) + XRx(3(X, X) - CAI)  
RA = RxG(X,A) 
(11) 
I n  addition, the (q,p)-subsystem in (11) is  exact dy- 
namic feedback linearizable as 
4 = P ,  P = u  
where q, p ,  U E W 2  by applying the following invertible 
change of control 
7 = RX% - ;(A, A ) W  - A(LJ(x, A) - cxI)el 
Proof: By direct calculation. 0 
Now, we are ready to present our main result on E- 
stabilization of kinematic SE(2) vehicles. 
Proposition 2. (€-stabilization) A n y  desired position 
20 E R2 for  the kinematic SE(2) wehicle in (6) can be 
rendered globally exponentially €-stable by applying the 
following quasi-smooth dynamic state feedback 
7 = -Cpei l (q  - ZO) - C d v  - cdXe1 
- G(X, X)W - (;(A, A) - c x I ) i e l  (12) 
A = -  cx(X - E ) ,  X(0) > € 
where % , C ~ , C X  > 0 are constants and q is  defined in 
(10). I n  addition, w( t )  --+ 0 as t + 00. 
4692 
Proof: Fix an E > 0, clearly X = E is globally exponen- 
tially stable for the X-subsystem and X ( t )  2 E > 0, Vt 2 
0. Let qo = 20. Based on proposition 1, q = U. Thus, 
applying the following feedback 
U = -cp(q - 40) - cdp, cp, cd > 0 
guarantees global exponential stability of ( q , p )  = 
(zd,O). Substituting this feedback law in (1) gives the 
dynamic state feedback in the question. Since both 
p = i and X are exponentially vanishing, from Lemma 
0 
Remark 3. Based on dynamics of the kinematic SE(2) 
vehicle in (6) ,  if TI = . q  = 0, z remains invariant and 
in local coordinates 6 = 72. Therefore, the orienta- 
tion of the vehicle can be exponentially stabilized to 
any desired orientation without changing the position. 
Hence, it is sufficient to only stabilize the translational 
dynamics of a kinematic SE(2) vehicle as in Proposi- 
tion 2 and then switch to a local controller that changes 
the attitude of the vehicle to a desired attitude. 
2, it follows that w(t) + 0 as t -+ 00. 
Exponential etracking of a smooth desired output zd(.) 
can be obtained based on the following result. 
Proposition 3. (e-tracking) Let zd(t) : R + R2 be 
a C2 smooth desired trajectory. Then, the following 
dynamic state feedback law achieves global exponential 
e-tracking for the desired output (i.e. position) xd(') of 
the kinematic SE(2)  vehicle in (6) 
7 = - C p R i ' Z  - CdV - cdiel 
(13) 
- &(A, X)V - (&(A, A) - cxI)Ael 
+ Ril(Cpzd + cdxd + Zd) 
i = -cx(X-E) 
where cp,cd,cx > 0 are constants and q is defined in 
(10). 
Proof: Let us consider the following partial state- 
output feedback law 
'11 = -cp(q - zd)  - cd(p - i d )  f ?d, cp, cd > 0 
and define the output error as e = q - xd. Then, e 
satisfies the following output error dynamics 
e fcde f C p  = 0 
Since cp,cd > 0, the output error e globally exponen- 
tially converges to zero. This means that z(t) globally 
exponentially converges to a X-neighborhood of %d(t) 
where X exponentially converges to E. cl 
5 Example: A Mobile Robot 
Consider the mobile robot depicted in Fig. 1 [l, 21. 
The robot has two rolling wheels that can be controlled 
t Y  
Figure 1: A mobile robot 
independently using input torques. The dynamics of 
this nonholonomic mobile robot in coordinates is given 
in equation (6) .  
Figures 2 and 3 show the path of the mobile robot 
starting at position x = (2, 3)T for the initial orienta- 
tion angles 0 = k7r/4, k = 0,.  . . ,7. The trajectories for 
the position and input torques (controls) are shown in 
Figure 4. These results demonstrate that the controller 
aggressively stabilizes the origin for this nonholonomic 
mobile robot. Each trajectory exponentially converges 
to a point within a distance e = 0.01 from the desired 
equilibrium xd = 0. This is sufficiently close to the ori- 
gin for all practical purposes. The values of the param- 
eters in all simulations for the mobile robot were cho- 
sen as CA = l,cp = 1,cd = 2,X(0) = 0 . 5 , ~  = 0.01. The 
trace trajectories of this nonholonomic robot are shown 
in Fig. 5. Also, Fig. 6 demonstrates simulation results 
of the E-tracking for a nonholonomic robot starting at 
position (4,4) with orientation angle x / 2 .  The desired 
trajectory is an ellipse (z, y) = (3 sint, 4cost). Clearly, 
the robot very quickly converges to an e-neighborhood 
of the desired trajectory. 
6 Conclusion 
In this paper, we introduced a class of diffeomorphisms 
referred to as near-identity diffeomorphisms which al- 
low dynamic partial feedback linearization of the trans- 
lational dynamics of certain classes of SE(2) vehicles. 
This in turn led to global exponential 6-stabilization 
and €-tracking in position for these planar vehicles. The 
main features of the obtained control laws are that they 
are coordinate independent (i.e. require no switching 
of the local charts), and are directly designed for the 
dynamic model of a vehicle as compared to the kine- 
matic model. We applied our results to tracking and 
stabilization of a wheeled nonholonomic mobile robot. 
The simulation results demonstrate that the dynamic 
state feedback used for estabilization and etracking 
of this mobile robot is an aggressive control law (i.e. 
the solution converges exponential fast to the desired 
4693 
position/trajectory). 
References 
[l] A. Astolfi, “Discontinuous control of nonholo- 
nomic systems,” Systems & Control Letters, vol. 27, 
[2] A. Astolfi, “On the stabilization of the nonhole 
nomic systems,” The 33rd Conf. on Decision and Con- 
trol, pp. 3481-3484, Orlando, FL, 1994. 
[3] R. T. M’Closkey and R. M. Murray, “Exponen- 
tial stabilization of driftless nonlinear control systems 
via time-varying homogeneous feedback,” Proc. IEEE 
conf. on Decision and Control, pp. 943-948, San Anto- 
nio, TX, 1993. 
[4] K. Y. Pettersen and 0. Egeland, “Exponential 
stabilization of an underactuated surface vessel,” Proc. 
IEEE Conf. on Deision and Control, pp. 1682-1687, 
1996. 
[5] J. Hauser, S. Sastry, and G. Meyer, “Nonlin- 
ear control design for slightly non-minimum phase sys- 
tems,” Automatica, vol. 28, pp. 665-679, 1992. 
[6] P. Martin, S. Devasia, and B. Paden, “A different 
look at output tracking: control of a VTOL aircraft,” 
Proceedings of 33rd IEEE Conference on Decision and 
Control, pp. 2376-2381, 1994. 
[7] M. Milam and Murray R. M., “A testbed for 
nonlinear flight control techniques: the Caltech ducted 
fan,” Proc. of the IEEE Int. Conf. on Control Appli- 
cations, vol. 1, pp. 345-351, 1999. 
[8] M. Milam, K. Mushambi, and Murray R. M., 
“A new computational approach to real-time trajec- 
tory generation for constrained mechanical systems,” 
Proc. of the 39th IEEE Conf. on Decasion and Control, 
(91 R. Olfati-Saber, “Exponential etracking and E- 
stabilization of second-order nonholonomic S E (  2) ve- 
hicles using dynamic state feedback,” Proceedings of 
the American Control Conference, Anchorage, Alaska, 
May 2002. 
[lo] R. Murray and S. Sastry, “Nonholonomic mo- 
tion p1anning:steering using sinusoids,” IEEE Pans .  
on Automatic Control, vol. 38, no. 5, pp. 700-716, May 
1993. 
[ll] 3.-P. Pomet, “Explicit design of time-varying sta- 
bilizing control laws for a class of controllable systems 
without drift,” Systems & Control Letters, vol. 18, pp. 
[12] R. T. M’Closkey and R. M. Murray, “Exponen- 
tial stabilization of driftless nonlinear control systems 
using homogeneous feedback,” IEEE Trans. on Auto- 
matic Control, pp. 614-628, May 1997. 
pp. 37-45, 1996. 
vol. 1, pp. 845-551, 2000. 
147-158, 1992. 
[13] A. Bloch and S. Drakunov, “Stabilization and 
tracking in the nonholonomic integrator via sliding 
modes,” Systems & Control Letters, vol. 29, pp. 91-99, 
1996. 
[14] M.-C. Laiou and A. Astolfi, ‘LQuasi-smooth con- 
trol of chained systems,” Proc. of the American Control 
Conference, San Diego, CA, June 1999. 
[15] R. Olfati-Saber, Nonlinear Control of Un- 
demctuated Mechanical Systems with Application to 
Robotics and Aerospace Vehicles, Ph.D. thesis, Mas- 
sachusetts Institute of Technology, Department of Elec- 
trical Engineering and Computer Science, February 
2001,http://wuu.cds.caltech.edu/”olfati. 
1 
1 
0 5  
0 
0 0 5  1 1 5  2 2 5  
vLLl 0 5  0 0 5  1 1 5  2 2 5  
.i/ 1 I 
0 5  
0 
0 0 5  1 1 5  2 2 5  
I1 1 
Figure 2: Trajectories of the nonholonomic mobile 
robot in (zl,zz)-plane for initial position z = 
(2,3)T (U = 0) and orientation angles 0 = 
0, ~ 1 4 ,  ~ / 2 , 3 ~ / 4 .  
5 
1 1 
Figure 3: Trajectories of the nonholonomic mobile 
robot in (21, zz)-plane for initial position 3: = 
(2,3)T (U = 0) and orientation angles 0 = 
K, 5 ~ / 4 , 3 ~ / 2 , 7 ~ / 4 .  
4694 
- \  
3 -. 
2 5  \ 
30 
20- 
fn 2 10- 
P 
P 
t 0 
Q-lo-ll  
4- %= 
-20- 
Figure 4: Trajectories of the nonholonomic mobile robot 
in ( 2 1 ,  zz)-plane starting at z = (2, 3)T (w = 0) 
with angle 0 = a14 
1 
n 
’: 
I :  
:: 
I 
J 0-y- I I  
I 1  
I !  
, I  
0 
I 
z 0 -  
1 -  
2 -  
3- 
-4. 
-5 
-4 
~ 
51 
4 2 0 2 4 8 
Figure 5: Trace trajectories of a two-wheel nonholonomic 
robot with initial position (5,4) and heading 
angles (a) 6 = 0 and (b) 6 = n /2 .  
I 
0 2 4 8 8 10 12 14 18 18 
-4 
that=) 
5 10 15 
time (sec) 
-30; 
(b) 
3 
5r 
4 -  
3- 
2- 
1 ’  
D -  
1 -  
2 -  
3- 
-4 - 
Figure 6: (a),(b) Position trajectory and control for track- 
ing starting at the initial point (4,4,7r/2) and 
(c) The associated trace trajectories of the non- 
holonomic robot for initial condition in (a). 
4695 


Exponential ε-tracking and ε-stabilization of second-order nonholonomic SE(2) vehicles using dynamic state feedback

A different look at output tracking: control of a VTOL aircraft,”

A new computational approach to real-time trajectory generation for constrained mechanical systems,”

A testbed for nonlinear flight control techniques: the Caltech ducted fan,”

Discontinuous control of nonholonomic systems,”

Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift,”

Exponential stabilization of an underactuated surface vessel,”

Exponential stabilization of driftless nonlinear control systems using homogeneous feedback,”

Exponential stabilization of driftless nonlinear control systems via time-varying homogeneous feedback,”

LQuasi-smooth control of chained systems,”

Nonholonomic motion p1anning:steering using sinusoids,”

Nonlinear control design for slightly non-minimum phase systems,”

Nonlinear Control of Undemctuated Mechanical Systems with Application to Robotics and Aerospace Vehicles,

On the stabilization of the nonhole nomic systems,”

Stabilization and tracking in the nonholonomic integrator via sliding modes,”

http://authors.library.caltech.edu/5152/1/OLFacc02.pdf

Exponential ε-tracking and ε-stabilization of second-order nonholonomic SE(2) vehicles using dynamic state feedback

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

Caltech Authors - Main