Nonlinear Control of Wheeled Mobile Robots

145 Pages·2001·5.67 MB·English

by  Warren Dixon PhD

1
Model Development and Control
Objectives
1.1 Introduction
In this chapter, we describe the kinematic model for wheeled mobile robots
(WMRs) for the so-called kinematic wheel under the nonholonomic con-
straint of pure rolling and non-slipping. Based on the kinematic model, we
present differentiable, time-varying kinematic controllers for the regulation
and the tracking control problems. Through a Lyapunov-based stability
analysis, we demonstrate that the controllers yield global asymptotic reg-
ulation or tracking and that all signals remain bounded during closed-loop
operation. From the stability analysis, it is clear that the regulation problem
cannot be solved as a special case of the tracking problem due to restric-
tions on the desired trajectory. Motivated by the facts that the kinematic
controllers are limited to asymptotic results and that the tracking controller
does not solve the regulation problem as a special case, we introduce a uni-
fled kinematic control structure (i.e., the regulation problem is a special
case of the tracking control problem). Through a Lyapunov-based stability
analysis, we demonstrate that the unified kinematic controller yields global
exponential tracking and regulation and that all signals remain bounded
during closed-loop operation.
In addition to the kinematic control problem, we also examine incorporat-
ing the dynamic model in the overall control design. To this end, we present
the dynamic model and describe several associated properties. Based on the
2 .1 Model Development and Control Objectives
dynamic model, we investigate the use of standard backstepping techniques
to develop a torque control input for the unified control problem (versus
the velocity control input for kinematic controllers). Through a Lyapunov-
based stability analysis, we demonstrate that the unified controller yields
global exponential regulation; however, we illustrate how the proposed de-
sign breaks down for the tracking problem. In subsequent chapters, we
illustrate how the unified kinematic controller can be redesigned to incor-
porate the effects of the dynamics via standard backstepping techniques.
1.2 Kinematic Model Development
The kinematic model for the so-called kinematic wheel under the nonholo-
nomic constraint of pure rolling and non-slipping is given as follows ]71[
= s(q)v
(1.1)
where q(t), O(t) 3 E R are defined as
)2.1(
xc(t) and yc(t) denote the position of the center of mass (COM) of the
WMR along the X and Y Cartesian coordinate frames and O(t) E 1 R
represents the orientation of the WMR (see Figure 1.1), ~s(t) and ~¢(t)
denote the Cartesian components os0 of [ the linear ]0 velocity, denoted by ~v (t) E
R] ,1 0(t) E 1~] denotes the angular velocity, the matrix S(q) E 2×3~] is
defined as follows
S(q)= sin(cid:127) 0 (1.3)
0 1
and the velocity vector v(t) 2 E R is defined as
V = V[ 1 v2]T = [vl ]O T • (1.4)
Note that the COM and the center of rotation are assumed to coincide.
1.3 Regulation Problem
In this section we present the open-loop error system for the regulation
problem. Based on the open-loop error system, we present the differen-
tiable, time-varying feedback controller given in [20] and then examine the
stability of the resulting closed-loop error system through a Lyapunov-
based stability analysis.
3.1 Regulation Problem 3
cY
c X X ~
Figure 1.1. Wheeled Mobile Robot
1.3.1 Open-Loop Error System
The control objective for the regulation problem is to force the actual
Cartesian position and orientation to a constant reference position and
orientation. To quantify the regulation control objective, we define k(t),
~(t), 0(t) E ~i as the difference between the actual Cartesian position and
orientation and the reference position and orientation as follows
~ = x~ - x~ ~ = y~ - Yr~ 0 = 0 - 0~ (1.5)
where x~(t), yc(t), O(t) were defined in (1.2) and x~c, yr~, ~0 1 E N represent
the constant reference position and orientation. To facilitate the closed-
loop error system development and stability analysis, a global invertible
transformation was defined 0sion c[ [20] as foll ows 0nis
(1.6)
2e = -sinO cosO 0
3e 0 0 1
to relate the auxiliary error signals denoted by el(t), e2(t), e3(t) E 1 R to
the position and orientation regulation error signals ~(t), ~(t), 0(t) defined
4 .1 Model Development and Control Objectives
in (1.5). After taking the time derivative of (1.6) and using (1.1-1.4), the
kinematic model given in (1.1) can be rewritten in terms of the auxiliary
variables defined in (1.6) as follows
(1.7)
e2 "-~ --v2el •
a~ v2
Remark 1.1 Based on the inverse of the transformation defined in (1.6)
given as follows 0soc[ 0nis 0][ex]
j~ = sin 0 cos ~ 0 e2 (1.8)
0 0 1 ea
it is clear that if t~m el (t), e2 (t), e3(t) = ,O then miloo.--t ~(t), ~l(t), ~(t) = .O
1.3.2 Control Development
The regulation control objective is to design a controller for the transformed
kinematic model given by (1.7) that forces the actual Cartesian position
and orientation to a constant reference position and orientation. Based on
this control objective, a differentiable, time-varying controller was proposed
in [20] as follows
v2 -k2e3 + e~ sin(t) (1.9)
where kl, k2 E 1~] are positive constant control gains. After substituting
(1.9) into (1.7) for vl(t) and v2(t), the following closed-loop error system
is obtained
~ = -v2el • (1.10)
e3 -k2e3 + 2 e sin(t)
Remark 1.2 Note that the closed-loop dynamics for e3(t) given in (1.10),
represent a stable linear system subjected to an additive disturbance given
by the product e2(t)sin(t). If the additive disturbance is bounded 5.e., if
e2(t) C Coo), then it is clear that e3(t) C/:oo- Furthermore, if the additive
disturbance asymptotically vanishes 5.e., if oomLFt e2(t) = O) then it is clear
from standard linear control arguments [7] that Um e3 (t) = .O
t ~--- O0
3.1 Regulation Problem 5
1.3.3 Stability Analysis
Given the closed-loop error system in (1.10), we can now invoke Lemma
A.2, Lemma A.12, and Lemma A.14 of Appendix A to determine the sta-
bility result for the kinematic controller given in (1.9) through the following
theorem.
Theorem 1.1 The differentiable, time-varying kinematic control law given
in (1.9) ensures global asymptotic position and orientation regulation in the
sense that
(i.II)
lira 5:(t), (cid:127)(t), O(t) = .O
*-..'-t CO
Proof: To prove Theorem i.I, we define a non-negative function denoted
by 1/1 (t) e ~1 as follows
+
After taking the time derivative of (1.12), substituting (1.10) into the re-
suiting expression for el(t) and e2(t), and then cancelling common terms
the following expression is obtained
=-kle . )31.1(
Based on (1.12) and (1.13), it is clear that el(t), e2(t) E /:oo. Since
e2(t) E/~oo it is clear from Remark 1.2 that e3(t) E/:oo. Based on the fact
that el(t), e2(t), e3(t) E/:oo, we can utilize (1.9) and (1.10) to prove that
vl(t),v2(t),el(t),e2(t),e3(t)
e ~oo. Since ~l(t), e2(t), e3(t) e /:o~, we can
invoke Lemma A.2 of Appendix A to conclude that el(t), e2(t), e3(t) are
uniformly continuous. After taking the time derivative of (1.9) and utilizing
the aforementioned facts, we can show that ~)l(t), /J2(t) C/:oo, and hence,
vl(t) and v2(t)are uniformly continuous.
We have already proven that el(t), ~l(t) E £0o. If we can now prove that
el(t) E/:2, we can invoke Leinma A.12 of Appendix A to prove that
lim el(t) ---- .O (1.14)
t +---- ~C
To this end, we integrate both sides of (1.13) as follows
5/ 0/
Vl(t)dt = kl e~(t)dt. (1.15)
-
After evaluating the left-side of (1.15), we can conclude that
F
kl e2(t)dt = 1V (0) - 1V (co) < 1/1 (0) < c~ (1.16)
6 .1 Model Development and Control Objectives
where we utilized the fact that VI(0) _> 1V (oo) > 0 (see (1.12) and (1.13)).
Since the inequality given in (1.16) can be rewritten as follows
0/i
4(t)dt < V < o~ (1.17)
we can utilize Definition A.1 to conclude that et(t) E £2. Based on the
facts that el(t),~l(t) e L:oo and el(t) E /:2, we can now invoke Lemma
A.12 of Appendix A to prove the result given in (1.14).
After taking the time derivative of the product el(t)e2(t) and then sub-
stituting (1.10) into the resulting expression for the time derivative of el (t),
the following expression is obtained
d
t-d (ele2) = [e~v2] + el (e2 -- kle2). (1.18)
Given the facts that lim el(t) = 0 and the bracketed term in (1.18) is
t--*OO
uniformly continuous (i.e., e2(t) and v2(t) are uniformly continuous), we
can invoke Lemma A.14 of Appendix A to conclude that
lira d (el(t)e2(t)) = 0 lira e~(t)v2(t) = 0. (1.19)
t ---*oO a~: t ---+oo
From (1.19), it is clear that
lira e2(t)v2(t) = 0. (1.20)
After utilizing (1.9), (1.10), (1.14), and (1.20), we can conclude that
lim vl(t) = 0 lim ~l(t) = 0 lira ~2(t) = 0. (1.21)
t---*OO t---*oo t~Oo
To facilitate further analysis, we take the time derivative of the product
e2(t)v2(t) and utilize (1.7) and (1.9) to obtain the following expression
~/(~v~) d
= [4 cos(t)] + ~ (v~ + 24 sin(t)) - k~v~. (1.22)
Since the bracketed term in (1.22) is uniformly continuous, we can utilize
(1.20) and (1.21) and invoke Lemma A.14 of Appendix A to conclude that
t--.oo lim d dt (e2(t)v2(t)) = 0 mil~o.-t e~(t) cos(t) = 0. (1.23)
From the second limit in (1.23), it is clear that
lira e2(t) = 0. (1.24)
t OO~----
Based on (1.24), it is clear from Remark 1.2 that
lim e3(t) = 0. (1.25)
t--*OO
After utilizing (1.8), (1.14), (1.24), and (1.25), we obtain the global asymp-
totic regulation result given in (1.11). •
4.1 Tracking Problem 7
1.4 Tracking Problem
For the previous regulation problem, the control objective was to force the
actual Cartesian position and orientation to a constant reference position
and orientation. In contrast to the regulation problem, the control objective
for the tracking control problem is to force the actual Cartesian position
and orientation to track a time-varying reference trajectory. To quantify
the tracking control objective, we define ~(t), ~(t), 0(t) I C R as follows
= xc - xrc = Y cY - crY 0 = 0 - rO (1.26)
where the actual position and orientation, denoted by xc(t), yc(t), 0(t),
were defined in (1.2) and qr (t) = [ x~c(t) y~(t) Or(t) ]T C (I1 3 denotes
the time-varying reference position and orientation. In order to ensure that
the reference trajectory is selected to satisfy the pure rolling and nonslip-
ping constraint imposed on the actual WMR, the reference trajectory is
generated via a reference robot which moves according to the following
dynamic trajectory
rO = S(q~)vr (1.27)
where S(-) was defined in (1.3) and v~(t) = [ vlr(t) v2r(t) ]T E 2 R
denotes the reference time-varying linear and angular velocity. With regard
to (1.27), it is assumed that the signal v~(t) is constructed to produce the
desired motion and that v~(t), iJr(t), qr(t), and 0r(t) are bounded for all
time.
Remark 1.3 To illustrate one method for selecting vr(t) such that a de-
sired Cartesian path is generated, we first express the desired trajectory as
follows
yrc(t) = gr(Xrc(t)) )S2.1(
where gr (') E 1 R is a desired path selected to eb second order differentiable.
To facilitate further analysis, we divide the first row of (1.27) by the sec-
ond row and perform some algebraic manipulation to obtain the follovnng
expression
rgO
-- tan rO (1.29)
crXO
where (1.28) was utilized. After taking the time derivative of (1.29), we
obtain the follougng expression
d ( rgO ~ 02gr c~& = (1 + tan 2 0r) 0r- (1.30)
8 .1 Model Development and Control Objectives
After substituting (1.27) into (1.30) for &~c(t) and then rearranging the
resulting expression, we obtain the following expression
I {02gr~ a (1.31)
0r - (1 +tane0r) \~x2rc] vlrcosOr =V2r
where vlr(t) represents the desired linear velocity which can eb arbitrarily
selected. Based on (1.28) and (1.29), it is clear that at(O) should eb selected
as follows
)23.1(
yrc(0) = gr(x~o(0))
0r(0) 0--~rclxrc(0)
= ta-nI
where xrc( ) O is arbitrarily selected.
1.4.1 Open-Loop Error System
To develop the open-loop tracking error system, we take the time deriv-
ative of (1.6) and use (1.1-1.4), (1.26), and (1.27) to obtain the following
expression [20] 2ev+lv[ oeriv ]3e
e2 = --v2el -4- Vlr Sin e3 (1.33)
3}~ v2 -- v2r.
Remark 1.4 Note that if Vlr(t), v2r(t) = 0, then the open-loop regulation
error system given in (1.7) is recovered.
1.4.2 Control Development
The tracking control objective is to design a controller for the transformed
kinematic model given by (1.33) that forces the actual Cartesian posi-
tion and orientation to track the timewarying reference trajectory given in
(1.27). Based on this control objective, a differentiable time-varying con-
troller was proposed in [20] e3, [ as folklel+vlcos lows ]
= sin e3 (1.34)
2 V --Vlr e3 2 e -- k2e3 -F v2r
where kl, k2 E 1 E are positive constant control gains. After substituting
(1.34) into (1.33) for vl(t) and v2(t), we obtain the following closed-loop
1.4 Tracking Problem 9
error system
v2e2 - klel
-v2el q- wit sine3
(1.35)
sin 3 e
--Vlr e 2 -- k2e3.
e3
1.4.3 Stability Analysis
Given the closed-loop error system in (1.35), we can now invoke Lemma A.2,
Lemma A.12, and Lemma A.13 of Appendix A to determine the stability
result for the kinematic controller given in (1.34) through the following
theorem.
Theorem 1.2 Provided the reference trajectory (i.e., v~( t ), iJr( , ) t q~( , ) t and
O~(t)) is selected to be bounded for all time and that
lim vlr(t) ~ 0, (1.36)
t---*Oo
the kinematic control law given in (1.3~) ensures global asymptotic position
and orientation tracking in the sense that
lim ~(t), ~(t), ~(t)= 0. (1.37)
$~OO
Proof: To prove Theorem 1.2, we define a non-negative function denoted
by V2(t) E ~1 as follows
12 12
2V = ~e½ + ~e2 + ~e 3. (1.38)
After taking the time derivative of (1.38), substituting (1.35) into the re-
sulting expression for ~l(t), ~2(t), and ~3(t), and then cancelling common
terms, we obtain the following expression
V2 = -kle~ - k2e~. (1.39)
Based on (1.38) and (1.39), it is straightforward that el(t), e2(t), e3(t) E
/:co and that el(t),e3(t) E £2 (see (1.15-1.17)). Since el(t), e2(t), e3(t) e
£~,we can utilize (1.34), the assumption that v~(t) E /:~, and the fact
that
sin e3
lim ~ = 1 (1.40)
e3---*0 e 3
to prove that vl(t), v2(t) E/~. From these facts, we can utilize the closed-
loop tracking error dynamics given in (1.35) to prove that ~l(t), ~2(t),
~3(t) ~ L~; hence, by invoking Lemma A.2 of Appendix A, it is clear
10 .1 Model Development and Control Objectives
that el(t), e2(t),e3(t) are uniformly continuous. Since el(t), ~l(t), e3(t),
~3(t) E/:oo and el(t), e3(t) e £:2, we can invoke Lemma A.12 of Appendix
A to conclude that
lira el(t) ---- 0 lira e3(t) ---- .O (1.41)
t--~O0 QO*---$
In order to prove that lim e2(t) = 0, we first take the time derivative of
OO*--t
the closed-loop tracking error dynamics for e3(t) given in (1.35) as follows
e3 -- -~)lr 3e3enis e2 - vlr 3e3enis "2e - vlr ,k//e3 - coSeae~ sine3) e3e2 - k2~3.
(1.42)
Based on the fact that e2(t), ~2(t), e3(t), ~3(t) e/:oo, the assumption that
v~(t), 9~(t) e/:oo, and the fact that
lira ¢/ e3 cos e3 --- sin e3 )
(1.43)
3-0 \ e] = 0
we can conclude from (1.42) that ~3(t) C £oo; hence, from Lemma A.2 of
Appendix A, ~3(t) is uniformly continuous. Based on (1.41) and the fact
that ~3(t) is uniformly continuous, we can use the following equality
lim /0 ~ d (e3(T)) dT= lim e3(t) + Constant (1.44)
OO*---t ~TT oo*----t
to conclude that the left-side of (1.44) exists and is finite; hence, we can
now invoke Lemma A.13 of Appendix A to prove that
lira 3(t) = o. (1.45)
Oo*---t
Based on (1.41) and (1.45), it is straightforward from (1.35) that
t~oo lim vlrt ,t )~s~inee32(t[) "t" ) = 0. (1.46)
Finally, based on (1.36) and (1.40), we can conclude from (1.46) that
nm e2(t) = o. (1.47)
Based on (1.8), (1.41) and (1.47), the global asymptotic tracking result
given in (1.37) can now be directly obtained. •
Remark 1.5 Based on the restriction placed on the reference trajectory
given in (1.36), the regulation problem described in Section 1.3 cannot eb
solved with the tracking controller given in (1.34).