Leader–follower formation control of nonholonomic mobile robots with input constraints

Abstract

The paper deals with leader–follower formations of nonholonomic mobile robots, introducing a formation control strategy alternative to those existing in the literature. Robots’ control inputs are forced to satisfy suitable constraints that restrict the set of leader possible paths and admissible positions of the follower with respect to the leader. A peculiar characteristic of the proposed strategy is that the follower position is not rigidly fixed with respect to the leader but varies in proper circle arcs centered in the leader reference frame.

Full text

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Leader-Follower Formation Control of Nonholonomic
Mobile Robots with Input Constraints
Luca Consolinia, Fabio Morbidib, DomenicoPrattichizzob, Mario Tosquesc
aDept. of Information Engineering, University of Parma, Via Usberti 181/a, 43100 Parma, Italy
bDept. of Information Engineering, University of Siena, Via Roma 56, 53100 Siena, Italy
cDept. of Civil Engineering, University of Parma, Via Usberti 181/a, 43100 Parma, Italy
Abstract
The paper deals with leader-follower formations of nonholonomic mobile robots, introducing a formation control strategy
alternative to those existing in the literature. Robots control inputs are forced to satisfy suitable constraints that restrict the
set of leader possible paths and admissible positions of the follower with respect to the leader. A peculiar characteristic of the
proposed strategy is that the follower position is not rigidly fixed with respect to the leader but varies in proper circle arcs
centered in the leader reference frame.
Key words: Formation control; Autonomous mobile robots; Input constraints; Geometric conditions; Adaptive stabilization
1 Introduction
In the last decade formation control became one of
the leading research areas in mobile robotics. By for-
mation control we simply mean the problem of con-
trolling the relative position and orientation of the
robots in a group while allowing the group to move
as a whole. Different robot formation typologies have
been studied in the literature: ground vehicles (Das et
al., 2002; Fax and Murray, 2004; Marshall et al., 2004;
Lin et al., 2005; Ghabcheloo et al., 2006), unmanned
aerial vehicles (UAVs) (Singh et al., 2000; Koo and
Shahruz, 2001), aircraft (Giulietti et al., 2000; Fierro et
al., 2001), surface and underwater autonomous vehicles
(AUVs) (Skjetne et al., 2002; Edwards et al., 2004).
Three main approaches have been proposed to tackle
the robot formation control problem: behavior based,
virtual structure and leader following. In the beha-
vior based approach (Balch and Arkin, 1998; Lawton
et al., 2003) several desired behaviors (e.g. collision
avoidance, formation keeping, target seeking) are pre-
⋆This paper was not presented at any IFAC meeting.
Corresponding author. Tel.: +39 0521 906114; Fax: +39 0521
905723.
Email addresses: luca.consolini@polirone.mn.it
(Luca Consolini), morbidi@dii.unisi.it (Fabio Morbidi),
prattichizzo@dii.unisi.it (Domenico Prattichizzo),
mario.tosques@unipr.it (Mario Tosques).
scribed for each robot. The resulting action of each
robot is derived by weighing the relative importance
of each behavior. The main problem of this approach
is that the mathematical formalization is difficult and
consequently it is not easy to guarantee the conver-
gence of the formation to a desired configuration. The
virtual structure approach (Lewis and Tan, 1997; Do
and Pan, 2007) considers the formation as a single vir-
tual rigid structure so that the behavior of the robotic
system is similar to that of a physical object. Desired
trajectories are not assigned to each single robot but
to the entire formation as a whole. The behavior of
the formation in this case is exactly predictable but a
large inter-robot communication bandwidth is required.
In the leader-follower approach a robot of the forma-
tion, designed as the leader, moves along a predefined
trajectory while the other robots, the followers, are
to maintain a desired distance and orientation to the
leader (Das et al., 2002). Perhaps the main criticism to
the leader-follower approach is that it depends heavily
on the leader for achieving the goal and over-reliance on
a single agent in the formation may be undesirable, es-
pecially in adverse conditions (Fax and Murray, 2004).
Nevertheless leader-follower architectures are particu-
larly appreciated for their simplicity and scalability.
The leader-follower formation control of nonholonomic
mobile robots with bounded controlinputs is the subject
of this paper.

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We propose a leader-follower setup alternative to those
existing in the literature (Mariottini et al., 2007),
assuming that the desired angle between the leader
and the follower is measured in the follower frame in-
stead of the leader frame. As shown in (Consolini et
al., 2006), this approach guarantees lower control ef-
fort and smoother trajectories for the follower than the
usual leader-follower approaches defined on the leader
reference frame (Das et al., 2002). According to the
proposed setup we find suitable conditions on leader
velocity and trajectory curvature ensuring the follower
gets into and keeps the formation while satisfying its
own velocity constraints. As pointed out in Remark 1,
from a geometric point of view the follower position is
not fixed with respect to the leader reference frame but
varies in suitable circle arcs centered in the leader ref-
erence frame. The main contributions of this paper are
stated in two theorems. A first theorem gives a sufficient
and necessary condition on leader and follower velocity
bounds for the existence of a control law that allows the
follower to maintain the formation independently from
the trajectory of the leader. A second theorem provides
a sufficient condition and a control law for the follower
to asymptotically reach the formation for any initial
condition and any leader motion, while still respecting
the bounds.
Notation. The following notation is used in the paper:
R+= {t ∈ R | t ≥ 0}; ∀a,b ∈ R, a ∧ b = min{a,b},
a ∨ b = max{a, b}; ∀t > 0, sign(t) = 1; sign (0) = 0;
∀t < 0,
?x, y?
τ(θ) = (cosθ,sinθ)T, ν(θ) = (−sinθ,
∀x ∈ R2\{0}, arg(x) = θ where θ ∈ [0,2π) and
x = ?x?τ(θ); ∀w ∈ R2, w⊥= ?w?ν(arg(w)).
sign(t) = −1; ∀x, y ∈ Rn(n ≥ 1),
?n
=
i=1xiyi, ?x?=
??x, x?; ∀θ∈
R,
cosθ)T;
2Basic definitions
Consider the following definition of robot as a velocity
controlled unicycle model.
Definition 1 Let R = (x,y,θ)T∈ C1([0,+∞),R3).
R is called a robot with initial condition R ∈ R3(and
control (v,ω)T∈ C0([0,+∞),R2)) if the following sys-
tem is verified
?
˙ x = v cosθ, ˙ y = v sinθ,˙θ = ω
(x(0), y(0), θ(0)) = R.
(1)
If v(t) ?= 0, we set κ(t) = ω(t)/v(t) which is the (scalar)
curvatureof the path followedby the robot at time t. De-
note by P(t) = (x(t), y(t))Tthe position of R at time t,
θ(t) its heading, τ(θ(t)) the normalized velocity vector
and ν(θ(t)) the normalized vector orthogonal to τ(θ(t));
then {τ(θ(t)), ν(θ(t))} represents the robot reference
frame at time t.
R1
P1
θ1
τ(θ1)
φ
R0
θ0
τ(θ0)
P0
d
Fig. 1. (d,φ)-formation.
Definition 2 Let (V,K−,K+) ∈ R3and R be a
robot. We say that R satisfies the trajectory constraint
(V,K−,K+) if ∀t ≥ 0
0 < v(t) ≤ V, K−≤ κ(t) ≤ K+.
All along the paper we suppose that the following con-
dition is satisfied:
Assumption 1 (Physical constraint) Let Vp, K−
K+
pbe three constants. We suppose that every robot con-
sidered in this paper satisfies the trajectory constraint
(Vp,K−
p,
p,K+
p).
This constraint is assumed to be the mechanical limita-
tion common to all the robots considered in this paper.
The following definition introduces the notion of leader-
follower formation used in the paper.
Definition 3 Let d > 0, φ : |φ| <
R0= (PT
that R0and R1are in (d,φ)-formation with leader R0
at time t, if
π
2
and let
0,θ0)T, R1= (PT
1,θ1)Tbe two robots. We say
P0(t) = P1(t) + dτ(θ1(t) + φ),(2)
and, simply, that R0and R1are in (d,φ)-formation with
leader R0, if (2) holds for all t ≥ 0. Moreover we say that
R0and R1are asymptotically in (d,φ)-formation with
leader R0if
lim
t→∞P0(t) − (P1(t) + dτ(θ1(t) + φ)) = 0 .
With reference to Fig. 1, Definition 3 states that two
robots R0, R1are in (d,φ)-formation with leader R0if
the position P1of the follower R1is always at distance d
from the position P0of the leader R0and the angle be-
tween vectors τ(θ1) and P0−P1is constantly equal to φ,
that is the position P0of the leader remains fixed with
respect to the follower reference frame {τ(θ1), ν(θ1)}.
An equivalent way to state Definition 3 is the following:
set the error vector,
E(t) = P0(t) − (P1(t) + dτ(θ1(t) + φ)), (3)
2

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then R0 and R1 are in (d,φ)-formation (asymptoti-
cally) if and only if E(t) = 0, ∀t ≥ 0 (respectively,
lim
t→∞E(t) = 0).
3 Characterization of a necessary and sufficient
condition for the leader-follower formation
Problem 1 Set d > 0, φ : |φ| <
satisfying the trajectory constraint (V0,K−
beanotherrobot.Findnecessaryandsufficientconditions
on V0, K−
suitable initial conditions for R1such that R0and R1
are in (d,φ)-formation with leader R0(remark that v1
and ω1must be such that Assumption 1 is satisfied).
π
2. Let R0be a robot
0,K+
0) and R1
0, K+
0such that there exist controls v1, ω1and
The following theorem gives an answer to Problem 1.
Theorem 1 Set d > 0, φ : |φ| <π
2.
1) The following properties are equivalent:
A) For any robot R0= (PT
R0, satisfying the trajectory constraint (V0,K−
there exists an initial condition R1and controls v1, ω1
for the robot R1= (PT
in (d,φ)-formation with leader R0.
0, θ0)Twith initial condition
0,K+
0),
1, θ1)Tsuch that R0and R1are
B) The following properties hold:
−1
1
d≤ K−
0≤ K+
0≤ K+
0≤
0≤1
1
dcosφ, if φ ≥ 0
, if φ < 0
−
dcos φ≤ K−
d
(4)
? K−
0≤ K−
0≤ K+
0≤? K+
0
(5)
V0cos?0 ∧ (arcsin(K+
∧(φ − arcsin(K−
0dcosφ) − φ)∧
0dcosφ))?≤ Vpcosφ
(6)
where
? K±
2) If B) holds, for any robot R0satisfying the trajectory
constraint (V0,K−
(d,φ)-formation at time t = 0 and arcsin(K−
θ0(0) − θ1(0) ≤ arcsin(K+
0= (sign K±
p)?((K±
p)−1− dsinφ)2+ d2cos2φ?−1
2.
(7)
0,K+
0) and for any robot R1which is in
0dcosφ) ≤
0dcosφ), with controls,
v1 = v0(t)cos(θ0− θ1− φ)
cosφ
, ω1 = v0(t)sin(θ0− θ1)
dcosφ
(8)
then R0and R1are in (d,φ)-formation and ∀t ≥ 0,
arcsin(K−
0dcosφ) ≤ θ0(t)−θ1(t) ≤ arcsin(K+
0dcosφ) . (9)
R1
φ
φ
P1
Ad(δ,σ1,σ2)
σ1
σ2
P0
θ0
δ
R0
d
Fig. 2. The arc of circle Ad
?δ, σ1,σ2).
Remark 1 Property (9) shows the connection be-
tween the relative heading angle θ0− θ1, the curva-
ture of the path followed by the leader R0, the dis-
tance d and the visual angle φ. Moreover this im-
plies the following geometric property. Since robots
R0, R1 are in (d,φ)-formation, P0(t) − P1(t)
dτ(θ1(t) + φ) = dτ((θ0(t) +
therefore, defining, for any σ1≤ σ2, Ad(δ(t),σ1,σ2) =
dτ([δ(t) + σ1,δ(t) + σ2]) as the arc of circle centered
in the origin, radius d, angle of the reference axis
δ(t) and aperture (σ2− σ1), by (9), P0(t) − P1(t) ∈
Ad(θ0(t)+φ,−arcsin(K+
that is
?θ0(t) + φ + π,
−arcsin(K+
?see Fig. 2 where σ1 = −arcsin(K+
K−
0
et al., 2002), the followers are not rigidly disposed with
respect to the leader reference frame, but their relative
positions vary in time in suitable circle arcs and only
these remain stable with respect to the leader reference
frame
=
φ) + (θ1(t) − θ0(t))),
0dcosφ),−arcsin(K−
0dcosφ))
P1(t) ∈ P0(t) + Ad
0dcosφ),−arcsin(K−
0dcosφ)?.
0dcosφ), σ2 =
−arcsin(K−
0< 0 < K+
0dcosφ) and δ(t) = θ0(t)+φ+π, in the case
?. This shows that, differently from (Das
PROOF. A) ⇒ B). Suppose that R0and R1are in
(d,φ)-formation, the trajectory constraint
0 < v0(t) ≤ V0, K−
0≤ κ0(t) ≤ K+
0, ∀t ≥ 0(10)
is verified and (by Assumption 1) the physical constraint
0 < v1(t) ≤ Vp, K−
p≤ κ1(t) ≤ K+
p, ∀t ≥ 0(11)
holds for robot R1. Since E(t) = P0(t) − (P1(t) +
dτ(θ1(t) + φ)) = 0, differentiating it, we get that
˙P0=˙P1+ d˙θ1ν(θ1+ φ), that is v0τ(θ0) = v1τ(θ1) +
3

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dω1ν(θ1+ φ) which implies, multiplying by the ro-
tation matrix R(−θ1)
v0τ(θ0− θ1) = v1(1, 0)T+ dω1ν(φ) . Therefore it has
to be
ω1= v0sin(θ0− θ1)
dcosφ
?
=(τ(−θ1),ν(−θ1)), that
,(12)
v1= v0
cos(θ0− θ1) +sin(θ0− θ1)sinφ
cosφ
?
= v0cos(θ0− θ1− φ)
cosφ
.
(13)
Then (8) is true and
|θ0(t) − θ1(t)| < π/2, ∀t ≥ 0 ,(14)
sincev1hastobegreaterthanzeroby(11).Furthermore,
setting β(t) = θ0(t) − θ1(t), on [0, +∞)
v0
dcosφ
˙β = ω0− ω1=
?κ0dcosφ − sinβ?. (15)
Let β ∈ C1([0, +∞),R) be a solution of (15), then it is
easy to verify that the following properties hold:
a) if |κ0(t)d cosφ| ≥ c > 1, ∀t ≥ 0 then
limt→+∞|β(t)| = +∞ ;
b) let K be a constant such that |K|dcosφ ≤ 1 if
∀t ≥ 0, κ0(t) = K < 0 (> 0) and −π
(−π ≤ β(0) ≤
arcsin(Kdcosφ);
c) if−
(<)β(0) ≤ (<)arcsin(K+
≤ (<)β(t) ≤ (<)arcsin(K+
notation ≤(<) means that the property holds also when
all inequalities “≤” are replaced with “<”.
To verify (4) suppose that φ > 0 (the reasoning is anal-
ogous for φ ≤ 0). By contradiction, suppose that (4)
is false, then at least one of the following two inequal-
ities must be true: K+
0 >
pose that K+
0 >
of curvature K+
0with constant velocity V0. Then hy-
pothesis a) holds and limt→∞|β(t)| = +∞, therefore
β is unbounded which is impossible by (14). Suppose
that K−
0
< −1
then, by a) as before, a contradiction follows; sup-
pose that K−
being |β(0) − φ| <
we get that limt→+∞β(t) = arcsin(K−
property b) applied with K = K−
?
which implies limt→∞v1(t) < 0, since K−
1 < (K−
Suppose now that (5) does not hold, for instance that
2≤ β(0) ≤ π,
π
2, respectively), then limt→+∞β(t) =
1
dcosφ≤ K−
0≤ K+
0≤
0dcosφ),thenarcsin(K−
0dcosφ), ∀t ≥ 0, where the
1
dcosφ, arcsin(K−
0dcosφ) ≤
0dcosφ)
1
dcosφ,K−
0
< −1
d. Sup-
1
dcosφ, and that R0 follows a circle
d, if φ is such that K−
0dcosφ < −1,
0dcosφ ≥ −1, since −π
π
2by (14) and 0 < φ <
2≤ β(0) ≤ π,
π
2
0dcosφ), by
< 0. Therefore
0
limt→∞v1(t) = V0(1 − (K−
0dcosφ)2+ K−
0dsinφ),
< 0 and
0
0d)2which contradicts (11), therefore (4) holds.
K+
0>? K+
0=
sign K+
p
?
((K+
p)−1− dsinφ)2+ d2cos2φ
.(16)
Suppose for example that K+
soning for the other case is analogous), then (16) im-
0
≥ K−
0
≥ 0 (the rea-
plies that K+
Suppose that R0 follows a circle of curvature K+
with constant velocity V0. Then by the necessity hy-
pothesis there exist initial conditions such that R0
and R1 are in (d,φ)-formation and it has to be
limt→∞sinβ(t) = K+
˙β =
p <
??
(K+
0)−2− d2cos2φ + dsinφ?−1.
0
0dcosφ, since β is such that
0dcosφ−sinβ) and K+
Therefore, by (8): limt→+∞κ1(t) = limt→+∞
limt→∞sinβ(t)?d(cosβ(t)cosφ
= limt→∞
d
??
impossible by (11). By contradiction suppose that (6)
is false and set for simplicity α±= arcsin(K±
(remarkthat |K±
that 0 ∧ (α+−φ) ∧(φ−α−) = α+−φ, analogously we
reason if 0 ∧ (α+−φ) ∧ (φ−α−) = φ−α−. Let R0be
the robot which followsa circle with curvatureK and ve-
locity V0. By hypothesis there exists an initial condition
R1and controls v1, w1such that robot R1is in (d,φ)-
formation with leader R0. If β(0) = θ0(0) − θ1(0) is
different from π−α+then set K = K+
α+and limt→+∞v1(t) = V0
fore the first inequality of (11) is false. If β(0) =
θ0(0) − θ1(0) = π − α+, take K = K+
chosen such that
Since β(0) is now different from π − α+as be-
fore, limt→+∞v1(t) > Vp, which contradicts (11).
If 0 ∧ (α+− φ) ∧ (φ − α−) = 0 then take K =
(which is the solution of arcsin(Kdcosφ) − φ = 0);
since α−≤ φ ≤ α+, it follows that K−
and reasoning as before, we get a contradiction. There-
fore (6) holds. Finally to verify (9), remark that if R0
and R1are in (d,φ)-formation then it is necessary that
−
follows from property c).
B) ⇒ A). Let R0 be a robot which verifies traje-
ctory constraint (10) with V0, K−
ditions (4), (5) and (6). The aim is to show that there
exist initial conditions such that R1 with controls v1,
ω1 given by (8) is in (d,φ) formation with R0 and
constraint (11) is verified. In fact if R1 has the con-
trols v1and ω1, by the definition of E(t), given by (3):
˙E = v0τ(θ0) − (v1τ(θ1) + dν(θ1+ φ)ω1) = v0τ(θ0) −
(v0
cos φ
τ(θ1) + dν(θ1 + φ)v0
v0
cosφR(θ0)[(cosφ, 0)T− sin(θ0− θ1)ν(θ1− θ0+ φ) −
cos(θ0−θ1−φ)τ(θ1−θ0)] = 0 ,withR(θ) = (τ(θ), ν(θ)).
Choose the initial condition of the follower R1such that
R0 and R1 are in (d,φ)-formation at time t = 0 and
arcsin(K−
recall that |K±
V0
dcos φ(K+
0dcosφ ≤ 1,by (4).
ω1(t)
v1(t)=
+ sinβ(t)sinφ)?−1
1
?sign β(t)
?
1
sin2β(t)− 1 cosφ + sinφ)?−1
=
(K+
0)−2− d2cos2φ + dsinφ?−1> K+
pwhich is
0dcosφ)
0dcosφ| ≤ 1 by (4)). Suppose first of all
0, limt→+∞β(t) =
cos(α+−φ)
cosφ
> Vp , there-
0− ǫ with ǫ
>
V0cos(arcsin((K+
0−ǫ)dcosφ)−φ)
cosφ
Vp.
tanφ
d
0 ≤ K ≤ K+
0
1
dcosφ≤ K−
0≤ K+
0≤
1
dcos φ(by (4)), therefore (9)
0, K+
0verifying con-
cos(θ0−θ1−φ)
sin(θ0−θ1)
dcosφ
)=
0dcosφ) < θ0(0) − θ1(0) < arcsin(K+
0|dcosφ ≤ 1 by (4). By c), ∀t ≥ 0
0dcosφ),
4

Page 5

arcsin(K−
0dcosφ) < θ0(t) − θ1(t) < arcsin(K+
Consider φ ≥ 0 (the case φ < 0 is analogous), then θ0−
θ1−φ < arcsin(K+
θ0−θ1−φ > arcsin(K−
φ = −π
v1= v0
cosφ
> 0. Moreover from (17) and (6) it
follows directly that v1(t) ≤ Vp. Finally the curvature
of the path followed by R1is given by κ1(t) =
sin β
dcos(β−φ)=
0dcosφ) .
(17)
0dcosφ)−φ ≤ arcsin(1)−φ ≤π
0dcosφ)−φ ≥ −arcsin(cosφ)−
2, being −1 ≤ K−
cos(θ0−θ1−φ)
2and
0d, K+
0dcosφ ≤ 1, therefore
ω1(t)
v1(t)=
1
d[cotβ cosφ+sinφ]. Being the function
f(β) =



(d[cotβ cosφ + sinφ])−1
ifβ ?= 0, |β| ≤π
2
0if β = 0
monotone increasing, it follows by (5) that
sign? K−
+sinφ]?−1therefore sign? K−
+dsinφ?−1whichimpliesby(7)thatK−
0
?d[cot(arcsin(d? K−
1cosφ))cosφ + sinφ]?−1
??
0
(? K+
≤ω1(t)
v1(t)≤ sign? K+
+d sinφ?−1≤ω1(t)
0
?d[cot(arcsin(d? K+
0
v1(t)≤ sign? K+
1cosφ))cosφ+
(? K−
0)−2− d2cos2φ +
0)−2− d2cos2φ+
p≤ω1(t)
2
??
v1(t)≤ K+
p.
Therefore all the constraints are satisfied.
4Stabilization of the leader-follower formation
Problem 2 Set d > 0, φ : |φ| <
(PT
(V0,K−
conditions on V0, K−
of controls v1, ω1such that for any initial condition R1,
R0and R1are asymptotically in (d,φ)-formation with
leader R0 (remark that v1 and ω1 must be such that
Assumption 1 is satisfied).
π
2. Let R0 =
0, θ0)Tbe a robot satisfying the trajectory constraint
0,K+
0, K+
0) and R1be another robot. Find sufficient
0that guarantee the existence
The proposed control strategy consists of two steps. In
the first step the follower moves with maximum linear
and angular velocities until its direction is sufficiently
closetothat ofthe leaderin ordertosatisfythe condition
α−≤ θ0−θ1≤ α+, where α−and α+are related to the
maximum admissible curvatures K−
followed by the leader. In the second step, the follower
performs the control defined in the previous section with
an added stabilizing term in order to reduce the error
asymptotically to zero. The stabilizing term is chosen
accurately in order to satisfy constraint (11).
The basic geometric idea under the stabilization is the
following: the error vector E is decomposed with respect
to vectors τ(θ1) and ν(θ1+ φ) as shown in Fig. 3,
0, K+
0of the path
E =
?E, ν(θ1+ φ)⊥?
?τ(θ1), ν(θ1+ φ)⊥?τ(θ1) +
?E, τ(θ1)⊥?
?ν(θ1+ φ), τ(θ1)⊥?ν(θ1+ φ)
=?E, τ(θ1+ φ)?
cosφ
τ(θ1) +?E, ν(θ1)?
cosφ
ν(θ1+ φ) .
R1
P1
θ1
τ(θ1)
φ
R0
E
−Eν
τ(θ1)
ν(θ1+ φ)
Eτ
P0
Fig. 3. Error decomposition, where Eτ =
?E, ν(θ1)?
cos φ
.
?E, τ(θ1+φ)?
cos φ
and
Eν =
The stabilizing controller adds correcting terms pro-
portional to the error components to equations (8).
To ensure that the physical constraint of the follower
is satisfied, the correcting terms are multiplied by a
time-varying gain η(t) that must be chosen in a suit-
able way (see equation (29)). In the following theorem
S1denotes the quotient space R/R equipped with the
canonical topology being R the equivalence relation,
x ∼ y ⇔ x−y = 2nπ, n ∈ Z, where Z is the integer set.
Theorem 2 Set d > 0, φ : |φ| <π
Let R0= (PT
satisfying the trajectory constraint (V0,K−
pose that the following properties hold:
2and R0, R1∈ R3.
0,θ0) be a robot with initial condition R0,
0,K+
0). Sup-
0 < W0≤ v0(t)(18)
−1
1
d< K−
0≤ K+
0≤ K+
? K−
0<
0<1
1
dcosφ, if φ ≥ 0
, if φ < 0−
dcos φ< K−
d
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0< K−
0≤ K+
0<? K+
0dcosφ))?< Vpcosφ
p)−1− dsinφ)2+ d2cos2φ?−1
0
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V0cos?0 ∧ (arcsin(K+
∧(φ − arcsin(K−
where
0dcosφ) − φ)∧
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? K±
Then there exists ¯ ǫ > 0 suchthat for any ǫ : 0 < ǫ < ¯ ǫ, for
anyrobotR1= (PT
exist suitable controls v1, ω1 such that R0 and R1 are
asymptotically in (d,φ)-formation and ∃t ≥ 0 : ∀t ≥¯t
0= (sign K±
p)?((K±
2.
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1,θ1)Twithinitial condition R1,there
arcsin((K−
0− ǫ)dcosφ) ≤ θ0(t) − θ1(t) ≤
≤ arcsin((K+
0+ ǫ)dcosφ) .
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PROOF. Take ¯ ǫ > 0 such that
−1
1
d< K−
0− ¯ ǫ ≤ K+
0− ¯ ǫ ≤ K+
0+ ¯ ǫ <
0+ ¯ ǫ <1
1
dcosφ, if φ ≥ 0
, if φ < 0−
dcosφ< K−
d
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