Distributed Coordination Theory for Ground and Aerial Robot Teams by Ashton Roza A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto c Copyright 2019 by Ashton Roza
272
Embed
Distributed Coordination Theoryfor Ground and Aerial Robot ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Distributed Coordination Theory for Ground and Aerial
Robot Teams
by
Ashton Roza
A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy
Graduate Department of Electrical and Computer Engineering
This chapter presents extensive simulation trials to study the effectiveness of our
control solution presented in Chapter 7 for formation control of unicycles under
different realistic scenarios not captured by the main theoretical result in that
chapter. This includes performance in the presence of undirected state dependent
sensor graphs, relaxation of high gain requirements and robustness to unmodelled
effects including sensor noise, input noise, sampling and saturated inputs.
Chapter 1. Introduction 11
1.3 Statement of contributions
What follows is a list of significant original contributions made in this thesis.
1. Chapter 4
• Reduction theorem for almost global asymptotic stability in Theorem 4.1.3
based on lemmas 4.1.15, 4.1.16, 4.1.17.
2. Chapter 5
• The problem of rendezvous of flying robots (P1) using strictly local and dis-
tributed feedbacks is open. Theorem 5.1.1 is the first local and distributed
solution yielding global practical stability.
• This work has been published in the IEEE Transactions on Automatic Control
(TAC) in (Roza et al., 2017).
3. Chapter 6
• The solution presented in Theorem 6.1.1 for rendezvous of kinematic unicy-
cles (P2), is the first one for directed graphs containing a globally reachable
node involving feedbacks that are local and distributed, time-independent,
and continuously differentiable. Brockett’s necessary condition in Theorem
1 of (Brockett, 1983), implies that an equilibrium point cannot be asymp-
totically stabilized for under-actuated driftless systems, including kinematic
unicycles, using a continuously differentiable, time-invariant control law. As
a consequence, one cannot asymptotically stabilize simultaneously a desired
position and angle of a kinematic unicycle using a continuously differentiable,
time-invariant control law. One requires either discontinuous, time-varying, or
dynamic feedback. Our solution to rendezvous of kinematic unicycles (as well
as other problems) does not contradict Brockett’s necessary condition because
Chapter 1. Introduction 12
the control problem involves asymptotic stabilization of a set as opposed to
a particular equilibrium point in the state space. In particular, we do not
specify, a priori, a final position and angle of the unicycles in the ensemble.
Previous solutions to the rendezvous problem require either time-varying
or discontinuous feedback or are restricted to undirected sensing graphs. We il-
lustrate through simulations in Section 6.1.1 that the proposed time-independent,
continuously differentiable feedback has practical advantages over the time-
varying feedback in (Lin et al., 2005) in that it induces a more natural be-
haviour in the ensemble of unicycles. The feedback in (Lin et al., 2005) makes
the unicycle “wiggle” indefinitely, a behaviour which would be unacceptable in
practice. The feedback in (Zheng et al., 2011) is restricted to undirected sens-
ing graphs and it does not achieve rendezvous for directed graphs containing
a globally reachable node. The feedback in (Dimarogonas and Kyriakopou-
los, 2007) induces instantaneous changes in direction that are impossible to
achieve with realistic implementations.
• This work has been published in the IEEE Transactions on Control of Network
Systems (TCNS) in (Roza et al., 2018).
4. Chapter 7
• The problem of stabilizing static parallel formations (P3) by means of smooth,
local and distributed feedbacks is to date open. Theorem 7.1.1 presents an
almost semiglobal solution to this problem, i.e., a solution making the unicy-
cles achieve the desired objective for almost all initial conditions in any given
compact subset of their collective state space.
• The conditions in Theorem 7.1.1 are relaxed for parallel line formations in
Corollary 7.3.1. A special case of this setup is full synchronization of the
unicycles, a problem of note in its own right. In this case, our solution is to be
Chapter 1. Introduction 13
compared to the one in (Dimarogonas and Kyriakopoulos, 2007), which relies
on discontinuous control, but considers a more general class of sensor graphs
than ours.
• This work has been conditionally accepted for publication in the IEEE Trans-
actions on Automatic Control (TAC) in 2018.
5. Chapter 8
• For PFP, PFP-B, Theorem 8.1.1 is the first almost-global solution that does
not require communication.
• For CFP, Theorem 8.2.1 is the first almost global solution with local and
distributed feedbacks, not restricted to formations on a common circle.
• We present the first almost global solutions for LPP,CPP in Theorem 8.1.1
and Theorem 8.2.1 respectively for general classes of sensing graphs (digraphs
containing a globally reachable node for line paths and connected undirected
graphs for circle paths) that do not require communication.
6. Chapter 9
• For general formation path following (P5) all approaches in the literature
require at least one of the following assumptions:
– the path to be followed is parametrized by time (formation trajectory
tracking)
– each unicycle needs to compute its own separate path to follow (unrealistic
in practice),
– the digraph has a leader-follower topology,
– all unicycle control inputs require global information, e.g., the group cen-
tre of mass.
Chapter 1. Introduction 14
Theorem 9.1.1 presents the first solution to P5 along any smooth Jordan curve
for any digraph containing a globally reachable node and not requiring any
of the assumptions above. Due to the complexity of the problem, this solu-
tion possesses some drawbacks compared to the other chapters in this thesis.
Namely, it only has a local domain of attraction and requires communication
of auxiliary states. Resolving these issues can be studied in future research.
Chapter 2
Modelling
In this chapter we present models for kinematic unicycles and flying robots. We start with
a discussion on how the attitude of these vehicles can be represented as rotation matrices
in the special orthogonal groups SO(2) and SO(3). A large potion of the material has
been taken from an analogous discussion in the master’s thesis in (Roza, 2012). Next we
present the system equations for both models. For kinematic unicycles, positions evolve
in Euclidean space R2 while the attitude evolves in SO(2). For flying robots, on the
other hand, positions evolve in R3 and the attitude evolves in SO(3). The combination
of translations in Euclidean space Rk and attitudes in SO(k) for k ∈ 2, 3, known as the
configuration space, also has a group structure and is referred to as the special Euclidean
group and denoted SE(k). Finally, we will discuss how to model the sensing between
neighboring robots.
2.1 Attitude Representation as a Rotation Matrix
Consider a team of n robots and two right-handed orthogonal frames, I and Bi, either
in R2 (Figure 2.1) or R3 (Figure 2.2). Suppose that I is an inertial frame, while Bi is
attached to the i-th agent in an ensemble of robots labelled 1, . . . , n. The attitude of
15
Chapter 2. Modelling 16
Figure 2.1: Inertial and body frames in two dimensions.
I
Bi
ox
oy
oz
bix
biy
biz
Ri
Figure 2.2: Inertial and body frames in three dimensions.
robot i is defined to be the orientation of frame Bi with respect to the inertial frame I.
A rotation matrix R is a k × k matrix in the special orthogonal group SO(k) for
k ∈ 2, 3, defined as
SO(k) = R ∈ Rk×k : R⊤R = Ik = RR⊤, det(R) = 1.
In the above, Ik is the k× k identity matrix. The definition readily implies that rotation
matrices have the property that R−1 = R⊤. One can show, as its name suggests, that
SO(k) constitutes a group under the operation of matrix multiplication. Now consider
the coordinate frames in Figure 2.1 and Figure 2.2, and define the rotation matrix Ri of
Chapter 2. Modelling 17
frame Bi with respect to frame I as
Ri :=
bix · ox biy · oxbix · oy biy · oy
,
Ri :=
bix · ox biy · ox biz · oxbix · oy biy · oy biz · oybix · oz biy · oz biz · oz
(2.1)
for the cases of R2 and R3 respectively, where “·” denotes the dot product of two geometric
vectors. One can check that Ri as defined in the first equation of (2.1) is in SO(2) while
the second is in SO(3). Vice versa, a rotation matrix Ri uniquely identifies a relative
orientation of frame Bi with respect to frame I, since the columns of Ri are the coordinate
representations of the coordinate axes (bix, biy) and (bix, biy, biz) respectively in the frame
I. In conclusion, SO(k) can be viewed as the set of attitudes of a robot in Rk for
k ∈ 2, 3.
Rotation matrices can be made to act on Rk for k ∈ 2, 3 via multiplication, giving
rise to rotational transformations. That is, if vb in Rk is the coordinate representation of
a geometric vector v in frame Bi, its representation in the coordinates of I is Rivb.
The set SO(k) can be given the structure of a smooth manifold which is compact,
connected, and of dimension k for k ∈ 2, 3. One can show that the tangent space to
SO(k) at the identity element I is the set of skew-symmetric matrices,
so(k) := S ∈ Rk×k : S⊤ = −S.
The sets so(2) and so(3) are vector spaces isomorphic to R and R3 via the isomorphisms
Chapter 2. Modelling 18
s : R → so(2) and S : R3 → so(3) respectively,
s(ω) =
0 −ω
ω 0
,
S(Ω) =
0 −Ω3 Ω2
Ω3 0 −Ω1
−Ω2 Ω1 0
.
(2.2)
To simplify notation, in this thesis these isomorphisms will be represented with a su-
perscript × as ω× := s(ω) for ω ∈ R and Ω× := S(Ω) for Ω ∈ R3. We represent
the inverse operation with a subscript ×. For example, given a skew-symmetric matrix
M = −M⊤ ∈ so(3), we denote M× := [M32 M13 M21]⊤ where Mij is the (i, j)-th entry
of M . One can show that the set SO(k) constitutes a Lie group with corresponding
Lie algebra so(k) for k ∈ 2, 3. The tangent space at a generic element R ∈ SO(k) is
obtained through left-multiplication by R of matrices in so(k),
TRSO(k) = Rso(k) = RS : S ∈ so(k). (2.3)
This result restricts the structure of the kinematic equations of a rotating body as follows.
If one wants to describe the evolution of a rotation matrix using a vector field on SO(k),
R = f(R), the vector field must have the property that, for all R, f(R) ∈ TRSO(k).
Hence, in light of (2.3), the vector field in two and three dimensions must have the
structure
R = R s(ω), (2.4)
R = RS(Ω), (2.5)
respectively where ω ∈ R and Ω ∈ R3 is an input parameter that may depend on time.
One can show that ω and Ω are precisely the angular velocities of the rotating body in
Chapter 2. Modelling 19
R2 and R3 respectively. Because of this fact, and in light of the fact that s : R → so(2)
and S : R3 → so(3) are isomorphisms, we can think of elements of so(2) and so(3) as
angular velocities, and equations (2.4) and (2.5) are called the kinematic equation of a
rotating rigid body.
Some important properties of skew-symmetric matrices are given below.
s(ω)⊤ = − s(ω)
(s(ω)d) · d = 0, d ∈ R2
S(Ω)⊤ = −S(Ω)
S(Ω)d = Ω× d, d ∈ R3
S(RΩ) = RS(Ω)R⊤, R ∈ SO(3).
2.2 System model
Having reviewed the rotation matrix parametrization of attitude in two and three di-
mensions, we are ready to introduce the two vehicle classes investigated in this thesis.
The first class corresponds to ground-based mobile robots each modelled as kinematic
unicycles in SE(2). The second class corresponds to flying robots in SE(3).
Kinematic Unicycles
We begin by modelling a group of n kinematic unicycles. We fix an orthogonal frame
I = ox, oy in R2, and attach to unicycle i an orthogonal body frame Bi = bix, biy in
such a way that bix is the heading axis of the unicycle (as in Figure 2.1). We denote by
xi ∈ R2 the position of unicycle i in the coordinates of frame I. The attitude of body
frame Bi relative to I is represented by a rotation matrix Ri ∈ SO(2). Unicycle i’s wheels
point in the direction of its heading axis Rie1 which represents the instantaneous axis of
motion for unicycle i. Since the velocity xi cannot act along the second body axis Rie2,
Chapter 2. Modelling 20
Figure 2.3: Kinematic unicycle class. The right figure shows a differential drive robotthat is modelled as a unicycle.
this results in the nonholonomic constraint
xi · Rie2 = 0.
Letting θi ∈ S1 be the angle between vectors ox and bix and identifying S1 with the set
of real numbers modulo 2π, one can write the rotation matrix Ri in terms of θi with the
isomorphism
Ri = Ri(θi) =
cos θi − sin θi
sin θi cos θi
,
illustrating the fact that S1 ∼= SO(2). With these conventions and identifying the unit
circle with the set of real numbers modulo 2π, the model of unicycle i is
xi = uiRie1 (2.6)
θi = ωi, i ∈ n, (2.7)
Chapter 2. Modelling 21
where the pair (ui, ωi), composed of the linear and angular speeds of unicycle i, is the
control input. A kinematic unicycle model is illustrated in Figure 2.3 which includes an
image of an actual robotic unicycle in the lab referred to as a differential-drive robot.
The configuration space of each robot therefore consists of a translation in R2 and an
attitude in SO(2) and can be written as an element in the Special Euclidean Group
SE(2) =
R x
0 1
∈ R
3×3 : R ∈ SO(2), x ∈ R2
which is isomorphic to R2×S1 and just like SO(2) and SO(3), constitutes a group structure
with matrix multiplication as the group operation. The group identity e and inverse are
given respectively by
e =
I 0
0 1
,
R x
0 1
−1
=
R⊤ −R⊤x
0 1
.
(2.8)
We collect the translational and rotational states into the vectors x := (xi)i∈n ∈ R2n and
θ := (θi)i∈n ∈ Tn. The overall system state space is R2n×Tn which also inherits a group
structure through the Cartesian product.
The relative displacement of robot j with respect to robot i is xij := xj−xi while the
relative angles are given by θij = θj − θi. The rotation of robot j with respect to frame
i is defined by Rij := (Ri)
−1Rj, and it is a function of θij . If v ∈ R2 is the coordinate
representation of a vector in frame I, then we denote by vi := R−1i v the coordinate
representation of v in body frame Bi. The notation for the kinematic unicycle is given in
Table 2.1.
Next we will model the sensing between unicycles in the ensemble. In particular,
as the unicycles perform the desired control task, not every unicycle will be able to see
Chapter 2. Modelling 22
Table 2.1: Table of Notation for Kinematic Unicycle Model
Quantity Description
xi ∈ R2 inertial position of unicycle ix ∈ R2n (xi)i∈nRi ∈ SO(2) attitude of unicycle iθi ∈ S1 heading angle of unicycle iθ ∈ Tn (θi)i∈nωi ∈ R angular velocity of unicycle iri = R−1
i r coord. repr. of r in frame Bixij = xj − xi rel. displacement of robot j wrt robot iNi set of neighbors of robot iyi = (xij)j∈Ni vector of rel. pos. available to robot i
every other unicycle in the ensemble. Instead, each unicycle can only sense a subset of
neighboring unicycles. This sensing convention is naturally represented using the ideas
from graph theory that will be discussed in Section 4.2. We define the sensor graph
G = (V, E), where each node in the node set V represents a robot, and an edge in the
edge set E between node i and node j indicates that robot i can sense robot j. We assume
that G has no self-loops. Given a node i, its set of neighbours Ni represents the set of
vehicles that robot i can sense. For any j ∈ Ni robot i can sense the relative displacement
of robot j in its own body frame, i.e., the quantity xiij , as well as the relative heading
angle θij .
In a realistic scenario, the neighbor set Ni would be the set of robots within the
field of view of robot i. For instance, if each robot mounted an omnidirectional camera,
then one could define Ni to be the collection of robots that are within a given distance
from robot i. With such a definition, the sensor digraph G would be state-dependent,
making the stability analysis too hard at present. Relatively little research has been
done on distributed coordination problems with state-dependent sensor graphs. In this
context, in the simplest case when the robots are modelled as kinematic integrators, it
has been shown in (Lin et al., 2007b) that the circumcentre law of Ando et al. (Ando
et al., 1999) preserves connectivity of the sensor graph and leads to rendezvous if the
Chapter 2. Modelling 23
sensor graph is initially connected. Despite the simplicity of the robot model, the stability
analysis in (Lin et al., 2007b) is hard, and the control law is continuous but not Lipschitz
continuous.
In light of the above, in this thesis we assume that Ni is static for each i ∈ n (and
hence G is constant as well).
We now define the notion of local and distributed feedback. Define vectors yi :=
(xij)j∈Ni, yii := (xiij)j∈Ni, and ϕi := (θij)j∈Ni. The relative displacements and angles
available to robot i are contained in the vector (yii, ϕi).
Definition 2.2.1. A local and distributed feedback (ui, ωi) for robot i is a locally Lipschitz
function (yii, ϕi) 7→ (ui, ωi).
A local feedback is one in which all quantities are represented in the body frame
of robot i, while a distributed feedback is one in which only relative quantities with
respect to neighboring robots are accessible. In applications, a local and distributed
feedback for robot i can be computed with on-board cameras. No information needs to
be communicated between agents using a communication system or require centralized
information.
Most results in this thesis will require local and distributed feedbacks. However, in
some problems related to higher layer control specifications like formation flocking in
a particular direction and path following, a beacon or access to path information will
be required. Naturally the feedbacks will not be local and distributed in those cases as
information is required from the external environment.
Other Vehicle Models on the Plane
Controllers developed for kinematics unicycles can be extended, with some limitation, to
other models evolving on the plane as discussed below. This illustrates that the study of
kinematic unicycles is fundamental for solving a broad range of mobile robotics problems.
Chapter 2. Modelling 24
Figure 2.4: Bicycle class. Image from (Francis and Maggiore, 2016).
Bicycles . A model of a bicycle is illustrated in Figure 2.4. The position of bicycle
i’s back wheel in the inertial frame I is denoted xi ∈ R2. The angle of the back wheel is
θi ∈ S1 relative to I and the angle of the front wheel relative to the back wheel is γi ∈ S1
which is called the steering angle. The distance between the center of the two wheels is
B meters. The control inputs are the speed of the back wheel ui and the steering rate
ωi = γi. The equations of motion for the bicycle were derived in (Francis and Maggiore,
2016) as
xi = ui
cos θi
sin θi
,
θi =uiB
tan γi,
γi = ωi.
(2.9)
As stated in (Francis and Maggiore, 2016), this model “turns out to be quite useful
because it captures the essential features of a car with four wheels, only the front two
being steerable”. It is shown in (Francis and Maggiore, 2016) that a controller developed
for a kinematic unicycle can be adapted to the bicycle as long as one shows that the front
wheel of the bicycle is never orthogonal to the back wheel for closed loop solutions.
Dynamic Unicycles . A dynamic model for the unicycle is given in (El-Hawwary
Chapter 2. Modelling 25
and Maggiore, 2013a) as,
xi = viRi(θi)e1,
vi = ai,
θi = ωi,
ωi = αi, i ∈ n,
(2.10)
where, as for kinematic unicycles, xi ∈ R2 is the position and θi ∈ S1 is the angle.
In the dynamic model vi ∈ R is the speed state and ωi is the angular speed state of
unicycle i. The state of unicycle i can therefore be written as χi = (xi, vi, θi, ωi) ∈
R2 × R2 × S1 × R =: X . The control inputs are the translational acceleration ai and
angular acceleration αi. In (El-Hawwary and Maggiore, 2013a, Proposition V.1.), the
authors show that for uniformly bounded C1 functions ui(χi) and ωi(χi) the controller
ai = ˙ui(χi)− k(vi − ui(χi)),
αi = ˙ωi(χi)− k(ωi − ωi(χi))
(2.11)
globally asymptotically stabilizes the set
Γ = χ ∈ X : vi = ui(χi), ωi = ωi(χi)
for system (2.10). In the set Γ2, the states (xi, θi) in system (2.10) satisfy the equations
of a kinematic unicycle with inputs (ui, ωi)
xi = ui(χi)Ri(θi)e1,
θi = ωi(χi), i ∈ n.
(2.12)
Suppose we design the uniformly bounded C1 functions ui(χi) and ωi(χi) to solve a
particular control problem for kinematic unicycle model in (2.12) (such as rendezvous).
Then under some assumptions, the Reduction Theorem that will be stated formally in
Chapter 2. Modelling 26
Figure 2.5: Flying Robot Class.
Theorem 4.1.2 tells us that the feedbacks in (2.11) solve the same problem for the dynamic
unicycle model in (2.10). Therefore problems for dynamic unicycles can be reduced to
problems for kinematic unicycles in many cases. However, unicycles must measure their
own speeds which is not local and distributed.
Flying Robots
We now model a group of n flying robots. We fix a right-handed orthonormal inertial
frame I, common to all robots, and attach at the centre of mass of robot i a right-handed
orthonormal body frame Bi = bix, biy, biz (as in Figure 2.2). We denote by (xi, vi) the
inertial position and velocity of robot i. We let g denote the gravity vector in frame I.
The attitude of body frame Bi relative to I is represented by a rotation matrix
Ri ∈ SO(3). The unit vector qi := −Rie3, depicted in Figure 2.5, is referred to as
the thrust direction vector of robot i. We assume that a thrust force uiqi is applied at
the centre of mass of robot i. Notice that uiqi has magnitude ui, is directed opposite
to biz , and has constant direction in body frame Bi. This results in two nonholonomic
constraints on the acceleration when we ignore the effect of gravity
xi · Rie1 = 0
xi · Rie2 = 0.
Robot i is assumed to have an actuation mechanism that induces control torques τix, τiy, τiz
Chapter 2. Modelling 27
about its body axes. We let τi := (τix, τiy, τiz) be the torque vector, and Ωi denote the
angular velocity of the robot with respect to frame I.
Picking (xi, vi, Ri,Ωii) as the state for robot i, we obtain the equations of motion
xi = vi,
mivi = −uiRie3 +mig = Ti +mig,
(2.13)
Ri = Ri (Ωii)
×,
JiΩii = τi − Ωii × JiΩ
ii.
(2.14)
In the above, mi is the mass of robot i and Ji = J⊤i is its inertia matrix. We define the
(inertial) relative positions and velocities as xij := xj − xi, vij := vj − vi. The model
of a flying robot is illustrated in the Figure 2.5. This model is standard and is widely
used in the literature to model flying vehicles such as quadrotor helicopters. See, for in-
stance, (Hua et al., 2009). Sometimes researchers use alternative attitude representations,
prominently quaternions (Abdessameud and Tayebi, 2011) or Euler angles (Mokhtari
et al., 2006; Castillo et al., 2005). The model (2.13), (2.14) ignores aerodynamic effects
such as drag and wind disturbances (such effects are included in (Hua et al., 2009)). It
also ignores the dynamics of the actuators.
The state vector for each robot is (xi, vi, Ri,Ωi) ∈ R3 × R3 × SO(3) × R3. The
configuration space of each robot consists of its position xi ∈ R3 and an attitude Ri ∈
SO(3) and can be written as an element in the Special Euclidean Group
SE(3) =
R x
0 1
∈ R
4×4 : R ∈ SO(3), x ∈ R3
and analogous to SE(2), constitutes a group. The overall state space of (2.13), (2.14) is
R3n × R3n × SO(3)n × R3n.
In this thesis we adopt the convention that if r ∈ R3 is an inertial vector, the coordi-
Chapter 2. Modelling 28
Table 2.2: Table of Notation for Flying Robots
Quantity Description
mi, Ji mass and inertia matrix of robot ixi ∈ R3 inertial position of robot ix ∈ R3n (xi)i∈nvi ∈ R3 linear velocity of robot iv ∈ R3n (vi)i∈nRi ∈ SO(3) attitude of robot iΩi ∈ R3 angular velocity of robot iΩ ∈ R3n (Ωi)i∈nqi = −Rie3 thrust direction vector of robot iTi = −uiRie3 applied thrust vector of robot iri = R−1
i r coord. repr. of r in frame Bixij = xj − xi rel. displacement of robot j wrt robot ivij = vj − vi rel. velocity of robot j wrt robot iΩi ∈ R3 reference angular velocity of robot iNi set of neighbors of robot iyi = (xij , vij)j∈Ni vector of rel. pos. and vel. available to robot i
nate representation of r in frame Bi is denoted by ri, that is, ri := R−1i r. In particular,
the angular velocity of robot i in its own body frame is denoted by Ωii. Finally, the
reference angular velocity of vehicle i is denoted Ωi. The notation is summarized in
Table 2.2.
Example 2.2.2. This example has been taken from (Roza, 2012).
A well known vehicle that falls in the class of flying robots is the quadrotor helicopter,
see (Mokhtari et al., 2006) or (Abdessameud et al., 2012). Referring to Figure 2.6, a
quadrotor helicopter consists of four rotors connected to a rigid frame. The distance
from the centre of mass to the rotors is denoted by d. For robot i, each rotor produces
a thrust force fij, j ∈ 1, . . . , 4 parallel to the biz axis, and a reaction torque τrij of the
motor that drives it. To produce a thrust fij in the negative biz (i.e., upward) direction,
the two rotors on the bix axis rotate in the clockwise direction, while the rotors on the
biy axis rotate in the counter-clockwise direction.
The physical inputs are the reaction torques τrij of the motors. Using the development
Chapter 2. Modelling 29
Figure 2.6: Illustration of a quadrotor helicopter taken from (Roza, 2012).
from (Castillo et al., 2005), the rotor dynamics are given by IrzΩrij = −bΩ2rij+τrij where
Irz is the rotor moment of inertia about the rotor z-axis, Ωrij is the angular speed of
rotor j, and b is a coefficient of friction due to aerodynamic drag on the rotor. There
is also an approximate algebraic relationship between the rotor thrust and rotor speed
given by, fij = γΩ2rij , where γ is a parameter that can be experimentally determined. If
we assume steady-state rotor dynamics such that Ωrij = 0, then fij = (γ/b)τrij = cτrij
where c = γ/b is the algebraic scaling factor between the rotor thrust and the applied
motor torque. Using this fact, it is readily seen that the relationship between the control
inputs and the motor torques is given by
ui
τix
τiy
τiz
=
c c c c
0 −cd 0 cd
cd 0 −cd 0
1 −1 1 −1
τri1
τri2
τri3
τri4
.
In the above, the total thrust is equal to the summation of the four rotor thrusts; the
torque about the bix axis is proportional to the differential thrust of the two rotors on
the biy axis, fi4 − fi2; the torque about the biy axis is proportional to the differential
thrust of the two rotors on the bix axis, fi1 − fi3; and the torque about the biz axis is
Chapter 2. Modelling 30
equal to the summation of the four reaction torques which are equal and opposite to the
applied motor torques τrij . With the definition of (ui, τi) above, the quadrotor helicopter
is modelled with (2.13), (2.14).
The modelling of the sensor graph G = (V, E) for flying robots is the same as for
kinematic unicycles. If j ∈ Ni is a neighbor of unicycle i, then robot i can sense the
relative displacement and velocity of robot j in its own body frame, i.e., the quantities
xiij , viij. Define the vector yi := (xij , vij)j∈Ni. The relative displacements and velocities
available to robot i are contained in the vector yii := (xiij , viij)j∈Ni. It is also assumed
that robot i can can sense its own angular velocity in its own frame Bi and a local and
distributed feedback is defined below.
Definition 2.2.3. A local and distributed feedback (ui, τi) for robot i is a locally Lipschitz
function (yii,Ωii) 7→ (ui, τi).
In applications, a local and distributed feedback for robot i can be computed with
on-board cameras and rate gyroscopes. Note that in a local and distributed feedback,
we have not allowed unicycle i to measure the attitude of a neighbor j ∈ Ni in its
body frame R−1i Rj or its relative angular velocity represented in its own body frame
R−1i (Ωj − Ωi) = Ri
jΩjj − Ωii as these quantities are more difficult to obtain using sensors
measurements. In this thesis, such measurements are not required to solve the problem
of rendezvous. However, future extension of this result to formation control will most
certainly require these measurements.
Chapter 3
Coordination Problems
In this chapter we will introduce the coordination problems investigated in this thesis
for the kinematic unicycles in (2.6), (2.7) and flying robots in (2.13), (2.14). Instead of
considering the standard control problem of stabilizing equilibrium points, in this work,
each problem will be stated as a set stabilization problem with additional sensing con-
straints. The main motivation behind this is that most of the sets under consideration
simply do not reduce to equilibrium points, even in appropriately chosen error coordi-
nates. Therefore, they are most naturally defined as sets. This chapter uses stability
notions that will be introduced in the preliminaries in Chapter 4.
3.1 Rendezvous of Flying Robots
In this section, we define the rendezvous control problem for flying robots whose solution
is discussed in Chapter 5. The objective of this problem is to design local and distributed
feedbacks in the sense of Definition 2.2.3 to drive a group of n robots to the rendezvous
manifold,
Γ :=
(xi, vi, Ri,Ωii)i∈n ∈ R
3n × R3n × SO(3)n × R
3n : xij = vij = 0,
Ωii = Ωii(yi, Ri), i, j ∈ n
(3.1)
31
Chapter 3. Coordination Problems 32
where Ωii(yi, Ri) is a suitable function to be designed in Chapter 5. The rendezvous
manifold Γ is the subset of the state space in which all positions and velocities are equal.
There is no condition imposed on the relative attitudes and the control input for robot
i is a function of (yii,Ωii) where yii = (xiij , v
iij)j∈Ni are the relative displacements and
velocities of robot i with respect to its neighbors. The Rendezvous Problem for Flying
Robots (RP-F) is stated as follows.
Problem 1 (Rendezvous Problem for Flying Robots (RP-F)). Consider system (2.13), (2.14)
and sensor digraph G = (V, E) containing a globally reachable node. Design local and dis-
tributed feedbacks (u⋆i , τ⋆i ) : (y
ii,Ω
ii) 7→ (ui, τi) for all i ∈ n that globally practically stabilize
Γ under the constraint (u⋆i , τ⋆i )|Γ = (0, 0).
The goal of the rendezvous control problem is to achieve synchronization of the robot
positions and velocities to any desired degree of accuracy from any initial configura-
tion. The requirement (u⋆i , τ⋆i )|Γ = (0, 0) means that the robots are not actuated when
rendezvous is achieved.
3.2 Rendezvous of Kinematic Unicycles
In this section, we define the rendezvous control problem for kinematic unicycles whose
solution is discussed in Chapter 6. The objective of this problem is to design local and
distributed feedbacks in the sense of Definition 2.2.1 to drive a group of n robots to the
rendezvous manifold,
Γ :=
(xi, θi)i∈n ∈ R2n × T
n : xij = 0, i, j ∈ n
. (3.2)
The rendezvous manifold Γ is the subset of the state space in which all positions coincide
with one another. There is no condition placed on the relative heading angles. The
control input for unicycle i must be strictly a function of yii = (xiij)j∈Ni, the relative
Chapter 3. Coordination Problems 33
displacements between unicycle i and its neighbors. Unicycles are not permitted to
measure any information about their orientation, not even their orientation relative to
their neighbors ϕi = (θij)j∈Ni. The Rendezvous Problem for Kinematic Unicycles (RP-U)
is stated as follows.
Problem 2 (Rendezvous Problem for Kinematic Unicycles (RP-U)). Consider the sys-
tem of kinematic unicycles in (2.6), (2.7) and a sensor digraph G containing a globally
reachable node. Design local and distributed feedbacks (u⋆i , ω⋆i ) : yii 7→ (ui, ωi) for all
i ∈ n that globally asymptotically stabilize the rendezvous manifold Γ under the con-
straint (u⋆i , ω⋆i )|Γ = (0, 0).
The requirement (u⋆i , ω⋆i )|Γ = (0, 0) in the rendezvous problem (RP-U) means that the
unicycles do not move when they achieve rendezvous, i.e., they are stopped and do not
oscillate. This means that the unicycles do not consume any energy when rendezvous is
achieved as one would expect from a good control strategy.
3.3 Formation Control Problems for Kinematic Uni-
cycles
For kinematic unicycles in (2.6), (2.7), a number of formation control problems will be
introduced. A formation of n unicycles is a geometric pattern defined modulo roto-
translations by means of desired inter-agent displacements. Let the fixed vector d11i ∈ R2
denote the desired displacement of unicycle i relative to unicycle 1, measured in the frame
of unicycle 1, i.e., d11i := R−11 (xi − x1). In contrast to the rendezvous problem where the
robots converge to a common position, this specification is usually more useful in practice.
We collect all the fixed relative displacements in a vector d := (d11i)i∈2 :n ∈ R2(n−1) which
specifies the formation.
An example of a formation consisting of four unicycles defined in terms of the offset
vectors d11i, i ∈ 2 :n is illustrated in Figure 3.1. The two configurations of unicycles
Chapter 3. Coordination Problems 34
depicted in Figure 3.1, labelled 1 and 2, are related to one another through a rigid
roto-translation in the inertial frame I. That is, to arrive at configuration 2 first rotate
configuration 1 by ϕ radians about unicycle 1 followed by a translation of r units along
the dotted line illustrated in Figure 3.1. For both configurations, the offsets d11i ∈ R2
between unicycle 1 and unicycle i as measured in body frame 1, are identical and therefore
the two configurations represent the same formation. We correspondingly say that the
formation is invariant under roto-translations.
Figure 3.1: Formation in terms of fixed relative displacement vectors d11i, i ∈ 2 :n.
The labelling of the unicycles is done solely for the purpose of defining the formation,
and does not imply any attribution of priority to the unicycles. In the problem of stopping
formations discussed in Section 3.3.1 and formations with final parallel collective motion
discussed in Section 3.3.2, it will be assumed, without loss of generality, that unicycle 1
is chosen to be at the front of the formation so that d11i · e1 ≤ 0 for all i ∈ 2 :n. This is
the case in the example in Figure 3.1. For a given formation d, we define the formation
manifold as,
Γ :=
(x, θ) ∈ R2n × T
n : x1i = R1d11i, i ∈ n
. (3.3)
Note that the rendezvous manifold in (3.2) corresponds to the case that d = 0. Now
we present, in detail, the formation control problems for kinematic unicycles studied in
for all i ∈ n that almost globally asymptotically stabilize a subset of Γgp under the con-
straint that o(τ)|Γgp = wr(τ).
The exact quantities that unicycles need to measure are discussed in detail in Sec-
tion 9.1.
Chapter 3. Coordination Problems 46
3.4 Literature review
In this section we will discuss the literature related to multi-agent coordination. We will
first discuss the results for single and double integrators. These are the most basic vehicle
classes since they are fully actuated. Results for single and double integrators will be-
come important building blocks for constructing control solutions for the under-actuated
systems in this thesis. Next we discuss results that characterize relative equilibria for
teams of kinematic unicycles using local and distributed feedback. Finally, we review the
literature on rendezvous for kinematic unicycles and flying robots followed by results for
formations of kinematic unicycles that either stop or have final collective motions.
We remark that the control problems introduced in this section cannot be solved using
standard results for output synchronization of nonlinear heterogeneous systems (Wieland
et al., 2011; De Persis and Jayawardhana, 2014; Burger and De Persis, 2015; Liu and
Jiang, 2013; Isidori et al., 2014; Zhu et al., 2016). These results rely heavily on com-
munication of auxiliary states between neighboring agents in the network which is not
permitted in this work. Most importantly, the control feedbacks obtained using these
approaches require agents to measure quantities that are not local and distributed. Con-
sider, for example, the rendezvous problem for kinematic unicycles. In this case, the
outputs to be synchronized need to be chosen as yi = h(xi, θi) = xi for each robot. How-
ever, the final controller for robot i in these results requires measurement of the quantity
yj − yi = xj − xi 6= R−1i (xj − xi) where j is a neighbour of i. This quantity is not local
and distributed.
Coordination problems for single and double integrators . The problem
of rendezvous for networks of single and double integrators is well-established in the
literature. See for example (Ren and Beard, 2005; Moreau, 2004; Olfati-Saber and
Murray, 2004; Ren and Atkins, 2007). The majority of the literature on formation
control considers single and double-integrator models. A dominant approach for single-
integrator formation control is distance-based (Krick et al., 2009; Oh and Ahn, 2014;
Chapter 3. Coordination Problems 47
Smith et al., 2006), where it is required that the distances between robots take on desired
values. Often in this setting, the feedbacks are deduced from the gradient of a potential
function whose minimum specifies the desired formation modulo roto-translations. This
approach requires the sensing graph to be infinitesimally rigid, which is quite restrictive.
Other approaches define formations in terms of relative angles between neighbouring
robots, instead of distances, (Zhao et al., 2014; Eren, 2012), or in terms of a complex
Laplacian, (Lin et al., 2013; Lin et al., 2016; Lin et al., 2014). In this latter case,
formations are defined modulo scaling and roto-translations. Finally, formation flocking
of double-integrators is considered in (Deghat et al., 2016), where the authors stabilize a
formation and make sure that all robots in the formation achieve a common final velocity.
See also (Tanner et al., 2003).
Relative equilibria for kinematic unicycles . The papers (Justh and Krish-
naprasad, 2004; Sarlette et al., 2010) show that the only possible relative equilibria for
a team of unicycles with local and distributed control laws correspond to either parallel
(including stopping) or circular collective motions. In fact, we will be able to find local
and distributed solutions to the problems studied in this thesis for unicycle rendezvous,
stopping formations and parallel and circular formation flocking where the final collective
motion is either parallel or circular. However, it is not possible to obtain formation flock-
ing with any other type of collective motion, for example, around an oval shaped path
using strictly local and distributed feedbacks. For kinematic unicycles in three dimen-
sions, it is shown in (Justh and Krishnaprasad, 2004; Justh and Krishnaprasad, 2005)
that the only possible relative equilibria correspond to parallel, circular or helical forma-
tions. In (Scardovi et al., 2008), the authors propose distributed controllers to stabilize
relative equilibria but do not specify a particular formation. While kinematic unicycles
in three dimensions are not studied explicitly in this work, we suspect that extension of
the results from two dimensions to three dimensions would not cause any difficulties.
Kinematic unicycle rendezvous . In (Lin et al., 2005), the authors presented
Chapter 3. Coordination Problems 48
the first solution to this problem. The feedback in (Lin et al., 2005) is local and dis-
tributed, but it requires the use of time-varying feedbacks. In (Zheng et al., 2011) the
authors present a solution using a local and distributed, continuously differentiable, and
time-independent feedback. However the result yields rendezvous only when the sensing
graph is undirected and connected. The feedback in (Zheng et al., 2011) makes the uni-
cycles converge to a circular formation instead of rendezvous for some directed graphs
containing a globally reachable node. In (Zoghlami et al., 2013) the authors also consider
undirected graphs, and present a feedback to achieve rendezvous in finite time. In (Di-
marogonas and Kyriakopoulos, 2007) both positions and attitudes of the unicycles are
synchronized using a time-invariant distributed control. The authors assume an initially
connected sensing graph. The controller that is implemented, however, is discontinuous
and imposes excessive switching. In (Zheng and Sun, 2013) a time-independent, local
and distributed controller is presented. However, the authors make the assumption that
whenever two vehicles get sufficiently close together they merge into a single vehicle,
introducing a discontinuity in the control function. The same merging technique is used
in (Yu et al., 2012) for cyclic and tree graphs where each unicycle keeps its neighboring
vehicle within its windshield’s field of view in order to maintain graph connectivity and
achieve rendezvous. In (Ajorlou et al., 2015; Listmann et al., 2009) distributed solutions
are presented whereby the unicycles move toward the average position of their neighbors.
However, a unicycle’s feedback becomes undefined when it already lies at this average
position which includes the case when the unicycles are at rendezvous. Finally, in (Jafar-
ian, 2015) the authors solve the problem of practical rendezvous using a hybrid controller
in which the unicycles converge to an arbitrarily small neighborhood of one another for
undirected and connected graph topologies. The case of kinematic vehicles in three-space
is investigated in (Nair and Leonard, 2007; Dong and Geng, 2013; Hatanaka et al., 2012).
The authors of (Nair and Leonard, 2007; Dong and Geng, 2013) consider the problem of
full attitude and position synchronization, but assume fully actuated vehicles.
Chapter 3. Coordination Problems 49
Flying robot attitude synchronization and rendezvous . The problem of atti-
tude synchronization for flying robots is studied in (Ren, 2010; Abdessameud and Tayebi,
2009; Abdessameud and Tayebi, 2013). The proposed controllers do not require mea-
surements of the angular velocity, but they do require absolute attitude measurements.
In (Nair and Leonard, 2007), the authors use the energy shaping approach to design
local and distributed controllers for attitude synchronization. The same approach is
adopted in (Sarlette et al., 2009) to design two attitude synchronization controllers, both
distributed. The first controller achieves almost-global synchronization for directed con-
nected graphs. However, the controller design is based on distributed observers (Scardovi
et al., 2007), and therefore requires auxiliary states to be communicated among neighbor-
ing vehicles. It also employs an angular velocity dissipation term that forces all vehicle
angular velocities to zero in steady-state. The second controller in (Sarlette et al., 2009)
does not restrict the final angular velocities, and does not require communication, but it
requires an undirected sensing graph, and guarantees only local convergence.
For the rendezvous problem of flying robots, in (Wang, 2016) the authors consider
directed graphs containing a globally reachable node and develop an adaptive feedback
that is not local and distributed. In (Lee, 2012; Abdessameud and Tayebi, 2011) the au-
thors consider formation control for flying robots. However, again, the feedbacks are not
local and distributed. In (Abdessameud and Tayebi, 2011) the sensing graph is assumed
to be undirected, and communication among vehicles is required, while in (Lee, 2012)
the graph is balanced, and it is assumed that each vehicle has access to the thrust input
of its neighbors, therefore requiring once again communication between vehicles. Both
approaches in (Lee, 2012; Abdessameud and Tayebi, 2011) use a two-stage backstepping
methodology in which the first stage treats each vehicle as a point-mass system to which a
desired thrust is assigned. A desired thrust direction is then extracted and backstepping
is used to design a rotational control such that vehicle rendezvous or formation control
is achieved.
Chapter 3. Coordination Problems 50
Formations of kinematic unicycles . A formation controller for single integrator
robots can always be turned into a controller for kinematic unicycles if one considers a
point at a positive distance d in front of each unicycle. These points behave like single
integrators under an appropriate choice of feedback transformation, and can be driven to
a desired formation using the numerous techniques discussed earlier. However, although
the points converge to a formation, the unicycles themselves do not. Choosing a small
value of d reduces this error, but requires large control inputs. This results from the
fact that kinematic unicycles have a nonholonomic constraint which is not present in
the integrator model. For this reason, solutions for integrators do not adapt well into
solutions for kinematic unicycles. We now discuss solutions in the literature designed
specifically for kinematic unicycle formation control.
In (Dimarogonas and Kyriakopoulos, 2008), a discontinuous controller is presented
that stabilizes formations with synchronized heading directions, but unicycles require a
common sense of direction. The papers (Lin et al., 2005; Sepulchre et al., 2007; Tabuada
et al., 2005) discuss the feasibility of achieving various formations using local and dis-
tributed feedback. In (Lin et al., 2005), time-dependent solutions are presented in each
case. For general geometric patterns, unicycles require a common sense of direction.
Similarly, the solution in (Jin and Gans, 2017) is time dependent and requires measure-
ment of a common direction in addition to the velocity input of a neighbouring unicycle,
which can only be obtained if the unicycles can communicate these inputs with each
other. In (Liu and Jiang, 2013), a leader-follower approach is considered. The analysis
transforms the unicycle model into a system of double integrators through dynamic feed-
back linearization. The desired formation is attained for digraphs containing a spanning
tree, but each follower robot requires access to the acceleration of the leader through
communication. In (Oh and Ahn, 2013), each unicycle estimates its own position us-
ing dynamic extension, requiring communication among unicycles. The unicycles use
these estimated states to attain the desired formation globally. The rotational control,
Chapter 3. Coordination Problems 51
however, is time-dependent and oscillatory.
Kinematic unicycle formations with final collective motions . Results in
the literature for formation flocking remain quite limited. In (Reyes and Tanner, 2015)
it is stated that “although some links between provably convergent formation control
and flocking have been identified, the two behaviors have not been integrated into a
single design.” In (Reyes and Tanner, 2015) a solution is presented for parallel formation
flocking that requires knowledge of a common frame (i.e., beacon) and uses a dynamic
feedback linearization that requires communication of auxiliary states between agents.
In (Sepulchre et al., 2007), a local solution is presented for all-to-all undirected graphs.
However, only stability is shown and not asymptotic stability. For circular formation
flocking, (Sepulchre et al., 2007; Sepulchre et al., 2008) present asymptotically stable
solutions on a common circle with a focus on specific (M,N)-patterns. In (Chen and
Zhang, 2011) the authors present a controller with a repulsion function such that unicycles
achieve equal spacing around a common circle assuming a jointly connected graph, while
in (El-Hawwary and Maggiore, 2013a), the spacing of the unicycles can be freely chosen
beforehand. In these results, the feedbacks are local and distributed, however, the final
formation is restricted to lie on a common circle and the stability results in (Sepulchre
et al., 2007; Sepulchre et al., 2008; El-Hawwary and Maggiore, 2013a) are local.
A far more studied problem in the literature is formation path following. In a common
virtual structure approach (Peng et al., 2015; Loria et al., 2016; Sadowska et al., 2011) a
(possibly virtual) leader moves along the desired path at a desired speed and the other
unicycles converge to their corresponding place-holders with respect to the leader. In
this approach, at least one unicycle must have access to the virtual leader state. This
is trajectory tracking rather than path following and the path is not invariant in this
case. Another approach that is common in the literature (Do and Pan, 2007; Zhang and
Leonard, 2007; Ghabcheloo et al., 2009; Egerstedt and Hu, 2001; Chen and Tian, 2011;
Doosthoseini and Nielsen, 2015), assigns a different path Ci(si) to each unicycle i where
Chapter 3. Coordination Problems 52
si parametrizes the displacement of unicycle i along its path. Formation path following is
achieved whenever all unicycles lie on their corresponding paths and the path parameters
for all unicycles achieve consensus. To achieve this, each unicycle invokes a path following
controller in order to converge to its own path Ci(si) with a desired speed profile. In the
transient behaviour, the speed control is modified, slowing down or speeding up certain
unicycles, in order to synchronize the si states. While this approach does not require a
virtual leader, it does require communication of the quantities si between neighboring
unicycles and each unicycle needs to compute its own path to be followed. In (Brinon-
Arranz et al., 2014) unicycles converge to a common, compact, time-varying path with
uniform spacing. Each unicycle stores, on-board, the state of an exosystem which must
be communicated with neighboring unicycles. In (Reyes and Tanner, 2015) no virtual
leader is required in the solution, but rather, the formation centre of mass is driven to a
desired path. However, each unicycle must know the location of the formation centre of
mass, requiring all-to-all sensing. Finally, in (Consolini et al., 2012), the authors present
a solution for hierarchical, leader-follower topologies. Unicycles approximately achieve
formation as long as the path followed by the leader has sufficiently small curvature.
However, the stabilizing control of a unicycle depends on the linear speed inputs of its
neighbors.
There are several results in the literature that consider, specifically, formation circle
following. Most consider motion along a common circle (Sepulchre et al., 2007; Sepulchre
et al., 2008; Paley et al., 2008; Yu and Liu, 2016; Yu et al., 2018), while others, more
in line with this thesis, allow unicycles to travel around a common centre with different
radii and correspondingly, with different speeds. This is the problem studied in (Seyboth
et al., 2014) where the undirected sensing graph is assumed to be all-to-all and, by
changing a gradient potential function in the control law, one can achieve either phase
agreement or balancing. In (Zheng et al., 2015), on the other hand, the graph is a
ring and the spacing between neighboring unicycles are equal. Neither of these results,
Chapter 3. Coordination Problems 53
however, achieve arbitrary formations as in this thesis.
An important distinction among the results in the literature for formation path fol-
lowing is whether or not the formation rotates rigidly with the curve. A formation that
rotates rigidly with the curve is defined with respect to the curve’s frame of reference
rather than with respect to the inertial frame and, as such, will always point tangent to
the path as it follows it. On the other hand, most results define the formation based on
fixed offsets defined in the inertial frame I and the formation maintains a fixed orien-
tation relative to I as it traverses the curve. Each unicycle must be able to sense the
common inertial frame in this case. The distinction between formations defined with
respect to the path frame of reference versus the inertial frame is illustrated in Figure 3.6
for a diamond shaped formation composed of four unicycles.
Directionof motion
Directionof motion
Figure 3.6: (left) Diamond formation that rotates rigidly with the curve C and alwayspoints in the path’s tangent direction. (right) Diamond formation defined with respectto the inertial frame I that does not rotate as it traverses C.
Chapter 4
Preliminaries
This chapter contains fundamental definitions and results that will be applied on a reg-
ular basis throughout this thesis. First we will discuss a number of topics related to the
stability of sets and present corresponding propositions and theorems. Next we discuss
a number of relevant concepts in graph theory and review classes of gradient and homo-
geneous systems. Finally, we discuss a number of results in the literature for multi-agent
coordination in networks where the agents belong to elementary vehicle classes such as
single and double integrators and rotational integrators. These elementary tools will
be referred to as “control primitives”and will serve as building blocks to construct our
control solutions. This way of constructing feedbacks out of simpler building blocks is a
central theme to this thesis that will be seen time and again.
4.1 Basic Stability Theory
The primary objective in each chapter of this thesis will be to design feedbacks for each
agent in a multi-agent team evolving in state-space X , a smooth manifold, in order to
solve a coordination task. The feedbacks will be designed to satisfy a number of sensing
requirements and the control specification for each coordination problem, introduced in
Chapter 3, involves the stabilization of a desired closed subset of the state-space Γ ⊂ X
54
Chapter 4. Preliminaries 55
as opposed to stabilization of an equilibrium point. Solving the coordination task will
amount to “driving” the ensemble of agents to Γ. There are two main components to
this, the first relates to the stability of Γ and the second relates to attractivity. First, we
will discuss each of these concepts and present several formal definitions. Then, we will
present reduction theorems, powerful tools that will be used on numerous occasions in
this work.
4.1.1 Definitions
In this section we will define several notions related to the stability and attractivity
properties of a closed subset Γ ⊂ X . It will be shown how the definitions for stability
and attractivity can be combined to obtain an array of definitions for asymptotic stability
and practical stability. Finally, we will formalize the notion of stability of one set relative
to another. Note that the definitions presented in this section represent one choice and
are not fully inclusive. For a more detailed discussion of stability definitions, the reader
can refer to (Bhatia and Szego, 2002).
Let X ⊂ Rn be open. 1 If d : X × X → [0,∞) is the distance function between two
points in X and Γ ⊂ X is a closed subset of X , then we denote by ‖χ‖Γ := infψ∈Γ d(χ, ψ)
the point-to-set distance of χ ∈ X to Γ. If ε > 0, we let Bε(Γ) := χ ∈ X : ‖χ‖Γ < ε
and by N (Γ) we denote a neighborhood of Γ in X .
In the discussion that follows, consider a smooth dynamical system
Σ : χ = f(χ) (4.1)
with state space X and solutions defined for all time t ≥ 0, and let φ(t, χ0) denote the
solution at time t with initial condition χ(0) = χ0. A closed set Γ ⊂ X is said to be
positively invariant for Σ if for all χ0 ∈ Γ and all t > 0, φ(t, χ0) ∈ Γ.
1One can consider, more generally, a complete Riemannian manifold (X , g) with associated Rieman-nian distance function d : X × X → [0,∞) on X .
Chapter 4. Preliminaries 56
Figure 4.1: Illustration on the left shows stability of Γ - solutions with initial conditionsin N (Γ) remain inside Bε(Γ) for all time. Illustration on the right shows attractivity ofΓ with domain of attraction D(Γ) - solutions with initial conditions in D(Γ) converge toΓ.
Set Stability
Roughly, a closed subset Γ ⊂ X for system (4.1) is stable if solutions starting close to Γ re-
main close to Γ for all time. The precise definition for stability is given in Definition 4.1.1
and is illustrated on the left hand side of Figure 4.1.
Definition 4.1.1. The closed set Γ ⊂ X is stable for Σ if for any ε > 0, there exists a
neighborhood N (Γ) ⊂ X such that, for all χ0 ∈ N (Γ), φ(t, χ0) ∈ Bε(Γ), for all t > 0.
Stability is a key factor to ensure robustness in engineering applications. For example,
consider the problem of rendezvous where Γ represents the rendezvous set in which the
positions of all agents coincide. One would expect from a good control solution that
if initial robot positions are close to rendezvous, i.e., if the initial state is in a small
neighborhood of Γ then the corresponding closed-loop solution should not diverge too
much from Γ. If a set is not stable then it is called unstable.
Set Asymptotic Stability
The domain of attraction of a closed set Γ ⊂ X for system (4.1), denoted D(Γ), is the
set of initial conditions in the state space X from which solutions converge to the set
Chapter 4. Preliminaries 57
Γ as time approaches infinity. The notion of domain of attraction is formally stated in
Definition 4.1.2.
Definition 4.1.2. The domain of attraction of the closed set Γ ⊂ X for system Σ is the
set D(Γ) = χ0 ∈ X : limt→∞ ‖φ(t, χ0)‖Γ = 0.
Definition 4.1.3. The closed set Γ ⊂ X is (locally) attractive for Σ if D(Γ) is a neigh-
borhood of Γ; Γ is globally attractive for Σ if D(Γ) = X . Moreover, Γ is (locally) asymp-
totically stable for Σ if Γ is stable and locally attractive; Γ is globally asymptotically stable
for Σ if Γ is stable and globally attractive.
While it is desirable to have global results, asking for global asymptotic stability is
often too strong in many applications. This may be because this property is simply
too difficult to prove or, more fundamentally, it may not even be possible to design
continuous time-invariant control inputs to make a desired closed subset Γ ⊂ X globally
asymptotically stable due to a topological obstruction. In fact, this issue is quite common.
Take for example Theorem 1 in (Bhat and Bernstein, 2000), which implies that any
smooth time-invariant vector field on a compact manifold without boundary cannot have
any equilibrium that is globally asymptotically stable. Consequently, it is impossible to
globally asymptotically stabilize an equilibrium point in SO(2) and SO(3) via continuous
time-invariant feedback. For this reason, we will define a slightly weaker form of global
asymptotic stability called almost global asymptotic stability in which the domain of
attraction is not global, but rather, contains almost every point in the state-space.
Before giving a formal definition of almost global asymptotic stability, we need to
understand what a set of Lebesgue measure zero is. We will start by defining Lebesgue
measure zero sets in Rn and then extend this definition to smooth manifolds. Intuitively,
a set of Lebesgue measure zero is negligible in the sense that it occupies no volume in the
state space and the probability of randomly choosing a point in this set is therefore zero.
Define an open cube in Rn as the product of open intervals U = (a1, b1) × (a2, b2) · · · ×
Chapter 4. Preliminaries 58
(an, bn) where ai, bi ∈ R and ai < bi for all i ∈ n. Denote the volume of this cube by the
product of the intervals v(U) = (b1−a1)(b2−a2) . . . (bn−an). A set of Lebesgue measure
zero in Rn is one that can be covered by a collection of cubes occupying an arbitrarily
small volume.
Definition 4.1.4. A subset Γ ⊂ Rn has zero measure if for every ǫ > 0, there exist open
cubes U1, U2, . . . such that Γ ⊂ ⋃∞i=1 Ui and
∞∑
i=1
v(Ui) < ǫ.
In the case of a smooth manifold M , a set Γ ⊂M has Lebesgue measure zero in M if
it has zero measure under every smooth coordinate chart on M (Lee, 2013). Moreover,
by Proposition 6.8 in (Lee, 2013) M\Γ is dense in M . This means that the closure of
M\Γ satisfies M\Γ =M .
Definition 4.1.5. If M is a smooth n-dimensional manifold, a subset Γ ⊂ M has zero
measure inM if for every smooth coordinate chart (U, ϕ) forM , the subset ϕ(Γ∩U) ⊂ Rn
has measure zero.
We are now ready to define almost global stability properties.
Definition 4.1.6. The closed set Γ ⊂ X is almost globally attractive for Σ if X\D(Γ)
has Lebesgue measure zero. The set Γ is almost globally asymptotically stable for Σ if Γ
is stable and almost globally attractive.
The notion of almost global attractivity is illustrated in Figure 4.2. Next we give two
final definitions of attractivity and asymptotic stability which require a high gain control
k ∈ R. Consider the dynamical system
Σ(k) : χ = f(χ, k) (4.2)
Chapter 4. Preliminaries 59
Figure 4.2: Illustration of almost global attractivity of the set Γ. The set X\D(Γ) hasLebesgue measure zero.
and let φk(t, χ0) denote the solution at time t with initial condition χ(0) = χ0. For
system (4.2) the domain of attraction will depend on the parameter k in general and is
denoted by Dk(Γ).
Definition 4.1.7. The closed set Γ ⊂ X is semiglobally attractive with high-gain param-
eter k for Σ(k) if for each compact set K satisfying Γ ⊂ K ⊂ X , there exists k⋆ > 0 such
that for all k > k⋆, Γ is attractive for Σ(k) and K ⊂ Dk(Γ). The set Γ is semiglobally
asymptotically stable for Σ(k) if Γ is stable and semiglobally attractive.
Definition 4.1.8. A closed subset Γ ⊂ X is almost semiglobally attractive with high-gain
parameter k for Σ(k) if there exists a set N ⊂ X\Γ of Lebesgue measure zero such that
for each compact subset K satisfying Γ ⊂ K ⊂ (X\N), there exists k⋆ > 0 such that
for all k > k⋆, Γ is attractive for Σ(k) and K ⊂ Dk(Γ). The set Γ is almost semiglobally
asymptotically stable for Σ(k) if Γ is stable and almost semiglobally attractive. See
Figure 4.3.
The difference between global asymptotic stability and semiglobal asymptotic stability
of a closed subset Γ ⊂ X is that with the former, solutions converge to the set Γ from all
initial conditions, while with the latter, the domain of attraction can be made arbitrarily
large within the state-space by increasing the control gain k. The difference between
almost global asymptotic stability and almost semiglobal asymptotic stability is that
Chapter 4. Preliminaries 60
Figure 4.3: Illustration of almost semiglobal attractivity of the set Γ. For compact setsK1 ⊂ K2 ⊂ X\N , that do not contain the set N of Lebesgue measure zero, for all k ≥ k⋆1(left) and k ≥ k⋆2 > k⋆1 (right), solutions with initial conditions in the sets K1 and K2
respectively, converge to Γ. The domain of attraction of Γ approaches full measure asthe high gain k is increased.
Table 4.1: Asymptotic and Practical Stability Abbreviations
Definition num. Abb. Meaning
4.1.3 LAS or AS local asymptotic stability or asymptotic stability4.1.3 GAS global asymptotic stability4.1.6 AGAS almost-global asymptotic stability4.1.7 SGAS semi-global asymptotic stability4.1.8 ASGAS almost semi-global asymptotic stability4.1.9 GPS global practical stability
with the former, solutions converge to the set Γ with domain of attraction D(Γ) of full
measure, while with the latter, the domain of attraction approaches full measure with
increasing control gain k.
The abbreviations for each type of asymptotic stability are listed in Table 4.1.
In place of asymptotic stability, one can also consider the weaker notion of practical
stability.
Definition 4.1.9. The closed set Γ ⊂ X is globally practically stable for Σ(k) if for any
ε > 0, there exists k⋆ > 0 such that for all k > k⋆, Bε(Γ) has a subset containing Γ which
is globally asymptotically stable for Σ(k).
An illustration of global practical stability of the set Γ is shown in Figure 4.4. Notice
the duality between the concepts of semiglobal stability and global practical stability. If Γ
Chapter 4. Preliminaries 61
Figure 4.4: Illustration of global practical stability of the set Γ. For ǫ1 > ǫ2, for anyk ≥ k⋆1 (left) and k ≥ k⋆2 > k⋆1 (right), for any initial conditions in X , solutions convergeto the neighborhoods Bǫ1(Γ) and Bǫ2(Γ) respectively.
is asymptotically stable then it is necessarily practically stable without the requirement
of a high gain parameter k. The purpose of considering practical stability instead of
asymptotic stability is that in most engineering applications, it is sufficient that solutions
converge to an arbitrarily small neighborhood Bε(Γ) of Γ, and not Γ itself. The advantage
of considering notions of practical stability of a subset is that they are often significantly
easier to prove. The downside to practical solutions, however, is that they require a high
gain. If the neighborhood Bε(Γ) is to be very small, then the gain will typically be large.
Now consider an equilibrium p ∈ X for system (4.1). If the linearization of the system
equations in (4.1) at p has all eigenvalues in the open left half complex plane, then p is
exponentially stable. If an equilibrium p is exponentially stable, then for all initial condi-
tions χ0 in a small neighborhood of p, the error ‖φ(t, χ)− p‖ or ‖φk(t, χ)− p‖ converges
to zero at an exponential rate. If at least one eigenvalue at p is positive, the equilibrium
is called exponentially unstable as stated in Definition 4.1.10 taken from (Freeman, 2013).
Definition 4.1.10. An equilibrium p ∈ X is exponentially unstable for vector field f if
the differential dfp at p has at least one eigenvalue in the open right-half complex plane.
Relative and local set stability definitions
In this section we will generalize some of the definitions presented in the previous sections.
The definitions for relative and local set stability, taken from (El-Hawwary and Maggiore,
2013b), are given in Definition 4.1.11.
Chapter 4. Preliminaries 62
Definition 4.1.11. Let Γ1 ⊂ Γ2 be two subsets of X that are positively invariant for Σ.
Assume that Γ1 is compact and Γ2 is closed.
• Γ1 is stable relative to Γ2 for Σ if, for any ǫ > 0, there exists a neighborhood N (Γ1)
such that, φ(R+,N (Γ1) ∩ Γ2) ⊂ Bǫ(Γ1). The notions of relative set attractivity,
and asymptotic and practical stability are obtained analogously by restricting initial
conditions to lie in Γ2.
• Γ2 is locally stable near Γ1 if for all x ∈ Γ1, for all c > 0 and all ǫ > 0, there exists
δ > 0 such that for all x0 ∈ Bδ(Γ1) and all t⋆ > 0, if φ([0, t⋆], x0) ⊂ Bc(x) then
φ([0, t⋆], x0) ⊂ Bǫ(Γ2).
• Γ2 is locally attractive near Γ1 if there exists a neighbourhood N (Γ1) such that, for
all x0 ∈ N (Γ1), ‖φ(t, x0)‖Γ2 → 0 as t→ ∞.
The definitions for local stability of Γ2 near Γ1 and local attractivity of Γ2 near Γ1 are
illustrated in Figure 4.5 adapted from (El-Hawwary and Maggiore, 2013b). An explana-
tion of local stability of Γ2 near Γ1 was given in (El-Hawwary and Maggiore, 2013b) as
follows: “Given an arbitrary ball Bc(x) centred at a point x in Γ1, trajectories originating
in Bc(x) sufficiently close to Γ1 cannot travel far away from Γ2 before exiting Bc(x).”
Local attractivity of Γ2 near Γ1, on the other hand, means that all initial conditions
beginning in a neighborhood of Γ1 converge to Γ2 as t→ ∞. It can be seen immediately
that the following implications hold
• Γ2 is stable =⇒ Γ2 is locally stable near Γ1
• Γ1 is stable =⇒ Γ2 is locally stable near Γ1
• Γ2 is attractive =⇒ Γ2 is locally attractive near Γ1
The following illustrative example was taken from course notes (Maggiore, 2015). It
shows that Γ2 may be locally stable near Γ1 even if Γ2 itself is unstable, i.e., Γ2 is locally
stable near Γ1 ; Γ2 is stable.
Chapter 4. Preliminaries 63
Figure 4.5: Illustration of local stability of Γ2 near Γ1 (left) and local attractivity of Γ2
near Γ1 (right).
Example 4.1.12. Consider the system
x1 = −x1(1− x22)
x2 = x2,
let Γ1 = 0 be the origin, and let Γ2 be the x2 axis. We claim that Γ2 is unstable, but it
is locally stable near Γ1. Indeed, if x2(0) 6= 0, then x2(t) → ∞, so that eventually x1(t)
will have the same sign as x1(t) and tend to infinity. Thus Γ2 is unstable. On the other
hand, for any ball Bc(0), let ǫ > 0 be arbitrary. We see from the phase portrait that we
can find δ > 0 such that solutions originating in Bδ(Γ1) do not exit Bǫ(Γ2) as long as
they remain in Bc(0). Thus, Γ2 is locally stable near Γ1.
The definitions for exponential stability and instability of an equilibrium point can
similarly be defined relative to a subset Γ ⊂ X as stated in Definition 4.1.13 below.
Definition 4.1.13. Consider the dynamical system Σ, let Γ be a closed subset of X and
suppose Γ is invariant under the flow of the vector field f . Let f |Γ be the restriction of
the vector field f to Γ.
• If p ∈ Γ is an equilibrium of f , we say that p is exponentially stable relative to
Γ if p is exponentially stable for f |Γ, i.e., the differential d(f |Γ)p at p has all its
eigenvalues in the open left-half complex plane.
• If p ∈ Γ is an equilibrium of f , we say that p is exponentially unstable relative to Γ
Chapter 4. Preliminaries 64
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
x1
x2
Bǫ(Γ2)
Bc(0)Bδ(Γ1)
Figure 4.6: Example of local stability of Γ2 near Γ1 in Example 4.1.12. Image from (Mag-giore, 2015)
if p is exponentially unstable for f |Γ, i.e., the differential d(f |Γ)p at p has at least
one eigenvalue in the open right-half complex plane.
Theorem 4.1.1 below, taken from (Freeman, 2013), allows us to say when the domain
of attraction of a set of exponentially unstable equilibriaA is Lebesgue measure zero. This
is a desirable property if the states in A represent undesirable equilibrium configurations
of the state-space for the closed loop system that do not correspond with the control
objective.
Theorem 4.1.1 (Proposition 1 in (Freeman, 2013)). Suppose A is a set of equilibria for
system Σ, each equilibrium in A is exponentially unstable, and
D(A) =⋃
z∈A
D(z). (4.3)
Then D(A) is meager and has zero measure.
Condition (4.3) of Theorem 4.1.1 is satisfied if and only if a solution converging to
the set A, necessarily converges to one of the exponentially unstable equilibria in the set
Chapter 4. Preliminaries 65
A. A set is meager if it can be expressed as the union of countably many nowhere dense
subsets.
4.1.2 Reduction Theorems
In this section we introduce a powerful tool for analyzing the stability of sets. This tool
is called reduction and allows one to determine the stability and asymptotic stability
properties of a compact positively invariant subset of X , call it Γ1, in a hierarchical
fashion. In particular, if Γ1 satisfies a certain stability or asymptotic stability property
relative to another positively invariant set Γ2 containing it, then the reduction theorem
gives conditions guaranteeing that Γ1 also satisfies this property with respect to the entire
state-space X .
Reduction Theorem for Asymptotic Stability
Theorem 4.1.2 presents the Reduction Theorem for stability, local asymptotic stability,
and global asymptotic stability.
Theorem 4.1.2 (Reduction Theorem (El-Hawwary and Maggiore, 2013b; Seibert and
Florio, 1995)). Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X , be two closed sets that are positively
invariant for Σ, and suppose Γ1 is compact. Consider the following assumptions:
(i) Γ1 is LAS relative to Γ2;
(i’) Γ1 is GAS relative to Γ2;
(ii) Γ2 is locally stable near Γ1;
(iii) Γ2 is locally attractive near Γ1;
(iii)’ Γ2 is globally attractive;
(iv) all trajectories of Σ are bounded.
Chapter 4. Preliminaries 66
Figure 4.7: Illustration of the Reduction Theorem. The set Γ1 is asymptotically stablerelative to Γ2 and Γ2 is asymptotically stable. By reduction, Γ1 is asymptotically stable.
Then, the following implications hold: (i) ∧ (ii) =⇒ Γ1 is stable; (i) ∧ (ii) ∧ (iii) ⇐⇒
Γ1 is LAS; (i)’ ∧ (ii) ∧ (iii)’ ∧ (iv) ⇐⇒ Γ1 is GAS.
Note that one can replace condition (ii) with the stronger condition “Γ2 is stable”
and replace (iii) with the stronger condition “Γ2 is asymptotically stable”. Then it is
clear conditions (ii) and (iii) together are satisfied when Γ2 is asymptotically stable and
(ii) and (iii)’ together are satisfied when Γ2 is globally asymptotically stable. Reduction
is illustrated in Figure 4.7.
To help illustrate how the Reduction Theorem works, we provide a simple proof in
the special case of linear systems using tools learned in an undergraduate control course.
Proposition 4.1.14. Consider the linear system
x = Ax, (4.4)
where x ∈ Rn and A ∈ Rn×n. Suppose the origin Γ1 = 0 is an equilibrium point,
Γ2 is an m-dimensional A-invariant subspace and the following assumptions from the
Reduction Theorem hold
(i)’ Γ1 is (globally) asymptotically stable relative to Γ2
(ii),(iii)’ Γ2 is (globally) asymptotically stable.
Then Γ1 is (globally) asymptotically stable.
Chapter 4. Preliminaries 67
Proof. Since Γ2 is A-invariant, there exists a coordinate transformation z = (z1, z2) =
T−1x such that z1 ∈ Rm represents coordinates tangential to Γ2 and z2 ∈ Rn−m repre-
sents coordinates transversal to Γ2. Moreover, let A = T−1AT . By the Representation
Theorem for linear systems, it follows that
z1
z2
=
A11 A12
0 A22
z1
z2
. (4.5)
Restricted to the set Γ2, the system equations in (4.5) are given by
z1 = A11z1
and assumption (i)’ implies that the eigenvalues of A11 belong to the open left half plane.
Moreover, assumptions (ii),(iii)’ imply the origin of the system,
z2 = A22z2
in (4.5) is asymptotically stable and therefore the eigenvalues of A22 belong to the open
left half plane. We conclude that A in (4.5) has all eigenvalues in the open left half plane
and therefore Γ1 is asymptotically stable.
Reduction Theorem: Almost Global Asymptotic Stability
The next result presents a reduction theorem for almost global asymptotic stability. This
is a novel contribution of this thesis.
Theorem 4.1.3. Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X , be two closed sets that are positively
invariant for Σ, and suppose Γ1 is compact. Consider the following assumptions:
(i) Γ2 is an embedded submanifold of X which is globally asymptotically stable for Σ.
(ii) Γ1 is globally attractive relative to Γ2 and can be decomposed as a disjoint union
Chapter 4. Preliminaries 68
Figure 4.8: Illustration of reduction for almost global asymptotic stability. The figureon the left illustrates the dynamics on Γ2 which is globally asymptotically stable relativeto X . Γ2 contains three unstable equilibria A = z1, z2, z3 and a compact set K thatis almost globally asymptotically stable relative to Γ2. By the Reduction Thoerem theset K is also AGAS relative to X as shown in the figure on the right. The set D(A) isLebesgue measure zero and contains the initial condition χ0 whose solution converges tothe point z3 ∈ A.
Γ1 = A⊔K where A is a set of isolated equilibria which are exponentially unstable
relative to Γ2 and K is asymptotically stable relative to Γ2.
(iii) all trajectories of Σ are bounded.
Then K is almost globally asymptotically stable.
The Reduction Theorem for almost global asymptotic stability is illustrated in Fig-
ure 4.8. The proof of Theorem 4.1.3 relies on the three lemmas presented next. The
proof of Theorem 4.1.3 is given after the presentation of the lemmas.
Lemma 4.1.15. Let Γ be a closed embedded submanifold of X which is invariant under
a C1 vector field f : X → TX . Let (U, ϕ) be a smooth coordinate chart centred at p ∈ Γ
and let dfp be the matrix representation of the differential dfp : TpX → Tf(p)(TX ) in
coordinates. If p is an equilibrium of f , then the subspace Tϕ(p)ϕ(Γ ∩ U) of Tϕ(p)ϕ(U) is
(dfp)-invariant.
Proof. Let n = dimX , p ∈ Γ be such that f(p) = 0, and let vp ∈ TpΓ be arbitrary
with vector representation vp ∈ Tϕ(p)ϕ(Γ ∩ U) in coordinates. We need to prove that
Chapter 4. Preliminaries 69
dfpvp ∈ Tϕ(p)ϕ(Γ ∩ U). Since Γ is embedded, there exists an open set U ⊂ U containing
p and a smooth submersion h : U → Rn−dimΓ such that Γ∩ U = h−1(0). Let I = (−1, 1)
and σ : I → X be a smooth regular curve whose image is contained in Γ ∩ U , and such
that σ(0) = p and σ(0) = vp. Since Γ is invariant under the flow of f , f(σ(t)) ∈ Tσ(t)Γ
for all t ∈ I or, what is the same, dhσ(t)(f(σ(t))) = 0 for all t ∈ I. Equivalently,
dhσ(t)f(σ(t)) = 0 for all t ∈ I where f(σ(t)) is the vector representation of f(σ(t)) and
dhσ(t) is the matrix representation of the differential dhσ(t) : Tσ(t)U → Th(σ(t))Rn−dimΓ in
coordinates. The derivative of dhσ(t)f(σ(t)) with respect to t must be zero at t = 0,
d
dt
∣
∣
∣
t=0
[
dhσ(t)f(σ(t))]
= 0,
or(
d
dt
∣
∣
∣
t=0dhσ(t)
)
f(σ(0)) + dhσ(0)d
dt
∣
∣
∣
t=0f(σ(t)) = 0.
Since f(σ(0)) = f(p) = 0, using the chain rule we get dhσ(0)dfσ(0)vp = 0, proving that
dfpvp ∈ TpΓ.
Lemma 4.1.16. Let Γ be a closed embedded submanifold of X which is invariant under
a C1 vector field f : X → TX . If p ∈ Γ is an equilibrium of f which is exponentially
unstable relative to Γ, then p is exponentially unstable relative to X .
Proof. Let n = dimX and k = dimΓ. Since Γ is embedded, there exists a smooth
coordinate chart (U, ϕ) centred at p, i.e., ϕ(p) = 0, such that ϕ(Γ ∩ U) = x ∈
Rn : xk+1 = · · · = xn = 0 where x = (x1, . . . , xk, xk+1, . . . , xn) are local coordi-
nates (Lee, 2013). Since Γ is positively invariant, it follows that the restriction of the
vector field f(x) = (fi(x))i∈1,...,n to Γ ∩ U , represented in local coordinates, is given
by f |Γ∩U(x1, . . . , xk) = (fi(x1, . . . , xk, 0, . . . , 0))i∈1,...,k. Since p is exponentially unsta-
ble relative to Γ, d(f |Γ∩U)ϕ(p) has at least one eigenvalue in the open right-half complex
plane. To prove that p is exponentially unstable relative to X , it needs to be shown
that dfϕ(p) has at least one eigenvalue in the open right-half complex plane. It follows
Chapter 4. Preliminaries 70
from Lemma 4.1.15 that the tangent space Tϕ(p)ϕ(Γ ∩ U) is (dfϕ(p))-invariant in local
coordinates and therefore dfϕ(p) has the upper triangular form,
dfϕ(p) =
A1 A2
0 A3
where
A1 =
df1(x)dx1
. . . df1(x)dxk
......
dfk(x)dx1
. . . dfk(x)dxk
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
x=0
=
df1(x1,...,xk,0,...,0)dx1
. . . df1(x1,...,xk,0,...,0)dxk
......
dfk(x1,...,xk,0,...,0)dx1
. . . dfk(x1,...,xk,0,...,0)dxk
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
(x1,...,xk)=0
= d(f |Γ∩U)ϕ(p)
and therefore dfϕ(p) contains at least one eigenvalue in the open right-half complex plane.
Lemma 4.1.17. For the dynamical system Σ, suppose Γ ⊂ X is a closed embedded
submanifold of X which is globally asymptotically stable. If N ⊂ Γ is a set of Lebesgue
measure zero in Γ, then Γ\N is globally asymptotically stable.
Proof. To show that Γ\N is stable, it needs to be shown that for any ε > 0, there exists a
neighborhood N (Γ\N) such that, for all χ0 ∈ N (Γ\N), φ(t, χ0) ∈ Bε(Γ\N), for all t > 0.
Since Γ is globally asymptotically stable, there exists a neighborhood N1(Γ) such that,
for all χ0 ∈ N1(Γ), φ(t, χ0) ∈ Bε/2(Γ), for all t > 0. Since N is Lebesgue measure zero in
Γ, Γ\N is dense in Γ by Proposition 6.8 in (Lee, 2013) and for all χ ∈ N there exists a
point χ ∈ Γ\N such that χ ∈ Bǫ/2(χ). Stability follows by choosing N (Γ\N) = N1(Γ).
Analogously, global attractivity of Γ\N follows from global attractivity of Γ and
density of Γ\N in Γ.
The proof of Theorem 4.1.3 is presented below.
Chapter 4. Preliminaries 71
Proof of Theorem 4.1.3. By Lemma 4.1.16, the isolated equilibria in A, which are ex-
ponentially unstable equilibria relative to Γ2, are also exponentially unstable relative to
X . It holds from Theorem 2.37 in (Rudin, 1976) that since A is compact, if A were an
infinite subset of Γ2, then A would necessarily have a limit point χ ∈ Γ2. Since A is
closed, it contains all its limit points and therefore χ ∈ A. But this contradicts that all
points in A are isolated and therefore A must be finite. Suppose there are m equilibria
in A labelled χ1, . . . , χm. Then for all i ∈ 1, . . . , m, there exists ǫi > 0 such that
Bǫi(χi) ∩ A = χi and the minimum ǫ = minǫ1, . . . , ǫm exists. It follows that for all
i ∈ 1, . . . , m, Bǫ(χi) ∩ A = χi. The condition D(A) =⋃
z∈AD(z) in Theorem 4.1.1
holds if any solution φ(t, χ0) with initial condition χ0 ∈ X that converges to A, necessar-
ily converges to a particular point χ ∈ A. This must be the case since for any solution
φ(t, χ0) converging to A there exists a time T > 0 such that φ(t, χ0) ∈ Bǫ/2(A) for all
t > T . Therefore, there exists a point χ ∈ A such that φ(T, χ0) ∈ Bǫ/2(χ). For t > T it
is impossible for the solution to leave the vicinity of χ and converge to another point in
A located at least a distance ǫ away. Therefore the condition in (4.3) holds and D(A)
has zero measure by Theorem 4.1.1. It holds from analogous arguments that the set
D(A) ∩ Γ2 is Lebesgue measure zero in Γ2.
Denote W := X\D(A) which is a positively invariant domain of full measure and
define Γ2 := W ∩ Γ2, a set of full measure in Γ2. Then Γ2 is GAS relative to X by
Lemma 4.1.17. Since W is positively invariant, it follows that Γ2 is GAS relative to W .
Since K is stable relative to Γ2 and W is positively invariant, K is also stable relative to
Γ2 = W ∩ Γ2 and since Γ1 = A ∪ K is globally attractive relative to Γ2, it immediately
holds that K is globally attractive relative to Γ2. Therefore K is GAS relative to Γ2 and
Γ2 is GAS relative to W implying, by Theorem 4.1.2, that K is globally asymptotically
stable relative toW . This implies that K is almost globally asymptotically stable relative
to X .
Chapter 4. Preliminaries 72
4.1.3 Lyapunov and Krasovskii-LaSalle Theorems
In this section we state a number of standard stability results taken from (Maggiore,
2009). These include Lyapunov Stability Theorems and the Krasovskii-LaSalle Invariance
Principle which will be used often in this work. These results are stated here for the
reader’s convenience.
Consider the dynamical system
χ = f(χ), χ ∈ X (4.6)
where X ⊂ Rn is open and f is either locally Lipschitz on X or C1. Suppose that 0 ∈ X .
For a C1 function V (χ), LfV (χ) := (d/dχ)V (χ) · f(χ) is the Lie derivative of V along f .
Theorem 4.1.4 (Lyapunov’s direct method). Let χ = 0 be an equilibrium of (4.6) with
X ⊂ Rn, and suppose there exists a C1 function V : D → R, with D ⊂ X a domain
containing 0, such that
(i) V is positive definite at 0
(ii) The Lie derivative LfV is negative definite at 0
then 0 is an asymptotically stable equilibrium of (4.6).
The global version of the Lyapunov Theorem is given below.
Theorem 4.1.5 (Barbashin-Krasovskii). Let χ = 0 be an equilibrium of (4.6) with
X = Rn, and suppose there exists a C1 function V : Rn → R function such that V is
positive definite at 0, radially unbounded, and LfV is negative definite at 0. Then, χ = 0
is globally asymptotically stable.
Theorem 4.1.6 (Krasovskii-LaSalle Invariance Principle). Let Ω ⊂ D be a compact
positively invariant set. Let V : D → R be a C1 function such that for all χ ∈ Ω, LfV
Chapter 4. Preliminaries 73
is negative semi-definite. Let E = χ ∈ Ω : LfV = 0 and let N be the largest invariant
subset of E. Then for all χ0 ∈ Ω, φ(t, χ0) → N as t→ ∞.2
From Theorem 4.1.6, it is also possible to obtain global results. In particular, if
D = Rn, every level set of V is compact, and if for all χ ∈ Rn, LfV is negative semi-
definite, then for all χ0 ∈ Rn, φ(t, χ0) converges to N , the largest invariant set of E =
χ ∈ Rn : LfV = 0. The three theorems just stated have been presented with increasing
generality with the Krasovskii-LaSalle Invariance Principle above being the most general
as it does not require the function V to be positive definite and the attractor set is not
necessarily a point. The following proposition establishes global practical stability of an
equilibrium.
Proposition 4.1.18 (Lyapunov result for global practical stability). Let χ = 0 be an
equilibrium of (4.6) with X = Rn, and suppose there exists a C1 function V : Rn → R
such that V is positive definite at 0, has compact level sets, and LfV is negative outside
the set Vc := χ ∈ Rn : V < c. Then, Vc is globally asymptotically stable.
Proof. Stability follows because LfV is negative outside the sub-level set Vc of the Lya-
punov function V . For attractivity, consider any initial condition χ0 ∈ Rn. There exists
ǫ > c such that χ0 ∈ Vǫ. Since LfV is negative outside of Vc and Vǫ\Vc is a compact set,
d := maxχ∈Vǫ\Vc LfV (χ) < 0 is well-defined. Therefore, φ(t, χ0) ∈ Vc for all t > (ǫ− c)/d
proving attractivity of Vc.
Remark 4.1.7. The three theorems and proposition stated above all require V (χ) to be
a C1 function. The proofs still hold, however, under the milder assumption that V (χ)
and LfV (χ) are both continuous.
2The proof of the Krasovskii-LaSalle Invariance Principle (and in turn, the other theorems in thissection) also apply when the state space is a smooth Riemannian manifold (X , g). See, for example, theproof in (Khalil, 2002, Theorem 4.4).
Chapter 4. Preliminaries 74
Application to Gradient and Homogeneous Systems
We now review classes of gradient and homogeneous systems and show how they are well
suited for stability analyses using the Lyapunov and Krasovskii-LaSalle theorems just
discussed. The definition of a gradient system is as follows,
Definition 4.1.19. A gradient system on the state space X is a differential equation of
the form,
χ = −∇V (χ), χ ∈ X (4.7)
where V : X → R is a real valued, twice continuously differentiable function, and ∇V (χ)
is its gradient.
The main feature of a gradient system is that the vector field −∇V (χ) is orthogonal
to the level sets of the corresponding storage function V and point in the direction of
steepest decent of V . The stationary points of the system correspond to the set where
the gradient of V is zero, i.e., the critical points of V . The function V becomes a natural
candidate for a Lyapunov analysis. Taking the time derivative of V along system (4.7)
yields,
V =∂V
∂χχ = −‖∇V (χ)‖2 ≤ 0.
Consider any isolated critical point χ which is a local minimum of V . Then χ is an
equilibrium for (4.7) and it immediately follows from the Lyapunov’s direct method that
χ is asymptotically stable for (4.7). Gradient systems can also be analysed with the
Krasovskii-LaSalle Invariance Principle. In particular, if V is defined globally on the
domain D = Rn and its level sets are compact, then since the time derivative of V is
negative semi-definite and equals zero only in the set E = χ ∈ X : ∇V (χ) = 0 of the
critical points of V , we can conclude by Krasovskii-LaSalle that all solutions converge to
the largest invariant set of critical points of V .
Definition 4.1.20. Let U,W be finite-dimensional vector spaces and let V be a set. A
Chapter 4. Preliminaries 75
function f : U → W is homogeneous of degree r ≥ 0 if, for all λ > 0 and for all χ ∈ U ,
f(λχ) = λrf(χ). A function f : U×V →W , (χ,Υ) 7→ f(χ,Υ), is homogeneous of degree
r with respect to χ if for all λ > 0 and for all (χ,Υ) ∈ U × V , f(λχ,Υ) = λrf(χ,Υ).
By this definition, a homogeneous function of degree r is one that scales by powers
of r moving outward along rays going through the origin. Examples of homogeneous
functions are
• The function f(x, y) = atan2(y, x) is homogeneous of degree zero. Notice that f is
undefined at (x, y) = (0, 0);
• Linear functions are homogeneous of degree one directly by the scalar multiplication
property f(λχ) = λχ for any λ > 0;
• The real function f(x, y) = xy is homogeneous of degree two since f(λx, λy) = λ2xy
for any scalar λ > 0;
• The real function f(x, y) = x2y is homogeneous of degree two with respect to x
and one with respect to y.
If the function f(χ) is homogeneous of degree r and g(χ) is homogeneous of degree s
then the product h(χ) is homogeneous of degree r + s since
h(λχ) = f(λχ)g(λχ) = λr+sf(χ)g(χ) = λr+sh(χ).
Similarly, h(χ) = f(χ)/g(χ) is homogeneous of degree r − s. Consider the following
propositions related to homogeneous functions.
Proposition 4.1.21. For a compact set V , let f : Rn × V → R, (χ,Υ) 7→ f(χ,Υ) be a
continuous function, homogeneous of degree zero with respect to χ. Then f achieves a
maximum on Rn × V .
Chapter 4. Preliminaries 76
Proof. Since f(χ,Υ) is homogeneous of degree zero with respect to χ, it follows that for all
χ 6= 0, f(χ,Υ) = ‖χ‖0f(χ/‖χ‖,Υ) = f(χ/‖χ‖,Υ). Since f is continuous and (χ/‖χ‖,Υ)
lie on compact sets, f(χ/‖χ‖,Υ) has a maximum value on the domain (Rn\0) × V .
Moreover, f(0,Υ) = 0. Therefore, f achieves a maximum on Rn × V .
Proposition 4.1.22. For a compact set V , let f : Rn × V → R, (χ,Υ) 7→ f(χ,Υ) be
a function, homogeneous of degree r ≥ 1 with respect to χ and continuous on the set
(Rn\0)× V . Then f is continuous on Rn × V .
Proof. It needs to be shown that f is continuous on 0×V . Since f is homogeneous of de-
gree r ≥ 1, f(0,Υ) = 0. For all χ 6= 0, f(χ,Υ) satisfies |f(χ,Υ)| = ‖χ‖r|f(χ/‖χ‖,Υ)| ≤
‖χ‖rmax(χ,Υ)∈(Rn\0)×V |f(χ/‖χ‖,Υ)|. This maximum exists because (χ/‖χ‖,Υ) lie on
compact sets and |f(χ,Υ)| is continuous on (Rn\0)× V . Therefore, for all
δ <
(
ǫ
max(χ,Υ)∈(Rn\0)×Υ |f(χ/‖χ‖,Υ)|
)1/r
,
‖χ‖ < δ implies |f(χ,Υ)| ≤ ǫ and therefore f(χ,Υ) is continuous on the set 0 × V .
Now consider the dynamical system,
Σ : χ = f(χ), χ ∈ X (4.8)
where X is a finite-dimensional vector space. We say that Σ in (4.8) is a homogeneous
system of degree r if f is a homogeneous function of degree r.
Proposition 4.1.23 below shows that if we have a homogeneous system and a positive
definite homogeneous Lyapunov function V , then to prove global asymptotic stability of
the origin 0, it is enough to show negative definiteness of LfV on a compact set rather
than the entire state-space. This is beneficial because compact sets enjoy several useful
properties. For example, the fact that continuous real functions attain a maximum on
compact sets is an important feature that will be used in this thesis.
Chapter 4. Preliminaries 77
Proposition 4.1.23. Consider the homogeneous system of degree r > 0 in (4.8), and
assume that X = Rn. Suppose that there exists a continuous positive definite function
V : Rn → R, homogeneous of degree s ≥ 1 and suppose that LfV (χ) is continuous and
satisfies LfV (θ) < 0 for all θ ∈ χ ∈ Rn : ‖χ‖ = 1 ∼= Sn−1. Then χ = 0 is globally
asymptotically stable.
Proof. Since V (χ) is homogeneous of degree s ≥ 1, for all λ > 0 and for all χ ∈ X ,
V (λχ) = λsV (χ). It follows from Euler’s Theorem presented in Section 12.8 in (Allen,
1938) that the derivative DV (χ) := ∂V (χ)/∂χ is homogeneous of degree s−1. Moreover,
V (χ) is radially unbounded because it is both positive definite and homogeneous of degree
s ≥ 1. One can write χ = λθ where λ = ‖χ‖ and θ ∈ Sn−1 is a unit vector. The Lie
derivative LfV (χ) therefore satisfies
LfV (χ) = DV (λθ)f(λθ)
which by the homogeneity properties of DV and f becomes,
LfV (χ) = λr+(s−1)DV (θ)f(θ) = λr+(s−1)LfV (θ).
By assumption, for all θ ∈ Sn−1, LfV (θ) < 0. Therefore LfV (χ) ≤ 0 with equality
if and only if λ = ‖χ‖ = 0 and it follows that LfV (χ) is negative definite. By the
Barbashin-Krasovskii Theorem and Remark 4.1.7, the origin is globally asymptotically
stable.
The proofs in Chapter 5 and, particularly, Chapter 6 use similar arguments to Propo-
sition 4.1.24 below. This proposition is presented primarily as an aid to understanding
the logic in those proofs.
Proposition 4.1.24. Consider the homogeneous system of degree r > 0 in (4.8), and
assume that X = Rn. Let V : X → R be a continuous positive definite function,
Chapter 4. Preliminaries 78
homogeneous of degree s ≥ 2. Let g : X → R be a continuous function, homogeneous of
degree s− 1. Finally, let
W (χ) = α√
V (χ) + g(χ)
for α > 0. Then there exists α⋆ > 0 such that for all α > α⋆, W (χ) is positive definite.
Moreover, if LfW (χ) is continuous and LfW (θ) < 0 for all θ ∈ χ ∈ Rn : ‖χ‖ = 1 ∼=
Sn−1 and for all α > α⋆, then χ = 0 is globally asymptotically stable.
Proof. The function W (χ) is continuous and homogeneous of degree s− 1 because both√
V (χ) and g(χ) are such. First it is shown that there exists α⋆ > 0 such that for all
α > α⋆, W (χ) is positive definite. For χ 6= 0, V (χ) > 0 and one can write
W (χ) =√
V (χ)
(
α +g(χ)√
V (χ)
)
.
The function g(χ)/√
V (χ) is homogeneous of degree zero because the numerator and
denominator are both homogeneous of degree s− 1 and therefore, by Proposition 4.1.21,
it attains a maximum over X . Therefore, choosing α⋆ > maxχ∈X ‖g(χ)/√
V (χ)‖ implies
that β := α⋆ + g(χ)/√
V (χ) > 0. Then, for all χ 6= 0, W (χ) >√
V (χ)β > 0 since V (χ)
is positive definite and W (0) = 0 since W (χ) is homogeneous of degree s− 1 > 0. W (χ)
is radially unbounded because it is both positive definite and homogeneous of degree
s − 1 > 0. Moreover, if LfW (θ) < 0 for all θ ∈ χ ∈ Rn : ‖χ‖ = 1 ∼= Sn−1 and for
all α > α⋆, then it follows from Proposition 4.1.23 that χ = 0 is globally asymptotically
stable.
4.2 Graph Theory
Since graph theory is such a broad field, in this background section we will limit the
discussion solely to notions in graph theory that are pertinent to this work. For more in
depth detail on the notions introduced in this section and graph theory in general, we
2011), J2 = J1, J3 = 1.4J1, J4 = 1.2J1, J5 = 1.2J1. We pick aij = 0.3 for all j ∈ Ni
and γ = 30. The control gains k1 and k2 in (5.1) are chosen to be k1 = 2 and k2 = 0.45.
The initial conditions of the robots are shown in Table 5.1. The initial attitudes Ri(0)
of the robots are: up(right), side(ways) 1, side(ways) 2 and (upside)down respectively
given by:
1 0 0
0 1 0
0 0 1
,
1 0 0
0 0 −1
0 1 0
,
1 0 0
0 0 1
0 −1 0
,
1 0 0
0 −1 0
0 0 −1
.
Figure 5.4 shows the simulation without the presence of disturbances while Figure 5.5
shows the simulation when disturbances are present. The disturbances are: an additive
random noise with maximum magnitude of 0.25N on the applied force; an additive
random noise with maximum magnitude of 0.25N·m on the applied torque; an additive
measurement error for the angular velocity, with maximum magnitude of 0.25 rad/s; an
additive random noise on the quantity fi(yii) accounting for errors in measurements of
relative displacements and velocities of the vehicles. The direction of this vector has
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 105
3
1
2
4 5
Figure 5.3: Sensor digraph used in the simulation results.
been rotated within 0.25 rad and the magnitude is scaled between 0.75 to 1.25 times
the actual magnitude. The disturbances are updated 10 times per second. In both
cases of Figure 5.4 and Figure 5.5, the vehicles’ positions and velocities converge to a
neighborhood of one another.
In Figure 5.4 the vehicles remain within 0.25m of one another while in Figure 5.5 the
vehicles remain within 1m of one another at steady state. These neighborhoods can be
made even smaller by further increasing the control gains k1 and k2. However, this would
result in having higher control inputs. Metrics related to the thrust and torque inputs are
presented in Table 5.2. The first two rows show peak control norms and the last two show
the root mean square (rms) of the control norms. In these simulations we considered zero
gravity, i.e., g = 0. This was done to improve visibility of the simulation results. In the
presence of gravity, the vehicles would still converge to the same neighborhood of one
another, however at steady state they would accelerate in the direction of gravity since
gravity is not compensated through the control inputs in (5.1).
5.3 From Rendezvous to Formations
A notable omission from this thesis is a solution for formation control of flying robots.
The analogous problem for kinematic unicycles will be presented in Chapter 7 using
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 106
0 50 100 150 200−20
−10
0
10
20
time (s)
i x
0 50 100 150 200−10
−5
0
5
10
time (s)
i y0 50 100 150 200
−20
−10
0
10
time (s)
i z
0 50 100 150 2000
0.2
0.4
0.6
0.8
time (s)sp
eed
(m/s
)
Figure 5.4: Rendezvous control simulation without the presence of disturbances. At thetop-left, top-right and bottom-left: positions of the five robots expressed in the inertialframe I. At the bottom-right: linear speeds ‖vi‖, i ∈ 1 : 5.
strictly local and distributed feedbacks. For completeness, in this section we discuss how
one can transform the rendezvous controller for flying robots in (5.1) (or the solution
in (Roza et al., 2014) with hovering) into a formation controller. The final feedback,
however, does not meet the strict local and distributed sensing requirements since each
agent needs to know its own orientation in the inertial frame I.
Consider any desired configuration of n flying robots x = d where d = (d1, . . . , dn) ∈
R3n is centred, without loss of generality, about the origin, i.e.,∑n
i=1 di = 0. A set of
n flying robots is said to be in formation, when they satisfy the desired configuration d
modulo translations. That is, the position of the i-th robot satisfies xi = di + x where
x = (1/n)∑n
i=1 xi is the average position of the robots. Correspondingly, define the
formation manifold as
Γf :=
χ ∈ X : xi = di + x, vij = 0, Ωii = Ωii, i, j ∈ n
. (5.5)
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 107
0 50 100 150 200−50
0
50
time (s)
i x
0 50 100 150 200−10
−5
0
5
10
time (s)
i y0 50 100 150 200
−10
0
10
20
time (s)
i z
0 50 100 150 2000
0.5
1
1.5
time (s)sp
eed
(m/s
)
Figure 5.5: Rendezvous control simulation with the presence of disturbances. At thetop-left, top-right and bottom-left: positions of the five robots expressed in the inertialframe I. At the bottom-right: linear speeds ‖vi‖, i ∈ 1 : 5.
For all i ∈ n, define the fixed inertial vector δi := −di attached to robot i with endpoint
xi = xi+ δi. The collection of endpoints is denoted x := (xi)i∈n and xij := xj − xi. Then
the formation manifold in (5.5) can be rewritten as
Γf :=
χ ∈ X : xij = 0, vij = 0, Ωii = Ωii, i, j ∈ n
. (5.6)
This follows because xij = 0 implies that xij = dj − di and therefore, using the fact that∑n
i=1 di = 0,
xi − x =1
n
n∑
j=1
(xi − xj) =1
n
n∑
j=1
(di − dj) = di
as in (5.5). Therefore Γf reduces to the rendezvous manifold in (3.1) in terms of
(xi, vi, Ri,Ωii)i∈n quantities. It can be immediately shown that there is a diffeomorphism
between (xi, vi, Ri,Ωii)i∈n and (xi, vi, Ri,Ω
ii)i∈n where the latter can be treated as new
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 108
states where ˙xi satisfies
˙xi = xi + δi = xi = vi.
Since ˙xi is the same as xi, one achieves a formation controller that globally practically
stabilizes Γf by replacing xiij in (5.1) with xiij , satisfying
xiij = xiij + (dj − di)i.
The term (dj − di)i requires measurement of the fixed inertial vector dj − di in body
frame. This, in turn, requires each agent to know its own attitude in the inertial frame.
One could similarly adapt the rendezvous controller in (Roza et al., 2014) to a formation
controller that allows the desired formation to hover, but requires communication of
thrust inputs between neighboring robots.
5.4 Proofs
5.4.1 Proof of Theorem 5.1.1
The feedback in (5.1) is local and distributed because it is a smooth function of yii and
Ωii only. On the rendezvous manifold Γ, yi = 0 for all i ∈ n and it follows that the
requirement u⋆i |Γ = 0 holds and Ωii|Γ = 0. Moreover, since Ωii = Ωi
i|Γ = 0 for all i ∈ n
on Γ, it follows that the requirement τ ⋆i |Γ = 0 also holds. Now we need to show that the
feedback in (5.1) renders the rendezvous manifold Γ in (3.1) globally practically stable.
We begin by expressing the translational portion of the dynamics in coordinates relative
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 109
to robot 1, i.e., in terms of the variables (x1j , v1j)j∈2 :n,
x1 = v1,
v1 = − 1
m1
R1e3u1 + g,
x1j = v1j ,
v1j = − 1
mj
Rje3uj +1
m1
R1e3u1, j ∈ 2 :n,
(5.7)
Ri = Ri(Ωii)×,
JiΩii = τi − Ωii × JiΩ
ii, i ∈ n.
(5.8)
Since all relative states (xij , vij) can be expressed in terms of the variables above through
the identity (xij , vij) = (x1j − x1i, v1j − v1i), perfect rendezvous occurs if and only if the
This section presents the sufficiency proof of Theorem 6.1.1. Necessity was proved in (Lin
et al., 2005). This section also presents corresponding propositions, lemmas and claims
and their proofs.
We first observe that the closed loop system with inputs defined in (6.1) has no finite
escape times because ‖xi‖ = ‖uiRie1‖ for all i ∈ n is bounded by a quadratic function
of x and θ is bounded. For any sensor digraph G containing a globally reachable node,
there is a condensation digraph satisfying the properties in Proposition 4.2.3. The key
tool in our proof is this condensation digraph and the isolated node sets Lj defined in
Section 4.2.3. The same tool was employed in (Hatanaka et al., 2015) for pose synchro-
nization (synchronization of positions and attitudes) of fully actuated vehicles.
The dynamics of unicycles associated with an isolated node set Lj are independent of
the nodes outside of this set because, for any robot i ∈ Lj, the feedbacks in (6.1) depend
only on states of robots within Lj. Therefore, the dynamics of the collection of unicycles
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 129
−60 −40 −20 0 20−10
−5
0
5
10
15(a)
ix (m)
i y (m
)
−60 −40 −20 0 20−10
−5
0
5
10
15(c)
ix (m)
i y (m
)
−60 −40 −20 0 20−10
−5
0
5
10
15(b)
ix (m)
i y (m
)
Figure 6.4: Rendezvous control simulation for: (a) proposed feedback in (6.1), (b) feed-back in (Lin et al., 2005), and (c) feedback in (Zheng et al., 2011)
0 50 100 150−10
−5
0
5
u i (m
/s)
time (s)
0 50 100 150−0.2
0
0.2
0.4
0.6
omeg
a i (ra
d/s)
time (s)
Figure 6.5: Simulation control inputs for proposed feedback in (6.1)
in Lj,
xi = uiRie1 (6.4)
θi = ωi, i ∈ Lj (6.5)
define an autonomous dynamical system. Henceforth, the dynamics in (6.4), (6.5) are
denoted by ΣLj and we define the reduced rendezvous manifold
ΓLj :=
(xi, θi)i∈Lj : xik = 0, i, k ∈ Lj
.
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 130
Recall from Section 4.2.3 that the set L−1 is empty, which implies that the set ΓL−1is
also empty. We adopt the convention that ΓL−1is GAS for ΣL−1
.
The proof of Theorem 6.1.1 relies on an induction argument on the node sets Lj . Key
in the induction argument is the next result stating that if the vehicles in Lj−1 achieve
rendezvous, then so do the vehicles in Lj.
Proposition 6.2.1. Consider system (2.6), (2.7) and assume that the sensor digraph G
contains a globally reachable node and let ρ1 and ρ2 be such that 0 < ρ1 < ρ2. Suppose
that, for some integer j ≥ 0, the set ΓLj−1is globally asymptotically stable for the
dynamics ΣLj−1. Then there exists l⋆ > 0 such that for any k > l⋆ in (6.1) and for all
linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n, feedback (6.1) globally asymptotically stabilizes
ΓLj for the dynamics ΣLj .
In the statement of Proposition 6.2.1, the lower bound l⋆ on k depends on the sensor
digraph G, while in the statement of Theorem 6.1.1, k⋆ does not. In the proof of Theo-
rem 6.1.1 below we show that the lower bound l⋆ can in fact be made uniform over sensor
digraphs G containing a globally reachable node. The same comment holds for Corol-
lary 6.2.1 below. In Section 6.2.1, we use the above proposition to prove Theorem 6.1.1,
and in Section 6.2.2 we prove Proposition 6.2.1.
In the special case when G is strongly connected, we have L0 = V. Since, by definition,
L−1 = ∅, the set ΓL−1is GAS for ΣL−1
, and Proposition 6.2.1 yields the following
corollary.
Corollary 6.2.1. Consider system (2.6), (2.7) and assume that the sensor digraph G
is strongly connected and let ρ1 and ρ2 be such that 0 < ρ1 < ρ2. Then there exists
l⋆ > 0 such that for any k > l⋆ and for all linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n,
feedback (6.1) solves RP-U.
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 131
6.2.1 Proof of Theorem 6.1.1
To begin with, the feedback in (6.1) is local and distributed because it is a smooth function
of yii only. The proof in this section is performed in two steps. First we will show that for
all digraphs G = (V, E) containing a globally reachable node, for all parameter bounds
ρ1, ρ2 such that 0 < ρ1 < ρ2 there exists l⋆ > 0 such that for all k > l⋆, and for all linear
functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n, feedback (6.1) solves RP-U. We then show that the
lower bound on k can be made uniform over all sensor digraphs with n nodes containing
a globally reachable node.
Consider any digraph G = (V, E) containing a globally reachable node, parameter
bounds ρ1 and ρ2 such that 0 < ρ1 < ρ2, and the node sets Lj and Lj defined in
Section 4.2.3. By construction, the node sets Lj are isolated, the subgraph (V0, E0) is
strongly connected, and L0 = L0 = V0.
The proof is by induction. Since the subgraph (L0, E0) is strongly connected, by
Corollary 6.2.1, there exists l⋆0 such that choosing k > l⋆0 makes the set ΓL0globally
asymptotically stable for system ΣL0for all linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n.
Now consider Lj and suppose the reduced rendezvous manifold ΓLj−1is globally
asymptotically stable for system ΣLj−1. It holds from Proposition 6.2.1 that there exists
l⋆j such that choosing k > l⋆j makes the isolated node set ΓLj globally asymptotically
stable for system ΣLj for all linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n. By part (ii) of
Proposition 4.2.3, the condensation digraph C(G) contains a globally reachable node, so
there is a path from every node of C(G) to the unique root of C(G). By part (i) of the
same proposition, C(G) is acyclic, which implies that the paths connecting the nodes of
C(G) to the unique root of C(G) have a maximum length, L. Recall that, by definition,
LL =∑L
i=1 Li is the union of those strongly connected components Vi of V that are associ-
ated with nodes vi of the condensation digraph C(G) with the property that the maximum
path length from vi to the root v0 is ≤ L. As we argued earlier, the set of such nodes vi
equals the entire condensation digraph, implying that LL = V. Let l⋆ > maxl⋆0, . . . , l⋆L.
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 132
By induction, it must hold that choosing k > l⋆ makes ΓLL = Γ globally asymptotically
stable for system ΣLL = ΣV = Σ for all linear functions fi ∈ Fi(G, ρ1, ρ2), ∈ n. We
conclude that Γ is globally asymptotically stable.
To prove Theorem 6.1.1, it remains to show that the lower bound k can be taken to
be uniform over the set of digraphs with n nodes containing a globally reachable node.
In other words, we need to show that for all n, for all ρ1, ρ2 such that 0 < ρ1 < ρ2,
there exists k⋆ > 0 such that choosing k > k⋆ solves RP-U for all digraphs with n nodes
containing a globally reachable node and for all linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n.
Corresponding to n nodes, there is a finite number of directed graphs, m ≥ 1, containing
a globally reachable node. Enumerating these graphs by Gj, j ∈ 1 :m, the result from the
first part of the proof gives values (l⋆)j, such that choosing k > (l⋆)j, (6.1) solves RP-U
for all linear functions fi ∈ Fi(Gj , ρ1, ρ2), i ∈ n. Choosing k⋆ = max(l⋆)1, . . . , (l⋆)m
then solves RP-U for all digraphs Gj , j ∈ 1 :m for all linear functions fi ∈ Fi(Gj , ρ1, ρ2),
i ∈ n.
6.2.2 Proof of Proposition 6.2.1
Consider a digraph G = (V, E) containing a globally reachable node and let ρ1 and ρ2
be such that 0 < ρ1 < ρ2. Recall that the vertex set Lj ⊂ V is the union of those
vertex sets Vi that correspond to vertices vi in the condensation digraph C(G) associated
to strongly connected components with maximal path length j to the root node v0.
Without loss of generality, we assume there is a single vertex set Vi such that Lj = Vi.
In the case that Lj is the union of several vertex sets, one can repeat the argument of
this proof sequentially for each component. We denote A := Lj−1 and B := Lj = Viand therefore Lj = A ∪ B. Denote by r the number of unicycles in the set B. By
assumption, ΓA is globally asymptotically stable for the dynamics ΣA and the digraph
associated to the nodes in B is strongly connected. We need to show that ΓA∪B is
globally asymptotically stable for the dynamics ΣA∪B. The proof relies on the following
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 133
coordinate transformation.
Coordinate Transformation
Recall that x = (xi)i∈n ∈ R2n and θ = (θi)i∈n ∈ Tn. For each i ∈ n, define
Xi := fi(yi)/Ai, (6.6)
where Ai :=∑
j∈Niaij , and let X := (X1, . . . , Xn). We may express X as
X = (diag(1/A1, · · · , 1/An)L⊗ I2)x.
In the above, diag(. . .) is the diagonal matrix with diagonal elements inside the parenthe-
sis and L is the weighted Laplacian matrix of the sensor digraph associated with the gains
aij . X lies on the subspace of R2n given by Im(diag(1/A1, · · · , 1/An)L⊗ I2) ⊂ R2n. We
let X := Im(diag(1/A1, · · · , 1/An)L⊗ I2) ⊂ R2n. From now on, we will take into consid-
eration that X is a vector in R2n constrained to lie in the 2(n− 1) dimensional subspace
X representing the values of X which are well-defined. Since the sensor digraph contains
a globally reachable node, by Proposition 4.2.1 the matrix L⊗ I2 has rank 2(n− 1), and
Ker(L ⊗ I2) = span1⊗ e1, 1⊗ e2. Let x := [I2 · · · I2]x =∑
i xi, then the linear map
T : R2n → R2n × R2, x 7→ (X, x) is an isomorphism onto its image. Under the action of
T , the subspace x ∈ R2n : x1 = · · · = xn is mapped isomorphically onto the subspace
(X, x) ∈ ImT : X = 0. Since the feedbacks in (6.1) are local and distributed, it can
be seen that the dynamics of the closed-loop unicycles in (X, x, θ) coordinates are inde-
pendent of x. Moreover, as we have seen, in these coordinates the control specification is
the global stabilization of (X, x, θ) ∈ X × R2 × Tn : X = 0, a set whose description is
independent of x. In light of these considerations, for the stability analysis we may drop
the variable x, and show that the set Γ := (X, θ) ∈ X × Tn : X = 0 is GAS for the
(X, θ) dynamics.
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 134
Denote gi(yi) := ‖fi(yi)‖fi(yi). Let a = (aij)(i,j)∈E denote the vector containing the
parameters of the linear consensus controller fi ∈ Fi(Gi, ρ1, ρ2), i ∈ n. Given positive
bounds ρ1 and ρ2, by definition of Fi(G, ρ1, ρ2), the vector of controller gains a lies in the
compact set K(G, ρ1, ρ2) := x ∈ (R+)|E| : ρ1 ≤ xi ≤ ρ2, i ∈ n where R+ is the set of
positive real numbers and |E| is the cardinality of the edge set E . From here on we will use
the hat notation to refer to quantities represented in (X, θ) coordinates. Using (6.6), the
functions fi and gi and their body frame representations are given in (X, θ) coordinates
by
fi(a, Xi) = AiXi, gi(a, Xi) = Ai2‖Xi‖Xi
f ii (a, Xi, θi) = AiR−1i (θi)Xi, g
ii(a, Xi, θi) = Ai
2R−1i (θi)‖Xi‖Xi,
(6.7)
where we have made explicit the dependence of these functions on the parameter vector
a. We can use these functions to rewrite feedback (6.1) in new coordinates as ui =
gii(a, Xi, θi) · e1, ωi = −kf ii (a, Xi, θi) · e2. We remark that fi and fii are homogeneous of
degree one with respect to Xi. Similarly, gi and gii are homogeneous of degree two with
respect to Xi. The closed-loop unicycle dynamics in (X, θ) coordinates are given by
Xi =
∑
j∈Niaij((g
jj · e1)Rje1 − (gii · e1)Rie1)
Ai, (6.8)
θi = −kf ii · e2. (6.9)
We will refer to system (6.8), (6.9) as Σi.
In analogy with what we did earlier, for a set ofm nodes S ⊂ V we letXS := (Xi)i∈S ∈
XS and θS := (θi)i∈S ∈ TnS := Tm. XS lies on the subspace of R2m given by XS := πS(X) in
which πS : R2n → R2m is the projection map whereby πS(X) = (Xi)i∈S ∈ R2m. Moreover,
if S is an isolated node set, the systems Σi, i ∈ S determine an autonomous dynamical
system which we denote by ΣS. We also denote the reduced rendezvous manifold by
ΓS := (XS, θS) ∈ XS × TnS : XS = 0 . In new coordinates, it needs to be shown that the
set ΓA∪B is globally asymptotically stable for the dynamics ΣA∪B under the assumption
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 135
that ΓA is globally asymptotically stable for the dynamics ΣA.
Stability analysis
Let
V (γ,XB) =∑
i∈B
γiX⊤i Xi
Wtran(γ,XB) =√
V (γ,XB)
Wrot(a, XB, θB) =∑
i∈B
f ii (a, Xi, Ri) · e1,
(6.10)
where γi(a) > 0 are gains that are continuous functions of a ∈ K(G, ρ1, ρ2) defined in the
proof of Lemma 6.2.3 and let γ(a) := (γ1(a) . . . γr(a)). Consider the continuous function
W : Rr ×K × XB × TnB → R defined as
W (γ, a, XB, θB) = αWtran(γ,XB) +Wrot(a, XB, θB), (6.11)
where α > 0 is a design parameter. Just as in Chapter 5, the Lie derivative ofW (γ, a, XB, θB)
on the vector field in (6.8), (6.9) can be shown to be continuous. Using W (γ, a, XB, θB)
as a Lyapunov function, the proof follows very similar arguments to Proposition 4.1.24.
The next two lemmas are used in the subsequent analysis.
Lemma 6.2.2. Let γ(a) be any positive real-valued continuous function and consider
the continuous function W (γ, a, XB, θB) defined in (6.11). There exists α⋆ > 0 such that,
for all α > 2α⋆ and for all a ∈ K, the following properties hold:
(i) W ≥ 0 and W−1(0) = (XB, θB) : XB = 0.
(ii) For all c > 0, the sublevel set Wc := (XB, θB) : W (γ, a, XB, θB) ≤ c is compact.
(iii) α⋆√
V (γ,XB) < W (γ, a, XB, θB) < 2α√
V (γ,XB).
The proof is in the Section 6.2.3. From now on assume α > 2α⋆.
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 136
Lemma 6.2.3. Consider system (6.8), (6.9). There exist positive real-valued functions
γi(a) in (6.10) and l⋆ > 0 such that choosing k > l⋆ implies
There exists l⋆ > 0 such that choosing k > l⋆, the matrix above is negative definite and
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 147
therefore the first term satisfies,
V[
1⊤ β⊤]
−2M2
nI αM1
2I
αM1
2I −k
M4I
1
β
≤ −σV (γ,XB), (6.20)
σ > 0 for all a ∈ K. This concludes the proof of Lemma 6.2.3.
Proof of Claim 6.2.2
Recalling that V (γ,XB) =∑
i∈B γiX⊤i Xi with Xi = fi/Ai and defining bij :=
aijAi
2 , it
holds that,
∑
i∈B
∂V (γ,XB)
∂Xiai(a, XB) = 2
∑
i∈B
γifiAi
· ai(a, XB)
≤2∑
i∈B
γifi ·(
∑
j∈Ni∩B
bij(‖fj‖fj − ‖fi‖fi)−∑
j∈Ni∩A
bij‖fi‖fi)
≤2∑
i∈B
γi
(
∑
j∈Ni∩B
bij(−‖fi‖3 + ‖fj‖fj · fi)−∑
j∈Ni∩A
bij‖fi‖3)
≤∑
i∈B
γi∑
j∈Ni∩B
bij
(
−4
3‖fi‖3 +
4
3‖fj‖3
)
+∑
i∈B
γi∑
j∈Ni∩B
bij
(
−2
3‖fi‖3 + 2‖fj‖fj · fi −
4
3‖fj‖3
)
− 2∑
i∈B
γi∑
j∈Ni∩A
bij‖fi‖3.
The first term equals 43γ⊤Mh with h := (‖fi‖3)i∈B. M is the (r× r)-matrix whose (i, j)-
th component is∑
k∈Ni∩Bbik for i = j, bij for j ∈ Ni∩B and zero otherwise for i, j ∈ 1 : r
where it is assumed without loss of generality that B = 1 : r. The components of the
matrix M are continuous functions of a. Since B corresponds to a strongly connected
digraph, the zero eigenvalue of M is unique and all components of left eigenvectors
corresponding to the zero eigenvalue are nonzero and have the same sign (see Proposition
D.5 in (Hatanaka et al., 2015)). Further, it follows from Theorem 3.4.35 in (Abraham
et al., 1988) that there exists a left eigenvector associated to the zero eigenvalue of M ,
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 148
γ(a) = (γ1(a), . . . , γr(a)), which is a continuous function of a. Without loss of generality,
we can choose the vector γ(a) with positive components. Therefore,
∑
i∈B
∂V (γ,XB)
∂Xi
ai(a, XB) ≤∑
i∈B
γi∑
j∈Ni∩B
bij
(
−2
3‖fi‖3 + 2‖fj‖fj · fi −
4
3‖fj‖3
)
− 2∑
i∈B
γi∑
j∈Ni∩A
bij‖fi‖3 =: κ(γ, a, XB).
The term
κ1(γ, a, XB) :=∑
i∈B
γi∑
j∈Ni∩B
bij
(
−2
3‖fi‖3 + 2‖fj‖fj · fi −
4
3‖fj‖3
)
≤∑
i∈B
γi∑
j∈Ni∩B
bij
(
−2
3‖fi‖3 + 2‖fi‖‖fj‖2 −
4
3‖fj‖3
)
is less than or equal to zero with equality only when fi = fj for all i, j ∈ B and as such
κ(γ, a, XB) is less than or equal to zero with equality only when fi = fj for all i, j ∈ B.
Now we prove that κ(γ, a, XB) = 0 only if fi = 0 for all robots i ∈ B. In the case
that A is not empty, the inequality κ(γ, a, XB) ≤ −2∑
i∈B γi∑
j∈Ni∩Abij‖fi‖3 implies
κ(γ, a, XB) = 0 only if fi = 0 for any i ∈ B with a neighbor in A. As such, by the
previous arguments, κ(γ, a, XB) = 0 only if fi = 0 for all i ∈ B. On the other hand, if A
is empty, then B is isolated and strongly connected. Therefore κ(γ, a, XB) = κ1(γ, a, XB)
is equal to zero only if κ1(γ, a, XB) = 0 which is the case only if fi = fj for all i, j ∈ B.
For all XB ∈ XB (i.e., all well-defined values of XB ∈ R2r), this implies that there exists
x = (x1, . . . , xr) such that diag(A1, · · · , Ar)XB = (L⊗ I2)x ∈ span1⊗ e1, 1⊗ e2 where
L is the Laplacian matrix for the agents in B (isolated). Since B is strongly connected,
for all a ∈ K(G, ρ1, ρ2) there exists a unique vector γ (with positive entries) such that
γ⊤(L⊗ I2) = 0. Since γ⊤(L⊗ I2)x = γ⊤1⊗ (αe1 + βe2) for some α, β ∈ R, it holds that
γ⊤1 ⊗ (αe1 + βe2) = 0. Since all entries of γ are positive, this implies α = β = 0 and
(L⊗ I2)x = 0. Therefore x ∈ span1⊗ e1, 1⊗ e2 or, equivalently, fi = 0 for all i ∈ B.
Therefore κ(γ, a, XB) = 0 only if Xi = 0 for all i ∈ B and as such κ(γ, a, XB) is
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 149
negative definite. Note that κ(γ, a, XB) is homogeneous of degree three with respect to
XB because fi is homogeneous of degree one with respect to XB for all i ∈ B. This
completes the proof of the claim.
Chapter 7
Formations of Kinematic Unicycles
In this chapter we present a control solution solving the Parallel Formation Problem (PP).
The control inputs ui and ωi will be constructed by combining two control primitives dis-
cussed in Section 4.3: a uniformly bounded consensus controller for single-integrators
fi((xij)j∈Ni) in (4.11) and a rotational integrator consensus controller gi((θij)j∈Ni, η)
in (4.24). We will also discuss special cases of parallel line formations and full syn-
chronization of unicycles.
7.1 Solution to the Parallel Formation Problem (PP)
To begin, we define offset vectors rigidly attached to each unicycle. Let α > 0 be a design
parameter, and d = (d1i1)i∈2,...n ∈ F be a desired parallel formation. Define
α1 := α, β1 := 0,
αi := −d11i · e1 + α, βi := −d11i · e2, i ∈ 2 :n.
(7.1)
Referring to Figure 7.1, attach the offset vector δi := αiRie1 + βiRie2 to the body frame
of unicycle i, and let xi := xi+δi be the endpoint of the offset vector in the coordinates of
frame I. Note that the offset vector δi is defined in the body frame of robot i whereas the
150
Chapter 7. Formations of Kinematic Unicycles 151
analogous offset defined in Section 5.3 for formations of flying robots was defined relative
to the inertial frame. Defining the offsets in body frame will permit us to develop a local
and distributed solution for unicycle formation control. Define further
xij := xj − xi,
yi := (xij)j∈Ni, yki := (xkij)j∈Ni.
Figure 7.1: Representation of the offset vector δi.
We now show that PP reduces to synchronizing the unicycles’ heading angles and the
endpoints xi. To this end, suppose that θij = 0 and xij = 0 for all i, j ∈ n. Then,
0 = xi1i = [(xi + δi)− (x1 + δ1)]i
= xi1i + (δi − δ1)i
= xi1i − d11i = x11i − d11i.
The last identity follows from the fact that Ri = R1. We conclude that θij = 0 and
xij = 0 for all i, j ∈ n implies x11i = d11i, so that the unicycles satisfy the parallel formation
requirement. Vice versa, it is clear that if the unicycles form a parallel formation, then
θij = 0 and xij = 0 for all i, j ∈ n.
Chapter 7. Formations of Kinematic Unicycles 152
We have thus shown that PP amounts to the simultaneous synchronization of the
headings θi and the endpoints xi. We now present feedbacks that do just that. Let
fi(·) be a bounded integrator consensus controller as in (4.11), and gi(·) be an attitude
synchronizer as in (4.24). The feedbacks for unicycle i are chosen as follows,
ui =u⋆i (y
ii, ϕi) = fi(y
ii) · e1 + βiω
⋆i (y
ii, ϕi),
ωi =ω⋆i (y
ii, ϕi) =
1
αi
(
fi(yii) · e2 + kgi(ϕi, η)
)
, i ∈ n,(7.2)
where k > 0 is a high-gain parameter, and η = (η1, . . . , ηn), ηi := 1/αi. We are now
ready to present the main result of this chapter.
Theorem 7.1.1. Consider the collection of n unicycles in (2.6), (2.7) with controller (7.2),
where the functions fi(·), gi(·) are defined in (4.11), (4.24) and satisfy properties A1, A2
and B1-B3. Assume that sensor graph G is undirected and connected. For any pa-
rameters aij = aji > 0 in (4.11) and any parameters bij = bji > 0 in (4.24) satisfying
(bij)(i,j)∈E ∈ (R+)|E|\Nb as in Theorem 4.3.2, there exists α⋆ > 0 such that for any parallel
formation d = (d11i)i∈2 :n ∈ F, choosing α > α⋆maxi∈2 :n (−d11i · e1) in (7.1), the formation
manifold Γp is almost semiglobally asymptotically stable with high-gain parameter k.
The proof is presented in Section 7.5. Roughly speaking, the theorem states that
letting the offset α in (7.1) grow proportionally to the length of the formation (the quan-
tity maxi (−d11i · e1)), and choosing k in (7.2) to be sufficiently large, the controller (7.2)
ensures that almost all initial conditions in any given compact set are contained in the
domain of attraction of the formation manifold Γp . Another property of controller (7.2)
is that (u⋆i , ω⋆i )∣
∣
Γp= 0 for all i ∈ n, and therefore the unicycles come to a stop as Γp
is approached, as required in the statement of PP in Chapter 3. In the next section we
further discuss the controller (7.2).
Chapter 7. Formations of Kinematic Unicycles 153
7.2 Discussion of the Control Solution
As we mentioned earlier, the philosophy behind controller (7.2) is to convert PP into a
synchronization problem in which we make the offset vectors xi and the heading angles
θi converge to one another. This is illustrated in Figure 7.2, where the vectors xi, i ∈ n
all meet at a common point at a distance α in front of the formation, and all heading
directions are aligned.
Figure 7.2: Parallel formation where offset vectors xi meet at a common point at adistance α in front of the formation.
In (7.2), the terms containing fi(yii) aim to achieve consensus on the endpoints of the
offset vectors xi, while the terms containing gi(ϕi, η) aim to achieve consensus on the
unicycle angles. It will be shown in Section 7.5 that the choice of (7.2) achieves both
these, at times competing, objectives simultaneously by making use of gradient properties
of the systems (4.9) and (4.22) with inputs (4.11) and (4.24) respectively.
The block diagram in Figure 7.3 summarizes the design of feedbacks (ui, ωi)i∈n. From
its sensors, unicycle i obtains the vector (yii, ϕi) of its heading and displacement relative
to its neighbours. These quantities can be measured locally in unicycle i’s body frame
using, for example, on-board cameras. The offset extraction block takes as input the
vector (yii, ϕi) and outputs (yii, ϕi), where each component of yii = (xiij)j∈Ni is computed
as,
xiij = xiij + αjRije1 + βjR
ije2 −
[
αi βi]
⊤. (7.3)
Chapter 7. Formations of Kinematic Unicycles 154
PositionConsensus
AttitudeSynchroniztion
OffsetExtraction
Figure 7.3: Block diagram of the formation control system for robot i.
This computation requires that, in addition to (yii, ϕi), unicycle i has access to the
formation parameters (αj , βj)j∈Ni of its neighbours. These quantities must be stored
in memory on-board unicycle i before deployment. Moreover, in order to compute xiij
in (7.3), unicycle i must be able to identify its neighbours so as to use, for each j ∈ Ni,
the appropriate bias constants (αj, βj). Such identification can be achieved, for instance,
by means of visual markers. A consequence of using the constants (αj , βj) is that the
unicycle feedbacks are not identical and the formation is not invariant to a relabelling
of the agents. This is hardly surprising because, in our formulation of PP, we allow for
general, non-symmetric formations.
An important property of the feedback in (7.2) is that it is local and distributed, since
u⋆i and ω⋆i depend on (yii, ϕi). As a consequence of this feature, the asymptotic position
and orientation of the formation with respect to the inertial frame depend only on the
initial configuration of the unicycles.
Chapter 7. Formations of Kinematic Unicycles 155
7.3 Special cases: Line formations and full synchro-
nization
As a by-product of the formation control solution, we present corresponding solutions for
the special cases of parallel line formations and full synchronization.
A parallel line formation is a parallel formation satisfying d11i · e1 = 0 (and hence
αi = α for all i ∈ 2 :n). The set of all such formations will be denoted LF. Clearly,
LF ⊂ F. In the case of full synchronization, the unicycles have the same position and
orientation with respect to the inertial frame, i.e., d11i = 0 for all i ∈ 2 :n (and therefore
αi = α and βi = 0 for all i ∈ n). Full synchronization, therefore, corresponds to the
formation 0 ∈ F. Examples of a parallel line formation and full synchronization are
illustrated in Figure 7.4.
According to Theorem 7.1.1, in both of these cases it suffices that α satisfies the less
strict condition α > 0. This is advantageous, as it will be discussed in Chapter 10 that
large values of α can slow down the rate of convergence of the unicycles to the formation.
Arbitrarily choosing α = 1, the corresponding controller in (7.2) reduces to
u⋆i (yii, ϕi) = fi(y
ii) · e1 + βiω
⋆i (y
ii, ϕi),
ω⋆i (yii, ϕi) = fi(y
ii) · e2 + kgi(ϕi, 1), i ∈ n,
(7.4)
in which, xiij = xiij+Rije1−e1+βjRi
je2−βie2. Since the values αi = α = 1 for all i ∈ n are
equal, unicycle i only needs to store the quantities (βj)j∈Ni of its neighbours on-board.
The next corollary is a specialization of Theorem 7.1.1 to parallel line formations.
Corollary 7.3.1. Consider the collection of n unicycles in (2.6), (2.7) with controller (7.4),
where the functions fi(·), gi(·) are defined as in (4.11), (4.24) and enjoy properties A1,
A2 and B1-B3. Assume that sensor graph G is undirected and connected. For any
parameters aij = aji > 0 in (4.11), any parameters bij = bji > 0 in (4.24) satisfying
Chapter 7. Formations of Kinematic Unicycles 156
Figure 7.4: (a) shows an example of a parallel line formation while (b) shows an exampleof full synchronization, a special case of a parallel line formation.
(bij)(i,j)∈E ∈ (R+)|E|\Nb as in Theorem 4.3.2, and any parallel line formation d ∈ LF,
the formation manifold Γp is almost semiglobally asymptotically stable with high-gain
parameter k.
In the special case of full synchronization, βi = 0 for all i ∈ n, and the controller
in (7.4) reduces to
u⋆i (yii, ϕi) = fi(y
ii) · e1,
ω⋆i (yii, ϕi) = fi(y
ii) · e2 + kgi(ϕi, 1), i ∈ n,
(7.5)
in which, xiij = xiij +Rije1 − e1. Since the αi and βi parameters are equal for all agents,
unicycle i does not need to store any parameters of its neighbours on-board, the control
inputs are identical for all unicycles, so that in this case the configuration is invariant
to relabelling of agents. This is hardly surprising since the formation is symmetric in
this case. The controller in (7.5) can be viewed as an extension of the result for unicycle
rendezvous in Chapter 6. In fact, in Chapter 6 the controller was defined as
u⋆i (yii, ϕi) = ‖fi(yii)‖fi(yii) · e1,
ω⋆i (yii, ϕi) = −kfi(yii) · e2, i ∈ n,
(7.6)
Chapter 7. Formations of Kinematic Unicycles 157
in which fi(yii) =
∑
j∈Niaijx
iij is a linear single integrator consensus controller.
While the controller in (7.6) guarantees global rendezvous, in which only the unicy-
cle positions are synchronized, the controller in (7.5) guarantees almost semiglobal full
synchronization where both positions and angles of the unicycles are synchronized. The
control inputs in (7.5) and (7.6) are similar in structure. The main difference is that the
full synchronization controller in (7.5) has an additional term kgi(ϕi, η) responsible for
aligning the unicycle heading angles, not required for rendezvous. In fact, for unicycle i,
(7.5) depends on (yii, ϕi) while (7.6) depends only on yii.
7.4 Simulation Results
This section presents simulations for a group of five unicycles to illustrate our results.
The interaction function f(s) for the bounded integrator consensus control is chosen as
in (4.12) while the interaction function for the attitude synchronizer is chosen satisfying
assumptions B1, B2 and B3 as in Figure 4.16. The undirected sensing graph is cyclic with
connections as shown in Figure 7.5 and the desired triangular formation is specified by
d112 = (−10, 5), d113 = (−10,−5), d112 = (−20, 10) and d112 = (−20,−10), as illustrated in
Figure 7.6. We have chosen random initial unicycle positions on a 40m × 40m area with
1
2
3
5
4
Figure 7.5: Undirected graph G under consideration in the simulation results.
random initial angles. The corresponding plot of a simulation run is shown in Figure 7.7.
Chapter 7. Formations of Kinematic Unicycles 158
Figure 7.6: Triangular formation specified by the offset vectors d112 = (−10, 5), d113 =(−10,−5), d114 = (−20, 10) and d115 = (−20,−10).
Extensive simulation trials will be presented in Chapter 10 to study the effectiveness
of our control solution under different realistic scenarios not captured by the main result
in Theorem 7.1.1 including
• performance in the presence of state dependent sensor graphs in which each unicy-
cle’s neighbors are those that lie within a given radius of itself;
• performance for directed sensing graphs as opposed to undirected sensing graphs;
• performance when the high gain conditions on α and k are ignored;
• robustness of the approach to unmodelled effects including sensor noise, input noise,
sampling and saturated inputs
• extension of the control solution for kinematic unicycles to the dynamic unicycle
model in (2.10).
7.5 Proof of Theorem 7.1.1
We divide the proof of Theorem 7.1.1 in several steps. In Section 7.5.1, we derive the
closed-loop dynamics in (xi, θi)i∈n coordinates. In Sections 7.5.2 and 7.5.3, we propose
Chapter 7. Formations of Kinematic Unicycles 159
-80 -60 -40 -20 0 20 40
x (m)
-20
-10
0
10
20
30
40
50
60
70
80
90
y (m
)
Figure 7.7: Simulation for a triangle formation. Initial positions are indicated with andfinal positions are indicated with ×.
a Lyapunov function V for the closed-loop system, and carry out a Lyapunov analysis
yielding the property V ≤ 0. In Section 7.5.4, we show that, for sufficiently large α > 0,
the zero level set of V coincides with the formation manifold Γp on a neighbourhood
of Γp . This result will imply, via Lyapunov’s direct method, asymptotic stability of Γp .
A further Lyapunov analysis is employed to show that Γp is in fact almost semiglob-
ally asymptotically stable with high-gain parameter k. Each step of the proof will be
presented in its own subsection.
7.5.1 System dynamics in (xi, θi)i∈n coordinates
To simplify the analysis, we consider new coordinates (x, θ) = (xi, θi)i∈n under the dif-
feomorphism F : (x, θ) 7→ (x, θ) given by F ((xi, θi)i∈n) = (xi + δ(θi), θi)i∈n. Computing
the time derivative of xi yields,
˙xi = uiRie1 +Ri
0 −ωiωi 0
(αie1 + βie2)
= uiRie1 + αiωiRie2 − βiωiRie1
= (ui − βiωi)Rie1 + αiωiRie2,
Chapter 7. Formations of Kinematic Unicycles 160
from which we get
˙xi = (ui − βiωi)Rie1 + αiωiRie2
θi = ωi, i ∈ n.
(7.7)
Using Lemma 4.3.3(i) and the fact that the dot product is invariant to rotations, i.e,
R−1i fi(yi) · e1 = fi(yi) · Rie1, the feedbacks in (7.2) can expressed as follows:
u⋆i (yii, ϕi) = fi(yi) ·Rie1 + βiω
⋆i (y
ii, ϕi),
ω⋆i (yii, ϕi) =
1
αi(fi(yi) · Rie2 + kgi(ϕi, η)) .
(7.8)
Substituting ui = u⋆i (yii, ϕi) and ωi = ω⋆i (y
ii, ϕi) from (7.8) into (7.7) and using the
fact that fi(yi) = (f(yi) · Rie1)Rie1 + (f(yi) · Rie2)Rie2 yields the closed-loop system in
(xi, θi)i∈n coordinates,
˙xi = fi(yi) + kgi(ϕi, η)Rie2
θi =1
αi(fi(yi) · Rie2 + kgi(ϕi, η)) , i ∈ n.
(7.9)
Notice that the control inputs are defined precisely in terms of (x, θ) and so the equations
of motion in (7.9) constitute a dynamical system. The closed loop system in (7.9) has no
finite escape times because ‖ ˙x‖ is bounded by a linear function of x and θ is bounded.
The parallel formation manifold Γp in (3.4) in (x, θ) coordinates becomes,
Γp :=
(x, θ) ∈ R2n × T
n : x1i = 0, θ1i = 0, i ∈ n
. (7.10)
Chapter 7. Formations of Kinematic Unicycles 161
7.5.2 Lyapunov analysis
From Lemma 4.3.2, system (4.9) is gradient with nonnegative storage function Vt. In-
spired by (Mallada et al., 2016), define a Lyapunov function Vr(θ) as,
Vr(θ) :=1
2
n∑
i=1
∑
j∈Ni
bij
∫ θij
0
g(s)ds. (7.11)
Since G is connected, we have that Vr ≥ 0 and V −1r (0) = θ ∈ Tn : (∀i, k ∈ n) θi = θj.
Next, combine Vt(x) and Vr(θ) as follows:
V (x, θ) := Vt(x) + kVr(θ). (7.12)
Since Vt and Vr are nonnegative, V is nonnegative and V −1(0) = Γp .
Using (7.9), the time derivative of Vt(x) is given by,
Vt =
n∑
i=1
−fi · (fi + kgiRie2)
=n∑
i=1
(
−‖fi‖2 − (fi · Rie2)kgi)
.
(7.13)
Since g(θij) = −g(θji), we have
∂Vr∂θi
=1
2
∑
j∈Ni
bij∂
∂θij
(∫ θij
0
g(s)ds
)
∂θij∂θi
+1
2
∑
j∈Ni
bji∂
∂θji
(∫ θji
0
g(s)ds
)
∂θji∂θi
= −∑
j∈Ni
bijg(θij).
(7.14)
Chapter 7. Formations of Kinematic Unicycles 162
Using the above, identity (4.24), and the fact that ηi = 1/αi, we obtain
Vr = −n∑
i=1
∑
j∈Ni
bijαig(θij) (fi · Rie2 + kgi)
=
n∑
i=1
−gi (fi · Rie2 + kgi)
=n∑
i=1
(−(fi · Rie2)gi − kg2i ).
(7.15)
Combining (7.13) and (7.15), we get
V = Vt + kVr
=n∑
i=1
(
−‖fi‖2 − 2(fi · Rie2)(kgi)− (kgi)2)
=
n∑
i=1
(
−‖fi · Rie1‖2 − ‖fi · Rie2 + kgi‖2)
≤ 0.
(7.16)
7.5.3 Lyapunov analysis in relative coordinates
To further simplify the stability analysis, we perform another coordinate transformation
with the intention of quotienting out the dynamics of unicycle 1. More precisely, consider
the diffeomorphism
F : R2n × Tn → R
2(n−1) × R2 × T
(n−1) × S1,
F (x, θ) = (x, x11, θ, θ1),
where x := (x11i)i∈2 :n, θ := (θ1i)i∈2 :n. Using Lemma 4.3.3(i) and the fact that fi(yi) ·
Rie2 = fi(y1i ) · R1
i e2, the dynamics in (7.9) can be written in new coordinates as,
Chapter 7. Formations of Kinematic Unicycles 163
˙x11i =[
fi(y1i ) + kgi(ϕi, η)R
1i e2 − f1(y
11)− kg1(ϕ1, η)e2
]
−(
1
α1
(
f1(y11) · e2 + kg1(ϕ1, η)
)
)×
x11i
θ1i =1
αi
(
fi(y1i ) ·R1
i e2 + kgi(ϕi, η))
− 1
α1
(
f1(y11) · e2 + kg1(ϕ1, η)
)
˙x11 =f1(y11) + kg1(ϕ1, η)e2 − ω×
1 x11
θ1 =1
α1
(
f1(y11) · e2 + kg1(ϕ1, η)
)
,
(7.17)
where i ∈ 2 :n.
We remark that y1i = (x1ij)j∈Ni = (x11j − x11i)j∈Ni, ϕi = (θij)j∈Ni = (θ1j − θ1i)j∈Ni
and R1i are functions of the relative quantities (x, θ), and do not depend on the absolute
quantities x11 and θ1. It follows that system (7.17) has a decoupled subsystem with state
(x, θ) ∈ R2(n−1) × Tn−1. Moreover, Γp in new coordinates is given by
(x, x11, θ, θ1) : x11i = 0, θ1i = 0, i ∈ 2 :n
, (7.18)
which is also independent of absolute quantities x11 and θ1.
Based on these considerations, the variables x11 and θ1 may be dropped, yielding a
new dynamical system with state (x, θ) ∈ R2(n−1) × Tn−1. Proving almost semiglobal
asymptotic stability of Γp for system (7.9) is equivalent to proving that the equilibrium
point
Γp :=
(x, θ) = (0, 0) ∈ R2(n−1) × T
(n−1)
. (7.19)
is almost semiglobally asymptotically stable for the (x, θ) subsystem.
We now return to the Lyapunov analysis of Section 7.5.2, expressing V in relative
coordinates (x, θ). Using the fact that ‖xij‖ = ‖R−11 xij‖ = ‖R−1
1 (x1j−x1i)‖ = ‖x11j−x11i‖,
Chapter 7. Formations of Kinematic Unicycles 164
and θij = θ1j − θ1i, we have
Vt(x, θ) := Vt|(x,θ)=F−1(x,x11,θ,θ1)
=1
2
n∑
i=1
∑
j∈Ni
aij
∫ ‖x11j−x11i‖
0
f(s)ds
Vr(x, θ) := Vr|(x,θ)=F−1(x,x11,θ,θ1)
=1
2
n∑
i=1
∑
j∈Ni
bij
∫ θ1j−θ1i
0
g(s)ds.
(7.20)
The identities in (7.20) imply that V can indeed be expressed in terms of relative quan-
tities (x, θ), and in these coordinates it is given by V (x, θ) := Vt(x, θ) + kVr(x, θ).
Since V −1(0) = Γp, it follows that V −1(0) = Γp and therefore V is positive definite
at (x, θ) = (0, 0). For any c > 0, the sublevel set Vc = (x, θ) ∈ R2(n−1) ×T(n−1) : V ≤ c
is closed since V is continuous. Next we show that Vc is bounded, and hence compact.
In the set Vc,
aij
∫ ‖x1ij‖
0
f(s)ds ≤ c,
for all i ∈ n, j ∈ Ni. If ‖x1ij‖ > c2 where c2 is defined in A1, then this implies that
aijc1(‖x1ij‖ − c2) ≤ c where c1 is defined in A1 and therefore ‖x1ij‖ ≤ (c/c1aij) + c2 is
bounded. Since the undirected graph is connected, this proves boundedness of (x, θ).
Moreover, using a standard result in (Lee, 2013, Proposition 8.16), the time derivative
of V satisfies,
˙V =V |(x,θ)=F−1(x,x11,θ,θ1)
=
n∑
i=1
(
−‖fi(y1i ) · R1i e1‖2 − ‖fi(y1i ) · R1
i e2 + kgi(ϕi, η)‖2)
.(7.21)
Once again, since y1i , ϕi and R1i are functions of (x, θ), ˙V (x, θ) is independent of x11
and θ1. In light of (7.21), ˙V ≤ 0, with equality if and only if fi(y1i ) · R1
i e1 = 0 and
fi(y1i ) ·R1
i e2 = −kgi(ϕi, η) for all i ∈ n. Together, these conditions imply that on the set
Chapter 7. Formations of Kinematic Unicycles 165
E := (x, θ) : ˙V (x, θ) = 0 it holds that
fi(y1i ) = −kgi(ϕi, η)R1
i e2, ∀i ∈ n. (7.22)
7.5.4 Local asymptotic stability of Γp
In this section we show that there exists ǫ > 0 such that E ∩ (x, θ) : ‖θ‖ ≤ ǫ =
V −1(0) = Γp, implying that ˙V is negative definite, and Γp is locally asymptotically
stable by Lyapunov’s direct method.
Let (x, θ) ∈ E be arbitrary. By Lemma 4.3.3(iii), we have 0 =∑n
i=1 fi(y1i ) =
R−11
∑ni=1 fi(yi). Using (7.22), we get −∑n
i=1 gi(ϕi, η)R1i e2 = 0, and using (4.24), we
get
−n∑
i=1
ηi∑
j∈Ni
bijg(θij)R1i e2 = 0.
We have R1i e2 =
[
− sin(θ1i) cos(θ1i)
]
⊤, so
−n∑
i=1
ηi∑
j∈Ni
bijg(θij)
− sin(θ1i)
cos(θ1i)
= 0.
The first component of the above identity gives
n∑
i=1
ηi∑
j∈Ni
bijg(θij) sin(θ1i) = 0, (7.23)
which depends solely on relative angles θ. Expanding g(s) and sin(s) about s = 0, we
get
g(s) = g(0)s+ h1(s)s
sin(s) = s+ h2(s)s,
where lims→0 h1(s) = 0 and lims→0 h2(s) = 0. Moreover, g(0) > 0 by B3. Using the
Chapter 7. Formations of Kinematic Unicycles 166
above identities in (7.23) we get
n∑
i=1
ηi∑
j∈Ni
bij [g(0)θijθ1i + g(0)h2(θ1i)θijθ1i
+h1(θij)θijθ1i + h1(θij)h2(θ1i)θijθ1i] = 0.
(7.24)
Dividing by g(0), we have
n∑
i=1
ηi∑
j∈Ni
bijθijθ1i
[
1 + h2(θ1i) +h1(θij)
g(0)+h1(θij)h2(θ1i)
g(0)
]
= 0. (7.25)
Ignoring, for now, higher order terms h1(θij), h2(θ1i) in (7.25) and substituting in ηi =
1/αi, we getn∑
i=1
(θ1i/αi)∑
j∈Ni
bij [θ1j − θ1i] = 0. (7.26)
Define a weighted Laplacian matrix L by L(i, j) = −bij for i 6= j and L(i, i) =∑
j∈Nibij .
Then L is a symmetric Laplacian for the connected undirected graph G, and therefore
kerL = span1. Next, let
λi(α, d) :=maxi αiαi
=α +maxi(−d11i · e1)
α− d11i · e1,
λ = (λ1, . . . , λn),
(7.27)
and
D(λ) := diag(
λi)
i∈n.
The identity in (7.26) can now be rewritten as
− 1
maxi αi
[
0 θ⊤]
D(λ(α, d))L
0
θ
= 0. (7.28)
Chapter 7. Formations of Kinematic Unicycles 167
Letting
L(λ) := P⊤D(λ)LP, P =
01×(n−1)
I(n−1)
,
identity (7.28) implies
θ⊤L(λ(α, d))θ = 0. (7.29)
Denoting by M(λ) := (L(λ)+ L(λ)⊤)/2 the symmetric part of L, identity (7.29) becomes
θ⊤M(λ(α, d))θ = 0. (7.30)
We will show that for large α > 0, M(λ(α, d)) is positive definite. Referring to the
definition of λi in (7.27), note that
λi(α, d) → 1 asα
maxi(−d11i · e1)→ ∞, (7.31)
and λi(α, d) = 1 when −d11i ·e1 = 0 for all i ∈ n. In light of this observation, consider first
the case in which λ = 1, so that D(λ) = D(1) = In, the identity matrix. Then (7.28)
reduces to[
0 θ⊤]
L
0
θ
≥ 0,
with equality if and only if
[
0 θ⊤]
∈ span1 (since kerL = span1), which can occur
only if θ = 0. Owing to the equivalence of (7.28) and (7.30), we have that θ⊤M(1)θ ≥ 0,
with equality holding if and only if θ = 0, and thus M(1) is positive definite and, since
M(λ) is symmetric, all its principal leading minors mi(λ), i ∈ n, have the property that
mi(1) > 0, i ∈ n. Since the functions mi(λ) are continuous, there exists ε > 0 such that
for all λ ∈ Rn such that ‖λ − 1‖ < ε, mi(λ) > 0, i ∈ n. From (7.31), we deduce that
Chapter 7. Formations of Kinematic Unicycles 168
there exists α⋆ > 0 such that
α
maxi(−d11i · e1)> α⋆ =⇒ ‖λ(α, d)− 1‖ < ε
=⇒ mi(λ(α, d)) > 0 i ∈ n.
We have thus established the existence of α⋆ > 0 such that, for all α > α⋆maxi(−d11i ·e1),
the matrix M(λ(α, d)) is positive definite.
Now assuming that α satisfies the above bound so thatM(λ(α, d)) is positive definite,
we return to identity (7.25) including higher-order terms, and rewrite it as
θ⊤M(λ(α, d))θ + r(θ) = 0, (7.32)
where M(·) is as before and
r(θ) =n∑
i=1
ηi∑
j∈Ni
bijθijθ1i
[
h2(θ1i) +h1(θij)
g(0)+h1(θij)h2(θ1i)
g(0)
]
.
We will show, using similar arguments to (Francis and Maggiore, 2016, Proof of Theo-
rem 6.1), that there exists ǫ > 0 such that in an ǫ-neighborhood of θ = 0, identity (7.32)
holds only if θ = 0.
Condition (7.32) holds only if ‖θ⊤M(·)θ‖ = ‖r(θ)‖. Suppose for a moment that
limθ→0
‖r(θ)‖‖θ⊤M(·)θ‖
= 0. (7.33)
Then for sufficiently small θ, ‖r(θ)‖ ≤ ‖θ⊤M(·)θ‖/2, and the unique solution to (7.32)
is θ = 0, as desired. To show that (7.33) holds, express θ as θ = ‖θ‖φ where φ =
(φ1i)i∈2 :n ∈ Sn−1 is a unit vector. Correspondingly, θ1i = ‖θ‖φ1i for all i ∈ 2 :n. One
Chapter 7. Formations of Kinematic Unicycles 169
can then write,
θ⊤M(·)θ =‖θ‖2φ⊤M(·)φ,
r(θ) =‖θ‖2n∑
i=1
∑
j∈Ni
bijηiφijφ1i
[
h2(θ1i) +h1(θij)
g(0)+h1(θij)h2(θ1i)
g(0)
]
= ‖θ‖2h(θ, φ),
where h(·, ·) has the property that limθ→0 h(θ, φ) = 0. Then,
limθ→0
‖r(θ)‖‖θ⊤M(·)θ‖
= limθ→0
‖h(θ, φ)‖φ⊤M(·)φ = 0,
since limθ→0 h(θ, φ) = 0 and minφ∈Sn−1(φ⊤M(·)φ) > 0 because M(·) positive definite and
φ is a unit vector.
To summarize, there exists ǫ > 0 such that if ‖θ‖ ≤ ǫ, then (7.25) is zero only if
θ = 0, implying that gi = 0 for all i ∈ n. By (7.22) this implies that fi = 0 for
all i ∈ n and therefore, by Lemma 4.3.3(ii), xi = xj for all i, j ∈ n. It follows that
E ∩ (R2(n−1) × θ : ‖θ‖ ≤ ǫ) = Γp .
To summarize our findings so far, we have shown that V is positive definite, V −1(0) =
(x, θ) = (0, 0) = Γp , and ˙V is negative definite on (R2(n−1) × θ : ‖θ‖ ≤ ǫ), a
neighbourhood of Γp . By Lyapunov’s stability theorem, the equilibrium Γp is locally
asymptotically stable for the (x, θ) subsystem.
7.5.5 Almost semiglobal asymptotic stability of Γp
Having established that for α > 0 sufficiently large, the equilibrium Γp is asymptotically
stable for the (x, θ) subsystem, we now prove that Γp is almost semiglobally asymptoti-
cally stable with high-gain parameter k. The idea is to show that, for sufficiently large
k, for almost all initial conditions in any given compact set the solutions of the (x, θ)
subsystem enter in finite time and remain inside the set (R2(n−1) × θ : ‖θ‖ ≤ ǫ) on
which ˙V is negative definite, which implies that they converge to Γp.
Chapter 7. Formations of Kinematic Unicycles 170
Rewrite the dynamics of the θ subsystem in (7.17) as,
˙θ = kF (θ) + ∆(x, θ), (7.34)
where
Fi(θ) :=
(
1
αigi(ϕi, η)−
1
α1g1(ϕ1, η)
)
∆i(x, θ) :=1
αifi(y
1i ) · R1
i e2 −1
α1f1(y
11) · e2.
After the time scaling τ = kt, system (7.34) reads as
θ′ = F (θ) +1
k∆(x, θ), (7.35)
where prime denotes differentiation with respect to τ . In what follows, we denote by
Σ(0) the nominal system θ′ = F (θ), and by Σ(k) the perturbed system (7.35).
The vector field F coincides with the attitude synchronization dynamics of the col-
lection of rotational integrators in (4.22) with feedback (4.24), expressed relative to inte-
grator 1. Therefore, by Theorem 4.3.2, the equilibrium θ = 0 is almost globally asymp-
totically stable for Σ(0). Let D(0) be the domain of attraction of θ = 0 for Σ(0), a set
of full-measure.
The term (1/k)∆ acts as a perturbation in (7.35). Since, by assumption A2, the
functions fi(y1i ) · R1
i e2 and f1(y11) · e2 are uniformly bounded, the map ∆ is uniformly
bounded, i.e., there exists ∆ > 0 such that sup ‖∆‖ < ∆. The uniform bound on the
perturbation (1/k)∆ tends to zero as k → ∞.
Since θ = 0 is asymptotically stable for Σ(0), there exists r > 0 and a C1 positive
definite Lyapunov function W : Br(0) → R whose derivative along Σ(0), LFW : Br(0) →
R, is negative definite. We may assume, without loss of generality, that r ≤ ǫ. Let c > 0
be such that the sublevel set Wc := θ : W (θ) < c is contained in Br(0) ⊂ Bǫ(0), and let
ǫ′ > 0 be such that Bǫ′(0) ⊂Wc ⊂ Bǫ(0). Since LFW∣
∣
∂Wc< 0, Wc is positively invariant
Chapter 7. Formations of Kinematic Unicycles 171
for Σ(0). Moreover, letting
k0 =max∂Wc
‖∂W/∂θ‖min∂Wc
|LFW | ∆,
we have that for all k > k0 it holds that LF+(1/k)∆W∣
∣
∂Wc< 0, and thus Wc is positively
invariant for the perturbed system Σ(k) in (7.35).
Let θ ∈ D(0) be arbitrary. Then the solution of Σ(0) through θ converges to 0, and
let T > 0 be the first time when the solution enters Bǫ′(0). By continuity of solutions
with respect to initial conditions and bounded perturbations (Khalil, 2002, Theorem 3.4),
there exists µ > 0 and k ≥ k0 such that for all k > k, all solutions of Σ(k) through initial
conditions in Bµ(θ) are contained in Wc at time T . Since Wc is positively invariant for
Σ(k) and contained in Bǫ(0), all solutions of Σ(k) through Bµ(θ) enter in finite time and
remain inside the set Bǫ(0), for all k > k.
Let K ⊂ D(0) be an arbitrary compact set. The arguments above yield an open
cover of K by balls Bµ(θ) and associated gains k, where µ and k depend on θ. Taking a
finite subcover, we obtain points θi, i ∈ k ⊂ Tn−1, associated balls Bµi(θi), and gains
ki, i ∈ k. Letting k⋆ = maxi∈k ki, for all k > k⋆ all solutions of Σ(k) through points in
K enter and remain inside Bǫ(0).
Returning to the (x, θ) dynamics, the x subsystem has no finite escape times because
V is proper and nonincreasing along solutions. Then, from the results just obtained
we have that, for any compact subset K of D(0) (a set of full-measure in Tn−1), there
exists k⋆ > 0 such that, for all k > k⋆, all solutions of the (x, θ) subsystem through
initial conditions in R2(n−1) × K enter in finite time and remain inside the closed set
R2(n−1) × θ : ‖θ‖ ≤ ǫ. For any c > 0, the set Vc ∩ (R2(n−1) × θ : ‖θ‖ ≤ ǫ) is
compact because the sublevel set Vc is compact. Since ˙V is negative definite on this set,
all solutions through R2(n−1)×K converge to Γ. This proves that Γ is almost semiglobally
asymptotically stable.
Chapter 8
Formations of Kinematic Unicycles
with Parallel and Circular Collective
Motions
In this chapter we present control solutions to the following problems introduced in
Chapter 3:
• Parallel FormationFlocking Problem PFP andParallel FormationFlocking Problem
with a Beacon PFP-B
• Formation Line Path Following Problem LPP
• Circular Formation Flocking Problem CFP
• Formation Circle Path Following Problem CPP.
The control inputs ui and ωi for each control problem will be constructed out of one or
more of the following control primitives discussed in Section 4.3:
• a consensus controller for single-integrators fi((xij)j∈Ni) in (4.10),
• a line path following controller for single integrators h(x) in (4.16) and,
172
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 173
• a rotational integrator consensus controller gi((θij)j∈Ni, η) in (4.24).
Control solutions to PFP, PFP-B and LPP are presented in Section 8.1 and control solu-
tions to CFP and CPP are presented in Section 8.2.
8.1 Solutions to Formation Problems with Final Lin-
ear Motion (PFP, PFP-B and LPP)
In this section we present solutions to the formation problems with final linear motion.
Let α > 0 be a design parameter, and let d = (d1i1)i∈2 :n ∈ F be the desired formation.
Define the quantities αi, βi, δi and xi exactly as in Chapter 7 for formations that stop.
Also define yi := (xij)j∈Ni, yki := (xkij)j∈Ni as before. Recall the following set definitions
from Chapter 3
Γ =
(x, θ) ∈ R2n × T
n : x1i = R1d11i, i ∈ n
Γp = (x, θ) ∈ Γ : θi = θ1, i ∈ 2 :n
Γpf = Γp
Γpfb = (x, θ) ∈ Γ : θi = θp, i ∈ n
Γlp = (x, θ) ∈ Γpfb : x1 ∈ C(r0, p) .
By the same arguments as in Chapter 7, stabilizing Γpf = Γp in (3.5) reduces to synchro-
nizing the unicycles’ heading angles and the endpoints xi. For Γpfb there is the additional
requirement that θi = θp for all i ∈ n. Γlp follows by also imposing x1 ∈ C(r0, p). The
latter requirement can be replaced with x1 ∈ C(r0, p) since θ1 = θp and x1 ∈ C(r0, p)
imply x1 = x1 + αRie1 = x1 + αp ∈ C(r0, p). The converse can be shown as well.
For a single integrator consensus controller fi(·) defined in (4.10) choose the feedback
law as,
ui = u⋆i (yii, ϕi, µ
ii) = fi(y
ii) · e1 + βiω
⋆i (y
ii, ϕi, µ
ii) + µii · e1,
ωi = ω⋆i (yii, ϕi, µ
ii) =
1
αi
(
fi(yii) · e2 + µii · e2
)
, i ∈ n.(8.1)
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 174
where µi will be defined in the main theorem given below in Theorem 8.1.1 whose proof
is given in Section 8.4.
Theorem 8.1.1 (Solutions to PFP, PFP-B and LPP). Consider system (2.6), (2.7) with
directed sensor graph G containing a globally reachable node and any parameters aij > 0
for i ∈ n, j ∈ Ni.
1. (PFP) Suppose G is a hierarchical digraph. For any (d, w) ∈ PF, the local and dis-
tributed control inputs in (8.1) with α > 0, µ11 = we1, and µ
ii = (w/|Ni|)
∑
j∈NiRije1, i ∈
2 :n almost globally asymptotically stabilize the parallel formation flocking mani-
fold Γpf , thus solving PFP for the class of hierarchical digraphs.
2. (PFP-B) For any (d, p, w) ∈ PFB, the control inputs in (8.1), strictly functions of
(yii, ϕi, pi), with α > 0, and µii = wpi almost globally asymptotically stabilize the
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 179
1
2
3
5
4
Figure 8.1: Undirected graph G under consideration in the simulation results.
We have chosen random initial positions on a 40m × 40m area with random initial
angles. The corresponding simulation results for formations with parallel collective mo-
tions are shown in Figure 8.2 while those for formations with circular collective motions
are shown in Figure 8.3. For parallel formation flocking with no beacon in Figure 8.2(b),
although the sensor graph is not hierarchical, the local and distributed control in (8.1)
with µii = (w/|Ni|)∑
j∈NiRije1, i ∈ n still manages to stabilize Γpf .
8.4 Proofs
In Chapter 3 we introduced the sets Γpf ,Γpfb ,Γlp,Γcf ,Γcp to be almost globally asymp-
totically stabilized. As for stopping formations in Chapter 7, it will be useful to consider
(x, θ) = (xi, θi)i∈n as new coordinates. The parallel formation flocking manifold Γpf
in (3.5), expressed in (x, θ) coordinates, becomes
Γpf :=
(x, θ) ∈ R2n × T
n : x1i = 0, θ1i = 0, i ∈ 2 :n
(8.4)
which is the same as Γp in (7.10). The parallel formation flocking manifold with a beacon
Γpfb in (3.6) becomes,
Γpfb :=
(x, θ) ∈ Γpf : θi = θp, i ∈ n
. (8.5)
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 180
Figure 8.2: Simulation results for formations with final parallel collective motion: (a)formation flocking with beacon p = (1, 0) pointing in the direction of the positive x-axis (b) formation flocking with no beacon where µii = (w/|Ni|)
∑
j∈NiRije1, i ∈ n, (c)
formation path following for C(r0, p) = x ∈ R2 : x = r0 + sp, s ∈ R with r0 = (200, 0),p = (0, 1). Initial positions are indicated with and positions at the end of the simulationare indicated with ×.
To stabilize Γlp in (3.7) there is the additional requirement that x1 ∈ C(r0, p). Therefore,
the formation line path following manifold becomes
Γlp :=
(x, θ) ∈ Γpfb : x1 ∈ C(r0, p)
. (8.6)
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 181
-60 -40 -20 0 20 40
x (m)
-40
-20
0
20
40
60
y (m
)
(a)
0 50 100
x (m)
-100
-80
-60
-40
-20
0
20
40
60
80
100
y (m
)
(b)
Figure 8.3: Simulation for formations with circular collective motion (a) formation flock-ing, (b) formation path following around the point c = (100, 0). Initial positions areindicated with and positions at the end of the simulation are indicated with ×.
Finally, the sets Γcf ,Γcp in (3.8) and (3.9) expressed in terms of (x, θ) coordinates become
Γcf :=
(x, θ) ∈ R2n × T
n : x1i = 0, θ1i = ρi(d, β1), i ∈ 2 :n
,
Γcp :=
(x, θ) ∈ Γcf : x1 = c
,
respectively. The equations of motion in terms of (x, θ) coordinates is given in (7.7).
Recall that in CFP and CPP, αi = 0 for all i ∈ n. Notice that the control inputs
presented in Section 8.1 and Section 8.2 were defined precisely in terms of (x, θ) and so
the equations of motion in (7.7) constitute a dynamical system.
8.5 Proof of Theorem 8.1.1
We will begin by proving the results for PFP-B and LPP corresponding to the sets Γpfb
and Γlp respectively. We will then present the proof for PFP for hierarchical digraphs, cor-
responding to the set Γpf . To this end, consider the closed loop system (2.6), (2.7), (8.1)
with µi = wp for PFP-B and µi = h(xi) for LPP. It needs to be shown that the sets
Γpfb and Γlp in (8.5) and (8.6) respectively are almost globally asymptotically stable.
Since the dot product is invariant to a change of frame and fi(yii) = R−1
i fi(yi), it holds
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 182
that fi(yii) · e1 = Rifi(y
ii) · Rie1 = fi(yi) · Rie1, similarly fi(y
ii) · e2 = fi(yi) · Rie2 and
µii · e1 = µi · Rie1. The control inputs in (8.1) represented with respect to the inertial
frame therefore satisfy,
u⋆i (yii, ϕi, µ
ii) = fi(yi) · Rie1 + βiω
⋆i (y
ii, ϕi, µ
ii) + µi ·Rie1,
ω⋆i (yii, ϕi, µ
ii) =
1
αi(fi(yi) · Rie2 + µi · Rie2) , i ∈ n.
(8.7)
Substituting (ui, ωi) = (u⋆i , ω⋆i ) in (8.7) into (7.7) and using the fact that (fi(yi) ·
which is strictly a function of x⊥1 and x. Moreover, for PFP-B, x⊥1 ≡ 0 and therefore
˙x⊥1 = 0. For LPP, since c⋆(x1) lies on the line perpendicular to p⊥ at all times, it follows
that c⋆(x1) · p⊥ = 0. Therefore, the right hand sides of (8.10) are independent of x‖1. The
sets Γpfb and Γlp expressed in new coordinates both can be written as
(x‖1, x⊥1 , x, θ) ∈ R× R× R2(n−1) × T
n : x⊥1 = 0, x = 0, θi = θp, i ∈ n,
where the angle of p in the inertial frame is denoted by θp. This set is also independent
of x‖1. Therefore, x
‖1 can be dropped leaving the remaining states as (x⊥1 , x, θ) ∈ R ×
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 184
R2(n−1) × Tn. The sets Γpfb and Γlp reduce to the point
K := (x⊥1 , x, θ) ∈ R× R2(n−1) × T
n : x⊥1 = 0, x = 0, θi = θp, i ∈ n.
For PFP-B, µi = µ1 = wp for all i ∈ 2 :n and system (8.10) reduces to
˙x⊥1 = 0
˙x1i = fi(yi)− f1(y1)
θi =1
αi(fi(yi) · Rie2 + wp · Rie2).
(8.11)
For LPP, since c⋆(xi)− xi is parallel to p⊥, it holds that µi = [k0(c
⋆(xi)− xi) · p⊥]p⊥+wp
for i ∈ n. Moreover, since c⋆(xi) lies on C(r0, p) for all i ∈ n, it follows that (c⋆(xi) −
c⋆(x1)) ·p⊥ = 0 for all i ∈ 2 :n. These two facts imply that µi−µ1 = [k0(c⋆(xi)− c⋆(x1)) ·
p⊥ − k0(xi − x1) · p⊥]p⊥ = (k0xi1 · p⊥)p⊥ in (8.10) for all i ∈ 2 :n. Therefore for LPP,
system (8.10) reduces to
˙x⊥1 = (f1(y1) + µ1) · p⊥
˙x1i · p = (fi(yi)− f1(y1)) · p
˙x1i · p⊥ = (fi(yi)− f1(y1)) · p⊥ + k0xi1 · p⊥
= (fi(yi, xi1)− f1(y1, 0)) · p⊥
θi =1
αi(fi(yi) ·Rie2 + µi ·Rie2), i ∈ n,
(8.12)
where the state x has been split into components parallel and perpendicular to p and
fi(yi, xi1) := fi(yi) + k0xi1 represents the consensus control law fi(yi) with an additional
edge added from unicycle i to unicycle 1. Therefore, both fi(yi) and fi(yi, xi1) are integra-
tor consensus control laws. In (8.12), fi(yi)·p =∑
j∈Niaij(x1j ·p−x1i ·p) is strictly a func-
tion of the elements of x in the direction of p and fi(yi) ·p⊥ =∑
j∈Niaij(x1j ·p⊥− x1i ·p⊥)
is strictly a function of the elements of x in the direction of p⊥. It follows from the
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 185
equations of motion for flocking and path following in (8.11), and (8.12) respectively and
the absence of finite escape times that
Γ3 := (x⊥1 , x, θ) ∈ R2 × R
2(n−1) × Tn : x = 0
is globally asymptotically stable in both cases. This implies boundedness of x and, in
turn, boundedness of fi(yi) for all i ∈ n. For PFP-B, x⊥1 ≡ 0 is bounded while for LPP,
˙x⊥1 satisfies
˙x⊥1 = f1(y1) · p⊥ + µ1 · p⊥ ≤ B − k0x⊥1 ,
where B = sup ‖f1(y1)‖ < ∞. This implies boundedness of x⊥1 and boundedness of the
states (x⊥1 , x, θ). On the set Γ3, the equations of motion of (x⊥1 , x, θ) in (8.11), (8.12)
both reduce to,
˙x⊥1 = µ1 · p⊥ = −k0x⊥1˙x1i = 0
θi =1
αi(wp · Rie2) =
w
αisin(θp − θi), i ∈ n.
(8.13)
In the case of flocking, x⊥1 ≡ 0 which implies ˙x⊥1 = 0. For system (8.13), x⊥1 → 0 as
t→ ∞ and the compact set
Γ2 := (x⊥1 , x, θ) ∈ Γ3 : x⊥1 = 0, i ∈ n,
diffeomorphic to Tn, is an embedded submanifold of R2 × R2(n−1) × Tn and is globally
asymptotically stable relative to Γ3. By Theorem 4.1.2, Γ2 is globally asymptotically
stable.
The point K can therefore be written as
K = (x⊥1 , x, θ) ∈ Γ2 : θi = θp, i ∈ n.
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 186
To complete the proof it needs to be shown that K is almost globally asymptotically
stable. In K, the control inputs in (8.7) satisfy u⋆i = w and ω⋆i = 0 for all i ∈ n as
desired. Notice that the rotational system in (8.13) is decoupled for each unicycle i ∈ n.
As a consequence of Proposition 4.3.4, the set Γ1 = A ∪ K, where A is the finite set of
isolated equilibria given by
A = (x⊥1 , x, θ) ∈ Γ2 : θi ∈ θp, θp + π, i ∈ n\K,
is globally attractive relative to Γ2. Moreover K is asymptotically stable relative to Γ2
whereas the equilibria in A are exponentially unstable relative to Γ2. In the set A, at
least one unicycle has a heading angle 180 offset from the beacon while the remaining
unicycle headings are aligned with the beacon. It follows from Theorem 4.1.3, setting
Γ2 = Γ2 and Γ1 = Γ1, that K is almost globally asymptotically stable. This concludes
the proof of PFP-B and LPP.
Now we present the proof for PFP assuming hierarchical digraphs. Consider the closed
loop system (2.6), (2.7), (8.1) with µ11 = e1 and µii = (w/|Ni|)
∑
j∈NiRije1, i ∈ 2 :n. The
leader’s control inputs are given by,
u⋆1(y11, ϕ1, µ
11) = w, ω⋆1(y
11, ϕ1, µ
11) = 0 (8.14)
and the control inputs of the follower unicycles are given by,
u⋆i (yii, ϕi, µ
ii) = fi(y
ii) · e1 + βiω
⋆i (y
ii, ϕi, µ
ii) +
w
|Ni|∑
j∈Ni
Rje1 ·Rie1,
ω⋆i (yii, ϕi, µ
ii) =
1
αi
(
fi(yii) · e2 +
w
|Ni|∑
j∈Ni
Rje1 · Rie2
)
,
(8.15)
where∑
j∈NiRje1 · Rie1 =
∑
j∈Nicos(θij) and
∑
j∈NiRje1 · Rie2 =
∑
j∈Nisin(θij). The
feedbacks (8.14), (8.15) for unicycle i are local and distributed since they depend only
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 187
on (yii, ϕi).
It needs to be shown that Γpf is almost globally asymptotically stable. The closed
loop equation for the leader is ˙x1 = wR1e1, θ1 = 0. This implies that the leader moves
in a straight line at the desired speed w in the direction of its initial heading. In this
case, since the heading vector p := R1e1 is constant, we can consider the angle θp := θ1
as a parameter rather than a state. The closed loop equations of motion for the follower
unicycles with respect to the inertial frame are given as before in (8.8).
Define the vertex set Lj ⊂ V to be the set of unicycles in layer j of the hierarchical
digraph and Lj := ∪ji=1Li. The equations of motion of unicycles associated with a node
set Lj are independent of the nodes outside of this set because, for any i ∈ Lj, the
feedbacks u⋆i (·) and ω⋆i (·) in (8.14), (8.15) depend only on states of unicycles within Ljitself. Therefore, the dynamics of unicycles in the isolated set Lj in terms of relative
and u⋆i = w, ω⋆i = 0 for all i ∈ n. In the arguments that follow it will be shown that
Γpf is almost globally asymptotically stable, which solves the formation control problem.
An induction approach based on reduction will be employed. Consider first the isolated
set L2. The leader unicycle with heading vector p = R1e1 is the unique neighbor for all
follower unicycles in L2 and the control inputs in (8.15) reduce to
u⋆i (yii, ϕi, µ
ii) = fi(yi) ·Rie1 + βiω
⋆i (y
ii, ϕi, µ
ii) + wp · Rie1,
ω⋆i (yii, ϕi, µ
ii) =
1
αi(fi(yi) · Rie2 + wp ·Rie2) , i ∈ L2
(8.17)
which has the same form as formation flocking with a beacon and, by the same arguments
as before, the set ΓL2is almost globally asymptotically stable in the state space X2. Now
for k ∈ N, consider the isolated node set Lk−1 and assume the set ΓLk−1is almost globally
asymptotically stable in Xk−1 coordinates with domain of attraction Xk−1\Nk−1 where
Nk−1 is a set of Lebesgue measure zero. Next it will be shown that this implies ΓLk is
almost globally asymptotically stable in Xk coordinates. Inserting ΓLk−1into Xk, implies
Γ3 := (x, θ)Lk ∈ Xk : x1i = 0, θi = θp, i ∈ Lk−1
is almost globally asymptotically stable because the closed loop system in (8.8) is globally
Lipschitz and therefore has no finite escape times. The domain of attraction of Γ3 is given
by Xk\Nk, where
Nk := (x, θ)Lk ∈ Xk : (x, θ)Lk−1∈ Nk−1
is the insertion of Nk−1 into Xk and remains a set of Lebesgue measure zero. All unicycles
in the set Lk have neighbors strictly in Lk−1 and the equation of motion for (x)Lk in (8.16)
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 189
can be written as
˙x1i = (fi − f1) +w
|Ni|∑
j∈Ni
Rje1 − wp, i ∈ Lk,
where f1(y1) = 0 and the term (w/|Ni|)∑
j∈NiRje1−wp vanishes in Γ3 because Rje1 = p
for all j ∈ Lk−1. This term is bounded since it is a continuous function of the bounded
quantities (θ)Lk−1. Then, it must hold that (x)Lk is bounded because the origin is expo-
nentially stable for the nominal linear system
˙x1i = (fi − f1), i ∈ Lk.
Therefore, all closed loop solutions remain bounded. In the set Γ3, the equations of
motion for unicycles in Lk in (8.16) reduce to
˙x1i = fi(yi)− f1(y1),
θi =1
αi(fi(yi) · Rie2 + wp · Rie2) , i ∈ Lk,
(8.18)
which has the same form as (8.11) with the redundant state x‖1 ≡ 0 dropped. It follows
that the compact set
Γ2 := (x, θ)Lk ∈ Γ3 : x1i = 0, i ∈ Lk
is globally asymptotically stable relative to Γ3 and reduction in Theorem 4.1.2 implies
that Γ2 is globally asymptotically stable relative to Xk\Nk, a positively invariant set of
full measure. We conclude that Γ2 is almost globally asymptotically stable in Xk. In the
set Γ2, rotational dynamics satisfy θi = (w/αi) sin(θp − θi) for i ∈ Lk and the point
K := ΓLk = (x, θ)Lk ∈ Γ2 : θi = θp, i ∈ Lk
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 190
is almost globally asymptotically stable relative to Γ2 with an additional finite number
of exponentially unstable isolated equilibria
A = (x, θ)Lk ∈ Γ2 : θi ∈ θp, θp + π, i ∈ Lk\K.
The set Γ1 = A∪K is globally attractive with respect to Γ2 and applying Theorem 4.1.3,
setting Γ2 = Γ2 and Γ1 = Γ1, the point K is almost globally asymptotically stable relative
to Xk\Nk or, equivalently, almost globally asymptotically stable relative to Xk. It follows
by induction that Γpf is almost globally asymptotically stable. This concludes the proof
of PFP for hierarchical digraphs.
8.6 Proof of Theorem 8.2.1
Consider the point zi := xi + γiRie2 offset γi units along unicycle i’s second body axis.
Then under the feedback transformation ui = ui + γiωi, the system (zi, θi) acts as a
unicycle with control inputs (ui, ωi), that is,
zi = uiRie1 − γiωiRie1 = uiRie1
θi = ωi.
Denote the collection of these offsets by z = (zi)i∈n.
In this proof, we convert the problems CFP and CPP in (x, θ) coordinates into
analogous problems for formations on a common circle in (z, θ) coordinates satisfying
the desired angular spacings θ1i = ρi(d, β1) for i ∈ 2 :n. To begin, for any radius
r > 0 let γi = βi − r, define zi := zi + rRie2 for all i ∈ n and denote z = (zi)i∈n.
In the sets Γ1 := (z, θ) ∈ R2n × Tn : zi = z1, θ1i = ρi(d, β1), i ∈ 2 :n and
Γ2 := (z, θ) ∈ Γ1 : z1 = c, the unicycles in (z, θ) coordinates lie on a common cir-
cle of radius r with angular spacing θ1i = ρi(d, β1) for all i ∈ 2 :n. In Γ2, the centre of
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 191
Figure 8.4: Desired formation of unicycles (x, θ) (shaded) is achieved when (z, θ) (notshaded) lie on a common circle of radius r with the desired spacing θ1i = ρi(d, β1) for alli ∈ n
the circle is c ∈ R2. First it will be shown that the control inputs in (8.2) almost globally
asymptotically stabilize Γ1 and Γ2 for CFP and CPP respectively. Then, it will be shown
that Γ1 and Γ2 are precisely the representations of Γcf and Γcp in new coordinates, where
(z, θ) represents the radial projection of the unicycle formation in original coordinates
(x, θ) onto a common circle C0 of radius r (see Figure 8.4).
Using the control inputs in (8.2) with ui = u⋆i (yii, ϕi, νi) and ωi = ω⋆i (y
ii, ϕi, νi), (ui, ωi)
become,
ui = ui − γiωi = ui + (βi − r)ω⋆i − γiω⋆i = ui
ωi =uir+Kϕ(νi)νi, i ∈ n.
(8.19)
Using the fact that zi = zi + rRie2 = xi + γiRie2 + rRie2 = xi + βiRie2 = xi, it follows
that νi = −(∑
j∈Nizij) · Rie1 for CFP and νi = −(c − zi) · Rie1 for CPP. Moreover
ui = rw + kgi((θij − ρij(d, β1))j∈Ni, 1). From Theorem 8.2.3, there exist k⋆, K⋆ > 0 such
that for all K ∈ (0, K⋆), k ∈ (0, k⋆), the sets Γ1 and Γ2, in which all unicycles (z, θ)
lie on a common circle of radius r with desired angular spacings, are almost globally
asymptotically stable for CFP and CPP respectively.
Using the fact that zi = xi for all i ∈ n, the sets Γ1 and Γ2 can be written in (x, θ)
coordinates as (x, θ) ∈ R2n×Tn : x1i = 0, θij = ρi(d, β1), i ∈ n and (x, θ) ∈ R2n×Tn :
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 192
x1i = 0, θij = ρi(d, β1), x1 = c, i ∈ n respectively which correspond to Γcf and Γcp
respectively. Therefore, Γcf and Γcp are almost globally asymptotically stable for CFP
and CPP respectively. This concludes the proof for CFP and CPP.
8.7 Proof of Theorem 8.2.3
This proof makes use of Theorem V.5. in (El-Hawwary and Maggiore, 2013a) where
ui = rw + k∑
j∈Nisin(θij − ρij(d, β1)), a Kuramoto consensus controller, has been re-
placed by an almost global rotational integrator consensus controller ui = rw+kgi((θij−
ρij(d, β1))j∈Ni, 1). Substituting (ui, ωi) = (u⋆i (·), ω⋆i (·)) in (8.3) into (7.7) with αi = 0,
βi = r for all i ∈ n yields,
˙xi = −rKϕ(νi)νiRie1
θi =uir+Kϕ(νi)νi, i ∈ n.
(8.20)
The equations on the right hand side of (8.20) are bounded and therefore the closed
loop system has no finite escape times. In coordinates relative to unicycle 1, (x, θ) where
x := (x1i)i∈2 :n ∈ R2(n−1) and θ := (θ1i)i∈2 :n ∈ Tn−1, define the sets
Γ2 = (x, θ) ∈ R2(n−1) × T
n−1 : x = 0
K = (x, θ) ∈ Γ2 : θ1i = ρi(d, β1), i ∈ 2 :n,
where K corresponds to Γcf in relative coordinates and is a point. Using the same
arguments as in Proposition V.2. in (El-Hawwary and Maggiore, 2013a), there exist
K⋆ ∈ (0, (w/2)), k⋆ > 0 such that for all K ∈ (0, K⋆) and all k ∈ (0, k⋆) the set
Γ2, diffeomorphic to Tn−1 and an embedded submanifold of R2(n−1) × Tn−1, is globally
asymptotically stable relative to R2(n−1) × Tn−1. This implies boundedness of x, while θ
is bounded because it lies on a bounded set. In the set Γ2, νi = 0 for all i ∈ n and the
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 193
We have chosen the path C to be the stable limit cycle of a Van der Pol oscillator
Chapter 9. General Formation Path Following 202
1
2
3
5
4
Figure 9.4: Digraph G under consideration in the simulation results.
Figure 9.5: Formation specified by the offset vectors d1 = (15, 0), d2 = (5,−8), d3 =(5, 8), d4 = (−10,−3) and d5 = (−10, 3).
with equations of motion given by
y = c0z
z = µ(1− (c0y)2)c0z − c0y,
(9.8)
for x = (y, z) ∈ R2, µ = 1.5 and c0 = 0.04. This set is illustrated in Figure 9.6(a) by
the solid line. Correspondingly, we choose the single integrator path following controller
h(x) in (4.15) as,
h(x) =w(c0z, µ(1− (c0y)
2)c0z − c0y)
‖(c0z, µ(1− (c0y)2)c0z − c0y)‖(9.9)
where the vector field in (9.8) is normalized and scaled to the desired speed w. We choose
Chapter 9. General Formation Path Following 203
the initial unicycle positions as x1(0) = (−25,−20)m, x2(0) = (−15,−15)m, x3(0) =
(−27,−25)m, x4(0) = (−20,−20)m, and x5(0) = (−18,−38)m which are indicated by
the circles in Figure 9.6(b). For the control inputs in (9.5), (9.6) choose w = 1, k = 10,
γ = 1. For unicycles i ∈ 1, 2, 3, for which di · e1 > 0, let φi(xi, σi) = tan−1 ζi + π while
for unicycles i ∈ 4, 5, for which di · e1 < 0, let φi(xi, σi) = tan−1 ζi. The corresponding
simulation results are shown in Figure 9.6. Figure 9.6(b) illustrates the individual paths
traversed by the five unicycles as the formation converges to and moves around C. Clearly
the paths followed by each unicycle are significantly different from one another in order
to maintain the desired formation. One advantage of this approach is that unicycles do
not need to compute their individual paths, but rather, only require knowledge of the
common curve C. The path traversed by x1, unicycle 1’s estimate of the formation origin,
is illustrated by the dashed line in Figure 9.6(a) which converges to the curve C as desired
and moves along it with the speed w.
-50 0 50
x (m)
-80
-60
-40
-20
0
20
40
60
80
y (m
)
(b)
-60 -40 -20 0 20 40 60
x (m)
-80
-60
-40
-20
0
20
40
60
80
y (m
)
(a)
Figure 9.6: Simulation results: (a) illustrates the curve C corresponding to the stable limitcycle of the Van der Pol oscillator with µ = 1.5, c0 = 0.04 and shows the time evolutionof x1 illustrated by the dashed line, (b) illustrates the individual paths traversed by thefive unicycles as the formation moves around C. Initial positions are indicated with and positions at the end of the simulation are indicated with ×.
Chapter 9. General Formation Path Following 204
9.3 Proof of Theorem 9.1.1
In this section we will drop the arguments for a number of functions for simplicity of
notation: φi(xi, σi) → φi, φ′i(y
ii, xi, σi) → φ′
i, ζi(xi, σi) → ζi, ξi(xi, σi) → ξi. Before
presenting the proof for Theorem 9.1.1, the formation path following problem will be
reformulated in terms of new states (x, σ, φ) related to the original coordinates under the
diffeomorphism F : R2n × Tn × Tn → R2n × Tn × Tn given by
where we have used the identity sin ξi sinψi+cos ξi cosψi = cos(σi− θr). Substituting φi
from (9.21) into (9.23) yields
ξi = − w
‖δi‖[sin ξi cosψi + tanψi cos(σi − θr)− cos ξi sinψi] + θr − γ sin ξi − θr
= − w
‖δi‖
[
sin ξi cosψi +sinψicosψi
(sin ξi sinψi + cos ξi cosψi)− cos ξi sinψi
]
− γ sin ξi
= − w
‖δi‖
[
sin ξi cosψi +sin2 ψicosψi
sin ξi + cos ξi sinψi − cos ξi sinψi
]
− γ sin ξi
= −(
γ +w
‖δi‖cosψi +
w
‖δi‖sin2 ψicosψi
)
sin ξi
(9.24)
and choosing
γ > maxi∈n
(
w
‖δi‖
∣
∣
∣
∣
cosψi +sin2 ψicosψi
∣
∣
∣
∣
)
=: γ⋆
implies that the set
Γ1 := Γgp
is asymptotically stable relative to Γ2. Figure 9.7 illustrates the sets Γ1, Γ2 and Γ3 just
discussed. Based on reduction in Theorem 4.1.2, Γgp is asymptotically stable as long as
two assumptions hold in a neighborhood of Γgp: (i) cos(σi − θh(xi)) 6= 0 for all i ∈ n so
that φi and φ′i in (9.6) are well defined and (ii) αi = ‖δi‖ cosφi is bounded away from
Chapter 9. General Formation Path Following 211
zero for all i ∈ n (equivalently φi is bounded away from ±π/2) ensuring that ω⋆i in (9.4)
is well defined.
Figure 9.7: Conceptual illustration of the reduction sets Γ1, Γ2 and Γ3. The set Γ3 isasymptotically stable, the compact set Γ2 is asymptotically stable relative to Γ3 andthe point Γ1 is asymptotically stable relative to Γ2. Reduction implies the set Γ1 isasymptotically stable as illustrated by the solution starting at χ0 off the set Γ3.
Starting with (i), since ψi 6= ±π/2 there exists ρ > 0 such that in the set Γgp,
cos(σi − θh(xi)) = cos(σi − θr) = cosψi is bounded away from 0 by ρ. Since Γgp is
compact, there exists an ǫ1 > 0 neighborhood Bǫ1(Γgp) in which cos(σi − θh(xi)) is
bounded away from 0 by ρ/2, i.e., | cos(σi − θh(xi))| > ρ/2.
To show (ii), in the set Γgp, φi in (9.21) reduces to
φi = tan−1
[
−‖δi‖w
θrcosψi
+ tanψi
]
∈ (−π/2, π/2).
Since ψi 6= ±π/2 and θr is bounded (since C is compact), there exists ρ > 0 such that
φi is bounded away from ±π/2 by ρ in Γgp, i.e., |φi ± π/2| > ρ. Since Γgp is compact
there exists an ǫ2 < ρ/4 neighborhood of Γgp, Bǫ2(Γgp), in which φi is bounded away
from ±π/2 by ρ/2, i.e., |φi ± π/2| > ρ/2. In addition, in Bǫ2(Γgp), |φi−φi| < ρ/4. This
implies that |φi ± π/2| > ρ/4 in Bǫ2(Γgp).
Therefore choosing ǫ = min(ǫ1, ǫ2), conditions (i) and (ii) hold in the set Bǫ(Γgp).
Therefore Γgp is asymptotically stable.
Chapter 10
Unicycle Formation Simulation
Trials
In this chapter, we present extensive simulation trials to study the effectiveness of our
control solution presented in Chapter 7 for formation control of unicycles under different
realistic scenarios not captured by the main theoretical result in Theorem 7.1.1. The
simulations will study the following items:
• performance in the presence of state dependent sensor graphs in which each unicy-
cle’s neighbors are those that lie within a given radius of itself
• performance for directed sensing graphs as opposed to undirected sensing graphs
• performance when the high gain conditions on α and k are ignored
• robustness of the approach to unmodelled effects including sensor noise, input noise,
sampling, and saturated inputs
• extension of the control solution for kinematic unicycles to the dynamic unicycle
model in (2.10)
We will not present extensive simulations for formations with final collective motion
discussed in Chapter 8 and Chapter 9. However, we predict similar outcomes since the
Figure 10.3: Formation measure. The unicycles at the initial time t0 are not shaded whilethe unicycles at time tf are shaded. The average position of the unicycles are indicatedwith . In the top figure, x(t0) and x(tf ) are the same and therefore µd = 0. In thebottom figure ‖x(tf ) − x(t0)‖ 6= 0 and therefore drift is present which is roughly twicethe size of the formation itself.
• Collision Measure. The collision measure µc is the minimum distance attained
between any two unicycles between the initial time t0 and the final time tf when
the formation is achieved and is defined as
µc := mint∈[t0,tf ],i,j∈n
‖xij(t)‖.
Assuming robots are 0.5m in radius or less, we say that a collision occurs if µc < 0.5.
• Input Measures. Finally, there are four input measures that correspond to the
supremum and mean of the control input magnitudes from the initial time t0 to the
formation time tf , maximized over i ∈ n. These are given by
C # of tests that begin with a connected graphD # of tests that begin with a disconnected graphF # of tests that achieve formationCF # of tests that begin with a connected graph
and end in formation (µf < µf)DF # of tests that begin with a disconnected graph
and end in formation (µf < µf)tf formation timeµd drift measureµc collision measureNc # of tests in which a collision occursµu,1 , µω,1 , µu,2 , µω,2 input measures
time and study the effect this has. The undirected graph that will be used for the base
case simulations is shown in Figure 10.6.
Now we present the simulation results for the base case in Table 10.3. The column
called F/N (%) represents the percentage of simulations in which formation is achieved,
i.e., there exists a time tf after which µf < 0.05. The column called Nc/N (%) represents
the percentage of simulations in which a collision occurs. The remaining columns cor-
respond to the performance measures. The quantity tfµu,2 upper bounds the distance
travelled by any single robot in the ensemble and tfµω,2 upper bounds the number of
radians that any robot rotates before formation is achieved. The quantity tfµω,2/2π
therefore upper bounds the number of revolutions spun by any robot. The values pre-
sented in Table 10.3 are averages taken over all N iterations and we draw the following
conclusions:
• F/N (%)= 100 indicates that all simulations achieve formation.
• the average drift of the formation is 76m, i.e., 3.4 times the formation size.
• there is a collision 68 percent of the time which is high. To avoid this, one would
have to design a high level collision avoidance layer. It turns out that the colliding
µf (formation threshold) 0.05k 15α 5fixed or state-dependent graphs fixedundirected or directed graph undirectedinput saturation nosampling and disturbances no
1
2 3
4 5
Figure 10.6: Undirected graph used in the base case.
10.2.1 Study of µf
In this section, we will show how changing the formation threshold µf affects the per-
formance measures µd and tf . The average results for N = 500 simulations are shown
in Table 10.4 where the bold values correspond to the base case. Both µd and tf are
inversely proportional to µf . We can see that the smaller the threshold values of µf , the
further the formation needs to drift. Also the time to achieve smaller formation threshold
values grows very rapidly. To speed up convergence in a neighborhood of the formation
(where the rate of convergence is slow), one can always add a gain K to the controller
in (7.2) as follows,
u⋆i (yii, ϕi) = Kfi(y
ii) · e1 + βiω
⋆i (y
ii, ϕi),
ω⋆i (yii, ϕi) =
K
αi
(
fi(yii) · e2 + kgi(ϕi, η)
)
, i ∈ n.(10.1)
This will increase the rate of convergence by a factor of K. However, this will also
increase the magnitude of the control inputs by a factor of K and one may therefore
Figure 10.7: Illustration of configurations satisfying different values of µf . The centre ofthe circles represent the desired unicycle positions while the actual unicycle positions areindicated with ×.
• The time to reach formation tf increases with α.
• All four input measures are inversely proportional to α. The maximum angular
velocity is as high as 994 rad/s when α = 0.1. This would certainly require satura-
tion.
• The lower the value of α, the more oscillatory the response. The most revolutions
that a unicycle makes is 1.58 when α = 0.1.
The main conclusions from the simulation results in Table 10.6 for varying k are:
• For k < 15 (the base value), some simulations fail. Only 0.4 percent of simulations
Figure 10.8: Illustration of a formation where one agent is disconnected from the rest ofthe group. Regardless of this, the rest of the group achieves formation among themselves.Initial positions are indicated with and final positions are indicated with ×.
• For this particular digraph, tf is 2.23 times the value of the base case. This makes
sense because the undirected graph in the base case has 2.4 times the number of
edge connections. The input measures are also higher in the base case for the same