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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
University of Toronto
c© Copyright 2019 by Ashton Roza
Abstract
Distributed Coordination Theory for Ground and Aerial Robot Teams
Ashton Roza
Doctor of Philosophy
Graduate Department of Electrical and Computer Engineering
University of Toronto
2019
This thesis investigates distributed coordination problems for two important classes
of robots. One class corresponds to ground-based mobile robots, each modelled as a
kinematic unicycle. The second corresponds to flying robots, each propelled by a thrust
vector and endowed with an actuation mechanism producing torques about three or-
thogonal body axes. The following coordination problems are studied in this thesis:
rendezvous, formation control, linear and circular formation flocking and formation path
following.
For rendezvous of kinematic unicycles, a smooth, time-independent control law is
presented that drives the unicycles to a common position from arbitrary initial conditions,
under the assumption that the sensing digraph is time-invariant and contains a globally
reachable node. The proposed feedback is very simple and is local and distributed. For
rendezvous of flying robots, a control strategy is presented that makes the centres of
mass of the vehicles converge to an arbitrarily small neighborhood of one another. The
convergence is global, and each vehicle can compute its own control input using local and
distributed feedback.
For formation control, the objective is to make an ensemble of kinematic unicycles
achieve pre-defined inter-agent spacings with parallel heading angles. We consider scenar-
ios where the formation either stops or moves with a final collective motion. In the latter
case, problems of linear and circular formation flocking and formation path following are
ii
studied. A control law is presented in each case that solves the problem for almost all
initial conditions. For stopping and flocking formations, the proposed control laws are
local and distributed while for formation path following, the control laws additionally
require each agent to measure its displacement from the path. The idea used to solve
the formation control problems is to rigidly attach an offset vector to the body frame of
each unicycle. It is shown that stabilizing the desired formation amounts to achieving
consensus of the endpoints of the offset vectors, and simultaneously synchronizing the
unicycles’ heading angles. Extension of formation control to flying robots using strictly
local and distributed feedback is not addressed in this work and remains a challenging
open problem.
iii
Acknowledgements
I would like to sincerely thank my supervisors Manfredi Maggiore and Luca Scardovi
for their continued dedication and availability during the course of my PhD program.
The time they have dedicated to meeting with me on a weekly basis, reviewing my work
and making suggestions has been greatly appreciated. They have been a great source of
wisdom and inspiration. I would also like to thank my family for their continued support
during my PhD program.
iv
Contents
1 Introduction 1
1.1 Coordination Problems Investigated in This Thesis . . . . . . . . . . . . 7
1.2 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Statement of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Modelling 15
2.1 Attitude Representation as a Rotation Matrix . . . . . . . . . . . . . . . 15
2.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Coordination Problems 31
3.1 Rendezvous of Flying Robots . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Rendezvous of Kinematic Unicycles . . . . . . . . . . . . . . . . . . . . . 32
3.3 Formation Control Problems for Kinematic Unicycles . . . . . . . . . . . 33
3.3.1 Formation Control . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2 Formation with Final Parallel Collective Motion . . . . . . . . . . 36
3.3.3 Formation with Circular Collective Motion . . . . . . . . . . . . . 39
3.3.4 General Formation Path Following . . . . . . . . . . . . . . . . . 42
3.4 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Preliminaries 54
4.1 Basic Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
v
4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.2 Reduction Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.3 Lyapunov and Krasovskii-LaSalle Theorems . . . . . . . . . . . . 72
4.2 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.1 Basic Definitions in Graph Theory . . . . . . . . . . . . . . . . . 79
4.2.2 Classes of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.3 Graph Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Control Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.1 Control Primitives for Single Integrators . . . . . . . . . . . . . . 87
4.3.2 Control Primitives for Double Integrators . . . . . . . . . . . . . . 91
4.3.3 Control Primitives for Rotational Integrators . . . . . . . . . . . . 92
4.3.4 Control Primitives for Rotating bodies in SO(3) . . . . . . . . . . 96
5 Rend. of Flying Robots with Loc. and Dist. Feedbacks 98
5.1 Solution of the Rendezvous Control Problem (RP− F) . . . . . . . . . . 98
5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 From Rendezvous to Formations . . . . . . . . . . . . . . . . . . . . . . . 105
5.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.4.1 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . 108
5.4.2 Proof of Lemma 5.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4.3 Proof of Lemma 5.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 123
6.1 Solution of the Rendezvous Control Problem (RP− U) . . . . . . . . . . 123
6.1.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2.1 Proof of Theorem 6.1.1 . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.2 Proof of Proposition 6.2.1 . . . . . . . . . . . . . . . . . . . . . . 132
vi
6.2.3 Proof of Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7 Formations of Kinematic Unicycles 150
7.1 Solution to the Parallel Formation Problem (PP) . . . . . . . . . . . . . 150
7.2 Discussion of the Control Solution . . . . . . . . . . . . . . . . . . . . . . 153
7.3 Special cases: Line formations and full synchronization . . . . . . . . . . 155
7.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.5 Proof of Theorem 7.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.5.1 System dynamics in (xi, θi)i∈n coordinates . . . . . . . . . . . . . 159
7.5.2 Lyapunov analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.5.3 Lyapunov analysis in relative coordinates . . . . . . . . . . . . . . 162
7.5.4 Local asymptotic stability of Γp . . . . . . . . . . . . . . . . . . . 165
7.5.5 Almost semiglobal asymptotic stability of Γp . . . . . . . . . . . . 169
8 Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 172
8.1 Solutions to Formation Problems with Final Linear Motion (PFP, PFP-B
and LPP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.2 Solutions to Formation Problems with Final Circular Motion (CFP and
CPP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.2.1 Formation control of unicycles on a common circle . . . . . . . . . 177
8.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.5 Proof of Theorem 8.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.6 Proof of Theorem 8.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.7 Proof of Theorem 8.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9 General Formation Path Following 195
9.1 Solution to the General Formation Path Following Problem (GPP) . . . . 195
9.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
vii
9.3 Proof of Theorem 9.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
9.3.1 Proof of Theorem 9.1.1 . . . . . . . . . . . . . . . . . . . . . . . . 205
10 Unicycle Formation Simulation Trials 212
10.1 Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
10.2 Simulation Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.2.1 Study of µf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
10.2.2 Study of high gain parameters α and k . . . . . . . . . . . . . . . 222
10.2.3 Study of state-dependent undirected graphs . . . . . . . . . . . . 224
10.2.4 Study of directed graphs . . . . . . . . . . . . . . . . . . . . . . . 226
10.2.5 Study of input saturation . . . . . . . . . . . . . . . . . . . . . . 228
10.2.6 Study of disturbances and sampling . . . . . . . . . . . . . . . . . 230
10.2.7 Study of dynamic unicycles . . . . . . . . . . . . . . . . . . . . . 232
11 Conclusions and future research 234
11.1 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Bibliography 239
viii
Notation and Acronyms
(v1, . . . , vn) in (R)n column vectors v = [v1 · · · vn]⊤ in Rn are identified
with n-tuples (v1, . . . , vn) in (R)n
v · w := v⊤w Euclidean inner product of v, w ∈ Rn
‖v‖ := (v · v)1/2 the Euclidean norm of v
R+ the set of positive real numbers
e1, e2 the natural basis of R2
e1, e2, e3 the natural basis of R3
1 the vector of ones in Rn
⊗ the Kronecker product of matrices
S1 the unit circle, which we identify with the set of
real numbers modulo 2π
Sn the n-dimensional unit sphere
Tn the n-torus Tn := S1 × · · · × S1 (n times)
A\B the set-theoretic difference of the sets A and B
A the closure of the set A
|A| the cardinality of the set A
Bε(Γ) the ε neighborhood of the closed set Γ
N (Γ) a neighborhood of the closed set Γ
n 1, . . . , n
k :n k, . . . , n
ix
(xj)j∈I If I = i1, . . . , in is an index set, the ordered list of
elements (xi1 , . . . , xin) is denoted by (xj)j∈I
Lfϕ(x) If f(x) is a vector field and ϕ(x) is a C1 function
x 7→ Lfϕ(x) := (d/dx)ϕ(x) · f(x) is the Lie derivative
of ϕ along f
Notation for Kinematic Unicycle Model
Notation Explanation
xi ∈ R2 inertial position of unicycle i
x ∈ R2n (xi)i∈n
Ri ∈ SO(2) attitude of unicycle i
θi ∈ S1 heading angle of unicycle i
θ ∈ Tn (θi)i∈n
ωi ∈ R angular velocity of unicycle i
ri = R−1i r coordinate representation of r in frame Bi
xij = xj − xi relative displacement of robot j wrt. robot i
Ni set of neighbors of robot i
yi = (xij)j∈Ni vector of relative positions available to robot i
Notation for Flying Robots
Notation Explanation
mi, Ji mass and inertia matrix of robot i
xi ∈ R3 inertial position of robot i
x ∈ R3n (xi)i∈n
x
vi ∈ R3 linear velocity of robot i
v ∈ R3n (vi)i∈n
Ri ∈ SO(3) attitude of robot i
Ωi ∈ R3 angular velocity of robot i
Ω ∈ R3n (Ωi)i∈n
qi = −Rie3 thrust direction vector of robot i
Ti = −uiRie3 applied thrust vector of robot i
ri = R−1i r coordinate representation of r in frame Bi
xij = xj − xi relative displacement of robot j wrt. robot i
vij = vj − vi relative velocity of robot j wrt. robot i
Ωi ∈ R3 reference angular velocity of robot i
Ni set of neighbors of robot i
yi = (xij , vij)j∈Ni vector of relative positions and velocities available to robot i
Notation in Chapters 7 to 9
Notation Explanation
φi ∈ S1 auxiliary state of robot i
σi θi − φi
φi reference angle for φi
C path to follow
c⋆(x) orthogonal projection of x onto the line path C
τ ∈ S1 path parameter for C
(o(τ), r(τ), s(τ)) Frenet-Serret frame on C at τ ∈ S1
π : C → S1 maps a point on C to τ ∈ S1
θr : S1 → S1 angle of the tangent vector r(τ) relative to I
d11i ∈ R2 desired displacement of robot i wrt robot 1
xi
di ∈ R2 desired displacement of robot i wrt the path frame
ψi atan2(di · e2,−di · e1), the angle of di with respect to
the −r(τ) axis
w > 0 desired path following speed
c ∈ R2 circle center for path following
ρi ∈ S1 desired heading offset of robot i wrt robot 1
p ∈ R2 beacon in I
θp ∈ S1 angle of p in I
δi ∈ R2 offset vector attached to frame Biαi component of δi parallel to Rie1
βi component of δi perpendicular to Rie1
xi = xi + δi endpoint of the offset vector δi
τi = π(xi) the path parameter τ corresponding to xi ∈ C
fi((xij)j∈Ni) integrator consensus controller
gi((θij)j∈Ni, η) rotational integrator consensus controller
h(x) integrator path following controller
θh : R2 → S1 angle of vector field h relative to the inertial frame
θ′h : R2 → R derivative of θh as defined in (9.3)
θ′′h : R2 → R second derivative of θh as defined in (9.3)
ξi(σi, xi) σi − θh(xi) + ψi
xii
Acronyms
Next we present a list of acronyms for each control problem.
Problem Acronym Explanation
Rendezvous RP-F rendezvous problem for flying robots
RP-U rendezvous problem for kinematic unicycles
Γ rendezvous manifold
Formation PP parallel formation problem
Control F formation specification
Γp parallel formation manifold
Parallel PFP parallel formation flocking problem
Formation PFP-B parallel formation flocking problem with a beacon
Flocking PF parallel formation flocking specification
PFB parallel formation flocking specification with a beacon
Γpf parallel formation flocking manifold
Γpfb parallel formation flocking manifold with a beacon
Formation LPP formation line path following problem
Line Path LP formation line path following specification
Following Γlp formation line path following manifold
Circular CFP circular formation flocking problem
Formation CF circular formation flocking specification
Flocking Γcf circular formation flocking manifold
Formation CPP formation circle path following problem
Circle Path CP formation circle path following specification
Following Γcp formation circle path following manifold
General GPP general formation path following problem
Formation GP general formation path following specification
xiii
Path Following Γgp general formation path following manifold
xiv
List of Tables
2.1 Table of Notation for Kinematic Unicycle Model . . . . . . . . . . . . . . 22
2.2 Table of Notation for Flying Robots . . . . . . . . . . . . . . . . . . . . . 28
4.1 Asymptotic and Practical Stability Abbreviations . . . . . . . . . . . . . 60
4.2 Neighbors sets in Figure 4.9 . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Table of Control Primitives . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1 Simulation Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 Control Effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.1 Simulation Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 128
10.1 Data Collected from Simulations . . . . . . . . . . . . . . . . . . . . . . . 219
10.2 Base Case Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
10.3 Base Case Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 222
10.4 Simulation Results Varying µf . . . . . . . . . . . . . . . . . . . . . . . . 222
10.5 Simulation Results Varying α . . . . . . . . . . . . . . . . . . . . . . . . 224
10.6 Simulation Results Varying k . . . . . . . . . . . . . . . . . . . . . . . . 225
10.7 Simulation Results Varying the Sensing Radius . . . . . . . . . . . . . . 226
10.8 Simulation Results for the digraph in Figure 10.9 . . . . . . . . . . . . . 228
10.9 Simulation Results with Input Saturation . . . . . . . . . . . . . . . . . . 229
10.10Simulation Parameters in Section 10.2.6 . . . . . . . . . . . . . . . . . . 231
10.11Simulation Results with Perturbations and Sampling . . . . . . . . . . . 231
xv
10.12Simulation Results for Dynamic Unicycles . . . . . . . . . . . . . . . . . 233
xvi
List of Figures
1.1 Quadrotor fire inspection. The center of the formation moves along the
path in the direction indicated. . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Example of snow removal on a highway. . . . . . . . . . . . . . . . . . . . 3
2.1 Inertial and body frames in two dimensions. . . . . . . . . . . . . . . . . 16
2.2 Inertial and body frames in three dimensions. . . . . . . . . . . . . . . . 16
2.3 Kinematic unicycle class. The right figure shows a differential drive robot
that is modelled as a unicycle. . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Bicycle class. Image from (Francis and Maggiore, 2016). . . . . . . . . . 24
2.5 Flying Robot Class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Illustration of a quadrotor helicopter taken from (Roza, 2012). . . . . . . 29
3.1 Formation in terms of fixed relative displacement vectors d11i, i ∈ 2 :n. . . 34
3.2 formation line path following . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 circular formation flocking with centre of rotation c . . . . . . . . . . . . 40
3.4 The Frenet-Serret frame with origin o(τ) on the smooth Jordan curve C. 43
3.5 Formation path following. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 (left) Diamond formation that rotates rigidly with the curve C and always
points in the path’s tangent direction. (right) Diamond formation defined
with respect to the inertial frame I that does not rotate as it traverses C. 53
xvii
4.1 Illustration on the left shows stability of Γ - solutions with initial conditions
in N (Γ) remain inside Bε(Γ) for all time. Illustration on the right shows
attractivity of Γ with domain of attraction D(Γ) - solutions with initial
conditions in D(Γ) converge to Γ. . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Illustration of almost global attractivity of the set Γ. The set X\D(Γ) has
Lebesgue measure zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Illustration of almost semiglobal attractivity of the set Γ. For compact
sets K1 ⊂ 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 as the high gain k is increased. 60
4.4 Illustration of global practical stability of the set Γ. For ǫ1 > ǫ2, for any
k ≥ k⋆1 (left) and k ≥ k⋆2 > k⋆1 (right), for any initial conditions in X ,
solutions converge to the neighborhoods Bǫ1(Γ) and Bǫ2(Γ) respectively. . 61
4.5 Illustration of local stability of Γ2 near Γ1 (left) and local attractivity of
Γ2 near Γ1 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Example of local stability of Γ2 near Γ1 in Example 4.1.12. Image from (Mag-
giore, 2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.7 Illustration of the Reduction Theorem. The set Γ1 is asymptotically sta-
ble relative to Γ2 and Γ2 is asymptotically stable. By reduction, Γ1 is
asymptotically stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
xviii
4.8 Illustration of reduction for almost global asymptotic stability. The figure
on the left illustrates the dynamics on Γ2 which is globally asymptotically
stable relative to X . Γ2 contains three unstable equilibria A = z1, z2, z3
and a compact set K that is almost globally asymptotically stable relative
to Γ2. By the Reduction Thoerem the set K is also AGAS relative to X
as shown in the figure on the right. The set D(A) is Lebesgue measure
zero and contains the initial condition χ0 whose solution converges to the
point z3 ∈ A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.9 Example of a directed (left) and an undirected (right) graph. . . . . . . . 79
4.10 Example of a subgraph G ′ = (V ′, E ′) of the directed graph in Figure 4.9
with V ′ = 1, 2, 3, 5 and E ′ = (5, 1), (5, 2), (2, 3), (3, 5). . . . . . . . . . 80
4.11 Reverse directed spanning tree with root node 1. . . . . . . . . . . . . . . 81
4.12 Directed graph G containing a reverse directed spanning tree and globally
reachable node, node 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.13 Hierarchical sensing graph . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.14 Directed graph G containing a reverse directed spanning tree. The strongly
connected components G0, . . . ,G3 are boxed . . . . . . . . . . . . . . . . . 85
4.15 Condensation digraph C(G) associated with the graph G in Figure 4.14
(left) and reverse directed spanning tree contained in C(G) (right). . . . . 85
4.16 Illustration of properties B1, B2 and B3. . . . . . . . . . . . . . . . . . . 95
5.1 Block diagram of the rendezvous control system for robot i. The outer
loop assigns a desired thrust vector fi(yii). The inner loop thrust control
uses fi(yii) to assign the vehicle input ui while the rotational control uses
fi(yii) to assign the torque input τi. The vector yii contains the relative
displacements and velocities of vehicles that robot i can sense, measured
in the body frame of robot i. . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Illustration of the control input ui and reference angular velocity Ωi in (5.1).102
xix
5.3 Sensor digraph used in the simulation results. . . . . . . . . . . . . . . . 105
5.4 Rendezvous control simulation without the presence of disturbances. At
the top-left, top-right and bottom-left: positions of the five robots ex-
pressed in the inertial frame I. At the bottom-right: linear speeds ‖vi‖, i ∈
1 : 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.5 Rendezvous control simulation with the presence of disturbances. At the
top-left, top-right and bottom-left: positions of the five robots expressed
in the inertial frame I. At the bottom-right: linear speeds ‖vi‖, i ∈ 1 : 5. 107
6.1 Block diagram of the rendezvous control system for robot i. . . . . . . . 125
6.2 Illustration of the control inputs ui = u⋆i (yii) and ωi = ω⋆i (y
ii) in (6.1). . . 127
6.3 Sensor digraph used in the simulation results. . . . . . . . . . . . . . . . 128
6.4 Rendezvous control simulation for: (a) proposed feedback in (6.1), (b)
feedback in (Lin et al., 2005), and (c) feedback in (Zheng et al., 2011) . . 129
6.5 Simulation control inputs for proposed feedback in (6.1) . . . . . . . . . . 129
7.1 Representation of the offset vector δi. . . . . . . . . . . . . . . . . . . . . 151
7.2 Parallel formation where offset vectors xi meet at a common point at a
distance α in front of the formation. . . . . . . . . . . . . . . . . . . . . . 153
7.3 Block diagram of the formation control system for robot i. . . . . . . . . 154
7.4 (a) shows an example of a parallel line formation while (b) shows an ex-
ample of full synchronization, a special case of a parallel line formation. . 156
7.5 Undirected graph G under consideration in the simulation results. . . . . 157
7.6 Triangular formation specified by the offset vectors d112 = (−10, 5), d113 =
(−10,−5), d114 = (−20, 10) and d115 = (−20,−10). . . . . . . . . . . . . . 158
7.7 Simulation for a triangle formation. Initial positions are indicated with
and final positions are indicated with ×. . . . . . . . . . . . . . . . . . . 159
8.1 Undirected graph G under consideration in the simulation results. . . . . 179
xx
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 simulation
are indicated with ×. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.3 Simulation for formations with circular collective motion (a) formation
flocking, (b) formation path following around the point c = (100, 0). Initial
positions are indicated with and positions at the end of the simulation
are indicated with ×. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.4 Desired formation of unicycles (x, θ) (shaded) is achieved when (z, θ) (not
shaded) lie on a common circle of radius r with the desired spacing θ1i =
ρi(d, β1) for all i ∈ n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9.1 Illustration of the angle φi and angle σi = θi − φi. . . . . . . . . . . . . . 197
9.2 Illustration of the set Λ in which unicycle i can lie anywhere on the dashed
circle. The desired position of unicycle i is marked by ⋆. . . . . . . . . . 197
9.3 Illustration of angle condition σi = θr(τi)− ψi. . . . . . . . . . . . . . . . 198
9.4 Digraph G under consideration in the simulation results. . . . . . . . . . 202
9.5 Formation specified by the offset vectors d1 = (15, 0), d2 = (5,−8), d3 =
(5, 8), d4 = (−10,−3) and d5 = (−10, 3). . . . . . . . . . . . . . . . . . . 202
9.6 Simulation results: (a) illustrates the curve C corresponding to the stable
limit cycle of the Van der Pol oscillator with µ = 1.5, c0 = 0.04 and shows
the time evolution of x1 illustrated by the dashed line, (b) illustrates the
individual paths traversed by the five unicycles as the formation moves
around C. Initial positions are indicated with and positions at the end
of the simulation are indicated with ×. . . . . . . . . . . . . . . . . . . . 203
xxi
9.7 Conceptual illustration of the reduction sets Γ1, Γ2 and Γ3. The set Γ3 is
asymptotically stable, the compact set Γ2 is asymptotically stable relative
to Γ3 and the point Γ1 is asymptotically stable relative to Γ2. Reduction
implies the set Γ1 is asymptotically stable as illustrated by the solution
starting at χ0 off the set Γ3. . . . . . . . . . . . . . . . . . . . . . . . . . 211
10.1 Illustration of the average of two angles (a) (θ1, θ2) = (0, 3π/2) rad and
(b) (θ1, θ2) = (0, π) rad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
10.2 Illustration of the formation measure. . . . . . . . . . . . . . . . . . . . . 216
10.3 Formation measure. The unicycles at the initial time t0 are not shaded
while the unicycles at time tf are shaded. The average position of the
unicycles are indicated with . In the top figure, x(t0) and x(tf) are the
same and therefore µd = 0. In the bottom figure ‖x(tf )− x(t0)‖ 6= 0 and
therefore drift is present which is roughly twice the size of the formation
itself. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
10.4 Interaction function f(s) and g(s). . . . . . . . . . . . . . . . . . . . . . 218
10.5 Formation specified by d112 = (−10, 5), d113 = (−5,−5), d114 = (−20, 10)
and d115 = (−15,−10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
10.6 Undirected graph used in the base case. . . . . . . . . . . . . . . . . . . . 221
10.7 Illustration of configurations satisfying different values of µf . The centre of
the circles represent the desired unicycle positions while the actual unicycle
positions are indicated with ×. . . . . . . . . . . . . . . . . . . . . . . . 223
10.8 Illustration of a formation where one agent is disconnected from the rest
of the group. Regardless of this, the rest of the group achieves formation
among themselves. Initial positions are indicated with and final positions
are indicated with ×. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
10.9 Directed ring-coupled graph. . . . . . . . . . . . . . . . . . . . . . . . . . 227
10.10Typical input profile during simulation. . . . . . . . . . . . . . . . . . . . 228
xxii
10.11Input profile during simulation for µu,1 = 5 m/s and µω,1 = π/2 rad/s. . 230
10.12Simulation result in the presence of disturbances and sampling rate of 100Hz.232
10.13Simulation result in the presence of disturbances and sampling rate of 10Hz.232
10.14Simulation result for dynamic unicycles. . . . . . . . . . . . . . . . . . . 233
xxiii
Chapter 1
Introduction
The goal of this thesis is to develop control algorithms to coordinate the motion of
autonomous teams of robots in order to achieve some desired collective goal. As an
example, consider a team of autonomous quadrotor helicopters equipped with on-board
cameras given the task of monitoring a forested area during the dry season, searching for
any sign of fire. The control objective would be to make the vehicles organize an efficient
formation that follows a desired path at a desired velocity such that the entire dry area
is scanned. This could be achieved, for example, by flying in a parallel fashion between a
series of straight line segments, circling about each way-point with a desired radius and
for a desired time duration. This scenario is illustrated in Figure 1.1.
Typical coordination problems include attitude synchronization, rendezvous, flock-
ing, and formation control, and find application in many areas such as manufactur-
ing or repair tasks, sensor networks, search and rescue, forest fire control missions, or
surveillance/inspection. Depending on the situation, some tasks may be better suited
for ground-based robot teams, while others may be better suited for flying robots. A
common objective in coordination control is to make the ensemble of robots stabilize
a specific spatial configuration characterized by fixed desired inter-agent displacements.
For example, flocking birds or aircraft formations at an air show are often coordinated
1
Chapter 1. Introduction 2
Figure 1.1: Quadrotor fire inspection. The center of the formation moves along the pathin the direction indicated.
in a characteristic triangular configuration with constant inter-agent displacements. A
special case is called rendezvous when the desired inter-agent displacements all equal
zero, i.e., the robots’ positions coincide.
It is often the case that one would also like the formation to move with a desired
collective motion while maintaining the desired inter-agent spacing at all times. Two
types of collective motions that we consider in this thesis are flocking and path following.
Flocking formations have linear or circular collective motions but the roto-translation
of the line or centre of the circle are not defined a priori. Rather, they depend on the
initial conditions of the agents. The study of parallel and circular flocking for kinematic
unicycles is useful for gaining insight into how coordination seen in nature occurs with
very limited information. In particular, parallel and circular flocking arise in nature with
flocks of birds and schools of fish where no specific path is agreed on ahead of time.
On the other hand, in path following, the formation follows a specific path defined
a priori such as in the forest fire example in Figure 1.1. A key requirement in path
following is that the path be invariant. Namely, if the formation is initialized on the
path, it must stay on the path at all times. This is in contrast with a common trajectory
tracking approach, in which the formation is made to follow a reference point that moves
Chapter 1. Introduction 3
along the path according to a specific time parametrization. This latter approach is
undesirable in practice because if the formation is initialized on the path but not at the
same location as the reference point, then it could leave the path, and may collide with
off-course objects. There are many practical applications for formation path following.
For example, one may like to have formations of robots follow a series of linear and
circular path segments painted or taped on the floor of a warehouse to retrieve and stock
merchandise. Each robot is only able to measure, using on-board cameras, its relative
position and orientation to other agents as well as its displacement to the closest point
on the path (which may also be measured using on-board cameras). Another example is
a team of automatic snowploughs with the task of snow removal on highways. Typically,
the snowploughs form a diagonal line formation, as illustrated in Figure 1.2, passing snow
accumulated at each plough from one vehicle to the other until it clears to the shoulder
of the highway. In Figure 1.2 snow is being passed from unicycle 1 to unicycle 4 before
clearing the highway. Each plough only observes the vehicle to its left and right in the
process. In nature, formation circle path following is observed, for example, with fish
encircling a food source at a specific location.
Figure 1.2: Example of snow removal on a highway.
The work in this thesis solves a number of coordination problems for two important
Chapter 1. Introduction 4
classes of robots.
The first class of robots corresponds to ground-based mobile robots modelled as kine-
matic unicycles. Each robot moves in the two-dimensional plane, can rotate freely with a
desired angular speed, but its velocity is restricted to be parallel to its heading direction.
With three degrees-of-freedom and two actuators, each robot is under-actuated with
degree of under-actuation one and its configuration space is SE(2). The second class
of robots corresponds to flying robots. Each robot moves in three-dimensional space
and is endowed with an actuation mechanism producing torques about three orthogo-
nal body axes, but its thrust vector is restricted to be parallel to its thrust axis. With
six degrees-of-freedom and four actuators, each robot is under-actuated with degree of
under-actuation two and its configuration space is SE(3). A quadrotor helicopter is an
example of such a robot. Together, these two classes of robots can be employed in a
broad range of multi-agent coordination applications both on land and in the air.
One of the main difficulties in solving coordination problems for these two classes
of robots arises from nonholonomic constraints which only permit vehicles to move or
accelerate instantaneously along a single body axis.
Another difficulty in solving the coordination problems arises from the constraints
that we impose on sensing and the sharing of information. For kinematic unicycles, each
agent is permitted only to sense its heading angle and position relative to a subset of
neighbors, measured in its own body frame. From a practical standpoint, this infor-
mation can be sensed locally by each agent using a fixed on-board camera. We do not
permit the unicycles to have access to global information like the group centre of mass or
centralized information that is broadcast to the group from a central station. Moreover,
the agents cannot communicate with each other, in the sense of requiring an on-board
communication system using transmitters and receivers to send and receive electromag-
netic signals. This important distinction between sensing and communication will be
used throughout this work. A feedback meeting these sensing requirements is called lo-
Chapter 1. Introduction 5
cal and distributed, and from a practical standpoint it has the beneficial property that
it can be implemented on each unicycle using information sensed locally from a fixed
on-board camera. In the case of flying robots, each agent typically mounts a camera and
an inertial measurement unit (IMU) that includes a three-axis rate-gyroscope, so that
the robot is able to sense, in the coordinates of its own frame, the relative displacements
and velocities of nearby vehicles, and its own angular velocity. Robots can neither access
their own inertial position and velocity, nor their own attitude.
In this work, as is standard in the literature, the sensing between robots is modelled
as a graph, and is assumed to be static, meaning that the neighbors of each vehicle
remain fixed at all time. In most of the problems considered, unless stated otherwise,
the sensing digraph is assumed to be any digraph containing a globally reachable node.
This is a necessary connectivity condition to solve multi-robot coordination problems for
fixed graphs. The assumption of fixed sensing graphs is questionable in practice, but
is made to render the problem mathematically treatable. In a more realistic situation,
the sensing would be state-dependent, and each unicycle would only sense neighboring
unicycles lying within a given radius of itself. This remains a challenging open problem
for much simpler classes of robot models, such as double-integrators. While it will not be
formally proven that the results in this thesis work for state-dependent graphs, we will
present extensive simulation results to show that our solutions also work considerably
well for state-dependent graphs when the sensing radius is sufficiently large. We will also
run extensive simulation results to illustrate robustness of the approach to unmodelled
effects including sensor noise, input noise and collisions.
Problems will be defined in terms of two specifications. The first specification cor-
responds to the desired formation of the multi-agent system that is stabilized using a
local and distributed control. A second specification enforces a desired final motion of
the ensemble and, in the case of path following, may use cameras or a GPS system to
acquire information about the path. An advantage of this setup is that it is robust to
Chapter 1. Introduction 6
losses of the GPS signal, in which case the robots will maintain the formation and the
path control can resume once the signal is recovered.
Based on the discussion above, at the most basic level, the aim of this work is
to present solutions to multi-agent coordination problems for classes of nonholonomic
agents in a broad spectrum of ground and air applications. This area has been actively
researched in the last decade. Much of the current research results, however, suffer in
that they apply ad-hoc methodologies and solutions with local domains of operation.
The main contribution and most difficult aspect of this research is its strong emphasis
on enforcing local and distributed feedbacks. Except for problems with a path follow-
ing element, there is no reliance on external landmarks or measurement of a common
inertial frame. The feedbacks are completely independent of time, and do not require a
communication system, thus avoiding the need for synchronized clocks and eliminating
issues surrounding communication delays that ultimately limit performance. This also
makes the solutions presented in this work cost-effective in applications. Moreover, the
feedback expressions that we propose are compact, which makes them very efficient to
compute in real-time on-board each robot. The feedbacks are also highly compartmen-
talized due to the fact that they are constructed out of simpler control primitives such as
consensus controllers for single or double integrators and rotational integrators, making
them easy to interpret. Another important characteristic of the solutions presented in
this thesis is that they are, in most cases, almost global or semi-global, meaning that the
collective motions will be achieved for large sets of initial conditions. Finally, the control
solutions presented in this work are robust against the loss of agents. As long as the
graph maintains a globally reachable node after the loss, the control strategies will still
work. This is in contrast to leader-follower based approaches where the loss of the leader
robot compromises the correct functioning of the control strategies.
Based on these considerations, our solutions very much emulate the distributed co-
ordination behaviour seen in nature by animals, for example, in a herd or flock. These
Chapter 1. Introduction 7
animals are able to accomplish incredible feats of coordination just by sight and by sensing
the type of information that an IMU unit can provide such as angular velocities. These
formations in nature often arise without requiring much communication beyond sight (al-
though some vocalization may occur) or the sensing of surrounding landmarks (although
it could certainly be influenced by the presence of a food source) and the animals cer-
tainly do not need to measure time. Therefore, there is also an important philosophical
element at the core of this work as it tries to understand and make connections with
coordination phenomena in nature.
Each result in this thesis has had a contribution to the current literature and most
of the results have been published in scholarly journals as discussed in Section 1.3.
1.1 Coordination Problems Investigated in This The-
sis
In this section, we briefly outline the multi-agent coordination problems that will be
studied in this thesis. A common goal in all of the control problems is to stabilize a
pre-specified configuration and, in addition, to optionally enforce a final collective mo-
tion with domain of attraction as large as possible. The difference between rendezvous
and formation control is that with rendezvous, the desired configuration corresponds
to the case in which all inter-agent displacements are zero and relative heading angles
are unconstrained. For the formation problems, on the other hand, the inter-agent dis-
placements are non-zero and relative heading angles will be constrained. The difference
between stopping formations and those with a final collective motion is that, with the
former, the configuration does not translate or rotate in the inertial frame, while with
the latter, it does.
P1. Rendezvous of flying robots (Chapter 5). The objective of the rendezvous
problem for flying robots is to design smooth, local and distributed feedbacks for
Chapter 1. Introduction 8
each flying robot so as to drive the group to an arbitrarily small neighborhood of
one another from arbitrary initial conditions.
P2. Rendezvous of kinematic unicycles (Chapter 6). The objective of the ren-
dezvous problem for kinematic unicycles is to design smooth, local and distributed
feedbacks for each kinematic unicycle so as to drive the group to a common location
from arbitrary initial conditions.
P3. Formations of kinematic unicycles (Chapter 7). The objective of the for-
mation control problem for kinematic unicycles is to design smooth, local and dis-
tributed feedbacks for each kinematic unicycle so as to drive the ensemble, for
almost all initial conditions in any given compact set, to a parallel formation, i.e.,
one in which the unicycles’ headings are all parallel, and their relative displacement
vectors take on appropriate values corresponding to a desired geometric pattern.
Moreover, it is required that the unicycles come to a stop when they meet the
formation requirement.
The utility of stopping formations is manifest in problems where vehicles are re-
quired to distribute themselves over a terrain in order, for instance, to form an
antenna array or a sensor network. More generally, the problem investigated in
this chapter is a conceptual gateway to the problem of inducing collective motions
in formations.
P4. Formations of kinematic unicycles with final linear and circular col-
lective motions (Chapter 8). The objective of the formation control problem
with final linear and circular collective motions is to design smooth feedbacks for
each kinematic unicycle so as to drive the ensemble, for almost all initial condi-
tions, to a formation in which relative displacement vectors take on appropriate
values corresponding to a desired geometric pattern. Moreover, it is required that
the unicycles move with a final collective parallel or circular motion when they
Chapter 1. Introduction 9
meet the formation requirement. For formation flocking we require the feedbacks
be local and distributed. For parallel formation flocking with a beacon, unicycles
are permitted also to measure a common beacon in body frame corresponding to
the flocking direction. For formation path following unicycles are permitted also to
compute their perpendicular projection onto the desired path.
P5. General formation path following for kinematic unicycles (Chapter 9).
The objective of the general formation path following problem is to design smooth
feedbacks for each kinematic unicycle so as to asymptotically stabilize a desired
formation defined with respect to the Frenet-Serret frame on a smooth Jordan
curve, possibly having high curvature. Moreover, it is required that the formation
follows the Jordan curve with a desired speed, rotating rigidly with respect to the
curve. We relax the sensor requirements, allowing for some minimal communication
and GPS measurements, due to the complexity of the problem.
1.2 Thesis organization
The main body of this thesis is organized into the following chapters.
• Chapter 2: Modelling
In this chapter we present models for kinematic unicycles and flying robots.
• Chapter 3: Coordination Problems
In this chapter we will introduce the control problems investigated in this thesis
for teams of flying robots and kinematic unicycles.
• Chapter 4: Preliminaries
This chapter contains fundamental definitions and results that will be applied on
a regular basis throughout this thesis.
Chapter 1. Introduction 10
• Chapter 5: Rendezvous of Flying Robots by means of Local and Distributed Feed-
backs
This chapter presents a solution to P1.
• Chapter 6: Rendezvous of Kinematic Unicycles by means of Local and Distributed
Feedbacks
This chapter presents a solution to P2.
• Chapter 7: Formations of Kinematic Unicycles
This chapter presents a solution to P3.
• Chapter 8: Formations of Kinematic Unicycles with Parallel and Circular Collective
Motions
In this chapter, we present almost global solutions for four sub-problems of P4:
parallel formation flocking with and without a beacon (PFP-B, PFP), circular for-
mation flocking (CFP), formation line path following (LPP) and formation circle
path following (CPP).
• Chapter 9: General Formation Path Following of Kinematic Unicycles
This chapter presents a solution to P5.
• Chapter 10: Unicycle Formation Simulation Trials
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
Chapter 3. Coordination Problems 35
this thesis. Section 3.3.1 discusses formation control, Section 3.3.2 discusses formations
with parallel collective motion, Section 3.3.3 discusses formations with circular collective
motion, and finally Section 3.3.4 discusses formation path following of Jordan curves.
3.3.1 Formation Control
In this section, we define the formation control problem. The corresponding solution to
this problem is presented in Chapter 7.
Let d ∈ R2(n−1) be a desired formation and without loss of generality, choose unicycle
1 to be at the front of the formation so that d11i · e1 ≤ 0 for all i ∈ 2 :n. Let
F := d ∈ R2(n−1) : d11i · e1 ≤ 0, i ∈ 2 :n
be the corresponding set of all formations of n unicycles. For any d ∈ F, the objective
of the formation control problem is to design local and distributed feedbacks to drive a
group of unicycles to the parallel formation manifold,
Γp := (x, θ) ∈ Γ : θi = θ1, i ∈ 2 :n . (3.4)
The parallel formation manifold Γp is the subset of the state space in which the unicycles
have parallel headings, and their relative displacements meet the formation specification.
This type of formation is referred to as a parallel formation, i.e., formations in which the
unicycles’ headings are parallel to each other: θij = 0 for all i, j ∈ n. The formation
illustrated in Figure 3.1 is an example of a parallel formation. The Parallel Formation
Problem (PP) is stated as follows.
Problem 3 (Parallel Formation Problem (PP)). Consider the system of kinematic uni-
cycles in (2.6), (2.7) and a connected, undirected sensor graph G. For any choice of
d ∈ F, design local and distributed feedbacks (u⋆i , ω⋆i ) : (yii, ϕi) 7→ (ui, ωi) for all i ∈ n
Chapter 3. Coordination Problems 36
to almost semi-globally asymptotically stabilize the formation manifold Γp under the con-
straint (u⋆i , ω⋆i )|Γp
= (0, 0).
The requirement (u⋆i , ω⋆i )|p = (0, 0) means that the unicycles do not move when
they are in formation. The problem of full synchronization in which both positions and
headings synchronize for all unicycles is a special case of PP in which d = 0.
3.3.2 Formation with Final Parallel Collective Motion
Now we introduce two formation control problems with final parallel collective motion:
parallel flocking and line path following. For flocking, the formation moves along a
straight line whose roto-translation with respect to the inertial frame is not defined a
priori. The specific line followed depends on the initial conditions of the agents. On the
other hand, in line following, the formation follows a specific line path defined a priori in
R2. The solutions to these problems are presented in Chapter 8.
Parallel Formation Flocking
Let d ∈ F be a desired formation and w > 0 be a desired flocking speed. The 2-tuple
(d, w) is a parallel formation flocking specification, and
PF := (d, w) ∈ F× R : w > 0
represents the set of all parallel flocking formations of n unicycles. For any (d, w) ∈ PF,
the objective of the formation control problem is to design local and distributed feedbacks
to drive a group of unicycles to a desired formation corresponding to the parallel formation
flocking manifold,
Γpf := Γp (3.5)
which coincides with the parallel formation manifold in Section 3.3.1. However, unlike PP
where the formation is constrained to stop on the set Γp, for parallel formation flocking
Chapter 3. Coordination Problems 37
they are required to move at the desired steady-state speed w. Formally, the Parallel
Formation Flocking Problem (PFP) is stated as follows.
Problem 4 (Parallel Formation Flocking Problem (PFP)). Consider the system of kine-
matic unicycles in (2.6), (2.7) and sensor digraph G containing a globally reachable node.
For any choice of (d, w) ∈ PF, design local and distributed feedbacks (u⋆i , ω⋆i ) : (y
ii, ϕi) 7→
(ui, ωi) for all i ∈ n to almost globally asymptotically stabilize Γpf , under the constraint
(u⋆i , ω⋆i )|Γpf
= (w, 0).
Alternatively, suppose that each unicycle is permitted to measure, with respect to its
own body frame, a common inertial vector p of unit norm. That is, each unicycle can
measure the vector pi = R−1i p. The vector p is referred to as a beacon and specifies a
desired flocking direction. Let θp ∈ S1 be its angle with respect to the inertial frame.
This is not a local and distributed quantity as the beacon p is defined in the inertial
frame. The 3-tuple (d, p, w), is a parallel formation flocking specification with a beacon,
and
PFB := (d, p, w) ∈ F× R2 × R : ‖p‖ = 1, w > 0
represents the set of all parallel flocking formations of n unicycles with a beacon. The
set Γpf is replaced with the parallel formation flocking manifold with a beacon,
Γpfb := (x, θ) ∈ Γ : θi = θp, i ∈ n (3.6)
in which not only do the unicycles have the same heading angle, therefore constituting
a parallel formation, but the heading coincides with the angle of the beacon θp. The
Parallel Formation Flocking Problem with a Beacon (PFP-B) is stated as follows.
Problem 5 (Parallel Formation Flocking Problem with a Beacon (PFP-B)). Consider
the system of kinematic unicycles in (2.6), (2.7) and sensor digraph G containing a glob-
ally reachable node. For any (d, p, w) ∈ PFB, design control inputs (u⋆i , ω⋆i ) : (y
ii, ϕi, p
i) 7→
Chapter 3. Coordination Problems 38
(ui, ωi) for all i ∈ n, that almost globally asymptotically stabilize the set Γpfb under the
constraint (u⋆i , ω⋆i )|Γpfb
= (w, 0).
PFP-B will be solved for digraphs containing a globally reachable node, while PFP will
be solved for the specific class of hierarchical digraphs. Simulation results in Section 8.3
suggest that this local and distributed solution may also apply to more general graph
topologies.
Formation Line Path Following
Let d ∈ F be a desired formation and C(r0, p) = x ∈ R2 : x = r0 + sp, s ∈ R ⊂ R2
be a line to be followed by one unicycle in the ensemble (without loss of generalization,
unicycle 1) where r0, p ∈ R2, and p is a unit vector pointing tangent to C(r0, p). Let
w > 0 be a desired path following speed. Assume that unicycle 1 is at the front of the
formation so that d11i · e1 ≤ 0 for all i ∈ 2 :n. The 4-tuple (d, r0, p, w) is a formation line
path following specification, and
LP := (d, r0, p, w) ∈ F× R2 × R
2 × R : ‖p‖ = 1, w > 0
represents the set of all line following formations of n unicycles. The control inputs (ui, ωi)
of unicycle i will be chosen as functions of the local and distributed quantities (yii, ϕi)
and the direction of the path pi measured in body frame. The feedback for unicycle i
will also depend on the displacement vector between xi and its orthogonal projection
c⋆(xi) in C(r0, p), measured in body frame. This quantity is given by (c⋆(xi) − xi)i =
(r0 − xi)i − ((r0 − xi)
i · pi)pi. For any (d, r0, p, w) ∈ LP, the objective of the formation
control problem is to design such feedbacks in order to drive a group of unicycles to the
formation line path following manifold,
Γlp := (x, θ) ∈ Γpfb : x1 ∈ C(r0, p) (3.7)
Chapter 3. Coordination Problems 39
which coincides with Γpfb with the additional requirement that unicycle 1 lies on the path
C. An example of a configuration of unicycles in Γlp is illustrated in Figure 3.2. The
Formation Line Path Following Problem (LPP) is stated as follows.
Problem 6 (Formation Line Path Following Problem (LPP)). Consider the system of
kinematic unicycles in (2.6), (2.7) and sensor digraph G containing a globally reachable
node. For any (d, r0, p, w) ∈ LP, design feedbacks (u⋆i , ω⋆i ) : (yii, ϕi, p
i, (c⋆(xi) − xi)i) 7→
(ui, ωi) for all i ∈ n to almost globally asymptotically stabilize Γlp under the constraint
(u⋆i , ω⋆i )|Γlp
= (w, 0).
direction ofmotion
Figure 3.2: formation line path following
3.3.3 Formation with Circular Collective Motion
In this section, we state two formation control problems with final circular collective
motion: circular flocking and circle path following. For flocking, the formation moves
around a circle of desired radius whose center is not defined a priori. Rather, it depends
on the initial conditions of the agents. On the other hand, in circle path following, the
formation follows a specific circle path whose center is defined a priori. The solutions
Chapter 3. Coordination Problems 40
to these problems are presented along with the linear collective motion problems in
Chapter 8.
Circular Formation Flocking
The objective of the circular formation flocking control problem is to design local and
distributed feedbacks (ui, ωi) for all i ∈ n to make a group of unicycles converge to
a desired formation d ∈ R2(n−1) that encircles, with angular speed w > 0, a center
point c ∈ R2 which is not defined a priori and depends on the initial configuration of
the unicycles. For one unicycle in the ensemble (without loss of generality, unicycle 1)
specify, a priori, the radius of rotation β1. The 3-tuple (d, β1, w) is a circular formation
flocking specification, and
CF := (d, β1, w) ∈ R2(n−1) × R× R : β1 > 0, w > 0
represents the set of all circular flocking formations of n unicycles.
Figure 3.3: circular formation flocking with centre of rotation c
Chapter 3. Coordination Problems 41
When the desired formation d is achieved, each unicycle i ∈ 2 :n must move along a
concentric circular orbit of radius βi =√
(d11i · e1)2 + (β1 − d11i · e2)2 with angular speed
w. The radii βi, i ∈ n are illustrated in Figure 3.3. Unlike in the case of parallel
flocking, where all unicycle headings converge to a common direction, in the case of
circular formation flocking, the heading directions will not align in general. Instead,
the heading of unicycle i is offset relative to the heading of unicycle 1 by the angle
ρi(d, β1) = atan2(d11i · e1, β1 − d11i · e2) illustrated in Figure 3.3 where ρ2 = −ρ4 and
ρ1 = ρ3 = 0.
Inspired by (El-Hawwary and Maggiore, 2013a), define the offset vector δi = βiRie2
for each unicycle i rigidly attached to its body frame and perpendicular to the heading
direction Rie1. The vector xi = xi + δi represents the perceived centre of a circle of
radius βi that unicycle i would trace out if its translational speed were wβi and its
angular speed were w. On the set Λ := (x, θ) ∈ R2n × Tn : x1i = 0, i ∈ n unicycles
lie on concentric circles of desired radii (βi)i∈n about a common centre. Relative to this
set, the desired formation is achieved if, in addition, θ1i = ρi(d, β1) for all i ∈ 2 :n. The
circular formation flocking manifold is therefore given by,
Γcf := (x, θ) ∈ Λ : θ1i = ρi(d, β1), i ∈ 2 :n . (3.8)
The Circular Formation Flocking Problem (CFP) is stated as follows.
Problem 7 (Circular Formation Flocking Problem (CFP)). Consider the system of
kinematic unicycles in (2.6), (2.7) and a connected, undirected sensor graph G. For any
(d, β1, w) ∈ CF, design local and distributed feedbacks (u⋆i , ω⋆i ) : (y
ii, ϕi) 7→ (ui, ωi) for all
i ∈ n to almost globally asymptotically asymptotically stabilize Γcf under the constraint
(u⋆i , ω⋆i )|Γcf
= (wβi, w).
Chapter 3. Coordination Problems 42
Formation Circle Path Following
For the formation circle path following problem, the desired formation d ∈ R2(n−1) follows
a particular circular orbit with centre c ∈ R2 defined a priori. The 4-tuple (d, β1, c, w) is
a formation circle path following specification, and
CP := (d, β1, c, w) ∈ R2(n−1) × R× R
2 × R : β1 > 0, w > 0
represents the set of all circle following formations of n unicycles. The control inputs
(ui, ωi) of unicycle i are chosen as functions of the local and distributed quantities (yii, ϕi)
and the centre of the desired circle c relative to xi measured in body frame, i.e., (c−xi)i.
The formation circle path following manifold is defined as
Γcp := (x, θ) ∈ Γcf : x1 = c (3.9)
where the only difference from Γcf is that the perceived centers xi = xi + δi not only
coincide, but they coincide at the desired circle center c. The Formation Circle Path
Following Problem (CPP) is stated as follows.
Problem 8 (Formation Circle Path Following Problem (CPP)). Consider the system of
kinematic unicycles in (2.6), (2.7) and a connected, undirected sensor graph G. For any
(d, β1, c, w) ∈ CP, design feedbacks (u⋆i , ω⋆i ) : (y
ii, ϕi, (c− xi)
i) 7→ (ui, ωi) for all i ∈ n that
almost globally asymptotically stabilize the formation circle path following manifold Γcp
under the constraint (u⋆i , ω⋆i )|Γcp
= (wβi, w).
3.3.4 General Formation Path Following
The formation control problems introduced up to this point have considered either forma-
tions that stop or have final parallel or circular collective motions. These basic motions
can be combined to address path following problems where a formation of unicycles
Chapter 3. Coordination Problems 43
travels between a series of way-points along straight lines and, in turn, stops at each
way-point or encircles each way-point with a desired radius. However, such a scheme
is not appropriate, for example, in a situation where the formation must follow a more
general path with high curvature such as a winding road to deliver packages.
To allow for such situations, the goal of the general formation path following problem
that will be introduced in this section and solved in Chapter 9 is to develop a controller to
make unicycles obtain formation about a desired smooth Jordan curve C ∈ J , where J is
the set of smooth Jordan curves in R2, and follow it at a desired speed w > 0. The set C,
diffeomorphic to S1, has a regular parametrization o : S1 → C, o(τ) where we will refer to
τ as the path parameter. Let π : C → S1 be the inverse of o(τ) that maps a point in C to a
point τ ∈ S1. For every τ ∈ S1, there is a Frenet-Serret frame (o(τ), r(τ), s(τ)) where o(τ)
is the origin of the frame, r(τ) = (d/dτ)o(τ)/‖(d/dτ)o(τ)‖ is the tangent vector to the
curve at τ pointing in the desired direction of motion and s(τ) is the counter clockwise
rotation of r by π/2 rad. Define the rotation matrix with columns corresponding to the
axes of the Frenet-Serret frame at τ ∈ S1 given by R0(τ) = [r(τ) s(τ)]. The Frenet-Serret
frame is illustrated in Figure 3.4.
Figure 3.4: The Frenet-Serret frame with origin o(τ) on the smooth Jordan curve C.
For general formation path following we will no longer define formations with the
Chapter 3. Coordination Problems 44
collection of fixed offsets (d11i)i∈n relative to the body frame of unicycle 1. Instead the
formation will be defined relative to the Frenet-Serret frame on C. In particular, the
unicycles are said to achieve formation at τ ∈ S1 if each unicycle i ∈ n lies at a fixed
desired displacement di ∈ R2 relative to the Frenet-Serret frame (o(τ), r(τ), s(τ)), i.e.,
R0(τ)−1(xi− o(τ)) = di. The corresponding point about which the formation is achieved
o(τ) = π−1(τ) ∈ C is referred to as the formation origin. This is illustrated in Figure 3.5
for a rectangular formation composed of four unicycles labelled x1, . . . , x4. Denote the
desired, fixed angle of di with respect to the −r(τ) axis by ψi = atan2(di · e2,−di · e1).
We make the assumption that di · e1 6= 0 for all i ∈ n, i.e., in formation, no unicycle lies
on the s(τ) axis corresponding to ψi = ±π/2. The angles ψ1, ψ2 are illustrated for the
example in Figure 3.5.
formationorigin
Figure 3.5: Formation path following.
We collect the above relative displacements in a vector d := (di)i∈n ∈ R2n. A forma-
tion specification d is defined rigidly with respect to the path’s frame of reference and is
not invariant to translations and rotations. The 3-tuple (d, C, w) is a general formation
path following specification, and
GP := (d, C, w) ∈ R2n × J × R : di · e1 6= 0, w > 0, i ∈ n
Chapter 3. Coordination Problems 45
represents the set of all general path following formations of n unicycles. In the general
formation path following problem, each unicycle stores on-board an additional auxiliary
state φi ∈ S1 with dynamics
φi = ωi ,piv , i ∈ n, (3.10)
where ωi ,piv denotes its angular speed. Let φ := (φi)i∈n ∈ Tn be the collection of the
auxiliary states of all unicycles. A more detailed description of (3.10) will be given in
Chapter 9. For any (d, C, w) ∈ GP, define the general formation path following manifold
as
Γgp :=
(x, θ, φ) ∈ R2n × T
n × Tn : (∃τ ∈ S
1)
R0(τ)−1(xi − o(τ)) = di, φi = φi(xi(xi, θi, φi), θi − φi), i ∈ n
,
(3.11)
where φi(xi(xi, θi, φi), θi − φi) is a desired angle for φi, to be designed in Chapter 9. In
Γgp, there exists τ ∈ S1 such that the unicycles have achieved formation at τ . For any
(d, C, w) ∈ GP, the objective of the General Formation Path Following Problem (GPP)
is to develop feedbacks that asymptotically stabilize a subset of Γgp and such that the
formation origin o(τ), moves at the desired speed w along C in the direction of r(τ).
Problem 9 (General Formation Path Following Problem (GPP)). Consider the system
of kinematic unicycles in (2.6), (2.7) with auxiliary state in (3.10) and sensor digraph G
containing a globally reachable node. For any (d, C, w) ∈ GP, design distributed feedbacks
(u⋆i , ω⋆i , ω
⋆i ,piv) : (xi, θi, φi, (xj, θj , φj)j∈Ni) 7→ (ui, ωi, ωi ,piv)
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
Chapter 4. Preliminaries 79
Table 4.2: Neighbors sets in Figure 4.9
Node i Ni-Directed Graph Ni-Undirected Graph
1 2 2, 52 3 1, 3, 53 4, 5 2, 4, 54 5 3, 55 1, 2, 3 1, 2, 3, 4
refer the reader to (Godsil and Royle, 2001).
4.2.1 Basic Definitions in Graph Theory
We denote a graph by the pair G = (V, E), where V is a set of nodes labelled as 1, . . . , n
and E = (i, j) ∈ V ×V : i connects to j is the set of edges such that (i, j) ∈ E if node i
connects to node j. A directed graph or digraph (undirected graph), is a graph where the
edges are ordered (unordered) pairs in which (i, j) 6= (j, i) ((i, j) = (j, i)). An example of
an undirected and directed graph are illustrated in Figure 4.9. Edges in a directed graph
are indicated with arrows whereas edges in an undirected graph are indicated with lines.
The set of neighbors of node i is the set Ni := j ∈ V : (i, j) ∈ E. The neighbor sets in
Figure 4.9 are listed in Table 4.2:
1
2
3
5
4
1
2
3
5
4
Figure 4.9: Example of a directed (left) and an undirected (right) graph.
Given positive numbers aij > 0, i ∈ n, j ∈ Ni, the associated weighted Laplacian
matrix of G is the matrix L = D − A, where the i-th diagonal entry of D is the sum∑
j∈Niaij , and A is the matrix whose element Aij is aij if j ∈ Ni, and 0 otherwise. The
Chapter 4. Preliminaries 80
Laplacian matrix for the directed and undirected graphs in Figure 4.9 for unitary weights,
i.e., aij = 1 for all j ∈ Ni, are given by
L =
1 −1 0 0 0
0 1 −1 0 0
0 0 2 −1 −1
0 0 0 1 −1
−1 −1 −1 0 3
L =
2 −1 0 0 −1
−1 3 −1 0 −1
0 −1 3 −1 −1
0 0 −1 2 −1
−1 −1 −1 −1 4
respectively.
A graph G ′ = (V ′, E ′) is a subgraph of G if V ′ ⊂ V and E ′ ⊂ E . An example of a
subgraph of the directed graph G in Figure 4.9 is shown in Figure 4.10.
1
2
3
5
Figure 4.10: Example of a subgraph G ′ = (V ′, E ′) of the directed graph in Figure 4.9 withV ′ = 1, 2, 3, 5 and E ′ = (5, 1), (5, 2), (2, 3), (3, 5).
A graph G is called static if it is constant for all time and time varying, denoted G(t),
if the set of nodes and/or edges varies with time t.
4.2.2 Classes of Graphs
In this section, we present several important classes of graphs. An undirected graph
G = (V, E) is said to be connected if for any two nodes i, j ∈ V there exists a path from
node i to node j by traversing the edges in the graph. A complete graph is an undirected
graph with an edge between every pair of nodes, that is, for any i, j ∈ V, (i, j) ∈ E . In
Chapter 4. Preliminaries 81
the case of directed graphs, a graph is strongly connected if for any two nodes i, j ∈ V
there exists a path from i to j. The undirected graph in Figure (4.9) is connected while
the directed graph is strongly connected. By removing the edge (5, 1), the directed graph
in Figure (4.9) is no longer strongly connected.
A directed spanning tree is a digraph consisting of n− 1 edges such that there exists
a unique directed path from a node, called the root, to every other node. A reverse
directed spanning tree is a graph which becomes a directed spanning tree by reversing the
directions of all its edges. We identify the root of a reverse spanning tree with the root of
its associated spanning tree. Therefore, in a reverse directed spanning tree the root node
can be sensed indirectly by all other nodes in the graph by means of directed paths. In
this sense, the root node can propagate information indirectly through the entire graph.
An example of a reverse directed spanning tree is illustrated in Figure 4.11.
5 6
7 8 9
32
1
4
Figure 4.11: Reverse directed spanning tree with root node 1.
A digraph G contains a reverse directed spanning tree if it has a subgraph G ′ = (V ′, E ′)
satisfying V ′ = V which is a reverse directed spanning tree. A node is globally reachable
if there exists a path from any other node to it. For a digraph G, existence of a globally
reachable node is equivalent to having a directed spanning tree in the reverse graph. An
example of such a digraph is illustrated in Figure 4.12 where node 1 is a globally reachable
Chapter 4. Preliminaries 82
node. Strongly connected graphs necessarily contain a globally reachable node. In fact,
every node is globally reachable by definition.
1 2
3 4 5
6
78
9
10
11 12
Figure 4.12: Directed graph G containing a reverse directed spanning tree and globallyreachable node, node 1.
If a digraph does not contain a reverse directed spanning tree, then there does not
exist any node in the graph that can propagate information to the entire network. This
can be interpreted as a lack of connectivity in the graph. In fact, Proposition 4.2.1 relates
the presence of a globally reachable node directly to the eigenvalues of the corresponding
weighted Laplacian.
Proposition 4.2.1 ((Ren and Beard, 2005; Lin et al., 2005)). The following conditions
are equivalent for a digraph G:
(i) G contains a globally reachable node.
(ii) For any set of positive gains aij > 0, i, j ∈ 1, . . . , n the associated weighted
Laplacian matrix L of G is positive semi-definite, has rank n − 1, and KerL =
span1.
A hierarchical digraph, defined in Definition 4.2.2, is a class of digraphs that contains
a globally reachable node.
Chapter 4. Preliminaries 83
Definition 4.2.2. A digraph G with n nodes containing a globally reachable node is
called a hierarchical digraph with m ∈ N layers if there exists a partition of V, P =
Lℓℓ∈1,...,m into m nonempty subsets called layers such that for all ℓ ∈ 1, . . . , m and
for all i ∈ Lℓ, if (i, j) ∈ E is an edge from node i to node j then j ∈ ⋃k∈1,...,ℓ−1 Lk.
In a hierarchical digraph G, the nodes are divided into layers in which layer 1 corre-
sponds to a single root. Nodes in layer i have neighbors only in layers j < i as illustrated
in Figure 4.13. As such, the root node is globally reachable and has no neighbors. As
an example, this type of graph can model the behaviour seen in flocking birds that only
observe other birds in front of themselves in a flocking formation.
Figure 4.13: Hierarchical sensing graph
4.2.3 Graph Decomposition
In this section it will be shown how a digraph containing a globally reachable node can
be decomposed into so-called strongly connected components. The strongly connected
components can be treated as nodes in a higher level graph known as a condensation
digraph. We now give a number of formal definitions.
A set of nodes S ⊂ V is an isolated component if it has no outgoing edges, i.e.,
for any edge (i, j) ∈ E , if i ∈ S then j ∈ S. A subgraph G ′ is an induced subgraph
Chapter 4. Preliminaries 84
of G if for any two vertices i, j ∈ V ′, (i, j) ∈ E ′ if (i, j) ∈ E . A strongly connected
component G ′ of G is a maximal strongly connected induced subgraph of G. In other
words, there does not exist any other strongly connected induced subgraph of G containing
G ′. Letting G0 = (V0, E0), . . . ,Gr = (Vr, Er) be the strongly connected components of G,
the condensation digraph of G, denoted C(G) = (VC(G), EC(G)), is defined as follows. The
vertex set VC(G) is the set of nodes vii∈0,...,r where the node vi is a contraction of the
vertex set Vi of the i-th strongly connected component Gi. The edge set EC(G) contains
an edge (vi, vj) if there exist vertices i′ ∈ Vi and j′ ∈ Vj such that (i′, j′) ∈ E . The
following properties of the condensation digraph are found in (Hatanaka et al., 2015).
Proposition 4.2.3 ((Hatanaka et al., 2015)). Consider a digraph G containing a globally
reachable node. The condensation C(G) satisfies the following properties:
(i) C(G) is acyclic, i.e., there is no path in C(G) beginning and ending at the same
node.
(ii) C(G) contains a reverse directed spanning tree T with a unique root v0 ∈ VC(G).
(iii) There exists at least one vertex vi ∈ VC(G) such that v0 is the only neighbor of vi.
An example of a digraph G containing a reverse directed spanning tree is shown
in Figure 4.14. The strongly connected components are boxed. The resulting acyclic
condensation digraph C(G) is shown in Figure 4.15. The vertex v0 in the figure is the
unique root of the reverse directed spanning tree in C(G).
As in (Hatanaka et al., 2015), we define the vertex set Lj ⊂ V to be the union
of those vertex sets Vi that correspond to vertices vi in the condensation digraph with
the property that the maximal path length from vi to the root v0 is equal to j. By
this definition, L0 := V0. We let L−1 := ∅. Defining the vertex set Lj := ∪ji=0Li, by
construction, the neighbors of any vertex in Lj are contained in Lj−1. Therefore each
Chapter 4. Preliminaries 85
1 2
3 4 5
6
78
9
10
11 12
G0
G3
G1
G2
Figure 4.14: Directed graph G containing a reverse directed spanning tree. The stronglyconnected components G0, . . . ,G3 are boxed
v0
v1
v2
v3
v0
v1
v2
v3
Figure 4.15: Condensation digraph C(G) associated with the graph G in Figure 4.14 (left)and reverse directed spanning tree contained in C(G) (right).
node set Lj is isolated. For the example in Figure 4.15, we have L0 = 1, 2, 3, 4, 5, 6,
L1 = 10 ∪ 11, 12 and L2 = 7, 8, 9.
4.3 Control Primitives
The control solutions presented in each section of this thesis will combine one or more
of the control primitives presented in this section. In the case of kinematic unicycles,
the control primitives are based on two kinematic models in two dimensions: single
integrators in R2 and rotational integrators in SO(2). In the case of flying robots, the
control primitives are based on double integrators in R3 and rotating bodies in SO(3).
Chapter 4. Preliminaries 86
The single and double integrators evolve in Euclidean space and represent positions
whereas the rotational integrators in SO(2) and rotating bodies in SO(3) represent angular
quantities. The control primitives considered in this work are listed in Table 4.3.
In the consensus problems, the goal is to globally asymptotically stabilize the set
where states of all agents coincide, be they translational or rotational. In this problem,
each agent can only sense a subset of neighboring agents. Throughout this section, the
sensing convention is defined by a sensor graph G = (V, E) discussed in Section 4.2, where
each node in the set V represents an agent, and an edge in the edge set E between node
i and node j indicates that agent i can sense agent j. Given a node i ∈ V, recall that its
set of neighbours Ni represents the set of agents that agent i can sense.
In the integrator path following problem, the goal is to drive a single integrator to a
smooth path in R2 and follow it with the desired speed and direction.
In the problems of rotational integrator and rotating body equilibrium stabilization,
the goal is to almost globally asymptotically stabilize an equilibrium in SO(2) and SO(3)
respectively.
While single and double integrators have a state-space which is a non-compact man-
ifold, rotational integrators and rotating bodies evolve on compact manifolds without
boundary. This fact leads to a topological obstruction in which global asymptotic con-
sensus or equilibrium stabilization cannot be achieved using continuous, time-invariant
control (Bhat and Bernstein, 2000). In this case, almost global asymptotic stability is
the best one can do with continuous time-invariant control. As a result of this, one may
expect that a control solution built out of a rotational integrator control primitive might
inherit this characteristic.
Chapter 4. Preliminaries 87
Table 4.3: Table of Control Primitives
Unicycle
Single integrator consensus (4.10)Uniformly bounded single integrator consensus (4.11)Single integrator path following (4.15)Rotational integrator equilibrium stabilization (4.21)Rotational integrator consensus - Kuramoto (4.23)Rotational integrator consensus (4.24)
Flying Robot
Double integrator consensus (4.19)Rotating body equilibrium stabilization (4.26)
4.3.1 Control Primitives for Single Integrators
Single integrator consensus controllers
Consider n single integrators
xi = vi, i ∈ n, (4.9)
where xi ∈ R2 and vi is the control input of subsystem i. Let xij := xj − xi and
x = (xi)i∈n ∈ R2n.
• The feedback
vi = fi((xij)j∈Ni) :=∑
j∈Ni
aijxij , (4.10)
where aij > 0 for all j ∈ Ni, is an integrator consensus controller. If G contains
a globally reachable node then the set x ∈ R2n : xi = xj , i, j ∈ n is globally
asymptotically stable for system (4.9), (4.10). See (Ren and Beard, 2005; Moreau,
2004; Olfati-Saber and Murray, 2004).
• The feedback
vi = fi((xij)j∈Ni) :=∑
j∈Ni
aijf(‖xij‖)‖xij‖
xij , (4.11)
is a uniformly bounded integrator consensus controller if aij = aji > 0 and f : R →
R, the interaction function, is a locally Lipschitz function satisfying:
Chapter 4. Preliminaries 88
A1: sf(s) > 0 for all s 6= 0, f(0) = 0, and there exist c1, c2 > 0 such that
|f(s)| > c1 for all |s| > c2.
A2: sup |f(s)| <∞.
Notice that the conditions on c1 and c2 in A1 imply that f(s) cannot tend to
zero as s tends to infinity. Each element (xij/‖xij‖)f(‖xij‖) of the sum in (4.11)
is continuous at xij = 0 because f(s) is a continuous function and f(0) = 0 by
assumption A1. We will omit the simple proof of the fact that each fi((xij)j∈Ni) is
Lipschitz continuous.
Examples of suitable interaction functions are f(s) = tanh(s) and
f(s) =
s, if |s| ≤ 1
s/|s| if |s| > 1.
(4.12)
The following proposition shows that feedback (4.11) globally asymptotically sta-
bilizes the consensus subspace x ∈ R2n : xi = xj , i, j ∈ n for any connected,
undirected sensor graph G.
Proposition 4.3.1. Consider system (4.9) with feedback (4.11). Assume that
f(s) satisfies assumptions A1 and A2 and the sensor graph G is undirected and
connected. For any parameters aij = aji > 0, the consensus set x ∈ R2n : xi =
xj , i, j ∈ n is globally asymptotically stable.
Proof. Consider system (4.9) with feedback (4.11). The feedback fi in (4.11) for
unicycle i points into the convex hull formed by its neighbours. By Corollary
3.9 in (Lin et al., 2007a) the group of unicycles for system (4.9) achieves global
consensus.
In addition, the following two lemmas will be used later in the thesis.
Chapter 4. Preliminaries 89
Lemma 4.3.2. If the sensor graph G is undirected and connected, then for any
parameters aij = aji > 0, system (4.9) with feedback (4.11), where f(s) satisfies
assumptions A1 and A2, is a gradient system, x = −∇Vt(x), with nonnegative
storage function
Vt(x) =1
2
n∑
i=1
∑
j∈Ni
aij
∫ ‖xij‖
0
f(s)ds. (4.13)
Moreover, V −1t (0) = x ∈ R2n : xi = xj , i, j ∈ n.
Proof. Assumption A1 implies that the function xij 7→∫ ‖xij‖
0f(s)ds is nonnegative,
and it attains its global minimum when xij = 0. Since G is connected, Vt attains a
global minimum when xij = 0 for all i, j ∈ n, and therefore Vt is positive definite.
We now show the gradient property, i.e., (∂/∂xi)Vt = −fi((xij)j∈Ni)⊤. We have
∂Vt∂xi
=1
2
∑
j∈Ni
aij∂
∂‖xij‖
(
∫ ‖xij‖
0
f(s)ds
)
∂‖xij‖∂xij
∂xij∂xi
+1
2
∑
j∈Ni
aji∂
∂‖xji‖
(
∫ ‖xji‖
0
f(s)ds
)
∂‖xji‖∂xji
∂xji∂xi
=1
2
∑
j∈Ni
aijf(‖xij‖)(
xij⊤
‖xij‖
)
(−1)
+1
2
∑
j∈Ni
ajif(‖xji‖)(
xji⊤
‖xji‖
)
(1)
= −∑
j∈Ni
aijf(‖xij⊤‖)‖xij‖
xij = −fi((xij)j∈Ni)⊤.
Lemma 4.3.3. Assume G is undirected and connected, and consider system (4.9)
with feedback (4.11), where f(s) satisfies assumptions A1 and A2. For any param-
eters aij = aji > 0, the following three properties hold:
(i) R−1fi((xij)j∈Ni) = fi((R−1xij)j∈Ni) for all i, j ∈ n, R ∈ SO(2).
(ii) x ∈ R2n : fi((xij)j∈Ni) = 0, ∀i ∈ n = x ∈ R2n : xi = xj , i, j ∈ n.
Chapter 4. Preliminaries 90
(iii)∑
i fi(·) = 0.
Proof. To show (i), we use the fact that ‖R−1xij‖ = ‖xij‖. Then,
R−1fi(·) =∑
j∈Ni
aijf(‖R−1xij‖)‖R−1xij‖
R−1xij = fi((R−1xij)j∈Ni).
To show (ii), assume fi((xij)j∈Ni) = 0 for all i ∈ n. Then system (4.9) is at a
fixed point. By Proposition 4.3.1, the set x ∈ R2n : xi = xj , i, j ∈ n is globally
asymptotically stable, so it contains all fixed points. Therefore, xij = 0 for all
i, j ∈ n. Conversely, if xij = 0 for all i, j ∈ n, then it follows by definition that
fi = 0 for all i ∈ n. Finally, property (iii) follows by summing over the functions
fi in (4.11) and using the identities aij = aji, xij = −xji.
Path following controllers
Consider a smooth simple (no self intersections) curve C ⊂ R2 and final speed w > 0. If
x = σ(s) is an arc length parametrization of C, then define r(x) = r(σ(s)) = (d/ds)σ(s).
Consider the single integrator,
x = v, (4.14)
where x ∈ R2 and v ∈ R2 is the control input. The feedback,
v = h(x), (4.15)
where h : R2 → R2 is an integrator path following controller for C satisfying the following
properties:
A1: h is globally Lipschitz, i.e., there exists c > 0 such that for all x1, x2 ∈ R2, ‖h(x1)−
h(x2)‖ ≤ c‖x1 − x2‖,
A2: h(x) = wr(x) for all x ∈ C,
Chapter 4. Preliminaries 91
A3: the set C is asymptotically stable for (4.14), (4.15).
In this thesis, we will consider paths that are either straight lines of the form C(r0, p) =
x ∈ R2 : x = r0 + sp, s ∈ R ⊂ R2 for r0, p ∈ R2 or smooth Jordan curves. Single inte-
grators are essentially the simplest model that one can design a path following controller
for. For any smooth line or Jordan curve, there always exists a path following controller
satisfying properties A1-A3 above. To obtain such a controller for Jordan curves, one
can use a feedback linearization technique (Nielsen et al., 2010).
For a line C(r0, p) define an integrator line following controller
h(x) = k0(r0 − x)− k0((r0 − x) · p)p+ wp (4.16)
with k0 > 0. The path following controller in (4.16) can be written alternatively as,
h(x) = k0(c⋆(x)− x) + wp (4.17)
where c⋆(x) is the orthogonal projection of the point x onto C(r0, p), i.e., (c⋆(x)−x)·p = 0.
It is easy to verify that h in (4.16) satisfies properties A1-A3. Therefore, this controller
makes x converge to the path C(r0, p) and follow it in the direction of the beacon p with
steady state speed w.
4.3.2 Control Primitives for Double Integrators
Double integrator consensus
Consider a collection of n double-integrators
xi = vi
vi = Ti, i ∈ n,
(4.18)
Chapter 4. Preliminaries 92
where xi ∈ R3, vi ∈ R3 and Ti ∈ R3 is the control input of subsystem i. Let xij := xj−xi,
vij := vj − vi, x := (xi)i∈n ∈ R3n and v := (vi)i∈n ∈ R3n.
The feedback
Ti = fi((xij, vij)j∈Ni) :=∑
j∈Ni
aij
(
xij + γvij
)
, i ∈ n, (4.19)
where aij, γ > 0 is a double-integrator consensus controller. The following theorem is
taken from Ren et al. in (Ren and Atkins, 2007, Theorems 4.1, 4.2) and says that for
sufficiently large γ, if the sensing digraph contains a globally reachable node, then the
system of double integrators achieves consensus.
Theorem 4.3.1 ((Ren and Atkins, 2007)). Consider system (4.18) with feedback (4.19)
and sensor digraph G containing a globally reachable node with corresponding weighted
Laplacian matrix L with weights aij > 0. Suppose that γ is chosen to satisfy
γ > maxµi 6=0
√
√
√
√
2
|µi| cos(
π2− tan−1 −Re(µi)
Im(µi)
)
where (µi)i∈n are the eigenvalues of −L. Then the set
(x, v) ∈ R3n × R
3n : xi = xj , vi = vj , i, j ∈ n
is globally asymptotically stable.
4.3.3 Control Primitives for Rotational Integrators
Rotational integrator equilibrium stabilization
Consider a rotational integrator
θ = ω, (4.20)
Chapter 4. Preliminaries 93
where θ ∈ S1 and ω is the control input. A feedback
ω = g(θ) := −k0 sin(θ), (4.21)
where k0 > 0, is a rotational integrator equilibrium stabilizer. The following proposition
says that the equilibrium θ = 0 is almost globally asymptotically stable. We do not claim
originality for this result.
Proposition 4.3.4. Consider system (4.20) with feedback (4.21). The union of the two
equilibria θ = 0 and θ = π is globally attractive. The first equilibrium is almost globally
asymptotically stable while the second is exponentially unstable.
Proof. This follows by a standard Lyapunov analysis using the Lyapunov function V =
1 − cos(θ). The equilibrium θ = π is exponentially unstable since computing the lin-
earization at θ = π gives
d
dθ(−k0 sin θ)|θ=π = −k0 cos θ|θ=π = −k0 cosπ = k0 > 0.
Rotational integrator consensus
Consider a collection of rotational integrators,
θi = ωi, i ∈ n (4.22)
where θi ∈ S1, θ := (θi)i∈n ∈ Tn, θij := θj − θi and ωi is the control input of subsystem i.
• A feedback,
ωi = gi((θij)j∈Ni) :=∑
j∈Ni
bij sin(θij) (4.23)
Chapter 4. Preliminaries 94
with bij = bji > 0 is a Kuramoto consensus controller. This is the well-known
Kuramoto model for attitude synchronization of angles in S1 (Kuramoto, 1984)
with zero natural frequencies.
Proposition 4.3.5 ((Lin et al., 2007a)). Consider system (4.22) with feedback (4.23)
and connected, undirected sensor graph G. The set θ ∈ Tn : θi = θj , i, j ∈ n is
asymptotically stable with domain of attraction containing
Sπ := θ ∈ Tn : |θij | < π, i, j ∈ n ,
the set where all vehicle angles lie in the same half plane on an open arc of π
radians.
Proof. Without loss of generality, assume θi ∈ (−π/2, π/2) ⊂ S1 for all i ∈ n which
is diffeomorphic to the open segment (−π/2, π/2) ∈ R. If the angle of a unicycle
i is greater than that of all its neighbors then gi will be negative and its angle
will decrease. The opposite is true for a unicycle whose angle is less than all its
neighbors. Therefore, for all unicycles i ∈ n, the input gi points into the convex
hull formed by its neighbours. By (Lin et al., 2007a) the group of unicycles for
system (4.22) achieves consensus with the feedback in (4.23) on (−π/2, π/2).
• A feedback
ωi = gi((θij)j∈Ni, η) := ηi∑
j∈Ni
bijg(θij), (4.24)
is a rotational integrator consensus controller if (i) ηi > 0 where η := (ηi)i∈n, (ii)
bij = bji > 0, and (iii) g : S1 → R is a continuously differentiable interaction
function satisfying the following three assumptions (Mallada et al., 2016):
B1: sg(s) > 0 for all s ∈ (−π, π)\0, g(0) = g(π) = 0.
B2: g(s) is an odd function: g(−s) = −g(s) for s ∈ (−π, π).
Chapter 4. Preliminaries 95
B3: g(s) > 0, ∀s ∈ (− πn−1
, πn−1
) and g(s) < 0, ∀s ∈ (−π,− πn−1
) ∪ ( πn−1
, π).
A sample interaction function satisfying B1-B3 is shown in Figure 4.16.
Figure 4.16: Illustration of properties B1, B2 and B3.
The Kuramoto consensus controller corresponds to the choice ηi = 1 for all i ∈ n
and g(s) = sin(s), and satisfies properties B1-B2, but does not satisfy property
B3 for n > 2. In (Mallada et al., 2016), Mallada-Freeman-Tang showed that a
feedback enjoying properties B1-B3 almost globally stabilizes the set θ ∈ Tn :
θi = θj , i, j ∈ n for almost all gains bij = bji > 0. The following result is a special
case of Theorem 2 in (Mallada et al., 2016).
Theorem 4.3.2 ((Mallada and Tang, 2013; Mallada et al., 2016)). Let G be
an undirected and connected sensor graph and consider system (4.22) with feed-
back (4.24) satisfying assumptions B1-B3, expressed in states relative to agent one,
i.e., θ := (θ1i)i∈2 :n,
θ1i = gi((θij)j∈Ni, η)− g1((θ1j)j∈N1, η), i ∈ 2 :n.
There exists a setNb ⊂ (R+)|E| of Lebesgue measure zero such that for any collection
of gains (bij)(i,j)∈E ∈ (R+)|E|\Nb, and for any ηi > 0, i ∈ n, the origin θ = 0
is almost globally asymptotically stable. Moreover, there exists a compact set of
Chapter 4. Preliminaries 96
isolated equilibria A such that 0 ∪ A is globally attractive and the equilibria in
A are exponentially unstable.
The above result follows from Theorem 2 in (Mallada et al., 2016). In particu-
lar, system (4.22) with feedback (4.24) satisfies the model in equations (14)-(16)
in (Mallada et al., 2016) by letting χi(s) = ηis, letting ζ be the identity function,
and eliminating the integrator state γi.
The main difference here compared to the solution in (Mallada et al., 2016, Theorem
2) is that, in (Mallada et al., 2016), each system has an additional constant bias.
The integrator state γi is used in (Mallada et al., 2016) to compensate for this
bias. In this thesis, system (4.22) has zero bias, so the integrator state γi is not
needed. Accounting for this small difference, the proof of Theorem 4.3.2 follows
from minimal modifications to the proof of (Mallada et al., 2016, Theorem 2).
4.3.4 Control Primitives for Rotating bodies in SO(3)
Rotating body equilibrium stabilization
Consider the dynamic system for a rotating axis Γ ∈ S2 in three dimensions with sym-
metric inertia matrix J ,
Γ = Γ× ω,
Jω = τ − ω × Jω
(4.25)
where ω ∈ R3 is the body’s angular velocity and τ is the torque control input. A feedback,
τ = τd(Γ, ω) (4.26)
with smooth function τd : R3 × R3 → R3 is a rotating body equilibrium stabilizer if the
closed-loop system (4.25), (4.26) has an AGAS equilibrium point (Γi, ωi) = (e3, 0).
An example of a thrust direction controller is presented in (Chaturvedi et al., 2011)
Chapter 4. Preliminaries 97
as,
τd(Γ, ω) = kq (e3 × Γ)−Kωω, (4.27)
where kq is positive and Kω is a symmetric and positive definite matrix.
Proposition 4.3.6 (Theorem 2 in (Chaturvedi et al., 2011)). Feedback (4.27) is a ro-
tating body equilibrium stabilizer.
Chapter 5
Rendezvous of Flying Robots with
Local and Distributed Feedbacks
In this chapter, we present a solution to the rendezvous control problem for flying robots
(RP− F) introduced in Section 3.1. The feedbacks will be constructed out of the double-
integrator consensus control primitive fi((xij , vij)j∈Ni) in (4.19).
5.1 Solution of the Rendezvous Control Problem (RP− F)
We use a consensus controller for double-integrators fi((xij , vij)j∈Ni) = fi(yi) in (4.19) to
construct the feedbacks
ui =u⋆i (y
ii,Ω
ii) = −mifi(y
ii) · e3,
τi =τ⋆i (y
ii,Ω
ii)
=Ωii × JiΩii − k1Ji
(
(Ωii × fi(yii))× e3
)
− k21k2[
Ωii − k1(fi(yii)× e3)
]
, i ∈ n.
(5.1)
The result below states that for sufficiently large k1, k2 > 0, the feedbacks in (5.1)
solve RP-F if the network of robots has a sensor digraph containing a globally reachable
node.
98
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 99
ConsensusControl
RotationalControl
robot
Sensors
ThrustControl
Figure 5.1: Block diagram of the rendezvous control system for robot i. The outer loopassigns a desired thrust vector fi(y
ii). The inner loop thrust control uses fi(y
ii) to assign
the vehicle input ui while the rotational control uses fi(yii) to assign the torque input τi.
The vector yii contains the relative displacements and velocities of vehicles that robot ican sense, measured in the body frame of robot i.
Theorem 5.1.1. RP-F is solvable for system (2.13), (2.14) if and only if the sensor
digraph G contains a globally reachable node, in which case a solution is the following.
Let fi(yi), i ∈ n, be a double-integrator consensus controller in (4.19) satisfying the
conditions in Theorem 4.3.1. The local and distributed feedback in (5.1) where k1, k2 > 0
are control parameters, makes the rendezvous manifold (3.1) globally practically stable,
that is, for any ε > 0, there exist k⋆1 > 1, k⋆2 > 0 such that for all k1 > k⋆1, k2 > k⋆2, the
set Bε(Γ) has a globally asymptotically stable subset containing Γ.
The proof of Theorem 5.1.1 is presented in Section 5.4.1.
Explanation of the proposed controller
Consider the block diagram in Figure 5.1, we now explain in detail the operation of its
two nested loops. We begin with the observation that a double-integrator consensus
controller fi(yi), i ∈ n also makes the system
xi = vi
vi = fi + g, i ∈ n
(5.2)
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 100
reach rendezvous, since the addition of the gravity vector g is common to all agents and
does not affect the relative dynamics. Now compare system (5.2) to the translational
dynamics of the flying robots,
xi = vi
vi = − 1
mi
uiRie3 + g, i ∈ n.(5.3)
If it were the case that fi = −(1/mi)uiRie3, systems (5.2) and (5.3) would be identical.
Then, setting −uiRie3 = mifi(yi) in (5.3) would solve RP-F. Inspired by this observation,
the outer loop of the block diagram in Figure 5.1 assumes that −uiRie3 is the control
input of (5.3) and computes a desired double-integrator force mifi(yi) which becomes a
reference signal for the inner loop.
We now explore in more detail the operation of the inner loop. First we observe
that fi(yi) satisfies Rifi(yii) =
∑
j∈Niaij(Rix
iij + γRiv
iij) =
∑
j∈Niaij(xij + γvij) = fi(yi).
Moreover, using the fact that dot products are invariant under rotations, we have
u⋆i (yii,Ω
ii) = −mifi(y
ii) · e3 = mi(Rifi(y
ii)) · (−Rie3) = mifi(yi) · qi,
where qi is the thrust direction vector. Thus, the thrust magnitude is the projection
of the desired thrust mifi(yi) onto the thrust direction vector—see Figure 5.2. Now let
Ωii(yi, Ri) := R⊤
i k1 (fi(yi)× Rie3) = k1 (fi(yii)× e3). Then we have from (5.1) that
τ ⋆i (yii,Ω
ii)=Ωii × JiΩ
ii − k1Ji
(
(Ωii × fi(yii))× e3
)
− k21k2(
Ωii −Ωii(yi, Ri)
)
. (5.4)
For simplicity of notation, we drop the arguments of Ωii(yi, Ri). We will show in the
proof of Theorem 5.1.1 that the torque inputs τi make Ωii converge to an arbitrarily small
neighborhood of Ωii, i ∈ n. Thus, Ωi
i can be seen as a reference angular velocity for the
inner loop. Using the fact that, for all a, b ∈ R3 and all R ∈ SO(3), R(a×b) = (Ra)×(Rb),
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 101
we have
Ωi(yi, Ri) = RiΩii = Rik1
(
fi(yii)× e3
)
= k1(
(Rifi(yii))× (Rie3)
)
= k1(fi(yi)×−qi) = k1(qi × fi(yi)).
Thus Ωi is perpendicular to the plane formed by the thrust direction vector qi and the
desired thrust force mifi(yi)—see Figure 5.2. Since the angular velocity vector identifies
an instantaneous axis of rotation, it follows that if Ωi = Ωi, then robot i rotates about
Ωi according to the right-hand rule. Referring to Figure 5.2, we see that such a rotation
closes the gap between uiqi and mifi(yi), and the speed of rotation is proportional to
sinϕ, where ϕ is the angle between uiqi and mifi(yi) marked in the figure. When the gap
is closed, we have ui = ‖mifi(yi)‖, qi = mifi(yi)/‖mifi(yi)‖, and thus uiqi = mifi(yi).
In conclusion, the inner loop assigns (ui, τi) to make Ωi approximately converge to Ωi in
the sense of global practical stability, so that uiqi = −uiRie3 approximately converges to
mifi(yi), which is computed by the outer loop.
While the intuition behind the proposed controller is simple, the proof that the inter-
play between the two nested loop results in global practical stability of the rendezvous
manifold is rather delicate, and it crucially relies on the homogeneity of the functions
fi(yi), i ∈ n.
Remark 5.1.2. Theorem 5.1.1 proves global practical stability of the rendezvous man-
ifold Γ. The reason that the stability is practical and not asymptotic is roughly as
follows. In order to achieve rendezvous of the flying robots, uiqi is driven approximately
tomifi(yi). What’s important is not so much the difference in magnitude of these vectors
but rather the difference in angle between them. In Figure 5.2, one can see that Ωi acts
to reduce this angle with a rate proportional to the magnitude of Ωi. Since Ωi is a linear
function of fi(yi), as the robots approach consensus Ωi converges to zero at the same rate
as fi(yi). This leads to increasing inaccuracy in closing the gap between the vectors uiqi
and mifi(yi) insomuch that in a very small neighborhood of rendezvous, Ωi is so small
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 102
Figure 5.2: Illustration of the control input ui and reference angular velocity Ωi in (5.1).
that it fails to make the translational dynamics act as double integrators. More detailed
reasoning is provided in Remark 5.4.1.
Features of the proposed controller
(i) The feedback of Theorem 5.1.1 is static. It does not depend on auxiliary states
that require communication between neighboring robots.
(ii) The feedback of Theorem 5.1.1 is local and distributed in the sense of Defini-
tion 2.2.3. Interestingly, it does not require sensing of relative attitudes, which
can be computed using on-board cameras, but are harder to compute than relative
displacements.
(iii) On the rendezvous manifold Γ there is no prespecified thrust direction qi for robot i
and the robot thrust directions do not need to align at rendezvous. This is desirable
if one wants to employ the proposed controller in a hierarchical control setting to
enforce additional control specifications.
(iv) In (Roza et al., 2014), we developed a solution for almost global asymptotic stability
of the set Γ combining a uniformly bounded double integrator consensus controller
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 103
fi(yi) with a rotating body equilibrium stabilizer τd(Γ, ω) in (4.26). The proposed
controller in this chapter has a number of advantages over this previous work. Un-
like (Roza et al., 2014), the inner control loop does not require any derivatives of
the reference thrust force fi(yi). In (Roza et al., 2014), the expressions resulting
from such derivatives are large and may pose difficulty in real-time computation
of the control law. More importantly, the computation of such derivatives requires
communication of the thrust control inputs u⋆i between neighboring robots, a prob-
lem that has been overcome in the present approach. The approach in (Roza et al.,
2014) requires that robots have access to a common inertial vector and its relative
attitude with respect to its neighbors. This requirement is absent in this chapter.
(v) The proposed control law in this chapter does not guarantee hovering of the robots.
While the robots converge to each other, nothing can be said about the motion of
the ensemble. This cannot be otherwise, for it would be impossible to solve RP-F
with hovering without additional sensors. One would need some measurement of
the gravity vector, for example provided by a three-axis accelerometer. The point
of view of these authors is that the proposed solution of RP-F with strictly local
and distributed feedbacks will serve as a layer in a hierarchy of higher-level control
specifications such as hovering and path following. That said, if all agents can
measure gravity, the solution in (Roza et al., 2014) allows the robots to hover in
steady state or have a desired final thrust in the direction of a common inertial
vector sensed by all robots.
5.2 Simulation Results
We consider a group of five robots with the sensor digraph in Figure 5.3. The robot
masses and inertia matrices are: m1 = 3 Kg, m2 = 3 Kg, m3 = 3.4 Kg, m4 = 3.2 Kg,
m5 = 3.2 Kg and J1 := diag (0.13, 0.13, 0.04)Kg·m2, as in (Abdessameud and Tayebi,
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 104
Table 5.1: Simulation Initial Conditions
Vehicle i xi(0) (m) vi(0) (m/s) Ri(0)1 (0,−10, 10) (0, 0, 0) side 12 (0, 10, 10) (0, 0, 0) side 23 (0, 0, 0) (0, 0, 0) down4 (−10, 0,−10) (0, 0, 0) up5 (10, 0,−10) (0, 0, 0) up
Table 5.2: Control Effort
Figure 5.4 Figure 5.5maxi supt |ui(t)| (N) 20.4 17.21maxi supt ‖τi(t)‖ (N·m) 15.27 16.47maxi rms(|ui(t)|) (N) 1.72 4.31maxi rms(‖τi(t)‖) (N·m) 1.43 2.24
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
vector (x, v) := (x1j , v1j)j∈2 :n is zero. Denoting,
X := (x, v) ∈ X := R3(n−1) × R
3(n−1),
R := (R1, . . . , Rn) ∈ R := SO(3)n,
Ω := (Ω11, . . . ,Ω
nn) ∈ Ω := R
3n,
the new collective state is (x1, v1, X,R,Ω) ∈ R3 × R3 × X× R × Ω. The meaning of the
new state is this: X contains all translational states (positions and velocities) relative to
robot 1, R contains all the attitudes, and Ω contains all body frame angular velocities.
Due to the identity (xij , vij) = (x1j − x1i, v1j − v1i), the vector yi = (xij , vij)j∈Ni
is a linear function of X which we will denote yi = hi(X). Similarly, the vector yii =
(xiij , viij)j∈Ni is a function of X and R, linear with respect to X . We will denote this
function yii = hii(X,R).
Using the definitions above, we may now express fi(yii) and Ωi
i(yi, Ri) = k1(fi(yii)×e3)
(the latter function was discussed in Section 5.1) in terms of states. Accordingly, we define
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 110
gi : X → R3, gii : X× R → R3 and Ω : X× R → Ω as follows:
gi(X) := (fi hi)(X),
gii(X,R) := R−1i gi(X) = (fi hii)(X,R),
Ω(X,R) :=(
Ωii(hi(X), Ri)
)
i∈n.
(5.9)
We remark that gi is linear and gii is linear with respect to its first argument.
Finally, we can rewrite the rendezvous manifold in new coordinates as,
(x1, v1, X,R,Ω) ∈ R3 × R
3 × X× R× Ω : X = 0, Ω = Ω(X,R). (5.10)
Since the feedbacks in (5.1) are local and distributed, it can be seen that the dynamics of
the closed-loop system in (X,R,Ω) coordinates are independent of (x1, v1). The norm of
the right hand side of the translational system in (5.3) grows at most linearly with (x, v)
and therefore (x, v) and, in particular, (x1, v1) have no finite escape times. Moreover, as
we have seen, in these coordinates the set in (5.10) is independent of (x1, v1). In light
of these considerations, for the stability analysis we may drop the variables (x1, v1), and
show that the set
Γ = (X,R,Ω) ∈ X× R× Ω : X = 0, Ω = Ω(X,R). (5.11)
is globally practically stable for the (X,R,Ω) dynamics which will imply that Γ is globally
practically stable as well.
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 111
Lyapunov function
Consider n double-integrators with control fi(yi), expressed in X coordinates:
x1j = v1j
v1j = fj(yj)− f1(y1) = gj(X)− g1(X), j ∈ 2 :n.
(5.12)
By Theorem 4.3.1, the origin of this linear time-invariant system is globally asymptot-
ically stable. Thus, there exists a quadratic Lyapunov function V : X → R, V (X) =
X⊤PX , where P is a symmetric positive definite matrix, such that the derivative of V
along the vector field in (5.12) is negative definite.
Let J ∈ R3n×3n be the block-diagonal matrix with the i-th block equal to Ji, and
consider the continuous function W : X× R× Ω → R defined as
W (X,R,Ω) = αWtran(X) +Wrot(X,R,Ω), (5.13)
where α > 0 is a parameter to be assigned later and
Wtran(X) =√
V (X) +1
2V (X),
Wrot(X,R,Ω) =
n∑
i=1
gii(X,R) · e3 +1
2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R)).
The function Wtran(X) has been chosen as the summation of two terms:√
V (X) and
12V (X). The first is homogeneous of degree one while the second is homogeneous of
degree two with respect to X . The resulting time derivatives will possess these same
homogeneity properties, a fact that will be crucial in the Lyapunov analysis for proving
that W (X) ≤ 0. Taking the derivative of Wtran(X) yields
dWtran(X)
dX=
1
2√
V (X)
dV (X)
dX+dV (X)
dX
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 112
which is continuous everywhere except at V (X) = 0 which is the set X = 0. Therefore
Wtran is C1 everywhere except at X = 0. In light of Remark 4.1.7, this is not an issue
for doing a Lyapunov analysis since we show next that the Lie derivative of Wtran is
continuous. To this end, Wtran satisfies
Wtran = LfWtran(X,R) =dWtran(X)
dX· f(X,R)
where f(X,R) corresponds to the vector field for relative coordinates X = (xij , vij)j∈2 :n
in (5.7) which is homogeneous of degree one with respect to X . LfWtran(X,R) is contin-
uous for X 6= 0 because dWtran(X)/dX and f(X,R) are such. Moreover, LfWtran(X,R)
is the sum of two functions, one (1/2√V )dV (X)/dX · f(X,R) homogeneous of degree
one and the other dV (X)/dX · f(X,R) homogeneous of degree two with respect to X .
Proposition 4.1.22 implies that LfWtran(X,R) is continuous at X = 0.
Lemma 5.4.1. Consider the continuous function W defined in (5.13). Then
α⋆ := sup(X,R)∈(X\0)×R
n∑
i=1
|gii(X/√
V (X), R) · e3| <∞,
and for all α > α⋆, the following properties hold:
(i) W ≥ 0 and W−1(0) = Γ.
(ii) For all c > 0, for all k1 > 0, the sublevel set Wc := (X,R,Ω) :W (X,R,Ω) ≤ c is
compact.
(iii) For all ε > 0, there exists δ > 0 such that Wδ ⊂ Bε(Γ).
The proof is in Section 5.4.2.
From now on we assume α > α⋆. In light of the lemma, if we show that W is
nonincreasing outside a certain compact region of the state space, then all trajectories
of (5.7), (5.8) with feedback (5.1) are bounded, ruling out finite escape times. Moreover,
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 113
in light of parts (i) and (iii) of the lemma, to prove that Γ is practically stable it suffices
to prove that for every δ > 0, there exists a gain vector (k1, k2) such that Wδ is globally
asymptotically stable. Therefore, by Proposition 4.1.18 and Remark 4.1.7 we need to
show that W ≥ δ =⇒ W < 0.
Define the quantities
ρ(X) :=√
V (X), µ(X) := X/√
V (X).
Since the numerator and denominator are both homogeneous of degree one, the function
µ(X) is homogeneous of degree zero with respect to X . Therefore, the image satisfies
µ(X\0) = µ(S6(n−1)−1). Since µ(X) is continuous and the set S6(n−1)−1 is compact, the
image, denoted S1 := µ(X\0), is compact.
Dropping the arguments of ρ(X) and µ(X) for notational convenience, the functions
Wtran(X) and Wrot(X,R,Ω) can be expressed as Wtran(X) = ρ+ ρ2
2and
Wrot(X,R,Ω) = ρ
n∑
i=1
gii(µ,R) · e3 +1
2
(
Ω−Ω(ρµ,R))
⊤J(
Ω−Ω(ρµ,R))
.
In writing the above, we used the identity X = ρµ and the fact that the function gii(X,R)
is linear with respect to X , implying that gii(ρµ,R) = ρgii(µ,R).
Stability analysis
Let δ > 0 be arbitrary. We have W ≤ α(ρ+ ρ2/2) + ρ sup(µ,R) |gii(µ,R) · e3|+ (1/2)(Ω−
Ω)⊤J(Ω − Ω). Using the definition of α⋆ in Lemma 5.4.1 and the fact that α > α⋆, we
get
W ≤ α(2ρ+ ρ2/2) + (1/2)(Ω−Ω)⊤J(Ω−Ω).
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 114
It readily follows that there exists ∈ (0,min1, δ) such that
Λ := (X,R,Ω) : ρ ∈ (0, ), ‖Ω−Ω(ρµ,R)‖2 < ⊂Wδ.
We will show that there exist α > 0 and a gain vector (k1, k2) such that W < 0 outside
the set Λ. This will imply that W ≥ δ =⇒ W < 0, proving that Wδ is globally
asymptotically stable.
Lemma 5.4.2. Consider the closed-loop system (5.7), (5.8) with feedback (5.1). If
k1 > 1, k2 > 0, then there exist scalars M1, . . . ,M4 > 0 such that the derivatives of ρ
and Wrot along the closed-loop system satisfy the following inequalities:
ρ ≤ρ[
−M2 +M1
n∑
i=1
‖gii(µ,R)× e3‖]
,
Wrot ≤ρ2n∑
i=1
[
−k1‖gii(µ,R)× e3‖2 +M4
k2
]
+ ρM3 −k21k22
‖Ω−Ω(ρµ,R)‖2.(5.14)
The proof is in Section 5.4.3.
From now on we let k1 > 1. Using the inequalities in Lemma 5.4.2, we get
W ≤(ρ+ ρ2)
[
−αM2 + αM1
n∑
i=1
‖gii(µ,R)× e3‖]
+ ρ2n∑
i=1
[
−k1‖gii(µ,R)× e3‖2 +M4
k2
]
+ ρM3 −k21k22
‖Ω−Ω(ρµ,R)‖2.
Denote βi(µ,R) := ‖gii(µ,R) × e3‖, and β(µ,R) := (β1(µ,R), . . . ,βn(µ,R)). For no-
tational convenience, we omit the arguments of the functions β and Ω. With these
definitions, the inequality above may be rewritten as
W ≤ (ρ+ ρ2)(
−αM2 + αM11⊤β)
+ρ2(
−k1‖β‖2+M4n
k2
)
+ ρM3 −k21k22
‖Ω−Ω‖2.
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 115
For every k2 > nM4/M3, we have
W ≤ (ρ+ ρ2)(
−αM2 +M3 + αM11⊤β)
− ρ2k1‖β‖2 −k21k22
‖Ω−Ω‖2.
If we further pick α > maxα⋆, 3M3/M2, we have
W ≤ (ρ+ ρ2)(
−2M3 + αM11⊤β)
− ρ2k1‖β‖2 −k21k22
‖Ω−Ω‖2.
Splitting the term −ρ2k1‖β‖2 into two parts and collecting terms for ρ and ρ2, we
obtain
W ≤ρ2(
−2M3 + αM11⊤β − k1
2‖β‖2
)
+ ρ
(
−2M3 + αM11⊤β − ρ
k12‖β‖2
)
− k21k22
‖Ω−Ω‖2.
Consider now the expression
M3 − αM11⊤β +
k1
2‖β‖2=
[
1⊤ β⊤]
M3
nI −αM1
2I
−αM1
2I k1
2I
1
β
.
If k1 > 2n(αM1/2)2/(M3), the above quadratic form is positive definite, implying that
M3 − αM11⊤β +
k1
2‖β‖2 ≥ 0. (5.15)
Since < 1, we also have M3 − αM11⊤β + (k1/2)‖β‖2 ≥ 0. Using the latter inequality,
we get a further upper bound for W ,
W ≤ −ρ2M3 + ρ
(
−2M3 + αM11⊤β − ρ
k12‖β‖2
)
− k21k22
‖Ω−Ω‖2. (5.16)
Using (5.16), we now prove that outside Λ, W < 0. In other words, when either ρ ≥
or ‖Ω−Ω‖2 ≥ , W < 0.
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 116
Remark 5.4.1. If the derivative W were negative definite, then the rendezvous manifold
Γ would be globally asymptotically stable. However, this is not guaranteed in (5.16). The
reason is as follows. Suppose ρ is very small and ‖Ω−Ω‖ = 0. Then all terms multiplied
by ρ2 become negligible and what remains in (5.16) is, W ≤ ρ(
−2M3 + αM11⊤β)
. As
we have no control over the value of the constants M1 and M3 in the equation above, W
can be greater than zero if the second term dominates the first.
Suppose first that ρ ≥ . Then from (5.16) we have
W ≤ −ρ2M3 + ρ
(
−2M3 + αM11⊤β − k1
2‖β‖2
)
− k21k22
‖Ω−Ω‖2.
By inequality (5.15) we conclude that
W ≤ −ρ2M3 − ρM3 −k21k22
‖Ω−Ω‖2 < 0.
Next, suppose that ‖Ω−Ω‖2 ≥ . Then from (5.16),
W ≤ −ρ2M3 + ραM11⊤β − k21k2
2
≤ −ρ2M3 + ραM1M5 −k21k22,
where M5 := max(µ,R)∈S1×R1⊤β(µ,R). The maximum exists because β is continuous
and S1 × R is a compact set. If k2 > (αM1M5/k1)2/ then W < 0.
We have therefore proved that, if
• α > maxα⋆, 3M3/M2,
• k1 > max1, 2n(αM1/2)2/(M3), and
• k2 > maxnM4/M3, (αM1M5/k1)2/,
then W > δ implies that W < 0. Therefore, for any initial condition, the solution
of (5.7), (5.8) with feedback (5.1) is bounded and the set Wδ is globally asymptotically
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 117
stable.
5.4.2 Proof of Lemma 5.4.1
Recall the definition of W (X,R,Ω), and assume that X 6= 0,
W =α
(
√
V (X) +1
2V (X)
)
+
n∑
i=1
gii(X,R) · e3 +1
2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R))
=√
V (X)
(
α +
∑ni=1 g
ii(X,R) · e3
√
V (X)
)
+α
2V (X) +
1
2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R)).
Since gii(X,R) is linear with respect to X , we have
W =√
V (X)
(
α +
n∑
i=1
gii (µ(X), R) · e3)
+α
2V (X)+
1
2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R)),
where µ(X) = X/√
V (X) is continuous on X\0 and bounded as follows
‖µ(X)‖ =‖X‖√
V (X)=
‖X‖√X⊤PX
≤ 1√
λmin(P ).
Since gii is continuous, µ(X) is bounded, and R ∈ R, a compact set, it follows that the
function∑n
i=1 |gii (µ(X), R) · e3| has a bounded supremum. Accordingly, let
α⋆ = sup(X,R)∈(X\0)×R
n∑
i=1
∣
∣gii (µ(X), R) · e3∣
∣ .
For all α > α⋆, we have W (X,R,Ω) > W (X,R,Ω),
W (X,R,Ω) :=α
2V (X) +
1
2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R)) ≥ 0.
We derived the bound above for X 6= 0, but since gii(0, R) = 0 (by linearity of gii with
respect to X), the bound also holds for X = 0. The above inequality implies that W ≥ 0
and W−1(0) ⊂ W−1(0). But W = 0 if and only if V (X) = 0 (i.e., X = 0) and Ω = Ω.
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 118
Thus W−1(0) ⊂ Γ. Moreover, in Γ, (X,Ω −Ω) = 0 and therefore Γ ⊂ W−1(0), proving
part (i) of the lemma.
For part (ii), note that for all c > 0, Wc ⊂ W ≤ c. Since W is a positive definite
quadratic form in the variables (X,Ω −Ω), its sublevel sets are compact in (X,Ω−Ω)
coordinates. Thus if (X,R,Ω) ∈ Wc, X and Ω −Ω(X,R) are bounded. For all k1 > 0,
since Ω(X,R) = k1(gii(X,R)×e3) is continuous and R ∈ R, a compact set, Ω is bounded,
implying that Ω is also bounded. Therefore the set Wc is bounded. Continuity of W
implies that Wc is compact. This concludes the proof of part (ii) of the lemma.
For part (iii), let ε > 0 be arbitrary. Since W is a positive definite quadratic form in
the variables (X,Ω−Ω), there exists δ > 0 such thatW (X,R,Ω) ≤ δ implies ‖X‖ ≤ ε/2,
and ‖Ω − Ω(X,R)‖ ≤ ε/2. This implies that (X,R,Ω) ∈ Bε(Γ). Thus, Wδ ⊂ W ≤
δ ⊂ Bε(Γ), as required. This concludes the proof of Lemma 5.4.1.
5.4.3 Proof of Lemma 5.4.2
Rewrite the dynamics of X in (5.7) as
x1j = v1j
v1j = [gj(X)− g1(X)] +Rj
[
(gjj (X,R) · e3)e3 − gjj (X,R)]
−R1
[
(g11(X,R) · e3)e3 − g11(X,R)]
.
To get the identities above, we added and subtracted in (5.7) the ideal force feedbacks
fj(yj) = gj(X) and f1(y1) = g1(X), and we replaced uj and u1 in (5.7) by the assigned
feedbacks in (5.1). Finally, we used the identity Rigii = gi.
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 119
Taking the time derivative of√
V (X) along the above vector field we get
d
dt
√
V (X) =1
2√
V (X)
[
−X⊤QX +
n∑
j=2
∂V
∂v1jRj
(
(gjj(X,R) · e3)e3 − gjj (X,R))
−n∑
j=2
∂V
∂v1jR1
(
(g11(X,R) · e3)e3−g11(X,R))
]
.
The first term in the bracket is the derivative of V (X) along the nominal vector field (5.12),
and Q = Q⊤ is a positive definite matrix. Letting M2 = λmin(Q)/(2λmax(P )) and using
the fact that the Euclidean norm is invariant under rotations, we have
d
dt
√
V (X) ≤ −M2
√
V (X) +1
2√
V (X)·
[
n∑
j=2
∥
∥
∥
∥
∂V
∂v1j
∥
∥
∥
∥
(
‖(gjj(X,R) · e3)e3 − gjj (X,R)‖+ ‖(g11(X,R) · e3)e3 − g11(X,R)‖)
]
.
We claim that ‖(gii(X,R) · e3)e3 − gii(X,R)‖ = ‖gii(X,R) × e3‖. Indeed, writing gii =
(gii · e3)e3 + gii − (gii · e3)e3, we have gii × e3 = (gii − (gii · e3)e3) × e3. Since the vector
gii − (gii · e3)e3 is perpendicular to e3, ‖(gii − (gii · e3)e3)× e3‖ = ‖gii − (gii · e3)e3‖, so that
‖gii × e3‖ = ‖gii − (gii · e3)e3‖. This proves the claim. Using the identity just derived, we
get
d
dt
√
V (X) ≤−M2
√
V (X)
+1
2√
V (X)
[
n∑
j=2
∥
∥
∥
∥
∂V
∂v1j
∥
∥
∥
∥
(
‖gjj (X,R)× e3‖+ ‖g11(X,R)× e3‖)
]
.
Rewriting this in terms of (ρ, µ), we get
ρ ≤ −M2ρ+1
2ρ
[
n∑
j=2
∥
∥
∥
∥
∂V
∂v1j(ρµ)
∥
∥
∥
∥
(
‖gjj(ρµ,R)× e3‖+ ‖g11(ρµ,R)× e3‖)
]
.
Since the functions gii are linear with respect to their first argument, and the partial
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 120
derivatives of the quadratic form V are linear functions, by the homogeneity of the norm
we have
ρ ≤ −M2ρ+ρ
2
[
n∑
j=2
∥
∥
∥
∥
∂V
∂v1j(µ)
∥
∥
∥
∥
(
‖gjj (µ,R)× e3‖+ ‖g11(µ,R)× e3‖)
]
.
The functions ‖∂V/∂v1j‖ are continuous. The quantity µ belongs to S1, a compact set.
Therefore ‖∂V/∂v1j‖ has a maximum,
ρ ≤−M2ρ+maxµ∈S1j∈2 :n
∥
∥
∥
∥
∂V
∂v1j(µ)
∥
∥
∥
∥
ρ
2
[
n∑
j=2
(
‖gjj (µ,R)× e3‖)
+ (n− 1)‖g11(µ,R)× e3‖]
≤−M2ρ+maxµ∈S1j∈2 :n
∥
∥
∥
∥
∂V
∂v1j(µ)
∥
∥
∥
∥
ρ
2(n− 1)
n∑
j=1
‖gjj(µ,R)×e3‖.
Letting M1 := max µ∈S1j∈2 :n
‖∂V/∂v1j‖ (n− 1)/2, we get the first inequality in (5.14).
We now turn to the second inequality in (5.14). Recall the definition of Wrot,
Wrot(X,R,Ω) =
n∑
i=1
gii(X,R) · e3 +1
2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R)).
The derivative of Wrot along the vector field in (5.7), (5.8) is
Wrot =n∑
i=1
[
(
d
dtgii
)
· e3 + (Ωii −Ωii(X,R)) ·
(
τi − Ωii × JiΩii − Ji
(
d
dtΩii
))
]
.
To express (d/dt)gii, recall that gii(X,R) = R−1
i fi(hi(X)). Then,
d
dtgii =
(
d
dtR−1i
)
fi(hi(X)) +R−1i
d
dt(fi(hi(X))) .
The function fi(hi(X)) is linear. Its derivative along the vector field (5.7), (5.8) with
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 121
feedback (5.1) is given by
d
dtfi(hi(X)) =
∑
j∈Ni
(
aijd
dt(xj1 − xi1) + bij
d
dt(vj1 − vi1)
)
=∑
j∈Ni
(
aij(vj1 − vi1) + bij
(
− 1
mjRje3uj +
1
miRie3ui
))
,
which is linear with respect to X because ui = −gii(X,R) · e3 is such. We will denote
it hi(X,R), hi(X,R) := (d/dt)fi(hi(X)). Consistently with our notational convention
in Table 2.2, we will let hii(X,R) := R−1i hi(X,R). The function hii(X,R) is linear with
respect to X . Returning to the derivative of gii, we have
d
dtgii = −(Ωii)
×R−1i fi(hi(X)) +R−1
i hi(X,R)
= −Ωii × gii(X,R) + hii(X,R).
Similarly, since Ωii(X,R) = k1(g
ii(X,R) × e3), we have d
dtΩii = k1 (−Ωii × gii + hii) × e3.
Substituting the above identities in the expression for Wrot and since τ ⋆i = Ωii × JiΩii −
k1Ji((Ωii × gii)× e3)− k21k2(Ω
ii −Ωi
i), we get
Wrot =
n∑
i=1
[
−(Ωii × gii) · e3 + hii · e3 − k1(Ωii −Ωi
i) · Ji(hii × e3)− k21k2‖Ωii −Ωii‖2]
.
Using the property of the triple product that (Ωii × gii) · e3 = (gii × e3) · Ωii, we obtain
Wrot =
n∑
i=1
[
−(gii × e3) · Ωii + hii · e3 − k1(Ωii −Ωi
i) · Ji(hii × e3)− k21k2‖Ωii −Ωii‖2]
.
Adding and subtracting the term (gii × e3) ·Ωii and collecting the term Ωii −Ωi
i, we have
Wrot =n∑
i=1
[
−(gii×e3) ·Ωii+hii ·e3−((gii×e3)+k1Ji(hii×e3)) ·(Ωii−Ωi
i)−k21k2‖Ωii−Ωii‖2]
.
Substituting in the first term inside the bracket Ωii = k1(g
ii×e3), taking norms, and using
Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 122
the fact that k1 ≥ 1, we arrive at the inequality
Wrot ≤n∑
i=1
[
− k1‖gii × e3‖2 + ‖hii · e3‖+ k1κi(X,R)‖Ωii −Ωii‖ − k21k2‖Ωii −Ωi
i‖2]
,
where κi(X,R) := ‖gii(X,R)× e3‖+ ‖Ji(hii(X,R)× e3)‖. Note that κi(X,R) is homoge-
neous with respect to X because gii and hi are linear with respect to X and the norm is
a homogeneous function.
Splitting the term −k21k2‖Ωii − Ωii‖2 into two parts and noticing that the function
k1κi(X,R)‖Ωii − Ωii‖ − (k21k2/2)‖Ωii − Ωi
i‖2 is quadratic in the variable ‖Ωii − Ωii‖ with
maximum κ2i (X,R)/(2k2), we get
Wrot ≤n∑
i=1
[
− k1‖gii × e3‖2 + ‖hii · e3‖ −k21k22
‖Ωii −Ωii‖2 +
κ2i (X,R)
2k2
]
.
Now rewriting this in terms of (ρ, µ), we get
Wrot ≤n∑
i=1
[
− k1‖gii(ρµ,R)× e3‖2 + ‖hii(ρµ,R) · e3‖ −k21k22
‖Ωii −Ωii‖2 +
κ2i (ρµ,R)
2k2
]
.
Using the homogeneity with respect to X of ‖gii × e3‖, ‖hii · e3‖, and κi, we get
Wrot ≤n∑
i=1
[
− k1ρ2‖gii(µ,R)× e3‖2 + ρ‖hii(µ,R) · e3‖ −
k21k22
‖Ωii −Ωii‖2 + ρ2
κ2i (µ,R)
2k2
]
.
Since ‖hii(µ,R) · e3‖ and κ2i (µ,R) are continuous functions over the compact set S1 ×
R, they each have a maximum. Letting M3 = n · max(µ,R)∈S1×R
i∈n(‖hii(µ,R) · e3‖), M4 =
max(µ,R)∈S1×R
i∈n
(
κ2i (µ,R)
2
)
, we conclude that
Wrot ≤ ρ2n∑
i=1
[
−k1‖gii(µ,R)× e3‖2 +M4
k2
]
+ ρM3 −k21k22
n∑
i=1
‖Ωii −Ωii‖2,
as required. This concludes the proof of Lemma 5.4.2.
Chapter 6
Rendezvous of Kinematic Unicycles
with Local and Distributed
Feedbacks
In this chapter, we present a solution to the rendezvous control problem for kinematic
unicycles (RP− U) introduced in Section 3.2. The feedbacks will be constructed out of
the integrator consensus control primitive fi((xij)j∈Ni) in (4.10).
6.1 Solution of the Rendezvous Control Problem (RP− U)
Let Fi(G, ρ1, ρ2) denote the set of integrator consensus controllers fi((xij)j∈Ni) = fi(yi)
in (4.10) such that aij > 0 for all j ∈ Ni and 0 < ρ1 < aij < ρ2 with sensor digraph G.
We use f(yi) to construct the feedbacks
ui = u⋆i (yii) = ‖fi(yii)‖fi(yii) · e1,
ωi = ω⋆i (yii) = −kfi(yii) · e2, i ∈ n.
(6.1)
The result below states that for sufficiently large k, the feedbacks in (6.1) solve RP-U
123
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 124
if the network of unicycles has a sensor digraph containing a globally reachable node.
Theorem 6.1.1. RP-U is solvable for system (2.6), (2.7) if and only if the sensor digraph
G contains a globally reachable node, in which case a solution is the following. For each
positive integer n, and each ρ1, ρ2 such that 0 < ρ1 < ρ2, there exists k⋆ > 0 such that
for all k > k⋆, for all sensor digraphs G with n nodes containing a globally reachable
node, and for all linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n, feedback (6.1) solves RP-U.
Theorem 6.1.1 states that, if the sensor digraph G contains a globally reachable node,
then for a sufficiently large gain k, controller (6.1) solves RP-U. The lower bound on k is
uniform over all sensor digraphs with n nodes, and all consensus controllers fi with gains
aij in a fixed compact interval [ρ1, ρ2]. In practice this implies that one can tune the
controller (6.1) without knowing the sensor digraph G nor the controller parameters aij .
All that is required is to know bounds on these parameters. Moreover, it is clear that
on the rendezvous manifold Γ, yi = 0 for all i ∈ n and it follows that the requirement
(u⋆i , ω⋆i )|Γ = (0, 0) holds.
Remark 6.1.2. Consider a single kinematic unicycle (x1, θ1) ∈ R2 × S1 with inputs
(u1, ω1). As a consequence of Theorem 6.1.1, the controller
u1 = −‖x1‖x1 · R1e1,
ω1 = x1 ·R1e2
globally asymptotically stabilizes the set (x1, θ1) ∈ R2×S1 : x1 = 0. Note that no high
gain is required in this case.
The necessity portion of Theorem 6.1.1 was proved in (Lin et al., 2005). The suffi-
ciency part, namely the fact that the feedback (6.1) solves RP-U, is proved in Section 6.2.
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 125
The feedback in (6.1) is very similar in form to the one in (Zheng et al., 2011) given by,
u⋆i (yii) = fi(y
ii) · e1,
ω⋆i (yii) = fi(y
ii) · e2, i ∈ n.
(6.2)
with aij = 1 for all j ∈ Ni. The main difference in (6.1) is the extra multiplicative factor
‖fi(yii)‖ in the control u⋆i (yii) and the control gain k chosen sufficiently large. While the
feedback in (6.2) achieves rendezvous for undirected and connected graphs, the solution
cannot be generalized to the broad class of directed graphs considered in this chapter in
that, as shown in (Zheng et al., 2009), when the sensor digraph is a directed ring, the
feedback (6.2) drives the unicycles to a circular formation instead of achieving rendezvous.
Our solution in (6.1) can be viewed as an adaptation of the controller in (6.2) that allows
for rendezvous with directed graph topologies containing a globally reachable node.
In the presence of link failures in the sensor digraph, as long as the resulting graph
after the last failure maintains a globally reachable node, the presented solution in (6.1)
will still achieve rendezvous. It remains an open problem to solve RP-U with saturated
feedback. If the rotational control gain k > k⋆ results in feedbacks that are too large,
the magnitude of the gains aij can be reduced appropriately to avoid actuator saturation
over any compact set of initial conditions. This, however, would slow the convergence
rate.
ConsensusControl
RotationalControl
robot
Sensors
SpeedControl
Figure 6.1: Block diagram of the rendezvous control system for robot i.
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 126
The proposed control architecture is illustrated in the block diagram of Figure 6.1.
There are two nested loops. The outer loop treats each robot as a single-integrator driven
by the linear consensus controller,
xi = fi(yi), i ∈ n. (6.3)
The set (xi)i∈n ∈ R2n : xij = 0, i, j ∈ n is globally asymptotically stable for (6.3) if
the sensing digraph has a globally reachable node (Moreau, 2004). The signal fi(yi(t)) is
computed in the body frame Bi, and used as a reference signal for the inner-loop thrust
and rotational controllers that assign the unicycle control inputs in (6.1). The intuition
behind these controllers is shown in Figure 6.2. The speed input ui is the dot product
ui = u⋆i (yii) = ‖fi(yii)‖fi(yii) · e1. This is the projection of the reference ‖fi(yi)‖fi(yi)
onto the heading axis bix of robot i. The angular speed, on the other hand, is propor-
tional to the dot product between the reference fi(yi) and the second body axis biy. In
Figure 6.2, one can see that ωi = ω⋆i (yii) = −kfi(yii) · e2 = −k‖fi(yi)‖ sin(φi) acts to
reduce the angle φi between bix and fi(yi) with a rate proportional to the magnitude
of fi. Together, these control inputs drive the robot velocity uibix approximately to the
reference ‖fi(yi)‖fi(yi). The convergence is approximate because the control inputs do
not depend on the time derivative of fi. It is the difference in angle between uibix and
‖fi(yi)‖fi(yi) as opposed to the difference in magnitude that is important for obtaining
rendezvous. Since ‖fi(yi)‖fi(yi) is homogeneous of degree two, as the robots approach
consensus, ω⋆i (yii) converges to zero slower than u⋆i (y
ii). This allows ω
⋆i (y
ii) to exert suffi-
cient control authority even as the robots converge to consensus, closing the gap between
the vectors uibix and ‖fi(yi)‖fi(yi).
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 127
Figure 6.2: Illustration of the control inputs ui = u⋆i (yii) and ωi = ω⋆i (y
ii) in (6.1).
6.1.1 Simulation Results
We consider a group of five robots with sensor digraph in Figure 6.3. For the feedback
in (6.1), we pick aij = 0.05 for all j ∈ Ni. The control gain k is chosen to be k = 0.2. The
initial conditions of the robots are shown in Table 6.1. The simulation is presented in
Figure 6.4(a) and the control inputs for the five vehicles are plotted in Figure 6.5 showing
reasonable linear and rotational speeds. In Figure 6.4 the trajectories corresponding to
the feedback in (Lin et al., 2005), the feedback in (Zheng et al., 2011), and our solution are
compared. Notice that some curves in the figure have cusps although the actual solution
is C1. These cusps, however, are present only because the state is being projected into
the inertial (ix, iy) plane, i.e., the angular states θ are not illustrated. The simulations
are run with the initial conditions in Table 6.1 and the sensing digraph in Figure 6.3. It
can be seen that the proposed feedback has practical advantages over the time-varying
feedback in (Lin et al., 2005). The proposed feedback induces a more natural behaviour
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) causes the unicycles to converge to a circular formation instead
of rendezvous. Rendezvous is guaranteed only for undirected graphs in this case.
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 128
3 1
2
4
5
Figure 6.3: Sensor digraph used in the simulation results.
Table 6.1: Simulation Initial Conditions
Vehicle i xi(0) (m) θi(0) (rad)1 (0, 10) 02 (−10,−10) 2π/53 (−50, 10) 4π/54 (−10, 0) 6π/55 (10, 0) 8π/5
6.2 Proofs
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
d
dtW (γ, a, XB, θB) ≤ −σV (γ,XB) + Φ(a, XA, θ), σ > 0, (6.12)
for all a ∈ K where Φ(a, XA, θ) is continuous with respect to its arguments and Φ(a, 0, θ) =
0.
The proof of Lemma 6.2.3 is presented in Section 6.2.3. Select the positive real-valued
functions γi(a) as in Lemma 6.2.3.
Consider l⋆ in Lemma 6.2.3. We will now show that choosing k > l⋆ implies ΓA∪B
is globally asymptotically stable for ΣA∪B. The proof will make use of the reduction
theorem (Theorem 4.1.2). We will first show that all solutions of the closed-loop sys-
tem are bounded. The rotation matrices live in a compact set, therefore we only need
to show that the states XA∪B = (Xi)i∈A∪B are bounded. Since A is isolated, ΣA is an
autonomous subsystem and by assumption, ΓA = (XA, θA) ∈ XA ×TnA : XA = 0 (com-
pact), is globally asymptotically stable. Therefore, XA is bounded. From the inequality
W (γ, a, XB, θB) ≥ α⋆√
V (γ,XB) in part (iii) of Lemma 6.2.2, to show boundedness of
V (γ,XB), it suffices to show thatW (γ, a, XB, θB) is bounded. Boundedness of V (γ,XB),
in turn, implies boundedness of XB. From the bound on the derivative of W in (6.12),
and by Lemma 6.2.2 we obtain
d
dtW (γ, a, XB, θB) ≤ −σW (γ, a, XB, θB)
2
(2α)2+ Φ(a, XA, θ),
σ > 0. Since XA is bounded and θ ∈ Tn lies on a compact set, it holds that Φ(a, XA, θ)
is bounded and therefore W is bounded, which implies that XB is bounded. Therefore
XA∪B is bounded, as claimed. Now define the set, Λ := (XA∪B, θA∪B) ∈ XA∪B × TnA∪B :
XA = 0. Since the set ΓA is globally asymptotically stable for system ΣA and XA∪B is
bounded, it holds that Λ is globally asymptotically stable for ΣA∪B.
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 137
To show that the set ΓA∪B, which is compact, is globally asymptotically stable for
the system ΣA∪B, it suffices to show that ΓA∪B is globally asymptotically stable relative
to Λ. On the set Λ, Φ(a, XA, θ) is equal to zero and the derivative ofW is therefore given
by
d
dtW (γ, a, XB, θB) ≤ −σW (γ, a, XB, θB)
2
(2α)2, σ > 0.
By Lemma 6.2.2, all level sets of W (γ, a, XB, θB) are compact and W−1(0) = (XB, θB) :
XB = 0. This implies ΓA∪B is globally asymptotically stable relative to the set Λ. By
Theorem 4.1.2, ΓA∪B is globally asymptotically stable for ΣA∪B. This completes the
proof.
6.2.3 Proof of Lemmas
Throughout this section we will make use of functions µi and µ defined as follows. Define
the continuous functions µ(γ,XB) := XB/√
V (γ,XB), and µi(γ,XB) := Xi/√
V (γ,XB), i ∈
B. Since the numerator and denominator are both homogeneous of degree one, these
functions are both homogeneous of degree zero with respect to XB. Therefore, the im-
ages satisfy µ(γ,XB\0) = µ(γ, S2r−1 ∩ XB) and µi(γ,XB\0) = µi(γ, S2r−1 ∩ XB), where
S2r−1 is the unit sphere in R2r where r is the number of agents in B. Since µ, µi and
γ(a) are continuous functions and the sets K and S2r−1 ∩ XB are compact, the images
µ(γ(K),XB\0) and µi(γ(K),XB\0) are compact sets.
Proof of Lemma 6.2.2
Recall the definition of W (γ, a, XB, θB),
W = α√
V (γ,XB) +∑
i∈B
f ii (a, Xi, θi) · e1 =√
V (γ,XB)
(
α +
∑
i∈B fii (a, Xi, θi) · e1
√
V (γ,XB)
)
.
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 138
Using the fact that f ii (a, Xi, θi) is homogeneous with respect to its second argument, we
have
W =√
V (γ,XB)
(
α +∑
i∈B
f ii (a, µi(γ,XB), θi) · e1)
.
Since f ii is continuous, µi(γ,XB) is bounded, and θB ∈ TnB, a compact set, it follows that
the function∑
i∈B
∣
∣
∣f ii (a, µi(γ,XB), θi) · e1
∣
∣
∣
has a bounded supremum. Accordingly, let
α⋆ = sup(a,XB ,θB)∈K×XB×Tn
B
∑
i∈B
∣
∣
∣f ii (a, µi(γ(a), XB), θi) · e1
∣
∣
∣.
For all α > 2α⋆, we have W (γ, a, XB, θB) ≥ W (γ, a, XB, θB) := α⋆√
V (γ,XB) ≥ 0.
This inequality implies that W ≥ 0 and W−1(0) ⊂ W−1(0). But W = 0 if and only if
V (γ,XB) = 0 (i.e., XB = 0). Thus W−1(0) ⊂ (XB, θB) : XB = 0. Conversely, on the
set (XB, θB) : XB = 0, XB = 0 and hence W = 0, and therefore (XB, θB) : XB =
0 ⊂W−1(0). It follows that W−1(0) = (XB, θB) : XB = 0 proving part (i).
For part (ii), note that for all c > 0, Wc ⊂ W (γ, a, XB, θB) ≤ c. Since the sublevel
sets of W are compact and θB ∈ TnB, a compact set, the set Wc is bounded. Continuity
of W implies that Wc is compact.
For part (iii), it has already been shown that W (γ, a, XB, θB) ≥ α⋆√
V (γ,XB). It
also holds that
W =√
V (γ,XB)
(
α +∑
i∈B
f ii (a, µi(γ,XB), θi) · e1)
≤√
V (γ,XB) (α+ α) ≤ 2α√
V (γ,XB).
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 139
Proof of Lemma 6.2.3
We first compute inequalities for Wtran and Wrot for system (6.8) and (6.9). We then
combine them to derive (6.12). Consider unicycle i ∈ B. The dynamics of Xi in (6.8)
are split into two terms, for neighboring robots j ∈ Ni ∩A and j ∈ Ni ∩ B respectively,
Xi =∑
j∈Ni∩A
aij(ujRje1 − uiRie1)
Ai+
∑
j∈Ni∩B
aij(ujRje1 − uiRie1)
Ai. (6.13)
For simplicity of notation, we drop the arguments of gi(a, Xi) and gii(a, Xi, θi). Adding
and subtracting the term,
∑
j∈Ni∩Baij(gj − gi)−
∑
j∈Ni∩Aaij gi
Ai
to (6.13) yields,
Xi =
∑
j∈Ni∩Baij(gj − gi)−
∑
j∈Ni∩Aaij gi
Ai+
∑
j∈Ni∩Baij(ujRje1 − uiRie1)
Ai
−∑
j∈Ni∩Baij(gj − gi)
Ai+
∑
j∈Ni∩AaijujRje1
Ai+
∑
j∈Ni∩Aaij(gi − uiRie1)
Ai
=
∑
j∈Ni∩Baij(gj − gi)−
∑
j∈Ni∩Aaij gi
Ai+
∑
j∈Ni∩Baij(ujRje1 − gj)
Ai
−∑
j∈Ni∩Baij(uiRie1 − gi)
Ai+
∑
j∈Ni∩AaijujRje1
Ai+
∑
j∈Ni∩Aaij(gi − uiRie1)
Ai.
Replacing uj and ui by the assigned feedbacks in (6.1) and using the identity Rigii = gi
then,
Xi = ai(a, XB) + bi(a, XB, θ) + ci(a, XB, θ) + di(a, XA, θ),
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 140
where,
ai(a, XB) :=
∑
j∈Ni∩Baij(gj − gi)−
∑
j∈Ni∩Aaij gi
Ai
bi(a, XB, θ) :=
∑
j∈Ni∩BaijRj((g
jj · e1)e1 − gjj )
Ai−∑
j∈Ni∩BaijRi((g
ii · e1)e1 − gii)
Ai
ci(a, XB, θ) :=
∑
j∈Ni∩AaijRi(g
ii − (gii · e1)e1)
Ai
di(a, XA, θ) :=
∑
j∈Ni∩Aaij(g
jj · e1)Rje1
Ai.
The time derivative of Wtran =√
V (γ,XB) in (6.10) yields,
Wtran =1
2√V
[
∑
i∈B
∂V (γ,XB)
∂Xi(ai(a, XB) + bi(a, XB, θ) + ci(a, XB, θ))
]
+1
2√V
∑
i∈B
∂V (γ,XB)
∂Xidi(a, XA, θ).
(6.14)
The derivative of the first term is considered in Claim 6.2.2.
Claim 6.2.2. There exist gains γi(a) > 0 in (6.10) that are continuous functions of a
and a continuous function κ(γ, a, XB), negative definite and homogeneous of degree three
with respect to XB, such that
∑
i∈B
∂V (γ,XB)
∂Xiai(a, XB) ≤ κ(γ, a, XB).
The proof of Claim 6.2.2 is presented at the end of this chapter. Let the gains γi be
as in Claim 6.2.2. The derivative of the remaining terms in the square brackets of (6.14)
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 141
satisfies,
∑
i∈B
∂V (γ,XB)
∂Xi
(bi(a, XB, θ) + ci(a, XB, θ))
≤∑
i∈B
1
Ai
∂V (γ,XB)
∂Xi
[
∑
j∈Ni∩B
aij∥
∥(gjj · e1)e1 − gjj∥
∥
+∑
j∈Ni∩B
aij∥
∥(gii · e1)e1 − gii∥
∥+∑
j∈Ni∩A
aij∥
∥gii − (gii · e1)e1∥
∥
]
≤∑
i∈B
1
Ai
∂V (γ,XB)
∂Xi
[
∑
j∈Ni∩B
aij∥
∥(gjj · e1)e1 − gjj∥
∥+∑
j∈Ni
aij∥
∥(gii · e1)e1 − gii∥
∥
]
.
We claim that ‖(gii(a, Xi, θi) · e1)e1 − gii(a, Xi, θi)‖ = |gii(a, Xi, θi) · e2|. Indeed, writing
gii = (gii · e1)e1 + gii − (gii · e1)e1, we have gii · e2 = (gii − (gii · e1)e1) · e2. Since the
vector gii − (gii · e1)e1 is parallel to e2, |(gii − (gii · e1)e1) · e2| = ‖gii − (gii · e1)e1‖, so that
|gii · e2| = ‖gii − (gii · e1)e1‖. Then,
∑
i∈B
∂V (γ,XB)
∂Xi(bi(a, XB, θ) + ci(a, XB, θ))
≤∑
i∈B
∥
∥
∥
∥
∂V (γ,XB)
∂Xi
∥
∥
∥
∥
(
∑
j∈B
∣
∣gjj · e2∣
∣+ n∣
∣gii · e2∣
∣
)
which is homogeneous of degree three with respect to XB since ∂V (γ,XB)/∂Xi is homo-
geneous of degree one and gii is homogeneous of degree two with respect to XB for all
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 142
i ∈ B. The last term in (6.14) satisfies,
1
2√
V (γ,XB)
∑
i∈B
∂V (γ,XB)
∂Xidi(a, XA, θ)
≤ 1
2√
V (γ,XB)
∑
i∈B
1
Ai
∂V (γ,XB)
∂Xi
∑
j∈Ni∩A
aij(gjj · e1)Rje1
≤ 1
2√
V (γ,XB)
∑
i∈B
1
Ai
∥
∥
∥
∥
∂V (γ,XB)
∂Xi
∥
∥
∥
∥
∑
j∈Ni∩A
aij‖gjj‖
≤∑
i∈B
sup(a,XB)∈K×XB
1
2√
V (γ(a), XB)
∥
∥
∥
∥
∂V (γ(a), XB)
∂Xi
∥
∥
∥
∥
∑
j∈Ni∩A
‖gjj‖
:= Φtran(XA, θ).
(6.15)
The bounded supremum of
1√
V (γ(a), XB)
∥
∥
∥
∥
∂V (γ(a), XB)
∂Xi
∥
∥
∥
∥
exists because this term is homogeneous of degree 0 with respect to XB and γ(a) is a
continuous function of a. Moreover, XA = 0 implies that ‖gjj‖ = 0 for all j ∈ A and
hence Φtran(0, θ) = 0. Everything together, (6.14) yields,
Wtran ≤1
2√
V (γ,XB)
[
κ(γ, a, XB) +∑
i∈B
∥
∥
∥
∥
∂V (γ,XB)
∂Xi
∥
∥
∥
∥
(
∑
j∈B
∣
∣gjj · e2∣
∣+ n∣
∣gii · e2∣
∣
)]
+ Φtran(XA, θ).
(6.16)
Since κ(γ, a, XB) is homogeneous of degree three with respect to XB. We can write,
κ(γ, a, XB) =
√
V (γ,XB)V (γ,XB)√
V (γ,XB)V (γ,XB)κ(γ, a, XB)
=√
V (γ,XB)V (γ,XB)κ
(
γ, a,XB
√
V (γ,XB)
)
=√
V (γ,XB)V (γ,XB)κ (γ, a, µ(γ,XB)) .
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 143
Analogous operations can be performed with the remaining term in the square bracket
of (6.16) yielding,
Wtran ≤V (γ,XB)
2
[
κ(γ, a, µ(γ,XB)) +∑
i∈B
∥
∥
∥
∥
∂V (γ, µ(γ,XB))
∂Xi
∥
∥
∥
∥
·(
∑
j∈B
∣
∣gjj (a, µj(γ,XB), θj) · e2∣
∣+ n∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣
)]
+ Φtran(XA, θ).
Since κ is continuous and negative definite, a lies on a compact set K and µ(γ(a), XB)
lies on a compact set S1, it follows that κ(γ(a), a, µ(γ(a), XB))/2 has bounded maximum
−M2 < 0. Similarly, the function
∥
∥
∥
∥
∂V (γ(a), µ(γ(a), XB))
∂Xi
∥
∥
∥
∥
has a maximum. Letting
M1 := n max(a,ψ)∈K×S1
i∈B
∥
∥
∥
∥
∂V (γ(a), ψ)
∂Xi
∥
∥
∥
∥
yields,
Wtran ≤V[
−M2 +M1
2n
∑
i∈B
(
∑
j∈B
∣
∣gjj (a, µj(γ,XB), θj) · e2∣
∣+ n∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣
)]
+ Φtran(XA, θ)
≤V[
−M2 +M1
2n
∑
i∈B
(
n∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣+ n∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣
)
]
+ Φtran(XA, θ)
≤V[
−M2 +M1
∑
i∈B
∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣
]
+ Φtran(XA, θ)
(6.17)
for all a ∈ K. This proves the first inequality. We now turn to the second. Recall the
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 144
definition of Wrot, Wrot(a, XB, θB) =∑
i∈B fii (a, Xi, θi) · e1. The time derivative of Wrot
along the vector field in (6.8), (6.9) is Wrot =∑
i∈B
(
ddtf ii
)
·e1. To express (d/dt)f ii , recall
that f ii (a, Xi, θi) = R−1i fi(a, Xi). Then, d
dtf ii =
(
ddtR−1i
)
fi + R−1i
dfidt. We will denote the
derivative of fi(a, Xi) = AiXi by,
hi(a, X, θ) := (d/dt)fi(a, Xi)
= Ai (ai(a, XB) + bi(a, XB, θ) + ci(a, XB, θ) + di(a, XA, θ))
where the first three terms are homogeneous of degree two with respect to XB and the last
term is homogeneous of degree two with respect to XA. Consistently with our notational
convention, we will let hii(a, X, θ) := R−1i hi(a, X, θ). Returning to the derivative of f ii ,
we haved
dtf ii = −(ωi)
×R−1i fi(a, Xi) +R−1
i hi(a, X, θ)
= −
0 −ωiωi 0
f ii (a, Xi, θi) + hii(a, X, θ).
We substitute the above identity in the expression for Wrot,
Wrot =∑
i∈B
−e⊤1
0 −ωiωi 0
f ii (a, Xi, θi) + hii(a, X, θ) · e1
=∑
i∈B
(
(f ii (a, Xi, θi) · e2)ωi + hii(a, X, θ) · e1)
.
Substituting the feedback ωi = −k(f ii (a, Xi, θi) · e2) and taking norms, we arrive at the
inequality
Wrot ≤∑
i∈B
[
− k∣
∣
∣f ii (a, Xi, θi) · e2
∣
∣
∣
2
+ hii(a, X, θ) · e1]
.
This gives,
Wrot ≤[
−k∑
i∈B
∣
∣
∣f ii (a, Xi, θi) · e2
∣
∣
∣
2
+ ℓ(a, XB, θ)
]
+ Φrot(a, XA, θ)
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 145
where
ℓ(a, XB, θ) :=∑
i∈B
AiR⊤i (ai(a, XB) + bi(a, XB, θ) + ci(a, XB, θ)) · e1
and Φrot(a, XA, θ) :=∑
i∈B AiR⊤i di(a, XA, θ) · e1. Note that
∑
i∈B
∣
∣
∣f ii (a, Xi, θi) · e2
∣
∣
∣
2
and
ℓ(a, XB, θ) are homogeneous of degree two with respect toXB. The function Φrot(a, XA, θ)
does not depend on XB and Φrot(a, 0, θ) = 0. This yields,
Wrot ≤V (γ,XB)
[
−k∑
i∈B
∣
∣
∣f ii (a, Xi/
√
V (γ,XB), θi) · e2∣
∣
∣
2
+ ℓ(a, XB/√
V (γ,XB), θ)
]
+ Φrot(a, XA, θ)
≤V (γ,XB)
[
−k∑
i∈B
∣
∣
∣f ii (a, µi(γ,XB), θi) · e2
∣
∣
∣
2
+ ℓ(a, µ(γ,XB), θ)
]
+ Φrot(a, XA, θ).
|ℓ(a, µ(γ(a), XB), θ)| has a bounded supremum. Letting
M3 = sup(a,ψ,θ)∈K×S1×Tn
(|ℓ(a, ψ, θ)|) ,
we conclude that,
Wrot ≤V (γ,XB)
[
−k∑
i∈B
∣
∣
∣f ii (a, µi(γ,XB), θi) · e2
∣
∣
∣
2
+M3
]
+ Φrot(a, XA, θ) (6.18)
for all a ∈ K. By using the inequalities (6.17) and (6.18) we now bound the derivative
of W to derive (6.12). Notice that
W =αWtran + Wrot
≤V (γ,XB)
[
−αM2 + αM1
∑
i∈B
∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣
−k∑
i∈B
∣
∣
∣f ii (a, µi(γ,XB), θi) · e2
∣
∣
∣
2
+M3
]
+ Φ(a, XA, θ),
Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 146
where Φ(a, XA, θ) := αΦtran(XA, θ) + Φrot(a, XA, θ). Choose α > 3M3/M2. This implies,
W ≤ V
[
−2M3 + αM1
∑
i∈B
∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣− k∑
i∈B
∣
∣
∣f ii (a, µi(γ,XB), θi) · e2
∣
∣
∣
2]
+ Φ(a, XA, θ).
(6.19)
Since f ii (a, Xi, θi) is homogeneous with respect to Xi, we have,
f ii (a, µi(γ,XB), θi) =
√
‖gii(a, µi(γ,XB), θi)‖‖gii(a, µi(γ,XB), θi)‖
gii(a, µi(γ,XB), θi).
Plugging the last expression into (6.19) yields
W ≤ V
[
−2M3 + αM1
∑
i∈B
∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣
−k∑
i∈B
1
‖gii(a, µi(γ,XB), θi)‖∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣
2
]
+ Φ(a, XA, θ).
Since gii(a, µi(γ,XB), θi) is a continuous function of its arguments and µi(γ,XB) is com-
pact, |gii(a, µi(γ,XB), θi)| has a maximum M4. This implies,
W ≤V[
−2M2 + αM1
∑
i∈B
∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣
−k∑
i∈B
1
M4
∣
∣gii(a, µi(γ,XB), θi) · e2∣
∣
2
]
+ Φ(a, XA, θ).
Denote βi(a, µi(γ,XB), θi) := |gii(a, µi(γ,XB), θi) · e2|, and β := (βi(a, µi(γ,XB), θi))i∈B.
Then,
W ≤ V
[
−2M2 + αM11⊤β − k
M4|β|2
]
+ Φ(a, XA, θ)
= V[
1⊤ β⊤]
−2M2
nI αM1
2I
αM1
2I −k
M4I
1
β
+ Φ(a, XA, θ).
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
parallel formation flocking manifold Γpfb , thus solving PFP-B.
3. (LPP) For any (d, r0, p, w) ∈ LP the control inputs in (8.1), strictly functions of
(yii, ϕi, pi, (c⋆(xi) − xi)
i), with α > 0, µii = h(xi)i and h(·) a line path following
controller for single integrators defined in (4.16) almost globally asymptotically
stabilize the formation line path following manifold Γlp , thus solving LPP.
The terms in (8.1) containing fi(yii) have the objective of achieving consensus on
endpoints xi while those containing µii have the objective of synchronizing the headings
θi in addition to stabilizing the desired path C(r0, p) in the case of path following. Note
that for path following, the controller in (8.1) will enforce unicycle 1 to follow the line
C(r0, p). If the line is to be followed by unicycle j as opposed to unicycle 1, one simply
changes the value of βi to βi = −d11i · e2 + d11j · e2 for all i ∈ n.
The control inputs in (8.1) for unicycle i are therefore strictly functions of (yii, ϕi),
(αj , βj)j∈Ni and µii where µii is a function of pi in the case of PFP− B. In the case of
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 175
LPP, µii (with k0 = 1 for simplicity) satisfies
µii = h(xi)i = (c⋆(xi)− xi)
i + wpi
= (r0 − xi)i − ((r0 − xi)
i · pi)pi + wpi
= (r0 − xi)i − αie1 − βie2 − ((r0 − xi)
i · pi)pi + ((αie1 + βie2) · pi)pi + wpi
= (c⋆(xi)− xi)i − αie1 − βie2 + ((αie1 + βie2) · pi)pi + wpi
which is a function of pi and (c⋆(xi)−xi)i, the displacement between xi and its orthogonal
projection c⋆(xi) on the path represented in body frame.
8.2 Solutions to Formation Problems with Final Cir-
cular Motion (CFP and CPP)
In this section we present solutions to the formation problems with final circular motion.
Recall from Section 3.3.3 that xi = xi+ δi = xi+βiRie2 where βi is the radius of circular
motion of unicycle i. Recall the following set definitions from Chapter 3
Γcf =
(x, θ) ∈ R2n × T
n : x1i = 0, θ1i = ρi(d, β1), i ∈ 2 :n
Γcp = (x, θ) ∈ Γcf : x1 = c
where Γcf corresponds to the set where the perceived centres xi coincide and the relative
headings satisfy θ1i = ρi(d, β1) for i ∈ 2 :n. Γcp follows by also imposing that the
perceived centres xi coincide at the desired circle center c. For gains r,K, k > 0, choose
the feedback law as,
ui = u⋆i (yii, ϕi, νi) = ui + (βi − r)ω⋆i (y
ii, ϕi, νi)
ωi = ω⋆i (yii, ϕi, νi) =
uir+Kϕ(νi)νi, i ∈ n
(8.2)
where
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 176
• ui := rw + kgi((θij − ρij(d, β1))j∈Ni, 1) where ρij(d, β1) := ρj(d, β1)− ρi(d, β1)
• gi(·) is an almost global rotational integrator consensus controller in (4.24)
• ϕ(νi) := 1/(1 + ‖νi‖) is a decentralized saturation function, and
• νi will be defined in the main theorem for CFP and CPP given below in Theo-
rem 8.2.1.
Theorem 8.2.1 (Solutions to CFP and CPP). Consider system (2.6), (2.7) with con-
nected, undirected sensor graph G and any parameters aij = aji > 0 in (4.10) and
bij = bji > 0 in (4.24) satisfying (bij)(i,j)∈E ∈ (R+)|E|\Nb as in Theorem 4.3.2.
1. (CFP) For any (d, β1, w) ∈ CF, there exist K⋆, k⋆ > 0 such that for any K ∈
(0, K⋆), k ∈ (0, k⋆), for any r > 0, the local and distributed control inputs in (8.2)
with νi = −fi(yii) · e1 and fi(·) a single integrator consensus controller defined
in (4.10) almost globally asymptotically stabilize the circular formation flocking
manifold Γcf , thus solving CFP.
2. (CPP) For any (d, β1, c, w) ∈ CP there exist K⋆, k⋆ > 0 such that for any K ∈
(0, K⋆), k ∈ (0, k⋆), for any r > 0, the control inputs in (8.2) with νi = −(c−xi)i·e1,
strictly functions of (yii, ϕi, (c − xi)i), almost globally asymptotically stabilize the
formation circle path following manifold Γcp, thus solving CPP.
The proof of Theorem 8.2.1 is given in Section 8.4. The control inputs in (8.2) for
unicycle i are strictly functions of (yii, ϕi), the formation parameters (βj)j∈Ni and νi. In
the case of flocking, νi is a function of yii and (βj)j∈Ni while in the case of path following,
νi is a function of (c − xi)i = (c − xi)
i − βie2. The feedbacks in (8.2) are based on the
control solution in (El-Hawwary and Maggiore, 2013a), however, unlike (El-Hawwary
and Maggiore, 2013a), they allow for possibly different radii βi of each unicycle so that
formations are not restricted to a common circle. Also, the term gi in ui corresponds
to an almost global rotational integrator consensus controller in (4.24) rather than a
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 177
Kuramoto model in (4.23), allowing for an almost global result. An adaptation of the
controller in (El-Hawwary and Maggiore, 2013a) is discussed in Section 8.2.1.
Remark 8.2.2. The solutions presented above for line and circle path following assume
that all unicycles in the ensemble sense the desired path. A more general result can be
obtained for the case where only a non-empty subset of unicycles S ⊂ V sees the path
assuming that at least one unicycle in S is a globally reachable node for G. For undirected
graphs, S can be any non-empty subset of V. For line path following choose the feedback
law as in (8.1) with µii = wpi for i /∈ S and µii = h(xi)i = k0(c
⋆(xi) − xi)i + wpi for
i ∈ S. Measurement of pi is still required by all agents in this case. For circle path
following choose the feedback law as in (8.2) with νi = −f(yii) · e1 for i /∈ S, and
νi = −f(yii) · e1 − (c− xi)i · e1 for i ∈ S.
8.2.1 Formation control of unicycles on a common circle
The solutions in Theorem 8.2.1 are based on an intermediate result presented in this
section that makes unicycles converge almost globally to a common circle of desired radius
r > 0, i.e., βi = r for all i ∈ n, and with desired inter-agent spacings θ1i = ρi(d, β1) ∈ S1
for i ∈ 2 :n. For n kinematic unicycles in (2.6), (2.7) consider the feedbacks,
ui = u⋆i (yii, ϕi, νi) = ui
ωi = ω⋆i (yii, ϕi, νi) =
uir+Kϕ(νi)νi, i ∈ n
(8.3)
where xi = xi + rRie2, ϕ(νi) and ui are defined as in (8.2) and νi is defined in Theo-
rem 8.2.3 below. The feedbacks in (8.3) are a minor modification of those in (El-Hawwary
and Maggiore, 2013a) where the only difference is that the term gi in ui is an almost global
rotational integrator consensus controller in place of the Kuramoto consensus controller
used in (El-Hawwary and Maggiore, 2013a). The result in Theorem 8.2.3 shows that the
set where the n unicycles lie on a common circle of radius r with desired angular spacings
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 178
θ1i = ρi(d, β1) for i ∈ 2 :n is almost globally asymptotically stable. In (El-Hawwary and
Maggiore, 2013a), the corresponding result only shows asymptotic stability. Naturally,
the proof of Theorem 8.2.3, presented in Section 8.7, will use the same arguments as
in (El-Hawwary and Maggiore, 2013a) just with the additional requirement of showing
an almost global domain of attraction. Part 1 of Theorem 8.2.3 concerns flocking while
part 2 concerns path following where a specific circle centre c ∈ R2 is specified a priori
which was not considered explicitly in (El-Hawwary and Maggiore, 2013a).
Theorem 8.2.3. Consider the system of unicycles in (2.6), (2.7), radius r > 0, desired
angular speed w > 0, a set of desired angular offsets (ρi(d, β1))i∈2 :n ∈ Tn−1, an integrator
consensus controller fi(·) in (4.10) and a rotational integrator consensus controller gi(·)
in (4.24). Assume that the sensor graph G is connected and undirected and choose
bij = bji > 0 in (4.24) satisfying (bij)(i,j)∈E ∈ (R+)|E|\Nb as in Theorem 4.3.2. Then there
exist K⋆ ∈ (0, (w/2)) and k⋆ > 0 such that for all K ∈ (0, K⋆) and all k ∈ (0, k⋆) the
feedback laws in (8.3) with
1. νi = −f(yii)·e1 almost globally asymptotically stabilize the set Γcf with (u⋆i , ω⋆i )|Γcf
=
(wr, w)
2. νi = −(c − xi)i · e1, almost globally asymptotically stabilize the set Γcp with
(u⋆i , ω⋆i )|Γcp
= (wr, w).
8.3 Simulation Results
In this section, we present simulation results for formation control with parallel and cir-
cular final collective motions using the feedbacks presented in Section 8.1 and Section 8.2.
We consider a group of five unicycles with undirected graph shown in Figure 8.1 and with
desired triangular formation specified by d112 = (−10, 5), d113 = (−10,−5), d112 = (−20, 10)
and d112 = (−20,−10), illustrated in Figure 7.6.
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) ·
Rie1)Rie1 + (fi(yi) · Rie2)Rie2 = fi(yi) and (µi · Rie1)Rie1 + (µi · Rie2)Rie2 = µi yields,
˙xi = fi(yi) + µi
θi =1
αi(fi(yi) · Rie2 + µi · Rie2) , i ∈ n.
(8.8)
Since the closed loop system in (8.8) is globally Lipschitz, it has no finite escape times.
The path following controller in (4.16) can be written as in (4.17), i.e.,
h(xi) = k0(c⋆(xi)− xi) + wp, (8.9)
where c⋆(xi) is the orthogonal projection of the point xi onto C(r0, p). Therefore, in the
case of LPP, µi = h(xi) = k0(c⋆(xi) − xi) + wp. In the case of PFP-B, we adopt the
convention that c⋆(xi) = xi such that h(xi) = k0(c⋆(xi)− xi)+wp = wp. Therefore, with
this convention, µi = h(xi) also in this case. Letting p⊥ be a unit vector perpendicular
to p, consider the new states x‖1 := x1 · p, x⊥1 := (x1 − c⋆(x1)) · p⊥ = (r0 − x1) · p⊥,
x := (x1i)i∈2 :n ∈ R2(n−1) and θ = (θi)i∈n ∈ Tn defined under the diffeomorphism
F : R2n × Tn → R× R× R
2(n−1) × Tn
F (x, θ) = (x‖1, x
⊥1 , x, θ)
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 183
with smooth inverse yielding
x1 = x‖1p− (x⊥1 − r0 · p⊥)p⊥
(xi)i∈2 :n = (x1i + (x‖1p− (x⊥1 − r0 · p⊥)p⊥))i∈2 :n
θ = (θi)i∈n.
The system equations in (8.8) can be expressed in terms of new coordinates as
˙x‖1 = (f1(y1) + µ1) · p
˙x⊥1 = (f1(y1) + µ1) · p⊥ − c⋆(x1) · p⊥
˙x1i = fi(yi)− f1(y1) + µi − µ1
θi =1
αi(fi(yi) ·Rie2 + µi · Rie2), i ∈ n.
(8.10)
Note that fi(yi) is strictly a function of x, Ri is strictly a function of θ and it can be
shown that
µi = −k0[(x1 − c⋆(x1)) · p⊥]p⊥ − k0x1i + k0(x1i · p)p+ wp
= −k0x⊥1 p⊥ − k0x1i + k0(x1i · p)p+ wp
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
translational and absolute rotational coordinates (x, θ)Lj := ((x1i)i∈Lj\1, (θi)i∈Lj) ∈ Xj ,
given by
˙x1i = fi +w
|Ni|∑
j∈Ni
Rje1 − wp, i ∈ Lj\1
θi = ωi, i ∈ Lj,(8.16)
define an autonomous dynamical system on the reduced state-space Xj . Correspondingly,
one can define the reduced formation flocking manifold for Lj as ΓLj = (x, θ)Lj ∈ Xj :
x1i = 0, θi = θp, i ∈ Lj. The new coordinates are obtained under a differeomorphism
F : R2n × Tn → R2 × R2(n−1) × Tn, F (x, θ) = (x1, x, θ) and the state x1 was dropped
because it does not appear in the dynamics in (8.16) or the reduced flocking manifold
ΓLj . Since fi(yi) = 0 and p = Rie1 for all i ∈ Lj on ΓLj , the control inputs satisfy u⋆i = w
and ω⋆i = 0 for all i ∈ Lj.
Letting m be the total number of layers in the hierarchical digraph, the set Γpf
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 188
expressed in (x, θ)Lm ∈ Xm = R2(n−1) × Tn coordinates becomes
Γpf := ΓLm =
(x, θ)Lm ∈ R2(n−1) × T
n : x1i = 0, θi = θp, i ∈ Lm
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
equations of motion can be written as
˙x1i = 0
θ1i = (k/r)gi((θij − ρij(d, β1))j∈Ni, 1)− (k/r)g1((θ1j − ρ1j(d, β1))j∈N1, 1).
(8.21)
By Theorem 4.3.2, system (8.21) has a globally attractive set of isolated equilibria Γ1 =
A ∪ K relative to Γ2 in which the point K is asymptotically stable and the equilibria
in A are exponentially unstable. Then by Theorem 4.1.3, setting Γ2 = Γ2 and Γ1 =
Γ1, K is almost globally asymptotically stable. Therefore in original coordinates Γcf
corresponding to a formation flocking on a common circle of radius r, is almost globally
asymptotically stable.
Replace νi with νi = −(c − xi) · Rie1 and consider the system dynamics in (8.20).
Define the Lyapunov function V =∑n
i=1 ‖c− xi‖2 with derivative given by
V =n∑
i=1
rKϕ(νi)νi(c− xi) · Rie1
=
n∑
i=1
−rKϕ(νi)‖νi‖2 ≤ 0.
Since
θi =uir+Kϕ(νi)νi,
we have
θi ≥infθ∈Tn ui(θ)
r−K.
There exists k⋆ > 0 such that choosing k ∈ (0, k⋆) implies infθ∈Tn ui > rw/2, supθ∈Tn ui <
∞ for all i ∈ n and K < w/2. Then there exist ζ1, ζ2 > 0 such that 0 < ζ1 < θi < ζ2 for
all i ∈ n, for all time and initial conditions. Stability of Γ2 := (x, θ) ∈ R2n × Tn : xi =
c, i ∈ n holds from the fact that Γ2 corresponds to the minimum of V . Moreover, the
level sets of V are compact in (x, θ) coordinates and since V ≤ 0, the Krasovskii-LaSalle
invariance principle implies that νi → 0 as t → ∞. Using the fact that θi is bounded
Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 194
away from zero and using ideas from the proof of Proposition V.2. in (El-Hawwary and
Maggiore, 2013a), (xi − c) → 0 as t → ∞. Together with stability, this implies that Γ2
is globally asymptotically stable and, in turn, boundedness of the states (x, θ).
In the set Γ2, νi = 0 for all i ∈ n, the equation of motion for θi reduces to θi =
w + (k/r)gi((θij − ρij(d, β1))j∈Ni, 1), and the set Γcp is almost globally asymptotically
stable relative to Γ2. It can be shown using Theorem 4.1.3, as in the case of Γcf , that
Γcp is almost globally asymptotically stable.
Chapter 9
General Formation Path Following
In this chapter, we present a solution to the general formation path following control
problem (GPP) introduced in Section 3.3.4. The feedbacks will be constructed by com-
bining two control primitives discussed in Section 4.3: an integrator consensus controller
fi((xij)j∈Ni) in (4.10) and a path following controller for single integrators h(x) in (4.15).
In particular, h(x) in this chapter will be chosen as a single integrator path following
controller for smooth Jordan curves.
9.1 Solution to the General Formation Path Follow-
ing Problem (GPP)
In this section we present the solution to GPP. As with the other formation control
problems, we attach an offset vector δi(θi, φi) of fixed length to the body frame of each
unicycle i ∈ n. In particular, we let the auxiliary state −φi in (3.10) be the angle of
δi relative to the heading angle θi of unicycle i and let ‖δi‖ = ‖di‖ as illustrated in
Figure 9.1. Define the endpoint of the offset vector by xi(xi, θi, φi) := xi + δi(θi, φi) and
let x := (xi)i∈n. The endpoint xi(xi, θi, φi) will act as a local estimate of the formation
origin for unicycle i. Also as in the other formation control problems, we decompose
195
Chapter 9. General Formation Path Following 196
the vector δi into a component αi along the heading axis and βi along the perpendicular
axis of unicycle i, i.e., δi := αiRie1 + βiRie2. The offsets αi(φi), βi(φi) are given by
αi(φi) := ‖δi‖ cosφi and βi(φi) := −‖δi‖ sinφi. Unlike in the other formation control
problems, the components αi(φi) and βi(φi) will not be constant as they depend on the
auxiliary state φi which, in turn, depends on the curvature of the path to be followed
which is not constant in general. Define the angle σi := θi−φi as illustrated in Figure 9.1
and denote σ := (σi)i∈n. The angle σi represents the angle of the offset vector δi in inertial
frame.
Our first goal is to make the estimates xi achieve consensus and converge to C so
that all agents possess knowledge of a common formation origin. This configuration
corresponds to the set
Λ =
(x, θ, φ) ∈ R2n × T
n × Tn : x1i = 0, x1 ∈ C, i ∈ 2 :n
.
In the set Λ, the quantity τi := π(xi) is well-defined for all i ∈ n because xi ∈ C
for all i ∈ n, and the endpoint x1 coincides with the origin of a Frenet-Serret frame
(o(τ1), r(τ1), s(τ1)). Relative to this set, we will stabilize the set
Γ′gp :=
(x, θ, φ) ∈ Λ : R0(τ1)−1(xi − o(τ1)) = di, φi = φi(xi, σi), i ∈ n
⊂ Γgp, (9.1)
where Γgp is defined in (3.11). In the set Γ′gp, the formation is achieved at τ1. In the
set Λ, the condition ‖R0(τ1)−1(xi − o(τ1))‖ = ‖xi − o(τ1)‖ = ‖di‖ for Γ′
gp is satisfied
for all i ∈ n. However, the formation is not yet achieved as illustrated in Figure 9.2
where unicycle i can lie anywhere on a circle of radius ‖di‖ about o(τ1) but the condition
R0(τ1)−1(xi−o(τ1)) = −R0(τ1)
−1δi = di for Γ′gp is not satisfied. Or what is the same, the
angle between −r(τ1) and −δi is not necessarily equal to ψi. This requirement can be
written as an additional constraint on Λ as illustrated in Figure 9.3 in which the angle
σi must equal σi := θr(τi) − ψi where θr(τi) is the angle of r(τi) in inertial frame. The
Chapter 9. General Formation Path Following 197
manifold Γ′gp can therefore be expressed alternatively as follows,
Γ′gp := (x, θ, φ) ∈ Λ : σi = θr(τi)− ψi, φi = φi(xi, σi), i ∈ n . (9.2)
Figure 9.1: Illustration of the angle φi and angle σi = θi − φi.
Figure 9.2: Illustration of the set Λ in which unicycle i can lie anywhere on the dashedcircle. The desired position of unicycle i is marked by ⋆.
Let fi(·) be a single integrator consensus controller in (4.10) and h(·) be a path
following controller for C in (4.15) such that r(x) := r π(x) defines the direction of
Chapter 9. General Formation Path Following 198
Figure 9.3: Illustration of angle condition σi = θr(τi)− ψi.
motion for x ∈ C. The angle of the vector field h at the point x ∈ R2 is denoted
θh(x) := atan2(h(x) · e2, h(x) · e1).
The partial derivatives of θh(x) relative to the components of x are well defined as long
as ‖h(x)‖ 6= 0. This is never the case in a neighborhood of C since, by assumption,
‖h(x)‖ = w 6= 0 for all x ∈ C. We can therefore define functions
θ′h(x) :=dθh(x)
dx· h(x)
θ′′h(x) :=dθ′h(x)
dx· h(x).
(9.3)
Letting
ω⋆i (yii, xi) :=
1
αi
(
kfi(yii) · e2 + h(xi)
i · e2)
, (9.4)
choose the feedback law for unicycle i as,
u⋆i (yii, xi) = kfi(y
ii) · e1 + βiω
⋆i (y
ii, xi) + h(xi)
i · e1,
ω⋆i (yii, xi, σi, φi) = ω⋆i ,piv(y
ii, xi, σi, φi) + ω⋆i (y
ii, xi),
ω⋆i ,piv(yii, xi, σi, φi) = −k0 sin(φi − φi(xi, σi)) + φ′
i(yii, xi, σi),
(9.5)
Chapter 9. General Formation Path Following 199
where k, k0 > 0 and the functions φi(xi, σi) and φ′i(y
ii, xi, σi) are defined next. Let
ζi(xi, σi) :=‖δi‖w
−θ′h(xi) + γ sin ξi(xi, σi)
cos(σi − θh(xi))+ tanψi,
ξi(xi, σi) := σi − θh(xi) + ψi,
where γ > 0. Define the functions φi(xi, σi) and φ′i(y
ii, xi, σi) in (9.5) respectively as
φi(xi, σi) =
tan−1 ζi(xi, σi), if di · e1 < 0
tan−1 ζi(xi, σi) + π, if di · e1 > 0,
φ′i(y
ii, xi, σi) =
1
1 + ζi(xi, σi)2‖δi‖w
(−θ′′h(xi) + γ(ω⋆i (yii, xi)− θ′h(xi)) cos ξi(xi, σi)
cos(σi − θh(xi))+
−θ′h(xi) + γ sin ξi(xi, σi)
cos2(σi − θh(xi))sin(σi − θh(xi))(ω
⋆i (y
ii, xi)− θ′h(xi))
)
.
(9.6)
There are two possibilities for φi(xi, σi) in (9.6), depending on the sign of di · e1. The
first entails (xi, σi)φi7−→ (−π/2, π/2), the second (xi, σi)
φi7−→ (π/2, 3π/2). This will ensure
that the unicycles follow the curve moving in the forward direction. The terms in (9.5)
containing fi(yii) have the objective of achieving consensus on the estimates (xi)i∈n of the
formation origin while those containing h(xi)i have the objective of making xi converge
to and follow the path C at the desired speed and direction. Finally, the control input
ω⋆i ,piv(yii, xi, σi, φi) is chosen to achieve the angle condition σi = θr(τi)− ψi. The control
inputs in (9.5), (9.6) are functions of
• relative positions to neighboring unicycles yii = (xiij)j∈Ni,
• the integrator path following controller at xi represented in body frame, h(xi)i
• the angle of δi relative to the angle of h(xi), σi − θh(xi),
• the quantities θ′h(xi) and θ′′h(xi).
Chapter 9. General Formation Path Following 200
The quantity yii can be written as
xiij =R−1i
xj +Rj
αj
βj
− xi −Ri
αi
βi
=R−1i
xij + ‖δj‖RiR
−1i Rj
cosφj
− sinφj
− ‖δi‖Ri
cosφi
− sinφi
=
xiij + ‖δj‖Ri
j
cosφj
− sin φj
− ‖δi‖
cosφi
− sin φi
.
(9.7)
To compute (9.7), unicycle i requires measurement of quantities xiij , Rij , ‖δj‖ and
Rij(cosφj,− sinφj) for all j ∈ Ni. We can alternatively write,
Rij
cosφj
− sinφj
= Ri
j
δjj‖δj‖
=δij
‖δj‖.
To obtain the quantity Rij(cosφj , sinφj), unicycle j must either communicate the angle
φj or have a device on board that indicates the vector δjj visually in its body frame so that
unicycle i can measure δij using cameras. For example, using a string of lights around
the unicycle body. The main theorem for the formation path following problem is given
in Theorem 9.1.1 whose proof is given in Section 9.3.
Theorem 9.1.1. Consider system (2.6), (2.7), (3.10) with sensor digraph G containing a
globally reachable node. Let fi(·) be a single integrator consensus controller in (4.10) with
any parameters aij > 0 for i ∈ n, j ∈ Ni, and h(·) be a path following controller in (4.15)
for the smooth Jordan curve C. For any (d, C, w) ∈ GP, there exist gains k⋆, γ⋆ > 0 such
that choosing k > k⋆ and γ > γ⋆, the control inputs in (9.5), (9.6) asymptotically stabilize
the manifold Γ′gp ⊂ Γgp, thus solving the formation path following control problem.
Remark 9.1.2. The control input for ω⋆i (x, σi, φi) has a singularity when αi = ‖δi‖ cosφi =
Chapter 9. General Formation Path Following 201
0 and the signals φi,φ′i in (9.6) have singularities when cos(σi − θh(xi)) = 0. It will be
shown in the proof of Theorem 9.1.1 that in a neighborhood of Γ′gp neither of these singu-
larities occur and therefore do not affect the asymptotic stability result. To remove the
presence of singularities in the control inputs, one can employ bump functions without
affecting asymptotic stability.
Remark 9.1.3. If one sets φ′i(x, σi) = 0 in (9.5), the result for asymptotic stability in
Theorem 9.1.1 becomes a result for practical stability where instead of solutions converg-
ing exactly to Γ′gp they converge to a neighborhood of Γ′
gp that can be made arbitrarily
small by choosing the control gain k0 in (9.5) sufficiently large. This has the advantage
of significantly simplifying the control input ω⋆i ,piv(yii, xi, σi, φi) in (9.5) which no longer
depends on θ′′h(xi) which may be a large expression. For example, for h(x) defined in (9.9)
for a Van der Pol oscillator, one can compute θ′h(x) as,
θ′h(x) = −c0wµy3z + y2 + 2µyz3 − µyz + z2
((y + µz(y2 − 1))2 + z2)3/2
with y = c0x · e1 and z = c0x · e2. Whereas the expression for θ′′h(x) is roughly five times
this length.
9.2 Simulation Results
In this section, we present simulation results for formation path following using the
feedbacks presented in (9.5). We consider a group of five unicycles with the directed
sensing graph shown in Figure 9.4 with unitary weights. Each edge is unidirectional
except for the bidirectional edge between unicycles 2 and 4. The desired formation is
specified by d1 = (15, 0), d2 = (5,−8), d3 = (5, 8), d4 = (−10,−3) and d5 = (−10, 3)
about o(t1) as illustrated in Figure 9.5.
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
F (x, θ, φ) = ((xi + δ(θi, φi))i∈n, θ − φ, φ) = (x, σ, φ)
and with inverse
F−1(x, σ, φ) = ((xi − δ(σi + φi, φi))i∈n, σ + φ, φ) = (x, θ, φ).
The time derivative of σi is given by σi = θi−φi which, after setting ωi = ω⋆i ,piv(yii, xi, σi, φi)+
ω⋆i (yii, xi) and ωi ,piv = ω⋆i ,piv(y
ii, xi, σi, φi), becomes σi = ω⋆i (y
ii, xi). The time derivative of
xi = xi + αiRie1 + βiRie2 is given by
˙xi = uiRie1 + αiRi
0 −θiθi 0
e1 + βiRi
0 −θiθi 0
e2 + αiRie1 + βiRie2
= uiRie1 + αiθiRie2 − βiθiRie1 + αiRie1 + βiRie2.
Substituting θi = ω⋆i ,piv(yii, xi, σi, φi) + ω⋆i (y
ii, xi) yields
˙xi = u⋆iRie1 + αiω⋆i ,pivRie2 − βiω
⋆i ,pivRie1 + αiω
⋆iRie2 − βiω
⋆iRie1 + αiRie1 + βiRie2.
Chapter 9. General Formation Path Following 205
Substituting the quantities αi = ‖δi‖ cosφi, βi = −‖δi‖ sinφi, α = −‖δi‖ sinφiω⋆i ,piv and
β = −‖δi‖ cosφiω⋆i ,piv , one computes
˙xi = u⋆iRie1 + αiω⋆iRie2 − βiω
⋆iRie1,
which is independent of ω⋆i ,piv and the closed-loop dynamics for the (x, σ, φ) system are
given by
˙xi = (u⋆i (yii, xi)− βiω
⋆i (y
ii, xi))Rie1 + αiω
⋆i (y
ii, xi)Rie2,
σi = ω⋆i (yii, xi),
φi = ω⋆i ,piv(yii, xi, σi, φi).
(9.10)
Notice that the control inputs presented in (9.5) are defined precisely in terms of (yii, xi, σ, φ)
and so the equations of motion in (9.10) constitute a dynamical system.
The manifold Γ′gp represented in (x, σ, φ) coordinates becomes
Γgp :=
(x, σ, φ) ∈ R2n × T
n × Tn : x1i = 0, x1 ∈ C,
σi = θr(π(xi))− ψi, φi = φi(xi, σi), i ∈ n ,(9.11)
where φi : R2 × S1 → (−π/2, π/2) ∪ (π/2, 3π/2) is a desired angle of φi in (9.6). It
needs to be shown that Γgp is asymptotically stable for system (9.10). Moreover, the
state x1 should move at the desired speed w > 0 along C in the direction of r(τ1).
9.3.1 Proof of Theorem 9.1.1
Consider the closed loop system (2.6), (2.7), (3.10), (9.5), (9.6). It needs to be shown
that the set Γgp is asymptotically stable. Since the dot product is invariant to a change
of frame and fi(yii) = R−1
i fi(yi), it holds that fi(yii) · e1 = Rifi(y
ii) · Rie1 = fi(yi) · Rie1,
similarly fi(yii) ·e2 = fi(yi) ·Rie2, h(xi)
i ·e1 = h(xi) ·Rie1 and h(xi)i ·e2 = h(xi) ·Rie2. The
control inputs (u⋆i , ω⋆i ) in (9.5) represented with respect to the inertial frame therefore
Chapter 9. General Formation Path Following 206
satisfy,
u⋆i = kfi(yi) ·Rie1 + βiω⋆i + h(xi) ·Rie1,
ω⋆i =1
αi(kfi(yi) · Rie2 + h(xi) · Rie2) , i ∈ n
(9.12)
where k > 0, which will be chosen later, scales the primitive fi(yi) in the controller. Sub-
stituting (9.12) into (9.10) and using the fact that (fi(yi)·Rie1)Rie1+(fi(yi)·Rie2)Rie2 =
fi(yi) and (h(xi) · Rie1)Rie1 + (h(xi) ·Rie2)Rie2 = h(xi) yields
˙xi = kfi(yi) + h(xi)
σi =1
αi(kfi(yi) · Rie2 + h(xi) · Rie2)
φi = ω⋆i ,piv , i ∈ n.
(9.13)
The system equations in (9.13) can be expressed in terms of new states: x1 ∈ R2,
x := (x1i)i∈2 :n ∈ R2(n−1), σ := (σi)i∈n ∈ Tn and φ := (φi)i∈n ∈ Tn as
˙x1 = kf1(y1) + h(x1)
˙x1i = kfi(yi)− kf1(y1) + h(xi)− h(x1)
(9.14)
σi =1
αi(kfi(yi) · Rie2 + h(xi) · Rie2)
φi = ω⋆i ,piv ,
(9.15)
where xi = x1i + x1. The set Γgp expressed in (x1, x, σ, φ) coordinates becomes
Γgp :=
(x1, x, σ, φ) ∈ R× R2(n−1) × T
n × Tn : x1i = 0,
x1 ∈ C, σi = θr(π(x1))− ψi, φi = φi(x1i + x1, σi), i ∈ n .
It needs to be shown that Γgp is asymptotically stable which is sufficient to solve the
formation path following control problem. The x dynamics in (9.14) can be written as
˙x = −kLx+ h(x1, x), (9.16)
Chapter 9. General Formation Path Following 207
where h(x1, x) = (h(xi) − h(x1))i∈2 :n acts as a vanishing perturbation as x → 0 and L
is the positive definite reduced Laplacian matrix of L where L(i, j) = −aij for i 6= j
and L(i, i) =∑
j∈Niaij . Moreover, using the fact that h(xi) is globally Lipschitz with
Lipschitz constant c one can write
‖h(x1, x)‖ = ‖(h(xi)− h(x1))i∈2 :n‖
≤n∑
i=2
‖h(xi)− h(x1)‖ ≤n∑
i=2
c‖x1i‖ ≤ c(n− 1)‖x‖.(9.17)
By (Khalil, 2002, Lemma 9.1), there exists k⋆ > 0 such that choosing k > k⋆ implies
that the set
Γ3 := (x1, x, σ, φ) ∈ R× R2(n−1) × T
n × Tn : x = 0
is globally exponentially stable provided that x1 has no finite escape times. System (9.14),
(9.15) is globally Lipschitz (since fi(·) and h(·) are globally Lipschitz) and therefore has
no finite escape times.
In the set Γ3, it holds that h(x1, x) = 0, fi(yi) = 0 for all i ∈ n and the equations
in (9.14) reduce to,
˙x1 = h(x1),
˙x1i = 0,
σi =1
αi(h(xi) · Rie2),
φi = ω⋆i ,piv .
(9.18)
Two important things take place on the set Γ3 (i) xi → C and (ii) φ → φi. Property (i)
follows directly from the fact that on Γ3, ˙xi = h(xi) for all i ∈ n, where h(xi) is a single
integrator path following controller. To show property (ii), consider the derivatives of
Chapter 9. General Formation Path Following 208
θh(xi) and θ′h(xi) which on Γ3 satisfy,
θh(xi) =∂θh(xi)
∂xi· h(xi) = θ′h(xi)
θh(xi) = θ′h(xi) =∂θ′h(xi)
∂xi· h(xi) = θ′′h(xi).
(9.19)
Moreover, recalling that
ζi =‖δi‖w
−θ′h(xi) + γ sin ξicos(σi − θh(xi))
+ tanψi,
in Γ3 the function φi in (9.6) has the time derivative φi given by
φi =1
1 + ζ2iζi
=1
1 + ζ2i
‖δi‖w
(
−θ′h(xi) + γ cos ξiξicos(σi − θh(xi))
− −θ′h(xi) + γ sin ξicos2(σi − θh(xi))
d
dtcos(σi − θh(xi))
)
=1
1 + ζ2i
‖δi‖w
(−θ′′h(xi) + γ(ω⋆i − θ′h(xi)) cos ξicos(σi − θh(xi))
+−θ′h(xi) + γ sin ξicos2(σi − θh(xi))
sin(σi − θh(xi))(ω⋆i − θ′h(xi))
)
=φ′i,
(9.20)
where φ′i was defined in (9.6). We have used the fact that the time derivative of ψi is zero
and the derivative of ξi is given by ξi = σi − θh(xi) + ψi = ω⋆i − θ′h(xi). Then, from (9.5)
the equation of motion for φi is given by
φi = −k0 sin(φi − φi) + φi,
for which the set φ ∈ S1 : φ = φ is asymptotically stable. Therefore, the compact set
Γ2 = (x1, x, σ, φ) ∈ Γ3 : x1 ∈ C, φi = φi, i ∈ n
Chapter 9. General Formation Path Following 209
is asymptotically stable relative to Γ3. It holds from Theorem 4.1.2 that the set Γ2 is
asymptotically stable. In the set Γ2, xi = x1 ∈ C for all i ∈ n and x1, traverses C
with speed w. It follows that π(xi) is well-defined and h(xi) = wr(xi) = wr(π(xi))
by property A2 for integrator path following controllers in (4.15). Therefore θh(xi) =
θh(x1) ≡ θr(π(x1)) =: θr (correspondingly θh(xi) = θr) for all i ∈ n. Therefore ξi =
σi − θh(xi) + ψi = σi − θr + ψi,
φi = φi =
tan−1(
‖δi‖w
−θr+γ sin ξicos(σi−θr)
+ tanψi
)
, if di · e1 < 0
tan−1(
‖δi‖w
−θr+γ sin ξicos(σi−θr)
+ tanψi
)
+ π, if di · e1 > 0
(9.21)
and ω⋆i is given by
ω⋆i =1
αi(h(xi) · Rie2) =
‖h(xi)‖αi
sin(θh(xi)− θi) =w
αisin(θr − θi).
Now we write out the dynamics of ξi = σi− θr +ψi in Γ2 which we’d like to converge
to zero. Taking the derivative of ξi yields
ξi = σi − θr = ω⋆i − θr =w
αisin(θr − θi)− θr
= − w
‖δi‖ cosφisin((θi − φi − θr + ψi) + φi − ψi)− θr
= − w
‖δi‖ cosφisin(ξi + φi − ψi)− θr
= − w
‖δi‖ cosφi
sin(ξi + φi − ψi)− θr,
where we used the fact that the time derivative of ψi is zero, φi = φi and αi = ‖δi‖ cosφi =
‖δi‖ cosφi. Using the identity sin(ξi + φi − ψi) = sin ξi cos(φi − ψi) + cos ξi sin(φi − ψi)
implies,
ξi = − w
‖δi‖ cosφi
[sin ξi cos(φi − ψi) + cos ξi sin(φi − ψi)]− θr
= − w
‖δi‖
[
sin ξicos(φi − ψi)
cosφi
+ cos ξisin(φi − ψi)
cosφi
]
− θr.
(9.22)
Chapter 9. General Formation Path Following 210
The following identities can be derived using standard trigonometric identities,
cos(φi − ψi)
cosφi
=cosφi cosψi + sinφi sinψi
cosφi
= cosψi + tanφi sinψi
sin(φi − ψi)
cosφi
=sinφi cosψi − cosφi sinψi
cosφi
= tanφi cosψi − sinψi,
which after substitution into (9.22) yields
ξi = − w
‖δi‖[sin ξi cosψi + tanφi(sin ξi sinψi + cos ξi cosψi)− cos ξi sinψi]− θr
= − w
‖δi‖[sin ξi cosψi + tanφi cos(σi − θr)− cos ξi sinψi]− θr,
(9.23)
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
212
Chapter 10. Unicycle Formation Simulation Trials 213
solutions in these chapters are based on a similar methodology. For the simulation results
in this chapter, unless stated otherwise, we consider the five-unicycle formation illustrated
in Figure 7.6 with initial angles (θi)i∈n chosen randomly in Tn and the positions (xi)i∈n
chosen randomly in a 60× 60 meter grid.
10.1 Performance Measures
In this section we introduce a number of performance measures that will be used to
evaluate and compare the effectiveness of the control solution in Chapter 7 under various
scenarios considered in the simulation results. These include a formation measure, drift
measure, collision measure and input measures and are described below.
• Formation Measure. The most important measure of performance for the forma-
tion control problem is whether or not the unicycles achieve the desired formation.
Any test trial that does not meet this condition is considered a failure. This aspect
will be captured by the formation measure. The formation measure µf : X → R is
a function of the state χ ∈ X that satisfies two conditions
1. it is positive semi-definite
2. it equals zero if and only if the formation is achieved, i.e., µ−1f (0) = Γp,
where Γp is the parallel formation manifold defined in (3.4). These conditions would
suggest that a natural candidate for µf(χ) is the Lyapunov function V (χ) defined
in (7.12). However, recall that V was designed to meet the two conditions above
under the assumption of fixed connected undirected graphs. This assumption will
not necessarily hold when we consider the scenario of state-dependent graphs. For
example, if the initial positions of the robots are too sparsely separated to sense one
another, i.e., the edge set E is empty, then all the neighbor sets (Ni)i∈n are empty
and V = 0 even though the formation is not achieved. This violates Condition
Chapter 10. Unicycle Formation Simulation Trials 214
2. The formation measure should therefore be defined independent of the graph,
control gains or any other factors that don’t intrinsically specify the spatial config-
uration of the robots. Moreover, a Lyapunov function has the requirement that its
derivative be negative definite along system trajectories which is not necessary for
choosing the formation measure. It follows that designing µf will allow for a lot
more flexibility compared to designing the Lyapunov function.
We proceed to the design of µf . We have seen in Chapter 7 that achieving formation
amounts to the simultaneous synchronization of the heading angles (θi)i∈n and
the endpoints (xi)i∈n. One choice of formation measure could therefore simply
be µf(χ) =∑n
i=1
∑nj=1(‖xij‖ + ‖ sin(θij)‖). However, we opt for a more intuitive
choice. Consider a collection of n unicycles. The average endpoint position ¯x ∈ R2
and angle θ ∈ S1 are given by,
¯x :=1
n
n∑
i=1
xi,
θ := atan2
(
n∑
i=1
sin θj ,n∑
i=1
cos θj
)
.
This choice of average for angular quantities on S1 is standard and has a singu-
larity when (∑n
i=1 sin θj ,∑n
i=1 cos θj) = (0, 0). For example, in Figure 10.1 (a) the
average angle is θ = −π/4 while in (b) the average angle is undefined. Note that
(1/n)∑n
i=1 θi does not correspond with the average angle and leads to erroneous
results. For example, in Figure 10.1 this calculation yields (a) 3π/4 rad and (b)
π/2 rad.
Consider Figure 10.2 where the set of endpoints (xi)i∈n are indicated by the black
points drawn within the dotted circle and their average position ¯x is indicated in
the figure by . The angle θ represents the average heading angle of the unicycles
with respect to the inertial frame (only unicycle i is illustrated in the figure). The
Chapter 10. Unicycle Formation Simulation Trials 215
Figure 10.1: Illustration of the average of two angles (a) (θ1, θ2) = (0, 3π/2) rad and (b)(θ1, θ2) = (0, π) rad.
actual position of unicycle i has been illustrated at position xi with heading angle
θi − θ with respect to the average. The desired formation is achieved in the set
χ ∈ X : xi = ¯x, θi = θi, i ∈ n
,
where all offset vectors and heading angles coincide with the average. The desired
position of unicycle i in this set is denoted by xi ,des in Figure 10.2. The error
between xi and xi ,des is bounded by
‖xi − xi ,des‖ < ‖xi − ¯x‖ + 2√
α2i + β2
i
∥
∥
∥
∥
sin
(
θi − θ
2
)∥
∥
∥
∥
=: ei(χ).
Correspondingly, define the formation measure as the maximum distance maxi∈n ei,
normalized by the size of the formation maxi∈2 :n ‖d11i‖, that is,
µf(χ) :=maxi∈n ei(χ)
maxi∈2 :n ‖d11i‖.
Note that normalizing by the size of the formation allows us to use the same mea-
Chapter 10. Unicycle Formation Simulation Trials 216
sure to compare the performance of the control strategy for arbitrary formations.
We say that a formation is achieved if there exists a time tf after which the for-
mation measure satisfies µf < µf where µf > 0 is called the formation threshold.
Otherwise, the test is considered a failure. In the simulation trials, we have chosen
µf = 0.05.
Figure 10.2: Illustration of the formation measure.
• Drift Measure. In an ideal scenario, a formation should be achieved without
translating or drifting too much from its initial configuration. For example, if the
unicycles are constrained to move within an enclosed building, drifting too much
might mean that some unicycles collide with the walls before the formation is
achieved. We capture the aspect of drift with a drift measure which compares the
average position of unicycles at the initial time x(t0) = (1/n)∑
i∈n x(t0) to the
average position of unicycles at a final time x(tf ) = (1/n)∑
i∈n x(tf ) where tf is
the time when the formation is achieved, i.e., when the formation measure satisfies
µf < µf . The drift measure is therefore defined as
µd := ‖x(tf )− x(t0)‖
Chapter 10. Unicycle Formation Simulation Trials 217
and is illustrated in Figure 10.3.
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
µu,1 := maxi∈n supt∈[t0,tf ] |ui(t)| (m/s)
µω,1 := maxi∈n supt∈[t0,tf ] |ωi(t)| (rad/s)
Chapter 10. Unicycle Formation Simulation Trials 218
µu,2 := maxi∈n meant∈[t0,tf ](|ui(t)|) (m/s)
µω,2 := maxi∈n meant∈[t0,tf ](|ωi(t)|) (rad/s).
10.2 Simulation Trials
In this section, we present numerous simulation trials to understand the effectiveness of
our control solution for stopping formations under different parameter choices. Each trial
is composed of N = 500 simulations. We consider the five-unicycle formation illustrated
in Figure 7.6 as the formation under study and for each simulation, initial angles (θi)i∈n
are chosen randomly in Tn and the positions (xi)i∈n are chosen randomly in a 60 × 60
meter grid. We choose the interaction function f(s) in (4.11) and rotational interaction
function g(s) in (4.24) as shown in Figure 10.4. Let aij = 30 and bij = αi + αj for all
j ∈ Ni and ηi = 1/αi. For each test, we compute the quantities in Table 10.1 and let
t0 = 0.
-5 -4 -3 -2 -1 0 1 2 3 4 5
s
-1
-0.5
0
0.5
1
f(s)
-4 -3 -2 -1 0 1 2 3 4
s
-1
-0.5
0
0.5
1
g(s)
X: 0.7884Y: 0.9987
Figure 10.4: Interaction function f(s) and g(s).
We begin by studying a base case that will serve as a reference of comparison between
test trials. The control parameters, i.e., (µf , k, α) used for the base case are presented
in Table 10.2. In the subsequent sections, we will vary each of the parameters one at a
Chapter 10. Unicycle Formation Simulation Trials 219
Table 10.1: Data Collected from Simulations
Quantity Meaning
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
Chapter 10. Unicycle Formation Simulation Trials 220
agents are precisely those sharing common αi values. If one considers, instead, the
formation in Figure 10.5, where no two unicycles share a common αi value then
there is a collision only 11 percent of the time.
Figure 10.5: Formation specified by d112 = (−10, 5), d113 = (−5,−5), d114 = (−20, 10) andd115 = (−15,−10).
• the average time to achieve formation is 220 seconds.
• the maximum speed input is 76.4238 m/s and the maximum angular speed input is
17.1043 rad/s. These are both large but can be resolved by using input saturation
as we discuss later.
• the maximum displacement of any single unicycle in the ensemble is 129.7534m and
the maximum number of revolutions of any single unicycle is 4.1278/2π = 0.66.
Therefore the solution has very little oscillation which is a desirable property.
Chapter 10. Unicycle Formation Simulation Trials 221
Table 10.2: Base Case Parameters
Quantity Base Value
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
Chapter 10. Unicycle Formation Simulation Trials 222
Table 10.3: Base Case Simulation Results
F/N (%) µd (m) Nc/N (%) tf (s)
100 76.4330 68.20 220.3635
µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)76.4238 17.1043 0.5888 0.0187 129.7534 4.1278
Table 10.4: Simulation Results Varying µf
µf µd (m) tf (s)
0.01 115.0104 5794.50.02 99.0958 1403.70.05 76.4330 220.36350.1 60.0024 54.07230.15 51.3388 24.10760.2 43.0169 13.15820.25 39.2520 8.9690
need to add input saturation. We illustrate formations in Figure 10.7, relative to the
frame of unicycle 1, corresponding to different values of µf .
10.2.2 Study of high gain parameters α and k
Based on Theorem 7.1.1, we need to choose α and k as high gain parameters. In this
section we show that these gains do not actually need to be chosen too large to still
achieve formation. The simulation results varying α are presented in Table 10.5 and the
results varying k are presented in Table 10.6. The values of the base case are in bold.
The main conclusions from the simulation results in Table 10.5 for varying α are:
• For α < 5 (the base value), some simulations fail. Only 2.2 percent of simulations
fail when α is chosen as low as 0.1.
• drift is proportional to α.
• collisions always remain above 60 percent.
Chapter 10. Unicycle Formation Simulation Trials 223
-10 0 10 20 30-20
-10
0
10
20µ
f = 0.02
-10 0 10 20 30-20
-10
0
10
20µ
f = 0.01
-10 0 10 20 30-20
-10
0
10
20µ
f = 0.05
-10 0 10 20 30-20
-10
0
10
20µ
f = 0.1
-10 0 10 20 30-20
-10
0
10
20µ
f = 0.15
-10 0 10 20 30-20
-10
0
10
20µ
f = 0.2
-10 0 10 20 30-20
-10
0
10
20µ
f = 0.25
-10 0 10 20 30-20
-10
0
10
20µ
f = 0.3
-10 0 10 20 30-20
-10
0
10
20µ
f = 0.35
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
fail when k is chosen as low as 1.
• drift is proportional to k.
Chapter 10. Unicycle Formation Simulation Trials 224
Table 10.5: Simulation Results Varying α
α F/N (%) µd (m) Nc/N (%) tf (s)
0.1 97.80 65.8306 61.76 196.41900.5 98.00 69.6150 66.33 192.79161 99.40 68.6228 64.59 187.50875 100 76.4330 68.20 220.363510 100 89.1039 70.00 315.933725 100 138.9444 69.20 853.3169
α µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)0.1 85.4778 994.4596 0.7337 0.0508 144.1049 9.97320.5 84.3687 660.0438 0.7010 0.0312 135.1392 6.00961 81.4894 197.9675 0.6934 0.0290 130.0245 5.43435 76.4238 17.1043 0.5888 0.0187 129.7534 4.127810 74.8656 7.6092 0.4822 0.0120 152.3422 3.776425 73.5076 3.2741 0.2501 0.0034 213.4184 2.9089
• collisions attain a minimum of 45 percent when k = 50.
• The time to reach formation attains a minimum of 92.59s when k = 50.
• All four input measures are proportional to k.
• The most revolutions that a unicycle makes is 1.07 when k = 100.
The simulations reveal that we cannot choose α and k much less than the base values
of α = 5 and k = 15 without running a risk of failure (i.e., formation may not be
achieved). It is also important not to choose these values too large, especially α, as this
can increase drift and/or tf .
10.2.3 Study of state-dependent undirected graphs
In this section, we study our control solution under state-dependent undirected graphs
where each unicycle’s neighbors are those that lie within a given radius of itself. We vary
the sensing radius from 20m to 45m. The results are presented in Table 10.7. The column
called C/N (%) represents the percentage of simulations that begin connected, CF/C
(%) is the percentage of initially connected graphs that achieve formation, and DF/D
Chapter 10. Unicycle Formation Simulation Trials 225
Table 10.6: Simulation Results Varying k
k F/N (%) µd (m) Nc/N (%) tf (s)
1 99.60 73.4022 67.07 2994.85 99.80 73.8609 74.75 609.732915 100 76.4330 68.20 220.363550 100 125.2349 45.80 92.594075 100 168.9605 50.60 99.6497100 100 217.0383 53.60 129.4235
k µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)1 73.2869 9.0937 0.0423 0.0016 126.7413 4.79635 73.7068 10.1675 0.2139 0.0081 130.4128 4.931015 76.4238 17.1043 0.5888 0.0187 129.7534 4.127850 103.6473 46.4832 2.0498 0.0531 189.7969 4.921175 127.2770 65.3029 2.2921 0.0555 228.4063 5.5290100 156.9870 91.0888 2.1674 0.0521 280.5187 6.7436
(%) is the percentage of initially disconnected graphs that achieve formation. Naturally,
the sensing radius has to be large enough to maintain connection when unicycles lie in
formation.
The main conclusions from the simulation results in Table 10.7 for state-dependent
undirected graphs are:
• the proportion of initially connected undirected graphs goes from 5 percent to 95
percent as the sensing radius goes from 20m to 45m.
• the proportion of initially connected undirected graphs that achieve formation goes
from 59 percent to 98 percent as the sensing radius goes from 20m to 45m.
• the proportion of initially disconnected undirected graphs that achieve formation
goes from 11 percent to 70 percent as the sensing radius goes from 20m to 45m.
• the sensing radius does not have a large effect on the drift.
• tf is inversely proportional to the sensing radius due to increased connectivity as
the sensing radius increases.
Chapter 10. Unicycle Formation Simulation Trials 226
Table 10.7: Simulation Results Varying the Sensing Radius
rad.(m) C/N (%) CF/C (%) DF/D (%) µd (m) Nc/N (%) tf (s)
20 5.80 58.62 11.25 74.5940 81.43 227.979825 22.20 68.47 24.68 72.4965 72.67 163.533630 45.40 79.30 44.69 70.5644 72.19 112.720835 72.20 89.20 45.32 71.9680 73.25 112.302240 86.20 94.90 52.17 70.2627 71.46 100.828145 95.40 97.90 69.57 69.6455 65.42 101.9338
rad. µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)20 65.9074 15.7912 0.7856 0.0312 179.0938 7.122325 76.2224 16.2460 1.0513 0.0402 171.9238 6.579330 85.0675 19.1963 1.2967 0.0482 146.1645 5.434535 94.0355 19.9240 1.2721 0.0442 142.8599 4.961540 101.7096 21.7516 1.3289 0.0436 133.9915 4.398145 107.6055 23.6756 1.3108 0.0427 133.6198 4.3515
• the input measures are typically proportional to the sensing radius.
• The maximum number of revolutions that a unicycle makes is inversely proportional
to the sensing radius.
One benefit of the control solution is that it is robust to the disconnection of agents.
This is illustrated in Figure 10.8 where a single agent begins too far from the rest of the
group to sense anyone. Despite this, the formation is still attained among the remaining
agents.
10.2.4 Study of directed graphs
We present simulation results for the directed ring-coupled graph in Figure 10.9 as op-
posed to the undirected graph in Figure 10.6. The main conclusions from the simulation
results in Table 10.8 are:
• 99 percent of the simulations achieved formation.
• The drift and collision measures are not changed significantly compared to the base
case.
Chapter 10. Unicycle Formation Simulation Trials 227
-80 -60 -40 -20 0 20 40 60 80
x (m)
-100
-80
-60
-40
-20
0
20
y (m
)
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
reason.
1
2 3
4 5
Figure 10.9: Directed ring-coupled graph.
Chapter 10. Unicycle Formation Simulation Trials 228
Table 10.8: Simulation Results for the digraph in Figure 10.9
Graph F/N (%) µd (m) Nc/N (%) tf (s)
Fig 10.6 100 76.4330 68.20 220.3635Fig 10.9 99.00 79.7923 63.03 493.0773
Graph µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)Fig 10.6 76.4238 17.1043 0.5888 0.0187 129.7534 4.1278Fig 10.9 38.0586 11.6481 0.2805 0.0102 138.2853 5.0263
10.2.5 Study of input saturation
We have seen in the base case that the input measures, µu,1 = 76.4238 m/s and µω,1 =
17.1043 rad/s, are unacceptably high for most applications. Typical profiles of the inputs
ui and ωi are plotted in Figure 10.10. The large control inputs are only present at the
start of the simulation and decrease rapidly to more reasonable values as the formation
is reached. This suggests that we may be able to saturate the inputs without affecting
performance too much. In this section, we saturate the inputs so that µu,1 and µω,1
are bounded by different values corresponding to the first two columns in Table 10.9.
Table 10.9 presents the simulation results where the bold quantities correspond to the
base case.
0 20 40 60 80 100
time (s)
0
10
20
30
40
|ui| (
m/s
)
0 20 40 60 80 100
time (s)
0
0.5
1
|ωi| (
rad/
s)
Figure 10.10: Typical input profile during simulation.
Chapter 10. Unicycle Formation Simulation Trials 229
Table 10.9: Simulation Results with Input Saturation
µu,1 (m/s) µω,1 (rad/s) F/N (%) µd (m) Nc/N (%) tf (s)
0.2 π/8 100 80.5316 64.60 624.76211 π/4 100 78.3436 65.60 275.70205 π/2 100 76.2906 67.60 216.513010 π 100 78.2598 72.60 213.968315 3π/2 99.80 77.5945 69.34 216.593120 2π 100 75.3784 72.00 211.3089
76.4238 17.1043 100 76.4330 68.20 220.3635
µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)0.2 π/8 0.1687 0.01 105.4219 6.21861 π/4 0.4080 0.0199 112.4854 5.48355 π/2 0.5836 0.0230 126.3490 4.987910 π 0.5896 0.0209 126.1483 4.468015 3π/2 0.5959 0.0210 129.0616 4.555420 2π 0.6095 0.0208 128.7945 4.4015
76.4238 17.1043 0.5888 0.0187 129.7534 4.1278
The profiles of the inputs ui and ωi for µu,1 = 5 m/s and µω,1 = π/2 rad/s are
plotted in Figure 10.11. The main conclusions from the simulation results in Table 10.9
for varying input saturation values are:
• the saturation did not lead to any significant formation failures (however, there was
a single failure out of 500 tests for the case where µu,1 = 15 and µω,1 = 3π/2).
This suggests that the solution is very robust to input saturation.
• the saturation did not lead to any significant change in drift and collisions.
• There was no significant increase in the formation time tf unless µu,1 < 1 m/s and
µω,1 < π/4 rad/s.
• There was no significant change in the mean control magnitudes µu,2 and µω,2
unless µu,1 < 1 m/s and µω,1 < π/4 rad/s where the values decrease.
Chapter 10. Unicycle Formation Simulation Trials 230
0 10 20 30 40 50 60
time (s)
0
2
4
6
|ui| (
m/s
)
0 10 20 30 40 50 60
time (s)
0
0.5
1
1.5
2
|ωi| (
rad/
s)
Figure 10.11: Input profile during simulation for µu,1 = 5 m/s and µω,1 = π/2 rad/s.
10.2.6 Study of disturbances and sampling
In this section, we present a simulation result including a number of disturbances as well
as input sampling not included in the theoretical results. Due to slow simulation speed,
only a single test is performed instead of 500 in the other trials. Moreover, the inputs will
be saturated in which µu,1 ≤ 5m/s and µω,1 ≤ π/2 rad/s. The simulation parameters
are listed in Table 10.10. The disturbances are:
• an additive random noise with maximum magnitude of 0.25m/s on the input ui;
• an additive random noise with maximum magnitude of 0.25 rad/s on the input ωi;
• an additive random noise with maximum magnitude of 0.25 rad on the quantity
gi((θij)j∈Ni, η) accounting for errors in measurements of relative headings;
• an additive random noise on the quantity fi(yii) accounting for errors in measure-
ments of relative displacements of the vehicles. The direction of this vector has been
rotated within 0.25 rad and the magnitude is scaled between 0.75 to 1.25 times the
actual magnitude.
Chapter 10. Unicycle Formation Simulation Trials 231
Table 10.10: Simulation Parameters in Section 10.2.6
Quantity Base Value
k 15α 5fixed or state-dependent graphs fixedundirected or directed graph undirectedinput saturation yessampling and disturbances yes
Table 10.11: Simulation Results with Perturbations and Sampling
µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s)5 π/2 3.9174 1.2296
Each unicycle samples its input 100 times per second. The simulation results are pre-
sented in Figure 10.12. The simulation results were run for 750s and the unicycles
achieved the formation measure µf = 0.1123 and no collisions. This shows that the
system still works well with perturbations and a sampling rate of 100Hz. In this time,
the formation drifted 74m and the input measures satisfied the values in Table 10.11.
In particular, the average input values µu,2 and µω,2 are significantly higher than in the
base case due to the disturbances. Reducing the sampling rate to 10Hz no longer has
acceptable performance as illustrated in Figure 10.13 where the unicycles achieve the for-
mation measure µf = 0.6455 after 750s and the unicycles move with noisy trajectories.
Chapter 10. Unicycle Formation Simulation Trials 232
Figure 10.12: Simulation result in the presence of disturbances and sampling rate of100Hz.
Figure 10.13: Simulation result in the presence of disturbances and sampling rate of10Hz.
10.2.7 Study of dynamic unicycles
In this section, we present simulation results for the dynamic unicycle model presented
in (2.10). We select the control inputs as
ai = ˙ui(χi)− k(vi − ui(χi)),
αi = ˙ωi(χi)− k(ωi − ωi(χi))
where ai is the translational acceleration input, αi is the rotational acceleration input
and (ui, ωi) are the feedbacks for formation control of kinematic unicycles with the same
parameters as the base case. Moreover, we select k = 1. Denote
Chapter 10. Unicycle Formation Simulation Trials 233
Table 10.12: Simulation Results for Dynamic Unicycles
F/N (%) µd (m) Nc/N (%) tf (s)
98 90.7796 58.33 175.2707
µa,1 (m/s2) µα,1 (rad/s2) µa,2 (m/s2) µα,2 (rad/s2) tfµu,2 (m) tfµω,2 (rad)2 π/2 0.6354 0.1268 334.4075 12.4730
µa,1 := maxi∈n supt∈[t0,tf ] |ai(t)| (m/s2)
µα,1 := maxi∈n supt∈[t0,tf ] |αi(t)| (rad/s2)
µa,2 := maxi∈n meant∈[t0,tf ](|ai(t)|) (m/s2)
µα,2 := maxi∈n meant∈[t0,tf ](|αi(t)|) (rad/s2).
The acceleration inputs are saturated such that µa,1 ≤ 2m/s2 and µα,1 ≤ π/2 rad/s2. The
simulation results are presented in Figure 10.14 and Table 10.12. Compared to the base
case, the drift increases, the collisions decrease to 58 %, the formation time decreases
unexpectedly as a result of input saturation and the maximum number of revolutions
made by any agent has increased to 12.4730/2π = 1.96.
-60 -40 -20 0 20 40 60 80
x (m)
-20
0
20
40
60
80
100
y (m
)
Figure 10.14: Simulation result for dynamic unicycles.
Chapter 11
Conclusions and future research
The work discussed in this thesis has been motivated by applications in multi-agent co-
ordination both for land and aerial based applications. We have studied fundamental
coordination problems including rendezvous, full synchronization, and formation con-
trol with and without final collective motions. We have focussed on two robot models:
ground-based kinematic unicycles in SE(2), and flying robots in SE(3). The main source
of difficulty in controlling these systems arises due to nonholonomic constraints that re-
strict the instantaneous velocities or accelerations of robots to a single direction defined
in their body frame.
The coordination problems have two goals. The first goal is to stabilize, using lo-
cal and distributed feedback, a fixed pre-defined configuration of agents modulo roto-
translations. In addition, there may be a higher-level control specification that enforces
a desired final motion of the ensemble. In the case of path following, achieving such
motion will require cameras or a GPS system allowing the robots to detect the desired
path. All the control problems addressed in this thesis have been formulated as set sta-
bilization problems with additional constraints on the inputs to ensure the final desired
collective motion.
In order to prove the results throughout this thesis, we have relied on numerous sta-
234
Chapter 11. Conclusions and future research 235
bility theorems stated in Section 4. Of particular importance are the reduction theorems
which have been employed consistently throughout the thesis. Many of the results would
have been far too difficult to prove without reduction, especially to the degree of globality
sought out in this work. All the control solutions have been constructed out of a number
of control solutions, or building blocks, solving analogous multi-agent coordination tasks
for much simpler integrator models. This framework has led to control solutions that
are quite intuitive. The control solutions presented also enjoy the following desirable
properties, often absent from results in the literature
• The feedbacks are completely independent of time thus avoiding the need for syn-
chronized clocks
• The feedbacks do not require a communication system eliminating issues surround-
ing communication delays
• Feedback expressions solving all the control problems are short and efficient
• Solutions are typically almost global or semi-global
• Solutions work for graphs containing a globally reachable node (undirected or di-
rected graphs depending on the problem)
• The control solutions presented in this work are robust to the loss of agents.
11.1 Future Research
Below is a list of possible future research directions.
• The reduction theorem for almost global asymptotic stability in Theorem 4.1.3
assumes that the set of equilibria in A, exponentially unstable relative to Γ2, are
isolated. It would be interesting to study whether the result still holds in cases
when the equilibria in A are not necessarily isolated.
Chapter 11. Conclusions and future research 236
• The result presented for rendezvous of flying robots in Chapter 5 shows global
practical stability of the rendezvous manifold. As in the case of kinematic unicycles
in Chapter 6, it may be possible to improve this result to global asymptotic stability.
In a preliminary conference paper (Roza et al., 2014) we presented an almost global
solution, however the feedback was not local and distributed and therefore was
excluded from this thesis.
• The proposed control law in Chapter 5 does not guarantee hovering of the robots.
While the robots converge to each other, nothing can be said about the motion of
the ensemble. The point of view of these authors is that the proposed solution of
the rendezvous problem with strictly local and distribution will serve as a layer in a
hierarchy of higher-level control specifications such as hovering and path following.
In a preliminary conference paper (Roza et al., 2014) we did provide a control
solution, combining a uniformly bounded double integrator consensus controller
and a rotating body equilibrium stabilizer in (4.26), for rendezvous of flying robots
where all agents did hover in steady state. However, this result was excluded from
this thesis because it required communication of thrust inputs between neighboring
agents as well as measurement of relative attitudes and angular velocities between
neighboring agents, none of which was required in Chapter 5.
• A common drawback of all the solutions that we have presented in this thesis
is the assumption of fixed, undirected graphs. This assumption is questionable
in practice and a future research direction might be extension of the developed
approach to state-dependent sensor graphs. 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. If the camera is not omni-directional,
then the sensing will also depend on the robot heading angles. Stability is hard to
analyse for state dependent graphs. It needs to be shown any formation starting
Chapter 11. Conclusions and future research 237
with in a connected configuration are guaranteed to stay connected for all time
and achieve the goals. We believe that solutions in the literature for consensus of
double-integrators with time-dependent sensor digraphs could be extended to rigid
bodies using the framework in this thesis. However the Lyapunov function used in
the analysis would need to be modified extensively. Another common drawback is
the absence of collision avoidance in the control scheme or system perturbations.
• In Chapter 7, the solution requires two high gain parameters k and α. We have seen
in simulation that increasing these gain too large will have negative consequence
on the rate of convergence of the formation. In extensive simulations, we have seen
that these gains do not typically need to be very large for the controller to work
well. While this is encouraging, a future direction would be to characterize exactly
how large these gains need to be, or find another control solution that does not
require them altogether.
• In Chapter 8, we presented a solution to PFP for the class of hierarchical digraphs.
Based on the simulation results, this controller may also work for any sensor di-
graph containing a globally reachable node. Also, the solutions to PP, CFP and
CPP assume undirected sensor graphs but may also work for any sensor digraph
containing a globally reachable node.
• The result for formation path following for Jordan curves in Chapter 9 has a number
of drawbacks compared to the results for formation line and circle path following in
Chapter 8 that might be improved in the future. The asymptotic stability is local
as opposed to almost global, it requires communication of pivot angles and every
agent is required to see the common path. These assumptions weren’t necessary in
Chapter 8.
• A notable omission from this work is a discussion of formation control of flying
robots using local and distributed feedbacks, analogous to the work in Chapter 7
Chapter 11. Conclusions and future research 238
for kinematic unicycles. As we discussed in Section 5.3, one could easily modify
the solution for rendezvous in Chapter 5 (or the preliminary result in (Roza et al.,
2014)) to a solution for formation control by making the robots converge to fixed
offsets from one another defined in inertial frame instead of converging exactly to
one another. However, an approach like this assumes that each agent can measure
their own attitude in the inertial frame which is not a local and distributed quantity.
Instead, inspired by the work on formation control of kinematic unicycles in Chap-
ter 7, one might try to attach fixed body referenced offset vectors to each robot,
xi = xi + αiRie1 + βiRie2 + γiRie3, such that the formation is achieved when the
endpoints xi achieve consensus and all robot attitudes synchronize. One might
stabilize this formation by appropriately combining a double integrator consensus
controller with a consensus controller for rotating bodies in SO(3). A special case
of this solution would be full synchronization of flying robots. We have attempted
to do this but without success. One difficulty of adapting the approach in 7 to the
dynamic case of flying robots is its reliance on control primitives that are gradient
systems.
After the formation control problem for flying robots is addressed, one could then
look at including high level specifications, some of which may require GPS or other
sensors, like hovering (requiring each agent to measure gravity), formation path
following and flocking, analogous to the work in Chapter 8 for kinematic unicycles.
• The results in this thesis could be extended in the future to other classes of robots
not considered in this work. For example, fixed-wing aircraft or boats. They could
also be extended to allow for formations that are not necessarily fixed, but change
with time. Such dynamic formations are seen in swarms of fish, for example.
• Experiments using the control theory in this thesis will be performed as part of
future undergraduate student projects.
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