communication loss management and analysis for multiple ......lqr: linear quadratic regulator lqg:...

Communication Loss Management and Analysis for MultipleSpacecraft Formation Flying Missions

by

Mohamed Elnabelsya

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Aerospace Science and EngineeringUniversity of Toronto

Copyright c© 2010 by Mohamed Elnabelsya

Abstract

Communication Loss Management and Analysis for Multiple Spacecraft Formation

Flying Missions

Mohamed Elnabelsya

Master of Applied Science

Graduate Department of Aerospace Science and Engineering

University of Toronto

2010

This thesis presents a method for managing periods of communication loss between

multiple spacecraft in formation flying (MSFF), and analyzes the effects of this method

on the stability of the formation keeping control algorithm. The controller of interest in

this work in an adaptive nonlinear controller, where synchronization is also incorporated

to force the position tracking errors to converge to zero at the same rate. The com-

munication loss compensation technique proposed in this thesis is to use the previously

communicated data in lieu of the lost data, which is an effective and computationally-

efficient technique that is advantageous for small satellites. The performance parameter

of interest in this research is the maximum rate of communication loss that an MSFF sys-

tem can withstand before going unstable, and this is analyzed theoretically and through

simulations. Finally, experiments involving multiple robots in formation with communi-

cation loss are conducted, and the results are presented.

ii

Dedication

I dedicate this thesis to my Family - my mother Ihsan, my father Gamal, and my two

little sisters Reem and Nada.

iii

Acknowledgements

This work would not have been possible without the support of many in my life. I

would like to thank God for giving me the strength to complete yet another endeavour

in my life, and to thank my family for their continuous love and support.

I would like to thank Professor Hugh H. T. Liu for his tremendous support and for

believing in me throughout my research period. His continuous guidance was an integral

part of this work’s completion, and his high expectations and inspiring supervisory style

have made this an exceptional learning experience. I always sensed how much he cares

for his students and how his main goals are to help them become great researches and

contribute strongly to their fields of studies, and for that I will be forever grateful.

I would like to thank Professor Chris Damaren for creating two excellent courses -

Spacecraft Dynamics and Control I and II - and for his great teaching style and careful

selection of the material to cover. Those were two of the most educational courses I have

ever taken, and I know I will continuously refer back to this material long after I leave

UTIAS.

I would like to thank Keith Leung for guiding me throughout the experimental phase

of my research. Without his patience and continuous assistance, the experiments would

have taken a much longer time to get completed. I would like to also thank Valentin

Peretroukhin for assisting me with some of the experimental work.

Finally, I would like to thank all my friends at UTIAS for making my stay a truly

wonderful and unforgettable experience.

iv

Contents

1 Introduction 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 System Modelling 6

2.1 MSFF Position Dynamics Model . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Adaptive Control Formulation . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Integrating Synchronization Control . . . . . . . . . . . . . . . . . . . . . 9

2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Adaptive Control Simulation . . . . . . . . . . . . . . . . . . . . . 11

2.4.2 Adaptive Synchronization Control Simulation . . . . . . . . . . . 14

3 Communication Loss Stability Analysis 18

3.1 Asynchronous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.2 Controller Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.2 Linear System Simulation . . . . . . . . . . . . . . . . . . . . . . 27

v

3.2.3 Nonlinear System Simulation . . . . . . . . . . . . . . . . . . . . 29

4 ADS Stability Analysis with Synchronization Control 31

4.1 ADS Model with Synchronization . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.1 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.2 Linear System Simulation . . . . . . . . . . . . . . . . . . . . . . 35

4.2.3 Nonlinear System Simulation . . . . . . . . . . . . . . . . . . . . 38

5 Robots in Formation Analysis and Experiments 42

5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1 UTIAS Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.2 Test Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Controller Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3.1 Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3.2 Smooth Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.5 Code Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.5.1 Formation Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.5.2 Communication Loss Code . . . . . . . . . . . . . . . . . . . . . . 56

5.6 Experimental Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6 Conclusion and Future Work 63

6.1 Thesis Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Thesis Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A Formation Code for Follower 1 66

vi

List of Tables

5.1 A050166 Formatted as a Square Array . . . . . . . . . . . . . . . . . . . 49

5.2 Differentiation Filter Example . . . . . . . . . . . . . . . . . . . . . . . . 50

vii

List of Figures

2.1 Schematic representation of an MSFF system . . . . . . . . . . . . . . . 6

2.2 Relative Trajectory of the MSFF System with No Synchronization . . . . 12

2.3 Tracking Error of Follower Spacecraft 1 with No Synchronization . . . . . 12

2.4 Internal Sync Error of Follower Spacecraft 1 with No Synchronization . . 12





2.9 Tracking Error of Follower Spacecraft 4 with No Synchronization . . . 13


2.11 External Sync Error of the x Axis with No Synchronization . . . . . . . . 13

2.12 External Sync Error of the y Axis with No Synchronization . . . . . . . . 13

2.13 External Sync Error of the z Axis with No Synchronization . . . . . . . . 14

2.14 Relative Trajectory of the MSFF System with Synchronization . . . . . . 14

2.15 Tracking Error of Follower Spacecraft 1 with Synchronization . . . . . . . 15

2.16 Internal Sync Error of Follower Spacecraft 1 with Synchronization . . . . 15





viii



2.23 External Sync Error of the x Axis with Synchronization . . . . . . . . . . 16

2.24 External Sync Error of the y Axis with Synchronization . . . . . . . . . . 16

2.25 External Sync Error of the z Axis with Synchronization . . . . . . . . . . 16

3.1 ADS Communication Loss Model . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Relative Trajectory of the Linearized System with 98% Communication Loss 28

3.3 Tracking Error of the Linearized System with 98% Communication Loss . 28


3.5 Tracking Error of the Linearized System with 99% Communication Loss . 28

3.6 Relative Trajectory of the Nonlinear System with 98% Communication Loss 29

3.7 Tracking Error of the Nonlinear System with 98% Communication Loss 29

3.8 Relative Trajectory of the Nonlinear System with 99% Communication Loss 29

3.9 Tracking Error of the Nonlinear System with 99% Communication Loss 29


4.2 Tracking Error of Follower Spacecraft 1 with 89% Communication Loss . 36











ix









5.1 A Vicon Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Lab Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3 An iRobot Create Roomba with Markers and a Laptop . . . . . . . . . . 44

5.4 Simplified Robot Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.5 Schematic representation of the MRF Configuration . . . . . . . . . . . . 58

5.6 No Communication Loss, Robots’ Trajectories . . . . . . . . . . . . . . . 58

5.7 No Communication Loss, Follower 1 Tracking Error . . . . . . . . . . . . 58




5.11 50% Communication Loss, Robots’ Trajectories . . . . . . . . . . . . . . 59

5.12 50% Communication Loss, Follower 1 Tracking Error . . . . . . . . . . . 59








x







xi

List of Acronyms

MSFF: Multiple spacecraft in formation flying

LGQ: Linear quadratic Gaussian

MIMO: Multiple-input multiple-output

ADS: Asynchronous dynamical systems

NCS: Networked control systems

BMI: Bilinear matrix inequality

LQR: Linear quadratic regulator

LQG: Linear quadratic Gaussian

UTIAS: University of Toronto institute for aerospace studies

MRF: Multiple robots in formation

xii

Chapter 1

Introduction

Multiple spacecraft in formation flying (MSFF) is defined as a set of multiple spacecraft

where their states are coupled through a common control law [44]. MSFF have been a

topic of major interest since the 1960s when some projects required multiple spacecraft

to perform rendezvous manoeuvres [48]. Advancement in this field did not progress in

a significant manner since then until interest was renewed in the mid 1990s with the

launch of Earth Orbiter-1, the first autonomous formation flying mission. Ever since,

major strides and development have occurred and as research continues, the technology

would reach the level where MSFF can be integrated into many proposed space projects

[52].

MSFF has many advantages, paramount amongst them is the ability to perform

special missions which would not be possible using a single large spacecraft. Formation

flying can enhance a space mission’s performance by distributing mission tasks over many

small spacecraft, and if accurate MSFF is achieved, multiple satellites can be arranged as

to create a single large entity; this would result in space instruments with unprecedented

abilities, such as higher resolution imagery and interferometry. MSFF could also be

utilized to optimize planetary surveillance, exploration, and observation missions, and

because they have the ability to achieve a longer mission life, MSFF could generate

greater scientific return [48] [52] [7] [57].

Having multiple spacecraft perform a mission increases the redundancy and reliability

of the system if something goes wrong, and a defected component would mean sending

a replacement satellite, rather than redesigning and building the whole system. It also

allows the whole system to be reconfigured by simply sending the appropriate commands,

and combined with the structural flexibility of an MSFF system, this means the same

1

Chapter 1. Introduction 2

group of spacecraft could be used for various different missions interchangeably. Sensing

capabilities of the system also expand with MSFF, and maintenance becomes more man-

ageable and cost efficient. Finally, autonomous formation reduces ground maintenance

requirements, lowering the operational costs [48] [28] [57] [59].

1.1 Literature Review

For autonomous MSFF to be realistic and successful in space missions, communication

between the spacecraft also needs to be autonomous. A ground-based command and

control system for relative positioning of the spacecraft will be very complex and may

not be able to provide sufficiently rapid corrective control commands for manoeuvres,

formation reconfiguration, or collision avoidance [24]. Of the ongoing research currently

taking place for autonomous MSFF, the main areas being looked at are formation sensing

(where technology is used to develop more accurate sensors) and formation control (where

control algorithms are derived for the spacecraft to maintain the desired formation) [22].

Research on formation sensing attempts to improve the autonomy of spacecraft naviga-

tion through the use of wide area differential and differential carrier phase GPS (global

positioning system.) On the formation control front, work is being conducted on the use

of absolute vs. relative control architectures, including the communication requirements

and formation trajectories generated by these architectures. This thesis will focus on

MSFF control.

Five main architectures for controlling MSFF have been studied and examined in

the literature: multiple-input multiple-output (MIMO), virtual structure, behavioural,

cyclic, and leader-follower [44]. MIMO control techniques require the MSFF system to

be modeled as a single plant with multiple inputs and multiple outputs, and some of

the control techniques used with MIMO architectures include linear quadratic Gaussian

(LQG) and directed graphs [18] [47] [51]. Virtual structure architecture assumes the

MSFF system is a rigid body, where the overall motion of the system include rigid body

motions with expansions and contractions; Controlling the virtual structure of an MSFF

mission requires fitting a formation template to the position of the spacecraft within

the formation at every time step [29] [30]. Behavioural architecture combines different

controllers, each with different objectives, to achieve an overall state for the MSFF system

[4] [5]. In the cyclic architecture, the controller is formed by connecting controllers of

the spacecraft in the formation non-hierarchically, and some of the controllers derived for


this architecture include potential field and rule-based controllers [34] [40] [50]. Finally,

the leader-follower approach is similar to the cyclic approach in that the controller is

formed by connecting controllers of the spacecraft within the formation, however this

is done hierarchically. Various control laws have been studied for the leader-follower

architecture including sliding mode control, model predictive control, adaptive control,

intelligent control, synchronization control and H∞ control [41] [33] [9] [53] [20] [56][58]. The leader-follower architecture is the most developed and examined of the five

architectures mentioned [44].

The dynamics and control of a single spacecraft is well established and understood,

mainly due to the rigid body assumption which denotes that no relative motion exists

between 2 points on the body. In MSFF, the ultimate goal is for the spacecraft to act like

a single rigid body in their movement and manoeuvres, however in reality a formation

of spacecraft behaves like a deformable body due to relative position tracking errors and

synchronization errors, and so control forces are required to restore the spacecraft to

their desired formation [35]. An adaptive nonlinear controller has been proposed that

guarantees tracking errors of the relative positions converge to zero, which adopts the

leader-follower approach, and simulation results have proven the effectiveness of this

technique [9]. This controller however does not force follower satellites to track their

trajectories at the same rate, thus during manoeuvres the geometric configuration gets

distorted. A synchronization technique and algorithm can be integrated into the adaptive

nonlinear control strategy which guarantees that the synchronization error of the relative

positions also converge to zero [20]. This means that the formation configuration can be

kept during manoeuvres, which is highly desirable in MSFF, and again simulation results

have proven the effectiveness of this technique.

Precise MSFF requires the follower spacecraft to sense changes in the states of the

leader spacecraft, and so inter-satellite communication becomes integral to the success of

autonomous MSFF missions. This means that the response of the system when commu-

nication loss occurs needs to be analyzed and understood. Temporarily and periodic loss

of communication can occur at any time due to many factors, including solar winds and

component failures [22]. Communication loss analysis has been studied for some dynamic

systems, but very little work has been conducted on studying the effects of communi-

cation loss on MSFF. The majority of work on communication loss analysis have been

conducted on linear time-invariant systems, and most of the published work on communi-

cation loss has dealt with multi-agent systems [38] [36] [17] [37], asynchronous dynamical


systems (ADS) [1] [2] [39], Markov jump linear systems [10] [15] [6], and networked con-

trol systems (NCS) [21] [26] [3] [55]. In [39], stability criteria are established for nonlinear

systems, however applying these criteria to non-trivial dynamic systems is challenging.

Studying the different communication loss analysis methods proposed in those papers

can lead to techniques for examining the effects of communication loss in MSFF.

1.2 Objectives

Various controllers have been proposed for MSFF, including adaptive nonlinear control

[9]. The advantage of using an adaptive controller is that the relative positions of the

spacecraft are guaranteed to converge for a suitably smooth trajectory even in the pres-

ence of constant, slow varying, or unknown parameters (i.e. the masses of the spacecraft).

This controller is developed utilizing a leader-follower architecture, where the equations

of dynamics describe the states of the follower spacecraft relative to the leader spacecraft.

Since the control law depends on the relative states of the follower spacecraft with respect

to the leader, it is important to understand how the system behaves during periods of

communication loss, and how robust the controller is to periodic packet loss between the

spacecraft. In [20], a synchronization algorithm is integrated into the nonlinear adaptive

controller to force the position tracking errors in the MSFF system to converge to zero

at the same rate, thus maintaining the geometric configuration of the spacecraft when

executing manoeuvres. This synchronization algorithm will require multiple spacecraft

within the formation to communicate their states to each other, and so communication

loss will have an even stronger effect on the system when synchronization is incorporated

into the controller.

The objective of this paper is to propose a technique to managing MSFF systems

during periods of communication loss, and to investigate the effects of this technique on

the stability of the system. The communication loss compensation technique proposed in

this thesis is to use the previously communicated data in place of the lost data, and since

this technique is computationally efficient, it becomes advantageous for small satellites

not equipped with high on-board processing budgets. The stability of MSFF systems

denotes the ability of the follower spacecraft to converge to their desired trajectories, thus

maintaining the desired configuration of the formation. Another objective is to devise

an experimental plan for ground robots in formation, and to analyze the behaviour of

the system under communication loss both experimentally and theoretically using the


proposed communication loss management technique. The reason the experiments will

be conducted using ground robots is because it is not feasible to perform experimental

testing on real satellites (due to time constraints, high risk levels, and high costs) and the

university of Toronto institute for aerospace studies (UTIAS) is not yet equipped with

hardware suitable for satellite formation testing.

1.3 Overview

The rest of this thesis is presented in 4 more chapters. Chapter 2 presents the system

model and the formulation for the adaptive nonlinear controller with synchronization,

and simulation results showing the effectiveness of the controller. Chapter 3 introduces

the proposed communication loss management technique, analyzes the stability of MSFF

system with an approximation of the adaptive controller, then contrasts the theoretical

analysis with simulation results. In Chapter 4, MSFF systems with synchronization

are again analyzed using the proposed communication loss management technique, and

the theoretical analysis is contrasted with simulation results. Chapter 5 introduces the

experimental setup for ground robots in formation by presenting the system model, and

analyzing the system under communication loss to compare the experimental outcome

with the theoretical results. Finally, concluding remarks and future work are discussed

in Chapter 6.

Chapter 2

System Modelling

2.1 MSFF Position Dynamics Model

In this section, the system model for MSFF will be derived. A drawing depicting a leader-

follower system is shown in Fig. 2.1 on the following page. The inertial coordinate system

(X, Y, Z) is attached to the center of the Earth, R(t) is the position vector from the origin

of the inertial coordinate system to the leader spacecraft, the relative coordinate system

(xl, yl, zl) is attached to the leader spacecraft such that xl is in the opposite direction

of the velocity vector, yl is along the direction of R(t), and zl completes the right hand

coordinate frame, and ρ is the position vector from the origin of the relative coordinate

system to the follower spacecraft [9].

The nonlinear position dynamics of the leader and follower spacecraft with respect to

Figure 2.1: Schematic representation of an MSFF system

6

Chapter 2. System Modelling 7

the inertial coordinate system (X, Y, Z) are:

ml··R+ml (M +ml)G

(R/ ‖R‖3

)+ Fdl = ul (2.1)

mf

(··R+

··ρ)

+mf (M +mf )GR + ρ

‖R + ρ‖3+ Fdf = uf (2.2)

where ml,mf are the masses of the leader and the follower, Fdl, Fdf are disturbance force

vectors, ul(t), uf (t) are the control input vectors of the leader and follower spacecraft, M

= 5.97 × 1024kg is the Earth’s mass, and G = 6.672 × 10−11m3/kg.s2 is the universalgravity constant. Because M >> ml,mf , equations (2.1) and (2.2) can be simplified as:

ml··R+mlMG

(R/ ‖R‖3

)+ Fdl = ul (2.3)

mf

(··R+

··ρ)

+mfMGR + ρ

‖R + ρ‖3+ Fdf = uf (2.4)

The dynamic equations describing the position of the follower spacecraft relative to the

leader spacecraft can then be written as:

mf··ρ+mfMG

(R + ρ

‖R + ρ‖3− R‖R‖3

)+mfml

ul + Fdf −mfml

Fdl = uf (2.5)

Equation (2.5) can be written in terms of the relative coordinate system (xl, yl, zl) as:

mf··q+C (ω)

·q+N (q, ω,R, ul) + Fd = uf (2.6)

where q(t) = [x, y, z]T is the relative position vector, and C(ω) is the Coriolis-like matrix

given by:

C (ω)∆= 2mfω

0 −1 01 0 0

0 0 0

(2.7)N(.) is a nonlinear term defined as:

N(q, ω,

·ω,R, ul

)∆=

mfMG

x‖R+q‖3 −mfω

2x+mf·ω y +

mfmlulx

mfMG(y+‖R‖‖R+q‖3 −

1‖R‖2

)−mfω2y −mf

·ω x+

mfmluly

mfMGz

‖R+q‖3 +mfmlulz

(2.8)

where Fd is the total, constant disturbance force defined as:

Fd∆=Fdf − (mf/ml)Fdl (2.9)


The left hand side of equation (2.6) can be linearly parameterized as:

mf··q+C (ω)

·q+N

(q, ω,

·ω,R, ul

)+ Fd

= W(ξ,·q, q, ω,

·ω,R, ul

)θ

(2.10)

where W(.) is the regression matrix composed of known functions, ξ is a dummy variable,

and θ is the system’s constant parameter vector. W(.) and θ are defined as:

W(ξ,·q, q, ω,

·ω,R, ul

)∆=

ξx − 2ω

.y−ω2x− .ω y + x‖R+q‖3 ulx 1 0 0

ξy − 2ω.x−ω2y + .ω x+

(y+‖R‖‖R+q‖3 −

1‖R‖2

)uly 0 1 0

ξz +z

‖R+q‖3 ulz 0 0 1

(2.11)

and

θ∆=[mf mf/ml Fdx Fdy Fdz

]T(2.12)

2.2 Adaptive Control Formulation

The main objective now is to design a control input uf so that q (t)→ qd (t) as t→∞ .The performance of the control can be evaluated using the tracking error e(t) defined as:

e (t)∆= qd (t)− q (t) (2.13)

To account for the system parametric uncertainty due to instrumentation error, an adap-

tation law for on-line estimation of the unknown parameters is introduced and defined

as:≈θ (t)

∆= θ −

∧θ (t) (2.14)

where≈θ denotes the parameter estimation error vector and

∧θ is the dynamic estimate

of θ . The filtered tracking error r(t) is defined as:

r (t)∆=·e (t) + Λe (t) (2.15)

where Λ is a constant, diagonal, positive-definite, control gain matrix. Differentiating

equation (2.15) and multiplying it by mf gives:

mf·r = mf

( ··qd +Λ

·e)−mf

··q (2.16)

Equation (2.16) can be substituted into equation (2.6) to get:

mf·r = mf

( ··qd +Λ

·e)+C (ω)

·q+N (q, ω,R, ul)+Fd−uf = W

(··qd +Λ

·e,·q, q, ω,R, ul

)θ−uf

(2.17)


The control input uf (t) is then designed as:

uf = W (.)∧θ+Mr (2.18)

where M is a constant, diagonal, positive definite matrix.∧θ (t) is updated using the

algorithm:·∧θ = ΓW

T (.) r (2.19)

where Γ is a constant, diagonal, positive-definite, adaptation gain matrix. Equations

(2.15), (2.18) and (2.19) constitute the proposed adaptive nonlinear controller. Finally,

by differentiating equation (2.14) with respect to time, the parameter

·≈θ can be described

as:·≈θ = −ΓW T (.) r (2.20)

By defining the Lyapunov function:

V (r,≈θ)

∆=

1

2rTmfr +

1

2

≈θT Γ−1

≈θ (2.21)

it is shown in [9] that global asymptotic convergence of the position and velocity tracking

errors can be accomplished by the adaptive controller. In addition, global asymptotic

convergence is also proved for an adaptive controller through the passivity theorem [8]

[27].

2.3 Integrating Synchronization Control

As mentioned in Chapter 1, for the formation to keep its configuration during a manoeu-

vre, synchronization is needed between the spacecraft in an MSFF system. To integrate

the synchronization technique into the proposed MSFF adaptive controller, first the syn-

chronization error, which identifies the performance of the synchronization controller, is

defined as [20]:

Ξ (t) = Te (t) (2.22)

where T is a generalized synchronization transformation matrix. Ξ (t) represents how

one trajectory converges with respect to another. It is now important to introduce the

concepts of internal and external synchronization. If for a specific follower spacecraft e(t)

in Eq. 2.22 is defined as:


e(t) =

ex

ey

ez

then the synchronization error is titled internal synchronization, since the synchronization

occurs between the axes of the same follower spacecraft. If however for a system with n

follower spacecraft e(t) is defined as:

e(t) =

e1x

e2x...

enx

then the synchronization is between the same axis of different follower spacecraft, and

this is termed external synchronization (in the example above, the synchronization is

for the x-axis, and can similarly be written for the y and z axes). The objective then

becomes to achieve Ξ (t)→ 0 as t→∞ . The following synchronization transformationmatrix is chosen:

T =

2 −1 −1−1 2 −1

. . . . . . . . .

−1 2 −1−1 −1 2

(2.23)

This synchronization transformation denotes that each follower spacecraft will be syn-

chronized with the follower spacecraft before it and the spacecraft after it. The coupled

position error e* which contains the tracking error e(t) and the synchronization error

Ξ (t) is defined as:

e∗ (t) = e (t) + BTTt∫

0

Ξdτ (2.24)

where B is a positive coupling gain matrix. The coupled filtered tracking error r∗(t) is

defined as:

r∗ (t) = ė∗ (t) + Λe∗ (2.25)

where Λ is a constant, diagonal positive-definite, control gain matrix. The control input

uf is then defined as:

uf (t) = W (.)∧θ (t) + Mr (t) + KsT

TΞ (t) (2.26)


where Γ is a constant, diagonal, positive-definite, adaptation gain matrix. Equations

(3.24, 3.25, 3.26) constitute the adaptive synchronization controller, and by defining the

following Lyapunov function:

V (r,≈θ,Ξ)

∆=

1

2rTmfr +

1

2

≈θT Γ−1

≈θ +

1

2ΞTKsΞ +

1

2

t∫0

TTΞdτ

T BΛKs t∫

0

TTΞdτ

(2.27)

global asymptotic convergences to zero of e(t) and Ξ (t) are shown in [20].

2.4 Simulation Results

In this section, the effectiveness of the adaptive nonlinear synchronization controller is

demonstrated through Matlab simulations of an MSFF system with one leader and four

follower spacecraft. The goal of the controller is to drive the follower spacecraft from

their initial position to a desired final positions relative to the leader spacecraft, and to

maintain the desired final position. Having a follower spacecraft maintain a constant

position relative to the leader spacecraft is termed an along-track trajectory, and in the

simulations presented here all four follower spacecraft have an along-track trajectory.

The parameters of the MSFF system, adopted from [20] are as follows:

mf1 = mf2 = mf3 = mf4 = 410kg M = diag[0.13, 0.12, 0.09]

µ = 9.86× 1014m3/s2 Λ = diag[0.04, 0.04, 0.04]R = [0, 4.224× 107, 0]Tm B = diag[0.0008, 0.0008, 0.0008]

ml = 1550kg Fd = [−1.025, 6.248,−2.415]T × 10−5Nqd1 = [100, 0, 0]

Tm q1(0) = [150, 10, 20]Tm

qd2 = [0,−100, 0]Tm q2(0) = [−10,−130,−20]Tmqd3 = [−100, 0, 0]Tm q3(0) = [−140, 10,−20]Tmqd4 = [0, 100, 0]

Tm q4(0) = [30, 160, 20]Tm

q̇d1 = q̇d2 = q̇d3 = q̇d4 = [0, 0, 0]Tm/s q̇1(0) = q̇2(0) = q̇3(0) = q̇4(0) = [0, 0, 0]

Tm/s

ω = 7.272× 10−5rad/s

2.4.1 Adaptive Control Simulation

The first simulation is run for the MSFF described above using only an adaptive con-

troller, without synchronization. The following figures outline the results of the simula-

tions:


Figure 2.2: Relative Trajectory of the MSFF System with No Synchronization

Figure 2.3: Tracking Error of Follower

Spacecraft 1 with No Synchronization

Figure 2.4: Internal Sync Error of Follower













Figure 2.10: Internal Sync Error of Fol-

lower Spacecraft 4 with No Synchroniza-

tion

Figure 2.11: External Sync Error of the x

Axis with No Synchronization

Figure 2.12: External Sync Error of the y

Axis with No Synchronization


Figure 2.13: External Sync Error of the z Axis with No Synchronization

As can be observed from the simulation results, the adaptive nonlinear controller

succeeds in forcing the tracking errors of the follower spacecraft to converge to zero. The

internal synchronization for the follower spacecraft and external synchronization of the

three axes are also presented to verify the effectiveness of the synchronization algorithm

after it gets incorporated into the simulation.

2.4.2 Adaptive Synchronization Control Simulation

The same MSFF system is simulated once again using the full adaptive nonlinear syn-

chronization controller, and the results are as follows:

Figure 2.14: Relative Trajectory of the MSFF System with Synchronization



Spacecraft 1 with Synchronization


lower Spacecraft 1 with Synchronization














Figure 2.23: External Sync Error of the x

Axis with Synchronization

Figure 2.24: External Sync Error of the y

Axis with Synchronization

Figure 2.25: External Sync Error of the z Axis with Synchronization


As can be observed from the simulations results, the adaptive synchronization control

law is able to force the internal and external synchronization errors to converge to zero

faster than the adaptive controller alone, and the tracking errors of the four spacecraft

still converge to zero. Since the external synchronization error converge to zero at a faster

rate, this means the MSFF geometric configuration is maintained for a longer time, a

feature desirable in MSFF systems.

This chapter defines an MSFF system model, and presents the formulation for an

adaptive nonlinear controller and an adaptive nonlinear synchronization control which

guarantees convergence in the presence of constant, slow varying, or unknown parameters.

Simulation results show the adaptive controller’s ability to force the tracking errors of

the follower spacecraft to converge to zero, and the adaptive synchronization controller’s

ability to decrease the synchronization errors and force them to converge faster than the

adaptive control alone. Chapter 3 will introduce ADS and show how they can be used

to model communication loss within a networked system.

Chapter 3

Communication Loss Stability

Analysis

In this chapter, the foundation for the theoretical analysis of communication loss between

spacecraft will be presented. In particular, the behaviour and stability of a follower space-

craft will be analyzed when communication is lost between it and the leader spacecraft.

It is assumed that the leader spacecraft communicates its states (position and velocity

data) to the follower spacecraft within the MSFF system, and the follower spacecraft

then calculate their relative states and determine the appropriate control forces; thus,

when communication is lost between a follower and the leader, the follower will not be

aware of its relative states and will not be able to calculate the required control forces. It

is also assumed that the communication loss is stochastic in nature (since external events

such as solar winds and flares could occur at any time), and thus the amount of time

where communication is lost within the mission life is presented as a probability value.

To manage periods of communication loss, it is proposed that the follower spacecraft

use the last communicated data in lieu of the lost data to calculate the required con-

trol forces. The main advantage of this technique is that it is computationally simple,

thus becoming a very attractive method for small spacecraft with limited computational

resources. In the literature studied, authors have used various methods for modelling

communication loss within a system, including Bernoulli processes [45], Markov chain

models [25] [43], and asynchronous dynamical systems (ADS) [1] [2]. In the following

section, an MSFF system with communication loss will be modelled as an ADS, and the

stability of the system when the proposed communication loss management technique is

employed will be examined and analyzed.

18

Chapter 3. Communication Loss Stability Analysis 19

3.1 Asynchronous Dynamical Systems

ADS are systems that incorporate discrete and continuous dynamics. The discrete dy-

namics are driven by external events with fixed rates, and the continuous dynamics are

governed by differential equations (or difference equations for the discretized form) [1]

[2]. ADS with rate constraints on events can be used to model communication loss in a

networked control system, and stability criteria of such systems have been studied and

presented in literature [1] [2] [60]. The following figure, which has been studied in [60],

illustrates a networked control system with data packet loss:

Figure 3.1: ADS Communication Loss Model

This example shall be used to model an MSFF system with communication loss. As

can be seen in Fig. 3.1, communication loss is modeled as a switch that closes randomly

with a certain probability, represented by the constant term as rate r; it is important

to clarify that the rate r does not indicate the frequency at which the switch closes,

but rather the probability that at any given time, the switch will be closed. The closed

position of the switch (position S1) represents the dynamics of the system when the

communication exists, thus the states of the leader spacecraft are transmitted. The

open position of the switch (position S2) represents the dynamics of the system when

transmission is lost between the spacecraft. Since the compensation technique being

proposed during periods of communication loss is to use the last communicated data,

then the dynamics of the switch can be modeled as follows:

S1 → x(kh) = x(kh)S2 → x(kh) = x((k − 1)h)

(3.1)


3.1.1 Stability Criteria

The main attraction of using ADS to model communication loss is that the stability

criteria for ADS have been studied and are available in the literature. These criteria

and any associated theorems can therefore be used to analyze and study the stability of

MSFF systems with communication loss to understand the behaviour of those systems

when data transfer between the spacecraft is not always available. This subsection will

highlight these main criteria, which are fully presented in [1] [2] [60].

Given the ADS in Fig. 3.1 where the equations of dynamics for the continuous states

are linear and time invariant, if there exists a Lyapunov function V(x) and scalars a1, a2

such that:

ar1a1−r2 > a > 1 (3.2)

where r is the switching rate as mentioned previously and

V (x(k + 1))− V (x(k)) ≤ (a−2s − 1)V (x(k)), s = 1, 2 (3.3)

then the ADS is exponentially stable, with decay rate greater than a. This criteria

guarantees the ADS to be stable on the whole. If for s the discrete state dynamics can

be represented in the following form:

x((k + 1)h) = Φsx(kh) (3.4)

then the scalars a1, a2 and a Lyapunov function in the following form:

V (x(kh)) = xT (kh)Px(kh) (3.5)

where P is a positive-definite symmetric matrix can be formulated as a bilinear matrix

inequality (BMI) problem by rewriting Eq. 3.2 and Eq. 3.3 as

r log a1 + (1− r) log a2 > 0 (3.6)

and

ΦTs PΦs ≤ a−2s P (3.7)

where a BMI is defined as a function of the form

F (x, y) = F0 +m∑i=1

xiFi+n∑j=1

yjGj+m∑i=1

n∑j=1

xiyjHij > 0


where Gj and Hij are symmetric matrices with the same dimensions as Fi, x ∈


Theorem: If the closed-loop system (Φ − ΓK) of the system in Fig. 3.1 with nocommunication loss is stable, then:

- If the open-loop system(Φ) is marginally stable, then the system is exponentially

stable for all 0 < r ≤ 1- If the open-loop system is unstable, then the system is exponentially stable for all

1

1− γ1/γ2< r ≤ 1 (3.13)

where

γ1 = log [λ2max(Φ− ΓK)]

γ2 = log [λ2max(Φ)]

(3.14)

As already stated, the conditions of the theorem are necessary for the stability of an

ADS, and for all r values that don’t fall within the specified range, the system is unstable.

Using the communication loss management technique proposed, this theorem allows for

calculating the maximum rate of communication loss that the system can tolerate. If

the expected rate of communication loss for a particular MSFF system is less than the

rate determined by the theorem, then the system is guaranteed to maintain its desired

formation, otherwise the system will become unstable and the follower spacecraft would

not converge to their desired trajectories. The setback of using this theorem is that

it is developed for linear discrete systems, and assumes a simple gain feedback control

law. Since this study is meant as a preliminary analysis of MSFF stability under the

effects of communication loss, the equations of dynamics for MSFF systems presented

in Chapter 2 shall be linearized and discretized, and the adaptive control law shall be

approximated as a constant gain control law in order to use the ADS analysis theorem.

Future studies shall relax these constraints to fully analyze the nonlinear system with

the adaptive control law.

Linearization can be performed on the nonlinear dynamics of MSFF systems using

the first-order terms of their Taylor series expansion. The following represents the general

equation for linearizing a multivariable function:

f(X) ≈ f(X0) +∇f |p · (X −X0) (3.15)

where X is the states vector and X0 is the linearization point of interest. For the MSFF

nonlinear equations presented in Chapter 2, assuming ul = 0, the linearized equations of


dynamics for MSFF systems around the point (0,0,0) are as follows:

mf q̈ + Cl(.)q̇ +Nl(.)q + Fd = uf (3.16)

where

Nl(.) =

mf

(µR3− ω2

)mf ω̇ 0

−mf ω̇ −mf(

2µR3

+ ω2)

0

0 0mfµ

R3

(3.17)

Cl(.) =

0 2mfω 0

−2mfω 0 00 0 0

(3.18)The augmented states vector can be represented as:

ql =

qq̇

(3.19)then:

q̇l =

[03×3] [I3×3][−Nl(.)/mf] [−Cl(.)/mf] ql +

[03×3][I3×3/mf

]u−

Fdx

Fdy

Fdz

(3.20)

The adaptive control law from Chapter 2 is defined as:

uf = Wl(.)θ̂ + Mr (3.21)

where the regressor matrix for the linearized equations is:

W (.) =

−Λẋ+ 2ωẏ +

(µR3− ω2

)x+ ω̇y 1 0 0

−Λẏ − 2ωẋ−(

2µR3− ω2

)y − ω̇x 0 1 0

−Λż +(µR3

)z 0 0 1

(3.22)

and∧θ is the estimate for the linear system’s constant parameter vector θ defined as:

θ∆=[mf Fdx Fdy Fdz

]T(3.23)


3.1.2 Controller Estimate

Since the stability criteria and the theorem stated above are based upon the assumption

that the control law for the plant is in the form −K −x(kh), the objective is now tosimplify the adaptive control law as a constant feedback control law to properly apply

the theorem and analyze the effects of communication loss on an MSFF system. To do

so, the MSFF system is first assumed to be in a circular orbit ( therefore·ω = 0), and

then the parameter variable∧θ is assumed to be a constant (the validity of this assumption

stems from the definition of the parameter variable, which is required to be a constant

or slow-varying). To properly select the values for the slow-varying parameters, possible

values for those parameters would be tested to determine the value that results in the

largest r when the theorem in section 3.1.1 is used; this constitutes the worst-case value

of that parameter, and for other values the system would still be stable. For instance,

when the mass of the spacecraft is tested, it is determined that the worst-case value for

that parameter is its minimum value, or the value of the spacecraft with no fuel, and so

once the stability criteria are derived for that worst-case value, the system will still be

stable for other values of the parameter. While this gives the system robust attributes,

it is important to point out that by keeping the∧θ value a constant, the controller ceases

to be adaptive. The control law can now be approximated as:

uf = Wl(.)θ + Mr = −Klql (3.24)

where Kl is a constant matrix for the linearized system. The control law is now a

constant feedback law, where the gain K is an approximation to the system’s behaviour

under adaptive control law, and the theorem can be used.

3.1.3 Discretization

The final step that needs to be taken before the theorem could be applied is to discretize

the MSFF system. While discretizing the state space model is a straight forward task, the

discrete equivalent of the K matrix is also required before the theorem could be applied;

this is because the continuous equivalent of the K matrix in a feedback control law will

not necessarily be the same in the discrete form, as is observed in the difference between

the continuous-time and discrete-time Riccati equations used to calculate the feedback

control law for linear-quadratic-regulator (LQR) and linear-quadratic-Gaussian (LQG)


controllers [49]. In this subsection, the bilinear equivalence of the linearized MSFF state

space model will be derived, along with the discrete form of the constant Kl matrix.

The full derivation of the bilinear equivalence of a continuous system is available in

[16] and will be highlighted here. Starting from the continuous state space equation in

the following form:

ẋ = Ax+Be

u = Cx+De(3.25)

the Laplace transform is taken:

sX = AX +BE

U = CX +DE

and s can be substituted for by the z-transform equivalent of the trapezoid/bilinear rule:

2(z − 1)h(z + 1)

X = AX +BE (3.26)

U = CX +DE (3.27)

The equations are then transformed back to the time domain:

x(k + 1)− x(k) = Ah2

[x(k + 1) + x(k)] +Bh

2[e(k + 1) + e(k)] (3.28)

The k + 1 terms are now grouped together and defined as w(k + 1):

x(k+ 1)− Ah2x(k+ 1)− Bh

2e(k+ 1) = x(k) +

Ah

2x(k) +

Bh

2e(k)

∆=√hw(k+ 1) (3.29)

where the√h is a factor introduced to balance the gain of the discrete equivalence

between the input and output [16]. Writing w at time k and solving for x yields:

x(k) =

(I − Ah

2

)−1√hw(k) +

(I − Ah

2

)−1Bh

2e(k) (3.30)

Eq. (3.30) is now substituted into Eq. (3.29):

w(k + 1) =

(I +

Ah

2

)(I − Ah

2

)−1w(k) +

(I − Ah

2

)−1B√he(k) (3.31)

Finally, the output equation for the bilinear equivalence can be obtained by substituting

Eq. (3.30) into Eq. (3.27):

u(k) =√hC

(I − Ah

2

)−1w(k) +

D + C(I − Ah

2

)−1Bh

2

e(k) (3.32)


Since the state space representation of the linearized MSFF system is in the following

form:

ẋ = Ax+Bu (3.33)

then the bilinear equivalence is in the following form:

w(k + 1) = Φw(k) + Γu(k) (3.34)

whereΦ = (I + Ah

2)(I − Ah

2)−1

Γ = (I − Ah2

)−1B√h

(3.35)

Since the output control law is of the form

u = −Kx (3.36)

then to get the discrete equivalence of the K matrix, x(k) is substituted with the w(k)

equivalence:

u(k) = −Kx(k) = −K(I + AT2

)−1[w(k)− BT2u(k)] (3.37)

u(k) = −[I −K(I + AT2

)−1BT

2]−1K(I +

AT

2)−1w(k) (3.38)

Therefore the discrete form of the constant K matrix, or Kd is in the following form:

[I −K(I + AT2

)−1BT

2]−1K(I +

AT

2)−1 (3.39)

Now the MSFF system can be fully discretized, and the theorem for determining the

maximum rate of communication loss a system can tolerate can be applied.

3.2 Numerical Example

A one-leader-one-follower example using one of the 4 follower spacecraft used in the

simulations from Chapter 2 is analyzed in this section. This example will be simplified and

discretized using the techniques discussed in this chapter, and the theorem for analyzing

the stability of the system in the presence of communication loss will be applied. In

addition, a simulation of the linearized system and the nonlinear system in the presence

of communication loss will be shown to verify the results obtained from the theoretical

analysis.


3.2.1 Theoretical Analysis

The parameters of the system that will be analyzed are as follows:

mf = 410kg q(0) = [150, 10, 20]Tm

µ = 9.86× 1014m3/s2 q̇d = [0, 0, 0]Tm/sR = [0, 4.224× 107, 0]Tm q̇(0) = [0, 0, 0]Tm/sω = 7.272× 10−5rad/s M = diag[0.13, 0.12, 0.09]qd = [100, 0, 0]

Tm Λ = diag[0.04, 0.04, 0.04]

Fd = [−1.025, 6.248,−2.415]T × 10−5N

Using a 1 second sampling period, this system is discretized as outlined in subsection

3.1.3, and when the stability criteria theorem is applied to this example, r is calculated to

be 0.01; since r represents the rate at which the switch in Fig. 3.1 closes, this means that

the MSFF system will remain stable using the proposed communication loss management

technique for communication loss rates that are less than 99%. The effective sampling

period for this example is then calculated to be just under 100 seconds, or in other words

if the sampling period is chosen to be for example 99 seconds, and the system experiences

no communication loss, it will remain stable and the follower spacecraft will converge to

its desired states. This suggests that another use for the theorem is to determine the

appropriate sampling period for the system (i.e. how often the leader spacecraft needs to

transmit its states to the follower spacecraft) especially if the expected communication

loss rate is known ahead of time.

3.2.2 Linear System Simulation

As a validation to the results produced by the theorem, the linearized system with the

constant feedback control law was simulated using the proposed communication loss

management technique, and the figures below show the simulation results for 98% and

99% communication loss rates:

As can be observed from Fig. 3.2-3.5, the linearized MSFF system maintains stability

when the communication loss rate is 98% and the follower spacecraft is able to track its

desired trajectory, but the system is unstable for the case when the rate is 99%. This

complies with the results obtained from the theoretical analysis of the system.


Figure 3.2: Relative Trajectory of the Lin-

earized System with 98% Communication

Loss

Figure 3.3: Tracking Error of the Lin-


Loss



Loss

Figure 3.5: Tracking Error of the Lin-


Loss


3.2.3 Nonlinear System Simulation

The theoretical analysis in this section is conducted for the linearized MSFF system, and

uses an approximation of the adaptive controller. It remains to be seen how close the

analysis on the linearized system matches the behaviour of the nonlinear system. To do

so, the full nonlinear system with adaptive control will be simulated as in Chapter 2,

with communication loss introduced to the system, and using the proposed management

technique. The theoretical analysis of the linearized system indicate that the system

will remain stable if the communication loss rate is less than 99%, and below are the

simulation results for the MSFF system under 98% and 99% communication loss rates:

Figure 3.6: Relative Trajectory of the Non-

linear System with 98% Communication

Loss

Figure 3.7: Tracking Error of the Nonlinear

System with 98% Communication Loss

Figure 3.8: Relative Trajectory of the Non-

linear System with 99% Communication

Loss

Figure 3.9: Tracking Error of the Nonlinear

System with 99% Communication Loss

As can be observed from the plots in Fig. 3.6-3.9, the nonlinear MSFF system main-


tains stability when the communication loss rate is 98% and the follower is again able to

track its desired trajectory, but the system is unstable for the case when the rate is 99%.

This complies with the theoretical analysis conductd on the linearized system, and sug-

gests that the analysis of the simplified system is a good approximation of the behaviour

of the nonlinear system with the adaptive control law in the presence of communication

loss.

This chapter outlines the communication loss management technique proposed by this

thesis, which is to use the latest communicated data in lieu of the lost data. The chapter

also introduces ADS, and demonstrates how the can be used to model communication

loss within MSFF systems. In order to use the ADS stability criteria to analyze the

effect of communication loss on MSFF systems, the system model is linearized and the

controller is simplified as a constant feedback control law. Simulation results of the

linearized and nonlinear systems are also presented and contrasted with the theoretical

analysis. Chapter 4 will perform similar theoretical analysis on MSFF systems with

synchronization control between the follower spacecraft.

Chapter 4

ADS Stability Analysis with

Synchronization Control

In Chapter 3, a theoretical approach to analyzing the behaviour of an MSFF system

when communication is lost between the leader and a follower is presented. However, to

incorporate synchronization control into MSFF systems, follower spacecraft are required

to communication with other follower spacecraft in addition to the leader. This chapter

will extend the theoretical analysis from the previous chapter to account for the case when

synchronization is integrated into the controller. Using the linearization and controller

simplification techniques from Chapter 3, the system model for an MSFF system with

synchronization control will be derived, and a numerical example for the theoretical

analysis will be provided. The theoretical analysis of the numerical example will then be

contrasted with simulation results of the linearized and the nonlinear MSFF system as

was done in the previous chapter.

4.1 ADS Model with Synchronization

Similar to the approach taken in Chapter 3, the main goal again becomes to represent the

MSFF system with synchronization control as a linear time invariant system of equations

in the following form:

ẋ(t) = Ax(t) +Bu(t) (4.1)

where the control force u can be approximated as a constant gain feedback law in the

31

Chapter 4. ADS Stability Analysis with Synchronization Control 32

following form:

u(t) = −Kx(t) (4.2)

The linearized system model for a 1-leader 1-follower system shown in Eq. 3.20 can

be extended for n-follower spacecraft systems as shown below:

q̇l =

[03×3] [I3×3][03×(6×(n−1))

][−Nl(.)/mf] [−Cl(.)/mf] [03×(6×(n−1))]

[03×3][03×(6×1)

][I3×3]

[03×(6×(n−2))

][03×(6×1)

] [−Nl(.)/mf] [−Cl(.)/mf] [03×(6×(n−2))]. . . . . . . . . . . .

[03×3][03×(6×(n−1))

][I3×3][

03×(6×(n−1))] [−Nl(.)/mf] [−Cl(.)/mf]

ql+

[03×(3×n)

][I3×3/mf

] [03×(3×(n−1))

][03×(3×n)

][03×3]

[I3×3/mf

] [03×(3×(n−2))

]. . . . . . . . .[

03×(3×n)]

[03×(3×(n−1))

] [I3×3/mf

]

u− (4.3)

[ [Fdx1 Fdy1 Fdz1

] [Fdx2 Fdy2 Fdz2

] . . . [ Fdxn Fdyn Fdzn ]]T

where

q =[x1 y1 z1 ẋ1 ẏ1 ż1 x2 y2 z2 ẋ2 ẏ2 ż2 · · · xn yn zn ẋn ẏn żn

]T(4.4)

and

u =[ufx1 ufy1 ufz1 ufx2 ufy2 ufz2 · · · ufxn ufyn ufzn

]T(4.5)

The control input (Eq. 2.26) can be simplified in the same manner highlighted in the

previous chapter to be approximated as a feedback control law in the form of Eq. 4.2,

for which K will be defined as:


K =

V1 0 0 V2 V3 0 V4 0 0 V5 0 0 V0 V6 0 0 V7 0 0

0 V8 0 V9 V10 0 0 V11 0 0 V12 0 V0 0 V13 0 0 V14 0

0 0 V15 0 0 V16 0 0 V17 0 0 V18 V0 0 0 V19 0 0 V20

V6 0 0 V7 0 0 V1 0 0 V2 V3 0 V4 0 0 V5 0 0 V0

0 V13 0 0 V14 0 0 V8 0 V9 V10 0 0 V11 0 0 V12 0 V0

0 0 V19 0 0 V20 0 0 V15 0 0 V16 0 0 V17 0 0 V18 V0... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

V4 0 0 V5 0 0 V0 V6 0 0 V7 0 0 V1 0 0 V2 V3 0

0 V11 0 0 V12 0 V0 0 V13 0 0 V14 0 0 V8 0 V9 V10 0

0 0 V17 0 0 V18 V0 0 0 V19 0 0 V20 0 0 V15 0 0 V16

(4.6)

for the synchronization transformation matrix

T =

2 −1 0 0 −1−1 2 −1 0 0

. . . . . . . . .

0 0 −1 2 −1−1 0 0 −1 2

(4.7)

where:

V0 = 01×(6×(n−3))

V1 =(µR3− ω2

)mf +M1(2B − 1)Λ

V2 = M1(2B − 1)− ΛmfV3 = 2ωmf

V4 = V6 = −BΛM1V5 = V7 = −BM1


V8 = M2(2B − 1)Λ−(

2µR3

+ ω2)mf

V9 = −2ωmfV10 = M2(2B − 1)− ΛmfV11 = V13 = −BΛM2V12 = V14 = −BM2V15 = M3(2B − 1)Λ + µR3mfV16 = M3(2B − 1)− ΛmfV17 = V19 = −BΛM3V18 = V20 = −BM3

The MSFF system with synchronization incorporated into the controller is now in

the form of Eq. 4.1 and 4.2, and the stability criteria for ADS can be applied; more

specifically, the theorem for determining that maximum rate of communication loss the

system can withstand can now be applied to analyze the behaviour of the MSFF system

when communication is lost between the leader and follower spacecraft, as well as between

follower spacecraft and other follower spacecraft.

4.2 Numerical Example

The five-satellite system from Chapter 2 will be used to conduct a stability analysis in

the presence of communication loss. First, the theoretical analysis will be conducted

using the ADS stability criteria for the linearized model with the simplified controller

that includes synchronization between the follower spacecraft, and then simulation re-

sults for the linearized system will be contrasted with the theoretical analysis. Finally,

the nonlinear system with the adaptive synchronization control from Chapter 2 will be

simulated again, this time with communication loss, to determine how the behaviour of

the system compares with the theoretical analysis conducted on the linearized system.

4.2.1 Theoretical Analysis

The parameters of the 1-leader four-follower system are shown below:


mf1 = mf2 = mf3 = mf4 = 410kg M = diag[0.13, 0.12, 0.09]

µ = 9.86× 1014m3/s2 Λ = diag[0.04, 0.04, 0.04]R = [0, 4.224× 107, 0]Tm B = diag[0.0008, 0.0008, 0.0008]ω = 7.272× 10−5rad/s Fd = [−1.025, 6.248,−2.415]T × 10−5Nqd1 = [100, 0, 0]

Tm q1(0) = [150, 10, 20]Tm

qd2 = [0,−100, 0]Tm q2(0) = [−10,−130,−20]Tmqd3 = [−100, 0, 0]Tm q3(0) = [−140, 10,−20]Tmqd4 = [0, 100, 0]

Tm q4(0) = [30, 160, 20]Tm

q̇d1 = q̇d2 = q̇d3 = q̇d4 = [0, 0, 0]Tm/s q̇1(0) = q̇2(0) = q̇3(0) = q̇4(0) = [0, 0, 0]

Tm/s

After discretizing Eq. 4.3 and Eq. 4.6 using the bilinear equivalence with a 1 second

sampling period and applying the ADS stability theorem, r is calculated to be 0.1, which

means that the system shall remain stable for communication loss rates less than 90%.

Since all the follower spacecraft in this example are similar to the one studied in section 3.2

(with the exception of the initial position and the final desired positions of the follower

spacecraft) it can be observed that addting synchronization attributes to the MSFF

system increased its sensitivity to communication loss. With the system in which each

follower is only required to communicate with the leader, the theoretical analysis shows

the system to be capable of maintaining stability for communication loss rates lower

than 99%, but with the system where the follower spacecraft also communicate with

other follower spacecraft, the MSFF system is now only capable of maintaining stability

for communication loss rates lower than 90%. This means that missions which would

potentially use synchronization techniques need to assess the probability of packet loss

occuring between the spacecraft and determine whether the system would remain stable

under the worst-case conditions before integrating synchronization into the controller.

4.2.2 Linear System Simulation

To validate the theoretical results, the linearized system was simulated using the pro-

posed communication loss management technique. The figures below show the simulation

results for 89% and 90% communication loss rates:

As can be observed, the simulation results match the results obtained from the theo-

retical analysis. The MSFF system maintains stability using the proposed management

technique when the communication loss rate is 89%, but becomes unstable when the




Loss


Spacecraft 1 with 89% Communication

Loss



Loss



Loss

Figure 4.5: Tracking Error of Follower Spacecraft 4 with 89% Communication Loss




Loss



Loss



Loss



Loss



communication loss rate reaches 90%, which complies with the results from the theorem.

It remains to be seen how close those results are to the nonlinear system with adaptive

synchronization control.

4.2.3 Nonlinear System Simulation

The following figures show the simulation results of the full nonlinear MSFF system with

adaptive synchronization control when the system experiences communication loss. The

proposed communication loss management technique is once again used, and the results

below are for 94% and 95% communication loss rates (The reason for including these 2

cases is explained shortly):



Loss



Loss



Loss



Loss





Loss



Loss



Loss



Loss



As can be observed from the simulation results, the nonlinear system behaves better

than the theoretical analysis indicated; while the theorem applied to the linearized system

indicated the system will maintain stability for communication loss rates lower than 90%,

the nonlinear system with adaptive synchronization control is able to maintain stability

for communication loss rates lower than 95%. This shows the limitations to using the

ADS theorem for analyzing communication loss in MSFF systems in that the theorem

applies to linear systems, and the analysis on the linearized MSFF system may not

match the results of the nonlinear system. This suggests than until a nonlinear approach

to analyzing communication loss in MSFF systems is developed, nonlinear simulations

should accompany any theoretical analysis conducted on the linearized system to have

more confidence in the generated results.

Similar to the linearized systems, it can be observed that once synchronization is

integrated into the adaptive controller, the system’s stability becomes more sensitive to

communication rate. When the follower spacecraft only communicated with the leader,

the nonlinear simulations from section 3.2.3 show that the system remains stable for

communication loss rates lower than 99%, while when the follower spacecraft are required

to also communicate with other follower spacecraft, the stability of the system can be

maintained for communication loss rates lower than 95%. This again suggests that MSFF

systems should only incorporate synchronization if the communication loss rates expected

to be experienced by the system in space are less than the limits provided by theoretical

and simulation analysis.

This chapter presents the theoretical analysis on MSFF systems with synchronization

control that experience communication loss. Similar to the approach taken in Chapter


3, the MSFF system is linearized, and the controller is simplified as a constant feedback

control law. Simulation results are presented for the linearized and the nonlinear system,

and contrasted with the theoretical analysis. Chapter 5 will present the framework for

performing experimental work using multiple robots in formation using the proposed

communication loss management technique.

Chapter 5

Robots in Formation Analysis and

Experiments

In addition to the theoretical and simulation work conducted for this research project,

experimental work was also conducted to examine the effects of the proposed communi-

cation loss management technique in a setup involving mobile hardware. Due to equip-

ment and budget limitations at the (UTIAS), experimental work with multiple satellites

was not feasible, and so other alternatives needed to be considered. It was decided

that experiments involving multiple robots in formation would be a feasible setup for

demonstrating the effectiveness of the proposed management technique when the robots

experience communication loss. In this chapter, a breakdown of the experimental setup

and equipment will provided, along with a system model and a controller formulation.

Theoretical analysis on the multiple robots in formation (MRF) will be conducted using

the ADS stability analysis theorem, and the results will be compared with the outcome

from the experimental work.

5.1 Experimental Setup

5.1.1 UTIAS Resources

Using the facilities available at UTIAS, an experimental setup was drafted to illustrate the

practical implementation of the proposed management technique when communication

between the leader and the follower robots is lost. A 10-camera Vicon motion capture

system (Fig. 5.1) is available and installed in one of the laboratories with an area of about

42

Chapter 5. Robots in Formation Analysis and Experiments 43

Figure 5.1: A Vicon Camera

15m × 8m (Fig. 5.2). The Vicon system is capable of providing position data of therobots in the x, y, and z direction, as well as orientation data in terms of roll, pitch, and

yaw angle measurements. Since the robots will be moving in a 2-D plane, only position

data in the x and y direction will be used, in addition to the yaw angle measurements.

The data from the Vicon system is provided at 100Hz with millimeter accuracy. Five

identical robots built from the iRobot Create (two-wheel differential drive) platform are

also available and can be used to create a one-leader four-follower formation scenario.

The robots are equipped with markers placed in different configurations (allowing the

Vicon system to uniquely identify them), and are also equipped with a mounting platform

containing a laptop computer running Player (Fig. 5.3); Player is a software that provides

a network between the robots and the Vicon system, allowing for data acquisition and

logging, and capable of performing motion control. While the Vicon system is able to

log the position and orientation data of the robots, Player is able to access this data and

perform any required calculations to determine the linear and angular velocities to pass

to the robots. The follower robots will require to obtain the data of the leader robot to

determine their control inputs, and thus loss of the leader’s information from the Vicon

system constitutes communication loss between the leader and follower robots.


Figure 5.2: Lab Area

Figure 5.3: An iRobot Create Roomba with Markers and a Laptop


5.1.2 Test Scenario

The goal of creating this scenario is to run a simple MRF experiment using the pro-

posed communication loss management technique where ADS theoretical stability anal-

ysis could be performed, and show how well the theoretical analysis compares with the

experimental results. To simplify the theoretical analysis, the test scenario will involve

the leader robot only traveling in a straight line, without rotating or changing directions;

this will eliminate the need to use angular dynamics in the system model and will facili-

tate the application of the ADS stability analysis theorem. While it is acknowledged that

this simplification does not result in a comprehensive test scenario, it is satisfactory since

the objective is to have a proof of concept for the effectiveness of the proposed packet loss

compensation technique, and illustrate the application of the ADS theoretical analysis

on a real test case. Details of the experimental configuration are presented in the final

section of this chapter. The objective of the control law will be to maintain a desired

relative displacement between the leader robot and the follower robots as the leader trav-

els forward with a constant linear velocity. Communication loss can be induced between

the leader and follower robot (as will be demonstrated in section 5.15) and the proposed

communication loss management technique will be used to compensate for the lost data.

The results can then be contrasted to the theoretical analysis of the system to determine

how effective the ADS stability analysis theorem is in determining the maximum rate of

communication loss the system can withstand before becoming unstable.

5.2 System Model

Many papers in the literature have studied the modeling of mobile robots, which include

first order [14] [13] [12] and second order [31] [32] [11] kinematics modeling. As stated

earlier, only a special case will be considered, where the leader is traveling in a straight

line at a constant velocity. Thus, the system model can be simplified as represented as

Newton’s second law [23]:

For a point mass traveling in a straight line, the system can be modelled as follows:

Fx = mẍ

Fy = mÿ(5.1)


Figure 5.4: Simplified Robot Model

ẍ = Fxm

ÿ = Fym

(5.2)

which can be expressed in state-space form:

ẋ

ẏ

ẍ

ÿ

=

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

x

y

ẋ

ẏ

+

0 0

0 0

1 0

0 1

uxuy

(5.3)

where

ux =Fxm

uy =Fym

(5.4)

This state-space model will be used for the theoretical analysis on MRF during periods

of communication loss.

5.3 Controller Formulation

5.3.1 Control Law

Due to the limitations of the equipment and sensors available at UTIAS (no rate sensors,

no accelerometers), a simple and proved control law is desired to keep the MRF system

in formation. As such, different papers discussing previous experiments involving MRF


were studied, and the control law used in [42] was adopted the purpose of conducting the

experiments. The control law is detailed as follows:

For an MRF system with i follower robots, let (rxi, ryi), θi, and (vi, ωi) represent the

Cartesian position, orientation, and linear and angular speed of the ith robot, then the

kinematic equations for the robots can be written as:

ṙxi = vi cos(θi) (5.5)

ṙyi = vi sin(θi) (5.6)

θ̇i = ωi (5.7)

The kinematic equations are feedback linearized for a fixed point off the center of the

wheel axis denoted (xi, yi) where:

xi = rxi + d cos(θi) (5.8)

yi = ryi + d cos(θi) (5.9)

Let

viωi

= cos(θi) sin(θi)−1d

sin(θi)1d

cos(θi)

uxiuyi

(5.10)then:

ẋiẏi

= uxiuyi

(5.11)Letting (xdi ,y

di ) denote the desired xi and yi for the follower robots, a simple control

law in the following form:

uxi = ẋdi − kxi(xi − xdi ) (5.12)

uyi = ẏdi − kyi(yi − ydi ) (5.13)


where kxi > 0 and kyi > 0 is used to guarantee that xi(t) → xdi (t) and yi(t) → ydi (t) ast→∞.

The main advantage of the control law above and the simplified experimental case is

that only the velocity of the leader and the follower robots are required for the calcula-

tions, and so the derivative of the data acquired from the Vicon system need to be taken

once, rather than twice had the control law required acceleration measurements. This,

however, is still a challenging task; due to the high frequency at which the data are pro-

vided by the Vicon system (100Hz) and the noise associated with those measurements,

calculating the slope between two consecutive data points in the following form

x2 − x1t2 − t1

(5.14)

will not be a practical numerical differentiation method and will yield unsatisfactory

calculations for the appropriate control forces (This has been tested and confirmed).

Thus, a proper technique for determining the velocity of the robots is needed before the

control law can be implemented.

5.3.2 Smooth Differentiation

Pavel Holoborodko is a mathematician who derived his own noise-robust differentiation

techniques for different cases that are particularly beneficial for carrying out experiments

with noisy data where differentiation is required [19]. His differentiation scheme possesses

the following characteristics:

- Precise on low frequencies

- Smooth and guaranteed suppression of high frequencies

- Computationally favorable structure

The full derivation of the algorithm is available in [19]. For a filter of length N , let

the coefficients be {ck} and the time step be h, then

f ′(x) ≈ 1h

M∑k=1

ck.(fk − f−k) (5.15)

where

ck =1

22m+1

2mm− k + 1

− 2mm− k − 1

(5.16)


m =N − 3

2(5.17)

M =N − 1

2(5.18)

The number sequence formed by the expression in the square brackets belongs to a

self-dual family of strip graphs, and can be found in the Encyclopedia of Integer Sequences

under the identity A050166 [46]. Table 5.1 displays some entries from A050166 formatted

as a square array, which can be read diagonally.

Table 5.1: A050166 Formatted as a Square Array

1 1 1 1 1 1 1 1 1 1

2 4 6 8 10 12 14 16 18

5 14 27 44 65 90 119 152

14 48 110 208 350 544 798

42 165 429 910 1700 2907

132 572 1638 3808 7752

429 2002 6188 15504

1430 7072 23256

4862 15194

16796

Thus, for n = 2, Table 5.2 shows the differentiation filter equations for different values

of N .

The general equation presented in Eq. 5.5 assumes that the data is received at an

equally spaced interval h, however it was observed that the data output from the Vicon

system is not always steady, and while the average output is set for 100Hz, sometimes the

interval between a set of data and another set of data can take up to 1/5th of a second.

Thus, the following form of the algorithm (which is designed for irregularly spaced data)

is more applicable:

f ′(x) ≈M∑k=1

ck.fk − f−kxk − x−k

.2k (5.19)


Table 5.2: Differentiation Filter Example

N Smooth noise-robust differentiators (n=2, exact on 1, x, x2)

52(f1 − f−1) + f2 − f−2

8h

75(f1 − f−1) + 4(f2 − f−2) + f3 − f−3

32h

914(f1 − f−1) + 14(f2 − f−2) + 6(f3 − f−3) + f4 − f−4

128h

1142(f1) + 48(f2 − f−2) + 27(f3 − f−3) + 8(f4 − f−4) + f5 − f−5

512h

Another advantage of Eq. 5.19 is that in addition to being a centered filter (which

is beneficial for offline calculations of derivatives) it can also be used as a forward (one-

sided) filter, which is needed for real-time online calculations of the derivative. One of

the disadvantages of using one-sided filters however is that it takes longer to achieve the

same noise suppression as a centered filter [19], and thus more data points are needed at

every calculation of the derivative to achieve valid results. The algorithm described in

this subsection shall be used for online derivative calculations during the experimental

validation of the proposed communication loss management technique.

5.4 Theoretical Analysis

The MRF system can be analyzed using the ADS theorem to determine the behaviour

of the system under periods of stochastic communication loss. The space-state model of

the system is shown in Eq. 5.3, where the A m

communication loss management and analysis for multiple ......lqr: linear quadratic regulator lqg:...

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