GLOCAL CONTROL FOR MECANUM-WHEELED

VEHICLE WITH SLIP COMPENSATION

BY

JIRAYU UDOMSAKSENEE

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF MASTER OF

ENGINEERING (INFORMATION AND COMMUNICATION

TECHNOLOGY FOR EMBEDDED SYSTEMS)

SIRINDHORN INTERNATIONAL INSTITUTE OF TECHNOLOGY

THAMMASAT UNIVERSITY

ACADEMIC YEAR 2018

Ref. code: 25615922040554LTX

GLOCAL CONTROL FOR MECANUM-WHEELED

VEHICLE WITH SLIP COMPENSATION

BY

JIRAYU UDOMSAKSENEE

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF MASTER OF

ENGINEERING (INFORMATION AND COMMUNICATION

TECHNOLOGY FOR EMBEDDED SYSTEMS)

SIRINDHORN INTERNATIONAL INSTITUTE OF TECHNOLOGY

THAMMASAT UNIVERSITY

ACADEMIC YEAR 2018

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Acknowledgements

This research is financially supported by Thailand Advanced Institute of

Science and Technology (TAIST), National Science and Technology Development

Agency (NSTDA), Tokyo Institute of Technology and Sirindhorn International

Institute of Technology (SIIT) under the Excellent Thai Students (ETS) program, and

Thammasat University (TU).

I am gratefully indebted to my thesis advisor, Asst. Prof. Dr. Itthisek

Nilkhamhang of SIIT at TU, for his helpful advice and consistency support along this

thesis. We worked hard together during day and night on my thesis. He is very kind to

let Mr. Hendi Wicaksono brief me on the fundamental concept of the glocal control

system at the beginning of my thesis. In addition to that, he also introduced me to Prof.

Shinji Hara of Tokyo University, the expert of glocal control concept that I have an

opportunity to learn more and obtain his advice on my thesis.

I also would like to thank Assoc. Prof. Masaki Yamakita and Yamakita lab

members, especially Mr. Rin Takano, for their lectures, helpful advice, and kind

hospitality during my ten-week stay at Tokyo Institute of Technology.

I would like to gratefully thank Assoc. Prof. Dr. Waree Kongprawechnon.

With her kind advice and support, I have a chance to be a part of TAIST program. In

addition, I have also got her continuing support and valuable advice on this thesis. I

also would like to acknowledge valuable comments from Dr. Pished Bunnun, the

Chairperson of Examination Committee.

Last but not least, I would also like to thank my friends namely Mr. Apisit

Pinitnanthakorn, Mr. Peammawat Chantevee, and Ms. Panatda Nalinnopphakhun for

providing additional information.

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Abstract

GLOCAL CONTROL FOR MECANUM-WHEELED VEHICLE WITH SLIP COMPENSATION

by

JIRAYU UDOMSAKSENEE

Bachelor of Engineering, Sirindhorn International Institute of Tecnology, Thailand, 2015

Master of Engineering, Sirindhorn International Institute of Tecnology, Thailand, 2018

This thesis proposes a hierarchical decentralized controller for a mecanum-

wheeled vehicle represented as a homogenous multi-agent system. The equations of

motion for each individual mecanum wheel and the entire vehicle with slip are analyzed

to construct a linear time-varying interconnected model. The global objective is the

position trajectory tracking of the vehicle as a result of the effective forces produced by

each wheel. The local objective is the driving force and slip controls of each wheel,

with consideration of the interconnection between all agents. The proposed hierarchical

linear quadratic regulator (LQR) control ensures satisfaction of both global and local

objectives, according to the concepts of glocal control. Simulation results of a

mecanum-wheel vehicle are shown that verify the performance and validity of the

method.

Keywords: Mecanum wheel, glocal control, decentralized hierarchical control, slip

control

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Table of Contents

Chapter Title Page

Signature Page i

Acknowledgements ii

Abstract iii

Table of Contents iv

List of Figures vi

List of Tables viii

1 Introduction 1

1.1 Motivation 1

1.1.1 Practical Applications 2

1.1.2 Problems of Mecanum-Wheeled Vehicle 5

1.1.3 Anti-Slip Control System for Conventional-Wheeled Vehicles 5

1.1.4 Anti-Slip Control System for Mecanum-Wheeled Vehicles 6

1.2 Objective 7

1.3 Thesis Scope 7

1.4 Thesis Structure 8

2 Literature Review 9

2.1 Mecanum-wheel Vehicle 9

2.1.1 Fundamental Vehicle Dynamics. 12

2.2 Anti-Slip Control Techniques 14

2.2.1 Slip Ratio 11

2.2.2 Anti-Slip Control via Kinematic Control 15

2.2.3 Anti-SlipControl via Driving Force Control 15

2.3 Hierarchically Decentralized Optimal Control 14

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2.3.1 Homogeneous Multi-Agent Dynamical Systems 17

2.3.2 Multi-Agent Dynamical Systems with physical interconnection 21

3 Dynamical Modelling of Mecanum-Wheel Vehicle 24

3.1 Fundamental Vehicle Dynamics 24

3.2 Slip Ratio 26

3.3 Driving Force Dynamics 26

3.4 Group and Layering 28

3.4.1 Upper Layer: Position Control 29

3.4.2 Lower Layer: Driving Force Control 29

3.5 Hierarchical Decentralized Structure 29

4 Hierarchically Decentralized Optimal Control 31

4.1 Trajectory Tracking Controller 32

4.2 Hierarchically Decentralized Optimal Control 32

4.3 Performance Index 34

4.4 Hierarchical State Feedback LQR design 36

4.5 Numerical Simulation 37

4.5.1 Linear Translation along x-y Axis 40

4.5.2 Trajectory with Diagonal Cornering 53

5 Conclusions and Recommendations 58

References 60

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List of Figures

Figures Page

1.1 Omnidirectional robots: (a) mecanum drive (b) omni drive (c) swerve drive. 2

1.2 Mecanum wheel. 3

1.3 NASA OmniBot 2. 4

1.4 Industrial robots: (a) Airtrax ATX-3000 industrial forklifts (b) Mecanum-

wheeled vehicle with container and trolley 2. 4

1.5 Medical mecanum-wheeled vehicle: OMNI 6, CIIPS wheelchair 7, iRW 8

(from left to right). 4

1.6 The interactive shopping trolley 9. 5

2.1 Mecanum wheel. 10

2.2 Mecanum-wheeled vehicle. 10

2.3 Single wheel model. 11

2.4 Mecanum-wheeled vehicle model. 12

2.5 Driving and slip forces developed on a roller of the i wheel. 12

2.6 Friction coefficient versus slip ratio 12. 14

2.7 Relaxation length versus slip ratio 22. 16

2.8 Block diagram of hierarchical networked control system. 19

2.9 Block diagram of state feedback controller. 19

3.1 Single wheel model. 25

3.2 Mecanum-wheeled vehicle model. 25

3.3 Driving and slip forces developed on a roller of the i wheel 25

3.4 Friction coefficient versus slip ratio 12. 27

3.5 Relaxation length versus slip ratio 22. 27

3.6 Hierarchical network system of the mecanum-wheeled vehicle. 29

4.1 Overall control system. 31

4.2 P-Controller. 32

4.3 Block diagram of hierarchical networked control system with physical

interconnection. 33

4.4 Block diagram of state feedback controller with physical interconnection. 33

4.5 Friction coefficient versus slip ratio of rubber rollers on dry tarmac surface. 38

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4.6 Relaxation length versus slip ratio 22. 38

4.7 Overall control system with state feedback controller. 39

4.8 Manually-tuned state feedback controller. 39

4.9 X-Y trajectory of manually-tuned state feedback controller. 40

4.10 Velocity and position responses of manually-tuned state feedback

controller. 41

4.11 Driving force responses of manually-tuned state feedback controller. 41

4.12 Slip responses of manually-tuned state feedback controller. 42

4.13 Torques generated by manually-tuned state feedback controller. 42

4.14 Position error of manually-tuned state feedback controller. 43

4.15 Driving Force error of manually-tuned state feedback controller. 43

4.16 X-Y trajectory of HLQR controller with global gains. 44

4.17 Velocity and position responses of HLQR controller with global gains. 45

4.18 Driving force responses of HLQR controller with global gains. 45

4.19 Slip responses of HLQR controller with global gains. 46

4.20 Torques generated by HLQR controller with global gains. 46

4.21 Position error of HLQR controller with global gains. 47

4.22 Driving Force error of HLQR controller with global gains. 47

4.23 X-Y trajectory of HLQR controller without global gains. 48

4.24 Velocity and position responses of HLQR controller without global gains. 49

4.25 Driving force responses of HLQR controller without global gains. 49

4.26 Slip responses of HLQR controller without global gains. 50

4.27 Torques generated by HLQR controller without global gains. 50

4.28 Position error of HLQR controller without global gains. 51

4.29 Driving Force error of HLQR controller without global gains. 51

4.30 X-Y trajectory of diagonal cornering. 54

4.31 Velocity and position responses of diagonal cornering. 55

4.32 Driving force responses of diagonal cornering. 55

4.33 Slip responses of diagonal cornering. 56

4.34 Torques generated during diagonal cornering. 56

4.35 Position error during diagonal cornering. 57

4.36 Driving Force error during diagonal cornering. 57

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List of Tables

Tables Page

2.1 Combination of wheel motion and resulting vehicle direction 19. 11

4.1 System parameters and description. 31

4.2 Vehicle ppecification. 37

4.3 Controller gains. 39

4.4 Weighting matrices. 44

4.5 Integral square error of the first path. 53

4.6 Control effort comparison. 53

4.7 Integral square error of diagonal cornering. 53

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Chapter 1

Introduction

1.1 Motivation

Nowadays, autonomous mobile robots are widely used in many industrial and

household applications. These robots are typically non-holonomic systems that have

limited maneuverability in tight, confined workspaces due to the minimum steering

angle. This may necessitate multiple readjustments of the orientation to navigate

through narrow pathways and around corners. In these situations, holonomic or omni-

directional wheeled robots would provide better performance and maneuverability 1.

This includes mobile robots that employ mecanum wheels shown in Fig. 1.1(a) with

the ability to move in any direction without changing the orientation of the vehicle.

Other mechanisms that allow for omni-directional movements include omni wheels and

swerve drives shown in Fig. 1.1(b) and Fig. 1.1(c), respectively. A comparison of these

omnidirectional wheel types is as follows 2. Both omni drives and mecanum wheels

have compact designs that simplify control system development when vehicle dynamic

and kinematic are available. However, mecanum drives provide more traction force and

higher load capacity compared to omni drives. In comparison, the swerve drive has a

much more complex design. Moreover, mecanum-wheeled vehicles do not suffer from

high friction and scrubbing caused by wheel steering that is common in swerve drives.

As our aim is to develop a control system of a robot designed primarily for industrial

applications with high load capacity, this research focuses on mecanum-wheeled

vehicles. However, mecanum drives suffer from slippage and require an adequate anti-

slip control system due to high sensitivity to floor conditions.

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(a) (b) (c)

Figure 1.1 Omnidirectional robots: (a) mecanum drive (b) omni drive (c) swerve drive.

The design of mecanum wheels looks like a conventional wheel with free-

moving rollers attached to the circumferences at angle of 𝛼 to the main rotational axis,

as shown in Fig. 1.2. The number of rollers can vary depending on the design and size

of the wheel. There are different types of rollers made from materials such as rubber

and polyurethane which are found in heavy duty applications 3. The angle between the

free-moving rollers and the rotational axis of the wheel is typically 45∘ and allows for

omnidirectional mobility of the vehicle.

1.1.1 Practical Applications

Omni-directional mobility allows mecanum-wheeled vehicles to operate well in

congested environments. Moreover, they are very suitable for applications that require

a high degree of maneuverability. For these reasons, mecanum-wheeled vehicles and

robots find practical applications in various fields, such as exploration, industrial, and

service 2.

Mecanum-wheeled vehicles are often used to support search-and-rescue

missions and planetary explorations. When navigating through unknown or rough

terrains, the omni-directional capabilities of mecanum wheels allow it to travel

efficiently pass obstacles and narrow spaces with increased maneuverability. An

example where these robots are used in place of human beings to explore hazardous

environments is the NASA OmniBot, shown in Fig. 1.3.

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In industrial applications, mecanum robots have been developed for

transporting and handling materials, as well as for automated inspection. Material

handling robots are used to transport cargo or workpieces inside of factories and

warehouses. These busy environments often have narrow or congested pathways that

require omni-directional movement. One example is the Airtrax ATX-3000 industrial

forklift shown in Fig. 1.4 (a) that is capable of carrying heavy loads in environment

with limited space. Alternatively, the mecanum wheels can be installed on a container

or trolley, as shown in shown in Fig. 1.4 (b), to move small goods 4. In 5, a mecanum-

wheeled vehicle installed with an ultrasonic scanner is developed for pipe inspection

on uneven surfaces while avoiding gravitational slip and also maintaining its velocity

and alignment.

Lastly, mecanum-wheeled vehicles are also used in robotics for health-care

applications and customer service. Powered wheelchairs utilizing mecanum wheels

have been developed to help the elderly, handicapped people or those who have

difficulties in walking. These wheelchairs allow the user to move around and assist their

daily-living activities. Three examples are shown in Fig. 1.5, and includes the Office

Wheelchair for High Maneuverability and Navigational Intelligence for People with

Severe Handicap (OMNI) 6, Center for Intelligent Information Processing Systems

(CIIPS) omni-directional wheelchair developed at the University of Western Australia

7, and intelligent Robotic Wheelchair (iRW) 8. Another example of service applications

is the Interactive Behavior Operated Trolley (InBOT) shopping cart shown in Fig. 1.6

can help customers shop and find desired products 9. Those customers can even control

the movement of the shopping cart without pushing it.

Figure 1.2 Mecanum wheel.

𝛼

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Figure 1.3 NASA OmniBot 2.

(a) (b)

Figure 1.4 Industrial robots: (a) Airtrax ATX-3000 industrial forklifts (b) Mecanum-wheeled vehicle with container and trolley 2.

Figure 1.5 Medical mecanum-wheeled vehicle: OMNI 6, CIIPS wheelchair 7, iRW 8 (from left to right).

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Figure 1.6 The interactive shopping trolley 9.

1.1.2 Problems of Mecanum-Wheeled Vehicle

However, mecanum-wheeled robots become difficult to control effectively

when moving over surfaces with very low or very high coefficient of friction, such as

an oily floor or rough concrete. In these conditions, undesired phenomenon such as

slipping or skidding can be observed due to the reduced traction of mecanum wheels

when compared with conventional wheels 10. This is caused by a reduction in the

effective driving forces due to the orientation of the rollers. To alleviate this problem,

several hardware solutions have been proposed. An alternative mecanum wheel was

developed by Bengt Ilon to solve the problem when the wheel is operating on an uneven

surface, where each roller is split into two parts and centrally mounted to ensure that

they will always touch the floor. Likewise, an improved design was proposed in 11 that

add extra twist mechanism capable of transforming itself by adjusting the orientation

of rollers so the robot can travel in a specific direction with higher driving force using

the same torque when compared to the traditional design.

1.1.3 Anti-Slip Control System for Conventional-Wheeled Vehicles

The problem of slip also occurs in vehicles with conventional wheel drives. To

compensate for this, improved control systems have been developed. In 12, a driving

force distribution controller for in-wheel motor electric vehicles (IWM-EVs) was

proposed to ensure straight motion when subjected to a strong wind disturbance. The

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main concept involved the implementation of driving force control in order to suppress

the slip ratio of electric vehicles. A similar work examines the effect of acceleration on

split-friction surfaces and utilizes glocal control with a decentralized hierarchical

structure 13. Lastly, 14 introduced a limiter with driving stiffness estimation in order to

maintain the driving force within controllable conditions.

1.1.4 Anti-Slip Control System for Mecanum-Wheeled Vehicles

Several dynamic and kinematic models have been developed for mecanum-

wheeled vehicles. A popular method for velocity control of omni-directional robots is

based on kinematic equations 15. Kinematic controllers use pure geometrical motion

and neglect forces and torques, thus simplifying analysis and design. However, omitting

these dynamic effects also decreases the performance of the system. Alternatively,

dynamic models can be obtained using Lagrangian equations 16 17 or combining

Newtonian mechanics with Lagrange method 18 to give a more accurate representation

of the vehicle. These previous researches mention the effect of wheel traction but do

not incorporate slip into either the kinematic or the dynamic models. An example of a

model of a mecanum-wheeled vehicle with slip is derived based on Newton’s law in

19. As mentioned previously, the problem of slip greatly effects conventional-wheeled

vehicles and is especially important for mecanum wheels that have reduced traction.

Currently, the number of research related to slip and slip control of omni-

directional vehicle is limited, especially for mecanum drives. A summary of anti-slip

techniques is provided in 20, where a velocity adjustment method is implemented in

order to control the wheel slip but resulted in instability of the closed loop control

system. Due to the limited numbers of researches related to mecanum-wheeled vehicles,

some implementations related to conventional non-holonomic robots can be adapted

for mecanum-wheeled robot. In 21, a well-known conventional wheel tire model, such

as George Rill’s tire model 22 and Pacejka’s tire model 23, is adapted to describe a

mecanum wheel with slip ratios. Velocity feedback control, which is independent from

platform kinematics, is developed to make braking of the mecanum-wheeled vehicle

safer. Another research explored slip control for mecanum-wheeled vehicles by using

position rectification control with kinematic-based symptomatic and preventive

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rectifications of position and orientation 24. Other methods include a fuzzy inference

control system based on kinematic modelling 25 and position corrective control 26, but

experimental results demonstrate considerable difficulty in achieving trajectory

tracking. These methods treat each mecanum wheel as separate agents and typically

neglect the physical interconnection between them. Therefore, this research proposes a

decentralized hierarchical controller that considers the physical interconnection of the

wheels through the vehicle chassis to achieve higher accuracy and performance by

encouraging collaborative control 12272829.

1.2 Objective

This paper investigates the dynamic model of each individual mecanum wheel

and equations of motion of the vehicle with slip in order to establish a decentralized

hierarchical structure. The upper layer consists of the global objective for position

control of the vehicle. The lower layer is used for driving force control of each wheel.

The interconnection between layers is determined and a glocal control strategy 30 based

on hierarchical LQR (HLQR) is proposed. Since the direction of motion of the vehicle

depends on the coordination between all mecanum wheels, each driven by independent

motors and subjected to different slip ratios, the concept of glocal control will be

applied to achieve consensus between all agents and improve performance. The validity

of the proposed controller is shown by simulation of a four-wheel mecanum robot.

1.3 Thesis Scope

1. Assume that the model is made for the linear translation only.

2. Assume that the viscous friction coefficient between shaft and bearing is

omitted.

3. Assume that all the states of the agents can be measured.

4. Only force adjustment controllers will be used in order to suppress slip.

5. Assume that the platform is travelling on a horizontal plane.

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1.4 Thesis Structure

The organization of the thesis is as follows. Chapter 2 presents the literature

review of this research. Chapter 3 the dynamic modelling of the mecanum-wheeled

vehicle. Chapter 4 presents the hierarchically decentralized optimal control. Chapter 5

presents the conclusion of this research.

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Chapter 2

Literature Review

This chapter provides an overview of dynamical modelling for mecanum-

wheeled vehicles, anti-slip control techniques, and hierarchically decentralized optimal

control for time-varying systems. A general description of the mechanics for a

mecanum-wheeled robot is given, followed by derivation of dynamical equations using

Newton’s Second Law of Motion. A review of anti-slip control techniques is conducted,

including a detailed analysis of slip and how to suppress it. The last section summarizes

existing control systems based on hierarchically decentralized optimal control that are

relevant to this thesis.

2.1 Mecanum-Wheeled Vehicle

Omni-directional mobility is required to provide better performance and

maneuverability for autonomous mobile vehicles (AMVs) in certain applications such

as material handling and transport. AMVs equipped with omni-directional drives, such

as mecanum wheels, can perform tasks that are not suitable for other non-holonomic

mobile vehicles, especially in narrow environments that require free translational and

rotational movements. Depending on the objective, high load capacity is also necessary.

Mecanum-wheels are invented by Bengt Ilon in 1973. The design of mecanum

wheels looks like a conventional wheel with free-moving rollers attached to the

circumferences at angle of 𝛼, which is typically 45∘ in practice, to the main rotational

axis, as shown in Fig. 2.1. The number of rollers can vary depending on the design and

size of the wheel. There are different types of rollers made from materials such as rubber

and polyurethane which are found in heavy duty applications 3. By installing such

wheels to a platform, omnidirectional mobility of the vehicle can be realized.

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Mecanum wheel.

Mecanum-wheeled vehicle.

Omni-directional mobility can be achieved when these wheels are mounted to

each corner of the platform as shown in Fig. 2.2. The number of the wheels are

identified differently in different researches. Fig. 2.4 shows the model of the vehicle

with a specific set of wheel numbers. The wheels are numbered starting from the front

right wheel to the back right wheel in a counter clockwise direction. However, 18

identified the number of the wheel in another way. Nevertheless, most of the mecanum-

wheeld vehicles studied are written in the form similar to Fig. 2.4 as the characteristic

of the wheels can be distinguished using even and odd numbers. Because of the

orientation that the rollers, which are appeared on the mecanum wheels, make with the

motor shafts, the direction of driving forces of each wheel making with the surface are

shifted as shown in Fig. 2.5. Hence, different combinations of motor torques applied to

the wheels result in different movement of the vehicle as each wheel has its own motor

and could be driven independently. The relationship of the vehicle movement and the

combination of the motor torque with the same wheel number identification as in Fig.

2.4 can be found in Table 2.1. Note that the table shows the direction of angular velocity

of each wheel where the arrows in the figure are in positive direction. + refers to positive

direction of motion. - refers to negative direction of motion. 0 means that the wheel is

not rotate at that moment.

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Table 2.1: Combination of wheel motion and resulting vehicle direction 19.

Wheel No. 1 2 3 4

Direction of the vehicle

North + + + +

+ + 0 0

0 0 + +

South - - - -

- - 0 0

0 0 - -

East - + - +

- 0 0 +

0 + - 0

West + - + -

+ 0 0 -

0 - + 0

North-east 0 + 0 +

South-west 0 - 0 -

North-west + 0 + 0

South-east - 0 - 0

Single wheel model.

𝑟

𝑇 𝜔

𝐹

𝐹

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 Mecanum-wheeled vehicle model.

Driving and slip forces developed on a roller of the 𝑖 wheel.

2.1.1 Fundamental Vehicle Dynamics

Several dynamic and kinematic models have been developed for mecanum-

wheeled vehicles. A popular method for velocity control of omni-directional robots is

based on kinematic equations 15. Kinematic controllers use pure geometrical motion

and neglect forces and torques, thus simplifying analysis and design. However, omitting

these dynamic effects also decreases the performance of the system. Alternatively,

dynamic models can be obtained using Lagrangian equations 16 17 or combining

Newtonian mechanics with Lagrange method 18 to give a more accurate representation

of the vehicle. These previous researches mention the effect of wheel traction but do

not incorporate slip into either the kinematic or the dynamic models. An example of a

model of a mecanum-wheeled vehicle with slip is derived based on Newton’s law in

19. As mentioned previously, the problem of slip greatly effects conventional-wheeled

vehicles and is especially important for mecanum wheels that have reduced traction.

In 19, Newton’s law has been applied in order to find the dynamic equation.

Fig. 2.4 shows a mecanum-wheeled vehicle of total mass 𝑚 with 𝑁 in-wheel motors

(IWMs), where 𝑁 is assumed to be 4. The 𝑖th wheel is depicted in Fig. 2.3 and has a

radius of 𝑟 , which is assumed to be the same for all wheels and will henceforth be

referred to simply as 𝑟. Around the circumference of the wheel are rollers, positioned

such that the rotational axis of each roller makes an angle of 𝛼 45° with the rotational

axis of the wheel. Fig. 2.4 shows the body of the vehicle with reference to the world

frame, where 𝑥, 𝑦, 𝑧 refers to the stationary coordinate axis and 𝑥′, 𝑦′, 𝑧′ refers to

the body-attached coordinate axis at the geometrical center of the vehicle. Here, 𝑖 is the

wheel number, 𝜑 is the orientation of the vehicle and 𝛽 is the direction of the vehicle

𝑦

𝑥

1 2

3

4 𝜑

𝛽 𝑦′

𝑥′

𝑣 𝑦

𝑥

𝑦’

𝑥’

𝛼 𝑆

𝐹 𝐹 𝐹

𝐹

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velocity with respect to the vehicle orientation. 𝐹𝑇𝑖 is the force developed on the roller

due to the motor torque 𝑇𝑖 at the circumference of the 𝑖 th mecanum wheel.. The

developed torque can be written as:

where 𝜔 and 𝜔 is rotational velocity and acceleration of the 𝑖 th mecanum wheel

respectively, 𝜇 is the coefficient of dynamic friction between wheel and ground that

depends on the material. 𝐹 is the frictional force proportional to the weight of the

vehicle and 𝐼 is the inertia constant of the wheel about its center of mass. Viscous

friction between motor shaft and bearings is neglected for simplicity.

The developed force 𝐹𝑇𝑖 is divided into effective driving force 𝐹𝑖 and slip force

𝑆𝑖, which is an ineffective force. The sum of all effective forces produced by each wheel

determine the total motion of the vehicle, whereas the ineffective force causes free-

rolling motion of the rollers and does not contribute significantly to vehicular dynamics

21. From Fig. 2.5, 𝐹 and 𝑆 can be expressed as follows:

𝐹 𝐹 𝑠𝑖𝑛𝛼 (2.2)

𝑆 𝐹 𝑐𝑜𝑠𝛼 (2.3)

By using Newton’s second law of motion, the sum of effective force

components along the 𝑥- and 𝑦-axis, 𝐹 and 𝐹 , in the world frame causes motion of

the mobile platform as follows:

𝑚𝑥 𝐹 (2.4)

𝑚𝑦 𝐹 (2.5)

where 𝑥 and 𝑦 are acceleration components of the mobile platform in the 𝑥- and 𝑦-axis

respectively.

𝑇 𝑟 𝐹 𝑟 𝜇 𝐹 𝐼 𝜔

(2.1)

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Friction coefficient versus slip ratio 12.

2.2 Anti-Slip Control Techniques

2.2.1 Slip Ratio

Slip can occur when there is insufficient friction between wheels and the

ground. The slip ratio of a wheel can be represented as follows:

𝜆𝑣 𝑣

max |𝑣 |, |𝑣| (2.6)

where 𝑣 is the velocity of 𝑖th wheel and 𝑣 is the linear velocity of the vehicle.

The above equation shows the relationship of slip ratio and velocity.

Alternatively, there is also a relationship between slip ratio and frictional force. A graph

of slip ratio vs friction coefficient in Fig. 2.6 shows that frictional force 𝐹 can be

defined as a function of slip ratio where 𝐹 𝜆 𝜇 𝜆 𝐹 . It can be seen that the

slip ratio depends on both the material of the surface and the wheels. In order to

suppress the slip the following relationship should be satisfied:

𝐹 𝐹 (2.7)

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From Fig. 2.6, the curve in the beginning of the slip ratio vs friction coefficient

graph is almost linear. For simplicity we can consider the prescribed part as a linear

graph start from the origin (0,0) to the point around the peak of friction coefficient

which the slippage could still be controlled. Our objective is to keep the slip ratio in

this linear region. Note that the product of the slope of this graph and the normal force

acting on each wheel is called the driving stiffness 𝑆 .

In order to avoid slip, many techniques could be implemented. The following

control techniques are typical ones those can be implemented for positioning control

20.

2.2.2 Anti-Slip Control via Kinematic Control

In kinematic control or velocity control, the motion of robot is controlled based

on its pure geometry without concerning about forces and torques which are causes of

motion. The relationship among wheels and vehicle velocities are modelled. The

velocities are then converted into either acceleration or position of both wheels and

vehicle. With (2.6) the slip ratio of the vehicle can be controlled by kinematic control.

By keeping slip ratio low, the slippage can be avoided.

2.2.3 Anti-SlipControl via Driving Force Control

The techniques of anti-slip control can be chosen based on the equipment

specification. As we assume that all the states are measurable, driving force control will

be implemented in this thesis. In driving force adjustment for anti-slip control, torques

are controlled instead of velocity. The idea is to lower torques in order to satisfy

inequality (2.7). By using Newton’s second law, dynamics model for driving force

control can be found.

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Relaxationl length versus slip ratio 22.

For the purpose of control, the first-order dynamic model of wheel force is

defined 13 using the tire dynamics model from 22:

𝜏 𝐹 𝐹 𝐹 (2.9)

where 𝜏𝑖 is the relaxation time constant that can be identified from the relaxation length

𝑙𝑟𝑖 as follows:

𝜏

𝑙

𝑟𝜔

(2.10)

The relationship between 𝑙𝑟𝑖 and 𝜆𝑖 is shown in Fig. 2.7. 𝐹 is the dynamic

driving force and 𝐹 is the steady-state driving force that can be represented by

𝐹 𝑆 𝜆 in the linear region of Fig. 2.6 where 𝑆 is the driving stiffness 12.

By taking the derivative of (2.6), the dynamics of slip ratio are obtained:

where

Here, 𝜔 is the nominal vehicle angular velocity and 𝑣 𝜔𝑟 .

𝜆𝜔

|𝓌||𝓌||𝓌|

𝜆𝑣

|𝓌|𝑟 (2.11)

𝓌𝜔 , max |𝑣 |, |𝑣| |𝑣 | acceleration𝜔, max |𝑣 |, |𝑣| |𝑣| deceleration

(2.12)

𝑙

𝐹 2 𝑘𝑁 𝐹 4 𝑘𝑁 𝐹 6 𝑘𝑁 𝐹 8 𝑘𝑁

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From (2.9), (2.11) and (2.12), the following linear-time varying interconnected

system is established for the driving force and slip dynamics:

Finally, the linear-time varying interconnected dynamical model of a vehicle

(2.13) is established. For the purpose of readability, the time notation 𝑡 will be omitted

from time-varying parameters from here onwards. By applying an anti-slip control

technique, the slip ratio in each wheel of the vehicle can be suppressed.

2.3 Hierarchically Decentralized Optimal Control

In order to apply the hierarchically decentralized control, the driving force

dynamics model of each agent is developed in the last subsection separately. As the

trend of networked dynamical system is increasing, the decentralized controller seem

to be more important. One of these decentralized controller is the hierarchically

decentralized controller. The hierarchically decentralized controller is a controller that

can customize the collaboration among agents. By varying gains and interconnection

structure, various results can be achieved. With the use of optimal control, optimal

performance can be ensured.

2.3.1 Homogeneous Multi-Agent Dynamical Systems

The controller used in the lower layer of this research is designed based on

hierarchically decentralized optimal control by the modeling following 27, 28, 29, and

12. From 27, hierarchical decentralized controller synthesis for homogeneous multi-

agent systems is discussed.

𝑥 𝑡 𝐴 𝑡 𝑥 𝑡 𝐵 𝑢 𝑡 𝐴 𝑡 𝑥 𝑡

𝑦 𝑡 𝐶 𝑥 𝑡 (2.13)

𝑥 𝑡 𝐹 𝑡 𝜆 𝑡 𝑢 𝑡 𝑇 𝑡

𝐴 𝑡

|𝓌|

|𝓌|

|𝓌|

, 𝐵 𝑡0

|𝓌|, 𝐶 𝑡 1 0

0 1, 𝐴 𝑡

0 0

|𝓌|0

where 𝑐 , 𝑐 .

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Hierarchical Decentralized Structure

Consider a homogeneous multi-agent system having 𝑁 agents, where the 𝑖 th

agent has following model:

where 𝐴 ∈ ℝ , 𝐵 ∈ ℝ , 𝐶 ∈ ℝ , which 0 𝑚 𝑛 and 𝑖 1, ⋯ , 𝑁 .

𝑥𝑖, 𝑥𝑗 ∈ ℝ𝑛𝑖 , 𝑢𝑖 ∈ ℝ𝑚𝑖 , and 𝑦𝑖 ∈ ℝ𝑝𝑖 are the state vector of the 𝑖 th and 𝑗th agent, the

input vector of the 𝑖th agent, and the output of the 𝑖th agent respectively. The subsystem

of the 𝑖th wheel is denoted as 𝐻 𝑠 . Note that the number of state of each subsystem 𝑛

are all the same and the number of input of each subsystem 𝑚 𝜇 for all 𝑖. (2.14) can

be either time varying or time invariant model.

This design is based on the homogeneous hierarchical system where the

interconnected systems are denoted as:

where 𝑥 𝑥 , ⋯ , 𝑥 ∈ ℝ , 𝑢 𝑢 , ⋯ , 𝑢 ∈ ℝ , 𝑦 𝑦 , ⋯ , 𝑦 ∈ ℝ ,

𝒜 𝐼 ⊗ 𝐴 ∈ ℝ , ℬ 𝐼 ⊗ 𝐵 ∈ ℝ , 𝒞 𝐼 ⊗ 𝐶 ∈ ℝ , 𝐼 is an

N-by-N identity matrix and ⊗ refers to the Kronecker product.

Consider the graph 𝒢 that represents the information structure among 𝑁 agents.

Nodes and edges ε represent each agent and the interconnection between two agents

respectively. The information structure is denoted by matrix 𝐾 consisting of elements

𝐾 as the weights for the information exchanges between agents and can be set

following to class 𝕂 :

𝑥 𝐴𝑥 𝐵𝑢 𝑦 𝐶𝑥

(2.14)

𝑥 𝒜𝑥 ℬ𝑢 𝑦 𝒞𝑥

(2.15)

𝕂 ≔ 𝐾 𝐾 ∈ ℝ |𝐾 0 when 𝑖 𝑗 and 𝑖, 𝑗 ∈ 𝜀 (2.16)

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Block diagram of hierarchical networked control system.

Block diagram of state feedback controller.

The information exchange is made by each agent sending out a state feedback

aggregated signal for information structure 𝑧 to cooperate with other connected agents

to realize the global objectives. Simultaneously, each agent receives an input aggregate

signal for information structure 𝑤 sent by other connected agents individually.

Each subsystem 𝐺 𝑠 is implemented with its individual local controller giving

𝑢ℓ, as an output as shown in Fig. 2.9, which is the state feedback design, having the

control input:

The control input for the entire hierarchical network with both layers is represented by:

where 𝑤 𝑤 , ⋯ , 𝑤 , 𝑧 𝑧 , ⋯ , 𝑧 , 𝑢ℓ 𝑢ℓ, , ⋯ , 𝑢ℓ, , 𝐾 ⊗ 𝐼 is the

interconnection among subsystems in Fig. 2.8, and ⊗ is the Kronecker product. Note

that even if the design is output feedback, (2.17) and (2.18) are still valid.

𝑢 𝑤 𝑢ℓ, (2.17)

𝑢 𝑤 𝑢ℓ 𝐾 ⊗ 𝐼 𝑧 𝑢ℓ (2.18)

𝑥

𝑦 𝑧 𝑤

𝑢

𝑢ℓ,

𝐺 𝑠

𝐹ℓ 𝐹𝑢

𝐻 𝑠

Upper layer

Lower layer

𝑤 𝑧

𝐺 𝑠 𝐺 𝑠

𝐺 𝑠

𝐾 ⊗ 𝐼

𝐺 𝑠

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State Feedback Design

In order to find local state feedback gain 𝐹ℓ and information structural state

feedback gain 𝐹 , we start with the optimal state feedback control law of the system

which can be represented as:

where state feedback gain ℱ is represented as follows:

where ℱ ∈ ℝ and local state feedback gain 𝐹ℓ ∈ ℝ𝜇 𝛿and information structural

state feedback gain 𝐹𝑢 ∈ ℝ𝜇 𝛿 . Hence, the state feedback design problem is to

determine 𝐹ℓ , 𝐹𝑢 , and 𝐾 such that the following performance index in the next

subsection is minimized. Note that the global feedback gain here is the information

structural state feedback gain 𝐹 that improves the cooperation among agents.

Performance Index

In order to ensure optimal performance, the following performance index have

to be minimized.

where 𝐽 relates to the local and global objectives and 𝐽 is a penalty for the control

input, 𝐽 ,ℒ is a local performance index composing of the individual penalties for the

states of subsystems, and 𝐽 ,𝒢 is a global performance index.

𝑢 ℱ𝑥 (2.19)

ℱ 𝐼 ⊗ 𝐹ℓ K ⊗ 𝐹 (2.20)

𝐽 𝐽 𝐽 (2.21) 𝐽 𝐽 ,ℒ 𝐽 ,𝒢 (2.22)

𝐽 𝑢 𝑡 ℛ𝑢 𝑡 𝑑𝑡

(2.23)

𝐽 ,ℒ 𝑥 𝑡 𝐼 ⊗ 𝑄 𝑥 𝑡 𝑑𝑡

(2.24)

𝐽 ,𝒢 𝑥 𝑡 𝐾 ⊗ 𝑄 𝑥 𝑡 𝑑𝑡

(2.25)

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where ℛ ≻ 0 ∈ ℝ is the weighting matrix corresponding to input,

𝑄 ≽ 0 ∈ ℝ is the local state weighting matrix and 𝑄 ≽ 0 ∈ ℝ is the

global state weighting matrix corresponding to the information structure.

Note that 𝐽 ,𝒢 is the extra term that does not exist in typical performance index

is added to improve the control performance as 𝐾 and 𝑄 are related to the

interconnection of the networked dynamical system.

By rewriting (2.22) as follows:

where 𝑡 is the initial time, 𝑡 is the finished time and the weighting matrices

ℛ ≻ 0 ∈ ℝ and 𝑄 ≽ 0 ∈ ℝ , ℛ and 𝑄 can be represented as follows:

where 𝑄 ≽ 0 ∈ ℝ , 𝑄 ≽ 0 ∈ ℝ , 𝑅 ≻ 0 ∈ ℝ , and 𝑅 ≻ 0 ∈ ℝ .

After ℛ and 𝑄 are found, ℱ can be computed by solving the Riccati equation

27.

The idea of the hierarchical decentralized d optimal control of homogeneous

multi-agent dynamical systems can be expanded into a more general case. In

heterogeneous case, a similar problem formulation with the use of Khatri Rao product

has been studied in 29.

2.3.2 Multi-Agent Dynamical Systems with physical interconnection

From the previous sections, the hierarchical decentralized controllers for multi-

agent dynamical system with information structure interconnection are studied. In this

section, the system with physical interconnection are introduced. An example of the

system with physical interconnection is an EV. While the wheels which are considered

as agents of the EV are installed to each side of the car body, the car body itself act as

𝐽 𝑥 𝑡 𝑄𝑥 𝑡 𝑑𝑡 𝑢 𝑡 ℛ𝑢 𝑡 𝑑𝑡 (2.26)

𝑄 𝐼 ⊗ 𝑄 K ⊗ 𝑄 (2.27)

ℛ 𝐼 ⊗ 𝑅 K ⊗ 𝑅 (2.28)

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a physical interconnection among the wheels. The hierarchical decentralized controller

for homogeneous multi-agent dynamical system with physical interconnection 12 is

studied below.

Hierarchical Decentralized Structure

From the homogeneous multi-agent dynamical systems section, the physical

interconnection term is added so the dynamical system becomes:

where 𝐴 ∈ ℝ , 𝐵 ∈ ℝ , 𝐶 ∈ ℝ , which 0 𝑚 𝑛 and 𝑖 1, ⋯ , 𝑁 .

𝑥𝑖, 𝑥𝑗 ∈ ℝ𝑛𝑖 , 𝑢𝑖 ∈ ℝ𝑚𝑖 , and 𝑦𝑖 ∈ ℝ𝑝𝑖 are the state vector of the 𝑖 th and 𝑗th agent, the

input vector of the 𝑖th agent, and the output of the 𝑖th agent respectively. 𝐴 ∈ ℝ is

the physical interconnection matrix between the 𝑖 th and the 𝑗 th agent. Note that the

number of state of each subsystem 𝑛 are all the same and the number of input of each

subsystem 𝑚 𝜇 for all 𝑖. (2.29) can be either time varying or time invariant model.

This design is based on the homogeneous hierarchical system where the time-

varying interconnected systems are denoted as:

where 𝑥 𝑥 , ⋯ , 𝑥 ∈ ℝ , 𝑢 𝑢 , ⋯ , 𝑢 ∈ ℝ , 𝑦 𝑦 , ⋯ , 𝑦 ∈ ℝ ,

𝒜 𝐼 ⊗ 𝐴 Γ ⊗ 𝐴 ∈ ℝ , ℬ 𝐼 ⊗ 𝐵 ∈ ℝ , 𝒞 𝐼 ⊗ 𝐶 ∈

ℝ and Γ ∈ ℝ is the inter-layer interaction matrix represented by 1 1 .

In this case there are also local state feedback gain 𝐹ℓ and information structural

state feedback gain 𝐹 as well as he previous two cases and the extra physical

interconnection state feedback gain 𝐹𝑝𝑢. In order to find these gains, we start with the

optimal state feedback control law of the system which can be represented as:

𝑥 𝐴 𝑥 𝐵 𝑢 𝐴 𝑥

𝑦 𝐶 𝑥

(2.29)

𝑥 𝒜𝑥 ℬ𝑢 𝑦 𝒞𝑥

(2.30)

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where state feedback gain ℱ is represented as follows:

where ℱ ∈ ℝ and local state feedback gain 𝐹ℓ ∈ ℝ𝜇 𝑛𝑖and information structural

state feedback gain 𝐹𝑢 ∈ ℝ𝜇 𝑛𝑖 and physical interconnection state feedback gain

𝐹𝑝𝑢∈ ℝ𝜇 𝑛𝑖 . Hence, the state feedback design problem is to determine 𝐹ℓ, 𝐹𝑢, 𝐹𝑝𝑢

and

𝐾 such that the following performance index in the next subsection is minimized. Note

that the global feedback gain here are the information structural state feedback gain 𝐹

and physical interconnection state feedback gain 𝐹𝑝𝑢 that improve the cooperation

among agents.

Performance Index

In order to ensure optimal performance, the following performance index have

to be minimized.

where 𝑡 is the initial time, 𝑡 is the finished time and the weighting matrices

ℛ ≻ 0 ∈ ℝ , 𝑄 ≽ 0 ∈ ℝ , and 𝑆 is 𝑃 𝑡 which is the unique positive

definite solution of the Riccati equation at the finished time.

where 𝑄 ≽ 0 ∈ ℝ , 𝑄 ≽ 0 ∈ ℝ , 𝑄 ≽ 0 ∈ ℝ ,

𝑅 ≻ 0 ∈ ℝ , 𝑅 ≻ 0 ∈ ℝ , and 𝑅 ≻ 0 ∈ ℝ .

After ℛ, 𝑄, and 𝑆 are found, ℱ can be computed by solving the Riccati equation.

𝑢 ℱ𝑥 (2.31)

ℱ 𝐼 ⊗ 𝐹ℓ K ⊗ 𝐹 Γ ⊗ 𝐹 (2.32)

𝐽 𝑥 𝑡 𝑄𝑥 𝑡 𝑑𝑡 𝑢 𝑡 ℛ𝑢 𝑡 𝑑𝑡 𝑥 𝑆𝑥 (2.33)

𝑄 𝐼 ⊗ 𝑄 K ⊗ 𝑄 Γ ⊗ 𝑄 (2.34)

ℛ 𝐼 ⊗ 𝑅 K ⊗ 𝑅 Γ ⊗ 𝑅 (2.35)

𝑆 𝐼 ⊗ 𝑆 (2.36)

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Chapter 3

Dynamical Modelling of Mecanum-Wheeled Vehicle

This chapter presents the mathematical model of a mecanum-wheeled vehicle.

The dynamical equations are developed based on the implementation of driving force

control for a conventional-wheeled vehicle with slip, as described in 12, and derivation

of the dynamical equations of a mecanum-wheeled vehicle as described in 19.

3.1 Fundamental Vehicle Dynamics

This paper studies a mecanum-wheeled vehicle of total mass 𝑚 with 𝑁 in-

wheel motors (IWMs), where 𝑁 is assumed to be 4. The 𝑖th wheel is depicted in Fig.

3.1 and has a radius of 𝑟 , which is assumed to be the same for all wheels and will

henceforth be referred to simply as 𝑟 . Around the circumference of the wheel are

rollers, positioned such that the rotational axis of each roller makes an angle of

𝛼 45° with the rotational axis of the wheel. Fig. 3.2 shows the body of the vehicle

with reference to the world frame, where 𝑥, 𝑦, 𝑧 refers to the stationary coordinate

axis and 𝑥′, 𝑦′, 𝑧′ refers to the body-attached coordinate axis at the geometrical center

of the vehicle. Here, 𝑖 is the wheel number, 𝜑 is the orientation of the vehicle and 𝛽 is

the direction of the vehicle velocity with respect to the vehicle orientation. 𝐹 is the

force developed on the roller due to the motor torque 𝑇 at the circumference of the 𝑖th

mecanum wheel. The developed torque can be written as:

where 𝜔 and 𝜔 are the rotational velocity and acceleration of the 𝑖th mecanum wheel

respectively, 𝜇 is the coefficient of dynamic friction between the wheel and ground.

𝐹 is the frictional force proportional to the weight of the vehicle and 𝐼 is the inertia

constant of the wheel about its center of mass. Viscous friction between motor shaft

and bearings is neglected for simplicity.

𝑇 𝑟 𝐹 𝑟 𝜇 𝐹 𝐼 𝜔

(3.1)

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Figure 3.2  Mecanum-wheeled vehicle.

Figure 3.3 Driving and slip forces developed on a roller of the 𝑖 wheel.

The developed force 𝐹 is divided into effective driving force 𝐹 and slip force

𝑆 , which is an ineffective force. The sum of all effective forces produced by each wheel

determine the total motion of the vehicle, whereas the ineffective force causes free-

rolling motion of the rollers and does not contribute significantly to vehicular dynamics

21. From Fig. 3.3, 𝐹 and 𝑆 can be expressed as follows:

𝐹 𝐹 𝑠𝑖𝑛𝛼 (3.2)

𝑆 𝐹 𝑐𝑜𝑠𝛼 (3.3)

By using Newton’s second law of motion, the sum of effective force

components along the 𝑥- and 𝑦-axis, 𝐹 and 𝐹 , in the world frame causes motion of

the mobile platform as follows:

where 𝑥 and 𝑦 are acceleration components of the mobile platform in the 𝑥- and 𝑦-axis

respectively.

Figure 3.1 Single wheel model.

𝑚𝑥 𝐹 𝐹 𝜔 sin 𝛼 1 𝜑 (3.4)

𝑚𝑦 𝐹 1 𝐹 𝜔 cos 𝛼 1 𝜑 (3.5)

𝑟

𝑇𝜔

𝐹

𝐹

𝑦

𝑥

1 2

3

4 𝜑

𝛽 𝑦′

𝑥′

𝑣 𝑦

𝑥

𝑦’

𝑥’

𝛼 𝑆

𝐹 𝐹 𝐹

𝐹

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3.2 Slip Ratio

In many wheel models, forces are considered as function of slip. As the ideal

roller generates force along its rotational axis, slip is considered only for the

corresponding direction 21. The slip ratio of a wheel is defined as:

𝜆𝑣 𝑣

max 𝑣 , |𝑣| (3.6)

where 𝑣 is the velocity of 𝑖th wheel in the direction of 𝐹 , i.e. 𝑣 𝑣 𝑠𝑖𝑛𝛼, 𝑣 is

the velocity of the wheel corresponding to the direction of the force exerted on the roller

that touches the ground at that moment, i.e. 𝑣 𝜔 𝑟 , and 𝑣 is the linear velocity of

the vehicle. Moreover, 𝑣 can be represented in the world frame as 𝑣 𝑣 𝑐𝑜𝑠𝛼

and 𝑣 𝑣 𝑠𝑖𝑛𝛼.

3.3 Driving Force Dynamics

For the purpose of control, the first-order dynamic model of the wheel force is

defined 13 using the tire dynamics model from 22:

𝜏 𝐹 𝐹 𝐹 (3.7)

where 𝜏 is the relaxation time constant that can be identified from the relaxation length

𝑙 as follows:

𝜏

𝑙

𝑟𝜔

(3.8)

The relationship between 𝑙 and 𝜆 is shown in Fig. 3.4. 𝐹 is the dynamic tire

force and 𝐹 is the steady-state tire force that can be represented by 𝐹 𝑆 𝜆 𝑏 where 𝑆 is the driving stiffness and 𝑏 is the 𝑦-intercept of the slip ratio vs friction

coefficient graph, shown in Fig. 3.5 12.

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Figure 3.4 Relaxation length versus slip ratio 22.

Figure 3.5 Friction coefficient versus slip ratio 12.

By taking the derivative of (3.6), the slip ratio dynamics are obtained:

where

Here, 𝜔 is the nominal vehicle angular velocity and 𝑣 𝜔𝑟 .

Applying (3.1) and along Newton’s Second Law of Motion, (3.9) becomes:

𝜆𝜔

|𝓌||𝓌||𝓌|

𝜆𝑣

|𝓌|𝑟

(3.9)

𝓌𝜔 , max 𝑣 , |𝑣| 𝑣 acceleration

𝜔, max 𝑣 , |𝑣| |𝑣| deceleration

(3.10)

𝑙

𝐹 2 𝑘𝑁 𝐹 4 𝑘𝑁 𝐹 6 𝑘𝑁 𝐹 8 𝑘𝑁

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Similarly, (3.7) is also be rearranged in the following form:

𝐹1𝜏

𝐹𝑆𝜏

𝜆𝑏𝜏

(3.12)

From (3.11) and (3.12), the following linear time-varying interconnected system

is established for the driving force and slip dynamics:

where 𝑐 , 𝑐 and 𝑏 𝑡 0 in the linear region of 𝜇 vs 𝜆 graph,

making 𝑓 𝑡 0 0 when 𝜏 0. For the purpose of readability, the time notation

𝑡 will be omitted from time-varying parameters from here onwards.

3.4 Group and Layering

From the model of a mecanum-wheel vehicle in Fig. 3.2, a hierarchical control

configuration consisting of two layers is established, as shown in Fig. 3.6 where 𝑊 𝑖

refers to the 𝑖 th wheel. The lower layer (LL) is a set that includes all wheels:

𝑊 𝑊 𝑖 , 𝑖 ∈ 1, 𝑁 . The upper layer (UL) combines information from all to drive

the vehicle. Utilizing this grouping and layering, the motion control objectives can be

obtained hierarchically as follows.

𝜆𝑟 𝑠𝑖𝑛𝛼𝐼 |𝓌|

𝐹|𝓌||𝓌|

𝜆1

𝐼 |𝓌|𝑇

∑ 𝐹

𝑚|𝓌|𝑟

(3.11)

𝑥 𝑡 𝐴 𝑡 𝑥 𝑡 𝐵 𝑢 𝑡 𝐴 𝑡 𝑥 𝑡 𝑓 𝑡

𝑦 𝑡 𝐶 𝑥 𝑡 (3.13)

𝑥 𝑡 𝐹 𝑡 𝜆 𝑡 𝑢 𝑡 𝑇 𝑡

𝐴 𝑡

|𝓌|

|𝓌|

|𝓌|

, 𝐵 𝑡0

|𝓌|, 𝐶 𝑡 1 0

0 1, 𝐴 𝑡

0 0

|𝓌|0

𝑓 𝑡 𝑏𝜏

0

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3.4.1 Upper Layer: Position Control

Position control of the vehicle with 𝑁 motors. This layer determines the

reference motion of the vehicle and sends the command signal to each wheel. It receives

information from all wheels and compute the position of the vehicle according to (3.1),

(3.4), and (3.5).

3.4.2Lower Layer: Driving Force Control

Motor torque command 𝑇 is generated for tracking the actual driving force with

the reference value 𝐹 ∗ by controlling the linear time-varying interconnected system in

(3.13). The driving force of the 𝑖th wheel can be computed by (3.1).

3.5 Hierarchical Decentralized Structure

Consider a mecanum-wheeled vehicle having 𝑁 4 wheels, where the 𝑖 th

wheel is the same agent in (3.13) as:

Figure 3.6 Hierarchical network system of the mecanum-wheeled vehicle.

𝐹∗ 𝐹∗

𝐹∗

𝐹∗

𝑭𝟏

𝑭𝟐

𝑭𝟑

𝑭𝟒

Upper Layer

Lower Layer

Vehicle

𝑊 1

𝑊 2 𝑊 3

𝑊 4

Graph 𝒢

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𝑥 𝐴 𝑥 𝐵 𝑢 𝐴 𝑥

𝑦 𝐶 𝑥

(3.14)

where 𝐴 ∈ ℝ , 𝐵 ∈ ℝ , 𝐶 ∈ ℝ , 𝐴 ∈ ℝ which 0 𝑚 𝑛 and

𝑖 1, ⋯ , 𝑁. 𝑥 , 𝑥 ∈ ℝ , 𝑢 ∈ ℝ , and 𝑦 ∈ ℝ are the state vector of the 𝑖th and 𝑗th

agent, the input vector of the 𝑖th agent, and the output of the 𝑖th agent respectively. The

subsystem of the 𝑖th wheel is denoted as 𝐻 𝑠 . Note that the number of input of each

subsystem 𝑚 𝜇 ∀𝑖 and the number of state of each subsystem 𝑛 𝛿 ∀𝑖 for

homogeneous system and (3.14) is a simplified version of (3.13) under the assumption

that the slip ratio is in the range of [-0.2, 0.2].

The homogeneous hierarchical network model of the mecanum-wheeled vehicle

can be established following to 31 and 32:

𝑥 𝒜𝑥 ℬ𝑢 𝑦 𝒞𝑥 (3.15)

where 𝑥 𝑥 , ⋯ , 𝑥 ∈ ℝ , 𝑢 𝑢 , ⋯ , 𝑢 ∈ ℝ , 𝑦 𝑦 , ⋯ , 𝑦 ∈ ℝ ,

𝒜 𝐼 ⊗ 𝐴 Γ ⊗ 𝐴 ∈ ℝ , ℬ 𝐼 ⊗ 𝐵 ∈ ℝ , 𝒞 𝐼 ⊗ 𝐶 ∈

ℝ , and Γ ∈ ℝ , where Γ is the inter-layer interaction matrix represented by

1 1 . Note that 1 1 ⋯ 1 ∈ ℝ .

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Chapter 4

Hierarchically Decentralized Optimal Control

This chapter proposes a control system for the mecanum-wheeled vehicle with

a hierarchical structure as established in Chapter 3. The multi-level control objective is

to achieve trajectory tracking of the vehicle and driving forces, while maintaining the

slip ratios of all wheels. To satisfy both global and local requirements, the system

integrates a proportional (P) controller for vehicle trajectory tracking with a hierarchical

LQR (HLQR) controller for driving force and slip control of each wheel. The overall

control system is shown in Fig. 4.1, where all parameters are described in Table 4.1.

Table 4.1: System parameters and description.

Parameters Description

𝑥∗, 𝑦∗ Reference position of the vehicle in 𝑥- and 𝑦-axis

𝑥, 𝑦 Actual position of the vehicle in 𝑥- and 𝑦-axis

𝐹∗ Reference driving force

𝐹 Actual driving force

𝐹 ∗ Feedback reference driving force from P-controller

𝐹 ∗ Feedforward reference driving force from position reference

𝜆 Actual slip ratio

𝑇 Generated torque

𝑒 , 𝑒 Position error of the vehicle in 𝑥- and 𝑦-axis

𝑒 Driving force error calculated from 𝐹∗ 𝐹

𝑒 Slip ratio error calculated from 𝜆∗ 𝜆 where 𝜆∗ 0

Figure 4.1 Overall control system.

𝐹 ∗ 𝑥, 𝑦 𝐹∗ HLQR-

Controller P-Controller

Position Reference

𝑒 , 𝑒

𝐹 ∗

𝑇 Plant

𝑥∗, 𝑦∗ 𝐹, 𝜆

𝑒 , 𝑒

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4.1 Trajectory Tracking Controller

The upper layer consists of the P-controller for position control, with a

proportional error gain 𝐾 , that tracks the reference position 𝑥∗, 𝑦∗ by minimizing the

position error 𝑒 , 𝑒 . The P-controller generates a feedback reference driving force

for the vehicle and is shown in Fig. 4.2.

4.2 Hierarchically Decentralized Optimal Control

By combining the controllers described in Chapter 2, 27 and 12, a hierarchically

decentralized optimal controller for the mecanum-wheeled vehicle is developed. As the

vehicle can be represented as a homogenous system with physical interconnection,

treating it as a networked dynamical system has the potential to improve control

performance.

Consider the hierarchical model described in Chapter 3 and the graph 𝒢 in Fig.

3.6 that represents the information structure, which can be represented by a positive

semidefinite matrix 𝐾 consisting of weights 𝐾 among 𝑁 wheels. Nodes and edges 𝜀

represent each wheel and the interconnection between two wheels respectively. In order

to ensure that every wheel exchanges information with all the others wheel, 𝐾 is set

as follows:

𝐾1, when 𝑖 𝑗 and 𝑖, 𝑗 ∈ 𝜀

𝐾 , when 𝑖 𝑗 (4.1)

P-Controller

Figure 4.2 P-Controller.

𝑥∗, 𝑦∗ 𝐾 𝑒 , 𝑒 𝐹 ∗

𝑥, 𝑦

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Figure 4.3 Block diagram of hierarchical networked control system with physical interconnection.

Figure 4.4 Block diagram of state feedback controller with physical interconnection.

The information exchange shown in Fig. 4.3 and Fig. 4.4 occurs when each

wheel sends out 𝑧 and 𝑧 , which are the state feedback aggregated signals for

information structure and physical interconnection, respectively, to cooperate with

other connected wheels in realizing the global objectives. Simultaneously, each wheel

receives the signals 𝑤 and 𝑤 , which are input aggregate signals for information

structure and physical interconnection, respectively, from other connected wheels in

the system. In this scenario, the global objective of the lower level is driving force

control, where the responses from one wheel works toward ensuring that all wheels

satisfy their objectives.

Each subsystem 𝐺 𝑠 is implemented with a local controller giving 𝑢ℓ, as

shown in Fig. 4.4. The control input for each wheel is:

The control input for the entire hierarchical network with both layers is represented by:

𝑢 𝑤 𝑤 𝑢ℓ, (4.2)

𝑢 𝑤 𝑤 𝑢ℓ 𝐾 ⊗ 𝐼 𝑧 Γ ⊗ 𝐼 𝑧 𝑢ℓ (4.3)

𝑧 𝑤

Γ ⊗ 𝐼𝜇 Upper layer

Lower layer 𝑤 𝑧 𝐺 𝑠

𝐺 𝑠

𝐺 𝑠

𝐾 ⊗ 𝐼𝜇

𝐺 𝑠

𝑥 𝑦

𝑧 𝑤

𝑢 𝑢ℓ,

𝐺 𝑠

𝐹ℓ 𝐹𝑢

𝐻𝑖 𝑠

𝐹𝑢𝑝

𝑤

𝑧

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where 𝑤 𝑤 , ⋯ , 𝑤 , 𝑧 𝑧 , ⋯ , 𝑧 , 𝑤 𝑤 , ⋯ , 𝑤 ,

𝑧 𝑧 , ⋯ , 𝑧 , 𝑢ℓ 𝑢ℓ, , ⋯ , 𝑢ℓ, , 𝐾 ⊗ 𝐼 and Γ ⊗ 𝐼 represent the

information structure and physical interconnection respectively, 𝐾 is the information

structure matrix described in (4.1), Γ is inter-layer interaction matrix represented by

1 1 , 𝐼 is an identity matrix of size 𝜇 𝜇, where 𝜇 is the number of input of each

subsystem in (3.14) and ⊗ is the Kronecker product.

Assuming that all system states can be measured, a hierarchical state feedback

controller shown in Fig. 4.4 is used. The objective is to determine the lower layer state

feedback gains that includes the local state feedback gains 𝐹ℓ and the global feedback

gains. The global feedback gains consist of the information structure state feedback

gains 𝐹 and the physical interconnection state feedback gains 𝐹 that improve the

cooperation among wheels. In order to determine these values, the optimal state

feedback control law of the system is represented as:

where the state feedback gain ℱ is obtained as follows:

where ℱ ∈ ℝ and 𝐹ℓ ∈ ℝ , 𝐹 ∈ ℝ and 𝐹 ∈ ℝ are the local state

feedback gains, information structure state feedback gains, and physical

interconnection state feedback gains respectively. Hence, the state feedback design

problem is to determine 𝐹ℓ , 𝐹 , and 𝐹 . Note that 𝐼 is an identity matrix of size

𝑁 𝑁, where 𝑁 is the number of agents and 𝛿 is the number of state of each subsystem

in (3.14).

4.3 Performance Index

In order to ensure optimal performance, the following performance index 𝐽 is

𝑢 ℱ𝑥 (4.4)

ℱ 𝐼 ⊗ 𝐹ℓ K ⊗ 𝐹 Γ ⊗ 𝐹 (4.5)

𝐽 𝐽 𝐽 (4.6) 𝐽 𝐽 ,ℒ 𝐽 ,𝒢 (4.7)

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where 𝐽 relates to the local and global objectives and 𝐽 is a penalty for the control

input, 𝐽 ,ℒ is a local performance index composing of the individual penalties for the

states of the subsystems, and 𝐽 ,𝒢 is a global performance index. This can be represented

as a typical quadratic performance index as:

where 𝑡 is the initial time and 𝑡 is the final time. The weighting matrices

ℛ ≻ 0 ∈ ℝ and 𝑄 ≽ 0 ∈ ℝ are described as follows:

where 𝑄ℓ ≽ 0 ∈ ℝ is the local state weighting matrix , 𝑄 ≽ 0 ∈ ℝ is the

global state weighting matrix corresponding to the information structure,

𝑄 ≽ 0 ∈ ℝ is the global state weighting matrix corresponding to the physical

interconnection, 𝑅ℓ or 𝑅 ≻ 0 ∈ ℝ is the local input weighting matrix,

𝑅 ≻ 0 ∈ ℝ is the global input weighting matrix corresponding to the

information structure, and 𝑅 ≻ 0 ∈ ℝ is the global input weighting matrix

corresponding to the information structure. This allows (4.8) to be rewritten as:

Note that 𝐽 ,𝒢 is an additional term that does not exist in typical performance

index functions, introduced to improve the control performance, as 𝐾, Γ, 𝑄 and 𝑄

are related to the interconnection of the networked dynamical system of the meacanum

wheels.

𝐽 𝑥 𝑡 𝑄𝑥 𝑡 𝑑𝑡 𝑢 𝑡 ℛ𝑢 𝑡 𝑑𝑡 (4.8)

𝑄 𝐼 ⊗ 𝑄ℓ 𝐾 ⊗ 𝑄 Γ ⊗ 𝑄 (4.9)

ℛ 𝐼 ⊗ 𝑅ℓ 𝐾 ⊗ 𝑅 Γ ⊗ 𝑅 (4.10)

𝐽 𝑢 𝑡 ℛ𝑢 𝑡 𝑑𝑡 (4.11)

𝐽 ,ℒ 𝑥 𝑡 𝐼 ⊗ 𝑄ℓ 𝑥 𝑡 𝑑𝑡 (4.12)

𝐽 ,𝒢 𝑥 𝑡 𝐾 ⊗ 𝑄 Γ ⊗ 𝑄 𝑥 𝑡 𝑑𝑡 (4.13)

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4.4 Hierarchical State Feedback LQR design

The optimal state feedback gain ℱ can be calculated as:

where 𝒫 ∈ ℝ is the unique positive definite solution of the Riccati equation:

If 𝒜, ℬ, 𝒞, ℛ, 𝑄 belong to some operator algebra 33 or semigroup then the

solution 𝒫 also belong to that algebra or semigroup and it can be verified that ℱ has the

same property. However, for the homogeneous hierarchical network model of the

mecanum-wheeled vehicle given in (3.15), the dimensions of ℬ are not compatible with

the other matrices and therefore does not belong to the same operator algebra or

semigroup. In this case, an alternative method can be used according to 27. The

following steps are implemented in order to design the hierarchical decentralized state-

feedback controller.

Step 1: Local LQR Design

Select the weighting matrices for the local objectives, 𝑄ℓ ∈ ℝ and

𝑅ℓ ∈ ℝ such that 𝑄ℓ/ , 𝐴 is observable and 𝑅ℓ ≻ 0 for 𝑖 1, ⋯ , 𝑁. Then solve

the corresponding local Riccati equation:

𝑃ℓ𝐴 𝐴 𝑃ℓ 𝑃ℓ𝐵 𝑅ℓ𝐵 𝑃ℓ 𝑄ℓ 0 (4.16)

for the unique positive definite solution 𝑃ℓ ∈ ℝ .

Step 2: Setting Upper Layer Interactions

Design a positive semidefinite matrix 𝐾 ∈ ℝ according to (4.1).

Step 3: Global LQR Design

Set the global weighting matrices 𝑅 ≻ 0 ∈ ℝ , 𝑄 ≽ 0 ∈ ℝ ,

𝑅 ≻ 0 ∈ ℝ and 𝑄 ≽ 0 ∈ ℝ for 𝐽 ,𝒢 as follows:

𝑄 𝑃ℓ𝐵 𝑅 𝐵 𝑃ℓ (4.17)

𝑄 𝑃ℓ𝐵 𝑅 𝐵 𝑃ℓ 𝑃ℓ𝐴 𝐴 𝑃ℓ (4.18)

where these weighting matrices are introduced to establish cooperation among agents.

ℱ ℛ ℬ 𝒫 (4.14)

𝒫𝒜 𝒜 𝒫 𝑄 𝒫ℬℛ ℬ 𝒫 0 (4.15)

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Step 4: State Feedback Gain Calculation

Set the state feedback gains 𝐹ℓ, 𝐹 and 𝐹 as follows:

𝐹ℓ 𝑅ℓ𝐵 𝑃ℓ (4.19)

𝐹 𝑅 𝐵 𝑃ℓ (4.20)

𝐹 𝑅 𝐵 𝑃ℓ (4.21)

4.5 Numerical Simulation

The validity of the proposed controller will be tested by simulation using vehicle

parameters as defined in Table 4.2 following to 14. It is assumed that the mecanum

wheel uses rubber rollers and is moving on a dry tarmac surface. The graph of slip ratio

vs friction coefficient is constructed by Pacejka’s Magic Formula 23. For simplicity,

the slip region |𝜆𝑖| 0, 0.2 is considered a linear graph, as shown in Fig. 4.5, with a

driving stiffness of 39600. It is desired to maintain the slippage within this region.

The relaxation time 𝜏 can be determined from the 𝑙𝑟𝑖 vs 𝜆𝑖 graph shown in Fig.

4.6 as:

𝜏

𝑙

𝑟𝜔𝒊 (4.22)

Since the vehicle mass is 850 kg, the normal forces acting on each wheel on a

horizontal plane is 𝐹 𝑚𝑔 4⁄ 2084.625 N, where 𝑔 9.81 m/s2.

Table 4.2: Vehicle specification.

Vehicle mass 𝑚 850 kg

Wheel radius 𝑟 0.302 m

Wheel inertia 𝐼 1.24 kgꞏm2

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Figure 4.5 Friction coefficient versus slip ratio of rubber rollers on dry tarmac surface.

Figure 4.6 Relaxation length versus slip ratio 22.

The simulation considers only linear translation on the 𝑥-𝑦 plane. A second-

order low pass filter with a time constant of 0.2 is used to ensure the second derivative

of the reference trajectory, suppress maximum overshoot of the response, and simplify

the tuning of gains.

The hierarchical decentralized controller for homogeneous multi-agent

dynamical system with physical interconnection is implemented for the trajectory

tracking problem. The model of the mecanum wheels are set according to (3.15) in

Chapter 3. As this is a linear time-varying interconnected system, the behaviour of the

𝑙

𝐹 2 𝑘𝑁 𝐹 4 𝑘𝑁 𝐹 6 𝑘𝑁 𝐹 8 𝑘𝑁

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time-varying variables in the model have to be known in order to implement LQR

controller. The block diagram of the control system and the state feedback controller

are shown in Fig. 4.7 and 4.8. The gains of this control system are determined through

manual tuning and are given in Table 4.3.

The simulations consider two reference trajectories that examine the

performance of the proposed controller under varying conditions. These trajectories

include linear translation along 𝑥-𝑦 axis and trajectory with diagonal cornering. More

detail of these trajectories will be discussed later in each subsection.

Table 4.3: Controller gains.

Proportional Gain for the P-controller 𝐾 100,000

Gain for the driving force error 𝐾 300,000

Gain for the driving force 𝐾 0.001

Gain for the slip ratio error 𝐾 100,000

Figure 4.7 Overall control system with state feedback controller.

Figure 4.8 Manually-tuned state feedback controller.

𝐹 ∗ 𝑥, 𝑦 𝐹∗ State FB-Controller P-Controller

Position Reference

𝑒 , 𝑒

𝐹 ∗

𝑇 Plant

𝑥∗, 𝑦∗ 𝐹, 𝜆

𝑒 , 𝑒

𝐹∗ 𝑒

𝑇

𝐹

𝜆∗

𝜆

𝐾 1𝑠

𝐾

𝐾𝑒

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4.5.1 Linear Translation along x-y Axis

The first trajectory contains only linear translation on the 𝑥 -𝑦 axis and no

changes in vehicle orientation. The reference position and velocity trajectory plots are

shown in Fig. 4.9 as x-y plot and Fig. 4.10 as time-domain plot. A second-order low

pass filter with a time constant of 0.2 is used to ensure the second derivative of the

reference trajectory remains bounded, suppress maximum overshoot of the response,

and simplify the tuning of controller gains. This trajectory is used to compare the

performances of the manually-tuned state feedback controller gains described in Fig.

4.7 and 4.8 with HLQR-optimized gains, with and without global terms shown in Fig.

4.1.

Manually-Tuned State Feedback Controller

The results of the manually-tuned state feedback controller are presented in Fig.

4.9 to 4.15. The driving force and position tracking errors are low and the slip ratios are

maintained within the specified range of 0.2, 0.2 . However, it is noted that high-

frequency oscillations are observed in the response that can lead to undesired

phenomena and increase the control effort.

Figure 4.9 X-Y trajectory of manually-tuned state feedback controller.

Start/Stop

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Figure 4.10 Velocity and position responses of manually-tuned state feedback controller.

Figure 4.11 Driving force responses of manually-tuned state feedback controller.

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Figure 4.12 Slip responses of manually-tuned state feedback controller.

Figure 4.13 Torques generated by manually-tuned state feedback controller.

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Figure 4.14 Position error of manually-tuned state feedback controller.

Figure 4.15 Driving force error of manually-tuned state feedback controller.

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Hierarchical LQR Controller with Global Gains

The hierarchical LQR controller with global gains is implemented using the

weighting matrices shown in Table 4.4 and a proportional gain of 𝐾 100,000. The

results are shown in Fig. 4.16 to 4.22. The position and driving force tracking

performance remains very good and the slip ratios are maintained within the specified

range of 0.2, 0.2 . It is noted that the choice of weighting matrices may have

significant on the performance of the system.

Table 4.4: Weighting matrices.

Local state weighting matrix 𝑄ℓ 300*𝐼

Local input weighting matrix 𝑅ℓ 0.1

Global input weighting matrix corresponding to information structure

𝑅 0.5

Global input weighting matrix corresponding to physical interconnection

𝑅 0.5

Figure 4.16 X-Y trajectory of HLQR controller with global gains.

Start/Stop

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Figure 4.17 Velocity and position responses of HLQR controller with global gains.

Figure 4.18 Driving force responses of HLQR controller with global gains.

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Figure 4.19 Slip responses of HLQR controller with global gains.

Figure 4.20 Torques generated by HLQR controller with global gains.

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Figure 4.21 Position error of HLQR controller with global gains.

Figure 4.22 Driving force error of HLQR controller with global gains.

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Hierarchical LQR Controller without Global Gains

The positive effects of the global gains are studied by removing them from the

LQR controller. This is accomplished by setting the weights 𝑅 and 𝑅 to zero, while

all other weighting matrices remain the same (𝑄ℓ and 𝑅ℓ). The results shown in Fig.

4.23 to 4.27 demonstrate the importance of information exchange offered by the global

gains. The position and driving force tracking performance is significantly decreased

and there is a higher amount of slip, thought the slip ratio does remain within the

specified range.

Figure 4.23 X-Y trajectory of HLQR controller without global gains.

Start/Stop

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Figure 4.24 Velocity and position responses of HLQR controller without global gains.

Figure 4.25 Driving force responses of HLQR controller without global gains.

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Figure 4.26 Slip responses of HLQR controller without global gains.

Figure 4.27 Torques generated by HLQR controller without global gains.

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Figure 4.28 Position error of HLQR controller without global gains.

Figure 4.29 Driving force error of HLQR controller without global gains.

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Result Comparison and Analysis

The performance of the proposed HLQR with and without global gains are

compared to the manually-tuned state feedback controller for analysis and validation.

The maximum overshoot percentage error (𝑒 ) is defined as:

where 𝑒 is the error between the reference and actual value, 𝐿 is the length of the

trajectory, which is 0.5 m along each axis. The maximum overshoot percentage error

for position tracking of the HLQR controller with global gains is [0.0454% 0.0912%

along the 𝑥- and 𝑦-axis, respectively. This is compared to the manually-tuned state

feedback controller that achieves a better position tracking performance of

[0.00078% 0.00078% along the 𝑥- and 𝑦-axis, respectively. The integral square

error (ISE) of both controllers are shown in Table 4.5. Note that suffices 𝑝 and 𝑛 refer

to pico (10 ) and nano (10 ) respectively. Again, the performance in terms of

position tracking error of the manually-tuned state feedback controller is slightly better

but remains comparable to the HLQR controller with global gains. However, the results

clearly show that the proposed HLQR controller with global gains performs the best at

handling the slip ratio and achieves the least driving force error. Table 4.6 compares

the control efforts of HLQR and manually-tuned state-feedback controllers, measured

by the integral square (IS) of the torque signal supplied to each wheel. The HLQR

controller with global gains require 6.85% and 7.07% less control effort for the 1st/3rd

and 2nd/4th wheels, respectively. It is also observed that the manually-tuned state

feedback controller produces high frequency oscillations that can lead to a variety of

undesired phenomena, such as actuator saturation and loss of system stability 34.

Moreover, manually determining the state feedback gains is a complicated process

requiring trial and error. HLQR therefore offers a systematic way to optimize the

performance. The comparison of HLQR results with and without global gains highlight

the importance of information exchange that helps to achieve better control of the

system.

𝑒|max 𝑒 |

𝐿∗ 100

(4.23)

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Table 4.5: Integral square error of the first path.

Manually-Tuned State Feedback Controller

HLQR Controller

Driving Forces Error

Position Error Driving Forces Error

Position Error

Wheel No.

Value 𝑁

Axis Value 𝑚

Wheel No.

Value 𝑁

Axis Value 𝑚

1 61.7375 x 69.92 𝑝 1 18.4262 x 87.06 𝑛

2 60.9979 y 103.05 𝑝 2 18.4299 y 8.19 𝑛

3 61.7375

3 18.4262

4 60.9979 4 18.4299

Table 4.6: Control effort comparison.

Manually-Tuned State Feedback Controller

HLQR Controller

Wheel No. Control Effort 𝑁 ∗ 𝑚

Wheel No. Control Effort 𝑁 ∗ 𝑚

1 1337 1 1245.4

2 1338.9 2 1244.2

3 1337 3 1245.4

4 1338.9 4 1244.2

4.5.2 Trajectory with Diagonal Cornering

The second reference trajectory consists of diagonal cornering and movement.

Only the results of the proposed HLQR controller with global gains are presented.

Similar to the first simulation, a second-order low pass filter with a time constant of 0.2

is used to ensure the second derivative of the reference trajectory remains bounded,

suppress maximum overshoot of the response, and simplify the tuning of controller

gains. All weighting matrices and 𝐾 remain the same. The reference position and

velocity of this new path are shown in Fig. 4.30 as x-y plot and 4.31 as time-domain

plot, where once again the orientation is kept constant. The results of position and

driving force control of the mecanum-wheeled vehicle are given in Fig. 4.30 to 4.36.

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It is noted that during diagonal linear translation, only driving forces from the

even-numbered wheels are required to move in the southwest direction. In order to

achieve a similar amount of acceleration as when moving along the 𝑥- and 𝑦-axis, more

driving forces must be produced by the 2nd and 4th wheel, while the driving forces of

the 1st and 3rd wheel is maintained around zero. This leads to an increase in the slip ratio

and torque generated at the even-numbered wheels. However, the proposed HLQR

controller is able to track this path without significant errors while maintaining the slip

ratio below 0.2 as specified. The results also show low driving force and position

tracking error. The maximum overshoot percentage error is [0.35% 0.1% along the

𝑥- and 𝑦-axis, respectively, while the integral square error is shown in Table 4.7.

Moreover, it is observed that high frequency oscillation is not present in the result.

Table 4.7: Integral square error of diagonal cornering.

Driving Forces Error Position Error

Wheel No. Value 𝑁 Axis Value 𝑚

1 13.5393 x 54.64 𝑛

2 22.0176 y 4.3698 𝑛

3 13.5393

4 22.0176

Figure 4.30 X-Y trajectory of diagonal cornering.

Start

Stop

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Figure 4.31 Velocity and position responses of diagonal cornering.

Figure 4.32 Driving force responses of diagonal cornering.

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Figure 4.33 Slip responses of diagonal cornering.

Figure 4.34 Torques generated during diagonal cornering.

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Figure 4.35 Position error during diagonal cornering.

Figure 4.36 Driving force error during diagonal cornering.

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Chapter 5

Conclusions and Recommendations

In conclusion, this research studied the control of a mobile robot with mecanum

wheels that provide omnidirectional mobility to the platform. These wheels contain a

number of rollers attached at the circumference, typically at an angle of 45° to the wheel

axes. With these rollers, the direction of traction forces are rotated and enable the

vehicle omnidirectional mobility. Even though omnidirectional mobility is a useful

property for many practical applications, an important drawback of this kind of wheel

is that it is sensitive to slippage.

To achieve accurate control of the vehicle, a new multi-level controller is

developed. The upper layer consists of a traditional proportional control to handle

position tracking. The lower layer treats the mecanum-wheeled vehicle as a multi-agent

system of wheels physically interconnected by the platform chassis. A hierarchical

LQR controller is used to achieve driving force and slip ratio control of each wheel.

This also involves information exchange among the wheels that simultaneously

improves the global performance.

Numerical simulation results verify and validate that proposed control system

is applicable to a mecanum-wheeled vehicle in both linear translation along x-y axis

and translation with diagonal cornering. It is found that the control input for a pair of

even wheels required during the south-west translation are as twice as south- or west-

translation while the control input for a pair of odd wheels remain around zero. The

difference in amount of control input required result in higher amount of slip ratio

compared to the linear translation along x-y axis case. However, in both cases the slip

ratios are suppressed in the linear region where slippage does not occur. A small amount

of driving force and position errors are observed. The control inputs do not oscillate so

much enable the system to avoid undesired phenomena such as actuator saturation and

loss of system stability. In consequence, the control efforts are lessened to the smallest

amount. By comparing the responses of the traditional state feedback controller and the

proposed controller, the proposed controller gives better performances in many ways

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including handling the slip ratio with the best performance, achieving the least driving

force error and control efforts, and avoiding the appearance of high frequency

oscillation responses.

As part of the recommendation for future work, experimental testing should be

realized to determine the feasibility and practicality of this control system. Rotational

motions should also be included in the model and control design. A study on split or

variable friction surfaces should be considered. Moreover, it is suggested that robust

control theory be implemented to improve the performance of the system.

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