中 華 大 學

中 華 大 學 碩 士 論 文

具全向輪驅動單球輪機械人之分層滑模控制 Hierarchical Sliding Mode Control of A Spherical

Robot Driven By Omni Wheels

系所別：電機工程學系碩士班 學號姓名：M09801010 許博翔 指導教授：黃啟光 博士

中 華 民 國 101 年 2 月

i

摘要

本論文提出分層滑模來控制全向輪球型機器人，此球型機器人採用全向輪的十字型組 合，可沿著第一和第二的方向行駛。分層滑模控制基本上是兩個層次結構，根據里亞 普若夫穩定性定理，推導出各子系統的等效控制和總控制。熟悉的分層滑模定位控制 是根據狀態來進行切換，容易導致機器人身體太快直立，但球輪仍等速前進的情形。

甚至，當球輪靜止但不在規劃位置，且機器人身體垂直靜止時，非常不容易達到定位 控制。所以，本論文提出週期性的分層滑模控制來改善這些問題，採用時間切換來放 鬆身體收斂的要求。週期的長短基本上皆能達到定位控制的要求，重點是，週期越短 則其軌跡越平順。若採用較容易硬體實現的長週期切換，則可配合身體角度狀態的三 次方來設定安全收斂角度，此方法在設定角度外收斂較快，反之，在設定角度內收斂 較慢；希望機器人在幾乎達到所指定的位置時，身體才達到垂直靜止的姿態。

關鍵字：全向輪球型機器人、分層滑模控制、週期性的分層滑模控制。

ii

Abstract

The hierarchical sliding mode control (HSMC) has been proposed to achieve the position control of a spherical robot driven by Omni wheels. The two-direction movement can be accomplished by the cross type arrangement of driving wheels. Based on the two-layer architecture of HSMC and the Lyapunov stability theorem, at first, the equivalent control of each subsystem is deduced, and then the total control law is derived. The switching scheme of the HSMC is dependent on the state of the spherical robot, so it usually causes that the body converges too fast to be upright and still before the spherical wheel becomes still and at the desired position. In this thesis, a periodic hierarchical sliding mode control (PHSMC) is proposed to solve the spherical wheel which is at the constant speed or is still but not in correct position situation. Basically, the periodic switching scheme instead of state can slow down the body convergent rate. The position control can be achieved by various duration of the period. The shorter the duration is, the smoother the trajectory is.

For the long duration of the periodic switching, the hardware requirement is easy at the cost of slow body convergent rate. Therefore, the cubic of body angle term, which can result in fast convergent rate outside the designated body angle but in slow convergent rate within the designated body angle, is proposed to be included. By adding the cubic term, the body can become still and upright only when the spherical wheel reaches the desired position with slow speed.

Keywords: Spherical robot driven by Omni wheels, hierarchical sliding mode control, periodic hierarchical sliding mode control.

iii

致謝

本篇論文能夠如期完成，學生首要感謝黃啟光教授的悉心指導。對於學術研究方 面的啟發，讓我瞭解正確的研究方法和態度。同時更對研究產生興趣，只要肯努力，

最終，也會得到相對應的成果。並且在研究期間給予學生很大的鼓勵和幫助，在此致 上衷心的感激與謝忱。

另外，也特別感謝學長黃崑書和溫兆源，還有同學王彥翔以及學弟邱仲威，資料 上的協助和提供學術上寶貴的意見，並帶給我在研究時期的歡樂與珍貴回憶。在此一 併致謝。最後感謝家人的支持，在我求學的期間不斷的給予鼓勵，才能一心一意的完 成學業，並且順利完成論文。

iv

Contents

摘要……….……….………...i

Abstract….……….……….ii

致謝………..…….……….………...iii

Contents ……….……….………..iv

List of Tables……….…vi

List of Figures………….……….vii

1. Introduction……….………...1

1.1 Motivation……….……….1

1.2 Literature Survey……….………...3

1.3 Structure of the Thesis……….………...4

2. System Description and Model………..5

3. Hierarchical Sliding Mode Control………....9

3.1 Hierarchical Sliding Mode Control………9

3.2 Period Hierarchical Sliding Mode Control………...12

4. Simulations and Discussions………14

4.1 HSMCs……….14

4.2 Comparisons between HSMC and HSMC1……….17

4.3 PHSMCs………...25

4.4 Comparisons between PHSMC and PHSMC1……….30

4.5 Effect of the Designated Body Angle………...38

4.6 Effect of Switching Duration………...42

5. Conclusions and Future Research………47

5.1 Conclusions………..47

v

5.1 Future Research………48 Reference………..………49

vi

List of Tables

Table 4.1.1: Parameters of the body and spherical wheel..………..………14 Table 4.1.2: Parameters of HSMC and HSMC1………..15

vii

List of Figures

2.1.1 Coordinates of the spherical robot………5

3.1.1 Structure of hierarchical sliding mode control………10

4.1.1 The first axial responses of the HSMC.………..15

4.1.2 The second axial responses of the HSMC………..16

4.1.3 The first axial responses of the HSMC1………...16

4.1.4 The second axial responses of the HSMC1……….17

4.2.1 The first axial body angle comparison between HSMCs.……..……….20

4.2.2 The first axial body anglular velocity comparison between HSMCs………..20

4.2.3 The first axial spherical wheel angle comparison between HSMCs………...21

4.2.4 The first axial spherical wheel angular velocity comparison between HSMCs……..21

4.2.5 The second axial body angle comparison between HSMCs………...22

4.2.6 The second axial body anglular velocity comparison between HSMCs……….22

4.2.7 The second axial spherical wheel angle comparison between HSMCs…………...23

4.2.8 The second axial spherical wheel angular velocity comparison between HSMCs...23

4.2.9 Responses of the HSMC for the one axis system within 200 seconds………24

4.2.10 Responses of the HSMC1 for the one axis system within 200 seconds………24

4.3.1 The first axial responses of the PHSMC…………...26

4.3.2 The second axial responses of the PHSMC...26

4.3.3 The first axial responses of the PHSMC1...27

4.3.4 The second axial responses of the PHSMC1...27

4.3.5 The first axial responses of the PHSMC...28

4.3.6 The second axial responses of the PHSMC...28

4.3.7 The first axial responses of the PHSMC1...29

viii

4.3.8 The second axial responses of the PHSMC1...29

4.4.1 The first axial body angle comparison between PHSMCs...31

4.4.2 The first axial body angular velocity comparison between PHSMCs...31

4.4.3 The first axial spherical wheel angle comparison between PHSMCs...32

4.4.4 The first axial spherical wheel angular velocity comparison between PHSMCs...32

4.4.5 The second axial body angle comparison between PHSMCs...33

4.4.6 The second axial body angular velocity comparison between PHSMCs...33

4.4.7 The second axial spherical wheel angle comparison between PHSMCs...34

4.4.8 The second axial spherical wheel angular velocity comparison between PHSMCs...34

4.4.9 Control inputs of PHSMC...35

4.4.10 Control inputs of PHSMC1...35

4.4.11 Responses of the PHSMC for the one axis system...36

4.4.12 Responses of the PHSMC1 for the one axis system...36

4.4.13 The responses of the PHSMC for the one axis system...37

4.4.14 The responses of the PHSMC1 for the one axis system...37

4.5.1 The first axial body angle responses of the PHSMC1 under various designated angles………..38

4.5.2 The first axial body angular velocity responses of the PHSMC1 under various designated angles...39

4.5.3 The first axial spherical wheel angle responses of the PHSMC1 under various designated angles... ...39

4.5.4 The first axial spherical wheel angular velocity responses of the PHSMC1 under various designated angles...40

4.5.5 The second axial body angle responses of the PHSMC1 under various designated angles...40

ix

4.5.6 The second axial body angular velocity responses of the PHSMC1 under various designated angles...41 4.5.7 The second axial spherical wheel angle responses of the PHSMC1 under various designated angles...41 4.5.8 The second axial spherical wheel angular velocity responses of the PHSMC1 under various designated angles...42 4.6.1 The first axial body angle responses of the PHSMC1 under various switching durations...43 4.6.2 The first axial body angular velocity responses of the PHSMC1 under various

switching durations...43 4.6.3 The first axial spherical wheel angle responses of the PHSMC1 under various switching durations...44 4.6.4 The first axial spherical wheel angular velocity responses of the PHSMC1 under various switching durations...44 4.6.5 The second axial body angle responses of the PHSMC1 under various switching

durations...45 4.6.6 The second axial body angular velocity responses of the PHSMC1 under various

switching durations...45 4.6.7 The second axial spherical wheel angle responses of the PHSMC1 under various switching durations...46 4.6.8 The second axial spherical wheel angular velocity responses of the PHSMC1 under various switching durations...46

1

Chapter 1 Introduction

1.1 Motivation

In recent years, the mobile Pendubot robots have become the focus of the study. They can perform high complexity work or even in high-risk working environment, and they can let people have a better quality of life now. In the future, the robot will become increasingly important and plays an important role in the human life.

The two-wheel Robotic Mobility Platform (RMP) such as the Segway attracted significantly many motives in robotic transportation researches [1]. A single wheel with inverse mouse-ball drive can achieve the static and dynamic stability, and it has been evolved by Carnegie Mellon University (CMU) [2-6]. Their whole system layout including actuator mechanism and control sub-system has been presented. Then some important results including dynamic balancing, station keeping, and point-to-point motion were also discussed. Especially, their papers pointed out that their single- wheel can move directly in any directions, unlike balancing 2-wheel platforms which must turn into the new direction before moving, and. Therefore, a balancing rolling machine whose body is supported by a single Omni-directional spherical wheel has been proposed by them. However, the conflicted demand of both a high-friction and low-friction material inherited in the CMU robot became the major concern to be compromised.

A spherical robot is the target of this thesis, and its working principle can be virtually expressed as a spherical wheel driven by two perpendicular pairs of Omni wheels. Each pair has two parallel Omni wheels linked to the same actuator. The single Omni-directional

2

spherical wheel can solve the conflicting demand of both a high-friction and low-friction material inherited in the CMU robot. The motivation of the presented one is initially inspired by the single wheel with inverse mouse-ball drive which is developed by CMU, so their important characteristics such as static and dynamic balancing ability are also feasible for our presented one. Furthermore, for the presented one, the weight or load of robot can be directly transferred to the contact surface which is the crucial point related to the normal contact force between the driving part and the spherical wheel in the similar scheme.

Thereby, the applicability and robustness of the invented one can be greatly enriched by this distinct and particular trait. The presented one is also insensitive to the dust, and the spherical wheel can be made by the rubber same as the material of car tires. The above two important features suggest that the presented one assembled with a rubber spherical wheel can be promisingly adopted into the automobile industry. Moreover, the high efficiency and better balance of driving force associated with the presented one can improve its quality.

The dynamical model has been derived based on Euler Lagrange approach in the paper [7]. The derived model indicates that two pairs of Omni wheels to drive the spherical wheel is directly control the attitude of robot body [8-14], but it can be replaced by the acceleration of the spherical wheel. This scenario implies that the zero acceleration or constant speed of spherical wheel will not affect the attitude of the robot body with zero tilt.

But a constant speed is not necessary to be a zero speed which is required for a position control. Therefore, the position control or the point-to-point motion of the spherical robot will be the main focus in the thesis.

3

1.2 Literature Survey

In recent years, there have been a lot of interests in under-actuated systems which have fewer actuators than the degrees of freedom to be controlled. Many studied on control of under-actuated systems model have published. Xin and Kaneda [15] presented a robust controller for Acrobot and the swing-up control has been verified by simulation results.

Fantoni, Lozano, and Spong [16] solved the control of an under-actuated two-link Pendubot based on energy approach of the system. The sliding-mode control (SMC) [17-18]

belonging to a kind of variable-structure control system, is a nonlinear feedback control whose structure is intentionally changed to achieve the desired performance. Because SMC becomes less sensitive to system parameter variations and noise disturbances, one can consider using SMC to implement the control of the under-actuated systems.

Wang [19] presented a stable hierarchical sliding mode control method for a class of second-order under-actuated systems which have two subsystems. Wang [19] also proved the switch law of combination parameter by different condition of sliding surfaces, and all of the sliding surface can converge to zero asymptotically. In [20-25], a stable hierarchical sliding mode controller was designed for a class of second-order under-actuated systems.

First, the under-actuated system is separated into two subsystems, and the sliding surface of each subsystem is defined. A second-layer sliding surface is defined to combine these two first-layer sliding surfaces. The hierarchical sliding mode control law is derived by using the Lyapunov theorem. The control law can be derived to drive the subsystems toward their sliding surfaces where are asymptotically stable of all the sliding surfaces.

Powerful hierarchic SMC (HSMC) became very popular in position control of under-actuated systems. However, these applications to the proposed spherical robot will easily result in an undesired constant velocity due to the fast convergence of the body of

4

robot. Therefore, in the thesis, we will propose a periodic switching scheme instead of the state switching scheme, that is, periodic hierarchic SMC (PHSMC).

1.3 Structure of the Thesis

This thesis is organized as follows. In Chapter 2, we discuss the system description and mode of the single spherical wheel robot. The Hierarchical sliding mode control will be presented in Chapter 3. In Chapter 4, simulations and discussions based on the proposed control laws are conducted by using MATLAB. The conclusions and future researches are given in Chapter 5.

5

Chapter 2

System Description and Model

In order to derive the dynamical model of the proposed CWWU, we begin with the spherical wheel: radius

R

; mass

m

_sw; moment of inertia

I

_sw; rotating angles



₁ and



₂; angular velocities



₁



and



₂



. Similarly, for driving wheels,

r

is the radius;

I

_dw is the moment of inertia;

R 

₁

/ r

and

R 

₂

/ r

are the rotating angles. The exerting torques

1

,

2

 

of driving wheels will be expressed as the effective ones for the spherical wheel and the body as

( / ) R r

²



₁,

( / ) R r

²



₂. The total mass of driving wheels will be considered as part of body mass, and the equivalent moment inertia

I

_d1

 I

_dw1

( / ) R r

² and

2 2 2

( / )

d dw

I  I R r

. For the body with angles



₁ and



₂, and mass

m

_b centered at a distance from the center of spherical wheel, then its vertical position of the mass center can be derived as

R 

1 sin



²



₁



sin²



₂ , as shown in Figure 2.1;

I

_x,

I

_y, and

I

_xy are the moment inertia of the body;



₁ and



₂ are the angular velocities of the body along both directions;

g

is the gravity acceleration.

R

j k

2

2

R2



^{0,R2 sin2,R 1sin22}



Figure 2.1.1: Coordinates of the spherical robot

6

The dynamic model based on the Euler-Lagrange approach can be expressed as follows [13]:

1

1 2 1

1 1

1

2 1

2 2

2 2 2 2

2

0

0

R

U r

R M r

U R

r R r



 

 

  

  

 



  

    

   

      

        

           

        

           

  

     

   

 

(2.1)

where

1 12

1 1

12 2

2 2

0 0

0 0

0 0

0 0

I I

N J

M I I

N J

  

  

 

   

  

 

,

1 1 2 12

1 2 1 2

1

1

1 12 2 2

2 1 1 2

2 2

1 1

( )

2 2

0 0

1 1

( )

2 2

0 0

I I I I

N

I I I I

N

   



   



   

    

     

 

  

  

 

       

  

     

 

  

  

 

(2.2)

The associated definitions and their partial derivatives are summarized as below.

2 2

2 1 2

1 2 2

1 2

2 2

2 1 2

2 2 2

1 2

2 1 1 2 2

12 2 2

1 2

2

1 1

2

2 2

1 1

2 2

cos cos 1 sin sin ;

cos cos 1 sin sin ;

sin cos sin cos 1 sin sin ;

( ) ;

( ) ;

cos ; cos ;

x b

y b

xy b

sw sw b d

sw sw b d

b b

I I m

I I m

I I m

J I m m R I

J I m m R I

N m R

N m R

 

 

 

 

   

 





 

 

 

 

 

 

   

   





7

2 2

1 2 2 1 1 2 2

2 2 2

1 1 1 2

2 2

1 2 2 1 1 2 2

2 2 2

2 2 1 2

2 2 2 2

12 2 1 2 1 2

2 2 2 2 2

1 1 2

12

2

sin cos cos sin

2 ;

(1 sin sin )

sin cos cos sin

2 ;

(1 sin sin )

cos cos sin sin

sin cos ;

(1 sin sin )

b

b

b

b

I I

m

I I

m

I m

I m

   

   

   

   

   

 

  



   

   

 

 

   

 

    

  



2 2 2 2

2 1 2 1 2

1 1 2 2 2

1 2

1

1 1

2

2 2

cos cos sin sin

sin cos ;

(1 sin sin )

sin ; sin ;

b

b

N m R

N m R

   

 

 

 

 



 

  



  



Or the nonlinear under-actuated system expression will be used in the next chapter.

     

     

     

     

1 2

2 1 11 1 12 2

3 4

4 2 21 1 22 2

5 6

6 3 31 1 32 2

7 8

8 4 41 1 42 2

x x

x f X b X u b X u

x x

x f X b X u b X u

x x

x f X b X u b X u

x x

x f X b X u b X u



  



  



  



  

(2.3)

where

X  [ x x x x x x x x

_{1 2 3 4 5 6 7 8}

]

^T

 [        

_{1 1 1 1}

^

_{2 2 2 2}

^ ]

^T is state variables vector;

1

( ),

4

( )

f X f X

and

b

₁₁

( ), X b

₄₂

( ) X

are the following nonlinear functions.

2 2

2 1 1 2 12

1 2 1 2 1 2

1 2 12 2 1 1 2

2 2

12 1 12 2 2

1 1 2 2

2

1 2 12 2 1 1 2

2 12

2 2

1 2 12 1 1 2 12 2

1 1

( ) ( ( ) )

( ) 2 2

1 1

(( ) )

( ) 2 2

( ) ( )

I I I I I

f X I I I

I I I I I

I I I

I U I U

I I I I I I

   

   

   

   

 

   

     

     

   

   

     

 

 

     

(2.4)

8

2 2

1 2 1 1 2 12

2 2 1 2 1 2

1 1 2 12 2 1 1 2

2 2 2

1 12 1 12 2 2 1

1 1 2 2 1

2

1 1 2 12 2 1 1 2 1 1

1 2 1 12

2 2

1 1 2 12 1 1 1 2 12

1 1

( ) ( ( ) )

( ) 2 2

1 1 1

(( ) )

( ) 2 2

( ) (

N I I I I I

f X J I I I

N I I I I I N

J I I I J

N I U N I

J I I I J I I I

   

   

    

    



   

     

     

    

    

      

  

    

) ₂

U







(2.5)

2 2

12 1 1 2 12

3 2 1 2 1 2

1 2 12 2 1 1 2

2 2

1 1 12 2 2

1 1 2 2

2

1 2 12 2 1 1 2

12 1

2 2

1 2 12 1 1 2 12 2

1 1

( ) ( ( ) )

( ) 2 2

1 1

(( ) )

( ) 2 2

( ) ( )

I I I I I

f X I I I

I I I I I

I I I

I U I U

I I I I I I

   

   

   

   

 

   

     

     

   

   

     

 

 

     

(2.6)

2 2

2 12 1 1 2 12

4 2 1 2 1 2

2 1 2 12 2 1 1 2

2 2 2

2 1 1 1 2 12 2

1 2 1 2 2

2

2 1 2 12 2 1 1 2 2 2

2 12 2 1

2

2 1 2 12 1 2 1 2 12

1 1

( ) ( ( ) )

( ) 2 2

1 1 1

( ( ) )

( ) 2 2

( ) (

N I I I I I

f X J I I I

N I I I I I N

J I I I J

N I U N I

J I I I J I I I

   

   

    

    



   

     

     

    

     

      

  

    

²) ₂

U







(2.7)

2 12

11 2 12 2

1 2 12 1 2 12

1 2 1 12

21 2 22 2

1 1 2 12 1 1 1 2 12

12 1

31 2 32 2

1 2 12 1 2 12

2 12 2 1

41 2 42

2 1 2 12

( ) ; ( )

( ) ( )

( ) ( 1) ; ( )

( ) ( )

( ) ; ( )

( ) ( )

( ) ; ( ) (

( )

I R I R

b X b X

I I I r I I I r

N I R N I R

b X b X

J I I I J r J I I I r

I R I R

b X b X

I I I r I I I r

N I R N I

b X b X

J I I I r J

 

   

  

   

 

   

 

 

_{2 1 2 12}² ₂

1 )

( )

R I I I  J r

 

(2.8)

9

Chapter 3

Hierarchical Sliding Mode Control

3.1 Hierarchical Sliding Mode Control

We now consider a nonlinear under-actuated system expressed as (3.1):

     

     

     

     

1 2

2 1 11 1 12 2

3 4

4 2 21 1 22 2

5 6

6 3 31 1 32 2

7 8

8 4 41 1 42 2

x x

x f X b X u b X u

x x

x f X b X u b X u

x x

x f X b X u b X u

x x

x f X b X u b X u



  



  



  



  

(3.1)

where

X   x x

1

,

2

x

8



^T is the state variable vector,

f X

₁

( )

,

f X

₂

( ) f X

₄

( )

and

11

( )

b X

,

b

₁₂

( ) X b

₄₂

( ) X

, are the continuous nonlinear functions,

u

₁,

u

₂ are the inputs of driving wheels to control the body and the spherical wheel.

The fundamental sliding surface structure of the hierarchical sliding mode control (HSMC) and the proposed periodic hierarchical sliding mode control (PHSMC) are drawn in Figure 3.1.1. We extend it to be a two-parallel HSMC, and each subsystem is for each direction of the proposed spherical robot in which the coupling factor between two axes are ignored in the thesis.

For the first axis, the first sliding surface

s

₁

 c x

_{1 1}

 x

₂ where

c

₁

 0

is designed for a stable body. When enter the sliding surface, it implies that

s

₁

 0

or

x

₂

  c x

₁ . The second stable sliding surface,

s

₂

 c x

_{2 3}

 x

₄ where

c

₂

 0

, is composited of the spherical

10

wheel state variables

( , x x

_{3 4}

)

. The sliding surface

S

_A

 k s

_{1 1}

 k s

_{2 2} is a combination of two stable sliding surfaces. Similarly, for the second axis, the combination of two sliding surfaces

S

_B

 k s

_{3 3}

 k s

_{4 4} in which

s

₃

 c x

_{3 5}

 x

₆ with

c

₃

 0

is also aimed for a stable body, and the stable spherical wheel is expressed as

s

₄

 c x

_{4 7}

 x

₈ with

c

₄

 0

.

s

1

x

1

x

₂

s

A

s

3

x

5

x

₆

s

B

s

2

x

3

x

₄

s

4

x

7

x

₈

Figure 3.1.1: Structure of hierarchical sliding mode control.

Control inputs

u

₁ or

u

₂ are composited of three terms to be

u

₁

 u

_eq1

 u

_eq2

 u

_sw1 and

u

₂

 u

_eq3

 u

_eq4

 u

_sw2. We consider the derivative of the first sliding surface.

1 1 1 2

1 2 1 11 1 12 2

1 2 1 11

(

_eq1 _eq2 _sw1

)

12 2

s c x x

c x f b u b u

c x f b u u u b u

    

   

     

(3.2)

By substituting ₁ ^{1 2 1}

11 eq

c x f

u b

 



into the above equation, we have

1 11

(

_eq2 _sw1

)

12 2

s   b u  u  b u

(3.3) Similarly, for ₂ ^{2 4 2}

22 eq

c x f

u b

 



and the derivative of the second sliding surface, we have

2 21

(

_eq1 _sw1

)

22 2

s   b u  u  b u

(3.4) We define

S

_A

 k s

_{1 1}

 k s

_{2 2}, and the following switching law is needed for the HSMC to avoid the case that

S

_A

 0

but

s

₁

 s

₂



0,

11

20 1 2

1 1 2 2 2

20 1 2

0

A 0

k s s

S k s k s k

k s s

 

       

where

k

₂₀

 0

and

k

₁

 0

. (3.5)

It is also needed for the second axial sliding surface

S

_B.

3 3 4 4

S

B

 k s  k s

₄ ^{40 3 4}

40 3 4

0 0

k s s

k k s s

 

    

where

k

₄₀

 0

and

k

₃

 0

. (3.6)

For the first axial cost functional

¹

²

A

2

A

V  S

, it derivative is



1 1 2 2



1 11 2 1 12 2 2 21 1 1 22 2

1 11 2 2 21 1 1 12 2 22 1 2

( ( ) ) ( ( ) )

( )( )

A A A A

A eq sw eq sw

A eq eq sw

V S S S k s k s

S k b u u b u k b u u b u

S k b u k b u k b k b u u

      

 

       

 

      

(3.7)

We ignore the coupling term

u

₂ to simplify the above equation as

1 11 2 2 21 1

(

1 12 2 22

)

1

A A eq eq sw

V   S   k b u  k b u  k b  k b u  

(3.8)

Therefore, by designing

 

 

1 11 2 2 21 1

1

2 21 1 11

( )

eq eq A A

sw

k b u k b u sign S kS

u k b k b



   

 

, the negative

derivative can be ensued as follows:

2 2

( )

A A A A A A

V     sign S S  kS    S  kS

(3.9) Thus the control input along the first direction

u

₁

 u

_eq1

 u

_eq2

 u

_sw1 can be expressed as

 

   

   

1 1 2 1 2 2 4 2

1

2 21 1 11 2 21 1 11 2 21 1 11

( _A) _A

k c x f k c x f sign S kS

u k b k b k b k b k b k b



    

  

  

(3.10)

Similarly, consider 1 ²

B 2 B

V  S

where

S

_B

 k s

_{3 3}

 k s

_{4 4}, and we also ignore the input

u

1 in the derivative of the second axial cost functional

V

_B. The equivalent controls are

3 6 3

3

31 eq

c x f

u b

 



and ₄ ^{4 8 4}

42 eq

c x f

u b

 



, and the second axial switching control is

designed as

 

 

3 32 4 4 42 3

2

4 42 3 32

( )

eq eq B B

sw

k b u k b u sign S kS

u k b k b



   

 

(3.11)

12

Then with the second axial control input

u

₂

 u

_eq3

 u

_eq4

 u

_sw2 or

 

   

   

3 3 6 3 4 4 8 4

2

4 42 3 32 4 42 3 32 4 42 3 32

(

_B

)

_B

k c x f k c x f sign S kS

u k b k b k b k b k b k b



    

  

  

, the negative derivative of the

second axial cost function can be obtained as

V

_B

    S

_B

 kS

_B².

3.2 Period Hierarchical Sliding Mode Control

The conventional HSMC needs the state switching Equations 3.5 and 3.6 to ensure that all the sliding surfaces

s s s

₁

,

₂

,

₃, and

s

₄ in the bottom level of HSMC structure, as shown in Figure 3.1.1, will be entered. However, for the proposed spherical wheel robot, the HSMC will easily result in constant speed problem. In order to achieve the position control, the state switching scheme should be modified as the following periodic time switching, denoted as the period hierarchical sliding mode control (PHSMC).

20

2 2

20

( ) ( 2 ) 0

2

k t P

k t k t P

k P t P

  

       

(3.12)

40

4 4

40

( ) ( 2 ) 0

2

k t P

k t k t P

k P t P

  

       

(3.13)

The stability of the HSMC has been proved in [22-27], but the stability of the PHSMC will no longer be ensured, so the designated body angle (DBA) scheme based on the property of cubic power can be added to enhance the stability of the proposed PHSMC. If the designated body angle



_DBA is chosen, then coefficients _{1 2}

¹⁸⁰

DBA

z z

   

are specified.

The basic four sliding surfaces are modified as

3

1 1 1 2

(

1 1

)

s  c x   x z x

(3.14)

13

2 2 3 4

s  c x  x

(3.15)

3

3 3 5 6

(

2 5

)

s  c x   x z x

(3.16)

4 4 7 8

s  c x  x

(3.17)

The associated four equivalent inputs and two switching inputs are expressed as

2

1 2 1 1 1 1 2

1

11

3( )

eq

c x f z x z x

u b

  



(3.18)

2 4 2

2

21 eq

c x f

u b

 



(3.19)

2

3 6 3 2 5 2 6

3

32

3( )

eq

c x f z x z x

u b

  



(3.20)



4 8 4



4

42 eq

c x f

u b

 



(3.21)

 

 

1 11 2 2 21 1

1

2 21 1 11

( )

eq eq A A

sw

k b u k b u sign S kS

u k b k b



   

 

(3.22)

 

 

3 32 4 4 42 3

2

4 42 3 32

( )

eq eq B B

sw

k b u k b u sign S kS

u k b k b



   

 

(3.23)

The control inputs

u

₁,

u

₂ for the proposed PHSMC1 are summarized as follows.

 

   

   

2

1 1 2 1 1 1 1 2 2 2 4 2

1

2 21 1 11 2 21 1 11 2 21 1 11

(3 ) (

_A

)

_A

k c x f z x z x k c x f sign S kS

u k b k b k b k b k b k b

      

  

  

(3.24)

 

   

   

2

3 3 6 3 2 5 2 6 4 4 8 4

2

4 42 3 32 4 42 3 32 4 42 3 32

(3 ) (

_B

)

_B

k c x f z x z x k c x f sign S kS

u k b k b k b k b k b k b

      

  

  

(3.25)

14

Chapter 4

Simulations and Discussions

4.1 HSMCs

Four different HSMCs including HSMC, HSMC1, PHSMC and PHSMC1 are proposed in the thesis, and in the section, their simulations will be conducted by

MATLAB

TM and the results will be discussed. Parameters of the body and spherical wheel are listed in Table 4.1.1, and parameters for both HSMC and HSMC1 are outlined in Table 4.1.2. In simulations, the initial attitudes of body are



₁

(0)   / 5

and



₂(0)



/ 6, with angular velocities



₁

(0)  

₂

(0)  0

; the angles and angular velocities of spherical wheel are all zeros, that is,



₁

(0)  

₁

 (0)  0

and



₂

(0)  

₂

 (0)  0.

For both HSMCs, their first and second axial responses of the spherical wheel robot are depicted in Figures 4.1.1-4. It notes that the constant speed of the spherical wheel problem can be observed, that is, when the body is still

 

₁



₂

 0

and upright

 

₁



₂

 0

, the spherical wheel velocities

1

60 rad/sec

  

and



₂

  50 rad/sec

for HSMC;



₁

  40 rad/sec

and



₂

  30 rad/sec

for HSMC1.

Table 4.1.1: Parameters of the body and spherical wheel Radius and mass of a hollow sphere

Radius and mass of thin solid disk type driving wheels

0.1

_sw

0.25

R  m and m  kg

0.1

_dw

0.2

r  m and m  kg

Radius and mass of a solid cylinder body

R

_b

 0.1 m and m

_b

 7 kg

Height of a solid cylinder body

h  2  0.4 m

15

Table 4.1.2: Parameters of HSMC and HSMC1

K 7

1 3

c  c

2

2 4

c  c 1

1 3

k  k

10

2 4

k  k 1

 ₀

1 2

z  z 180 /15 

Figure 4.1.1: The first axial responses of the HSMC.

0 5 10

-0.5 0 0.5 1

t(sec)

 1(rad)

Body angular position

0 5 10

-1.5 -1 -0.5 0 0.5

t(sec)

1(rad/sec)

Body angular velocity

0 5 10

0 200 400 600

t(sec)

 1(rad)

Spherical wheel angle

0 5 10

0 20 40 60

t(sec)

 1(rad/sec)

Spherical wheel angular velocity

16

Figure 4.1.2: The second axial responses of theHSMC.

Figure 4.1.3: The first axial responses of the HSMC1.

0 5 10

-0.2 0 0.2 0.4 0.6

t(sec)

2(rad)

Body angular position

0 5 10

-1.5 -1 -0.5 0 0.5

t(sec)

2(rad/sec)

Body angular velocity

0 5 10

0 200 400 600

t(sec)

2(rad)

Spherical wheel angle

0 5 10

0 20 40 60

t(sec)

 2(rad/sec)

Spherical wheel angular velocity

0 5 10

-0.5 0 0.5 1

t(sec)

1(rad)

Body angular position

 0.26 rad

0 5 10

-1.5 -1 -0.5 0 0.5

t(sec)

1(rad/sec)

Body angular velocity

0 5 10

0 100 200 300 400

t(sec)

1(rad)

Spherical wheel angle

0 5 10

0 20 40 60

t(sec)

 1(rad/sec)

Spherical wheel angular velocity

17

Figure 4.1.4: The second axial responses of the HSMC1.

4.2 Comparisons between HSMC and HSMC1

The main difference between HSMC and HSMC1 is that the additional parameters

1 2

180 15

z  z  

for the cubic term of HSMC1 are to set the designated body angle to be

15

which can result in a fast convergent rate outside the designated angle but in a slow convergent rate within the designated angle. Therefore, for the HSMC1, the body angle convergent trajectories should be different around the designated angle 15∘ (0.26 rad) by checking Figures 4.2.1 and 4.2.5. As expected, the body angle responses of HSMC1 are faster than those of HSMC due to the additional terms. Moreover, for both axes, the body and spherical wheel angular velocity resposes of HSMC are smoother than those of HSMC1, as referred to Figures 4.2.2, 4.2.4, 4.2.6, 4.2.8.

Unfortunately, both HSMCs encounter the constant speed problem. For both HSMCs

0 5 10

-0.2 0 0.2 0.4 0.6

t(sec)

2(rad)

Body angular position

 0.26 rad

0 5 10

-1.5 -1 -0.5 0 0.5

t(sec)

2(rad/sec)

Body angular velocity

0 5 10

0 100 200 300 400

t(sec)

2(rad)

Spherical wheel angle

0 5 10

0 10 20 30 40

t(sec)

 2(rad/sec)

Spherical wheel angular velocity