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控制 Control of single spherical wheel robot driven

by omni wheels

系 所 別：電機工程學系碩士班 學號姓名：M09601017 溫兆源 指導教授：黃啟光 博士

中 華 民 國 100 年 8 月

這篇論文主要研究的重點是提出適用於全向輪驅動球輪之球形機器人的控制方 法。此球形機器人之動態模組是從尤拉方程式衍生，因此，此機器人在固定速度下 保持垂直的角度功能，提出分為兩大演算法之七種控制法則。第一種演算法為可變 結構系統控制(VSSC)，其特性是可調整參數讓它到沿滑動平面達到上述的功能。第 二種為非線性迴授控制，它與 VSSC 不同的是該輸入是平滑的。藉由這兩個類型的 模擬已經可以證實可讓球型機械人保持垂直且平穩等速的移動。

關鍵字：全向輪、球輪、可變結構系統控制、非線性迴授控制

Abstract

This thesis mainly discusses the control of a spherical robot using Omni wheels to drive a spherical wheel. The dynamical model is derived from Euler Lagrange approach.

Therefore, seven different control methods are presented which can achieve a constant speed at a vertical balance altitude. The proposed control methods can be categorized into two algorithms. The first algorithm is the variable structure system control (VSSC) in which the time needed to enter the sliding surface or to reach the stable point can be adjusted by parameters. The second one is the nonlinear feedback with smoothing inputs which are different from those of switching input of VSSC. The constant speed of the spherical robot with vertical balance altitude can be achieved by both algorithms and will be verified by simulations.

Keywords: Omni wheels; Spherical wheel; Variable structure system control; Nonlinear feedback control;

Contents

摘要………..i

Abstract………ii

Contents………..iii

List of Figures…...………..iv

1 Introduction………....1

1.1 Trend of Robot Mechanism……….………1

1.2 Literature Survey………...………....2

1.3 Novel Robot Mechanism………..………...4

1.4 Organization of the Thesis………5

2 System Description and Modeling……….…6

2.1 System Description of CMU………6

2.2 System Description of CWWU………8

2.3 Model of CWWU………...10

2.4 Augmented Model of CWWU……….14

3 Control Methods..………16

3.1 VSSCs Control………...…..………..16

3.2 Nonlinear Feedback Control…...………..……….18

4 Simulations and Discussions………....21

4.1 VSSCc Simulations….……….……….21

4.2 Nonlinear Feedback Control Simulations……...…………..………..29

5 Conclusions and Future Research...………..34

5.1 Conclusions ………...34

5.2 Future Research………...34

List of Figures

2.1 Omni wheel……….………6

2.2 CMU_Ballbot Eric and Anish…….………....7

2.3 The mechanism of a CWWU…………..………...10

2.4 Coordinates of the spherical robot……….………….….…...11

4.1 Convergent rates of normalized cost function of the proposed VSSC1..………...22

4.2 Angular displacement and velocity of the spherical wheel based on VSSC1…...22

4.3 Cost functions of the proposed VSSC1-4 with the convergent rate

a

=10….…...23

4.4 The virtual inputs of the proposed VSSC1-3 and the real input of VSSC4 along the first axis based on the convergent rate

t

=3sec and

a

=10...…………...24

4.5 The virtual inputs of the proposed VSSC1-3 and the real input of VSSC4 along the first axis based on the convergent rate

t

=3sec and

a

=10...………….24

4.6 Trajectories of VSSC1 along the first axis with

t

=3secand

a

=10……….25

4.7 Trajectories of VSSC2 along the first axis with

t

=3secand

a

=10…….…...25

4.8 Trajectories of VSSC3 along the first axis with

t

=3secand

a

=10…....…….…26

4.9 Trajectories of VSSC4 along the first axis with

t

=3secand

a

=10.………26

4.10 Trajectories of VSSC1 along the second axis with

t

=3sec and

a

=10………27

4.11 Trajectories of VSSC2 along the second axis with

t

=3sec and

a

=10…….…27

4.12 Trajectories of VSSC3 along the second axis with

t

=3sec and

a

=10.……...28

4.13 Trajectories of VSSC4 along the second axis with

t

=3sec and

a

=10……....28

4.14 Normalized cost function of the proposed NF…...………...29

4.15 Trajectories of NF1 along the first axis……….……….30

4.16 Trajectories of NF2 along the first axis….……….30

4.18 Trajectories of NF1 along the second axis……….……….……31 4.19 Trajectories of NF2 along the second axis…..……….…...32 4.20 Trajectories of NF3 along the second axis...………..……….……32 4.21 The virtual inputs of the proposed NF1-2 and the real input of NF3 along the first axis based on the smoothing parameter

a

=25………...………...33 4.22 The virtual inputs of the proposed NF1-2 and the real input of NF3 along the second axis based on the smoothing parameter

a

=25………..………..33

Introduction

1.1 Trend of Robot Mechanism

In 1994, a two-wheeled robot using the inverse pendulum control to eliminate the third castor wheel has been developed in Japan [1]. They also presented a one-wheeled balance robot in which the ellipsoid shape wheel is adopted and driven with an axle along the major axis [2]. Recently, the two-wheel RMP robotic platforms based on the Segway are the interesting research topic in robotic locomotion [3]. About Segway, use of gyro and tilt sensors is popular. When using only single gyro sensor, its un-avoided accumulation error can be reduced by an embodied IIR (Infinite Impulse Response) filter that can realize error compensation of gyro sensor under feedback scheme [4].

Thereafter, a single wheel with inverse mouse-ball drive can achieve the static and dynamic stability has been developed by Carnegie Mellon University (CMU) [5-6]. The overall design of the system, such as actuator mechanism and control system was presented. Then some initial results including dynamic balancing, station keeping, and point-to-point motion are also discussed. Most of all, their papers pointed out that unlike balancing 2-wheel platforms which must turn before driving in any direction, and the single-wheel can move directly in any direction. Therefore, they are the first group to propose a balancing rolling machine whose body is supported by a single Omni-directional spherical wheel. However, for the CMU robot, the conflict demand of

both a high-friction and low-friction material at the same time for the spherical ball becomes the major concern to be compromised.

1.2 Literature Survey

In [6], a modified robot mechanics, driven by one dc motor at a time, has been presented with a simplified dynamic equation. Then an LQR (linear quadratic regulator) control law has been tried to achieve stabilization and basic path tracking of the robot.

However, the capabilities of the LQR controller fall short of robust balancing and agile mobility that is crucial to operate a single wheel effectively in human environments.

Therefore, under this driving scheme, they derived a dynamic model of the robot moving in a flat terrain which considers unknown friction, parameters variations and exogenous disturbances, and a sliding–mode control method with backstepping scheme can accomplish robust balancing and agile path tracking of the robot. The merit and performance of their proposed control method have been verified by simulations.

The VSSC and the nonlinear feedback control, both controls can be applied to control the spherical wheel via driving Omni wheels. Under parametric variations and disturbances, the strong robustness of feedback stability of VSSC attracts the interests of control engineers as a practical means to make robust controllers for robotic arm, motor drives, etc. [7-8]. These studies confirmed that the robust nature of VSSC is guaranteed by a sliding mode. The sliding mode is a special case of VSSC which can keep an invariant trajectory while the plant is under various environments. This robust nature of sliding mode is good and suitable for motion controls because its algorithm is very simple and easy to be implemented in real time computer control systems. However, the main drawback of sliding mode is that the resulting control input is discontinuous on the

switching surface and, consequently, the control input chatters up to an infinite frequency theoretically. Such a control might not even be directly applicable to the plant, so some researches overcame this problem by using a proper continuous function to replace the undesired discontinuous function [9-10]. This one is a nonlinear control in which earlier Russia of the scholars and engineers are using this one to motor control. And they found an important and interesting behavior of the system, is sliding mode, in the control of the structure. VSSC has many advantages and different applications, such as, the stability control in power systems, airplane landing control, all types of motor position or speed control, robot control, and control of hydraulic servo valve. VSSC has the following characteristics:

(1) After the system enters the sliding mode, the effect of parameter variations and external disturbances can be greatly suppressed. Therefore, accurate identification of parameters associated with the controlled system is not required.

(2) The control structure can be easily implemented with fast response, and it is also suitable for time-varying systems and nonlinear systems.

Nonlinear feedback can decouple the nonlinear coupling terms in nonlinear systems, and the earlier work was done by Hemami and Camana [11]. They applied the nonlinear feedback control technique to a simple locomotion system and obtained decoupled subsystems [11-12]. The nonlinear feedback becomes feasible because of advanced digital integrated circuits. Recently, the research work on nonlinear system control still attracted much attention, and the related activity in this area has been reported in [13].

The work carried by Mutoh and Nikiforuk [14] provides a unique nonlinear feedback technique which ensures system stability with a region of feedback gains, and further justified by Willems and Aeyels [15]. However, this technique requires the knowledge of plant dynamics for constructing the gain sectors. In practice, this information sometimes

needs some burdens of parameter identification process to obtain [16]. Analysis and synthesis of nonlinear feedback control systems have received great attention in the control system community for the past several decades, due to the nonlinear nature of physical systems. While the early work has focused on asymptotic stability analysis of nonlinear systems, various versions of the small gain theorem has been used to show BIBO (bounded-input bounded-output) stability of nonlinear feedback systems [17-19].

Recent work on ISS (input-to-state stability) has extended BIBO stability to asymptotic stability [20-23]. On the other hand, singular perturbation theory [24] has also had its role in design of nonlinear feedback control systems.

1.3 Novel Robot Mechanism

The working principle of the presented combination of Omni wheel and spherical wheel unit (CWWU) in the paper can be virtually expressed as a spherical wheel driven by two perpendicular pairs of Omni wheels. Each pair has to parallel Omni wheels linked to the same actuator. It indicates that the CWWU needs two actuators. The motivation of the presented one is initially inspired by the single wheel with inverse mouse-ball drive developed by CMU, so their static and dynamic balancing ability is also one of important characteristics of the presented one. However, 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 which is same material as the car tire. 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.

1.4 Organization of the Thesis

This thesis is organized as follow. In Chapter 2, we discuss the system description and mode of the single Omni wheel. The control laws of the spherical robot to achieve zero titled altitude under constant speed are presented in Chapter 3. In Chapter 4, simulations and discussions based on the proposed control laws are conducted by using MATLAB. The conclusions are given in Chapter 5.

Chapter 2

System Description and Model

2.1 System Description of CMU

Single Omni wheel has one serious no braking ability, as shown in Figure 2.1, so multi-Omni-wheeled robots are then adopted and developed. Usually, the robot formed by the multi-Omni-wheel has one major disadvantage, that is, the low efficiency of driving force. If a straight line path is planned for a three-Omni-wheel base robot to follow, then only one half of the driving force can be used owing to an equiangular triangle arrangement of these three Omni wheels. This kind of arrangement will create undesired cancellation of the driving force, so the low efficiency of the driving force is incurred by the cancellation. Furthermore, for the case of four-Omni-wheel base robot, it is likely unsuitable for non-smooth floor or obstacles. Because the suspension of one of these four Omni wheels incidentally occurs due to uneven floor, it can result in malfunction of the robot.

Figure 2.1: Omni wheel

( http://www.phxhs.k12.az.us/education/club/club.php?sectiondetailid=27681)

Drive mechanism of the CMU is basically the inverse of a mouse-ball drive, that is, rollers drive the ball to produce motion, as shown in Figure 2.2. The ball is actuated by a pair smooth stainless steel rollers placed orthogonally at the sphere’s equator. These rollers are linked to two high torque DC servomotors through timing belts. There are two spring-loaded passive idler rollers which are opposite the drive rollers, and idler rollers can apply force at the ball’s equator to maintain contact between the drive rollers and the ball. This arrangement can reduce the underside effect caused by un-avoided slippage.

For example, if one roller is being driven, the orthogonal roller must be slipping. This unwanted simultaneous demand of both a high-friction and low-friction material for the ball becomes a compromise to be made. Because it is always desirable to have high friction between the ball and the floor, the high friction can avoid the serious slippage on the floor to cause the difficulty to balance the robot, or even worse the malfunction of the robot. Therefore, they pointed out that the drive works well but the initial ball eventually wore out. Three commercial low friction, Omni-directional ball transfer devices are adopted to support the entire Ballbot body rests on top of the ball. This design is smoothly to transfer the weight or load of robot through the ball into the floor.

Figure 2.2: CMU_Ballbot Eric and Anish ( http://www.msl.ri.cmu.edu/projects/ballbot/ )

2.2 System Description of CWWU

The CWWU is the abbreviation of the combination of Omni wheel and spherical wheel unit. Basically the CWWU is a novel spherical wheel driven by Omni wheels, so it combines the advantages of both types of wheels. Figure 2.3 depicts the drive mechanism of the CWWU base robot. There are two pairs of Omni wheels (A and B, C and D), and each pair of Omni wheels is designed to be driven by an actuator, such as a high torque DC motor or Step motor with an encoder. These two pairs of Omni wheels are arranged perpendicularly each other. The spherical wheel is actuated by these two pair of Omni wheels. It can be made similar to the CMU robot in which the ball is a 200 mm diameter hydro-formed steel shell covered with a 3.2 mm thick urethane outer layer. However, the preferred one is made by rubber likes the tire of cars. Thus, the proposed CWWU base robot can be easily applied in the robotic locomotion, and it can work well in the automobile environment in the future. A pre-determined distance between two Omni wheels in the same pair can affect the normal force exerted on the spherical wheel by the pair of Omni wheels. In other words, it relies on whether the friction force between the spherical wheel and the pair of Omni wheels is large enough to avoid the undesired slippage. The advantages of the CWWU base robot are summarized as follows.

There are no idle rollers working in the opposite moving direction likes the CMU robot, so the proposed one does not encounter the serious wore-out problem of the ball.

Demands of both a high-friction and low-friction material for the ball is not required for the proposed one.

For the CMU robot, three commercial low friction omni-directional ball transfer devices are adopted to support the entire Ballbot body rests on top of the ball. The

additional the low friction may sometimes become serious when the weight or load of the robot is increasing significantly.

Both Omni wheels in the same pair are designed to drive the spherical ball simultaneously unlike the CMU robot which only one drive roller. So the efficiency and balance of driving force of the CWWU robot is better than that of the CMU one.

The driving force always exerts on the spherical wheel along the tangent direction, so the maximum torque can be attained by the proposed one.

The contact surface between spherical wheel and driving pair of Omni wheels is larger than that of the CMU robot. So the undesired slippage can be migrated easily by the proposed one.

The small idle wheels of the Omni wheel take turn to contact the spherical wheel, so the proposed one is capable of reducing the unwanted effect of dust as compared with the CMU robot. In other words, the proposed one can be applied to dusty environments.

The weight of robot can be effectively transferred into the normal contact force between the spherical wheel and the driving Omni wheel pairs. This contact force is essential to reduce the possibility of undesired slippage.

Different loadings can be easily adopted by the proposed one. If the loading is increasing, then the contact force is also proportionally increasing.

As mentioned above, a pre-determined distance between two Omni wheels in the same pair can affect the normal force exerted on the spherical wheel by the pair of Omni wheels.

It notes that the shorter pre-determined distance is, the higher the normal force is.

Therefore, different applications can be designed flexibly based on various loading of the robot with a suitable distance.

The spherical can be made by rubber likes the material of tire using by cars. It implies that the proposed one can not only be easily applied in the robotic locomotion, but also

work well in the automobile environment in the near future. However, the CMU robot is so sensitive to the deformation of its ball which may shift the tight contact surface between roller and the ball to be non-tight one. Moreover, in the worst case, a serious deformation can cause the malfunction of the robot.

The proposed one has good braking ability, and it ability can be enhanced by the pre-determined distance. In case of increasing the weight or load of robot, the requirement braking ability is also increasing. Fortunately, the braking ability is also upgraded proportionally to weight or load for the proposed one.

Figure 2.3: The mechanism of a CWWU

2.3 Model of CWWU

Here, we begin with the related notations of the spherical wheel robot. Omni wheel radius is R , and drive wheels radius is r . Rotating angles

φ

₁ and

φ

₂ are the spherical wheel along the first and second directions, respectively, and

φ

₁^' as well as

φ

₂^' are the angular velocities. Its mass is

m

_SW and the moment inertia is

I

_SW. Then,

R φ

₁/

r

and

2/

R φ r

are the rotating angles of the driving wheels along the first and second directions.

The exerted torque of the driving wheels are denoted as

τ

₁ and

τ

₂with the moment inertia

I

_dw1 and

I

_dw2 , respectively, and by the radial ratio the effective torques and effective moment inertia on the spherical wheel can be transferred as −

R τ

₁/

r

, −

R τ

₂/

r

,

2

1 ( / ) 1

d dw

I

=

R r I

, and

I

_d2 =( / )

R r I

² _dw2. For the body angles along the vertical axis

θ

₁ and

θ

₂, and the body mass

m

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

R

+ℓ 1−

S

₁²−

S

₂² , where

S

₁^≜sin

θ

₁,

2 sin 2

S

^≜

θ

,

C

₁^≜cos

θ

₁, and

C

₂ ^≜cos

θ

₂, as shown in Figure 2.4. Moreover,

ω

₁ and

ω

2 are the angular velocities and

α

₁ as well as

α

₂ are angular accelerations of the body along both directions.

φ

2

θ

2

ℓ

R φ

2

j

 k



2

2 2 2

(0, R φ − ℓ S , ℓ 1 − S )

R

Figure 2.4: Coordinates of the spherical robot (Similarly, the other direction is same)

The moving distance of the ball along the ground is the same as the rotating distance of drive wheels along the circumstance of the ball, that is,

R θ

₁=

r φ

₁ and

R θ

₂=

r φ

₂. Then spherical wheel position

P

_sw



can be written as equation (2.1), and its velocity

v

_sw is also derived as equation (2.2).

P

_sw =

R i φ

₁+

R φ

₂

j

(2.1)

v

^sw=

R i φ

^1′+

R φ

^2′

j

(2.2) The kinetic energy of the spherical wheel can be written as equation (2.3).

1 ₁² ₂^{2 2} 1 ² ₁² ₂²

[ ( ) ] ( )( )

2 2

sw sw sw sw sw sw

K

=

I φ

′ +

φ

′ +

m v

=

I

+

m R φ

′ +

φ

′ (2.3)

The position of body Pb

and its velocity vb

can be derived as equations (2.4) and

(2.5).

2 2

1 1 2 2 1 2

( ) ( ) ( 1 )

P

b =

R φ

−

S i

+

R φ

−

S



j

+

R

+ −

S

−

S k



ℓ ℓ ℓ (2.4)

1 1 1 2 2 2

1 1 1 2 2 2 2 2

1 2

( ) ( )

b 1

S C S C

v R C i R C j k

S S

ω ω

φ

′

ω φ

′

ω

^{− −}

= − + − +

− −

  

 ℓ ℓ ℓ (2.5)

where

i



, j , k are the vectors along three directions.

Its potential energy and kinetic energy are written as equations (2.6) and (2.7).

U

=

m g

_b ℓ 1−

S

₁²−

S

₂² (2.6) 1 ₁² ₂² _{1 2} ²

[ 2 ]

b 2 x y xy b b

K

=

I ω

+

I ω

−

I ω ω

+

m v

(2.7)

where

I

_x,

I

_y, and

I

_xy are the body moment inertia. After substituting equation (2.5) into equation (2.7), we have equations (2.8).

2 2 2 2

2 1 2 2 2 1 2 2

1 2

2 2 2 2

1 2 1 2

2 1 1 2 2 2 2 2

1 2 1 2 1 1 1 2 2 2

2 2

1 2

1 1

[ ] [ ]

2 1 2 1

[ ] 1 ( ) .

1 2

b x b y b

b xy b b b

C C C C

K I m I m

S S S S

S C S C

m I m R m R C m R C

S S

ω ω

ω ω φ φ φ ω φ ω

= + + +

− − − −

′ ′ ′ ′

+ − + + − −

− −

ℓ ℓ

ℓ ℓ ℓ

(2.8)

The kinetic energy of drive wheels can be written as equation (2.9).

2 2

2 2 2 2

1 2 1 2 2 2 1 1 2 2

1 1 1 1

2 2 2 2

dw dw dw d d

R R

K I I I I

r φ

′

r φ

′

φ

′

φ

′

= + = + (2.9)

where

2

1 2 1

dw d

I R I

r

= and

2

2 2 2

dw d

I R I

r

= .

The total kinetic energy is

2 2 2 2

2 2 2 2 1 2 2 2 1 2 2

1 2 2 2 1 2 2 2

1 2 1 2

2 1 1 2 2 2 2 2 2 2

1 2 1 2 1 1 1 2 2 2 1 1 2 2

2 2

1 2

2 1 1

1 1 1

( )( ) [ ] [ ]

2 2 1 2 1

1 1 1

[ ] ( )

1 2 2 2

1 2

sw b dw

sw sw x b y b

b xy b b b d d

K K K K

C C C C

I m R I m I m

S S S S

S C S C

m I m R m R C m R C I I

S S

I

φ φ ω ω

ω ω φ φ φ ω φ ω φ φ

ω

= + +

′ ′

= + + + + + +

− − − −

′ ′ ′ ′ ′ ′

+ − + + − − + +

− −

= +

ℓ ℓ

ℓ ℓ ℓ

2 2 2

2 2 12 1 2 1 1 2 2 1 1 1 2 2 2

1 1 1

2

I ω

−

I ω ω

+2

J φ

′ +2

J φ

′ −

N φ ω

′ −

N φ ω

′ where

2 2 2 2

2 1 2 2 1 2 2 1 1 2 2

1 2 2 2 2 2 12 2 2

1 2 1 2 1 2

2 2 2 2

1 1 2 2 1 1 2 2

, ,

1 1 1

; ; ; ;

x b y b xy b

sw sw b d sw sw b d b b

C C C C S C S C

I I m I I m I I m

S S S S S S

J I m R m R I J I m R m R I N m R C N m R C

= + = + = −

− − − − − −

= + + + = + + + = =

ℓ ℓ ℓ

ℓ ℓ

Therefore,

the

Euler-Lagrange equation

L

=

K

−

U

=

K

_sw+

K

_b+

K

_dw−

U

can be

formulated as

equation (2.10).

1 _{1 1}² 1 _{2 2}² _{12 1 2} 1 _{2 2}^'2 _{1 1}^' _{1 2 2}^' ₂

2 2 2

L

=

I ω

+

I ω

−

I ω ω

+

J ϕ

−

N φ ω

−

N φ ω

−

U

(2.10)

Potential energy of the body

U

=

m g

_b ℓ 1−

S

₁²−

S

₂² and its partials are

1 1

2 2

1 1 2

2 2

2 2

2 1 2

1

1

b

b

S C U m g

S S

S C U m g

S S

θ

θ

−

∂ =

∂ − −

−

∂ =

∂ − −

ℓ

ℓ

where g is the gravity acceleration.

The dynamical model can be derived by

d L L

dt q

^∂ −^∂

q

=

τ

∂ ′ ∂ (2.11) where q corresponds to

θ θ

₁, ₂,

φ

₁, and

φ

₂, and its derivative q′ is related to

1, 2, 1

ω ω φ

′ , and

φ

₂′ . Based on equations (2.10) and (2.11) with

ω

₁′ =

α

₁ and

ω

₂′ =

α

₂ ,

the dynamical model of the body are

_{1 1 12 2} 1 ¹ ₁^{2 1} _{1 2} ¹² 1 ² ₂² ₁

( )

2 2

I I I I U R

I I

α α ω ω ω ω r τ

θ θ θ θ θ

∂ ∂ ∂ ∂ ∂

− + + − + + = −

∂ ∂ ∂ ∂ ∂ (2.12)

_{2 2 12 1} ² ₂^{2 2} _{1 2} ^{12 1} ₁² ₂

2 1 1 2 2

1 1

( )

2 2

I I I I U R

I I

α α ω ω ω ω r τ

θ θ θ θ θ

∂ ∂ ∂ ∂ ∂

− + + − + + = −

∂ ∂ ∂ ∂ ∂ (2.13)

Finally, the dynamical models of the spherical wheel for both axes are _{1 1 1 1} ¹ ₁² _{1 2 2 2 2} ² ₂² ₂

1 2

N R

;

N R

J N J N

r r

φ α ω τ φ α ω τ

θ θ

∂ ∂

′′− − = ′′− − =

∂ ∂ (2.14)

where the associated partial derivatives are summarized as below.

2 2 2 2

2 2

1 2 1 1 2 2 1 2 1 1 2 2

2 2 2 2 2 2

1 1 1 2 2 2 1 2

2 2 2 2 2 2 2 2

2 2

12 1 2 1 2 12 1 2 1 2

2 2 2 2 2 1 1 2 2 2

1 1 2 2 1 2

1 2

1

1 2

2 ; 2 ;

(1 ) (1 )

; ;

(1 ) (1 )

;

b b

b b

b b

I I S C C S I I S C C S

m m

S S S S

I C C S S I C C S S

m S C m S C

S S S S

N N

m R S m R

θ θ θ θ

θ θ

θ θ

∂ ∂ ∂ ∂

= = = =

∂ ∂ − − ∂ ∂ − −

∂ + ∂ +

= − = −

∂ − − ∂ − −

∂ ∂

= − = −

∂ ∂

ℓ ℓ

ℓ ℓ

ℓ ℓ

2;

S

2.4 Augmented Model of CWWU

Augment equations (2.12-2.13) as the below simplified form, where

1 2

[( / ) ( / ) ]

R r τ R r τ

^T becomes the input of driving wheels to control the body to be vertical attitude.

2 1 1

1 2

2 2

2

M α ω α ω ω

ω

 

   

= −

   

   

 

∏

















∂

∂

∂

∂

2 1

θ θ U U

-

















2 1

τ τ

r R r R

(2.15)

where



 





−

= −

2 12

12 1

I I

I

M I

,

∏

















∂

− ∂

∂

− ∂

∂ + ∂

∂

∂

∂ +∂

∂

∂

∂

− ∂

∂

− ∂

=

2 2

1 2

1 12

2 1

2 12

1 2

2 1

1 1

2 ) 1

2 (1

2 ) (1 2

1

θ θ

θ θ

θ θ θ

θ

I I

I I

I I I

I

By cancelling both of torques, that is, substituting equation (2.14) into equation (2.15), we can have the second simplified form as equation (2.16). It can be found that

'' '' 1 1 2 2

[

J φ J φ

]^T becomes the virtual control input term for controlling the body to be

vertical attitude.

2 1

1

1 1 1

2 2 1 2

2 2 2 2

2

2

U M J

J U

ω θ

α φ

α ω ω φ

ω θ

∂ 

  ∂  ′′

 =   − − 

    ∂   ′′

     ∂   

∏

(2.16)

where



 





−

−

−

= −

2 2 12

12 1

1

2

I I N

I N

M I

,

1 1 1 2 12

1 1 2 1 2

2

1 12 2 2 2

2 1 1 2 2

1 1

( )

2 2

1 1

( )

2 2

N I I I I

I I I N I

θ θ θ θ θ

θ θ θ θ θ

∂ ∂ ∂ ∂ ∂

 

− − +

 ∂ ∂ ∂ ∂ ∂ 

 

= ∂ ∂ ∂ ∂ ∂ 

+ − −

 

∂ ∂ ∂ ∂ ∂

 

∏

In this Chapter, the dynamical model of the spherical robot has been derived based on the Euler Lagrange approach, and it is different from previous ones [25-27]. The derived model is more difficult as compared with the previous ones, because the trajectory of the spherical ball, that is,

J

_{1 1}

φ

′′ and

J

_{2 2}

φ

′′ will directly affect the attitude of the body by observing equation (2.16). It implies that the constant speed control is feasible, but point to point with zero attitude of the body will become very difficult.

Chapter 3

Control Methods

3.1 VSSCs Control

In this thesis, we propose four VSSCs methods of the spherical wheel. The sliding mode control (SMC) of VSC for the spherical robot is proposed. Two sliding surfaces are designed along both directions for reducing the dimension of the system. The convergence of the body attitude can be adjusted by two designing positive real parameters

a and

₁

a . A positive Lyapunov function or cost function is selected as

₂

1 _{1 1 1} ² 1 _{2 1 2} ²

( ) ( )

2 2

L

vss =

ω

+

a θ

+

ω

+

a θ

(3.1) Then, its derivative is

dL

^vss ( _{1 1 1})( _{1 1 1}) ( _{2 2 2})( _{2 2 2})

a a a a

dt

=

ω

+

θ α

+

ω

+

ω

+

θ α

+

ω

(3.2) By designing

α

₁+

a

₁

ω

₁= −

sign

(

ω

₁+

a

_{1 1}

θ ω

)( ₁+

a

_{1 1}

θ

)²ⁿ (3.3)

α

₂+

a

₂

ω

₂ = −

sign

(

ω

₂+

a

_{2 2}

θ

)(

ω

₂+

a

_{2 2}

θ

)²ⁿ (3.4) where n is non-negative integer

By equations (3.3) and (3.4), equation (2.16) can be written as

2 1 1 1 2

1 1 1 1 1 1 1 1 1

2 2 2 1 2

2

2 2 2 2 2 2 1 2

2 2 2

2

( )( )

( )( )

n

n

J U

sign a a a

M sign a a a U

J ω φ

ω θ ω θ ω θ

ω θ ω θ ω ω ω

ω φ

θ

 ∂ 

   ′′+∂ 

 − + + − =   − 

   

 ∂ 

− + + −

     ′′ +∂ 

∏

(3.5)

Then a negative derivative of the cost function can be obtained as

2 1 2 1

1 1 1 1 1 1 2 2 2 2 2 2

2 1 2 1

1 1 1 2 2 2

( )( ) ( )( )

( ) ( ) 0

n n

vss

n n

dL sign a a sign a a

dt

a a

ω θ ω θ ω θ ω θ

ω θ ω θ

+ +

+ +

= − + + − + +

= − + − + ≤

(3.6)

Therefore, the first VSSC1 virtual control law can be written as follows:

2 2 1

1 1 1 1 1 1 1 1 1 1 1

2 2 2 1 2

2 2 2 2 2 2 2 2 2 2 2

2

2

( )( )

( )( )

n

n

U

J sign a a a

J M sign a a a U

ω θ

φ ω θ ω θ ω

φ ω θ ω θ ω ω ω

ω θ

∂ 

   

′′  + + +  ∂

 =  +   − 

 ′′ + + +   ∂ 

       ∂ 

∏

(3.7)

For the VSSC2, we introduce the term (1−

e

^−at) with positive a to reduce the magnitude of initial switching input of the VSSC1, that is,

2 2 1

1

1 1 1 1 1 1 1 1 1 1

2 2 2 1 2

2 2 2 2 2 2 2 2 2 2 2

2

2

( )( ) (1 )

( )( ) (1 )

n at

n at

U

J sign a a e a

J M sign a a e a U

ω θ

φ ω θ ω θ ω

φ ω θ ω θ ω ω ω

ω θ

−

−

∂ 

   

′′  + + − +  ∂

 =  +   − 

 ′′ + + − +   ∂ 

       ∂ 

∏

^(3.8)

The negative derivative of the cost function can be derived as

( )

2 1 2 1

1 1 1 1 1 1 2 2 2 2 2 2

2 1 2 1

1 1 1 2 2 2

(1 ) ( )( ) ( )( )

(1 ) ( ) ( ) 0

at n n

vss

at n n

dL e sign a a sign a a

dt

e a a

ω θ ω θ ω θ ω θ

ω θ ω θ

− + +

− + +

 

= − −  + + + + + 

= − − + + + ≤

(3.9)

For the VSSC3, we also introduces the term (1−

e

^−at) with positive

a

to reduce the magnitude of the total virtual input of the VSSC1, that is,

2 2 1

1

1 1 1 1 1 1 1 1 1 1

2 2 2 1 2

2 2 2 2 2 2 2 2 1 2 2

2

2

( )( )

(1 )

( )( )

n at

n

U

J sign a a a

e M

J sign a a a U

ω θ

φ ω θ ω θ ω

φ ω θ ω θ ω ω ω

ω θ

−

   ∂ 

  

′′  + + +  ∂

 = −   +  − 

 ′′  + + +   ∂ 

       ∂ 

∏

^(3.10)

The derivative of the cost function as equation (3.11) eventually becomes negative.

( )

[ ]

( )

2 1 2 1

1 1 1 2 2 2

2 1

1

1 1 1 1

1 1 1 2 2 2 2 2 1 2 2

2 2 2

2

2

2 1 2 1

1 1 1 2 2 2

(1 ) ( ) ( )

( ) ( )

at n n

vss

at

n n

vss t

dL e a a

dt

U

e a a a M M

a U

dL a a

dt

ω θ ω θ

ω θ

ω θ ω θ ω ω ω

ω ω

θ

ω θ ω θ

− + +

− − −

+ +

→∞

= − − + + +

   ∂ 

    ∂ 

  

+ + +  +  − ∂∂ 

⇒ = − + + +

∏

(3.11)

The virtual inputs of VSSC1-3 related to the acceleration of the spherical wheel. For smoothing the real input of driving wheels, VSSC4 based on equation (2.15) imposes the same (1−

e

^−at) term on the input of driving wheels as

2 2 1

1

1 1 1 1 1 1 1 1 1

1 2 2

2 2 2 2 2 2 2 2 2

2 2

2

( )( )

(1 )

( )( )

n at

n

R U

sign a a a

r e M

R sign a a a U

r

τ ω θ ω θ ω ω θ

ω θ ω θ ω ω ω

τ ω

θ

−

 ∂ 

      

 = − −  + + + +  −∂ 

    + + +    ∂ 

        

 

   ∂ 

∏

^(3.12)

The derivative of the cost function becomes

( )

[ ]

( )

2 1 2 1

1 1 1 2 2 2

2 1

1

1 1 1 1

1 1 1 2 2 2 1 2

1 2 2

2

2

2 1 2 1

1 1 1 2 2 2

(1 ) ( ) ( )

( ) ( )

at n n

vss

at

n n

vss t

dL e a a

dt

U

e a a a M M

a U

dL a a

dt

ω θ ω θ

ω θ

ω θ ω θ ω ω ω

ω ω

θ

ω θ ω θ

− + +

− − −

+ +

→∞

= − − + + +

   ∂ 

  

    ∂

  

+ + +  +  − ∂∂ 

⇒ = − + + +

∏

(3.13)

3.2 Nonlinear Feedback Control

In this method, the convergence of the body attitude can be adjusted by two designing parameters to result in stable subsystems. This algorithm with smoothing

inputs which is different from the non-continuous type switching input of VSSC.

Equation (2.16) with two designed stable subsystems

α

₁+

b

₁

ω

₁+

k

_{1 1}

θ

and

2

b

2 2

k

2 2

α

+

ω

+

θ

can be written as

2 1 1 1

1

1 1 1 1 1 1 1 1 1

2 2 1 2 2

2 2 2 2 2 2 2 2 2

2 2 2

2

0

J U

b k b k

M M

b k U b k

J ω φ

θ

α ω θ ω θ

α ω θ ω ω ω θ

ω φ

θ

 ∂ 

− ′′−

   

∂

+ + +

 = ∏   + +  =

 + +     ∂   + 

    − ′′−∂   

(3.14)

The nonlinear feedback (NF1) is implemented as

2 1

1

1 1 1 1 1 1

2 1 2 2

2 2 2 2 2 2

2

2

U

J b k

J U M b k

ω θ

φ ω θ

φ ω ω ω θ

ω θ

∂ 

   

′′ ∂ +

 = ∏   − +  

 ′′   ∂   + 

     ∂   

(3.15)

For the same reason to smooth the initial magnitude of the inputs, we introduce the smoothing term (1−

e

^−at) into the NF1 to be NF2.

2 1

1

1 1 1 1 1 1

2 1 2 2

2 2 2 2 2 2

2

2

(1 ^at)

U

J b k

e M

J U b k

ω θ

φ ω θ

φ ω ω ω θ

ω θ

−

 ∂  

 

   

′′ ∂ +

 = − ∏  − +  

 ′′    ∂   + 

     ∂   

(3.16)

The system based on the NF2, as shown in equation (3.17), eventually becomes stable.

2 1

1

1 1 1 1 1 1 1 1 1

2 2 1 2 2

2 2 2 2 2 2 2 2 2

2

2 at

U

b k b k

M e M

b k U b k

ω θ

α ω θ ω θ

α ω θ ω ω ω θ

ω θ

−

 ∂  

 

   

∂

+ + +

 = ∏  − +  

 + +     ∂   + 

     ∂   

(3.17)

Based on equation (2.15) containing the real input of the driving wheels, the two stable subsystems can be implemented as

2

1 1

1

1 1 1 1 1 1 1 1 1

1 2

2 2 2 2 2 2 2 2 2 2

2 2

2

0

U R

b k r b k

M M

b k U R b k

r

ω θ τ

α ω θ ω θ

α ω θ ω ω ω θ

ω τ

θ

∂   

  ∂   

+ + +

 =   − − +  =

     

+ + ∂  +

     ∂     

∏

^(3.18)

be designed as

2 1 1

1 1 1 1 1

1 2

2 2 2 2

2

2 2

2

(1 ^at)

R U

b k

r e M

b k

R U

r

τ ω θ ω θ

ω ω ω θ

τ ω

θ

−

 ∂  

       

 = −   −∂ +  + 

     ∂   + 

       

 

   ∂  

∏

(3.19)

The system with NF, as shown below, gradually becomes stable, and the convergent rate depends on the positive parameter a .

2 1

1

1 1 1 1 1 1 1 1 1

1 2

2 2 2 2 2 2 2 2 2 2

2

2 at

U

b k b k

M e M

b k U b k

ω θ

α ω θ ω θ

α ω θ ω ω ω θ

ω θ

−

  ∂  

 

  ∂  

+ + +

  =    −   +   

 + +      ∂   +  

          ∂      

∏

(3.20)

Chapter 4

Simulations and Discussions

4.1 VSSCs Simulations

In order to verify the performance of the proposed control laws, we carry out several numerical simulations using MATLAB^TM. In these simulations, the spherical wheel is assumed to be made as a hollow sphere with radius

R

=0.1

m

and mass

m

_sw =0.25

kg

; the driving wheels are the thin solid disks, each one with the same radius

r

=0.1

m

and the same mass

m

_dw =0.2

kg

; the body is a solid cylinder in which the radius

R

_b=0.1

m

, mass

m

_b =7

kg

and height

h

=2ℓ=0.4

m

. In order to control the convergent rate, we introduce parameters

σ

₁ and

σ

₂ into the VSSC virtual control law, as shown in below.

2 2 1

1

1 1 1 1 1 1 1 1 1 1 1

2 2 2 1 2

2 2 2 2 2 2 2 2 2 2 2 2

2

2

( )( )

( )( )

n

n

U

J sign a a a

J M sign a a a U

ω θ

φ σ ω θ ω θ ω

φ σ ω θ ω θ ω ω ω

ω θ

∂ 

   

′′  + + +  ∂

 =  +   − 

 ′′ + + +   ∂ 

       ∂ 

∏

(4.1)

The cost function becomes

2 1 2 1

1 1 1 1 1 1 1 2 2 2 2 2 2 2

2 1 2 1

1 1 1 1 2 2 2 2

( )( ) ( )( )

( ) ( ) 0

n n

vss

n n

dL sign a a sign a a

dt

a a

σ ω θ ω θ σ ω θ ω θ

σ ω θ σ ω θ

+ +

+ +

= − + + − + +

= − + − + ≤

(4.2)

Based on the settings as

φ

₁(0)=

φ

₂(0)=

φ

₁′(0)=

φ

₂′(0)= , 0

θ

₁(0)= −

π

/ 12,

θ

₂(0)=

π

/ 6,

1(0) 2(0) 0

ω

=

ω

= ,

n

= , 0

a

₁=

a

₂ = . The convergent rate can be adjusted by selecting 5

σ

1 and

σ

₂, and the normalized cost functions are shown in Figure 4.1. In Figure 4.1, parameters

σ

₁=0.43 and

σ

₂=0.87 are selected for the convergent rate

t

=3sec. For the convergent rate

t

=5sec, parameters are chosen as

σ

₁=0.26 and

σ

₂=0.52. The constant speed control

φ

₁′(0)=

φ

₂′(0)=constant can be observed from Figure 4.2.

0 1 2 3 4 5 6 7 8 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cost Function of the VSSC1

t(sec) L V S S /L V S S (0 )

a1=5,t=3sec, σ 1=0.43, σ 2=0.87 a1=5,t=5sec, σ 1=0.26, σ 2=0.52

Figure 4.1: Convergent rates of normalized cost function of the proposed VSSC1.

0 2 4 6 8

-100 -50 0

t(sec)

φ 1

0 2 4 6 8

0 50 100 150 200

t(sec)

φ 2

0 2 4 6 8

-15 -10 -5 0

t(sec)

φ ′ 1

0 2 4 6 8

0 10 20 30

t(sec)

φ ′ 2

t=3sec t=5sec

t=3sec t=5sec

t=3sec t=5sec

t=3sec t=5sec

Figure 4.2: Angular displacement and velocity of the spherical wheel based on VSSC1

Similarly, we also introduce

σ

₁ and

σ

₂ into VSSC2-4. Normalized cost functions of VSSC1-4 are depicted in Figure 4.3, and the associated virtual input or real input of driving wheels are drawn in Figures 4.4 and 4.5. Figure 4.3 indicates that the slope of initial portion of cost function for VSSC2 is flat as referred to equation (3.9) when

0 sec

t

= , and it is positive for both VSSC3-4, as referred to equations (3.11) and (3.13).

It notes that for the VSSC2 the smoothing term is applied to the sign term (switching) only which occurring after 3 seconds, so the cost functions and the virtual inputs are similar for VSSC1 and VSSC2. However, the virtual inputs of VSSC3 or the real input of VSSC4 start from zero which are different from that of VSSC1 and VSSC2, as observed from Figures 4.4 and 4.5. The associated trajectories related to the sliding surfaces are depicted in Figures 4.6-4.13.

0 1 2 3 4 5 6 7 8

0 0.5 1 1.5 2 2.5 3

Cost Function of the VSSC

t(sec) L V S S /L V S S (0 )

VSSC1 VSSC2 VSSC3 VSSC4

Figure 4.3: Cost functions of the proposed VSSC1-4 with the convergent rate

a

=10.

0 2 4 6 8 -5

0 5

t(sec)

J 1 φ ′′ 1

The virtual inputs of the proposed VSSC1

0 2 4 6

-4 -2 0 2

t(sec)

J 1 φ ′′ 1

The virtual inputs of the proposed VSSC2

0 2 4 6 8

-10 -5 0 5

t(sec)

J 1 φ ′′ 1

The virtual inputs of the proposed VSSC3

0 2 4 6 8

-5 0 5

t(sec)

τ 1

The real inputs of the proposed VSSC4

Figure 4.4: The virtual inputs of the proposed VSSC1-3 and the real input of VSSC4 along the first axis based on the convergent rate

t

=3sec and

a

=10.

0 2 4 6 8

-5 0 5 10

t(sec)

J 2 φ ′′ 2

The virtual inputs of the proposed VSSC1

0 2 4 6

-5 0 5

t(sec)

J 2 φ ′′ 2

The virtual inputs of the proposed VSSC2

0 2 4 6 8

-10 0 10 20

t(sec)

J 2 φ ′′ 2

The virtual inputs of the proposed VSSC3

0 2 4 6 8

-5 0 5 10

t(sec)

τ 2

The real inputs of the proposed VSSC4

Figure 4.5: The virtual inputs of the proposed VSSC1-3 and the real input of VSSC4 along the second axis based on the convergent rate

t

=3sec and

a

=10.

-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 -0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

θ 1 ω 1

Trajectories of VSSC1 VSSC1

Sliding Surface

START

END

Figure 4.6: Trajectories of VSSC1 along the first axis with

t

=3secand

a

=10.

-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

θ 1 ω 1

Trajectories of VSSC2 VSSC2

Sliding Surface

END

START

Figure 4.7: Trajectories of VSSC2 along the first axis with

t

=3secand

a

=10.

-0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 -1

-0.5 0 0.5 1 1.5 2 2.5

θ ₁ ω 1

Trajectories of VSSC3 VSSC3

Sliding Surface

START END

Figure 4.8: Trajectories of VSSC3 along the first axis with

t

=3secand

a

=10.

-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0

-0.5 0 0.5 1 1.5 2

θ 1 ω 1

Trajectories of VSSC4 VSSC4

Sliding Surface

START END

Figure 4.9: Trajectories of VSSC4 along the first axis with

t

=3secand

a

=10.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.5

0 0.5 1 1.5 2 2.5

θ 2 ω 2

Trajectories of VSSC1 VSSC1

Sliding Surface

START END

Figure 4.10: Trajectories of VSSC1 along the second axis with

t

=3sec and

a

=10.

0 0.1 0.2 0.3 0.4 0.5 0.6

-0.5 0 0.5 1 1.5 2 2.5

θ 2 ω 2

Trajectories of VSSC2 VSSC2

Sliding Surface

START END

Figure 4.11: Trajectories of VSSC2 along the second axis with

t

=3sec and

a

=10.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.5

0 0.5 1 1.5 2 2.5 3 3.5 4

θ 2 ω 2

Trajectories of VSSC3 VSSC3

Sliding Surface

END

START

Figure 4.12: Trajectories of VSSC3 along the second axis with

t

=3sec and

a

=10.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

θ 2 ω 2

Trajectories of VSSC4 VSSC4

Sliding Surface

END START

Figure 4.13: Trajectories of VSSC4 along the second axis with

t

=3sec and

a

=10.

4.2 Nonlinear Feedback Control Simulations

For the same initial conditions, the convergence of the body attitude based on NF1-3 can be adjusted by four designing parameters

b b k

₁, ₂, ₁, and

k

₂. For all three cases, the settings of equations (3.15), (3.16), and (3.19) are listed as follows:

φ

₁(0)=

φ

₂(0)= , 0

1(0) 2(0) 0

φ

′ =

φ

′ = ,

θ

₁(0)= −

π

/ 12 ,

θ

₂(0)=

π

/ 6 ,

ω

₁(0)=

ω

₂(0)=0 ,

a

₁=

a

₂ =5 , 25

a

= ,

b

₁=

b

₂ = and 5

k

₁=

k

₂= . Figure 4.14 shows the normalized cost functions 3 in which only the slope of NF1 is negative. For comparison, trajectories of NFs along the first axis related to the sliding surface

ω

₁+5

θ

₁ are shown in Figures 4.15-4.17, and those related to the sliding surface

ω

₂+5

θ

₂ along the second axis are depicted in Figures 4.18-4.20. The virtual inputs of NF1-2 and real input of NF3 for both axes are drawn in Figures 4.21 and 4.22. Only the inputs of NF1 do not start from zero which is impractical in real case.

0 1 2 3 4 5 6 7 8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Cost Function of the NF

t(sec) L V S S /L V S S (0 )

NF1

NF2

NF3

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0

0.05 0.1 0.15

θ 1 ω 1

Trajectories of the NF1

NF1

START

END

Figure 4-15: Trajectories of NF1 along the first axis.

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

θ 1 ω 1

Trajectories of the NF2

NF2

START

END

Figure 4-16: Trajectories of NF2 along the first axis.

-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 -0.2

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

θ 1 ω 1

Trajectories of the NF3

NF3

START

END

Figure 4-17: Trajectories of NF3 along the first axis.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0

θ 2 ω 2

Trajectories of the NF1 START NF1 END

Figure 4.18: Trajectories of NF1 along the second axis.

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.6

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

θ 2 ω 2

Trajectories of the NF2

NF2

START END

Figure 4.19: Trajectories of NF2 along the second axis.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

θ 2 ω 2

Trajectories of the NF3

NF3 END

START

Figure 4.20: Trajectories of NF3 along the second axis.