Summary of Engineering Doctoral Dissertation: Researching and developing some control laws for a wheeled mobile robot in the presence of slippage

The objective of the thesis is to propose some new control methods to compensate for the negative effects of pattern uncertainty, external noise and wheel slip. Analyze and build dynamic and kinetic models of mobile robots when model uncertainties exist. Demonstrating the correctness and effectiveness of new control methods by the Lyapunov stability standard and Barbalat lemma. Advanced domestic and foreign control methods for mobile robots in the presence of uncertainty models, external noise, and wheel slip. Then propose new control methods.

MINISTRY OF EDUCATION

AND TRAINING

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY

-

NGUYEN VAN TINH

RESEARCHING AND DEVELOPING SOME CONTROL LAWS FOR A WHEELED MOBILE ROBOT IN THE PRESENCE OF SLIPPAGE

ENGINEERING DOCTORAL DISSERTATION Major: Control and Automation Technology

Code: 9.52.02.16

SUMMARY OF ENGINEERING DOCTORAL

DISSERTATION

Ha Noi, 2018

This work is completed at:

Graduate University of Science and Technology Vietnam Academy of Science and Technology

Supervisor 1: Dr Pham Minh Tuan

Reviewer 1: ………

………

Reviewer 2: ………

………

Reviewer 3: ………

………

This Dissertation will be officially presented in front of the Doctoral Dissertation Grading Committee, meeting at: Graduate University of Science and Technology Vietnam Academy of Science and Technology At ………… hrs …… day …… month…… year ……

This Dissertation is available at:

1 Library of Graduate University of Science and Technology

2 National Library of Vietnam

ABSTRACT

The necessary of the thesis

In these years, there is a growing recognition that mobile robots have the capability to operate

in a wide area and further the ability to manipulate in an automatic and smart way without any actions

taken by human Hence, this project concentrated on researching and developing some control laws

for wheeled mobile robots

The researching problems of this thesis

The author concentrated on radical control methods in order to deal with wheel slipping whenever there exist slippage, model uncertainties, and external disturbances

Object of study

So as to easily demonstrate the validity and performance of the proposed control methods, the object of study was selected to be one three-wheel mobile robot To be specific, this robot consists

of two differential driving wheels and one caster wheel used to make gravity balance

The purpose of researching

Proposing a number of radical control approaches so as to cope with the negative effects of model uncertainties, external disturbances, and above all slippage

Approaches of study

The approach of study is illustrated as the following order:

 Analyzing and building the kinematic and dynamic model of the mobile robot with the occurrence of model uncertainties, external disturbances, and above all slippage

 Researching, analyzing state-of-the-art control methods which were designed both domestic and foreign for this topic After that, some radical control methods were proposed

 Proving the correctness and efficiency of the proposed control approaches via Lyapunov standard and Barbalat lemma

 Demonstrating the above-mentioned control methods through Matlab/Simulink tool

Scientific and practical benefits of this project

Scientific benefits: Building novel control approaches for a wheeled mobile robot with the

purpose of compensating for the negative effects of model uncertainties, external disturbances, and above all slippage

Practical benefits: the proposed control methods in this project could be applied for wheeled

mobile robots operating in warehouses with the slippery floor and/or in orchards with wet land

Structure of the thesis

Chapter 1: Overviewing domestic and foreign studies in recent years, and then showing a

process by which the kinematic and dynamic model of a wheeled mobile robot are established in the presence of model uncertainties, external disturbances, and slippage

Chapter 2: Designing an adaptive control law based a three-layer neural network

Chapter 3: Designing a robust adaptive backstepping control law based a Gaussian wavelet

network

Chapter 4: Designing a backtepping control law ensuring finite-time convergence at

dynamic level

CHAPTER 1 OVERVIEWING AND MATHEMATIC MODELS

1.1 Problem statement

Motion control problem is fairly important in the field of mobile robot because the performance of control laws affects the efficiency of the application of mobile robots in production and life Thus, this problem is chosen as the goal of this project

These days, motion control problem for wheeled mobile robots has attracted the consideration of researchers all over the world Needless to say, a wheeled mobile robot is one of the system subjected to nonholonomic constraint [1] Furthermore, it is one multi input – multi output nonlinear system [2] It is thanks to the recent advances in control theory as well as engineering, there were a large number of different control methods applied such as sliding mode control [3-4], robust control [5], … These control laws were under the assumption of pure rolling and no slippage

Notwithstanding, in application practice, the violation of the above assumption can still happen That is to say, there exists slippage [12-13]

Slippage is one of the key factors making the visible degradation of control performance Therefore, in such circumstances, so as to heighten control performance, a controller must be capable

of compensating slippage

1.2 Domestic study

In Vietnam, until now, there have been reports researching autonomous vehicles such as the group of the authors from University of Transport and Communications in Hanoi studying swarm robots [14-15] One group of authors at Hanoi University of Science and Technology researched on building a mathematical model for one four-wheel electrical car considering the interaction between wheels and road [18] Nevertheless, there have been still not many the studying results of addressing slippage for wheeled mobile robots to be published

1.3 Foreign study

There have been reports researching on control problems compensating slippage for wheeled mobile robots It is due to slippage that the performance of closed system deteriorates and even the state of the system is unstable Frequently, so as to cope with slippage, the values of friction parameter and sideslip angle must be always measured in real-time accurately Specifically, the authors in [12] addressed slippage through compensating slip-ratios of wheels Gyros and accelerometers were utilized in [13] so as to compensate slippage in real-time The study in [19] reported a robust controller by which both slip-kinematic and slip-dynamic models were taken care thanks to the framework of differential flatness

r

r b

Where  R, L are angular coordinates of the right and left wheel respectively

Thereby, showing the kinematic model as follows [4]:

M M

cossin

x y

The nonholonomic constraint always assures the two following factors:

 The direction of the linear motion is always perpendicular to the wheel shaft

 Both the linear and angular motion of this robot fully depend on the pure rolling of the differential driving wheels

Specifically, the mathematical model of this constraint is shown as follows [32]:

Caster wheel

GM

Platform

θ

Figure 1.1 One wheeled mobile robot and slippage.

M M

cos sinsin cos

x y

with M1 and M2 being upper and lower bound of M and satisfying M2M10

Property 2: matrix M2B v  is skew-symmetric, that is to say

1.6 Conclusion for Chapter 1

The attention and attempt of researchers all over the world for compensating slippage has increasingly become more prevalent than ever before However, most the studies were conducted under the assumption that the sideslip angle and friction parameter always are measure exactly in real time

It goes without saying that accelerates and velocities are always directly measured via affordable and feasible sensors Yet, it is difficult and expensive to measure the sideslip angle and friction parameter [40]

Taking into account all the factors mentioned above, this project is going to offer radical control approaches so as to compensate slippage for a wheeled mobile robot without measuring the sideslip angle and friction parameter In stark contrast, the negative effects of slippage are going to deal with in an indirect way via the proposed controllers here

Moreover, the kinematic and dynamic model of the wheeled mobile robot subjected to slippage, model uncertainties, and external disturbances were established successfully These model are going to be used for designing control laws in next chapters This researching result was published

in the number 3 published material

CHAPTER 2 DESIGNING AN ADAPTIVE CONTROL LAW BASED ON A

THREE-LAYER NEURAL NETWORK 2.1 Problem statement

Due to the fact that the control law in Chapter 3 was designed under such an ideal, the applicability of that control method is very limited Therefore, in this chapter 4, one radical control method is proposed under a more practical condition in order to heighten the applicability in comparison to the method in Chapter 3

To be specific, such a more practical condition involve the following factors:

 There exist model uncertainties and external disturbances

 The velocities and accelerates of slippage are not measured

Let D(xD ,y D) be a target which is moving in a known desired trajectory (see Figure 3.1) Without loss generality, the motion equation of D can be supposed as follows:

cos( ) sin( )

D D

, TD, R, , x0, y0 are constant parameters, and time t varies from zero to infinity

We assume that the tool location is at point P So, the requirement of the position tracking control problem is to control the WMR so that P has to track D with the position tracking errors being

uniformly ultimately bounded

Remark 2.1: In Figure 2.1, we denote (xP, yP) as the position of P Let (xP, yP, ) be the actual

posture of the WMR, and (xPd, yPd, d) be the desired one of the WMR The presence of both the longitudinal and lateral slips makes it impossible to control the WMR in the way that the actual

posture (xP, yP, ) tracks the desired one (xPd, yPd, d) with an arbitrarily good tracking performance [32] Instead of this, it is fully possible to control the WMR with the purpose of making the actual

position (xP, yP) track the desired one (xPd, yPd) with an arbitrarily good tracking performance [32]

2.2 Structure of the three-layer neural network (NN)

Admittedly, artificial neural networks have the ability of approximating nonlinear and sufficiently smooth functions with arbitrary accuracy In this subsection, a three-layer NN is introduced briefly [8] As illustrated in Figure 2.2, the output of the NN can be computed as

w

 

  

W and V   v ij are the NN weight matrices (𝐳) = [𝟏,(𝒛𝟏),(𝒛𝟐), … ]𝐓 with 𝐳 =

[𝒛𝟏, 𝒛𝟐, … ]𝐓 Next, () is the activation function of the NN In this paper, the activation function

is chosen to be the sigmoid kind as (𝒛) = 𝟏/(𝟏 + 𝐞𝐱𝐩 (−𝒛))

Let 𝐟(𝐱): 𝐑𝐍𝟏 → 𝐑𝐍𝟑 be a smooth function There exist optimal weight matrices W and V

so that:

𝐟(𝐱) = 𝐖𝐓(𝐕𝐓𝐱) + , (2.3) where is the vector of optimal errors

Assumption 2.1:  is bounded Especially, ‖‖ ≤ 𝒃 where 𝒃 expresses an upper bound of  Let 𝐟̂(𝐱, 𝐖̂ , 𝐕̂) = 𝐲̂(𝐱, 𝐖̂ , 𝐕̂) = 𝐖̂(𝐕̂𝐓𝐱) denote an estimation of f(x), where 𝐖̂ , 𝐕̂ are estimation matrices of 𝐖 and 𝐕, respectively, and they are provided by an online weight tuning algorithm to be revealed subsequently

2.3 Expressing the vector filtered tracking errors (FTE)

Let O-XY be the global coordinate system, M-XY be the body coordinate system which is attached to the platform of the WMR (see Figure 2) The coordinate of the target is represented in M-XY as follows:

cos sinsin cos

cos sinsin cos

x y

Remark 2.2: If 1 ≠ 0, then h is an invertible matrix

Let us define the position tracking error vector as

In order to tackle this problem via the novel proposed control method, first of all, the scheme

of entire closed loop system is proposed as Figure 2.3

The vector FTE is defined as follows:

where Λ is one diagonal, positive-definition, and constant matrix It can be chosen arbitrarily

2.4 Structure of the controller

where K is a 2 × 2 diagonal, constant, positive definite matrix and is chosen arbitrarily 𝐟̂(𝐱, 𝐖 ̂ , 𝐕̂) is

the output of the NN in order to approximate 𝐟(𝐱)

In this work, let us propose the online weight tuning algorithm for the NN weights as follows:

Theorem 1 For the WMR subject to wheel slip as in Eq (1.8), let the control input be given

by Eq (2.19) and the online weight tuning algorithm be provided by Eqs (2.24) and (2.25) Then, according to Lyapunov theory and LaSalle extension, the stability of the closed-loop system is assured to achieve the desired tracking performance where  as well as the vector of the weight

errors are uniformly ultimately bounded [8] and can be kept arbitrarily small

Figure 2.2 structure of the three-layer neural network

Figure 2.3 Scheme of the whole closed loop control system

2 3cos 0, 20,5 3sin 0, 2

Figure 2.4 the timelines of slip velocities

2.7 Conclusion for chapter 2

All in all, in this chapter, an adaptive tracking controller based on a three-layer NN with the online weight updating algorithm was developed to let the WMR track a desired trajectory with one desired tracking performance It has been clear that the convergence of both the position tracking errors and the NN weight errors to an arbitrarily small neighborhood of the origin was ensured by the standard Lyapunov criteria and LaSalle extension The results of the Matlab simulations illustrated the validity and efficiency of the proposed control method

velocities of wheel slip

longitudinal slip of the right wheel longitudinal slip of the left wheel lateral slip

controller

WMR subject

to slippag

Bảng 2.1 Các tham số của rô bốt di động [21]

I G

The inertia of moment of the platform about the vertical axis

2)

I W The inertia of moment of the wheel about the rotational axis 0,0025 (kg.m2)

I D The inertia of moment of the wheel about the diameter axis 0,005 (kg.m2)

Figure 2.5 control performance comparison between two control methods in example 2.1

Figure 2.6 Comparison of tracking errors between two control methods in example 2.1

Figure 2.7 the torques of the proposed method in example 2.1

CHAPTER 3 DESIGNING A ROBUST ADAPTIVE BACKSTEPPING CONTROL LAW

BASED A GAUSSIAN WAVELET NETWORK 3.1 Problem statement

Even though the control method in chapter 2 illustrated the efficiency to cope with model

uncertainties and external disturbances, the control accuracy, namely the tracking error vector e, still

not small enough in compared to the expectation of tasks demanding high-accuracy The reason may be:

 There was the classification in a clear way for particular tasks Especially, what control terms are used to deal with the negative effects of slippage at the kinematic level and/or model uncertainties as well as external disturbances at the dynamic level?

 There was not robust control term, so the stable criterion is only UUB Specifically, the tracking errors were only ensured to converge to a near-zero compact set rather than asymptotic convergence to zero

Therefore, in this chapter, one novel robust adaptive tracking control method based on the backstepping technique [8] is proposed for a wheeled mobile robot so as to compensate slippage, model uncertainties and external disturbances The scheme of this control system is shown in Fig 3.1

In particular, this system consists of two closed control loops The outer loop comprises the kinematic controller In this kinematic controller, the kinematic robust term is utilized for compensating the harmful influence of slippage the output of kinematic is the same as the input of the inner dynamic control loop The inner loop is composed of the dynamic controller Here, the Gaussian wavelet network is employed in order to approximate unknown nonlinear functions due to

no prior acknowledgement of the dynamic model of this WMR The dynamic robust term is useful

to cope with the negative effects of model uncertainties and external disturbances

3.2 Structure of the Gaussian wavelet network

Let us consider a Gaussian wavelet network as Fig 5.2 The outputs of this network with p

wavelet basis functions are shown as follows:

x is the input vector; w shows a weight with j = 1,…, p; j j  x denotes

a multidimensional wavelet function as follows

 and c ij are the dilation and translation parameters, respectively

Thanks to the strong approximation ability of the Gaussian wavelet network [41], given any smooth function f x , there exists an optimal weight matrix W, an optimal vector ξ, and an optimal vector c such that

where ε describes one vector of optimal approximation errors

3.3 Designing kinematic control law

First of all, the first derivative of (2.6) is computed as follows

D D

cos sinsin cos

x y

Assumption 3.3: x D, y D are twice differential

Assumption 3.4: slip velocities are bounded Thus, there exists a known positive  so that

 

χ

Since the velocities of slippage are not measured, χ in (3.7) is uncertain Thus, the kinematic

in this method is proposed as follows:

D 1

D 0

cos sinsin cos

t c

x d

where vc is the desired vector of the vector of angular velocities of the differential driving wheels

v; r is the kinematic robust term proposed as follows:

  e

r

where  is the gain of the kinematic robust

3.4 Designing the dynamic control law

The dynamic control law is proposed as follows

ˆ , ˆ, ,ˆ ˆ