REPORT TOPIC MODELING AND SIMULATING THE THREEWHEELED MOBILE ROBOT

Wheeled Mobile Robot Wheeled Mobile Robots WMR are dynamic systems where an appropriatetorque needs to be applied to the wheels to obtain desired motion of platform.Motion control algori

HA NOI UNIVERSITY OF SCIENCE AND

TOPIC: MODELING AND SIMULATING THE

THREE-WHEELED MOBILE ROBOT.

INSTRUCTOR: Ph.D Vũ Thị Thúy Nga

Mục lục

Table of Contents

i

Part 1: Introduction 1

1.1 Wheeled Mobile Robot 1

1.2 Some typed of Wheeled Mobile Robot 1

1.2.1 Differential Drive 1

1.2.2 Bicycle Drive 1

1.2.3 Tricycle Drive 1

1.2.4 Car Drive 2

1.2.5 Omni Robot 2

1.3 Choosing WMR three wheels 3

Part 2: Modeling the Moblie Robot Three Wheels 3

2.1 Kinetic Model 3

2.2 Dynamic Model 7

Part 3:Controlling Methods Wheel Mobile Robot 16

3.1 Control Overview 16

3.2 Basic approaches 17

3.2.1 Directional and translational control 17

3.2.2 Basic approaches 19

3.3 Orbit Following Control 22

3.3.1 following the trajectory using basic approaches 22

3.3.2 Analysis feedforward and feedback elements 22

3.3.3 Linearization of feedback 23

3.3.4 Development of tracking kinetic trajectory 25

3.3.5 Linear Controller 27

Mục lục

Part 5: Control System On Matlab Simulink 345.1 Knematic Model 34

5.2 Dynamic Model 365.3 Kynematic Controller 38

5.4 Dynamic Controller 395.5 Simulink Result 39

5.5.1 First Case 395.5.2 Second case 405.5.3 Third Case 425.5.4 Fouth case 43Reference 45

Part 1: Introduction

1.1 Wheeled Mobile Robot

Wheeled Mobile Robots (WMR) are dynamic systems where an appropriatetorque needs to be applied to the wheels to obtain desired motion of platform.Motion control algorithms therefore need to consider the system’s dynamicproperties Usually this problem is tackled using cascade control schemas with theouter controller for velocity control and the inner torque(force, motor current,etc.)

The outer controller determines the required system velocities for the system

to navigate to the reference pose or to follow the reference trajectory While theinner faster controller calculates the required torques to achieve the systemvelocities determined from the outer controller

1.2 Some Typed Of Wheeled Mobile Robot

1.2.1 Differential Drive

A different wheeled robot is a mobile

robot is a mobile robot whose movement is

based on separately driven wheels placed on

their side of the robot body

It can change its direction by varying

the relative rate of rotation of its wheels and

hence does not require an additional steering

motion Robots with such a driven typically

have on or more castor wheels to prevent the

vehicle from tilting

1.2.2 Bicycle Drive

Bicyce Drive also have 2 wheels as

Differential Drive, but its wheels are arrange

in a straight line, and usually only one wheel

is active and other can control the steering

angle, similar like a bicycle

Tricycle Drive is a combination of WMR mentioned above, it has 3 wheels:

2 rear wheels are arranged coaxially and front wheel issteering one Two of three wheels are attached to the actuator

to control and the remaining one is free or to control desiredspeed or angle

1.3 Choosing WMR three wheels

Robot nonholonomic

Two Drive Rear

Wheel

Front Drive Wheel

ICR: instantaneous center of rotation of vehicle R(t): instantaneous radius of vehicle’s trajectory

v : longitudinal vehicle velocity

r : wheel’s radius

The model is placed in a general coordinate system  X g ,Y g , and the coordinate system of motion associated with WMR X m ,Y m  The state vector

of the vehicle in the general coordinate system is:

Because the movement of the vehicle through transmission of 2 rear wheelsthat moniting the WMR direction; therefore, the vehicle's longitudinal velocity isdetermined by:

The equation of external kinematic of WMR on the general

coordinate system is determined as follows:

In order to eleminate the complex in presentation, we temporarily ignore thedependence of the quantity into time, the above equation can be rewritten as:

Or we can write as following matrix:

In which the directions of motion are:

System equation 2.5 is rewritten as follows:

Beside, the WMR can only move along the wheels, not drift Therefore, the motion

of the WMR is constrained:

Where:

The relationship between (x2,y2) with (x,y) as following:

Therefore;

Subtituting 2.10 into 2.8 we have :

From 2.11, we have constrain matrix A as following:

Thus we have:

From 2.7 anf 2.13 we have: A.S.v =0

Or:

2.2 Dynamic Model

Model 3-wheel mobile robot is considered within 2 rear wheels as active driven

by two engines, font wheel as free one These actuators will generate torques

 R and L

The dynamic model of WMR’s motion is described by the Lagrange

formula as follows:

Where:

: the difference between the kinetic and potential energy of

the system

P : the power lost due to friction

k : the index of the general coordinate component q k

g k : gravity analysis in the direction of q k

        k : component Lagrange associated wih he jth

constrain according to q k

a jk : constrain cefficient according to q k

The dynamic model 2.13 can be rewritten in matrix as follows:

M qq  V q,q  F q  G q d   Eq.u  AT q. (2.16)Where:

q : general coordinate vector

M q : positive definte matrix of mass and

Eq : matrix tránition form

actuation space to coordinate space

u : input torque vector

AT q : motion constraint coefficient matrix

      : constrain force vector

In order to simplify 2.16 , we temporarily

ignore the dependence on q Then 2.16

Phần 2: Mô hình hóa đối tượng Mobile Robot 3 bánh

Multiplying both sides of 2.18 with ST we have:

ST MS.v  ST M.S.v  ST V  F  G  d  ST E.u (2.19)

Equation 2.19 becomes:

Thus:

From 2.7 and 2.21, we can describe the state space

Consider for the dynamic modeil, within m, J are mass anf torque around ICR

respectively; and ignore disturbance component in formula 2.15

Phần 2: Mô hình hóa đối tượng Mobile Robot 3 bánh

Meanwhile, we consider that WMR just run on even surface That means its

potential energy is constant and q k component disappeared in equation 2.15

The WMR’s kinetic energy is:

The Lagrangian function is calculated by:

We have:

Phần 2: Mô hình hóa đối tượng Mobile Robot 3 bánh

, and

In the ideal condition, ignoring the friction that means

lost power equal 0 (P=0)

Thus the equation 2.15 becomes:

In order to 2 rear wheels can run with longitudinal velocity

v R ;v L needed torques  R , L

The combined force to pull the WMR moving is ;

considering in two components x and y we have:

Incoordinate system of motion associated with WMR, traction force exerted

by the two wheels on the vehicle tends to make the vehicle rotate about its axis with torque M determined by:

M  L (R L)

2r

Subtituting 2.27, 2.28 and 2.29 into 2.26 we have:

Knematic model 2.30 is writen in the matrix as follow:

M q.q  Eu  AT 

Where:

From that, state space 2.22 can be represented as following:

Because of the non-linear of system, in order to design controller, we

normally separated model into 2 parts, including:

Knematic model:

Phần 2: Mô hình hóa đối tượng Mobile Robot 3 bánh



Where matrixs M,E, V are defined by 2.32

Part 3: Controlling Methods Wheel Mobile Robot.

3.1 Control Overview

As mentioned above, to facilitate control design, we often divide the statemodel of the vehicle into two component models, kinematics and dynamics.Therefore, the construction of control options is also based on these two models

To design the controller for this WMR, we normally use the classic controlscheme, using structure of 2 control loops, that is cascade control strategy In detail,the outer loop is speed control loop (the kinematic control loop); and the inner loop isthe torque control loop (dynamic control loop) The outer loop kinematics controllercalculates the speed value to become the set value for the inner loop kinematicscontroller The block diagram of the system is described as follows:

Beside, the above control system can also be represented in another wayaccording to the EMR method as follows:

3.2 Basic approaches

3.2.1 Directional and translational control.

Let the angle of direction at time t is φ(t) và and desired angle is φref(t), thecontrol angle deviation :

In this case, the variable to be controlled is φ(t) and in order to control φ(t)

to reach the desired value we need to control the bias ( ) to zero

The direction angle of the robot is expressed through the system of

differential equations of the direction angle as follows:

In this case, the control variable is selected as  t , it is determined corresponding

to the angular deviation ( ) through proportional controller as follow:

The equation (3.1) becomes:

This equation is approximated to:

18

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In order to follow a given trajectory or given state; we need translational means control longitudinal vehicle The suitable solution is a control that is

control-proportional to the distance from current position to the reference position:

3.2.2 Basic approaches

In this section, by combining the angular control and the translationalcontrol mentioned in the previous section, we will have some basic approaches

in practice to control the robot to a reference state

In this case, WMR is asked for reach to reference position without requiringthe final direction angle The direction angle of the robot is controlled continuouslyfollowing the direction of reference location, the direction from robot’s currentpositon to reference called r, it is determined:

Vận tốc dài và góc lái của bánh trước được điều khiển như sau:

The longitudinal velocity and steering direction of the front wheel are controlled

However, the above control law will encounter some problems In detail, theabove speed is always positive, so the robot may accidentally pass through thereference point, this will cause the speed to continue increasing (because thedistance to the reference point continue increasing) Otherwise, when passingthrough the reference point, the reference direction angle will suddenlyreversed The solution here is changing the formula of velocity and steeringangle as follows:

a Controlling to a reference pose through the intermediate point.

Unlike the previous part, in this case, in addition to going to the requiredposition, the robot also needs to reach a given angle of direction Basically, this isquite easy to do because the control rules above can be applied again

Ý tưởng của phương pháp này là sử một điểm trung gian x t , y t  được đặt cách

điểm tham chiếu một khoảng r sao cho hướng từ điểm trung gian về phía điểm

tham chiếu trùng với hướng tham chiếu:

The idea of that method is to use an intermediate point  x t , y t  placed a distance r from the reference point

Thuật toán điều khiển robot sẽ được chia làm 2 giai đoạn: Ở giai đoạn 1, robot được điều khiển đi tới điểm trung gian Khi khoảng cách từ vị trí của robot tới điểm

trung gian đủ nhỏ ( ) thì robot sẽ chuyển sang giai đoạn

2, điều khiển robot về điểm tham chiếu

The control algorithm will devided into 2 stages In first stage, the robot is driven to the internediate point When the distance form the robot’s position to thereferene point small enough ( ) the robot will change to second stage, driven robot to reference point

b Từ trạng thái tham chiếu tới quỹ đạo tham chiếu

In here, the trajectory is divided into many small segments and the terminals will be

the references states In other words, a reference orbit includes a series of references states Ti

with i  1, 2,3 n The robot have to move sequentially from T1,T2 ,… until Tn

3.3 Orbit Following Control

3.3.1 Following the trajectory using basic approaches.

Tương tự như phần 3.2.2.c, ta có thể hình dung quỹ đạo tham chiếu như một

vị trí chuyển động tham chiếu Mỗi thời điểm trích mẫu, điểm tham chiếu lại đượcxác định bằng điểm hiện tại của quỹ đạo tham chiếu theo thời gian x ref t , y ref

t    Việc điều khiển tới điểm tham chiếu này được áp dụng luật điều khiển nhưcông thức (3.2)

Similar to section 3.2.2.c, we can consider the reference trajectory as reference movement positions Each sampling times, the reference point is re-determined bycurrent point of reference orbit in time x ref  t , y ref  t    

3.3.2 Analysis feedforward and feedback elements

The approach described in section 3.3.1 is easy to implement, but it has thedisadvantage that it is very susceptible to disturbance in the control loops, so afeedforward disturbance compensation component is required

Firstly, we consider a planar differential system A system is said to be aplanar differential if there is a set of flat outputs and all the states and systeminputs can be rewritten as functions of planar outputs and a finite number of timederivatives its space Consequently, if a system is planar differential then everysystem variable can be computed from flat outputs without integration That is,from the reference trajectory we can calculate the control inputs

We can easily see that the 3 wheeled mobile robot object in question is a planar differential system Indeed:

Thus, formula (3.3) is the formula for calculating reference inputs corresponding to areference trajectory This is also an open-loop controller for the kinematic model toensure the robot follows the set trajectory.

3.3.3 Linearization of Feedback

The idea of this method is to perform transformations to linearize the input to the system making the system between the new input and the output linear Since then, the design of linear controllers for the system becomes possible The feedback linearization design process is as follows:Chọn các đầu ra phẳng thích hợp, số lượngđầu ra phải bằng số lượng đầu vào

- Selecting the appropriate flat outputs, the number of outputs should be equal to the number of inputs

- Differentiate the outputs and check for the occurrence of the inputs, repeating until all inputs appear

- The system of equations is solved for the highest derivative of each input

In the case of the 3-wheel mobile robot, as demonstrated from the previous section,

it is a planar differential system The flat output of the system is x(t) and y(t) Thefirst derivative of the flat inputs is:



  y  v.sin  

The equation has sufficient input vs and The equation is rewriten as follows:

So with this transformation, the input of the system becomes

u1 u2 T   x yT and state model z  x x y yT is described as :

Or z= Az +Bu

Then weed need to design the controller for this new linear system The reference

trajectory is given by , then the reference trajectory can be determined

3.3.4 Development of tracking kinetic trajectory.

The position deviation of the robot in general coordiante system with thereference position is:

The position deviation of the robot in the coordinate system of motionassociated with WMR is:

Derivative of deviation:

Where vref and wref are linear reference velocity:

vfb and wfb are signals that are determined depending on the control law wechoose The error equation is rewritten

3.3.5 Linear Controller

Realize zero-error ( e x = e y = e =0) is an equilibrium when both components respond

From the nonlinear error model in 3.3.4, we linearize around the zero-error point: