The career episode discusses my findings and problem-solving

approaches during the major group project, “Mathematical Modelling for Robot

Localization using Laser Beams” for the academic course of Design

Optimization for Manufacturing (49928). I conducted this project in the first

semester with the group of three while pursuing Masters of Engineering in

Manufacturing in the reputed University of Technology of Sydney.

I undertook this project to develop a mathematical model using MATLAB and

excel solver for the localization of mobile robots using the laser beam. I

performed the project to determine the location of the robot, especially for

the robots in a manufacturing environment using a non- conventional approach to

mathematical optimization. This project was conducted under the guidance and

supervision of my university supervisor, Professor Shoudong Huang.

PART II

1.2

The industrial robots have been used in a manufacturing

environment for quite some time. Currently, the mobile robots have been popular

in the industries where these robots flexibly travel around the shop floor to

perform a wide range of tasks such as material handling, welding, assembly line

activities etc. However, the robot localization and navigation becomes crucial

for such robots to operate in a dynamic environment as that of the

manufacturing. Different approaches are used for the robot localization

including RFID, barcodes, QR codes etc. The laser beams are one of the most

reliable and commonly used localization schemes for the robots. In the project,

I have attempted to perform the mathematical modelling for the localization of

robot with the use of the laser beam. I applied the optimization problem

solving approach to formulate a simple and reliable mathematical model that

could be used for the determination of robot location of a robot equipped with

a laser scanner.

1.3

The main aim of the project was to develop a mathematical

model for localization of robot using laser beams. The secondary objectives of

the project were as follows:

·

To implement the optimization technique for determination of

robot location

· To

develop a simple and reliable localization scheme that can be implemented for

mobile robots

·

To implement accuracy and precision in the developed

mathematical model

·

To infer the scope of the proposed approach for use in the

manufacturing environments

1.4

This project was

undertaken in a group where I was the team leader. The hierarchy of the project

is as illustrated below:

University of

Technology of

Sydney

Design

Optimization

For Manufacturing

Project

Supervisor

Prof.Shoudong Huang

Shail Jani

Mihir Patel

Ajay Trivedi

Fig.

Project Hierarchy

1.5

I was the appointed

team leader for the project, and I performed major responsibilities of the

project which are as listed below:

· To

generate the problem statement, case scenario and methodology for the proposed

approach

·

To generate the laser beam data and major assumptions for

the formulation of the problem

·

To perform the analysis of the problem and develop the

mathematical model

·

To use the MATLAB and excel solver for the optimization of

the model

· To

obtain the solutions and draw inferences on the scope and limitations of the

developed model

PART III

1.6

In the initial phase of the project, I identified the

problem statement for the conceptualization of mathematical model that would be

developed. I developed the problem scenario for the formulation of the

optimization approach in the mathematical model. I decided to develop, test and

implement the localization scheme on a robot that was equipped with a laser

scanner which could obtain six laser beams at certain fixed angles with respect

to the robot heading. The length of the laser beam would show the distance to

the obstacle in the direction. For the problem, I assumed that the environment

consisted only of one line segment. I prepared the following graph for the

conceptualization and the determination of the problem-solving approach.

Fig.

Problem Formulation

I discussed with my team members and decided on the solution

methodology that would use the mathematical modelling approach. I decided first

to generate the laser beam data with a fixed robot location and orientation and

with the assumption that the location is unknown, I will derive the desired

optimization model. I decided to use another robot location to generate laser

beam data and use the developed model for the localization to determine the

accuracy and determine the limitation of the approach. (ANZSCO CODE 233512 TASK

1, 2)

1.7

I started the mathematical modelling with the formulation of

the scope of the problem. I had to determine the robot location depending on

the position of the laser beams. I established the problem scope and

assumptions to optimize the problem. For the assumptions, I fixed the location

of the robot with the fixed coordinates being, (p, q, s) = (2.5, 2.5, 4.5). I

took (x, y, t) to be the location with respective x, y, z-axes and ‘a’ to the

angle of degree. Since the robot could only obtain the beams at six angles, I

took r1, r2,r3, r4, r5, r6 as the ray of laser for respective angles. I assumed

ten variables in the problem which were the location variable: p, q, s, angle

variable: a and the ray of the lasers at different angles. I also set the

constraint for the problem which was zero for this case. (ANZSCO CODE 233512

TASK 1, 2, 3)

1.8

After the problems assumptions and scope of the problem, I

established the objective function with the assumption that the robot location

is fixed. I obtained the objective function which is as follows:

(c1-d1)

2+ (c2-d2) 2+

(c3-d3) 2+ (c4-d4) 2+

(c5-d5) 2+ (c6-d6) 2

(ANZSCO CODE 233512 TASK 1, 2)

For the problem analysis, I calculated the distance for the

different angles mathematically for all the angles as d1, d2, d3, d4, d5 and

d6. I also formulated the condition for the angles which is as shown below:

Angle Condition for Infinite rays

conformation – t

Return value

of

Return value of d as

Angle (if)

Computed Value

Angle (else)

20

of d

-180

or t<-180
20
-144

or t<-144
20
-108

or t<-108
20
-72

or t<-72
20
-36

or t<-36
20
0

or t<0
20
Table. Condition of
Angle
I also prepared the following table for the determination of
v which is the distance obtained in y-axis with respect to the line segment.
if v is (+ve)
Range length ('r)
if
y>3

if Y<=3 Return value of d as if 3>y+v>0

else

20

return value of d as

return value of d as

calculated

20

if v is

(-ve)

Range length (‘r)

if y<0 if y>=0

Return value of d as 20

if

3>y+v>0

Else

return value of

d as

return value of d as 20

calculated

Table. Condition for

Line segment with intersection

1.9

After the formulation of the objective function, assumptions

and the condition, I used the MATLAB solver to subject the model to the

optimization. With the code laser beam.m, I obtained the minimal distances as

follows:

d1=20; (?)

d2=20; (?)

d3=20; (?)

d4=20; (?)

d5=1.518697689

d6=2.121320344

I also obtained the solution using the Excel solver in the

spreadsheet Laser beam length- Data worksheet.xlsx. The solution obtained in

both cases was the same. Hence I determined the approach was correct and

accurate. (ANZSCO CODE 233512 TASK 1, 2)

1.10

I was able to

develop the mathematical model and apply the optimization technique to obtain

the solution. Based on the problem analysis and solution approach, I was able

to find the distance

coordinates for the localization of the robot. I determined

the accuracy of the modelling approach as the solutions using both MATLAB and

excel solver were same. For the problem and the modelling, I inferred the

following limitations and the solution:

·

If v is (+ve) Range length (‘r) then if y>3 then the

value of d is 20 or if y<=3
·
Calculated value has to be taken into consideration, or else
return the value as 20.
·
If v is (-ve) range length ('r) then if y<0 return value
of d as 20; if y>=0 and if 3>y+v>0

·

Return the calculated value, or else d is 20 m.

(ANZSCO

CODE 233512 TASK 1, 2)

1.11

I had few setbacks during the project while using MATLAB for

optimization. I came to the point where the solver has reached the point where

the objective function was less than the objective limit tolerance. I relaxed

the objective limit tolerance with optimooptions to reduce the tolerance value.

I also had a problem with using the Excel solver where I

received the message that the solution might not be entirely correct and there

was a possibility of a better solution to the modelling problem. I adjusted the

convergence setting in the Solver option in the Excel so as to work at a higher

level of precision.

1.12

I used the mathematical modelling and optimization technique

for the localization of the mobile robots. The use of optimization technique

for localization using the laser beams was the creative approach I applied in

the project.

1.13

I performed the project in the team of three under the

supervision of the project supervisor. In the course of the project, I

regularly contacted my supervisor to take his consultations and suggestions. I

followed the rules and regulations of the university throughout the project. I

prepared the reports of my approach, methodology and findings which I submitted

to the supervisor. In the project, I was the team leader, and hence I assumed

major responsibilities. I allocated the team duties and held team discussions

to determine the approach to solving and modelling. I took helpful suggestions

from the university professors which I mediated to my team. In the project, I

used the MATLAB and Excel solver for obtaining the solution through

optimization technique. I followed the MathWorks tutorial during

the project.

PART IV

1.14

I was able to develop a simple and reliable

mathematical model for the localization of mobile robots using the laser beams.

In the project, I used the optimization technique to determine the robot

location. This method was checked for accuracy, and I was able to formulate a

model for the robot localization. I used a simple approach that can be modified

further for different robots for their localization. This approach can be used

for the location of the mobile robots in the dynamic environment of the

industrial and manufacturing floors. The solution methodology I used in the

project can be customized for different robots for obtaining accurate and

precise localization schemes in industri