The leader for the project, and I performed

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.



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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.




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




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







For Manufacturing






Prof.Shoudong Huang






Shail Jani

Mihir Patel


Ajay Trivedi






Project Hierarchy




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


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






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.
























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)



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)




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:


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 


Return value of d as


Angle (if)

Computed  Value

Angle (else)





of d












or t<-180 20             -144=36
or t<-144 20             -108=72
or t<-108 20             -72=108
or t<-72 20             -36=144
or t<-36 20             0=180
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




return value of d as


return value of d as










if v is




Range length (‘r)







if y<0 if y>=0







Return value of d as 20





return  value  of 
d  as


return value of d as 20









Table. Condition for
Line segment with intersection




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

d1=20; (?)

d2=20; (?)

d3=20; (?)

d4=20; (?)




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)




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.


CODE 233512 TASK 1, 2)




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.




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.




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.





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


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