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Trajectory Tracking, Simulation and Shaping of
Moving Land Vehicle Using MATLAB, INS and
GPS
Othman Maklouf
Aeronautical Engineering Department
University of Tripoli
Tripoli, Libya
omaklouf@yahoo.com
Aya Abulsayen
Communication Department
College of Electronic Technology
Tripoli, Libya
aya.abulsayen@gmail.com
Amira Ghanem
Communication Department
College of Electronic Technology
Tripoli, Libya
ameraghanem664@gmail.com
Abstract— Car-like robot trajectory tracking and Motion
planning became increasingly important, because modern
navigation technologies allowed navigating even in urban
terrains. These car-like robots can be used for military and
civilian domain, they may use in minefields and hazards areas
application. The aim of this paper is to discuss and analysis the
main sources of errors affecting the trajectory tracking of land
vehicle navigation application. It will introduce the main
navigation systems which generally includes the INS and GPS.
A MATLAB with SIMULINK codes are generated through
which it has been possible to perform several types of
simulations.
Keywords—INS; IMU; GPS; car-like robot.
I. INTRODUCTION
Robots became more and more present in our daily life.
They used in the military, aerospace, and many other
applications. Several techniques have been used for the
purpose of the robot trajectory determination. Recently,
global positioning system (GPS) is the most know tracking
system used in such area. It can track the movement of a
vehicle by providing information on exact location. All GPS
positions are based on measuring the distance from the
satellites to the GPS receiver on the earth. In certain areas
the GPS signal is either not present or very inaccurate
because of multipath interference.
Inertial navigation system (INS) is a navigation system
that computes vehicle position, velocity, and attitude
relative to a reference frame by means of dead-reckoning
(DR) techniques[1][2][3][4][5].
II.INERTIAL NAVIGATION SYSTEM (INS)
INS is a self-contained navigation technique in which
measurements provided by the inertial sensors
(accelerometers and gyroscopes) are used to track, the
position and orientation of moving objects relative to a
known starting point without the need for external
references. Due to its highly positioning accuracy specially
for higher technology inertial sensors, INS systems are
generally found in almost all forms of long distance
aircrafts, submarines, guided missiles, and robots
applications [4][5]. The INS computation process is more
complicated as it sounds because any errors in the
accelerometer or gyroscope measurements will lead to
errors in the determined position, velocity and attitude.
Inertial measurement unit (IMU) is the main component of
the INS, typically IMU consists of three accelerometers and
three rate gyros, all of them placed in three axis
Fig. 1 Inertial navigation algorithm.
The accelerometers measuring the linear acceleration,
and the rate gyros measuring the angular velocity. Fig. 1
shows the concept of the inertial navigation algorithm.
Recent advances in the construction of micro
electromechanical systems (MEMS) have made it possible
to manufacture small size and low cost IMUs [5].
III.GLOBAL POSITIONING SYSTEM
GPS is a worldwide radio-navigation system satellite-
based navigation system developed by the U.S. Department
of Defense (DoD) formed from a constellation of 24
satellites and their ground stations. GPS satellite signal has
two codes modulated upon it, the C/A code and the P-code.
[5][6]. The technique used in the GPS is however CDMA
(Code Division Multiple Access). Each satellites signal are
defined with a unique code that allows the user to separate
them from the others. The GPS consists basically of three
segments: the space segment, the control segment, and the
user segment. The space segment consists of 24 satellites
arranged in 6 orbital planes with an inclination angle of 55º
relative to the earth equator. The control segment monitors
the health of the orbiting satellites and uploads navigation
data. The user segment consists of receivers specifically
designed to receive, decode, and process the GPS satellite
signals.
IV.INS SIMULATION USING MATLAB AND
SIMULINK
A MATLAB with SIMULINK code shown in Fig. 2 is
designed to simulate the INS algorithm. A car like robot
model is used to generate a predefined reference trajectory
which will be tracked using INS. Several simulation runs
under different scenarios were performed to evaluate the
2019 19th international conference on Sciences and Techniques of Automatic
control & computer engineering (STA), Sousse, Tunisia, March 24-26, 2019
STA2019_Paper_150_ CRS
978-1-7281-1292-3/19/$31.00 ©2019 IEEE
196
performance of the designed navigation algorithm. The
simulation contains four steps and can be summarized as
follow:
• Reference trajectory generation.
• INS simulation without sensor errors.
• INS simulation with accelerometer bias and gyro
bias taken as a case study to evaluate their effects on the
accuracy of INS estimated trajectory.
Fig. 2. INS Simulation Block diagram using Simulink with
MATLAB.
A. Car Like Robot Modeling And Simulation
The kinematic model of the car like robot is derived by
employing the non-holonomic constraints. These constraints
hold under the assumption that there is no slippage at the
wheel. The constraints related to an automobile are those of
the vehicle’s velocity[7][8]. The general form of the non-
holonomic constraint is
(1)
Where and are the velocities of a wheel within a
given (u, w) coordinate system, and θ is the angle of the
wheel with respect to the x-axis. For small angles of
steering, the car can be modeled as a bicycle, as shown in
Fig. 3.[4]
Fig. 3. General Coordinates for a car-like robot. [7]
Denote (x, y) as being the position of the center of gravity, θ
as the orientation of the vehicle with respect to the x-axis,
and Ø as the steering angle between the front wheel and the
body axis. The dynamic equations can be derived with a
couple added assumptions and can be summarized in “(2).”
(2)
Where FD is the driving force, applied at the rear axle,
along the u-axis, and u1 and u2 are the input signals.
A Simulink model of the car like robot will be built up
using the five equations given in “(2)”, and can be
represented using the Simulink Modeling as given in Fig.4.
Fig. 4. Simulink model of the car like robot
B. Reference Trajectory.
The Simulink model of the car like robot represented in
Fig. 4 is used to generate a predefined reference trajectory
in local level frame. Shaping of this trajectory is chosen
with a starting and ending points as shown in Fig. 5. This
reference trajectory has been considered in all analysis and
comparison studies during the simulation. Fig. 6 shows the
velocities components of the reference trajectory.
Fig. 5. Shaping of the reference trajectory in coordinates of local level
frame.
Fig. 6. Velocities components of the reference trajectory.
197
V. ERROR ANALYSIS
In this section the effect of the sensor bias in the
measurements of the acceleration and angular velocity on
the INS derived trajectory will be carried out. The reference
trajectory created in the previous section is applied here as
an input for the designed INS algorithm. Several simulation
runs have been conducted to study how the sensor bias can
be degrade the accuracy of navigation system. First INS
simulation is conducted without sensor errors. The
simulation results shows that INS derived trajectory closely
matches up reference trajectory as shown in Fig 7. The
differences should be due only to imperfect numerical
integration.
-1500 -1000 -500 0500 1000 1500 2000 2500
-500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
East (m)
North(m)
Ref.
INS
Fig 7: Comparesion between the reference and INS derived trajectory.
The main sources of sensor errors which include
accelerometer bias and gyro bias have been chosen in this
work study. TABLE 1 gives the suggested values of the
selected errors applied in this simulation.
TABLE 1; The Values of the Errors.
A. Accelerometer Bias Effect
To evaluate the effect of the accelerometer bias on the
derived INS trajectory, three values have been suggested.
The suggested values are 0.0005μg, 0.005μg and 0.05μg.
First, 0.0005μg (High grade IMU) is set into the program.
Fig. 8 shows a comparison between the reference trajectory
and the INS derived trajectory. It is clear that there is a very
small difference between the two trajectories, this is more
obviously shown in Fig. 9, as error in horizontal and vertical
position. It is clear that this difference is due to the improper
measurement of the accelerometer which in turn, results in
accurate computation in position.
-1500 -1000 -500 0500 1000 1500 2000 2500
-500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
East (m)
North(m)
Ref. Traj.
INS Traj.
Fig 8: Comparison between reference and INS derived trajectories given
Acc bias = 0.0005 μg .
050 100 150 200 250 300
-50
0
50
100
Time
Error(m)
Error in East Direction
050 100 150 200 250 300
-50
0
50
Time
Error(m)
Error in North Direct ion
Fig 9: Error in the horizontal and vertical position given Acc . bias =
0.0005 μg .
Second, 0.005μg (Medium grade IMU) is set into the
program. Fig 10 shows a comparison between the reference
trajectory and the INS derived trajectory. Due to an
increasing in accelerometer bias it would be expect that, the
difference between the two trajectories as well as the error
in both horizontal and vertical positions, will increase. This
is clear in Figs (10 and 11), where the values are listed in
these figures
Fig 10. Comparesion between the reference and INS derived trajectory
given Acc bias= 0.005 μg .
The Errors
The values
Minimum
Case Ι
Intermediate
Case ΙΙ
Maximum
Case ΙΙΙ
Accele.
Bias(μg)
0.0005
0.005
0.05
Gyro
bias(rad/hr) 0.00015 0.0015 0.015
198
050 100 150 200 250 300
-500
0
500
1000
Time
Error(m)
Error in East Direction
050 100 150 200 250 300
-500
0
500
1000
Time
Error(m)
Error in North Direct ion
Fig 11. Error in the horizontal and vertical position given Acc. bias =
0.005 μg.
Finally, when the bias increases to 0.05μg (Low grade
IMU) in this case the low cost MEMES IMU is considered,
the main features of this type of IMU is its highly output
measurements error. Fig 12 shows the comparison between
INS derived trajectory with highly input accelerometer bias
and the reference trajectory. It is clear that a highly
deviation is obtained compared with the reference one. The
error in both horizontal and vertical positions, is further
increased as illustrated in Fig 13.
-5000 -4000 -3000 -2000 -1000 01000 2000 3000 4000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
East (m)
North(m)
Ref Traj
INS Traj
Fig 12 Comparison between the reference and INS derived trajectories
given Acc bias = 0.05μg.
050 100 150 200 250 300 350
-5000
0
5000
10000
Time
Error(m)
Error in East Direc tion
050 100 150 200 250 300 350
-5000
0
5000
10000
Time
Error(m)
Error in North Direction
Fig 13. Error in the horizontal and vertical position given Acc. bias =
0.05 μg..
B. The Gyro Bias Effect
This section will introduce the effect of the gyro bias on
the accuracy of using INS for tracking robot trajectory.
Different values of gyro bias would be set to the program
for several simulation runs. The selected values of gyro bias
are 0.00015 rad/hr, 0.0015 rad/hr and 0.015 rad/hr which
represented the High, medium, and low grade IMU
respectively. Fig. 14 shows a comparison between the
reference trajectory and the derived INS trajectory as a
result of a 0.00015 rad/hr gyro bias. The resulted drift will
affect the accuracy of the transformation between the body
and navigation frames. This in turn will result in small
deviation between the two trajectories. Fig. 15 shows the
errors in the horizontal and vertical position.
-1500 -1000 -500 0500 1000 1500 2000 2500
-500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
East(m )
North(m)
Ref Traj
INS Tr aj
Fig 14. Comparison between the reference and INS derived
trajectories given gyro bias = 0.00015 rad/h..
010 20 30 40 50 60 70 80
0
5
10
15
Time(ms)
Error(m)
Error in east direc tion
010 20 30 40 50 60 70 80
-60
-40
-20
0
Run time in ms
Error(m)
Error in north direct ion
Fig 15. Error in horizontal and vertical position given gyro bias =
0.00015 rad/h.
Increasing gyro bias to 0.015 rad/hr the deviation
between the two trajectories will be increased and this is
illustrated in Fig. 16 , as a result for this increasing between
the two trajectories the error in the horizontal and vertical
position, are increased. as shown in Fig 17. When the bias in
the Gyro is further increased to 0.015 rad/h the INS derived
trajectory became highly deviated from the reference
trajectory as shown in Fig 18, this will be more obviously in
the horizontal and vertical position as illustrated in Fig 19.
-2000 -1500 -1000 -500 0500 1000 1500 2000 2500
-500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
East(m )
North(m)
Ref. Traj.
INS Tr aj.
Fig 16. Comparison between the reference and INS derived
trajectories given gyro bias = 0.0015 rad/hr
010 20 30 40 50 60 70 80
0
50
100
Run time in ms
Error(m)
Error in east direc tion
010 20 30 40 50 60 70 80
-600
-400
-200
0
Run time in ms
Error(m)
Error in north direct ion
Fig. 17. Error in the horizontal and vertical position.
199
-1500 -1000 -500 0500 1000 1500 2000 2500
-1000
0
1000
2000
3000
4000
5000
East(m )
North(m)
Ref. Traj.
INS Tr aj.
Fig 18 Comparison between the reference and INS derived
trajectories given gyro bias = 0.015 rad/hr.
010 20 30 40 50 60 70 80
0
1000
2000
3000
Run time in ms
Error(m)
Error in East Direction
010 20 30 40 50 60 70 80
-1500
-1000
-500
0
Run time in ms
Error(m)
Error in North Direct ion
Fig 19. Error in the horizontal and vertical position.
Reveals to the simulation results and analysis of the
above given figures. It is clear that the error is increased
when the sensor errors increased from the minimum to the
maximum.
VI. GPS SIMULATION
In this section GPS simulation will be considered in
order to investigate the performance of GPS as a very
common navigation system used for trajectory tracking. The
effect of the different error sources that degraded the
accuracy of GPS accuracy will also be introduced. The GPS
receiver is simulated using Simulink with MATLAB. The
structure of the navigation algorithm based on GPS is
presented in Fig 20.
Fig. 20. Simulink block diagram of GPS receiver
VII. GPS SIMULATION RESULTS
The same reference trajectory adopted in simulating INS
is also suggested for simulating GPS. Fig. 21 Shows a
comparison between the reference trajectory and GPS
estimated trajectory. Obviously, the two trajectories are very
similar. Closely looked to this figure it can be clearly
notified that the GPS derived trajectory suffering from
discontinuity due to the low data rate associated with the
satellite-based navigation systems.
There are several main factors affecting the GPS
accuracy, including receivers devices, the measurements
accuracy of satellite locations and method of distribution
them in the space. The GPS estimated trajectory resulting
due to the all sources of errors are illustrated in the Fig 22.
Reveals to Fig. 22, it is clear that the GPS estimated
trajectory hovering around the reference trajectory and did
not match the reference trajectory exactly. Fig. 23 clearly
indicated that, GPS provide less accuracy in vertical
positioning than for horizontal positioning, the vertical
position error reached Approximately 50 m, while the
horizontal position error reached error Approximately 45 m
in simulation results.
-1500 -1000 -500 0500 1000 1500 2000 2500
-500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
East (m)
North(m)
GPS Traj.
Ref. Traj.
Fig. 21. Reference and GPS trajectories.
-1500 -1000 -500 0500 1000 1500 2000 2500
-500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
East (m)
North(m)
Ref. Traj.
GPS Traj.
Fig. 22. Reference and GPS trajectories with effect of all error.
1920 1940 1960 1980 2000 2020 2040 2060
3880
3900
3920
3940
3960
3980
4000
4020
4040
4060
North(m)
GPS Traj.
Ref. Traj.
1960 1980 2000 2020 2040 2060 2080 2100 2120
3600
3700
3800
3900
4000
4100
4200
East(m)
North(m)
Ref. Traj.
GPS Traj.
200
In recent years, GPS devices have become so accurate
that the error rates dropped to view meters, the error can be
reduced by development of programs and receiver hardware.
0 2 4 6 8 10 12 14
x 10
4
-50
0
50
Time
Error(m)
GPS Error in Horiz ontal Position
0 2 4 6 8 10 12 14
x 10
4
-50
0
50
100
Time
Error(m)
GPS Error In Verti cal Position
Fig 23. Error in the horizontal and vertical distance.
VIII. CONCLUSION
INS is an autonomous navigation system because it
requires no information outside the system. Due to its high
output data rate INS suitable to use for trajectory
determination of high speed moving objects. Using low
grade IMU with low accuracy inertial sensors may results
in navigation solution with unbounded error that will
increase over time. The system therefore seems to be drift
with time. In case of good environmental conditions GPS
can provide an adequate positioning to the user, which will
not always occur as the signal from the satellites can be
blocked by different obstacles and high buildings. Error
analysis leads to the fact that for enhancing the performance
of the INS it becomes necessary to integrate the INS with
GPS in order to enhance the performance of the whole
system.
IX. REFERENCES
[1]. Titterton D.H. and Weston, J.L. “Strapdown inertial navigation
technology;” Peter Peregrinus Ltd., London, UK, 1997.
[2]. Bekir, E. “Introduction to Modern Navigation Systems”, World
Scientific Publishing Co. Pte. Ltd., USA, 2007.
[3]. Bose, P. “Modern Inertial Sensors and Systems”, Prentice-Hall, ISBN-
13: 978-8120333536, India, 2008.
[4]. Farrell, J. “Aided Navigation: GPS with High Rate Sensors”, McGraw-
Hill, ISBN: 978-0-07-149329-1, USA, 2008.
[5]. Mohander S. Grewal, Lawrance R. Weill, Angus P. Anderws “Global
positioning system inertial navigation system and integration”
Copyright © 2nd Edition 2007, A John Wiley &Sons,
[6]. A. Noureldin et al., Fundamentals of Inertial Navigation, Satellite-
based Positioning and their Integration, DOI: 10.1007/978-3-642-
30466-8_1,Springer-Verlag Berlin Heidelberg 2013.
[7]. Eric N Moret., “Dynamic Modeling and Control of a Car-Like Robot”
M.Sc. Thesis, Virginia Polytechnic Institute and State University”,
(2003).
[8]. Lukas Krammer., “Motion Planning for Car-like Robots” M.Sc. Thesis,
Technical University Wien”, Wien, 23.07.2010.
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