A Solution to Partial Observability in Extended Kalman Filter Mobile Robot Navigation

TELKOMNIKA, Vol.16, No.1, February 2018, pp. 134~141
ISSN: 1693-6930, accredited A by DIKTI, Decree No: 58/DIKTI/Kep/2013
DOI: 10.12928/TELKOMNIKA.v16i1.9025  134
Received October 17, 2017; Revised December 25, 2017; Accepted January 26, 2018
A Solution to Partial Observability in Extended Kalman
Filter Mobile Robot Navigation
Hamzah Ahmad*, Nur Aqilah Othman, Mohd Syakirin Ramli
Faculty of Electrical & Electronics Engineering,Universiti Malaysia Pahang,Pekan Campus,26600 Pekan,
Pahang Darul Makmur, 09-4246024, Malaysia
*Corresponding author, e-mail: hamzah@ump.edu.my
Abstract
Partial observability in EKF based mobile robot navigation is investigated in this paper to find a
solution that can preventerroneous estimation.By only considering certain landmarks in an environment,
the computational costin mobile robotcan be reduced but with an increase of uncertainties to the system.
This is known as suboptimal condition ofthe system. Fuzzy Logic technique is proposed to ensure that the
estimation achieved desired performance even though some of the landmarks were excluded for
references. The Fuzzy Logic is applied to the measurement innovation of Kalman Filter to correct the
positions of both mobile robot and any observed landmarks during observations. The simulation results
shown that the proposed method is capable to secure reliable estimation results even a number of
landmarks being excluded from Kalman Filter update process in both Gaussian and non-Gaussian noise
conditions.
Keywords:partial observability,mobile robot,navigation,fuzzy logic,estimation
Copyright © 2018 Universitas Ahmad Dahlan. All rights reserved.
1. Introduction
Mobile robot navigation is one of the fascinating research which supports a truly
autonomous mobile robot behavior. It is one of the aspects that really means in nowadays
applications especially when a robot needs to perform a task to replace human operation. Robot
has been known to functionally works in explorations, underwater navigation, mining and
military by performing different kind of tasks. Depends on the applications, the mobile robot may
require several conditions to be satisfied about its design and specifications. In navigation, one
of its challenges is known as Simultaneous Localization and Mapping problem where mobile
robot has to identify its location with reference to any of landmarks being observed and at the
same time building a map based on its observations. This problem addressed several issues to
be solved such as operating in dynamic environment, computational cost, uncertainties and
data association [1]-[2].
Three main approaches are available in SLAM which are the mathematical modeling,
the behavioral model and the probabilistic techniques. Among these three popular methods,
probabilistics has captured most of the researcher interest where a lot of techniques have been
proposed based on probabilistics [1]. EKF as one of the celebrated probabilistics approach in
SLAM made a significant achievement as it is the most successful method to be applied online.
In EKF based navigation, there a some issues to be solved and one of it is the computational
cost. The computational cost increased if more landmarks are observed and processed during
mobile robot movements. However, this problem do not influence the estimation performance
but instead will require higher cost to build the robot.
This paper analyzes the aspect of partial observability as one of the problem in EKF
based SLAM [3]-[4]. A sensor such as infrared sensor, sonar sensor or laser range finder
attached to the mobile robot can obtained information about the environment regarding its
relative measurements i.e relative angle and distance. The information generally contains
various data. Unfortunately, these data can be erroneous due to unexpected conditions. This
may due to the sensor fault or system limitations that a mobile robot is equipped with. Owing to
these circumstances, mobile robot can accidentally lost its way and cannot perform localization
and mapping accurately. Interestingly, this problem sometimes is very difficult to be recognized
as mobile robot can intermittently experience such problem without prior notice. One way to

TELKOMNIKA ISSN: 1693-6930 
A Solution to Partial Observability in Extended Kalman Filter Mobile Robot… (Hamzah Ahmad)
135
demonstrate this condition is by considering a mobile robot which is able to recognize several
landmarks at a time compared to other different observation time. Even though this problem is
critical, still under investigation and further improvement, the situation has motivates a case to
determine the system performance when only several identified landmarks are referred during
mobile robot observations. By doing this, it is expected that the system can reduce its
computational cost and at the same time guarantees good estimation results [5], [6].
In SLAM, the partial observability problem is categorized and analyzed into three main
aspects; unstable partially observable, partially observable and observable SLAM problem[7]. It
is claimed that the partially observability has makes the mobile robot depends heavily on its
initial conditions [8], [9]. Due to partial observability, the performance of observation can
become worse as the updated state covariance of the system cannot guaranteed to be
converge even if the mobile robot has done multiple observations on its surroundings. EKF-
SLAM has been one of the influential technique and been analyzed thoroughly in different
perspectives [10]-[13]. The technique is also been widely applied in real-life applications for
navigation purposes.
This paper is organized as follow. Section 2 describes the proposed technique of EKF
based SLAM and Fuzzy Logic technique. Section 3 demonstrates the simulation results and
analysis of the proposed method amd finally section 4 concludes the paper.
2. Extended Kalman Filter and Fuzzy Logic Approaches in SLAM
2.1. Extended Kalman Filter in SLAM
SLAM normally consists of two general processes which are the kinematics or process
step and the measurement step. The process stage defines the mathematical analysis of the
model focusing on the kinematics of the mobile robot with consideration of the landmarks
location being recognized. SLAM uses the mobile robot velocity and angular velocity to
determine its location. The measurement step on the other hand calculates the relative angle
and distance between mobile robot and any observed landmarks. Both of process and
measurement stages utilizes the mobile robot sensors to determine the positions and map
building. The process step is as follows.
(1)
where is an augmented state consisting of the mobile robot heading angle and x,y positions
with landmarks x,y locations. A and B are the transition matrix and control matrix respectively. U
is the control input of mobile robot velocity and angular velocity, while ω is the process noise of
the mobile robot. The measurement process takes the following mathematical equation.
(2)
is the measurement matrix containing information about relative distance and relative
angle between mobile robot to each landmark. H is the measurement matrix and ν is the
measurement noise occurred during mobile robot observations. EKF works based on the above
two SLAM model to estimate the mobile robot and landmarks positions. There are two main
stages to be computed which are predict and update stages. Table 1 shows the EKF steps to
perform the estimation. Tables and Figures are presented center, as shown below and cited in
the manuscript.
Table 1. Kalman Filter Algorithm
Kalman Filter
Prediction State model ̂ ̂
State covariance
Update State model ̂ ̂ ̂
State covariance
Kalman Gain

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TELKOMNIKA Vol. 16, No. 1, February 2018 : 134 – 141
136
As shown in Table 1 above, ̂ is the estimated augmented state. is the state
covariance while the sign of “+”, “-“ defines the priori and posteriori estimation at time k. Kalman
Gain plays important role in Kalman Filter where it leads to better estimation. Remarks that, A,
B, H are the linearized matrices as this study considers EKF as it filter for estimation. The
process noise and measurement noise covariances are presented by Q and R and these two
noises are uncorrelated throughout the analysis.
From Table 1, the updated state model consists of element ̂ which
determines the measurement innovation of the system. If the value of this measurement
innovation is too big, then Kalman Gain attempts to reduce it so that the updated state model
remains holding a very small error during mobile robot observations. From above table 1, this is
also means that the measurement innovation must be kept very small at all time. In normal EKF,
this cannot be guarantees at all time as sometimes due to environment conditions, there is a
possibility of sensor fault which in turns exhibits higher error to the readings. As one of the
family of Bayes Filter, EKF only depends on its latest update. Hence, the estimation may
becomes erroneous if no further action being made to the system. Having said that, Fuzzy Logic
is being proposed to control the amount of information gathered by the sensor. If the
measurement innovation accidentally produced higher amount of noises, then the Fuzzy Logic
reduced it for better estimation results.
2.2. Fuzzy Logic Technique
As mentioned on previous subsection, Fuzzy Logic is proposed to reduced the noises
or error developed by the measurement innovation computations. Figure 1 presents the design
of Fuzzy Logic to the proposed system. To design the Fuzzy Logic, there are three main phases
to be considered i.e Fuzzification, Rule Evaluation and Defuzzification[14]. Fuzzification
describes the conditions that needs to be assessed which are in this case the measured relative
angle and measured relative distances between mobile robot and landmarks. It is essential to
have prior knowledge on the system performance before beginning to design Fuzzy Logic. The
input and output of Fuzzy Logic is same as the main objectives of Fuzzy Logic is to reduce the
measurement innovation. By knowing these situation, then the fuzzy sets and membership
function are selected. Mamdani technique is chosen with three fuzzy sets each for both input
and output.
The rule evaluation determines the best configuration of the designed rule to the whole
system. It computes the fuzziness of the system based on the rules. Defuzzification is then
evaluates using Center of Gravity to obtain the reduced measurement innovation to be
calculated with Kalman Gain in achieving small error to the updated system. The fuzzy sets and
triangular memberships are included in Figure 2 for references. Triangular membership was
selected as it compute faster and regularly used for design compared to other available
membership types. The fuzzy sets are defined based on preliminary results of simulations.
Figure 1. Modified measurement innovation calculation with the presence of fuzzy logic
controller to the proposed system

137
(a) (b)
(c) (d)
Figure 2. Input to the fuzzy logic (a) Measured relative angle (b) Measured relative distance.
Output of the fuzzy logic (c) Calculated relative angle (d) Calculated relative distance
3. Results and Analysis
This section explains the results of research and at the same time is given the
comprehensive discussion about our proposed technique. The simulation settings are described
in Table 2 for reference. The parameters are chosen based on our previous experimental
analysis. Global coordinate system is referred as the reference for the system to operate. The
simulation considers two different environmental noises which are Gaussian and non-Gaussian
noise to assess the EKF performance in both situations where suboptimal conditions take place
during observation. The environment is also assume to be planar and mobile robot received all
the observation during its measurements.
3.1. EKF based SLAM with suboptimal conditions
This subsection explains the effect of having an EKF as estimator in navigation
especially in SLAM problem. It is undeniable, a normal EKF do the estimation nicely in
Gaussian noise conditions but with a full computational cost of analysis. By doing some
modifications on the state covariance update as shown in Table 1, the computational cost can
be reduced hence produces faster computations. Figures 3 presents the performance of normal
EKF and EKF with partial observability respectively. Mobile robot movements estimation as well
as landmarks estimation determines that the EKF with partial observability conditions has better
estimation. Remark that in partial observability, only certain landmarks are considered. Only
three number of the landmarks are treated for reference while other landmarks are excluded
0.060.15
1
0
0.18 0.23 0.38
Degreeofmembership
0.040.05
1
0
0 0.02 0.05
Degreeofmembership
7-5
1
0
12 35
De
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ee
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m
be
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ip
-10
-0.05 -0.04
1
0
0.05 0.08
Degreeofmembership
0.01-0.08

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138
from measurement innovation calculations. The mobile robot also holds very high confidence
about its location compared to the normal conditions as depicted in Figure 3(b).
An interesting results happen when the environment conditions are changed to include
non-Gaussian noise. If the mobile robot operates in non-Gaussian noise, then the results
becomes not as promising as the previous case and this is shown in Figure 4. The estimation
exhibits erroneous results especially for the landmarks estimation even when the mobile robot
has higher confidence about its location(refer to Figure 4(b)). Notice that, the state covariance
for suboptimal conditions are also producing lower uncertainties that eventually makes EKF
becomes optimistics about its estimation [5]. For this case, minor modifications on the approach
especially on measurement innovations must be made to guarantee better results.
Table 2. Simulation Parameters
Variable Values
Process Noise 10e-7
Measurement Noise
Angular noise=0.001
Distance noise=0.001
Random noise
Process noise
(minimum -0.002, maximum 0.001)
Measurement noise
(minimum -0.15,
maximum 0,01)
Landmarks locations
[ -20; 60;
60; 120;
-30; 180;
-200; 230;
-220; 160;
200; 160;
-190; -40;
170; -50;
20; 240;
-10; -200;
-80; -60;
10; 130
3 ; 4];
Initial state covariance
Mobile robot=0.001
Landmarks=100
Mobile Robot initial position (x,y, theta)=(0,0,0)
3.2. EKF based SLAM in Suboptimal Condition with Fuzzy Logic
As shown in previous subsection, EKF becomes erroneous especially when non-
Gaussian noise exist during mobile robot observations. In order to overcome such inconsistency
in estimation, Fuzzy Logic is proposed to correct the estimation. The Fuzzy Logic is then
attached after the measurement innovation step to ensure that the updated state estimation
becomes better and producing less error.
Preliminary results in Figure 5 have shown that by applying Fuzzy Logic, the landmarks
estimation has become way better than the one being predicted by the EKF with suboptimal
condition especially about its landmarks estimations. Additional simulations with different mobile
robot movements are also being carried to obtain the consistency of the proposed technique.
The results in figure 6 depicts that the proposed technique guarantees a good estimation can be
achieved for different mobile robot movements compared to the normal EKF estimations. It is
worth to mention that, if the mobile robot operates in different environment, Fuzzy Logic must be
modified and re-design to suit with the new environment. Otherwise, the results can be not
promising as expected. Based on the observations and literatures [9], the angle measurement
plays an important role to guarantee good estimation.

139
(a) (b)
Figure 3 (a) Comparison between Normal EKF(red dash line) and EKF with suboptimal
conditions(blue dash line).(b) State covariance performances for mobile robot and landmarks
(a) (b)
Figure 4 (a) Comparison between Normal EKF(red dash line) and EKF with suboptimal
conditions(blue dash line) in non-Gaussian noise conditions. (b) State covariance performances
for mobile robot and landmarks
(a) (b)
Figure 5. Comparison between suboptimal EKF(blue line) and Fuzzy EKF(black line) in non-
Gaussian noise conditions for both (a) mobile robot and landmarks estimation and (b) Updated
state covariance performances

 ISSN: 1693-6930
140
(a)
(b)
Figure 6. The performance of the proposed fuzzy EKF in suboptimal condition(blue) in various
mobile robot (a), (b) movements compared to normal EKF performance
4. Conclusion
This paper has presented an analysis of Fuzzy Logic in Extended Kalman Filter
navigation to prevent error in partial observability problem. Two different noise conditions are
being analyzed to determine any inconsistency of the proposed method. Based on the results,
the Fuzzy Logic technique can further improved the estimation results especially for the
landmarks estimation for both gaussian and non-gaussian noise environments even with
different mobile robot movements. Note that, as the proposed technique is only designed for a
static landmarks conditions, the system performance can varies if more dynamic environment is
taken into account for observation and require modifications on Fuzzy Logic design.
Acknowledgements
The author would like to thank Ministry of Higher Education and Universiti Malaysia
Pahang for their continuous supports in this research through RDU160145 and RDU160379.
Special thanks to Namerikawa Laboratory in Keio University for their participation and advises.
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