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Hindawi Journal of Robotics Volume 2022, Article ID 1999082, 13 pages https://doi.org/10.1155/2022/1999082 Research Article Comparative Study of Two Localization Approaches for Mobile Robots in an Indoor Environment Eman Alhamdi and Ramdane Hedjar Department of Computer Engineering at the College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia Correspondence should be addressed to Eman Alhamdi; 437202950@student.ksu.edu.sa Received 9 March 2022; Accepted 4 May 2022; Published 2 June 2022 Academic Editor: L. Fortuna Copyright © 2022 Eman Alhamdi and Ramdane Hedjar. €is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In the last years, mobile robot localization has been developed signi‡cantly due to the need for accurate solutions to determine the position and orientation of the wheeled mobile robot (WMR) in a given environment. Many diŽerent sensors have been used to solve the problem. For instance, ultrasonic sensors, laser, or infrared sensors are also used to determine the pose of the WMR. However, sensors are sensitive to noise measurements and disturbances, which can distort the acquired information. For this reason, adequate algorithms should be used to reduce these uncertainties and determine the optimal pose of the WMR. In this research work, we focus on the comparative study of the most used algorithms, using landmarks as sensors, which are the extended Kalman ‡lter and particle ‡lter. Further, for an eŽective comparison, the simulation results were conducted and analyzed using diŽerent performance criteria. €e simulations results showed better estimation performance achieved by the particle ‡lter being compared to the extended Kalman ‡lter when the sensors are subject to non-Gaussian noises. without any collision with obstacles to the target. Locali- 1. Introduction zation in mobile robotics is the ability of the robot to de- €e mobile robot is a device that can move from one place to termine its position and orientation concerning the known another until it reaches the target autonomously inside an reference frame in the environment. €e main way to ‡gure unknown or partially known environment. In recent years, out the location of the WMR is by taking data from diŽerent the applications of mobile robotics have increased, and the sensors to estimate the next position, knowing the current robotics ‡eld has matured quickly with more and more position. robots entering practical new services. Indeed, we can ‡nd €e pose of the mobile robot is described by two co- the mobile robot in diŽerent applications performing dif- ordinates (x, y) and the orientation (θ), which is given by ‡cult tasks on the military side as well as on the civil side. €erefore, to carry out any task perfectly, the mobile Pose y . (1) robot must be equipped with wheels and sensors to move and interact with the environment. €erefore, the funda- mental components of the mobile robot are the sensors to be utilized for localization and navigation. €erefore, to move €ere are two types of navigation: indoor navigation and easily and safely, the robot should determine its position and outdoor navigation. We can ‡nd many techniques in the orientation precisely. literature used to determine the pose of the WMR. €e To this end, the mobile robot should perform the fol- classical one is the dead-reckoning method or simply lowing tasks: localization, path planning, and navigation [1]. odometry. €is technique, ‡rst used in marine vessels, in- tegrates the velocity and the time to determine the change of €e importance of safe navigation is shown by its movement 2 Journal of Robotics the particle ﬁlter by using two methods: local localization by the position. But the obtained position is inaccurate due to the accumulation of encoders errors. For outdoor naviga- dead rocking and global localization by using landmarks and vision sensor. In paper [10] the authors proposed a new tion, the global position system (GPS) is the approach used to determine the position and orientation of the vehicle. In method in indoor wireless local area networks using an indoor navigation, encoders, proximity sensors (infrared, adaptive local search particle ﬁlter because of the diﬃculty to sonar, ultrasound, and laser), and landmarks are utilized to predict the state due to interference and reﬂection. *e estimate the position and the orientation of the mobile authors in [11] have investigated the localization problem robot. using the two diﬀerent particle ﬁltering strategies: (SIR) *ese sensors are fundamental inputs to the robot; sampling importance resampling strategy and (APF) aux- however, two main problems can be faced in using these iliary variable particle ﬁlter in the RoboCup soccer envi- sensors: noise and aliasing. *ese uncertainties can distort ronment. Because of the need for an accurate estimation, which is reliable indoor noisy environments, it is commonly the estimation of the position and orientation of the robot. To remedy this problem, many methods have been devel- provided by the particle ﬁlter. In [12] the authors have proposed a novel hybrid particle/FIR ﬁltering algorithm to oped in the literature. Among them, we can ﬁnd the ex- tended Kalman Filter that minimizes the eﬀect of noise and improve the reliability of particle ﬁlters. uncertainties during the estimation process and the particle ﬁlter. 3. Kinematic Model of the Wheeled Mobile Robot 2. Related Work To determine the equations of the estimator, the kinematic Noura et al. [2] have presented the simultaneous localization nonlinear model of the wheeled mobile robot is used, and it and mapping (SLAM) method with the extended Kalman is given as Filter for a WMR in the case of a small-scale environment. x _ � v cos(θ), *e idea of SLAM is related to observing the full environ- ⎧ ⎪ ment in the closed-loop; they have noted that the SLAM is y � v sin(θ), (2) eﬃcient when the extended Kalman ﬁlter is applied. *e ⎩ θ � ω, result has shown that this technique is more consistent for a high number of iterations of the algorithm, but for a large where (x, y) is the position of the WMR and θ is the ori- scale and complicated environment, it will be slower. entation, v is the linear speed, and ω is the steering angle. Zhou et al. [3] presented a new approach to automati- *e dead-reckoning approach uses the previous position cally detect a dynamic object in order to improve vision- and orientation at the time t � k in order to calculate the based localization and mapping in crowded environments. future pose at t � k + 1. Using the ﬁrst-order Taylor ap- *ey have combined accessible simulation data to train a proximation of the future position, the previous equation CNN that generalizes well the unseen environments. *e becomes [4] dynamic objects can be detected, even if no segmentation is x(k + 1) � x(k) + v(k) T cos(θ(k)), provided. ⎡ ⎢ Faisal et al. [4] have investigated the use of the fuzzy logic ⎢ ⎢ y(k + 1) � y(k) + v(k) T sin(θ(k)), (3) ⎢ s control approach to solving the navigation problem of a ⎢ WMR in an unknown and dynamic environment. *e θ(k + 1) � θ(k) + ω(k) T , online navigation technique is based on fuzzy logic approach for moving the mobile robot safely in an unknown dynamic where T � t − t is the sampling time. It is preferable to s k+1 k environment, while the localization is performed using the use the distance travelled by the mobile robot during T and dead-reckoning algorithm in order to estimate the actual this can be performed using pulses measured from the conﬁguration of the mobile in the scene. *e authors in [5] encoders on each DC motor of the WMR. have combined EKF and PF for better calibration of the kinematic parameters of the robot. *e work in [6] proposed an extended Kalman ﬁlter as a 3.1. Sensors: Landmarks. As human being needs the ﬁve new method for localization by combining it with an in- senses in life; also, the robot needs sensors to detect the frared system to reduce further the estimation errors. *e objects in the surrounding environment and interact with authors in [7] used the extended Kalman ﬁlter to solve the them. *ere are diﬀerent types of sensors that are used by localization problem when the measurement data are mobile robots. *is work deals with localization problems, available at a speciﬁc time. and the sensor used for this task is a landmark sensor where Because of the poor localization of the laser sensor that is the location is known in the environment (X , Y ). *is L L used in the indoor environment, the authors in [8] solved location may be artiﬁcial or physical. It is used to determine th this problem by using map-based particle ﬁlter localization the position and orientation of the robot. Assume the i in the indoor environment. *e main drawback of the landmark at the known position (x , y ) at the time k, the L,I L,i particle ﬁlter to determine accurate localization that it needs distance, and the bearing from the position of the mobile high time computation. So, in [9] the authors suggested a robot to the landmark is given by the following equation low computational mobile robot localization algorithm with [13]: Journal of Robotics 3 �������������������� 2 2 violated. So, the ﬁrst-order linearization has to be used for x − x + y − y k l,i k l,i ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ the dynamic model and the measurement model. Further, ⎜ ⎟ ⎝ ⎠ h � . (4) some sensors may have a strong nonlinear measurement a tan 2 y − y , x − x − θ l,i k l,i k k model. *erefore, the linearization of the nonlinear equations increases the estimation errors. Consequently, when the above assumptions are not suitable in practice, 3.2. Extended Kalman Filter. *e extended Kalman ﬁlter the particle ﬁlter is an appropriate algorithm to be used for (EKF) is the nonlinear version of the Kalman ﬁlter which the localization of a WMR. *e particle ﬁlter is a nonlinear linearizes the nonlinear model around the estimate of the ﬁlter that is one of the Sequential Monte Carlo algorithms current mean and covariance. It is used to fuse the inner to set the dynamic movement of the robot. It is a set of position estimation and outer measurement and to locate the samples called particles that approximate the probability mobile robot with four-wheeled wheel encoders. *e EKF distributions. A particle ﬁlter gives a perfect way to estimate has been a popular choice to deal with mobile robot the state position of the robot [2]. Each particle assigned has problems with sequential localization and mapping. *e a weight that is proportional to probability. Normalize the eﬃciency of EKF relies on the minimization of covariance weights of samples by the principle of importance sampling matrices for the system and measurement noise. Inaccurate [14]. *us, the particle ﬁlter is a nonlinear ﬁlter for a state knowledge of this statistical data can cause inaccuracy in the estimate that represents the noise model in a sample. It is an estimation performance [2]. *e EKF was developed in 1970 estimation algorithm based on the Bayesian algorithm to as the initial approach to estimate the state of a nonlinear approximate the probability density function (pdf) by system [14]. It is an approximation method for nonlinear tracking the variable over time with non-Gaussian and state estimation based on Taylor series expansion to over- multimodel pdf. *is technique is based on the construc- come the nonlinearity system and measurement models. tion of a sample-based representation of the pdf. *ere are EKF linearizes the nonlinear functions around the estimated multiple particles of variables that will be used, each part is trajectory [15]. It is well known that the EKF is prone to associated with weight, which speciﬁes the quality of the divergence, mainly for the bad initial estimate and high noise speciﬁc particle. An estimation of the variable is obtained [3]. by weighting the sum of all particles. *e particle ﬁlter *e discrete-time model of the WMR represented can be algorithm is repeated in nature and works into states: written as prediction and update. In each action, the particle is modiﬁed to the existing X � f X , U + w , k+1 k k k (5) model in the prediction step and added noise to simulate the Z � h X + v , k k k eﬀect of noise in the variable. *en, every particle’s weight is evaluated again based on sensory information in the update where X � x y θ is the state vector to be estimated, step. *e particle with a small weight is eliminated; this U � μ ω is the control vector, Z is the measured process is called resampling [17]. *e variable of interest, output, and (w, v) is the system and measurement dis- represented as a set of N samples, is given as follows: turbances deﬁned previously. *e EKF is based on the k k k linearization of the nonlinear equations (11) and (12). S � x , w for j � 1, . . . , N, (7) i j j Further, it is assumed that the initial state and noises are Gaussian and uncorrelated to each other. *e prediction where j is a particle and every particle has a copy of the and update of the EKF are given as follows (see [16]): variable and weight w , which deﬁnes the contribution of the particle to estimate the variable. (i) X � f(X , U ), k+1 k k *e N samples or particles (x ) for j � 1, . . . ., N, are T j (ii) P � A P A + Q k+1 k k k k k drawn from a known or estimated distribution P(x /y ) at − 1 T T (iii) K � P H (H P H + R) , k k k k k k k time k. Each sample has associated a weight w for j � (iv) X � X + K (Z − h(X )), N k+1 k+1 k k+1 k+1 1, . . . ., N, with w � 1 (normalization). *en, the pdf j�1 j (v) P � (1 − K H ) P k k k+1 k k k+1 P(x /y ) can be empirically approximated as where the dynamic and the output matrices are k N k k k k P ≈ zf X , U w δx − x dx . (8) k k k j j A � , k j�1 zX (6) *erefore, the best estimate of the state x , using the zh X Bayesian estimation rule, [21, 23] is H � . zX k k N k ∞ x x k k k k x � E � x P dx ≈ w x . (9) j j k k −∞ y y j�1 3.3. Particle Filter. In the EKF estimation, it is assumed that the noise measurements, due to sensors and from the Note that the variable of interest in our case at the environment, have a Gaussian probability density function discrete-time t � k, is the pose and the orientation of the (pdf). However, in practice, this assumption may be mobile robot deﬁned as 4 Journal of Robotics EKF with Gaussian Noise x-measured, and x-filtered 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position y-measured, and y-filtered 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position θ-measured, and θ-filtered 3 2 1 0123 2 x-position 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 1: State of the WMR with the EKF (one landmark and x-position Gaussian noise). Figure 2: EKF performance with one landmark and Gaussian noise. k (10) x � x(k) y(k) θ(k) . *e ﬁrst stage is the prediction, which uses the model to Estimated error on x-position simulate the eﬀect of an action on the set of particles when a suitable noise is added. While in updating stage, using in- formation gotten from sensing to update the particle weight 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 helps to explain the moving robot’s pdf. Given particle x-position distribution, ﬁrst, we should take an action for the robot pose; then, three methods will apply to evaluate the esti- Estimated error on y-position mated pose. First, calculate the weighted mean, select the best particle, and then the weighted mean in a small window 0 1 around the best-selected particle. 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position 3.3.1. Particle Filter Algorithm [17]. *e particle ﬁlter al- Estimated error on θ-position gorithm works in two stages: predict and update. *e main idea to describe the ﬁlter is as follows: 0.2 0.1 In this work, the previous algorithms will be used to estimate the position and the orientation of the WMR. An 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 appropriate measurement model will be identiﬁed according to the sensors that will be used with the assumptions on the x-position nature of diﬀerent pdfs. Figure 3: *e induced estimation error from the EKF (one A particle ﬁlter or Bayes ﬁlter is a probabilistic method landmark and Gaussian noise). that allows recursively to estimate an unknown probability density function; therefore, this is also known as recursive Bayesian estimation. It relies on mathematical models and Table 1: *e mean value of numerical reading of the EKF with (one continuous measurements to estimate the pdf, and it is used landmark-Gaussian noise). to calculate the probabilities of multiple beliefs to allow a e (m) e (m) e (deg) Time (sec) robot to infer its position and orientation by allowing the x y θ robots to continuously update their most likely position 0.076 0.092 0.010 0.010 s within a coordinate system, based on the most recently acquired sensor data. *ere are a lot of factors that aﬀect the accuracy of localization. *e primary factor is the unwanted widely used in natural and social sciences; non-Gaussian data in the environment which is called noise. It is not easy models expose challenges in ﬁtting data that often have to get high accuracy localization in a noisy environment; distinct spiky and compulsive features that diﬀer from therefore, data need to be ﬁltered to estimate it correctly. *e Gaussian distributions, which are known as non-Gaussian Gaussian distribution (also called normal distribution) is distributions [4]. y-position y-position y-position y-position y-position y-position y-position Journal of Robotics 5 Table 2: *e mean of the trials for the EKF (one landmark and Gaussian noise). Trials T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Mean e (m) 0.076 0.112 0.083 0.078 0.075 0.066 0.055 0.084 0.114 0.105 0.084 e (m) 0.092 0.135 0.108 0.090 0.089 0.085 0.063 0.098 0.171 0.159 0.010 e (deg) 0.010 0.014 0.011 0.008 0.005 0.005 0.004 0.009 0.019 0.018 0.010 Time (sec) 0.010 0.008 0.012 0.008 0.009 0.009 0.007 0.008 0.010 0.008 0.008 s EKF with Non-Gaussian Noise Estimated error on x-position 50 5 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position Estimated error on y-position 2 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position Estimated error on θ-position 4 3 2 1 01234 10 x-position 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4: WMR’s state with the EKF (one landmark and non- x-position Gaussian noise). Figure 6: *e induced estimation error for the EKF (one land- mark-non-Gaussian noise). x-measured, and x-filtered Table 3: *e mean value of numerical readings of the EKF with one 0 landmark and non-Gaussian noise. 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 e (m) e (m) e (deg) Time (sec) x y θ x-position 43.81 57.24 7.73 0.018 y-measured, and y-filtered it is possible to evolve the mobile robot in a closed-loop 50 100 where the circle shape is followed by using, for instance, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 model predictive control, where the prediction can be x-position conducted by the method used in [18]. θ-measured, and θ-filtered 4.1. EKF. For the ﬁrst ﬁlter, we used diﬀerent sensors, seen 0 previously, with Gaussian and non-Gaussian noises. To 5 apply the EKF, we have to use the linearized model of the 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 nonlinear system. *erefore, the state matrix of the line- x-position arized model is given by [22, 25] Figure 5: EKF performance with one landmark and non-Gaussian 1 0 −v(k)T cos(θ(k)) s noise. dF X , u A � � 0 1 v(k)T sin(θ(k)) . (11) k s dX 4. Simulation Results and Discussion 0 0 1 To accomplish the simulation part, the WMR is required to As it was mentioned previously, the source of the noise evolve in an open loop, without the controller, and achieve a can be the sensor itself or any hardware imperfection. *e circle shape where all variables change. To this end, the work described in [19] deals with a control strategy to ensure velocity and the steering angle are taken constant. Note that the optimal working condition and reduce the eﬀect of y-position y-position y-position y-position y-position y-position y-position 6 Journal of Robotics Table 4: *e mean of the trials with the EKF (one landmark and non-Gaussian noise). Trials T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Mean e (m) 43.81 0.366 21.35 53.90 21.96 1.88 10.93 38.57 3.57 113.51 21.81 e (m) 57.24 0.424 20.81 93.37 17.07 1.55 20.22 49.74 3.51 109.60 29.32 e (deg) 7.73 0.085 2.80 10.74 2.41 0.225 1.94 6.80 0.485 16.72 3.69 Time (sec) 0.018 0.010 0.008 0.019 0.012 0.016 0.012 0.015 0.014 0.015 0.013 s th Note that the result obtained in the 10 trial has been discarded since it is considered an outlier value. To improve the accuracy of the ﬁlter, another landmark is added in the space environment with a diﬀerent position than the ﬁrst one. Table 5: *e mean value of numerical reading of the EKF (two x-measured, and x-filtered landmarks and Gaussian noise). e (m) e (m) e (deg) Time (sec) x y θ 1 0.013 0.022 0.017 0.035 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x-position y-measured, and y-filtered Estimated error on x-position 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 x-position 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x-position θ-measured, and θ-filtered Estimated error on y-position 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x-position 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x-position Figure 7: EKF performance with two landmarks and Gaussian noise. Estimated error on θ-position 0.5 1 hidden dynamics induced by imperfections. In this work, 1.5 two diﬀerent kinds of noises have been considered: Gaussian 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 and non-Gaussian noises. x-position Figure 8: *e induced estimation error from the EKF (two landmarks and Gaussian noise). 4.1.1. One Landmark (1) Gaussian Noise. In Figure 1, it is described as the path (2) Non-Gaussian Noise. In Figure 4, we can see an increase when the EKF is applied with Gaussian noise and one in the error results with one path. As represented in Figure 4, landmark sensor. the black curve with noise appears that the ﬁlter is ineﬀective As shown in Figure 2, it is depicted as the measured and in decreasing the noise. ﬁltered signals for the three variables [X, Y, θ]. *ey are close As shown in Figure 5, there are three diﬀerent readings to each other for each reading. Globally, the simulation for measured and ﬁltered results. *is ﬁgure shows the time results show a slight increase in the error estimation at the (mn) for each measured and ﬁltered item. It can be seen the end of the simulation. two curves are not identical, especially at time t � 0.5 mn. Figure 3 shows the estimation error for each variable. Figure 6 shows the reading for the estimation errors. Here, we can see the diﬀerences in the reading in the results Here, we can see the diﬀerences between the measured of each three variables. Table 1 resumes the numerical curves and the ﬁltered curves. *e ﬁgure shows how the reading of the errors for one trial. error curves go up and down for the non-Gaussian noise. For the case of Gaussian noise, to analyze the sensitivity *e mean of numerical readings is shown in Table 3. of the ﬁlter, we run ten trials, and we take the average of all Table 4 shows the error reading for ten trials as well as the variable values and processing time. Table 2 below shows the average value for each variable. mean error for each run and the mean for all trials. y-position y-position y-position y-position y-position y-position Journal of Robotics 7 Table 6: *e mean of trials with the EKF (two landmarks and Gaussian noise). Trials T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Mean e (m) 0.013 0.012 0.014 0.013 0.012 0.011 0.013 0.012 0.012 0.013 0.012 e (m) 0.022 0.022 0.027 0.023 0.022 0.020 0.023 0.020 0.022 0.023 0.022 e (deg) 0.017 0.017 0.024 0.020 0.019 0.016 0.019 0.016 0.017 0.018 0.039 Time (sec) 0.035 0.027 0.018 0.019 0.030 0.019 0.019 0.020 0.019 0.020 0.022 s x-measured, and x-filtered Table 7: *e mean of numerical readings with the EKF (two landmarks and non-Gaussian noise). e (m) e (m) e (deg) Time (sec) x y θ 0.522 0.511 0.216 0.037 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x-position y-measured, and y-filtered they are close to each other in each reading. *is ﬁgure shows clearly when the system is subject to the Gaussian noise with two landmarks and the estimation performance is reasonable. 5 Figure 8 shows the behavior of the estimation errors. 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Here, we can see the diﬀerences between the state and its x-position estimate. *e numerical readings for one trial are shown in θ-measured, and θ-filtered Table 5. In Table 6 below we have all the ten-trial readings and the global mean value for each variable. 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (2) Non-Gaussian Noise. Figure 9 illustrates the three dif- x-position ferent readings for measured and ﬁltered results. We remark Figure 9: EKF performance with two landmarks and non-Gaussian that for this case (non-Gaussian noise), the EKF does not noise. achieve good estimation performance. *is fact has been mentioned in the previous chapters, where it is mentioned that the EKF is speciﬁc to the case of Gaussian noise. Estimated error on x-position Figure 10 shows the reading for errors estimate. Here, we can see the diﬀerences in the estimation errors. *e esti- 5 mation performances are not acceptable. Table 7 shows the 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 numerical readings for one trial. x-position To get the mean value of ten trials, we execute the al- gorithm ten times and collect all results that are shown in Estimated error on y-position Table 8. 4.2. Particle Filter. In this part, the particle ﬁlter is used to 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 estimate the pose of the WMR. As for the EKF, two diﬀerent x-position sensors are used. Estimated error on θ-position 4.2.1. One Landmark 5 (1) Gaussian Noise. In Figure 11, the green path shows the 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 behaviour of the ﬁltered state, and the red path shows the x-position measured state. It appears clearly that the ﬁlter is eﬀective in decreasing the negative eﬀect of the noise. Figure 10: *e induced estimation error from the EKF (two landmarks and non-Gaussian noise). Figure 12 shows the evolution of the three variables with their estimates. It appears that the PF estimator copes with the case of one landmark and Gaussian noise. 4.1.2. Two Landmarks Figure 13 shows the readings for the error estimates for the case of the PF with one landmark and Gaussian (1) Gaussian Noise. As shown in Figure 7, there are three noise. For the position estimation, the error is slightly high diﬀerent variation results for three diﬀerent readings which during the starting of the algorithm, and after that, the are for measured and ﬁltered results for the variables [X, Y, θ]; errors have decreased. For the last reading (estimation y-position y-position y-position y-position y-position y-position 8 Journal of Robotics Table 8: *e mean of the trials of the EKF (two landmarks and non-Gaussian noise). Trials T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Mean e (m) 0.522 5.87 0.656 0.562 0.276 0.434 1.37 0.472 3.04 0.150 1.27 e (m) 0.511 4.15 0.838 0.609 0.259 0.578 0.863 0.443 3.18 0.216 1.16 e (deg) 0.216 6.55 0.902 0.667 0.108 0.589 1.14 0.211 4.19 0.110 1.46 Time (sec) 0.037 0.040 0.023 0.027 0.025 0.036 0.028 0.026 0.025 0.025 0.029 s 6 Estimated error in mesured x and filtered x 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position Estimated error in mesured y and filtered y 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position Estimated error in mesured θ and filtered θ 3 2 1 0123 4 0.1 x-position 0.05 Mesured State 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Filtered State x-position Figure 11: *e state of the WMR using the PF with one landmark Figure 13: Localization error with the PF (one landmark and and Gaussian noise. Gaussian noise). x-measured, and x-filtered Table 9: *e mean of the experiments with the PF (one landmark and Gaussian noise). e (m) e (m) e (deg) Time (sec) x y θ 2 0.031 0.027 0.010 2.77 s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position y-measured, and y-filtered average of all variables. We remark that in all ten cases, the diﬀerence is very small. In Table 10 below, we have all the ten-trial readings and the global mean value for each variable. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position (2) Non-Gaussian Noise. Figure 14 depicts the simulation θ-measured, and θ-filtered results for the case of the PF with one landmark and non- Gaussian noise. It appears a good tracking performance of the estimator for this case. Figure 15 illustrates the state variables and their esti- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mates. *ese results show that the ﬁlter is eﬀective in this x-position case in reducing the eﬀect of the noise on the state variable. Figure 16 shows the readings for error estimates. Here Figure 12: PF performance with one landmark and Gaussian noise. also, the starting error estimate for the position is slightly high, and afterward, it decreases to an acceptable value. *e error of the orientation), we can see the variation of this mean of numerical readings ae shown in Table 11. error around 0.010. Table 9 shows the numerical readings For the non-Gaussian noise cases to measure the sen- for one trial. sitivity of the ﬁlter, we run the algorithm ten times, and we For the Gaussian noise case, to measure the sensitivity of get the mean of all variable values. It is noted that in all ten the ﬁlter, we run the algorithm ten times, and we take the cases, the diﬀerence is very small as shown in Table 12. y-position y-position y-position y-position y-position y-position y-position Journal of Robotics 9 Table 10: *e mean of the trials with the PF (one landmark and Gaussian noise). Trials T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Mean e (m) 0.031 0.014 0.012 0.011 0.014 0.017 0.019 0.029 0.010 0.012 0.016 e (m) 0.027 0.019 0.014 0.017 0.012 0.017 0.012 0.015 0.017 0.028 0.016 e (deg) 0.010 0.003 0.004 0.003 0.003 0.003 0.003 0.009 0.003 0.003 0.004 Time (sec) 2.77 2.71 2.67 2.67 2.68 2.73 2.77 2.72 2.73 2.72 2.45 s Estimated error in mesured x and filtered x 5 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position Estimated error in mesured y and filtered y 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position Estimated error in mesured θ and filtered θ 3 2 1 01234 0.04 x-position 0.03 0.02 Mesured State 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Filtered State x-position Figure 14: *e state of the WMR with the PF (one landmark-non- Figure 16: Estimation error for the PF with one landmark and non- Gaussian noise). Gaussian noise. x-measured, and x-filtered Table 11: *e mean of the estimation errors for the PF (one 0 landmark and non-Gaussian noise). 1 e (m) e (m) e (deg) Time (sec) x y θ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.013 0.013 0.003 2.74 s x-position y-measured, and y-filtered Figure 18 shows the readings for the error estimates. For this case, the error position is high at the beginning of the simulation. After that, the errors decrease and evolve around 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 zero. On the other hand, the orientation variation does not x-position converge to zero or around the origin as shown in Table 13. In Table 14 below, we have all the ten-trial readings and θ-measured, and θ-filtered the global mean value for each variable. (2) Non-Gaussian Noise. Figure 19 shows the variations of the position and the orientation of the WMR and their 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 estimated signals by the PF for the case of two landmarks x-position and non-Gaussian noise. Here also the tracking perfor- Figure 15: PF performance with one landmark and non-Gaussian mances are acceptable. noise. Figure 20 shows the reading for error estimates. Here also, we note the same behaviour of the estimator for case 4.2.2. Two Landmarks two landmarks with Gaussian noise. *is means that the estimation performance of the PF is not inﬂuenced by the (1) Gaussian Noise. Figure 17 shows the variations in the type of distribution of the noise. Table 15 resumes the state of the WMR and the estimated of these variables by the numerical reading of the errors for one trial. PF in the case of two Landmarks and Gaussian noise. *e Table 16 below shows the results of executing the al- tracking performances are acceptable. gorithm ten times. y-position y-position y-position y-position y-position y-position y-position 10 Journal of Robotics Table 12: *e mean of diﬀerent trials with the PF (one landmark and non-Gaussian noise). Trials T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Mean e (m) 0.013 0.020 0.016 0.014 0.010 0.011 0.010 0.018 0.012 0.009 0.013 e (m) 0.013 0.026 0.011 0.014 0.011 0.014 0.022 0.013 0.011 0.019 0.015 e (deg) 0.003 0.008 0.004 0.004 0.002 0.002 0.004 0.005 0.003 0.009 0.004 Time (sec) 2.74 2.75 2.75 2.73 2.74 2.75 2.71 2.75 2.75 2.73 3.01 s x-measured, and x-filtered 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position y-measured, and y-filtered 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position θ-measured, and θ-filtered 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position Figure 17: PF performance with two landmarks and Gaussian noise. Estimated error in mesured x and filtered x Table 13: *e mean of the estimation errors of the PF (two landmarks and Gaussian noise). 0.5 e (m) e (m) e (deg) Time (sec) x y θ 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.020 0.014 0.495 3.00 s x-position Estimated error inmesured y and filtered y tracking performances since the Kalman ﬁlter is built 0.5 under the assumption of Gaussian noises (state and measurement). Further, the particle ﬁlter achieves good performances for both cases of noises (Gaussian and 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 non-Gaussian). Indeed, the particle ﬁlter is built for any x-position distribution of the noise signals (state and Estimated error in muesured θ and filtered θ measurement). 0.05 On the other hand, the particle ﬁlter takes more time for processing than the EKF. *is is due to the sampling 0.05 process. We can note also that the accuracy of the PF 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 can be reduced further when the number of samples is x-position increased and this also will increase the time processing Figure 18: *e induced ﬁltering error with the PF (two landmarks further. and Gaussian noise). 4.3. <e Performance Metric. Performance evaluation of the Extended Kalman Filter estimation algorithms utilizes diﬀerent criteria. *ese cri- Particle Filter teria depend on the type of evaluation (performance opti- Figures 21 and 22 depict the mean value of a set of trials mization, estimation, and analysis). Since the ﬁltering for each ﬁlter. We can see that if the WMR is subject to problem developed in this work is data-dependent, its es- the Gaussian noise, the EKF achieves acceptable timation performance is random which needs to evaluate in y-position y-position y-position y-position y-position y-position Journal of Robotics 11 Table 14: *e mean of diﬀerent trials with the PF (two landmarks and Gaussian noise). Trials T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Mean e (m) 0.020 0.013 0.026 0.015 0.021 0.019 0.014 0.032 0.011 0.027 0.019 e (m) 0.014 0.021 0.018 0.017 0.014 0.015 0.013 0.016 0.010 0.011 0.014 e (deg) 0.495 0.389 1.04 0.762 0.533 0.579 0.256 1.07 0.011 0.344 0.547 Time (sec) 3.00s 3.05 3.21 3.03 3.03 3.03 3.04 3.05 3.21 3.04 3.06 s x-measured, and x-filtered Table 15: *e mean of the PF (two landmarks with non-Gaussian noise). e (m) e (m) e (deg) Time (sec) 1 x y θ 0.010 0.010 0.003 3.09 s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position (i) MSE (mean square error) y-measured, and y-filtered (ii) MMSE (minimum mean square error) (iii) RMSE (root mean square error) (iv) MAE (mean absolute value) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 In this work, the AEE (average Euclidian error) is x-position adopted for the evaluation of the estimator. AEE criterion θ-measured, and θ-filtered uses the concept of the Euclidian distance, and it is deﬁned as the length of curve segments between two points in Eu- clidian space [20]: M M 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1/2 1 1 AEE(X) � X � X X , (12) x-position i i i M M i�1 i�1 Figure 19: PF performance with two landmarks and non-Gaussian noise. where X � X − X denotes the estimation error, X is the i i i i unknown, and X is its estimate. In this work, we will proceed as follows. Estimated error in mesured x and filtered x th For each j trial, the mean error is calculated as μ(j) � (1/N) e(i), for i � 1, . . . , N, where i�1 0.5 3×1 3×1 μ(j) ∈ R is the mean, e(i) � X(i) − X(i) ∈ R is the 1 state estimation error, X(i) is the position and the orien- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 tation of the mobile robot at iteration i, and X(i) is its x-position th estimate. N is the number of iterations during the j trial or Estimated error in mesured y and filtered y run. For M (in our case M � 10) random runs or experiments, 0.5 the average Euclidian error of the estimator is 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 M M 1 1 1/2 x-position AEE(Est.Error) � μ(j) � μ(j) μ(j) . M M j�1 j�1 Estimated error in muesured θ and filtered θ (13) 0.1 0.05 *e smaller the AEE, the more accurate the estimator. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-position 4.3.1. <e Evaluated Cases Figure 20: *e induced estimation error from the PF (two landmarks and non-Gaussian noise). Case 1. One Landmark In this case, the AEE value for each set of simulations is given in Table 17. the statistical sense. *ere are many criteria used in the literature to evaluate the estimation performance based on Case 2. Two Landmarks the norm or distance between the teal vector and its estimate. In this case, the AEE for a set of trials is given in Table 18. In the literature, we can ﬁnd the following criteria: y-position y-position y-position y-position y-position y-position 12 Journal of Robotics Table 16: *e mean of ten trials with the PF (two landmarks-non-Gaussian noise). Trials T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Mean e (m) 0.010 0.015 0.015 0.011 0.010 0.012 0.011 0.020 0.012 0.014 0.013 e (m) 0.010 0.011 0.014 0.014 0.030 0.015 0.015 0.016 0.015 0.010 0.015 e (deg) 0.003 0.004 0.007 0.003 0.004 0.003 0.005 0.011 0.004 0.001 0.004 Time (sec) 3.09 3.41 3.08 3.36 3.12 3.09 3.08 3.43 3.09 3.05 3.18 s Extended Kalman Filter LANDMARK1 LANDMARK1 LANDMARKE LANDMARKE -G -N 2-G 2-N TIME 0.008 0.013 0.022 0.024 THETA 0.01 3.69 0.039 1.46 Y 0.01 29.32 0.022 1.16 X 0.084 21.81 0.012 1.27 Figure 21: Simulation results analysis of the EKF. Partical Filter LANDMARK LANDMARK LANDMARK LANDMARK 1-G 1-N 2-G 2-N 2.45 3.01 3.06 3.18 Time Theta 0.004 0.004 0.543 0.004 Y 0.016 0.015 0.014 0.013 X 0.016 0.013 0.014 0.013 Figure 22: Simulation results analysis of PF. 5. Conclusion Table 17: AEE for one landmark. Mobile robot navigation has received the greatest research attention for many years. *e success of the safe navigation AEE Gaussian noise Non-Gaussian noise task depends closely on the localization accuracy. *erefore, EKF 0.138 48.38 many methods of localization have been developed signif- PF 0.025 0.020 icantly because of the need to ﬁnd accurate solutions to Here, PF outperforms the EKF in the estimation of the position and ori- determine the position and orientation of the WMR in a entation of a WMR for the Gaussian noise in contrast to the non-Gaussian. given environment. *is work focuses on the comparative study of the most used ﬁlters for the localization of mobile robots. 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Journal of Robotics – Hindawi Publishing Corporation
Published: Jun 2, 2022
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