Computational Intelligence Approaches to Robotics, Automation, and Control
View this Special IssueResearch Article  Open Access
Fei Yu, Qian Sun, Chongyang Lv, Yueyang Ben, Yanwei Fu, "A SLAM Algorithm Based on Adaptive Cubature Kalman Filter", Mathematical Problems in Engineering, vol. 2014, Article ID 171958, 11 pages, 2014. https://doi.org/10.1155/2014/171958
A SLAM Algorithm Based on Adaptive Cubature Kalman Filter
Abstract
We need to predict mathematical model of the system and a priori knowledge of the noise statistics when traditional simultaneous localization and mapping (SLAM) solutions are used. However, in many practical applications, prior statistics of the noise are unknown or timevarying, which will lead to large estimation errors or even cause divergence. In order to solve the above problem, an innovative cubature Kalman filterbased SLAM (CKFSLAM) algorithm based on an adaptive cubature Kalman filter (ACKF) was established in this paper. The novel algorithm estimates the statistical parameters of the unknown system noise by introducing the SageHusa noise statistic estimator. Combining the advantages of the CKFSLAM and the adaptive estimator, the new ACKFSLAM algorithm can reduce the state estimated error significantly and improve the navigation accuracy of the SLAM system effectively. The performance of this new algorithm has been examined through numerical simulations in different scenarios. The results have shown that the position error can be effectively reduced with the new adaptive CKFSLAM algorithm. Compared with other traditional SLAM methods, the accuracy of the nonlinear SLAM system is significantly improved. It verifies that the proposed ACKFSLAM algorithm is valid and feasible.
1. Introduction
The simultaneous localization and mapping (SLAM) is that the mobile robot builds up the environment map in the unknown environment by utilizing the sensors onboard; at the same time the robots location is computed by the same map [1–3]. The SLAM was first proposed by Smith, Self, and Cheeseman in 1987, estimating robots poses and the features of the environment simultaneously using extend Kalman filter (EKF) [4]. Since then, the SLAM has been implemented in a number of different domains from indoor robots to outdoor, underwater, ground, and airborne systems [5, 6]. The SLAM algorithm has received extensive attention.
SLAM problem involves unknown and uncertain environment description and sensor noise; therefore its essence is a probabilistic estimation issue which is widely accepted by numerous researchers now [4]. At present, the most typical and widely used SLAM algorithm is EKFbased SLAM (EKFSLAM) introduced in [7–9]. But the precision of EKFSLAM is limited because the Taylor expansion will cause truncation errors; on the other hand the EKF needs to calculate the fussy Jacobian matrix which increases the computational load. To solve the problems of the EKF, the unscented Kalman filter (UKF) proposed by Julier and Uhlmann [10] was used in SLAM [11]. Unlike the EKF, the UKF uses a set of chosen samples to represent the state distribution. The UKFbased SLAM (UKFSLAM) algorithm can not only avoid the calculation of the Jacobian and Hessian matrices but also obtain higher approximation accuracy with the unscented transformation (UT). However, for highdimensional systems, the computation load is still heavy; thus, the filter converges slowly. In 2009, Arasaratnam and Haykin [12, 13] proposed a more accurate nonlinear filtering solution based on a cubature transform named the cubature Kalman filter (CKF) which can avoid linearization of the nonlinear system by using cubature point sets to approximate the mean and variance. The thirdorder accuracy of the system can be achieved with this method. What is more, the computation complexity of the CKF is lower than the one of the UKF. Because of its high accuracy and low calculation load, the CKF is widely used in attitude estimation and navigation [14]. CKFSLAM was introduced in [5], which achieved high positioning accuracy compared with traditional SLAM algorithms.
However, the above filters are all based on the framework of the Kalman filter (KF); it can only achieve a good performance under the assumption that the complete and exact information of the noise distribution has to be known as a prior. But in practice, the prior noise statistic is usually unknown totally or may be always timevarying because the model and the noise of system are influenced by the drift errors of the robots component, the kinematic model of the robot, and the uncertainty of the outdoor environment. With the uncertain or timevarying noise statistic, the state estimation will have large errors, or, even, the filters will be possible to diverge [15]. For SLAM, it has been shown that the performance can become significantly worse and the estimated results practical diverge with an incorrect a priori knowledge of the noise statistics [16, 17]. Therefore, to solve this problem, many researchers have proposed the filter algorithm and the adaptive filter algorithm which have been focused on in recent years. Although the filter has numerous advantages, such as being robust, independent of the noise statistic, and easy to set the filter parameters, the low filter accuracy is still its fatal flaw [18–20]. Hence, we only focus on the adaptive filter algorithm in this work.
The core idea of the adaptive filter algorithm is that, while the whole system is filtering, estimating and modifying the parameters of the system model and the statistic of the system noise which are unknown or timevarying, to decrease state estimation errors and improve the estimation accuracy. There are numerous adaptive algorithms, including the distribution test method, the maximum a posteriori method, the correlation test method, and the SageHusa adaptive method [21, 22]. Therein, the SageHusa adaptive method which is suboptimal unbiased maximum a posteriori (MAP) noise estimator is one of the widely used adaptive filtering methods [23], since it has advantages that the recursive formula is simple and the principle is clear and easy to implement. Therefore we use this kind of adaptive filtering to estimate unknown system noise here.
In this paper a novel adaptive filtering algorithm based on the CKFSLAM, named adaptive CKFSLAM (ACKFSLAM), for mobile robot outdoors, was proposed. In this proposed algorithm, we introduced the SageHusa adaptive filtering method into the traditional CKFSLAM algorithm. The system noise and the statistics of the observation noise were estimated on real time and modified to reduce the errors of the system model by using the adaptive estimator of the timevarying noise. What is more, it also prevented the filter divergence and improved the estimation accuracy significantly. The rest of the paper was organized as follows. The description of the SLAM and the traditional CKFSLAM algorithm were presented in Section 2. Section 3 showed the adaptive EKF algorithm and then a new ACKFSLAM algorithm was proposed. Numerical examples in different scenarios along with specific analysis were given in Section 4. Section 5 concluded this paper.
2. SLAM
2.1. Description of the SLAM
The SLAM is always described as follows: the mobile robot starting in an unknown location without previous knowledge of the environment builds a map using its onboard sensors while, simultaneously, using this same map to determine the location of the robot within this map [2]. Figure 1 illustrates the characteristic data acquisition and processing of the SLAM.
The essence of the SLAM is the filtering estimation of the system in the whole path. The system states are composed of the robots pose (position, orientation) and the landmarks states observed in the environment. Let the robot states be represented by its pose with an estimated vector and the covariance matrix , defined as
Without loss of generality, let us assume that there are a set of 2D static point features observed in the map; the position estimations of these features are given by their estimated vector and the covariance matrix as
In the SLAM, the total state vector is composed of the robots states and the landmarks states . Hence the estimations of the total state vector and the corresponding total error covariance matrix are given as wherein denotes the robotmap correlation.
The key of the SLAM is to determine the posterior probability density of the state vector which is and the implication is to obtain the joint probability density of the robot pose and the map when the control input and observation are known. We can obtain the following equation by utilizing the Bayes formula: wherein is the input control of the robot at time which drives from pose at time to pose at time ; is observation to the landmark at time ; is the motion model which is the conditional probability of at time when and are known; is the observation model which is the conditional probability of at time when and the landmark collection matrix are known.
2.2. CKFSLAM
In this subsection the principle of the classical CKF is introduced. Consider the general discretetime nonlinear statespace model as follows: wherein and are the state vector and the measurement vector at time , respectively; and are specific known nonlinear functions; and and are the noise vectors from two independent Gaussian processes with their means being and and their covariance matrices being and , respectively. The CKF is proposed to solve this nonlinear filtering problem on the basis of the sphericalradial cubature criterion. Firstly it approximates the mean and variance of the probability distribution through a set of 2 ( is the dimension of the nonlinear system) cubature points with the same weight, propagates the above cubature points through the nonlinear functions, and then calculates the mean and variance of the current approximate Gaussian distribution by the propagated cubature points [12]. A set of 2 cubature points are given by , where is the th cubature point and is the corresponding weight: wherein . Under the assumption that the posterior density at time is known, the steps involved in the timeupdate and the measurementupdate of the CKF are summarized as follows [12].
Timeupdate:(1)factorize (2)evaluate the cubature points (3)evaluate the propagated cubature points (4)estimate the predicted state (5)estimate the predicted error covariance
Measurementupdate:(1)factorize (2)evaluate the cubature points (3)evaluate the propagated cubature points (4)estimate the predicted measurement (5)estimate the innovation covariance matrix (6)estimate the crosscovariance matrix
With the new measurement vector , the estimation of the state vector and its covariance matrix at time can be obtained as follows.(1)Estimate the Kalman gain (2)Estimate the updated state (3)Estimate the corresponding error covariance
The CKFSLAM algorithm uses cubature rule and cubature point sets to compute the mean and variance of the probability distribution without any linearization of the SLAM system. Thus, the CKFSLAM algorithm does not demand to calculate Jacobian and Hessian matrices so that the truncation errors can be avoided. Hence, the estimation accuracy can reach the third order or higher. Furthermore, the computational complexity is alleviated to a certain extent than the UKF [13].
3. SLAM Algorithm Based on Adaptive CKF
3.1. Adaptive EKF
When the system noises are unknown or timevarying, the filtering algorithm cannot be recursively carried out in a common way. One effective solution is the adaptive filtering algorithm. The adaptive filtering technology has become a focus of the research attempting to solve the filter divergence problem caused by the inaccurate statistical properties of the noise and the mathematical model itself. The SageHusa adaptive filtering proposed by Sage and Husa is one of the most widely used adaptive filtering algorithms. Estimating the statistical parameters of the virtual noises online, along with the recursive estimate of the state, this method can reduce the error of the model and improve the accuracy of the whole system validly.
Nowadays, the adaptive EKF based on SageHusa is the most widely used adaptive algorithm. However, the estimated accuracy is low because of the truncation errors of the Taylor expansion in the EKF. Considering the advantages of the CKF, an adaptive CKF algorithm was proposed to improve the estimated accuracy in this work on the basis of the adaptive EKF. First of all, the adaptive EKF was introduced in detail here. Suppose that the discrete state equation and the observation equation of the nonlinear system are shown as (5). On the basis of [24], the SageHusa estimator based adaptive EKF algorithm is shown as wherein is the error covariance matrix of the state , is the filter gain matrix, and are the nonlinear states function and observation function with respect to states, respectively, is the difference between the measurement and the prediction, and covariance matrix of the , , , , and is obtained by recurrence of the timevarying noise statistics estimator: wherein denotes the innovation sequence; denotes the forgetting factor whose value is often set between 0.95 and 0.99. The memory span is limited utilizing the forgetting factor. As a result, the old information is forgotten little by little and the new information plays the lead role in estimating.
It has been analyzed that adaptive filtering algorithms cannot estimate the process and the measurement noise simultaneously in [25]. However, theoretical derivations and simulations in [26] show that when the measurement noise is already known, the process noise can be obtained by the iteration. Usually, the measurement noise statistic is relatively well known compared to the system model noise. So we can use this adaptive estimator to estimate the system noise and enhance the filtering accuracy.
3.2. Adaptive CKF
It is well known that the truncation errors of the Taylor expansion in the EKF will reduce the estimated accuracy or even diverge the filter. However, the new algorithm named the CKF can increase the estimated accuracy effectively, which was referred to in Section 2.2. So analogous with the adaptive EKF, the SageHusa noise statistics estimator was embedded into the CKFSLAM. And the adaptive CKFSLAM algorithm was obtained, which combines the advantages of the CKFSLAM and the SageHusa adaptive method. Without loss of generality, we still consider the nonlinear system shown as (5) and the detailed algorithm of the ACKFSLAM is given as follows.
Timeupdate:
Measurementupdate:
With the new measurement vector , the estimation of the state vector and its covariance matrix at time can be obtained by the following equations:
The improved ACKFSLAM algorithm proposed in this paper can be used in the situation when the noise statistical character of the system is absolutely or approximately unknown to make sure the filter works well, enhancing the stability of the filter.
4. Experiments and Analysis
4.1. Experiment Modeling
In order to verify the effectiveness and feasibility of the new SLAM algorithm proposed in this paper, a large number of simulations are tested with the simulation environment issued by Tim Bailey from University of Sydney. The simulation environment which is a 250 m × 200 m outdoor area includes the movement path and 135 static landmark points. The mobile robot moves along the path from (0, 0) anticlockwise as it is shown in Figure 2.
The motion model of the mobile robot is shown as follows: wherein denotes the pose of the mobile robot at time ; denotes the pose of the mobile robot at time ; denotes the sampling time; denotes the velocity of the mobile robot at time ; denotes the azimuth angle at time ; denotes the wheel base between the two axes; denotes the error of the system at time and . The observation model is shown as follows: wherein denotes the detected position corresponding to the th feature; denotes the distance between the th detected feature and the mobile robot; denotes the distance between the th detected feature and the heading of the mobile robot; denotes the measurement error at time and .
The experiment parameters are set as follows. The initial states of the mobile robot ; the sampling interval ; the velocity ; the velocity error ; the azimuth error ; the maximum angular rate is 0.2; the maximum distance of measurement is 30; the distance error ; the angular error .
4.2. Experimental Results and Analysis
Under the above conditions, fifty Monte Carlo simulations were performed for four SLAM algorithms, including the EKFSLAM, CKFSLAM, adaptive EKFSLAM (AEKFSLAM), and adaptive CKFSLAM (ACKFSLAM). And then the root mean square errors (RMSE) of the estimated results were compared.
Figures 3 and 4 show the comparisons of the EKFSLAM and the CKFSLAM; Figures 5 and 6 show the comparisons of the AEKFSLAM and the ACKFSLAM, wherein Figures 4 and 6 are partial enlargements of the rectangle regions in Figures 3 and 5, respectively. System noise and observation noise are shown as follows:
From Figures 3 to 6, we obviously know that the estimated error of the ACKFSLAM is much smaller than that of the CKFSLAM and EKFSLAM. It means that the ACKFSLAM algorithm can provide higher estimated accuracy of the nonlinear SLAM system than the other three solutions.
Figures 7 and 8 show the RMSEs of the estimated position errors of the ACKFSLAM, AEKFSLAM, CKFSLAM, and EKFSLAM.
From the results shown in Figures 7 and 8, we learn that the RMSE values of the position estimation errors of the ACKFSLAM algorithm are within 3 m. The precision of AEKFSLAM is nearly the same as CKFSLAM, and both of them are lower than ACKFSLAM algorithm. The RMSE values of the AEKFSLAM are within 8 m while the corresponding RMSEs of the CKFSLAM are within 10 m. The EKFSLAM algorithm has the lowest estimation precision and the RMSEs are within 17 m. So the precision of ACKFSLAM is higher than the other three algorithms. Comparing between these four algorithms, the estimation precisions of the standard CKFSLAM and ACKFSLAM are higher than those of standard EKFSLAM and AEKFSLAM because the cubature points are used in the CKF instead of the nonlinear transformation in the EKF. As a result, the errors can be reduced and the filtering precisions can be enhanced availably. We also can see that the estimation errors increase rapidly in the last part of the standard EKFSLAM and CKFSLAM. However, the estimation results of the ACKFSLAM and AEKFSLAM always keep stable. It is because the system process noise and observation noise are estimated and modified on real time to enhance the filter precision using the adaptive algorithm. These coincide with the theoretical analysis totally.
Table 1 shows the run time of the four algorithms. The calculated cost of the EKFSLAM is the least one, the ones of the AEKFSLAM and the CKFSLAM are almost equal, and the one of the ACKFSLAM is the most. That is because of the fact that the EKF only uses the Taylor expansion to linearize the nonlinear system which is extremely simple. But the CKF algorithm approximates the mean and variance by a set of 2 cubature points, so it costs a longer time than the EKF. As the adaptive algorithm is adopted in adaptive CKFSLAM algorithm, it costs much longer time than the CKFSLAM algorithm.

In order to verify the validity of the ACKFSLAM sufficiently, some different simulations were done with different environment noise. Suppose that the observation noise obeys the mixture Gauss distribution as , wherein
Figure 9 illustrates the estimated errors of the ACKFSLAM and the AEKFSLAM algorithms and Figure 10 is the partial enlargement of the rectangle region in Figure 9. As can be seen from Figures 9 to 11, the mobile robot motion trajectory still can be estimated with a high precision by the ACKFSLAM even though different environment noise is added.
(a)
(b)
To verify the effectiveness and superiority of the new adaptive SLAM algorithm, when the system noises are timevarying, another simulation scenario was set in which the process and the observation noises are timevarying. On the basis of [2], the parameters are set as Table 2.

Figure 12 illustrates the estimation errors of the ACKFSLAM, AEKFSLAM, and CKFSLAM algorithms when the system noises are timevarying and Figure 13 is the partial enlargement of the rectangle region in Figure 12. Figure 14 is the RMSEs of the position errors correspondingly. As can be seen from Figures 12 to 14, the estimated errors of the CKFSLAM are the biggest and the ones of the ACKFSLAM and the AEKFSLAM are much smaller. That is because of the fact that when the system noises are timevarying, standard CKFSLAM cannot accurately estimate them. However, the adaptive CKFSLAM and adaptive EKFSLAM can estimate and modify the parameters of the system model and the statistic of the system noises online to decrease the estimated errors and improve the accuracy effectively, owing to the SageHusa adaptive estimator. Additionally, comparing between the two adaptive methods, the estimated results of the ACKFSLAM are much better than that of the AEKFSLAM.
(a)
(b)
From the simulation results and the theoretical analysis above, we can see that the ACKFSLAM has higher estimation accuracy with a better numerical performance in the entire motion process than other algorithms. The effectiveness and superiority of the ACKFSLAM are verified.
5. Conclusions and Future Work
In this paper, a novel nonlinear SLAM algorithm based on the CKF and the SageHusa adaptive estimator was proposed in order to solve the problem that the noise statistic of the system is unknown or timevarying in real world and to increase the accuracy of the estimation. Firstly, the SageHusa adaptive noise estimator was introduced and the superiority of the CKF solution was analyzed theoretically. Secondly, we focused on proposing a new adaptive SLAM based on the CKF method through combining the advantages of the SageHusa adaptive estimator and the CKFSLAM solution, and the novel ACKFSLAM algorithm proposed here can estimate and correct the statistical character of the noises in real time and decrease the estimated errors. To verify the new SLAM algorithm, numerical simulations in different scenarios were carried out. The results showed that the proposed adaptive SLAM algorithm based on CKF can not only be applied to the systems in which the noise distribution is unknown or timevarying, but also significantly improve the estimation accuracy of the nonlinear SLAM system than traditional SLAM algorithms. The ACKFSLAM algorithm provides a new method for simultaneous localization and mapping in an unknown environment. However, the calculation burden of the ACKF algorithm is still huge. Reference [27] presented graphbased SLAM algorithms which can reduce the calculation cost significantly using a graph whose nodes correspond to the poses of the robot at different points in time and whose edges represent constraints between the poses. As a result, we plan to work on the adaptive graphbased SLAM algorithm to reduce the computation time at no sacrifice of the accuracy in the future.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Fundamental Research Funds for the Central Universities (HEUCFL1411002) and the National Natural Science Foundation of China (51379047).
References
 W. Zhang, M. Zhu, and Z. Chen, “An adaptive SLAM algorithm based on strong tracking UKF,” Robot, vol. 32, no. 2, pp. 190–195, 2010. View at: Google Scholar
 H. Wang, G. Fu, J. Li, Z. Yan, and X. Bian, “An adaptive UKF based SLAM method for unmanned underwater vehicle,” Mathematical Problems in Engineering, vol. 2013, Article ID 605981, 12 pages, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 E. Guerra, R. Munguia, Y. Bolea, and A. Grau, “Validation of data association for monocular slam,” Mathematical Problems in Engineering, vol. 2013, Article ID 671376, 11 pages, 2013. View at: Publisher Site  Google Scholar
 H. DurrantWhyte and T. Bailey, “Simultaneous localization and mapping: part I,” IEEE Robotics and Automation Magazine, vol. 13, no. 2, pp. 99–108, 2006. View at: Publisher Site  Google Scholar
 K. P. B. Chandra, D. W. Gu, and I. Postlethwaite, “Cubature kalman filter based localization and mapping,” World Congress, vol. 18, no. 1, pp. 2121–2125, 2011. View at: Google Scholar
 A. Nüchter, H. Surmann, K. Lingemann, J. Hertzberg, and S. Thrun, “6D SLAM with an application in autonomous mine mapping,” in Proceedingsof the IEEE International Conference on Robotics and Automation (ICRA '04), vol. 2, pp. 1998–2003, May 2004. View at: Google Scholar
 A. Chatterjee and F. Matsuno, “A Geese PSO tuned fuzzy supervisor for EKF based solutions of simultaneous localization and mapping (SLAM) problems in mobile robots,” Expert Systems with Applications, vol. 37, no. 8, pp. 5542–5548, 2010. View at: Publisher Site  Google Scholar
 S. Huang and G. Dissanayake, “Convergence and consistency analysis for extended Kalman filter based SLAM,” IEEE Transactions on Robotics, vol. 23, no. 5, pp. 1036–1049, 2007. View at: Publisher Site  Google Scholar
 G. Sun, M. Wang, and L. Wu, “Unexpected results of Extended Fractional Kalman Filter for parameter identification in fractional order chaotic systems,” International Journal of Innovative Computing, Information and Control, vol. 7, no. 9, pp. 5341–5352, 2011. View at: Google Scholar
 S. J. Julier and J. K. Uhlmann, “A new extension of the kalman filter to nonlinear systems,” in Proceedings of the International Symposium on Aerospace/Defense, Sensing, Simulation, and Controls, vol. 3, no. 26, p. 3.2, Orlando, Fla, USA, 1997. View at: Google Scholar
 X. Yan, C. Zhao, and J. Xiao, “A novel fastslam algorithm based on iterated unscented kalman filter,” in Proceedings of the IEEE International Conference on Robotics and Biomimetics (ROBIO '11), pp. 1906–1911, 2011. View at: Google Scholar
 I. Arasaratnam and S. Haykin, “Cubature kalman filters,” IEEE Transactions on Automatic Control, vol. 54, no. 6, pp. 1254–1269, 2009. View at: Publisher Site  Google Scholar
 I. Arasaratnam, S. Haykin, and T. R. Hurd, “Cubature Kalman filtering for continuousdiscrete systems: theory and simulations,” IEEE Transactions on Signal Processing, vol. 58, no. 10, pp. 4977–4993, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 W. Gao, Y. Zhang, and J. Wang, “A strapdown interial navigation system/beidou/doppler velocity log integrated navigation algorithm based on a cubature kalman filter,” Sensors, vol. 14, no. 1, pp. 1511–1527, 2014. View at: Google Scholar
 X. Tang, J. Wei, and K. Chen, “Squareroot adaptive cubature kalman filter with application to spacecraft attitude estimation,” in Proceedings of the 15th International Conference on Information Fusion, pp. 1406–1412, 2012. View at: Google Scholar
 R. K. Mehra, “On the identification of variances and adaptive Kalman filtering,” IEEE Transactions on Automatic Control, vol. 15, no. 2, pp. 175–184, 1970. View at: Google Scholar  MathSciNet
 F. R. Fitzgerald, “Divergence of the Kalman filter,” IEEE Transactions on Automatic Control, vol. 16, no. 6, pp. 736–747, 1971. View at: Google Scholar
 P. Batista, C. Silvestre, and P. J. Oliveira, “Kalman and H infinity optimal filtering for a class of kinematic systems,” in Proceedings of the 17th World Congress, International Federation of Automatic Control (IFAC '08), July 2008. View at: Publisher Site  Google Scholar
 Z. Wang, B. Shen, and X. Liu, “${H}_{\infty}$ filtering with randomly occurring sensor saturations and missing measurements,” Automatica, vol. 48, no. 3, pp. 556–562, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 J.H. Wang, C.L. Song, X.T. Yao, and J.B. Chen, “Sigma point Hinfinity filter for initial alignment in marine strapdown inertial navigation system,” in Proceedings of the 2nd International Conference on Signal Processing Systems (ICSPS '10), vol. 1, pp. V1580–V1584, July 2010. View at: Publisher Site  Google Scholar
 L. Zhao and X. Wang, “An adaptive UKF with noise statistic estimator,” in Proceedings of the 4th IEEE Conference on Industrial Electronics and Applications (ICIEA '09), pp. 614–618, May 2009. View at: Publisher Site  Google Scholar
 G. Sun, M. Wang, L. Huang, and L. Shen, “Generating multiscroll chaotic attractors via switched fractional systems,” Circuits, Systems, and Signal Processing, vol. 30, no. 6, pp. 1183–1195, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 J. Wang, J. Liu, and B.G. Cai, “Study on information fusion algorithm in embedded integrated navigation system,” in Proceedings of the International Conference on Intelligent Computation Technology and Automation (ICICTA '08), vol. 2, pp. 1007–1010, October 2008. View at: Publisher Site  Google Scholar
 H. Wang, J. Wang, X. Bian, and G. Fu, “SLAM of AUV based on the combined EKF,” Jiqiren/Robot, vol. 34, no. 1, pp. 56–64, 2012. View at: Publisher Site  Google Scholar
 S. Kawamura and N. Sakagami, “Planning and control of robot motion based on timescale transformation,” in Advances in Robot Control, pp. 157–178, Springer, New York, NY, USA, 2006. View at: Google Scholar
 S. Mei, F. Liu, and A. Xue, The Semi Tensor Product Methods in the Power System Transient Analysis, Tsinghua University Press, 2010.
 G. Grisetti, R. Kummerle, C. Stachniss, and W. Burgard, “A tutorial on graphbased SLAM,” IEEE Intelligent Transportation Systems Magazine, vol. 2, no. 4, pp. 31–43, 2010. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2014 Fei Yu et al. This 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.