sensors Article
An Accurate GPS-IMU/DR Data Fusion Method for Driverless Car Based on a Set of Predictive Models and Grid Constraints Shiyao Wang 1 , Zhidong Deng 1 and Gang Yin 2, * 1
2
*
State Key Laboratory of Intelligent Technology and Systems, Tsinghua National Laboratory for Information Science and Technology, Department of Computer Science, Tsinghua University, Beijing 100084, China;
[email protected] (S.W.);
[email protected] (Z.D.) State Key Laboratory of Coal Mine Disaster Dynamics and Control, College of Resources and Environmental Science, Chongqing University, Chongqing 400030, China Correspondence:
[email protected] Tel.: +86-23-6510-4511
Academic Editor: Ana M. Bernardos Received: 8 December 2015; Accepted: 4 February 2016; Published: 24 February 2016
Abstract: A high-performance differential global positioning system (GPS) receiver with real time kinematics provides absolute localization for driverless cars. However, it is not only susceptible to multipath effect but also unable to effectively fulfill precise error correction in a wide range of driving areas. This paper proposes an accurate GPS–inertial measurement unit (IMU)/dead reckoning (DR) data fusion method based on a set of predictive models and occupancy grid constraints. First, we employ a set of autoregressive and moving average (ARMA) equations that have different structural parameters to build maximum likelihood models of raw navigation. Second, both grid constraints and spatial consensus checks on all predictive results and current measurements are required to have removal of outliers. Navigation data that satisfy stationary stochastic process are further fused to achieve accurate localization results. Third, the standard deviation of multimodal data fusion can be pre-specified by grid size. Finally, we perform a lot of field tests on a diversity of real urban scenarios. The experimental results demonstrate that the method can significantly smooth small jumps in bias and considerably reduce accumulated position errors due to DR. With low computational complexity, the position accuracy of our method surpasses existing state-of-the-arts on the same dataset and the new data fusion method is practically applied in our driverless car. Keywords: local navigation; GPS-IMU/DR integrated navigation; multimodal data fusion; maximum likelihood estimation; driverless car
1. Introduction Autonomous navigation is one of the most key technologies for driverless cars. Accurate positioning and orientation estimation of vehicles is generally regarded as the basis of many sophisticated modules such as environmental perception, path planning, and autonomous decision-making of driverless cars under complex urban scenarios. Different from stand-alone GPS that is increasingly a popular navigation system, an enhanced differential GPS (DGPS) receiver with phase carrier signal measurements may run in operating modes of real time kinematics (RTK–DGPS), which has the highest absolute position accuracy of up to a few centimeters. In DGPS, mobile GPS device continuously receives correction data from ground-based reference station over transmitter of shorter range, aiming to compensate location inaccuracies [1]. DGPS systems operating under complicated urban scenarios, however, occasionally lose broadcast signals and probably acquire inaccurate localization data due to many unpredictable factors such as buildings’ occlusion, signal attenuation, and a diversity of Sensors 2016, 16, 280; doi:10.3390/s16030280
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electronic interference. In general, it works well in a limited range in terms of pseudorange correction principle. In addition, atmospheric visibility of satellite, potential environmental effects, and multipath may have negative impact on precision and reliability of GPS itself [2]. Two widely used multipath mitigation methods, i.e., high-resolution correlator (HRC) and multipath mitigation technique (MMT), and a new coupled amplitude delay lock loops (CADLL) method, wich is based on multipath signal amplitude, code phase, and carrier phase, are evaluated in [3]. They may fail under dynamic multipath scenario or when multipath is stronger than line-of-sight (LOS). Except for GPS, DR that employs vehicle kinematic model and incremental measurements of wheel encoder is often viewed to play a crucial role in precise short-term navigation of driverless cars [4]. As one of the autonomous relative navigations, the DR technique is capable of continually providing position information. A major disadvantage of using DR for navigation is that they typically suffer from accumulated error because of wheel slippage and wheel imperfection [5]. Actually, localization accuracy can maintain only within a very short range. As a result, substantial efforts have been made to improve long-term precision and robustness through slip estimation [6]. Several complementary navigation systems, including GPS, IMU, and DR, are usually combined through a variety of information fusion methods, typically such as Kalman filter (KF) [7]. In fact, GPS or GPS–IMU can provide absolute position and orientation, even if it contains discontinuous data and/or random drifts. Contrarily, as a local navigation system, DR is able to conduct accurate localization within a certain distance or duration. However, position errors will be accumulated with increase of distance. Undoubtedly, integration of GPS–IMU and DR is a natural selection to accurately navigate driverless car. In the last few decades, a lot of multimodal data fusion methods for meeting reliable, robust, and decimeter-level requirements for driverless cars, e.g., the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), has emerged. The EKF simplifies nonlinear filtering and is used for state estimation in [8–17]. Among this literature, several types of additional sources of information, including on-board motion sensors, cameras or LiDAR vision systems, and road map databases, are adopted to compensate for EKF-based navigation systems. References [8–11] improve accuracy of localization by integration of different navigation systems such as IMU, GPS, and DR. Ma et al. [12] combine stereo-camera sensor, IMU, and leg odometry by virtue of EKF. In [13,14], both accurate digital map and camera are integrated to improve location accuracy. Moreover, four EKF-based state estimation architectures are evaluated in [15], including nonlinear model (NLM) [16] and error model (ERM) [17], each with/without a complementary filter [18,19]. The experimental results show that NLM with a complementary filter has superior localization performance, which will be adopted to make comparison with our model in Section 3. Unlike the EKF, the UKF employs unscented transform to address approximation issues of the EKF, which is also extensively exploited in multimodal data fusions [20–22]. Actually, there still exist some problems even if the above two kinds of methods have been widely applied. The deficiencies of the KFs including EKF and UKF were specifically pointed out in [23]. For example, considering that there are uncertainties or unknown statistical characteristics for process and/or measurement noises, it is very hard to perform reliable multimodal data fusion. Hence, the above-mentioned fusion methods are not sufficient to establish robust and accurate state estimation. To the best of our knowledge, there have been no reports on multimodal data fusion methods based on a set of predictive models and occupancy grid constraints. In this paper, we propose a novel data fusion method for precise localization problem of driverless car using a set of ARMA predictive models and occupancy grid constraints. It is only based on on-board GPS–IMU and DR navigation data. First, a set of ARMA models with different structural orders are used for concurrent predictions, avoiding prior selection of the order of ARMA models. Second, both grid constraints and spatial consensus check on all predictive data and current measurements are conducted to remove outliers, resulting in stationary stochastic process. Third, the standard deviation of fused data can be controlled by grid size. Finally, the extensive experimental results achieved on field tests under real urban scenarios show that the proposed multimodal data fusion method can not only smooth small jumps in bias due to satellite signal occlusion or multipath but decrease
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accumulated location errors caused by DR. Most importantly, the localization precision of our method outperforms existing state-of-the-art methods in terms of the identical test dataset. This paper is organized as follows. Section 2 proposes our novel data fusion method. The experimental results and performance evaluation are provided in Section 3. Section 4 draws conclusions. 2. Accurate GPS–IMU/DR Data Fusion Method 2.1. Occupancy Grid Constraints-Based Local Navigation Suppose that vehicle displacement at a sampling period provided by GPS–IMU integrated navigation system or DR system is denoted as r∆xk , ∆yk sT , k = 0,1,2¨ ¨ ¨¨. Under normal circumstances, the time series t∆x1 , ∆x2 , ¨ ¨ ¨, ∆xk , ¨ ¨ ¨u and t∆y1 , ∆y2 , ¨ ¨ ¨, ∆yk , ¨ ¨ ¨u are considered as a collection of stationary stochastic processes, which implies that current state is only dependent on previous one-step or multi-step states without nonstationarity. In this paper, we use a couple of ARMAs for modeling such covariance stationary time series data. Notice that we adopt position increments r∆x, ∆ysT instead of absolute positions [x, y]T so as to avoid multilinear problems. In general, the selection of ARMA model order is viewed as the first step prior to parameter estimation. A comprehensive survey on methods for determining the order of ARMA can also see [24]. In the popular traditional methods, optimal model could be found on the basis of certain criterions after completing estimates of model parameters, e.g., final prediction error (FPE) [25], Akaike information criterion (AIC) [26], and minimum description length (MDL) [27,28]. In [29], eigenvectors of covariance matrix of input data rather than parameter estimation are employed to determine the model order. Using Bayesian framework, a new method for jointly estimating model order and parameters is presented in [30]. In fact, there do not exist any generic methods on the best order selection problem, although improvements have been proceeding [31], among which some policy is used to be closer to real structural model at the cost of computational complexity. Thus, the determination of model order is really regarded as one of the most difficulties. In this paper, we present a novel method that has no requirements for determining model order, where a set of ARMA models with multiple different orders are utilized for position predictions that are eventually evaluated and screened by occupancy grid constraints and spatial consensus check, together with current measurements. This leads to reasonable selection of the best structural parameter. The flowchart of the proposed multimodal data fusion method is shown in Figure 1, where ( ( ∆x1G , ∆x2G , ¨ ¨ ¨, ∆xkG and ∆y1G , ∆y2G , ¨ ¨ ¨, ∆ykG represent the eastern and northern displacements ( ( delivered by GPS–IMU, respectively. Similarly, ∆x1D , ∆x2D , ¨ ¨ ¨, ∆xkD and ∆y1D , ∆y2D , ¨ ¨ ¨, ∆ykD denote the eastern and northern displacements given by DR, respectively. The two groups of time series are used to predict current positions though a set of ARMA models with different orders. Considering that ARMA models contains multiple orders, as described as the p-order ARMA (p = 1,2,¨ ¨ ¨,n) in Figure 1, a total of 2n predictions can be yielded. With the addition of current measurements by GPS–IMU and DR, we make further use of grid constraints and spatial consensus check to have removal of outliers, in order to fulfill data fusion for resulting stationary stochastic processes.
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Figure 1. 1. The The flowchart flowchart of of the the proposed proposed data Figure data fusion fusion method. method.
2.2. Prediction Using Using aa Set 2.2. Prediction Set of of ARMA ARMA Models Models with with Different Different Orders Orders After raw data data from from GPS–IMU GPS–IMU and and DR, DR, it it is is required required to to establish set of of ARMA After collecting collecting raw establish aa set ARMA models with multiple structural orders for prediction of localization. In this paper, we adopt 1st order, models with multiple structural orders for prediction of localization. In this paper, we adopt 1st 2nd order, 3rdand order predictive models,models, respectively. WithoutWithout loss of generality, ARMA order, 2nd and order, 3rdARMA order ARMA predictive respectively. loss of generality, predictive model for GPS–IMU data candata be expressed as, ARMA predictive model for GPS–IMU can be expressed as, G
G
G
G
G
G
G
G G x G x 1`1y yGkp `θ1x ∆xkG “ Φ1x ∆yk kG´ Φk1 `k2¨ ¨¨ ` Φ `p θ p ∆xkG´ p p ∆y 2 ∆y 2 ∆x 1 ∆x p k´ k1k´1 2` θk2 p¨ ¨ ¨k k´22y k´2 ` G
G
G
G
G
G
(1a) (1a)
G
1 1 (1b) ykkG´1 ' y ¨ ¨ ` 1 '1 xkG´k11 `θ'21 2x ' x1 G' ∆ykG “ Φ1 1 ∆y ` 1Φy ∆ykG' Φ1 p' ∆ypkGy ∆xk2kG´ 2k1 ´22` ¨ k2 ´ pk`p θ 1 ∆x 2 ` ¨ ¨ ¨ p` θ kp p∆xk´k p ` ε k (1b)
where p∆x ∆y (xkG , ykGkG)q denotes prediction at time step kk using using previous p position increments or T G G G G G G G G displacements , ∆x , p = 1,2,..., n (e.g., 3). ykGk1´, 1,y∆y 2,,¨ y¨ k¨,G ∆y xkGk´2 1, , kx´kG2p, ]¨T¨ ¨,, ∆x p k´= p s1,2, , n (e.g., n n== 3). displacements δk,pk , p “ [r∆y ´k p1,,∆x k 2k,´ p , kx T
T 1 1 , ¨ ¨ ¨Φ1 , θ 1 , θ 1 , ¨ ¨ ¨, θ 1T s represents the 1 T Specifically, ¨ ¨ ¨Φ p , θ1 , θ2 , ¨ ¨ ¨, θ p sor or θ' “ 11,,Φ22, , [ rΦ '1 , 1 ,'2Φ, ' p ,p '1 ,1 '2 ,2, ' p ]p represents the Specifically, θ “ [rΦ 2 p , 1 , 2 , , p ] 2p-dimensional vector constituted by unknown parameters, ε k or ε1k indicates noise, and the superscript 2p-dimensional vector constituted by unknown parameters, k or 'k indicates noise, and the “G” denotes position increments from GPS–IMU data. If it is from DR, the superscript is expressed superscript ‘G’ denotes position increments from GPS–IMU data. If it is from DR, the superscript is by “D”. Assume that ε k is statistically independent and distributed with Gaussian distribution with expressed by ‘D’. Assume that k is statistically independent and distributed with Gaussian zero mean and variance of σ2 . Consequently, likelihood estimation of parameter vector θ is stated 2 distribution with zero mean and variance of . Consequently, likelihood estimation of parameter as follows: vector is stated as follows: The noise probability density function ε k is given by The noise probability density function k is given by 1 pε k q2 2 ppε k q “ expr´ (2) ( k ) 2 s 1 p( ) p2πσ2 q1{2 exp[ 2σ ] (2) k
(2 2 )1/2
2 2
Apparently, we have Apparently, we have 2 ˇ (p∆y ykGkG´ TθTkδ, pk,p )2 q 11 G ˇG pp∆y , θ q “ expr´ p ( k , p , ) exp[ ] s k yˇδ k k,p 2σ2 2 (222q)11/{22 2 p2πσ
(3) (3)
G The left left side side of ofEquation Equation(3) (3)indicates indicatesthat thatthis thisisisthe the probability distribution function of G, yi.e., The probability distribution function of ∆y k , k ˇ ˇy G , ), ~ NT( T 2, 2 ) . i.e., pp∆ypkG(ˇδ k , θk q, , p θ „ Npθ δk,p ,kσ , p q. k,p
Note that the probability of data is a function of Y, which contains all the ykG for a fixed value of , i.e. Y [y1G , y2G , , ykG ]T . If it is considered as a function of , then the likelihood function can be described below, 4
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Note that the probability of data is a function of Y, which contains all the ∆ykG for a fixed value of T
θ, i.e., Y “ r∆y1G , ∆y2G , ¨ ¨ ¨, ∆ykG s . If it is considered as a function of θ, then the likelihood function can be described below, Sensors 2016, 16, 280 Lpθq “ Lpθ; Y, δq “ ppY |δ , θq (4) where δ indicates matrix that contains Lall ( )the Lδ(k,p ; Y. , ) p(Y , ) (4) Considering that ∆ykG is statistically independent, the likelihood function is rewritten as where indicates matrix that contains all the k , p . G 2 ˇ Considering that independent, the as źmyk is statistically źm p∆yG ´ θisTrewritten δ q 1 likelihood function
Lpθq “
k “1
ˇ pp∆ykG ˇδk,p , θq “ m
k “1 m
L( ) k 1 p ( y k , p , ) k 1 G k
1{2 p2πσ 1 2q
(2 2 )1/ 2
expr´
exp[
k.p
k
22
( y k .2σ p) G k
T
2 2
]
(5)
s
(5)
As a result, θ should be estimated so as to make data have high probability. Instead of directly As aLpθq, result, should be estimated so as toofmake data have high probability. Instead of as maximizing wecan also make maximization any strictly increasing function of Lpθq such L ( ) , we can also make maximization of any strictly increasing function of directly maximizing log-likelihood function logLpθq, i.e., L( ) such as log-likelihood function log L( ) , i.e., m ˇ ÿ m Gˇ logLpθq “ logpp∆y (6) G k log L( ) log p (y ˇδk,p , ), θq
k
k“ 1 k 1
k, p
(6)
log L([26], ) [26], we can optimally of parameters of ARMA By maximizing By maximizing thethe logLpθq we can optimally findfind thethe set set of parameters θ of ARMA models. models. Correspondingly, unknown of parameters order, 2nd 3rd predictive order ARMA Correspondingly, unknown parameters 1st order, of 2nd1storder, and 3rdorder, orderand ARMA models predictive models can on-line be estimated. can on-line be estimated. 2.3. Accurate Data Fusion Method 2.3. Accurate Data Fusion Method At time step k, two current measurements from GPS–IMU and DR, respectively, together with At time step k, two current measurements from GPS–IMU and DR, respectively, together with six predictions delivered by the above-mentioned ARMA predictive models with 1st order, 2nd six predictions delivered by the above-mentioned ARMA predictive models with 1st order, 2nd order, order, and 3rd order, are all projected onto identical occupancy grid map for data fusion. Owing to and the 3rdfact order, are all projected onto identical occupancy grid map for data fusion. Owing to the that there always exist measurement noises and prediction errors caused probably by fact incorrect that theremodel always exist measurement and prediction errors caused probably by incorrect orders and nonlinearities,noises this paper presents a novel policy of eliminating outliers model orders and nonlinearities, this paper presents a novel policy of eliminating outliers through occupancy grid constraints and spatial consensus check (shown in Algorithm l). In through the occupancy andwe spatial consensus (shown in Algorithm l). In of the accurate data accurategrid dataconstraints fusion method, only choose thosecheck grid cells that contain the majority predictions andmethod, current measurements, including that cells of falling the frontier. With theof grids map, the number fusion we only choose those grid thaton contain the majority predictions and current of points falling into the grid on is counted. In this case, data of falling into an measurements, including thatsame of falling the frontier. With the dense grids map, thesolely number of points falling gridisofcounted. H × W are retained as inliers, sparse datainto scattered in other grid gridsofare into occupancy the same grid In this case, dense datawhile of solely falling an occupancy HˆW classifiedasasinliers, outliers. Our empirical data illustrates the selection of both Has and W equal to 0.2 m are retained while sparse data scattered in that other grids are classified outliers. Our empirical leads to the best performance. After eliminating outliers, we conduct refinement of inliers through data illustrates that the selection of both H and W equal to 0.2 m leads to the best performance. After Algorithm 1. Specifically, by iteratively filtering noise inside grids, the best localization of centroids eliminating outliers, we conduct refinement of inliers through Algorithm 1. Specifically, by iteratively could be estimated among inliers. filtering noise inside grids, the best localization of centroids could be estimated among inliers. Figure 2 shows the grid-based data fusion scheme, where the current navigation data Figure 2 shows the grid-based data where the current navigation datamodels measured measured by GPS–IMU and DR, as wellfusion as thescheme, predictions obtained using multiple ARMA by GPS–IMU DR, as as the predictions usingthe multiple ARMA models presented and in Section 2.2well are all indicated in blackobtained points, while grey points stand for thepresented outliers in Section are all black points, while thethe grey points stand forevaluated the outliers that should be that2.2 should be indicated eliminatedin and the red points denote data fusion results in accordance with ourand method. eliminated the red points denote the data fusion results evaluated in accordance with our method.
Figure 2. 2. Occupancy fusionmethod method GPS–IMU Figure Occupancygrid gridbased based data data fusion forfor GPS–IMU andand DR.DR.
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Algorithm 1. Spatial Consensus Check !´ ¯ ¯ ` ˘ ´ ` ˘) Input: ∆xkG´n , ∆ykG´n , ¨ ¨ ¨ , ∆xkG , ∆ykG , ∆xkD´n , ∆ykD´n , ¨ ¨ ¨ , ∆xkD , ∆ykD ` ˘( Output: ∆x k , ∆yk for i = 1 to n do G G G G G G G G ∆ xrki “ ∅G xi1 ∆xk´1 ` ¨ ¨ ¨ ` ∅xii ∆xk´i ` θ xi1 ∆yk´1 ` ¨ ¨ ¨ ` θ xii ∆yk´i D D D D D ∆ xrki “ ∅xi1 ∆xkD´1 ` ¨ ¨ ¨ ` ∅xii ∆xkD´i ` θ xi1 ∆ykD´1 ` ¨ ¨ ¨ ` θ xii ∆ykD´i G G G G G ∆yrki “ ∅yi1 ∆xkG´1 ` ¨ ¨ ¨ ` ∅yii ∆xkG´i ` θyi1 ∆ykG´1 ` ¨ ¨ ¨ ` θyii ∆ykG´i D D D D D ∆yrki “ ∅yi1 ∆xkD´1 ` ¨ ¨ ¨ ` ∅yii ∆xkD´i ` θyi1 ∆ykD´1 ` ¨ ¨ ¨ ` θyii ∆ykD´i
end 1 1 n1 “ n2 “ n for i = 1 to Max_iters do ∆x k “
G ` ¨ ¨ ¨ ` ∆x D ` ¨ ¨ ¨ ` ∆x rG 1 ` ∆ xrk1 rD 1 ` ∆xkG ` ∆xkD ∆ xrk1
∆yk “
kn2
kn1
1
1
n1 ` n2 ` 2 G ` ¨ ¨ ¨ ` ∆y D ` ¨ ¨ ¨ ` ∆y rD 1 ` ∆ykG ` ∆ykD rG 1 ` ∆yrk1 ∆yrk1 kn2
kn1
1
1
n1 ` n2 ` 2 Ñ
1
1
Compute the distance L of all n1 ` n2 ` 2 points ˆ ˙ to centroid points. ∆x1 ,
Ñ L1
Ñ
´ L1 ˘ “ ` 1 1 n1 ` n2 ` 2 ´ 1 sum
L
For a point ∆y , denotes ` 1 ˘ || ∆x , ∆y1 || ´ p∆x, ∆yq ą threshold, then if L1 Remove this point. 1 1 Update n1 or n2 . end `
˘ 1
end
After iteratively filtering noise inside grids, we get the final increments at time step k by averaging all the inliers. 3. Experimental Results To evaluate navigation performance of our method, we performed extensive on-site navigation experiments on our driverless car shown in Figure 3. In this section, we first make analysis of raw navigation data given by GPS–IMU and DR. Along the ground truths of autonomously driving trajectories, we then investigated position errors for stand-alone GPS–IMU, DR and our data fusion method. The ground truth is found through tight integration of NovAtel GPS receiver and IMU in the open air when the GPS mode is RTK. In addition, on the basis of datasets provided by [15], we conducted comparative study of the proposed method with state-of-the arts such as state dependent Riccati equation (SDRE) navigation filtering [15,16], which is an alternative to the EKF. Evaluation of four Kalman filtering based state estimation architectures is given in [15], which demonstrates that the NLM outperforms the other three architectures in the experiments provided in [15]. Hence, our method will be made comparison with SDRE in Section 3.2. Finally, the multimodal data fusion results in the presence of interrupted GPS–IMU signals will be further discussed.
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Figure 3. 3. Driverless byTsinghua Tsinghua University. Figure Driverlesscar cardeveloped developed by University.
3.1. Analysis of Raw Navigation Data 3.1. Analysis of Raw Navigation Data Let first us first examine characteristics of of raw raw navigation by by GPS–IMU that that is is Let us examine thethecharacteristics navigationdata datacollected collected GPS–IMU installed on our driverless car. Wekept keptaarecord record of data over 10 min whenwhen installed on our driverless car. We of position positionand andorientation orientation data over 10 min the driverless car parking at three different locations (also see Table 1). It is observed from Table 1 the driverless car parking at three different locations (also see Table 1). It is observed from Table 1 that there is significant change in x-coordinates at the identical location #1 over 10 min, the maximum that there is significant change in x-coordinates at the identical location #1 over 10 min, the value of which is up to 6.102 m with the standard deviation of 1.428. Note that location #1 is just maximum value of which is up to 6.102 m with the standard deviation of 1.428. Note that location #1 under overpass with heavy traffic flow. In this situation, GPS–IMU data become instable due to severe is just under overpass with heavy traffic flow. In this situation, GPS–IMU data become instable due satellite signal occlusion and multipath effects. However, position data of (x,y) recorded at locations #2 to severe multipath However, position data of (x,y) recorded at and #3satellite always signal remain occlusion very stableand because they areeffects. sampled in an open site. locations #2 and #3 always remain very stable because they are sampled in an open site. Table 1. Raw records of position data over 10 min.
Table 1. Raw records of position data over 10 min. (X[m], Y[m]) (X[m], Y[m])
#1 #2 #3 #1 #2 locations numbered #1, #2, #3#3 The driverless car parked at three different
car parked at three different numbered (X_min, Y_min) The driverless (´2521.839, 1990.442) (´215.399, 190.123) locations (´0.013, ´0.001) #1, #2, (X_max, Y_max) (´2515.737, 1990.554) (´215.393, 190.134) (0.099,0.397) (X_min, Y_min) (−2521.839, 1990.442) (−215.399, 190.123) (−0.013, −0.001) (X_average, Y_average) (´2516.651, 1990.461) (´215.396, 190.129) (0.038, 0.255) (X_max, Y_max) (−2515.737, 1990.554) (−215.393, 190.134) (X_stdev, Y_stdev) (1.428, 0.022) (0.001,0.002) (0.037,(0.099,0.397) 0.128)
#3
(X_average, Y_average) (−2516.651, 1990.461) (−215.396, 190.129) (0.038, 0.255) (X_stdev, Y_stdev) (1.428, 0.022) (0.001,0.002) (0.037, 0.128) Meanwhile, our driverless car is equipped with two wheel encoders (Nemicon SBH-1024-2MD), which can precisely produce the rolling turns of wheels. We then calculate trajectory distances of the Meanwhile, our driverless car isof equipped with two wheel encoders (Nemicon SBH-1024-2MD), two wheels and take the averaging them as DR outputs. Unlike GPS–IMU data, DR data look like which can stationary precisely but produce the rolling turns of wheels. rather have unusual accumulated errors. We then calculate trajectory distances of the
two wheels and take the averaging of them as DR outputs. Unlike GPS–IMU data, DR data look like 3.2.stationary Comparative Study rather but have unusual accumulated errors. Figure 4 shows four moving trajectories of driverless cars on a real route containing curved roads,
3.2. Comparative Study measurements, DR outputs, and data fusion obtained using our methods, as including GPS–IMU well as the ground truth. Apparently, the former three trajectories are roughly consistent with the Figure 4 shows four moving trajectories of driverless cars on a real route containing curved ground truth. To gain more clear insights about performance, we have four local trajectories enlarged, roads, GPS–IMU measurements, outputs, and obtained using our methods, as including shown in Figure 5. Notice that it has theDR same legend as in data Figurefusion 4.
as well as the ground truth. Apparently, the former three trajectories are roughly consistent with the ground truth. To gain more clear insights about performance, we have four local trajectories enlarged, as shown in Figure 5. Notice that it has the same legend as in Figure 4.
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Figure 4. The four global trajectories recorded by vehicle. Figure le. Figure 4. 4. The The four four global global trajectories trajectories recorded recorded by vehic vehicle.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Figure 5. 5. Locally Locally enlarged enlarged trajectories. trajectories. (a) beginning; (b) the vicinity vicinity of of the the Figure (a) In In the the vicinity vicinity of of the the beginning; (b) In In the Figure 5. Locally enlarged trajectories. (a) In the vicinity of the beginning; (b) In the vicinity of the first curved curved segmentation; segmentation; (c) (c) In In the the vicinity vicinity of of the the second second curved curved segmentation; segmentation; (d) (d) In In the the vicinity vicinity of of first first curved segmentation; (c) In the vicinity of the second curved segmentation; (d) In the vicinity of the destination. destination. the the destination.
Figure 5 demonstrates that the fusion results we achieved are almost between the GPS–IMU Figure 55 demonstrates wewe achieved areare almost between the GPS–IMU raw Figure demonstratesthat thatthe thefusion fusionresults results achieved almost between the GPS–IMU raw data and DR data, keeping smooth and steady, even in curved segmentation. Compared to the data and DR data, keeping smooth and steady, even in curved segmentation. Compared to the ground raw data and DR data, keeping smooth and steady, even in curved segmentation. Compared to the ground truth, our data fusion results are much closer to the actual localization physically going with truth, our data fusion resultsresults are much closer closer to thetoactual localization physically going withwith the ground truth, our data fusion are much the actual localization physically going the driverless car. To demonstrate the relative localization performances, Figure 6 (left) shows the driverless car. To the relative localization performances, Figure 6Figure (left) shows average the driverless car.demonstrate To demonstrate the relative localization performances, 6 (left)the shows the average position error curves of the three trajectories derived from GPS–IMU, DR, and our method, position error curves of the three trajectories derived from GPS–IMU, DR, and our method, respectively. average position error curves of the three trajectories derived from GPS–IMU, DR, and our method, respectively. The final average position error is only 0.18 m for the fusion data when the DR has The final average error isposition only 0.18 m for data the DRdata has already an DR errorhas of respectively. The position final average error is the onlyfusion 0.18 m forwhen the fusion when the already an error of 0.49 m. The difference between the fusion data and the reference is also plotted in 0.49 m. The difference the fusion data and the fusion reference is also in Figure 6 (right). already an error of 0.49between m. The difference between data and plotted the reference is also plottedThe in Figure 6 (right). The left curves in Figure 6 indicate average position errors. In other words, position Figure 6 (right). The left curves in Figure 6 indicate average position errors. In other words, position 8 8
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Sensors 2016, 16, 280 errors will be averaged over time, which can clearly reflect accumulation of errors. However, the left curves in Figure 6 indicate average position errors. In other words, position errors will be averaged right ones represent position errors at a specific time in our experiment. The overall position errorstime, willwhich be averaged overreflect time,accumulation which can clearly reflect accumulation errors. However, the over can clearly of errors. However, the rightofones represent position root–mean-square (RMS)position errors are calculated to be time 0.47 m for GPS–IMU 0.58 m overall for DR,position and 0.25 m right ones represent errors at a specific in our experiment. The errors at a specific time in our experiment. The overall position root–mean-square (RMS) errors are for the proposed method, respectively. Actually, our method still needs to be root–mean-square are calculated beand 0.47 mfusion for GPS–IMU 0.58 mmethod, for DR,respectively. and 0.25further m calculated to be 0.47(RMS) m for errors GPS–IMU 0.58 m for to DR, 0.25 m for the proposed improved turning-about, even if be the fusion results aremethod much better than of either for thewhen proposed method, respectively. Actually, our fusion still needs to ifthat be Actually, our fusion method still needs to further improved when turning-about, even thefurther fusion GPS–IMU or data for straight roads. It isresults caused by or big azimuth of DR improved when turning-about, if the fusion are for much better than thatroads. oferror either results areDR much better than thatoreven ofcurved either GPS–IMU orprobably DR data straight curved It is GPS–IMU or DR by data straighterror or curved It is probably caused by big azimuth error of DR when crossing intersection. probably caused bigfor azimuth of DR roads. when crossing intersection.
when crossing intersection.
Figure 6. position errorcurves curves(Left) (Left) evaluated evaluated DR, andand our our datadata fusion 6. Average evaluated using Figure 6. Average position error usingIMU–GPS, IMU–GPS, DR, fusion method. position obtained the norm of difference between the ground truth and the Errors in are as method. Errors in position are obtained as the norm of difference between the ground truth and the fusion data at each time step (Right). step (Right). fusion data at each time step (Right).
In general, the EKF is proven to work very well among all data fusion methods. Using the
In general, EKF provento to work work very data fusion methods. Using the the In general, thethe EKF is isproven very well wellamong amongallall data fusion methods. Using datasets on real road and the code implementation provided by [15], we carried out a comparative datasets on real road and thecode codeimplementation implementation provided by [15], wewe carried outout a comparative datasets onofreal road and the provided by [15], carried a comparative study of our method method with with NLM NLM (SDRE (SDRE filter). filter). The are shown shown in in Figures Figures 77 and and 8. 8. Figure Figure 77 study our The results results are study of our method with NLM (SDRE filter). The results are shown in Figures 7 and 8.ofFigure 7 shows the trajectories obtained by different fusion methods and the average position errors the shows the trajectories obtained by different fusion methods and the average position errors of the shows the trajectories obtained by different methods and the average position errors of the proposed method and and NLM when when comparedfusion with the the ground truth. truth. Figure depicts our fusion fusion data proposed method NLM compared with ground Figure 88 depicts our data proposed method andvalues. NLM Note whenthat compared with thewas ground truth. Figure 8 depicts ourtracking fusion data with the the reference values. Note that the ground ground truth was obtained using top-down camera tracking with reference the truth obtained using top-down camera of thereference driverlessvalues. car. The The precision of visual visual tracking system was experimentally experimentally determined totracking be withof the Note that the ground truth was obtained using top-down camerato the driverless car. precision of tracking system was determined be 15 cm ± 12 cm within 15 m × 10 m outdoor area. The red solid curve represents the experimental of the driverless car. The precision of visual tracking system was experimentally determined 15 cm ˘ 12 cm within 15 m ˆ 10 m outdoor area. The red solid curve represents the experimentalto be results achieved with NLM, while our position position error is red marked incurve green solid solid line. It is easy easy to see see 15 cm ± 12 achieved cm within 15NLM, m × 10 m outdoor area. Theis solid represents experimental results with while our error marked in green line. It the is to from Figure 77with that our our method outperforms NLM. Theisposition position RMS errors of of the above-mentioned above-mentioned from Figure that method outperforms NLM. The RMS errors the results achieved NLM, while our position error marked in green solid line. It is easy to see methods are listed in Table 2. listed Table 2. outperforms NLM. The position RMS errors of the above-mentioned frommethods Figure 7are that ourinmethod methods are listed in Table 2.
Figure 7. 7. Trajectories obtained by our method and with with NLM NLM [15] [15] (Left). (Left). The evolving of average average position errors errors over over time time (Right). (Right).
Figure 7. Trajectories obtained by our method and with NLM [15] (Left). The evolving of average position errors over time (Right).
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Figure 8. The corrected position (Left) corresponding to the trajectory in Figure 7. Errors in position are obtained as the norm of difference between the ground truth and our fusion data at each time-step (Right).
Table 2. Performance comparison of our method with the nonlinear model model (NLM) (NLM) in in [15]. [15].
NLM[15] NLM[15] Our Method
North Position RMSE (m) 0.8241 0.8241 0.7973
Our Method
0.7973
Approach Approach
East Position RMSE (m) 0.3268 0.3268 0.3171
North Position RMSE (m)
East Position RMSE (m)
0.3171
Position RMSE Position RMSE(m) (m) 0.8865
0.8865 0.8581 0.8581
From Figures 7 and 8 and Table 2, it is readily observed that our method performs better than From in Figures 7 and 8outperforms and Table 2,classical it is readily observed our method performs the NLM [15], which NLM withoutthat complementary [16] andbetter ERM than [17]. the NLM in [15], which outperforms classical NLM without complementary [16] and ERM [17]. Additionally, we conduct an additional analysis of computational load for the above methods. Note Additionally, conduct an additional analysis using of computational load(MATLAB) for the above methods. Note that the cycle we duration of the NLM is estimated the simulation [15], while that of that the cycle duration of the NLM is estimated using the simulation (MATLAB) [15], while that of our new method is evaluated through the on-board system of our driverless car implemented by C++. our new method is evaluated through the on-board system of our driverless car implemented by And the results are listed in Table 3. Although the comparison is unfair, it illustrates that the proposed C++. And the have results are listed in Table 3.inAlthough the comparison is unfair, it load. illustrates that the method does superior performance terms of accuracy and computational proposed method does have superior performance in terms of accuracy and computational load. Table 3. Analysis of computational load.
3.3. Data Fusion Results with Interrupted GPS–IMU Signal
NLM + CF positionOur Method delivered by When satellite signal occlusion NLM or multipath error occurs, estimates Cyclebe duration 0.8 0.005 results obtained GPS–IMU might severely biased occasionally. Figure 8 1.0 shows the experimental (ms) using our fusion method, which indicates that the proposed method is capable of providing steady and continual vehicle trajectories even if there is bias or even big interruption in GPS–IMU signals. 3.3. Data Fusion Results with Interrupted GPS–IMU Signal
Table 3. Analysis of computational load.
When satellite signal occlusion or multipath error occurs, position estimates delivered by GPS–IMU might be severely biased occasionally. Figure the experimental results obtained NLM NLM 8+ shows CF Our Method using our fusion method, which indicates that the proposed method is capable of providing steady Cycle duration (ms) 0.8 1.0 0.005 and continual vehicle trajectories even if there is bias or even big interruption in GPS–IMU signals. It is readily observed from Figure 9 that there indeed exist small discontinuity jumps in bias due It is readily observed from Figure 9 that there indeed exist small discontinuity jumps in bias due to satellite signal occlusion or multipath, as indicated in the red marked GPS–IMU raw data, which is to satellite signal occlusion or multipath, as indicated in the red marked GPS–IMU raw data, which occasionally even up to 2 m in bias and intolerable for navigation systems of driverless cars. However, is occasionally even up to 2 m in bias and intolerable for navigation systems of driverless cars. our new data fusion method makes vehicle trajectories perfectly smooth, as shown in curves marked However, our new data fusion method makes vehicle trajectories perfectly smooth, as shown in in green and the localization accuracy is almost unaffected by a jump in GPS–IMU data. In fact, the curves marked in green and the localization accuracy is almost unaffected by a jump in GPS–IMU proposed adaptive data fusion method is practically applied in THU-IV2 driverless cars developed data. In fact, the proposed adaptive data fusion method is practically applied in THU-IV2 driverless by ourselves. cars developed by ourselves.
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Vehicle Position
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Figure 9. 9. The data fusion results when interrupted GPS–IMU signals are present for straight roads Figure The data fusion results when interrupted GPS–IMU signals are present for straight (Left) and curved roads (Left) and roads curved(Right). roads (Right).
Conclusions 4. 4.Conclusions thispaper, paper,we wepropose propose aa new new multimodal multimodal data InInthis data fusion fusion method methodfor foraccurate accuratenavigation navigationofof driverless cars based on a set of predictive models and occupancy grid constraints. The novelty driverless cars based on a set of predictive models and occupancy grid constraints. The novelty of of our our method includes: (1) we employ a set of ARMA models with different structural orders to method includes: (1) we employ a set of ARMA models with different structural orders to concurrently concurrently make location predictions, avoiding subjectivelythe determining the order of ARMA make location predictions, avoiding subjectively determining order of ARMA models; (2) both models; (2) both grid constraints and spatial consensus check are presented to have removal of grid constraints and spatial consensus check are presented to have removal of outliers, in order to outliers, in order to generate stationary stochastic process; and (3) the standard deviation of data generate stationary stochastic process; and (3) the standard deviation of data fusion can be controlled fusion can be controlled by size of grid in advance. To evaluate navigation performance of the by size of grid in advance. To evaluate navigation performance of the proposed method, we conduct a proposed method, we conduct a considerable amount of on-site experiments. The experimental considerable amount of on-site experiments. The experimental results demonstrate that our method results demonstrate that our method can not only smooth small jumps in bias due to satellite signal can not only smooth small jumps in bias due to satellite signal occlusion or multipath but also achieve occlusion or multipath but also achieve promising localization fusion precision. Although promising localization precision. Although accumulated position caused by DR are accumulated position fusion errors caused by DR are significantly reduced, thereerrors still remains a certain significantly there remains certainInvalue with the increase of in a sense. value with reduced, the increase of still distance in aasense. particular, degradation in distance fusion accuracy willIn particular, degradation in fusion accuracy will occur when GPS–IMU retains discontinuity jumps occur when GPS–IMU retains discontinuity jumps in bias for a longer period of time. In this case, a in policy bias for longer period of time. In this case, a policy on blocking GPS–IMU data sources should onablocking GPS–IMU data sources should be adopted. be adopted. Acknowledgments: The authors would like to thank the anonymous reviewers for their valuable comments Acknowledgments: The authors to would like to thank the anonymous for their that that considerably contributed improving this paper. This work reviewers was supported in valuable part by comments the National considerably contributed to improving this paper. This work was supported in part by the National Science Science Foundation of China (NSFC) under Grant Nos. 91420106, 90820305, and 60775040, and by the National Foundation of China (NSFC) under Grant Nos. 91420106, 90820305, and 60775040, and by the National High-Tech High-Tech R&D Program of China under Grant No. 2012AA041402. R&D Program of China under Grant No. 2012AA041402. Author Contributions: Zhidong Zhidong Deng thethe experiments; Shiyao WangWang performed the Author Contributions: Deng conceived conceivedand anddesigned designed experiments; Shiyao performed Zhidong Deng and Shiyao Wang analyzed thethe data; Yin analysis tools; theexperiments; experiments; Zhidong Deng and Shiyao Wang analyzed data;Gang Gang Yincontributed contributed analysis tools; Zhidong Deng, ShiyaoWang, Wang,and andGang GangYin Yinwrote wrotethe thepaper. paper. Zhidong Deng, Shiyao
Conflicts of of Interest: The Conflicts Interest: Theauthors authorsdeclare declareno noconflict conflictof ofinterest. interest.
References References 1. 1. Bouvet, D.;D.; Garcia, G. Improving the accuracy of dynamic localization systemssystems using RTK GPSRTK by identifying Bouvet, Garcia, G. Improving the accuracy of dynamic localization using GPS by identifying the GPS latency. In Proceedings of the IEEE International on the Robotics and the GPS latency. In Proceedings of the IEEE International Conference Conference on the Robotics and Automation Automation (ICRA), San CA,April USA,2000; 24–28Volume April 2000; Volume 3, pp. 2525–2530. (ICRA), San Francisco, CA,Francisco, USA, 24–28 3, pp. 2525–2530. Karim Al-Jewari, Y.H.; Ahmad, AlRawi, A.A.A. Impact of multipath interference and change of Al-Jewari, Y.H.; Ahmad, R.B.;R.B.; AlRawi, A.A.A. Impact of multipath interference and change of velocity 2. 2. Karim velocity on the reliability and precision of GPS. In Proceedings of the 2nd International Conference on the on the reliability and precision of GPS. In Proceedings of the 2nd International Conference on the Electronic Electronic Design (ICED), Penang,19–21 Malaysia, 19–21 August 2014; pp. 427–430. Design (ICED), Penang, Malaysia, August 2014; pp. 427–430. Chen,X.;X.;Dovis, Dovis,F.;F.;Peng, Peng,S.; S.; Morton, Morton, Y. Y. Comparative Comparative studies 3. 3. Chen, studies of of GPS GPS multipath multipathmitigation mitigationmethods methods performance.IEEE IEEETrans. Trans.Aerosp. Aerosp.Electron. Electron. Syst. Syst. 2013, 2013, 49, performance. 49, 1555–1568. 1555–1568. [CrossRef] Oryschuk,P.;P.;Salerno, Salerno,A.; A.;Al-Husseini, Al-Husseini, A.M.; A.M.; Angeles, Angeles, J. 4. 4. Oryschuk, J. Experimental Experimentalvalidation validationofofananunderactuated underactuated two-wheeled mobile robot. IEEE ASME Trans. Mechatron. 2009, 14, 252–257. two-wheeled mobile robot. IEEE ASME Trans. Mechatron. 2009, 14, 252–257. [CrossRef]
11
Sensors 2016, 16, 280
5.
6.
7.
8. 9. 10.
11.
12.
13.
14. 15. 16. 17. 18.
19.
20.
21.
22.
23.
12 of 13
Kleeman, L. Optimal estimation of position and heading for mobile robots using ultrasonic beacons and dead-reckoning. In Proceedings of the IEEE International Conference on the Robotics and Automation, Nice, France, 12–14 May 1992; Volume 3, pp. 2582–2587. Yi, J.; Zhang, J.; Song, D.; Jayasuriya, S. IMU-based localization and slip estimation for skid-steered mobile robots. In Proceedings of the IEEE/RSJ International Conference on the Intelligent Robots and Systems (IROS), San Diego, CA, USA, 29 October–2 November 2007; pp. 2845–2850. Jo, K.; Chu, K.; Sunwoo, M. GPS-bias correction for precise localization of autonomous vehicles. In Proceedings of the IEEE Intelligent Vehicles Symposium (IV), Gold Coast, Australia, 23–26 June 2013; pp. 636–641. Aghili, F.; Salerno, A. Driftless 3-D attitude determination and positioning of mobile robots by integration of IMU with two RTK GPSs. IEEE ASME Trans. Mechatron. 2013, 18, 21–31. [CrossRef] Arsenault, A.; Velinsky, S.A.; Lasky, T.A. A low-cost sensor array and test platform for automated roadside mowing. IEEE ASME Trans. Mechatron. 2011, 16, 592–597. [CrossRef] Ilyas, M.; Yang, Y.; Qian, Q.S.; Zhang, R. Low-cost IMU/odometer/GPS integrated navigation aided with two antennae heading measurement for land vehicle application. In Proceedings of the 25th Chinese Control and Decision Conference (CCDC), Guiyang, China, 25–27 May 2013; pp. 4521–4526. Salmon, D.C.; Bevly, D.M. An exploration of low-cost sensor and vehicle model solutions for ground vehicle navigation. In Proceedings of the IEEE/ION Position Location and Navigation Symposium (PLANS), Monterey, CA, USA, 5–8 May 2014; pp. 462–471. Ma, J.; Susca, S.; Bajracharya, M.; Matthies, L.; Malchano, M.; Wooden, D. Robust multi-sensor, day/night 6-DOF pose estimation for a dynamic legged vehicle in GPS-denied environments. In Proceedings of the IEEE International Conference on the Robotics and Automation (ICRA), Saint Paul, MN, USA, 14–18 May 2012; pp. 619–626. Tao, Z.; Bonnifait, P.; Fremont, V.; Ibanez-Guzman, J. Mapping and localization using GPS, lane markings and proprioceptive sensors. In Proceedings of the IEEE/RSJ International Conference on the Intelligent Robots and Systems (IROS), Tokyo, Japan, 3–7 November 2013; pp. 406–412. Gruyer, D.; Belaroussi, R.; Revilloud, M. Map-aided localization with lateral perception. In Proceedings of the IEEE Intelligent Vehicles Symposium, Dearborn, MI, USA, 8–11 June 2014; pp. 674–680. Simanek, J.; Reinstein, M.; Kubelka, V. Evaluation of the EKF-based estimation architectures for data fusion in mobile robots. IEEE ASME Trans. Mechatron. 2015, 20, 985–990. [CrossRef] Nemra, A.; Aouf, N. Robust INS/GPS sensor fusion for UAV localization using SDRE nonlinear filtering. IEEE Sens. J. 2010, 10, 789–798. [CrossRef] Shin, E.-H. Estimation Techniques For Low-Cost Inertial Navigation; UCGE Reports Number 20219; Department of Geomatics Engineering, University of Calgary: Calgary, AB, Canada, 2005. Kubelka, V.; Reinstein, M. Complementary filtering approach to orientation estimation using inertial sensors only. In Proceedings of the IEEE International Conference on the Robotics and Automation (ICRA), Saint Paul, MN, USA, 14–18 May 2012; pp. 599–605. Vasconcelos, J.F.; Cardeira, B.; Silvestre, C.; Oliveira, P.; Batista, P. Discrete-time complementary filters for attitude and position estimation: design, analysis and experimental validation. IEEE Trans. Control Syst. Technol. 2011, 19, 181–198. [CrossRef] Brink, K.; Soloviev, A. Filter-based calibration for an IMU and multi-camera system. In Proceedings of the IEEE/ION Position Location and Navigation Symposium (PLANS), Myrtle Beach, SC, USA, 23–26 April 2012; pp. 730–739. Mebarki, R.; Lippiello, V. Image moments-based velocity estimation of UAVs in GPS denied environments. In Proceedings of the IEEE International Symposium on the Safety, Security, and Rescue Robotics (SSRR), Hokkaido, Japan, 27–30 October 2014; pp. 1–6. Vincenzo Angelino, C.; Baraniello, V.R.; Cicala, L. High altitude UAV navigation using IMU, GPS and camera. In Proceedings of the 16th International Conference on the Information Fusion (FUSION), Istanbul, Turkey, 9–12 July 2013; pp. 647–654. Taylor, C.N. An analysis of observability-constrained Kalman filtering for vision-aided navigation. In Proceedings of the IEEE/ION Position Location and Navigation Symposium (PLANS), Myrtle Beach, SC, USA, 23–26 April 2012; pp. 1240–1246.
Sensors 2016, 16, 280
24. 25. 26. 27. 28. 29. 30.
31.
13 of 13
De Gooijer, J.G.; Abraham, B.; Gould, A.; Robinson, L. Methods for determining the order of an autoregressive-moving average process: A survey. Int. Stat. Rev. 1985, 53, 301–329. Akaike, H. Fitting autoregressive models for prediction. Ann. Inst. Stat. Math. 1969, 21, 243–247. [CrossRef] Akaike, H. Information theory and an extension of the maximum likelihood principle. In Selected Papers of Akaike; Parzen, E., Tanabe, K., Kitagawa, G., Eds.; Springer: New York, NY, USA, 1998; pp. 199–213. Schwarz, G. Estimating the dimension of a model. Ann. Stat. 1978, 6, 461–464. [CrossRef] Rissanen, J. Modeling by shortest data description. Automatica 1978, 14, 465–471. [CrossRef] Liang, G.; Wilkes, D.M.; Cadzow, J.A. ARMA model order estimation based on the eigenvalues of the covariance matrix. IEEE Trans. Signal Process. 1993, 41, 3003–3009. [CrossRef] Ehlers, R.S.; Brooks, S.P. Bayesian Analysis of Order Uncertainty in ARIMA Models. Available online: https://www.researchgate.net/profile/Stephen_Brooks3/publication/228531965_Bayesian_analysis_of _order_uncertainty_in_arima_models/links/00463516679960b00a000000.pdf (accessed on 15 January 2006). Cassar, T.; Camilleri, K.P.; Fabri, S.G. Order estimation of multivariate ARMA models. IEEE J. Sel. Top. Signal Process. 2010, 4, 494–503. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
