 Original article
 Open Access
A posteriori estimation of stochastic model for multisensor integrated inertial kinematic positioning and navigation on basis of variance component estimation
 Kun Qian^{1},
 Jianguo Wang^{1}Email authorView ORCID ID profile and
 Baoxin Hu^{1}
https://doi.org/10.1186/s4144501600055
© The Author(s) 2016
 Received: 2 August 2016
 Accepted: 7 September 2016
 Published: 4 November 2016
Abstract
Improving a priori stochastic models of the process and measurement noise vectors in Kalman Filer (KF) has always been a challenge. As one preferable technique to address this challenge, the variance component estimation (VCE) applied on the Kalman Filter’s process and measurement noise covariance matrix (Q & R) has been proved in plenty of applications. Unsurprisingly, VCE was expected to reestablish the stochastic model about the random errors in the IMU’s measurements in a multisensor integrated positioning and navigation system applying Kalman Filter. However, in the conventional error statesbased GPS aided inertial navigation system (GPS/INS), the stochastic model tuning is difficult for the IMU’s measurements due to the amalgamation of the observables from inertial sensor and other aiding sensors. This paper proposes a generic method for the stochastic model tuning about the random errors in IMU measurements together with other sensors. The core of this novel approach is based on an innovative multisensor integration strategy which deploys upon the vehicle’s generic kinematic model and takes the IMU’s output as raw measurements in Kalman Filter (IMU/GNSS Kalman Filter). As a result, the statistical orthogonality between random error vectors of any two sensors enables the separate but parallel statistics collection of each individual random error source. Given these independent statistics corresponding to each error source, the VCE technique iteratively tunes all stochastic model of the process and measurement noise vectors. The success of the VCE algorithm is shown through a real dataset involving GPS and inertial sensors.
Keywords
 Variance component estimation
 Kalman filter
 GPS/INS
 IMU/GNSS KF
 Stochastic model
Introduction
The solution optimality of the Kalman filter (KF) relies on the appropriate stochastic model, which is commonly, also here specifically, about the variancecovariance (VC) matrices Q and R associated with the process and measurement noise vectors. The determination of these two covariance matrices in Kalman filter has been actively pursued by plenty of researchers since the advent of Kalman filter. Despite of the variation in numerous applicationbased algorithms, their methodologies can be roughly classified into two categories: VC matrix estimation (VCME) and variance component (scale factor) estimation (VCE). Mehra (1970, 1972) published his pioneering VCME work on the estimation of Q and R based on the system innovations in steadystate KF. Four relevant methods briefed in his work are: Bayesian, maximum likelihood, correlation and covariance matching. Mehra’s work has still been of important directive significance to various recent researches on VCME (Dunık and Šimandl 2008; Bavdekar, et al. 2011; Bulut et al. 2011; Matisko and Havlena 2013; etc.).

Kalman filter is constructed epoch wise by applying the least squares principle, which utilizes all of the random information as three groups of statistically independent measurements: the predicted state vector as a group of pseudomeasurements, the zero mean process noise vector also as a group of pseudomeasurements, and the raw measurement vector, whose residual or correction vectors can directly be calculated as the projection of the system innovation vector.

The redundancy distribution associated with the above mentioned three groups of measurements, and the redundant index for each of the individual independent measurements can be calculated epoch wise after the reliability theory transplanted from Least Squares into Kalman filter.
Given the redundancy contribution indexes and residuals for the individual variance components, the simplified VCE algorithm has accordingly been developed (Förstner 1979; Wang 1997; Wang 2009; etc.). For more details on its practical applications, one can refer to (Wang, et al. 2009; Gopaul, et al. 2010; Wang, et al. 2010; etc.).
However, in a traditional multisensoraided inertial navigation system based on the errorstate Kalman filter (socalled indirect Kalman filter), the variance component estimation technique depending on the measurement residuals is not applicable for individual sensor’s stochastic model tuning because the error measurements involve two sensors and their measurement residuals cannot be used to construct the stochastic model for individual sensor. This paper addresses the abovementioned problem in the errorstatebased INS system by taking the advantages of the novel multisensor integration strategy developed in (Qian et al. 2013, 2015; Wang et al. 2014) and succeeds in the stochastic model tuning for noise vectors, especially the posteriori stochastic model for the measurements from an IMU. The new integration strategy is designed to take a 3D kinematic trajectory model as the system model and therefore, allow applying the raw measurements from all of the sensors, inclusive of the IMUs, directly in measurement updates.
This paper is divided into six sections. This introduction is followed by Section VCE using redundancy contribution, which reviews variance component estimation using redundancy contribution in Least Square and Kalman Filter. Then, Section Multisensor integration for kinematic navigation and positioning using IMU/GNSS KF describes the direct Kalman filter used in full tightlycoupled multisensor integrated kinematic positioning and navigation. Section VCE realization under the novel multisensor integration strategy details the implementation of the VCE algorithm with certain discussions. The results from real road test data are provided to show the success of the proposed VCE algorithm in Section Road test results. Section Conclusion and future work concludes the paper.
Methods
The rigorous variancecovariance component estimation in Least Squares proposed by Helmert (1907) was simplified to a VCE algorithm based on measurement redundant contribution (Förstner 1979), which becomes popular in real applications (Cui et al. 2001; Bähr et al. 2007; etc.). Furthermore, Wang (1997) transplanted it into Kalman filter through an alternative derivation of the Kalman filtering algorithm by applying three groups of the independent random information: the purelypredicted state vector, the process noise vector and the real measurement vector. This derivation can project the system innovation vector into the residuals associated with these above used three groups of the measurements and also makes possible the calculation of their redundancy contribution. So, the VCE process as in (Förstner 1979) can be realized in Kalman filter. For the needs of further development, the relevant details will briefly be reviewed below.
VCE using redundancy contribution in LS
 L :

the measurement vector
 v :

the measurement residual vector
 B :

the design matrix
 F :

the nonlinear observation equations
 x ^{(0)} :

the approximate of the parameter vector x
 \( \delta \widehat{x} \) :

the correction vector for x ^{(0)}.
 L _{ i } :

n _{ i } × 1 vector of the ith type of the measurements
 v _{ i } :

the residual vector of L _{ i }
 B _{ i } :

the design matrix associated with L _{ i }
The alternative derivation of Kalman filter
 x _{ k } :

the n _{x} × 1 state vector
 w _{ k } :

the n _{w} × 1 process noise vector
 Λ _{ k } :

the coefficient matrix of w _{ k }
 z _{ k } :

the n _{z} × 1 measurement vector
 ε _{ k } :

the measurement noise vector
 Φ _{ k } :

the state transition matrix
 H _{ k } :

the design matrix
 (1)
the raw measurement vector l _{ z } = z _{ k } with its variance matrix \( {D}_{l_z}={R}_k \),
 (2)
the zero mean process noise vector w _{ k } as a pseudo measurement vector l _{ w } = w _{ k } with its variance matrix \( {D}_{l_w}={Q}_k \),
 (3)
the predicted state vector x _{ k/k − 1} from the previous epoch as another pseudomeasurement vector l _{ w } = x _{ k/k − 1} = Φ _{ k } x _{ k − 1} with its variance matrix \( {D}_{l_x}={\varPhi}_k{D}_{x_{x1}}{\varPhi}_k^T \), where \( {D}_{x_{x1}} \) is the variance matrix of x _{ k − 1},
One of the significant contributions made by this alternate derivation of KF was about to handle the process noise vector separately, which has made possible the simultaneous estimation of the variance components associated with the process noise vector w _{ k } and the measurement noise vector ε _{ k }.
Global VCE for Q and R in KF using redundancy contribution
Accordingly, the individual variance component in vector w _{ k } or ε _{ k } can be estimated by analogy with (2.13) and (2.14).
Discussion
 1)
Deployment of the vehicle’s kinematic model for the kinematic states, i.e., the velocity, acceleration, attitude and angular rate vectors as the core of the system model in Kalman filter,
 2)
Utilization of direct Kalman filter instead of indirect Kalman filter,
 3)
Application of all of the multisensor measurements through the measurement update in Kalman filter, especially the use of the IMU measurements instead of their use only through the free inertial navigation calculation between two aiding epochs in the traditional inertial navigation.
Thus, the novel multisensor integration strategy allows for equal treatment of all the sensors no matter which sensor is used as an aiding sensor. The structure of Kalman filter, e.g., for measurement models of IMU and GNSS receivers, is briefly introduced below.
State vector
 r :

the position vector in the navigation frame
 \( \begin{array}{ccc}\hfill {\mathbf{v}}_{nb}^b,\hfill & \hfill {\mathbf{a}}_{nb}^b,\hfill & \hfill {\boldsymbol{\upomega}}_{nb}^b\hfill \end{array} \) :

the velocity, acceleration and angular rate vector in body frame
 \( \begin{array}{ccc}\hfill \uptheta =\Big[p\hfill & \hfill \gamma \hfill & \hfill \psi \hfill \end{array}\Big]{}^T \) :

Euler angle vector (pitch, roll, heading)
 \( \begin{array}{cc}\hfill {\mathbf{b}}_g,\hfill & \hfill {\mathbf{b}}_a\hfill \end{array} \) :

the gyro and accelerometer’s bias vector
 \( \begin{array}{cc}\hfill {\mathrm{s}}_g,\hfill & \hfill {\mathrm{s}}_a\hfill \end{array} \) :

the gyro and accelerometer’s scale factors.
Measurement equations for GNSS observables
Furthermore, the measurement noises of the DD pseudoranges, carrier phases and range rates are denoted by \( {\varepsilon}_{\nabla \varDelta {P}_{AB}}^{jk} \), \( {\varepsilon}_{\nabla \varDelta {\phi}_{AB}}^{jk} \) and \( {\varepsilon}_{\nabla \varDelta Do{p}_{AB}}^{jk} \). Certainly, the ambiguity parameters \( \nabla \varDelta {N}_{AB}^{jk} \) will also be added to the state vector in (3.1) before they can be fixed as integer numbers.
Measurement equation for GPS heading measurement
 l _{ h } :

the DGPS heading measurement
 α :

the heading state
 ε _{ h } :

the heading measurement noise.
Measurement equations for IMU measurements
 \( {\upomega}_{ie}^n \) :

the Earth’s rotation rate in the local ENU navigation frame
 \( {\upomega}_{en}^n \) :

the craft rate in the local ENU navigation frame
 \( {\upomega}_{nb}^n \) :

the rotation rate of IMU body frame with respect to the local ENU frame
 \( {\mathrm{a}}_{nb}^n \) :

the acceleration of IMU body frame with respect to the local ENU frame
 b_{ g } :

the bias vector including start off biases and bias residuals for gyroscopes
 b_{ a } :

the bias vector including start off biases and bias residuals for accelerometers
 S_{ g } :

the scalar and misalignment error matrix of size 3 × 3 for gyroscopes
 S_{ a } :

the scalar and misalignment error matrix of size 3 × 3 for accelerometers
 ε_{ g } :

the Gaussian white noise vector for gyroscope measurements, and
 ε_{ a } :

the Gaussian white noise vector for accelerometer measurements.
 \( {\mathrm{C}}_n^b \) :

Direction cosine matrix that transforms the vector from navigation frame to IMU body frame.
In (3.6) and (3.7), the mechanical installation errors (misalignment of sensor axes) are ignored due to their considerably small influence on the state vector in comparison with other major errors in the lowcost IMU used in our land vehicle system.
VCE realization under the novel multisensor integration strategy
Variance components for the process noise vector in IMU/GNSS KF
Variance components for the measurement vector in IMU/GNSS KF
Measurement noise model for GNSS observables
Measurement noise model for the double differenced GNSS measurements has been studied for years (Collins and Langley 1999; Wieser et al. 2005; Luo et al. 2009; Wang et al. 2013; Takasu 2013). The commonly applied approach is that the measurement variance changes along with the elevation angle of the satellite.
 \( {\sigma}_P^2 \) :

variance component for pseudorange measurement at zenith
 \( {\sigma}_{Dop}^2 \) :

variance component for Doppler velocity measurement at zenith
 \( {\sigma}_{L1}^2 \) :

variance component for L1 carrier phase measurement at zenith.
 \( {\sigma}_{L2}^2 \) :

variance component for L2 carrier phase measurement at zenith. and
 \( {\sigma}_h^2 \) :

variance component for DGPS heading measurement.
Measurement noise model for IMU measurements
As introduced in Multisensor integration for kinematic navigation and positioning using IMU/GNSS KF section, the novel IMU/GNSS integration strategy allows to apply the IMU measurements directly through KF measurement update. This significant change on multisensor integration strategy makes possible estimate the variance components for individual IMU measurements about their random noises for the very first time.
Implementation discussion
Weakly observable process noise components
It has been found that small redundancy indices may result in the divergence of the estimates of the corresponding variance components. In IMU/GNSS KF, the redundancy contribution indices for the IMU bias drifts and scale factor drifts are quite small with the test data acquired by our land vehicle. A similar situation was given by Wang et al. (2010). In order to avoid a potential divergence, twelve variance components for four weakly estimable process noise sub vectors: w _{ bg}, w _{ ba}, w _{ sg} and w _{ sa } and are fixed in the VCE process.
Iterative VCE process
Results
Overview
The road test presented in this section, was performed on April 1, 2012 using the York University Multisensor Integrated kinematic positioning and navigation System (YUMIS) developed at the Earth Observation Laboratory of York University (Qian et al. 2012).
VCE results and analysis
The variances with the process noise factors
Figures 4 and 5 show the global VCE estimates of kinematic process noise component: jerk vector (\( {\mathbf{j}}_{nb}^b \)) and angular acceleration vector (\( {\overset{.}{\boldsymbol{\upomega}}}_{nb}^b \)) in the vehicle’s body frame. As can be seen, the estimates are convergent at the 4^{th} iteration.
However, it is noted that the global VCE results for jerks and angular accelerations achieved from this dataset around 10 min are definitely not universal to describe the generic land vehicle’s dynamics because this specific trajectory was mainly in straight line motion on the flat road and only experienced three sharp turns. Therefore, the results based on this specific short trajectory were used only to demonstrate the success of the proposed VCE algorithm in the multisensor integrated kinematic positioning and navigation. And further detailed research will be performed for practical utilization of the proposed VCE technique.
The variances with the IMU measurements
The initial variances for gyro measurements at the 2^{nd} iteration remained at the 1^{st} iteration (Fig. 6) as their redundant indices were relatively small (<0.1) due to the relative large initial angular acceleration process noise due to the large initial angular acceleration process noise (300 ^{0}/s^{2}).
Similarly, due to the weak redundancy contribution (<0.1), the measurement noise estimate for the accelerometer measurement at Y axis remained unchanged after the 2^{nd} iteration as well (Fig. 7). Therefore, its estimates will be maintained after the 2^{nd} iteration.
The variances with the GPS measurements
Figures 8 and 9 give the iterative VCE estimates for the GPS measurements.
In order to avoid the unstable quality of the raw Doppler velocity measurements, the time differenced L1 carrier phases were substituted for the double differenced Doppler measurements in IMU/GNSS KF. Accordingly, the accuracy of the Doppler velocity measurements is directly related to L1 carrier phase accuracy. As a result, the variance component estimation of the Doppler velocity measurements is skipped because its variance is easily achieved through error propagation.
The variance DGPS heading measurement noise
The DGPS heading measurements were derived from two GPS rovers on the vehicle’s roof using their L1 carrier phases. In general, the variance estimation for this heading measurement should be completed in more robust way directly involving the raw L1 carrier phase observables. However, in our implementation, the DGPS heading measurements were only valid when at least 5 L1 carrier phases reached the fixed ambiguities in order to guarantee a stable heading accuracy. Thus, the variance of the DGPS heading could be simplified through single numerical value. Correspondingly, the single variance factor is applicable to the DGPS heading measurement in IMU/GNSS KF.
Performance comparison and analysis
Navigation solution comparison
First, one can conclude that there was no statistical inconsistency between two solutions because the differenced navigation state vectors were always within their 3σ envelops.
In addition, the consistent smaller (finer) navigation solution standard deviation in the 7^{th} iteration compared to the initial navigation solution verifies the positive influences of VCE effort on IMU/GNSS KF, which is due to the scaleddown Q and R from the conservatively large initial values.
Analysis of the IMU Measurement residuals
Due to fact that the variance component estimation of the IMU measurements have been realized practically in the multisensor integrated navigation system, more discussions will be performed here. Because the land vehicle’s dynamics is usually exposed in vertical (Z) axial gyro and the forward (Y) axial accelerometer, the variance estimates for their observables theoretically cast more significant influences on their measurement residuals than the ones in the other axes. Figure 12 shows the normalized histograms of their measurement residuals.
Analysis of the GNSS measurement residuals
The a posteriori histograms fit the standardized normal distribution curves much better than that of a priori ones individually, which tells that the a posteriori estimated variances for the GNSS measurements were realistic.
Conclusion and future work
This paper proposed a simplified VCE algorithm for multisensor integrated kinematic positioning and navigation. First, the sensor measurements were processed through a fully tightlycoupled integration strategy (IMU/GNSS KF). In doing so, the random errors in IMU measurements could be statistically separated from the other error sources, which has made possible conducting the variance component estimation for the six IMU measurements along with the process noise components including jerks and changes of angular rates. Second, the simplified VCE using the redundancy contribution of individual independent measurements or measurement groups was applied and formulated for the individual components in the process noise vector and the measurement vector including the IMU and the double differenced GNSS measurements. The results showed that the stochastic models of the process and measurement noise in IMU/GNSS KF is improved through VC estimation.
Declarations
Acknowledgements
The authors would like to acknowledge the Natural Sciences and Engineering Research Council (NSERC) of Canada for the financial support under its RGPIN Program.
Authors’ contributions
This paper proposes a generic method for the stochastic model tuning about the random errors in IMU measurements together with other sensors. The core of this novel approach is based on authors’ innovative multisensor integration strategy which deploys upon the vehicle’s generic kinematic model and takes the IMU’s output as raw measurements in Kalman Filter (IMU/GNSS Kalman Filter). As a result, the statistical orthogonality between random error vectors of any two sensors enables the separate but parallel statistics collection of each individual random error source. Given these independent statistics corresponding to each error source, the VCE technique iteratively tunes all stochastic model of the process and measurement noise vectors. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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