Conference Proceedings
Reference
A Framework for Inertial Sensor Calibration using Complex Stochastic Error Models, in the proceedings of the Position Location and
Navigation Symposium (PLANS), 2012 IEEE/ION
STEBLER, Yannick, et al .
Abstract
Modeling and estimation of gyroscope and accelerometer errors is generally a very challenging task, especially for low-cost inertial MEMS sensors whose systematic errors have complex spectral structures. Consequently, identifying correct error-state parameters in a INS/GNSS Kalman filter/smoother becomes difficult when several processes are superimposed. In such situations, the classical identification approach via Allan Variance (AV) analyses fails due to the difficulty of separating the error-processes in the spectral domain.
For this purpose we propose applying a recently developed estimation method, called the Generalized Method of Wavelet Moments (GMWM), that is excepted from such inconveniences. This method uses indirect inference on the parameters using the wavelet variances associated to the observed process. In this article, the GMWM estimator is applied in the context of modeling the behavior of low-cost inertial sensors. Its capability to estimate the parameters of models such as mixtures of GM processes for which no other estimation method succeeds is first demonstrated through simulation studies. The GMWM [...]
STEBLER, Yannick, et al . A Framework for Inertial Sensor Calibration using Complex Stochastic Error Models, in the proceedings of the Position Location and Navigation Symposium (PLANS), 2012 IEEE/ION . 2012
DOI : 10.1109/PLANS.2012.6236827
Available at:
http://archive-ouverte.unige.ch/unige:26832
Disclaimer: layout of this document may differ from the published version.
1 / 1
A Framework for Inertial Sensor Calibration Using Complex Stochastic Error Models
Yannick Stebler1, St´ephane Guerrier2, Jan Skaloud1, and Maria-Pia Victoria-Feser2
1´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Geodetic Engineering Laboratory(TOPO), CH-1015 Lausanne, Switzerland
2University of Geneva, Research Center for Statistics and Department of Management Sciences, CH-1211 Geneva, Switzerland
Abstract—Modeling and estimation of gyroscope and ac- celerometer errors is generally a very challenging task, especially for low-cost inertial MEMS sensors whose systematic errors have complex spectral structures. Consequently, identifying correct error-state parameters in a INS/GNSS Kalman filter/smoother becomes difficult when several processes are superimposed. In such situations, the classical identification approach via Allan Variance (AV) analyses fails due to the difficulty of separating the error-processes in the spectral domain. For this purpose we propose applying a recently developed estimation method, called the Generalized Method of Wavelet Moments (GMWM), that is excepted from such inconveniences. This method uses indirect inference on the parameters using the wavelet variances associated to the observed process. In this article, the GMWM estimator is applied in the context of modeling the behavior of low-cost inertial sensors. Its capability to estimate the parameters of models such as mixtures of GM processes for which no other estimation method succeeds is first demonstrated through simulation studies. The GMWM estimator is also applied on signals issued from a MEMS-based inertial measurement unit, using sums of GM processes as stochastic models. Finally, the benefits of using such models is highlighted by analyzing the quality of the determined trajectory provided by the INS/GNSS Kalman filter, in which artificial GNSS gaps were introduced.
During these epochs, inertial navigation operates in coasting mode while GNSS-supported trajectory acts as a reference. As the overall performance of inertial navigation is strongly dependent on the errors corrupting its observations, the benefits of using the more appropriate error models (with respect to simpler ones estimated using classical AV graphical identification technique) are demonstrated by a significant improvement in the trajectory accuracy.
Index Terms—AV, GMWM, estimation, model identification, inertial, stochastic, MEMS
I. INTRODUCTION
Stochastic modeling is a challenging task for low-cost Micro-Electro-Mechanical System (MEMS) inertial sensors whose errors can have complex spectral structures. This makes the tuning process of the INS (Inertial Navigation System)/GNSS (Global Navigation Satellite System) Kalman Filter (KF) often sensitive and difficult. For example, first- order Gauss-Markov (GM) processes are very commonly used in inertial sensor models. But the estimation of their param- eters is a non-trivial task if the error structure is mixed with other types of noises. Such an estimation may be attempted by computing and interpreting Allan Variance (AV) plots.
Alternatively, a recent research (see [1]) has proposed to automate this process by maximizing the likelihood func- tion of the assumed state-space models of interest using a constrained version of the expectation-maximization (EM) algorithm (see [2]). This research has demonstrated that the EM-based approach is able to estimate models on which the AV identification fails. Nevertheless, in more complex cases where highly nonlinear likelihood surfaces are present (e.g.
sum of several GM processes), an EM-based algorithm is likely to converge to a wrong maximum. In such situations, the classical identification via AV analyzes also fails due to the difficulty of separating the GM processes in the spectral domain. For this reason we propose employing a recently developed estimation method, called the Generalized Method of Wavelet Moments (GMWM) for such purpose. GMWM has been developed for the estimation of composite stochastic processes (i.e. stochastic processes that can be written as sums of different stochastic processes) and is excepted from the previously mentioned inconveniences. The theoretical basis of this estimation approach was first introduced in [3]. This method relies on Wavelet Variances (WV) which can be seen a generalization of AV.
In this article, the GMWM estimator is applied in the context of MEMS-Inertial Measurement Unit (IMU) when characterizing the stochastic processes of its individual sensors (i.e. gyroscopes and accelerometers). The model-parameters are represented by several GM processes. The resulting mix- ture reaches such complexity that no other estimation method is successful in recovering the underlying parameters. The performance of the GMWM estimator is first demonstrated using a synthetic error signal issued from a process model.
Then, the estimator is applied on signals supplied from a real MEMS-IMU, using sums of GM processes as stochastic models. Finally, the benefits of using correct model parameters is highlighted by analyzing the quality of the determined trajectory provided by the INS/GNSS Kalman filter (KF).
For that purpose, artificial outages are introduced in the GNSS observations during which inertial navigation operates in coasting mode while GNSS-supported trajectory acts as a reference. As the overall performance of inertial navigation is strongly dependent on the uncompensated errors corrupt- ing its observations, the benefits of using appropriate error models (with respect to simpler ones estimated using classical
AV graphical identification technique) is demonstrated by a significant improvement in the trajectory accuracy.
This paper is organized as follows. Sec. I introduces the employed notation and the adopted convention. In Sec. II, a review of the main existing sensor calibration techniques together with their limitations is provided. The GMWM es- timator as well as the WV are introduced in Sec. III. A simulation study is then presented in Sec. IV that compares the GMWM with existing methods. Sec. V illustrates the benefits of the GMWM with real inertial signals. Finally, a few concluding remarks and perspectives of future work are outlined in Sec. VI.
NOTATION ANDCONVENTIONS
LetF✓ be the parametric model associated to an univariate Gaussian time series {Yt, t 2 Z} that is stationary or non- stationary but with backward difference1 of order d and let {yt, t= 1,2, ..., T} be the corresponding observed outcome.
Particularly, this outcome could be the signal of an accelerom- eter or gyroscope acquired during static conditions (i.e. input signal is a constant) where the mean has been removed. The remaining signal represents therefore the varying error which we are interested to model. (Note that a residual constant error may be later modeled as a random-bias within a Kalman filter).
Let ✓ 2 ⇥ ✓ <p be the parameter vector of the model of interest where ⇥ is an open subset of <p. Moreover, let ✓0
be the parameter vector of the true value and✓ˆits estimator.
The GMWM approach offers an alternative to the estimation based on the likelihood. It is straightforward to implement and often the only feasible estimation method when using complex models. The GMWM is generally based on Haar wavelet filters whose coefficients variances are equal to the half of the AV computed at the associated scales. It uses the unique relationship that exists (under some conditions) between a hypothetical model F✓ and the WV, denoted ⌫2 (see Sec. III-A for details), implied by it. Intuitively, the WV implied by F✓ is a function (or a mapping) defined as
✓ 7! ⌫2(✓), 8✓ 2 ⇥. The GMWM approach inverses this mapping and tries to approximate the point ✓( ˆ⌫2)where ⌫ˆ2 is the WV computed on the data at hand. Indeed, the GMWM aims to find the value of ✓ implied by ⌫ˆ2. The task is in practice realized by minimizing the distance between ⌫ˆ2 and
⌫2(✓). The solution of this optimization problem corresponds to the closest possible approximation of the observed WV (or AV) by the WV (or AV) implied by the parametric modelF✓. Tab.Isummaries the different notation and conventions used in the article. In addition, the random processes considered here to model IMU error behavior are (Gaussian) White Noise (WN), Random Walk (RW), first-order Gauss-Markov (GM), Bias Instability (BI), Rate Ramp (RR) and Quantization Noise (QN) which are presented in the frequency and in the time domain in App. C(see Tab. VII).
1The first order backward difference ofYtisYt(1)=Yt Yt 1 and the backward difference of orderdisYt(d)=Yt(d 1) Yt(d11)
TABLE I
NOTATION AND CONVENTIONS
SY(·) Power Spectral Density Function (PSD) of{Yt} Wj Vector ofjth level of wavelet coefficients evaluated
by Maximal Overlap Discrete Wavelet Transform (MODWT)
Wj,t tth element ofWj
{Wj,t} jth level of MODWT coefficients for stochastic process{Yt}
⌫2 Theoretical Wavelet Variance (WV) vector whose elements are{⌫⌧2j}where⌧j= 2j 1
⌫2(✓) WV implied by✓assuming thatF✓ corresponds to the true data generating function
ˆ
⌫2 Estimated Wavelet Variance (WV) based on the MODWT estimator, see (3)
2¯
y(⌧) AV at scale⌧
y?t(✓) Time series of the lengthN simulated underF✓
✓(0) Initial value of✓provide to an estimation procedure such as the GMWM
II. MOTIVATIONS
The most commonly used method for model identification and sensor calibration is the well known variance analysis technique based for example on the Allan Variance (AV) [4]
or other metrics such as the Hadamard Variance (HV) or the Total Variance (TV) [5–8]. This approach is a well established method for identifying stochastic processes affecting the out- put of a sensor. It can also be used to estimate the parameters of some model that is believed to describe the stochastic processes of interest. Although this method was originally intended for studying the stability of oscillators, it has been successfully applied to problems dealing with different types of sensors, among which stands the modeling of inertial sensor errors [9–14]. The AV at scale⌧(denoted as 2y¯(⌧)) is defined as:
2
¯ y(⌧) =1
2Eh
(¯yt(⌧) y¯t ⌧(⌧))2i
where y¯t(⌧) is the sample average of ⌧ consecutive obser- vations. The AV can be expressed in the frequency domain through the relationship between y2¯(⌧)and the Power Spectral Density (PSD) Syy(f) of the intrinsic processes [12] which links the parameter vector ✓ to 2y¯(⌧). This relationship is due to the known form of the PSD function characterizing different noise processes and enables to express✓as a function of 2¯y(⌧) (a detailed discussion on how to express this link can be found in [11]). In general, only five basic processes are considered with the AV: QN, WN, BI, RW and RR. These processes correspond to linear regions in a “ ¯y(⌧)v.s.⌧” log- log plot. Therefore,✓is usually estimated by performing linear regression of (visually) identified linear regions in such plots.
Unfortunately, this methodology suffers from sever drawbacks:
• When more than one process has to be estimated, the re- sulting estimator is in most cases not consistent. Loosely speaking, this implies that the distribution of✓ˆdoes not tend to concentrate near✓0as the sample goes to infinity.
An example of this statement is given in AppendixA.
• The estimated values of ✓ are in general obtained as functions of the parameters estimated by linear regression on linear regions in a “ y¯(⌧)v.s. ⌧” log-log plot. Such
methodology is only justified assuming that the error vec- tor (of the linear regression) ✏is such thatvar [✏] = 2✏I.
However, it was proven in [3] that var [✏] = ⌃ 6= ✏2I and an estimator for⌃were derived (see (5)). Therefore, Ordinary Least-Squares (OLS) should not be used in this context and more complex methods such as Iteratively Reweighted Least-Squares (IRLS) should be preferred to obtain reliable and efficient estimates ✓.ˆ
• Inference (confidence intervals, tests, etc.) about the es- timated parameters is in many cases impossible. Indeed, the system parameters are indirectly estimated through functions of coefficients estimated by linear regression (say ˆ). The standard solution for deriving the (asymp- totic) distribution of✓ˆfrom the distribution ˆis achieved though a first order approximation. In statistics, this approach is called the delta method. However, to apply this method it is required that the function 7! ✓ is one-to-one and that ˆ is a consistent estimator. Unfortu- nately, this is generally not the case here. Consequently, deriving the (asymptotic) distribution of ✓ˆ is in general not possible.
• The conventional AV methodology is limited to models composed of processes characterized by linear regions in a “ y¯(⌧)v.s.⌧” log-log plot and therefore this approach is far from being general.
As an alternative to the AV approach, [1] proposed esti- mating✓ˆby maximizing the log-likelihood of the state space model associated to the model F✓ of interest using the EM algorithm [2]. This approach is more general than the AV approach and works very well with relatively simple models.
Unfortunately, when the model complexity increases, this methodology becomes numerically challenging as it becomes very sensitive to the initial approximation of parameters and the convergence to global minimum is not guaranteed.
III. ESTIMATION THEORY FOR COMPOSITE STOCHASTIC
PROCESSES
This section provides a short introduction to the theory of WV and presents the GWMW estimator which relies on it.
A. Wavelet Variance Estimation
As pointed out by [15], the WV can be interpreted as the variance of a process after it has been subject to an approximate bandpass filter. Indeed, the WV can be built using wavelet coefficients issued from a modified Discrete Wavelet Transform (DWT) (see e.g. [16, 17]) called the Maximal Overlap DWT (MODWT) (see [15,18]). The wavelet coefficients are built using wavelet filters{˜hj,l}, j= 1, . . . , J which for j= 1and for the MODWT must satisfy
LX1 1 l=0
˜h1,l= 0,
LX1 1 l=0
˜h21,l= 1 2 and
X1 l= 1
˜h1,l˜h1,l+2m= 0 where˜h1,l = 0forl <0 andl L1,L1 is the length of˜h1,l, andm is a nonzero integer. LetHe1(f) =PL1 1
l=0 ˜h1,le i2⇡f l be the transfer function of ˜h1,l. The jth level wavelet filters
{h˜j,l} of length Lj = (2j 1)(L1 1) + 1can be obtained by computing the inverse discrete Fourier Transform of
Hej(f) =He1(2j 1f)
j 2
Y
l=0
ei2⇡2lf(L1 1)He1(12 2lf) The MODWT filter is actually a rescaled version of the DWT filter hj,l, i.e.˜hj,l =hj,l/2j/2. Filtering an infinite sequence {Yt;t2Z}using the filters{˜hj,t}yields the MODWT wavelet coefficients
Wj,t =
Lj 1
X
l=0
˜hj,lyt l, t2Z (1) The WV at dyadic scales ⌧j = 2j 1, are defined as the variances of{Wj,t}, i.e.
⌫⌧2j =var Wj,t (2) Notice that the WV are assumed not to depend on time. The condition for this property to hold is that the time series at hand is either stationary or non-stationary but with stationary backward differences of order d satisfying d L1/2. In addition,{˜hj,l}must be based on a Daubechies wavelet filter (see [17,19]). This is due to the fact that Daubechies wavelet filters of widthL1 contain an embedded backward difference filter of orderL1/2.
A consistent estimator for⌫⌧2j as defined in (2) is given by the MODWT estimator defined in [20] (see also [21])
ˆ
⌫2(⌧j) = 1 M(Tj)
X
t2Tj
Wj,t2 (3)
with Wj,t = PLj 1
l=0 h˜j,lyt l, t 2 Tj and where Tj is the set of time indices for which the MODWT coefficients are free of end effects, and M(Tj) = T Lj + 1 their number. Moreover, [20] show that under suitable conditions pM(Tj)⇣
ˆ
⌫2(⌧j) ⌫⌧2j⌘
is asymptotically normal with mean 0 and variance
SWj(0) = 2 Z 1/2
1/2
SW2j(f)df
= 2 Z 1/2
1/2|Hj(f)|4SF2✓(f)df
where| · | denotes the modulus.SWj(0)can be estimated by means of
SbWj(0) =
M(Tj)
X
⌧= M(Tj)
2 4 1
M(Tj) X
t2Tj
Wj,tWj,t+|⌧|
3 5
2
These results were extended to the multivariate case in [3]
who have demonstrated that under some regularity conditions the asymptotic distribution of⌫ˆ2 is given by
pT ⌫ˆ2 E[ ˆ⌫2] 7TD!
!1 N(0,V⌫ˆ2) whereV⌫ˆ2 = [ kl2]k,l=1,...,J with
2
kl= 2⇡Skl(0) (4)
and where Skl(f) = 2⇡1 P1
⌧= 1 kl(⌧)e if⌧ are the
cross spectral densities with cross-covariances kl(⌧) = cov(Wk,t2 , Wl,t+⌧2 ). The estimation of kl2 is in general not straightforward and [3] proposed the following estimator
ˆkl2 = 1 2
M(Tkl)
X
⌧= M(Tkl)
"
1 M(Tkl)
X
t2Tkl
Wk,tWl,t+⌧
#2
+ 1
2
M(Tkl)
X
⌧= M(Tkl)
"
1 M(Tkl)
X
t2Tkl
Wk,t ⌧Wl,t
#2
(5) A particular choice for the wavelet filter is given by the Haar wavelet filter which first DWT filter (j = 1) is
{h1,0= 1/p
2, h1,1= 1/p
2} (6)
with lengthL1= 2. If the process is stationary with backward differences of order d > 1 other wavelet filters such as Daubechies can be used [19]. Note that when WV is evaluated with Haar wavelet filters, it is actually equal to half the AV [4].
B. The GMWM Estimator
Under the settings defined in sectionIII-Athe series{Wj,t} are stationary with PSD SWj(f) = |Hej(f)|2SF✓(f). This implies that the variance of wavelet coefficients’ series is equal to the integral of its PSD [20], i.e.
⌫⌧2j = Z 1/2
1/2
SWj(f)df= Z 1/2
1/2|Hej(f)|2SF✓(f)df (7) Therefore, there exist a mapping
✓7!⌫2(✓) (8)
Such a mapping defines the theoretical WV implied by the parametric modelF✓. The connexion between the WV and✓is exploited in [3] to define an estimator for✓ by trying in some sense to inverse (8). This inverted map is used to compute the estimator✓ˆ=✓( ˆ⌫2)where⌫ˆ2 is the estimated WV. Finding explicitly an inverse mapping is in general impossible since this mapping is in most cases implicit. However, it is possible to inverse the map in a specific point such as⌫ˆ2by calibrating the value of ✓ in order to match ⌫2(✓) with its empirical counterpart⌫ˆ2. Therefore, the GMWM estimator proposed by [3] is the solution of the following optimization problem
✓ˆ= argmin
✓2⇥
⌫ˆ2 ⌫2(✓) T⌦ ⌫ˆ2 ⌫2(✓) (9) in which ⌦, a positive definite weighting matrix, is chosen in a suitable manner (see below) such that (9) is convex. The consistency of ✓ˆwas proven in [3] (see Theorem 2) and has the following distribution
pT⇣
✓ˆ ✓0
⌘ D 7 !T
!1 N 0,V✓ˆ (10) where
V✓ˆ=BV⌫ˆ2BT and
B= DT⌦D 1DT⌦
and whereD=@⌫2(✓)/@✓T,V⌫ˆ2 is the asymptotic covari- ance matrix of ⌫ˆ2 as defined in (4). When ⌦ = I, then V✓ˆ = DTD TDTV⌫ˆD DTD 1. The most efficient estimator is (asymptotically) obtained by choosing⌦=V⌫ˆ1, leading then toV✓ˆ= (DTV⌫ˆ1D) 1. In practice, the matrix Dis computed at ✓.ˆ
The analytical expressions of the WV⌫2(✓)used in (9) us- ing the Haar wavelet filter (6) can be computed for several well known models such as AR(p), sums of AR(p), ARMA(p,q) and others using the general results of [22] on the AV. In addition, the analytical WV of sums of independent processes correspond to the sum of the WV of individual processes within the model. Indeed, when the process is made up of the sum of independent processes, i.e.Yt=P
kXt(k), (7) can be expanded to
⌫⌧2j = Z 1/2
1/2|Hej(f)|2 X
k
SX(k)(f)
!
df (11)
= X
k
⌫k,⌧2 j (12)
with SX(k) the PSD and ⌫k,⌧2 j its WV at scale ⌧j of Xt(k). Therefore, when an analytical expression for ⌫2(✓) is avail- able, the estimator defined in (9) can be seen as a Generalized Method of Moments (GMM) estimator (see [23] for details) based on theJ⇥1 moment conditions:
E[g(Yt,✓)] =E⇥ ˆ
⌫2 ⌫2(✓)⇤
=0
However, when analytical expressions for⌫2(✓)are not avail- able or too complicated to compute, one can resort to simula- tions to approximate⌫2(✓). This places the GMWM estimator in the framework of indirect inference [24–26]. Indeed,⌫2(✓) can be approximated by computing the WV, denoted⌫?2(✓), of a simulated series {y?t(✓), t= 1, . . . , R·T}, R 1from F✓. Alternatively, we can compute R WV estimates ⌫ˆr?2(✓) on simulated series {y?(r)t (✓), t = 1, . . . , T} and obtain
ˆ
⌫?2(✓) =R1PR
r=1⌫ˆr?2(✓). AsRincreases, the quality of the approximation increases. When R ! 1, the approximation is perfect and ⌫ˆ?2(✓) can be used in (9) instead of ⌫ˆ2(✓).
The properties of such an estimator are described in [25], but also rely on the conditions set in [3]. In particular, for R sufficiently large, V✓ˆ ⇡ BVˆBT. A schematic illustration of the estimation algorithm is given in Figure1.
IV. SIMULATIONS
As the first step of the validation procedure of the GMWM estimator several simulations were performed. Composite stochastic processes {Yt : t = 1, . . . , N} associated with models F✓ of increasing complexity were simulated at a sampling interval of t, which is assumed to have physically meaningful units (e.g. deg/s, µg/s). The GMWM approach was applied to estimate ✓ and these results were compared to alternative estimation methods (i.e. AV and EM based approaches).
✓ˆ
Data Hsimulation for a given✓
F✓
Model:
Step 2
Auxiliary parameters
Step 1 Step 3
Step 4 Calibration
y1, . . . , yT y1, . . . , yh·T
Analytical expressions for available?⌫2(✓)
No Yes
ˆ
⌫2⌧j= 1 M(Tj)
X
t2Tj
Wj,t2
ˆ
⌫2
ˆ
⌫2 ⌫ˆ?2(✓)
Replace by⌫ˆ?2(✓)
Indirect inference GMM
⌫2(✓)
Fig. 1. Principle of the GMWM estimator (adapted from [27])
A. Sum of a Gaussian White Noise, a Random Walk and a Rate Ramp Process (WN-RW-RR)
This scenario is taken from [1] to compare the performance of the GMWM estimator with AV and unconstrained EM approach. The model F✓ is a sum of RW, WN, and a RR and can be written as:
Yt=Yt,W N +Yt,RW +Yt,RR (13) where Yt,W N, Yt,RW andYt,RR are defined in Appendix C (see Tab. VII). Therefore, the parameters to estimate are
✓={ W N2 , 2RW,!} (14) The true parameters were set to
✓0={0.04,4·10 4,0.003} (15) In Fig.2, the GMWM estimates are compared to the estimates computed using the unconstrained EM algorithm and to what would be obtained by fitting lines (using OLS) to the linear regions of the log-log plot of AV. The latter correspond either to WN (slope is 1/2), RW (slope is 1/2) or RR (slope is 1). As noted by [1], the EM based approach is very sensitive to the initial values of ✓. When ✓(0) is “far” from ✓0 this approach is likely to converge to a local minima. TheEMand EM* columns correspond to the results obtained by the EM approach when started at
✓(0) = {0.25,10 8,0.0} (16)
✓⇤(0) = {25,100,10} (17) respectively. The values of ✓(0) for the GMWM estimator were set to 1.0 and several tests revealed an insensitivity on the choice of ✓(0) in terms of convergence. Indeed, the GMWM algorithm converged correctly when initiated with
Fig. 2. Performance comparison between the GMWM (GW) and the EM algorithm started at good (EM) and a bad (EM*) initial values, and the AV technique (AV) for 200 simulated signals issued from a mixture of WN, RW and RR processes. The true parameters are marked by horizontal lines.
✓⇤(0). The Root-Mean-Square Error (RMSE) as well as the relative RMSE (R-RMSE) of the different estimation methods are listed in Tab.II. The EM approach with✓(0) provides the best results but the GMWM is considerably better thanEM*
or the AV based approach.
TABLE II
RMSE AND RELATIVE RMSE(R-RMSE)OF THE EM,EM?,AV AND GMWM ESTIMATORS FOR200SIMULATED PROCESSES OF SIZEN= 6000FROM MODEL WN-RW-RR.
EM EM? AV GMWM
RMSE R-RMSE RMSE R-RMSE RMSE R-RMSE RMSE R-RMSE
W N2 7.67·10 4 1.92·10 2 4.02·10 2 1.00 9.09·10 4 2.27·10 2 8.59·10 4 2.15·10 2
2
RW 1.17·10 5 2.93·10 2 3.24·10 4 8.09·10 1 4.45·10 4 1.11 5.99·10 5 1.5·10 1
! 2.52·10 4 8.4·10 2 2.37·10 4 7.88·10 2 1.18·10 3 3.94·10 1 5.29·10 4 1.76·10 1 TABLE III
RMSE AND RELATIVE RMSE(R-RMSE)OF THE EM,EM?AND GMWM ESTIMATORS FOR200SIMULATED PROCESSES OF SIZEN= 6000FROM MODEL WN-GM-RR.
EM EM? GMWM
RMSE R-RMSE RMSE R-RMSE RMSE R-RMSE
GM 2.44·10 3 3.05·10 1 7.47·10 3 9.33·10 1 3.31·10 3 4.14·10 1
2GM 5.53·10 2 2.21·10 1 3.02 1.21·101 9.67·10 2 3.87·10 1
2W N 1.26·10 2 1.97·10 2 1.3·10 2 2.03·10 2 1.38·10 2 2.15·10 2
! 5.49·10 4 5.49·10 1 1.68·10 3 1.68 1.81·10 4 1.81·10 1
B. Sum of a Gaussian White Noise, a First-Order Gauss- Markov and a Rate-Ramp Process (WN-GM-RR)
This scenario is also taken from [1] to compare the per- formance of the GMWM estimator with the constrained EM algorithm. The model F✓ is a sum of a GM process, a WN, and a RR that can be written as
Yt=Yt,W N +Yt,GM+Yt,RR (18) where Yt,W N, Yt,GM andYt,RR are defined in Appendix C (see Tab. VII). Therefore, we have
✓={ , GM2 , 2W N,!} (19) In practice, estimating the parameters of such a model is diffi- cult, not to say impossible, using the classical AV identification technique and is therefore omitted from this comparison.
Fig. 3 compares the performance of the GMWM estimator against the constrained EM (EM*) and the constrained EM with prior removal of the drift through least-square adjustment (EM), when applied on 200 signals issued from the following parameter values:
✓0={0.008,0.25,0.64,0.001} (20) The EM can fairly well estimate such a model but on the condition that ! is removed by OLS adjustment prior to estimation. This makes the comparison possible yet unfair with respect to the GMWM algorithm that operates the orig- inal signal without preprocessing. Nevertheless, the GMWM method was able to estimate correctly ✓ without any prior manipulation, which is a clear advantage regarding inference on✓. Moreover, the performance of the GMWM estimator isˆ comparable to the EM approach (with prior drift removal) for , GM2 and W N2 but is sensibly better regarding!. RMSE and R-RMSE values are listed in Tab.III. Note that the initial values in✓(0) were all set to 1.0.
Fig. 3. Performance comparison between the GMWM (GW), the EM algorithm with prior estimation of!by OLS (EM) and without (EM*) for 200 simulated signals issued from a sum of WN, GM and RR process. The true values of the parameters are marked by horizontal lines.
C. Sum of Three First-Order Gauss-Markov Processes Three mixed GM processes are impossible to discriminate using the method of AV analysis. Also, the EM approach systematically diverges in such complex scenarios. Therefore, an attempt to retrieve the correct values of the individual model parameters is carried only with the GMWM estimation.
The composite stochastic process we wish to estimate can be expressed as
Yt= X3 k=1
Yt,GM(k)
Fig. 4. Performance of the GMWM (GW) algorithm for 200 simulated signals issued from a sum of three GM processes. The true parameters are marked by horizontal lines.
whereYt,GM(k) is a GM process as defined in AppendixC(see Tab.VII) with parameters{ k, GM.k2 }. Thus, the goal of this simulation is to estimate the following parameter set:
✓={ 1, GM,12 , 2, 2GM,2, 3, GM,32 }
To assess the performance of the GMWM in this context, two hundred series Yt with N = 106 were simulated under the following true parameter values:
✓0={0.008,2.5·10 6,0.05,4.5·10 6,2.0,29.50·10 6} Fig. 4 depicts the values ✓ˆ and reveals that the GMWM technique is able to retrieve correctly the parameters of such complex model. Again, the initial values of ✓ were all set to 1.0, and with exception of 2 runs out of 200, the GMWM estimator converged (i.e. the success of convergence without aiding was 99%). In these 2 cases, a grid search algorithm was employed to provide a “better” initial guess of✓0 to the GMWM and convergence occurred. Note also that the same simulation was repeated by setting all initial parameters to various values, and no significant difference was observed with respect to the results in Fig. 4. RMSE and R-RMSE values are listed in Tab. IV.
It should be noted that the sum of three GM processes can be reparametrized as an ARMA(3,2) process (see e.g.
[28, 29]), so that one could in principle estimate the latter instead the former. However, when one of the GM process lies very near a unit root (as it is often the case with inertial sensors and in the simulation at hand), the estimation of the associated ARMA model is rarely possible. Moreover, even if the estimation of the ARMA model is possible, the results shall be inverted to GM-like representation since in many cases, and in particular in the simulation at hand, a sum of several GM models explains better the real underlying process (see [30,
31]). Indeed, to recover the sum of GM process-parameters from an estimated ARMA process together with their standard errors, several conditions need to be satisfied:
• The roots of the processes must lie outside the unit circle.
• The Jacobian matrix of the transformation between the two parametrizations must be invertible in order to apply the delta method (see [32,33]).
With the signals generated here, the estimated processes have roots that are near the unit circle. Moreover, the Jacobian ma- trix of the transformation evaluated at the✓ˆis not invertible. In that case at least, estimating an ARMA process and converting the estimated model and performing an inference to the sum of GM processes is infeasible.
V. EXPERIMENTS
Static data were collected during4.5hours at 100 Hz under constant temperature conditions from aXSens MTi-GMEMS- based IMU. After mean removal, the signals at hand contain measurement errors namely driven by stochastic processes issued from an unknownF✓. First, classical calibration proce- dure using AV will be performed on the X-axis gyroscope and the Z-axis accelerometer. Second, the GMWM estimator will be applied on the same signals using a more sophisticated model. Finally, the two resulting models will be used in an Extended Kalman Filter (EKF) that performs INS/GNSS integration and the quality of the obtained navigation solution will be evaluated with respect to the reference.
A. Modeling using Allan Variance
Classical stochatic calibration procedure consists in com- puting the AV and estimating the process parameters by OLS on the linear parts of the AV curve. Fig. 5 shows the Allan deviation (i.e. the root AV) curves of both signals under study.
Three random processes were identified. First, 2W N can be identified on the left part of the curve that slopes at ⌧ = 1 by 1/2. Second, a bias instability (BI) due to flicker noise in the measurements can be identified at the lowest point in the curve (see AppendixC for details). Flicker noise is often approximated using GM processes (or more generally AR processes) that overbound it [34]. In the case of the gyroscope signal only, the GM parameters were iteratively hand-tuned according to BI2 andTBI, while for the accelerometer, only a RW model was used. The RW parameter RW2 can be deduced from a slope of 1/2 on the right-part of AV plot. However,
TABLE IV
RMSE AND RELATIVE RMSE(R-RMSE)OF THEGMWMESTIMATOR FOR 200SIMULATED PROCESSES OF SIZEN= 106FROM MODEL GM-GM-GM.
RMSE R-RMSE
1 6.62·10 4 8.28·10 2
12 1.97·10 7 7.89·10 2
2 2.19·10 3 4.38·10 2
22 1.74·10 7 3.87·10 2
3 9.05·10 3 4.53·10 3
32 4.59·10 8 1.56·10 3
Fig. 5. Allan deviation of XSens MTi-G gyroscope (upper panel) and ac- celerometer (lower panel) with identified WN, BI and RW process parameters.
uncertainty in the AV estimation decreases as a function of⌧.
Thus, the modelsF✓ for the signals under study are:
Yt=Yt,W N +Yt,RW
for the gyroscopes, and
Yt=Yt,W N +Yt,GM
for the accelerometers (where Yt,W N, Yt,RW and Yt,GM are defined accordingly to Appendix C, Tab.VII). The estimated values in ✓ˆ are depicted in Fig. 5. The dashed lines are the results of linear regressions on parts of the curves for deducing the process parameters.
B. KF-(Self)-Tuning Approach
According to [35], the parameters obtained by AV analysis are considered as an initial approximation which is further tuned manually within a KF by analyzing errors in inertial coasting (e.g. by invoking artificial outages of GNSS obser- vations). The AV is therefore often used as starting point for further tuning, which is performed by analyzing position drift during GNSS outages. In this section, a KF-tuned of such type is considered. In fact, this model is derived from the research of [36] which used MTi-G IMUs for reconstructing trajectories of skiers. This model was typically tuned on datasets of short duration (i.e. a few minutes) using reference trajectories provided by integrating L1/L2 carrier-phase differential GNSS with a tactical-grade IMU. Moreover, this tuning was itera- tively repeated for each new dataset. Both sensors are modeled using the same composite process which can be written as:
Yt=Yt,W N +Yt,GM
where Yt,W N andYt,GM are defined according to Appendix C, Tab. VII. The values of the ✓ are listed in Tab. V and
TABLE V
KF-(SELF)-TUNED MODELS USED FOR THEXSens MTi-G ACCELEROMETERS AND GYROSCOPES
Process Parameter Value Unit
Gyroscope WN W N 720 deg/h/p
Hz
GM g 180 deg/h/p
Hz 0.001 (s) Accelerometer WN W N 100000 µg/p
Hz
GM g 40000 µg/p
Hz 0.01 (s)
represent typical values used in the experiments of [36].
C. Modeling using GWMM
The GMWM estimator was employed on both signals under the settings defined in Sec.III-B. The Haar Wavelet Deviation
ˆ
⌫ resulting from the MODWT filtering (see (1) and (2)) and on the observed sensors outputs {yt} are shown in Fig. 6 as black circles. The associated error bars are issued from the diagonal elements ofV⌫ˆ2 which was computed using (5). We considered the following models to describe the sensors errors:
Yt=Yt,W N + Xg k=1
Yt,GM(k) , g5
In other words, up to 5 GM processes summed with a WN process were considered as hypothetical models possibly representing the underlying stochastic process{yt}. The final choice of g was done according to the estimated values of g2 and . If two GM processes revealed no significant difference between their parameter values, one GM process was suppressed from F✓. The retained size ofg wasgg= 3 for the gyroscopes, and ga = 2 for the accelerometers. The estimated values of✓resulting from the GMWM estimator are presented in Tab.VI. The squared roots of the WV implied by the ✓ˆare shown as grey lines in Fig.6. They represent⌫( ˆ✓) and are the result of (9).
Note that the results presented here are based on analytical expressions for⌫2(✓) using Haar wavelet filters. In the case of a WN process we have that:
⌫2(✓) =
2W N
⌧ , ✓= W N2
GM process can be re-parametrized as an autoregressive model of order 1, denoted AR(1). Indeed,
Yt= Yt 1+W, Wiid⇠N(0, 2) (21) where =e tand 2= 2GM(1 e2 t)is equivalent to a GM as defined in AppendixC. Using the analytic expression for⌫2(✓)defined for an AR(1) we can deduce its counterpart for GM process which is given by:
⌫2(✓) =
⇣⌧
2 3 ⌧22 + 4⌧⌧/2+1 ⌧+1⌘
2
⌧2
8(1 )2(1 2) (22)
with✓=⇥ 2
GM
⇤T
.
TABLE VI
GMWM-BASED MODELS USED FOR THEXSens MTi-GACCELEROMETERS AND GYROSCOPES
Process Parameter Estimates Unit
Gyroscope WN W N 140 deg/h/p
Hz
GM g 113 deg/h/p
Hz 0.0005 (s)
GM g 65 deg/h/p
Hz 0.05 (s)
GM g 206 deg/h/p
Hz 10 (s)
Accelerometer WN W N 140 µg/p
Hz
GM g 5 µg/p
Hz 0.04 (s)
GM g 13 µg/p
Hz 0.002 (s)
Fig. 6. Result of the GMWM estimation represented by the matching of the wavelet deviation⌫(✓)issued from✓ˆon the wavelet deviation⌫ˆof the gyroscope (upper panel) and accelerometer (lower panel) error signals.
D. Validation
Validating and comparing the quality of the estimated mod- els in practice is a non-trivial task for several reasons. First, the true F✓ influencing the sensor-error behaviour is not known.
Second, the parameter calibration was performed on signals acquired in static environment of a constant temperature. A hypothesis that at least part of the error may vary with dynamic and/or changing environmental conditions is certainly realistic.
This fact could make the comparison less relevant as the actual stochastic behaviour may be masked by unmodeled effects that were absent during the calibration phase. (Note:
extension for considering dynamically-dependent sensor errors is discussed in Sec. VI). For these reasons, the following validation procedure is used here:
• A precise navigation solution for a given trajectory is computed using signals from high-grade sensors (typi-
cally tactical/navigation-grade IMUs, L1/L2 carrier-phase differential GNSS positioning).
• The real static error signals acquired by the sensor under study (in our case the signals of the XSens MTi-G) are added to the synthetic inertial signals emulated along the reference trajectory.
• Artificial outages in GNSS position/velocity observations are added to the dataset which is subsequently pro- cessed by a MEMS-IMU/GNSS integration implementing a closed-loop EKF. The in-house developed software enables flexible design in augmenting the error-states of the EKF by the sensor models under study.
• The quality of the model is judged by analyzing the actual navigation error as well as KF-predicted accuracy during inertial coasting mode.
Fig. 7 shows an extract of a trajectory issued from a helicopter flight performing Airborne Laser Scanning. The laser data were georeferenced using a trajectory obtained by integrating observations from a tactical-grade IMU (Litton LN200) and aJavad LegacyL1/L2 GNSS receivers (rover and master station). This trajectory will serve as a reference (dotted black line) to which the navigation solutions computed with the three sensor models (i.e. AV, KF-tuned and GMWM) are compared. It can be seen that the GMWM-based model (black- dashed line in Fig. 7) limits significantly the error growth during the GNSS-signal outage of 1 min duration as compared with the other two estimated models. Moreover, the effect of an incorrect error feedback (i.e. the closed-loop architecture) is revealed by the large positioning deviations for the other two estimated models. Beside the positioning error, the EKF estimated covariance matrixPneeds to be evaluated for each model. Fig. 8 depicts the true navigation errors (full lines) during the GNSS outage period together with the predicted navigation precision, i.e. the root of the diagonal elements of P), estimated by the EKF (dashed lines) for the East, North and Vertical components, respectively. The estimated precision overbounds the true error for both, the self-tuned and the GMWM-based models on all three components. However, the estimated precision is clearly underestimated on the East component when using the AV-based model. This confirms the note in [35] that AV analysis often underestimates the real sensor errors.
A particular attention is given to the KF-tuned model which revealed similar performances to the GMWM-based approach during several tests. For example, Fig. 9 shows true errors (in planimetry and altimetry) from a trajectory acquired with a car, where reference data were provided by a navigation- grade INS (Ixsea) integrated withJavad DeltaL1/L2/Glonass GNSS receivers. Two GNSS outages were introduced: the first (first column, duration of 50 sec.) after 6 min. and the second (second column, duration of 40 sec.) after 18 min. of operation. It can be seen that the GMWM-based model does not perform significantly better than the KF-tuned model during the first GNSS-outage period. However, in the second outage where dynamic is more complex the GMWM- based model outperformed that obtained by KF-tuning. This can be explained by the fact that the self-tuned model was
Fig. 7. Comparison between a reference trajectory (black dotted line) issued from a mapping flight in which a GNSS outage was introduced, with EKF solutions implementing the AV-based (light-grey line), the self-tuned (dark- grey line) and the GMWM-based (black dashed line) models under study.
designed for runs of short duration during which the correlated noise is not observed. Such model maybe relevant for some applications as those considered in [36]. However, if longer runs are considered, the effects of correlations cannot be neglected and the modeling by composite GM processes becomes appropriate and superior to simplified models.
VI. CONCLUSION ANDPERSPECTIVES
This paper proposes a new framework for stochastic cali- bration of inertial sensors using the GMWM estimator. This method enables the estimation of complex error models for which the AV-based technique fails and the EM-algorithm does not converge. Indeed, the simulations presented in Sec.
IV demonstrates that the GMWM approach revealed the capability of estimating composite stochastic models, such as sum of several GM processes, which was not the case for the classical estimation approaches.
The GMWM estimator was further applied for modeling error signals acquired with a real MEMS-based IMU in static environment and constant temperature. The estimated GMWM-based model, together with a second model based on AV-analysis, and a third KF-tuned model designed through experience in previous research, were then implemented in an EKF. The EKF was run on an emulated data where synthetic inertial observations along a real trajectory were corrupted by errors observed in static conditions. Since the main purpose was to validate the estimated model, this strategy prevented possible masking effect due to dynamically-dependent sensor errors that were not present in model establishment. The main conclusions from these experiments are that the GMWM es- timator was able to model complex-composite models which, under the hypothesis that error structure does not depend
Fig. 8. True navigation errors along the East (upper panel), North (middle panel) and Vertical (lower panel) components of the EKF forward solutions implementing the models under study, together with the associated filter estimated precision (dashed lines).
Fig. 9. True planimetric (upper panel) and altimetric (lower panel) navigation error during two GNSS-free periods (first and second column) of EKF solutions implementing the GMWM-based (black line) and self-tuned (grey line) model.
on dynamics, improved considerably (in comparison to the traditional modeling) the obtained navigation accuracy.
Future work aims to set up and conduct experiments which enable the construction and analysis of error signals acquired in dynamic environment. The observability of some processes, and therefore the justification of employing complex stochastic models for MEMS-inertial sensors, can only then be fully verified.
REFERENCES
[1] Stebler, Y., Guerrier, S., Skaloud, J., Victoria-Feser, M.P.: Constrained expectation-maximization algorithm for stochastic inertial error model- ing: study of feasibility. Measurement Science and Technology22(8), 085,204 (2011). URLhttp://stacks.iop.org/0957-0233/22/i=8/a=085204 [2] Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological)39(1), pp. 1–38 (1977). URLhttp:
//www.jstor.org/stable/2984875
[3] Guerrier, S., Stebler, Y., Skaloud, J., Victoria-Feser, M.: Wavelet variance based estimation for composite stochastic processes. Submitted working paper (2012). URLhttp://archive-ouverte.unige.ch/unige:18544 [4] Allan, D.: Statistics of Atomic Frequency Standards. In: Proceedings
of the IEEE, vol. 54, pp. 221–230 (1966). URL http://www.ino.it/
⇠azavatta/References/Allan.pdf
[5] Howe, D., Beard, R., Greenhall, C., Vernotte, F., Riley, W., Peppler, T.:
Enhancements to GPS Operations and Clock Evaluations using a Total Hadamard Deviation. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control52(8), 1253–1261 (2005). URLhttp://tf.boulder.
nist.gov/general/pdf/2005.pdf
[6] Vernotte, F., Lantz, E., Gagnepain, J.: Oscillator Noise Analysis: Mul- tivariance Measurement. IEEE Transactions on Instrumentation and Measurement42(2), 342–350 (1993)
[7] Vernotte, F., Lantz, E., Groslambert, J., Gagnepain, J.: A New Multi- Variance Method for the Oscillator Noise Analysis. In: Proceedings of the Frequency Control Symposium (1999)
[8] Vernotte, F., Zalamansky, G.: A Bayesian Method for Oscillator Sta- bility Analysis. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control46(6), 1545–1550 (1999)
[9] Guerrier, S.: Integration of Skew-Redundant MEMS-IMU with GPS for Improved Navigation Performance. Master thesis, ´Ecole Poly- technique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland (2008). URL http://129.194.42.64/www/dms/hec/assistants/guerrier/
RedundantIMU SG.pdf
[10] Guerrier, S.: Improving Accuracy with Multiple Sensors: Study of Redundant MEMS-IMU/GPS Configurations. In: Proceedings of the 22nd International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS 2009), pp. 3114–3121. Savannnah, GA, USA (2009)
[11] Hou, H.: Modeling Inertial Sensors Errors using Allan Variance. Mas- ter’s thesis, Geomatics Engineering, University of Calgary (2004). URL http://www.ucalgary.ca/engo webdocs/NES/04.20201.HaiyingHou.pdf [12] El-Sheimy, N., Hou, H., Niu, X.: Analysis and Modeling of Inertial
Sensors using Allan Variance. IEEE Transactions on Instrumentation and Measurement57(1), 140–149 (2008)
[13] Strus, J., Kirkpatrick, M., Sinko, J.: Development of a High Accuracy Pointing System for Maneuvering Platform. In: Proceedings of the 20th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS 2007). Fort Worth, TX, USA (2007) [14] Xing, Z., Gebre-Egziabher, D.: Modeling and Bounding Low Cost In-
ertial Sensor Errors. In: Position, Location and Navigation Symposium, 2008 IEEE/ION, pp. 1122–1132 (2008). DOI 10.1109/PLANS.2008.
4569999
[15] Percival, D., Guttorp, P.: Long-Memory Processes, the Allan Variance and Wavelets. Wavelets in Geophysics4, 325–344 (1994)
[16] Mallat, S.: A wavelet tour of signal processing. Academic press (1999) [17] Percival, D., Andrew, T.: Wavelet Methods for Time Series Analysis.
Cambridge Series in Statistical and Probabilistic Mathematics. Cam- bridge University Press, 32 Avenue of the Americas, New York, NY 10013-2473, USA (2000)
[18] Greenhall, C.: Recipes for Degrees of Freedom of Frequency Stability Estimators. In: IEEE Transactions on Instrumentation and Measure- ments, vol. 40, pp. 994–999. IEEE (1991)
[19] Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)
[20] Serroukh, A., Walden, A.T., Percival, D.B.: Statistical properties and uses of the wavelet variance estimator for the scale analysis of time series. Journal of the American Statistical Association 95(449), pp.
184–196 (2000). URLhttp://www.jstor.org/stable/2669537
[21] Percival, D.P.: On estimation of the wavelet variance. Biometrika82(3), pp. 619–631 (1995). URLhttp://www.jstor.org/stable/2337538 [22] Zhang, N.F.: Allan variance of time series models for measurement data.
Metrologia45(5), 549 (2008). URLhttp://stacks.iop.org/0026-1394/45/
i=5/a=009
[23] Hansen, L.P.: Large sample properties of generalized method of mo- ments estimators. Econometrica 50(4), pp. 1029–1054 (1982). URL http://www.jstor.org/stable/1912775
[24] Smith, A.: Estimating nonlinear time series models using simulated vector autoregressions. Journal of Applied Econometrics8, 563–584 (1993). URLhttp://www.econ.yale.edu/smith/2285075.pdf
[25] Gourieroux, C., Monfort, A., Renault, E.: Indirect Inference. Journal of Applied Econometrics8, 85–118 (1993). URLhttp://ideas.repec.org/a/
jae/japmet/v8y1993isps85-118.html
[26] Gallant, A.R., Tauchen, G.: Which moments to match? Econometric Theory 12(4), pp. 657–681 (1996). URLhttp://www.jstor.org/stable/
3532789
[27] Genton, M.G., Ronchetti, E.: Robust indirect inference. Journal of the American Statistical Association98(461), pp. 67–76 (2003). URLhttp:
//www.jstor.org/stable/30045195
[28] Granger, C.W.J., Morris, M.J.: Time series modelling and interpretation.
Journal of the Royal Statistical Society. Series A (General)139(2), pp.
246–257 (1976). URLhttp://www.jstor.org/stable/2345178
[29] Ter¨asvirta, T.: The invertibility of sums of discrete ma and arma processes. Scandinavian Journal of Statistics4(4), pp. 165–170 (1977).
URLhttp://www.jstor.org/stable/4615673
[30] Kittel, C.: Elementary Statistical Physics. John Wiley & Sons (1958) [31] Drexler, K.: Nanosystems- Molecular machinery, manufacturing, and
computation. John Wiley & Sons, New York (1992)
[32] Rao, C.R.: Linear Statistical Inferentcea and Its Applications (2nd ed.).
John Wiley, New York (1973)
[33] Benichou, J., Gail, M.H.: A delta method for implicitly defined random variables. The American Statistician 43(1), pp. 41–44 (1989). URL http://www.jstor.org/stable/2685169
[34] Xing, Z., Gebre-Egziabher, D.: Modeling and bounding low cost inertial sensor errors. In: Position, Location and Navigation Symposium, 2008 IEEE/ION, pp. 1122 –1132 (2008). URL http://ieeexplore.ieee.org/
stamp/stamp.jsp?tp=&arnumber=4569999
[35] Niu, X., Nasser, S., Goodall, C., El-Sheimy, N.: A universal approach for processing any mems inertial sensor configuration for land-vehicle navigation. Journal of Navigation60(2), 233–246 (2007)
[36] Waegli, A.: Trajectory Determination and Analysis in Sports by Satellite and Inertial Navigation. Ph.D. thesis, Lausanne (2009). DOI 10.
5075/epfl-thesis-4288. URLhttp://library.epfl.ch/theses/?nr=4288,http:
//library.epfl.ch/theses/?nr=4288
APPENDIXA
EXAMPLE OF THE INCONSISTENCY OF THE ALLAN VARI-
ANCEMETHODOLOGY
This Section presents an example of inconsistency of the classical AV methodology. Although it is not formally demonstrated here, we believe that AV based estimators are inconsistent when applied to composite stochastic processes.
To illustrate this statement, assume that we observe a process {Yt} which can be written asYt=Yt,W N+Yt,RW using the same notation as in App. C. Assume further that the process Yt,W N is independent of Yt,RW and let ✓ = [ W N2 RW2 ]T be the vector of parameters that specifies the model. The theoretical AV of this system can be easily obtained and is given by:
2
¯
y(⌧) =6 W N2 + 2⌧2+ 1 RW2 6⌧
In this case, the AV methodology would consist in applying a linear regression on the k first scales in order to estimate
