6th International Symposium on Mobile Mapping Technology, Presidente Prudente, São Paulo, Brazil, July 21-24, 2009
USING ALLAN VARIANCE TO DETERMINE THE CALIBRATION MODEL OF INERTIAL SENSORS FOR GPS/INS INTEGRATION Songlai Han a, b Jinling Wang b Nathan Knight b a
School of Optoelectronics Science and Engineering, National University of Defense Technology, 410073, Changsha, Hunan, China -
[email protected] b School of Surveying and Spatial Information System, The University of New South Wales, 2052, Sydney, NSW, Australia -
[email protected],
[email protected]
KEY WORDS: Sensor, Modeling, Integration, Navigation, GPS/INS
ABSTRACT: In this research, Allan Variance analysis is used to identify the stochastic error sources existing in inertial sensors and to determine the corresponding noise parameters. According to the noise parameters, the power spectral density (PSD) function of the stochastic error sources can be determined. The differential equation descriptions for individual stochastic errors are then derived for two circumstances: rational spectral PSD and non-rational spectral PSD. Then, a unified calibration model in the form of differential equation for multiple stochastic errors is derived. By incorporating the unified calibration model into the traditional INS error equation, the Kalman Filter for GPS/INS integration is augmented. At last, the actual test data are processed to elaborate and exemplify the approach proposed.
1. INTRODUCTION Inertial navigation system (INS) is a time-dependent system, whose performance will degrade with operation time. It has been clear that there are several stochastic error sources existing in inertial sensors (IEEE STD 647, 2006). The statistical moments of these stochastic errors, such as random walk, increase with time, which leads to the accumulation of inertial errors with time. In the long run, even for strategic level inertial system, this kind of error accumulation is unbearable, and the stochastic errors should be calibrated using external aiding facilities. GPS is the primary choice. Before the calibration procedure can be carried out, the error sources in inertial sensors should be identified first. There are several approaches to perform stochastic error identification: (1) Autocorrelation function approach (Brown and Hwang, 1997) (2) ARMA model approach (Nassar, 2003) (3) PSD approach (IEEE STD 1293, 1998) and (4) Allan Variance approach (IEEE STD 647, 2006). According to Brown and Hwang (1997), the autocorrelation function approach is not practical in real data analysis, because it requires a very long test time which may be longer than the lifetime of the tested sensors. The ARMA model approach is model-sensitive according to IEEE STD 952 (1997). Also according to IEEE STD 952 (1997), PSD approach and Allan Variance approach are the preferred methods for the stochastic error identification of the inertial sensor. In this research, the Allan Variance approach is used to identify the stochastic errors existing in inertial sensors. According to Allan Variance analysis, there are mainly five stochastic error sources existing in inertial sensors (IEEE STD 647, 2006): quantization noise, angle/velocity random walk, flicker noise, angular rate/acceleration random walk and ramp noise. All or some of them may appear in a specific inertial sensor, which should be determined by real data analysis. If we want to calibrate these stochastic errors during GPS/INS integration, they should be modelled as a unified calibration model that can be easily coupled into the INS error equations. In this research, we are trying to find a unified model in the form of differential equation for the blending of multiple stochastic errors. The PSDs of the stochastic errors can be
determined through Allan Variance analysis. If a stochastic error has rational spectrum, namely the PSD of the stochastic error can be expressed as the ratio of two polynomials of square frequency, then the stochastic error can be described using a differential equation driven by unit noise (Papoulis, 1991). If a non-rational spectrum exists for a stochastic error, an approach is proposed in this paper to find an approximate differential equation, also driven by unit white noise, to describe the corresponding stochastic error. Finally, each of the stochastic errors is described using a unique differential equation driven by unit white noise. It can be proved that multiple differential equations driven by independent white noises are equivalent to a single differential equation driven by a unit white noise. So, multiple stochastic errors can be described using a unified model in the form of differential equation. In the remaining part of this paper, Allan Variance analysis and its application to the identification of stochastic error sources in inertial sensors are introduced first. Then, the differential equation descriptions for individual stochastic errors are derived according to their PSDs. Further, the unified model in the form of differential equation for the blending of multiple stochastic errors is derived. Then, the augmentation, based on the unified calibration model of multiple inertial errors, to the Kalman Filter of GPS/INS integration is elaborated. Finally, the actual data from the inertial sensors of a tactical level INS are processed to determine the calibration model of the inertial sensors and the corresponding augmented Kalman Filter for GPS/INS integration is developed.
2. IDENTIFICATION OF STOCHSTIC ERROR SOURCES IN INERTIAL SENSORS As mentioned in Section 1, Allan Variance analysis (IEEE STD 647, 2006) is a preferred method for the identification of stochastic error sources in inertial sensors, hence it is used here to identify the error sources existing in the gyro and accelerometer.
2.1 Allan Variance: A Review
2
σ total (τ ) ≅ ∑ Anτ n / 2
Allan Variance, a time domain analysis technique, is an accepted IEEE standard for gyro specifications (IEEE STD 647, 2006). The method was initially developed by David Allan of the National Bureau of Standards to quantify the error statistics for a Caesium beam frequency standard (Allan, 1966). The method, in general, can be applied to analyse the error characteristics of any precise measurement instruments. The computation of Allan Variance is straightforward. Let us take the analysis to the angular rate data outputting from gyro as an example. Given the angular rates are recorded at a constant time interval τ 0 , a collection of N data points can be reformed to be K = N / M clusters where M is the number of samples per cluster. Then, compute the average for each cluster:
ωk (M ) =
1 M
M
∑ ω ki ;
k = 1," , K
(1)
σ A2 (τ M ) = ≅
1 2
(ω k +1 ( M ) − ω k ( M ) )2
K −1 1 2 ∑ (ω k +1 ( M ) − ω k ( M ) ) 2( K − 1) k =1
(2)
denotes the ensemble average and it is approximated where by the time average and τ M = M τ 0 is the correlation time. The accuracy in the estimate of the root Allan Variance increases with the additional number of cluster averages. In general, the accuracy ε of the computation for K cluster averages is given by IEEE STD 647 (2006):
ε=
Noise Types
1 2( K − 1)
(3)
2.2 Stochastic Error Sources in Inertial Sensors According to Allan Variance analysis, generally there are five error sources existing in inertial sensors (IEEE STD 647, 2006): Quantization Noise, Angle/Velocity Random Walk, Flicker Noise, Angular Rate/Acceleration Random Walk, and Ramp Noise. The noise parameters of these stochastic errors can also be determined by Allan Variance analysis, and these parameters can be used to determine the PSDs of the corresponding noises. The noises, their Allan Variances and PSDs are listed in Table 1. If the stochastic errors listed in Table 1 are statistically independent to each other, then the total Allan Variance for gyro or accelerometer can be expressed as: 2 σ total = σ Q2 + σ N2 + σ B2 + σ N2 + σ R2
The root Allan Variance is then given by:
AV ( σ 2 (τ ) )
Noise Para.
PSD ( S( f ) )
Q
(2 π f ) 2 Q 2T
Quantization Noise
3Q 2
Angle/Velocity Random Walk
N2 τ
N
N2
Flicker Noise
2 B 2 ln 2 π
B
⎛ B 2 ⎟⎞ 1 ⎜⎜ ⎟ ⎟ ⎜⎜⎝ 2 π ⎟⎠ f
Angular Rate /Acceleration Random Walk
K 2τ 3
K
⎛ K ⎞⎟2 1 ⎜⎜ ⎟ ⎜⎝ 2 π ⎠⎟ f 2
Ramp Noise
R 2τ 2 2
R
i =1
Finally, the Allan Variance from the cluster averages is computed as follows:
(5)
−2
τ2
R2 (2 π f ) 3
Table 1. Stochastic Error Sources in Inertial Sensors
After the Allan Variances are obtained for different correlation times τ , the coefficients An , which are linked with the noise parameters listed in Table 1, can be computed in the least squares sense. Finally, the PSDs can be obtained for each of the stochastic noises.
3. UNIFIED CALIBRATION MODEL FOR MULTIPLE STOCHASTIC ERRORS IN INERTIAL SENSOR Allan Variance analysis can only determine the characteristics of each individual error source, but all these error sources need to be mixed together into a unified calibration model to calibrate them using Kalman Filter. In the following subsections, the differential equation descriptions of the stochastic errors in inertial sensors are derived first. Then, a special consideration for quantization noise is discussed. Finally, the unified calibration model is derived for multiple stochastic errors.
3.1 Differential Equation Descriptions of the Stochastic Errors in Inertial Sensors The Allan Variance and thereafter determined PSDs of the stochastic errors in inertial sensors are listed in Table 1. Now, what we want to do is using the PSDs to determine the corresponding differential equation description of the stochastic errors. Generally, a stochastic process is generated by passing a unit white noise through a shaping filter (Brown and Hwang, 1997). A shaping filter is a linear system which is used to shape a unit white noise into a given spectral function. Conversely, given the PSD of a stochastic process, the transfer function of the shaping filter can also be obtained by the following equation (Brown and Hwang, 1997):
(4) S x (ω ) = G ( jω )
2
(6)
where S x ( ω ) is the PSD of a stochastic process x ; G ( jω ) is the transfer function of the shaping filter; ω is the circular frequency and equal to 2 π f . According to linear system theory, the following relationship holds:
For angular rate/acceleration random walk, the PSD is rational spectrum and can be rewritten, using circular frequency instead of frequency, as follows: S rw ( ω ) =
Y ( jω ) = G ( jω ) X ( jω )
(7)
where Y ( j ω ) is the Fourier transformation of system output, and X ( j ω ) is the Fourier transformation of system input. Carry out the inverse Fourier transformation in both sides of the above equation, and the differential equation description of the linear system can be obtained. Let us consider quantization noise and angle/velocity random walk first. According to equation (6) and the PSD of quantization noise, the transfer function of the shaping filter for quantization noise is:
K2 ω2
According to equation (6), the transfer function of the corresponding shaping filter is:
G rw ( jω ) =
K jω
(12)
(8)
So the differential equation description for quantization noise is:
d qn (t ) = Q T u (t )
(11)
So the differential equation description for angular rate random walk and acceleration random walk can be formulated as: d rw (t ) = Ku1 (t )
G qn ( jω ) = jω Q T
(10)
(9)
where u (t ) is a unit white noise. So if quantization noise is directly incorporated into INS error equations (Goshen-Meskin and Bar-Itzhack, 1992), it will be a kind of noise source which is the derivative of white noise. According to Kalman Filtering theory, the optimal estimation can only be performed on differential equations driven by white noises (Brown and Hwang, 1997). So quantization noises cannot be directly incorporated into the INS error equations used by GPS/INS integration. The special consideration about quantization noise will be discussed in the next subsection. Angle random walk and velocity random walk are white noises in angular rate and acceleration respectively. According to Kalman Filtering theory, they can be directly incorporated into the INS attitude error equation and velocity error equation (Brown and Hwang, 1997). So it is not necessary to develop the differential equation descriptions for quantization noise and angle/velocity random walk. Then, for the remaining three noises, two situations should be treated separately: rational spectrum and non-rational spectrum. If the PSD is a rational spectrum (the ratio of two polynomials of ω 2 or f 2 ), such as the PSD of angular rate/acceleration random walk, G ( j ω ) will be a rational function of j ω (Papoulis, 1991). Under this circumstance, G ( j ω ) represents a finite order linear system, and the corresponding system differential equation is straightforward (Brown and Hwang, 1997). If the PSD is a non-rational spectrum, such as the PSDs of flicker noise and ramp noise, G ( j ω ) will be an irrational function of j ω and represent an infinite order linear system. Under this circumstance, an acceptable rational approximation should be found for G ( j ω ) .
where u1 (t ) is a unit white noise. For flicker noise and ramp noise, the PSDs are non-rational spectra, so rational transfer functions can not be obtained. Here, we choose the first order Gauss-Markov process to approximate the actual flicker noise process and the second order GaussMarkov process to approximate the actual ramp noise process. The approximate transfer functions, Gˆ ( jω ) , derived from Gauss-Markov processes and the actual transfer functions, G ( jω ) , derived from PSDs are compared with each other to estimate the magnitude errors resulted from this approximation. For flicker noise, the first order Gauss-Markov process is used as the approximation: d fn (t ) + β d fn (t ) = β Bu 2 (t )
(13)
where β is the reciprocal correlation time needed to be determined; B is the flicker noise parameter listed in Table 1; u 2 (t ) is a unit white noise. The approximate transfer function is obtained from equation (13) by performing Fourier transform:
Gˆ fn ( jω ) =
βB jω + β
(14)
Considering (6) and the flicker noise PSD listed in Table 1, the actual irrational transfer function of flicker noise is: G fn ( jω ) =
B jω
(15)
The relative magnitude error in the unit of dB between Gˆ fn ( jω ) and G fn ( j ω ) is:
ε fn = 10 * log
Gˆ fn ( jω ) − G fn ( jω ) G fn ( jω )
(16)
Insert equations (14) and (15) into equation (16), and the error is then rewritten as:
ε fn = 10 * log
ω 1+ ω
−1
2
β
(17)
this indirectly. However, (Savage, 2002) only considered the situation where the INS error equations are implemented in inertial frame. Considering most of the INSs are implemented in local level geographical frame, the counterpart results are derived here in the local level geographical frame. The psi-angle error equation of INS is (Goshen-Meskin and Bar-Itzhack, 1992):
δ n = δ v − ω en × δ n
(22a)
δ v = − ω inn + ω ien × δ v − ψ × f b + δ g n + Cbn ∇ b
(22b)
(
)
ψ =
2
From equation (17), it can be found the magnitude error ε fn is smaller than 0dB in the whole frequency scope. Considering the low frequency nature of flicker noise, if β is chosen to be 3.97Hz, the corresponding magnitude error will be smaller than -3dB in the frequency band [0.1Hz, 10Hz]. For ramp noise, the second order Gauss-Markov process is used as the approximation: drr (t ) + 2ω 0 drr (t ) + ω 02 d rr (t ) = Ru3 (t )
−ω inn
δ vqb respectively, then a new velocity error and attitude error
considering quantization noises can be defined as follows:
δ vˆ = δ v − Cbnδ vqb
(23a)
− Cbn δα qb
(23b)
ψˆ = ψ
R
Gˆ rr ( jω ) =
−ω 2 + j 2ω 0ω + + ω 02 R G rr ( jω ) = ( jω )1.5
ε rr = 10 * log
ω3 ω + ω 04 4
−1
(19)
Perform differential operation on both sides of equation (23), and then insert them into equation (22). Considering: b C bn = Cbn (ω ib ×) − (ω inn ×)Cbn
From equation (21), it is noted that the magnitude error ε rr is also smaller than 0dB in the whole frequency scope. Considering ramp noise is also a low frequency noise, if ω 0 is chosen to be 0.01rad/s, magnitude error ε rr will be smaller than -0.5dB in the frequency band [0.1Hz, 10Hz]. So, till now, the differential equation descriptions for angular rate/acceleration random, flicker noise walk and ramp noise are obtained and formulated in equations (12), (13) and (18) respectively. Then, what we need to do is to find the equivalent stochastic model for blending these equations, but before this, we need to first deal with the quantization noise issue mentioned at the beginning of this subsection.
3.2 Special Considerations for Quantization Noise Because quantization noise cannot be incorporated into the INS error equation directly, (Savage, 2002) proposed a method to do
(24)
The new INS error equations obtained are:
(20)
(21)
(22c)
Although quantization noises are the derivatives of white noises in terms of angular rate and acceleration, they are white noises in terms of angle and velocity (Savage, 2002). Given the quantization noises of a gyro and an accelerometer are δα qb and
(18)
where R is the ramp noise parameter listed in Table 1; ω 0 is the undammed natural frequency of this second order system and needs to be determined; and u3 (t ) is unit white noise. Similar with the afore analysis to flicker noise, the approximate transfer function, the actual transfer function and the magnitude error can be obtained as follows:
×ψ +
Cbn ε b
δ n = δ vˆ − ω en × δ n + Cbnδ vqb
(
(25a)
)
δ vˆ = − ω inn + ω ien × δ vˆ − ψˆ × f b + δ g n + Cbn (∇ b − δ vqb ) + C bn f b × δα qb − b ⎡ (ω ien ×)C bn + Cbn (ω ib ×) ⎤ δ vqb ⎣ ⎦ ψˆ = −ω inn × ψˆ + C bn (ε b − δα qb ) − Cbn (ω ibb ×)δα qb
(25b)
(25c)
Through equation (25), the quantization noises of gyro and accelerometer are incorporated into the INS error equations. The following three items are the equivalent white noises of quantization noises for the position error equation, the velocity error equation and the attitude error equation respectively:
C bnδ vqb C bn
f
b
× δα qb
−
⎡ (ω ien ×)C bn ⎣
(26a) +
b C bn (ω ib ×) ⎤ δ vqb ⎦
b − Cbn (ω ib ×)δα qb
(26b) (26c)
After this special consideration, quantization noises in the gyro and the accelerometer are transformed into the equivalent white noises and incorporated into the INS error equations.
a2 =
2ω 0 β + ω 02
a3 = βω 0 2 a4 = 0
(31b) (31c) (31d)
3.3 The Unified Calibration Model Till now, the quantization noise and angle/velocity random walk have been incorporated into the INS error equations in the form of white noise. Next, what we want to do is to find the equivalent error model for the blending of the remaining three stochastic errors: angular rate/acceleration random walk, flicker noise and ramp noise. This equivalent error model will act as the calibration model of the inertial sensor, when INS is integrated with GPS. Angular rate/acceleration random walk, flicker noise and ramp noise may or may not concurrently exist in the same sensor. This should be determined by Allan Variance analysis of the actual data. Here, the most general case, where all the three noises exist in the same sensor, will be considered to thoroughly elaborate the problem. The differential equation descriptions of angular rate/acceleration random walk, flicker noise walk and ramp noise are formulated in equations (12), (13) and (18) respectively, and for convenience they are rewritten here using the differential operator as follows: Ku1 (t ) D β Bu 2 (t ) d fn (t ) = D+β Ru3 (t ) d rr (t ) = 2 D + 2ω 0 D + ω 02 d rw (t ) =
(27a) (27b) (27c)
where D is the differential operator. Define a new function z (t ) as follows: z (t ) = d rw (t ) + d fn (t ) + d rr (t )
(32)
where ω (t ) is a unit white noise. According to the definition of WSS, all coefficients in equation (32) are determined by the coefficients in equation (29):
b0 2 = K 2 + β 2 B 2
(33a)
b1 − 2b0 b2 = K 2 β 2 + R 2
(33b)
2
b2 2 − 2b1b3 = K 2ω 04 + β 2 B 2ω 04 + β 2 R 2
(33c)
b3 2 = K 2 β 2ω 04
(33d)
Combining equations (30) and (32), the equivalent stochastic error model of the blending of angular rate/acceleration random walk, flicker noise and ramp noise is obtained as:
z (4) (t ) + a1 z (3) (t ) + a 2 z (2) (t ) + a3 z (1) (t ) + a 4 z (0) (t ) = b0ω (3) (t ) + b1ω (2) (t ) + b2ω (1) (t ) + b3ω (0) (t )
(34)
(28)
D( D + β )( D 2 + 2ω0 D + ω02 ) z (t ) = D( D 2 + 2ω0 D + ω02 ) β Bu2 (t ) +
b0ω (3) (t ) + b1ω (2) (t ) + b2ω (1) (t ) + b3ω (0) (t )
4. AUGMENTATION FOR KALMAN FILTERING OF GPS/INS INTEGRATION
Insert equation (27) into equation (28) and after rearranging yields:
( D + β )( D 2 + 2ω0 D + ω02 ) Ku1 (t ) +
It can be proved that the right hand side of equation (29) is equivalent to the following expression in the sense of wide sense stationary (WSS) in terms of the definition of WSS (Papoulis, 1991):
(29)
In this section, the primary task is to derive an alternative form for INS error equations, which is augmented by the inertial sensors’ equivalent stochastic model obtained in the previous section. This alternative form can be used in Kalman Filtering to estimate and calibrate the inertial sensors’ error. The discrete INS error equation is derived from equation (25) and is formulated as: Y ( k + 1) = φYY ( k , k + 1)Y ( k ) + E ( k )
(35)
D( D + β ) Ru3 (t )
where Y ( k ) is a vector composed of nine elements accounting for position errors, velocity errors and attitude errors; E ( k ) represents the stochastic error existing in the system and can be divided into two parts:
The left hand side of equation (29) is expanded as:
z (4) (t ) + a1 z (3) (t ) + a 2 z (2) (t ) + a3 z (1) (t ) + a 4 z (0) (t )
(30) E ( k ) = ωY ( k ) + Z ( k )
(36)
where the coefficients are computed as follows: a1 = β +
2ω 0
(31a)
where ωY ( k ) is the white noise part of the stochastic error, which accounts for the quantization noise and angle/velocity random walk noise; and Z ( k ) is the colored noise part of the
stochastic error, which accounts for the flicker noise, the angular rate/acceleration random walk noise and the ramp noise. Reviewing equation (25), Z ( k ) is formulated as: Z ( k ) = [01×3
z1 ( k ) "
z 6 ( k )]T
where φ x1 x1 ( k , k + 1) = e F1Δ t I + F1Δ t , and
is the sampling
Q x1 ( k ) ≈ Q x1 (τ ) Δ t
(37)
where zi ( k ), i = 1," ,3 represent the stochastic errors of three gyros; and zi ( k ), i = 4," , 6 represent the stochastic errors of three accelerometers. Each of them has the continuous form the same as equation (34). What we want to do next is to find the expression of Z ( k ) according to equation (34), and use it to augment the Kalman Filering of GPS/INS integration. Taking z1 ( k ) as an example, its continuous counterpart z1 (t ) fulfils equation (34), so the differential equation of z1 (t ) can be realized in the state space as follows:
Δt
interval of discretization. The discrete process noise covariance matrix is (Brown and Hwang, 1997): (42)
The state space realization of zi ( k ), i = 2," , 6 can be obtained according to the above procedure for z1 ( k ) . Inserting zi ( k ), i = 1," , 6 into equation (37), the following relationship is obtained: ⎡0 ⎤ Z ( k ) = ⎢ 3× 24 ⎥ X ( k ) H ⎣ 6× 24 ⎦
(43)
T
x1 (t ) = F1 x1 (t ) + G1ω x1 (t )
(38a)
z1 (t ) = H 1 x1 (t )
(38b)
where ω x1 (t ) is unit white noise the same as that in equation (34); F1 , G1 and H 1 are defined as follows:
⎡0 ⎢1 F1 = ⎢ ⎢0 ⎢ ⎣0
0 0 1 0
0 0 0 1
According to the above values of F1 , G1 and H 1 , equation (38) is the observable canonical form realization of equation (34) (Ogata, 2001). Observable canonical form means that the state vector x1 (t ) is completely observable under the knowledge of z1 (t ) , which is a preferable property in state estimation. The corresponding process noise covariance matrix can be calculated as: (39)
Considering the value of G1 and that ω x1 (t ) is unit white noise, Q x1 (τ ) is: ⎡ b32 ⎢ ⎢b b Q x1 (τ ) = ⎢ 2 3 ⎢ b1b3 ⎢ ⎣ b0 b3
b3b2
b3b1
b22
b2 b1
b1b2
b12
b0 b2
b0 b1
b3 b0 ⎤ ⎥ b2 b0 ⎥ ⎥ b1b0 ⎥ ⎥ b02 ⎦
H 6× 24 = diag ( H 1 H i = [0
0
H2
"
0 1] , i = 1," , 6
H6)
(44)
Inserting Z ( k ) into equation (36) and then into equation (35), the augmented INS error equations is obtained as follows:
T − a4 ⎤ ⎡ b3 ⎤ ⎡0 ⎤ ⎢ ⎥ ⎥ ⎢0 ⎥ b2 − a3 ⎥ , G = ⎢ ⎥ , H1 = ⎢ ⎥ − a 2 ⎥ 1 ⎢ b1 ⎥ ⎢0 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ b − a1 ⎦ ⎣ 0⎦ ⎣1 ⎦
E ⎡ (G1ω x1 (t ))(G1ω x1 (t − τ ))T ⎤ = Q x1 (τ )δ (τ ) ⎣ ⎦
where X ( k ) = ⎡ x1T ( k ) " x6T ( k ) ⎤ ; 0 3× 24 is a zero matrix ⎣ ⎦ with the dimensions of 3 and 24 respectively; and H 6× 24 is a diagonal block matrix:
⎡ Y ( k + 1) ⎤ ⎡φYY ( k , k + 1) φ XY ( k , k + 1) ⎤ ⎡ Y ( k ) ⎤ ⎢ X ( k + 1) ⎥ = ⎢ φ XX ( k , k + 1) ⎥⎦ ⎢⎣ X ( k ) ⎥⎦ 0 ⎣ ⎦ ⎣ ⎡ ω (k ) ⎤ +⎢ Y ⎥ ⎣ω X ( k ) ⎦
(45)
where φYY ( k , k + 1) is the state transfer matrix of the original system, which is the same with that in equation (35); and φ XY ( k , k + 1) is: ⎡ 0 3× 24 ⎤ ⎥ ⎣ H 6× 24 ⎦
φ XY ( k , k + 1) = ⎢
(46)
which is the same as that in equation (43); and φ XX ( k , k + 1) is a diagonal block matrix:
φ XX ( k , k + 1) = diag (φ x1 x1 ," , φ x6 x6 )
(47)
The process noise covariance matrix for equation (45) is: (40) ⎡ Q Q (k ) = ⎢ Y ⎣ 0 24×9
To be incorporated into equation (35), the discrete form of equation (38) is needed: x1 ( k + 1) = φ x1 x1 ( k , k + 1) x1 ( k ) + G1ω x1 ( k )
(41a)
z1 ( k ) = H 1 x1 ( k )
(41b)
0 9× 24 ⎤ Q X ⎥⎦
(48)
where QY is the process noise covariance matrix accounting for the quantization noise and angle/velocity random walk noise in the original system; and Q X is a diagonal block matrix: Q X = diag (Q x1
" Q x6 )
(49)
In this section, the approach proposed in the above sections is applied to an actual inertial navigation system to determine the calibration model for the inertial sensors. Boeing CMIGITS-II, a tactical level navigation system, was used in our test to collect static inertial data up to six hours. Allan Variance approach was applied to the data set to analyse the stochastic noise sources and determine the noise parameters. Differential equation descriptions of the stochastic noises were then obtained according to the corresponding noise parameters. Finally, the unified calibration model to multiple stochastic noises and the augmented Kalman Filter were derived for the inertial sensors in CMIGITS-II. According to Allan Variance Analysis, it was noted that quantization noise and ramp noise were not significant in the tested gyros; and ramp noise was not significant in the tested accelerometers. All of the noise parameters can be computed according to equation (5) and Table 1 in the least squares sense, but considering the limited space, they were not listed here. Now, the differential equation descriptions of the stochastic noises will be derived. According to the above analysis, there are three stochastic noises to be considered for the gyros in this test: angle random walk, flicker noise and angular rate random walk; there are four stochastic noises to be considered for the accelerometers: quantization noise, velocity random walk, and flicker noise and acceleration random walk. We will take gyro x as an example to elaborate the procedure. According to subsection 3.1, we do not need to develop the differential description for quantization noise and angle/velocity random walk. According to (12) and (13), the differential equation description of flicker noise and angular rate random walk in gyro x are:
According to equation (40) the process noise covariance matrix is: ⎡ K 2β 2 Q gx (τ ) = ⎢ ⎢K β K 2 + β 2B2 ⎣
K β K 2 + β 2B2 ⎤ ⎥ K 2 + β 2 B 2 ⎥⎦
(53)
The discrete form of equation (52) is:
0 ⎞ ⎛ 1 x gx ( k + 1) = ⎜ x ( k ) + G gx ω gx ( k ) 1 β Δ t ⎟⎠ gx t Δ − ⎝ z gx ( k ) = [ 0 1] x gx ( k )
(54a) (54b)
The discrete process noise covariance matrix is: ⎡ K 2 β 2 Δt Q gx ( k ) ≈ ⎢ ⎢ K β K 2 + β 2 B 2 Δt ⎣
K β K 2 + β 2 B 2 Δt ⎤ ⎥ (55) ( K 2 + β 2 B 2 ) Δ t ⎥⎦
Trajectory Comparison 200 DGPS Reference General Model New Model
150
100
North (m)
5. APPLICATION TO ACTUAL INERTIAL SENSORS
50
0
-50
d fn (t ) + β d fn (t ) = β Bu 2 (t ) d (t ) = Ku (t ) rw
(49a)
-100
(49b)
1
-150 -600
-500
-400
-300
-200
-100
0
100
East (m)
The unified calibration model for flicker noise and angular rate random walk is just a lower order version of equation (34) and its coefficients can be obtained by inserting R = 0 and ω 0 = 0 into equation (31). So the unified calibration model for gyro x is:
(2) (1) z gx (t ) + β z gx (t ) =
(1) (0) K 2 + β 2 B 2 ω gx (t ) + K βω gx (t )
(51)
According to equations (38) and (51), the state space realization of the unified calibration model for gyro x is: x gx (t ) = Fgx x gx (t ) + G gx ω gx (t )
(52a)
z gx (t ) = H gx x gx (t )
(52b)
Figure 1. Trajectory Comparison: DGPS Reference, General Model and New Model
The discrete calibration models for other gyros and accelerometers can be obtained by following the above procedure. All of these discrete calibration models should be organised together, and then the augmented Kalman Filter model as equation (45) can be obtained and ready for GPS/INS integration application. Initial test results show this unified calibration model improves system performance impressively, especially during GPS outages. Figure 1 shows the navigation results comparison among DGPS reference, general model and new model, in which a two-minute GPS outage is included. Because of the limited space, only the trajectory comparison is shown here.
6. CONCLUDING REMARKS where the coefficient matrices are:
⎡0 Fgx = ⎢ ⎣1
0 ⎤ , G gx = ⎡ K β ⎢⎣ − β ⎥⎦
T
K 2 + β 2 B 2 ⎤ , H gx = [ 0 1] ⎦⎥
In this research, the calibration model of inertial sensors using Allan Variance analysis has been discussed in the following aspects: A. Identifying the error sources in inertial sensors and determining the corresponding noise parameters.
B.
C.
D.
Deriving the differential equation descriptions for individual stochastic errors. For the stochastic errors with non-rational PSDs, an approach is proposed to determine the approximate differential equation descriptions for the corresponding stochastic errors and to evaluate the error introduced by this approximation. Deriving a unified calibration model for the blending of multiple stochastic errors. Individual stochastic errors can be described using different differential equations driven by independent white noises. In the sense of wide stationary, the blending of these differential equations is equivalent to a single differential equation driven by a unit white noise. This equivalent differential equation acts as the unified calibration model. Developing an augmented Kalman Filter accounting for the unified calibration model. This augmented Kalman Filter can be used to calibrate the stochastic errors in inertial sensors during GPS/INS integration.
ACKNOWLEDGMENTS The first author is sponsored by the Chinese Scholarship Council for his studies in Australia. The first author would like to thank his supervisor, Dr. Jinling Wang, for his guidance, advice and support and his fellow Ph.D student, Nathan Knight, for his valuable advice and help. REFERENCES Allan, D.W., 1966, Statistics of Atomic Frequency Standards. Proc. IEEE, 54:221-230. Brown, R.G., Hwang, Patrick Y. C., 1997, Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises and Solutions, John Wiley & Sons, New York, pp, 105-111, 299-304. Goshen-Meskin, D., Bar-Itzhack, I., Y., 1992, Unified Approach to Inertial Navigation System Error Modeling, Journal of Guidance, Control, and Dynamics, 15(3): 648-653. HAN, S., WANG, J., 2008. Monitoring degree of observability in GPS/INS integration. Int. Symp. on GPS/GNSS, Tokyo, Japan, 25-28 November, 414-421. IEEE STD 647, 2006, IEEE Standard Specification Format Guide and Test Procedure for Single-Axis Laser Gyros, pp, 6880. IEEE STD 1293, 1998, IEEE Standard Specification Format Guide and Test Procedure for Linear, Single-Axis, Nongyroscopic Accelerometers, pp, 166-181. IEEE STD 952, 1997, IEEE Standard Specification Format Guide and Test Procedure for Single-Axis Interferometric Fiber Optic Gyros, pp, 62-73. Nassar, S., 2003, Improving the Inertial Navigation System Error Model for INS and INS/DGPS Applications, Doctor Thesis, University of Calgary. Ogata, K., 2001, Modern Control Engineering 3rd edition, Prentice Hall, New Jersey.
Papoulis A., 1991, Probability, Random Variables, Stochastic Processes, McGraw-Hill, New York, pp, 401-425. Savage, P., G., 2002, Analytical Modeling of Sensor Quantization in Strapdown Inertial Navigation Error Equations, Journal of Guidance, Control, and Dynamics, 25(5): 833-842. Wang, J., Garratt, M., Lambert, A., Wang, J.J., Han, S., & Sinclair, D., 2008, Integration of GPS/INS/Vision sensors to navigate Unmanned Aerial Vehicles. XXI Congress of the Int. Society of Photogrammetry and Remote Sensing, Beijing, P.R. China, 3-11 July, 963-970.
