EKF-RAIM算法：全球导航卫星系统的接收机自主完整性监测
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"基于RAIM的卡尔曼滤波在GNSS接收机中的应用"
全球导航卫星系统（Global Navigation Satellite System, GNSS）接收机的接收机自主完整性监测（Receiver Autonomous Integrity Monitoring, RAIM）算法是确保导航数据准确性和可靠性的关键技术。传统的RAIM算法通常基于加权最小二乘法，而本文提出了一种新的方法，将RAIM与扩展卡尔曼滤波器（Extended Kalman Filter, EKF）相结合，以提高监测效率和实时性。
卡尔曼滤波是一种优化的递归估计方法，能够处理带有噪声的动态系统。在GNSS接收机中，EKF用于融合来自多个卫星信号的数据，以提供最优的位置、速度和时间估计。然而，由于卫星信号可能存在误差或干扰，因此需要RAIM来检测和修正这些错误。
文章的创新之处在于，在假设存在阶跃和上升斜坡故障剖面的情况下，设计了一种计算EKF平均位置误差的新方法。这种方法能够更精确地评估EKF的性能，并据此设定保护水平。保护水平是确保导航数据质量的关键指标，它定义了允许的最大误差边界，超过这个边界则发出警告，防止用户做出基于错误信息的决策。
通过仿真对比，EKF RAIM算法表现出了与传统加权最小二乘RAIM算法相当的性能。尽管可能无法实现显著的性能提升，但这一方法为实时、高效地计算基于距离残差的RAIM保护水平提供了基础。这尤其对于那些依赖EKF的高级导航算法，如矢量跟踪接收机，具有重要意义。矢量跟踪接收机利用EKF来追踪多个卫星信号，从而提高定位精度和稳定性。
SUSMITA BHATTACHARYYA 和 DEMOZ GEBRE-EGZIABHER 的研究为GNSS接收机的RAIM提供了新的理论和技术支持，使得在复杂环境下的导航系统能够更可靠地运行，保障了用户的安全和导航服务的质量。这种结合EKF的RAIM方法有望在未来的GNSS接收机设计和更新中得到广泛应用。
expressions of ˜κ
−
k
and ˜κ
+
k
are given by
˜
κ
−
k
= ˜κ
−
k
− ˜κ
−
k
= [0] (5)
˜
κ
+
k
= ˜κ
+
k
− ˜κ
−
k
= K
k
ρ
T
k
˙ρ
T
k
T
(6)
where K
k
represents the Kalman gain, and [ρ
T
k
˙ρ
T
k
]
T
is
the linearized measurement or innovations vector. The
measurement noise covariance matrix at time t
k
for ρ
k
is
given by W
k
, which is defined in relation to the discussion
of (1) earlier. The measurement noise covariance matrix
for pseudorange rate residuals vector ˙ρ
k
is denoted by
W
k
,andW
k
is determined by the frequency locked loop
(FLL) jitter defined in [21]. The initial state error
covariance matrix is the identity matrix with a dimension
of 8 ×8. The process noise covariance matrix Q is chosen
manually by trial and error so that the state vector reaches
steady state quickly or has a desirable transient response.
The a posteriori x-, y-, and z-axis position errors are as
follows:
q
k
= κ
k
(
l
)
− ˜
κ
+
k
(
l
)
(7)
where q stands for x, y, z and l = 1, 3, 5, respectively.
Substituting for
κ
k
(l)and ˜κ
+
k
(l) in (7) gives the
following:
q
k
=
κ
k
(
l
)
− ˜
κ
−
k
(
l
)
−
˜
κ
+
k
(
l
)
− ˜
κ
−
k
(
l
)
=
κ
k
(
l
)
− ˜
κ
+
k
(
l
)
(8)
If
+
k
is the a posteriori state error covariance matrix at t
k
,
then the variances of the x-, y-, and z-axis position errors
are given by σ
2
q
k
=
+
k
(l, l); again, q stands for x, y, z and
l = 1, 3, 5, respectively. Thus, under fault-free conditions
and at time t
k
, the vertical position error (VPE) is
Gaussian distributed with zero mean and variance σ
2
z
k
.For
purposes of calculating the horizontal protection level
(HPL), which is discussed later, the horizontal position
error (HPE) under fault-free conditions is usually
approximated to be Gaussian distributed with zero mean
and variance (σ
2
x
k
+ σ
2
y
k
)[11]. To justify this
approximation, it is shown in [5] that beyond a certain M,
the probability that the HPE is greater than a value (>M)
is bounded above by the corresponding probability of a
Gaussian random variable with zero mean and variance
(σ
2
x
k
+ σ
2
y
k
). This implies that the HPL calculated using the
Gaussian distribution assumption would bound the actual
HPL (obtained from the true distribution) because the
HPL is larger than M. Thus, the use of a Gaussian
distribution for the HPE in place of a complex
non-Gaussian one is justified. Moreover, the Gaussian
assumption does not exhibit numerical problems that may
arise with the actual non-Gaussian distribution.
Next, bounds on the means of VPE and HPE are
calculated assuming step and ascending ramp fault
profiles. These bounds are required for protection level
computations.
1) Bounds on Means of VPE and HPE Under Fault:To
derive the position error bounds under fault, it is assumed
that a failure has occurred in satellite i. The failure started
at t
m
; b
k−m +1
is the error (or bias) in the pseudorange
measurement at t
k
(>t
m
) because of the fault, and
˙
b
k−m+1
is the corresponding error magnitude in the pseudorange
rate measurement. Because of the failure, the ith
pseudorange measurement error at t
k
is distributed as
N (b
k−m+1
, W
k
(i, i)). With these assumptions, the mean
μ
k
of the a posteriori state estimate error (
κ
k
) is derived
at t
k
.LetF
k−1
and C
k
be the state transition and
measurement model matrices of the EKF, respectively.
Now using the filter time and measurement update
equations, the mean of
κ
k
can be written as follows [22]:
μ
k
=
(
I − K
k
C
k
)
F
k−1
μ
k−1
− K
k
0 ...0 b
k−m+1
...0
˙
b
k−m+1
...0
T
Replacing (I − K
k
C
k
)F
k−1
with A
k
,
μ
k
= A
k
μ
k−1
− K
k
f
i, k−m+1
(9)
where f
i, k−m+1
is a column vector such that
f
i, k−m+1
(i) = b
k−m+1
, f
i, k−m+1
(i + n) =
˙
b
k−m+1
and
f
i, k−m+1
(j) = 0forj = i and j = i + n, where n is the
number of visible satellites. It is assumed that the filter
estimate was unbiased under no fault conditions, implying
that μ
m−1
= 0,μ
m
=−K
m
f
i, 1
and so on. Thus, μ
k
becomes
μ
k
= A
k
A
k−1
... A
m+1
−K
m
f
i, 1
+ A
k
... A
m+2
−K
m+1
f
i, 2
+ ...+ A
k
−K
k−1
f
i, k−m
+
−K
k
f
i, k−m+1
(10)
A more compact representation of (10) is
μ
k
=
m+1
j=k
A
j
−K
m
f
i, 1
+
m+2
j=k
A
j
−K
m+1
f
i, 2
+ ...
+ A
k
−K
k−1
f
i, k−m
+
−K
k
f
i, k−m+1
Grouping all terms together,
μ
k
=−
k−
(
m+1
)
j=0
k−j
ι=k
A
ι
K
k−j −1
f
i, k−m−j
− K
k
f
i, k−m+1
(11)
Substituting B
k−j
for
k−j
ι=k
A
ι
K
k−j −1
,
μ
k
=−
k−
(
m+1
)
j=0
B
k−j
f
i, k−m−j
− K
k
f
i, k−m+1
(12)
Since for an asymptotically stable filter, all absolute
eigenvalues of (A
j
) ∀j are less than (1 − δ), where δ is
BHATTACHARYYA & GEBRE-EGZIABHER: KALMAN FILTER–BASED RAIM FOR GNSS RECEIVERS 2447
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