无迹卡尔曼滤波（UKF）原理与应用解析
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收藏 162KB PDF 举报 "这篇资源是关于无迹卡尔曼滤波（Unscented Kalman Filter，简称UKF）的原始文章，深入解析了UKF的理论基础和实际应用案例。文章作者Simon Julier、Jeffrey Uhlmann和Hugh F. Durrant-Whyte在IEEE Transactions on Automatic Control期刊2000年3月刊上发表，探讨了一种用于非线性系统滤波和估计的新方法，旨在解决扩展卡尔曼滤波器（Extended Kalman Filter，EKF）在处理非线性问题时的局限性。"
**无迹卡尔曼滤波（UKF）详解**
无迹卡尔曼滤波是一种用于非线性系统的概率滤波方法，其核心思想是在非线性变换过程中保留概率分布的统计特性，而不是像EKF那样仅线性化模型。UKF通过采样技术（也称为"无迹变换"）来近似高维概率分布的均值和协方差，从而提供更精确的滤波结果。
1. **无迹变换原理**
- UKF的核心是选择一组特定的样本点，这些样本点被称为"sigma点"，它们能够充分代表概率分布的特征，并且在非线性函数作用下仍然能保持对均值和协方差的准确描述。
- 相比EKF中的雅可比矩阵线性化，UKF无需计算导数，降低了计算复杂度，尤其对于复杂的非线性系统，UKF更容易实现。
2. **滤波过程**
- UKF的滤波过程分为预测和更新两个阶段，与传统的卡尔曼滤波器类似。
- 在预测阶段，sigma点被非线性系统模型作用，然后通过加权平均重新生成新的均值和协方差估计。
- 在更新阶段，UKF利用观测数据对预测状态进行校正，同样采用sigma点来处理非线性观测模型。
3. **优缺点比较**
- UKF相对于EKF的主要优势在于它提供了更精确的滤波结果，尤其是在非线性度较高或系统模型不确定性较大的情况下。
- UKF的计算复杂度相对较低，因为不需要计算导数。
- 与Gauss第二阶滤波相比，UKF的实现更为简便。
4. **应用实例**
- 文章中通过导弹检测和跟踪、移动机器人导航等例子展示了UKF的有效性。
- 非线性滤波在自动化控制、目标检测、机器人定位等多个领域有广泛应用，UKF因其优势而成为这些领域的首选算法之一。
5. **索引术语**
- 协方差矩阵：描述随机变量不确定性的重要统计量。
- 估计：对未知系统状态的预测或推断。
- 过滤：通过消除噪声来提取有用信号的过程。
- 非线性滤波器：处理非线性系统状态估计的工具。
- 预测方法：基于已知信息对未来状态的预测算法。
无迹卡尔曼滤波器是一种强大的非线性滤波工具，通过无迹变换克服了EKF的局限性，提高了滤波精度，且在实现上相对简单，具有广泛的应用前景。在实际工程中，尤其是在需要精确状态估计的非线性系统中，UKF是一种值得考虑的解决方案。
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 3, MARCH 2000 477
Technical Notes and Correspondence_______________________________
A New Method for the Nonlinear Transformation of Means
and Covariances in Filters and Estimators
Simon Julier, Jeffrey Uhlmann, and Hugh F. Durrant-Whyte
Abstract—This paper describes a new approach for generalizing the
Kalman filter to nonlinear systems. A set of samples are used to param-
eterize the mean and covariance of a (not necessarily Gaussian) proba-
bility distribution. The method yields a filter that is more accurate than
an extended Kalman filter (EKF) and easier to implement than an EKF or
a Gauss second-order filter. Its effectiveness is demonstrated using an ex-
ample.
Index Terms—Covariance matrices, estimation, filtering, missile detec-
tion and tracking, mobile robots, nonlinear filters, prediction methods.
I. INTRODUCTION
The problem of generalizing the Kalman filter paradigm for non-
linear applications is considered. This work is motivated by the well-
known limitations of the extended Kalman filter (EKF), which simply
linearizes all nonlinear models so that the traditional linear Kalman
filter can be applied. Although the EKF (in its many forms) is a widely
used filtering strategy [1], [2], over 30 years of experience with it has
led to a general consensus within the tracking and control community
that it is difficult to implement, difficult to tune, and only reliable for
systems that are almost linear on the time scale of the update intervals.
As is well known, the optimal solution to the nonlinear filtering
problem is infinite dimensional [3] and a large number of suboptimal
approaches havebeen developed[4],[5]. These methods can be broadly
classed as numerical Monte Carlo [6] methods or analytical approxi-
mations [7]–[9]. However, the application of these methods to high-di-
mensioned systems is rarely practical, and it is a testament to the con-
ceptual simplicity of the EKF that it is still widely used.
In this paper, a new linear estimator [10]–[12] is developed. It yields
performance equivalent to the Kalman filter for linear systems, yet gen-
eralizes elegantly to nonlinear systems without the linearization steps
required by the EKF. We show analytically that the expected perfor-
mance of the new approach is superior to that of the EKF and, in fact,
lies between those of the modified, truncated second-order filter [13]
and the Gaussian second-order filter[14]. However, the algorithm is not
restricted to Gaussian distributions. We demonstrate the performance
benefits in an example application, and we argue that the ease of im-
plementation and more accurate estimation features of the new filter
recommend its use over the EKF in virtually all applications.
II. B
ACKGROUND
We seek the minimum-mean-squared error (MMSE) estimate of the
state vector of the nonlinear discrete time system
x
(
k
+1)=
f
[
x
(
k
)
;
u
(
k
)
;
v
(
k
)
;k
]
Manuscript received November 16, 1994; revised December 1, 1995,
September 1, 1996, and June 1, 1998. Recommended by Associate Editor, V.
Solo.
S. Julier is with IDAK Industries, Jefferson City, MO USA (e-mail:
sjulier@idak.com).
J. Uhlmann is with the Robotics Research Group, Oxford University, U.K.
H. F. Durrant-Whyte is with the Department of Mechanical and Mechatronic
Engineering, Sydney, Australia (e-mail: hugh@mech.eng.usyd.edu.au).
Publisher Item Identifier S 0018-9286(00)02145-0.
z
(
k
)=
h
[
x
(
k
)
;
u
(
k
)
;k
]+
w
(
k
)
(1)
where
x
(
k
)
is the state of the system at timestep
k;
u
(
k
+1)
is the input
vector,
v
(
k
)
is the noise process caused by disturbances and modelling
errors,
z
(
k
)
is the observation vector, and
w
(
k
)
is additive measure-
ment noise. It is assumed that the noise vectors
v
(
k
)
and
w
(
k
)
are zero
mean and
E
[
v
(
i
)
v
T
(
j
)] =
ij
Q
(
i
)
E
[
w
(
i
)
w
T
(
j
)] =
ij
R
(
i
)
;
8
i; j
E
[
v
(
i
)
w
T
(
j
)] =
0
:
The MMSE estimate of
x
(
k
)
is the conditional mean. Let
^
x
(
i
j
j
)
be the mean of
x
(
i
)
conditioned on all of the observations up to and
including time
j
^
x
(
i
j
j
)=
E
[
x
(
i
)
j
Z
j
]
where
Z
j
=[
z
(1)
;
111
;
z
(
j
)]
:
The covariance of this estimate is de-
noted
P
(
i
j
j
)
:
The Kalman filter propagates the first two moments of the distri-
bution of
x
(
k
)
recursively and has a distinctive “predictor-corrector”
structure. Given an estimate
^
x
(
k
j
k
)
;
the filter first predicts what the
future state of the system will be using the process model. Ideally, the
predicted quantities are
^
x
(
k
+1
j
k
)=
E
[
f
[
x
(
k
)
;
u
(
k
)
;
v
(
k
)
;k
]
j
Z
k
]
P
(
k
+1
j
k
)=
E
[
f
x
(
k
+1)
0
^
x
(
k
+1
j
k
)
g
2f
x
(
k
+1)
0
^
x
(
k
+1
j
k
)
g
T
j
Z
k
]
:
The expectations can be calculated only if the distribution of
x
(
k
)
conditioned on
Z
k
is known. In general, the distribution cannot be de-
scribed by a finite number of parameters and most practical systems
employ an approximation of some kind. It is conventionally assumed
that the distribution of
x
(
k
)
is Gaussian at any time
k:
Two justifica-
tions are made. First, only the mean and covariance need to be main-
tained. Second, given just the first two moments the Gaussian distribu-
tion is the entropy maximizing or least informative distribution [15].
The estimate at time
k
+1
is given through updating the prediction
by the linear update rule
^
x
(
k
+1
j
k
+1)=
^
x
(
k
+1
j
k
)+
W
(
k
+1)
(
k
+1)
P
(
k
+1
j
k
+1)=
P
(
k
+1
j
k
)
0
W
(
k
+1)
P
(
k
+1
j
k
)
1
W
T
(
k
+1)
(
k
+1)=
z
(
k
+1)
0
^
z
(
k
+1
j
k
)
W
(
k
+1)=
P
x
(
k
+1
j
k
)
P
0
1
(
k
+1
j
k
)
:
The EKF exploits the fact that the error in the prediction,
~
x
(
i
j
j
)=
x
(
i
)
0
^
x
(
i
j
j
)
;
can be attained by expanding (1) as a Taylor Series about
the estimate
^
x
(
k
j
k
)
:
Truncating this series at the first order yields the
approximate linear expression for the propagation of state errors as
~
x
(
k
+1
j
k
)
r
f
x
~
x
(
k
j
k
)+
r
f
v
v
(
k
)
where
r
f
x
is the Jacobian of
f
[
1
]
with respect to
x
(
k
)
;
and
r
f
v
is that
with respect to
v
(
k
)
:
Using this approximation, the state prediction
equations are
^
x
(
k
+1
j
k
)=
f
[
^
x
(
k
j
k
)
;
u
(
k
)
;
0
]
0018–9286/00$10.00 © 2000 IEEE
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