无迹卡尔曼滤波器详解：超越扩展卡尔曼的优势与应用

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"这篇文档是关于无迹卡尔曼滤波器( Unscented Kalman Filter, UKF)的介绍，主要讨论了其在非线性估计中的应用，以及相对于扩展卡尔曼滤波器(EKF)的优势。作者Eric A. Wan和Rudolph van der Merwe在文中揭示了EKF存在的问题，并提出了UKF作为改进方案。UKF由Julier和Uhlman首次提出，其核心是对高斯随机变量的处理方式进行了改进，避免了EKF中的线性化误差。" 正文: 无迹卡尔曼滤波器(无迹卡尔曼)是一种用于非线性系统状态估计的算法，它是对经典扩展卡尔曼滤波器的一个重要补充。扩展卡尔曼滤波器虽然广泛应用于各种非线性问题，但由于依赖于系统的线性化，可能会导致误差积累，尤其是在系统非线性程度较高时。无迹卡尔曼滤波器的出现，正是为了克服EKF的这些局限性。在传统的扩展卡尔曼滤波中，非线性系统的状态分布通常通过第一阶泰勒展开进行近似，即线性化处理。这种方法在非线性程度较弱的情况下可能有效，但当非线性较强时，这种线性化会导致估计精度下降，甚至产生错误的结果。无迹卡尔曼滤波器的核心思想是使用“无迹变换”来更准确地代表高斯随机变量在非线性函数下的传播。与EKF的局部线性化不同，UKF通过选择一组特定的采样点（称为“无迹点”或“sigma点”），并将这些点通过非线性函数进行变换，然后再重新组合成新的概率分布。这种方式可以更好地保留原始分布的信息，从而提高估计的精度。 UKF的具体步骤包括：(1) 生成sigma点，这些点既包含均值和协方差信息，又考虑到非线性函数的影响；(2) 应用非线性函数于这些点上；(3) 使用这些变换后的点来计算预测的均值和协方差；(4) 更新过程类似于EKF，但使用了经过非线性变换的sigma点信息；(5) 最后，通过观测值更新滤波器状态。在实际应用中，无迹卡尔曼滤波器被广泛用于许多领域，如机器人定位、传感器融合、动态系统建模、参数估计等。例如，在神经网络的权重学习中，UKF可以用来估计网络参数，同时在期望最大化(EM)算法中，它可以同时估计状态和参数。无迹卡尔曼滤波器提供了一种更为精确的非线性系统状态估计方法，它避免了扩展卡尔曼滤波器的线性化误差，适用于非线性程度较高的系统。对于初学者来说，理解和掌握UKF的原理和应用，对于提升在非线性估计领域的技能非常有益。

The Unscented Kalman Filter for Nonlinear Estimation

Eric A. Wan and Rudolph van der Merwe

Oregon Graduate Institute of Science & Technology

20000 NW Walker Rd, Beaverton, Oregon 97006

ericwan@ece.ogi.edu, rvdmerwe@ece.ogi.edu

Abstract

The Extended Kalman Filter (EKF) has become a standard

technique used in a number of nonlinear estimation and ma-

chine learning applications. These include estimating the

state of a nonlinear dynamic system, estimating parame-

ters for nonlinear system identiﬁcation (e.g., learning the

weights of a neural network), and dual estimation (e.g., the

Expectation Maximization (EM) algorithm) where both states

and parameters are estimated simultaneously.

This paper points out the ﬂaws in using the EKF, and

introduces an improvement, the Unscented Kalman Filter

(UKF), proposed by Julier an d Uhlman [5]. A central and

vital operation performed in the Kalm a n Filter is the prop-

agation of a Gaussian random variable (GRV) through the

system dynamics. In the EKF, the state distribution is ap-

proximated by a GRV, which is then propagated analyti-

cally through the ﬁrst-order linearization of the nonlinear

system. This can introduce large err ors in the true posterior

mean and covariance of the transformed GRV, which may

lead to sub-optimal performance and sometimes divergence

of the ﬁlter. The UKF addresses this problem by using a

deterministic sampling approach. The state distribution is

again approximated by a GRV, but is now represented using

a minimal set of carefully chosen sample points. These sam-

ple points completely capture the true mean and covariance

of the GRV, and wh e n propagated through the true non-

linear system, cap tu res the posterior mean and covariance

accurately to the 3rd order (Taylo r series expansion) for any

nonlinearity. The EKF, in contrast, only achieves ﬁrst-order

accuracy. Remarkably, the computational complexity of the

UKF is the same order as that of the EKF.

Julier and Uhlman demonstrated the substantial perfor-

mance gains of the UKF in the context of state-estimation

for nonlinear control. Machine learning problems were not

considered. We extend the use of the UKF to a broader class

of nonlinear estimation problems, including nonlinear sys-

tem identiﬁcation, training of neural networks, an d dual es-

timation problems. Our preliminary results were presented

in [13]. In this paper, the algorithms are further developed

and illustrated with a number of additional examples.

This work was sponsored by the NSF under grant grant IRI-9712346

1. Introduction

The EKF has been applied extensively to the ﬁeld of non-

linear estimation. General application areas may be divided

into state-estimation and machine learning. We further di-

vide machine learning into parameter estimation and dual

estimation. The framework for these areas ar e brieﬂy re-

viewed next.

State-estimation

The basic framework for the EKF involves estimation of the

state of a discrete-time nonlinear dynamic system,

(1)

(2)

where represent the unobserved state of the system and

is the only observed signal. The process noise drives

the dynamic system, and the observation noise is given by

. Note that we are not assuming add itivity of the noise

sources. The system dynamic model and are assumed

known. In state-estimation, the EKF is the standard method

of choice to achieve a recursive (approximate) maximum-

likelihood estimation of the state . We will review the

EKF itself in this context in Section 2 to help motivate the

Unscented Kalman Filter (UKF).

Parameter Estimation

The classic machine learning problem involves determining

a nonlinear mapping

(3)

where is the input, is the output, and the nonlinear

map is parameterized by the vector . The nonlinear

map, for example, may be a feedforward or recurrent neural

network ( are the weights), with numerous applications

in regression, classiﬁcation, and dynamic modeling. Learn-

ing corresponds to estimating the parameters . Typically,

a training set is provided with sample pairs consisting of

known input and desired outputs, . The err or of

the machine is deﬁned as , and the

goal of learning involves solving for the parameters in

order to minimize the expected squared err or.

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