卡尔曼滤波器解析_scipy卡尔曼滤波器资源-CSDN文库

需积分: 10 102 浏览量 2019-01-06 20:14:57 上传评论 1 收藏 776KB PDF 举报

资源推荐

资源详情

资源评论

A Gentle Introduction to Predictive Filters

Siome Klein Goldenstein

Abstract: Predictive ﬁlters are essential tools in modern science. They perform

state prediction and parameter estimation in ﬁelds such as robotics, computer vi-

sion, and computer graphics. Sometimes also called Bayesian ﬁlters, they apply the

Bayesian rule of conditional probability to combine a predicted behavior with some

corrupted indirect observation.

When we study and solve a problem, we ﬁrst need its proper mathematical formula-

tion. Finding the essential parameters that best describe the system is hard, modeling

their behaviors over time is even more challenging. Usually, we also have an in-

spection mechanism, that provides us indirect measurements, the observations, of the

hidden underlying parameters. We also need to deal with the concept of uncertainty,

and use random variables to represent both the state and the observations.

Predictive ﬁlters are a family of estimation techniques. They combine the system’s

dynamics uncertain prediction and the corrupted observation. There are many differ-

ent predictive ﬁlters, each dealing with different types of mathematical representations

for random variables and system dynamics.

Here, the reader will ﬁnd a dense introduction to predictive ﬁlters. After a general in-

troduction, we have a brief discussion about mathematical modeling of systems: state

representation, dynamics, and observation. Then, we expose some basic issues related

with random variables and uncertainty modeling, and discuss four implementations

of predictive ﬁlters, in order of complexity: the Kalman ﬁlter, the extended Kalman

ﬁlter, the particle ﬁlter, and the unscented Kalman ﬁlter.

1 Introduction

In applied sciences, we use mathematical models to describe, understand, and deal

with real objects. Applications are varied, and may involve guiding a mobile exploratory

vehicle, understanding a human face and its expressions, or controlling a nuclear power plant

reactor. A good mathematical representation lets us infer properties of the real entity, and

understand beforehand the possible implications of our interactions with it.

The state vector of a model is the set of parameters that uniquely describe a conﬁg-

uration of the object. The set of parameters that model an object is usually not unique, and

each situation leads to a different set of design choices. When the object’s parameters are not

Instituto de Computação, UNICAMP

Caixa Postal 6176, Campinas - SP, 13084-971 Brazil

siome@ic.unicamp.br

A Gentle Introduction to Predictive Filters

static, we also need a mathematical model for their change over time. We call a system, the

mathematical representation of states and dynamics.

Sometimes, the parameters we choose to represent a system are intrinsic to the prob-

lem but impossible to be measured, such as the temperature in the core of a nuclear power

plant. Fortunately, we can usually measure other quantities that are related to the unknown,

but essential parameter, such as the temperature of the cooling water leaving the core. These

indirect measurements are called observations, and can be used to infer information about

the hidden parameter.

Real life is hard, and full of errors. The observations we make are not precise, and they

may not explain all that is happening in the system. Also, our mathematical representation

of the model is not exact, giving slightly incorrect results. To cope with all these factors, we

need to incorporate the notion of uncertainties. There are many mathematical descriptions

for uncertainties, but in this paper we use probabilities.

Predictive ﬁlters estimate the optimal state of a system. First, they use the mathemat-

ical model of the system dynamics to propagate the state’s values and uncertainties. Later,

they combine this preliminary estimate and as much as it is possible from the observation.

There are several predictive ﬁlters, each appropriate for a different type of uncertainty repre-

sentation and dynamic modeling.

The Kalman ﬁlter is the simplest example of a predictive ﬁlter. It represents uncertain-

ties as Gaussian random variables, fully described by a mean and a covariance matrix, and

models the system with linear dynamics and observations. Since Gaussians are preserved

under linear transformation, the Kalman ﬁlter’s implementation uses only linear algebra op-

erations.

If we need more complex models to describe a system, we may have to use nonlin-

ear functions. The extended Kalman ﬁlter linearizes the system around the current state to

propagate and estimate covariance matrices.

We can represent the random variables as a collection of samples, and allow nonlinear

operations. In this case, we can use a particle ﬁlter, which is powerful and ﬂexible. Unfortu-

nately, it needs an a number of samples exponentially related to the state’s dimension.

Here, the last predictive ﬁlter we talk about is the unscented Kalman ﬁlter. It uses

a small number of specially selected particles to propagate random variables over nonlinear

transformations.

In this paper, we follow the same type of structure of Maybeck’s classical book [20].

In Section 2, we deal with the problem of building a mathematical model that describes the

state of the system (Section 2.1), its dynamics (Section 2.2), and how to inspect it at every

moment (Section 2.3). Then, in Section 3, we review uncertainties: probabilities and random

2 RITA • Volume • Número • 2004

A Gentle Introduction to Predictive Filters

variables. Finally, in Section 4, we describe the Bayesian concepts of predictive ﬁlters, and

look at some of its concrete implementations: the Kalman ﬁlter (Section 4.1), the extended

Kalman ﬁlter (Section 4.2), the particle ﬁlter (Section 4.3), and the unscented particle ﬁlter

(Section 4.4). In Section 5, we conclude this paper with a series of words of caution, as well

as a series of pointers to where the reader can ﬁnd more information.

2 System Representation

Predictive ﬁltering techniques rely on a mathematical representation of the system. In

this section, we discuss how to model a system’s parameters and their dynamical aspects, and

how to describe our indirect ability to inspect them.

A system is just like a black box, controlled by a set of parameters. To manipulate

and understand a system, we need a mathematical model that describes these parameters and

their behavior over time.

There are three general guidelines to modeling: ﬁrst, the mathematical model should

be as precise as possible – the system’s behavior model should be as close as possible to

reality. Second, the model should be tractable – we need to handle the mathematical model

in a efﬁcient way. Finally, the model should be as simple as possible – there is no need to

describe a detail of the system, if this detail holds no consequence to the desired application.

Usually, these three guidelines are incompatible, and we have to strike a balance. In the next

three sections we describe the ideas behind state description, the observation procedure, and

the dynamic behavior.

2.1 State Description

We use several real-valued variables to describe a system state.

If we use n parame-

ters to describe the system, we group them in a vector

∈ R

that uniquely identiﬁes the system’s situation at time t. We now look at two examples.

Example 1: Simple Robot Lets consider a mobile robot, that can turn around its axes and

move in the forward direction. We know that the robot cannot leave the ﬂoor, so we only

need two coordinates to describe its position. We also need a third parameter to describe its

facing direction. This is illustrated in Figure 1.

It is possible, and sometimes more adequate, to model systems with graphs and discrete variables, but that is beyond

the scope of this paper.

RITA • Volume • Número • 2004 3

剩余28页未读，继续阅读

评论收藏

内容反馈

chi_mian

粉丝: 10
资源: 5

卡尔曼滤波器解析

卡尔曼滤波器

卡尔曼滤波器简介

卡尔曼滤波

卡尔曼滤波器的通俗易懂介绍

很易懂的卡尔曼滤波器讲解，附带Simulink模型哦

Poor Man&#039;s T-SQL Formatter 1.6.10

一个应用实例详解卡尔曼滤波及其算法实现

卡尔曼滤波详解，内部培训PPT，非常详细

卡尔曼滤波器MATLAB源代码

卡尔曼滤波器介绍.doc

卡尔曼滤波器原理介绍

卡尔曼滤波器中过程噪声矩阵的解析计算，用于拟合密集厚散射体中磁场中的弯曲轨迹

永磁同步电机扩展卡尔曼滤波器.rar

最简单的卡尔曼滤波器

MVS算法比较与评估

中国科学院大学国科大计算机视觉董秋雷期末考试试卷

基于扩展卡尔曼滤波器的永磁同步电机转速和磁链观测器

最新资源

Poor Man's T-SQL Formatter 1.6.10