优化集成导航系统中卡尔曼滤波器的高性能调谐

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"这篇论文主要探讨了如何提升现代综合导航系统的性能，这些系统处理来自惯性传感器、全球导航卫星系统（GNSS）以及创新辅助方法的观测数据。论文重点在于统计一致性的导航解，即真实导航误差统计与预测导航误差统计的一致性，这是基于创新的故障检测和隔离方法可靠运行的重要前提。主要关注点是卡尔曼滤波器的优化调整，以提高导航精度和可靠性。" 在集成导航系统中，卡尔曼滤波是一种关键的数据融合算法，用于结合多种传感器数据以获得更准确的位置、速度和姿态估计。这篇由Benjamin Braun撰写的博士论文，来自慕尼黑工业大学飞行系统动力学研究所，深入研究了如何通过优化卡尔曼滤波器来增强这种系统的性能。卡尔曼滤波是一种最优估计理论中的方法，它基于贝叶斯定理，能够动态地更新对系统的状态估计，同时考虑到测量噪声和系统模型的不确定性。在集成导航系统中，卡尔曼滤波器的作用是融合来自惯性测量单元（IMU）的高频率但存在漂移的测量和低频率但精确的GNSS数据。通过调整滤波器参数，可以改善系统对各种误差源（如传感器噪声、信号多路径效应等）的鲁棒性，确保统计一致性和导航解的精度。论文可能涵盖了以下几个方面： 1. **滤波器设计**：探讨了适应不同导航需求的滤波器结构，如扩展卡尔曼滤波（EKF）或无迹卡尔曼滤波（UKF），并分析它们在处理非线性系统时的性能。 2. **参数优化**：详述了如何通过数学建模和仿真来确定最优的滤波器增益、过程噪声和测量噪声协方差，以达到最佳性能。 3. **故障检测与隔离**：阐述了统计一致性如何支持基于创新的故障检测和隔离方法，以识别和处理传感器故障或异常情况。 4. **实际应用与实验**：可能包括了实车测试或飞行试验，验证优化后的滤波器在实际环境中的表现。此外，论文还可能讨论了如何处理来自其他辅助传感器（如磁力计、轮速计或地形匹配）的数据，以及这些数据如何进一步增强系统的整体性能和可靠性。总体来说，这篇论文对理解和改进基于卡尔曼滤波的集成导航系统有重要的参考价值，对于从事导航技术、自动驾驶或航空航天领域的工程师和技术人员来说是一份宝贵的资源。

Nomenclature

xiv

Gravity

Gravitation vector

Gravity vector

Normal gravity

/ne

North/east deflection of the vertical

δγ

Gravity disturbance

Gravity potential

Gravity disturbing potential

Inertial navigation system

φλ

WGS84 latitude/longitude

WGS84 height

Cartesian position

Kinematic velocity

Roll angle

Pitch angle

Heading angle



Orientation quaternion

Earth rate vector

Transport rate vector

,MN

Meridian/normal curvature radii

Mean Earth radius

Global navigation satellite system

Speed of light

Frequency, wave length

Signal travel time

Pseudorange

Integrated carrier phase

Satellite position

Satellite velocity

Line-of-sight vector

Elevation angle

t∆

Satellite clock error

oc oe

Time of clock/ephemeris

0 /1/ 2f

Clock correction parameter

t∆

Receiver clock error

Ionospheric delay

Refraction

Obliquity factor

Ionospheric intersection angle

αβ

Klobuchar coefficients

Sunspot number

Tropospheric delay

Non-modeled errors

Modeling & navigation filter

Continuous-time system matrix

Discrete-time system matrix

Continuous-time input matrix

Discrete-time input matrix

Observation matrix

Feed-through matrix

Kalman gain

Covariance matrix

Input noise covariance matrix

Measurement noise cov. matrix

Innovation

Innovation covariance

State vector

State error vector

White Gaussian input noise

Autoregressive coefficients

Moving average coefficients

( )

Transfer function

( )

Discrete-time transfer function

[ ]

Discrete-time impulse response

Time

,tf

Time, frequency

tf∆

Sample time, sample rate

Covariance propagation time

Signal analysis

One-sided PSD of the signal

Two-sided PSD of the signal

Correlation function of the signal

Autocorrelation coefficient

Correlation time

Spatial correlation length

avar,y

Allan variance

mvar,y

Modified Allan variance

hvar,y

Hadamard variance

Number of samples

Window size

Averaging time

Number of windows

1 Introduction

1.1 Motivation

The thesis on hand deals with aspects related to integrated navigation systems for aerospace

applications, that is with the accurate and robust estimation of position, velocity and orientation of an

aircraft for flight guidance and control purposes.

The traditional field of navigation is presumed to be fully understood and the comprehensive theory

is well established. Conventional navigation systems are technically mature and are widely used in all

kinds of aircraft, for example civil transport aircraft, military aircraft and high-end general aviation

aircraft like modern business jets. Conventional systems are predominantly freely running navigation

grade inertial navigation systems (INS), which are solely aided by barometric height measurements to

account for the inherently unstable height channel. Although GPS has evolved to a generally accepted

means of navigation for safety critical applications by now, it is not before the two-thousands that

first certified global navigation satellite navigation (GNSS) aided inertial navigation systems have been

developed and applied in safety critical applications. Standalone GNSS receivers for area navigation,

which are augmented by satellite based augmentation systems (SBAS), however, have already found

their way into aeronautics and GNSS will replace the traditional terrestrial navigation aids in the

medium term.

It is thus legitimate to ask how this thesis can contribute to the elaborated theory of navigation about

which everything seems to have already been said. Convincing reasons will be given shortly.

In the last two decades new sensors that are promising for the purpose of navigation came on the

market. Especially lower cost microelectromechanical (MEMS) accelerometers and gyroscopes have

achieved the required maturity in the meantime to be seriously applied in small attitude and heading

reference (AHRS) and navigation systems. The performance of the latest MEMS gyroscopes can

indeed keep up with lower grade fiber optical gyroscopes. Moreover, multi-frequency multi-

constellation GNSS receivers with more than hundred tracking channels are available, which are rarely

larger than two credit cards. Small, powerful embedded computing systems with low power

consumption enable more complex algorithms. Together with the new sensors and the modern

computing modules, new innovative aiding methods like for example image based navigation or new

ranging methods using small laser scanners or lidar came up. This recent development opens up a

multitude of new fields of application like unmanned aerial vehicles for civil and military applications

or low-cost navigation systems and AHRS for low-end general aviation aircraft.

The central fusion and assessment of the single measurements with respect to their integrity gain in

importance and are probably more important than ever. Therefore, it is worth to have once again a

closer look on the core algorithms of an integrated navigation system and the corresponding error

estimation techniques.

The thesis intends to contribute to the better understanding of the principal sensor and modeling

error characteristics, their impact on the navigation solution error and the corresponding covariance

1 Introduction

estimation of the navigation filter. The focus is to a smaller extent on the enhancement of the accuracy

of the navigation solution in its classical sense but rather on the improvement of its statistical

consistency

. The navigation filter is said to be statistically consistent if the estimated navigation state is

equal to the expected value of the true navigation state and the predicted covariance corresponds to

the covariance of the true navigation state. The statistically correct prediction of the navigation state

error covariance is prerequisite for the correct assessment of the navigation solution, which in turn is

crucial for the reliable detection and isolation of measurement failures.

Fault detection and isolation by an integrity monitor is an indispensable part of an integrated

navigation system. Fault detection methods often judge the integrity of measurements by any form of

innovation testing (for example the simple chi-square test). For this, the innovation covariance

containing the state error covariance as well as the measurement error covariance is required as

accurate as possible. The better the nominal system inherent errors are understood and represented

in the navigation filter, the easier and the earlier even smaller exceptional faults can be detected.

The applications that make use of the outputs of the navigation computer like flight guidance and

control rely on the correctness and statistical consistency of the output values and the corresponding

uncertainty information. The flight management in unmanned aerial systems, for example, decides

based on the navigation state estimate covariance whether the nominal system works properly and is

trustworthy or whether a failure exists and it has to be switched to a backup system. Particularly if

aiding measurements are absent for longer times or if one or more navigation states cannot be

observed by the aiding measurements for a longer period, accurate and coherent prediction of the

statistics is essential for continuous operation.

Navigation grade INS feature only very slow error growth because the measurement errors of the

incorporated inertial measurement unit (IMU) are comparatively small. The errors are benign and play

only a minor role. In contrast, the small new generation lower grade sensors are affected by larger

errors. The nature of the encountered errors is most often not purely white, as assumed by the Kalman

filter, but colored with different correlation times. In common navigation system designs stochastic

measurement errors that are not estimated by the filter are mostly modeled as white noise for the sake

of simplicity or against better judgment. Time correlated noise is substituted by white noise and the

disregarded correlation is compensated by higher uncertainty. This might be problematic because the

navigation solution will be severely distorted if the magnitude of the substitute white noise is assumed

too small, resulting in persistent state error estimate offsets. Since the innovation is consequently

affected, too, there is the risk that the integrity monitor marks aiding measurements unjustifiably as

faulty. Otherwise, if the white noise density is chosen sufficiently large, the navigation solution will

not be biased but one gives valuable system performance away because the predicted state error

variance is larger than it could be.

Therefore, due to the mentioned reasons, it is desirable that the navigation filter is designed such that

all measurements, the inertial as well as the aiding measurements, and modeling errors like the gravity

model error are processed in a statistically correct manner.

Moreover, the integration of sensor measurement and modeling error models into the navigation filter

has an interesting side effect: it can enormously simplify the filter tuning process. Filter tuning refers

to the reasonable adjustment of the process noise and aiding measurement error covariance matrices

by the system designer. Filter tuning is in general a difficult task and requires some experience in

integrated navigation systems. It is an iterative process, which needs simulations on the one hand and

real, recorded measurement data for example from flight experiments on the other hand. The system

1.2 Contributions of the Thesis

designer is relieved from a great part of the filter tuning work since magnitudes and correlations of

many errors are automatically represented by the error models.

1.2 Contributions of the Thesis

The basic ways to increase the statistical consistency of the navigation solution are in principal well-

known and are addressed in the respective literature. Most textbooks as well as the relevant

certification standards are however restricted to naming only the necessity of measures but do not

describe how to actually accomplish it since this would go far beyond the scope of the textbooks. The

thesis goes a step beyond by elaborating the underlying theory and operationally relevant aspects

comprehensively and consistently for typical GPS/INS integrated navigation systems.

The derived models and methods can be used on the one hand for simulation, which is inevitable in

the predesign phase of a new navigation system and which is in fact demanded by the certification

regulations, and on the other hand for the real-time algorithms on the embedded system.

Furthermore, some interesting aspects concerning the widely-used basic algorithms of an integrated

navigation system are discussed, which cannot be found in this form and detail in the standard

textbooks, but may help the system designer to achieve a higher level of understanding.

For many presented methods templates are given with the formulas to be implemented in the

embedded firmware, which underlines the operational focus of the thesis.

The following questions and solution proposals shall be discussed in detail in this thesis:

How to describe sensor, model and observation errors by appropriate mathematical models,

which can be integrated into the navigation filter?

Physically motivated models for the typical sensor measurement and modeling errors in the field of

navigation, that is IMU errors, gravity model error, random vibration, GPS pseudorange errors

comprising satellite clock and orbit errors, ionospheric and tropospheric errors, receiver clock error

and receiver noise, are derived. This involves the identification of the model structure and

subsequently the estimation of the model parameters from real data. Common analysis techniques for

stochastical errors in time and frequency domain are used to describe their characteristics.

How to efficiently increase the consistency of the navigation solution with reasonable

additional computational effort?

The following measures are proposed:

• Accurate and robust inertial navigation algorithms. It is a prerequisite for statistical consistency that

the numerical error of the chosen inertial navigation algorithm is many orders smaller than all other

involved errors in all imaginable flight situations and even under unfavorable conditions like

permanent coning and sculling motion because the numerical error is not taken into account in the

navigation filter covariance prediction. Different algorithms for integrating IMU shall be compared

with respect to their robustness against noise and vibration.

• Statistically correct navigation filter initialization. The effect of a statistically incorrect initial

covariance matrix on the filter consistency in the settling period immediately after initialization

shall be clarified. Often, the filter is initialized with a diagonal, over-bounded navigation state error

covariance matrix. Cross-covariances for example between the initial orientation errors and the

1 Introduction

IMU bias states are mostly neglected. Algorithms for the filter initialization on ground and for the

initialization in-flight shall be presented.

• Proper consideration of out-of-sequence measurements. Since modern integrated navigation

systems generally process measurements from various sources, the system has to handle different

sample rates and different delays. Aiding observations may be only available out-of-sequence. A

navigation filter that treats these delays and disorder in a statistically correct manner shall be derived

in order to account for the true innovation of each new measurement. It shall be clarified under

which circumstances the increased effort is actually necessary.

• Navigation filter state augmentation. The core of an integrated navigation system is an error state

space filter (for example conventional Kalman filter) that estimates the navigation state errors. The

derived reality models of the bias-like sensor, model and observation errors shall be integrated into

the navigation error filter by augmenting the navigation filter states by the reality model states.

Time and cross correlations are henceforth correctly accounted for at the expense of higher

computational effort.

• Avoiding state augmentation of noise-like errors. A larger portion of the reality model errors are

noise-like or are not observable. If these errors are not estimated but only considered by the

navigation filter in a statistical sense, the computational effort can be strongly reduced.

• Efficient reality modeling. The number of reality model states has to be decreased in order to

increase the efficiency. The order of each reality model has to be adequately chosen such that the

main error characteristic is well represented. It has to be carefully analyzed which errors are

dominant and which time correlations have to be considered. The more details of the errors are

depicted the more additional states, which in turn increases the computational costs.

How to quickly assess steady-state navigation filter performance and how to encounter

noticed stability issues?

An efficient tool to quickly calculate the expected stationary covariance of the navigation filter given

the process noise covariance matrix, measurement covariance matrix and the measurement frequency

is presented. This method is convenient for the specification of integrated navigation systems at the

very beginning of the design phase, integrated navigation accuracy predictions and for navigation data

fusion testing. Since the navigation filter for delayed measurements tends to become unstable, a

thorough mathematical analysis that explains the destabilization mechanisms of the Kalman filter in

general is required and appropriate countermeasures have to be found.

Even if the computational power of embedded computers continuously increases all presented

measures and methods are traded off against efficiency in order to keep the numerical effort as lean

as possible avoiding computations with minor effect on the navigation solution and save spare

computational capacity for other tasks.

1.3 Look at the Relevant Certification Regulations in Effect

Since the methods to be presented are part of the safety critical avionics of an aerospace system, they

have to meet certain requirements for certification. It is therefore worth to have a brief look at the

relevant certification rules in effect first of all and to keep them in mind when discussing innovative

methods. The manufacturer has to certify the navigation system according to the applicable Technical

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