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Using Inertial Sensors for Position

and Orientation Estimation

Manon Kok

, Jeroen D. Hol

†

and Thomas B. Sch¨on

‡

Delft Center for Systems and Control, Delft University of Technology, the Netherlands

E-mail: m.kok-1@tudelft.nl

†

Xsens Technologies B.V., Enschede, the Netherlands

E-mail: jeroen.hol@xsens.com

‡

Department of Information Technology, Uppsala University, Sweden

E-mail: thomas.schon@it.uu.se

• Please cite this version:

Manon Kok, Jeroen D. Hol and Thomas B. Sch¨on (2017), ”Using Inertial Sensors for Position and

Orientation Estimation”, Foundations and Trends in Signal Processing: Vol. 11: No. 1-2, pp 1-153.

http://dx.doi.org/10.1561/2000000094

Abstract

In recent years, microelectromechanical system (MEMS) inertial sensors (3D accelerometers and

3D gyroscopes) have become widely available due to their small size and low cost. Inertial sensor

measurements are obtained at high sampling rates and can be integrated to obtain position and

orientation information. These estimates are accurate on a short time scale, but suﬀer from inte-

gration drift over longer time scales. To overcome this issue, inertial sensors are typically combined

with additional sensors and models. In this tutorial we focus on the signal processing aspects of

position and orientation estimation using inertial sensors. We discuss diﬀerent modeling choices and

a selected number of important algorithms. The algorithms include optimization-based smoothing

and ﬁltering as well as computationally cheaper extended Kalman ﬁlter and complementary ﬁlter

implementations. The quality of their estimates is illustrated using both experimental and simulated

data.

At the moment of publication Manon Kok worked as a Research Associate at the University of Cambridge, UK. A

major part of the work has been done while she was a PhD student at Link¨oping University, Sweden.

arXiv:1704.06053v2 [cs.RO] 10 Jun 2018

Contents

1 Introduction 3

1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Using inertial sensors for position and orientation estimation . . . . . . . . . . . . . . . . 6

1.3 Tutorial content and its outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Inertial Sensors 9

2.1 Coordinate frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Angular velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Speciﬁc force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Sensor errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Probabilistic Models 14

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Parametrizing orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Rotation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.2 Rotation vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.3 Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.4 Unit quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Probabilistic orientation modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.2 Alternative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Measurement models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.1 Gyroscope measurement models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.2 Accelerometer measurement models . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.3 Modeling additional information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Choosing the state and modeling its dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6 Models for the prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.7 Resulting probabilistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7.1 Pose estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7.2 Orientation estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Estimating Position and Orientation 31

4.1 Smoothing in an optimization framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 Gauss-Newton optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.2 Smoothing estimates of the orientation using optimization . . . . . . . . . . . . . . 33

4.1.3 Computing the uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Filtering estimation in an optimization framework . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 Filtering estimates of the orientation using optimization . . . . . . . . . . . . . . . 36

4.3 Extended Kalman ﬁltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3.1 Estimating orientation using quaternions as states . . . . . . . . . . . . . . . . . . 38

4.3.2 Estimating orientation using orientation deviations as states . . . . . . . . . . . . . 39

4.4 Complementary ﬁltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Evaluation based on experimental and simulated data . . . . . . . . . . . . . . . . . . . . 43

4.5.1 General characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5.2 Representing uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.5.3 Comparing the diﬀerent algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.6 Extending to pose estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Calibration 59

5.1 Maximum a posteriori calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Maximum likelihood calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 Orientation estimation with an unknown gyroscope bias . . . . . . . . . . . . . . . . . . . 61

5.4 Identiﬁability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Sensor Fusion Involving Inertial Sensors 64

6.1 Non-Gaussian noise distributions: time of arrival measurements . . . . . . . . . . . . . . . 64

6.2 Using more complex sensor information: inertial and vision . . . . . . . . . . . . . . . . . 66

6.3 Including non-sensory information: inertial motion capture . . . . . . . . . . . . . . . . . 67

6.4 Beyond pose estimation: activity recognition . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Concluding Remarks 70

8 Notation and Acronyms 72

A Orientation Parametrizations 75

A.1 Quaternion algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A.2 Conversions between diﬀerent parametrizations . . . . . . . . . . . . . . . . . . . . . . . . 76

B Pose Estimation 78

B.1 Smoothing in an optimization framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B.2 Filtering in an optimization framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B.3 Extended Kalman ﬁlter with quaternion states . . . . . . . . . . . . . . . . . . . . . . . . 79

B.4 Extended Kalman ﬁlter with orientation deviation states . . . . . . . . . . . . . . . . . . . 79

C Gyroscope Bias Estimation 80

C.1 Smoothing in an optimization framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

C.2 Filtering in an optimization framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

C.3 Extended Kalman ﬁlter with quaternion states . . . . . . . . . . . . . . . . . . . . . . . . 81

C.4 Extended Kalman ﬁlter with orientation deviation states . . . . . . . . . . . . . . . . . . . 81

Chapter 1

Introduction

In this tutorial, we discuss the topic of position and orientation estimation using inertial sensors. We

consider two separate problem formulations. The ﬁrst is estimation of orientation only, while the other is

the combined estimation of both position and orientation. The latter is sometimes called pose estimation.

We start by providing a brief background and motivation in §1.1 and explain what inertial sensors are

and give a few concrete examples of relevant application areas of pose estimation using inertial sensors.

In §1.2, we subsequently discuss how inertial sensors can be used to provide position and orientation

information. Finally, in §1.3 we provide an overview of the contents of this tutorial as well as an outline

of subsequent chapters.

1.1 Background and motivation

The term inertial sensor is used to denote the combination of a three-axis accelerometer and a three-

axis gyroscope. Devices containing these sensors are commonly referred to as inertial measurement units

(IMUs). Inertial sensors are nowadays also present in most modern smartphone, and in devices such as

Wii controllers and virtual reality (VR) headsets, as shown in Figure 1.1.

A gyroscope measures the sensor’s angular velocity, i.e. the rate of change of the sensor’s orientation.

An accelerometer measures the external speciﬁc force acting on the sensor. The speciﬁc force consists of

both the sensor’s acceleration and the earth’s gravity. Nowadays, many gyroscopes and accelerometers

are based on microelectromechanical system (MEMS) technology. MEMS components are small, light,

inexpensive, have low power consumption and short start-up times. Their accuracy has signiﬁcantly

increased over the years.

There is a large and ever-growing number of application areas for inertial sensors, see e.g. [7, 59,

109, 156]. Generally speaking, inertial sensors can be used to provide information about the pose of any

object that they are rigidly attached to. It is also possible to combine multiple inertial sensors to obtain

information about the pose of separate connected objects. Hence, inertial sensors can be used to track

human motion as illustrated in Figure 1.2. This is often referred to as motion capture. The application

areas are as diverse as robotics, biomechanical analysis and motion capture for the movie and gaming

industries. In fact, the use of inertial sensors for pose estimation is now common practice in for instance

robotics and human motion tracking, see e.g. [86, 54, 112]. A recent survey [1] shows that 28% of the

contributions to the IEEE International Conference on Indoor Positioning and Indoor Navigation (IPIN)

make use of inertial sensors. Inertial sensors are also frequently used for pose estimation of cars, boats,

trains and aerial vehicles, see e.g. [139, 23]. Examples of this are shown in Figure 1.3.

There exists a large amount of literature on the use of inertial sensors for position and orientation

estimation. The reason for this is not only the large number of application areas. Important reasons are

also that the estimation problems are nonlinear and that diﬀerent parametrizations of the orientation

need to be considered [47, 77], each with its own speciﬁc properties. Interestingly, approximative and

relatively simple position and orientation estimation algorithms work quite well in practice. However,

careful modeling and a careful choice of algorithms do improve the accuracy of the estimates.

In this tutorial we focus on the signal processing aspects of position and orientation estimation using

inertial sensors, discussing diﬀerent modeling choices and a number of important algorithms. These

algorithms will provide the reader with a starting point to implement their own position and orientation

estimation algorithms.

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魏水华

2023-07-26

：这份文件系统地介绍了如何使用IMU数据进行位置和姿态估计，很容易理解。
易烫YCC

2023-07-26

：这份文件不仅提供了理论知识，还提供了一些实际案例，为读者提供了更直观的理解。
Orca是只鲸

2023-07-26

：文件内容很实用，对于IMU数据的处理提供了一些实际可行的解决方案。
巧笑倩兮Evelina

2023-07-26

：作者在文件中运用了简洁的语言，没有过多的技术术语，对IMU数据的处理进行了详尽的解释。
赶路的稻草人

2023-07-26

：对于刚刚接触这个领域的人来说，这份文件解释得很清晰，非常有帮助。

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