PREPRINT 1 GNSSMEMS-INS Integration for Drone Navigation using EKF on Lie Groups

2025-05-02 0 0 4.17MB 30 页 10玖币

侵权投诉

PREPRINT 1

GNSS/MEMS-INS Integration for Drone

Navigation using EKF on Lie Groups

Marcos R. Fernandes, Giorgio M. Magalh˜

aes, Yusef C´

aceres, Jo˜

ao B. R. do Val

Abstract

Building upon the theory of Kalman Filtering on Lie Groups, this paper describes an Extended

Kalman Filter and Smoother for Loosely Coupled Integration of GNSS/INS tailored for post-processing

applications. The approach employs a dynamic model on a matrix Lie Group that aggregates position,

velocity, attitude, and the IMU biases as a single element of a Lie group. The development was motivated

by a drone-borne Differential Interferometric SAR (DinSAR) application, which requires high-precision

navigation information for short-ﬂight missions using low-cost MEMS sensors. The ﬁlter and the Rauch-

Tung-Striebel (RTS) smoother are both implemented and validated. The paper also presents a novel

algorithm to initialize the heading value as an alternative to gyro-compassing or magnetometer-based

alignments. The Mahalanobis Distance and the

𝜒2

-test are employed during the ﬁlter update step to

address the practical issue of outlier rejection for the GNSS measurements. The paper uses synthetic

data to compare classic navigation schemes based on multiplicative quaternions and Euler angles. Finally,

real data experiments demonstrate that the Kalman Filter based on Lie Groups performs better DinSAR

processing than state-of-the-art commercial software.

Index Terms

Inertial Navigation, Kalman Filtering, Lie Groups, Loosely Coupled Integration, DinSAR.

I. INTRODUCTION

EMOTE sensing applications based on Unmanned Aerial Vehicles (UAV), such as 3D

mapping [1] and Differential Interferometric SAR (DinSAR) [2], require high-ﬁdelity

position and attitude information [3].

This work was supported in part by the Coordena

c¸˜

ao de Aperfei

c¸

oamento de Pessoal de N

ıvel Superior (CAPES), Brazil Finance

Code 001 under Grant 88887.342183/2019-00; in part by the Conselho Nacional de Desenvolvimento Cient

ıﬁco e Tecnol

ogico

(CNPq) under Grant 303352/2018; and in part by FAPESP under Grant 2016/08645-9. The authors are with the School of

Electrical and Computer Engineering, University of Campinas – UNICAMP, Campinas, SP, Brazil.

{

marofe,jbosco

}

@unicamp.br,

October 7, 2022 DRAFT

arXiv:2210.02983v1 [cs.RO] 6 Oct 2022

PREPRINT 2

A Strap-down Inertial Navigation System (INS) provides position, velocity, and attitude

(PVA) information at high rates but with unqualiﬁed errors since it relies on integrating inertial

measurements from noisy sensors, such as accelerometers and gyroscopes. On the other hand, a

Global Navigation Satellite-based System (GNSS) can provide position and velocity with strict

error bounds but at a lower frequency. A Real-Time Kinematic Differential GNSS (RTK-GNSS)

could be employed for high-precision applications to achieve centimeter-level precision using

code and phase measurements [4]. A GNSS/INS integrated navigation system combines each

sensor’s strengths to get better PVA estimates.

In the literature, different types of GNSS/INS integration have appeared, such as Loosely

Coupled (LC), Tightly Coupled (TC), and Ultra-Tightly Coupled (UTC) [5]. Due to its simplicity

and low computational burden, the LC structure has been vastly used. This integration scheme

takes the position provided by the GNSS and combines it with the IMU measurements.

The sensors are inevitably affected by biases, and white noise [6]. Besides, with advances in

Micro Electromechanical Systems (MEMS) technology, commercial off-the-shelf MEMS-based

inertial sensors are available, which are lighter and low-cost but also require adequate noise

modeling. In addition, a good alignment process is indispensable to achieving centimeter-level

precision.

Over the past decades, algorithms based on the Kalman Filter have been intensively investigated

for GNSS/INS integration. A few examples are [7]–[9]. The classical Extended Kalman Filter

(EKF) is the most common GNSS/INS integration technique, and to achieve high-order approx-

imations, unscented or particle ﬁlters offer alternatives. They might provide better estimation

but require more computational resources than the EKF. For this reason, the EKF remains the

reference ﬁlter within the aerospace industry, cf. [10].

Several parameterizations apply to represent the attitude of a vehicle, such as Euler Angles,

Quaternions, Rotation vectors, and Rotation Matrices, cf. [5]. The challenge in choosing an

adequate representation of attitude is that some representations have singularities or add constraints.

Early applications of Kalman ﬁltering use the three-parameter Euler-angle representation.

However, the kinematic equations for Euler angles involve trigonometric functions, which make

the model highly non-linear, cf. [11]. Besides, the angles can become undeﬁned for some attitudes,

the so-called gimbal-lock situation.

Alternatively, quaternions are especially appealing for attitude representation since no singulari-

ties are present and the kinematic equation is linear, cf. [12]. However, the quaternion must satisfy

October 7, 2022 DRAFT

PREPRINT 3

a normalization constraint to represent rotations, which is disregarded by the measurement update

step of the standard EKF, cf. [13]. To circumvent such constraint, the Multiplicative Quaternion

Extended Kalman Filter (MEKF); see [11], [14] was proposed using a multiplicative form of

quaternion update that preserves the unity norm. The advantages of quaternion parameterization

have led to its frequent use in attitude determination systems. In fact, since the early 1980s,

the quaternion has been the most widely used attitude parameterization, cf. [12]. Nevertheless,

most of these techniques consider models on the Euclidean space and the noises as Gaussian

distributed.

Recently, the framework based on Lie Group theory has attracted the attention of sensor fusion

communities for rigid-body-related data fusion applications. For instance, a Discrete Extended

Kalman Filter on Lie Groups (D-LIE-EKF) appears in [15], [16], which generalizes the usual

Kalman Filter framework when the system dynamic or measurement model can be cast as a Lie

Group element.

Lie groups can properly model rigid bodies that undergo simultaneous translations and rotations,

such as vehicles following curved trajectories that do not suffer from singularities. Also, they

allow one to implement non-linear estimation tools using linear representations in Lie Algebra.

Physical systems modeling, such as the satellite attitude dynamics, ﬁxed-wing unmanned aircraft

systems, and multi-rotor ﬂying vehicles, to name a few, are examples that ﬁt as Lie groups

naturally.

Classically, the system evolves in the Euclidean space, and the disturbance is taken for granted

as additive Gaussian noise. Within D-LIE-EKF, the noise is Gaussian distributed, but acting in

the Lie Algebra, which induces a Concentrated Gaussian distributed in the Lie Group, cf. [17].

The robotics community has grown in the awareness that the use of probability distributions

deﬁned adequately on Lie groups is an essential tool for pose estimation of rigid bodies. For

instance, [18] s that a banana-shape distribution of the unknown position of a differential-drive

robot displays a banana-shaped distribution which one can obtain with the aid of the exponential

map of the Lie group SE(2).

In this work, we exploit the generalization of EKF on Lie groups to implement a Loosely

Coupled Integration of RTK-GNSS/MEMS-INS for drone-borne remote sensing applications such

as DinSAR. The Double Direct Isometries Lie Group

SE2(3)

( see [19]) is adopted to embed the

attitude, velocity, and position states combined with the translation group

𝑇(6)

to accommodate

the accelerometer and gyroscope biases. In addition, the RTS smoothing on Lie Groups (see

October 7, 2022 DRAFT

PREPRINT 4

[20]) is also implemented. The RTS allows one to leverage all information available to obtain the

state estimates for each time step, a proposal suitable for post-processing applications requiring

higher precision and accuracy.

In the context of navigation, Lie groups have been applied to industrial UAV systems [10],

real-time UAV helicopter navigation [21], and also land vehicle navigation [22]. However, none of

these previous studies aimed at developing a complete navigation system to meet the requirements

of aerial mobile mapping, including ﬁltering and smoothing. Most current works focus on land

vehicle navigation and ﬁltering solutions only.

Moreover, using simulated data, we show that the proposed ﬁlter and smoother outperform

conventional quaternion and Euler-based approaches. Secondly, we use a real dataset to show that

the Lie Group-based ﬁltering and smoothing yield better performances for DinSAR processing

compared to the state-of-the-art commercial software Inertial Explorer®.

In summary, the main contributions of this paper are three-fold.

The development of D-LIE-EKF and RTS smoother for loosely coupled integration of

GNSS/INS tailored for post-processing applications that require high precision and accuracy.

ii)

The proposal of a novel strategy to initialize the heading that does not rely on gyro-compassing

or magnetometer, allowing the use of low-cost MEMS inertial sensors.

iii)

The performance evaluation of Lie Group-based GNSS/INS integration against conventional

methods using simulated data and comparison of DinSAR processing using real dataset

against the commercial software Inertial Explorer®.

To our best knowledge, this is the ﬁrst implementation of loosely coupled RTK-GNSS/INS

integration tailored for post-processing applications, combining Kalman ﬁltering and RTS

smoothing on Lie Groups, applied to drone-borne remote sensing applications.

This paper is organized as follows. Section

II-A

brieﬂy introduces the mathematical concepts

for understanding the Kalman Filter algorithm on Lie Groups. Section II describes the modeling

of dynamic systems and random variables on Lie Groups, followed by the description of ﬁltering

and smoothing methods. Section III develops the navigation and sensors model, and Section IV

describes the integration scheme. After, experimental evaluations using both simulated and real

data appear in Section V. Concluding remarks are provided in Section VI.

The following notation is adopted:

𝑥𝛾

𝛽𝛼

for a vector

𝑥

represents the coordinates of some

kinematic property (position, velocity, etc.) of frame

𝛼

w.r.t. frame

𝛽

, expressed in the frame

𝛾

The navigation frame is the North-East-Down (NED) local-level frame, abbreviated by the index

October 7, 2022 DRAFT

PREPRINT 5

𝑛

; the body frame is indicated by

𝑏

, the inertial frame by

𝑖

, whereas the Earth Centered Earth

Fixed (ECEF) frame is indexed by 𝑒.

II. FILTERING AND SMOOTHING ON LIE GROUPS

A. Brief Review of Lie Group Theory

A Lie group is a mathematical structure that combines the concept of a differentiable manifold

with the concept of a group. For rigid-body applications, usually, the analysis is restricted to

a subgroup of the General Linear Group

𝐺𝐿(𝑛, R)

, also called matrix Lie groups [23]. The

𝐺𝐿(𝑛, R)

is the group of all invertible

𝑛×𝑛

matrices of real numbers with group operation

given by the matrix product. In this paper, we also restrict our consideration to connected and

unimodular Lie groups, as it is required for the way the uncertainty on Lie groups is modeled.

An important structure in the Lie group theory is the Lie algebra, a vector space equipped

with a bracket product

J·,·K

called the Lie bracket. It is always possible to ﬁnd a Lie algebra

associated with a Lie group [23]. For matrix Lie groups, in particular, the associated Lie algebra

is the vector space of matrices with Lie bracket given by the commutator

J𝑋, 𝑌 K=𝑋𝑌 −𝑌 𝑋

The importance of the Lie algebra lies in that most of the properties of the Lie group come from

properties in the Lie algebra.

The exponential map

exp𝐺:g→𝐺

, which for a matrix Lie group reduces to the usual matrix

exponential function, provides a connection between elements of the Lie group and elements of

the Lie algebra.

In general, the exponential map is not bijective; however, it is possible to show that there exist

open neighborhoods of the identity element on the Lie group and of the zero element on the Lie

algebra, for which the exponential map is a diffeomorphism

. In these open sets, one deﬁnes the

logarithm map log𝐺:𝐺→gas the inverse of the exponential map.

Since the Lie algebra is a vector space, it is possible to represent any element as a linear

combination of its basis. Therefore, instead of manipulating the elements

𝑋

as matrices for

A diffeomorphism is an isomorphism between differentiable manifolds. A differentiable invertible function between manifolds

with the differentiable inverse.

October 7, 2022 DRAFT

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

PREPRINT1GNSS/MEMS-INSIntegrationforDroneNavigationusingEKFonLieGroupsMarcosR.Fernandes,GiorgioM.Magalhaes,YusefC´aceres,JoaoB.R.doValAbstractBuildinguponthetheoryofKalmanFilteringonLieGroups,thispaperdescribesanExtendedKalmanFilterandSmootherforLooselyCoupledIntegrationofGNSS/INStailoredforpost-p...

展开>> 收起<<

PREPRINT 1 GNSSMEMS-INS Integration for Drone Navigation using EKF on Lie Groups.pdf

共30页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

PREPRINT 1 GNSSMEMS-INS Integration for Drone Navigation using EKF on Lie Groups

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: