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A Tightly-Integrated Magnetic-Field aided Inertial Navigation System
紧密集成的磁场辅助惯性导航系统

Chuan Huang 黄川Dept. of Electrical Engineering
电子工程系
Linköping University 林克平大学Linköping, Sweden 瑞典林雪平chuan.huang@liu.se

Gustaf Hendeby 古斯塔夫-亨德比Dept. of Electrical Engineering
电子工程系
Linköping University 林克平大学Linköping, Sweden 瑞典林雪平gustaf.hendeby@liu.se

Isaac Skog 艾萨克-斯科格Dept. of Electrical Engineering
电子工程系
Linköping University 林克平大学Linköping, Sweden 瑞典林雪平isaac.skog@liu.se

Abstract 摘要

A tightly integrated magnetic-field aided inertial navigation system is presented. The system uses a magnetometer sensor array to measure spatial variations in the local magneticfield. The variations in the field are - via a recursively updated polynomial magnetic-field model - mapped into displacement and orientation changes of the array, which in turn are used to aid the inertial navigation system. Simulation results show that the resulting navigation system has three orders of magnitude lower position error at the end of a 40 seconds trajectory as compared to a standalone inertial navigation system. Thus, the proposed navigation solution has the potential to solve one of the key challenges faced with current magnetic-field simultaneous localization and mapping (SLAM) systems - the very limited allowable length of the exploration phase during which unvisited areas are mapped.
介绍了一种紧密集成的磁场辅助惯性导航系统。该系统利用磁力计传感器阵列测量当地磁场的空间变化。通过递归更新的多项式磁场模型,将磁场变化映射为阵列的位移和方向变化,进而用于辅助惯性导航系统。仿真结果表明,与独立的惯性导航系统相比,由此产生的导航系统在 40 秒轨迹结束时的位置误差要低三个数量级。因此,所提出的导航解决方案有可能解决目前磁场同步定位和绘图(SLAM)系统所面临的主要挑战之一,即绘制未访问区域的探索阶段的允许长度非常有限。

Index Terms-inertial navigation, magnetic field, polynomial model, error state Kalman filter

I. INTRODUCTION I.引言

The magnetic-field is omnipresent and stable vector field, which can be a highly informative and reliable source for localization if measured accurately [1]; an example of the magneticfield variations inside a building is shown in Fig. 1. The distorted earth magnetic-field and magnetized materials in the environment provide fingerprints highly correlated to position. Hence, the magnetic-field is a viable and robust information source for localization in Global Navigation Satellite System (GNSS) denied environments, such as indoors and underwater [2], [3].
磁场是无处不在的稳定矢量场,如果测量准确,它可以成为一个信息量大且可靠的定位源[1];图 1 显示了建筑物内磁场变化的一个例子。扭曲的地球磁场和环境中的磁化材料提供了与位置高度相关的指纹。因此,在室内和水下等拒绝全球导航卫星系统(GNSS)的环境中,磁场是一种可行且稳健的定位信息源[2],[3]。
Indeed, magnetic-field based simultaneous localization and mapping (SLAM), where magnetic-field measurements are fused with the navigation solution from a inertial navigation system (INS), has turned out be one of the most promising techniques for scalable indoor localization [4]. However, when using low-cost inertial sensors the error growth rate of the inertial navigation system is typically in the order of 10 meters per minute [5]. Therefore, the allowable length of the exploration phases, where new areas are mapped, is extremely limited when using low-cost inertial sensors. Hence, to increase the usability of current magnetic-field based SLAM solutions we need robust magnetic-field odometry techniques that reduce navigation error drift rate.
事实上,基于磁场的同步定位和测绘(SLAM),即磁场测量与惯性导航系统(INS)的导航解决方案相融合,已成为最有前途的可扩展室内定位技术之一[4]。然而,在使用低成本惯性传感器时,惯性导航系统的误差增长率通常在每分钟 10 米左右 [5]。因此,在使用低成本惯性传感器时,绘制新区域地图的探索阶段的允许长度极为有限。因此,为了提高当前基于磁场的 SLAM 解决方案的可用性,我们需要能够降低导航误差漂移率的稳健磁场里程测量技术。
Fig. 1. Illustration of the magnetic-field magnitude variations inside a building. The field was measured with an magnetometer array, whose location was tracked by camera-based tracking systems. The field measurement was then interpolated and the field magnitude was projected on the floor.
图 1.建筑物内磁场大小变化示意图。磁场由磁力计阵列测量,其位置由摄像跟踪系统跟踪。然后对磁场测量结果进行内插,并将磁场大小投射到地板上。
Thanks to the recent sensor technology development, highperforming and affordable magnetometer vector-sensor arrays can be constructed. These arrays allows for snap-shot "images" of the magnetic-field to be collected, which in turns enables faster and richer feature learning in the SLAM process. Further, magnetometer array measurements must comply with easy to model physical laws. This allows us to model them in such a way that position translation and orientation change are encoded, making it perfect for complementing or correcting an INS. To that end, in this paper, we present a method for tightly integrated magnetic-field aided inertial navigation. The resulting navigation system has, compared to a pure inertial navigation system, an significantly reduced error growth rate. Hence, the proposed navigation method has the potential to greatly extend the allowable length of the exploration phases in magnetic-field based SLAM system.
得益于近年来传感器技术的发展,高性能、价格合理的磁强计矢量传感器阵列得以构建。这些阵列可以收集磁场的快照 "图像",从而在 SLAM 过程中实现更快、更丰富的特征学习。此外,磁强计阵列测量必须符合易于建模的物理定律。这样,我们就可以对其进行建模,从而对位置平移和方向变化进行编码,使其成为补充或修正 INS 的完美工具。为此,我们在本文中提出了一种紧密集成磁场辅助惯性导航的方法。与纯惯性导航系统相比,该导航系统的误差增长率显著降低。因此,所提出的导航方法有可能大大延长基于磁场的 SLAM 系统中探索阶段的允许长度。
The idea behind magnetic-field based odometry is that the velocity of a magnetometer vector sensor array can be
基于磁场的里程测量背后的理念是,磁强计矢量传感器阵列的速度可以

estimated via the differential equation
通过微分方程估计
The equation relates the rate of change of the magnetic field to the rotation rate of the array , the Jacobian of the magnetic field with respect to position , and the velocity . With a magnetometer array, such as that in Fig. 2, the Jacobian can be estimated from spatially distributed measurements and the velocity can be determined.
该方程将磁场 的变化率与阵列 的旋转速率、磁场相对于位置的雅各布系数 以及速度 联系起来。利用磁强计阵列(如图 2 所示),可以根据空间分布的测量结果估算出雅各比,并确定速度。
In [6]-[10], the differential equation (1) was used to develop magenetic-field aided INS solutions. The resulting implementations achieve much lower error growth rate compared to stand-alone inertial navigation systems. In a more recent work [11], a model-based approach to magnetic-field odometry was proposed, in which a polynomial model describing the local magnetic-field was developed. In the proposed modelbased odometry method the velocity was viewed as a model parameter to be estimated, which allowed estimation theory to be used to analyse the problem. Presented experiential results showed that the model-based odometry approach achieved a higher accuracy compared to approaches based upon directly solving (1). The model-based odometry approach was further explored in [12], where it was used to estimate both the translation and orientation change of the array.
在[6]-[10]中,微分方程(1)被用于开发磁场辅助 INS 解决方案。与独立的惯性导航系统相比,由此实现的误差增长率要低得多。最近的一项研究[11]提出了一种基于模型的磁场里程测量方法,其中开发了一个描述本地磁场的多项式模型。在所提出的基于模型的测距方法中,速度被视为一个需要估算的模型参数,因此可以使用估算理论来分析问题。实验结果表明,与直接求解(1)的方法相比,基于模型的里程测量法获得了更高的精度。文献[12]进一步探讨了基于模型的里程测量方法,并将其用于估计阵列的平移和方位变化。

B. Contributions B.捐款

We will in this paper, encouraged by the promising results on model-based magnetic-field odometry shown in [12], present a method for model-based magnetic-field aided inertial navigation. More precisely, we will: (a) derive a tightly integrated magnetic-field aided inertial navigation system using a recursively updated polynomial model; and (b) evaluate its performance using Monte Carlo simulations. All the data and code used to produce the presented results are made available at https://github.com/Huang-Chuan/magnetic-field-odometry.
受 [12] 中基于模型的磁场里程测量法所取得的可喜成果的鼓舞,我们将在本文中介绍一种基于模型的磁场辅助惯性导航方法。更准确地说,我们将(a) 利用递归更新的多项式模型推导出一个紧密集成的磁场辅助惯性导航系统;(b) 利用蒙特卡罗模拟评估其性能。用于得出上述结果的所有数据和代码均可在 https://github.com/Huang-Chuan/magnetic-field-odometry 网站上查阅。

II. SYSTEM MODELING II.系统建模

A moving platform with an inertial measurement unit (IMU) and a magnetometer sensor array consisting of sensors is considered; see Fig. 2. Our focus is on estimating the position, velocity, and orientation of the platform from the data generated by the sensors. To that end, in this section a state-space model that can be used to fuse the inertial and magnetic-field measurement will be derived.
我们考虑了一个带有惯性测量单元(IMU)和由 传感器组成的磁力计传感器阵列的移动平台;见图 2。我们的重点是从传感器生成的数据中估计平台的位置、速度和方向。为此,我们将在本节中推导出一个可用于融合惯性和磁场测量的状态空间模型。

A. Inertial Navigation Equations
A.惯性导航方程

The navigation equations for an inertial navigation system using low-cost sensor and moving at moderate velocities are given by [13]
使用低成本传感器并以中等速度移动的惯性导航系统的导航方程为 [13]
Fig. 2. Example of magnetic-field quiver plot, that is, a magnetic-field image, overlaid on the magnetic-field sensor array used to capture the field. Two arrows are missing due to broken sensors.
图 2.磁场颤动图示例,即磁场图像与用于捕捉磁场的磁场传感器阵列的叠加图。由于传感器损坏,缺少两个箭头。
where 其中
Here the superscript and subscript and denote the navigation frame and body frame at timestep , respectively. , and (3D rotation group) denote the position, velocity, and orientation (parameterized as a unit quaternion), respectively. The specific force and gravity are denoted by and , respectively and the sampling interval is denoted by . The body angular velocity is denoted by . Moreover, is the rotation matrix that transforms a vector in the body frame to the navigation frame . Furthermore, denotes quaternion multiplication, is the operator that maps a axis-angle to a quaternion, and is the operator that maps a quaternion to a rotation matrix (see [14] for details on quaternion algebra).
这里的上标和下标 分别表示时间步 时的导航帧和身体帧。 , 和 (三维旋转组)分别表示位置、速度和方向(参数为单位四元数)。比力和重力分别用 表示,采样间隔用 表示。机体角速度用 表示。此外, 是旋转矩阵,用于将身体框架 中的矢量转换到导航框架 中。此外, 表示四元数乘法, 是将轴角映射到四元数的算子, 是将四元数映射到旋转矩阵的算子(有关四元数代数的详情,请参阅 [14])。

B. Inertial Measurement Unit Measurement Model
B.惯性测量单元测量模型

Acceleration and gyroscope measurements, denoted by and , are modeled as the true values affected by sensor bias and noise, i.e.,
加速度和陀螺仪测量值(分别用 表示)被模拟为受传感器偏差和噪声影响的真实值,即
Here and denote accelerometer and gyroscope bias, respectively. Further, and denote accelerometer and gyroscope noise, respectively. The accelerometer and gyroscope measurement noises are modeled as white Gaussian noises with covariance and , respectively. Here, denote an identity matrix of dimension .
分别表示加速度计和陀螺仪偏差。此外, 分别表示加速度计和陀螺仪噪声。加速度计和陀螺仪测量噪声分别建模为协方差为 的白高斯噪声。这里, 表示维数为 的标识矩阵。
The IMU sensor biases are modeled as random walk processes, i.e.,
IMU 传感器的偏差被模拟为随机漫步过程,即
where and are Gaussian white noise with covariance matrix and , respectively.
其中 分别为高斯白噪声,其协方差矩阵分别为

C. Magnetometer Array Measurement Model
C.磁强计阵列测量模型

The magnetometer sensor array measurement is denoted as
磁强计传感器阵列测量值表示为
where denotes the measurement from the magnetometer in the array. Further, locally the magnetic-field at location in the body coordinate frame at time is modeled as
其中 表示阵列中 磁强计的测量值。此外,在人体坐标系中, 位置在时间 的局部磁场 建模为
where is a regression matrix defined in [11] and is the coefficients of the polynomial model; for a order polynomial the model has 3 unknown parameters [12]. Thus, the magnetometer sensor array measurement is modeled as
其中 是[11]中定义的回归矩阵, 是多项式模型的系数;对于 阶多项式,模型有 3 个未知参数[12]。因此,磁强计传感器阵列测量的模型为
where denotes the location of the magnetometer in the array expressed in the body coordinate frame at time . Further, denotes the measurement noise which is assumed to be white and Gaussian distributed with covariance matrix .
其中 表示 磁强计在阵列中的位置,以时间 的身体坐标系表示。此外, 表示测量噪声,假定为白色高斯分布,协方差矩阵为

D. Polynomial model recursive updates
D.多项式模型递归更新

The polynomial model in (6) describes the local magneticfield in the body frame at time . Given the change in pose
(6) 中的多项式模型描述了时间 时身体框架中的局部磁场。鉴于姿势的变化
where 其中
a way to update the polynomial model is needed. To that end, assume that is valid within the volume , centered at the origin of the array at time . Then if there exists a location such that , the magnetic field at location can be expressed as
需要一种更新多项式模型的方法。为此,假设 在体积 内有效,以时间 的阵列原点为中心。那么,如果存在一个位置 ,使得