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Magnetic-field Based Odometry - An Optical Flow Inspired Approach

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

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

Felix Trulsson 费利克斯-特吕尔松Dept. Underwater Technology
FOI Swedish Defence Research Agency
FOI 瑞典国防研究局
Kista, Sweden 瑞典基斯塔felix.trulsson@foi.se

Abstract 摘要

An optical flow inspired magnetic-field based odometry estimation process is presented. The estimation process is based upon taking "image" like measurements of the magneticfield using a magnetometer array. From the measurements a model of the local field is learned. Using the learned model the pose change that gives the smallest prediction error of the measurement at the next time instant is calculated. Two models for describing the magnetic-field are presented, and the performance of the odometry estimation process when using the two models is evaluated. The evaluation shows that at a high signal-to-noise ratio the pose change can be estimated with an error of only a few percentage of the true pose change. Further, the evaluation shows that the uncertainty of the estimate can be consistently estimated. Thus, the proposed odometry estimation process can be used to reduce the navigation error growth rate of, for example, inertial navigation systems by providing reliable odometry information when passing by magnetized objects.
本文介绍了一种受光流启发的基于磁场的里程估算过程。该估算过程基于使用磁强计阵列对磁场进行类似 "图像 "的测量。从测量结果中可以学习到局部磁场的模型。利用学习到的模型,可以计算出下一个时间瞬间测量误差最小的姿势变化。本文介绍了两种描述磁场的模型,并对使用这两种模型进行里程估算的性能进行了评估。评估结果表明,在信噪比较高的情况下,姿势变化的估计误差仅为真实姿势变化的几个百分点。此外,评估结果表明,估算结果的不确定性也能得到一致的估计。因此,当经过磁化物体时,建议的测距估算过程可通过提供可靠的测距信息来降低惯性导航系统等的导航误差增长率。


Vector-fields, such as the earth's magnetic-field and gravityfield, are highly informative sources for localization. These exemplified vector-fields are omnipresent and stable, and variations in the fields, if measured accurately, provide a fingerprint highly correlated to the measurement location [1]; an example of the magnetic-field variations inside a building is shown in Fig. 1. Hence, these fields constitute a viable and robust information source for localization in Global Navigation Satellite System (GNSS) denied environments, such as indoors or underwater [2], [3]. Indeed, recent research on magneticfield based simultaneous localization and mapping (SLAM) using low-cost magnetometers and inertial sensors has shown impressive results and opened up a potential path toward scalable and accurate pedestrian indoor localization [4], [5]. However, a fundamental limitation of these magnetic-field based SLAM solutions is the requirement to frequently revisit mapped areas for mitigation of the inertial navigation errors. The allowable length of the exploratory phases depends on the uniqueness of the features and the navigation error growth rate of the inertial navigation process. Hence, to increase the usability of current magnetic-field based SLAM solutions there is a need for techniques that: (i) enable faster and more unique magnetic-field feature learning, and (ii) reduce the inertial navigation error growth rate. The latter is important not only in
矢量场,如地球磁场和重力场,是极具信息价值的定位来源。这些示例性矢量场无处不在且稳定,如果测量准确,场的变化可提供与测量位置高度相关的指纹[1];图 1 显示了建筑物内部磁场变化的示例。因此,在室内或水下等拒绝全球导航卫星系统(GNSS)的环境中,这些磁场是一种可行且稳健的定位信息源[2]、[3]。事实上,最近利用低成本磁力计和惯性传感器进行的基于磁场的同步定位和绘图(SLAM)研究已经取得了令人印象深刻的成果,并为实现可扩展的精确行人室内定位开辟了一条潜在的道路[4],[5]。然而,这些基于磁场的 SLAM 解决方案的一个基本限制是需要经常重访测绘区域以减少惯性导航误差。探索阶段的允许长度取决于特征的独特性和惯性导航过程的导航误差增长率。因此,为了提高当前基于磁场的 SLAM 解决方案的可用性,需要采用以下技术:(i) 实现更快、更独特的磁场特征学习,以及 (ii) 降低惯性导航误差增长率。后者的重要性不仅体现在
@ 2021 IEEE
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。
magnetic-field based SLAM, but also for inertial sensor based navigation systems in general.
基于磁场的 SLAM,以及基于惯性传感器的一般导航系统。
Today, thanks to the last decade's sensor technology development, high-performing and affordable magnetometer vectorsensor arrays can be constructed. Similarly to how a camera can take an image of the surrounding environment, these sensor arrays can take an image-like measurement of a vectorfield; see Fig. 2 for an example of a magnetic-field "image". And just as in computer-vision based localization systems, these magnetic-field measurements can be used for odometry and localization via feature tracking and feature matching. However, unlike visual imaging, magnetic-field imaging is not impaired by smoke, dust, fog, etc. Further, in contrast to the hard to model structure of visual images, the imaged magneticfield must comply with easy to model physical laws [6]. Therefore, this paper will present a model-based estimation method for magnetic-field "image" based odometry.
如今,得益于过去十年传感器技术的发展,高性能、价格合理的磁强计矢量传感器阵列已经可以制造出来。与照相机拍摄周围环境图像的方式类似,这些传感器阵列可以对矢量场进行类似图像的测量;磁场 "图像 "示例见图 2。就像在基于计算机视觉的定位系统中一样,这些磁场测量值可通过特征跟踪和特征匹配来进行里程测量和定位。然而,与视觉成像不同的是,磁场成像不会受到烟、尘、雾等的影响。此外,与难以建模的视觉图像结构不同,成像的磁场必须符合易于建模的物理规律 [6]。因此,本文将提出一种基于模型的估算方法,用于基于磁场 "图像 "的里程测量。
The concept of magnet-field based odometry was first proposed in [7], where it was based upon the differential equation
基于磁场的里程测量概念首次提出于 [7],其基础是微分方程
The differential equation relates the rate of change of the magnetic field to the rotation rate (assumed to be measured by a gyroscope triad in [7]), the Jacobian of the magnetic field with respect to the location , and the velocity . From the measurements of a magnetometer array the Jacobian can be estimated and the differential equation solved. That is, the velocity can be estimated.
微分方程将磁场 的变化率与旋转速率 (在 [7] 中假定由陀螺仪三元组测量)、磁场 相对于位置 的雅各布系数以及速度 联系起来。根据磁强计阵列的测量结果,可以估算出雅各布 ,并求解微分方程。也就是说,可以估算出速度
In subsequent works [8] and [9] by the same authors, as well as in the recent papers [10]-[13], the differential equation (1) was used to develop magnetic-field based odometry aided inertial navigation system solutions. The result is a navigation system with a much slower error growth rate than a pure inertial navigation system; theoretically, the position error should grow linearly with time, instead of cubically. Indeed, the experimental results presented in [9] show that in an environment where there are sufficient variations in the magnetic field, such a magnetic odometry aided inertial navigation system can achieve a position error proportional to only a few percentage of the distance traveled.
Recently, a model-based approach to the magnetic-field odometry problem was proposed [14]. By introducing a model of the local magnetic-field and fitting the model to two consecutive array measurements, i.e., two consecutive field images, the translational and rotational motion of the array between the measurements can be estimated. In relation to computer-vision odometry, the model-based approach can be viewed as performing sub-pixel interpolation using the model and then doing optical-flow calculations for estimating the translational and rotational motion between the images [15].
最近,有人提出了一种基于模型的方法来解决磁场测距问题[14]。通过引入本地磁场的模型 ,并将该模型拟合到两个连续的阵列测量值(即两个连续的磁场图像),可以估算出测量值之间阵列的平移和旋转运动。就计算机视觉里程测量而言,基于模型的方法可视为利用模型 进行子像素插值,然后进行光流计算,以估计图像之间的平移和旋转运动[15]。
Viewing the magnetic-field odometry problem as a model estimation problem has several benefits. First, estimation theory can be applied to analyze the properties of the magneticfield odometry problem and to derive various estimators. Second, the translational and rotational motion of the array can theoretically be estimated without any gyroscopes, which makes it possible to perform dead-reckoning using only an array of magnetometers. Finally, and possibly most importantly, the experimental results presented in [14] indicates that modelbased magnetic-field odometry may, at low signal-to-noise ratios, provide higher accuracy than odometry approaches based upon directly solving (1).

B. Contributions B.捐款

Based upon the listed attractive properties of the modelbased odometry estimation process, we will in this paper extend the initial findings reported in [14] and present an optical flow inspired approach to magnetic-field odometry using magnetic-field "images". More precisely we will:
基于所列举的基于模型的里程估算过程的诱人特性,我们将在本文中扩展 [14] 中报告的初步研究成果,并提出一种受光流启发的方法,利用磁场 "图像 "进行磁场里程估算。更准确地说,我们将
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。
  1. present two magnetic-field models suitable for modelbased magnetic-field odometry and describe how the parameters of these models can be learned; and
  2. evaluate the odometry accuracy when using the two magnetic-field models to process magnetic-field measurements observed in a typical indoor environment.
All the data and code used to produce the presented results can be downloaded from www.openshoe.se.
所有数据和生成结果所使用的代码均可从 www.openshoe.se 下载。

II. Proposed Odometry Estimation Process

Next the proposed model-based odometry estimation process will be presented. To support the reader in assimilating the mathematical description of the estimation process a conceptional illustration of the process is presented in Fig. 3.
接下来将介绍拟议的基于模型的里程估算过程。为了帮助读者理解估算过程的数学描述,图 3 展示了估算过程的概念图示。
Consider a sensor array with three-axis magnetometers and let
考虑一个带有 三轴磁力计的传感器阵列,并假设
denote the measurements from these sensors at time instant . Here denotes the measurement from the :th magnetometer in the array. Further, let the local magneticfield model be defined such that it describes the magnetic-field at location with respect to the array coordinate frame at time . Moreover, let the pose change of the array between time instant and be denoted by
表示这些传感器在时间瞬间 的测量值。 表示阵列中 :第 1 个磁力计的测量值。此外,让本地磁场模型 被定义为描述位置 的磁场,相对于时间 的阵列坐标系。此外,让阵列在时间瞬间 之间的姿态变化用以下公式表示
Here and denote the displacement and orientation change, respectively.
The array measurements at the next time instant, i.e., , can then be modeled as
下一个时间瞬间(即 )的阵列测量结果可以建模为
Field measurements at time instant:

Input: Array geometry & model type

Field measurements at time instant:
Nonlinear least squares estimation:
Output:  输出:
Fig. 3. Conceptional illustration of the proposed odometry estimation process. From the magnetic-field measurements a model of the local magnetic-field is learned. Using the learned model the pose change that gives the smallest, in the weighted least squares sense, prediction error of the measurement is calculated. The covariance of the pose change estimate is also calculated.
图 3.拟议的测距估算过程概念图。从磁场测量结果 中可以学习到当地磁场的模型 。利用学习到的模型,计算出姿态变化 ,该姿态变化在加权最小二乘法意义上对测量 的预测误差最小。姿势变化估计值的协方差 也会被计算出来。
where 其中
Here denotes the directional cosine matrix parameterized by . Further, denotes the location of the :th magnetometer triad within the array. Moreover, denotes the magnetometer array observation noise, which is assumed white with covariance . Here denotes an identity (zero) matrix of dimension .
表示由 参数化的方向余弦矩阵。此外, 表示 :th 磁强计三元组在阵列中的位置。此外, 表示磁强计阵列观测噪声,假定为白色,协方差为 表示维数为 的同位(零)矩阵。
In the definition of the measurement model (4a) it has been assumed that a perfect magnetic-field model is available. However, is typically unknown and must learned on the fly. Moreover, the learned model will generally not be perfect. If the model estimate is unbiased and the model error small, then the error can be accounted for in the measurement model (4a) by substituting the observation noise covariance with the modified covariance
在测量模型(4a)的定义中,假定有一个完美的磁场模型 。然而, 通常是未知的,必须即时学习。此外,学习到的模型通常也不完美。如果模型估计值 无偏且模型误差较小,则可以用修正后的协方差代替观测噪声协方差 ,从而在测量模型 (4a) 中考虑到误差。
where 其中
Note that the modified covariance matrix depends on the pose change as it takes into account the uncertainties of the learned magnetic-field model when extrapolating the measured field into new regions.
请注意,修改后的协方差矩阵 取决于姿态变化,因为它在将测量磁场外推到新区域时,考虑了所学磁场模型 的不确定性。
Thus, given a magnetic-field model estimate and the measurement model (4a), the weighted least square estimate of the pose change can then be obtained as
因此,给定磁场模型估计值 和测量模型 (4a),姿势变化的加权最小二乘法估计值 可得
where 其中
Further, the covariance of the estimate is given by [16]
此外,估计值的协方差 由以下公式给出 [16]
where and 其中
Next, two magnetic-field models and methods to learn the model parameters, will be presented.
接下来,将介绍两个磁场模型 和学习模型参数的方法。


If the array moves within a static magnetic-field with no free current then, according to Maxwell's equations, the field should be both curl- and divergence-free [6]. Hence, the magnetic-field model should satisfy the conditions
如果阵列在没有自由电流的静态磁场中移动,那么根据麦克斯韦方程,磁场应该是无卷曲和发散的[6]。因此,磁场模型 应满足以下条件
Next, a polynomial and a Gaussian process model satisfying these conditions will be presented.

A. Polynomial model A.多项式模型

A curl- and divergence-free polynomial magnetic-field model can be obtained by the linear model
with the regression matrix and parameter vector selected as in [14]. For an :th order polynomial the model has unknown parameters. Since the model is linear the magnetic-field model can be estimated from the measurements as [16]
回归矩阵 和参数向量 的选择与文献[14]相同。对于 :三阶多项式,模型有 个未知参数。由于模型是线性的,因此磁场模型可以根据测量结果进行估算,