Magnetic-field Based Odometry - An Optical Flow Inspired Approach 基于磁场的测距--一种受光流启发的方法
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
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
Felix Trulsson 费利克斯-特吕尔松Dept. Underwater Technology 水下技术部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。
I. INTRODUCTION I.引言
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。
A. Related work A.相关工作
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. 在同一作者的后续著作[8]和[9]以及最近的论文[10]-[13]中，微分方程（1）被用于开发基于磁场的里程测量辅助惯性导航系统解决方案。其结果是，导航系统的误差增长率比纯惯性导航系统慢得多；理论上，位置误差应随时间线性增长，而不是立方增长。事实上，文献[9]中的实验结果表明，在磁场变化足够大的环境中，这种磁性里程计辅助惯性导航系统的位置误差只占飞行距离的几个百分点。
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). 将磁场测距问题视为模型估计问题有几个好处。首先，估算理论可用于分析磁场里程测量问题的特性，并推导出各种估算器。其次，理论上不需要任何陀螺仪就能估算出阵列的平移和旋转运动，这使得仅使用磁力计阵列就能进行死区重定位成为可能。最后，可能也是最重要的一点，文献[14]中的实验结果表明，在信噪比较低的情况下，基于模型的磁场里程测量法比直接求解（1）的里程测量法精度更高。
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。
present two magnetic-field models suitable for modelbased magnetic-field odometry and describe how the parameters of these models can be learned; and 介绍两种适合基于模型的磁场里程测量的磁场模型，并说明如何学习这些模型的参数；以及
evaluate the odometry accuracy when using the two magnetic-field models to process magnetic-field measurements observed in a typical indoor environment. 评估使用两种磁场模型处理在典型室内环境中观测到的磁场测量结果时的定位精度。
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 II.拟议的测距估算流程
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 和 之间的姿态变化用以下公式表示
Here and denote the displacement and orientation change, respectively. 和 分别表示位移和方向变化。
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: 瞬间的实地测量结果：
Input: Array geometry & model type 输入：阵列几何形状和模型类型
Field measurements at time instant: 瞬间的实地测量结果：
Nonlinear least squares estimation: 非线性最小二乘估计
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
and
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
and
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 和
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。
III. MAGNETIC-FIELD MODELS III.磁场模型
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. 接下来，将介绍满足这些条件的多项式和高斯过程模型。
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 通过线性模型可以得到一个无卷曲和发散的多项式磁场模型
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， 如下[16]
where the model parameter estimate is given by 其中，模型参数估计值为
Further, the cross-covariance of the two estimates and is given by [16] 此外，两个估计值 和 的交叉协方差由 [16] 给出。
where 其中
The polynomial order, i.e., , can be selected using, e.g., the Akaike information criterion (AIC) [17]. Further, the computational complexity of the learning process is approximately . 多项式阶数，即 ，可以使用 Akaike 信息准则（AIC）[17]等方法进行选择。此外，学习过程的计算复杂度约为 。
B. Gaussian process model B.高斯过程模型
A non-parametric method for modeling magnetic-fields using a curl- and divergence-free Gaussian process was presented in [6]. Adapted to the notation used within this paper, the method can be summarized as follows. The magnetic-field and the magnetization of the material in which the field is observed can jointly be modeled as 文献[6]提出了一种使用无卷曲和发散高斯过程的非参数磁场建模方法。根据本文使用的符号，该方法可概述如下。磁场和磁场观测材料的磁化可共同建模为
Here denotes the magnetization of the material and denotes a zero-mean Gaussian process with kernel function . The kernel function describes the correlation of the Gaussian process between point and point , i.e., , and is in the considered case defined as 表示材料的磁化， 表示核函数为零均值的高斯过程 。核函数描述了 点和 点（即 ）之间的高斯过程的相关性，在所考虑的情况下定义为
where 其中
and
Here, and denote the magnitude variations and the length scale of the radial base function part of the kernel. Moreover, denotes the magnitude variations of the linear part of the kernel. These three hyperparameters can either be set based upon prior knowledge about the underlying magnetic-field or learned from the field observations as described in [18]. 和 表示核的径向基函数部分的幅度变化和长度尺度。此外， 表示核的线性部分的幅度变化。这三个超参数既可以根据有关底层磁场的先验知识来设定，也可以如文献[18]所述从磁场观测中学习。
Given the Gaussian process model in (12) the magnetic field at the locations can be predicted from the measurements via the following four steps. First, define the two vectors 鉴于 (12) 中的高斯过程模型，可通过以下四个步骤从测量结果 预测 处的磁场 。首先，定义两个向量
describing the magnetic-field and magnetization at the prediction and measurement location, respectively. Further, in- 分别描述预测位置和测量位置的磁场和磁化。此外，在
troduce the ideal pseudo magnetization measurement vectors and defined as 引出理想伪磁化测量矢量 和 ，定义为
respectively. Moreover, let and since the magnetization at all physically feasible locations of the array must be zero. 分别为零。此外，由于阵列所有物理可行位置的磁化率必须为零，因此让 和 。
Second, assume that the measurement noise is Gaussian distributed and calculate the posterior mean and covariance of given the measurements and . That is, calculate [18] 其次，假设测量噪声 是高斯分布的，并根据测量结果 和 计算 的后验均值和协方差。也就是说，计算 [18]
and
where the covariance matrices , and can be calculated using the kernel definition in (13a). 其中，协方差矩阵 , 和 可以通过 (13a) 中的核定义计算出来。
Third, update the posterior mean and covariance using the knowledge that also should be zero. That is, 第三，利用 也应为零的知识更新后验均值和协方差。也就是说
and
where 其中
Finally, the magnetic-field estimate is given by removing the parts of the posterior mean and covariance relating to the magnetization. That is, 最后，通过去除与磁化有关的后验均值和协方差，得到磁场估计值。也就是说
and
Here the operator cuts out the sub block corresponding to rows to and columns to of matrix . The computational complexity of the learning process is approximately . 在这里，运算符 会切出与矩阵 的行 至 和列 至 相对应的子块。学习过程的计算复杂度约为 。
IV. Evaluation IV.评估
To evaluate the performance of the proposed odometry estimation process when using the two magnetic-field models the following simulation experiments were conducted. 为了评估拟议的里程估算程序在使用两种磁场模型时的性能，我们进行了以下模拟实验。
A. Simulation setup and evaluation metrics A.仿真设置和评估指标
Magnetic-field data was collected using a magnetometer within a volume of approximately in the room shown in Fig. 1. While collecting the data the position and orientation of the magnetometer was tracked by a high-end camera-based tracking system. A high-order dipole reference model was then fitted to the data. The field of the resulting reference model is shown in Fig. 4. Using the reference model, MonteCarlo simulations where an array moved according to three different trajectories through the field, were conducted. The three trajectories were: 使用磁强计在图 1 所示的房间内约 的范围内收集磁场数据。在收集数据的过程中，磁强计的位置和方向由高端摄像跟踪系统进行跟踪。然后根据数据拟合出一个高阶偶极子参考模型。参考模型的磁场如图 4 所示。利用参考模型，进行了蒙特卡罗模拟，阵列按照三种不同的轨迹穿过磁场。这三种轨迹分别是
An in-plane trajectory where the array moved without any rotations in a straight horizontal line, and with the -plane of the array aligned with the -plane of the reference model. 平面内轨迹，即阵列在水平直线上不做任何旋转，阵列的 - 平面与参考模型的 - 平面对齐。
An out-of-plane trajectory where the array moved without any rotations in a straight vertical line perpendicular to the -plane of the array, and the -plane of the array was aligned with the -plane of the reference model. 平面外轨迹，即阵列在垂直于 - 平面的垂直直线上不做任何旋转地移动，阵列的 - 平面与参考模型的 - 平面对齐。
A spiral trajectory where the array moved in an upward spiral with a constant radius and the -axis of the array pointing in the tangent of the movement and the roll being zero. 螺旋运动轨迹，即阵列以恒定半径向上螺旋运动，阵列的 - 轴指向运动的切线，滚动为零。
The trajectories are shown in Fig. 4. In all the trajectories the displacement between the measurements was . For the spiral trajectory the change in orientation was . Further, the geometry of the simulated array was the same as for the one shown in Fig. 2. That is, the array consisted of 30 magnetometers placed in a 6 times 5 grid with and spacing in the and axis directions, respectively. Moreover, the noise variance of the magnetometers was set to . This corresponds to the noise level of the low-cost magnetometer RM3100 from PNI Sensor, US. 轨迹如图 4 所示。在所有轨迹中，两次测量之间的位移 是 。 对于螺旋轨迹， 。此外，模拟阵列的几何形状与图 2 所示相同。也就是说，阵列由 30 个磁力计组成，放置在 6 乘 5 的网格中， 和 轴方向的间距分别为 和 。此外，磁强计的噪声方差设置为 。这相当于美国 PNI 传感器公司生产的低成本磁强计 RM3100 的噪声水平。
Using the simulated array measurements the pose change of the array was estimated using (3) and the root-mean-squareerrors (RMSE) 使用模拟阵列测量结果，利用 (3) 和均方根误差（RMSE）估算阵列的姿态变化。
and
were calculated. Here, the superscript denotes the simulation iteration at which the estimate was calculated and is 的计算结果。这里，上标 表示计算估计值时的模拟迭代次数， 为
Fig. 4. Illustration of the magnetic-field and trajectories used in the simulations. The field model is based on real measurements to which a high-order dipole model has been fitted. 图 4.模拟中使用的磁场和轨迹示意图。磁场模型基于实际测量结果，并拟合了一个高阶偶极子模型。
the total number of iterations in the Monte Carlo simulation. Furthermore, the mean values 是蒙特卡罗模拟的迭代总数。此外，平均值
and
of the uncertainties estimated by the estimator were computed. Moreover, as a measure of how much the field varied across the array the signal-to-noise ratio (SNR) 的不确定性进行了计算。此外，为了衡量整个阵列的场变化程度，信噪比 (SNR)
where 其中
was calculated. 计算得出。
When learning the magnetic-field models the following settings were used. In the case of the polynomial model the only hyperparameter is the polynomial order , which was automatically selected using AIC's method with a maximum allowed polynomial order . In the case of the Gaussian process model all hyperparameters except the noise variance were set to fixed values. The values , and were used as they gave the overall best estimation performance. The noise variance was estimated by maximizing the marginalized likelihood function as described in [6]. Noteworthy, theoretically it is possible to estimate all hyperparameters from the marginalized likelihood function, but in practice it prove to be difficult as the marginalized likelihood function contains multiple local optima. 学习磁场模型时使用了以下设置。在多项式模型中，唯一的超参数是多项式阶数 ，它是用 AIC 方法自动选择的，允许的最大多项式阶数为 。在高斯过程模型中，除噪声方差 外，所有超参数都被设为固定值。使用 和 的原因是它们能提供最佳的整体估计性能。噪声方差 是通过最大化边际似然函数估算的，如文献[6]所述。值得注意的是，理论上可以通过边际似然函数估计所有超参数，但在实践中却很困难，因为边际似然函数包含多个局部最优值。
B. Results and discussion B.结果和讨论
The results from the Monte Carlo simulations are shown in Fig 5-7. From the figures it can be seen that the estimation accuracy is highly dependent on the SNR. At a high SNR the displacement and orientation change can be estimated with an accuracy of and , respectively. This corresponds to a few percentage of the true pose change. Further, it can be seen that the polynomial and Gaussian process models perform approximately equally well, except for trajectory #2 (the out-of-plane trajectory), where the Gaussian process model performs poorly for all SNR; the poor performance is likely due to the flexibility of the Gaussian process model, which makes it worse at extrapolating than the polynomial model. Moreover, when the Gaussian process model is used, the proposed odometry estimation process generally becomes overconfident in the accuracy of the estimated pose change. When the polynomial model is used there is a good agreement between the estimated and true uncertainty in the pose change. 蒙特卡罗模拟的结果如图 5-7 所示。从图中可以看出，估计精度在很大程度上取决于信噪比。在高信噪比条件下，位移和方向变化的估计精度分别为 和 。这相当于真实姿态变化的几分之一。此外，可以看出多项式过程模型和高斯过程模型的表现大致相同，除了轨迹 2（平面外轨迹），高斯过程模型在所有信噪比下的表现都很差；表现差的原因可能是高斯过程模型的灵活性，这使得它的外推能力比多项式模型差。此外，当使用高斯过程模型时，拟议的里程估算过程通常会对估计姿势变化的准确性过于自信。当使用多项式模型时，姿势变化的估计不确定性和真实不确定性之间有很好的一致性。
Gaussian process models is the state-of-the-art technique for representing the magnetic-field map in magnetic-field SLAM solutions [5]. However, the presented results indicate that different magnetic-field models should be used for the odometry and mapping process if the proposed magneticfield odometry estimation process are to be combined with existing magnetic-field SLAM solutions. Considering that the polynomial model is a linear model of small dimension, the computational complexity of adding the odometry estimation process to existing magnetic-field SLAM solutions is low (approximately floating point operations) and should not hinder real-time implementations. 高斯过程模型是在磁场 SLAM 解决方案中表示磁场图的最先进技术[5]。然而，本文介绍的结果表明，如果要将建议的磁场里程估算过程与现有的磁场 SLAM 解决方案相结合，则应在里程估算和绘图过程中使用不同的磁场模型。考虑到多项式模型是一个小维度的线性模型，在现有的磁场 SLAM 解决方案中添加测距估计过程的计算复杂度很低（约为 浮点运算），应该不会妨碍实时实施。
V. Conclusions & Future Research V.结论与未来研究
A model-based magnetic-field odometry estimation process has been presented. Further, the performance of the proposed odometry estimation process when using a polynomial model and a Gaussian process model to describe the local magneticfield has been evaluated. The evaluation results shows that at high SNR the pose change can be estimated with an error of less than a few percentage of the true pose change. Moreover, the evaluation shows that with the polynomial model the uncertainty of the estimate can also be consistently predicted. Thus the proposed magnetic-field odometry technique constitutes a complimentary source of navigation information that can be used to reduce the position error growth rate of inertial navigation systems or other dead-reckoning systems. This is done by providing accurate and reliable odometry information when passing by magnetized objects, such as metal structures inside walls. When comparing the polynomial model and Gaussian process model, in general the polynomial model provides the highest odometry accuracy and the most consistent uncertainty estimates. Furthermore, the learning of the polynomial model has much lower computational complexity than the Gaussian 提出了一种基于模型的磁场测距估算程序。此外，还评估了在使用多项式模型和高斯过程模型描述局部磁场时，所提出的测距估计流程的性能。评估结果表明，在高信噪比情况下，姿势变化的估计误差小于真实姿势变化的几个百分点。此外，评估结果表明，使用多项式模型还可以一致地预测估计值的不确定性。因此，建议的磁场里程测量技术是导航信息的补充来源，可用于降低惯性导航系统或其他死飞系统的位置误差增长率。当经过磁化物体（如墙内的金属结构）时，磁场里程测量技术可提供准确可靠的里程测量信息。在比较多项式模型和高斯过程模型时，一般来说，多项式模型能提供最高的里程测量精度和最一致的不确定性估计。此外，学习多项式模型的计算复杂度远远低于高斯过程模型。
Fig. 5. Estimation error for trajectory #1, i.e., the in-plane trajectory. 图 5.1 号轨迹（即平面内轨迹）的估计误差。
process model, which makes it more suitable for usage in real-time applications. Our future research will be focused on developing methods for tight integration between model-based magnetic-field odometry and inertial navigation systems. 过程模型，使其更适合用于实时应用。我们未来的研究重点是开发基于模型的磁场里程测量与惯性导航系统紧密结合的方法。
ACKNOWLEDGMENT 致谢
This work has been funded by the Security Link project FOI-2010-1753:4 Magnetic-Field based Speed Aided Inertial Navigation and the Swedish Research Council project 202004253 Tensor-field based localization. 这项工作得到了安全链接项目 FOI-2010-1753:4 基于磁场的速度辅助惯性导航和瑞典研究理事会项目 202004253 基于张量场的定位的资助。
REFERENCES 参考文献
[1] M. Angermann, M. Frassl, M. Doniec, B. J. Julian, and P. Robertson, "Characterization of the indoor magnetic field for applications in localization and mapping," in Int. Conf. on Indoor Positioning and Indoor Navigation (IPIN), Sydney, NSW, Australia, Nov. 2012. [1] M. Angermann、M. Frassl、M. Doniec、B. J. Julian 和 P. Robertson，"用于定位和绘图的室内磁场特征"，国际室内定位和室内导航会议（IPIN），澳大利亚新南威尔士州悉尼，2012 年 11 月。室内定位与室内导航会议（IPIN），澳大利亚新南威尔士州悉尼，2012 年 11 月。
[2] T. N. Lee and A. J. Canciani, "MagSLAM: Aerial simultaneous localization and mapping using earth's magnetic anomaly field," Navigation, Mar. 2020. [2] T. N. Lee 和 A. J. Canciani，"MagSLAM：利用地球磁场异常进行空中同步定位和绘图"，《导航》，2020 年 3 月。
[3] H. F. Rice, V. Benischek, and L. Sczaniecki, "Application of atom interferometric technology for GPS independent navigation and time solutions," in IEEE/ION Position, Location and Navigation Symposium (PLANS), Monterey, CA, Jun. 2018. [3] H. F. Rice、V. Benischek 和 L. Sczaniecki，"原子干涉测量技术在独立于 GPS 的导航和时间解决方案中的应用"，IEEE/ION 位置、定位和导航研讨会（PLANS），加利福尼亚州蒙特雷，2018 年 6 月。
[4] P. Robertson, M. Frassl, M. Angermann, M. Doniec, B. J. Julian, M. Garcia Puyol, M. Khider, M. Lichtenstern, and L. Bruno, "Simultaneous localization and mapping for pedestrians using distortions of the local magnetic field intensity in large indoor environments," in Int. Conf. on Indoor Positioning and Indoor Navigation (IPIN), Montbeliard, France, Oct. 2013. [4] P. Robertson、M. Frassl、M. Angermann、M. Doniec、B. J. Julian、M. Garcia Puyol、M. Khider、M. Lichtenstern 和 L. Bruno，"在大型室内环境中利用本地磁场强度的扭曲实现行人的同时定位和绘图"，国际室内定位和室内导航会议（IPIN），法国蒙特贝利亚，2013 年 10 月。室内定位与室内导航会议（IPIN），法国蒙特贝利亚，2013 年 10 月。
Fig. 6. Estimation error for trajectory #2, i.e., the out-of-plane trajectory. 图 6.2 号轨迹（即平面外轨迹）的估计误差。
[5] M. Kok and A. Solin, "Scalable magnetic field SLAM in 3D using gaussian process maps," in International Conference on Information Fusion (FUSION), Cambridge, UK, Jul. 2018. [5] M. Kok 和 A. Solin：《使用高斯过程图的三维可扩展磁场 SLAM》，国际信息融合会议（FUSION），英国剑桥，2018 年 7 月。
[6] N. Wahlström, M. Kok, T. B. Schön, and F. Gustafsson, "Modeling magnetic fields using Gaussian processes," in IEEE Int. Conf on Acoustics, Speech and Signal Processing (ICASSP), Vancouver, BC, May 2013. [6] N. Wahlström、M. Kok、T. B. Schön 和 F. Gustafsson，"使用高斯过程建模磁场"，IEEE Int. Confon on Acoustics, Speech and Signal Processing (ICASSP)，温哥华，不列颠哥伦比亚省，2013 年 5 月。Conf on Acoustics, Speech and Signal Processing (ICASSP), Vancouver, BC, May 2013.
[7] D. Vissiere, A. Martin, and N. Petit, "Using distributed magnetometers to increase IMU-based velocity estimation into perturbed area," in 46th IEEE Conf. on Decision and Control, New Orleans, LA, Dec. 2007. [7] D. Vissiere、A. Martin 和 N. Petit，"使用分布式磁力计提高基于 IMU 的扰动区域速度估计"，第 46 届电气和电子工程师学会决策与控制会议，洛杉矶新奥尔良，2007 年 12 月。
[8] E. Dorveaux, T. Boudot, M. Hillion, and N. Petit, "Combining inertial measurements and distributed magnetometry for motion estimation," in Proc. of American Control Conf., San Francisco, CA, Jun. 2011. [8] E. Dorveaux、T. Boudot、M. Hillion 和 N. Petit，"结合惯性测量和分布式磁力测量进行运动估计"，美国控制会议论文集，加利福尼亚州旧金山，2011 年 6 月。
[9] E. Dorveaux and N. Petit, "Presentation of a magneto-inertial positioning system: navigating through magnetic disturbances," in Int. Conf. on Indoor Positioning and Indoor Navigation (IPIN), Guimaraes, Portugal, Sep. 2011. [9] E. Dorveaux 和 N. Petit，"磁惯性定位系统介绍：通过磁干扰导航"，国际室内定位与室内导航会议（IPIN），葡萄牙吉马良斯，2011 年 9 月。Indoor Positioning and Indoor Navigation (IPIN), Guimaraes, Portugal, Sep.
[10] C. I. Chesneau, M. Hillion, and C. Prieur, "Motion estimation of a rigid body with an EKF using magneto-inertial measurements," in Int. Conf. on Indoor Positioning and Indoor Navigation (IPIN), Alcala de Henares, Spain, Oct. 2016. [10] C. I. Chesneau、M. Hillion 和 C. Prieur，"使用磁惯性测量的 EKF 刚体运动估计"，国际室内定位与室内导航会议（IPIN），西班牙阿尔卡拉德埃纳雷斯，2016 年 10 月。室内定位与室内导航（IPIN）研讨会，西班牙阿尔卡拉德埃纳雷斯，2016 年 10 月。
[11] M. Zmitri, H. Fourati, and C. Prieur, "Improving inertial velocity estimation through magnetic field gradient-based extended Kalman filter," in Int. Conf. on Indoor Positioning and Indoor Navigation (IPIN), Pisa, Italy, Oct. 2019. [11] M. Zmitri, H. Fourati, and C. Prieur, "Improving inertial velocity estimation through magnetic field gradient-based extended Kalman filter," in Int.室内定位与室内导航会议（IPIN），意大利比萨，2019 年 10 月。
[12] M. Zmitri, H. Fourati, and C. Prieur, "Magnetic field gradient-based ekf for velocity estimation in indoor navigation," Sensors, vol. 20, no. 20, 2020. [12] M. Zmitri、H. Fourati 和 C. Prieur，"基于磁场梯度的用于室内导航速度估计的 EKF"，《传感器》，第 20 卷第 20 期，2020 年。
[13] _ , "Bilstm network-based extended kalman filter for magnetic field gradient aided indoor navigation," IEEE Sensors Journal, vol. 10.1109/JSEN.2021.3091862, 2021.
[14] I. Skog, G. Hendeby, and F. Gustafsson, "Magnetic odometry - a modelbased approach using a sensor array," in Int. Conf. on Information Fusion (FUSION), Cambridge, UK, Jul. 2018. [14] I. Skog, G. Hendeby, and F. Gustafsson, "Magnetic odometry - a modelbased approach using a sensor array," in Int.信息融合会议（FUSION），英国剑桥，2018 年 7 月。
Fig. 7. Estimation error for trajectory #3, i.e., the spiral trajectory. 图 7.3 号轨迹（即螺旋轨迹）的估计误差。
[15] A. Marhaban and M. Saripan, "Review of visual odometry: types, approaches, challenges, and applications." SpringerPlus, vol. 5, no. 1897, Oct. 2016 [15] A. Marhaban and M. Saripan, "Review of visual odometry: types, approaches, challenges, and applications." SpringerPlus, vol. 5, no.SpringerPlus，第5卷，第1897期，2016年10月
[16] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall, 1993. [16] S. M. Kay, Fundamentals of Statistical Signal Processing：Estimation Theory.Prentice Hall, 1993.
[17] L. Ljung, System Identification - Theory For the User, 2nd ed. Prentice-Hall, 1999. [17] L. Ljung，System Identification - Theory For the User，2nd ed. Prentice-Hall，1999。Prentice-Hall, 1999.
[18] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning. MIT Press, 2006. [18] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning.麻省理工学院出版社，2006 年。
Air is non-magnetic and thus has zero magnetization. 空气没有磁性，因此磁化率为零。
Data sheet available at: https://www.pnicorp.com/wp-content/uploads/ RM3100-Breakout-Board-Product-Sheet.pdf 数据表见： https://www.pnicorp.com/wp-content/uploads/ RM3100-Breakout-Board-Product-Sheet.pdf
