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Magnetic Odometry - A Model-Based Approach Using A Sensor Array
磁性测距--使用传感器阵列的基于模型的方法

Isaac Skog, Gustaf Hendeby, and Fredrik Gustafsson
艾萨克-斯科格、古斯塔夫-亨德比和弗雷德里克-古斯塔夫松Dept. of Electrical Engineering, Linköping University, Linköping, Sweden
瑞典林雪平，林雪平大学电子工程系e-mail: firstname.lastname@liu.se
电子邮件：firstname.lastname@liu.se

Abstract 摘要

A model-based method to perform odometry using an array of magnetometers that sense variations in a local magnetic field is presented. The method requires no prior knowledge of the magnetic field, nor does it compile any map of it. Assuming that the local variations in the magnetic field can be described by a curl and divergence free polynomial model, a maximum likelihood estimator is derived. To gain insight into the array design criteria and the achievable estimation performance, the identifiability conditions of the estimation problem are analyzed and the Cramér-Rao bound for the one-dimensional case is derived. The analysis shows that with a second-order model it is sufficient to have six magnetometer triads in a plane to obtain local identifiability. Further, the Cramér-Rao bound shows that the estimation error is inversely proportional to the ratio between the rate of change of the magnetic field and the noise variance, as well as the length scale of the array. The performance of the proposed estimator is evaluated using real-world data. The results show that, when there are sufficient variations in the magnetic field, the estimation error is of the order of a few percent of the displacement. The method also outperforms current state-of-theart method for magnetic odometry.
本文介绍了一种基于模型的方法，利用磁力计阵列感测当地磁场的变化来进行测距。该方法不需要事先了解磁场，也不需要绘制磁场图。假设磁场的局部变化可以用一个无卷曲和发散的多项式模型来描述，就可以得到一个最大似然估计器。为了深入了解阵列设计标准和可实现的估计性能，对估计问题的可识别性条件进行了分析，并推导出了一维情况下的 Cramér-Rao 约束。分析表明，在二阶模型中，一个平面上有六个磁强计三元组就足以获得局部可识别性。此外，Cramér-Rao 界值表明，估计误差与磁场变化率和噪声方差之间的比率以及阵列的长度尺度成反比。利用实际数据对所提出的估计器的性能进行了评估。结果表明，当磁场变化足够大时，估计误差仅为位移的百分之几。该方法还优于目前最先进的磁性里程测量方法。

I. INTRODUCTION I.引言

A model-based method to perform odometry using an array of magnetometers that sense variations in a local magnetic field is presented. In contrast to magnetic field based simultaneous localization and mapping (SLAM) and fingerprinting solutions, as represented by [1] and [2], the proposed method requires no prior knowledge of the magnetic field, nor does it compile any magnetic field map over time. Hence, the proposed method can e.g., be used to increase the positioning accuracy during the exploration phase of a magnetic SLAM process or in applications where the computational and memory resources prevent the construction and storage of a magnetic field map.
本文提出了一种基于模型的方法，利用磁力计阵列感知局部磁场的变化来进行测距。与[1]和[2]所代表的基于磁场的同步定位与绘图（SLAM）和指纹识别解决方案相比，所提出的方法不需要事先了解磁场，也不需要随时间推移编制任何磁场图。因此，在磁场 SLAM 过程的探索阶段，或在计算和内存资源无法构建和存储磁场图的应用中，建议的方法可用于提高定位精度。
The idea that the velocity of an object can be estimated by equipping it with a magnetometer array that observes the variations in the local magnetic field was introduced in [3]. The idea is based upon the differential equation
文献[3]中提出了一个想法，即通过在物体上安装磁力计阵列来观测当地磁场的变化，从而估算出物体的速度。这一想法基于微分方程
which relates the rate of change of the magnetic field to the rotation rate of the array (assumed to be measured by a gyroscope triad in [3]), the Jacobian of the magnetic field , and the velocity ; all quantities are expressed with respect to the array coordinate frame. With an assembly of spatially distributed magnetometers, also known as a magnetometer array, the Jacobian can be estimated
它将磁场 的变化率与阵列 的旋转速率（在 [3] 中假定由陀螺仪三元组测量）、磁场的雅各比 和速度 联系起来；所有量都是相对于阵列坐标系表示的。通过空间分布的磁强计组件（也称为磁强计阵列），可以估算出雅各布
Fig. 1. Sensor array made out of 32 MPU 9150 InvenSense sensor modules, each holding a magnetometer triad. Details regarding the array can be found at http://www.openshoe.org.
图 1.由 32 个 MPU 9150 InvenSense 传感器模块组成的传感器阵列，每个模块包含一个磁力计三元组。有关阵列的详细信息，请访问 http://www.openshoe.org。
and the differential equation solved. That is, the velocity can be estimated. Note that this is the velocity in the array frame and without further information regarding the orientation of the array it can only be used to determine the speed of the array in a fixed frame of reference. An example of a sensor array that can be used to perform magnetic odometry is shown in Fig. 1.
并求解微分方程。也就是说，可以估算出速度 。请注意，这是在阵列框架内的速度，如果没有关于阵列方位的进一步信息，只能用于确定阵列在固定参考框架内的速度。图 1 显示了一个可用于进行磁性里程测量的传感器阵列示例。
In paper [3] and in the subsequent work [4] and [5] by the same authors, as well as in the recent paper [6], the differential equation (1) is used to develop a speed aided inertial navigation system. The result is a positioning system with much slower error growth rate than a pure inertial navigation system; theoretically, the errors should grow linearly with time, instead of cubically. Indeed, the experimental results presented in [5] 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 a few percent of the distance traveled.
在论文[3]、同一作者的后续工作[4]和[5]以及最近的论文[6]中，微分方程 (1) 被用于开发速度辅助惯性导航系统。其结果是，与纯惯性导航系统相比，定位系统的误差增长率要慢得多；理论上，误差应随时间线性增长，而不是立方增长。事实上，文献[5]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行驶距离的百分之几。
In this paper, the velocity of the array is not estimated by directly solving (1). Instead the velocity estimation is viewed as a model parameter estimation problem where a signal model that has the velocity as a free parameter is fitted to the observed data. This allows the application of estimation theoretical tools to analyze the properties of the magnetic odometry problem and to derive various estimators for the displacement. Further, the proposed method makes it straightforward to use higher order models to describe the magnetic field variations, which enable a more accurate displacement estimation. Moreover, in the proposed method both the translational and rotational
在本文中，阵列的速度不是通过直接求解（1）来估算的。相反，速度估算被视为一个模型参数估计问题，即把速度作为自由参数的信号模型与观测数据进行拟合。这样就可以应用估算理论工具来分析磁性里程测量问题的特性，并推导出各种位移估算器。此外，建议的方法可以直接使用高阶模型来描述磁场变化，从而实现更精确的位移估计。此外，在建议的方法中，平移和旋转
Fig. 2. Illustration (in one dimension) of the idea behind the modelbased approach to displacement (speed) estimation using variations in the magnetic field. Given the measurements and from the magnetometer sensor array, a local model for the magnetic field is constructed and the displacement that best fits this model is estimated.
图 2.利用磁场变化进行位移（速度）估算的基于模型方法背后的理念说明（一维）。根据磁强计传感器阵列的测量结果 和 ，构建磁场的局部模型，并估算出最适合该模型的位移 。
motion of the array can be estimated without any gyroscopes, which makes it possible to perform dead-reckoning using only an array of magnetometers.
在没有陀螺仪的情况下也能估算出阵列的运动，因此只需使用磁力计阵列就能进行死区重定位。
The outline of the paper is as follows. First, a signal model for the magnetic field measurements is presented in Section II. Secondly, in Section III, a maximum likelihood estimator for the displacement of the array is presented. The identifiability conditions and Cramér-Rao for the estimation problem are also analyzed. Thirdly, in Section IV, the performance of the estimator is experimentally evaluated and compared to the case when the displacement is estimated by directly solving (1). Finally, in Section V, the results are summarized and conclusions drawn.
论文大纲如下。首先，第二节介绍了磁场测量的信号模型。其次，在第三节中介绍了阵列位移的最大似然估计器。此外，还分析了估计问题的可识别性条件和克拉梅尔-拉奥（Cramér-Rao）。第三，在第四节中，对估计器的性能进行了实验评估，并与直接求解（1）估计位移的情况进行了比较。最后，在第五部分，对结果进行总结并得出结论。

II. Signal Model II.信号模型

In this section, a model for the measurements obtained from a magnetometer array moving through a spatially varying magnetic field, as illustrated in Fig. 2, is derived.
如图 2 所示，本节将推导出磁力计阵列在空间变化磁场中移动时的测量模型。

A. Prerequisite A.先决条件

With reference to Fig. 2, assume a sensor array consisting of magnetometer triads that collects the measurement sets and , at time instant and , respectively. Here denotes the measurement at time instant of the magnetometer triad. Further, assume that during the time interval the array has undergone the translation and the rotation described by the Euler angles .
参照图 2，假设传感器阵列由 磁强计三元组组成，分别在时间瞬 间 和 收集测量集 和 。这里 表示 磁强计三元组在时间瞬间 的测量值。此外，假设在 时间间隔内，阵列经历了平移 和欧拉角描述的旋转 。
Next, assume that within the volume , centered at the origin of the array at time , the magnetic field can be described by the model . Here and denote the position within the volume and the model parameters, respectively. Further, assume that the displacement is such that for it holds that . Here denotes the position of the sensors in the array and denotes the rotation matrix parameterized by the Euler angles . The measurements in the two sets and can then be modeled as
接下来，假定在以时间 的阵列原点为中心的体积 内，磁场可以用模型 来描述。这里 和 分别表示体积 内的位置和模型参数。此外，假设位移是这样的：对于 ， 。 表示 传感器在阵列中的位置， 表示由欧拉角参数化的旋转矩阵 。这样，两组 和 中的测量值就可以建模为
Here denotes the measurement error of the sensor triad.
表示 传感器三元组的测量误差。
If the array moves in a static magnetic field with no free current then, according to Maxwell's equations, the field should be both curl and divergence free [7]. Hence, the magnetic field model per design, or via the choice of model parameter values, should satisfy the conditions
如果阵列在没有自由电流的静态磁场中运动，那么根据麦克斯韦方程，磁场应该是无卷曲和发散的[7]。因此，根据设计或通过选择模型参数值，磁场模型 应满足以下条件
for all . Next, a polynomial model will be designed that fulfills these conditions.
对于所有 。接下来，我们将设计一个满足这些条件的多项式模型。

B. Polynomial Magnetic Field Model
B.多项式磁场模型

One way to create an order polynomial model of a curl free field is to describe the scalar potential as an order polynomial and then define the magnetic field as the gradient of the potential. That is, the scalar potential is modelled by the polynomial
创建 无卷积场的多项式模型的一种方法是将标量电势 描述为 次多项式，然后将磁场定义为电势梯度。也就是说，标量势的模型是多项式
where is a vector whose elements are given by the product for , subject to . Further, is an arbitrary constant and the model parameters make up the polynomial coefficients. The number of model parameters is in this case . The magnetic field is then given by
其中， 是一个向量，其元素由 的乘积 给出，受 约束。此外， 是一个任意常数，模型参数 构成多项式系数。在这种情况下，模型参数的数量为 。磁场的计算公式为
where . By design the magnetic field described by the model in (5) is curl free ; hence requirement (3a) is fulfilled.
其中 。根据设计，(5) 中模型描述的磁场是无卷曲的 ；因此满足了要求 (3a)。
To ensure that the magnetic field is divergence free, the parameters must be chosen such that
为确保磁场不发散，参数 的选择必须满足以下条件
Here is an order polynomial in three variables. Therefore, for the equality in (6) to hold the coefficients for each power in the sum of the three polynomials must sum to zero. Hence, the constraint in (6) can equivalently be written as a system of equations. Let denote this system of equations, then a curl and divergence free order polynomial model for the magnetic field is given by
这里， 是三个变量的 阶多项式。因此，要使 (6) 中的等式成立， ，三个多项式之和中每个幂的系数之和必须为零。因此，(6) 中的约束条件可以等价地写成一个 等式系统。让 表示这个方程组，那么无卷曲和发散的 阶多项式磁场模型如下所示
This model can be reparameterized as a lower order model by introducing the matrix whose columns
通过引入矩阵 ，可将该模型重新参数化为低阶模型，其列为
spans the null space of , and then setting , where . (The null space matrix can be found via a QRfactorization of the matrix , see e.g., [8].) The curl and divergence free order polynomial model for the magnetic field, expressed in terms of the new parameters , is then defined by
跨越 的空空间，然后设置 ，其中 。（空空间矩阵可通过矩阵 的 QR 因子化找到，参见 [8] 等）。无卷曲和发散的 阶多项式磁场模型，用新参数 表示，定义如下

C. Measurement Model C.测量模型

Given the magnetic field model in (8), the measurement sets and can be described by the measurement model
鉴于 (8) 中的磁场模型，测量集 和 可用测量模型描述
where 其中
and
Here, or depending upon whether the rotation is assumed to be known or not. Assuming that the magnetometers have been calibrated, the measurement error can be assumed to be zero-mean Gaussian distributed with covariance matrix . Here denotes the variance of the measurement noise.
这里的 或 取决于是否假定旋转 是已知的。假定磁强计已经校准，测量误差可以假定为零均值高斯分布，协方差矩阵为 。 表示测量噪声的方差。

D. Identifiability Analysis
D.可识别性分析

The measurement model in (9a) has free parameters; recall, was the order of the polynomial model. Thus, a necessary condition for the parameters to be uniquely identifiable is that the number of magnetometer triads
(9a) 中的测量模型有 个自由参数；记得 是多项式模型的阶数。因此，参数唯一可识别的必要条件是磁强计三元组的数量
Hence, for a second order polynomial model, i.e., , the array must consist of at least three magnetometer triads if the rotation is assumed to be known. If the rotation is unknown, at least four magnetometer triads are needed.
因此，对于二阶多项式模型，即 ，如果假定旋转是已知的，阵列必须至少由三个磁力计三元组组成。如果旋转情况未知，则至少需要四个磁强计三元组。
A nonlinear model of the kind in (9a) is said to be locally identifiable if the matrix
如果矩阵
has full rank [9]. Due to the triangular structure of the matrix, . Next, utilizing this structure, a sufficient condition, with respect to the array geometry, for the problem to be locally identifiable in the case when will be studied.
具有全秩 [9]。由于矩阵的三角形结构， 。接下来，我们将利用这一结构，研究在 的情况下，与阵列几何有关的问题局部可识别的充分条件。
It is possible to show, via straightforward but lengthy calculations, that with six magnetometer triads placed in the planar grid
通过简单但冗长的计算，我们可以得出这样的结论：在平面网格中放置六个磁强计三元组后
where , the matrix has full rank. This implies that the coefficients , which describe the magnetic field, can be identified from the observations alone. By not restricting the sensors to be in a plane, it is possible to reduce the number of sensor triads needed to five by, e.g., changing the location of the fifth triad to and removing the sixth triad. It is worth noting that in [10] it is shown that for a linear model, i.e., , it sufficient to have three non-collinear magnetometer triads to estimate the model parameters .
其中 ，矩阵 具有全秩。这意味着描述磁场的系数 可以仅从观测结果 中确定。通过不限制传感器必须在一个平面内，可以将所需的传感器三元组数量减少到五个，例如，将第五个三元组的位置改为 ，并去掉第六个三元组。值得注意的是，文献[10]表明，对于线性模型，即 ，只需三个非共线磁力计三元组就能估算出模型参数 。
Regarding the identifiability of , it is possible to show that, given the parameters describing the field , the displacement of a single magnetometer triad is identifiable if the magnetic field is sufficiently exciting, i.e., any small change in location causes a change in the magnetic field. Since the orientation of a rigid object can be determined from the position of three non-collinear points on the object, this also implies that the orientation of the array can be determined from the measurements . Hence, with a second order model, it is sufficient to have an array with six magnetometer triads in a planar geometry to obtain local identifiablity for all parameters in the signal model.
关于 的可识别性，可以证明，给定描述磁场的参数 ，如果磁场足够激 励，即位置的任何微小变化都会引起磁场的变化，则单个磁强计三元组的位移是可识别的。由于刚性物体的方位可通过物体上三个非共线点的位置确定，这也意味着阵列的方位可通过测量结果确定 。因此，利用二阶模型，只需在平面几何中设置一个包含六个磁力计三元组的阵列，即可获得信号模型中所有参数的局部可识别性。

III. Estimator and Performance Bound
III.估算器和性能边界

For a separable linear model such as the one in (9a) with Gaussian distributed measurement noise, the maximum likelihood estimator and the Cramér-Rao bound for the nonlinear parameters are readily available; the linear parameters are nuisance parameters.
对于像 (9a) 中这种具有高斯分布测量噪声的可分离线性模型，非线性参数 的最大似然估计值和克拉梅尔-拉奥（Cramér-Rao）约束是现成的；线性参数则是干扰参数。

A. Maximum Likelihood Estimator
A.最大似然估计法

The maximum likelihood estimator is given by [11] as
最大似然估计值由 [11] 求得，即
where the projection matrix . The maximization in (13) can be solved using standard numerical techniques such as the Gauss-Newton or Levenberg-Marquardt method. Efficient ways to calculate the Jacobian matrix needed in the implementation of these methods can be found in [12].
其中投影矩阵为 。(13) 中的最大化可以用标准数值技术求解，如高斯-牛顿法或 Levenberg-Marquardt 法。计算这些方法所需的雅各布矩阵的有效方法可参见 [12]。

B. Cramér-Rao Bound B.克拉梅尔-拉奥约束

The Cramér-Rao bound is given by [11] as
Cramér-Rao 界值由 [11] 得出如下
where the Fisher information matrix is
其中费雪信息矩阵为
and . As the bound depends on the true model parameters, it cannot be evaluated directly from a sequence of real-world measurements. However, the bound can be approximated by using the estimated parameters, and thereby used as an indicator of the accuracy of the displacement estimate. Further, the bound can be used to gain insight into the basic properties of the displacement estimation problem. To do so, the bound for the one dimensional case will be analyzed next.
和 。由于该界限取决于真实的模型参数，因此无法直接根据一系列实际测量结果进行评估。不过，可以通过使用估计参数来近似计算该界限，并以此作为位移估计准确性的指标。此外，该界限还可用于深入了解位移估计问题的基本特性。为此，接下来将分析一维情况下的边界。
Consider an array consisting of single axis magnetometers equidistantly mounted in a straight line; refer to Fig. 2 for an illustration. Assume that the array moves along a straight line and is aligned with the direction of motion. Further, assume that the magnetic field locally can be modelled by the linear model, i.e., . Then the Cramér-Rao bound in (14) simplifies to
考虑一个由 单轴磁力计组成的阵列，这些磁力计等距安装在一条直线上；插图见图 2。假设阵列沿直线运动，并与运动方向保持一致。此外，假设局部磁场可用线性模型建模，即 。那么 (14) 中的 Cramér-Rao 约束简化为
As can be seen, the bound is proportional to the ratio between the noise variance and the rate of change of the field. Hence, if the field does not change, the accuracy degrades quickly.
可以看出，界限与噪声方差和场变化率之间的比率成正比。因此，如果场没有变化，精度就会迅速下降。
From a positioning perspective, it is generally more interesting to talk about the displacement estimation error in terms of percentages of the traveled distance than in terms of the variance of the distance estimate. Normalizing the variance with the squared displacement and simplifying yields
从定位的角度来看，用已走距离的百分比来表示位移估计误差通常比用距离估计方差来表示更有意思。用位移平方对方差进行归一化处理并简化后得出
That is, for long displacements the error in terms of a percentage of the traveled distance is inversely proportional to the "signal-to-noise" ratio and the length of the array. Hence, to optimize the estimation accuracy the size of array should be as large as possible, but still small enough so that the model of the magnetic field is valid.
也就是说，对于长位移，以移动距离百分比表示的误差与 "信噪比 "和阵列长度成反比。因此，为了优化估计精度，阵列的尺寸应尽可能大，但仍应足够小，以便磁场模型有效。

IV. EXPERIMENT AND RESULTS
IV.实验和结果

An example of the variations in the magnetic field inside an office building is shown in Fig. 3. Visual inspection shows that on a length scale of approximately the field variations can be well approximated using a second or third order polynomial model. This indicates that an array in which the sensors are distributed to cover a square with a side of a few decimeters should be used if aiming to obtain magnetic odometry using the proposed method. Unfortunately, we currently only have
图 3 显示了办公楼内磁场变化的一个例子。目测结果表明，在大约 的长度范围内，磁场变化可以用二阶或三阶多项式模型很好地近似。这表明，如果要使用建议的方法获得磁性里程测量，应使用传感器分布在边长为几分米的正方形上的阵列。遗憾的是，我们目前只有
Fig. 3. Example of the variations in the local magnetic field inside an office building. The mean value of each field components has been removed to better visualize the variations in the field.
图 3.办公楼内局部磁场变化示例。各磁场成分的平均值已被去除，以便更好地显示磁场的变化。
access to the array depicted in Fig. 1, in which the sensors are distributed over a square with a side of approximately .
在图 1 所示的阵列中，传感器分布在边长约为 的正方形上。
The following experiment was conducted with an artificially created magnetic field. The array was mounted on a linear actuator that, in steps of , was used to move the array along a straight line. At each step a set of magnetometer readings was collected, where each was calculated as the average value of 800 time samples of magnetometer data; the resulting noise standard deviation . To get faster variations in the magnetic field, three small magnets were placed at random along the path of the linear actuator. The resulting magnetic field as a function of the position of the array is depicted in Fig. 4.
下面的实验是在人工制造的磁场中进行的。阵列安装在一个线性推杆上，推杆以 的步长沿直线移动阵列。在每一步 ，收集一组磁强计读数 ，其中每个 都是根据 800 个磁强计数据时间样本的平均值计算得出的；由此得出的噪声标准偏差 。为了获得更快的磁场变化，沿着线性致动器的路径随机放置了三块小磁铁。由此产生的磁场与阵列位置的函数关系如图 4 所示。
The experiment was repeated five times. The data from each experiment was divided into two parts, each part containing the data from the sixteen sensors on the top and the bottom of the array, respectively. That is, from each experiment two data sets containing data from two planar arrays were obtained. The resulting ten data sets where subsequently processed using the proposed estimator together with a second order model for the magnetic field variations; the rotation was assumed to be known. The mean and standard deviation of the individual displacement estimation errors as a function of the position of the array was then calculated. The results are shown Fig. 5. Also shown is the Cramér-Rao bound (calculated using the estimated parameter values) in terms of the values. Further, for comparison, the estimation accuracy obtained when directly solving (1) using Euler's method was also evaluated. That is, the displacement was estimated as
实验重复了五次。每次实验的数据分为两部分，每部分分别包含阵列顶部和底部 16 个传感器的数据。也就是说，每次实验都能得到两个数据集，包含来自两个平面阵列的数据。随后，利用建议的估算器和磁场变化二阶模型对得到的十组数据进行处理；假定旋转 是已知的。然后计算了作为阵列位置函数的各个位移估计误差的平均值和标准偏差。结果如图 5 所示。图 5 还显示了 值的 Cramér-Rao 约束（使用估计参数值计算）。此外，为了进行比较，还评估了使用欧拉法直接求解 (1) 时获得的估计精度。也就是说，位移的估计值为
where the Jacobian of the field was estimated using a second order polynomial model. The result is shown by the dashed black line in Fig. 5.
其中，场 的雅各布因子是用二阶多项式模型估算的。结果如图 5 中黑色虚线所示。
The results show that when there are large variations in the magnetic field the displacement can be estimated using the proposed method with sub-millimeter accuracy. This corresponds to a few percent of the true displacement of . When the magnetic field flattens out, i.e., the field starts to lose sufficient excitation, towards the end of the trajectory the performance of the proposed estimator declines quickly and the estimation error is in the order of
结果表明，当磁场发生较大变化时，使用所提出的方法可以以亚毫米级的精度估算出位移。这相当于 真实位移的百分之几。当磁场趋于平缓，即磁场开始失去足够的激励时，在轨迹末端，所提出的估算方法的性能迅速下降，估算误差在以下数量级
Fig. 4. Field measured by the sensor array during the experiment.
图 4.实验期间传感器阵列测量到的场强。
Fig. 5. Mean and standard deviation (in terms of the bound) of the estimation error for the proposed method. Also shown is the mean estimation error of the direct method, which is based upon solving differential equation (1). The Cramér-Rao bound for the estimation error is also displayed.
图 5.拟议方法估计误差的平均值和标准偏差（以 bound 表示）。图中还显示了直接方法的平均估计误差，该方法基于微分方程 (1) 的求解。还显示了估计误差的 Cramér-Rao 边界。
of the true displacement. Further, the results show that the estimation accuracy obtained when using the estimator in (18) is substandard. Moreover, it can be seen that the estimation accuracy predicted by the Cramér-Rao bound is in agreement with the empirically calculated estimation.
的真实位移。此外，结果表明，使用（18）中的估计器得到的估计精度不达标。此外，可以看出克拉梅尔-拉奥约束预测的估计精度与经验计算的估计精度是一致的。

V. Summary, Discussion, and Conclusions
V.总结、讨论和结论

A model-based approach to magnetic odometry, i.e., estimation of the displacement of a magnetometer array moving through a spatially varying magnetic field, has been presented. Further, based upon a curl and divergence free polynomial model for the magnetic field, a maximum likelihood estimator for the displacement has been derived. Moreover, an identifiability analysis of the signal model when using a second order polynomial shows that it is sufficient to have magnetometer triads mounted in a plane to identify all the model parameters. This greatly simplifies the practical design of the array, as it can be constructed on a single printed circuit board.
介绍了一种基于模型的磁性里程测量方法，即估算磁力计阵列在空间变化磁场中的位移。此外，根据磁场的无卷曲和发散多项式模型，还推导出了位移的最大似然估计器。此外，在使用二阶多项式时，对信号模型的可识别性分析表明，只需将磁力计三元组安装在一个平面上，即可识别所有模型参数。这大大简化了阵列的实际设计，因为它可以安装在一块印刷电路板上。
The performance of the estimator has been experimentally evaluated using a miniaturized sensor array with the form factor and an artificially created magnetic field with a rate of change of up to . The results show that where there are sufficient variations in the magnetic field the displacement can be estimated with an error of a few percent. Translated into a typical indoor environment where magnetic field variations on the order of a few micro-Tesla per decimeter are common, the results indicate that an odometry accuracy of a few percentage of the traveled distance should be achievable when using an array built out of low-noise magnetometer sensors and which has a length scale of 1-3 decimeters. The results also shows that the proposed method clearly outperforms the accuracy of the current standard method, which is based upon solving (1) directly.
使用外形尺寸为 的微型传感器阵列和变化率高达 的人工磁场，对估算器的性能进行了实验评估。结果表明，在磁场变化足够大的情况下，位移估算误差仅为百分之几。在典型的室内环境中，每平方分米的磁场变化通常为几个微特斯拉，结果表明，如果使用由低噪声磁强计传感器组成的阵列，其长度范围为 1-3 平方分米，则可以实现占行驶距离几个百分点的测距精度。结果还表明，拟议方法的精度明显优于直接求解 (1) 的现行标准方法。
Another benefit of the proposed method is that the formulation fits well into a Bayesian filtering framework. This makes it easy to combine the magnetic odometry measurements with, e.g., inertial measurements, to create a system that would be able to provide accurate localization in a wide variety of indoor environments.
所提方法的另一个优点是，其公式非常适合贝叶斯滤波框架。这样就可以很容易地将磁性里程测量与惯性测量等结合起来，创建一个能够在各种室内环境中提供精确定位的系统。
Future work will therefore be focused on evaluating the proposed method using a full size sensor array that is observing only the natural variations in magnetic field. Further, as the proposed method can also be used with other magnetic field models than the one presented, the method will be tested using, e.g., the curl and divergence Gaussian process model in [7]. Moreover, as the identifiability analysis shows that the rotation of the array can also be determined, it will be investigated how well this can be estimated.
因此，未来的工作将侧重于使用仅观测磁场自然变化的全尺寸传感器阵列来评估所提出的方法。此外，由于所提出的方法还可用于其他磁场模型，因此将使用 [7] 中的卷曲和发散高斯过程模型等对该方法进行测试。此外，由于可识别性分析表明阵列的旋转也可以确定，因此将研究其估算效果如何。

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