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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 ,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 收集测量集 。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 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 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 可用测量模型描述
where 其中
and