MAINS: A Magnetic Field Aided Inertial Navigation System for Indoor Positioning
MAINS：用于室内定位的磁场辅助惯性导航系统

Index Terms-indoor positioning, magnetic field, error-state Kalman filter, aided navigation
索引术语--室内定位、磁场、误差状态卡尔曼滤波器、辅助导航

The outdoor magnetic field is omnipresent, relatively stable, and almost homogenous. Due to these properties, it has been used in navigation for a long time, where the magnetic field is mainly used as a heading reference to correct errors from integrating noisy gyroscope measurements [1]. However, those techniques cannot be applied without modifications for indoor applications because the indoor magnetic field is not homogenous. An example of the variations in the magnitude of the magnetic field inside a building is shown in Fig. 1. The correlation between the position and the magnetic field can be seen. Therefore, the inhomogeneous magnetic field can be used as a reliable source for localization in Global Navigation Satellite System (GNSS) denied environments, such as indoors or underwater [2]. Indeed, recent years have witnessed many
室外磁场无处不在、相对稳定且几乎均匀。基于这些特性，磁场在导航领域的应用由来已久，磁场主要用作航向参考，以校正噪声陀螺仪测量的积分误差[1]。然而，由于室内磁场并不均匀，因此这些技术在室内应用时不能不做修改。图 1 举例说明了建筑物内磁场大小的变化。可以看出位置和磁场之间的相关性。因此，在室内或水下等拒绝全球导航卫星系统（GNSS）的环境中，不均匀磁场可用作定位的可靠来源[2]。事实上，近年来许多

This work has been funded by the Swedish Research Council (Vetenskapsrådet) project 2020-04253 "Tensor-field based localization".
这项工作得到了瑞典研究理事会（Vetenskapsrådet）2020-04253 "基于张量场的定位 "项目的资助。

Chuan Huang and Gustaf Hendeby are with Dept. of Electrical Engineering, Linköping University, (e-mail: chuan.huang @liu.se; gustaf.hendeby@liu.se).
Chuan Huang 和 Gustaf Hendeby 现就职于林雪平大学电子工程系（电子邮箱：chuan.huang @liu.se; gustaf.hendeby@liu.se）。

Hassen Fourati and Christophe Prieur are with the GIPSA-Lab, CNRS, Inria, Grenoble INP, University Grenoble Alpes, 38000 Grenoble, France (email: hassen.fourati@gipsa-lab.fr; christophe.prieur@gipsa-lab.fr)
Hassen Fourati和Christophe Prieur是法国格勒诺布尔阿尔卑斯大学格勒诺布尔国家科学研究中心（CNRS）GIPSA实验室的成员（电子邮箱：hassen.fourati@gipsa-lab.fr; christophe.prieur@gipsa-lab.fr）。

Isaac Skog is with Dept. of Electrical Engineering, Uppsala University, Uppsala, Sweden, and Dept. of Electrical Engineering Linköping University, Linköping, Sweden, and the Div. of Underwater Technology, Swedish Defence Research Agency (FOI), Kista, Sweden (e-mail: isaac.skog@ angstrom.uu.se).
Isaac Skog 现为瑞典乌普萨拉市乌普萨拉大学电气工程系、瑞典林雪平市林雪平大学电气工程系以及瑞典基斯塔市瑞典国防研究局 (FOI) 水下技术部人员（电子邮件：isaac.skog@ angstrom.uu.se）。

Fig. 1. Illustration of the magnetic-field magnitude variations inside a building. The field near the floor was measured with a magnetometer, 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。地板附近的磁场是用磁力计测量的，磁力计的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。

successful applications in magnetic-field-based positioning, among which the magnetic-field-based SLAM has turned out to be a promising approach [3]-[5]. It enables the user to construct a magnetic field map while navigating and having drift-free positioning, provided revisiting the same region is possible. However, this technology relies heavily on the precision of the used odometric information. Otherwise, the position drift can be significant, making it challenging to reliably recognize the visited place and complete "loop closure" [6]. For instance, when an inertial navigation system with lowcost inertial sensors is used for doing the odometry, the error growth rate is typically on the order of 10 meters per minute [7], which means the system needs to revisit the same region within a minute to prevent the position drift from becoming too large to complete loop closure. Therefore, the permissible 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 odometry techniques that have a low position drift rate.
在基于磁场的定位领域，有许多成功的应用，其中基于磁场的 SLAM 是一种很有前途的方法 [3]-[5]。它能让用户在导航时构建磁场图，并在可能重访同一区域的情况下实现无漂移定位。然而，这项技术在很大程度上依赖于所使用的里程测量信息的精确性。否则，位置漂移可能会很大，从而难以可靠地识别到访地点并完成 "闭环"[6]。例如，当使用带有低成本惯性传感器的惯性导航系统进行里程测量时，误差增长率通常为每分钟 10 米[7]，这意味着系统需要在一分钟内重访同一区域，以防止位置漂移过大而无法完成闭环。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，我们需要具有低位置漂移率的强大里程测量技术。

With this limitation in mind, the concept of combining inertial measurements and distributed magnetometry has been proposed and realized [8]. An array of magnetometers enables
考虑到这一局限性，人们提出并实现了将惯性测量与分布式磁强计相结合的概念[8]。磁强计阵列可以
the calculation of the gradients of the magnetic field, from which the body velocity can be estimated. On the other hand, the authors in [2] adopted a model-based approach, treating pose changes in subsequent timestamps as the parameters of the model to be estimated. The estimated pose change from the magnetic field measurements can be used to aid an inertial navigation system to reduce the position growth rate. In this paper, we present a method for tightly integrated magneticfield-aided inertial navigation. The resulting navigation system has, compared to a pure inertial navigation system, a 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-fieldbased SLAM systems.
通过计算磁场梯度，可以估算出身体速度。另一方面，[2]中的作者采用了基于模型的方法，将后续时间戳中的姿势变化视为待估算模型的参数。从磁场测量中估算出的姿势变化可用于帮助惯性导航系统降低位置增长率。在本文中，我们介绍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。

Numerous methods for magnetic-field-based indoor positioning and navigation have been proposed. Current solutions are roughly categorized into three types: fingerprint-based methods, magnetic field SLAM, and magnetic field odometry. Fingerprint-based methods [9]-[11] generally rely on a premeasured magnetic field map to work, which greatly limits its usability. Therefore, we mainly discuss the latter two types, which are not constrained by a prior map.
人们提出了许多基于磁场的室内定位和导航方法。目前的解决方案大致分为三类：基于指纹的方法、磁场 SLAM 和磁场里程计。基于指纹的方法[9]-[11]一般依赖于预先测量的磁场地图，这大大限制了其可用性。因此，我们主要讨论后两类方法，它们不受事先地图的限制。

One of the first 2D magnetic field SLAM methods was proposed in [12], where the magnetic field strength map was constructed in hierarchical hexagonal tiles of different bin sizes. To generalize the motions to and cope with the complexity of representing the map, the authors in [3] selected a reduced-rank Gaussian process to represent the magnetic field map and used a Rao-blackwellized particle filter to estimate both the map and the location of the system. Later, in [5], the authors successfully achieved drift-free positioning using foot-mounted sensors with similar techniques. To achieve real-time processing, they used an EKF filter to update the magnetic field map based on its gradients [4]. Contrary to the "stochastic" loop closure mechanism in [3]-[5], where it reduces the uncertainty of the map in the revisited region, a "deterministic" one was employed in [13], where attitudeinvariant magnetic field information was used to detect loop closure and constraints on position estimates were formed. The aforementioned magnetic field SLAM solutions can achieve drift-free long-term large-scale positioning as long as frequent loop closure is possible.
最早的二维磁场 SLAM 方法之一是在 [12] 中提出的，在该方法中，磁场强度图是以不同粒度的分层六边形瓦片构建的。为了将运动推广到 并应对表示磁场图的复杂性，[3] 中的作者选择了秩降低的高斯过程来表示磁场图，并使用 Rao-blackwellized 粒子滤波器来估计磁场图和系统的位置。随后，在 [5] 中，作者使用类似的技术成功地利用脚踏式传感器实现了无漂移定位。为了实现实时处理，他们使用 EKF 滤波器根据梯度更新磁场图[4]。与[3]-[5]中的 "随机 "闭环机制相反，[13]中采用了 "确定 "闭环机制，即利用姿态不变的磁场信息来检测闭环，并形成位置估计的约束条件。只要能够频繁闭环，上述磁场 SLAM 解决方案就能实现无漂移的长期大尺度定位。

Magnetic field odometry, on the other hand, is a lightweight solution to provide odometric information with a magnetometer array. It does not construct a magnetic field map but provides odometric information based on local magnetic field properties. The seminal work [8] derived an equation that relates the body velocity and the gradient of the magnetic field, which can be calculated from the measurements from a set of spatially distributed magnetometers. Later, in [14], [15], the authors proposed an observer to estimate body velocity, proved its convergence, and showcased its usability for indoor localization. In subsequent works [16], [17], the authors incorporated inertial sensor biases and magnetic disturbance in the model and designed a filter based on the error-state Kalman filter (ESKF). To address the issue of the noisy magnetic field gradient, the authors in [18], [19] derived a differential equation for high-order derivatives of the magnetic field. They then developed a filtering algorithm comprising a primary filter which is used to estimate the gradient of the field. Later, the same authors [20], [21] proposed an AI-based solution where a Long Short-Term Memory (LSTM) model is used to create a pseudo-measurement of the inertial velocity of the target. The pseudo-measurement is used to handle adversarial situations where the states' observability is affected by the low gradient of the field and/or the target's velocity close to zero. The methods developed have demonstrated promising prospects in magnetic field odometry. However, they are susceptible to noise when computing the gradient, and observability issues arise from a weak magnetic field gradient or low target speed. To address this problem, [22] matches the waveforms from a pair of magnetometers in a sliding time window to reduce the influence of the temporary disappearance of the magnetic field gradient. Experiments show that the proposed method in [22] performs similarly to wheel odometry in magnetic-rich environments.
另一方面，磁场里程测量法是一种利用磁力计阵列提供里程测量信息的轻量级解决方案。它不构建磁场地图，而是根据当地磁场特性提供测距信息。开创性工作[8]推导出了一个人体速度与磁场梯度之间的关系式，该方程可通过一组空间分布式磁强计的测量值计算得出。随后，在 [14] 和 [15] 中，作者提出了一种估计人体速度的观测器，证明了其收敛性，并展示了其在室内定位中的可用性。在随后的作品[16]、[17]中，作者将惯性传感器偏差和磁干扰纳入模型，并设计了一个基于误差状态卡尔曼滤波器（ESKF）的滤波器。为了解决磁场梯度噪声问题，[18]、[19] 中的作者推导出了磁场高阶导数的微分方程。然后，他们开发了一种滤波算法，其中包括一个用于估计磁场梯度的初级滤波器。后来，同一作者[20]、[21]提出了一种基于人工智能的解决方案，即使用长短期记忆（LSTM）模型创建目标惯性速度的伪测量值。伪测量用于处理状态的可观测性受到场的低梯度和/或目标速度接近于零的影响的不利情况。所开发的方法在磁场里程测量中展现出了广阔的前景。然而，这些方法在计算梯度时容易受到噪声的影响，并且在磁场梯度较弱或目标速度较低时也会出现可观测性问题。为了解决这个问题，[22] 在一个滑动时间窗口中匹配一对磁力计的波形，以减少磁场梯度暂时消失的影响。实验表明，[22] 提出的方法在磁场丰富的环境中与车轮里程测量法的性能类似。

In a more recent work [23], a polynomial model was proposed to describe the local magnetic field and to develop a magnetic field odometry method. Presented experiential results showed that the model-based odometry approach can give a higher accuracy at low signal-to-noise ratios, compared to approaches in [8]. The model-based odometry approach was further explored in [2], where it was used to estimate both the translation and orientation change of the array. In the subsequent work [24], the authors included the magnetic field model in the state-space description of an INS system and developed a tightly-integrated magnetic-field-aided INS. Simulation results showed that it has a much slower drift rate than stand-alone INS.
最近的一项研究[23]提出了一个多项式模型来描述局部磁场，并开发了一种磁场测距方法。实验结果表明，与[8]中的方法相比，基于模型的里程测量方法在信噪比较低的情况下精度更高。文献[2]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。在随后的工作[24]中，作者将磁场模型纳入了 INS 系统的状态空间描述中，并开发了一种紧密集成的磁场辅助 INS。仿真结果表明，它的漂移率比独立的 INS 慢得多。

The contribution of this work is two-fold. Firstly, it extends the groundwork laid out in [24] by providing a thorough derivation and a comprehensive exposition of the proposed algorithm. Additionally, the performance of the proposed algorithm using real-world data was assessed and benchmarked against the state-of-the-art. Secondly, we have made the datasets used in our experiments and the source code for the proposed algorithm MAINS publicly available, in the hope that it will facilitate further research within the area of magnetic-field-based positioning. Both the datasets and the source code are available at https://github.com/Huang-Chuan/ MAINSvsMAGEKF.
这项工作有两方面的贡献。首先，它扩展了 [24] 中的基础工作，对所提出的算法进行了彻底的推导和全面的阐述。此外，我们还利用真实世界的数据对所提算法的性能进行了评估，并与最先进的算法进行了比较。其次，我们公开了实验中使用的数据集和拟议算法 MAINS 的源代码，希望能促进基于磁场定位领域的进一步研究。数据集和源代码均可在 https://github.com/Huang-Chuan/ MAINSvsMAGEKF 上获取。

Consider the problem of estimating the position and orientation of the sensor platform in Fig. 2, which consists of an inertial measurement unit (IMU) and an array of 30 magnetometers. To that end, a state-space model will be presented to realize a tightly-integrated magnetic-field aided inertial navigation system (INS).
图 2 中的传感器平台由一个惯性测量单元（IMU）和一个由 30 个磁力计组成的阵列组成。为此，将提出一个状态空间模型，以实现紧密集成的磁场辅助惯性导航系统（INS）。

Fig. 2. The sensor board used in the experiment. It has 30 PNI RM3100 magnetometers and an Osmium MIMU 4844 IMU mounted on the bottom side.
图 2.实验中使用的传感器板。它底部安装了 30 个 PNI RM3100 磁强计和一个 Osmium MIMU 4844 IMU。

Let the INS navigation state , the inertial measurements , and the process noise be defined as
让 INS 导航状态 、惯性测量 和过程噪声 分别定义为

respectively. Here, , and denote the position, velocity, and orientation (parameterized as a unit quaternion) at time , respectively. The superscript indicates that the vector is represented in the navigation frame. Moreover, and denote the accelerometer and gyroscope bias, respectively. Further, and denote the accelerometer and gyroscope measurements, respectively. Lastly, and denote the accelerometer and gyroscope measurement noise, respectively, and and denote the random walk process noise for the accelerometer and gyroscope biases, respectively. For an INS that uses low-cost sensors and moves at moderate velocities such that the effects of the transport rate, earth rotation, etc., can be neglected, the navigation equations are given by [25]
分别表示时间 时的位置、速度和方位（参数为单位四元数）。这里， 和 分别表示时间 时的位置、速度和方向（参数为单位四元数）。上标 表示矢量以导航框架表示。此外， 和 分别表示加速度计和陀螺仪偏置。此外， 和 分别表示加速度计和陀螺仪的测量值。最后， 和 分别表示加速度计和陀螺仪测量噪声， 和 分别表示加速度计和陀螺仪偏差的随机漫步过程噪声。对于使用低成本传感器并以中等速度移动的 INS，可以忽略传输速率、地球自转等的影响，其导航方程为 [25] 。

where 其中

and

Here, the subscript denotes the body frame at time , and denotes the rotation matrix that rotates a vector from the -frame to the -frame. Further, denotes the sampling period. Moreover, and denote the specific force and angular velocity, respectively. The vector denotes the local gravity. Furthermore, denotes quaternion multiplication, and is the operator that maps an axis-angle to a quaternion. Lastly, is modeled as a zeromean white Gaussian noise process with covariance matrix , where denotes the covariance matrix of the corresponding noise component and blkdiag (.) is an operator that creates a block diagnal matrix.
这里，下标 表示 时间的身体框架， 表示将矢量从 - 框架旋转到 - 框架的旋转矩阵。此外， 表示采样周期。此外， 和 分别表示比力和角速度。矢量 表示本地重力。此外， 表示四元数乘法， 是将轴角映射到四元数的算子。最后， 被模拟为具有协方差矩阵 的泽罗米白高斯噪声过程，其中 表示相应噪声分量的协方差矩阵，blkdiag (.) 是创建分块诊断矩阵的算子。

The magnetic field is a three-dimensional vector field whose properties are described by Maxwell's equations. Let be a model of the magnetic field at the location , parameterized by the parameter . When there is no free current in the space , the magnetic-field model should fulfill
磁场是一个三维矢量场，其特性由麦克斯韦方程组描述。让 成为 所在位置的磁场模型，参数为 。当空间 中没有自由电流时，磁场模型 应满足以下条件

for all . A polynomial magnetic field model, which fulfills (3), is given by [23]
。符合（3）的多项式磁场模型由 [23] 给出。

Here is the regression matrix defined in 23 . and is the coefficient of the polynomial model; for a order polynomial the model has unknown parameters [2]. Note that the model (4) can be defined in either the body frame or navigation frame. Within this paper, it will be defined in the body frame. Next, a procedure for transforming the model from the frame to the frame will be presented.
这里 是 23 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 个未知参数[2]。请注意，模型 (4) 可以在车身框架或导航框架中定义。本文将在车身框架内对其进行定义。接下来，将介绍将模型从 帧转换到 帧的程序。

Let the magnetic field model (4) be associated with the frame, which means accepts locations expressed in -frame and outputs magnetic field vector in the same frame. Then denotes the magnetic field model associated with the -frame, i.e.,
让磁场模型（4）与 框架相关联，即 接受以 - 框架表示的位置 ，并在同一框架中输出磁场矢量。那么 表示与 - 框架相关的磁场模型，即：

When the body frame moves, the coefficient evolves over time along with the changes in the body frame. Let the relative body frame change from to be encoded by
当车身框架移动时，系数 会随着车身框架的变化而变化。假设从 到 的相对身体框架变化由以下方式编码

where and denote the translation and orientation change from the body frame to , respectively. Then the dynamics of can be found by expressing the magnetic field vector at a given location with the models at two consecutive times and aligning them in the same frame. The basic idea is illustrated in Fig. 3, from which the following two equalities can be identified
其中 和 分别表示从本体框架 到 的平移和方向变化。然后，通过用连续两个时间的模型来表示给定位置的磁场矢量，并将它们在同一帧中对齐，就可以找到 的动态。基本思路如图 3 所示，从中可以确定以下两个等式

Fig. 3. A 2D illustration of the geometric relationship between the body frames at two consecutive times. The applicable region of the magnetic field model at time is in blue, and the black dot indicates the location where the two models output the corresponding magnetic field in their own coordinate frames.
图 3.两个连续时间体帧之间几何关系的二维图示。 时磁场模型的适用区域 为蓝色，黑点表示两个模型在各自坐标系中输出相应磁场的位置。

Here denotes the rotation matrix that rotates a vector from the -frame to the -frame. The rotation matrix and translation are given by
表示旋转矩阵，它将一个向量从 - 框架旋转到 - 框架。旋转矩阵 和平移 由以下公式给出

Here is an operator that maps a vector in to a skewsymmetric matrix such that .
这里 是一个算子，它将 中的一个向量映射为一个偏对称矩阵，这样 。

Substituting the generic magnetic model in (7) with the proposed polynomial model (5) yields the equality
将 (7) 中的通用磁场模型代入所提出的多项式模型 (5)，可以得到相等的结果

Note that for a given and represents 3 linear equations. Since is of dimension , which is greater than 3 , it is necessary to use more than one location vector to solve the equation system. In general, location vectors can be used to construct the equation system, i.e.,
请注意，给定的 和 表示 3 个线性方程。由于 的维数为 ，大于 3，因此需要使用多个位置向量 来求解方程组。一般来说，可以使用 位置向量来构建方程组，即：

Note that the location vectors can be chosen such that has full column rank, and they are fixed. Finally, it holds that
需要注意的是，位置向量 的选择可以使 具有全列秩，而且它们是固定的。最后，可以认为

where denotes the Moore-Penrose inverse of .
其中 表示 的摩尔-彭罗斯倒数。

Since the magnetic field model should describe the magnetic field locally, the update in 111 , which implicitly expands the applicable space of the model, will inevitably introduce additional modeling errors. To account for those errors, with a slight abuse of notation, the update of the polynomial coefficients of the magnetic field as the body frame change is modeled as
由于磁场模型应在局部描述磁场，因此 111 中的更新隐含地扩展了模型的适用空间 ，这将不可避免地带来额外的建模误差。为了考虑这些误差，略微滥用一下符号，磁场多项式系数在体框变化时的更新模型为

where is assumed to be a white Gaussian noise process with zero mean and covariance matrix . Hence, the dynamic model for the magnetic-field subsystem is
其中， 假设为均值为零且协方差矩阵为 的白高斯噪声过程。因此，磁场子系统的动态模型为

where 其中

Note that is a function of , and .
注意 是 和 的函数。

Given the magnetic field model in (5), the measurement from the sensor in the magnetometer array at time can be modeled as
考虑到 (5) 中的磁场模型， 时磁力计阵列中 传感器的测量结果 可模拟为

where denotes the location of the magnetometer in the array. Further, denotes the measurement error, which includes both the measurement noise and the imperfections of the magnetic-field model. The error 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。此外， 表示测量误差，其中包括测量噪声和磁场模型的缺陷。误差假定为白色高斯分布，协方差矩阵为 。

Given the presented navigation equations and the magneticfield model, the dynamics and observations of the full system can be described by the following state-space model.
根据所给出的导航方程和磁场模型，整个系统的动态和观测结果可以用以下状态空间模型来描述。

Let the state vector , the process noise vector , and the measurement noise be defined as
假设状态矢量 、过程噪声矢量 和测量噪声 定义如下

respectively. Combining the models in (2), (13), and (15) gives the state-space model
分别为将 (2)、(13) 和 (15) 中的模型组合起来，就得到了状态空间模型

where 其中

and

Here, the process noise covariance and the measurement noise covariance are
这里，过程噪声协方差 和测量噪声协方差 分别为

Fig. 4. The flowchart of the state estimation algorithm.
图 4 状态估计算法流程图状态估计算法流程图。

The quaternion in the state vector does not belong to Euclidean space. Therefore, standard nonlinear filter algorithms, such as the extended Kalman filter and unscented Kalman filter, cannot be applied to the state-space model in (17) without appropriate modifications. The error state Kalman filter (ESKF) presented in [25] circumvents the problem by the use of "error quaternion", which is a small perturbation around the "estimated quaternion" expressed in . With reference to Fig. 4 the algorithm works by propagating an estimated state via the state transition model in (17a) and using a complementary Kalman filter to estimate the error state , which is then used to correct the estimated state . Since the ESKF is a standard algorithm, only the error propagation model that is unique to the state-space model in (17), will here be derived.
状态向量中的四元数 不属于欧几里得空间。因此，标准的非线性滤波算法，如扩展卡尔曼滤波器和无符号卡尔曼滤波器，在没有适当修改的情况下无法应用于 (17) 中的状态空间模型。文献[25]中提出的误差状态卡尔曼滤波器（ESKF）通过使用 "误差四元数 "规避了这一问题，"误差四元数 "是以 表示的 "估计四元数 "周围的小扰动。参照图 4，该算法的工作原理是通过 (17a) 中的状态转换模型传播估计状态 ，并使用互补卡尔曼滤波器估计误差状态 ，然后用误差状态修正估计状态 。由于 ESKF 是一种标准算法，因此这里将只推导 (17) 中状态空间模型所独有的误差传播模型。

The error state is defined as
误差状态 定义为

Here, for the position, velocity, sensor biases, and magnetic field model parameters, the standard additive error definition is used (e.g., ). On the other hand, the orientation error satisfies the equation . The true state and the estimated state relate to each other via
在此，对于位置、速度、传感器偏差和磁场模型参数，采用标准的加法误差定义（如 ）。另一方面，方位误差 满足等式 。真实状态 和估计状态 通过以下方式相互关联

where the operator is defined by
其中，算子 的定义是

The dynamics of has been derived in [25] and are given by
的动力学参数已在 [25] 中得到，其计算公式为

where 其中

Here, and .
在此， 和 。

To the first order, the errors in (13) propagate according to
在一阶，(13) 中的误差根据以下公式传播

where 其中

However, instead of expressing the error development in terms of , we would like to express it in terms of the orientation error , velocity error , accelerometer bias estimation error , and gyroscope bias estimation error . To do so, note that from we have
不过，我们希望用方向误差 、速度误差 、加速度计偏差估计误差 和陀螺仪偏差估计误差 来表示误差发展，而不是用 来表示。为此，我们可以从 中得到