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2.1. Magnetic navigation using observers for non-linear dynamical systems
2.1.利用非线性动力系统的观测器进行磁导航

Objective Let us consider a rigid body strapped with inertial and magnetic sensors. Information on its trajectory is desired, and it is assumed that dead-reckoning solutions are an option. What technique is available to achieve this?
目标 让我们考虑一个绑有惯性传感器和磁传感器的刚体。我们希望获得有关其轨迹的信息，并假定死区重定位解决方案是一种选择。有什么技术可以实现这一目标？
Strapdown inertial navigation Strapdown inertial navigation is defined by the use of gyrometers and accelerometers strapped onto the rigid-body whose localization is sought. It consists in integrating kinematics differential equations (see Refs. Savage 2000, Woodman 2007]) from their measurements. This formulation is well suited for being used in conjunction with so-called observers for nonlinear systems; therefore, this work is limited to this scope.
捆绑式惯性导航捆绑式惯性导航是指将陀螺仪和加速度计捆绑在需要定位的刚体上。惯性导航包括根据测量结果积分运动学微分方程（见参考文献 Savage 2000，Woodman 2007]）。这种方法非常适合与所谓的非线性系统观测器结合使用；因此，这项工作仅限于此范围。
Magnetic navigation Usually, in this context, the magnetic field is only used as a heading reference in order to correct integration results. However, on earth, the magnetic field is generally not homogeneous (see. Fig. 2.1). This is so whatever the scale at which it is considered, as is evident from Ref. [Dorveaux, 2011. Table 2.1]. In particular, indoor environments are rich in ferromagnetic materials distorting the magnetic field, at a level measurable with low-cost magnetic sensors, see, e.g., Ref. de Vries et al. 2009], and compromising the use of the magnetic field as a heading reference. Nevertheless, this property makes it a variable of interest with regards to positioning instead, see, e.g., Refs. Gozick et al. 2011, Haverinen 2014, Haverinen and Kemppainen [2009], Shen et al. [2016], Storms et al. [2010], and more recently Ref. Kok and Solin [2018].
磁导航 通常，在这种情况下，磁场只用作航向参考，以校正积分结果。然而，在地球上，磁场通常不是均匀的（见图 2.1）。正如参考文献[Dorveaux, 2011.[多尔沃，2011 年。 表 2.1]。特别是，室内环境中含有大量铁磁性材料，这些材料会扭曲磁场，而低成本的磁传感器就可以测量到这种扭曲，参见参考文献 de Vries 等人，2009 年]，从而影响了磁场作为航向参考的使用。尽管如此，磁场的这一特性使其成为定位方面的一个重要变量。Gozick等人[2011]、Haverinen[2014]、Haverinen和Kemppainen[2009]、Shen等人[2016]、Storms等人[2010]以及最近的参考文献。Kok 和 Solin [2018]。
Usual, map-based approaches There is no single way of accounting for the magnetic field in the framework of observers for non-linear dynamical systems. In the usual fingerprinting/mapping approaches, the magnetic field inhomogeneity as a function of space is used to reconstruct the absolute trajectory by comparing it to a pre-existing map (see for example Ref. Storms [2009]). The various techniques revolve around mapping, for example, the intensity of the magnetic field in one dimension (in corridors for example), two, or three dimensions, then taking one or a sequence of measurements and matching them to the map.
基于地图的常规方法 在非线性动力学系统的观测器框架中，没有一种单一的磁场计算方法。在通常的指纹/映射方法中，磁场不均匀性作为空间函数，通过与预先存在的映射进行比较来重建绝对轨迹（例如，见参考文献 Storms [2009]）。各种技术都围绕着绘制一维（例如走廊）、二维或三维的磁场强度图，然后进行一次或一系列测量，并将其与地图进行匹配。
Dead-Reckoning approaches By contrast with map-based approaches, in MagnetoInertial Dead-Reckoning (MIDR) (see, e.g., Ref. Dorveaux 2011]), which is the topic of this work, the magnetic field is considered in that in the body frame, it is a function of time. Its temporal evolution is linked to velocity through the spatial gradient of the magnetic field, which enables trajectory reconstruction in a dead-reckoning framework when the latter is known.
死重法 与基于地图的方法不同，在磁惯性死重法（MIDR）中（参见参考文献 Dorveaux 2011]），也就是本研究的主题中，磁场被认为是体帧中时间的函数。磁场的时间演化通过磁场的空间梯度与速度相关联，当已知磁场的空间梯度时，就能在死重定位框架中重建轨迹。
Importance of the choice of model This model difference results in two entirely different approaches: magnetic fingerprinting involves a magnetic map, whereas MIDR
选择模型的重要性 这种模型差异导致两种完全不同的方法：磁性指纹识别涉及磁性图谱，而 MIDR
Figure 2.1: Total magnetic field intensity map, according to the WMM 2015 (World Magnetic Model)
图 2.1：根据 WMM 2015（世界磁模型）绘制的总磁场强度图
https://www.ngdc.noaa.gov/geomag/WMM/data/WMM2015/WMM2015_F_MERC.pdf
does not. Magnetic fingerprinting provides absolute positioning in the map, whereas MIDR does not since no map is kept during the estimation process, but it provides velocity instead. Therefore it is clear that the choice of model for the same physical system is a crucial step that needs to be carefully considered.
而 MIDR 则不然。磁性指纹识别可提供地图中的绝对定位，而 MIDR 则不然，因为在估算过程中不会保留地图，而是提供速度。因此，为同一物理系统选择模型显然是需要仔细考虑的关键步骤。
Questions Given the importance of modeling choices, it is worth spending time considering the underlying model that one will later try to observe. That is to say, the following questions must be answered:
问题 鉴于模型选择的重要性，值得花时间考虑以后要尝试观察的基本模型。也就是说，必须回答以下问题：
  • What information do magnetic field measurements provide about the trajectory?
    磁场测量提供了哪些有关轨迹的信息？
  • What is the adequate state-space model that must be observed?
    必须观察到的适当状态空间模型是什么？
These questions will be answered more specifically in the scope of low-cost strapdown Microelectromechanical Systems (MEMS) inertial navigation in indoor environments.
这些问题将在室内环境中的低成本带式微机电系统（MEMS）惯性导航范围内得到更具体的回答。
Method Our approach is based on known MIDR models in the literature which we review critically with respect to observability. Possible model candidates are deduced from these references, and the distinguishability of trajectories from each other given available measurements is studied. To that end, the following references are considered: Ref. Dorveaux [2011, the Ph.D. Thesis of E. Dorveaux on the subject, Refs. Vissière et al. [2007ab], seminal papers on the subject, and Ref. Batista et al. 2013, which is about observability in a previously identified corner case. This chapter then serves as a justification of the work regarding observer design and experimental evaluation that was undertaken mainly in Ch. 4 and Refs. Chesneau et al. 2016. 2017, but also of the rest of this thesis to a lesser extent.
方法 我们的方法基于文献中已知的 MIDR 模型，我们对这些模型的可观测性进行了严格审查。从这些参考文献中推导出可能的候选模型，并研究在现有测量条件下轨迹之间的可区分性。为此，我们考虑了以下参考文献：参考文献Dorveaux[2011，E. Dorveaux 关于此主题的博士论文，Refs.Vissière等人[2007ab]，关于这一主题的开创性论文，以及 Ref.Batista等人的论文[2013]，后者是关于先前确定的角情况下的可观测性。本章是对第 4 章和参考文献中有关观测器设计和实验评估工作的论证。Chesneau et al.2017，但在较小程度上也涉及本论文的其余部分。
Outline In Sec. 2.2 notations are defined, and a problem statement is formulated. In Sec. 2.3, results from Ref. Dorveaux 2011] are recalled, and reasons for considering alternative modeling architecture are provided. In Sec. 2.4 Refs. Vissière et al. 2007a|b. are briefly reviewed, and a minimal dynamical model is proposed and studied. Sec. 2.5 discusses the implications of using low cost inertial sensors in this context, and Sec. 2.6 concludes this chapter.
概述 第 2.2 节定义了术语，并提出了问题陈述。在第 2.3 节中，回顾了参考文献 Dorveaux 2011]中的结果，并提出了考虑其他建模架构的理由。Dorveaux 2011]的结果，并说明考虑其他建模架构的原因。在第 2.4 节中，参考文献Vissière et al. 2007a|b. 文献进行了简要回顾，并提出和研究了一个最小动力学模型。第 2.5 节讨论了在此背景下使用低成本惯性传感器的意义，第 2.6 节对本章进行了总结。

2.2. Problem statement 2.2.问题陈述

Let us come back to the considered rigid body that is moving in some navigation frame. It is equipped with strapdown MEMS inertial sensors, and with a magnetic measurement unit, such that both the vector components and the local spatial gradient of the magnetic field are measured at some fixed point of the rigid body. Moreover, let us assume that the magnetic field is stationary, that is to say, it is time-invariant in the navigation frame. The problem under consideration in this chapter is to provide a model that connects measurements and trajectory information together and determine what trajectories are distinguishable by only said measurements: those provided by the Inertial Measurement Unit (IMU) and those of the magnetic field. The underlying notion is that of observability.
让我们回到在某个导航框架中运动的刚体。它配备了带式 MEMS 惯性传感器和磁场测量单元，这样就可以在刚体的某个固定点测量磁场的矢量分量和局部空间梯度。此外，我们假设磁场是静止的，也就是说，它在导航框架中是时间不变的。本章要解决的问题是提供一个模型，将测量值和轨迹信息联系在一起，并确定哪些轨迹仅通过上述测量值（惯性测量单元（IMU）提供的测量值和磁场测量值）就能区分。其基本概念是可观测性。

2.2.1. Notations and definitions
2.2.1.符号和定义

Frames of reference 参考框架

Let us consider three reference frames. Let be a reference frame moving with the rigid body and an associated orthonormal basis. Sensors are fixed in this reference frame. Let be the navigation frame, with an associated orthonormal basis. This is the frame in which the trajectory is estimated. Let us denote an inertial frame of reference, and an associated orthonormal basis.
让我们考虑三个参照系。让 作为与刚体一起运动的参照系， 作为相关的正交基础。传感器固定在该参照系中。让 成为导航框架， 是相关的正交基础。这是估算轨迹的参照系。我们将 作为惯性参照系， 作为相关正交基。
(Cartesian) coordinates in (resp. and ) are denoted with the exponent (resp. and ). For example,
( （以及 和 ）中的（笛卡尔）坐标用指数 （以及 和 ）表示。例如
denotes the coordinates of a vector in .
表示矢量 在 中的坐标。
In the following, is assumed fixed with respect to the earth.
在下文中，假定 相对于地球是固定的。

Trajectory 轨迹

Let be a fixed point of the rigid body. By convention, let us define its Cartesian coordinates in body frame as . An attitude matrix representing the orientation of the rigid body in is defined such that in terms of matrix relations, for any point whose coordinates in are , the change of coordinates between and reads
让 成为刚体的一个固定点。按照惯例，我们将其在车身框架中的直角坐标定义为 。代表刚体在 中方位的姿态矩阵 的定义是：对于任何点 ，其在 中的坐标为 ，则 和 之间的坐标变化矩阵关系为
The trajectory of the rigid body is defined as the evolution of with time, where denotes the special Euclidean group that represents rigid body motion. Let us remark that since , then . The time variable is denoted .
刚体的轨迹定义为 随时间的演化，其中 表示表示刚体运动的特殊欧几里得群。我们注意到，既然 ，那么 。时间变量表示为 。
Eq. (2.1) implies 公式 (2.1) 意味着
Therefore, for any pair of reference frames
因此，对于任何一对参照系
The notation is then an alias of .
因此， 是 的别名。

Velocity 速度

The velocity vector of in the navigation frame is defined as
在导航框架中的速度矢量 定义为
This same 3-D vector can also be expressed in body frame while still representing the velocity vector of in the navigation frame,
同样的三维矢量也可以用体帧表示，同时仍然代表导航帧中 的速度矢量、
One must be careful that this definition of is not objective in that it is bound to the navigation frame . See also Ref. Matolcsi and Ván 2006. This is deliberate, since the trajectory reconstruction problem is bound to : it would not make sense to define the change of frame in such a way that would become independent of . However, doing so might lead to some degree of confusion when computing timederivatives not carefully. For instance, in general,
必须注意的是， 的定义并不客观，因为它受导航框架 的约束。另见参考文献。Matolcsi 和 Ván 2006。这是有意为之的，因为轨迹重构问题与 绑定：如果以 独立于 的方式来定义框架变化，就没有意义了。不过，这样做可能会在计算时间二乘法时造成一定程度的混乱。例如，在一般情况下、

Angular velocity 角速度

Let us denote the angular velocity of with respect to . It is defined such that
表示 相对于 的角速度。其定义为
where denotes the antisymmetric matrix
其中 表示反对称矩阵
The navigation frame is fixed with respect to the Earth, which is not strictly speaking an inertial frame, because it is spinning. Thus, is not measurable directly with strapdown gyrometers. Instead, they are sensitive to the angular velocity vector of with respect to that is denoted . Similarly, the angular velocity vector of with respect to is denoted . The relationship between , and is then
导航框架 相对于地球是固定的，而地球严格来说并不是惯性框架，因为它在旋转。因此， 无法通过带式陀螺仪直接测量。相反，它们对 相对于 的角速度矢量（表示为 ）很敏感。同样， 相对于 的角速度矢量表示为 。因此， 、 和 之间的关系为
Often, MEMS gyrometers are not considered sensitive enough to measure angular velocities whose magnitude is comparable with the earth's angular velocity 1 This is why Eq. 2.5 is often used as if were measured directly, or as if the earth were not spinning.
通常情况下，MEMS 陀螺仪的灵敏度不足以测量与地球角速度大小相当的角速度1，这就是为什么公式 2.5 经常被当作直接测量 或地球不旋转来使用的原因。

Acceleration 加速度

Let us denote the acceleration vector of in the navigation frame,
让我们把 表示为 在导航框架中的加速度矢量、
This same 3-D vector can also be expressed in body frame while still representing the same quantity
同样的三维矢量也可以用身体框架来表示，但仍然代表相同的量
Let us denote the local effective gravity vector in the navigation frame. Models of such as the World Gravity Model (WGM) include the effect of the Earth's rotation and the resulting centrifugal force.
让我们用 表示导航框架中的本地有效重力矢量。 的模型，如世界重力模型（WGM），包括地球自转的影响和由此产生的离心力。
Proper acceleration is defined as the acceleration of with respect to a nonrotating reference frame that would be only subjected to gravity, that is to say (see Ref. Savage [2000]),
适当加速度 被定义为 相对于只受重力作用的非旋转参照系的加速度，也就是说（见参考文献 Savage [2000]）、
where represents the Coriolis acceleration. Equation (2.7) is only valid because of the choice of a Cartesian coordinatization, by neglecting the angular acceleration of , and by including the centrifugal force into . Proper acceleration can also be expressed in body frame while still representing the same quantity,
其中 代表科里奥利加速度。等式 (2.7) 之所以有效，是因为选择了笛卡尔坐标系，忽略了 的角加速度，并将离心力包含在 中。适当的加速度也可以用体帧来表示，但仍然代表相同的量、
Strapdown accelerometers are sensitive to .
带式加速度计对 很敏感。

Magnetic field 磁场

Let us denote the magnetic field. It is a vector field, which in general, depends on time and space. At any point in space, the change of frame between and is assumed to be expressed a
让我们用 表示磁场。它是一个矢量场，一般来说取决于时间和空间。在空间中的任何一点 ，假定 和 之间的框架变化表示为
Admitting that the magnetic field is always regular enough, the Jacobian matrix at of the magnetic field is defined as
假设磁场总是足够规则，那么 磁场的雅各布矩阵定义为
and
with the change of frame
随着框架的变化
Magnetic sensors are assumed to provide direct and accurate measurements of and . In this chapter, these notations are respectively shortened to and whenever these notations are non-ambiguous.
假定磁传感器可直接准确测量 和 。在本章中，如果这些符号不明确，则分别简称为 和 。

State-space model 状态空间模型

Let us recall that the ideal outcome would be the ability to compute the trajectory and by using an observer for non-linear dynamical systems. Thus, the first step is to write the state-space model that needs to be observed in the form
让我们回顾一下，理想的结果是能够通过使用非线性动态系统的观测器来计算轨迹 和 。因此，第一步是将需要观测的状态空间模型写成以下形式
where is the state vector, is the known output vector of the system, and is the known input vector. In the following, denotes the manifold to which values of belong, denotes the open subset of to which values of the input vector belong, and denotes the open subset of to which values of the measured outputs belong.
其中 是状态向量， 是系统的已知输出向量， 是已知输入向量。在下文中， 表示 的值所属的流形， 表示 的开放子集，输入向量 的值所属的流形， 表示 的开放子集，测量输出的值所属的流形。
If it is assumed that measurements can directly access and , then one must be able to deduce , then the trajectory.
如果假设测量结果可以直接获取 和 ，那么就必须能够推导出 ，然后再推导出轨迹。

Observer and observability
观察者和可观察性

Let us recall the definition of an observer from Ref. Besançon 2007.
让我们回顾一下参考文献中关于观察者的定义。贝桑松 2007。
Definition 1 (Observer Besançon, 2007, Sec. 1.1.2, Def. 1]). Considering a system (2.9), an observer is given by an auxiliary system:
定义 1（《贝桑松观测器》，2007 年，第 1.1.2 节，定义 1]）。考虑到系统 (2.9)，观测器由辅助系统给出：
such that: 这样
(i)  (i)
(ii) as
(ii) 作为
Let us also recall the definition of observability from Ref. Besançon 2007. Reproducing its notations, denotes the solution of the state equation Eq. 2 (2.9) under the application of input on and satisfying .
让我们再回顾一下参考文献中关于可观测性的定义。贝桑松 2007》中的可观测性定义。根据这一定义， 表示在 上输入 并满足 的条件下，状态方程公式 2 (2.9) 的解。
Definition 2 (Indistinguishability Ref. [Besançon, 2007, Sec. 1.2.1, Def. 2]). A pair is indistinguishable for a system (2.9) if:
定义 2（不可区分性 参考文献[贝桑松，2007，第 1.2.1 节，定义 2]）。对于系统 (2.9) 而言，一对 在以下条件下是不可区分的：
A state is indistinguishable from if the pair is indistinguishable.
如果一对 不可区分，则 与 不可区分。
Definition 3 (Observability Ref. [Besançon, 2007, Sec. 1.2.1, Def. 3]). A system (2.9) is observable [resp. at ] if it does not admit any indistinguishable pair [resp. any state indistinguishable from .
定义 3 (Observability Ref. [Besançon, 2007, Sec. 1.2.1, Def. 3]).如果系统 (2.9) 不存在任何不可区分的对子[即任何与 不可区分的状态]，则该系统是可观察的[respect. at ]。

2.2.2. The modeling problem
2.2.2.建模问题

The whole point of this chapter is that the choice of dynamical model is neither completely straightforward nor neutral. There is more than one possible model, some of whom leading to algorithms quite different from dead-reckoning. Therefore, this work solely focuses on MIDR, as defined below.
本章的重点在于，动力学模型的选择既不完全直接，也不完全中立。可能的模型不止一种，其中一些模型导致的算法与死循环算法截然不同。因此，这项工作只关注下文定义的 MIDR。

MIDR solutions MIDR 解决方案

MIDR is defined by
MIDR 的定义是

  • The choice of the magnetic field vector (or , or the same quantity in any other frame of reference or any reparameterization) as a state variable and as an output variable (in and ). This is a model of measurements, therefore, no persistent map of the magnetic field is compiled;
    选择磁场矢量 （或 ，或在任何其他参照系或任何重参数化中的相同量）作为状态变量和输出变量（在 和 中）。这是一个测量模型，因此没有编制持续的磁场图；
  • The ability to use direct measurements of the spatial derivative of the magnetic field ;
    利用直接测量磁场空间导数的能力 ；
  • The use of (or , or the same quantity in any other frame of reference or any reparameterization) in the components of corresponding to the timederivative of (expressed in any frame of reference, or any reparameterization). More precisely, includes an instance of the following equation,
    （或 ，或在任何其他参照系或任何重参数化中的相同量）在 的组成部分中的使用，对应于 （在任何参照系或任何重参数化中表示）的时差。更确切地说， 包括以下方程的实例、
If is stationary in , that is to say, depends only on the space variable, then (2.11) reduces to
如果 在 中是静止的，也就是说， 只取决于空间变量，那么 (2.11) 可以简化为
Equations (2.11) and (2.12) are central in Chapters 2 and 4 as the starting point of all dynamical models that are used.
方程 (2.11) 和 (2.12) 是第 2 章和第 4 章的核心内容，是所有动力学模型的起点。
Why this restriction? According to Sec. 2.2.1 measured quantities are , and . Since the trajectory is defined as the orientation and position of the rigid body as a function of time, that is to say , a natural way of forming the state-space model is to combine the strapdown inertial navigation equations (2.3), (2.6), (2.7), with the magnetic field equation (2.8). For instance, as discussed in Sec. 2.1 one could choose to put and into the state vector, use and to form the input-vector, and use and as outputs. Assuming that the magnetic field is mapped beforehand, the function could then be formed using this map, thus forming our state-space model. Such an approach would be a hybridization between magnetic fingerprinting and inertial navigation, augmented with measurements of the magnetic gradient, and not dead-reckoning.
为何有此限制？根据第 2.2.1 节，测量量为 、 和 。由于轨迹被定义为刚体的方位和位置与时间的函数关系，即 ，因此形成状态空间模型 的自然方法是将束带惯性导航方程 (2.3)、(2.6) 和 (2.7) 与磁场方程 (2.8) 结合起来。例如，如第 2.1 节所述，可以选择将 和 放入状态矢量，用 和 组成输入矢量，用 和 作为输出。假定磁场事先已被映射，那么函数 就可以通过这个映射形成，从而形成我们的状态空间模型。这种方法是磁场指纹识别与惯性导航的混合，并通过测量磁场梯度进行增强，而不是死循环。
Higher order spatial derivatives Let us remark that the definition of MIDR can stated by replacing and with their respective spatial derivatives, with an equation of the form
让我们注意到，将 和 替换为各自的空间导数，就可以得到 MIDR 的定义，其方程形式为
If higher order spatial derivatives are measurable directly, both approaches could even be combined, with clear advantages of doing so (see Ref. Skog et al. [2018]).
如果高阶空间导数可以直接测量，甚至可以将两种方法结合起来，这样做的优势显而易见（见参考文献 Skog 等人[2018]）。

2.2.3. Problem formulation
2.2.3.问题的提出

Now that notations, MIDR and observability are defined, the problem is answering the following two questions:
既然定义了符号、MIDR 和可观测性，那么问题就在于回答以下两个问题：
  1. What dynamical system must be chosen?
    必须选择什么样的动力系统？
  2. How observable is each possible choice of dynamical system?
    每种可能选择的动力系统的可观测性如何？
These two questions are essential because answering them is making sure that the right tool is used for the job. They highlight that observability is bound to a choice of dynamical system. They also emphasize that the modeling step is an essential part of solving an observation problem for our physical system. For example, questions such as "what order is the system", or "is it observable" only make sense once a model is chosen, and the choice of model can be questioned.
这两个问题至关重要，因为回答这两个问题就能确保使用正确的工具来完成工作。它们强调了可观测性与动力系统的选择息息相关。他们还强调，建模步骤是解决物理系统观测问题的重要组成部分。例如，只有选择了模型，"系统是什么阶 "或 "是否可观测 "等问题才有意义，而模型的选择也会受到质疑。
Modeling choices are based on the following criteria: one must be able to evaluate the model, it must accurately represent reality, and its observability properties must allow dead-reckoning. Let us start with equation 2.12 , and study models in order of increasing complexity.
模型的选择基于以下标准：我们必须能够对模型进行评估，模型必须准确地代表现实，模型的可观测性必须允许死循环。让我们从方程 2.12 开始，按复杂程度递增的顺序研究模型。

2.3. Minimal dynamical system for velocity computation
2.3.速度计算的最小动力系统

This section recalls some relevant results of Ref. Dorveaux 2011 in the context of modeling and observability studies, and discusses its choice of model. Said model is provided in Sec. 2.3.1 An observability result is provided in Sec. 2.3.2 Then, the choice is discussed in Sec. 2.3 .3 ,
本节回顾了参考文献 Dorveaux 2011 在建模和可观测性研究方面的一些相关成果，并讨论了其模型的选择。Dorveaux 2011 在建模和可观测性研究方面的一些相关结果，并讨论其对模型的选择。第 2.3.1 节提供了上述模型，第 2.3.2 节提供了可观测性结果，第 2.3.3 节讨论了模型的选择、

2.3.1. Model 2.3.1.模型

Let us assume that the magnetic field is stationary in the navigation frame, that the navigation frame and inertial frame are identical, and that an unbiased strapdown 3 -axis gyrometer is available. The state-space model is chosen as
假设磁场在导航框架内静止不动，导航框架和惯性框架完全相同，并且有一个无偏带式三轴陀螺仪。状态空间模型 选为
Using our notations, the following system reproduces the one presented and studied in Ref. Dorveaux, 2011, Sec. 2.1.2, Eq. (2.8)], which is the simplest MIDR model of the state of the art,
使用我们的符号，下面的系统重现了参考文献 Dorveaux, 2011, Sec.Dorveaux, 2011, Sec. 2.1.2, Eq. (2.8)]中提出和研究的系统，它是目前最简单的 MIDR 模型、
If Eq. 2.15, is injected into Eq. 2.9, the following equation is obtained,
如果将公式 2.15 引入公式 2.9，就会得到下面的方程、
which is equivalent to . In other words, this model is only accurate for straight line trajectories at constant velocity in the navigation frame . The unavailability of reliable accelerometers can nevertheless justify this strong assumption.
相当于 。换句话说，该模型仅适用于导航框架中的匀速直线轨迹 。不过，由于无法获得可靠的加速度计，这一强有力的假设也是合理的。

2.3.2. Observability 2.3.2.可观察性

The model Eqs. (2.14)- 2.15 ) is proven observable in Ref. [Dorveaux 2011, Appx. C], since as long as is non-singular, there exists a converging non-linear observer that reconstructs both the magnetic field and the velocity,
参考文献[Dorveaux 2011, Appx C]证明了公式 (2.14)- 2.15 ) 模型的可观测性。[Dorveaux 2011, Appx. C]，因为只要 是非矢量的，就存在一个可以同时重建磁场和速度的收敛非线性观测器、
where and are constant gains.
其中 和 为恒定增益。
Remark The convergence speed of towards in the above observer seems to decrease unnecessarily when the gradient is small. This motivates replicating the proof of convergence provided in Ref. [Dorveaux 2011, Sec. 2.2] with a simpler observer,
备注 当梯度较小时，上述观测器中 向 的收敛速度似乎会不必要地降低。这促使我们复制参考文献[Dorveaux 2011, Sec 2.2]中的收敛证明。[Dorveaux 2011, Sec. 2.2] 中提供的收敛性证明、
Proof. Consider the Lyapunov function
证明考虑 Lyapunov 函数
Its time derivative can be written under the form
它的时间导数可以写成以下形式
because and are orthogonal.
因为 和 是正交的。
Since , then .
既然 ，那么 。
In order to complete the proof, let us show, following Ref. [Dorveaux 2011, Sec. 2.2], that the invariant set defined by is reduced to
为了完成证明，让我们按照参考文献[Dorveaux 2011, Sec.[多尔沃 2011，第 2.2 节]，由 定义的不变集简化为
which, by LaSalle's invariance principle, implies that this set is globally asymptotically stable.
根据拉萨尔不变性原理，这意味着这个集合是全局渐近稳定的。
From Eq. (2.23), it is obvious that in this set,
根据公式 (2.23)，很明显，在这一组中、
Using Eqs. (2.15 and (2.17) to take the time derivative of the previous equation yields immediately,
利用公式 (2.15) 和 (2.17) 对上一方程进行时间导数计算，立即得出结果、
Because by assumption is non-singular, it follows that in this set,
因为假设 是非奇数，所以在这个集合中、
which concludes the proof.
证明结束。
The advantage of this observer over the one proposed in Ref. Dorveaux, 2011 . Sec. 2.2] is that converges towards independently of .
与参考文献中提出的观测器相比，该观测器的优势在于：......Dorveaux, 2011 .2.2 节]中提出的观测器的优势在于， 与 无关，而是向 收敛。

2.3.3. Discussion: What about attitude estimation?
2.3.3.讨论：如何进行姿态估计？

Not only is the model Eqs. (2.14)- 2.15) only accurate for straight line trajectories at constant velocity, which makes sense with unreliable accelerometers but is very restrictive, but it is also incomplete.
公式 (2.14)- 2.15) 不仅仅适用于匀速直线轨迹，这对不可靠的加速度计来说是合理的，但却有很大的局限性，而且也不完整。
Attitude estimation is missing This observable state is not sufficient in order to compute a trajectory. Although is observable, the minimum requirement for computing the trajectory would be the ability to integrate the velocity in the navigation frame, Eq. 2.3). To achieve that, at least an estimate of the attitude matrix is missing, and necessary to obtain in the navigation frame, Eq. (2.4). This issue is indeed discussed at the end of Ref. [Dorveaux, 2011, Sec. 2.2]. In this reference, the proposed design solution is to estimate the attitude matrix in a separate state-observer as illustrated Fig. 2.2 since there is plenty of literature on the subject, see Refs. Michel et al. 2015, Makni 2016 and references therein.
缺少姿态估计 这种可观测状态不足以计算轨迹。虽然 是可观测的，但计算轨迹的最低要求是能够在导航框架中对速度进行积分（公式 2.3）。要做到这一点，至少需要对姿态矩阵 进行估算，并在导航框架中获得 （式 (2.4)）。这个问题在参考文献[Dorveaux, 2011, Sect.[Dorveaux, 2011, Sec. 2.2]。本参考文献提出的设计方案是在图 2.2 所示的单独状态观测器中估算姿态矩阵 ，因为这方面的文献很多，见参考文献 Michel 等人，2015 年；Makak 等人，2015 年。Michel et al. 2015, Makni 2016 及其中的参考文献。
The model is restrictive Attitude estimation algorithms using MEMS inertial sensors usually assume that the mean value of the acceleration vector is zero. Resorting to them makes sense considering that this assumption was implicitly taken. With this assumption, the gravity vector can be recovered from the measured proper acceleration vector. This is akin to using accelerometers as tilt sensors. This assumption is generally not true; there is no reason why all trajectories should follow straight lines at constant velocity. When gyrometers are available, they can be used to mitigate effects of nonzero with filtering (see Refs. Mahony et al. 2005, 2008), 4. However, after browsing through the literature, it becomes clear that research on attitude estimation algorithms using MEMS inertial sensors mainly aims at mitigating the effects of accelerations, which are considered as disturbances. The trajectory itself being seen as a disturbance is an undesirable property highlighting a representativeness problem.
模型具有限制性 使用 MEMS 惯性传感器的姿态估计算法通常假定加速度矢量 的平均值为零。考虑到这一假设是隐含的，采用这些算法是合理的。根据这一假设，重力矢量可以从测量到的适当加速度矢量中恢复。这类似于将加速度计用作倾斜传感器。一般来说，这一假设是不正确的；没有理由说所有轨迹都应该以恒定的速度沿着直线运行。如果有回旋仪，则可以通过滤波来减轻非零 的影响（见参考文献 Mahony 等人，2005 年，2008 年），4。然而，浏览文献后发现，使用 MEMS 惯性传感器的姿态估计算法研究主要是为了减轻加速度的影响，因为加速度被视为干扰。轨迹本身被视为干扰是一个不可取的特性，凸显了代表性问题。

2.4. Velocity and attitude estimation in an inertial frame
2.4.惯性框架中的速度和姿态估计

Let us consider attitude estimation jointly with velocity estimation, with the assumption that the navigation frame is inertial, implying . In fact, the seminal papers on MIDR Refs. Vissière et al. 2007a|b], do combine velocity and attitude estimation. Using our notations, the reduced state vector of the dynamic system in Ref. [Vissière et al. 2007a, Sec.II] whose observability is studied is equivalent to
让我们把姿态估计与速度估计结合起来考虑，假设导航框架是惯性的，这意味着 。事实上，关于 MIDR 的开创性论文 Refs.Vissière等人，2007a|b]，确实结合了速度和姿态估计。使用我们的符号，参考文献[Vissière et al.[Vissière 等人，2007a，第 II 章] 中研究的可观测性相当于
Figure 2.2: From Ref. Dorveaux, 2011, Fig. 2.3, "Trajectory reconstruction with a Magneto-Inertial NAVigation (MINAV) system"
图 2.2：摘自参考文献Dorveaux, 2011, 图 2.3，"磁惯性导航（MINAV）系统的轨迹重建"。
In Ref. Dorveaux [2011], velocity and attitude estimation are considered separately. A state-of-the-art attitude observer using accelerometers and gyrometers is implemented. The magnetic field can be used as a heading source. Velocity estimation is achieved in a separate observer, using gyrometers and magnetic field measurements. Outputs from both observers are combined in order to obtain velocity in the navigation frame, and integrated in order to compute a trajectory in a dead-reckoning framework.
在参考文献Dorveaux[2011]分别考虑了速度和姿态估计。利用加速度计和陀螺仪实现了最先进的姿态观测器。磁场可用作航向源。速度估算在另一个观测器中使用陀螺仪和磁场测量来实现。将两个观测器的输出结合起来，以获得导航框架中的速度，并进行整合，从而在死区重定位框架中计算轨迹。
with an output vector that reads
输出向量为
where stands for gyrometer biases, stands for accelerometer biases, and are defined according to Eqs. and (2.5) with , and vectors to represent coordinates in body frame of additional 3-axis magnetometers. It is worth noting that Ref. Vissière et al. 2007a does study the observability of this model but,
其中， 代表陀螺仪偏置， 代表加速度计偏置， 和 根据公式 和 (2.5) 与 进行定义，矢量 至 代表附加三轴磁力计在车身框架中的坐标。值得注意的是，参考文献Vissière et al. 2007a 的确研究了这一模型的可观测性，但是、
  • Contrary to Ref. Dorveaux [2011], this work does not completely fit into the definition of MIDR in Sec. 2.2.2 While it does use an instance of Eq. (2.11), direct measurements of the magnetic field gradient are not considered available. Instead, the gradient is computed by the observer. Two configurations are mentioned at the end of Ref. [Vissière et al. 2007a. Sec.III.C] about experimental results: one with a single 3 -axis magnetometer, and one with four 3-axis magnetometers. Using a single 3 -axis magnetometer is not sufficient to provide direct measurements of , and is outside the scope of this work;MIDR was probably not yet outlined at the time of writing.
    与参考文献2.2.2 中的 MIDR 定义。虽然它使用了公式 (2.11) 的一个实例，但并未考虑直接测量磁场梯度。相反，梯度是由观测者计算得出的。参考文献[Vissière et al.[Vissière 等人，2007a. Sec.III.C]的实验结果：一种是使用单个三轴磁力计，另一种是使用四个三轴磁力计。使用单个三轴磁力计不足以提供 的直接测量结果，也不在本文的研究范围之内；在撰写本文时，MIDR 可能还未被概述。
  • No clear justification is provided about the representativeness of the associated dynamic model ; The single-3-axis-magnetometer configuration is probably the only reason why is considered as a part of the state vector, also when observability is assessed.
    对于相关动态模型 的代表性，没有提供明确的理由；单三轴磁力计配置可能是 被视为状态矢量一部分的唯一原因，在评估可观测性时也是如此。
  • The model description is completely invariant with respect to a change of heading, which casts some doubts over conclusions of Ref. Vissière et al. 2007a Sec.II] concerning heading observability.
    模型描述在改变航向时完全不变，这让人对参考文献中关于航向可观测性的结论产生了一些怀疑。Vissière 等人，2007a 第 II 章] 有关航向可观测性的结论产生了一些疑问。

2.4.1. Proposed model 2.4.1.拟议模型

Let us take an intermediate approach between Refs. Vissière et al. 2007a and Dorveaux 2011 under the following guidelines,
让我们在参考文献 Vissière et al.Vissière et al. 2007a 和 Dorveaux 2011 之间的中间方法、
  • Quantities for which direct measurements are available and no representative dynamic model can be evaluated are removed from the state vector;
    从状态向量中删除无法直接测量且无法评估代表性动态模型的量；
  • Attitude is included into the state vector.
    姿态包含在状态向量中。
Since direct measurements of are available and no representative model for is available, the choice is made to use it as an input like in Ref. Dorveaux 2011 instead of as a state like in Ref. Vissière et al. 2007a. Then, strapdown inertial navigation equations are included in the model like in Ref. Vissière et al. 2007a
由于 的直接测量值可用，而 的代表性模型不可用，因此我们选择像参考文献 Dorveaux 2011 那样将其作为输入。Dorveaux 2011，而不是参考文献 Vissière et al.Vissière 等人，2007a。然后，像参考文献 Vissière 等人 2007a 中那样，在模型中加入带下惯性导航方程。Vissière 等人，2007a
instead of using a constant-velocity assumption like in Ref. Dorveaux [2011]. Since direct measurements of and are assumed available, they are not included in the state vector contrary to Ref. Vissière et al. 2007a. The resulting system reads
而不是像参考文献那样使用恒定速度假设。多尔沃[2011]。由于假定可以直接测量 和 ，所以与参考文献相反，没有将它们包括在状态矢量中。Vissière 等人，2007a。结果系统如下
The state dynamics function combines Eqs. (2.12), (2.7) and (2.5) into
状态动力学函数 将公式 (2.12)、(2.7) 和 (2.5) 整合为

2.4.2. Observability 2.4.2.可观察性

Is the system formed by Eqs. observable? Two cases can now be examined: the case of constant inputs, and the time-varying case.
所构成的系统是可观测的吗？现在我们可以研究两种情况：恒定输入和时变输入。

Case of constant inputs: local weak observability
恒定输入情况：局部弱可观测性

Let us examine whether the system formed by Eqs. (2.28) to (2.31) is locally weakly observable. Let us recall the definition that we use.
让我们来看看由式 (2.28) 至 (2.31) 形成的系统是否是局部弱可观测的。让我们回顾一下我们使用的定义。
Definition 4 (Local weak observability Ref. [Besançon, 2007, Sec. 1.2.1, Def. 5]). A system (2.9) is locally weakly observable [resp. at ] if there exists a neighborhood of any [resp. of ] such that for any neighborhood of resp. ] contained in , there is no indistinguishable state from resp. in when considering time intervals for which trajectories remain in .
定义 4（局部弱可观测性 参考[贝桑松，2007，第 1.2.1 节，定义 5]）。如果存在任何 [resp. of ] 的邻域 ，且对于 [resp. of ] 的任何邻域 [resp. at ]，则系统 (2.9) 是局部弱可观测的[res. at ]。 在考虑轨迹停留在 的时间间隔时， 中不存在与 或 无法区分的状态。
There is a systematic way of checking whether our system is locally weakly observable. First, the following definitions are recalled.
有一种系统的方法可以检验我们的系统是否是局部弱可观测的。首先，请回顾以下定义。
Definition 5 (Observation space Ref. [Besançon, 2007, Sec. 1.2.1, Def. 6]). The observation space for a system (2.9) is defined as the smallest real vector space (denoted by ) of functions containing the components of and closed under Lie derivation along for any constant (namely such that for any , , where .
定义 5（观测空间 参考[贝桑松，2007，第 1.2.1 节，定义 6]）。 系统 (2.9) 的观测空间定义为 函数的最小实向量空间（用 表示），其中包含 的分量，并在任何常数 （即对于任何 , , .
Definition 6 (Observability rank condition [resp. at ] Ref. [Besançon, 2007, Sec. 1.2.1, Def. 7]). A system (2.9) is said to satisfy the observability rank condition [resp. at ] if:
定义 6（可观测性等级条件 [resp. at ] 参考文献[贝松，2007 年，第 1.2.1 节，定义 7]。[贝桑松，2007 年，第 1.2.1 节，定义 7]）。如果满足以下条件，则称系统 (2.9) 满足可观测等级条件 [resp. at ]：
where is the set of with .
其中 是 与 的集合。
Then, the following theorem is recalled.
然后，再回顾一下下面的定理。
Theorem 2.4.1 (Ref. Besançon, 2007, Sec.1.2.1, Thm.1]). A system (2.9) satisfying the observability rank condition at is locally weakly observable at . More generally, a system (2.9) satisfying the observability rank condition is locally weakly observable.
定理 2.4.1（参考文献：Besançon, 2007, Sec.1.2.1, Thm.1]）。在 满足可观测性等级条件的系统 (2.9) 在 是局部弱可观测的。更一般地说，满足可观测性等级条件的系统 (2.9) 是局部弱可观测的。
Conversely, a system (2.9) locally weakly observable satisfies the observability rank condition in an open dense subset of .
反之，局部弱可观测的系统 (2.9) 在 的开放稠密子集中满足可观测性等级条件。
In order to compute the system's observation space, the way to take Lie derivatives on the group of rotations needs to be defined, because . To that end, let us choose any element
为了计算系统的观测空间，需要定义在旋转组 上取李导数的方法，因为 。为此，让我们任选一个元素
The exponential mapping from the Lie algebra to the Lie group defined by
从李代数 到李群 的指数映射定义如下
can be used as a local parameterization of around for the Rodrigues rotation vector in a neighborhood of 0 . Let us redefine as
可以作为 在 周围的局部参数化，用于 0 附近的罗德里格斯旋转矢量 。让我们将 重新定义为
Eq. 2.31) becomes 公式 2.31）变为
by invariance of the vector product by a direct isometry. This equation can be transformed into (see Ref. Jiang and Lin 1991 for full derivation)
通过直接等距的向量积不变性。这个方程可以转化为（完整推导见参考文献 Jiang 和 Lin 1991）
which, at simplifies further into
，进一步简化为
The system's observation space is spanned by
系统的观测空间跨度为
Thus, 因此、
It is clear that because is rank 2 . More precisely, is unobservable in the direction of , which means that heading is not locally weakly observable. Moreover, velocity is locally weakly observable if and only if is non-singular. If is singular, tilt might not even be observable; in the worst case, , and only the magnetic field is observable.
显然， ，因为 是秩 2。更确切地说， 在 方向上是不可观测的，这意味着航向不是局部弱可观测的。此外，当且仅当 是非奇异值时，速度才是局部弱可观测的。如果 是奇异的，那么倾斜可能都无法观测；在最糟糕的情况下， 、 ，只有磁场是可观测的。

Time-varying case 时变案例

Let us consider once more the system formed by Eqs. (2.28) to (2.31). Let us prove that if the spatial gradient of the magnetic field is non-singular, then the system is observable up to a heading difference.
让我们再次考虑由式 (2.28) 至 (2.31) 构成的系统。让我们证明，如果磁场的空间梯度是非奇异的，那么系统在航向差以内是可观测的。
Let us denote 记为
Let us denote and two solutions of Eqs. (2.28) to (2.31) with their respective initial conditions and for a given input such that
让我们把 和 分别表示为方程 (2.28) 至 (2.31) 的两个解，它们的初始条件分别为 和 ，对于给定的输入 ，使得
It is assumed that is non singular and that , for . Let us prove that
假定 是非奇数， ，为 。让我们证明
The equality is obviously true, because . The equality also holds, under the assumption that is of class at least and that is non-singular, because
等式 显然为真，因为 。假设 至少属于 类，且 是非奇数，那么 的等式也成立，因为
for all . Taking the time-derivative of Eq. (2.36) and using Eq. (2.30) simplifies into
对于所有 。对式 (2.36) 取时间导数并利用式 (2.30) 简化为
Because of Eq. (2.31),
由于公式 (2.31)、
It follows that the above equation holds if and only if , which concludes the proof.
因此，当且仅当 时，上式成立，证明结束。

2.4.3. Conclusion 2.4.3.结论

Tilt and velocity are observable if is non-singular In the simple model proposed above, if is non-singular, then everything but heading is observable, even for accelerated trajectories. While this result holds in the case of constant inputs, varying inputs do not make heading any more observable, because this issue stems from the symmetry of the dynamic model with respect to a change of heading. This symmetry comes from the assumption that is inertial (see Appx. A. Regarding the rest of the state, however, looking deeper at the time-varying case might be useful in cases where is singular.
如果 是非矢量的，则倾角和速度都是可观测的 在上文提出的简单模型中，如果 是非矢量的，则除了航向之外的一切都可观测，即使是加速轨迹也是如此。虽然这一结果在输入不变的情况下成立，但输入的变化并不会使航向变得更可观测，因为这一问题源于动态模型与航向变化的对称性。这种对称性来自于 是惯性的假设（见附录 A）。不过，在 是奇异的情况下，深入研究时变情况可能会对其他状态有所帮助。
The above condition might be restrictive According to Ref. Batista et al. [2013, the condition that be non-singular is unnecessarily restrictive. To support this statement, this reference provides several models that differ from the one discussed in Sec. 2.3 only by the definition of : constant or constant . The idea is that if the magnetic field is constant in one direction in , and if is known at all times, without hypothesis about the evolution of , movements along this direction are indistinguishable. However, if it is known that "rotates", unobserved components of the velocity vector at a given date (along the direction of constant field) can be recovered by using measurements at a later date, once they have turned to another direction. This behavior is not captured using the local weak observability criterion, for neither does it consider variations of input nor does it consider observability over a time interval.
参考文献 Batista 等人[2013]认为，上述条件可能具有限制性。Batista 等人[2013]认为， 非矢量的条件是不必要的限制。为了支持这一说法，该参考文献提供了几个模型，这些模型与第 2.3 节中讨论的模型仅在 ：恒定 或恒定 的定义上有所不同。我们的想法是，如果磁场在 的一个方向上是恒定的，而且如果 在任何时候都是已知的，那么在不假设 的演变的情况下，沿着这个方向的运动是无法区分的。但是，如果已知 是 "旋转 "的，那么在某一特定日期（沿恒定磁场方向）的速度矢量中未观测到的分量，一旦转向另一方向，就可以通过以后的测量来恢复。本地弱可观测性准则无法捕捉到这种行为，因为它既不考虑输入的变化，也不考虑时间间隔内的可观测性。

2.5. Low-cost inertial sensors: sensor biases, observability implications
2.5.低成本惯性传感器：传感器偏差、可观测性影响

One of the main drawbacks of low-cost MEMS inertial sensors is that their output is not accurate. In particular, the mean value of measurement noise, or bias, cannot be neglected in inertial navigation applications where performance is required (see, e.g., Ref. Woodman [2007]). Bias can be seen as an unknown constant parameter; but since it can change with time, it is arguably appropriate to consider it as part of the state vector.
低成本 MEMS 惯性传感器的主要缺点之一是输出不准确。特别是在要求性能的惯性导航应用中，测量噪声的平均值或偏差不容忽视（参见参考文献 Woodman [2007]）。偏差可视为一个未知的常量参数，但由于它会随时间变化，因此将其视为状态矢量的一部分也是合适的。

2.5.1. Proposed model 2.5.1.拟议模型

Let us consider joint attitude, velocity and inertial sensors biases estimation, with the assumption that (the Earth is not rotating). The resulting system reads
让我们考虑联合姿态、速度和惯性传感器的偏差估计，假设 （地球不自转）。结果系统如下
with inertial sensors measurements modeled by
惯性传感器测量的模型为
The state dynamic function reads
状态动态函数 如下
Remark This system is bilinear in the state because of gyrometer biases.
备注 由于陀螺仪的偏差，该系统的状态是双线性的。

2.5.2. Observability 2.5.2.可观察性

This section studies the system formed by Eqs. . The aim is to prove that it is not observable in the sense of Def. 33 such that even if heading is ignored
本节研究由式 形成的系统。目的是证明该系统在定义的意义上是不可观测的。33 的意义上的不可观测性，即使忽略航向
and the gradient of the magnetic field is non-singular, there exists indistinguishable trajectories. To do that, Def. 3 is used directly, as in Sec. 2.4.2 It is equivalent to check whether or not given and , there exists a unique solving the system.
而磁场梯度是非正弦的，则存在不可区分的轨迹。为此，可以直接使用定义3 可直接使用，如第 2.4.2 节所述，这等同于检查给定 和 时，是否存在唯一的 来求解系统。
Let and be two solutions of Eqs. (2.37)- 2.44 for for a given input such that
设 和 是公式 (2.37)-2.44 的两个解，即 对于给定输入 ，使得
Let us assume that is non singular for and that , for . Let us denote
我们假设 对 而言是非奇异的， ，对 而言是非奇异的。记为
If there exists a solution for different from identically, then the system is not observable.
如果存在 与 完全不同的解，则系统不可观测。
The equation forces . Then, Equation 2.40 yields
方程 迫使 。然后，方程 2.40 得出
It appears that this time, contrary to the system studied in Sec. 2.4.2, simply taking the time derivative of the output does not provide an unambiguous value of the velocity. Let us find out whether this ambiguity remains in the rest of the model, by looking for constant solutions for and .
与第 2.4.2 节中研究的系统相反，这一次仅仅求输出的时间导数似乎并不能提供明确的速度值。让我们通过寻找 和 的恒定解，看看模型的其余部分是否还存在这种模糊性。
To that end, , and are assumed to describe a static trajectory. In particular, they are all constant, and
为此，假定 , 和 描述的是静态轨迹。特别是，它们都是常数，并且
Equation (2.41 now yields
方程 (2.41) 现在得出
In order for to be identically zero, it is enough that be identically zero, or equivalently, that always leave invariant. Because of Eq. 2.42 ,
为了使 同等于零，只要 同等于零就足够了，或者说， 始终保持 不变。由于公式 2.42 、
In order for to leave invariant, it is enough to choose any , and such that
为了使 离开 不变，只需选择任意的 , 和 ，使得
In conclusion, one can choose such that Equation (2.47) is always true, because then, the right hand side of Eq. (2.49) can be arbitrarily set to 0 identically by the appropriate choice of . A family of trajectories described by the same inputs and outputs as a static one in has successfully been found. The reader can check that they are shaped as vertical helices. Thus, the model formed by Eqs. 22.37 is not observable in the sense of Def. 3
总之，我们可以选择 ，使式 (2.47) 始终为真，因为这样一来，式 (2.49) 的右边就可以通过适当选择 任意地设为 0。在 中，我们成功地找到了与静态轨迹相同的输入和输出所描述的轨迹族。读者可以查看这些轨迹的垂直螺旋形状。因此，由式 22.37 形成的模型 在定义的意义上是不可观测的。3

2.5.3. Conclusion and remarks
2.5.3.结论和评论

Unobservability of static trajectories If one takes into account inertial sensor biases, there exist some indistinguishable trajectories. The unobservable space is large and spans the attitude matrix, gyrometer and accelerometer biases, and velocity. Static trajectories (constant attitude, zero velocity) belong to this set of trajectories. This is problematic because this means that for a trajectory to be observable, it is necessary, but not sufficient, that the tracked object be continually moving. Imposing this constraint is impractical in a realistic use case.
静态轨迹的不可观测性 如果考虑到惯性传感器的偏差，就会存在一些无法区分的轨迹。不可观测空间很大，跨越姿态矩阵、陀螺仪和加速度计偏差以及速度。静态轨迹（姿态恒定、速度为零）就属于这一组轨迹。这就有问题了，因为这意味着要观测一个轨迹，被跟踪物体必须持续运动，但这还不够。在实际应用中，施加这种限制是不切实际的。
Implications on the choice of inertial sensors This issue stems entirely from inertial sensor biases uncertainty because otherwise, no such issue occurs (see. Sec. 2.4). In practice, prior knowledge of biases should be exploited to help limit trajectory reconstruction uncertainty. This means that inertial sensors must be chosen for their bias stability.
对选择惯性传感器的影响 这个问题完全源于惯性传感器偏差的不确定性，否则就不会出现这种问题（见第 2.4 节）。在实践中，应利用对偏差的事先了解来帮助限制轨迹重建的不确定性。这意味着必须选择偏差稳定的惯性传感器。
Implications on attitude estimation Attitude and heading observers, also in the static case, display the same observability issue as this model: accelerometer biases and tilt errors are indistinguishable, and gyrometer biases interpreted as vertical (causing a heading drift) are unobservable. Knowing the magnetic field has not improved the situation. For gyrometer biases, this issue can be solved whenever the term can be neglected, in approaches such as Ref. Afzal et al. 2011b that rely on the fact that the direction of the magnetic field is constant (but unknown) in the inertial frame in such cases.
对姿态估计的影响 同样在静态情况下，姿态和航向观测器显示出与该模型相同的可观测性问题：加速度计偏差和倾斜误差无法区分，而被解释为垂直（导致航向漂移）的陀螺仪偏差则无法观测。了解磁场并不能改善这种情况。对于陀螺仪偏差，只要可以忽略 项，这个问题就可以得到解决，如参考文献中的方法。阿夫扎尔等人 2011b 的方法，在这种情况下，磁场方向在惯性框架中是恒定的（但未知）。
Unobservability of heading Despite the fact that static trajectories (having constant input and output) belong to an indistinguishable set, trajectories outside of this set (e.g., with non-constant inputs) might belong to indistinguishable sets of lower dimension. However, for symmetry reasons, one still cannot expect heading to be observable: in particular, for a given input, one can change the initial heading with no effect on the output. To solve this problem, the model has to be changed.
航向的不可观测性 尽管静态轨迹（输入和输出恒定）属于不可区分集合，但该集合之外的轨迹（如输入非恒定）可能属于维度更低的不可区分集合。然而，由于对称性的原因，我们仍然不能期望航向是可观测的：特别是，对于给定的输入，我们可以改变初始航向，但对输出没有影响。要解决这个问题，必须改变模型。
Higher order magnetic models Information about velocity and gyrometer biases may be recovered using more magnetic sensors. For example, if the second spatial derivative of the magnetic field is known and used as an input, then can be added to the state vector and moved from the input vector to the output vector because an equation similar to 2.40 can be written. In the special case where this second spatial derivative is zero, then it is expected that only rotates at the rate . Measuring then gives access to the full angular velocity vector , solving the unobservability problem in the above-mentioned static case for velocity and gyrometer biases. If the second spatial derivate is not zero, one may expect the observability issue to remain solved; more work on this subject is needed to draw a definite conclusion in the general case. Ref. Skog et al. 2018, published during the redaction of this document, does prove this result in a discrete-time Bayesian filtering framework.
高阶磁场模型 可以使用更多的磁传感器来恢复速度和陀螺仪偏差的信息。例如，如果已知磁场的二次空间导数并将其作为输入，那么 就可以添加到状态矢量中，并从输入矢量移动到输出矢量，因为可以写出类似 2.40 的方程。在二阶空间导数为零的特殊情况下，预计 只以 的速率旋转。测量 就可以获得完整的角速度矢量 ，从而解决上述静态情况下速度和陀螺仪偏差的不可观测问题。如果第二个空间导数不为零，则可观测性问题仍可解决；要在一般情况下得出明确结论，还需要做更多相关工作。参考文献Skog等人2018年在本文编辑期间发表的论文确实在离散时间贝叶斯滤波框架中证明了这一结果。
Remark Ref. [Dorveaux 2011, Appx. D] proves the observability of accelerometer biases under the assumption that the attitude matrix is known. This assumption is not taken in this section, hence the different conclusion.
备注 参考文献[Dorveaux 2011, Appx. D]证明了加速度计偏差的可观测性，前提是姿态矩阵 已知。本节没有采用这一假设，因此得出了不同的结论。

2.6. Conclusion 2.6.结论

Chosen models In this chapter, a few existing models of the state of the art have been reviewed, and several models suitable for MIDR have been proposed. The two candidates considered as the best are proposed in Sec. 2.4 and 2.5 respectively. They are considered as the best models in that they are the most representative, compared with existing models proposed in the state of the art, and for their straightforwardness.
所选模型 在本章中，我们回顾了现有的一些先进模型，并提出了几个适合 MIDR 的模型。第 2.4 和 2.5 节分别提出了两个最佳候选模型。它们之所以被认为是最佳模型，是因为与现有的技术模型相比，它们最具代表性，而且简单明了。
Observability issues Neither of these models is observable in the sense of Def 3 More precisely, the heading is unobservable in both cases. Moreover, it has been proven that with low-cost MEMS sensors, static trajectories belong to an unobservable set of helical trajectories of dimension at least 4 , with attitude and the component of gyrometer biases corresponding to a heading drift both unobservable, with their associated values of velocity and accelerometer biases. These issues may be solved with hardware for which second-order spatial derivatives of the magnetic field can be measured directly. Since such hardware was never going to be available during this work, this case was not studied in this chapter.
更确切地说，在这两种情况下，航向都是不可观测的。此外，事实证明，使用低成本的 MEMS 传感器，静态轨迹属于不可观测的螺旋轨迹集，其维度至少为 4，姿态和与航向漂移相对应的陀螺仪偏差分量都是不可观测的，速度和加速度计偏差的相关值也是不可观测的。这些问题可以通过可以直接测量磁场二阶空间导数的硬件来解决。由于在这项工作中不可能有这样的硬件，因此本章没有研究这种情况。
Heading estimation issues Previous works on MIDR have mainly ignored the issue of (true or magnetic) heading estimation, in spite of the fact that it is one of the primary sources of error in dead-reckoning applications, even in Pedestrian Dead Reckoning (PDR) Solving this problem requires a model that is not symmetrical with respect to a change of initial heading. With accurate inertial sensors, this can be done by using a rotating Earth model. With low-cost MEMS sensors, this is for now unreachable, and the problem stays open.
航向估计问题 以前的 MIDR 研究主要忽略了（真实或磁性）航向估计问题，尽管它是惯性导航应用中的主要误差来源之一，甚至在行人惯性导航（PDR）中也是如此。使用精确的惯性传感器，可以通过使用旋转地球模型来解决这个问题。但对于低成本的微机电系统传感器来说，目前还无法实现这一目标，因此这一问题仍悬而未决。
Priorities Previous works have mostly focused on the ability to recover velocity and on cases in which the magnetic field gradient is singular. This study, in particular, Sec. 2.5 shows that this issue is secondary before the observability of inertial sensor biases, attitude and heading. These difficulties arise even before taking into account other measurement errors, such as the effect of uncertainties on the spatial gradient of the magnetic field on estimation, which remain to be addressed in later chapters. Despite all these difficulties, there remains some hope that generic trajectories be observable, and that an observer may be able to compute them.
优先事项 以前的工作主要集中在恢复速度的能力和磁场梯度为单数的情况。本研究，特别是第 2.5 节表明，在惯性传感器偏差、姿态和航向的可观测性之前，这个问题是次要的。甚至在考虑其他测量误差（如磁场空间梯度的不确定性对估算的影响）之前，这些困难就已经出现了，这些问题将在后面的章节中讨论。尽管存在这些困难，但仍有希望观测到一般轨迹，而且观测者可能能够计算出这些轨迹。
Remark Let us remark that position has not been considered, because, in all of these models, the magnetic field output does not depend on it, but on time. Therefore, a change in initial position has no influence on the output. This characterizes deadreckoning. A method must then be chosen in order to integrate the velocity, in a way that is robust to unknown initial conditions or to time-intervals during which observability is lost.
备注 在所有这些模型中，磁场输出并不取决于位置，而是取决于时间，因此没有考虑位置。因此，初始位置的变化对输出没有影响。这就是死循环的特点。因此，必须选择一种对速度进行积分的方法，这种方法对未知的初始条件或失去可观测性的时间间隔具有鲁棒性。
  1. The earth's angular velocity is
    地球的角速度为
  2. This change of frame is not objective either
    这种框架的变化也不客观
    Relativity and electrodynamics allow writing 2 (2.8) as an approximation in in absence of electric field and in otherwise, where is the speed of light. Even for as high as , neglecting this term is quite enough for use in pedestrian applications. By contrast, the same change of variable would not be valid with the electric field.
    相对论和电动力学允许将 2 (2.8) 写成近似值，在没有电场的情况下以 表示，在其他情况下以 表示，其中 是光速。即使 高达 ，忽略这一项也足以在行人应用中使用。相比之下，同样的变量变化在有电场的情况下是无效的。
  3. Let us also remark that in this context, the magnetic field is usually used as a heading reference in outdoor environments. More sophisticated algorithms are used indoors to account for ubiquitous magnetic disturbances, see Refs. Afzal et al. 2011b, Renaudin and Combettes 2014.
    我们还要指出，在这种情况下，磁场通常用作室外环境中的航向参考。在室内则使用更复杂的算法来考虑无处不在的磁场干扰，参见参考文献：Afzal 等人，2011b；Renaudin 和 Combettes，2014 年。Afzal 等人，2011b；Renaudin 和 Combettes，2014。