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2.1. Magnetic navigation using observers for non-linear dynamical systems

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
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?
  • What is the adequate state-space model that must be observed?
These questions will be answered more specifically in the scope of low-cost strapdown Microelectromechanical Systems (MEMS) inertial navigation in indoor environments.
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

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 )中的(笛卡尔)坐标用指数 (以及 )表示。例如
denotes the coordinates of a vector in .
表示矢量 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
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 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
The notation is then an alias of .
因此, 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, 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
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 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
with the change of frame
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 。在本章中,如果这些符号不明确,则分别简称为

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 。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 ,那么就必须能够推导出 ,然后再推导出轨迹。

Observer and observability

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 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

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 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 ;
  • 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 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