Basis (linear algebra)

[基]：线性代数里面的概念

In mathematics, a set B of vectors in a vector space V is called a basis1 if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components2 or coordinates3 of the vector with respect to B. The elements of a basis are called basis vectors4. An ordered basis is also called a frame5, a word commonly used, in various contexts, for referring to a sequence of data allowing defining coordinates.

Coordinate System 坐标系统，是描述物质存在的空间位置（坐标）的参照系，通过定义特定基准及其参数形式来实现。坐标是描述位置的一组数值，为了描述或确定位置，必须建立坐标系统，坐标只有存在于某个坐标系统才有实际的意义与具体的位置。

百度百科存在问题。In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold 6 such as Euclidean space.

这里的坐标系统应该只包含原点、方向、坐标（单位长度，数值），而不包含参考点。坐标系统的分类如下：

  • Number line
  • Cartesian coordinate system
  • Polar coordinate system
  • Cylindrical and spherical coordinate systems

Cartesian Coordinates 笛卡尔坐标系

仿射坐标系 Affine Coordinate System

Moving Frame 活动标架，是微分流形的一组局部基。n维微分流形M中开集U上的n个线性无关光滑向量场所构成的模笋(U)的一组基，称为活动标架。这样一组基是局部的，而整体上可能是不存在的。

参考系：被选作参考的物体或物体组（坐标原点）

坐标系：目的在于将矢量变为分量形式（数），以便定量计算

参考坐标系：Frame of refernce (or reference frame)

惯性参考坐标系：Inertial frame of reference

coordinate frames

inertial frame

vehicle frame

body frame

stability frame: Stability frame helps us rigorously define angle of attack and is useful　for analyzing stability of aircraft

Wind frame: Wind frame helps us rigorously define side-slip angle Side-slip angle is nominally zero for tailed aircraft

1
备注：文中是英文链接；后面引用是中文链接及相关解释。

地球参考框架与坐标系——同济大学测量系

坐标系定义要素：原点origin；指向orientation。

这里所述的概念不是很清晰，故不再说明。

常用的四大坐标系

  • 惯性坐标系（i系）。原点位于地球中心，坐标轴相对于恒星无转动，轴向定义为OX；、OY；、OZ；。其中OZ，的方向与地球极轴的方向一致（假定极轴方向保持不变），OXi、OYi在地球赤道平面内。
  • 地球坐标系（e系）。原点位于地球中心，坐标轴与地球固连轴向定义为OXe、OYe、OZe，其中OZe沿地球极轴方向，OXe 轴沿格林尼治子午面和地球赤道平面交线。地球坐标系相对于惯性坐标系绕Ozi轴以角速度众转动。
  • 导航坐标系（n系）。是一种当地地理坐标系，原点位于导航系统所处的位置P点，坐标轴指向北、东和当地垂线方向（向下）。导航坐标系相对于地球固连坐标系的旋转角速率w_en取决于P点相对于地球的运动，通常称为转移速率。
  • 载体坐标系（b系）。它是一个正交坐标系，轴向分别沿安装有导航系统的运载体的横滚轴、俯仰轴和偏航轴。

注释：因此水下机器人所用的惯性坐标系其实是导航坐标系，只是忽略地球的转动，所以也就将惯性坐标系、地球坐标系和导航坐标系进行的统一，一般统称为惯性坐标系。 哥氏加速度：从运动学知，当动点相对某一动参考系作相对线运动，同时该动系又在作转动运动时，则动点会受到哥氏加速度。

reference frame 数学：参考标架 物理学：参考坐标系 地球科学：参考框架

2022-10-31

Source 1:

Reference Frame: A reference frame specifies a reference for the calculation of velocities and accelerations of a system

A coordinate system is essentially a measuring stick used to define kinematic and dynamic quantities. Coordinate systems usually consist of an origin and a set of three mutually orthogonal unit vectors that specify three directions. Associated with each of these directions is a “scale” that permits measurement of quantities in the coordinate system. You can specify all kinematic and kinetic vector quantities as projections of the vector along the three unit vectors , as shown in Figure 5. {i}=(x,y,z)

Chapter 18 Reference Frames and Coordinate Systems

Federation of American Scientists, 美国科学家联盟

Frames of reference are characterized by several major properties: an origin, reference lines, reference planes, and stability.

https://www.fsd.ed.tum.de/research/

Principles of Naval Weapons Systems David R. Frieden; Gene P. Bender et al Published by US Naval Institute Press, 1985

Principles of Naval Weapon Systems Craig M. Payne Published by Naval Inst Pr, 2006

Principles of Naval Weapons Systems Edited by CDR Joseph Hall, USN

  • Coordinate System = A System of Coordinates → Coordinates
  • Frame of reference = A Reference Frame →

Frames of reference are characterized by several major properties: an origin, reference lines, reference planes, and stability. A truly stable or inertial reference frame is one that is held fixed or nonrotating with respect to distant stars. In this type of reference frame, target motion is very easily described since all changes in target position within the frame are due only to motion of the target itself. An unstable or non-inertial reference frame is one that is constantly rotatingin a random manner; therefore, changes in target position within such a reference frame are the result of both target motion and reference frame rotation.

Coordinate conversion 坐标转换，类似于单位转换，能量转换

Coordinate conversion is the process of changing from one system of coordinates that describe a point within a reference frame to another system of coordinates that describe the same point in the same reference frame.

Coordinate Transformation (Reference Frame Rotation) 坐标变换，参考系（坐标系）旋转

The next step in the process of stabilizing the target coordinates is transforming target coordinates from one reference frame to another using the same coordinate system. 数学变换方法 具体地讲，将复杂的问题通过变换转化成简单的问题；将难的问题通过变换转化成容易的问题；将未解决的问题通过变换转化成已解决或较易解决的问题。它是解决数学问题中常用的最基本的方法之一变换的形式有：传递形式的变换、符号表达方式的变换、空间关系的变换等。

Reference Frame Translation 参考系平移

Reference frame translation can be defined as "the location of a point (target) with respect to two different reference frames whose origins are physically displaced from one another."

参考系的旋转和平移，类似于刚体的旋转与平移。

Power Transmission, Transformation, and Conversion 能量的传输，传递，转换 A “simple machine,” such as a lever, transforms the magnitudes of the forces and velocities at its points of action but does not change the mechanical power, product of force and velocity. Modern machinery both transforms power and converts power from one type to another.

坐标系是数学上的概念，而参考系是物理上的概念。 Coordinate system 坐标系 coordinate transformation：（wiki上的结果与上述不同） The relationship between different systems is described by coordinate transformations, which give formulas for the coordinates in one system in terms of the coordinates in another system. Spatial reference system 空间参考系 A spatial reference system (SRS) or coordinate reference system (CRS) is a framework used to precisely measure locations on the surface of the Earth as coordinates. It is thus the application of the abstract mathematics of coordinate systems and analytic geometry to geographic space.

以下这段文字是集合了测量系统和空间参考系的一个组合坐标系统 These standards acknowledge that standard reference systems also exist for measuring elevation using vertical datums and time (e.g. ISO 8601), which may be combined with a spatial reference system to form a compound coordinate system for representing three-dimensional and/or spatio-temporal locations. There are also internal systems for measuring location within the context of an object, such as the rows and columns of pixels in a raster image, Linear referencing measurements along linear features (e.g., highway mileposts), and systems for specifying location within moving objects such as ships. The latter two are often classified as subcategories of engineering coordinate systems.

Reference Frame In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points―geometric points whose position is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers).

The idea of a reference frame is really quite different from that of a coordinate system. Frames differ just when they define different spaces (sets of rest points) or times (sets of simultaneous events).

Frames of reference and coordinate systems - YouTube

欧几里得变换（Euclidean transformation）

欧几里得变换（Euclidean transformation）详解

物理学咬文嚼字之六十六：参照系？坐标系!

参照框架引入物理学是一种物理的必然，而坐标系作为一种数学手段为单一参照框架下对运动的描述，以及为不同参照框架中运动甚至物理定律的变换研究，提供了手段。理解了参照框架的物理属性与坐标系的数学属性，物理学的天空可能会清朗一些，至少相对论的文本看起来不再那么云山雾罩。

Are ECI and ECEF both frames and/or coordinate systems? Is there a difference?

A reference frame (or simply "frame") is specified by an ordered set of three mutually orthogonal, possibly time dependent, unit-length direction vectors. A reference frame has an associated center. In some documentation external to SPICE, this is called a “coordinate frame.” A coordinate system specifies a mechanism for locating points within a reference frame.

这里给出的结果与我所想的问题一致：any reference frame can locate the point with any coordinate system. 在物理参考系下可以通过几何、代数的方法确定一个点（矢量）；有大小和方向的向量/建立坐标系。

矢量与向量

矢量：有大小和方向的物理量。 向量：数学当中的概念， 是注意区别物理中的矢量与自由向量的区别。比如力矢量不同于自由向量，它不仅包括大小方向，还有作用点。数学的大小相等方向相同的2个自由向量是相等的，因为自由向量可以平移。但是大小相等方向相同的2个力，如果作用点不同，那么它们不相等。 矢量是流形上一点的切空间上的向量，同时强调切点(箭尾)和向量本身（箭尾->箭头），视野是整个流形。向量的视野限于一个给定基点的切空间，且此给定基点是作为上下文被省略，这样只需要用点来表示了。

参考书籍： ADAMS mss_Mechanical System Simulation Knowledge Base JSBSim Reference Manual Unmanned_Rotorcraft_Systems_(Guowei_Cai,Ben_M._Chen,Tong_Heng_Lee(auth.))(z-lib.org) 无人驾驶旋翼飞行器系统 蔡国玮

机器人运动学基础

  1. 基，在线性代数中，基（也称为基底）是描述、刻画向量空间的基本工具。↩︎

  2. 分量。↩︎

  3. 坐标。↩︎

  4. 基矢量。↩︎

  5. 标架 ，标架亦称坐标系。几何学的基本概念，n 维仿射空间中的一个定点 O 连同一组有序基 e1,e2,...,en合在一起，称为空间的一个仿射标架或仿射坐标系，记为 {Q;e1,e2,...,en}。可以为复数，指多种不同的坐标系。↩︎

  6. 流形，是局部具有欧几里得空间性质的空间，在数学中用于描述几何形体。物理上，经典力学的相空间和构造广义相对论的时空模型的四维伪黎曼流形都是流形的实例。高维整体，二维的三角形、正方形、圆形可有三维空间的三棱锥、立方体、球得到，也就是说二维的这些图形是三维的局部特征。↩︎