李群中的三维旋转群

数学中的李群与李代数李群与李代数在数学中扮演着重要的角色。

本文将对李群与李代数的基本概念进行介绍，并讨论它们之间的关系。

一、李群（Lie Group）李群是一种同时具有群结构和流形结构的数学对象。

群结构指的是李群上定义了乘法运算，同时存在单位元、逆元等性质。

而流形结构则是指李群在每个点附近都具有局部同胚于欧几里得空间的性质。

举个简单的例子，旋转矩阵群SO(3)就是一个李群。

它由所有的旋转矩阵组成，而旋转矩阵的乘法运算便构成了群运算。

此外，SO(3)也是一个三维实流形，因为它在每个点附近都可以通过欧几里得空间进行局部的描述。

李群的定义使得我们可以在其上定义微分结构，进而研究其微分几何性质。

比如，我们可以定义李群上的切空间和切丛，进而研究其在每个点上的切向量和切空间的结构。

二、李代数（Lie Algebra）李代数是李群的切空间上的代数结构。

它通常用于描述李群的局部性质。

李代数由向量空间和李括号这两个部分构成。

向量空间是李代数的基础，它的元素被称为李代数的生成元或向量场。

李代数的生成元通常用一组基向量来表示，这些向量之间通过线性组合构成一个线性空间。

李括号则定义了李代数中向量场之间的运算。

对于两个向量场X和Y，李括号[X, Y]被定义为它们的Lie导数的对易子。

李代数的一个经典例子是三维旋转群的李代数so(3)。

它由三个无限小旋转生成元构成，通常记作J₁, J₂和J₃。

它们之间的李括号满足以下关系：[J₁, J₂] = J₃, [J₂, J₃] = J₁, [J₃, J₁] = J₂。

三、李群与李代数的关系李群与李代数之间存在着密切的联系。

事实上，对于任意一个李群，都可以构造出与之对应的李代数。

这个李代数被称为李群的切代数，它反映了李群局部性质的信息。

具体地，李群的切代数可以通过计算李群上的左不变矢量场的李括号来得到。

左不变矢量场在李群的每个点上都是不变的，因此它在整个李群上构成了一个矢量场。

反过来，给定一个李代数，也可以构造出与之对应的李群。

第4章李群李代数⼀、概述1. 李群和李代数的核⼼思想封结⼳逆法则；法则；可以理解为专门⽤于矩阵旋转的东西，符合封结⼳逆1. 可以理解为专门⽤于矩阵旋转的东西，符合，李代数可以理解为旋转向量旋转向量；；李群可以理解为旋转矩阵旋转矩阵，李代数可以理解为2. 李群可以理解为3. 李群是连续群，李代数可以表出李群的导数，所以李代数表⽰的是李群的局部性质；4. 进⽽我们可以理解为：旋转向量表达了旋转矩阵的局部（旋转发⽣那⼀瞬间的领域内）性质；5. 由拉格朗⽇中值定理可知：导数控制函数。

李代数控制李群，\phi控制R；【1】也就是说想要估计出函数值，我们可以研究该函数的导数，⽤来描述某个点领域内性质。

故⽽我们需要建⽴对李群的求导模型，通过分析导数的性质来估计出相机在这⼀时刻（领域内）的位姿。

但是我们知道群是指只有⼀个运算的集合（我们选择矩阵乘法），所以李群不对加法封闭【2】，但是我们知道李代数是建⽴在向量空间上的，⽀持加法运算。

所以我们需要⼀种让李群映射到李代数的机制，然后通过对李代数求导，求出李群的导数。

不过，对李代数求导后的结果⾮常复杂，所以我们需要寻找另外⼀种求导⽅式【3】，这就是我们接下来所要介绍的内容。

【注】【1】：某个名牌⼤学考研的复试题——你知道导数的作⽤是什么吗？【2】：李群也是⼀种群。

甭跟我扯什么鳄鱼不是鱼、⽇本⼈不是⼈。

【3】：对谁求导不重要，因为我们总可以通过这个导数控制相同的函数。

2. 李群的两种求导模型（都是映射到了李代数空间）1. BCH公式线性化（将李群的变化与李代数的变化联系起来）；；（复杂）求导模型；（复杂）2. 对李代数求导的对李代数求导的求导模型1. 需要求出左右雅可⽐矩阵的逆；扰动模型；（精简）；（精简）对微扰动求导的扰动模型3. 对微扰动求导的1. 不需要求出左右雅可⽐矩阵的逆；3. 这两种求导模型都是会有误差存在的4. 李群和李代数的基础符号1. 特殊正交群SO(3)，特殊欧式群SE(3)；2. 特殊正交群上的李代数\mathfrak{so}(3)，这⾥我们具象化为三维\phi向量或者反对称阵\widehat{\phi}；3. 特殊欧式群上的李代数\mathfrak{se}(3)，这⾥我们具象化为六维\xi向量或者四维⽅阵\widehat{\xi}；\rho表⽰三维空间中的平移，\phi表⽰三维空间中的旋转。

李群李代数在数据融合算法中的应用分析作者：袁治晴来源：《电脑知识与技术》2019年第08期摘要：数据融合是提升机器人、无人驾驶、无人机等应用能力的重要手段，一直是前沿技术中研究的一个热点，关于数据融合算法的分析设计，学术界和工程界对此方面进行了长期的研究与讨论，而数据融合算法结合李群李代数一直是此领域研究的一个热点，本文将对李群李代数应用于数据融合的算法进行分析，展开对基于李群李代数的扩展卡尔曼滤波系统模型的分析，并实验对比得出将李群李代数应用于数据融合算法在提升精度方面起着的重要一环。

关键词：数据融合；李群李代数；扩展卡尔曼滤波中图分类号：TP311 文献标识码：A文章编号：1009-3044（2019）08-0192-021 引言数据融合一直是研究领域的一个热点，是针对一个系统使用多种传感器这一特定问题而展开的一种关于数据处理的研究。

数据融合技术是多学科交叉的新技术，涉及概率统计、信息论、人工智能、模糊数学等理论。

提升精度是数据融合设计中的关键一环，而李群李代数理论正好提供了这样的工具，随着数据融合技术广泛应用于机器人、自动驾驶等计算机视觉领域，李群李代数对数据融合优化改进在越来越重要。

本文的研究方向是对数据融合中滤波算法优化改进进行分析，研究基于李群李代数扩展卡尔曼滤波算法并进行试验对比，进一步证明李群李代数在提升数据融合算法的精度占据重要地位。

2 李群李代数在数据融合系统模型中研究2.1 引言由于该系统模型算法步骤推导过程复杂，受篇幅限制，只给出具体的算法步骤来分析。

首先分析经典的扩展卡尔曼滤波EKF，接着介绍李群李代数基本知识，着重分析李群李代数在数据融合扩展卡尔曼滤波EKF算法的应用分析，为后续进行试验验证对比证明结论奠定了基础。

2.2 扩展卡尔曼滤波卡尔曼滤波算法的核心是动态调整权值，本质是通过预测结合测量来估计当前系统的状态，Kalman Filter用预测和测量进行状态估计，并根据卡尔曼增益K来决定用哪个来估计，当系统为线性高斯模型时，滤波器能给出最优的估计，实际系统存在不同程度的非线性，通过线性化方法将非线性系统转换为近似的线性系统，即为扩展卡尔曼滤波EKF，核心思想是围绕滤波值将非线性函数展开成泰勒级数并略去二阶及以上项，得到一个近似的线性化模型，然后应用卡尔曼滤波完成状态估计。

物理学中的群论——三维转动群主讲翦知渐群论-三维转动群第四章三维转动群三维转动群的表示4.1 维转动群的表示§拓扑群和李群42§4.2轴转动群SO (2)§4.3 三维转动群SO (3)§4.4二维特殊幺正群SU (2)§4.1拓扑群和李群连续群的基本概念1拓扑群无限群分为分立无限群和连续无限群有关有限群的理论对于分立无限群来说几乎全部成立定义4.1 连续群的维数, a2, …, a n所标明连续群G的元素由一组实参数a1其中至少有一个参数在某一区域上连续变化，且该组参数对标明群的所有元素是必需的而且足够的则该组参数中连续参数的个数l 称为连续群的维数。

在具体的群中，参数的取法可能不唯一例子如下的线性变换T(a,b)x'= T(a,b)x = ax +b,a,b∈(-∞,+∞), a≠0构成的集合，定义其上的乘法为：T(a1,b1)T(a2,b2)x = T(a1a2, a1b2+b1)x，b b T封闭律是显然的逆元素为T-1(a,b) = T(1/a, -b/a) ，单位元是T(1,0)结合律也容易证明因此{T(a,b)}构成个连续群。

构成一个连续群。

由于群元素的连续性质，需要在群中引入拓扑由于群元素的连续性质需要在群中引入简单说拓扑是个集子集族简单地说，拓扑是一个集合以及它的子集族拓扑学研究的是某个对象在连续变形下不变的性质为简单起见，我们仅讨论其元素可与l 维实内积空间的某个子有对应关系的群有一一对应关系的群集Sl该子集称为参数空间定义4.2 拓扑群群元的乘法法则和取逆法则在群的所有元素处都连续的群，称为拓扑群定义4.3 简单群和混合群拓扑群G的任意两个元素x1和x2在参数空间中如果能用一条或者多条道路连接（道路连通），则该群的参数空间是连通的，该群称为连通群或简单群。

若群的参数空间形成不相连结的若干片，则该群称为混合群。

前者如三维转动群SO(3)，后者如三维实正交群O(3)。

Euler anglesThisarticle is about the Euler anglesused in mathematics .For the use of thetermin physics and aerospace engi-neering ,see Rigid body dynamics .For chained rotations,chained rotations .Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body .[1]To describe such an orientation in 3-dimensional Euclidean space three parameters are required.They can be given in several ways,Euler angles being one of them;see charts on SO(3)for others.Euler angles are also used to describe the orientation of a frame of reference (typ-ically,a coordinate system or basis )relative to another.They are typically denoted as α,β,γ,or φ,θ,ψ.Euler angles represent a sequence of three elemental rota-tions ,i.e.rotations about the axes of a coordinate system .For instance,a first rotation about z by an angle α,a sec-ond rotation about x by an angle β,and a last rotation again about y ,by an angle γ.These rotations start from a known standard orientation.In physics,this standard initial orientation is typically represented by a motionless (fixed ,global ,or world )coordinate system ;in linear al-gebra,by a standard basis .Any orientation can be achieved by composing three el-emental rotations.The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations )or about the axes of a rotating coordi-nate system,which is initially aligned with the fixed one,and modifies its orientation after each elemental rotation (intrinsic rotations ).The rotating coordinate system may be imagined to be rigidly attached to a rigid body.In this case,it is sometimes called a local coordinate system.Without considering the possibility of using two different conventions for the definition of the rotation axes (intrin-sic or extrinsic),there exist twelve possible sequences of rotation axes,divided in two groups:•Proper Euler angles (z-x-z,x-y-x,y-z-y,z-y-z,x-z-x,y-x-y )•Tait–Bryan angles (x-y-z,y-z-x,z-x-y,x-z-y,z-y-x,y-x-z ).Tait–Bryan angles are also called Cardan angles ;nau-tical angles ;heading,elevation,and bank ;or yaw,pitch,and roll .Sometimes,both kinds of sequences are called “Euler angles”.In that case,the sequences of the first group are called proper or classic Euler angles.1Proper Euler anglesyzProper Euler angles representing rotations about z ,N ,and Z axes.The xyz (original)system is shown in blue,the XYZ (ro-tated)system is shown in red.The line of nodes (N )is shown in green.1.1Classic definitionEuler angles are a means of representing the spatial ori-entation of any reference frame (coordinate system or basis )as a composition of three elemental rotations start-ing from a known standard orientation,represented by another frame (sometimes referred to as the original or fixed reference frame,or standard basis ).The reference orientation can be imagined to be an initial orientationfrom which the frame virtually rotates to reach its ac-tual orientation.In the following,the axes of the origi-nal frame are denoted as x ,y ,z and the axes of the rotated frame are denoted as X ,Y ,Z .In geometry and physics ,the rotated coordinate system is often imagined to be rigidlyattached to a rigid body .In this case,it is called a “local”coordinate system,and it is meant to represent both the position and the orientation of the body.The geometrical definition (referred sometimes as static)of the Euler angles is based on the axes of the above-mentioned (original and rotated)reference frames and an 121PROPER EULER ANGLESadditional axis called the line of nodes.The line of nodes (N)is defined as the intersection of the xy and the XY coordinate planes.In other words,it is a line passing through the common origin of both frames,and perpen-dicular to the zZ plane,on which both z and Z lie.The three Euler angles are defined as follows:•α(orφ)is the angle between the x axis and the N axis.•β(orθ)is the angle between the z axis and the Z axis.•γ(orψ)is the angle between the N axis and the X axis.This definition implies that:•αrepresents a rotation around the z axis,•βrepresents a rotation around the N axis,•γrepresents a rotation around the Z axis.Ifβis zero,there is no rotation about N.As a conse-quence,Z coincides with z,αandγrepresent rotations about the same axis(z),and thefinal orientation can be obtained with a single rotation about z,by an angle equal toα+γ.1.2Alternative definitionThe rotated frame XYZ may be imagined to be initially aligned with xyz,before undergoing the three elemental rotations represented by Euler angles.Its successive ori-entations may be denoted as follows:•x-y-z,or x0-y0-z0(initial)•x’-y’-z’,or x1-y1-z1(afterfirst rotation)•x″-y″-z″,or x2-y2-z2(after second rotation)•X-Y-Z,or x3-y3-z3(final)For the above-listed sequence of rotations,the line of nodes N can be simply defined as the orientation of X after thefirst elemental rotation.Hence,N can be simply denoted x’.Moreover,since the third elemental rotation occurs about Z,it does not change the orientation of Z. Hence Z coincides with z″.This allows us to simplify the definition of the Euler angles as follows:•α(orφ)represents a rotation around the z axis,•β(orθ)represents a rotation around the x’axis,•γ(orψ)represents a rotation around the z″axis.1.3ConventionsDifferent authors may use different sets of rotation axes to define Euler angles,or different names for the same angles.Therefore any discussion employing Euler angles should always be preceded by their definition.[2]Unless otherwise stated,this article will use the convention de-scribed above.The three elemental rotations may occur either about the axes xyz of the original coordinate system,which is assumed to remain motionless(extrinsic rotations),or about the axes of the rotating coordinate system XYZ, which changes its orientation after each elemental rota-tion(intrinsic rotations).The definition above uses in-trinsic rotations.There are six possibilities of choosing the rotation axes for proper Euler angles.In all of them,thefirst and third rotation axes are the same.The six possible sequences are:1.z-x’-z″(intrinsic rotations)or z-x-z(extrinsic rota-tions)2.x-y’-x″(intrinsic rotations)or x-y-x(extrinsic rota-tions)3.y-z’-y″(intrinsic rotations)or y-z-y(extrinsic rota-tions)4.z-y’-z″(intrinsic rotations)or z-y-z(extrinsic rota-tions)5.x-z’-x″(intrinsic rotations)or x-z-x(extrinsic rota-tions)6.y-x’-y″(intrinsic rotations)or y-x-y(extrinsic rota-tions)Euler angles between two reference frames are defined only if both frames have the same handedness.1.4Signs and rangesAngles are commonly defined according to the right hand ly,they have positive values when they rep-resent a rotation that appears clockwise when looking in the positive direction of the axis,and negative values when the rotation appears counter-clockwise.The oppo-site convention(left hand rule)is less frequently adopted. About the ranges:•forαandγ,the range is defined modulo2πradians.A valid range could be[−π,π].•forβ,the range coversπradians(but can't be said to be moduloπ).For example could be[0,π]or [−π/2,π/2].3 Angles of a given frameas columns of a matrixof the theoretical ma-Hence the three Eulerthe same result algebra,which is more with unit vectors(X,Y, be seen that:for Y3,projecting itfirstz and the line of nodes. isπ/2−βand cos(π/2−function,inverse cosine functionargument.In this ge-of the solutions is valid. as a sequence of rota-but there will be only is because the sequence frame is not unique if the[3]be useful to represent42TAIT–BRYAN ANGLES2Tait–Bryan anglesTait–Bryan angles.z-y′-x″sequence(intrinsic rotations;N coin-cides with y’).The angle rotation sequence isψ,θ,Ф.Note that in this caseψ>90°andθis a negative angle.The second type of formalism is called Tait–Bryan an-gles,after Peter Guthrie Tait and George H.Bryan.The definitions and notations used for Tait-Bryan angles are similar to those described above for proper Euler an-gles(Classic definition,Alternative definition).The only difference is that Tait–Bryan angles represent rotations about three distinct axes(e.g.x-y-z,or x-y’-z″),while proper Euler angles use the same axis for both thefirst and third elemental rotations(e.g.,z-x-z,or z-x’-z″).This implies a different definition for the line of nodes. In thefirst case it was defined as the intersection between two homologous Cartesian planes(parallel when Euler angles are zero;e.g.xy and XY).In the second one,it is defined as the intersection of two non-homologous planes (perpendicular when Euler angles are zero;e.g.xy and YZ).Tait–Bryan angles.z-x’-y″sequence(intrinsic rotations;N coin-cides with x’)2.1ConventionsThe three elemental rotations may occur either about the axes of the original coordinate system,which remains motionless(extrinsic rotations),or about the axes of the rotating coordinate system,which changes its orientation after each elemental rotation(intrinsic rotations). There are six possibilities of choosing the rotation axes for Tait–Bryan angles.The six possible sequences are:1.x-y’-z″(intrinsic rotations)or x-y-z(extrinsic rota-tions)2.y-z’-x″(intrinsic rotations)or y-z-x(extrinsic rota-tions)3.z-x’-y″(intrinsic rotations)or z-x-y(extrinsic rota-tions)4.x-z’-y″(intrinsic rotations)or x-z-y(extrinsic rota-tions)5.z-y’-x″(intrinsicrotations)or z-y-x(extrinsic rota-tions):the intrinsic rotations are known as:yaw, pitch and roll6.y-x’-z″(intrinsic rotations)or y-x-z(extrinsic rota-tions)2.2Alternative namesThe principal axes of an aircraftTait-Bryan angles,following z-y’-x″(intrinsic rotations) convention,are also known as nautical angles,because they can be used to describe the orientation of a ship or aircraft,or Cardan angles,after the Italian mathemati-cian and physicist Gerolamo Cardano,whofirst described in detail the Cardan suspension and the Cardan joint. They are also called heading,elevation and bank,or yaw,pitch and roll.Notice that the second set of terms is also used for the three aircraft principal axes.3.2Extrinsic rotations 53Relationship with physical mo-tionsSee also:Givens rotations and Davenport rotations3.1IntrinsicrotationsAny target orientation can be reached,starting from a known reference orientation,using a specific sequence of intrinsic rotations,whose magnitudes are the Euler angles of the target orientation.This example uses the z-x′-z″sequence.Intrinsic rotations are elemental rotations that occur about the axes of the rotating coordinate system XYZ ,which changes its orientation after each elemental rotation.The XYZ system rotates,while xyz is fixed.Starting with XYZ overlapping xyz ,a composition of three intrinsic rotations can be used to reach any target orientation for XYZ .The Euler or Tait-Bryan angles (α,β,γ)are the amplitudes of these elemental rotations.For instance,the target orien-tation can be reached as follows:•The XYZ system rotates by αabout the Z axis (which coincides with the z axis).The X axis now lies on theline of nodes.•The XYZ system rotates about the now rotated X axis by β.The Z axis is now in its final orientation,and the X axis remains on the line of nodes.•The XYZ system rotates a third time about the new Z axis by γ.The above-mentioned notation allows us to summarizethis as follows:the three elemental rotations of the XYZ-system occur about z ,x ’and z ″.Indeed,this sequence is often denoted z-x’-z″.Sets of rotation axes associated with both proper Euler angles and Tait-Bryan angles are commonly named using this notation (see above for de-tails).Sometimes,the same sequence is simply called z-x-z ,Z-X-Z ,or 3-1-3,but this notation may be ambiguous as it may be identical to that used for extrinsic rotations.In this case,it becomes necessary to separately specify whether the rotations are intrinsic or extrinsic.Rotation matrices can be used to represent a sequence of intrinsic rotations.For instance,R =X (α)Y (β)Z (γ)represents a composition of intrinsic rotations about axes x-y’-z″,if used to pre-multiply column vectors ,whileR =Z (γ)Y (β)X (α)represents exactly the same composition when used to post-multiply row vectors .See Ambiguities in the def-inition of rotation matrices for more details.3.2Extrinsic rotationsExtrinsic rotations are elemental rotations that occur about the axes of the fixed coordinate system xyz .The XYZ system rotates,while xyz is fixed.Starting with XYZ overlapping xyz ,a composition of three extrinsic rota-tions can be used to reach any target orientation for XYZ .The Euler or Tait-Bryan angles (α,β,γ)are the ampli-tudes of these elemental rotations.For instance,the tar-get orientation can be reached as follows:•The XYZ system rotates about the z axis by α.The X axis is now at angle αwith respect to the x axis.•The XYZ system rotates again about the x axis by β.The Z axis is now at angle βwith respect to the z axis.•The XYZ system rotates a third time about the z axis by γ.In sum,the three elemental rotations occur about z ,x andz .Indeed,this sequence is often denoted z-x-z (or 3-1-3).Sets of rotation axes associated with both proper Euler angles and Tait–Bryan angles are commonly named usingthis notation (see above for details).Rotation matrices can be used to represent a sequence of extrinsic rotations.For instance,R =Z (γ)Y (β)X (α)64GIMBAL MOTION RELATIONSHIPrepresents a composition of extrinsic rotations about axes x-y-z ,if used to pre-multiply column vectors ,while R =X (α)Y (β)Z (γ)represents exactly the same composition when used to post-multiply row vectors .See Ambiguities in the def-inition of rotation matrices for more details.3.3Conversion betweenintrinsic and ex-trinsic rotations A rotation represented by Euler angles (α,β,γ)=(−60°,30°,45°),using z-x’-z″intrinsic rotationsThe same rotation represented by (γ,β,α)=(45°,30°,−60°),using z-x-z extrinsic rotationsAny extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elementalrotations,and vice versa.For instance,the intrinsic rota-tions x-y’-z″by angles α,β,γare equivalent to the extrin-sic rotations z-y-x by angles γ,β,α.Both are represented by a matrixR =X (α)Y (β)Z (γ)if R is used to pre-multiply column vectors ,and by a ma-trixR =Z (γ)Y (β)X (α)if R is used to post-multiply row vectors .See Ambiguities in the definition of rotation matrices for more details.3.3.1The proof of the conversion in the pre-multiply caseThe rotation matrix of the intrinsic rotation sequence x-y’-z″can be obtained by the sequential intrinsic element rotations from the right to the left:R =Z ′′Y ′X.In this process there are three frames related in the in-trinsic rotation sequence.Let’s denote the frame 0as theinitial frame,the frame 1after the first rotation aroundthe x axis ,the frame 2after the second rotation around the y’axis and the frame 3as the third rotation around z″axis.Since a rotation matrix can be represented among these three frames,let’s use the left shoulder index to denote the representation frame.The following notation means therotation matrix that transforms the frame a to the frame b and that is represented in the frame c :cR a →b .An intrinsic element rotation matrix represented in thatframe where the rotation happens has the same value as that of the corresponding extrinsic element rotation ma-trix:R 1→0=X,1R 2→1=Y,2R 3→2=Z.The intrinsic element rotation matrix Y’and Z″repre-sented in the frame 0can be expressed as other forms:Y ′=0R 2→1=0R 1→01R 2→10R −11→0=XY X −1Z ′′=0R 3→2=0R 1→01R 3→20R −11→0=X (1R 2→12R 3→21R −12→1)X−1=XY ZY −1X −1The two equations above are substituted to the first equa-tion:R =Z ′′Y ′X=(XY ZY −1X −1)(XY X −1)X =XY ZY −1(X −1X )Y (X −1X )=XY Z (Y −1Y )=XY Z Therefore,the rotation matrix of an intrinsic element ro-tation sequence is the same as that of the inverse extrinsicelement rotation sequence:R =Z ′′Y ′X =XY Z.4Gimbal motion relationshipEuler basic motions are defined as the movements ob-tained by changing one of the Euler angles while leav-ing the other two constant.Euler rotations are never ex-pressed in terms of the external frame,or in terms of the co-moving rotated body frame,but in a mixture.Theyconstitute a mixed axes of rotation system,where the first angle moves the line of nodes around the external axis z ,the second rotates around the line of nodes and4.2Intermediate frames7the third one is an intrinsic rotation around an axisfixed in the body that moves.These rotations are called precession,nutation,and intrinsic rotation(spin).As an example,consider a top. The top spins around its own axis of symmetry;this cor-responds to its intrinsic rotation.It also rotates around its pivotal axis,with its center of mass orbiting the piv-otal axis;this rotation is a precession.Finally,the top can wobble up and down;the inclination angle is the nutation angle.While all three are rotations when applied over individual frames,only precession is valid as a rotation operator,and only precession can be expressed in general as a matrix in the basis of the space.4.1Gimbal analogyZZNXxLeft:A three axes z-x-z gimbal where the external frame and external axis x are not shown and axes Y are perpendicular to each gimbal ring.Right:A simple diagram showing the Euler angles and where the axes Y of intermediate frames are located.If we suppose a set of frames,able to move each with respect to the former according to just one angle,like a gimbal,there will exist an externalfixed frame,onefi-nal frame and two frames in the middle,which are called “intermediate frames”.The two in the middle work as two gimbal rings that allow the last frame to reach any orientation in space.In these conditions,each Euler rotation works on one of the rings,independently from the rest.4.2Intermediate framesThe gimbal rings indicate some intermediate frames. They can be defined statically too.Taking some vectors i,j and k over the axes x,y and z,and vectors I,J,K over X,Y and Z,and a vector N over the line of nodes,some intermediate frames can be defined using the vector cross product,as following:•origin:[i,j,k](where k=i×j)•first:[N,k×N,k]•second:[N,K×N,K]•final:[I,J,K]These intermediate frames are equivalent to those of the gimbal.They are such that they differ from the previous one in just a single elemental rotation.This proves that:•Any target frame can be reached from the reference frame just composing three rotations.•The values of these three rotations are exactly the Euler angles of the target frame.86PROPERTIES5Relationship to other representa-tionsMain article:Rotation formalisms in three dimensions§Conversion formulae between formalismsEuler angles are one way to represent orientations.There are others,and it is possible to change to and from other conventions.5.1Rotation matrixAny orientation can be achieved by composing three el-emental rotations,starting from a known standard ori-entation.Equivalently,any rotation matrix R can be decomposed as a product of three elemental rotation ma-trices.For instance:R=X(α)Y(β)Z(γ)is a rotation matrix that may be used to represent a com-position of intrinsic rotations about axes x-y’-z″.How-ever,both the definition of the elemental rotation ma-trices X,Y,Z,and their multiplication order depend on the choices taken by the user about the definition of both rotation matrices and Euler angles(see,for instance, Ambiguities in the definition of rotation matrices).Un-fortunately,different sets of conventions are adopted by users in different contexts.The following table was built according to this set of conventions:1.Each matrix is meant to operate by pre-multiplyingcolumn vectors(see Ambiguities in the definition of rotation matrices)2.Each matrix is meant to represent an active rota-tion(the composing and composed matrices are sup-posed to act on the coordinates of vectors defined in the initialfixed reference frame and give as a re-sult the coordinates of a rotated vector defined in the same reference frame).3.Each matrix is meant to represent the compositionof intrinsic rotations(around the axes of the rotating reference frame).4.Right handed reference frames are adopted,and theright hand rule is used to determine the sign of the anglesα,β,γ.For the sake of simplicity,the following table uses the following nomenclature:1.1,2,3represent the anglesα,β,γ.2.X,Y,Z are the matrices representing the elementalrotations about the axes x,y,z of thefixed frame(e.g.,X1represents a rotation about x by an angleα).3.s and c represent sine and cosine(e.g.,s1representsthe sine ofα).4.Each matrix is denoted by the formula used to cal-culate it.If R=Z1X2Z3,we name it Z1X2Z3 .To change the formulas for the opposite direction of ro-tation,change the signs of the sine functions.To change the formulas for passive rotations,transpose the matri-ces(then each matrix transforms the initial coordinates of a vector remainingfixed to the coordinates of the same vector measured in the rotated reference system;same ro-tation axis,same angles,but now the coordinate system rotates,rather than the vector).5.2QuaternionsUnit quaternions,also known as Euler–Rodrigues param-eters,provide another mechanism for representing3D ro-tations.This is equivalent to the special unitary group description.Expressing rotations in3D as unit quaternions instead of matrices has some advantages:•Concatenating rotations is computationally faster and numerically more stable.•Extracting the angle and axis of rotation is simpler.•Interpolation is more straightforward.See for exam-ple slerp.•Quaternions do not suffer from gimbal lock as Euler angles do.5.3Geometric algebraOther representation comes from the Geometric alge-bra(GA).GA is a higher level abstraction,in which the quaternions are an even subalgebra.The principal tool in GA is the rotor R=[cos(θ/2)−Iu sin(θ/2)]whereθ= angle of rotation,(u)=rotation axis(unitary vector)and (I)=pseudoscalar(trivector in R3)6PropertiesSee also:Charts on SO(3)and Quaternions and spatial rotation9The Euler angles form a chart on all of SO(3),the special orthogonal group of rotations in 3D space.The chart is smooth except for a polar coordinate style singular-ity along β=0.See charts on SO(3)for a more complete treatment.The space of rotations is called in general “The Hypersphere of rotations ",though this is a misnomer:the group Spin(3)is isometric to the hypersphere S 3,but the rotation space SO(3)is instead isometric to the real pro-jective space RP 3which is a 2-fold quotient space of the hypersphere.This 2-to-1ambiguity is the mathematical origin of spin in physics .A similar three angle decomposition applies to SU(2),the special unitary group of rotations in complex 2D space,with the difference that βranges from 0to 2π.These are also called Euler angles.The Haar measure for Euler angles has the simple formsin(β).dα.dβ.dγ,usually normalized by a factor of 1/8π².For example,to generate uniformly randomized orienta-tions,let αand γbe uniform from 0to 2π,let z be uniform from −1to 1,and let β=arccos(z ).7Higher dimensionsIt is possible to define parameters analogous to the Euler angles in dimensions higher than three.[4]The number of degrees of freedom of a rotation matrix is always less than the dimension of the matrix squared.That is,the elements of a rotation matrix are not all com-pletely independent.For example,the rotation matrix in dimension 2has only one degree of freedom,since all four of its elements depend on a single angle of rotation.A rotation matrix in dimension 3(which has nine ele-ments)has three degrees of freedom,corresponding to each independent rotation,for example by its three Euler angles or a magnitude one (unit)quaternion.In SO(4)the rotation matrix is defined by two quater-nions ,and is therefore 6-parametric (three degrees of freedom for every quaternion).The 4×4rotation matri-ces have therefore 6out of 16independent components.Any set of 6parameters that define the rotation matrixcould be considered an extension of Euler angles to di-mension 4.In general,the number of euler angles in dimension D is quadratic in D;since any one rotation consists of choos-ing two dimensions to rotate between,the total number of rotations available in dimension D is N rot =(D 2)=D (D −1)/2,which for D =2,3,4yields N rot =1,3,6.A gyroscope keeps its rotation axis constant.Therefore,angles measured in this frame are equivalent to angles measured in the lab frame 8Applications8.1Vehicles and moving framesMain article:rigid body See also:axes conventionsTheir main advantage over other orientationdescrip-Industrial robot operating in a foundry.tions is that they are directly measurable from a gimbal mounted in a vehicle.As gyroscopes keep their rotation axis constant,angles measured in a gyro frame are equiva-lent to angles measured in the lab frame.Therefore gyros are used to know the actual orientation of moving space-craft,and Euler angles are directly measurable.Intrin-sic rotation angle cannot be read from a single gimbal,108APPLICATIONSso there has to be more than one gimbal in a spacecraft. Normally there are at least three for redundancy.There is also a relation to the well-known gimbal lock problem of mechanical engineering[5].Heading,elevation and bank for an aircraft with axes DIN9300 The most popular application is to describe aircraft at-titudes,normally using a Tait–Bryan convention so that zero degrees elevation represents the horizontal attitude. Tait–Bryan angles represent the orientation of the aircraft respect a reference axis system(world frame)with three angles which in the context of an aircraft are normally called Heading,Elevation and Bank.When dealing with vehicles,different axes conventions are possible.When studying rigid bodies in general,one calls the xyz system space coordinates,and the XYZ system body co-ordinates.The space coordinates are treated as unmov-ing,while the body coordinates are considered embedded in the moving body.Calculations involving acceleration, angular acceleration,angular velocity,angular momen-tum,and kinetic energy are often easiest in body coordi-nates,because then the moment of inertia tensor does not change in time.If one also diagonalizes the rigid body’s moment of inertia tensor(with nine components,six of which are independent),then one has a set of coordinates (called the principal axes)in which the momentof inertia tensor has only three components.The angular velocity of a rigid body takes a simple form using Euler angles in the moving frame.Also the Euler’s rigid body equations are simpler because the inertia tensor is constant in that frame.Polefigures displaying crystallographic texture of gamma-TiAl in an alpha2-gamma alloy,as measured by high energy X-rays.[6] 8.2Crystallographic textureIn materials science,crystallographic texture(or pre-ferred orientation)can be described using Euler angles. In texture analysis,the Euler angles provide a mathemat-ical depiction of the orientation of individual crystallites within a polycrystalline material,allowing for the quan-titative description of the macroscopic material.[7]The most common definition of the angles is due to Bunge and corresponds to the ZXZ convention.It is important to note,however,that the application generally involves axis transformations of tensor quantities,i.e.passive ro-tations.Thus the matrix that corresponds to the Bunge Euler angles is the transpose of that shown in the table above.[8]8.3OthersEuler angles,normally in the Tait–Bryan convention,are also used in robotics for speaking about the degrees of freedom of a wrist.They are also used in Electronic sta-bility control in a similar way.Gunfire control systems require corrections to gun-order angles(bearing and elevation)to compensate for deck tilt (pitch and roll).In traditional systems,a stabilizing gyro-scope with a vertical spin axis corrects for deck tilt,and stabilizes the optical sights and radar antenna.However, gun barrels point in a direction different from the line of sight to the target,to anticipate target movement and fall of the projectile due to gravity,among other factors.Gun mounts roll and pitch with the deck plane,but also require stabilization.Gun orders include angles computed from the vertical gyro data,and those computations involve Eu-ler angles.Euler angles are also used extensively in the quantum me-chanics of angular momentum.In quantum mechanics, explicit descriptions of the representations of SO(3)are very important for calculations,and almost all the work has been done using Euler angles.In the early history of quantum mechanics,when physicists and chemists had a。