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| {{Other uses|Rotation operator (disambiguation){{!}}Rotation operator}}
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| | |
| This article derives the main properties of rotations in 3-dimensional space.
| |
| | |
| The three [[Euler angles|Euler rotations]] are one way to bring a [[rigid body]] to any desired orientation by sequentially making [[rotations]] about axis' fixed relative to the object. However, this can also be achieved with one single rotation ([[Euler's rotation theorem]]). Using the concepts of [[linear algebra]] it is shown how this single
| |
| rotation can be performed.
| |
| | |
| ==Mathematical formulation==
| |
| Let
| |
| :<math>\hat e_1\ ,\ \hat e_2\ ,\ \hat e_3</math>
| |
| | |
| be a [[coordinate system]] fixed in the body that through a change in orientation is brought to the new directions
| |
| :<math>\mathbf{A}\hat e_1\ ,\ \mathbf{A}\hat e_2\ ,\ \mathbf{A}\hat e_3.</math>
| |
| | |
| Any [[euclidean vector|vector]]
| |
| :<math>\bar x\ =x_1\hat e_1+x_2\hat e_2+x_3\hat e_3</math>
| |
| | |
| rotating with the body is then brought to the new direction
| |
| :<math>\mathbf{A}\bar x\ =x_1\mathbf{A}\hat e_1+x_2\mathbf{A}\hat e_2+x_3\mathbf{A}\hat e_3</math>
| |
| | |
| i.e. this is a [[linear operator]]
| |
| | |
| The [[Matrix (mathematics)|matrix]] of this [[Operator (mathematics)|operator]] relative to the coordinate system
| |
| :<math>\hat e_1\ ,\ \hat e_2\ ,\ \hat e_3</math>
| |
| | |
| is
| |
| :<math>
| |
| \begin{bmatrix}
| |
| A_{11} & A_{12} & A_{13} \\
| |
| A_{21} & A_{22} & A_{23} \\
| |
| A_{31} & A_{32} & A_{33}
| |
| \end{bmatrix} =
| |
| \begin{bmatrix}
| |
| \langle\hat e_1 | \mathbf{A}\hat e_1 \rangle & \langle\hat e_1 | \mathbf{A}\hat e_2 \rangle & \langle\hat e_1 | \mathbf{A}\hat e_3 \rangle \\
| |
| \langle\hat e_2 | \mathbf{A}\hat e_1 \rangle & \langle\hat e_2 | \mathbf{A}\hat e_2 \rangle & \langle\hat e_2 | \mathbf{A}\hat e_3 \rangle \\
| |
| \langle\hat e_3 | \mathbf{A}\hat e_1 \rangle & \langle\hat e_3 | \mathbf{A}\hat e_2 \rangle & \langle\hat e_3 | \mathbf{A}\hat e_3 \rangle
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| As
| |
| :<math> \sum_{k=1}^3 A_{ki}A_{kj}= \langle \mathbf{A}\hat e_i | \mathbf{A}\hat e_j \rangle
| |
| = \begin{cases}
| |
| 0 & i\neq j, \\ 1 & i = j,
| |
| \end{cases}
| |
| </math>
| |
| | |
| or equivalently in matrix notation
| |
| | |
| :<math>
| |
| \begin{bmatrix}
| |
| A_{11} & A_{12} & A_{13} \\
| |
| A_{21} & A_{22} & A_{23} \\
| |
| A_{31} & A_{32} & A_{33}
| |
| \end{bmatrix}^T
| |
| \begin{bmatrix}
| |
| A_{11} & A_{12} & A_{13} \\
| |
| A_{21} & A_{22} & A_{23} \\
| |
| A_{31} & A_{32} & A_{33}
| |
| \end{bmatrix} =
| |
| \begin{bmatrix}
| |
| 1 & 0 & 0 \\
| |
| 0 & 1 & 0 \\
| |
| 0 & 0 & 1
| |
| \end{bmatrix}
| |
| </math>
| |
| the matrix is [[Orthogonal matrix|orthogonal]] and as a "right hand" base vector system is re-orientated into another "right hand" system the [[determinant]] of this matrix has the value 1. | |
| | |
| ===Rotation around an axis===
| |
| Let
| |
| | |
| :<math>\hat e_1\ ,\ \hat e_2\ ,\ \hat e_3</math>
| |
| | |
| be an orthogonal positively oriented base vector system in <math>R^3</math>. | |
| | |
| The linear operator
| |
| | |
| "Rotation with the angle <math>\theta</math> around the axis defined by <math>\hat e_3</math>"
| |
| | |
| has the matrix representation
| |
| | |
| :<math>
| |
| \begin{bmatrix}
| |
| Y_1 \\
| |
| Y_2 \\
| |
| Y_3
| |
| \end{bmatrix}
| |
| =
| |
| \begin{bmatrix}
| |
| \cos\theta & -\sin\theta & 0 \\
| |
| \sin\theta & \cos\theta & 0 \\
| |
| 0 & 0 & 1
| |
| \end{bmatrix}
| |
| \begin{bmatrix}
| |
| X_1 \\
| |
| X_2 \\
| |
| X_3
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| relative to this basevector system.
| |
| | |
| This then means that a vector
| |
| | |
| :<math>
| |
| \bar x=\begin{bmatrix}
| |
| \hat e_1 & \hat e_2 & \hat e_3
| |
| \end{bmatrix}
| |
| \begin{bmatrix}
| |
| X_1 \\
| |
| X_2 \\
| |
| X_3
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| is rotated to the vector
| |
| | |
| :<math>
| |
| \bar y=\begin{bmatrix}
| |
| \hat e_1 & \hat e_2 & \hat e_3
| |
| \end{bmatrix}
| |
| \begin{bmatrix}
| |
| Y_1 \\
| |
| Y_2 \\
| |
| Y_3
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| by the linear operator.
| |
| | |
| The [[determinant]] of this matrix is
| |
| | |
| :<math>
| |
| \det
| |
| \begin{bmatrix}
| |
| \cos\theta & -\sin\theta & 0\\
| |
| \sin\theta & \cos\theta & 0\\
| |
| 0 & 0 & 1
| |
| \end{bmatrix}=1
| |
| </math>
| |
| | |
| and the [[characteristic polynomial]] is
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \det\begin{bmatrix}
| |
| \cos\theta -\lambda & -\sin\theta & 0 \\
| |
| \sin\theta & \cos\theta -\lambda & 0 \\
| |
| 0 & 0 & 1-\lambda
| |
| \end{bmatrix}
| |
| &=\big({(\cos\theta -\lambda)}^2 + {\sin\theta}^2 \big)(1-\lambda) \\
| |
| &=-\lambda^3+(2\ \cos\theta\ +\ 1)\ \lambda^2 - (2\ \cos\theta\ +\ 1)\ \lambda +1 \\
| |
| \end{align}
| |
| </math>
| |
| | |
| The matrix is symmetric if and only if <math>\sin\theta=0</math>, i.e. for <math>\theta=0</math>
| |
| and for <math>\theta=\pi</math>.
| |
| | |
| The case <math>\theta=0</math> is the trivial case of an identity operator.
| |
| | |
| For the case <math>\theta=\pi</math> the [[characteristic polynomial]] is | |
| | |
| :<math>
| |
| -(\lambda-1){(\lambda +1)}^2
| |
| </math>
| |
| | |
| i.e. the rotation operator has the [[eigenvalue]]s
| |
| :<math>
| |
| \lambda=1 \quad \lambda=-1
| |
| </math>
| |
| | |
| The [[eigenspace]] corresponding to <math>\lambda=1</math> is all vectors on the rotation axis, i.e. all vectors
| |
| | |
| :<math>
| |
| \bar x =\alpha \ \hat e_3 \quad -\infty <\alpha < \infty
| |
| </math>
| |
| | |
| The [[eigenspace]] corresponding to <math>\lambda=-1</math> consists of all vectors orthogonal to the rotation axis, i.e. all vectors
| |
| | |
| :<math>
| |
| \bar x =\alpha \ \hat e_1 + \beta \ \hat e_2 \quad -\infty <\alpha < \infty \quad -\infty <\beta < \infty
| |
| </math>
| |
| | |
| For all other values of <math>\theta</math> the matrix is un-symmetric and as <math>{\sin\theta}^2 > 0</math> there is
| |
| only the eigenvalue <math>\lambda=1</math> with the one-dimensional [[eigenspace]] of the vectors on the rotation axis:
| |
| | |
| :<math>
| |
| \bar x =\alpha \ \hat e_3 \quad -\infty <\alpha < \infty
| |
| </math>
| |
| | |
| The rotation matrix by angle <math>\theta</math> around a general axis of rotation
| |
| <math>
| |
| \mathbf{k} = \left[\begin{array}{ccc}
| |
| k_1 \\
| |
| k_2 \\
| |
| k_3
| |
| \end{array}\right]
| |
| </math>
| |
| is given by [[Rodrigues' rotation formula]].
| |
| : <math>
| |
| R = I \cos\theta + [\mathbf{k}]_\times \sin\theta + (1 - \cos\theta) \mathbf{k} \mathbf{k}^\mathsf{T}
| |
| </math>,
| |
| | |
| where <math>I</math> is the [[identity matrix]] and <math>[\mathbf{k}]_\times</math> is the [[Hodge_star | dual 2-form]] of <math>\mathbf{k}</math> or
| |
| [[Cross_product | cross product matrix]],
| |
| | |
| : <math> [\mathbf{k}]_\times =
| |
| \left[\begin{array}{ccc}
| |
| 0 & -k_3 & k_2 \\
| |
| k_3 & 0 & -k_1 \\
| |
| -k_2 & k_1 & 0
| |
| \end{array}\right]
| |
| </math>.
| |
| | |
| Note that <math>[\mathbf{k}]_\times </math> satisfies <math>[\mathbf{k}]_\times \mathbf{v} = \mathbf{k}\times\mathbf{v} </math> for all <math>\mathbf{v}</math>.
| |
| | |
| ===The general case===
| |
| The operator
| |
| | |
| "Rotation with the angle <math>\theta</math> around a specified axis"
| |
| | |
| discussed above is an orthogonal mapping and its matrix relative to any base vector system is therefore an
| |
| [[orthogonal matrix]] . Furthermore its determinant has the value 1.
| |
| A non-trivial fact is the opposite, i.e. that for any orthogonal linear mapping in <math>R^3</math> having
| |
| determinant = 1 there exist base vectors
| |
| | |
| :<math>\hat e_1\ ,\ \hat e_2\ ,\ \hat e_3</math>
| |
| | |
| such that the matrix takes the "canonical form"
| |
| | |
| :<math>
| |
| \begin{bmatrix}
| |
| \cos\theta & -\sin\theta & 0 \\
| |
| \sin\theta & \cos\theta & 0 \\
| |
| 0 & 0 & 1\end{bmatrix}
| |
| </math>
| |
| | |
| for some value of <math>\theta</math>.
| |
| | |
| In fact, if a linear operator has the [[orthogonal matrix]]
| |
| | |
| :<math>
| |
| \begin{bmatrix}
| |
| A_{11} & A_{12} & A_{13} \\
| |
| A_{21} & A_{22} & A_{23} \\
| |
| A_{31} & A_{32} & A_{33}
| |
| \end{bmatrix}
| |
| </math>
| |
|
| |
| relative some base vector system
| |
|
| |
| :<math>\hat f_1\ ,\ \hat f_2\ ,\ \hat f_3</math>
| |
| | |
| and this matrix is symmetric, the "Symmetric operator theorem" valid in <math>R^n</math> (any dimension) applies saying | |
| | |
| that it has ''n'' orthogonal eigenvectors. This means for the 3-dimensional case that there exists a coordinate system
| |
| :<math>\hat e_1\ ,\ \hat e_2\ ,\ \hat e_3</math>
| |
| | |
| such that the matrix takes the form
| |
| :<math>
| |
| \begin{bmatrix}
| |
| B_{11} & 0 & 0 \\
| |
| 0 & B_{22} & 0 \\
| |
| 0 & 0 & B_{33}
| |
| \end{bmatrix}
| |
| </math>
| |
|
| |
| As it is an orthogonal matrix these diagonal elements <math>B_{ii}</math> are either 1 or −1. As the determinant is 1 these elements
| |
| are either all 1 or one of the elements is 1 and the other two are −1.
| |
| | |
| In the first case it is the trivial identity operator corresponding
| |
| to <math>\theta=0</math>. | |
| | |
| In the second case it has the form
| |
| | |
| :<math>
| |
| \begin{bmatrix}
| |
| -1 & 0 & 0 \\
| |
| 0 & -1 & 0 \\
| |
| 0 & 0 & 1
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| if the basevectors are numbered such that the one with eigenvalue 1 has index 3. This matrix is then of the desired form for <math>\theta=\pi</math>.
| |
| | |
| If the matrix is un-symmetric, the vector
| |
| :<math>
| |
| \bar E = \alpha_1\ \hat f_1 + \alpha_2\ \hat f_2 + \alpha_3\ \hat f_3
| |
| </math>
| |
| | |
| where
| |
| | |
| :<math>\alpha_1=\frac{A_{32}-A_{23} }{2} </math>
| |
| :<math>\alpha_2=\frac{A_{13}-A_{31}}{2}</math>
| |
| :<math>\alpha_3=\frac{A_{21}-A_{12}}{2}</math>
| |
| | |
| is non-zero. This vector is an eigenvector with eigenvalue
| |
| | |
| :<math>
| |
| \lambda=1
| |
| </math>
| |
| | |
| Setting
| |
| :<math>
| |
| \hat e_3=\frac{\bar E}{|\bar E|}
| |
| </math>
| |
| | |
| and selecting any two orthogonal unit vectors in the plane orthogonal to <math>\hat e_3</math>: | |
| | |
| :<math>\hat e_1\ ,\ \hat e_2</math>
| |
| | |
| such that
| |
| | |
| :<math>\hat e_1\ ,\ \hat e_2,\ \hat e_3</math>
| |
| | |
| form a positively oriented triple, the operator takes the desired form with
| |
| :<math>\cos \theta=\frac{A_{11}+A_{22}+A_{33}-1}{2}</math>
| |
| :<math>\sin \theta=|\bar{E}|</math>
| |
| | |
| The expressions above are in fact valid also for the case of a symmetric
| |
| rotation operator corresponding to a rotation with <math>\theta = 0</math>
| |
| or <math>\theta = \pi</math>. But the difference is that for <math>\theta = \pi</math>
| |
| the vector
| |
| :<math>
| |
| \bar E = \alpha_1\ \hat f_1 + \alpha_2\ \hat f_2 + \alpha_3\ \hat f_3
| |
| </math>
| |
| | |
| is zero and of no use for finding the eigenspace of eigenvalue 1, i.e. the
| |
| rotation axis.
| |
| | |
| Defining <math>E_4</math> as <math>\cos \theta</math> the matrix for the
| |
| rotation operator is
| |
| | |
| :<math>
| |
| \frac{1-E_4}{{E_1}^2+{E_2}^2+{E_3}^2}
| |
| \begin{bmatrix}
| |
| E_1 E_1 & E_1 E_2 & E_1 E_3 \\
| |
| E_2 E_1 & E_2 E_2 & E_2 E_3 \\
| |
| E_3 E_1 & E_3 E_2 & E_3 E_3
| |
| \end{bmatrix}
| |
| +
| |
| \begin{bmatrix}
| |
| E_4 & -E_3 & E_2 \\
| |
| E_3 & E_4 & -E_1 \\
| |
| -E_2 & E_1 & E_4
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| provided that
| |
| | |
| :<math>
| |
| {E_1}^2+{E_2}^2+{E_3}^2 > 0
| |
| </math>
| |
| | |
| i.e. except for the cases <math>\theta=0</math> (the identity operator) and <math>\theta=\pi</math>
| |
| | |
| ==Quaternions==
| |
| {{Main|Quaternions and spatial rotation}}
| |
| | |
| Quaternions are defined similar to <math>E_1\ ,\ E_2\ ,\ E_3\ ,\ E_4</math> with
| |
| the difference that the half angle <math>\frac{\theta}{2}</math> is used
| |
| instead of the full angle <math>\theta</math>.
| |
| | |
| This means that the first 3 components <math>q_1\ ,\ q_2\ ,\ q_3\ </math> are components of a vector defined from
| |
| | |
| :<math>
| |
| q_1\ \hat{f_1}\ +\ q_2\ \hat{f_2}\ +\ \ q_3\ \hat{f_1}\ =\ \sin \frac{\theta}{2}\quad \hat{e_3}=\frac{\sin \frac{\theta}{2}}{\sin\theta}\quad \bar E
| |
| </math>
| |
| and that the fourth component is the scalar | |
| :<math>
| |
| q_4=\cos \frac{\theta}{2}
| |
| </math> | |
| | |
| As the angle <math>\theta</math> defined from the canonical form is in the interval
| |
| :<math>0 \le \theta \le \pi</math>
| |
| | |
| one would normally have that <math>q_4 \ge 0</math>. But a "dual" representation of a rotation with quaternions | |
| is used, i.e.
| |
| :<math>q_1\ ,\ q_2\ ,\ q_3\ ,\ q_4\ </math>
| |
| | |
| and | |
| :<math>-q_1\ ,\ -q_2\ ,\ -q_3\ ,\ -q_4\ </math>
| |
| | |
| are two alternative representations of one and the same rotation.
| |
| | |
| The entities <math>E_k</math> are defined from the quaternions by
| |
| :<math> E_1=2 q_4 q_1</math>
| |
| :<math> E_2=2 q_4 q_2</math>
| |
| :<math> E_3=2 q_4 q_3</math>
| |
| :<math> E_4={q_4}^2 -({q_1}^2+{q_2}^2+{q_3}^2)</math>
| |
| | |
| Using quaternions the matrix of the rotation operator is
| |
| :<math>\begin{bmatrix}
| |
| 2({q_1}^2+{q_4}^2)-1 &2({q_1}{q_2}-{q_3}{q_4}) &2({q_1}{q_3}+{q_2}{q_4}) \\
| |
| 2({q_1}{q_2}+{q_3}{q_4}) &2({q_2}^2+{q_4}^2)-1 &2({q_2}{q_3}-{q_1}{q_4}) \\
| |
| 2({q_1}{q_3}-{q_2}{q_4}) &2({q_2}{q_3}+{q_1}{q_4}) &2({q_3}^2+{q_4}^2)-1 \\
| |
| \end{bmatrix}</math>
| |
| | |
| ==Numerical example==
| |
| | |
| Consider the reorientation corresponding to the [[Euler angle]]s
| |
| <math>
| |
| \alpha=10^\circ \quad \beta=20^\circ \quad \gamma=30^\circ \quad
| |
| </math>
| |
| relative a given base vector system
| |
| :<math>\hat f_1\ ,\ \hat f_2,\ \hat f_3</math> | |
| | |
| Corresponding matrix relative to this base vector system is (see [[Euler angles#Matrix orientation]])
| |
| | |
| :<math>
| |
| \begin{bmatrix}
| |
| 0.771281 & -0.633718 & 0.059391 \\
| |
| 0.613092 & 0.714610 & -0.336824 \\
| |
| 0.171010 & 0.296198 & 0.939693
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| and the quaternion is
| |
| :<math>
| |
| (0.171010,\ -0.030154,\ 0.336824,\ 0.925417)
| |
| </math>
| |
| | |
| The canonical form of this operator
| |
| :<math>
| |
| \begin{bmatrix}
| |
| \cos\theta & -\sin\theta & 0\\
| |
| \sin\theta & \cos\theta & 0\\
| |
| 0 & 0 & 1
| |
| \end{bmatrix}
| |
| </math>
| |
| with <math>\theta=44.537^\circ </math> is obtained with
| |
| :<math>\hat e_3=(0.451272,-0.079571,0.888832)</math>
| |
| | |
| The quaternion relative to this new system is then
| |
| :<math>
| |
| (0,\ 0,\ 0.378951,\ 0.925417) = (0,\ 0,\ \sin\frac{\theta}{2},\ \cos\frac{\theta}{2})
| |
| </math>
| |
| | |
| Instead of making the three Euler rotations
| |
| | |
| :<math>10^\circ,20^\circ,30^\circ</math>
| |
| | |
| the same orientation can be reached with one single rotation of size <math>44.537^\circ</math> around <math>\hat e_3</math>
| |
| | |
| ==References==
| |
| * {{citation |title=An Introduction to the Theory of Linear Spaces|first=Georgi|last= Shilov|author-link =Georgii Evgen'evich Shilov|publisher= Prentice-Hall|year=1961|id=Library of Congress 61-13845}}.
| |
| | |
| [[Category:Linear algebra]]
| |
| [[Category:Kinematics]]
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