Formulary


I. Coordinate Transformation, DCM

\overline{\bf{v}}: Vector in Euklidian space
{\bf{v}}: Coordinates of a vector \overline{\bf{v}} measured in the frame K = \{ {\bf{\bar{e}}}_x {\bf{\bar{e}}}_y{\bf{\bar{e}}}_z \}
{\bf{v}}': Coordinates of a vector \overline{\bf{v}} measured in the frame K' = \{ {\bf{\bar{e}}}'_x {\bf{\bar{e}}}'_y {\bf{\bar{e}}}'_z \}

\overline{\bf{v}} = v_x {\bf{\bar{e}}}_x + v_y {\bf{\bar{e}}}_y + v_z {\bf{\bar{e}}}_z = \sum_i v_i {\bf{\bar{e}}}_i

v_i = {\bf{\bar{e}}}_i \overline{\bf{v}} \equiv {\bf{v}}|_i

\overline{\bf{v}} = v_x {\bf{\bar{e}}}_x + v_y {\bf{\bar{e}}}_y + v_z {\bf{\bar{e}}}_z = v'_x {\bf{\bar{e}}}'_x + v'_y {\bf{\bar{e}}}'_y + v'_z {\bf{\bar{e}}}'_z
 

Coordinates {\bf{e}}'_i of basis vector {\bf{\bar{e}}}'_i measured in Frame K:

{\bf{\bar{e}}}'_i = \sum_j ( {\bf{\bar{e}}}_j {\bf{\bar{e}}}'_i ) {\bf{\bar{e}}}_j = \sum_j {\bf{e}}'_i|_j \: {\bf{\bar{e}}}_j \equiv \sum_j R_{ij}^T {\bf{\bar{e}}}_j    

{\bf{R}}^T = \begin{pmatrix} \;\;\; {\bf{e}}'_x^T \,\;\;\; \\ {\bf{e}}'_y^T \\ {\bf{e}}'_z^T \end{pmatrix},     {\bf{R}} = \begin{pmatrix} \\ {\bf{e}}'_x \; {\bf{e}}'_y \; {\bf{e}}'_z\\ \\ \end{pmatrix},     R_{ij}^T = {\bf{e}}'_i|_j = {\bf{\bar{e}}}_j {\bf{\bar{e}}}'_i = {\bf{\bar{e}}}'_i {\bf{\bar{e}}}_j

\begin{pmatrix} {\bf{e}}'_x \; {\bf{e}}'_y \; {\bf{e}}'_z \end{pmatrix} = {\bf{R}} \begin{pmatrix} {\bf{e}}_x \; {\bf{e}}_y \; {\bf{e}}_z \end{pmatrix}

{\bf{\bar{e}}}_j {\bf{\bar{e}}}'_i ={\bf{e}}'_i|_j\: : j-th coordinate of vector {\bf{\bar{e}}}'_i measured in K

{\bf{\bar{e}}}'_i {\bf{\bar{e}}}_j ={\bf{e}}_j|'_i\: : i-th coordinate of vector {\bf{\bar{e}}}_j measured in K‘
 

Transformation of coordinates of a vector \overline{\bf{v}}:

v_i = {\bf{\bar{e}}}_i \overline{\bf{v}} = {\bf{\bar{e}}}_i \sum_j v'_j {\bf{\bar{e}}}'_j = \sum_j ( {\bf{\bar{e}}}_i {\bf{\bar{e}}}'_j ) v'_j = \sum_j {\bf{e}}'_j|_i \: v'_j = \sum_j R_{ji}^T \: v'_j = \sum_j R_{ij} \: v'_j

{\bf{v}} = {\bf{R}} {\bf{v}}',      {\bf{v}}' = {\bf{R}}^T {\bf{v}}


II. Rigid Body

{\bf{R}}: Orientation of body-fixed coordinate frame as measured in the earth coordinate frame
{\bf{r}}_{earth}: Coordinates of a vector {\bf{r}} measured in the earth frame
{\bf{r}}_{body}: Coordinates of the same vector {\bf{r}} measured in the body frame
{\boldsymbol{\omega}}: Rotation rate vector of body measured in the body frame
{\bf{q}}|_{\bf{R}}: Unit quaternion associated to the rotation matrix {\bf{R}}

\dot{\bf{R}} = {\bf{R}} {\bf{\Omega}}_\times = {\bf{R}} \left( {\boldsymbol{\omega}} {\bf{J}} \right)

{\bf{r}}_{body}= {\bf{R}}^T {\bf{r}}_{earth}

\dot{\bf{r}}_{body}= {\bf{\Omega}}_\times^T {\bf{r}}_{body} = - ({\boldsymbol{\omega}}_{body} \times {\bf{r}}_{body})
 

Definition Pitch-Roll: 
{\bf{R}}(\epsilon_p,\epsilon_r) \equiv {\bf{R}}_y(\epsilon_p) {\bf{R}}_x(\epsilon_r) = \left(\begin{array}{ccc} c_p & s_p s_r & s_p c_r \\ 0 & c_r & - s_r \\ -s_p & c_p s_r & c_p c_r \end{array}\right)

{\bf{q}}|_{\bf{R}} \equiv {\bf{q}}_y(\epsilon_p) {\bf{q}}_x(\epsilon_r) = ( c_{\bar{r}} c_{\bar{p}} , s_{\bar{r}} c_{\bar{p}} , c_{\bar{r}} s_{\bar{p}} , - s_{\bar{r}} s_{\bar{p}} )^T

Definition Yaw-Pitch-Roll: 
{\bf{R}}(\epsilon_y,\epsilon_p,\epsilon_r) \equiv {\bf{R}}_z(\epsilon_y) {\bf{R}}_y(\epsilon_p) {\bf{R}}_x(\epsilon_r) = \left(\begin{array}{ccc} c_p c_y & s_p s_r c_y - c_r s_y & s_p c_r c_y + s_r s_y \\ c_p s_y & s_p s_r s_y + c_r c_y & s_p c_r s_y - s_r c_y \\ -s_p & c_p s_r & c_p c_r \end{array}\right)

{\bf{q}}|_{\bf{R}} \equiv {\bf{q}}_z(\epsilon_y) {\bf{q}}_y(\epsilon_p) {\bf{q}}_x(\epsilon_r) = ( c_{\bar{r}} c_{\bar{p}} c_{\bar{y}} + s_{\bar{r}} s_{\bar{p}} s_{\bar{y}} , s_{\bar{r}} c_{\bar{p}} c_{\bar{y}} - c_{\bar{r}} s_{\bar{p}} s_{\bar{y}} , s_{\bar{r}} c_{\bar{p}} s_{\bar{y}} + c_{\bar{r}} s_{\bar{p}} c_{\bar{y}} , c_{\bar{r}} c_{\bar{p}} s_{\bar{y}} - s_{\bar{r}} s_{\bar{p}} c_{\bar{y}} )^T

Definition II Roll-Pitch: 
{\bf{R}}^G(\epsilon_p,\epsilon_r) \equiv {\bf{R}}_x(\epsilon_r) {\bf{R}}_y(\epsilon_p) = \left(\begin{array}{ccc} c_p & 0 & s_p \\ s_p s_r & c_r & - c_p s_r \\ -s_p c_r & s_r & c_p c_r \end{array}\right)

{\bf{q}}|_{{\bf{R}}^G} \equiv {\bf{q}}_x(\epsilon_r) {\bf{q}}_y(\epsilon_p) = ( c_{\bar{p}} c_{\bar{r}} , c_{\bar{p}} s_{\bar{r}} , s_{\bar{p}} c_{\bar{r}} , s_{\bar{p}} s_{\bar{r}} )^T

Definition II Yaw-Roll-Pitch: 
{\bf{R}}^G(\epsilon_y,\epsilon_p,\epsilon_r) \equiv {\bf{R}}_z(\epsilon_y) {\bf{R}}_x(\epsilon_r) {\bf{R}}_y(\epsilon_p) = \left(\begin{array}{ccc} c_p c_y - s_p s_r s_y & - c_r s_y & s_p c_y + c_p s_r s_y \\ c_p s_y + s_p s_r c_y & c_r c_y & s_p s_y - c_p s_r c_y \\ -s_p c_r & s_r & c_p c_r \end{array}\right)

{\bf{q}}|_{{\bf{R}}^G} \equiv {\bf{q}}_z(\epsilon_y) {\bf{q}}_x(\epsilon_r) {\bf{q}}_y(\epsilon_p) = ( c_{\bar{p}} c_{\bar{r}} c_{\bar{y}} - s_{\bar{p}} s_{\bar{r}} s_{\bar{y}} , c_{\bar{p}} s_{\bar{r}} c_{\bar{y}} - s_{\bar{p}} c_{\bar{r}} s_{\bar{y}} , s_{\bar{p}} c_{\bar{r}} c_{\bar{y}} + c_{\bar{p}} s_{\bar{r}} s_{\bar{y}} , s_{\bar{p}} s_{\bar{r}} c_{\bar{y}} + c_{\bar{p}} c_{\bar{r}} s_{\bar{y}} )^T


III. Rotations

Rotation Matrices

{\bf{R}}_x = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos\epsilon & -\sin\epsilon \\ 0 & \sin\epsilon & \cos\epsilon \end{array}\right)    {\bf{R}}_y = \left(\begin{array}{ccc} \cos\epsilon & 0 & \sin\epsilon \\ 0 & 1 & 0 \\ -\sin\epsilon & 0 & \cos\epsilon \end{array}\right)    {\bf{R}}_z = \left(\begin{array}{ccc} \cos\epsilon & -\sin\epsilon & 0 \\ \sin\epsilon & \cos\epsilon & 0 \\ 0 & 0 & 1 \end{array}\right)
 

{\bf{R}}(\theta,\phi) \equiv {\bf{R}}_y(\theta) {\bf{R}}_x(\phi) = \left(\begin{array}{ccc} \cos\theta & \sin\phi \sin\theta & \cos\phi \sin\theta \\ 0 & \cos\phi & - \sin\phi \\ -\sin\theta & \sin\phi \cos\theta& \cos\phi \cos\theta \end{array}\right)
 

\begin{array}{rcl} {\bf{R}}(\Psi,\theta,\phi) &\equiv& {\bf{R}}_z(\Psi) {\bf{R}}_y(\theta) {\bf{R}}_x(\phi) \\[9px] &=& \left(\begin{array}{ccc} \cos\theta \cos\Psi & \sin\phi \sin\theta \cos\Psi - \cos\phi \sin\Psi & \cos\phi \sin\theta \cos\Psi + \sin\phi \sin\Psi \\ \cos\theta \sin\Psi & \sin\phi \sin\theta \sin\Psi + \cos\phi \cos\Psi & \cos\phi \sin\theta \sin\Psi - \sin\phi \cos\Psi \\ -\sin\theta & \sin\phi \cos\theta& \cos\phi \cos\theta \end{array}\right) \end{array}
 

{\bf{R}}_{\bf{n}}(\epsilon) = \left(\begin{array}{ccc} \cos\epsilon + (1-\cos\epsilon) n_x^2 & (1-\cos\epsilon)n_x n_y - \sin\epsilon\; n_z & (1-\cos\epsilon) n_x n_z + \sin\epsilon\; n_y \\ (1-\cos\epsilon) n_x n_y + \sin\epsilon\; n_z & \cos\epsilon + (1-\cos\epsilon) n_y^2 & (1-\cos\epsilon) n_y n_z - \sin\epsilon n_x \\ (1-\cos\epsilon) n_x n_z - \sin\epsilon\; n_y & (1-\cos\epsilon) n_y n_z + \sin\epsilon\; n_x & \cos\epsilon + (1-\cos\epsilon) n_z^2 \end{array}\right)

{\bf{R}}_{\bf{n}}(\alpha) {\bf{R}}_{\bf{n}}(\beta) = {\bf{R}}_{\bf{n}}(\beta) {\bf{R}}_{\bf{n}}(\alpha)

\operatorname{Tr}{\bf{R}}_{\bf{n}}(\epsilon) = 1 + 2 \cos\epsilon

{\bf{R}}_{\bf{n}}(\epsilon) {\bf{v}} = \cos\epsilon \; {\bf{v}} + \sin\epsilon \; {\bf{n}}\times{\bf{v}} + (1-\cos\epsilon )({\bf{n}}{\bf{v}}) \; {\bf{n}}

 
Generators

{\bf{J}}_x = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array}\right)    {\bf{J}}_y = \left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{array}\right)    {\bf{J}}_z = \left(\begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)

{\bf{J}}_\alpha |_{ij} = \epsilon_{i \alpha j} = - \epsilon_{\alpha i j}

\left[ {\bf{J}}_\alpha, {\bf{J}}_\beta \right] = \epsilon_{\alpha \beta \gamma} {\bf{J}}_\gamma

 
Definition:  {\bf{n}} {\bf{J}} \equiv \tilde{\bf{n}} \equiv [{\bf{n}}]_\times \equiv {\bf{N}}_\times \equiv {\bf{J}}_{\bf{n}}

{\bf{n}} {\bf{J}} = n_x {\bf{J}}_x + n_y {\bf{J}}_y + n_z {\bf{J}}_z = \sum_\alpha ({\bf{n}} \times {\bf{e}}_\alpha) \times {\bf{e}}_\alpha = \left(\begin{array}{ccc} 0 & -n_z & n_y \\ n_z & 0 & -n_x \\ -n_y & n_x & 0 \end{array}\right)

{\bf{n}} {\bf{J}} |_{ij} = - \sum_\alpha n_\alpha \epsilon_{\alpha i j}

\left[ {\bf{n}} {\bf{J}}, {\bf{m}} {\bf{J}} \right] = ( {\bf{n}} \times {\bf{m}} ) {\bf{J}}

({\bf{n}} {\bf{J}})^2 = {\bf{n}}{\bf{n}}^T - {\bf{1}},    ({\bf{n}} {\bf{J}})^3 = - {\bf{n}} {\bf{J}},    ({\bf{n}}{\bf{n}}^T)({\bf{n}} {\bf{J}}) = 0

({\bf{n}} {\bf{J}})({\bf{m}}{\bf{J}}) = {\bf{m}}{\bf{n}}^T - {\bf{1}}
 

({\bf{n}} {\bf{J}}) {\bf{r}} \equiv [{\bf{n}}]_\times {\bf{r}} = {\bf{n}} \times {\bf{r}}

{\bf{R}} [{\bf{n}}]_\times = [ {\bf{R}} {\bf{n}} ]_\times {\bf{R}}    {\bf{R}} ( {\bf{n}} \times {\bf{r}}) = ( {\bf{R}} {\bf{n}} ) \times ( {\bf{R}} {\bf{r}} )

 
Rotations and Generators

{\bf{R}}_{\bf{n}}(\epsilon) = \cos\epsilon\; {\bf{1}} + \sin\epsilon \; {\bf{n}}{\bf{J}} + (1-\cos\epsilon) {\bf{n}}{\bf{n}}^T = {\bf{1}} + \sin\epsilon \; {\bf{n}}{\bf{J}} + (1-\cos\epsilon) \left({\bf{n}}{\bf{J}}\right)^2

{\bf{R}}_{\bf{n}}(\delta) = {\bf{1}} + \delta {\bf{n}}{\bf{J}}

{\bf{R}}_{\bf{n}}(\epsilon) \left({\bf{n}}{\bf{J}}\right) = \left({\bf{n}}{\bf{J}}\right) {\bf{R}}_{\bf{n}}(\epsilon)

 
Rotation Derivatives

Definition:  {\bf{R}}'_{\bf{n}}(\epsilon) \equiv \frac{\partial}{\partial \epsilon} {\bf{R}}_{\bf{n}}(\epsilon)

{\bf{R}}'_{\bf{n}}(\epsilon) = {\bf{R}}_{\bf{n}}(\epsilon) \left({\bf{n}}{\bf{J}}\right) = \left({\bf{n}}{\bf{J}}\right) {\bf{R}}_{\bf{n}}(\epsilon)

{\bf{R}}_{\bf{n}}^{-1}(\epsilon) {\bf{R}}'_{\bf{n}}(\epsilon) = {\bf{R}}'_{\bf{n}}(\epsilon) {\bf{R}}_{\bf{n}}^{-1}(\epsilon) = {\bf{n}}{\bf{J}}

{\bf{R}}'_{x}(\epsilon) = {\bf{R}}_{x}(\epsilon) {\bf{J}}_{x},   {\bf{R}}'_{y}(\epsilon) = {\bf{R}}_{y}(\epsilon) {\bf{J}}_{y},   {\bf{R}}'_{z}(\epsilon) = {\bf{R}}_{z}(\epsilon) {\bf{J}}_{z}

 
Skew Symmetric Matrices

{\bf{S}}^{T} = -{\bf{S}},     {\bf{S}} + {\bf{S}}^{T} = 0

{\bf{S}}({\boldsymbol{\omega}}) = \left(\begin{array}{ccc} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{array}\right) = {\boldsymbol{\omega}} {\bf{J}}

{\bf{S}}({\boldsymbol{\omega}}) {\bf{r}} = {\boldsymbol{\omega}} \times {\bf{r}}

{\bf{S}}^T({\boldsymbol{\omega}}) = -{\bf{S}}({\boldsymbol{\omega}})

 
Rotations and Skew Symmetric Matrices

{\bf{S}}({\bf{R}}{\boldsymbol{\omega}}) = {\bf{R}} {\bf{S}}({\boldsymbol{\omega}}) {\bf{R}}^T,     {\bf{S}}({\bf{R}}^T{\boldsymbol{\omega}}) = {\bf{R}}^T {\bf{S}}({\boldsymbol{\omega}}) {\bf{R}}

{\bf{R}} {\bf{S}}({\boldsymbol{\omega}}) = {\bf{S}}({\bf{R}} {\boldsymbol{\omega}}) {\bf{R}},     {\bf{S}}({\boldsymbol{\omega}}) {\bf{R}} = {\bf{R}} {\bf{S}}({\bf{R}}^T {\boldsymbol{\omega}})

\dot{\bf{R}} = {\bf{S}}({\boldsymbol{\omega}}) {\bf{R}}

 
Exponential Map

e^{\epsilon {\bf{n}} {\bf{J}}} = e^{{\bf{S}}({\boldsymbol{\omega}})} = {\bf{R}}_{\bf{n}}(\epsilon)

e^{\epsilon {\bf{n}} {\bf{J}}} = {\bf{1}} + 2 \cos\frac{\epsilon}{2} \sin\frac{\epsilon}{2} \; {\bf{n}} {\bf{J}}} + 2 \sin^2\frac{\epsilon}{2} \; ({\bf{n}}{\bf{J}})^2 = {\bf{1}} + \sin\epsilon \; {\bf{n}} {\bf{J}}} + (1-\cos\epsilon) \; ({\bf{n}}{\bf{J}})^2

\exp({\bf{C}}) = \exp({\bf{A}})\exp({\bf{B}})   with   {\bf{C}}= {\bf{A}} + {\bf{B}} + \dfrac{1}{2} \left[{\bf{A}},{\bf{B}}\right] + \dfrac{1}{12} \left[{\bf{A}},\left[{\bf{A}},{\bf{B}}\right]\right] - \dfrac{1}{12} \left[{\bf{B}},\left[{\bf{A}},{\bf{B}}\right]\right] + \cdots

e^{\alpha {\bf{n}} {\bf{J}}} e^{\beta {\bf{m}} {\bf{J}}} = \exp\left[ \alpha{\bf{n}}{\bf{J}}} + \beta{\bf{m}}{\bf{J}} + \frac{1}{2} \alpha\beta ({\bf{n}} \times {\bf{m}}){\bf{J}} + \cdots \right]


IV. Quaternions

{\bf{q}}: Quaternion
q_0: Real part of quaternion {\bf{q}}
{\vec{q}}: Imaginary part of quaternion {\bf{q}}

{\bf{q}} = q_0 + q_1 {\bf{i}} + q_2 {\bf{j}} + q_3 {\bf{k}} = ( q_0, {\vec{q}} )^T = \left(\begin{array}{c} q_0 \\ {\vec{q}} \end{array}\right)

\operatorname{Re}[{\bf{q}}] = q_0
\operatorname{Im}[{\bf{q}}] = q_1 {\bf{i}} + q_2 {\bf{j}} + q_3 {\bf{k}} = {\vec{q}}

{\bf{q}}^* = q_0 - q_1 {\bf{i}} - q_2 {\bf{j}} - q_3 {\bf{k}} = \left(\begin{array}{c} q_0 \\ -{\vec{q}} \end{array}\right)

 
Quaternion Multiplications

{\bf{a}} {\bf{b}} = \left(\begin{array}{c} a_0 b_0 - a_1 b_1 - a_2 b_2 - a_3 b_3 \\ a_0 b_1 + a_1 b_0 + a_2 b_3 - a_3 b_2 \\ a_0 b_2 - a_1 b_3 + a_2 b_0 + a_3 b_1 \\ a_0 b_3 + a_1 b_2 - a_2 b_1 + a_3 b_0 \end{array}\right)

{\bf{a}} {\bf{b}} = \left(\begin{array}{c} a_0 b_0 - {\vec{a}} \cdot {\vec{b}} \\ a_0 {\vec{b}} + b_0 {\vec{a}} + {\vec{a}} \times {\vec{b}} \end{array}\right) = \left(\begin{array}{cc} a_0 & -{\vec{a}}^T \\ {\vec{a}} & a_0 {\bf{1}} + {\vec{a}}{\bf{J}} \end{array}\right) \left(\begin{array}{c} b_0 \\ {\vec{b}} \end{array}\right)

{\bf{a}} \cdot {\bf{b}} = a_0 b_0 + a_1 b_1 + a_2 b_2 + a_3 b_3

{\bf{a}} \times {\bf{b}} = \frac{1}{2}( {\bf{a}} {\bf{b}} - {\bf{b}} {\bf{a}} ) = \left(\begin{array}{c} 0 \\ {\vec{a}} \times {\vec{b}} \end{array}\right) = ( 0, a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1 )^T

|{\bf{q}}|^2 = {\bf{q}} {\bf{q}}^* = q_0^2 + q_1^2 + q_2^2 + q_3^2

{\bf{q}}^{-1} = \dfrac{1}{{\bf{q}}} = \dfrac{ {\bf{q}}^*}{ |{\bf{q}}|^2 }

 
Unit Quaternions

|{\bf{q}}|=1
{\bf{q}}^{-1} = {\bf{q}}^*}

 
Unit Quaternions and Rotations

{\bf{q}}|_{\bf{R}}: Unit quaternion associated to the rotation matrix {\bf{R}}
{\bf{R}}|_{\bf{q}}: Rotation matrix {\bf{R}} associated to the unit quaternion {\bf{q}}

{\bf{q}}|_{{\bf{R}}_{\bf{n}}(\epsilon)} = \left(\begin{array}{c} \cos(\frac{1}{2}\epsilon) \\ \sin(\frac{1}{2}\epsilon) \; {\vec{n}} \end{array}\right)

\begin{array}{rcl} {\bf{R}}|_{\bf{q}} &=& \left(\begin{array}{ccc} q_0^2 + q_1^2 - q_2^2 - q_3^2 & - 2 q_0 q_3 + 2 q_1 q_2 & 2 q_0 q_2 + 2 q_1 q_3 \\ 2 q_0 q_3 + 2 q_1 q_2 & q_0^2 - q_1^2 + q_2^2 - q_3^2 & - 2 q_0 q_1 + 2 q_2 q_3 \\ - 2 q_0 q_2 + 2 q_1 q_3 & 2 q_0 q_1 + 2 q_2 q_3 & q_0^2 - q_1^2 - q_2^2 + q_3^2 \end{array}\right) \\[24px] &=& 2 \left(\begin{array}{ccc} q_0^2 + q_1^2 - \frac{1}{2} & - q_0 q_3 + q_1 q_2 & q_0 q_2 + q_1 q_3 \\ q_0 q_3 + q_1 q_2 & q_0^2 + q_2^2 - \frac{1}{2} & -q_0 q_1 + q_2 q_3 \\ -q_0 q_2 + q_1 q_3 & q_0 q_1 + q_2 q_3 & q_0^2 + q_3^2 - \frac{1}{2} \end{array}\right) \\[24px] &=& \left( 2 q_0^2 - 1 \right) {\bf{1}} + 2 q_0 [\vec{q}]_\times + 2 \vec{q} \vec{q}^T \end{array}

 
{\bf{R}} {\vec{v}} = \operatorname{Im}[ {\bf{q}} {\bf{v}} {\bf{q}}^{-1} ],     where {\bf{v}} = (0,{\vec{v}})^T

{\bf{q}}|_{{\bf{R}}_2{\bf{R}}_1} = {\bf{q}}|_{{\bf{R}}_2}{\bf{q}}|_{{\bf{R}}_1}

\dfrac{ d\bf{q}}{dt} = \dfrac{1}{2} {\boldsymbol{\omega}} {\bf{q}},    {\boldsymbol{\omega}} = 2 \dfrac{ d\bf{q}}{dt} {\bf{q}}^{-1}     where {\boldsymbol{\omega}} = (0,{\vec{\omega}})^T    (for rigid body kinetics it’s \dot{ \bf{q}} = \dfrac{1}{2} {\bf{q}} {\boldsymbol{\omega}})

\delta{\bf{q}} \equiv {\bf{q}}|_{{\bf{R}}_{\bf{n}}(\delta)} = \left(\begin{array}{c} 1 \\ \frac{1}{2}\delta {\vec{n}} \end{array}\right)

 
Note: For numerical implementation see [1].

{\left(\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array}\right) \hat= \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array}\right) = R_x(\pi)

{\left(\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right) \hat= \left(\begin{array}{ccc} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array}\right) = R_y(\pi)

{\left(\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array}\right) \hat= \left(\begin{array}{ccc} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array}\right) = R_z(\pi)

 
Spherical Linear Interpolation, Slerp

{\bf{q}}(t) = \left( {\bf{q}}(1) {\bf{q}}(0)^{-1} \right)^t {\bf{q}}(0) = \dfrac{\sin \Omega t }{\sin \Omega}{\bf{q}}(1) + \dfrac{\sin \Omega(1-t) }{\sin \Omega} {\bf{q}}(0),     where \cos \Omega = {\bf{q}}(1) \cdot {\bf{q}}(0)


V. General

\cos(x \pm y ) = \cos(x) \cos(y) \mp \sin(x) \sin(y) \approx \cos(x) \mp \sin(x) \; y
\sin(x \pm y ) = \sin(x) \cos(y) \pm \cos(x) \sin(y) \approx \sin(x) \pm \cos(x) \; y

2 \cos^2(\frac{x}{2}) = 1 + \cos(x)
2 \cos(\frac{x}{2}) \sin(\frac{x}{2}) = \sin(x)
2 \sin^2(\frac{x}{2}) = 1 - \cos(x)

{\bf{a}} \times ( {\bf{b}} \times {\bf{c}} ) + {\bf{b}} \times ( {\bf{c}} \times {\bf{a}} ) + {\bf{c}} \times ( {\bf{a}} \times {\bf{b}} ) = 0
{\bf{a}} \times ( {\bf{b}} \times {\bf{c}} ) = {\bf{b}} ({\bf{a}} {\bf{c}}) - {\bf{c}} ( {\bf{a}} {\bf{b}} )

{\bf{a}} = {\bf{a}}_{||} + {\bf{a}}_\perp = ({\bf{a}} {\bf{n}}) {\bf{n}} - {\bf{n}} \times ({\bf{n}} \times {\bf{a}})

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