Chapter 2.2 Transformation between BODY and NED

Useful properties

Rotation matrices

$$R\ast R^T=R^T\ast R=I_3$$ $$\det (R)=1$$

The following notation is used to transform a vector from one coordinate frame to another:

$$v^{to}=R_{from}^{to}{v}^{from}$$

Skew- symmetry of a matrix and cross- product operator

$$S=-S^T$$

The vector cross product is defined by:

$$\lambda \times a:=S(\lambda )a$$

where

$$S(\lambda )=-S^T(\lambda )=\left[\begin{array}{ccc}0& -{\lambda }_3& {\lambda }_2\\ {\lambda }_3& 0& -{\lambda }_1\\ -{\lambda }_2& {\lambda }_1& 0\end{array}\right]$$

Smtrx.m

The inverse operator is denoted vex(.) such that

$$\lambda =vex(S(\lambda ))$$

vex.m

Euler angle transformation

$$R_{z,\psi }(\psi )=\left[\begin{array}{ccc}\cos \psi & -\sin \psi & 0\\ \sin \psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right]$$ $$R_{y,\theta }(\theta )=\left[\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right]$$ $$R_{x,\varphi }(\varphi )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & -\sin \varphi \\ 0& \sin \varphi & \cos \varphi \end{array}\right]$$

Linear velocity transformation (zyx convention)

Rzyx.m

Angular velocity transformation(zyx convention)

Tzyx.m

6 DOF kinematic equations

Unit quaternions

Linear velocity transformation

Rquat.m

Angular velocity transformation

Tquat

6 DOF kinematic equations 7 ODEs

Unit quaternion from euler angles

euler2q.m

Euler Angles from a unit quaternion

q2euler.m