Side note: Use of quaternion

• Relation \(\mathrm{R}'(t)=\hat\omega(t)\,\mathrm{R}(t)\) may lead to numerical drift from rotation matrix
• • ex. \(\mathrm{R}^{k+1}=(\mathrm{Id}+h\,\hat \omega^k)\,\mathrm{R}^k\;\;\) (explicit scheme)

• Using quaternion leads to more robust behavior

• - Quaternion expression: \(q'(t)=\frac{1}{2}\,q_{\omega}(t)\,q(t)\), with \(q_{\omega}(t)=(\omega(t)\,,\;\;0))\)

• - Quaternion is forced to keep a unit norm

• • ex. \(\left\{\begin{array}{l}q^{k+1}=q^k+\frac{1}{2}\,q_{\omega}^k\,q^k \\ q^{k+1}=q^{k+1}/\|q^{k+1}\| \end{array}\right.\)