quaternion has two definitions Hamilton and JPL

quaternion has two definitions Hamilton and JPL

definition

$\mathbf q = q_o + q_1i+q_2j+q_3z$

for Hamilton: $ijk = -1$

for JPL: $ijk = 1$

so:

for Hamilton:
$\mathbf R_{Hamilton} =\left[\begin{array}{ccc} 1-2 q_{2}^{2}-2 q_{3}^{2} & 2 q_{1} q_{2}-2 q_{0} q_{3} & 2 q_{1} q_{3}+2 q_{0} q_{2} \\ 2 q_{1} q_{2}+2 q_{0} q_{3} & 1-2 q_{1}^{2}-2 q_{3}^{2} & 2 q_{2} q_{3}-2 q_{0} q_{1} \\ 2 q_{1} q_{3}-2 q_{0} q_{2} & 2 q_{2} q_{3}+2 q_{0} q_{1} & 1-2 q_{1}^{2}-2 q_{2}^{2} \end{array}\right]$
for JPL:
$\mathbf R_{JPL} =\left[\begin{array}{ccc} 1-2 q_{2}^{2}-2 q_{3}^{2} & 2 q_{1} q_{2}+2 q_{0} q_{3} & 2 q_{1} q_{3}-2 q_{0} q_{2} \\ 2 q_{1} q_{2}-2 q_{0} q_{3} & 1-2 q_{1}^{2}-2 q_{3}^{2} & 2 q_{2} q_{3}+2 q_{0} q_{1} \\ 2 q_{1} q_{3}+2 q_{0} q_{2} & 2 q_{2} q_{3}-2 q_{0} q_{1} & 1-2 q_{1}^{2}-2 q_{2}^{2} \end{array}\right]$
so actually there is :(Conversion of the two kinds quaternions)
$\mathbf q_{Hamilton} ( q_o + q_1i+q_2j+q_3z)\leftrightarrow \mathbf q_{JPL}(q_o - q_1i-q_2j-q_3z)$

so in hybird case, there is wording of ${C_{q}}^T$

$C_q$ mean change the quaternions to Rotation matrix by Hamilton rules.

If the quaternions is in JPL, but change to R by Hamilton rules, it should to Transpose

ps:

library or files	Hamilton or JPL
Eigen	Hamilton
Ros	Hamilton
Indirect kalman filter for 3D Attitude Estimation	JPL
Ethzasl-msf	Hybird