Some doubts about Supplementary Material partE

[icameling]

Hello, I am wondering why the first equation in (E.12) is satisfied? Here is define of the question. Could you give some material about this equation? Thank you very mach!

[hyye]

For any quaternion, there is a left-hand side multiplication equivalent to a right-hand side multiplication. If the perturbation is only on the yaw angle, these two delta qs will be equal.

In the meantime, the estimated orientation in LIO is close to the ground-truth except for the yaw angle, which is a property of the inertial fusion.

[icameling]

Thanks for your quick reply. I am still confused about this equation.

Is _delta_L^W_ a normal perturbation quaternion?

Why _q_L^W left multiplication delat qz equal to q_L^W right multiplication a normal perturbation quaternion delta q_L^W_?

By the way, you say if the perturbation is only on the yaw angle, these two delta qs will be equal, Could you explain in detail? Hope for your reply. Thanks a lot!

[hyye]

1. It’s a normal perturbation on the right.
2. Since we want to find a constrained left perturbation which is close to a right perturbation.
3. My previous reply is not a good explanation. The yaw-major perturbation (drift) on the global pose can be corrected by a left perturbation.

[icameling]

So in LIO_Mapping, you observed the global pose of the yaw angle is not as accurate as the other 2 dimensions, then the rotation-constrationed maping is applyed to solve this problem. However, this situation may not be common. Our NDT-based lidar mapping suffers from the pitch-major error situation. Here is a part of the result of our NDT-based mapping. May be I misunderstand your meaning, or our mapping algorithm is not as perfect as yours.

And about the Eqution.12 in supplementary material, could I rewrite the eqution like the second way? How do you come up with this formula? I can't find material on this aspect of quaternion.

[hyye]

Since roll and pitch angles are observable in LIO (with inertial fusion), the accumulated error is in yaw angle major. This is different from your NDT-Lidar-only method.

The rotational error of the odometry is majorly in the global yaw angle (unobservable). Thus, I choose to correct the global yaw to make it align with the map. Intuitively, it should be on the left (it can be considered as “\delta q_W^{W’} * q_L^{W’}”). Of course, you can rewrite the equation, but the physical meaning may change, and I haven’t tried in that direction.

[icameling]

Thank you for your detailed explanation.:smile:

[shibowing]

@icameling 您好，我也最近再学习大佬的这篇工作，能否一起讨论一下哈，谢谢啦！ 微信： MachoEdward