The Jacobian of the inverse operation corresponds to the adjoint of the pose inverse rather than the adjoint of the pose. This is also discussed here: https://gtsam.org/2021/02/23/uncertainties-part3.html

I wrote this code precisely based on the page that you link, which clearly states that, if $\tilde{T}_{WB} = T_{WB} \text{Exp}( _{B}\eta)$ (Eq. 6), then $\Sigma_{W} = \text{Ad}_{\mathbf{T}_{WB}} \Sigma_B \text{Ad}_{\mathbf{T}_{WB}}^{T} $ (Eq. 11). This is the adjoint of $T_{WB}$, not of the inverse $T_{BW}$.

In addition (out of the scope of this PR), the covariance propagation of the composited pose (here: https://github.com/microsoft/lamar-benchmark/blob/main/scantools/capture/pose.py#L133) also does not feel correct to me (the Jacobians are different in my derivation). And the cross correlation, albeit being very important to covariance propagation at pose composition, is dropped in the computation.

This is implemented according to Eq. 15 of the same link. There is no Jacobian (I don't know that you are referring to) and we assume that the poses are not correlated.

The Jacobian of the inverse operation corresponds to the adjoint of the pose inverse rather than the adjoint of the pose. This is also discussed here: https://gtsam.org/2021/02/23/uncertainties-part3.html

I wrote this code precisely based on the page that you link, which clearly states that, if $\tilde{T}_{WB} = T_{WB} \text{Exp}( _{B}\eta)$ (Eq. 6), then $`\Sigma_{W} = \text{Ad}{\mathbf{T}{WB}} \Sigma_B \text{Ad}{\mathbf{T}{WB}}^{T}

$ (Eq. 11). This is the adjoint of $T_{WB}$, not of the inverse $T_{BW}`$.

In addition (out of the scope of this PR), the covariance propagation of the composited pose (here: https://github.com/microsoft/lamar-benchmark/blob/main/scantools/capture/pose.py#L133) also does not feel correct to me (the Jacobians are different in my derivation). And the cross correlation, albeit being very important to covariance propagation at pose composition, is dropped in the computation.

This is implemented according to Eq. 15 of the same link. There is no Jacobian (I don't know that you are referring to) and we assume that the poses are not correlated.

In Eq. (6) section the noise is defined in the body frame, so T_{WB} is exactly the pose inverse. And if we do some basic derivation, the derivative of -R^Tt with respect to t is -R^T rather than R.

Similarly, the Jacobian of the pose composition is not like what you wrote here, but is out of scope for this PR. In Eq. (15) the convention of covariance is also different from your assumption.

The covariance propagation via adjoint is valid because that you could write first-order derivatives (Jacobian) using adjoint. I don't understand what you meant by "there is no Jacobian".

Edit: this PR proposal is also wrong. The expression should be in other form. In the end the adjoint should not be relevant here due to the discrepancy between SE(3) and SO(3) x R^3. Will close this for now and open another one soon to fix it.