How to derive system model Jacobian for $SE_2(3)$?

[Chris7462]

I have been exploring the $SE_2(3)$ localization example and am curious about how the system model Jacobian is calculated, specifically how to compute $J^{u}_{x}$?

In the code example

manif/examples/se_2_3_localization.cpp

Lines 376 to 383 in de52f8b

	X = X.plus(u_est, J_x_x, J_x_u); // X * exp(u), with Jacobians
	// Prepare Jacobian of state-dependent control vector
	J_u_x.setZero();
	J_u_x.block<3, 3>(0, 3) = accLinCross;
	J_u_x.block<3, 3>(0, 6) = Eigen::Matrix3d::Identity()*dt;
	J_u_x.block<3, 3>(6, 3) = gCross;
	F = J_x_x + J_x_u*J_u_x; // chain rule for system model Jacobian

I understand that:

• $J^{x}_{x}=Ad_{Exp(\tau)}^{-1}$, as mentioned in Eq. (80) from manif paper
• $J^{x}_{u}=J_{r}(\tau)$, as per Eq. (81)

but I'm unsure about how to derive $J^{u}_{x}$.

Additionally, I've reviewd Barrau and Bonnabel's papaer, but it hasn't provided the clarity I was hoping for.

[joansola]

On lline 371 in the cpp file you have the expression of u_est. On line 376 you have X <- X (+) u_est, with Jacobians J_X_X and J_X_u.

It happens in this case that u_est is itself a function of X. Therefore, there exists a J_u_X which has to be computed and then used in the chain rule, line 383.

Now your question is about J_u_X. For this, you look at the code to copute u_est from X, and work out the derivative. This code is in lines 362--369, just above u_est.

The derivatives of these mathematical expressions I guess shoulde not be difficult. Work lines 362-369 line by line, using the formulas in the micro Lie paper, and then apply the chain rule to get J_u_X.

For example, if a line states accLin = dt*(R_k_est.transpose())*v_k_est + 0.5*dt*dt*acc_k_est, then compute the Jacobians: J_acclin_R, J_acciln_v, J_acclin_acc. They should be most quite easy. Look at the derivation rules in the paper, and in the Appendix related to SE(3) -- the Jacobians of SE_2(3) are block-wise equivalent to those of SE(3).

Once you have all lines, and the good names for all partial Jacobians, then apply the chain rule. Good names are such that if z=f(x,y) then you name J_z_x = df/dx and J_z_y=df/dy. That is, J_<output variablee name>_<input variable name>. Use good names otherwise the chain rule is impossible to get right.

Is this response enough?