Question about jacobian in se_3 localization example

[QiuYue-bit]

Hi , I have a question about the state transition matrix in se_3 localization demo.

In ESKF, the state transition matrix can be caculate by $\delta x_{k+1} = F \delta x_{k}$ ,in this system, propagation equation is , $X_{k+1} = X_{k} \oplus \tau $, thus F can be calculated by $\frac{\partial(X \cdot \exp (\tau ))}{\partial X} $, however the F I see in the code is the right Jacobian $J_{r}(\tau)$ , why?

manif/examples/se3_localization.cpp

Line 243 in ad00494

X = X.plus(u_est, J_x, J_u); // X * exp(u), with Jacobians

manif/examples/se3_localization.cpp

Line 245 in ad00494

P = J_x * P * J_x.transpose() + J_u * U * J_u.transpose();

[QiuYue-bit]

and the same problem in $SE_2(3)$,

manif/examples/se_2_3_localization.cpp

Lines 366 to 383 in ad00494

	Eigen::Vector3d accLin = dt*(R_k_est.transpose())*v_k_est + 0.5*dt*dt*acc_k_est;
	Eigen::Vector3d gLin = R_k_est.transpose()*g*dt;
	Eigen::Matrix3d accLinCross = manif::skew(accLin);
	Eigen::Matrix3d gCross = manif::skew(gLin);
	u_est << accLin, dt*omega_prev, dt*acc_k_est;
	u_est += u_noise;
	/// First we move - - - - - - - - - - - - - - - - - - - - - - - - - - - -
	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

also here J_u_x seems have bugs, i derive here and it seems work fine , i don't get why here is difference with previous implementation.
J_u_x.block<3, 3>(0, 3) = manif::skew(R_k_est.transpose()*v_k_est * dt) + 0.5*dt*dt * manif::skew(R_k_est.transpose()*g)
J_u_x.block<3, 3>(0, 6) = R_k_est.transpose()*dt;

[joansola]

Where do you see the Right Jacobian in the code? I just see J_x and J_u which are the Jacobians of the transition function wrt the state x and the control u.

Please point out where do you see the right Jacobian so that we can help.

For reference, the Jacobians wrt X and u of a function of the type

X+ = X (+) u

are to be found in the paper "micro Lie theory", eqns. (80, 81). In (81), you can find the right Jacobian.

[QiuYue-bit]

the X = X.plus(u_est, J_x, J_u); is rplus here ,and j_x corresponds to J_mout_m , which is t.rjac(); as $J_r(u_{est})$ here

manif/include/manif/impl/lie_group_base.h

Lines 465 to 479 in ad00494

	template <typename _Derived>
	template <typename _DerivedOther>
	typename LieGroupBase<_Derived>::LieGroup
	LieGroupBase<_Derived>::rplus(
	const TangentBase<_DerivedOther>& t,
	OptJacobianRef J_mout_m,
	OptJacobianRef J_mout_t) const
	{
	if (J_mout_t)
	{
	(*J_mout_t) = t.rjac();
	}
	return compose(t.exp(), J_mout_m, _);
	}

[joansola]

Sorry but this code says J_mout_t is Jr(u), not J_mout_m

And J_mout_t, being t the second argument of rplus(), is the right jacobian according to (81)

Whre is the problem?