Add function to compute the Jacobian derivative of additional frames rigidly attached to links

[xela-95]

Closes #208

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📚 Documentation preview 📚: https://jaxsim--210.org.readthedocs.build//210/

[xela-95]

I created this draft PR to share my code in an easy way with you.

However the test is failing for two reasons:

• some of the frames are not found by KinDynComputations
• for the other frames the resulting bias acceleration is quite different

[xela-95]

In b747a26 I commented out the test against iDynTree and tried to test directly $\dot J$ against the same quantity computing through AD. It seems that the inertial and body cases pass for all models, while the mixed case no.

[xela-95]

In 78b690d I also explicitly handled the conversion of the input representation of the jacobian via matrix ${} ^W T _I = diag({} ^W X _I, I _n)$ (with $I$ the active data representation). Unfortunately the test still fails for the [UR10-mixed] and [erogcub-mixed] cases, while it works for [box-mixed]. This is making me think of something related to joints that the box does not have, but I could not spot the bug here.

I suspect something here:

jaxsim/src/jaxsim/api/frame.py

Lines 400 to 423 in 78b690d

	case VelRepr.Mixed:
	W_H_F = transform(model=model, data=data, frame_index=frame_index)
	W_H_FW = W_H_F.at[0:3, 0:3].set(jnp.zeros((3, 3)))
	FW_H_W = Transform.inverse(W_H_FW)
	O_X_W = FW_X_W = Adjoint.from_transform(transform=FW_H_W)
	with data.switch_velocity_representation(VelRepr.Mixed):
	FW_J_WF_FW = jacobian(
	model=model,
	data=data,
	frame_index=frame_index,
	output_vel_repr=VelRepr.Mixed,
	)
	FW_v_WF = FW_J_WF_FW @ data.generalized_velocity()
	W_v_W_FW = jnp.zeros(6).at[0:3].set(FW_v_WF[0:3])
	W_vx_W_FW = Cross.vx(W_v_W_FW)
	# O_J̇_WF_I = FW_X_W @ (W_J̇_WL_I - W_vx_WFW @ W_J_WL_I)
	O_Ẋ_W = FW_Ẋ_W = -FW_X_W @ W_vx_W_FW
	O_J̇_WF_I = jnp.zeros(shape=(6, 6 + model.dofs()))
	O_J̇_WF_I += O_Ẋ_W @ W_J_WL_W @ T
	O_J̇_WF_I += O_X_W @ W_J̇_WL_W @ T
	O_J̇_WF_I += O_X_W @ W_J_WL_W @ Ṫ

@diegoferigo @flferretti any idea?

[diegoferigo]

I need some time to go through you code, but the following is definitely a bug. Let's see what happens as soon as you fix it.

[diegoferigo]

LGTM! Just minor suggestions.

[flferretti]

That's great @xela-95, thanks a lot! LGTM