Implements the effect of reflected inertia

[amcastro-tri]

what the title says. Only the math per Featherstone's book.

This branch does not contain unit testing.

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This change is 

[sherm1]

Reviewable status: 0 of 4 files reviewed, 1 unresolved discussion (waiting on @amcastro-tri)

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multibody/tree/multibody_tree.cc, line 893 at r1 (raw file):

  // Add the effect of reflected inertias.
  for (int v = 0; v < num_velocities(); ++v) {
    (*tau_array)(v) += reflected_inertia_(v) * known_vdot(v);

Oh, that's easy! Nice.

[amcastro-tri]

Reviewable status: 0 of 4 files reviewed, 1 unresolved discussion (waiting on @sherm1)

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multibody/tree/multibody_tree.cc, line 893 at r1 (raw file):

Previously, sherm1 (Michael Sherman) wrote…

Oh, that's easy! Nice.

yeah, I believe it's correct. I essentially wrote the Lagrangian of the system including the new kinetic energy terms due to the actuators. Since those only add a "diagonal mass matrix" independent of positions q, the Coriolis terms remain the same and the only thing that changes is the mass matrix that now includes those new diagonals.
That is, if D is the diagonal matrix that contains the reflected inertias such that 1/2⋅vᵀ⋅D⋅v is the additional kinetic energy due to the rotation of the rotors, then the updated equations of motion read:

(M+D)⋅v̇ + C(q,v) = τ(q,v)

C and whatever is on the right hand side remain unchanged.

I believe this analysis is correct, what do you think?

[sherm1]

Reviewable status: 0 of 4 files reviewed, 1 unresolved discussion (waiting on @mitiguy)

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multibody/tree/multibody_tree.cc, line 893 at r1 (raw file):

Previously, amcastro-tri (Alejandro Castro) wrote…

yeah, I believe it's correct. I essentially wrote the Lagrangian of the system including the new kinetic energy terms due to the actuators. Since those only add a "diagonal mass matrix" independent of positions q, the Coriolis terms remain the same and the only thing that changes is the mass matrix that now includes those new diagonals.
That is, if D is the diagonal matrix that contains the reflected inertias such that 1/2⋅vᵀ⋅D⋅v is the additional kinetic energy due to the rotation of the rotors, then the updated equations of motion read:

(M+D)⋅v̇ + C(q,v) = τ(q,v)

C and whatever is on the right hand side remain unchanged.

I believe this analysis is correct, what do you think?

Looks right to me. Of course a real model of the rotor would have mass and other inertias that would affect C and off-diagonal terms in M. But I think this is right for the approximation. Any thoughts on this @mitiguy ?