Topic/integration

[aescande]

This PR reworks the numerical integration for the non-trivial joints (planar, spherical and free):

• fix a sign error in the Magnus expansion and conditionally extend the expansion to 5 terms, depending on a user-defined precision,
• properly take into account that translation velocities are expressed in the successor frames (FS) by integrating the motion of this frame instead of considering it constant over the integration step,
• the error done during the integration is estimated and used to subdivide the integration step if necessary. Consequently, integration can be carried out on large intervals (tested up to 1 second),
• new tests are added, old ones are reworked
• detailed mathematical notes are added in the doc folder. Not yet linked to the doxygen.
• change name from eulerIntegration to integration or numericalIntegration to better reflect what's happening in the functions. Keep old names around but marked as deprecated.

This PR is not polished yet, but I'd like to have some general feedback first/while polishing. First of all the tests. I'm currently performing 4 types:

• comparing integration at constant velocity with analytical formula (for constant velocity it is possible to derive these formula, which is not the case for constant acceleration)
• comparing the integration at constant acceleration with step dt to N integrations at constant velocity with step dt/N, changing the velocity between those integrations. N is a large number (10000 right now)
• checking the self-consistency of the integration: one integration with step dt should yield the same result as N integrations with step dt/N.
• checking the consistency of the integration with the jacobian matrix in the following way: I take a point P on the body after the joint. Taking a constant velocity v for the joint, we should get the same result by (i) integrating the joint position and computing the new position of P (ii) integrating Jv, where J is the Jacobian matrix of the point.

The last test is an attempt to ensure that the integration is coherent with the library as a whole, because it could be possible to misunderstand the conventions adopted in RBDyn and write a wrong integration that could however pass the 3 first tests. The idea of using the Jacobian matrix is to get quantities in the world frame, where there is no specific convention, from whatever the original spaces are for the joint variables and velocities/accelerations. If anyone has a better idea, I'd be happy to hear it. Something related is done in EulerTestV2 in AlgoTest, but quantites are brought back in some variable space, which makes the test dependent on conventions, and makes it actually wrong for planar and free joints (although because of the very low accuracy required, the planar test is still passing - but I had to disable the one for free joint). I believe this test should be removed completely.

Another point is that the estimation of the error is adding some overhead. I didn't measure it for now, as it is up to a few dozens of scalar additions/multiplications, but it could be equivalent to the integration computation in itself. Does it seems ok? Or do I add a flag/versions without any precision check?

The PR solves #35, without addressing #65 (I could not get an integration coherent with the Jacobian matrix when changing the convention for the planar joint, and I don't get where I'm making a mistake).

[aescande]

@gergondet : there does not seems to be a mechanism for deprecation in RBDyn yet. Do you have any guidelines on how to proceed? Here are a few options I see:

• using [[deprecated]] (requires c++14)
• using some CMake facilities: https://stackoverflow.com/a/41225414/11611648
• writing some macro by hand (where?) as in https://stackoverflow.com/a/295229/11611648

[aescande]

Baring feedbacks, I'm done with the code so far. Code review can start :)