Add support for autodiff

[artivis]

Add support for the autodiff library

autodiff automatic differentiation made easier for C++ (web - github).

This PR only adds support for the autodiff::dual type.

Find a doc page here (mostly copied from the Ceres/Jet doc page).

The Jacobians produced by autodiff are compared against manif analytics ones in unit tests.

Also adds

• operator[](i) to both groups and tangents to access single elements of the underlying vector
• size() to both groups and tangents which return the underlying vector size (== RepSize)

Closes #196.
Closes #143.

[codecov]

Codecov Report

Merging #237 (c85ac91) into devel (317ff22) will increase coverage by 0.00%.
The diff coverage is n/a.

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+ Hits         1721     1722    +1     
  Misses         32       32           

[artivis]

@joansola, this is ready for review/approval.

[joansola]

Seems good to me. I have not tested it, though.

What exactly is the autodiff package?

[artivis]

What exactly is the autodiff package?

It's a modern (C++17) library for automatic differentiation that provides a scalar type and a few neat functions. Think ceres::Jet and the bits to evaluate the Jacobians.

Here is a striped down example (yet nearly compilable):

using dual = autodiff::dual;

// cost function
auto fun = [](const auto& measurement, const auto& state_i, const auto& state_j){
  return measurement - (state_j - state_i);
};

SE3<dual> xi, xj;
SE3Tangent<dual> meas_ij, e;

// cost function Jacs wrt underlying vector
Eigen::MatrixXd J_e_xi = autodiff::jacobian(fun, autodiff::wrt(xi), autodiff::at(meas_ij, xi, xj), e);
Eigen::MatrixXd J_e_xj = autodiff::jacobian(fun, autodiff::wrt(xj), autodiff::at(meas_ij, xi, xj), e);

// Local param Jacs
Eigen::MatrixXd J_xi_xiw0 = manif::autodiffLocalParameterizationJacobian<dual>(xi);
Eigen::MatrixXd J_xj_xjw0 = manif::autodiffLocalParameterizationJacobian<dual>(xj);

// cost function Jacs wrt local perturbation on the tangent space
Eigen::MatrixXd J_e_xiw0 = J_e_xi * J_xi_xiw0;
Eigen::MatrixXd J_e_xjw0 = J_e_xj * J_xj_xjw0;

Find more examples in the unit tests.

[joansola]

Great!