Compute Hessian of MbP dynamics function

[hongkai-dai]

If I have a function f(u, q, v, v̇, λ) = M * v̇ + C(q, v) - Bu - J'λ, I wish to compute the Hessian of this function f w.r.t u, q, v, v̇, λ. Some of the Hessian terms are easy to compute, but to compute ∂² f / ∂ q², it requires computing the Hessian of ∂² M / ∂ q² and ∂² C / ∂ q², which I don't know how to do yet.

I guess the easiest solution might be to instantiate MBP with nested AutoDiffScalar.

@sherm1

[sherm1]

Since f is a vector function, ∂f / ∂q is a matrix. That makes ∂²f / ∂q² a hypermatrix. Is that what you're looking for?

[hongkai-dai]

I should have been more clear, I mean ∂²f(i) / ∂q², namely the second order derivative of the i'th entry of f w.r.t q.

[sherm1]

We don't have a way to calculate just a single generalized force f(i) so the nested AutoDiff calculation would give { ∂²f(i)/∂q² | i ∈ 0..n-1 }. (That's a way to represent the 3D hypermatrix.) I'm not seeing an easier way to get that though.

You don't have u in the functional dependence of f. Should it be there?

[hongkai-dai]

You don't have u in the functional dependence of f. Should it be there?

Yes u should be there also.

We don't have a way to calculate just a single generalized force f(i) so the nested AutoDiff calculation would give { ∂²f(i)/∂q² | i ∈ 0..n-1 }. (That's a way to represent the 3D hypermatrix.) I'm not seeing an easier way to get that though.

Makes sense. I think if we use nested autodiff, we can still return f as an Eigen::Vector, where the i'th entry of this f contains the value of f, the gradient and the Hessian.

[sherm1]

Yes, each entry would contain its value, gradient, and Hessian.
Added u in the issue description.

BTW @edrumwri wrote a document on AutoDiffing that has a Hessian section.

Assigning to @rpoyner-tri for now as vaguely related to #10991 -- this is an application that would really stretch AutoDiff performance!

[rpoyner-tri]

Given where our thinking about autodiff is going, I'm handing this back to @hongkai-dai for further consideration.