Square root backward pass

[bjack205]

• Create a new branch off of newsolver
• Add a sqrt_backwardpass method in this file
• Add a use_sqrt_bp option to SolverOptions
• Add a top-level backwardpass! that calls the correct backwardpass based on the solver option

Resources:

• The old sqrt backward pass from much older version of TrajectoryOptimization: https://github.com/RoboticExplorationLab/Altro.jl/blob/newsolver/src/solver_opts.jl#L32
• The AL-iLQR Tutorial

[AVSurfer123]

@bjack205 Is there any past reference implementation for the square root backwards pass used in ALTRO? The DownDate operation in the tutorial is unclear, so was looking for the code. I'm trying to re-implement it in C++ and the link above no longer works.

[bjack205]

It's a rank 1 down date of a Cholesky factorization. Here is the Julia function:

https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.lowrankdowndate

Eigen also has this function.