"ENH: Implement 3-DOF Single Rail Button Flight Phase (Tip-off Analysis) - Preliminary"

[Rafit345]

Pull request type

• [x] Code changes (bugfix, features)
• [ ] Code maintenance (refactoring, formatting, tests)
• [ ] ReadMe, Docs and GitHub updates
• [ ] Other (please describe):

Checklist

• [x] Tests for the changes have been added (if needed)
• [x] Docs have been reviewed and added / updated
• [x] Lint (black rocketpy/ tests/) has passed locally
• [ ] All tests (pytest tests -m slow --runslow) have passed locally
• [ ] CHANGELOG.md has been updated (if relevant)

Current behavior

Refs #28

New behavior

This commit was made as a submission to the selective process deliverables challenge.

The method udot_rail2 functions as an intermediate flight phase before the rocket has fully left the guide rail, allowing for 3 degrees of freedom (linear motion along the rail, pitch and yaw).

Flight init includes a feature to run a simulation without udot_rail2. Numerical values enabling udot_rail2 are very close to 1 DOF flight. Flight phase transitions smoothly from 1 DOF rail phase to 3DOF and from 3 DOF to 6 DOF free flight.

Current equations of motion inside udot_rail2 rely heavily on udot_generalized, ensuring 3 DOF through vector operations.

--Summary-- Add preliminary udot_rail2 (3-DOF tip-off) support and safe, deterministic phase-insertion handling during rail → 6DOF transitions. Add a feature flag to enable/disable udot_rail2 on Flight init. Add a Hermite-root fallback to avoid hard failures when rail-exit root filtering returns no valid root (warn + midpoint fallback). Add comprehensive unit tests (alignment, no-roll, insertion-order, CSV comparisons) and sample CSV output for comparison runs with udot_rail2 enabled vs disabled.

Breaking change

• [ ] Yes
• [x] No

Additional information

Still working on the implementation of proper lagrangean expansion /derivation of equations of motion. Articles "Tip-off effect analysis of a vehicle moving along an inclined guideway by considering dynamic interactions" by Chou et al and "ANALYSIS OF MISSILE LAUNCHERS PART Q Tipoff Effects in Helical Rail Launchers" by Hosken et al are proving useful.

Some tests are still not passing (10/1561)