fix(trajectory): normalize quintic profile time

[Afloat16]

Summary

quintic(1, 2, 1000) starts near zero instead of 1, while a one-millisecond profile remains near 1.5 at both endpoints. The unscaled least-squares system loses numerical accuracy or rank and discards boundary constraints.

Evaluate the quintic in normalized time using a factored Hermite basis. Preserve endpoint velocity units, zero endpoint acceleration, supplied timestamps, sample shapes, the zero-duration result and existing coefficient-type errors. Promote low-precision samples before normalization and basis evaluation, retaining supported complex and extended sample types. Add independent polynomial/derivative, 70-digit float32, signed-velocity and native multi-axis controls. Use NumPy testing calls for the Boolean checks so they remain active under Python optimization.

Validation

• Exact base: 968ed048639c01c9032e256565731e2703de77fe. Updated tests on that base: 27 failed, 35 passed. Fixed trajectory tests: 76 passed, 14 subtests.
• Linux/Python 3.12.14: rerun supported native suite passes with 906 passed, 3 existing skips, 17 subtests. Rerun core without Swift: 897 passed, 12 expected skips, 17 subtests. Matching unchanged native-base extensions were reused and verified active.
• The 13 Boolean expressions are unchanged by AST comparison. All 62 parametrized test cases run directly under normal Python and -O: 172 actual Boolean checks pass and 172 forced-false calls raise AssertionError in each mode, without pytest assertion rewriting.
• Production code is unchanged from the previously validated revision. Its complete coverage run passes with 906 passed, 3 existing skips and 17 subtests, covering every executable quintic line. Coverage initially missed the existing C-versus-Python timing threshold; the unchanged complete serial repeat passes. That failure and the passing baseline timing control are retained. Coverage was not rerun for this test-only revision.
• Ruff formatting, updated-test lint and git diff --check pass. Rerun whole-project lint adds no diagnostics; existing diagnostics remain at 1211 versus 1214 on the base.
• The previous configured Sphinx HTML builds are reused for the unchanged production source. They succeed while repeating the baseline's 20 existing runblock errors at three ETS/ETS2 locations; those examples are not claimed as passing. Documentation was not rebuilt for this test-only revision.
• Numerical regressions exercise selected durations from 0.001 to 1,000,000. Other operating systems, Python versions and browser/Pyodide execution were not tested locally.

Related issue

None.

Checklist

• PR title follows Conventional Commits
• Tests pass locally, with existing skips listed above
• Added tests for this change
• Changed function and new tests use modern type hints
• Docstring updated in reST style
• Change is focused
• No project-specific data or notebooks added

[codacy-production]

Up to standards ✅

🟢 Issues 0 issues

Results:
0 new issues

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🟢 Metrics 65 complexity · 9 duplication

Metric	Results
Complexity	65
Duplication	9

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[codecov]

Codecov Report

❌ Patch coverage is 0% with 28 lines in your changes missing coverage. Please review.
✅ Project coverage is 0.00%. Comparing base (f28437f) to head (fd83877).

Files with missing lines	Patch %	Lines
src/roboticstoolbox/tools/trajectory.py	0.00%	28 Missing ⚠️

Additional details and impacted files

@@          Coverage Diff          @@
##            main    #736   +/-   ##
=====================================
  Coverage   0.00%   0.00%           
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  Files        143     143           
  Lines      14667   14693   +26     
=====================================
- Misses     14667   14693   +26     

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[petercorke]

Thanks for your interest in RTB, and for taking the time to create a PR. Once again I'm afraid your summary is just too cryptic. You declare that you are using it AI which is fine, just prompt it: to write summaries for somebody who is solo maintaining multiple large open source projects and likes clear/simple examples of the issue and a gentle onramp for the solution.

I see the problem but I'm reluctant to discard this function which I think has pedagogical value, as does its mode of failure. I'd propose this alternative:

• in existing quintic_func test the condition of X and raise an error for this case, and describe the issue clearly in the docstring
• create a parallel function quintic_hermite_func or whatever that implements your fix.

[petercorke]

Thanks for following up, it maybe that the hermit code becomes mainstream quintic_func one day and the old LS approach is kept commented out for pedagogy -- that algorithm is used in my book which is why I am keen to retain it. I didn't realise the conditioning went to hell quite so quickly. Thanks again.

[petercorke]

Thank you for your interest in RTB, and for this contribution, @Afloat16. You uncovered a real and quietly nasty bug that gives a very wrong answer under certain, not unreasonable, conditions.

It's taken me a while to figure out the right approach here, trying to balance correctness, pedagogy, variation from my book, and API simplicity. This lands in #805, merged in 97c6c0f. Your commits are in it, with your authorship. The main difference from your PR is that we kept the book's least-squares formulation as the default, but solve it in normalized time, which fixes the conditioning with a few lines instead of a new function. Your closed-form Hermite basis is kept in the file as method="hermite", with a credit to you, and it is now what the tests use as an independent check that the two methods agree.