Try other PyDy EoM formulations (seeking faster numerical execution)

[moorepants]

No description provided.

[moorepants]

With pydy/pydy#515 the slowdown seems to be cse now, but it finishes. Adjusting the code to (similar to this PR):

def test_accelerations():

    print('Deriving the symbolic dynamics model.')
    symbolics = derive.derive_equations_of_motion()
    print(symbolics)

    rhs = symbolics.kanes_method.mass_matrix.LUsolve(symbolics.kanes_method.forcing)
    rhs =sm.Matrix(symbolics.speeds).col_join(rhs)

    print('Generating the PyDy ODE numerical function.')
    pydy_rhs = generate_ode_function(
        rhs,
        #symbolics.kanes_method.forcing,
        symbolics.coordinates,
        symbolics.speeds,
        constants=symbolics.constants,
        #mass_matrix=symbolics.kanes_method.mass_matrix,
        #coordinate_derivatives=sm.Matrix(symbolics.speeds),
        specifieds=symbolics.specifieds,
        generator='cython',
        constants_arg_type='array',
        specifieds_arg_type='array',
    )

Gives these results:

In [1]: from pygait2d.tests.test_model import test_accelerations
/home/moorepants/miniconda/envs/gait2d-dev/lib/python3.13/site-packages/pydy/codegen/c_code.py:8: UserWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html. The pkg_resources package is slated for removal as early as 2025-11-30. Refrain from using this package or pin to Setuptools<81.
  from pkg_resources import parse_version

In [2]: test_accelerations()
Deriving the symbolic dynamics model.
Forming positions, velocities, accelerations and forces.
Initializing Kane's Method.
Forming Kane's Equations.
States (18): [qax(t), qay(t), qa(t), qb(t), qc(t), qd(t), qe(t), qf(t), qg(t), uax(t), uay(t), ua(t), ub(t), uc(t), ud(t), ue(t), uf(t), ug(t)]

Specifieds (9): [Fax(t), Fay(t), Ta(t), Tb(t), Tc(t), Td(t), Te(t), Tf(t), Tg(t)]

Constants (43): [g, ma, ia, xa, ya, mb, ib, lb, xb, yb, mc, ic, lc, xc, yc, md, id, xd, yd, hxd, txd, fyd, me, ie, le, xe, ye, mf, if, lf, xf, yf, mg, ig, xg, yg, hxg, txg, fyg, kc, cc, mu, vs]
Generating the PyDy ODE numerical function.
lambdifying the equations of motion
PyDy evaluation time: 0.0033768770008464344
Lambdify evaluation time: 0.11875866299669724
Autolev evaluation time: 0.0054086649979581125

PyDy may only be slightly faster than Autolev because the autolev function also computes the ground reaction forces and the stick figure points.

[moorepants]

I also just discovered that I had added an undocumented feature to SymPy ode_function_generator: `linear_sys_solver='sympy`` which gives nice performance, i.e. much shorter code generation and faster numerical evaluation.