Verify DH.Panda inertial parameters #688

The doubt

DH/Panda.py's inertial parameters (m, r, I per link) were added in PR #294 (Antun Škurić / askuric, merged 2022-04-07, closing #278). Per the PR's own description, the values were "just a quick transfer" from the MATLAB Robotics Toolbox's mdl_panda.m into the Python DH model.

That's a reasonable thing to do for m (mass is a scalar, frame-independent), but r (CoM offset) and I (inertia tensor) are only meaningful relative to a specific link frame -- and DH link frames have their own specific axis-placement convention (origin/axis choice determined by the a/d/alpha parameters and the standard vs. modified DH rules). There's no stated verification in the PR that RTB-MATLAB's DH link frame for each Panda joint is placed identically to RTB-Python's for the same a/d/alpha/mdh values -- the two toolboxes could in principle place a link frame's origin or axis orientation differently for the same nominal DH parameters, which would make a directly-copied r/I silently wrong despite being real, plausible-looking numbers. This is exactly the same class of bug flagged in #686 (the risk of copying DH-frame data into a URDF model without accounting for the frame difference) -- raised here reflexively about whether the original DH port itself was ever actually checked, since apparently it wasn't at the time.

Cross-check against an independent source

robot_descriptions' Panda (loaded today via Frankie(), see #686/#687) has its own, independently-sourced inertial data -- not derived from MATLAB at all, and expressed in the URDF's own (differently-conventioned) link frames. If DH.Panda()'s ported values are frame-correct, DH.Panda() and Frankie() should predict essentially the same dynamics for the same physical robot at the same joint angles, despite being two structurally different models built from two independent sources.

Gravity-load torque (rne(q, 0, 0), default gravity) at four configurations (zero, qr, and two random poses):

qz:      DH  [ 0.     -3.4344  0.     -3.2572  0.      1.6942  0.    ]
         RD  [ 0.     -3.4344  0.     -3.2572  0.      1.6942  0.    ]   diff: 0

qr:      DH  [ 0.    -16.72   -0.2691  19.3268  0.5998  1.7526 -0.0032]
         RD  [ 0.    -16.72   -0.2691  19.3268  0.5998  1.7526 -0.0032]  diff: 0

random1: DH  [ 0.     -7.2826 -2.9838  16.6589  0.7844  1.7237 -0.0263]
         RD  [-0.     -7.2826 -2.9838  16.6589  0.7844  1.7237 -0.0263]  diff: 0

random2: DH  [ 0.    -24.9307 -1.137   13.1356  0.3285  1.2961 -0.021 ]
         RD  [-0.    -24.9307 -1.137   13.1356  0.3285  1.2961 -0.021 ]  diff: 0

Agreement to displayed precision (4 dp) at every configuration tried, including two arbitrary ones -- not just the "nice" poses. This stresses m/r heavily (gravity torque is fundamentally a mass-times-moment-arm quantity) but only weakly exercises I (the inertia tensor only shows up under velocity/acceleration, not static gravity load).

Full inertia matrix M(q) (which does stress I) at q = [0.3, -0.5, 0.2, -1.8, 0.4, 1.5, 0.6]:

• Every off-diagonal term matches to the displayed precision (3 dp).
• Diagonal terms match to within ~1% (largest: M[0,0] = 0.530 (DH) vs. 0.535 (RD), a 0.9% difference) -- small enough to plausibly be rounding/minor-revision differences between the MATLAB-era source data and robot_descriptions' source, not the signature of a structural frame-convention error (which would typically show up as a much larger discrepancy, a sign flip, or a non-symmetric/non-matching structure).

Conclusion so far, and what's not yet covered

This is fairly strong evidence DH.Panda()'s ported parameters are correctly expressed in RTB-Python's own DH frame convention, despite the port never having been explicitly checked for that at the time. Not yet covered:

• Velocity-dependent (Coriolis/centrifugal) terms specifically, in isolation -- M(q) exercises I but not \dot{q}-dependent behavior.
• A true third, independent reference (e.g. Pinocchio or MuJoCo loading the same URDF robot_descriptions uses) rather than only cross-checking two RTB-internal models against each other -- if both happened to share some RTB-specific convention quirk, this check wouldn't catch it. (The #636 comment thread has a precedent for this kind of MuJoCo cross-check, worth reusing here.)
• Extreme/near-singular configurations.

xref #686 (the related URDF-Panda inertial-data gap, and where this doubt was first raised in the context of not porting DH data into URDF).