Principled Grammar Specification for Symbolic Regression of Robot Mass-Matrix Dynamics: Identification of sin(2q) Atom Requirements via Structural Analysis

We address a foundational question in symbolic regression of robot dynamics: what isthe minimal sufficient grammar of trigonometric atoms required to express the diagonalmass-matrix elements Mii(q) of a serial manipulator? Using Lasso regression on a principled atom library derived from rigid-body dynamics, we analyse all seven diagonal elementsof the Franka Panda composite rigid body mass matrix over 2500 joint configurations sampled to include near-singular regions. We show that the standard symbolic-regression atomlibrary — constants, single-angle trigonometric terms, and their pairwise products — explains 85% of variance in the most complex proximal-joint element (R2 = 0.852 on M11),and we identify sin(2q) singles together with their cross-products against single-angle trigas the principal missing atom class, raising explained variance to R2 = 0.939 with activeatom sparsity dropping from 47% to 22%. We argue this atom class arises naturally fromrotating off-diagonal inertia tensor components through joint angles, provide a completegrammar specification, and report negative results on triple products and double-angle pairwise products as further extensions (+0.034 and +0.004 R2 respectively, not justifying thesearch-space inflation). We additionally document a model-completeness diagnostic, ∆s,which correctly flags proximal-joint structural complexity even when discovery fails. Ourfindings inform grammar design for any symbolic-regression system targeting robot inversedynamics "Reproducibility artifacts (data, analysis scripts, Lasso JSONs) available at https://github.com/Qazi-pk/pir-jepa/tree/main/benchmarks/franka_mass."