MIME-VER-124 — RobotArmNode Inverse-Dynamics + PD Trajectory Tracking#

Date: 2026-04-30 Node under test: mime.nodes.actuation.robot_arm.RobotArmNode Algorithm ID: MIME-NODE-102 Benchmark type: Analytical (Mode 2 independent) Test file: tests/verification/test_robot_arm.py::test_ver124_pd_tracking Acceptance: max-per-joint RMS tracking error $< 0.5°$ over 2 s of a 0.5 Hz sinusoid

Goal#

Track a sinusoidal joint-space trajectory using the standard manipulator-control law (Sciavicco–Siciliano §6.6):

$$ \tau = M(q),\ddot q_{\mathrm{des}} + c(q,\dot q) + g(q) + K_p,e + K_d,\dot e, $$

with $e = q_{\mathrm{des}} - q$, and assert that the tracking RMS error stays below $0.5°$ across all 3 joints over a 2 s integration.

This benchmark stresses the full dynamics path: $M(q)$ from CRBA, $c+g$ from RNEA, and the forward-dynamics solve in update. A bug in any of those would manifest as drift or oscillation.

Configuration#

Parameter

Value

URDF

tests/control/fixtures/three_link_planar.urdf

Trajectory

$q_{\mathrm{des},i}(t) = A_i \sin(\omega t)$

Amplitudes

$A = (0.3, 0.2, 0.4)$ rad

Frequency

$\omega = 2\pi \cdot 0.5$ rad/s (0.5 Hz)

$K_p$ (per joint)

$(200, 200, 80)$ N·m/rad

$K_d$ (per joint)

$(20, 20, 8)$ N·m·s/rad

Joint friction

$0$ (overridden in test fixture)

Timestep

$10^{-3}$ s

Steps

2000 (2 s of simulated time)

JAX precision

x64 enabled at module load

Hot loop

jax.jit-compiled

The PD gains were chosen for a closed-loop natural frequency well above the 0.5 Hz disturbance band. The inverse-dynamics feed-forward is the principal contributor — without it, residual tracking error is $\approx 7°$ (verified during V&V development); with it, error drops below the $0.5°$ acceptance.

Procedure#

  1. Initialise $(q_0, \dot q_0)$ to the desired trajectory at $t=0$.

  2. JIT-compile a step closure that, given $(q, \dot q, t)$, computes $M(q), c+g, \tau_{\mathrm{ff}} = M\ddot q_{\mathrm{des}}+c+g$, the PD term $K_p e + K_d \dot e$, and calls arm.update.

  3. Run 2000 jitted steps, accumulating $\sum e^2$.

  4. Compute per-joint RMS = $\sqrt{\overline{e^2}}$ in degrees and assert the maximum component is below $0.5°$.

Result#

PASS. Per-joint RMS error stays within the acceptance.

Scope and Limitations#

  • One trajectory shape (single-frequency sinusoid). Higher-frequency components or non-smooth trajectories would test the integrator’s CFL-like stability boundary, which is not in scope here.

  • Joint friction is disabled in this benchmark to isolate the dynamics. With realistic friction, additional gain tuning would be required — that’s an experiment-folder concern, not a node-level V&V concern.

  • The inverse-dynamics feed-forward is computed at the current state rather than the desired state — both are valid approaches, but the current-state form is what the test uses.

Reproducibility#

  • Run: JAX_PLATFORMS=cpu .venv/bin/python -m pytest tests/verification/test_robot_arm.py::test_ver124_pd_tracking -x -q.