Study 01: How rank differs across gait phases

What we measure

For a deterministic actor mean and observation x, the per-state Jacobian relates small observation changes to action changes. We summarize its dimensionality with the entropy effective rank of the batch-averaged sensitivity map, then split the batch by gait phase: swing, when a foot is off the ground, and stance, when it bears load.

We measure the phase gap Δφ = erank(J_swing) − erank(J_stance). The global ranks of the two architecture families are nearly equal, at 9.3 and 9.2. Averaging over the gait cycle combines swing and stance into one value; evaluating them separately reveals how the learned input–output sensitivity depends on phase.

Phase gap per configuration and pooled swing, stance, and global Jacobian effective rank — (a) Phase gap Δφ on flat terrain: every SimBa configuration has higher rank during swing than stance; every MLP configuration has a negative gap. (b) Pooled swing, stance, and global (dashed) Jacobian erank per family. The nearly equal global values (9.3 vs 9.2) obscure the phase-dependent difference.

Untrained versus trained policy Jacobian matrices — Jacobian matrix for a deployed SimBa policy. (a) Before training, the structure is nearly random. (b) After training, basic input–output relationships emerge.

Results

Quantity	Value
Δφ, SimBa family (pooled)	+2.14
Δφ, MLP family (pooled)	−0.31
Mann–Whitney p	2.5e−6
Cliff's d	0.98
Global Jacobian erank, SimBa vs MLP	9.3 vs 9.2
Configurations with the expected sign	all SimBa positive, all MLP negative

The swing-dominant pattern also appears on the sim-to-real platform and on perceptive rough terrain, suggesting that it is not specific to a single simulator or terrain.