Compliance for Free: Learning Identifiable Impedance via Bilateral Teleoperation

We model contact at the end-effector with a diagonal, time-varying Cartesian impedance law:

$$ \mathbf{f}(t) = \mathbf{K}(t) \odot \mathbf{e}(t) + \mathbf{D}(t) \odot \dot{\mathbf{e}}(t), \qquad \mathbf{e}(t) = \mathbf{x}_{\mathrm{eq}}(t) \ominus \mathbf{x}_f(t), $$

where $\mathbf{x}_f(t)$ is the measured follower pose, $\mathbf{x}_{\mathrm{eq}}(t)$ is the operator's unobserved commanded equilibrium, and $\mathbf{f}(t)$ is the measured contact wrench. Given only the pair $(\mathbf{x}_f, \mathbf{f})$ — all a pose-only interface ever records — this is unidentifiable: for any axis and any positive stiffness, a compatible equilibrium exists that reproduces the observed force exactly. Two unknowns, one equation, every timestep, no amount of additional data collection fixes it.

Resolving it with an interface, not a model

Four-channel bilateral teleoperation actively couples a leader and a follower arm, so the leader pose is itself an independent, physically measured proxy for the operator's intended equilibrium:

$$ x_{\mathrm{eq}}(t) := x_l(t) \quad \text{(leader pose, independently measured)}. $$

Substituting this collapses the ambiguity — $e(t) = x_l(t) \ominus x_f(t)$ is now directly observed — and $(K(t), D(t))$ become identifiable by windowed regression on $(e, \dot e, f)$, subject to an identifiability mask that excludes (rather than imputes) timesteps with poor regression conditioning, insufficient excitation, or force below the per-axis sensorless noise floor.

At static poses, the residual between measured joint torque and a zero-payload gravity model is linear in the unknown tool's mass and mass-weighted center of mass, identified by least squares over a handful of static poses. A per-axis residual bias model (a random-Fourier-feature ridge regression over joint position and velocity) removes the remaining configuration-dependent drift. The per-axis noise floor $\sigma_{f,i}$ used by the identifiability mask is measured — not assumed — on held-out free-space sweep sessions.

A SmolVLA backbone (vision-language model frozen, action expert fine-tuned) takes a scene camera, a wrist camera, a language instruction (“wipe the {left, right} mark {normally, firmly}”), and a force-history token (a trailing 500 ms of 6-DoF wrench), and outputs an action chunk of absolute target pose and log stiffness, $[x_{\mathrm{eq}}, \log K]$, trained with flow-matching on pose and a masked Huber loss on $\log K$. A 1 kHz Cartesian impedance controller tracks this chunked output safely via temporal ensembling across overlapping chunks, log-space stiffness rate-limiting, and a passivity-preserving energy tank that freezes stiffness increases once depleted while always permitting decreases.

Applied to real bilateral wiping demonstrations, the identifiability mask retains 84.3–96.2% of in-contact timesteps on the three translational axes (Table 1) — every axis clears a 25% validity floor by a wide margin. The measured per-axis sensorless noise floor lies comfortably below the labels' working range on every axis, and extracted stiffness traces show clean, phase-separated dynamics through approach, contact, and retract.

Before training a policy on the extracted labels, we check they carry any signal at all: predicting $\log K_i(t)$ from a feature vector that deliberately excludes force. A learned predictor (random-Fourier-feature ridge) beats a per-axis constant baseline on every translational axis on a held-out, cross-session split — a 1.25× aggregate improvement in $\log K$ RMSE ($1.39\times$ on $f_x$, $1.03\times$ on $f_y$, $1.41\times$ on $f_z$) — rejecting the null hypothesis that the labels are pure noise, the necessary precondition for supervising a policy with them.

The central claim is that compliance extracted from bilateral demonstrations is a learnable, instruction-conditioned action. We test this as a three-link chain on the manner contrast (“firmly” vs.\ “normally”) on the in-plane axis $f_x$: does the adverb show up in the extracted label, then in the policy's own output, then in the arm's realized force?

All three links carry the instruction: the demonstrations encode the adverb in stiffness ($d=0.76$), the policy reproduces that dependence in its own output ($d=0.98$), and it survives all the way to realized contact force ($d=0.89$). The policy also commits to genuine anisotropy — a median in-plane/normal stiffness ratio of $2.50\times$ in the contact frame, far beyond the ${\approx}1.2\times$ in the pooled label distribution — so this is not regression to the label mean.

But predicting compliance is not by itself sufficient: of five policies, only ours modulates realized force with the instruction (Fig. below). B3 moves its commanded stiffness with manner ($d=2.26$ on $f_y$) but its realized force does not follow ($d=0.17$) — the rest of the policy still has to track what it predicts.

All policies are evaluated on a whiteboard-wiping task, $n=24$ rollouts per policy (3 seeds, a $2\times2$ referent$\times$manner instruction grid). B0: position output, fixed stiffness. B1: position output, oracle per-axis constant stiffness fit to our label distribution. B2: ForceVLA-style force-as-input, position output. B3: ForceVLA2/Force-Policy-style hybrid force+position output. B4 (ours): compliance output, force input.

Contact-force behavior is more informative than success alone: our policy attains the highest ink removal at a peak force 8.6 N lower and an RMS force 3.1 N lower than B0, tripping one protective stop in 24 rollouts against B0's six.

Before trusting any of the above, we verify the low-level controller actually responds to a commanded stiffness: a 20 s sinusoidal probe on all six axes shows realized stiffness tightly tracking the commanded signal, with the energy-tank trace confirming the passivity budget behaves as designed.

We freeze the force-history token at its first observed value for the remainder of the episode and re-run. All three force-consuming policies degrade when force is frozen (B4 by $-8.3$ points), consistent with the live force signal being causally used rather than ignored — though with $n=24$ per policy none of these drops individually reaches significance ($p \geq 0.73$).

Grid-searching, per axis, the single constant stiffness that best fits our label distribution gives unremarkable translational constants (141–158 N/m, well inside the controller's realizable range) but rotational constants that land on the extraction's lower bound — the rotational impedance in these demonstrations is softer than the controller's minimum commandable rotational stiffness. Deployed, B1 cannot hold orientation against the board, which is why it scores 0% and not evidence against constant impedance as such: the honest constant-impedance control in this study is B0 (hand-set, well-posed, 33.3%). The claim is narrower and, we think, sharper — a well-tuned constant impedance still cannot modulate force on instruction (Table 2).