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Impedance Cloning: Learning Equilibrium Point Parameters for Contact-Rich Manipulation

Hayato Takahashi, Ryoga Oishi, Yuki Kasuga, Toshiaki Tsuji

arXiv:2609.30842v1cs.RO

TL;DR

Contact-rich manipulation must regulate position and force when surface geometry differs from training conditions, while observable-state imitation lacks intrinsic compliance. Impedance Cloning learns equilibrium point and stiffness parameters from bilateral teleoperation demonstrations. Across wiping, pick-and-place, and grasping, the paper reports contact robustness, adaptation to unseen geometries, and improved torque tracking with ILBiT and Mamba.

  • Problem

    Observable-state imitation reproduces position, force, joint angles, or torques but lacks intrinsic compliance under unpredictable surface geometry changes.

    Source: §I · p. 1

  • Method

    Impedance Cloning uses a particle filter on bilateral teleoperation observations to estimate stiffness and equilibrium point parameters as imitation targets.

    Source: §I · p. 1, §III · p. 2

  • Results

    The method maintains stable contact under height perturbations where the trajectory-based baseline fails, adapts to unseen object geometries, and improves torque tracking with ILBiT and Mamba.

    Source: §I · p. 1, §V · p. 8

  • Takeaways & Limitations

    Imitating latent command parameters rather than observable outputs provides an adaptive mechanism for contact-rich manipulation under environmental uncertainty.

    Source: §I · p. 1, §V · p. 8

    The reported results do not measure muscle-level stiffness, and future work is proposed for dynamic tasks with multidirectional contacts.

    Source: §V · p. 8

Abstract

from arXiv · show

Contact-rich manipulation requires robots to regulate force against surfaces whose geometry deviates unpredictably from training conditions. Trajectory-based imitation learning, which reproduces observable outputs, breaks down under such shifts. We propose Impedance Cloning, which instead imitates the biomechanical priors that generate motion -- the stiffness and equilibrium point -- and thereby passively absorbs contact uncertainty. Because these parameters encode intent rather than outcome, they generalize across surface geometries where trajectory reproduction does not. We extract them from bilateral teleoperation demonstrations via a particle filter without force/torque sensors and evaluate the framework on two CRANE-X7 manipulators. In a wiping task with joint-space actions, the trajectory-based baseline loses contact below -6 cm, whereas the proposed method maintains a consistent 4-5 N contact force above -6 cm, with a gradual decrease below; with Cartesian-space actions, its force-height slope over 0 to +8 cm is 0.13 +/- 0.03 N/cm, versus 0.34-0.83 N/cm for fixed-impedance baselines. In a pick-and-place task with 10 diverse cups (100 trials), the proposed method succeeds in 84 trials, outperforming the fixed-impedance baseline (74/100) and performing comparably to a variable impedance control baseline (82/100) with one demonstration instead of ten. In a grasping task, the representation reduces torque tracking error with both ILBiT and Mamba backbones, confirming its generality across architectures.

I. INTRODUCTION

Observable-state imitation reproduces trained motions but lacks intrinsic compliance when contact geometry shifts. Impedance Cloning instead learns equilibrium point and stiffness parameters from bilateral demonstrations, and its evaluation reports robust contact across manipulation tasks and improved torque tracking with two model architectures.

I. INTRODUCTION

  • Conventional imitation methods encode position, force, joint angles, or torques, but lack intrinsic compliance when surface geometry deviates from training conditions.The paper contrasts this with variable-impedance approaches that often require estimating and reidentifying environmental dynamics and stiffness.Source: §I · p. 1
  • The method maintains stable contact under height perturbations where the trajectory-based baseline loses contact.The reported wiping comparison identifies loss of contact at −6 cm for the baseline.Source: §I · p. 1
  • Impedance Cloning replaces observable action targets with latent equilibrium point and stiffness parameters extracted from bilateral teleoperation demonstrations using a particle filter.The parameters can be replayed directly or generated by a policy trained on them.Source: §I · p. 1
  • The representation transfers across unseen surface geometries and object shapes in wiping, pick-and-place, and grasping tasks.The paper reports improved torque tracking with both Transformer-based ILBiT and SSM-based Mamba models.Source: §I · pp. 1–2

II. RELATED WORK

Prior work has explored observable-state imitation, variable impedance, and teleimpedance, but the paper positions equilibrium point parameters as a distinct imitation target. Its method estimates those latent parameters from bilateral teleoperation data and uses the leader robot both to measure and execute the reconstructed command.

A. Imitation Learning for Contact-Rich Manipulation

  • Conventional imitation learning represents observable robot states and lacks intrinsic compliance under environmental deviations.Prior work also used bilateral control to record position and force, while impedance-parameter actions had been explored in reinforcement learning.Source: A. Imitation Learning for Contact-Rich… · p. 2

B. Variable Impedance Control and Stiffness Adaptation

  • Unlike variance-based stiffness profiles that require repeated demonstrations of the same task, the proposed targets are operator stiffness and an independently estimated equilibrium point.The paper states these parameters can be estimated from a single demonstration and transferred without environmental re-identification.Source: B. Variable Impedance Control and… · p. 2

C. The Equilibrium Point Hypothesis in Motor Control and Robotics

  • The paper distinguishes its approach from teleimpedance and prior compliance estimates by treating equilibrium point and stiffness as imitation targets estimated from demonstration signals.It reports estimating them from either end-effector position and force or joint angle and torque, then reproducing them by replay or a trained policy.Source: C. The Equilibrium Point Hypothesis in… · p. 2

III. PROPOSED METHOD

  • Impedance Cloning comprises parameter estimation, imitation learning, and torque command generation.Bilateral teleoperation supplies the leader robot’s joint angle and estimated torque; the EP model converts that observable pair into the latent command, and the leader later executes it.Source: §III · p. 2

A. Definition of a Mathematical Model

The mathematical model treats the operator’s command as a spring between a virtual equilibrium configuration and the measured leader configuration. It is instantiated in joint and task spaces, with stiffness estimated separately for each degree of freedom or translational axis.

A. Definition of a Mathematical Model

  • The EP model represents the operator’s command as a spring between a virtual equilibrium configuration and the leader robot’s measured configuration.In the displayed task-space form, force is proportional to the difference between equilibrium and measured position.Source: A. Definition of a Mathematical Model · p. 2, Figure 1 · p. 3
  • In joint space, the model uses leader joint angle and estimated joint torque; in task space, it uses end-effector position and corresponding operational force.The task-space force is obtained from joint torque through the leader robot’s Jacobian.Source: A. Definition of a Mathematical Model · pp. 2–3
  • The model considers only per-axis stiffness and omits off-diagonal stiffness-matrix components, assigning an independent estimator to each joint or translational axis.The parameters are projected onto the leader’s coordinate frame, making frame choice a design choice for the action space rather than a model restriction.Source: A. Definition of a Mathematical Model · p. 3
  • The estimated parameter state is produced offline at each time step and used as the target for subsequent learning.The work describes particle-filter likelihood resampling and a weighted average for stable parameter estimates.Source: A. Definition of a Mathematical Model · p. 3

B. Parameter Estimation Algorithm

The particle filter estimates stiffness and equilibrium point from measured motion and torque by combining a slowly varying parameter assumption with sequential Bayesian estimation. Its transition and observation models constrain parameter drift and weight estimates against the measured torque.

B. Parameter Estimation Algorithm

  • At one instant, angle and torque provide one equation for two unknowns; the estimator resolves this by assuming command parameters vary slowly relative to measured configuration.Responses at different joint angles within a short window jointly constrain stiffness and equilibrium point.Source: B. Parameter Estimation Algorithm · p. 3
  • The particle filter uses a nonlinear transition that pulls equilibrium point toward the current angle more strongly at low stiffness, alongside small random walks of both parameters.The adaptive pull-back coefficient is stiffness-dependent, and system noise drives the parameter random walks.Source: B. Parameter Estimation Algorithm · p. 3
  • A lower bound prevents stiffness estimates from becoming negative, preserving their physical consistency.The hyperparameters were set empirically to balance physical-robot motion stability and responsiveness to environmental changes.Source: B. Parameter Estimation Algorithm · p. 3
  • Measured torque error under each particle's estimated state determines its likelihood, after which resampling and a weighted average produce the time-step estimate.The likelihood uses the squared error between measured torque and torque predicted from the particle's stiffness and measured angle.Source: B. Parameter Estimation Algorithm · p. 3

C. Implementation of the Imitation Learning Module

The implementation compares conventional trajectory actions with impedance-parameter actions while holding the execution scheme and, for learned models, the architecture and training procedure fixed. It evaluates replay-based and generative-model settings across tasks.

C. Implementation of the Imitation Learning Module

  • The proposed action representation uses impedance parameters instead of the baseline's joint, velocity, and torque trajectories, with both approaches sharing the same execution scheme for each backbone.Motion Copy directly replays recorded actions; learned models share architecture and training procedure, so the representation differs.Source: C. Implementation of the Imitation… · p. 3, Figure 2 · p. 4
  • For pick-and-place, the baselines include Motion Copy with fixed impedance and a variable-impedance controller that estimates stiffness inversely from positional variance across demonstrations.Cartesian wiping also tests Motion Copy at three fixed-impedance levels; ILBiT and Mamba serve as grasping backbones.Source: C. Implementation of the Imitation… · p. 4
  • The evaluation is designed to compare data-efficient replay performance with generative-model generalization under the two action representations.The supplied hyperparameter-table caption reports effective particle counts of 526–959 after resampling truncation, below nominal Np.Source: C. Implementation of the Imitation… · p. 4, Table I · p. 4

D. Formulation of Torque Command Generation Rules

During autonomous execution, follower responses provide command references for the leader, whose torque command includes an estimated human operational torque derived from the inferred impedance parameters. The follower itself runs only the standard feedback loop.

D. Formulation of Torque Command Generation Rules

  • The follower's position, velocity, and torque responses serve as the leader's command references, which are used with joint inertia and position, velocity, and force gains to formulate the leader torque command.The leader torque command also includes the estimated human operational torque as its first term.Source: D. Formulation of Torque Command… · p. 4
  • The estimated human operational torque is computed from stiffness and equilibrium point, leader response angle and velocity, and a constant damping coefficient.The damping coefficient is introduced only during execution; during demonstrations, the operator's arm supplies physical damping.Source: D. Formulation of Torque Command… · p. 4
  • The inferred human operational torque enters the leader alone, while the follower runs only the standard feedback loop, creating an asymmetric configuration for adaptive contact motion.The baseline instead runs the follower loop using replayed or predicted position, velocity, and torque references.Source: D. Formulation of Torque Command… · p. 4

IV. EXPERIMENTS

The experiments use bilateral CRANE-X7 manipulators to assess parameter validity and the method across wiping, pick-and-place, and grasping. The setup estimates joint torque with a reaction force observer rather than physical force/torque sensors.

IV. EXPERIMENTS

  • The experiment plan tests parameter validity, wiping robustness to height perturbations, pick-and-place generalization across objects, and grasping adaptability across sequence-modeling architectures.The three manipulation tasks are identified as Experiments 2–4, following parameter validation in Experiment 1.Source: §IV · p. 4
  • During autonomous execution, the architecture separates leader and follower signal flow; the leader control includes inferred-parameter torque computation, while the follower uses the standard feedback loop.The figure distinguishes the parameter-based red components from the black feedback loop.Source: Figure 3 · p. 5

A. Test Environment and System Configuration

  • Two 8-DOF CRANE-X7 manipulators form the bilateral system, with joint torques estimated by a reaction force observer without physical force/torque sensors.A whiteboard cleaner is held by the end-effector hand, and the target whiteboard is fixed to the workbench.Source: A. Test Environment and System… · p. 5, Figure 4 · p. 5

B. Dataset and Training Settings

The paper trains on bilateral-control demonstrations and evaluates with small datasets, using matched torque-based inputs to compare action representations. Its criteria measure wiping contact force, cup-transport success, and aligned torque-tracking error.

B. Dataset and Training Settings

  • The wiping dataset contains approximately 75,000 steps from 10 motion samples, while pick-and-place uses 10 demonstrations and grasping uses 16.For pick-and-place, Impedance Cloning and Motion Copy use one demonstration; VIC uses all ten.Source: B. Dataset and Training Settings · p. 5
  • ILBiT and Mamba provide torque-based comparison inputs, avoiding a direct comparison with camera-conditioned ACT or Diffusion Policy that would confound action representation with input modality.ILBiT takes joint-angle and torque sequences; Mamba is an SSM-based architecture.Source: B. Dataset and Training Settings · p. 5

C. Evaluation Criteria and Experimental Conditions

  • Wiping performance is assessed by average contact force during the contact interval, with board-height offsets varied across joint- and Cartesian-space conditions and five trials per condition.Joint-space offsets span −10 to +10 cm; Cartesian-space offsets span −8 to +8 cm.Source: C. Evaluation Criteria and… · pp. 5–6
  • Pick-and-place success counts cups transported without dropping across 10 types and 100 trials, while grasping torque error uses DTW-aligned RMSE against a time-normalized training reference.The torque reference averages training trajectories after normalization to 500 points and sign-flips follower values before comparison.Source: C. Evaluation Criteria and… · p. 6

D. Experiment 1: Validity of Parameter Estimation

The estimated equilibrium-point parameters reconstruct demonstrated torque, with stability dependent on suitable pull-back hyperparameters. In wiping, Impedance Cloning is less sensitive to height changes than fixed-impedance replay in the tested ranges, though contact is not maintained at every offset.

D. Experiment 1: Validity of Parameter Estimation

  • 0.071 NRMSE: the two-parameter estimator reconstructed measured torque, and estimates remained within a factor of 1.6 when hyperparameters were restricted to settings within 5% of the best reconstruction error.Across the unrestricted γ–β sweep, mean stiffness ratios ranged from 4.5 to 5.9; excessive pull-back degraded NRMSE to 0.275.Source: D. Experiment 1: Validity of Parameter… · p. 6

E. Experiment 2: Wiping Task — Robustness to Height Perturbation

  • From −6 to +10 cm in joint space, Impedance Cloning maintained approximately 4–5 N, while the baseline nearly lost contact at −6 cm and below.Below −6 cm, the proposed method’s force also decreased gradually, which the authors attribute to posture fluctuations and cleaner-edge catching.Source: E. Experiment 2: Wiping Task —… · p. 6
  • 0.13 ± 0.03 N/cm: the Cartesian-space method’s absolute force–height slope over 0 to +8 cm, versus 0.34 ± 0.06, 0.52 ± 0.09, and 0.83 ± 0.05 N/cm for fixed-impedance 0.5Kp, 0.75Kp, and Kp.At −6 cm and below, the proposed method lost contact; the authors suggest the restoring force reverses when wiping passes the equilibrium point.Source: E. Experiment 2: Wiping Task —… · p. 6
  • The wiping results support use of Impedance Cloning in both joint and Cartesian coordinate spaces, with lower sensitivity than fixed-impedance replay in the tested Cartesian positive offsets.The coordinate-space choice also shapes failure behavior: joint-space posture changes and Cartesian-space equilibrium-point crossing are associated with contact loss at lower offsets.Source: E. Experiment 2: Wiping Task —… · p. 6

F. Experiment 3: Pick-and-Place Task — Generalizability Across Object Types

Pick-and-place tests varied cup geometry and material while keeping object position fixed. Impedance Cloning improved overall success over fixed-impedance replay and performed comparably to VIC despite using one demonstration.

F. Experiment 3: Pick-and-Place Task — Generalizability Across Object Types

G. Experiment 4: Grasping Task

In grasping, replacing trajectory actions with equilibrium-point parameters reduced torque-tracking error across both tested sequence-modeling architectures. The conclusion presents these parameters as an effective representation for robust contact-rich manipulation, while delimiting the interpretation and identifying future extensions.

G. Experiment 4: Grasping Task

  • The grasping evaluation trained on 16 demonstrations of one cup and tested on 10 cups of varying stiffness and geometry, comparing direct position-and-force imitation with equilibrium-point actions under each backbone.The tested backbones were ILBiT and Mamba.Source: G. Experiment 4: Grasping Task · p. 7
  • Across 10 unseen containers, the equilibrium-point action space reduced mean torque-tracking error and variance for both ILBiT and Mamba; per-container gripper-joint comparisons were significant for ILBiT (p = 0.02) and Mamba (p = 0.002).Table III reports mean ± SD; its standard deviations capture variation across containers, not trials.Source: G. Experiment 4: Grasping Task · p. 7, Table III · p. 8

V. CONCLUSIONS

  • The paper concludes that biomechanically grounded upper-level parameters are effective for robust contact-rich manipulation, and reports less sensitivity to positive height offsets in its Cartesian formulation.The authors do not claim to measure muscle-level stiffness; they identify dynamic multidirectional contacts and coordinate-space selection as future work.Source: §V · p. 8

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