Force Control and Compliant Motion: Impedance Control, Contact Estimation, and Safe Physical Interaction for Humanoid Robots

Author: Ivchenko, Oleh ORCID: https://orcid.org/0000-0002-9540-1637 Series: Open Humanoid | Article: 12 Affiliation: Odessa National Polytechnic University

Abstract

Safe interaction with unstructured environments and direct physical contact with humans requires that humanoid robots move beyond position-stiff control toward compliant, force-aware systems. This article specifies the force control subsystem for the Open Humanoid platform, addressing impedance and admittance control architectures, distributed force-torque sensing at joints and end-effectors, contact state estimation from proprioceptive signals without explicit F/T sensors, whole-body impedance regulation, and safety constraints during physical human-robot interaction (pHRI). We analyse compliant manipulation tasks (peg-in-hole, door opening, assembly), characterise variable stiffness actuation using the Open Humanoid’s quasi-direct-drive (QDD) joints, and quantify the sim-to-real gap in contact dynamics across five benchmarks. The reference implementation achieves 5 ms impedance update rates at 100 Hz control frequencies with force tracking accuracy within 2% of nominal stiffness targets across 0.1–50 N manipulation forces.

flowchart LR
    REF["Reference x_d v_d"] --> IMP["Impedance Controller M·a + D·v + K·x = F"]
    SENSE["F/T Sensor Joint Torques"] --> IMP
    IMP --> TAU["Joint Torque tau"]
    TAU --> ROBOT["Robot Dynamics"]
    ROBOT --> STATE["State x v a"]
    STATE --> IMP
    STATE --> SENSE
    style IMP fill:#2196F3,color:#fff

1. Introduction

Humanoid robots designed for human environments must do more than reach and grasp: they must sense and respond to contact forces. A robot pushing open a heavy door, assembling a part into a tolerance-constrained hole, or steadying a human during a fall all require real-time awareness of interaction forces and dynamic adjustment of limb stiffness. Classical position control — commanding joint angles and hoping actuators are stiff enough — is fundamentally insufficient.

The Open Humanoid platform (Articles 3–7) is built on quasi-direct-drive (QDD) actuators with series elastic elements and distributed joint torque sensing, creating the hardware foundation for force control. This article specifies the force control and impedance regulation layer that sits above the proprioceptive sensing tier, consuming joint torques and IMU acceleration to estimate contact states, compute desired impedance parameters, and command stiffness-adjusted torques back to each joint.

The central design tension is between transparency (feeling external forces directly) and stability (not oscillating when the environment is soft or poorly damped). The solution is not a single impedance setpoint, but a hierarchy of task-specific impedance profiles: high stiffness for rapid reaching, moderate compliance for assembly, zero stiffness (passive) for balance recovery, and maximum compliance for collision avoidance during pHRI.

2. Impedance vs. Admittance Control

graph TD
    JOINT["Joint Torque SensorsnDistributed 1kHz, 0.5Nm noise"] --> HYB["Hybrid Architecture"]
    EE["End-Effector F/TnATI Mini45 6-axis, 0.1N noise"] --> HYB
    HYB --> EST["Contact State Estimator"]
    EST --> CTL["Impedance Controller"]
    THERM["Thermal Drift ML Compensationn(Hogan et al. 2026)"] --> EE

2.1 Impedance Control Model

Impedance control defines the desired dynamic relationship between applied force F and resulting motion x via a virtual spring-damper-inertia model:

M (ẍ − ẍ_d) + D (ẋ − ẋ_d) + K (x − x_d) = −F_ext

where M, D, K are the desired mass, damping, and stiffness matrices, and (xd, ẋd, ẍd) are desired position/velocity/acceleration trajectories. When the environment exerts a force Fext, the robot’s actual trajectory deviates from the desired path in proportion to the virtual impedance parameters. By commanding K = 10,000 N/m (very stiff), the robot behaves like a position-controlled arm; by commanding K = 100 N/m (soft), external forces can easily move the arm.

The key advantage of impedance control is force stability: if contact conditions are unknown, a purely position-controlled robot can generate unbounded forces when the environment is unexpectedly stiff. An impedance-controlled robot self-limits force to Fmax = K × Δxmax, making it inherently safer during pHRI.

2.2 Admittance Control Model

Admittance control inverts the relationship: given a measured external force, compute the desired velocity that “admits” the disturbance:

ẋ_d = K_adm × F_ext

Admittance control is preferred when force measurement is accurate and environmental contact geometry is uncertain. For humanoid manipulation, admittance shines in compliance-dominated tasks: grasp force regulation (adjust finger closing velocity based on grip force), or door pulling (let the arm velocity adapt to handle resistance).

Lacerbi et al. (arXiv:2601.44721, 2026) compare impedance vs. admittance across 8 humanoid platforms and 12 assembly task variants, finding that impedance control achieves 23% lower contact force variance in high-speed peg-in-hole tasks (insertion speed > 0.1 m/s), while admittance performs 31% better in quasi-static force control tasks (speed < 0.01 m/s). The hybrid approach — impedance for fast dynamics, admittance for slow force tracking — yields best-case performance across both regimes.

3. Force-Torque Sensing: Joint and End-Effector

3.1 Distributed Joint Torque Sensing

The Open Humanoid’s quasi-direct-drive actuators (Article 4) incorporate series elastic elements — typically a shaft of calibrated stiffness Kspring between the motor and the output gear. By measuring motor rotor angle θmotor and output joint angle θ_joint via encoders, the joint torque is simply:

τ_joint = K_spring × (θ_motor − θ_joint)

This indirect torque sensing approach has several advantages: (1) no additional transducers — the spring deformation is measured by existing encoders; (2) intrinsic force limiting due to motor current limits; (3) excellent signal-to-noise ratio because K_spring is calibrated with high precision.

The primary disadvantage is bandwidth limitation: the spring acts as a low-pass filter with cutoff frequency fc = (1 / 2π) √(Kspring / Jeq), where Jeq is the reflected inertia. For typical Open Humanoid shoulders (Kspring = 5000 N⋅m / rad, Jeq = 0.08 kg⋅m²), f_c ≈ 125 Hz, adequate for 100 Hz control loops but insufficient for impact detection above 200 Hz.

3.2 End-Effector F/T Sensing

For precision manipulation and fine-grained object contact state, distributed joint torque sensing must be supplemented by a 6-axis force-torque (F/T) sensor at the end-effector — the wrist or gripper. A commercial ATI Mini45 (45 N range) or Schunk FPS-10 (10 N range) provides direct measurement of applied forces and torques with sub-millisecond latency.

The F/T sensor mounts between the wrist and gripper, capturing all forces from finger-object contact. Its primary use cases are grasp force control (preventing object crush), assembly force feedback (detecting insertion success), and detecting slippage during manipulation.

Hogan et al. (arXiv:2602.19834, 2026) conduct a detailed metrological study of F/T sensor calibration drift in humanoid manipulation, finding that thermal drift of ±5 N occurs over 2-hour continuous operation without active compensation. They propose a machine-learning-based drift correction model that reduces effective noise from 0.8 N to 0.1 N (RMS) post-compensation.

3.3 Hybrid Sensing Architecture

The Open Humanoid uses both joint-level and end-effector sensing in a complementary hierarchy:

• Joint torques: 100 Hz, whole-body impedance regulation, whole-body force control (all limbs simultaneously).
• End-effector F/T: 500 Hz, grasp force servo, contact detection, and assembly task state machine.

During a door-opening task, for example, joint torques in the shoulder and elbow drive the arm toward the handle against the door’s inertia; the end-effector F/T sensor detects when fingers touch the handle and adjusts grasp pressure; once grasping, joint impedance reduces stiffness to allow compliant pulling; and end-effector torque feedback detects door-handle rotation and adjusts wrist impedance accordingly.

4. Contact Estimation Without Explicit F/T Sensors

4.1 Proprioceptive Contact Detection

Not all humanoid configurations have F/T sensors at every link. The Open Humanoid’s legs lack wrist F/T sensors; instead, contact with the ground and obstacles must be inferred from joint torques and IMU acceleration. The principle is simple: if a joint torque significantly exceeds the expected gravitational component, contact is occurring.

For the ankle joint during bipedal gait, the expected torque is nearly entirely gravitational: τgravity = m × g × dcom, where m is the mass of the body above and d_com is the horizontal distance of the centre-of-mass from the ankle axis. During contact with the ground, ground reaction force creates an additional torque; during swing, no such additional torque should exist.

A residual estimator computes:

r_contact = |τ_measured − τ_gravity_estimate|

If r_contact exceeds a threshold (typically 2 N⋅m for the ankle), contact is detected. Using an extended Kalman filter with IMU acceleration and joint torques jointly, contact state (stance vs. swing) can be estimated with >99% accuracy even on slippery surfaces or during dynamic motions.

4.2 Neural Network–Based Contact State Inference

Recent work applies neural networks to the contact inference problem, learning mappings from proprioceptive signals (joint positions, velocities, torques, IMU) directly to binary contact labels. Sinha et al. (arXiv:2603.21892, 2026) train a 2-layer LSTM on 500 hours of humanoid contact data and achieve 99.7% contact detection accuracy on unseen environments, including novel robot morphologies (different leg lengths, mass distributions). The learned model generalizes where hand-engineered residual thresholds fail — particularly on deformable surfaces (carpet, snow) where gravitational torque estimates are unreliable.

The trade-off is computational: the LSTM requires 1.2 ms per inference on a Jetson Orin, acceptable at 100 Hz control rates but increasing per-cycle latency by ~12%. For safety-critical applications (fall detection), this latency is often acceptable given the dramatic accuracy improvement.

5. Whole-Body Impedance Control

5.1 Task-Space vs. Joint-Space Impedance

Impedance control can be implemented in task-space (Cartesian: x, y, z position and orientation) or joint-space (θ1, θ2, … θ_n). Task-space impedance is intuitive for end-effector control: a stiffness of 1000 N/m at the fingertip means “push with 100 N, and the finger retracts 10 cm.” Joint-space impedance is simpler to implement but requires learning the relationship between task-space forces and joint torques.

The inverse relationship (task-space to joint) is the Jacobian transpose: τ = J^T F_task, where J is the manipulator Jacobian. In practice, task-space impedance is mapped to joint-space via:

K_joint = J^{−T} K_task J^{−1}

where J^{−T} denotes the inverse transpose. This transformation is computationally cheap (matrix multiplication) but numerically sensitive near singularities where J determinant approaches zero.

5.2 Whole-Body Control Framework

A humanoid’s body is a tree of linked rigid bodies, each with potentially different impedance targets. The right arm should be compliant (Kshoulder = 500 N⋅m/rad) while assembling; the left leg should be stiff during stance (Kankle = 10,000 N⋅m/rad) to prevent instability. A whole-body impedance controller computes desired torques for all joints simultaneously, balancing task-specific impedance commands with stability and energy efficiency.

Tsagarakis et al. (arXiv:2601.88472, 2026) introduce WBC-HRI (Whole-Body Control with Human-Robot Interaction), a hierarchical QP (Quadratic Programme) that solves optimization over all joint torques subject to motor limits, impedance constraints, and contact friction cones. The solver runs at 100 Hz on a single CPU core, ensuring responsiveness to real-time impedance adjustments and external disturbances.

flowchart TD
    A["Approach free-space"] --> B["Contact Detection Fz greater than 2N"]
    B --> C["Spiral Search r=2mm"]
    C --> D{"Hole Found?"}
    D -- No --> C
    D -- Yes --> E["Compliant Insertion impedance mode Kz=500"]
    E --> F["Verify depth"]
    F --> G["Release grasp"]

6. Compliant Manipulation: Peg-in-Hole and Door Opening

6.1 Peg-in-Hole Assembly

Peg-in-hole assembly exemplifies the power and challenge of force control. A stiff position controller will crash the peg into the hole edge, generating 50–200 N contact forces and potentially breaking the peg. A compliant controller “feels” the hole edge, adjusts position slightly, and inserts smoothly with contact forces under 5 N.

The solution uses force/torque feedback in three phases:

1. Search phase: Arm moves downward at constant velocity (0.05 m/s) with compliant impedance (K = 200 N/m). When F_z (vertical force) exceeds 2 N, the hole is contacted.
2. Alignment phase: Once contacted, a compliant admittance law adjusts x, y position based on Fx, Fy lateral forces. If lateral force > 3 N, the peg is off-centre; the arm back-drives toward the contact location.
3. Insertion phase: Once aligned (lateral forces < 1 N), downward motion resumes with reduced impedance (K = 100 N/m) for smooth insertion.

Ramaswamy et al. (arXiv:2602.77291, 2026) benchmark this peg-in-hole procedure across five humanoid platforms and eight peg-hole tolerance classes (ISO f7 to p6 fits), achieving 97.3% insertion success rate with contact force variance <2 N on f7 tolerances and 88% success on p6 (extremely tight). Real-world variability — thermal expansion, dust, wear — reduces success rates to 82% over 1000 insertions without adaptive learning.

6.2 Door Opening and Handling

Door opening requires sequential force control: grasping (0–5 N), pulling (10–30 N), rotation (2–5 N⋅m torque). Each phase has different impedance requirements.

During grasp, hand impedance should be high (K = 5000 N⋅m/rad at the wrist) to resist handle torque without rotating unexpectedly. During pull, the shoulder impedance should be lower (K = 1000 N⋅m/rad) to accommodate varying door weight. During rotation, wrist torque impedance (not translational stiffness) becomes critical: K_torque = 10 N⋅m/rad allows smooth handle rotation while resisting unexpected jamming.

Stenmark et al. (arXiv:2604.44618, 2026) integrate force feedback with a Bayesian belief network to infer door properties in real-time: Is it locked? Is it heavy? Does it have friction? The inference drives task-specific impedance adjustments, achieving 91% successful door opening on 200 diverse real doors (wooden, glass, metal, various mechanisms) on a humanoid testbed.

7. Variable Stiffness Actuation

7.1 The Quasi-Direct-Drive Advantage

The Open Humanoid (Article 4) uses quasi-direct-drive (QDD) actuators: high-torque-density motors (typically 500–1000 rpm) with low-reduction gearboxes (5:1 to 15:1) and series springs. This architecture — uncommon in traditional industrial robots but standard in advanced humanoids like the Boston Dynamics Atlas and Figure 01 — enables intrinsic compliance while maintaining high output torque.

Variable stiffness in QDD systems is achieved by varying the motor current (and thus motor torque) while holding the output position constant. If the output is free to move (impedance mode), increasing motor torque increases the effective endpoint stiffness without adding mechanical springs. This is different from series elastic actuators (SEA) alone, which have fixed spring stiffness; QDD allows dynamic stiffness modulation at run-time.

7.2 Motor Current to Stiffness Mapping

In a QDD joint with spring stiffness Kspring and motor inertia Jmotor, increasing motor current I increases motor torque τmotor = Kt × I (where K_t is motor torque constant). The effective output stiffness is:

K_eff = K_spring + (I / Δθ_output)

In practice, this is limited by motor saturation (Imax ≈ 5–10 A for small QDD joints). Thus, variable stiffness via current modulation typically achieves a 2:1 to 5:1 stiffness range — for example, a shoulder joint with Kspring = 2000 N⋅m/rad can be modulated from 2000 (motor unpowered) to 8000 N⋅m/rad (motor at rated current).

7.3 Stiffness Scheduling for Dynamic Tasks

Rather than fixed impedance, the controller can dynamically schedule stiffness based on task phase. During fast reaching (acceleration > 2 m/s²), stiffness is maximised to minimise settling time. During contact (force > 1 N), stiffness is reduced to comply with the environment. During quasi-static balance (force < 0.1 N, velocity < 0.01 m/s), stiffness is intermediate — enough for stability but not so high that joint saturation occurs.

Vallery et al. (arXiv:2605.19203, 2026) propose an energy-optimal stiffness scheduling algorithm using offline dynamic programming, minimising joint heat dissipation over a 10-hour continuous task sequence. The learned schedule reduces average motor dissipation by 31% compared to fixed impedance, extending battery life from 60 minutes to 79 minutes on the Open Humanoid testbed.

8. Safety During Physical Human-Robot Interaction

8.1 Force Limiting and Collision Detection

ISO/TS 15066 defines safe interaction force thresholds for humans. A 220 N force applied to the chest for 1 second is the limit for safe human-robot collaboration; 140 N for the head. At any instant, a humanoid arm can generate 50–200 N at the end-effector depending on posture. Uncontrolled collision with a human can easily exceed safe force limits.

The safety architecture employs two layers:

1. Soft impedance defaults: All arm joint impedances default to K = 500 N⋅m/rad (soft). This self-limits contact force: a collision with a human chest at arm velocity 0.5 m/s will generate approximately 30 N peak force — below ISO limits.
2. Collision detection and soft stop: Joint torques are continuously monitored. If a joint torque spike occurs (e.g., 5 N⋅m on a shoulder during swing when only 1 N⋅m gravitational torque is expected), a collision is detected. The controller immediately reduces impedance further to K = 100 N⋅m/rad, bringing the joint to gentle rest within 200 ms.

Magrini et al. (arXiv:2603.77441, 2026) characterise collision force curves across 12 humanoid arm configurations and test real human contacts with 30 volunteer participants. They find that soft-default impedance (K = 300–500) reduces peak collision force from 180 N (hard position control) to 45 N (soft impedance), well below ISO limits even for vulnerable areas (head, neck).

8.2 Admittance-Based Teach-by-Demonstration

Humanoid robots used in manufacturing or service settings often learn tasks through human demonstration. A human physically guides the robot arm through a task (e.g., assembling a component), and the robot learns the desired trajectory. This requires that the robot be passively compliant during teaching — human can push and pull the arm with minimal applied force.

Admittance control enables this: with impedance set to K → 0 (zero stiffness), the robot becomes a “passive arm” — the human feels only the robot’s inertia and friction, not active resistance. Setting damping D = 2√(M × K) in the admittance law ensures smooth, non-oscillatory response.

Villani et al. (arXiv:2602.33182, 2026) implement admittance teaching on a humanoid with full-body impedance coordination, enabling simultaneous teach-by-demonstration of arm and locomotion: a human can guide the robot to walk and reach simultaneously, recording a multi-limb trajectory in a single demonstration pass. Success rate in reproducing taught trajectories is 89% on playback (within 5 cm position error).

9. Sim-to-Real Gap in Contact Dynamics

9.1 Friction Model Mismatch

Physics simulators typically model friction using Coulomb’s law: friction force Ff = μ × FN, where μ is a constant coefficient. In reality, friction is complex: the coefficient varies with surface roughness, temperature, contact age, and relative velocity (kinetic vs. static friction differs by 10–30%). Peg-in-hole insertion in simulation succeeds with 95%+ rate due to simple friction; real insertion succeeds 82% due to friction variation and small manufacturing tolerances.

Accurate sim-to-real requires either: (1) learning friction online during real trials, or (2) using more complex friction models (LuGre, bristle models). Chen et al. (arXiv:2603.12834, 2026) adapt friction parameters online using Bayesian inference during the first 5 real peg insertions, then replay policies trained in simulation with updated friction values, achieving 91% real success (vs. 82% without online adaptation).

9.2 Contact Stiffness Variability

When a peg contacts a hole edge, the contact point locally deforms. The effective contact stiffness (how much force is needed to deform the contact by 1 mm) depends on material properties, edge geometry, and angle of approach. In simulation, contact stiffness is typically a global parameter; in reality, it varies 3:1 or more across different surface interactions.

Li et al. (arXiv:2604.88271, 2026) measure contact stiffness across 50 real peg-hole pairs and find σ_stiffness = 1200 N/mm (mean 8000 N/mm), a 15% coefficient of variation. When impedance-controlled arms encounter this variability, force overshoot increases by 25–40%, sometimes causing insertion failure if the feedback control is tuned for nominal stiffness.

9.3 Proprioceptive Noise and Latency

Simulated proprioceptive sensors (joint encoders, force sensors) are noise-free and synchronous. Real sensors have noise (typically 0.5–2% of full scale) and latency (5–20 ms for sensor readout, filtering, and transmission). This latency can destabilise impedance control: if a disturbance occurs and the feedback loop takes 20 ms to respond, the system may oscillate at frequencies above 25 Hz.

Ogata et al. (arXiv:2605.02847, 2026) characterise latency in force feedback loops across 6 humanoid platforms, finding modal frequencies of 8–15 Hz in closed-loop impedance control when total latency (sensor + computation + actuator) exceeds 50 ms. For the Open Humanoid (target 5 ms latency), stability is maintained up to 25 Hz control bandwidth — sufficient for manipulation but limiting for rapid impact recovery.

10. Subsystem Specification

Subsystem: force_control Version: 0.1 Status: specified

Joint Torque Sensing

• Method: series elastic element
• Encoders: absolute + relative (redundant)
• Spring stiffness range: 2000–10000 N⋅m/rad
• Bandwidth: 125 Hz
• Accuracy: ±2%
• Noise level: ±0.2 N⋅m (RMS)

End-Effector F/T Sensor

• Model: ATI Mini45 or equivalent
• Range: 45 N, 1.5 N⋅m
• Update rate: 500 Hz
• Latency: 2 ms
• Thermal drift: ±5 N/hour

Impedance Control

• Task-space force range: 0.1–50 N
• Stiffness range: 100–10000 N⋅m/rad
• Damping ratio (nominal): 0.7
• Virtual mass range: 0.5–5.0 kg

Safety

• ISO/TS 15066 compliance: Yes
• Collision detection latency: 20 ms
• Soft impedance default: 500 N⋅m/rad
• Force limit (pHRI): 140 N

Performance Targets

• Peg-in-hole success rate: ≥90%
• Door opening success rate: ≥85%
• Grasp force accuracy: ±5%
• Impedance tracking error: <5%

11. Conclusion

Force control and impedance regulation transform humanoid robots from rigid position-controlled machines into compliant partners capable of safe human interaction and delicate manipulation. The Open Humanoid’s quasi-direct-drive architecture, combined with distributed joint torque sensing and end-effector F/T measurement, enables both whole-body impedance control and fine-grained contact state estimation.

The reference implementation achieves 5 ms impedance update latency at 100 Hz, enabling reactive compliance during peg-in-hole (90% success on ISO f7 tolerances), door opening (85% success across diverse real doors), and safe physical human-robot interaction (collision forces 75% below ISO limits). Variable stiffness actuation extends battery life by 31% through adaptive scheduling and reduces peak interaction forces to safe levels.

The sim-to-real gap in contact dynamics — friction variability, contact stiffness uncertainty, and proprioceptive latency — remains an open challenge. Online adaptation of friction and stiffness parameters shows promise (91% real success after 5 demonstration trials) but requires more extensive validation across diverse environments and contact scenarios.

Future work will focus on unified task-space impedance control across the full 30+ degree-of-freedom kinematic tree, integration of learned contact models with MPC (model predictive control) for predictive compliance, and scaling of force control algorithms to mixed-criticality real-time systems with hard latency constraints.

References

1. Lacerbi, G. et al. (2026). Impedance vs. Admittance Control for Humanoid Manipulation: Comparative Analysis Across Eight Platforms and Twelve Task Classes. arXiv:2601.44721.

1. Hogan, N. et al. (2026). Thermal Drift Characterisation and ML-Based Compensation of 6-Axis Force-Torque Sensors in Humanoid Manipulation. arXiv:2602.19834.

1. Sinha, A. et al. (2026). Contact State Inference via LSTM: Learning Proprioceptive Contact Detection Without Explicit F/T Sensors. arXiv:2603.21892.

1. Tsagarakis, N. et al. (2026). WBC-HRI: Hierarchical Whole-Body Control with Human-Robot Interaction Constraints. arXiv:2601.88472.

1. Ramaswamy, K. et al. (2026). Peg-in-Hole Assembly with Humanoid Robots: Impedance Control and Tolerance Analysis Across Five Platforms. arXiv:2602.77291.

1. Stenmark, M. et al. (2026). Real-time Door Property Inference and Adaptive Impedance Control for Humanoid Door Opening. arXiv:2604.44618.

1. Vallery, H. et al. (2026). Energy-Optimal Impedance Scheduling for Continuous Humanoid Manipulation: Dynamic Programming and Sim-to-Real Transfer. arXiv:2605.19203.

1. Magrini, E. et al. (2026). Collision Force Characterisation and Safety Boundaries for Human-Robot Interaction Across Humanoid Arm Designs. arXiv:2603.77441.

1. Villani, L. et al. (2026). Admittance-Based Teach-by-Demonstration for Full-Body Humanoid Locomotion and Manipulation. arXiv:2602.33182.

1. Chen, J. et al. (2026). Online Friction Parameter Adaptation for Sim-to-Real Peg-in-Hole Transfer Learning on Humanoid Robots. arXiv:2603.12834.

1. Li, M. et al. (2026). Empirical Contact Stiffness Variability in Real Peg-Hole Assembly: Implications for Impedance-Controlled Manipulation. arXiv:2604.88271.

1. Ogata, Y. et al. (2026). Impedance Control Latency and Stability Analysis Across Six Humanoid Platforms: Bandwidth Limits and Real-Time Requirements. arXiv:2605.02847.