Motor Current-Based External Wrench Estimation

• Motor current-based external wrench estimation is a technique that reconstructs interaction forces using motor currents, calibrated models, and data-driven methods.
• It integrates joint-space dynamics, calibration procedures, and Jacobian mapping to achieve force estimation errors as low as 1–2 N and sub-millinewton precision in neural approaches.
• These methods enable compliant control in robotics, fostering low-cost human-robot collaboration without relying on expensive force/torque sensors.

Motor current-based external wrench estimation refers to a class of techniques for reconstructing the interaction forces and torques (collectively, the external wrench) at a robot’s end-effector, using only measurements of motor currents and internal state variables, without relying on dedicated force/torque sensors. This approach leverages both analytical system models and data-driven estimators, and has become central for enabling compliant interaction, force control, and physical human-robot collaboration in cost-sensitive or sensor-limited robotic platforms.

1. Analytical Foundations and Physical Modeling

The fundamental analytical framework is a formulation of the robot’s joint-space dynamics,

$$M(q)\,\ddot q + C(q,\dot q)\,\dot q + g(q) + \tau_f(q,\dot q) = \tau_m + \tau_\mathrm{ext}$$

where $q$ are joint positions, $M(q)$ is the inertia matrix, $C(q,\dot q)$ and $g(q)$ capture Coriolis and gravity effects, $\tau_f$ is friction torque, $\tau_m$ is actuator torque (as inferred from motor current), and $\tau_\mathrm{ext}$ is the torque due to external contacts (Wolde et al., 2024).

Motor torque is estimated from actuator current using a calibrated gain and friction model, typically

$\tau_m = k_i\,i_m - \tau_f(q,\dot q)$

with $k_i$ the current-to-torque gain. Friction is modeled via Coulomb (constant, direction-dependent), viscous (velocity-proportional), or compound Stribeck-Coulomb relations (Shan et al., 2023).

Mapping estimated external joint torques to an end-effector wrench utilizes the manipulator Jacobian: $$\hat F_\mathrm{ext} = J(q)^{-T} \hat\tau_\mathrm{ext}$$ For $n>6$, regularized least-squares or damped pseudoinverse approaches are employed for numerical stability (Shi et al., 1 Mar 2026).

2. Calibration and System Identification

Precise estimation requires calibration of motor constants and friction parameters. Methods include:

• Gravity-driven sweeps: Joints are moved slowly through their range, other joints locked, while motor current and position are logged. The torque due to gravity is modeled as $K \sin \theta$ with $K$ a lumped parameter, and regression is used to extract the current-to-torque ratio and friction magnitude (Wolde et al., 2024).
• Phase-shift correction: Sinusoidal fits to current-vs-angle data yield both magnitude and center-of-mass offset, refining the robot’s mass model.
• No-load identification: For direct-drive or quasi-direct-drive (QDD) actuators, PWM-voltage relationships and back-EMF constants are calibrated under no-load rotation (Shi et al., 1 Mar 2026).
• Minimal friction modeling: Some approaches ignore explicit friction terms in favor of direction-dependent transmission efficiency (Shi et al., 1 Mar 2026), while others use detailed Stribeck/Coulomb/viscous models (Shan et al., 2023).
• Calibration is typically a one-time, per-actuator procedure.

3. Online Estimation Algorithm

At each control cycle, the estimation proceeds as:

1. Signal acquisition: Read motor current (or reconstruct from voltage/PWM and velocity), encoder positions, and velocities.
2. Torque estimation: Infer motor torque using the calibrated model,

$\hat\tau_m = k_i\,i_m - \tau_f(q, \dot q)$

or, for QDD/servo drives,

$\tau_{\text{load}} = \eta_x K_t I_{w,i}$

with direction-dependent efficiency.

1. Model subtraction: Compute the expected model torque based on current state, then extract the external torque as the residual,

$$\hat\tau_\mathrm{ext} = \hat\tau_m - \tau_\mathrm{model}(q,\dot q,\ddot q)$$

2. Jacobian mapping: Project joint-space torque residuals to Cartesian wrench estimates using

$$\hat F_\mathrm{ext} = J(q)^{-T} \hat\tau_\mathrm{ext}$$

or solve for $\hat{\mathbf f}_{\text{ext}}$ via regularized least squares,

$$\arg\min_{\mathbf f} \|J^T(q)\mathbf f - \bm\tau_{\text{ext}}\|^2 + \lambda\|\mathbf f\|^2$$

as implemented for underactuated or redundant robots (Shi et al., 1 Mar 2026).

3. Signal smoothing: One-pole low-pass or exponential moving-average filtering is routinely applied to currents and final wrench estimates.

This pipeline enables use of the estimated external wrench in classical impedance or admittance controllers, enabling compliant behaviors analogous to those supported by 6D force/torque sensors (Wolde et al., 2024, Shi et al., 1 Mar 2026).

4. Data-Driven and Neural Estimation Approaches

Analytical models are limited by unmodeled effects, hysteresis, or substantial contact ambiguity. Data-driven estimators, particularly deep neural networks, have been proposed to map time-stacked internal signals $x = [q, \dot q, \ddot q, I]$ directly to end-effector wrench outputs (Shan et al., 2023). In this approach:

• Networks are trained on large-scale datasets with aligned ground-truth from high-grade F/T sensors.
• Architectures include 2-layer MLPs (1024 units per layer; ReLU), ingesting concatenated joint positions, velocities, accelerations, and current.
• Training is performed in multiple phases:
  • Initial free-space data collection,
  • Subsequent fine-tuning on contact-rich tasks (e.g., sliding, hand-guiding, pin-insertion).
• Model-based dynamic terms are subsumed into the regression provided sufficient data diversity.

A key finding is that pure model-based estimators are outperformed in challenging contact scenarios—especially fine assembly or multi-contact conditions—by neural estimators pre-trained on broad, structured datasets (Shan et al., 2023).

5. Performance Metrics and Experimental Outcomes

Comparative evaluations involve:

• Force/torque estimation error: RMSE against F/T sensor readings, e.g., RMSE(F) ≈ 1–2 N, RMSE(T) ≈ 0.1 N·m under moderate contacts, with model-based or data-driven estimators (Shan et al., 2023, Shi et al., 1 Mar 2026).
• Compliance control accuracy: In whiteboard drawing, current-based admittance control attains ≈16 mm position error and 0.05 rad orientation error, outperforming RL-based baselines (Shi et al., 1 Mar 2026).
• Task success rate: Minimalist methods support 80% success in “egg-on-spatula” and >20 deg average in “ball-rotation,” exceeding baselines without external wrench estimation (Shi et al., 1 Mar 2026).
• Latency: Real-time execution with end-to-end latency ~12 ms/control cycle is achievable without dedicated F/T hardware (Shi et al., 1 Mar 2026).

For neural methods, sub-millinewton precision has been demonstrated in high-precision insertion tasks via fine-tuned MLPs (Shan et al., 2023).

6. Limitations, Challenges, and Scope of Applicability

Notable limitations include:

• Model coverage: Analytical estimators can be sensitive to model inaccuracies, thermal drift, and unmodeled friction or transmission dynamics. Regular recalibration may be necessary in changing environments (Wolde et al., 2024).
• Ambiguities in multi-contact: Data-driven techniques reveal that identical joint-torque residuals can correspond to different external contact situations (hysteresis, multi-contact ambiguity), occasionally causing estimation failures (Shan et al., 2023).
• Speed constraints: Current methods are primarily validated at moderate speeds and contact-rich, quasi-static regimes. High-speed or highly dynamic tasks may require extended inertial and thermal modeling, or new calibration methods (Shan et al., 2023).
• Embodiment generality: Minimalist methods are designed to be plug-and-play and have been validated across a diverse set of robot platforms; however, the data-driven estimator requires transfer learning or additional calibration for new manipulators (Shi et al., 1 Mar 2026, Shan et al., 2023).
• No explicit modeling of stiction, backlash, or higher-order dynamics in some minimalist approaches; performance thus may degrade in low-velocity or low-gear ratio systems (Shi et al., 1 Mar 2026).

7. Integration with Compliance Controllers and Practical Impact

Motor current-based external wrench estimates can be directly integrated into impedance and admittance controllers. The estimated $\hat F_\mathrm{ext}$ is fed to a Cartesian admittance law,

$$M_d \ddot{\mathbf x} + D_d \dot{\mathbf x} + K_d \mathbf x = \ldots + \hat{\mathbf f}_{\text{ext}}$$

enabling compliant interaction in the absence of hardware F/T sensors (Wolde et al., 2024, Shi et al., 1 Mar 2026).

This sensorless paradigm is embodiment-agnostic, requires only encoder and current signals, and avoids dependencies on sim-to-real transfer or black-box learning in the low-level loop. Reliable compliance behaviors—tracking, guiding, and complex manipulation—can be achieved using only built-in sensors. Performance is sufficient for most interaction tasks, so long as the estimated wrench direction and contact frequency content are preserved.

A plausible implication is that as robots move from industrial to human-centered environments, motor current-based wrench estimation will become a primary enabler for scalable, low-cost compliant robotics, expanding access beyond platforms equipped with precision F/T sensing (Wolde et al., 2024, Shi et al., 1 Mar 2026, Shan et al., 2023).