External Wrench Estimation Algorithm

Updated 9 February 2026

External wrench estimation algorithms are methods that infer time-varying spatial forces and moments on robotic systems using sensor fusion and dynamic models.
They integrate model-based observers like EKF and momentum observers with learning-based techniques to enhance accuracy and robustness under diverse interaction scenarios.
Applications span humanoid locomotion, force-sensitive manipulation, and aerial robotics, achieving significant improvements in disturbance rejection and control performance.

An external wrench estimation algorithm computes or infers the time-varying spatial force and moment (the “wrench”) applied to a robotic system (e.g., by the environment, an object, or physical interaction), using a combination of kinematic, dynamic, proprioceptive, and, if available, force/torque sensor data. Accurately estimating the external wrench is essential for contact-rich manipulation, humanoid locomotion, aerial interaction, collision detection, grasp optimization, and force-sensitive control without reliance on dedicated 6D force/torque sensors.

1. Dynamic Principles and State-Augmented Observer Designs

Physical systems subjected to wrenches satisfy rigid-body Newton–Euler equations. In humanoid robots, the system evolves according to

$\dot h = \sum_{i=1}^N w_i + w_\mathrm{ext}$

where $h=[\ell^\top,\,k^\top]^\top\in\mathbb{R}^6$ is spatial momentum (linear and angular about the Center of Mass), $w_i$ are the measured contact wrenches, and $w_\mathrm{ext}$ is the unknown external wrench. Discretized process models propagate states: $\begin{aligned} c_{k+1} &= c_k + (\Delta t/m)\ell_k \ \ell_{k+1} &= \ell_k + \Delta t [m g + \sum_i f_{i,k} + f_{\mathrm{ext},k}] \ k_{k+1} &= k_k + \Delta t [\sum_i (p_i-c_k)\times f_{i,k} + \tau_{i,k} + \tau_{\mathrm{ext},k}] \ f_{\mathrm{ext},k+1} &= f_{\mathrm{ext},k} + w^f_k \ \tau_{\mathrm{ext},k+1} &= \tau_{\mathrm{ext},k} + w_k^\tau \end{aligned}$ where $(w^f_k,\,w^\tau_k)$ are process noise terms.

Extended Kalman Filters (EKFs) are constructed with state vectors $\mathbf{x}_k = [c_k,\,\ell_k,\,k_k,\,f_{\mathrm{ext},k},\,\tau_{\mathrm{ext},k}]$ . The measurement, typically comprising sensed contact wrenches, is modeled as $z_k = h(\mathbf{x}_k) + v_k$ . The estimator is recursively updated via EKF prediction and correction steps (Rotella et al., 2015).

2. Model-Based, Hybrid, and Learning-Based Approaches

Model-based methods treat external wrenches as unknown, possibly random-walk, components, estimating them by enforcing dynamic consistency through observers or filters. Examples include:

Momentum-Observer-Based Estimation: The residual dynamics between expected generalized momentum $\dot{p}$ and measured input torque are tracked by an observer. For floating-base systems, the momentum observer (MOB) signal $r$ yields $r\approx\tau_e + \tau_u$ where $\tau_u$ includes friction and model uncertainties. In complex systems, learning-based modules such as GRU-networks (in MOB-Net (Lim et al., 2024)) estimate and subtract $\hat\tau_u$ to yield a bias-corrected external torque estimate.
Hybrid Model–Learning Approaches: In aerial robots, a neural network is trained to predict residual terms $\phi_\theta$ in the continuous-time equations $\dot{v} = f_\text{fp}(v, R, \gamma) + \phi_\theta(v, R, \gamma)$ . A combined observer cancels the learned residual, so only true external disturbance wrenches remain (Alharbat et al., 10 Apr 2025).
Sensorless Learning on Internal Signals: In manipulation contexts, large MLPs or recurrent nets are trained on joint positions, velocities, accelerations, and motor currents to directly regress the wrench at the end-effector, bypassing explicit model-based mappings (Shan et al., 2023, Lim et al., 2023).

A summary of representative estimation paradigms:

Method	Sensor Inputs	Observer/Network Structure
EKF on momentum (Rotella et al., 2015)	F/T, kinematics, IMU	EKF, linearized momentum/CoM augmented state
MOB-Net (Lim et al., 2024)	Encoders, IMU, torques	MOB + modular GRU, per-limb
Learning-based (Shan et al., 2023)	Encoders, velocities, currents	Large MLP, optionally LSTM
Hybrid aerial (Alharbat et al., 10 Apr 2025)	IMU, velocities, prop speed	Neural ODE + momentum observer

3. Data Processing, Estimator Inputs, and Observability

Accurate external wrench estimation depends on the informativeness of sensed quantities and excitation. EKF-based algorithms require that contact wrenches excite all six spatial directions for full state observability; at least one non-coplanar contact or two non-collinear planar contacts are necessary (Rotella et al., 2015).

In sensorless estimators, raw signals such as joint positions, velocities (via finite differences and low-pass filtering), and motor currents form the feature vector. Inclusion of IMU signals (linear accelerations and angular rates) enhances the ability to infer external torques on floating bases (Shan et al., 2023, Lim et al., 2023). For learning-based estimators, careful dataset design—covering free-space, contact, sliding, and fine manipulation—is crucial for generalization and low RMSE.

In model-based approaches, process and measurement noise covariances in the EKF or observer are tuned to balance responsiveness and noise rejection. In learning-based schemes, regularization and data-driven fine-tuning address overfitting and domain transfer.

4. Algorithmic Workflow and Implementation

A canonical external wrench estimation algorithm for humanoids proceeds as:

Initialization:

State vector initialized with prior CoM, momentum, and zero external wrench.
Covariance matrix set to reflect uncertainty in each state.

At each time-step:

Sensor Acquisition: Contact force/torque vectors, positions, joint encoders, and (if used) IMU signals are read.
Predict Step: The dynamic model propagates the state and estimates external wrench evolution as a random walk.
Correction Step: The estimated contact wrenches (from model) are compared against sensor readings, and the innovation is used to correct the full state via the Kalman gain.
Extraction: External wrench estimate is simply the corresponding block in the augmented state vector.

Pseudocode for the EKF-based observer (Rotella et al., 2015):

for k in 1..K:
    # Prediction
    x_pred = f(x_est_prev, u_prev)
    P_pred = F_prev @ P_prev @ F_prev.T + Q
    # Correction
    y = z_k - h(x_pred)
    K = P_pred @ H.T @ np.linalg.inv(H @ P_pred @ H.T + R)
    x_est = x_pred + K @ y
    P = (np.eye(len(K)) - K @ H) @ P_pred
    # Extract external wrench
    f_ext_est, tau_ext_est = x_est[12:15], x_est[15:18]

For data-driven estimators, a forward pass through a trained MLP or GRU followed by exponential smoothing constitutes the estimation loop (Shan et al., 2023).

5. Application Domains and Performance Metrics

External wrench estimation is foundational in:

Locomotion feedback and disturbance rejection: EKF-based estimators reduce CoM drift by 80% and maintain stability under foot slippage given contact observability (Rotella et al., 2015, Lim et al., 2023, Lim et al., 2024).
Force-sensitive manipulation and assembly: Learned estimators achieve sub-3 N force RMSE and track wrench signals in tasks without F/T sensors (Shan et al., 2023).
Grasp analysis in elasticity-aware manipulation: The “stress-minimizing” metric computes the largest resistible external wrench subject to material fracture constraints, solved via repeated convex conic optimization and BEM precomputation (Pan et al., 2019).
Human–object interaction biomechanics: Marker-based external wrench estimation reconstructs object–hand wrenches and induced joint torques robustly across variable marker layouts (Herneth et al., 2024).
Aerial robotics, wind/contact discrimination: Hybrid model-based and learning observers reduce wrench estimation error by an order of magnitude for multicopters under wind and physical interaction (Tomić et al., 2018, Alharbat et al., 10 Apr 2025).

Performance is typically quantified by RMSE on each wrench axis, convergence time after disturbance (<50 ms for some EKF methods), and the ability to support downstream controllers (e.g., ZMP feedback, collision detection) without failure (Rotella et al., 2015, Lim et al., 2024, Lim et al., 2023).

6. Current Limitations and Future Directions

Existing algorithms can be limited by:

Observability breakdown: Static or low-rank contact configurations preclude full external wrench identification (Rotella et al., 2015).
Data-set dependency: Learning-based algorithms require representative training data—extrapolation to novel environments or contact scenarios remains an open question (Shan et al., 2023, Lim et al., 2023).
Model uncertainty: Observer-based approaches are susceptible to errors from unmodeled friction, parameter drift, and structural flexibilities; hybrid approaches attempt to mitigate this but require significant training and integration effort (Alharbat et al., 10 Apr 2025, Lim et al., 2024).
Contact discrimination in mixed interactive and aerodynamic environments: Robust separation of wind/drag from physical interaction remains challenging, though approaches using combined model-based, power, and particle filter techniques improve robustness (Tomić et al., 2018).

Ongoing research is extending external wrench estimators to deployable prosthetics, real-time biomechanical analysis, large-scale human-object motion datasets, and teleoperation contexts—all with the aim of enabling safe, sensorless, and physically consistent robot interaction.