End-Effector Force Estimator

• End-Effector Force Estimation is a framework that infers interaction forces using sensor data, physics-based models, and learning-based methods.
• It improves robotic control, compliance, and contact monitoring in applications from surgical systems to heavy machinery.
• Innovative techniques, including Kalman filtering and deep learning, address challenges like sensor drift, noise, and calibration.

An end-effector force estimator is a computational or algorithmic framework for inferring the interaction forces and/or torques exerted by a robot’s distal manipulator (typically the tool or end-effector) on its environment. This estimation is central to closed-loop force control, disturbance rejection, task compliance, contact monitoring, and skill assessment in robotics, soft manipulation, surgical systems, exoskeletons, and agricultural automation. Direct force-torque sensing at the end-effector (e.g., via a wrist sensor) offers the highest fidelity but entails challenges such as cost, fragility, mass, drift, and practical inapplicability in soft robotics or biocompatible scenarios. Consequently, a spectrum of indirect and multi-modal estimation methods has emerged, leveraging proprioceptive signals, physics-based modeling, and machine learning.

1. Principles and Function of End-Effector Force Estimation

At its core, force estimation seeks to reconstruct the external wrench (vector of forces and torques) at the end-effector from available sensor measurements, which may include joint torques/currents and motion, end-effector pose/orientation, environmental feedback, or various exteroceptive cues (e.g., vision, tactile, Hall-effect, EMG/IMU). The task is fundamentally an inverse problem—given measurable system variables and a (possibly imperfect) model of the robot and task environment, one solves for the unknown external force components that best explain the measurements.

Standard methods include:

• Direct model-based estimation: using the manipulator’s kinematic and dynamic equations (Newton-Euler or Lagrangian formalisms) to relate actuator forces/torques and measured states to the unknown external wrench, often requiring accurate knowledge of mass/inertia, gravity, friction, and occasionally payload (Werner et al., 13 Oct 2025).
• Observer-based estimation: applying Kalman filtering, generalized momentum observers, or disturbance observers for separating internal and external torque contributions (Nadeau et al., 2024).
• Learning-based estimation: training neural or statistical models (MLP, CNN, RNN, GRU) to regress external force from robot state, sensor readings, or fused multi-modal input, often overcoming deficiencies in modeling friction, compliance, or multi-point contacts (Chua et al., 2020, Shan et al., 2023).
• Direct sensing: interpreting raw data from custom force-torque transducers (e.g., magnetic Hall arrays) or tactile skins using calibration and analytic or learned inverse mappings, including uncertainty quantification (Tanaka et al., 2024).

2. Model-Based and Observer-Based Estimation

Physics-based approaches remain foundational for rigid and soft robots. For rigid-body manipulators, the wrench $F_{ee}$ is computed as:

$$M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) + \tau_{fric}(\dot{q}) = \tau_{motor} + J(q)^T F_{ee}$$

where all terms are defined as in (Werner et al., 13 Oct 2025, Shan et al., 2023).

Given measurements of actuator torques (via motor currents or cylinder pressures), joint angles, velocities, and accelerations (via encoders and possibly IMUs), one solves for $F_{ee}$:

$$F_{ee}(t) = -[J(q(t))]^{-T} \Big[\tau_{motor}(t) - M(q)\ddot{q} - C(q,\dot{q})\dot{q} - G(q) - \tau_{fric}(\dot{q})\Big]$$

Calibration procedures involve identification of inertial, frictional, gravity, and actuator parameters using task-specific experiments (e.g., applying known masses, free vibration analysis, hysteresis evaluation), as demonstrated in heavy excavation machinery (Werner et al., 13 Oct 2025).

For soft robots, model-based estimation necessitates advanced mechanics. Quasi-static finite element models (FEM) are used to capture geometric and material nonlinearities with actuation and boundary condition mapping, as exemplified in fiber-reinforced continuum arms (Cangan et al., 2022). Equilibrium equations are linearized at each step, and the inverse problem is formulated as a quadratic program:

$$\Delta \lambda^* = \underset{\Delta \lambda}{\operatorname{argmin}}~ \|W_{ra} \Delta \lambda - \Delta R_{real}\|^2_2$$

subject to actuation and regularization constraints, where $W_{ra}$ maps actuation to orientation error.

Kalman filter-based observer stages can further decouple the effect of sensor biases and compensate for drift in direct force-torque measurements; this is achieved via cascaded joint and bias filters in six-axis sensor systems (Nadeau et al., 2024).

3. Learning-Based and Data-Driven Estimation Approaches

Where model fidelity is limited or physical signals are insufficient, learning-based estimators provide a powerful alternative.

• Vision- and state-based networks, such as deep CNNs (e.g., ResNet-50) with robot state fusion, estimate distal interaction force from endoscopic images and robot kinematics, outperforming rigid-body analytical baselines in robot-assisted surgery (Chua et al., 2020).
• Multimodal CNNs learn to fuse EMG (time/frequency) and IMU time-series to estimate end-point force in dynamic human movement, attaining high $R^2$ across contraction regimes and outperforming SVM/ANN baselines on both intra- and inter-subject generalization (Hajian et al., 2022).
• Purpose-designed recurrent architectures (GRU, LSTM) with gating mechanisms are employed for uncertainty estimation and to address hysteretic, non-Markovian, or nonlinear system effects seen in tactile and magnetic force sensors and hybrid end-effectors (Tanaka et al., 2024).

Learning-based estimators demand extensive data: paired ground-truth force and sensor/robot state during task-relevant motions, with careful dataset splits and calibration on varied contact, manipulation, and environmental conditions (Shan et al., 2023).

Key findings across these works reveal that multi-modal input (e.g., robot state + vision, EMG + IMU) consistently increases generalization and accuracy compared to uni-modal approaches, particularly under shifts in material properties, toolings, and workspace locations (Chua et al., 2020, Hajian et al., 2022).

4. Sensor Modalities and Noise Compensation

End-effector force estimators may draw on a range of sensor modalities:

• Proprioceptive: joint encoders, motor currents, hydraulic pressures, IMU readings (link or end-effector mounted).
• Exteroceptive: high-frequency vision (monocular, stereo), tactile arrays, force-torque sensors, magnetic Hall-effect sensors, EMG/IMU in wearable scenarios.
• Hybrid: vision for contact detection, kinematics for pose, and joint torques for physical model parameterization (Yang et al., 2024).

Effective bias and drift correction is critical, especially for six-axis force-torque sensors susceptible to temperature, mechanical load, and environmental influences. State-space Kalman filtering frameworks continuously estimate both static and dynamic (linear drift) wrench offsets, provided inertial parameters are known and sufficient excitation occurs (Nadeau et al., 2024).

Noise from magnetic coupling, environmental field disturbances, or mechanical coupling is addressed by:

• Analytical shielding and optimal magnet–sensor geometry design (in Hall sensor systems) (Tanaka et al., 2024).
• Per-axis bias cross-calibration, gain normalization, and online filtering.
• Explicit uncertainty estimation via recurrent networks, enabling anomaly detection and robust operation under latent environmental variability.

5. Calibration, Validation, and Practical Considerations

Systematic calibration is fundamental:

• Model-based estimators require multi-stage routines: actuator/output relationships (e.g., hydraulic cylinder area, pressure–force mapping), inertial/frequency response (e.g., via power-spectral density minimization), friction, and gravity parameters, and sensor–environment couplings (Werner et al., 13 Oct 2025, Cangan et al., 2022).
• Learning-based estimators are typically pretrained on broad, unbiased datasets (e.g., random joint-space coverage under human and environment-induced forces), followed by fine-tuning on specific task datasets (e.g., sliding, hand-guiding, pin-insertion), yielding substantial error reduction in precision tasks (Shan et al., 2023).
• Sensing architectures using Hall-effect arrays or tactile skins undergo axis-wise calibration for gain, bias, and cross-sensitivity under controlled loading conditions (Tanaka et al., 2024).
• Experimental protocols for sensorless and model-based force estimation validate performance against precision reference devices (ATI F/T sensors, custom 1-axis gauges), with reported errors as low as 1.2% (soft arms) (Cangan et al., 2022), 6.6% (industrial excavators) (Werner et al., 13 Oct 2025), or RMSE < 0.02 N for bias-compensated six-axis F/T sensors (Nadeau et al., 2024).

Implementation considerations include filtering strategies (e.g., low-pass at 2–3 Hz for hydraulic excavators), real-time computation constraints, and observability (sufficient excitation for parameter and bias estimation).

6. Applications, Impact, and Limitations

End-effector force estimators are central to a wide array of domains:

• Robot manipulation (industrial, collaborative, assembly) with or without direct force sensing (Shan et al., 2023).
• Soft and continuum robotics under uncertain and highly compliant actuation (Cangan et al., 2022).
• Teleoperated and autonomous surgery, deployable in both vision-augmented and hybrid modalities (Chua et al., 2020, Yang et al., 2024).
• Wearable and rehabilitation robotics, merging EMG/IMU and exoskeletal control (Hajian et al., 2022).
• Mobile and heavy machinery (e.g., automated excavators), fusing hydraulic and inertial sensors with dynamic modeling for payload and grading control (Werner et al., 13 Oct 2025).
• Agricultural manipulation, e.g., design and control of soft or hybrid end-effectors for crop harvesting (S et al., 2022).

Limitations are mainly:

• Dependence on model accuracy in dynamic regimes and under complex, multimodal contacts.
• Observability constraints for bias estimation (lack of excitation).
• Challenges in sensor calibration, environmental drift, and transfer to new toolings or environments.
• Learning-based generalizability, especially as in-vivo surgical or agricultural environments may entail unmodeled factors compared to laboratory settings.

Ongoing research is pursuing combined dynamic observers (e.g., EKF/UKF), improved model fusion, attention mechanisms, and generalization strategies to further expand the accuracy and robustness of end-effector force estimators in real-world and deformable environments (Cangan et al., 2022, Hajian et al., 2022).