The starting point is the robot dynamic model: M(q)·q̈ + C(q,q̇)·q̇ + g(q) + τ_friction = τ_motor + τ_ext, where τ_ext is the sought torque from external forces. (1) Current-based method: motor torque is estimated from measured current via the torque constant (τ = Kt·i); subtracting the predicted dynamics and friction terms leaves an estimate of the external torque. (2) Disturbance observer (DOB): treats all deviations from the nominal model as a disturbance and estimates it through a low-pass filter of chosen bandwidth (Q-filter); a reaction torque observer separates the external-force component from that disturbance. (3) Generalized-momentum method (De Luca): a residual is defined r = K·(p − ∫(τ + Cᵀq̇ − g + r)dt), where p = M(q)·q̇ is the generalized momentum; the residual r is a first-order-filtered estimate of the external torque and requires neither q̈ nor inversion of M(q). The estimated external torque is mapped to a Cartesian force via the Jacobian transpose: F_ext = (Jᵀ)⁺·τ_ext. Observer bandwidth (gain K / Q-filter cutoff) sets the trade-off between response speed and rejection of noise and model error.
Dedicated force/torque sensors are expensive, fragile, add mass and mechanical compliance at the end effector, and require wiring and calibration. Sensorless force estimation provides contact awareness, collision detection, compliant control and safe human–robot interaction without those costs, using information already available in the drives (current/torque) and the robot model.
The model M(q)·q̈ + C(q,q̇)·q̇ + g(q) describing inertia, Coriolis/centrifugal forces and gravity; used to predict the torque required for contact-free motion.
Motor phase current converted to torque via the torque constant (τ = Kt·i); the primary sensing input replacing the F/T sensor.
Estimates the lumped disturbance (deviations from the nominal model) through a Q-filter; a reaction torque observer extracts the external component.
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External-torque estimator based on momentum p = M(q)·q̇, first-order filtered; requires neither accelerations nor inertia-matrix inversion.
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Compensation of friction (Coulomb, viscous) and transmission effects; the dominant error source in sensorless estimation.
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Maps the estimated external torque to a task-space force/torque: F_ext = (Jᵀ)⁺·τ_ext.
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Inaccurate identification of inertial/gravity parameters causes a systematic bias in the force estimate, mistaken for real contact.
Friction (especially in gearboxes) and motor torque ripple introduce speed- and position-dependent errors that degrade sensitivity to small forces.
High observer gain reduces contact-detection latency but amplifies current noise and model error, leading to false positives.
Mapping torque to Cartesian force via (Jᵀ)⁺ is ill-conditioned near singularities and for directions in the Jacobian null space.
Ohishi et al. introduce the disturbance observer for a DC drive, the basis for sensorless disturbance/torque estimation.
Murakami, Yu and Ohnishi extend the disturbance observer to multi-DoF manipulators, defining sensorless torque/force control.
De Luca et al. (DLR) introduce the generalized-momentum residual for collision detection without torque sensors — a standard in safe human–robot interaction.
Haddadin, De Luca and Albu-Schäffer publish a survey of collision detection, isolation and identification, systematizing model-based force estimation.
Time complexity: O(n). Space complexity: O(n).
Cost and accuracy depend on computing the M, C, g terms and the friction model at every real-time control-loop step.
Residual gain K / Q-filter cutoff — sets the trade-off between response speed and rejection of noise and model error.
Quality of inertial and gravity parameter identification; directly bounds estimation accuracy.
Type and completeness of the friction model (Coulomb, viscous, temperature/position-dependent) and its compensation.
Accuracy of the torque constant Kt and compensation of torque ripple and gear friction.
Control-loop rate determining the maximum achievable estimation bandwidth and contact-detection latency.
The estimator runs deterministically at every real-time control-loop step; all computations are active on each cycle.
Residuals and dynamics recursions can be partly parallelized across joints/links, but the control loop remains sequential in time (dependency between time steps).
Computation is lightweight (O(n)) and runs on a standard robot controller / microcontroller in a real-time loop; no accelerators required.
A vectorized CPU handles dynamics-model and filter evaluation well at high control-loop rates.
