Reaction Force Observer (RFOb)

• Reaction Force Observer (RFOb) is a model-based estimator that calculates interaction forces in servo systems without direct sensors, enabling high-precision sensorless force control.
• It integrates a cascaded disturbance observer (DOB) with an outer force controller to isolate contact forces, enhancing control bandwidth to approximately 20–30 Hz.
• Experimental validations show reduced transient sensitivity (around 40%), low overshoot (<5%), and improved stability through adaptive parameter tuning and robust discrete-time techniques.

A Reaction Force Observer (RFOb) is a model-based observer for estimating interaction (contact) forces in servo-actuated systems without the use of direct force sensors. RFObs extend the classical disturbance observer (DOB) framework by isolating and estimating the force transmitted between the plant and its environment. RFObs have been recognized for providing higher force-control bandwidth, improved stability, and enabling sensorless force control, even in the presence of plant/model mismatches and various external disturbances. The RFOb methodology has undergone detailed mathematical characterization, stability analysis, and practical validation—particularly in works on adaptive force control with real-time parameter estimation and robust performance across environments (Sariyildiz et al., 2019, Sariyildiz, 2023, Sariyildiz et al., 2021, Sariyildiz et al., 2019).

1. Mathematical Structure and Observer Synthesis

The RFOb builds upon low-order model-based DOB architectures. Starting with single-axis plant dynamics

$M\ddot{x}_m = K_F i_m - F_d,$

where $M$ is true inertia, $K_F$ is the thrust coefficient, $i_m$ is the applied current, and $F_d$ aggregates exogenous disturbances. The classical DOB, typically implemented as a first-order low-pass filter with bandwidth $\omega_{\rm DOB}$, estimates and compensates for $F_d$. Its estimate is

$$\hat{F}_d(s) = \frac{\omega_{\rm DOB}}{s + \omega_{\rm DOB}} \left[K_{F,n} i_m(s) - M_n s^2 X_m(s)\right],$$

with $K_{F,n}, M_n$ being the nominal parameters.

The RFOb generalizes this by utilizing identified model parameters ($\hat M, \hat K_F$) and focuses on reconstructing the external (interaction) force:

$M$0

The sum yields estimates for both environment interaction and internal frictions.

Key parameter definitions:

• Inertia/torque mismatch: $M$1, $M$2.
• Force estimation error: $M$3. Modeling errors ($M$4) corrupt estimates for frequencies above $M$5, sharpening the necessity for robust parameter calibration.

2. Integration in Force-Control Architectures

The RFOb operates in a cascaded observer structure:

• Inner Loop: Classical DOB compensates generalized disturbances, enhancing servo stiffness and disturbance rejection.
• Outer Loop: RFOB isolates the interaction force, to which a proportional force controller ($M$6) is applied, closing the force-regulation loop with the environment, often modeled as Kelvin–Voigt:

$M$7

Closed-loop characteristics (with phase compensation as needed) are described by

$M$8

where $M$9 captures cascaded bandwidths and controller gains, and is affected directly by environmental impedance and observer parameterization (Sariyildiz et al., 2019, Sariyildiz et al., 2021).

3. Stability and Robustness Analysis

Stability of RFOb-based force control is governed by frequency-domain criteria and design inequalities that preclude non-minimum phase (NMP) zeros and maintain permissible gain/phase margins:

• DOB Robustness Constraint: The product $K_F$0 ensures the sensitive trade-off (waterbed effect) between disturbance rejection and sensor-induced noise peaks, with $K_F$1 providing adequate damping.
• RFOb Loop NMP Zero Condition: For minimum-phase behavior, enforce $K_F$2. Violation ($K_F$3) induces right-half-plane zeros in the open loop, sharply degrading stability (Sariyildiz et al., 2019, Sariyildiz et al., 2021, Sariyildiz et al., 2019).
• Sampled-Data Implementation Constraints: In discrete time, both DOB and RFOB observer gains must satisfy $K_F$4, $K_F$5 to ensure all poles/zeros remain inside the unit circle (Sariyildiz, 2023).
• Force-Loop Gain Bound: $K_F$6 limits the achievable bandwidth without loss of stability (Sariyildiz, 2023).

Root-locus and frequency-response analyses support these constraints, with phase-lead compensation via bandwidth separation ($K_F$7) being especially effective for improving stability margin.

4. Adaptive and Practical Design Methodologies

Adaptive RFOb designs incorporate real-time parameter identification and dynamic retuning of observer and control gains (Sariyildiz et al., 2019):

• Recursive Least Mean Squares (RLMS) Estimation: Used for plant parameters $K_F$8 and environmental parameters $K_F$9 during respective contact/non-contact phases. RLMS with bounded projection assures convergence (plant ≈ 1 s, environment ≈ 0.5 s in practice).
• Online Tuning Rules: After each identification, observer gains ($i_m$0, $i_m$1), nominal/identified inertias, and $i_m$2 are updated according to robustness and stability constraints depending on the contact regime (pure stiffness, pure damping, or mixed).
• Implementation Recommendations: Identification should run at ≥1 kHz, parameter updates triggered only after stabilization (<5% change), and outputs lightly filtered (50–100 Hz) to avoid noise amplification.

A schematic summary of parameter adjustment:

Parameter	Function	Update Rule/Strategy
$i_m$3	Disturbance rejection bandwidth	High as possible; $i_m$4
$i_m$5	Force estimation bandwidth	2–3$i_m$6 above $i_m$7
$i_m$8, $i_m$9	Nominal/identified inertia	$F_d$0 ≥ largest $F_d$1; $F_d$2
$F_d$3	Force controller gain	Initial: $F_d$4; ramp up with care

5. Discrete-Time Effects and Waterbed Trade-Off

Digital implementation of RFOb architectures introduces additional constraints:

• The DOB and RFOb filters must be implemented using appropriate Tustin or backward-Euler approximations.
• Waterbed effect: Increasing either the observer gain (high $F_d$5, $F_d$6) or mismatched inertia/torque ratios improves low-frequency disturbance rejection but introduces sharp sensitivity peaks at higher frequencies.
• Stability is contingent on keeping $F_d$7 and $F_d$8 well below unity. Typical practice is $F_d$9 (Sariyildiz, 2023).
• Experiments confirm that excessive mismatch ($\omega_{\rm DOB}$0 too large) or observer gain ($\omega_{\rm DOB}$1) leads to non-minimum-phase zeros, bandwidth collapse, and instability.

6. Experimental Validation and Performance

Experiments consistently demonstrate the robustness and efficacy of the RFOb approach:

• Bandwidth Improvements: Force control bandwidth is extended from 3–5 Hz (sensor-based) to ~20–30 Hz (RFOb-based) (Sariyildiz et al., 2019).
• Impact Transient Sensitivity: Adaptive RFOb reduces step/transient sensitivity by ~40%.
• Force Tracking: Experiments with both stiff and compliant environments confirm that properly tuned RFOb tracks actual force measurements with low overshoot (<5%) and high bandwidth (>45° phase margin).
• Parameter Sensitivity: Violating design inequalities (e.g., $\omega_{\rm DOB}$2) causes oscillations and bandwidth collapse; adhering to them ensures overdamped, high-fidelity force regulation (Sariyildiz, 2023, Sariyildiz et al., 2019, Sariyildiz et al., 2021).

7. Practical Guidelines and Implementation Considerations

Several practical guidelines ensure successful RFOb deployment:

• Select velocity filtering bandwidth ($\omega_{\rm DOB}$3) as high as noise and encoder limits permit (5–10× control bandwidth).
• Nominal inertia should slightly exceed the largest expected true value; observer and force-filter bandwidths must obey robustness constraints.
• Avoid updating control parameters during large transients; freeze adaptation until estimation stabilizes.
• Use independent tuning of DOB and RFOb bandwidths, with preference for $\omega_{\rm DOB}$4 to gain phase lead (Sariyildiz et al., 2019, Sariyildiz et al., 2019).
• Digitally realize all observer blocks with matching discrete-time cut-off frequencies; verify closed-loop responses with root-locus/Nyquist criteria.

This synthesis of theoretical and practical knowledge underpins the modern usage of RFOb for high-bandwidth, sensorless force-control in robotic and mechatronic systems, supporting robust, adaptive interaction with uncertain environments across a spectrum of hardware platforms (Sariyildiz et al., 2019, Sariyildiz, 2023, Sariyildiz et al., 2021, Sariyildiz et al., 2019).