Transmission Force Observer (TFOB) Overview

• Transmission Force Observer (TFOB) is a method that quantifies force and torque transmission in Series Elastic Actuators using the Maximum Torque Transmissibility framework.
• It employs frequency-domain analysis to evaluate motor and load dynamics, comparing torque and velocity constraints to determine the effective transmission bandwidth.
• Experimental validations with varying gear ratios and parameters confirm the predicted cutoff frequencies and offer practical design guidelines for optimizing actuator performance.

A Transmission Force Observer (TFOB) quantifies the ability of an actuator system—specifically a Series Elastic Actuator (SEA)—to transmit force or torque from a motor, through elastic and mechanical elements, to the output load, subject to limitations imposed by both motor drives and the mechanical design. The concept of Maximum Torque Transmissibility (MTT) rigorously analyzes these constraints in the frequency domain, offering a unified framework for evaluating SEA performance with respect to full exploitation of the continuous motor torque over a specified bandwidth (Lee et al., 2019).

1. Dynamics and Open-Loop Transmissibility of SEAs

The dynamic model for force transmission in SEAs considers the motor and load as coupled inertia–damper systems, linked by a compliant element (spring) and a gear train. The relevant transfer functions are:

• Motor dynamics:

$P_{m}(s) = \frac{1}{J_{m}s^{2} + B_{m}s}$

• Load dynamics:

$P_{l}(s) = \frac{1}{J_{l}s^{2} + B_{l}s}$

where $J_{m}, B_{m}$ and $J_{l}, B_{l}$ are, respectively, the motor and load inertia and damping.

For a dynamic load, the open-loop transfer from motor torque $\tau_{c}$ to output torque $\tau_{\mathrm{out}}$ takes the form: $$P_{\mathrm{dyn}}(s) = \frac{N_{m}^{-1}K_{s}P_{m}(s)}{1 + K_{s}P_{l}(s) + N_{m}^{-2}K_{s}P_{m}(s)}$$ For static (high-impedance) loads, the transfer simplifies to: $$P_{\mathrm{stat}}(s) = \frac{N_{m}^{-1}K_{s}P_{m}(s)}{1 + N_{m}^{-2}K_{s}P_{m}(s)}$$ The transfer function from motor torque to motor velocity is given by: $$P_{V}(s) = \frac{P_{m}(s)[1+K_{s}P_{l}(s)]s}{1 + K_{s}P_{l}(s) + N_{m}^{-2}K_{s}P_{m}(s)}$$

2. Force-Feedback Control and Motor Requirements

Imposing force/torque tracking via a controller $C(s)$ (typically of the PD family) on the SEA output, the closed-loop system determines both the required continuous motor torque $T_{c}(s)$ and corresponding motor velocity $V_{m}(s)$: $$T_{c}(s) = \frac{N_{m}^{-1}C(s)}{1 + N_{m}^{-1}C(s)P(s)} T_{d}(s)$$

$V_{m}(s) = P_{V}(s)T_{c}(s)$

where $P(s) = P_{\mathrm{dyn}}(s)$ or $P_{\mathrm{stat}}(s)$ depending on load dynamics.

3. Maximum Torque Transmissibility (MTT) Definition

The MTT framework evaluates whether the SEA can deliver its maximum possible torque (as defined by gear ratio and motor limits) across a frequency range, expressing the system’s transmission capacity as a normalized frequency-dependent criterion:

3.1. Motor Torque-Based MTT

The maximum actuator output torque is prescribed by the motor’s continuous torque capability $T_{m.c}$ and gear ratio $N_{m}$: $T_{d}^{\max} = N_{m}T_{m.c}$ The frequency-domain transmissibility is thus: $$\mathrm{MTT}_{\tau}(s) = \frac{|T_{c}(s)|}{T_{m.c}} = \left| \frac{[1+K_{s}(P_{l} + N_{m}^{-2}P_{m})]C(s)}{1 + K_{s}[P_{l} + N_{m}^{-2}P_{m}(1+C(s))]} \right|$$ If $\mathrm{MTT}_{\tau}(j\omega) > 1$, the continuous motor torque limit is exceeded and the SEA cannot deliver the full output torque at frequency $\omega$.

3.2. Velocity-Based MTT

Similarly, the system is bounded by the motor’s maximum permissible velocity $V_{p}$: $$\mathrm{MTT}_{V}(s) = \frac{|V_{m}(s)|}{V_{p}} = \left| \frac{P_{m}(s)C(s)[1+K_{s}P_{l}(s)]s}{1 + K_{s}[P_{l}(s) + N_{m}^{-2}P_{m}(s)(1+C(s))]} \right|\frac{T_{m.c}}{V_{p}}$$ If $\mathrm{MTT}_{V}(j\omega) > 1$, the motor velocity constraint becomes the limiting factor.

4. Maximum Torque Frequency Bandwidth

The frequency beyond which the SEA cannot transmit full torque is of practical interest. The critical cutoff frequencies are:

• $\omega_{MT_{\tau}}$ where $\mathrm{MTT}_{\tau}(j\omega_{MT_{\tau}}) = 1$
• $\omega_{MT_{V}}$ where $\mathrm{MTT}_{V}(j\omega_{MT_{V}}) = 1$

The maximum-torque frequency bandwidth is

$$\omega_{MT} = \min(\omega_{MT_{\tau}}, \, \omega_{MT_{V}})$$

This formally demarcates the frequency range over which full actuator output is realizable, conditioned by both torque and velocity constraints.

5. Influence of Design Parameters on MTT and Bandwidth

Variations in physical and control parameters systematically affect SEA performance as captured by MTT and $\omega_{MT}$:

Parameter	Effect on MTT and $\omega_{MT}$	Critical Behaviors
Load inertia $J_l$	Increasing $J_l$ lowers bandwidths. The static-load limit ($J_l\rightarrow\infty$) minimizes $\omega_{MT}$.	Higher inertia reduces bandwidth, but extends torque transmission for given $K_s$.
Feedback gain $K_p$	Raising $K_p$ increases low-frequency tracking, but beyond a critical gain ($K_p^{\rm crit} = 1+N_m^{-2}(B_l/B_m)$), $\omega_{MT_{\tau}}\to 0$.	Excessive $K_p$ leads to loss of transmissibility even at low frequencies.
Gear ratio $N_m$	Larger $N_m$ raises torque output but reduces $\omega_{MT}$. For small $N_m$, $\mathrm{MTT}_{\tau}$ is limiting; for large $N_m$, $\mathrm{MTT}_V$ dominates.	Crossover in dominant limit as $N_m$ increases.
Spring stiffness $K_s$	Moderate increases in $K_s$ improve $\omega_{MT_{\tau}}$, but beyond a threshold (dependent on $J_l, B_m, B_l, N_m$), transmissibility drops abruptly.	Overly stiff springs degrade bandwidth.

Editor's term: Parameter mapping—summarizes these effects for practical design optimization.

6. Experimental Validation of the MTT Criterion

A Varying-Gear Transmission (VGT) testbed was constructed utilizing a Maxon BLDC motor ($T_{m.c}=0.0315$ Nm, $V_{p}=10.47$ rad/s) and a spring of $K_{s}=1.1$ Nm/rad, with gear ratios ranging from 1:1 to 36:1. Principal findings:

• In static-load conditions, experimental motor torque and velocity outputs matched the predicted $\mathrm{MTT}_{\tau}$ and $\mathrm{MTT}_{V}$ curves across frequency sweeps. For $N_m=1$, torque limit occurred at ~40 rad/s (velocity limit not reached); for $N_m=36$, velocity limit occurred at ~5 rad/s.
• Feedback gain sensitivity was validated: for $N_m=8$ and $K_p=2$, full-torque tracking at 1 Hz was achieved, but tracking failed at $K_p=4$, consistent with the derived $K_p^{\rm crit}$.
• Under dynamic loading (spring $K_s=20$ Nm/rad, loads $J_l=0.003$ and $0.007$ kg·m²), the high-inertia load achieved full-torque tracking at 5 Hz, while the low-inertia case failed at the exact frequency predicted by $\omega_{MT_{\tau}}$.

7. Practical Design Guidelines

The MTT criterion yields quantitative design strategies for SEAs:

• Set feedback gain $K_p < 1+N_m^{-2}(B_l/B_m)$ to avoid premature loss of torque transmissibility bandwidth.
• Select gear ratios to keep $\omega_{MT_{V}}$ above application requirements, avoiding excessive reduction that would prematurely impose velocity rather than torque limits.
• Calibrate spring stiffness to maximize natural frequency without surpassing the stiffness at which $\omega_{MT_{\tau}}$ collapses; this limit is a function of all principal system parameters.
• Factor in the load inertia $J_l$: higher inertia extends torque transmission range for fixed $K_s$, though at the cost of reduced overall actuator motion bandwidth.

Through direct computation of $\mathrm{MTT}_{\tau}(j\omega)$ and $\mathrm{MTT}_{V}(j\omega)$ for given actuator designs and controllers, designers can immediately determine the frequency bandwidth for full continuous motor torque transmission and identify cutoffs where mechanical, electrical, or control limits become prohibitive. The MTT framework unifies analysis of compliance, control, and actuation constraints in a single frequency-domain tool for SEA performance evaluation (Lee et al., 2019).