Maximum Torque Transmissibility (MTT)

• Maximum Torque Transmissibility (MTT) is a frequency-based criterion that defines a SEA's capacity to transmit its full rated torque across a specified bandwidth.
• It uses analytical transfer functions and force-feedback control models to formalize both torque- and velocity-based limitations and determine the maximum-torque bandwidth.
• MTT informs design guidelines by linking mechanical parameters and controller gains to practical performance, ensuring actuators operate within safe torque and speed limits.

Maximum Torque Transmissibility (MTT) is a frequency-dependent performance criterion for Series Elastic Actuators (SEAs), quantifying the actuator’s ability to transmit its full rated output torque across a prescribed dynamic range. Developed to address the mismatch between motor-side torque/velocity constraints and output-torque generation in SEAs, MTT formalizes both torque- and speed-based limitations and introduces the associated concept of a maximum-torque frequency bandwidth. This framework enables rigorous evaluation of how mechanical design, controller parameters, and load conditions cap the dynamic torque capability of SEAs (Lee et al., 2019).

1. Dynamic Model of Series Elastic Actuator

The SEA system comprises a motor coupled to a load via a compliant spring, typically with gear reduction. The dynamics on the motor and load sides are

$$P_m(s) = \frac{1}{J_m s^2 + B_m s}, \qquad P_l(s) = \frac{1}{J_l s^2 + B_l s}$$

where $J_m$, $B_m$ denote motor inertia and damping, and $J_l$, $B_l$ those of the load. With spring constant $K_s$ and gear ratio $N_m$, the transfer function from commanded motor torque $\tau_c$ to output torque $\tau_{out}$ takes the form:

• Dynamic-load case (finite $J_l$, $B_l$):

$$P_{\text{dynamic}}(s) = \frac{N_m^{-1} K_s P_m(s)} {1 + K_s P_l(s) + N_m^{-2} K_s P_m(s)}$$

• Static-load case ($J_l, B_l \to \infty$):

$$P_{\text{static}}(s) = \frac{N_m^{-1} K_s P_m(s)} {1 + N_m^{-2} K_s P_m(s)}$$

The transfer function for motor velocity per unit torque is

$$P_V(s) = \frac{P_m(s)[1 + K_s P_l(s)] s} {1 + K_s P_l(s) + N_m^{-2} K_s P_m(s)}$$

These relationships establish the foundation for analyzing SEA response to control inputs over frequency.

2. Force-Feedback Control and SEA Output

To achieve output torque regulation, SEAs are equipped with force-feedback controllers $C(s)$ wrapped around the open-loop plant $P(s)$. The control-loop relations are:

$$T_c(s) \equiv \tau_c(s) = \frac{N_m^{-1} C(s)} {1 + N_m^{-1} C(s) P(s)} T_d(s)$$

$V_m(s) = P_V(s) T_c(s)$

Here, $T_d(s)$ is the desired output torque trajectory. These transfer-functions directly determine the torque applied by the motor and the resulting motor speed required to meet $T_d$.

3. Formal Definition of Maximum Torque Transmissibility

MTT captures whether the actuation system can supply the full desired output torque $T_d^{max} = N_m T_{m.c}$ (where $T_{m.c}$ is the motor’s maximum continuous torque) without exceeding hardware constraints at all frequencies:

• Torque-based MTT:

$$MTT_\tau(s) = \frac{|T_c(s)|}{T_{m.c}} \bigg|_{T_d = T_d^{max}} = \left| \frac{[1 + K_s(P_l(s) + N_m^{-2}P_m(s))] C(s)} {1 + K_s[P_l(s) + N_m^{-2}P_m(s)(1+C(s))]} \right|$$

$MTT_\tau(j\omega) > 1$ indicates that motor torque capability is violated at frequency $\omega$.

• Velocity-based MTT:

$$MTT_V(s) = \frac{|V_m(s)|}{V_p} \bigg|_{T_d = T_d^{max}} = \left| \frac{P_m(s)\,C(s)\,[1 + K_s P_l(s)]\,s} {1 + K_s[P_l(s) + N_m^{-2}P_m(s)(1 + C(s))]} \frac{T_{m.c}}{V_p} \right|$$

$MTT_V(j\omega) > 1$ indicates that the maximum permissible motor speed $V_p$ is exceeded.

For the static-load limit ($P_l(s)\rightarrow\infty$):

$$MTT_\tau^{sta}(s) = \left| \frac{1 + N_m^{-2} K_s P_m(s)}{1 + N_m^{-2} K_s P_m(s)(1 + C(s))} C(s) \right|,\quad MTT_V^{sta}(s) = \left| \frac{P_m(s) C(s) s}{1 + N_m^{-2} K_s P_m(s)(1 + C(s))} \frac{T_{m.c}}{V_p} \right|$$

4. Maximum-Torque Frequency Bandwidth

At a given frequency $\omega$, if either $MTT_\tau(j\omega)$ or $MTT_V(j\omega)$ exceeds unity, the SEA cannot produce $T_d^{max}$ without motor saturation. Hence, two limiting bandwidths are defined:

• Torque-limit bandwidth $\omega_{MT_\tau}$: $|MTT_\tau(j\omega_{MT_\tau})| = 1$
• Speed-limit bandwidth $\omega_{MT_V}$: $|MTT_V(j\omega_{MT_V})| = 1$

The maximum-torque frequency bandwidth is then

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

This provides a concise, quantitative figure-of-merit for SEA design, directly indicating the maximum frequency at which full rated torque is realizable.

5. Influence of System and Control Parameters

MTT unifies analysis across mechanical and control domains:

• Load inertia ($J_l$): Increases in $J_l$ decrease plant bandwidth and shift both MTT curves downward, reducing $\omega_{MT}$; the static-load case represents the most restrictive scenario.
• Spring stiffness ($K_s$): Small $K_s$ yields compliant but bandwidth-limited transmission; larger $K_s$ initially increases $\omega_{MT}$, but above a threshold, high required loop gain causes rapid violation of $MTT_\tau$, driving $\omega_{MT_\tau}$ to zero.
• Gear ratio ($N_m$): Higher $N_m$ raises $T_d^{max}$ but scales down effective motor dynamics; as $N_m$ increases, $\omega_{MT_V}$ generally decreases faster than $\omega_{MT_\tau}$, making speed-limiting dominant for high $N_m$.
• Feedback gain (P- or PD-controller): Increasing proportional gain $K_p$ elevates low-frequency loop gain and narrows MTT roll-off; above a critical $K_p^* = 1 + \frac{B_l}{B_m} N_m^{-2}$, full-torque bandwidth vanishes. Derivative action shapes high-frequency roll-off but does not affect the DC limit.

The following table summarizes parameter effects:

Parameter	Effect on $\omega_{MT}$	Remark
$J_l$	Decreases $\omega_{MT}$ as $J_l$ increases	Reduces both torque and speed bandwidth
$K_s$	Non-monotonic; optimal $K_s$ maximizes	Overly large $K_s$: immediate torque violation
$N_m$	Increases $T_d^{max}$, lowers $\omega_{MT}$	High $N_m$: speed limit typically dominates
$K_p$	Excessive gain eliminates $\omega_{MT}$	Must satisfy $K_p < K_p^*$

6. Experimental Validation and Empirical Observations

Lee and Oh conducted comprehensive experiments using a rigidly mounted “Varying-Gear Transmission” SEA (Maxon BLDC motor, belt-pulley stages spanning $N_m \in \{1, 2.4, 4.5, 8, 15, 36\}$). Chirp inputs at $|τ_d| = T_d^{max}$ were used to measure normalized motor torque, speed, and torque tracking error. Experiments verified:

• For low $N_m$, the torque limit ($MTT_\tau = 1$) was crossed at $\omega \approx \omega_{MT_\tau}$, leading to immediate torque saturation and tracking failure.
• For high $N_m$, the speed-limit ($MTT_V = 1$) mode was violated first; at $\omega > \omega_{MT_V}$, motor velocity exceeded $V_p$, torque dropped, and tracking error increased.

These results empirically reaffirm both the computed MTT curves and the bandwidth-limiting nature of motor-side constraints (Lee et al., 2019).

7. Design Guidelines and Practical Implications

• Bandwidth selection: Choose $\omega_{MT}$ to comfortably exceed target application torque frequencies.
• Spring stiffness: Select $K_s$ to maximize $\omega_{MT}$ but remain below the value that forces $MTT_\tau(0) > 1$.
• Gear ratio: Select $N_m$ to balance torque amplification and bandwidth, ensuring $\omega_{MT}$ is not speed-limited.
• Controller gains: For P-control, ensure $K_p < K_p^*$ to permit nonzero bandwidth; adjust derivative gain to shape frequency response as needed.
• Load matching: For high-impedance environments, analyze with static-load approximations to ensure robustness.

These prescriptions enable optimized SEA designs that deliver full rated torque over the desired frequency spectrum, as quantified by MTT and its associated bandwidth metric (Lee et al., 2019).