---
title: Maximum Torque Transmissibility (MTT)
url: https://www.emergentmind.com/topics/maximum-torque-transmissibility-mtt
type: topic
---

# Maximum Torque Transmissibility (MTT)

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 [1902.05346].

## 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 [1902.05346].

## 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 [1902.05346].

Source: https://www.emergentmind.com/topics/maximum-torque-transmissibility-mtt