[Robot Hardware 03] - Actuators (2): Reducers
Robot hardware from a Physical AI perspective - reducers
The Role of a Reducer
A reducer is a mechanical device that converts a motor’s high-speed, low-torque output into the low-speed, high-torque output needed to drive a robot joint. Because many BLDC motors are designed to produce high power at high speed, reducers are widely used in multi-jointed robots that must drive heavy links slowly. Large-diameter torque motors can also be used in direct-drive or low-ratio QDD (Quasi-Direct Drive) configurations.
Robot drive architectures differ depending on how the reducer is connected. A motor and joint can be placed far apart using a belt or cable in a remote-actuation arrangement, but most multi-jointed robots use a modular form in which the reducer is connected directly to the motor shaft.
Reducer Types and Detailed Posts
The following reducer types are commonly used in robots.
Ideal Torque and Speed Conversion
Concentrating the motor, reducer, bearings, and sensors near the joint makes it easier to build a compact module. It also reduces long cable or belt paths, lowering transmission error and assembly complexity. Some robot reducer units integrate an output bearing, such as a cross-roller bearing, to support external moment loads directly. A component set may still require a separate output bearing, and an integrated structure does not mean that the transmission is infinitely stiff.
Let $N$ be the reduction ratio, defined as motor speed divided by output-shaft speed. An ideal reducer has the following relationship:
\[\omega_{out}=\frac{\omega_{in}}{N}\]The output torque during forward driving, when the motor drives the load through the reducer, can be approximated using the forward efficiency $\eta_f$:
\[T_{out}\approx\eta_f N T_{in}\]A reducer increases torque by lowering speed, but friction and deformation prevent all input power from reaching the output. Backdriving, in which the output shaft drives the motor in the reverse direction, cannot be described by simply inverting this equation. It requires a separate backward efficiency $\eta_b$, breakaway torque, and contact conditions to be considered.
Losses and Nonlinearities in Real Reducers
Forward Efficiency and Backward Efficiency
Transmission efficiency can depend on the direction of energy flow:
- Forward driving: the motor drives the load through the reducer.
- Backward driving: an external force drives the reducer and motor backward from the output shaft.
In many high-ratio mechanisms, backward efficiency is lower than forward efficiency because of differences in internal friction and contact forces. This is not a universal law of kinematics, however. The difference depends not only on reduction ratio but also on tooth geometry, preload, lubrication, load, and reducer architecture. A prototype 1:100 Bilateral Drive Gear reported 89.0% forward efficiency, 85.3% backward efficiency, and a 0.020 N·m backdrive starting torque, showing that high bidirectional efficiency is possible even at a high reduction ratio.[1]
Some reducers exhibit self-locking: reverse motion does not begin when the torque applied to the output shaft cannot overcome internal friction and contact conditions. Not every worm gear is self-locking; the result depends on lead angle, coefficient of friction, and lubrication. Self-locking should therefore be evaluated using backdrive starting torque and geometric conditions, not a single steady-state efficiency value. Even when the output shaft does not move, impact reactions and stresses can still be transmitted through the gears, bearings, and housing.
Output Impedance: Linear and Nonlinear Elements
Mechanical impedance describes the dynamic resistance observed when an external force moves a joint. A system linearized as inertia, viscous friction, and stiffness can be written as
\[Z(s)=\frac{T(s)}{\omega(s)}=J_{eq}s+B_{eq}+\frac{K_{eq}}{s}\]Here, $J_{eq}$ is equivalent inertia, $B_{eq}$ is equivalent viscous friction, and $K_{eq}$ is equivalent torsional stiffness. Real reducers also have stiction, Coulomb friction, backlash, and lost motion, which cannot be represented by $B_{eq}$ alone. A more general joint-torque relationship separates them as follows:
\[\tau_{ext}=J_{eq}\ddot q+B_{eq}\dot q+K_{eq}q +\tau_{friction}(\dot q)+\tau_{backlash}\]The first three terms are impedance components that can be linearized around an operating point. The last two are nonlinear elements that depend on speed, direction of motion, and gear-contact state. We should distinguish four properties when interpreting a reducer’s output impedance:
- Friction: creates the breakaway torque needed to start motion and causes energy loss during motion.
- Reflected inertia: resists rapid changes in angular acceleration.
- Torsional stiffness: determines elastic deformation in gear teeth, the flexspline, bearings, and housing.
- Backlash and lost motion: create nonlinear position error when the input direction changes and the output does not follow immediately.
If backward efficiency is low or stiction is large, a person must first overcome a large friction torque to move the robot. This can make the robot feel heavy, but it should be distinguished from the reflected inertia that increases with reduction ratio. Friction mainly obstructs the start and continuation of motion; inertia resists changes in angular acceleration.
Reflected Inertia: The Squared Reduction-Ratio Effect
The total equivalent inertia observed at the output shaft is approximately
\[J_{eq,out}=J_{load}+J_{gear,out}+N^2J_{motor}\]The motor rotor inertia $J_{motor}$ is therefore reflected to the output shaft as $N^2$ times its original value. At a 10:1 reduction ratio, the motor-rotor term is multiplied by 100; at 100:1, it is multiplied by 10,000. This factor applies only to the motor-rotor inertia term. It does not mean that the physical inertia of the robot link and reducer also increases by the same ratio.
Conversely, the load inertia viewed from the motor side is reduced as follows:
\[J_{load\rightarrow motor}=\frac{J_{load}}{N^2}\]High reduction ratios therefore have two sides. At the output shaft, they increase the reflected motor inertia and can make fast backdriving and physical transparency worse. At the motor, they make load changes and disturbances appear smaller, which can help static load support and position control. Reduction ratio is not a value that is always better when lower; it is a design variable that trades output authority and position holding against backdrivability and dynamic interaction.
Even with no friction, a sufficiently large rotor inertia and a high reduction ratio can make fast backdriving feel substantially resistant. Conversely, if rotor inertia is very small or motion is slow, the same reduction ratio may have little effect.
Torsional Stiffness, Backlash, and Lost Motion
A reducer is not a perfectly rigid body. Gear-tooth contact, the flexspline in a strain-wave gear, bearings, and the housing all deform under load. This torsional stiffness can itself be nonlinear and depend on torque.
- Backlash: geometric clearance that exists until contact is re-established after the gear-meshing direction changes
- Lost motion: the total position loss observed at the output during input reversal, including not only backlash but also elastic deformation, friction, and assembly error
- Torsional stiffness: the ratio of torsional-angle change to output-torque change, describing elastic deformation under load
These three quantities cannot substitute for one another. A small backlash can still coexist with a large torsional error under load if stiffness is low, and a strain-wave reducer with nearly zero nominal backlash can still have flexspline deformation and lost motion. Belts and cables can also achieve high stiffness with suitable preload and structural design. Remote actuation has the additional advantage of placing heavy motors near the torso, reducing mass and inertia in distal links.
Impact and Contact Loads
When a high reduction ratio is combined with low mechanical compliance, a fast impact can produce a large torque to accelerate the motor rotor and a large instantaneous load inside the transmission. If the load exceeds the reducer’s impact rating, it can damage the gear teeth, bearings, or housing.
An impact does not automatically mean that the gears will break. The actual result depends on impact speed and energy, transmission torsional stiffness, tooth impact caused by backlash, the reducer’s emergency-stop torque, load paths through bearings and housing, passive compliance in the links, and the controller’s contact detection and torque limiting.
Reduction Ratio and Robot Dynamics
We have seen that motor-rotor inertia is reflected to the output shaft in proportion to the square of the reduction ratio. This reflected inertia does more than affect the resistance felt when a person pushes the robot by hand. It is included directly in the inertia matrix used by the controller’s robot-dynamics model, changing both the torque required for a desired motion and the joint’s response to external force.
The joint-space dynamics of a general multi-jointed robot can be written as
\[M(q)\ddot q +C(q,\dot q)\dot q +G(q) +\tau_f(q,\dot q) = \tau +\tau_{ext}\]where
- $q$, $\dot q$, and $\ddot q$ are joint position, velocity, and acceleration
- $M(q)$ is the robot inertia matrix
- $C(q,\dot q)\dot q$ is the Coriolis and centrifugal term
- $G(q)$ is the gravity term
- $\tau_f$ is friction torque from reducers, bearings, and other components
- $\tau$ is the drive torque delivered by the actuator to the joint
- $\tau_{ext}$ is the external torque applied to the joint by the environment
In this equation, reflected inertia is not a separate external-force term. It is included as part of the inertia matrix $M(q)$ that multiplies joint acceleration $\ddot q$.
How Reflected Inertia Enters the Inertia Matrix
Consider an ideal rigid transmission with reduction ratio $N$. Let the motor rotation angle be $\theta_m$ and the output-joint angle be $q$:
\[\theta_m=Nq\]The motor angular velocity is therefore
\[\dot\theta_m=N\dot q\]For motor-rotor inertia $J_m$, the rotor’s kinetic energy is
\[T_m=\frac{1}{2}J_m\dot\theta_m^2 =\frac{1}{2}N^2J_m\dot q^2\]From the output-joint coordinate $q$, the motor rotor therefore appears to have an inertia of $N^2J_m$.
The equivalent inertia of a single joint can be simplified as
\[J_{eq}=J_{link}+J_{gear,out}+N^2J_m\]The same relationship extends to a multi-jointed robot in matrix form. If the motor-rotor inertia matrix is $\mathbf{J}_m$ and the reduction-ratio matrix is $\mathbf{N}$, the equivalent joint-space inertia can be written conceptually as
\[M_{eq}(q)=M_{robot}(q)+M_{gear}(q)+\mathbf{N}^{T}\mathbf{J}_m\mathbf{N}\]If each joint uses an independent reducer and $\mathbf{N}$ is diagonal, approximately $N_i^2J_{m,i}$ is added to the corresponding diagonal element of the joint inertia matrix.
In a cable differential or coupled transmission, one motor can connect to multiple joints, so the reduction-ratio relationship need not be diagonal. In that case, reflected motor inertia can affect not only diagonal elements but also off-diagonal elements representing coupling between joints.
Torque Required for a Desired Acceleration
Producing a desired joint acceleration $\ddot q_d$ requires torque proportional to the inertia. Ignoring the other dynamics for a moment,
\[\tau_{inertia}=M_{eq}(q)\ddot q_d\]As reflected inertia increases, more torque is therefore required to produce the same joint acceleration.
Conversely, if the actuator’s available joint torque is fixed, the achievable acceleration is limited by the inverse of the equivalent inertia:
\[\ddot q\approx M_{eq}^{-1}(q)\tau\]A high reduction ratio amplifies motor torque at the output shaft, but it also reflects motor-rotor inertia by the square of the ratio. Motor-torque amplification grows approximately with $N$, while reflected rotor inertia grows with $N^2$.
When reflected rotor inertia dominates the total inertia, continuing to increase the reduction ratio does not improve dynamic joint acceleration by the same factor. It helps produce large static torque, but can be disadvantageous in motions that require fast direction changes or high angular acceleration.
Joint Response to External Force
The same relationship applies when the environment applies force to the robot. Ignoring gravity and friction and considering only external torque,
\[\ddot q\approx M_{eq}^{-1}(q)\tau_{ext}\]For the same external torque, a larger equivalent inertia produces less joint acceleration. A joint with high reflected inertia therefore appears to resist moving away immediately when pushed or struck quickly from outside.
This should be distinguished from stiction. Stiction creates a torque threshold that must be overcome before the joint starts moving; reflected inertia makes it difficult to change the velocity of a joint that is already moving.
In other words, a reducer can limit the robot’s response to external force in two ways:
- Internal friction prevents a small external force from becoming joint motion.
- Reflected inertia lowers the acceleration response to rapidly changing external force.
Why High Reduction Ratios Help Position Control
High reduction ratios are not disadvantageous for every control objective.
Because the reducer amplifies motor torque, a small motor can support a large static load. The load inertia also appears smaller from the motor side:
\[J_{load\rightarrow motor}=\frac{J_{load}}{N^2}\]To the motor controller, link inertia changes and external disturbances are therefore transmitted at a reduced scale. This helps in tasks where the robot must support a large load and maintain an accurate position.
In systems such as industrial robots, where high-stiffness trajectory tracking matters more than yielding to external force, these properties are advantages. A high reduction ratio and high position-control gain can hold joint position firmly even when the external load changes.
The same properties can be disadvantages in robots whose main purpose is physical interaction. The motor’s authority over the load increases, but the physical transparency through which the environment moves the motor backward from the output shaft decreases.
Reduction ratio ultimately balances two directions of interaction:
- drive authority and position-holding ability from motor to load
- transparency of external force and motion from load to motor
Reflected Inertia in Model-Based Control
Model-based controllers such as computed-torque control and whole-body control use a robot-dynamics model to calculate the torque needed for a desired motion.
A feedforward torque for a desired joint acceleration can be written conceptually as
\[\tau_d=\hat M_{eq}(q)\ddot q_d +\hat C(q,\dot q)\dot q +\hat G(q) +\hat\tau_f(q,\dot q)\]The estimate $\hat M_{eq}$ should include not only link and reducer inertia but also reflected motor-rotor inertia.
If reflected inertia is omitted from the model, the controller behaves as if it were controlling a lighter robot than the real one. During fast acceleration and direction changes, it underestimates the required torque, causing tracking error and delayed response.
If reflected inertia is known accurately, feedforward torque can compensate for part of its effect. Including it in the model does not make its physical effect disappear, however.
The controller can calculate the required torque, but the actual actuator remains limited by:
- maximum motor and driver current
- continuous and peak torque
- supply voltage and back-EMF
- winding and driver thermal limits
- sensor and communication delay
- reducer friction and backlash
- mechanical resonances in the structure and transmission
Model-based control can compensate for known inertia and gravity, but it cannot create infinite torque or bandwidth.
Limits of a Rigid-Joint Model
The inertia-matrix expression above assumes that the motor and output joint are connected as a perfectly rigid body. In other words, it assumes the relationship $\theta_m=Nq$ always holds and ignores torsional deformation in the reducer.
Real reducers have finite torsional stiffness. When motor and load inertia are connected through reducer elasticity, the actuator behaves as a two-inertia system, not as one rigid body.
The motor-side coordinate $\theta_m$ and output-joint coordinate $q$ then need to be treated as independent states. The transmission torque caused by reducer deformation can be simplified as
\[\tau_s=K_s\left(\frac{\theta_m}{N}-q\right) +B_s\left(\frac{\dot\theta_m}{N}-\dot q\right)\]Here, $K_s$ and $B_s$ are torsional stiffness and damping referenced to the output shaft.
At low frequencies, where the reducer can be treated as rigid, motor reflected inertia can be combined into $M(q)$ as one inertia. As control frequency increases and elastic deformation becomes significant, the motor and output shaft begin to move differently.
Mechanical resonances and anti-resonances then appear. Raising controller bandwidth close to these frequencies can cause vibration, phase delay, and instability.
The control bandwidth of an actuator with a reducer is therefore not determined only by the motor driver’s current-control speed. Consider all of the following together:
- motor-to-load inertia ratio
- reducer torsional stiffness
- damping in the reducer and structure
- backlash and lost motion
- locations of motor-side and output-side sensors
- mechanical resonance and anti-resonance frequencies
- bandwidth of current, velocity, position, and torque controllers
Position Bandwidth and Torque Bandwidth
When discussing reduction ratio and control bandwidth, distinguish which control bandwidth is meant.
The current controller inside a motor driver can operate quickly even with a high reduction ratio. A high ratio amplifies motor torque and reduces load disturbances as seen from the motor side, which can help produce high position stiffness and precise trajectory tracking.
Accurately reproducing a small output torque or responding quickly to external force is a separate problem. Output torque contains reducer friction, torsional deformation, torque ripple, and backlash, and it may be difficult to estimate actual output torque accurately from motor-side current alone.
A high-ratio actuator can therefore have high position-control performance while also having low torque transparency and limited impedance-control bandwidth.
A low-ratio actuator needs a larger motor and higher current to produce the same joint torque, but the relationship between motor current and output torque is more direct and the mechanical response to external force can be faster.
Reduction Ratio Is a Design Variable for the Control Objective
Ultimately, neither a high nor a low reduction ratio is universally good or bad.
A high reduction ratio helps with:
- large output torque from a small motor
- high static load capacity
- precise position holding
- reduced load disturbances as seen from the motor
It can be disadvantageous for:
- low reflected inertia
- high backdrivability
- reproducing small contact torque
- fast compliance to external force
- high torque and impedance-control bandwidth
A low reduction ratio reverses these relationships, but demands a larger and heavier motor, higher current, and greater heat generation.
The reduction ratio is therefore not simply a number that says how many times motor torque will be amplified. It is a central system-design variable that determines how easily force and motion pass in both directions between motor and environment, and what kind of performance the controller can prioritize.
Two different approaches are used in robots where dynamic interaction matters. One is QDD, which lowers the reduction ratio itself. The other is SEA, which adds intentional elasticity between the reducer and the load.
QDD and SEA: Different Design Strategies
QDD and SEA are two representative strategies for reducing the high output impedance introduced by a reducer.
- QDD: Combines a high-torque-density motor with a low reduction ratio to reduce reflected inertia and friction. It helps achieve high backdrivability and torque transparency, but requires a larger torque motor and higher current, increasing mass and thermal-management demands while transmitting load disturbances more directly to the motor side. A QDD exoskeleton has reported both low backdrive torque and high torque bandwidth.[2]
- SEA: Places an intentional series elastic element between the reducer and load. It reduces peak impact load, measures force from spring deformation, and can store energy. It also introduces elastic resonance and position-accuracy issues; depending on spring stiffness, zero-motion force bandwidth and torque bandwidth at high torque can be limited.[3,4]
Both approaches can improve backdrivability and contact safety, but they have different trade-offs in motor size, thermal management, position accuracy, and torque bandwidth. Neither is a universal solution for every robot.
Specifications to Check When Selecting a Reducer
When comparing reducers, check more than reduction ratio and rated torque:
- reduction ratio
- rated and peak output torque
- emergency-stop or impact torque
- forward and backward efficiency
- backdrive starting torque
- backlash and lost motion
- torsional stiffness and allowable twist
- input and output inertia
- maximum input speed
- radial, axial, and moment capacity of the output bearing
- service life and allowable duty cycle
- lubrication method and operating-temperature range
Efficiency can depend on load and speed, and stiffness and lost motion can also depend on the test torque. When comparing numbers from different manufacturers, check whether measurement direction, load, speed, and temperature conditions are equivalent.
Next post: [Robot Hardware 04] - Actuators (3): QDD Actuators
References
[1] H. Matsuki, K. Nagano, and Y. Fujimoto, “Bilateral Drive Gear—A Highly Backdrivable Reduction Gearbox for Robotic Actuators,” IEEE/ASME Transactions on Mechatronics, vol. 24, no. 6, pp. 2661–2673, 2019. https://doi.org/10.1109/TMECH.2019.2946403
[2] S. Yu et al., “Quasi-Direct Drive Actuation for a Lightweight Hip Exoskeleton with High Backdrivability and High Bandwidth,” 2020. https://arxiv.org/abs/2004.00467
[3] G. A. Pratt and M. M. Williamson, “Series Elastic Actuators,” IEEE/RSJ International Conference on Intelligent Robots and Systems, 1995. https://www.cs.cmu.edu/~cga/legs/jh1c.pdf
[4] S. Oh and K. Kong, “Performance Analysis of Series Elastic Actuator Based on Maximum Torque Transmissibility,” 2019. https://arxiv.org/abs/1902.05346