[Robot Hardware 03] - Actuators (2): Reducers

Robot hardware from a Physical AI perspective - reducers

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Robot Hardware from a Physical AI perspective

A reducer is the mechanical component that converts a motor’s high-speed, low-torque output into the low-speed, high-torque motion needed at a robot joint. A typical BLDC motor spins quickly but cannot directly support heavy links, so multi-jointed robots almost always need reducers.

Robot actuation can be organized by how the motor is connected to the joint. Some systems use belts or cables to place the motor away from the joint, which is often called remote actuation. Most multi-jointed robots, however, use a compact module where the reducer is mounted directly on the motor shaft.

Common reducer types include:

  1. Harmonic Drive reducers
  2. Planetary gearboxes
  3. Cycloidal drives

This integrated structure is popular for two main reasons.

  1. Packaging efficiency: The drive train can be concentrated inside the joint, reducing the overall volume of the robot.
  2. High stiffness: Unlike belts and cables, there is no long compliant transmission element between the motor and the joint. Many reducers also include high-stiffness bearings, such as cross-roller bearings, that help support external moment loads.

From a robotics perspective, however, a reducer is not just a device that multiplies torque according to $T_{out} = N \times T_{in}$.

The reducer type, gear ratio $N$, and internal friction strongly shape the robot’s output impedance, backdrivability, and control bandwidth. Together with the rest of the transmission, the reducer defines much of the robot’s physical personality.

There are two useful ways to think about how reducers affect robot performance:

  1. Physical interaction
  2. Control performance

1. Physical Interaction: Output Impedance and Backdrivability

The physical “heaviness” or resistance you feel when pushing a robot is called mechanical impedance. It is closely tied to backdrivability, which is the ability of an external force at the output side to rotate the motor backward through the transmission.

Mechanical impedance $Z(s)$ can be written as the ratio between external torque $T(s)$ and resulting angular velocity $\omega(s)$:

\[Z(s) = \frac{T(s)}{\omega(s)} = J_{eq}s + B_{eq} + \frac{K_{eq}}{s}\]

Here, $J_{eq}$ is equivalent inertia, $B_{eq}$ is equivalent viscous friction, and $K_{eq}$ is equivalent stiffness. At a robot joint, the dominant contributors are usually reflected inertia from the gear ratio $N$ and internal friction.

Friction contributes to the $B_{eq}$ term and is one of the main reasons backdrivability gets worse. In reducers with large internal friction, such as worm gears, a small external force may not move the joint at all. The joint behaves as if its impedance were almost infinite until the force exceeds a threshold.

Friction is not just an energy loss. It also creates an asymmetry: the efficiency depends on which direction energy is flowing. This asymmetry is essential for understanding reducer dynamics.

Energy-Flow Asymmetry: Forward vs. Backward Driving

Energy flow can be divided into two cases.

  • Forward driving (FWD): The motor drives the load. Motor $\rightarrow$ Load. The efficiency is written as $\eta_f$.
  • Backward driving (BWD): The environment drives the motor through the load. Environment $\rightarrow$ Motor. The efficiency is written as $\eta_b$.

Because of the mechanical structure, backward-driving efficiency $\eta_b$ is usually lower than forward-driving efficiency $\eta_f$. As internal friction increases, $\eta_b$ drops quickly. Below a certain point, it becomes effectively zero and the reducer becomes self-locking. External impact energy then cannot flow back into the motor. Instead, it is dissipated inside the reducer or causes damage.

This efficiency asymmetry changes the robot’s apparent inertia. Apparent inertia does not mean the physical mass has changed. It means the ratio between an externally applied force $F$ and the acceleration $a$ that actually occurs, $F/a$, has changed.

A simple way to understand this is to imagine pushing a robot arm by hand.

Industrial Robot vs. Collaborative Robot

  1. FWD case: motor-driven motion When the motor moves the robot arm, friction is just a loss torque that the motor has to overcome. The controller can model the system inertia as the physical sum of motor rotor inertia and link inertia.

  2. BWD case: externally driven motion When the robot is powered off and a person pushes the arm, the external force must first overcome reducer friction before it can accelerate the motor rotor.

    • If $\eta_b$ is low, most of the external force is lost to friction, and only a small amount becomes kinetic energy.
    • The person applies a large force, but the robot barely accelerates.
    • In other words, $F \gg ma$, so the robot feels as if it has an enormous mass.

The lower the backward-driving efficiency, the larger the apparent inertia becomes compared with the real physical inertia. This is why robots with high-ratio gearboxes often feel rigid and high-impedance during impact. Instead of yielding and absorbing the shock, the transmission resists it, and damage can occur.

High reduction ratio does not always mean poor backdrivability. Some specially designed gears, such as bilateral drive gears, reduce friction enough to achieve high backward-driving efficiency even at ratios around 100:1. But even if friction is minimized, there is another limit that cannot be removed as long as a gear ratio exists.

Reflected Inertia

Imagine a perfect reducer with zero friction. Would it feel feather-light when you backdrive it?

The answer is no. It can still feel extremely heavy because of reflected inertia, which is amplified by the gear ratio $N$.

\[J_{reflected} = N^2 \cdot J_{motor}\]

Why Does It Feel Heavy?

Consider a robot arm with a 100:1 reducer.

If you move the output joint by 1 degree, the motor shaft inside has to rotate by 100 degrees. That means during a short push or impact, the motor rotor must accelerate 100 times more than the joint.

The motor rotor itself may be small and light, but accelerating it that quickly requires a large force. Your hand feels that required rotor acceleration as resistance, which makes the whole robot arm feel much heavier than it physically is.

This “fake weight” grows with the square of the gear ratio.

  • 10:1 reduction $\rightarrow$ feels 100 times heavier
  • 100:1 reduction $\rightarrow$ feels 10,000 times heavier

Friction vs. Inertia

Friction and reflected inertia feel different when you touch a robot.

  1. Friction
    • Feeling: sticky, stiff, or viscous.
    • Behavior: It creates a static threshold. Once you push hard enough, the joint starts moving, but it becomes sticky again when it stops.
  2. Reflected inertia
    • Feeling: heavy, like pushing a massive object.
    • Behavior: It resists changes in motion. Slow movement may be possible, but a fast tap or impact is strongly resisted.

Impact and Failure

In high-ratio systems, reducing friction can make slow motion feel smoother, but it does not solve the problem of fast impacts. The $N^2$ inertia amplification remains.

During an external impact, the rotor needs to accelerate and move out of the way for the impact energy to be absorbed. If reflected inertia is too large, the rotor resists acceleration and stays effectively fixed. The impact energy then has nowhere to go, so it often ends up damaging the weakest part of the reducer, such as gear teeth.

In short, for robots that need dynamic interaction, reducing friction is not enough. Reflected inertia grows with the square of the gear ratio, and that is a physical limit. Even a low-friction gearbox can still behave as a high-impedance transmission if the reduction ratio is large.

This is why different actuation mechanisms become important:

  1. QDD (Quasi-Direct Drive): Reduce the gear ratio itself to minimize reflected inertia.
  2. SEA (Series Elastic Actuator): Place an elastic element between the actuator and the load so impact energy can be stored in the spring instead of damaging the drive train.

I will cover QDD and SEA in more detail later.

Control Performance: Gear Ratio and Control Bandwidth

Reducers also affect the robot’s control bandwidth. A typical multi-jointed robot controller is designed from a dynamics model, often written in the Lagrangian form:

\[M(q)\ddot{q} + C(q, \dot{q})\dot{q} + G(q) = \tau\]

In that equation, the actuator and transmission are often compressed into the joint torque term $\tau$. In real hardware, however, torque has to pass through motor dynamics, current control, reducer friction, backlash, compliance, and reflected inertia before it reaches the joint.

As the reduction ratio increases, the joint can produce more torque, but the motor-side dynamics are also magnified. Friction and reflected inertia reduce how quickly the joint can respond to changes in command. That means the practical control bandwidth can be much lower than the ideal model suggests.

Next post: [Robot Hardware 04] - Actuators (3): QDD Actuators