Gear Reducers

Introduction

Motors typically operate at high speed with low torque, whereas robot joints require low speed with high torque. A gear reducer (also called a transmission) is a critical mechanical component connecting the motor to the load, converting speed and torque through gear mechanisms.

Basic Principles

Gear Ratio

The gear ratio \(N\) (also called the transmission ratio) is defined as the ratio of input speed to output speed:

\[ N = \frac{\omega_{in}}{\omega_{out}} \]

Torque Amplification

Under ideal conditions where power is conserved, the reducer amplifies torque:

\[ \tau_{out} = N \cdot \tau_{in} \cdot \eta \]

Where \(\eta\) is the transmission efficiency.

Speed Reduction

\[ \omega_{out} = \frac{\omega_{in}}{N} \]

Reflected Inertia

Load inertia reflected to the motor side:

\[ J_{equiv} = J_{motor} + \frac{J_{load}}{N^2} \]

Benefit of High Gear Ratios

The \(N^2\) term means that a high gear ratio significantly reduces the load inertia "felt" by the motor, making control easier. The trade-off is slower output shaft response.

Planetary Gear Reducer

Structure

        ┌─────────────────┐
        │   Ring Gear      │
        │  ┌──┐  ┌──┐    │
        │  │Pla│  │Pla│   │
Input ──→│  │net│──│net│   │──→ Output
(Sun)   │  │   │  │   │   │(Carrier)
        │  │   │  │   │   │
        │  └──┘  └──┘    │
        └─────────────────┘

• Sun gear: Central gear, connected to the motor input
• Planet gears: Revolve around the sun gear, typically 3–4 units
• Ring gear: Outer internal gear, usually fixed
• Carrier: Connects the planet gears, provides the output rotation

Characteristics

Parameter	Typical Values
Gear ratio	3:1 – 100:1 (single-stage 3–10, multi-stage can be cascaded)
Efficiency	90–97% (single-stage)
Backlash	1–3 arcmin (precision grade) / 5–15 arcmin (standard grade)
Size	Compact, coaxial input/output

Advantages and Limitations

• Advantages: Compact structure, coaxial design, high load capacity, flexible multi-stage cascading
• Limitations: Has backlash, noise increases with gear ratio, requires lubrication maintenance

Applications

• Industrial collaborative robots (e.g., some joints in the UR series)
• AGV drive wheels
• 3D printer extruders

Harmonic Drive

Structure

A harmonic drive consists of three concentric components:

Component	English	Function
Wave generator	Wave Generator	Elliptical cam + thin bearing, input
Flexspline	Flexspline	Thin-walled elastic cup-shaped gear, output
Circular spline	Circular Spline	Rigid internal gear, fixed

Operating Principle

1. The wave generator rotates, deforming the flexspline into an ellipse
2. The flexspline meshes with the circular spline at the major axis of the ellipse
3. The flexspline has 2 fewer teeth than the circular spline (e.g., flexspline 100 teeth, circular spline 102 teeth)
4. One revolution of the wave generator advances the flexspline by 2 teeth relative to the circular spline

Gear ratio calculation:

\[ N = \frac{Z_{rigid}}{Z_{rigid} - Z_{flex}} = \frac{Z_{rigid}}{2} \]

For example, with a circular spline of 100 teeth: \(N = 100/2 = 50:1\)

Characteristics

Parameter	Typical Values
Gear ratio	50:1 – 160:1 (single-stage)
Efficiency	65–85%
Backlash	<1 arcmin (near-zero backlash)
Repeatability	<5 arcsec
Lifespan	Related to flexspline fatigue

Advantages and Limitations

• Advantages: Extremely high gear ratio, zero backlash, high precision, compact size
• Limitations: Limited flexspline lifespan (elastic fatigue), lower efficiency, not backdrivable, expensive
• Typical Brands: Harmonic Drive (Japan), Laifual (China), Leader Drive

Applications

• Six-axis industrial robotic arms (especially wrist joints)
• Collaborative robot joints
• Aerospace mechanisms (e.g., satellite antenna drives)

RV Reducer

Structure

The RV (Rotary Vector) reducer combines a planetary gear stage with a cycloidal gear stage for two-stage reduction:

1. First stage: Involute planetary gears (preliminary reduction)
2. Second stage: Cycloidal gears (main reduction, provides high stiffness)

Characteristics

Parameter	Typical Values
Gear ratio	30:1 – 200:1
Efficiency	75–85%
Stiffness	Extremely high (about 3x that of harmonic drives)
Backlash	<1 arcmin
Load capacity	High (dual-stage support)

Advantages and Limitations

• Advantages: Extremely high stiffness, impact resistant, high load capacity, long lifespan
• Limitations: Bulky, heavy, expensive
• Typical Brands: Nabtesco, ZD Leader

Applications

• Industrial robot base and upper arm joints (heavy loads requiring high stiffness)
• J1–J3 joints on ABB, FANUC, KUKA industrial arms

Cycloidal Reducer

Principle

A cycloidal disc, driven by an eccentric shaft, performs a cycloidal motion and meshes with pin gear housing to produce speed reduction.

Characteristics

• Gear ratio: 6:1 – 87:1 (single-stage)
• Strong shock resistance
• Relatively simple structure
• Commonly used as the second stage in RV reducers

QDD (Quasi-Direct Drive)

Design Philosophy

QDD (Quasi-Direct Drive) uses a low gear ratio (4:1 – 9:1) reducer, preserving the motor's backdrivability.

Comparison with High Gear Ratio Approaches

Feature	High Gear Ratio (50–160:1)	QDD (4–9:1)
Output torque	High	Medium
Backdrivability	Poor (nearly impossible)	Excellent
Force control precision	Requires torque sensor	Current-based torque estimation
Control bandwidth	Low	High
Collision safety	Requires additional protection	Naturally compliant
Efficiency	Medium	High

Backdrivability Principle

Backdrivability depends on the reverse efficiency of the reducer. When reverse efficiency > 0, external forces can drive the motor:

\[ \eta_{backward} = \frac{2\eta_{forward} - 1}{1} \quad (\text{approximate}) \]

• Planetary gear \(\eta_{forward} = 95\%\) → \(\eta_{backward} \approx 90\%\) (backdrivable)
• Harmonic drive \(\eta_{forward} = 70\%\) → \(\eta_{backward} \approx 40\%\) (difficult to backdrive)
• Worm gear \(\eta_{forward} = 50\%\) → \(\eta_{backward} < 0\) (self-locking, not backdrivable)

Motor Requirements for QDD

Since the gear ratio is low, the motor itself must produce high torque:

• High pole pair count (e.g., 21 pole pairs, 42 poles)
• Large-diameter outer rotor
• High-quality permanent magnets (N52-grade NdFeB)
• High-efficiency FOC drive

Applications

• Legged robots: MIT Cheetah, Unitree Go2/B2, Unitree H1
• Collaborative robots: Force-controlled joints
• Exoskeletons: Assistive devices requiring compliance

Comprehensive Reducer Comparison

Feature	Planetary Gear	Harmonic Drive	RV Reducer	Cycloidal	QDD
Gear ratio range	3–100	50–160	30–200	6–87	4–9
Efficiency	90–97%	65–85%	75–85%	80–90%	90–97%
Backlash	Medium	Very low	Very low	Low	Low
Stiffness	Medium	Medium	Very high	High	Low
Backdrivability	Good	Poor	Poor	Medium	Excellent
Size/weight	Medium	Small	Large	Medium	Medium
Cost	Medium	High	High	Medium-low	Medium (requires large motor)
Noise	Medium	Low	Low	Medium	Low

Selection Decision

graph TD
    A[Reducer Selection] --> B{Backdrivability needed?}
    B -->|Yes| C[QDD 4-9:1]
    B -->|No| D{Extremely high precision needed?}
    D -->|Yes| E{Heavy load?}
    D -->|No| F[Planetary Gear]
    E -->|Yes| G[RV Reducer]
    E -->|No| H[Harmonic Drive]

    style C fill:#bfb,stroke:#333
    style F fill:#ffb,stroke:#333
    style G fill:#fbb,stroke:#333
    style H fill:#bbf,stroke:#333

Typical Robot Configurations

Robot Type	Large Joints	Small Joints (Wrist)	Reduction Strategy
Industrial 6-axis arm	RV reducer	Harmonic drive	High gear ratio
Collaborative robot	Harmonic drive	Harmonic drive	High gear ratio + torque sensor
Quadruped robot	QDD	QDD	Low gear ratio
Humanoid robot	QDD (legs) + Harmonic (arms)	Harmonic / direct drive	Hybrid approach

Summary

• Reducers achieve torque amplification via \(\tau_{out} = N \cdot \tau_{in}\)
• Planetary gears are versatile and suitable for most scenarios
• Harmonic drives offer zero backlash and high precision, making them the top choice for arm wrist joints
• RV reducers have the highest stiffness, suitable for heavy-load joints
• Low gear ratio QDD approaches trade torque for backdrivability and are mainstream for legged robots
• Selection must consider gear ratio, efficiency, precision, stiffness, backdrivability, and cost holistically

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