Gear Ratio and Torque Transmission Basics

Before getting into different options for actuators, I thought it would be a good idea to review the basics of gear ratios and torque transmission. Consider the following diagram, where a motor powers a very simplified robotic arm.

(Note: Diagrams drawn with my insane egad skills)

In this simplified diagram, the loads on the arm linkage are m_arm (mass of arm located at the linkages center of mass) and m_load (mass of the load the arm is lifting, located at the claw). Both of these loads create a resultant torque on the motor shaft. The equation for these torques are given by:

Lets take a closer look at the motor. As you can see, the motor shaft needs to be able to provide a minimum torque greater than the resultant total torque to move the arm. However, it's good practice to add some safety factor to this value to ensure there is more than enough torque required to move the arm as intended, rather than meeting the minimum requirements.

This torque can get pretty high depending on the radius of the loads, arm mass, load mass and arm position (with the highest required torque found when Θ = 90°). This is where gear reductions come in handy - as they allow for much higher torque transmission than an individual motor shaft! Consider the scenario where the motor shaft is connected to an output gear that rotates the simplified arm.

These two gears will have some form of gear ratio between them. It's important to note that the gear ratio formula may vary for some of the types of drives discussed in this document. However at it's most basic form, the gear ratio (G.R.) can be expressed as:

where:

t_out: output teeth
t_in: input teeth
N_in: input speed (rpm)
N_out: output speed (rpm)
T_out(ideal): ideal output torque (N•m)
T_in: input torque (N•m)

One important relationship to note here is that output speed is inversely proportional with gear ratio and that output torque is directly proportional to the gear ratio. This means that high gear ratio joints imply a higher output torque and a lower output speed. Both of these attributes are excellent for robotic applications. Having a higher output means that the arm can support and move greater loads, while lower output speeds allows for more precise positioning for operators.

As you probably noticed, in the gear ratio equation the output torque is labelled as ideal. In reality, the output torque is slightly less than the input torque, as all transmission types have some percentage of efficiency, as power is lost between the input and output shaft.

The instantaneous power of the gear (P) and the efficiency of the gear system (μ) are given by:

Another key relationship to note is that at the same instantaneous power, speed and torque are inversely proportional. As speed increases, torque decreases and vice versa.

What Gear Ratios do we Need?

This is a very important question to ask, but at the same time is a very difficult question to answer. I won't be able to provide any answers in this page but I will list the things that must be considered before properly answering this question.

How many linkages does our arm have, and how long are the linkages?
How much does each linkages weigh, and where is it's center of mass?
What is the rated torque of the motor driving the joint in question?
What kind of drive are you using (gear drive, belt drive, cycloid drive, etc.)
How many stages of reduction are you using in your drive?
How heavy of a load do you need to manipulate?
What safety factor are you applying?

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