Calculate servo torque from load, arm length, angle, safety factor, and linkage ratio, or reverse-solve maximum load and arm with common servo units.

Angle is measured above horizontal. Blank safety factor and torque ratio mean 1; a blank candidate omits comparison.

Static gravity estimate only. Acceleration, friction, shock, linkage losses and voltage-dependent servo ratings need separate assessment.

Factors and candidate comparison

1 adds no safety allowance. Select an appropriate factor for your application.

1 means direct drive; 0.5 means half the load torque is required at the servo. Include actual linkage effects.

Servo Torque Formula

The calculator above estimates static gravity torque: required servo torque = m × 9.80665 × r × cos(θ) × safety factor × servo/load torque ratio. Here θ is the arm angle above horizontal, so 0° gives the maximum gravity moment and 90° gives zero. The ratio means required servo torque divided by load torque, before the safety factor; it is 1 for direct drive. A blank safety factor or ratio means 1, adding no safety allowance.

It can also solve load mass or arm distance for available torque. At a vertical arm or zero inverse denominator it cannot identify a finite unique limit. This model excludes acceleration, friction, shock and unmodeled linkage losses. A candidate comparison is not a continuous-duty or safe-selection certification.

The separate dynamic direct-drive relation below is background, not an input mode of the widget. It combines inertia and gravity when the signs of both contributions act in the stated required direction; actual dynamics can require a signed torque balance and additional loads. Its angle φ is measured from vertical, unlike the widget’s angle above horizontal. See OpenStax’s torque definition. Torque-unit conversions follow NIST SI conversion guidance.

Tₛₑᵣᵥₒ = Iα + m g r sin(φ)
  • Where Tservo is the Servo Torque (N·m)
  • I  is the moment of inertia about the servo axis (kg·m²) 
  • α is the angular acceleration (rad/s²) 
  • m is the mass (kg) 
  • g is the acceleration due to gravity (9.81 m/s²)
  • r is the lever arm radius (m)
  • φ is the angle from vertical (gravity direction): φ = 0° gives zero gravity torque; φ = 90° gives maximum gravity torque

How to Calculate Servo Torque?

The following worked examples illustrate the dynamic direct-drive formula, with no separate safety or linkage factor. They use g ≈ 9.81 m/s² and are not calculator test cases because the widget models static gravity only. For a widget example, 2 kg at 10 cm, 0° above horizontal, safety factor 1.5 and ratio 1 requires 2 × 9.80665 × 0.10 × 1.5 = 2.941995 N·m = 30 kgf·cm. A 40 kgf·cm candidate is about 33.3333% above this static requirement.

Example Problem #1

  1. First, determine the moment of inertia (kg·m²). In this example, the moment of inertia (kg·m²) is determined to be 5.
  2. Next, determine the angular acceleration (rad/s²). For this problem, the angular acceleration (rad/s²) is measured to be 12.
  3. Next, determine the mass (kg). In this case, the mass (kg) is found to be 4.
  4. Next, determine the radius. For this problem, this is 5 m, and assume the arm is horizontal so φ = 90° (maximum gravity torque).
  5. Finally, calculate the Servo Torque using the formula above: 

Tservo = I*α + m * g * r * sin(φ)

Inserting the values from above and solving yields: 

Tservo = 5*12 + 4 * 9.81 * 5 * sin(90°) = 256.2 (N·m)

Example Problem #2

Using the same method as above, determine the variables required by the formula. For this example problem, these are:

moment of inertia (kg·m²) = 89

angular acceleration (rad/s²) = 4

mass (kg) = 58

radius (m) = 2.5

Assume the arm is horizontal so φ = 90° (maximum gravity torque). Entering these given values yields: 

Tservo = 89*4 + 58 * 9.81 * 2.5 * sin(90°) = 1778.45 (N·m)