Robot Flange Inertia calculator
Calculate the mass moment of inertia J_flange at the tool flange, the static load torque M_load and the payload m_total from a list of masses with centre-of-gravity distance and optional own inertia. The calculator compares all three quantities live against the robot's allowable data-sheet values and rates the design with a traffic light, live with every input.
Calculation
Model: rigid masses at the tool flange, inertia by the parallel-axis (Steiner) theorem about the flange axis, static load torque from weight. No dynamic acceleration torque, no stiffness or vibration analysis. Sizing aid - the limits of the specific robot data sheet govern.
Results
Calculating …
Calculation in your browser, inputs go to our server only when you export or save.
Formulas and fundamentals
Moment of inertia by the parallel-axis theorem
The moment of inertia of a load mounted at the flange about the flange (rotation) axis follows from the parallel-axis (Steiner) theorem: for each body J_i = J_own + m_i·r_i². Summed, this gives:
The own term J_own is the inertia of the body about its own centre-of-gravity axis parallel to the flange axis; the Steiner term m_i·r_i² accounts for the offset by the centre-of-gravity distance r_i. For a point mass or a body small compared with r_i the own term vanishes and simplifies to:
This is the lower bound of the true inertia.
Static load torque and payload
The static load torque at the flange arises from the weight of the total load acting at distance a from the flange centre:
With g = 9.81 m/s². It loads the flange axis regardless of motion and is decisive for far-reaching tools and a horizontal flange orientation. The payload is the sum of all mounted masses including gripper, adapter and workpiece:
Allowable data-sheet values
For an admissible design all three data-sheet criteria must be met simultaneously: J_flange ≤ J_allow (allowable mass moment of inertia about the flange axes), M_load ≤ M_allow (allowable load torque) and m_total ≤ m_allow (allowable payload). Under dynamic operation the acceleration torque must additionally be checked against the allowable axis torque:
This dynamic check is not included here.
Worked example
A point mass of 5 kg sits at distance r = 0.15 m from the flange axis. The moment of inertia is J = m·r² = 5·0.15² = 0.1125 kg·m².
If the load's centre of gravity is also 0.15 m from the flange centre, the static load torque is M_load = m·g·a = 5·9.81·0.15 = 7.36 Nm. The payload is 5 kg.
With data-sheet limits J_allow = 0.3 kg·m², M_allow = 20 Nm and m_allow = 10 kg, all three criteria are met at utilisations of 38, 37 and 50 percent - the design is green. Accounting for the gripper's real own inertia instead of a point mass raises J_flange accordingly.
Frequently asked questions
Why does the distance r enter the inertia quadratically?
Because the moment of inertia depends on the axis distance via r² (m·r²). Doubling the centre-of-gravity distance quadruples the inertia contribution. Far-reaching or outboard masses therefore dominate the flange inertia - a compact layout close to the flange axis helps more than reducing mass.
When may I treat the mass as a point mass?
When the body's own size is small compared with the centre-of-gravity distance r. Then J = m·r² is a good approximation and also the lower bound. For compact bodies close to the flange (small r) the own term J_own can dominate and should be added.
What is the difference between load torque and moment of inertia?
The static load torque M_load = m·g·a arises from weight and loads the flange axis even at standstill. The mass moment of inertia J_flange is kinematic and acts only under acceleration (M = J·α). Both are separate data-sheet limits and must be met independently.
Do I have to check the dynamic acceleration torque separately?
Yes. This calculator covers the three static data-sheet criteria. For fast motion, additionally verify M_a = J_flange·α against the allowable axis torque, with α from the motion profile. Excessive flange inertia lengthens cycle times and can excite vibration.
How do I determine the centre-of-gravity distance a?
a is the distance of the combined centre of gravity of the mounted load from the flange centre in the reach direction. For several masses it is the mass-weighted mean of the individual centres. The distance perpendicular to the weight governs the load torque.
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