Enter the link lengths, link masses and joint angles to estimate the static holding torque at each joint of a 1–3-link robot arm. Include the tool and workpiece in the tip load, then change the pose to see how the gravity load shifts between joints.
Save a baseline to compare results. The calculation covers gravity loads in a vertical plane; acceleration, friction and gearbox losses need a separate motor-sizing calculation.
Physical AI Lab / Engineering toolsHow much torque
holds this arm in place?
Change link lengths, masses and joint angles to estimate the torque each joint needs to hold the arm against gravity.
Arm configuration
Include both the tool and carried object. Their combined mass is treated as a point load at the arm tip.
Angle convention J1 is 0° when pointing horizontally right. J2 and J3 are relative to the preceding link. Positive angles are counterclockwise.
Arm geometry and loads
Side view in a vertical plane · lengths in mm
Required joint torque
Static holding torque against gravity · N·m
These results show static torque at the current pose. They exclude inertia during motion, friction, impacts, thermal limits and gearbox efficiency. Applying a margin multiplier does not replace motor sizing or payload validation.
Same mass.
Different pose, different torque.
Choose Reach horizontally and save a baseline. Change J1 to 90° to reduce the horizontal moment arm of gravity. The arm still has the same weight.
A lower torque at one joint does not mean all joints are unloaded. Compare the J1 and J2 values together.
Calculation method and scope
Model: A serial robot arm with 1–3 rigid links in a vertical plane. Joint rotation axes are perpendicular to the screen. You specify the mass and center of mass of each link.
Holding torque at joint j: τⱼ = g Σ mᵢ(xᵢ − xⱼ), with g = 9.80665 m/s². Only links downstream of that joint and the tip load are included. Positive holding torque is counterclockwise; negative is clockwise.
Link center of mass: The default 50% is the midpoint of a uniform link. If you know the combined center of mass of the actual structure, wiring and motors, adjust its position from the starting joint. Joint motor masses are not added automatically.
Use: Comparing poses and estimating gravity loads during early design. 3D motion, dynamics, manufacturer payload and inertia limits, fatigue life and safety require separate assessment. A result of 0 N·m means zero gravity moment at that joint in this pose.
Input ranges: Link length 1–10,000 mm; mass 0–1,000 kg; joint angle −180–180°; center of mass 0–100%; margin multiplier 1–5.
Further reading: Physical AI Lab · Motor sizing and duty cycle · Oriental Motor · Motor sizing calculations
Check the result with a 400 mm arm
Select 1 link. Enter 400 mm length, 2 kg link mass, 0° angle, a 50% center of mass, 0.5 kg tip load and a margin multiplier of 1. The result is 5.88 N·m. These are worked calculations you can reproduce above, not measurements from a physical robot.
The link contributes 2 × 9.80665 × 0.20 = 3.92266 N·m; the tip load adds 0.5 × 9.80665 × 0.40 = 1.96133 N·m. Their sum is 5.88399 N·m. Lengths are converted from millimeters to meters. This follows the force-times-lever-arm relationship explained in Oriental Motor’s sizing guide.
Why two joint torques change differently
Now select 2 links. Keep link 1 at 400 mm and 2 kg; set link 2 to 300 mm and 1 kg. Use 50% centers of mass, the same 0.5 kg tip load and multiplier of 1. Enter these angle pairs:
| J1 / J2 angle | J1 torque (N·m) | J2 torque (N·m) |
|---|---|---|
| 0° / 0° | 12.75 | 2.94 |
| 90° / 0° | 0.00 | 0.00 |
| 90° / −90° | 2.94 | 2.94 |
J2 is relative to link 1. At 90° / −90°, link 1 points up and link 2 points right. Both joints then see the same horizontal lever arms for link 2 and the tip load. At 90° / 0°, those lever arms are zero; the arm still has mass.
Use the result in your next calculation
For motor selection, carry the gravity term into the motor torque and duty-cycle calculation. To include acceleration and external forces, see inverse dynamics. If your starting point is a current reading, use the motor-current torque estimation guide; current-derived torque needs calibration and transmission losses accounted for.