How to calculate total moment around Z-axis when the armlengths are not the same for all motors on a hexacopter?

Question

I am trying to build a mathematical model of a hexacopter where armlengths (distance to the center of mass) are not the same for all motors. I have the datasheet of the motor I plan to use together with propeller showing the torque (mNm) values at specific speeds (RPM).

I wonder how I can calculate the total moment around Z-axis at given rotation speeds for the motors. My intuition tells me that motors that are more distant to center of mass should have greater effect on the total moment around Z-axis, therefore I need to multiply the armlengths with the torque value I read from the datasheet before summing them to get the total moment around Z-axis. However I cannot find any physical foundation to support my intuition. Also the calculations I have seen online just sum all torque values without taking armlengths into account.

Answer 1

Your intuition is right for forces but not for torques. A torque is a force multiplied by the lever arm. So for the X and Y axis you multiply the motor thrust by the arm length (CG to motor centre) to get the torque, then sum the torques.

For the Z axis, you just sum the torques. The length of the lever arm doesn't matter. You could think of it as taking the motor torque, then dividing by the distance to get the force, then multiplying by the distance to get the torque at the other end.