Thanks CraftyDan, who mentioned the keyword in the Electrohub CG thread that I needed to look deeper into this. Mind checking if I got it right though? 
Why would you want to calculate the center of thrust of a multirotor?
Knowing this helps a lot to build a multirotor that uses all of its motors as effective as possible. Because if you make sure the center of gravity (COG) is at this point, all of your motors can run at the same speed to achieve a stable hover.
If your COG is moved forwards, the front montors have to work harder to keep the multirotor level.
Let's go back to school
So, the task is to find the center of thrust for a set of motors. This is similar to finding the center of mass for a set of objects with the only difference being that our motors produce lift or „negative mass“. You might have heard about calculating this in your physics class.
Let's start with a very simple setup and have two motors, each producing the same amount of thrust t1 and t2 of 100g each. The distance between the centers of both motors is 10 cm.
To calculate the center of thrust between those two, we have to set a start point for measuring the required distances. To make this easy, we just use the center of motor 1 for the calculation. This results in two distances. The distance l1 to motor 1 being 0 cm and l2 to motor 2 being 10 cm.
The formula to find the center of thrust CT now becomes:
So the distance to the center of thrust from motor 1 is 5 cm – exactly in the middle as you might have expected.
This works no matter where we put the start point. If we had chosen the center between both motors, the distances would be l1 = -5 cm (to the left) and l2 = 5cm:
So with 0 cm it would be exactly at the chosen midpoint again.
Changing thrust
Let's play with the thrust and make motor 2 more powerful, perhaps by changing to a bigger prop. So t2 becomes 300 g, which changes the first formula to:
This made the center of thrust move 2,5 cm to the right. Just as we should've expected.
Extending to more motors
To extend this formula to more motors, we just do the following:
The part on the right just shows how lazy mathematicians are. ;-)
Without a lot of explaining text, let's just enter these values in the formula. Oh and all three motors have the same amount of thrust again.
So the center of thrust is 8,33 cm right of the center of motor 1.
Going multi-dimensional
How can we transfer this to a quadcopter now where the motors are not in one line? This is actually easier than I had first expected. Basically instead of one line, you use two by switching to an XY coordinate system. (Distances were chosen randomly.)
Now let's put these values in our formula for the X axis:
So the center of the thrust on the X axis is located exactly in the middle. Which you hopefully expected because quads usually are mirrored at the roll axis.
But what about the other axis?
The center of thrust on the pitch axis would be 4 cm in front of motors 3 and 4.
The combined center of thrust is the point of intersection between CTx and Cty.
Time to play
What can we do with this knowledge now? Well first you can try to match the center of trust and the center of gravity of the copter. Putting the flight controller at this point will help building an efficient and stable multirotor.
Yes, this is not necessary at all, because most flight controllers will compensate for any differences and still be able to fly nicely. But to me it's nice to know what would be the „perfect“ setup.
You can also try and shift the center of thrust now by modifying the thrust you get from a set of motors.
Perhaps you are building a dead cat style multirotor based on the Electrohub. The flight controller is supposed to be mounted a little to the back of the base plates. But what if you want it to be in the center of the plates for aesthetic reasons?
You could for example try and tilt the back motors a little so they produce less thrust downwards... This should move the center of lift more to the front (and give a more „swooshy“ yaw like with V- or A-tails).
But it's late and I really have to go to bed now. So I leave that calculation to someone else.
Why would you want to calculate the center of thrust of a multirotor?
Knowing this helps a lot to build a multirotor that uses all of its motors as effective as possible. Because if you make sure the center of gravity (COG) is at this point, all of your motors can run at the same speed to achieve a stable hover.
If your COG is moved forwards, the front montors have to work harder to keep the multirotor level.
Let's go back to school
So, the task is to find the center of thrust for a set of motors. This is similar to finding the center of mass for a set of objects with the only difference being that our motors produce lift or „negative mass“. You might have heard about calculating this in your physics class.
Let's start with a very simple setup and have two motors, each producing the same amount of thrust t1 and t2 of 100g each. The distance between the centers of both motors is 10 cm.
To calculate the center of thrust between those two, we have to set a start point for measuring the required distances. To make this easy, we just use the center of motor 1 for the calculation. This results in two distances. The distance l1 to motor 1 being 0 cm and l2 to motor 2 being 10 cm.
The formula to find the center of thrust CT now becomes:
So the distance to the center of thrust from motor 1 is 5 cm – exactly in the middle as you might have expected.
This works no matter where we put the start point. If we had chosen the center between both motors, the distances would be l1 = -5 cm (to the left) and l2 = 5cm:
So with 0 cm it would be exactly at the chosen midpoint again.
Changing thrust
Let's play with the thrust and make motor 2 more powerful, perhaps by changing to a bigger prop. So t2 becomes 300 g, which changes the first formula to:
This made the center of thrust move 2,5 cm to the right. Just as we should've expected.
Extending to more motors
To extend this formula to more motors, we just do the following:
The part on the right just shows how lazy mathematicians are. ;-)
Without a lot of explaining text, let's just enter these values in the formula. Oh and all three motors have the same amount of thrust again.
So the center of thrust is 8,33 cm right of the center of motor 1.
Going multi-dimensional
How can we transfer this to a quadcopter now where the motors are not in one line? This is actually easier than I had first expected. Basically instead of one line, you use two by switching to an XY coordinate system. (Distances were chosen randomly.)
Now let's put these values in our formula for the X axis:
So the center of the thrust on the X axis is located exactly in the middle. Which you hopefully expected because quads usually are mirrored at the roll axis.
But what about the other axis?
The center of thrust on the pitch axis would be 4 cm in front of motors 3 and 4.
The combined center of thrust is the point of intersection between CTx and Cty.
Time to play
What can we do with this knowledge now? Well first you can try to match the center of trust and the center of gravity of the copter. Putting the flight controller at this point will help building an efficient and stable multirotor.
Yes, this is not necessary at all, because most flight controllers will compensate for any differences and still be able to fly nicely. But to me it's nice to know what would be the „perfect“ setup.
You can also try and shift the center of thrust now by modifying the thrust you get from a set of motors.
Perhaps you are building a dead cat style multirotor based on the Electrohub. The flight controller is supposed to be mounted a little to the back of the base plates. But what if you want it to be in the center of the plates for aesthetic reasons?
You could for example try and tilt the back motors a little so they produce less thrust downwards... This should move the center of lift more to the front (and give a more „swooshy“ yaw like with V- or A-tails).
But it's late and I really have to go to bed now. So I leave that calculation to someone else.
