Gyroscopic effect in quadcopters

[user1234]

TL;DR Summary

In gyroscopic effect of a drone, why does body frame angular velocity equal to precession angular velocity which i have seen in all articles.

Hi everyone, this is my first post in the site. I post this thread below 'Mechanical Engineering' topic because i see most drone questions is posted under this subject but question can be more suitable to other subject so please let me know.

In quadcopter dynamic equations of Newton-Euler formation that I have encountered in all academic publications,
torque equation is like this:

The cause of gyroscopic effect is from the spinning of the four propellers
but i am confused about gyroscopic part of this equation, because, as you know,
in gyroscopic effect torque causes precession angular velocity perpendicular to spinning and torque vector.
In the equation above, it is written down as if the angular velocity [p,q,r] of the body frame is produced completely from gyroscopic torque caused by the spinning of propellers however, the torque applied to the rigid body itself also contributes to the angular velocity.

• Is it assumed that angular velocity created by gyroscopic effect is much higher than the ones created by other factors so do they assume that precession is equal to the body frame a.v. ?
• Gyroscopic torque is subtracted from total torque produced by propellers and the relation between movement of the body and applied torque to the body is established, but why is not generated angular velocity from the gyro effect considered because it will not be same with the situation where the same torque is applied but any gyro effect is not observed.

 

[.Scott]

Your question: why does body frame angular velocity equal to precession angular velocity which i have seen in all articles?

If I understand this correctly, you are saying that those articles are calculating the precession velocity of the quad-copter only on the angular velocity of the frame - and more specifically, the frame yaw rate. So, I take it they are not including the angular velocities (AV) of the propellers.

I have attached photos of 5 quad-copter models below. If you look carefully at the pitch of each blade, it is clear that adjacent propellers rotate in opposite directions. So the total AV of the propellers would be close to zero.

And the reason its close to zero is that anything that would change that total angular momentum would cause the copter to change its rates of yaw, pitch, or roll - but mostly yaw. The lift is shared by the two props rotating one way (say clockwise) and the other two (rotating counter-clockwise). Momentarily shifting that share of lift from one pair to the other will cause a change in the angular momentum of the frame. So if the frame is yawing to the right at a constant rate, the AV (and thus angular momentum) of the props is close to balanced. But to correct the yaw, you would have to unbalance the total prop AV for a second in the right direction and then restore the balance.

 

[user1234]

Your question: why does body frame angular velocity equal to precession angular velocity which i have seen in all articles?

If I understand this correctly, you are saying that those articles are calculating the precession velocity of the quad-copter only on the angular velocity of the frame - and more specifically, the frame yaw rate. So, I take it they are not including the angular velocities (AV) of the propellers.

I have attached photos of 5 quad-copter models below. If you look carefully at the pitch of each blade, it is clear that adjacent propellers rotate in opposite directions. So the total AV of the propellers would be close to zero.

And the reason its close to zero is that anything that would change that total angular momentum would cause the copter to change its rates of yaw, pitch, or roll - but mostly yaw. The lift is shared by the two props rotating one way (say clockwise) and the other two (rotating counter-clockwise). Momentarily shifting that share of lift from one pair to the other will cause a change in the angular momentum of the frame. So if the frame is yawing to the right at a constant rate, the AV (and thus angular momentum) of the props is close to balanced. But to correct the yaw, you would have to unbalance the total prop AV for a second in the right direction and then restore the balance.


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Thanks for reply, but i think you didn't understand me correctly.
First of all, the equation above is represented in the body frame that means the frame is fixed to center of mass of quadcopter and rotates with the quadcopter.
In your reply you are talking about angular velocity of yaw axis(z axis) which is produced by drag torque and represented with the term that is encircled with blue square below but i am talking about the phenomenon called gyroscopic effect which is produced by the spinning propellers and encircled with red square below.

 

[.Scott]

All four props are spinning about the yaw axis. But their total angular momentum is very near zero.
So, their total gyroscopic effect will be near zero.
The intent of most of the rest of my discussion was to describe why it would need to be kept close to zero - and what the exceptions would be.

 

[Lnewqban]

Welcome, @user1234 !

The gyroscopic effect should appear only for pitch and roll.
Is that correct?
If so, what all the articles that you have read tell about the interdependence of the additional movement (body frame angular velocity) induced by those two inputs?

 

[user1234]

Welcome, @user1234 !

The gyroscopic effect should appear only for pitch and roll.
Is that correct?
If so, what all the articles that you have read tell about the interdependence of the additional movement (body frame angular velocity) induced by those two inputs?

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Yes, gyro torque appear only for x and y axis, as seen from the formulae.
As for the your second question i didn't understand the phrasing "interdependence of additional movement", articles don't explain deeply what causes what, authors just state what external forces exist, and what they are etc. and skip to control strategies. All we got is the equation above(moments part of the equation).

 

[user1234]

All four props are spinning about the yaw axis. But their total angular momentum is very near zero.
So, their total gyroscopic effect will be near zero.
The intent of most of the rest of my discussion was to describe why it would need to be kept close to zero - and what the exceptions would be.

All four props are spinning about the yaw axis. But their total angular momentum is very near zero.
So, their total gyroscopic effect will be near zero.
But this does not solve our problem, if gyro effect is so small, why is precession angular velocity which is created by gyro effect responsible to produce body frame angular velocity.

 

[hutchphd]

why is precession angular velocity which is created by gyro effect responsible to produce body frame angular velocity.

It is not what produces this velocity. It is drag from rotors Suppose you increase ##\omega_1 , \omega_4## ? This will both increase lift and create clockwise rotation of the body (yaw) because of increased prop (rotor) drag. Suppose you decrease ##\omega_2, \omega_3## ? This will both decrease lift and create clockwise rotation of the body (yaw) because of decreased prop (rotor) drag.
So net yaw rotation in level flight. Voila!

 

[user1234]

It is not what produces this velocity. It is drag from rotors Suppose you increase ##\omega_1 , \omega_4## ? This will both increase lift and create clockwise rotation of the body (yaw) because of increased prop (rotor) drag. Suppose you decrease ##\omega_2, \omega_3## ? This will both decrease lift and create clockwise rotation of the body (yaw) because of decreased prop (rotor) drag.
So net yaw rotation in level flight. Voila!

I do already understand yaw movement of the quad what i try to understand is movement of the quad in x and y axes. If you look at the formulae you will see gyro precession a.v. is substituted with body coordinate frame angular velocity [p q r]. Does not that mean that gyro effect is responsible completely for body frame a.v in x and y axes?

 

[hutchphd]

I believe the x and y motions are due to creation of direct thrust imbalance. In fact this is easy to confirm if you have one: just rotate it off x axis and note the response. If you wish to discuss the formulae you must point me to specific equations.....preferably numbered....otherwise untenable for me. Do you understand how a boomerang returns That does use precession. Quite clever those bush people. There is very little net angular momentum here to work with here.. It is possible these folks are incorrect in their analysis, but I do not know. You must be much more specific.

 

[Lnewqban]

Yes, gyro torque appear only for x and y axis, as seen from the formulae.

Good!
That we can explain.

As for the your second question i didn't understand the phrasing "interdependence of additional movement",

Sorry, English is not my native language.

I meant, what is bothersome on control and/or stability of our drone?

In other words, how are the following related to each other, regarding cause and effect?:
*Gyroscopic effect of the drone.
*Body frame angular velocity.
*Precession angular velocity.

 

[DiegoTableros]

I want to understand that gyroscopic effect too! But it seems that nobody knows how to derive it properly.
My contribution is this:
I've studied some quadrotor models too and in my research I found that we need to consider each propeller as a another rigid body and use the Euler equation for rigid bodies for it; that will lead us to describe its torque in function of each propellers' AV and the precession angular velocity.
Unfortunately, my question is the same as yours, why we can consider the precession AV as the quadcopter AV too?
I'll be pending for your responses, thanks!

 

[user1234]

I want to understand that gyroscopic effect too! But it seems that nobody knows how to derive it properly.
My contribution is this:
I've studied some quadrotor models too and in my research I found that we need to consider each propeller as a another rigid body and use the Euler equation for rigid bodies for it; that will lead us to describe its torque in function of each propellers' AV and the precession angular velocity.
Unfortunately, my question is the same as yours, why we can consider the precession AV as the quadcopter AV too?
I'll be pending for your responses, thanks!

I posted a question that is ,i think, a simplified model of a quadcopter in stack exchange here but no one answered the question satisfyingly. I think quadcopter model with props and main frame has complex dynamic structure and in articles it is considered somehow a simple model. If i reach a strong conclusion i would post it here.

 

[user1234]

Here is my final conclusion: If a spinning object(wheel, propeller etc.) is also rotating around the axis perpendicular to the spinning axis then there must be some torque which is cross product of angular momentum of spinning wheel and angular velocity of precession. I mean it can't be said that in an axis there is some angular velocity of the wheel(propeller) and some part of it belongs to precession a.v. and some part of it belongs to the other factor. When the spinning wheel(propeller) rotates around precession axis, torque must be present to allow this rotation or in other words the spinning wheel creates a counter torque when it precess.
My words may not be most suitable to explain it but this must be solution.