Classic Inverted Pendulum - Equations of Motion
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- Опубликовано: 5 окт 2024
- In this video, we derive the full nonlinear equations of motion for the classic inverted pendulum problem. Although the Lagrange formulation is more elegant, this video uses a simpler Newtonian approach that anyone with sophomore-level engineering dynamics should be able to understand.
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Please make a 2nd part of this video explaining the control system for stabilizing the inverted pendulum! Very good work!!
Here it is: www.spumone.org/courses/control-notes/inverted-pendulum/
There are 4 more videos.
Man I love the love way you actually explain stuff, and not simply state equations! Fantastic explanation of accelerations!
Glad to help
Excellent video. Most control books don't review mechanics and all and just throw in the final result or vaguely deduct the equations needed to build the state-variables expression. The only thing I would have liked is that Newton's 2nd law for rotation had been used instead, working with torques and the moment of inertia of the pendulum in respect to the axis in the pendulum-cart joint. However, the point of this video is to achieve the equation in a simple way to avoid having to remember rigid body mechanics entirely and stuff like that, so I understand why you took the way you took. Thank you very much!
Where is part 2?? This was such a good refresher video. I loved this stuff in college and after only a few years of not using it I've lost a lot of it!
Very fundamental and generalized way of applying the Newton formula for deriving the system dynamics.. No need to By-Heart any advanced formulas... All are basic things. Excellent..
This was what I have been trying to find for weeks. Thank you very much :)
You're welcome
Wow great video. Had a minor memory lapse on centripetal acceleration but thank you for the great explanation!
Thanks for the excellent explanation!!! I didn't notice the time passing by.
You're welcome.
Is there a part two to this video taking the laplace transformation? that would be an absolute life saver right now
Great job. After a long time I understood inverted pendulum. Thanks a lot .
Excellent video. Tried solving a similar system using Euler Lagrangian method cos i though Newton's approach would be difficult. The way it is described here makes it look so simple.
Thank you! Best tutorial on yt
Nice video, I understood a lot from it. However, what if I want to write the equations of motion for an inverted pendulum with a fixed end and an inertia wheel on the other?
Thanks for your great video. It's easy to follow and understand. At about 16:00 you started to eliminate the tensions with multiplying cos(theta) to (3) and sin(theta) to (4). This seems to solve the problems the the tension very nicely, but I don't get the clue behind it.
Could you drop some explaining words, why this step is allowed? thanks a lot and keep going with your nice videos.
--alex
The tension force T is expressed in two equations. From (3), you can say T = AAAA and from(4) T = BBBB.
Now T = T or AAAA = BBBB. Now simplify this one equation. Multiplying (3) by cos and (4) by sin is to simplify the equation. You don't have to do it but you should.
I get a little confused here too. Beacause, Tsen(theta) in equation (3) is a vector in the î direction, and Tcos(theta) is a vector in the j direction. I don't get how summing these vectors, even when multiplying by any number, could be equal to zero. But regardless that, it was na awesome explanation! Thank you
I think we have to do:
(3)*cos*j and (4)*sin*i (**)
if (3) is in terms of i , (4) is terms of j and ixj = k.. so (**) is in terms of k
Your Video helps me a lot!
This was brilliant!
I will just say Its perfectly perfect. The best experience in learning inverted pendulum, by the way where is second part?
thank you very much. it is very useful for us .
Very nice video. Thanks a lot!
BROTHER...YOU ARE JUST GOD !!!!!!
Fantastic explanation thank you Sir ❤
After approximating sin(θ) = θ, cos(θ) = 1, and θ·θ'^2 = 0, where the prime indicates first derivative, and after substituting equation (5) in equation (6) and taking the Laplace transform, would the transfer function be Θ(s)/F(s) = [1/(Mc·L)]/[s^2 - g·(Mc·Mp)/(Mc·L)]?
should be "g(Mc+Mp)" not "g(Mc*Mp)" but yeah, thats the TF
This video wins BIGLY
Very helpful Thank you.
You're welcome.
Thank You very much.
NOTE: The direction of tension forces in the free body diagrams are drawn in the wrong direction. It should've been the opposite. Because otherwise, assuming no control force, we would expect the cart to accelerate upwards which doesn't make sense.
great video!
but why is the angle formula -cos(x)-sin(x) for the tangential acceleration? I mean why minus
8 yrs and still no reply
Thank you soooooooo muuuuuucccchhhh
Thank so much for this explanation!!! You saved me! hahaha I would never wonder to multiply those equations by sin and cossine in order to eliminate the tension T, hahaha. Great video!
What if I write Newton's second law for rotation of pendulum... Can you pls explain that
Excellent!!!!
Thank you so much .. i was looking for it...can you do a video on differential drive movile robot ..forming state space equations..
Excellent!
If I had a rod with a considerable mass and a moment of inertia, should I make a free body diagram for that rod and for the ball separately, or the forces due to the ball can be considered into the rod's free body diagram? Thanks!
Where can I go to help me substitute the F (force) term for the motor properties from the wheel motors. Planning to use a two wheeled cart, wheels adjacent to each other...
How do I convert from e_theta and e_r to i and j? I sort of get what's happening by looking at it, but not in any concrete way
we considered centripetal force when calculating the acceleration of the mass relative to cart but should not that be balanced by centrifugal acceleration? and can you suggest me any piece of reference to learn more
Why is the tension acting on the cart negative in the x-direction?
Because of the way he has the FBD of the cart drawn. Also, something in tension always pulls away from it's ends...
Nice video, good explanation. What software or library did you use to do the simulation of the system?
Working Model
Hi, thank you for your great video. If you could mention some of your reference used, it would be great. Thank
Thank you!
can we replace the x with lsin(theta) and have one degree of liberty?
how can i solve a question with lagrangian method when piont of support of a pendulum is oscillating horizontally
In practical when the cart moves left the pendulum moves right, so keeping this in mind the ac in the equation ap = ac + ap/c should be negative of the acceleration of cart?
Correct me if I am wrong?
Because I compared your answers with a text I was reading on this topic and there was a slight difference in the final answer due to this reason.
The equations in the video are correct.
hello, Brianno Coller thx, what are the changes for two wheels inverted pendulum? for example radius of the wheels etc....
what about the moment of inertia of the pendulum?
In this derivation, I modeled the pendulum as a point mass.... so it did not have a moment of inertia. It could be added without much difficulty.
@@bdcoller iam making a self balancing robot and im treating it as this pendulum on a cart example, can you please tell me how to calculate the moment of inertia for something like that ?
@@bdcoller What would be its inertia, then? And where would it be added in the equations?
Why we use not the Friction power on car
can anyone explain the accelection about the cirle e theta part?
You can read the section 11.5B ("Radial and Transverse Components") of the chapter 11 ("Kinematics of particles") of the textbook "Vector mechanics for engineers: Statics and dynamics" (12th edition), by Beer, Johnston, Cornwell & Self. They deduce the same equation in Eq. 11.46 of the textbook.
@@altuber99_athlete thank you so much.I'll refer the book you suggested.👍
You can support your code is not? thanks you!
Thanks for the video! I'm a bit unsure why you only showed tension in the rod. Shouldn't the rod also have a shear force perpendicular to tension? Otherwise it's just a string??
strings only support a tension force whereas rods support compression and tension. when the pendulum is above the horizontal the rod is in compression so the string would bend
nice video sir
what mean " Lθ double dot " and " L θ dot square "? They are simple variables-?
Are they******
And What mean the " L "?
What would be the angle needed, given the acceleration "a" of the linear moving part, so that the pendulum would be at rest?
forteleaerieneromane assuming that the translating part is moving infinitely to either left or right
why your final equations different from wikipedia?
can we not use an angle sensor to give us the angle of the rod. such as mpu6050
Yes. I'm not sure what your point is. Equations of motion remain the same.
If I want to make an autonomous inverted pendulum, using a microcontroller and control system, So will I need an angle sensor to find the angle of the rod, or just only these equations will be enough? thank you
in a quadcopter which has 4 motors,. an angle sensor is used that gives the angle of the quadcopter and the quacopter then sets different speeds of the 4 motors to keep the quad level.
@@Spectamin The equations by themselves are not enough to control the system. To balance the pendulum, you would need some sort of feedback. An encoder (angle sensor) or an IMU would be very valuable .
Thanks a lot
sir please make more vedio on equation of motion via Newton's second law
Hey, great video. I'm actually trying to model such a pendulum for a project. Could you please refer me to a video, or a website that explains why the acceleration of the pendulum is the acceleration of the cart + the acceleration of the pendulum relative to the cart?
Thanks!
+Denis Kartachov www.spumone.org/courses/dynamics-notes/rigid_body_kinematics/
Yes, that equation is correct. I just checked the textbook by Young, Freedman, Sears and Zemansky. They say that equation is valid only for Newtonian mechanics, but fails for objects traveling at near the speed of light (of course, neither the car nor the pendulum will ever move that fast.)
Why can we add (3)cos(theta)+(2)sin(theta) together?
Do you mean (3) cos(theta) + (4) sin(theta), like that shown at 19:00? If so, then it is important to know what this means. The (3) refers to equation (3) and the (4) refers to equation (4). So (3) cos(theta) means that I'm taking every term on both sides of equation (3) and multiplying by cos(theta). When I do so, I still have an equality (i.e. the stuff on the left side of the equation EQUALS the stuff on the right side. Similarly for (4) sin(theta). Then we simply take these two equalities and add them together: add the left sides of the two equations; add the right sides; the two sums should be equal. That's it.
On equation (5) (at 22:33) shouldn't it be -2MpL(theta double dot)?
Are you suggesting that I'm missing a 2? There should not be a 2. This is because of the trig identity: cos^2 + sin^2 = 1. It is described at 19:00.
can please tell me why you take centripetal acceleration outward .As much as i know it always towards center of circle. Or it is a centrifugal force . pls reply
between very nice video
It is not outward. In the video, the centripetal acceleration is written as -L thetaDot^2 e_r. Notice that it's in the -e_r direction... so it's inward.
Brianno Coller ok i got it
Why didn't apply moment of inertia? 🤔
It's because I treat the pendulum as a point mass on the end of a massless rod.
in 10:51 why is it minus sign?, shouldn't it be a plus sign? Help please My exam is on Thursday, November 29th.
It's because that term is a centripetal acceleration. Centripetal acceleration is toward the center. The basis vector e_r is outward. Therefore, the centripetal acceleration is in the minus e_r direction.
Ohh I see. Right, thank you so much, Brianno! God bless you! Regards from Innopolis, Tatarstan, Russia. Wednesday, November 28th, 2018.
is very good
oh man! beautifully explained. :-)
cool
Hey i neeed to take a laplace transform of the final equation (at 22:41) but I don't understand how can I take L transform for the MpL(theta')^2 * theta. as it has a derivative and the independent variable multiplied together.
PLEASE HELP ASAP.
you have to linearize before taking the Laplace transform.
+Brianno Coller rekt
Laplace transformation method is not applicable for nonlinear system as the case with this inverted pendulum cart system. If you really wanna do that, then you have to impose some assumptions so that the system is considered as a linear system. The physical meaning behind this notion is that when you start the experiment, the pendulum must be placed very close to the vertical position and there are no external forces that deviate it from that position, therefore, it is safe to assume that con(theta) = 1 and sin(theta) = theta.
Tqm
Beauty
Brianno rules
17:06 lmao