Classic Inverted Pendulum - Equations of Motion

Рет қаралды 153,014

9 жыл бұрын

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.

Пікірлер: 113

@theodoretsitsimis9973 8 жыл бұрын

Please make a 2nd part of this video explaining the control system for stabilizing the inverted pendulum! Very good work!!

@altuber99_athlete 4 жыл бұрын

Here it is: www.spumone.org/courses/control-notes/inverted-pendulum/ There are 4 more videos.

@sebascarra 9 жыл бұрын

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!

@osamaahmadieh4660 4 жыл бұрын

Man I love the love way you actually explain stuff, and not simply state equations! Fantastic explanation of accelerations!

@bdcoller 4 жыл бұрын

Glad to help

@clintathusaz 8 жыл бұрын

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!

@ahmetfurkankula337 4 жыл бұрын

This was what I have been trying to find for weeks. Thank you very much :)

@bdcoller 4 жыл бұрын

You're welcome

@senthilkumaranmahadevan4991 5 жыл бұрын

Great job. After a long time I understood inverted pendulum. Thanks a lot .

@ichoose111 6 жыл бұрын

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.

@alvarovargas2217 9 жыл бұрын

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?

@venkatyalamati3285 8 жыл бұрын

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..

@Rafisidqi 6 жыл бұрын

Hi, thank you for your great video. If you could mention some of your reference used, it would be great. Thank

@2994steven 5 ай бұрын

Is there a part two to this video taking the laplace transformation? that would be an absolute life saver right now

@ricardomiranda7737 7 жыл бұрын

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...

@pierluigimicarelli30 8 жыл бұрын

hello, Brianno Coller thx, what are the changes for two wheels inverted pendulum? for example radius of the wheels etc....

@laiafbin4495 5 жыл бұрын

Your Video helps me a lot!

@danteflores7596 7 жыл бұрын

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!

@youcefbourrai7556 4 жыл бұрын

Thank you! can we replace the x with lsin(theta) and have one degree of liberty?

@RajRoy-ok9uf 8 жыл бұрын

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

@fcvgarcia 8 ай бұрын

Thanks for the excellent explanation!!! I didn't notice the time passing by.

@bdcoller 8 ай бұрын

You're welcome.

@Mr-jq9xf 6 жыл бұрын

how can i solve a question with lagrangian method when piont of support of a pendulum is oscillating horizontally

@damianwiecaw492 2 жыл бұрын

Thank you! Best tutorial on yt

@rogerisanta 4 жыл бұрын

Very nice video. Thanks a lot!

@kalyan5968 3 жыл бұрын

Thank you so much .. i was looking for it...can you do a video on differential drive movile robot ..forming state space equations..

@ohmakademi 5 жыл бұрын

thank you very much. it is very useful for us .

@ertsberg 7 жыл бұрын

Nice video, good explanation. What software or library did you use to do the simulation of the system?

@bdcoller 7 жыл бұрын

Working Model

@evanparshall1323 3 жыл бұрын

Wow great video. Had a minor memory lapse on centripetal acceleration but thank you for the great explanation!

@lukescazzero8280 2 жыл бұрын

acceleration in the radial direction @10:50 should be divided by L, not multiplied. Right?

@vuthanhty1329 7 жыл бұрын

You can support your code is not? thanks you!

@DusanPavlicek78 10 ай бұрын

This was brilliant!

@matthewjames7513 6 жыл бұрын

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??

@nikaross7646 6 жыл бұрын

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

@TheShellify 9 жыл бұрын

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

@bandar1606 7 жыл бұрын

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.

@murilloprestesvilla9539 5 жыл бұрын

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

@shyennepinheiro4547 5 жыл бұрын

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

@MrGayAnalSexMan 3 жыл бұрын

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

@forteleaerieneromane 9 жыл бұрын

What would be the angle needed, given the acceleration "a" of the linear moving part, so that the pendulum would be at rest?

@forteleaerieneromane 9 жыл бұрын

forteleaerieneromane assuming that the translating part is moving infinitely to either left or right

@ricardomiranda7737 7 жыл бұрын

Excellent!

@shyennepinheiro4547 5 жыл бұрын

Excellent!!!!

@haroonkhan7217 4 жыл бұрын

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.

@bdcoller 4 жыл бұрын

The equations in the video are correct.

@RiaziMohandesi Жыл бұрын

Fantastic explanation thank you Sir ❤

@kayzoku1 8 жыл бұрын

great video! but why is the angle formula -cos(x)-sin(x) for the tangential acceleration? I mean why minus

@fatimaumer6835 2 ай бұрын

8 yrs and still no reply

@shantanubanerjeebackup_pho9184 5 жыл бұрын

sir please make more vedio on equation of motion via Newton's second law

@ahsannaseer9433 4 жыл бұрын

Thank You very much.

@usamaniaz2981 6 жыл бұрын

I will just say Its perfectly perfect. The best experience in learning inverted pendulum, by the way where is second part?

@m3tzeko 6 жыл бұрын

Why is the tension acting on the cart negative in the x-direction?

@ironpine5815 6 жыл бұрын

Because of the way he has the FBD of the cart drawn. Also, something in tension always pulls away from it's ends...

@deniskartachov4845 8 жыл бұрын

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!

@bdcoller 8 жыл бұрын

+Denis Kartachov www.spumone.org/courses/dynamics-notes/rigid_body_kinematics/

@altuber99_athlete 4 жыл бұрын

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.)

@mehmetali4626 2 ай бұрын

Very helpful Thank you.

@bdcoller 2 ай бұрын

You're welcome.

@altuber99_athlete 4 жыл бұрын

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)]?

@ethanboba5262 9 ай бұрын

should be "g(Mc+Mp)" not "g(Mc*Mp)" but yeah, thats the TF

@cyrusIIIII 7 жыл бұрын

why your final equations different from wikipedia?

@turkey343434 Жыл бұрын

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.

@hky9906 6 жыл бұрын

Thanks a lot

@jellederooij885 5 жыл бұрын

On equation (5) (at 22:33) shouldn't it be -2MpL(theta double dot)?

@bdcoller 5 жыл бұрын

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.

@prathmeshlonkar2387 4 жыл бұрын

BROTHER...YOU ARE JUST GOD !!!!!!

@saranashjain794 7 жыл бұрын

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

@bdcoller 7 жыл бұрын

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.

@saranashjain794 7 жыл бұрын

Brianno Coller ok i got it

@ravikiran5567 5 ай бұрын

What if I write Newton's second law for rotation of pendulum... Can you pls explain that

@mritunjaichandra2407 8 жыл бұрын

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.

@bdcoller 8 жыл бұрын

you have to linearize before taking the Laplace transform.

@drumbum7999 8 жыл бұрын

+Brianno Coller rekt

@bandar1606 7 жыл бұрын

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.

@jasonparker6138 3 жыл бұрын

This video wins BIGLY

@muhammadyaseen8536 5 жыл бұрын

nice video sir

@jfbarbosaboro 7 жыл бұрын

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!

@happykitty9074 7 ай бұрын

Thank you soooooooo muuuuuucccchhhh

@kimlazarovic5239 3 жыл бұрын

Why we use not the Friction power on car

@linkmaster959 5 жыл бұрын

Why can we add (3)cos(theta)+(2)sin(theta) together?

@bdcoller 5 жыл бұрын

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.

@ashwinchander1837 4 жыл бұрын

can anyone explain the accelection about the cirle e theta part?

@altuber99_athlete 4 жыл бұрын

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.

@ashwinchander1837 4 жыл бұрын

@@altuber99_athlete thank you so much.I'll refer the book you suggested.👍

@zak3037 8 жыл бұрын

is very good

@ignaciomora8317 4 жыл бұрын

what mean " Lθ double dot " and " L θ dot square "? They are simple variables-?

@ignaciomora8317 4 жыл бұрын

Are they****** And What mean the " L "?

@Spectamin 5 жыл бұрын

can we not use an angle sensor to give us the angle of the rod. such as mpu6050

@bdcoller 5 жыл бұрын

Yes. I'm not sure what your point is. Equations of motion remain the same.

@Spectamin 5 жыл бұрын

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

@Spectamin 5 жыл бұрын

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.

@bdcoller 5 жыл бұрын

@@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 .

@anjalir4971 2 жыл бұрын

cool

@shivamgoswami3624 5 жыл бұрын

oh man! beautifully explained. :-)

@robertoenriquechavezrodrig731 5 жыл бұрын

in 10:51 why is it minus sign?, shouldn't it be a plus sign? Help please My exam is on Thursday, November 29th.

@bdcoller 5 жыл бұрын

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.

@robertoenriquechavezrodrig731 5 жыл бұрын

Ohh I see. Right, thank you so much, Brianno! God bless you! Regards from Innopolis, Tatarstan, Russia. Wednesday, November 28th, 2018.

@sebaspalacio02 2 жыл бұрын

Tqm

@ISLAMguidanceful 4 жыл бұрын

what about the moment of inertia of the pendulum?

@bdcoller 4 жыл бұрын

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.

@ISLAMguidanceful 4 жыл бұрын

@@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 ?

@altuber99_athlete 4 жыл бұрын

@@bdcoller What would be its inertia, then? And where would it be added in the equations?