Equations of Motion for Inverted Pendulum

In summary, Kerry is working on a simulation for an inverted pendulum mounted on a cart using Lagrangian dynamics. They want to add viscous damping terms and a driving force to the equation, but are unsure if they can simply add these terms to the existing equation or if they need to re-derive it. After some re-arranging and substitution, Kerry believes they can add the terms to the \stackrel{..}{x} equation, but it may affect the accuracy of the \stackrel{..}{\theta} equation. They are seeking confirmation on their thought process.
  • #1
KLoux
176
1
I'm working on a simple simulation for an inverted pendulum mounted on a cart. I derived the equations of motion using Lagrangian dynamics, but I want to go back and add viscous damping terms to the linear motion of the cart, as well as the angular motion of the pendulum. I also want to add a driving force to the cart (my end goal is a non-linear controls simulation). Can I add these terms to the equation I initially derived, or do I need to re-derive the equations from the beginning? I think I know the answer to this, but I was hoping someone might have better advice - it's been a long time since I've done anything with Lagrange multipliers (or Lagrange dynamics at all, for that matter).

This is what I came up with for the conservative system: [tex]x[/tex] is the position of the cart, and [tex]\theta[/tex] is the angular position of the pendulum, with zero being straight up (vertical) from the cart. The cart has mass [tex]m_{c}[/tex] and the pendulum has mass [tex]m_{p}[/tex] and length [tex]l[/tex]. The below equation is a combination of the equations for both generalized coordinates.

[tex]\stackrel{..}{x}(m_{c} + m_{p} - cos(\theta)) + \stackrel{..}{\theta}(m_{p}l cos(\theta) - l/4) + sin(\theta)(g/2 - m_{p}l\stackrel{.}{\theta}^{2})=0[/tex]

My gut feeling is that I cannot simply add [tex]B_{c} \stackrel{.}{x} + B_{p} \stackrel{.}{\theta}[/tex] to the left hand side and set the equation equal to [tex]F(t)[/tex]...:frown: Am I wrong?

Thanks for your help!

-Kerry
 
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  • #2
OK, after some re-arranging and substitution, I get the following:

[tex]\stackrel{..}{x} = \frac{gtan(\theta)}{2} - \frac{l\stackrel{..}{\theta}}{4cos(\theta)}[/tex]
[tex]\stackrel{..}{\theta} = \frac{-2sin(\theta)(g(m_{c} + m_{p}) + 2m_{p}l\stackrel{.}{\theta}^{2}cos(\theta))}{l(4m_{p}cos^{2}(\theta) - m_{c} - m_{p})}[/tex]

I'm somewhat confident that I can add my damping and forcing terms to the [tex]\stackrel{..}{x}[/tex] equation, and the position of the cart will calculate correctly. I'm also somewhat confident that doing this will make the equation for [tex]\stackrel{..}{\theta}[/tex] incorrect, even if I don't add the rotational damping term to that equation. Similarly, adding the rotational damping will make the cart position calculation wrong. Since the pendulum position is affected by cart motion, the rotational damping term will also make the pendulum position calculation incorrect. Is this thought process right?

Thanks,

Kerry
 

1. How do equations of motion apply to an inverted pendulum?

Equations of motion describe the relationship between the position, velocity, and acceleration of a system. In the case of an inverted pendulum, these equations help us understand how the angle and position of the pendulum change over time.

2. What are the inputs and outputs of equations of motion for an inverted pendulum?

The inputs for equations of motion for an inverted pendulum are the initial conditions of the pendulum, such as its starting angle, velocity, and position. The outputs are the values for the angle, velocity, and position of the pendulum at any given time.

3. How are the equations of motion for an inverted pendulum derived?

The equations of motion for an inverted pendulum are derived using principles of physics, such as Newton's laws of motion and conservation of energy. These equations take into account the forces acting on the pendulum, such as gravity and friction.

4. Can the equations of motion be used to predict the behavior of an inverted pendulum?

Yes, the equations of motion can be used to predict the future behavior of an inverted pendulum. By plugging in different initial conditions or changing the parameters of the system, we can see how the pendulum will behave over time.

5. Are there any limitations to using equations of motion for an inverted pendulum?

Equations of motion for an inverted pendulum are based on simplified models and assumptions, so there may be limitations in accurately predicting the behavior of a real-life pendulum. Factors such as air resistance and imperfections in the pendulum's structure can also affect its motion.

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