# Equation of Motion of a Inverse Pendulum on a Cart

• silentwf
In summary, the conversation discusses the nonlinear and inherently unstable inverted pendulum and the goal of maintaining a specific angle using a feedback controller. The system includes a cart mass, a pendulum point mass, and a motor for a second input. The conversation also includes a question about deriving the equation of motion and using the x component to solve for the system. The expert summarizer advises that this approach is not sufficient and balanced, and that the entire system should be considered in deriving the nonlinear model and state-space equations.

## Homework Statement

The nonlinear, inherently unstable inverted pendulum is shown in
Figure 1.15. The goal is to maintain the pendulum angle θ(t) = 0
by using a feedback controller with a sensor (encoder or poten-
tiometer) for θ(t) and an actuator to produce an input force f (t).
The cart mass is m 1 , the pendulum point mass is m 2 , and we
assume that the pendulum rod is massless. There are two possi-
ble outputs, the pendulum angle θ(t) and the cart displacement
w(t). The classical inverted pendulum has only one input, the
force f (t). We will consider a second case, using a motor to
provide a second input τ(t) (not shown) at the rotary joint of
Figure 1.15. For both cases (they will be very similar), derive the
nonlinear model for this system, i.e., draw the free-body diagrams
and write the correct number of independent ordinary differential
equations.
(See attached file for picture)

## Homework Equations

$\Sigma F = ma$

## The Attempt at a Solution

So I only want to ask a concept question and not the whole solution. My question is in the derivation of the equation of motion. I referenced http://www.spumone.org/courses/control-notes/inverted-pendulum/ for an equation of motion, but I have one question. In the derivation of the equation, the author of the video combines the x and y components and solves for the entire system. My question is, can I solve only for the part X?
Here's what I did: (i used mb to signify mass of ball point, mc to signify mass of cart)
Looking at only the cart:
in the x direction $(m_{c}+m_{b})\ddot{w_{cart}}=f-T\sin{\theta}$

Looking only at the point mass:
in the x direction $T\cdot\sin{\theta}=f-m_{c}\cdot \ddot{w_{cart}} = -m_{b}L\ddot{\theta}\cos{\theta}-m_{b}L\dot{\theta}^{2}\sin{\theta}$

Since the Tension forces must equate on the point mass and on the cart, combining two equations give us:
$(m_{c}+m_{b})\ddot{w_{cart}}=f+ m_{b}L\ddot{\theta}\cos{\theta} + m_{b}L\dot{\theta}^{2}\sin{\theta}$

Now my question is, can I use this to derive my nonlinear model and use this as state space equations?
Also, in the case that I have theta as a function of cosine/sine, how would I express my state-space?

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sorry,I'm afraid that you can not.Since only X part is not enough to solve this problem.Besides it's not balanced

## 1. What is an inverse pendulum on a cart?

An inverse pendulum on a cart is a physical system that consists of a cart that can move horizontally on a track and a pendulum attached to the cart in an inverted position. This system is used to study the dynamics of systems with multiple degrees of freedom.

## 2. What is the equation of motion for an inverse pendulum on a cart?

The equation of motion for an inverse pendulum on a cart can be written as: m_c* x''(t) = F(t) - m_p* g* sin(theta) + m_p* l* theta''(t)* cos(theta) - b* x'(t) - d* theta'(t), where m_c is the mass of the cart, x(t) is the position of the cart, F(t) is the external force applied to the cart, m_p is the mass of the pendulum, g is the acceleration due to gravity, l is the length of the pendulum, theta(t) is the angle of the pendulum with respect to the vertical, b is the coefficient of friction between the cart and the track, and d is the damping coefficient of the pendulum.

## 3. What are the parameters that affect the motion of an inverse pendulum on a cart?

The motion of an inverse pendulum on a cart is affected by various parameters such as the mass of the cart and pendulum, the length of the pendulum, the external force applied to the cart, the coefficient of friction between the cart and the track, and the damping coefficient of the pendulum. Additionally, the initial conditions of the system, such as the initial position and velocity of the cart and the angle and angular velocity of the pendulum, also affect the motion.

## 4. How is the equation of motion derived for an inverse pendulum on a cart?

The equation of motion for an inverse pendulum on a cart can be derived using the principles of mechanics, specifically the Newton's second law of motion and the Euler-Lagrange equation. By considering the forces acting on the cart and pendulum, and the kinetic and potential energies of the system, the equation of motion can be obtained through mathematical analysis and calculations.

## 5. What is the significance of studying the equation of motion for an inverse pendulum on a cart?

The equation of motion for an inverse pendulum on a cart is a fundamental equation in the field of dynamics and control. It is used to understand the behavior of systems with multiple degrees of freedom and to design control systems for such systems. It is also relevant in various engineering applications, such as robotics, aerospace, and mechanical systems.