Equation of Motion: Inverted Pendulum

  1. 1. The problem statement, all variables and given/known data
    For a controls class, I am to develop a simulation using AdamsView of a pendulum attached to a link arm and driven by a motor.

    The pendulum must start from rest and eventually be balanced at the top of the link arm.

    I am stuck at deriving the equations of motion right now. We are to use Langrangian for developing the equations.


    2. Relevant equations
    These are the energy equations for the system
    The kinetic energy of the entire device is:
    K=1/2IdO'[t]2+1/4(2(2Il+L2m)B'[t]2+4Lmr cos[B[t]]B'[t]O'[t]+(2Il+L2m+2mr2-(2Il+L2m)cos[2[B[t]])O'[t]2)

    The potential energy is:
    P=gLm cos[B[t]]

    The work of the motor is:
    W=TO[t]

    Paramaters(meters, kilograms, seconds)


    Centroidal moment of inertia of the inverted link Il=0.0000343
    Centroidal moment of inertia of the driver link Id=0.000687
    Distance from the driver axis to the hinge of the inverted link, r=0.1079
    Distance from the end to the centroid of the inverted link, L=0.16
    Mass of the driving bar (the horizontal one), m=0.127
    Mass of the inverted link, md=0.0249
    O=theta
    B=beta

    I've attached a diagram of this.


    3. The attempt at a solution
    Now will deriving with respect to thetas give the equation of motion for the driving link and then with respect to my betas I will get the equation for the inverted link?

    I am also not sure what part the work of the motor plays in this. After deriving the equations, will they both be set equal to the work of the motor?

    Any help would be greatly appreciated. Thanks
     

    Attached Files:

  2. jcsd
  3. Here is the basic formula for deriving equations of motion with the Lagrangian technique

    [​IMG]
    [​IMG]

    Where T is the kinetic energy and V is the potential energy of the system

    Now using Mathematica I was able to derive the following equations:

    1) Deriving with respect to theta' and theta
    [​IMG]

    2) Deriving with respect to beta' and beta
    [​IMG]

    Again, what I am not sure of is whether or not to add these two equations together and come up with a single equation of motion for the entire system or whether each belongs to one of the arms.
    Also, where does the motor come into play? What I am thinking is that this is the force the two motion equations are equal to.

    Any input is appreciate. Thanks.
     
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?