Inverted double pendulum on a cart

Hello all . I am trying to solve the inverted double pendulum on a cart problem. I have all the equations of motion for the problem. Since i am new to matlab i am not sure about how to go about the problem. I initialized the parameters for the problem. I am attaching the picture of the equations of motion in matrix form . I need to find theta double dot.

I am also attaching the code for your reference.

if true
  tspan=[0 5]; %time spane
  ic=[0:0.1:pi];%initial conditions

   % parameters
mo=2;
m1=1;
m2=1;
l1=0.5;
L1=1;
l2=0.5;
L2=1;
g=9.8;
I1=0.0126;
I2=0.0185;

%equations of motion
d1=mo+m1+m2;
d2=m1*l1+m2*L1;
d3=m2*l2;
d4=m1*l1*l1+m2*L2*L2+I1;
d5=m2*L1*l2;
d6=m2*l2*l2+I2;
d7=(m1*l1+m2*L1)*g;
d8=m2*l2*g;

 D=[d1 d2*cos(x1) d3*cos(x3); d2*cos(x1) d4 d5*cos(x1-x3); d3*cos(x2) d5*cos(x1-x3) d6]
 C=[0 -d2*sin(x1)*x2 -d3*sin(x2)*x4; 0 0 d5*sin(x1-x3)*x4; 0 d5*sin(x1-x3)*x2 0]
 G=[0;-d7*sin(x1);-d8*sin(x3)]
 H=[1 0 0].'

 I= inv(D);
 J=I*(-G+H);

 [t,x]= ode45(pend,tspan,ic);

 plot (t,x), 
 grid on
 xlabel('Time(s)');
 legend('theta1', 'theta2')

end

[Roqye Rostmi]

Hi I am working on this problem too. i need simulate these equations of motion in matlab simulink but i don't know how! The equations that i have :

[Nikola Stojiljkovic]

@Roqye Rostmi Hello, have you worked the problem out? I am working on a project and i would very much appritiate some help, as i am on a tight deadline...

Answer 1

[Meeshawn Marathe]

Please go through your code. The states chosen are not correct. You would have to convert the equation of motion to a set of first order equations. Let x be the cart displacement and theta1 and theta2 be the angular deviations of the rods from the upright position. Your state vector should look like:

y =  [x1 x2 theta1 theta3 theta2 theta4]',

where,

x2=d(x1)/dt,
theta3=d(theta1)/dt, 
theta4=d(theta2)/dt

Hence the state equation becomes dy/dt = f(y,u). Now d(theta3)/dt and d(theta4)/dt can be found out by rearranging the equation (2). Use an ODE solver (ex: ode45) and pass the arguments appropriately. The odefun will have 6 state equations based on dy/dt = f(y,u).

[Aravind Krishnamoorthi]

Thank you Meeshwan will try it out.