Solve for Nonlinear State Space Equation

Trying to derive the Nonlinear steady state equation for a double pengelum. Solved for matrix [theta1;theta1dot;theta2;theta2dot]. Need in terms of [theta1dot;theta2doubledot;theta2dot;theta2doubledot]. Where theta1dot and theta2 dot would just be the variable but theta1doubledot and theta2doubledot would the values factored out of the respective equations.

solving for eqn 17 have matrix 18

output for equation 17 in terms of 20

clear; clc;

% Initalize variables
syms theta1(t) theta2(t) r1 r2 m1 m2 g u

% Initalize kinematic constraints
x1 = r1*sin(theta1);
y1 = -r1*cos(theta1);
x2 = x1+r2*sin(theta2+theta1);
y2 = y1-r2*cos(theta2+theta1);

% Initalize velocity equations
x1dot=diff(x1);
y1dot=diff(y1);
x2dot=diff(x2);
y2dot=diff(y2);

% Solve for potential energy
pe=m1*g*y1+m2*g*y2;

%Solve for kinetic energy
ke=1/2*m1*(x1dot^2+y1dot^2)+1/2*m2*(x2dot^2+y2dot^2);
simplify(ke);

% Solve for L=KE-PE
l=ke-pe;
simplify(l);

% Initalize variables
syms theta1dot theta2dot

% Solve for partial derivitive of L by theta1dot
eqn1=subs(diff(subs(l(t), diff(theta1(t)), theta1dot), theta1dot), theta1dot, diff(theta1(t)));
eqn1=simplify(eqn1);

% Solve for the time derivative of eqn1 
eqn1=diff(eqn1);
eqn1=simplify(eqn1);

% Solve for partial derivitive of L by theta1
a=simplify(diff(l,theta1));

% Solve for first equation of motion
eqn1=simplify(eqn1-a)-u

eqn1(t) = 

% Solve for partial derivitive of L by theta2dot
eqn2=subs(diff(subs(l(t), diff(theta2(t)), theta2dot), theta2dot), theta2dot, diff(theta2(t)));
eqn2=simplify(eqn2);

% Solve for the time derivative of eqn2 
eqn2=diff(eqn2);
eqn2=simplify(eqn2);

% Solve for partial derivitive of L by theta2
z=simplify(diff(l,theta2));

% Solve for second equation of motion
eqn2=simplify(eqn2-z)

eqn2(t) = 

[V,S] = odeToVectorField(subs(eqn1),subs(eqn2));
S=[S(3);S(4);S(1);S(2)]

S = 

V=[V(3);V(4);V(1);V(2)]

V = 

[Walter Roberson]

What is your question?

[Cason Cox]

How to you isolate theta1doubledot in eqn1 and theta2doubledot in eqn2?

[Walter Roberson]

I do not see any theta1doubledot in your code?

[Torsten]

I think

d^2/dt^2 theta1(t)
d^2/dt^2 theta2(t)

are meant.

But it's already solved for them after applying "odeToVectorField", isn't it ?

[Cason Cox]

Sorry, yes that is what I meant.

"odeToVectorField" solved for [theta1(t); d/dt theta1(t); theat2(t); d/dt theta2(t)] but I am trying to solve for the matrix with [d/dt theta1(t); d^2/dt^2 theta1(t); d/dt theta2(t); d^2/dt^2 theta2(t)]

[Torsten]

You are mistaken. "odeToVectorField" solved for d/dt theta1(t) , d^2/dt^2 theta1(t), d/dt theta2(t) , d^2/dt^2 theta2(t).

If it could solve theta1(t) and theta2(t), you wouldn't have to integrate the equations using an ODE integrator (the call to which is missing in your code) - you already had the solution.