Solve for Nonlinear State Space Equation

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Cason Cox 2022-12-6

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评论： Torsten 2022-12-7

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 =

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Cason Cox 2022-12-6

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 2022-12-7

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.

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