Error using dsolve for three second-order differential equations

I am trying to use dsolve to solve three differential equations of motion. They are all second order differential equations.

However, I am receiving the following error message:

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Error using mupadengine/feval_internal
No differential equations found. Specify differential equations by using symbolic functions.
Error in dsolve>mupadDsolve (line 341)
T = feval_internal(symengine,'symobj::dsolve',sys,x,options);

Error in dsolve (line 210)
sol = mupadDsolve(args, options);

Error in basicEOM (line 65)
S = dsolve(odes,conds)

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My full code is as follows:

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clc,clear


% 2 dimensional equations of motion for a quadcopter/drone
% neglecting air resistance, wind, other fluid dynamic implications, thermal implications, and complex geometric
% features
syms x(t) y(t) theta(t) thetaDot(t) thetaDDot(t) xDot(t) xDDot(t) yDot(t) yDDot(t) %x y and theta are the generalized coordinates. Three DOF system
g = -9.81; % accel due to gravity on Earth.
% alpha is the angle of the arm w.r.t. the body from the horizontal on the
% body
%beta is the angle between the two arms above the body --> 180 - 2*alpha
%Fl and Fr are the left and right thrust forces
% M is the mass of the body and I2 is the MOI of the body
% m is the mass of one arm and I1 is the MOI of one arm. L is the length of
% the arms
% assigning some basic values:
alpha = 25; %degrees
beta = 180 - 2*alpha;
Fl = 1;
Fr = 1;
M = 2;
m = 1;
I1 = 1;
I2 = 1;
L = 0.3;




thetaDot(t) == diff(theta(t),t);
thetaDDot(t) == diff(theta(t),t,2);
xDot == diff(x(t),t);
xDDot(t) == diff(x(t),t,2);
yDot(t) == diff(y(t),t);
yDDot(t) == diff(y(t),t,2); 

%equations of motion derived from langrage's equations:
ode1 = (I2 + 2*I1)*thetaDDot(t) - m*g*L/2 *(cosd(alpha+theta(t))) + cosd(beta+theta(t)+alpha) - (Fr-Fl)*cosd(alpha) == 0;
ode2 = (M+2*m)*xDDot(t) - (Fr+Fl)*sind(theta(t)) == 0;
ode3 = yDDot(t) + g - (Fr+Fl)*cosd(theta(t))/(M+2*m) == 0;

odes = [ode1;ode2;ode3];
%initial conditions
cond1 = theta(0) == 0;
cond2 = x(0) == 0;
cond3 = y(0) == 0;
cond4 = thetaDot(0) == 0;
cond5 = xDot(0) == 0;
cond6 = yDot(0) == 0;


conds = [cond1 cond2 cond3 cond4 cond5 cond6];

S = dsolve(odes,conds)

Thank

Answer 1

[Ameer Hamza]

There are a few issues in your code. Compare the following code with your code to find the current syntax of symbolic declaring ODEs in MATLAB

clc,clear
% 2 dimensional equations of motion for a quadcopter/drone
% neglecting air resistance, wind, other fluid dynamic implications, thermal implications, and complex geometric
% features
syms x(t) y(t) theta(t) %x y and theta are the generalized coordinates. Three DOF system
g = -9.81; % accel due to gravity on Earth.
% alpha is the angle of the arm w.r.t. the body from the horizontal on the
% body
%beta is the angle between the two arms above the body --> 180 - 2*alpha
%Fl and Fr are the left and right thrust forces
% M is the mass of the body and I2 is the MOI of the body
% m is the mass of one arm and I1 is the MOI of one arm. L is the length of
% the arms
% assigning some basic values:
alpha = 25; %degrees
beta = 180 - 2*alpha;
Fl = 1;
Fr = 1;
M = 2;
m = 1;
I1 = 1;
I2 = 1;
L = 0.3;
thetaDot(t) = diff(theta(t),t);
thetaDDot(t) = diff(theta(t),t,2);
xDot(t) = diff(x(t),t);
xDDot(t) = diff(x(t),t,2);
yDot(t) = diff(y(t),t);
yDDot(t) = diff(y(t),t,2);
%equations of motion derived from langrage's equations:
ode1 = (I2 + 2*I1)*thetaDDot(t) - m*g*L/2 *(cosd(alpha+theta(t))) + cosd(beta+theta(t)+alpha) - (Fr-Fl)*cosd(alpha) == 0;
ode2 = (M+2*m)*xDDot(t) - (Fr+Fl)*sind(theta(t)) == 0;
ode3 = yDDot(t) + g - (Fr+Fl)*cosd(theta(t))/(M+2*m) == 0;
odes = [ode1;ode2;ode3];
%initial conditions
cond1 = theta(0) == 0;
cond2 = x(0) == 0;
cond3 = y(0) == 0;
cond4 = thetaDot(0) == 0;
cond5 = xDot(0) == 0;
cond6 = yDot(0) == 0;
conds = [cond1 cond2 cond3 cond4 cond5 cond6];
S = dsolve(odes,conds)

However, MATLAB symbolic engine is taking too much time to solve this equation. I eventually terminated the execution after some time without getting any answer. If time is an issue, I recommend using a numerical solver, e.g., ode45.