Coupled Matrix Differential Equations - Quaternions

Hi! I am working on a project for a class where I must recreate the results of a paper of my choosing that relates to the course. I have chosen this paper: https://www.tandfonline.com/doi/abs/10.1080/00207170500079779 (downloadble PDF)

Another link: https://doi.org/10.1080/00207170500079779

I am working on recreating the first graph shown in the paper (graph of qBarc(t)) and, to my understanding, I must solve a set of coupled matrix differential equations. I believe I am setting it up correctly but do not know why I keep getting errors. Here is the code I have written:

%% Numeric Formulas: Design Example 1

syms q1c(t) q2c(t) q3c(t) q4c(t)

% Parameter Values (rad/s)
a1=0.51;
a2=0.251;
a3=0.151;
omega1=0.182;
omega2=0.424;
omega3=0.827;

% Commanded Angular Velocity Vector (Rates)
w1c=a1*sin(omega1*t);
w2c=a2*sin(omega2*t);
w3c=a3*sin(omega3*t);
wc=[w1c;w2c;w3c];

% Command Rate Derivatives
wcDot=[a1*omega1*cos(omega1*t);a2*omega2*cos(omega2*t);a3*omega3*cos(omega3*t)];

% Skew Symmetric Matrices
omegac=-[0 -w3c w2c;w3c 0 -w1c;-w2c w1c 0];

% Defining Commanded Quaternions (non-constant in this example, need to find)
qc=[q1c(t);q2c(t);q3c(t)];
qBarc=[qc;q4c(t)];

% Initial Values
qBarc0=[0.2000;0.1000;0.4472;0.8660];

% Command Quaternion Kinematic Equations
qDotc=diff(qc);
q4Dotc=diff(q4c);
qBarcDot=[qDotc;q4Dotc];
ode1 = qDotc==0.5*omegac.*qc+0.5*q4c(t)*wc;
ode2 = q4Dotc==-0.5*wc'*qc;

qc0 =[qBarc0(1);qBarc0(2);qBarc0(3)];
qc40 = qBarc0(4);
conds=qBarc0;

odes = {ode1,ode2};
[qcSol(t),q4cSol(t)]=dsolve(odes,conds);

% ALTERNATIVE WAY USING ODE45
% ode1=@(t,qc)[qcDot;0.5*omegac.*qc+0.5*q4c(t)*wc];
% ode2=@(t,q4c)[q4Dotc;-0.5*wc'*qc];
% odes = {ode1,ode2};
 
%[qcSol(t),q4cSol(t)] = ode45(odes,[0 60],conds);

I am not sure what I am doing wrong and I have been looking into as much as I can but cannot seem to solve for qBarc(t). Please let me know if anyone has an idea how to solve for this I would greatly appreciate it!

[Torsten]

I doubt that the symbolic toolbox can handle a system of ordinary differential equations written out in the way you did (namely as matrix equation), e.g.

ode1 = qDotc==0.5*omegac.*qc+0.5*q4c(t)*wc;

My guess is that you will have to write the equations one by one.

[Erin Hayes]

Thank you! Turns out I did have to do that and then solve them as regular coupled differential equations, thanks for the tip!