Converting a Non-Linear equation to canonical form

I have an equation of Motion for a 2 link robotic arm such as:

EOM2 =
 
 a2*(m2*lc2^2 + l1*m2*cos(q2)*lc2 + I2) - v2*(- (-l1*lc2*m2*sin(q2))*q_dot1 - (-l1*lc2*m2*sin(q2))*q_dot2) + a1*(m2*l1^2 + 2*m2*cos(q2)*l1*lc2 + m1*lc1^2 + m2*lc2^2 + I1 + I2) + g*m2*(lc2*cos(q1 + q2) + l1*cos(q1)) - q_dot2*v1*l1*lc2*m2*sin(q2) + g*lc1*m1*cos(q1)
  

I want to convert the EOM into a canonical form such that EOM = Y.θ where Y consists of symbolic variables like a,v and q. Here, a includes a1, and a2, v includes v1 and v2, and q includes q1, q2,, q_dot1, q_dot2, and any function that includes q like sin(q1) or sin(q1 +q2), cos(q2) and so on.

My approach was to first do it by hand and thus my Answer is:

% Y(a,v,q) Regressor matrix
Y1_Matrix = [a(1), cos(q(2))*a(1), cos(q(2))*a(2), a(1) + a(2), a(2), sin(q(2))*v(1)*q_dot(2),...
    sin(q(2))*(q_dot(1)+q_dot(2))*v(2), cos(q(1)), cos(q(1)+q(2))];
Y2_Matrix = [0, 0, sin(q(2))*v(1)*q_dot(1)+a(1)*cos(q(2)), a(1) + a(2), a(1) + a(2), 0, 0, 0, cos(q(1)+q(2))];

Y_Matrix = [Y1_Matrix;Y2_Matrix]; % 2X9

% Theta is vector
Theta_Vector = [m2*l1^2 + m1*lc1^2 + I1 + I2;
         2*m2*l1*lc2;
         l1*m2*lc2;
         m2*lc2^2;
         I2;
         -m2*l1*lc2;
         -m2*l1*lc2;
         g*m2*l1+g*lc1*m1;
         g*m2*lc2];    % 9X1
     
EOM3 = Y_Matrix*Theta_Vector      

I also found that there is a `collect()` function one could use to collect coefficients of variables such as:

collect(EOM2, [a(1), a(2), v(1), v(2), q(1), q(2), cos(q(1) + q(2)), cos(q(1)), sin(q(1)), cos(q(2)), sin(q(2))])

But the above approach becomes tedious when dealing with higher order robotic arm in future !!!

Questions:

1. I wanted to see if there is a more elegant and faster approach to do so, specifically with anything related to q where I have to keep track of so many symbolic variables !!
2. Also can I convert the collected coefficients directly into a Y.θ matrix*vector form without it reqriting myself ?

                                                                                                                                a2*(m2*lc2^2 + I2) + a1*(m2*lc2^2 + l1*m2*cos(q2)*lc2 + I2) + q_dot1*v1*l1*lc2*m2*sin(q2) + g*lc2*m2*cos(q1 + q2)

[Walter Roberson]

look at the first example of odeFunction which shows a workflow for converting to matrix times variables using \

[Japnit Sethi]

I tried using it in the form below:

vars = [q; q_dot; a; v; r]
[M,F] = massMatrixForm(EOM2, vars)

But I am recieving an error of form !

[Walter Roberson]

Please post enough code to be able to reproduce the problem.

[Japnit Sethi]

The complete code is as follows:

%Define the number of degrees of freedom
n_dof = 2;

% Define the generalized coordinates and their derivatives
q = sym('q',[n_dof,1],'real');
q_dot = sym('q_dot',[n_dof,1],'real');
q_ddot = sym('q_ddot',[n_dof,1],'real');

% Define the mass of each link
m1 = sym('m1','positive');
m2 = sym('m2','positive');

% Define the gravitational acceleration
g = sym('g','positive');

% Define the moment of inertia for each link
I1 = sym('I1','positive');
I2 = sym('I2','positive');

% Define the length of each link
l1 = sym('l1','positive');
l2 = sym('l2','positive');

% Define the distance between a hinge a the link's center of mass
lc1 = sym('lc1','positive');
lc2 = sym('lc2','positive');

a = sym('a',[2,1],'real');
v = sym('v',[2,1],'real');
r = sym('r',[2,1],'real');

EOM2 =
 
 a2*(m2*lc2^2 + l1*m2*cos(q2)*lc2 + I2) - v2*(- (-l1*lc2*m2*sin(q2))*q_dot1 - (-l1*lc2*m2*sin(q2))*q_dot2) + a1*(m2*l1^2 + 2*m2*cos(q2)*l1*lc2 + m1*lc1^2 + m2*lc2^2 + I1 + I2) + g*m2*(lc2*cos(q1 + q2) + l1*cos(q1)) - q_dot2*v1*l1*lc2*m2*sin(q2) + g*lc1*m1*cos(q1)

 
 % Y(a,v,q) Regressor matrix
Y1_Matrix = [a(1), cos(q(2))*a(1), cos(q(2))*a(2), a(1) + a(2), a(2), sin(q(2))*v(1)*q_dot(2),...
    sin(q(2))*(q_dot(1)+q_dot(2))*v(2), cos(q(1)), cos(q(1)+q(2))];
Y2_Matrix = [0, 0, sin(q(2))*v(1)*q_dot(1)+a(1)*cos(q(2)), a(1) + a(2), a(1) + a(2), 0, 0, 0, cos(q(1)+q(2))];
Y_Matrix = [Y1_Matrix;Y2_Matrix]; % 2X9
% Theta is vector
Theta_Vector = [m2*l1^2 + m1*lc1^2 + I1 + I2;
         2*m2*l1*lc2;
         l1*m2*lc2;
         m2*lc2^2;
         I2;
         -m2*l1*lc2;
         -m2*l1*lc2;
         g*m2*l1+g*lc1*m1;
         g*m2*lc2];    % 9X1
     
EOM3 = Y_Matrix*Theta_Vector      

[Walter Roberson]

First, you had some problems with your code.

%Define the number of degrees of freedom
n_dof = 2;

% Define the generalized coordinates and their derivatives
q = sym('q',[n_dof,1],'real');
q_dot = sym('q_dot',[n_dof,1],'real');
q_ddot = sym('q_ddot',[n_dof,1],'real');

% Define the mass of each link
m1 = sym('m1','positive');
m2 = sym('m2','positive');

% Define the gravitational acceleration
g = sym('g','positive');

% Define the moment of inertia for each link
I1 = sym('I1','positive');
I2 = sym('I2','positive');

% Define the length of each link
l1 = sym('l1','positive');
l2 = sym('l2','positive');

% Define the distance between a hinge a the link's center of mass
lc1 = sym('lc1','positive');
lc2 = sym('lc2','positive');

a = sym('a',[2,1],'real');
v = sym('v',[2,1],'real');
r = sym('r',[2,1],'real');

EOM2 = a(2)*(m2*lc2^2 + l1*m2*cos(q(2))*lc2 + I2) - v(2)*(- (-l1*lc2*m2*sin(q(2)))*q_dot(1) - (-l1*lc2*m2*sin(q(2)))*q_dot(2)) + a(1)*(m2*l1^2 + 2*m2*cos(q(2))*l1*lc2 + m1*lc1^2 + m2*lc2^2 + I1 + I2) + g*m2*(lc2*cos(q(1) + q(2)) + l1*cos(q(1))) - q_dot(2)*v(1)*l1*lc2*m2*sin(q(2)) + g*lc1*m1*cos(q(1));

 
 % Y(a,v,q) Regressor matrix
Y1_Matrix = [a(1), cos(q(2))*a(1), cos(q(2))*a(2), a(1) + a(2), a(2), sin(q(2))*v(1)*q_dot(2),...
    sin(q(2))*(q_dot(1)+q_dot(2))*v(2), cos(q(1)), cos(q(1)+q(2))];
Y2_Matrix = [0, 0, sin(q(2))*v(1)*q_dot(1)+a(1)*cos(q(2)), a(1) + a(2), a(1) + a(2), 0, 0, 0, cos(q(1)+q(2))];
Y_Matrix = [Y1_Matrix;Y2_Matrix]; % 2X9
% Theta is vector
Theta_Vector = [m2*l1^2 + m1*lc1^2 + I1 + I2;
         2*m2*l1*lc2;
         l1*m2*lc2;
         m2*lc2^2;
         I2;
         -m2*l1*lc2;
         -m2*l1*lc2;
         g*m2*l1+g*lc1*m1;
         g*m2*lc2];    % 9X1
     
EOM3 = Y_Matrix*Theta_Vector      ;

Secondly: you cannot express EOM2 in the form . Your EOM2 is nonlinear in q. EOM2 has a residue of g*m2*(lc2*cos(q1 + q2) + l1*cos(q1)) + g*lc1*m1*cos(q1) which cannot be expressed as an expression times the product of variables that are present in vars.

[c,t] = coeffs(EOM2, vars)

[Japnit Sethi]

Eom2 can be expressed in the form of Y.θ If I write it down by hand as I already know that this is a non linear set of equations where Y consists of coefficients anything that contains q as well!! This includes cos() and sin() variables !! I already mentioned it when I first wrote my question !!

Which is why I posted this question to see if anyone has a solution to this problem 'the matlab' way instead of writing down the equation by hand !!

[Walter Roberson]

Sorry, I will need you to post a link to your definition of "canonical form" so that we can study it. The definition you are using does not appear to correspond to any "canonical form" that I have encountered.

[Japnit Sethi]

There is no such quote on quote my definition of canonical form ! If you are so tangled up to that to the terminology ! I wanted a matrix representation of the equation of motion such that regressor matrix has all the variables of the non linear equations described above !!

[Walter Roberson]

Perhaps it would help to work through an example.

 E = 5*q1 - 2*q1*q2 - 3

How would you write that as Y*theta where theta = [q1;q2] ?

[Japnit Sethi]

In that scenario I don't necessarily want to split the q1 and q2 independently, I want to split it into 2 matrices such that Y(contains all q terms) and Theta (remaining constants): Y =[q1, q1*q2, 1] Theta = [5; -2; -3]

For my original question, my constants are m1,m2, l1, l2, lc1, lc2,I1, I2 and g

[Walter Roberson]

Your Y for this example does not contain only symbolic variables in combination: it also contains 1. That was not a permitted option in your original formulation.

[theta, Y] = coeffs(EOM2, vars)