How to use ODE solvers for a 2nd order differential equation with a Mass Matrix ?

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I would like to simulate a system which comes at the following form:
where the Mass Matrix (M) is a 2x2 matrix. I simulated the system by using the inverse of the M matrix (it is positive definite, so I can do that) but I would also like to simulate it by using the 'Mass' matrix option that can be provided to the ode solver. If I use the inverse of the matrix, then the system breaks down to 4 first order ODEs and it's beiing simulated. How can I break the system to first order ODEs and use the mass matrix option ? My issue is that as far as I know, ODE solvers solve first order ODEs. The q variable denotes angles of a 2-DOF robotic manipulator, so its dimension is 2x1.

Accepted Answer

Bjorn Gustavsson
Bjorn Gustavsson on 23 Nov 2020
As far as I understand all you have to is to do the same conversion from a system of second order ODEs to a twice as big system of first-order ODEs by introducing an angular velocity vector, this should give you a mass-matrix that has a 2-by-2 identity-matrix and your M in the upper left and lower right and zeros in the other quadrants:
[1 0 0 0
0 1 0 0
0 0 M11 M12
0 0 M21 M22]*[dq1dt;dq2dt;dv1dt;dv2dt] = [v1;v2;C(q,v)*v - G(q) + tau]
(in a very lazy formatting...)

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