Comparison of mjd_transitionFD on simple inverted pendulum model with MATLAB c2d

[dtorre38]

Hello,

I am interested in using mjd_transitionFD for my simulations, but I wanted to see how it compared with MATLAB's c2d. So I ran mjd_transitionFD and c2d on a simple inverted pendulum model. I used several of c2d's discretization methods and have tabulated the results below:

Math is according to https://en.wikipedia.org/wiki/Discretization

In comparison to MATLAB's ZOH method, mjd_transitionFD has some differences in the A matrix (A21) and the entire B matrix. I know that the MuJoCo API mentions that mjd_transitionFD produces "Finite differenced state-transition and control-transition matrices dx(t+h) = Adx(t) + Bdu(t)". I have also reviewed the humanoid standing balance, so I know that the A and B matrices which are produced are good enough for control, but if possible I wanted more insight into the inner workings of mjd_transitionFD.

-Daniel

[yuvaltassa]

1. I don't understand how you made this comparison. You're using a Matlab wrapper for MuJoCo, or you wrote your own dynamics and are hoping that they match MuJoCo's? My guess is that you did the latter and that there is a mismatch between the models.
2. The inner workings are right here.

[dtorre38]

I simply wrote my own dynamics and obtained all the values from the MJMODEL.txt file. Thank you for posting the link to the mjd_transitionFD code!

[yuvaltassa]

I would guess that your integrator is different from MuJoCo. The Euler integrator explained here, is not simple Euler. It is semi-implicit and has a fully implicit component (w.r.t joint damping).

[dtorre38]

I only ran mjd_transitionFD when initializing the controller in init_controller(), not for every integration time step. Similarly in MATLAB I did not run a simulation, I only converted the dynamics from continuous to discrete. For both I assumed the upright unstable fixed point. I wanted the initial set of values to get a sense of how the linearization performed in comparison to c2d.