Romeo walking Results about consim HOT 4 CLOSED

I've managed to repeat the same test as above with a time step of 10 ms, and Exponential is still stable with this large time step!

With dt=10 ms Exponential gives roughly the same integration error of Euler with dt=1/4 ms. This means that we need 40 times more time steps with Euler than with Exponential.

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I've found the maximum time step for the controller to remain stable (it is 40 ms), and I've used it as basic time step for the simulator. Here the new results:

Results are good even with large time steps. However, it should be noticed that this doesn't mean the simulator would work with a time step of 40 ms. In these tests we are indeed measuring the local error, which means that every time step (40 ms) the state of the system is reset to the ground truth, so the simulator cannot diverge. If instead I measure the global error (i.e. I don't reset the state to the ground truth) I can see that even Exponential diverges for time steps >= 5ms:

However, for dt=2.5 ms Exponential is stable, while Euler is not. For dt<=1.25 ms both Exponential and Euler are stable, but Exponential performs better.

Anyway I think it's better to focus on the local error, because the global error is dominated by the collision checking errors, which are rather independent of the integration scheme, and so it tends to "hide" the benefits of Exponential over Euler.

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New results for the same test. Here I'm showing the local integration error as a function of the computation time (per time step, which is 40 ms in this test):

These results are excellent. Euler never beats Exponential. Using 0, 1 or 2 matrix multiplications in expm we get peak performance of Exponential.

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The previous test was using a contact stiffness of 1e5 and a contact damping of 1e2, which is underdamped. If we use instead a critically damped contact (i.e. damping 5e2) the results are a bit better for Euler:

Euler is never the best, but sometimes it's better than the standard Exponential integrator. It seems that Exponential is most beneficial for underdamped contacts, probably because it's able to predict the contact force oscillations much better than Euler, which simply assumes constant forces during the time step.

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