L0 · From physics to data: why inertia and damping matter

The equation has two competing effects:

Larger J resists changes in velocity. The same torque produces less acceleration.
Larger b removes more energy when the joint is moving. Velocity decays faster when the applied torque is small or zero.

The fit input is a chirp: its frequency changes over time. This gives the experiment both slower and faster motion, which helps separate an acceleration effect (J) from a velocity-dependent effect (b). The validation input is a different multisine, so it tests the fitted model under a motion it did not see during fitting.

Open the static figure or run the notebook. Read the four trajectory panels from left to right:

fit position and validation position;
fit velocity and validation velocity.

The black True-system curve is the observation target. The orange Initial-model curve is the uncalibrated baseline. The blue Identified-model curve is the fitted result. Look for acceleration differences in the velocity panels and accumulated position differences in the position panels. A small residual means the identified model reproduces that observed quantity for that experiment.

Try one change

In the Marimo lesson, change Initial-model damping while leaving the truth and input unchanged. The orange curve changes immediately; the blue result stays fixed and is marked stale. Press Run identification to fit again from the new starting values. A successful fit should still converge to the same Oracle values.

Compare the new starting prediction with the completed fit, then check validation. This shows both how the starting guess changes the initial error and whether fitting still reaches a useful result.

Checkpoint

Which observation would be most directly affected by increasing inertia: the instantaneous acceleration or the sign of the applied torque? The answer is acceleration. A parameter is useful only when the experiment makes its effect observable in the data.