PYTHON · NUMPY

1/T³ and 1/T

Fit the scaling exponent for two published energy models and show they disagree.

01Challenge

Sweep the motion duration and fit the exponent for two energy definitions in the literature: an I²R dissipation model and a linear integral-of-torque model. Report both exponents.

02Model

Izumi derived, in 1997, that dissipated energy scales as 1/T³ in operating time, with the optimal joint-angle shapes self-similar across durations — one solution time-rescales instead of being re-solved. A model linear in torque cannot produce that exponent; it gives roughly 1/T. Both models appear in peer-reviewed work. They are not competing estimates of one quantity, and the difference decides how much slowing down is believed to be worth.

03Guided practice

Run the sweep and read the two fitted exponents. Check the I²R result against Izumi's −3.

04Feedback

PASS WHEN You recovered two different exponents from one motion.

If your run fails, check:

The quadratic loss term is what produces the cubic law; a linear model structurally cannot.
Self-similarity is the practical consequence: solve once, rescale in time.
A paper claiming 'X% saving from slowing down' is uninterpretable without its energy model.

05Retrieve & space

Why does an I²R model give 1/T³ where a linear-|τ| model gives 1/T?
What does self-similarity of the optimal shape buy you computationally?
Two papers report different savings for the same slowdown. Name the first thing to check.

06Mastery & project

Given an energy model, predict its duration-scaling exponent before running anything.

Take a published energy-optimal trajectory result and identify which model it used. State the exponent it implies.

GO FURTHER · THE LIVE PLAYGROUND

Open the instrument ↗ — Two axes on a shared DC link. Move the meter, slide the phase, watch the number change.

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PYTHON · NUMPY

1/T³ and 1/T

Fit both exponents from a duration sweep.

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the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you