PYTHON · NUMPY

Angle is the integral of angular velocity, so say so in the loss

Take the one result in the review where a physical prior on the transition moved real-robot success, recover the trial count behind its two percentages, and test the effect against the resolution of that counter.

01Challenge

Reconstruct the smallest trial count consistent with 82.2 percent and 94.4 percent, express the improvement in rollouts rather than percentage points, run an exact test on the resulting table, and compare the effect against the review's own variance term.

Predict firstA real-robot success rate is reported as moving from 82.2 percent to 94.4 percent. What is the smallest number of trials that can produce both of those figures?

02Model

A network that predicts a joint angle and an angular velocity as two independent outputs can be wrong in a way physics forbids: the angle it predicts need not be the integral of the velocity it predicts. Sato and colleagues, writing four pages in the Journal of the Robotics Society of Japan, volume 43 number 6, pages 603 to 606, impose that relation as a loss term inside an imitation-learning model built on bilateral control. Real-robot success moves from 82.2 percent to 94.4 percent, on 15 demonstrations for one task and 39 for the other, at 10 rollouts per distance including an untrained 24 cm. The setting is worth knowing because it is unlike the English-language default: the CRANE-X7 arm carries no torque sensor, disturbance torque comes from an observer and reaction torque from a second one, on a 500 Hz control loop, and the school calls this work deep predictive learning rather than world modelling, which is part of why an English query does not reach it. Two things then need checking, and this lab does both. Twelve point two percentage points at that trial count is 11 extra successful rollouts, so the effect is about 11 times the resolution of its own counter, which the appearance-randomisation tax elsewhere in the review is not. And the review's own variance term, a real-robot success rate moving across the full 0 to 100 percent range with task, props and model held constant, is 8.2 times this result, so the number is clean inside one laboratory and does not yet travel to another.

03Guided practice

1. See what the loss buys. Take one smooth joint trajectory, give two predictors the same noise budget, and spend it two ways: independently on each channel, or on velocity alone with the angle integrated from it. Print per-channel RMSE beside the consistency residual, which is the worst gap between the predicted angle and the integral of the predicted velocity. Note which predictor per-channel error prefers.

2. Recover the counter. Search every denominator up to 200 for one that renders both 82.2 percent and 94.4 percent to one decimal place. Take the smallest and print the two success counts.

3. Restate the result in rollouts. Divide 100 by that trial count to get the resolution of one rollout, then express the 12.2 point gain as a number of rollouts. This is the sentence you will quote when someone asks how big the effect is.

4. Apply the same test to a second result. The appearance-randomisation ladder in the same review reports a 5.34 point tax at 20 real episodes per task. Compute its effect-to-resolution ratio and compare. One of the two survives.

5. Put the ceiling on it. Divide the review's 0 to 100 point variance band by your 12.2 point effect. Report the exact test and that ratio in the same breath, because a result that is significant within a laboratory and smaller than the between-laboratory band is a real finding with a stated scope.

04Feedback

PASS WHEN PASS when your run prints 90 as the smallest trial count consistent with both percentages, 74/90 and 85/90 as the two success counts, 11 extra successful rollouts, an effect-to-resolution ratio of 10.98x against 1.07x for the randomisation tax, a Fisher exact two-sided p of 0.01848, and a variance band 8.2x the size of the 12.2 point effect.

If your run fails, check:

If no denominator satisfies both percentages, check that you are rounding to one decimal place before comparing. 74/90 is 82.222 percent and 85/90 is 94.444 percent.
The search returns 90, 107, 180 and 197. Take the smallest, and note that 90 is 9 distances at the review's stated 10 rollouts per distance.
Fisher's exact test needs no third-party library. math.comb gives the hypergeometric weights, and the two-sided p is the sum of every table at most as probable as the observed one.
If your consistency residual is zero for both predictors, you have integrated the true velocity rather than the predicted one.

05Retrieve & space

State the physical identity this loss term imposes on the two predicted channels.
What did real-robot success move from and to, and on how many demonstrations for the shorter of the two tasks?
At 90 rollouts, how many extra successful rollouts is the whole 12.2 point result?
What is the effect-to-resolution ratio for this result, and for the appearance-randomisation tax in the same review?
Which measured variance term in the review is larger than this entire effect, and what holds constant while it moves?

06Mastery & project

You can turn a reported percentage back into a trial count, compare an effect against the resolution of the counter that produced it, and state the scope a result holds inside rather than repeating its headline.

Add one term to a policy loss you have written that costs no new labels because it states a physical identity the outputs already have to satisfy, integration of a rate being the plainest one. Report three numbers separately: the residual before, the residual after, and the task effect, with the trial count beside the task effect.

GO FURTHER · THE LIVE PLAYGROUND

Sato et al., JRSJ 43(6):603-606 ↗ · The four-page open-access letter itself, in Japanese: imitation learning on bilateral control with the angle and angular-velocity relation imposed as a loss.
TR-2026-35, section 4 ↗ · The three Japanese letters that take the transition itself as subject, and the vocabulary finding that deep predictive learning does not translate as world model.
Stiffness inside the predicted next state, JRSJ 44(4):433-436 ↗ · The companion result: a scalar stiffness emitted inside a predicted t plus 20 ms next state, discriminating 15 from 10 Nm per rad from five demonstrations per bottle.

← A world model can look right and move wrong Four bands and a missing axis →

PYTHON · NUMPY

Angle is the integral of angular velocity, so say so in the loss

One constructed illustration of the loss term, then arithmetic on the review's own two percentages. Standard library only.

You read

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