PYTHON · NUMPY

Does a longer horizon ever make deliberation worth it?

A deliberative planner costs up to 1440 times a one-dot-product law. Sweeping its horizon from 0.12 s to 1.20 s on a cart that relocates in about a second, when does it track better?

Sweep the rollout horizon on two bodies and establish whether deliberation ever beats a one-dot-product law.

01Challenge

The deliberative path plans ahead and costs up to 1440 times the cheap one. Before you sweep it: at which horizon does it finally track better than a tuned linear law?

Predict firstA deliberative planner costs up to 1440 times a one-dot-product law. Sweeping its horizon from 0.12 s to 1.20 s on a cart that relocates in about a second, when does it track better?

02Model

A plan is only worth making if it reaches far enough to matter. A cart that needs about a second to relocate cannot be helped by a plan that sees a tenth of one, because inside that window the cheap law has already made the same move. So the horizon must at least cover the body's own maneuver time, and that is a number you read off the mechanism. But clearing it is necessary rather than sufficient. On a near-linear plant a well-tuned linear law is already close to optimal, and there is simply no dividend left for lookahead to collect at any price.

03Guided practice

Run the sweep on both bodies. Compare tracking at each horizon against the cheap arm's tracking. Note the compute multiplier in the last column throughout.

04Feedback

PASS WHEN You gave deliberation every horizon it asked for and it never presented a bill worth paying.

If your run fails, check:

On the stiff body the deliberative arm tracks 0.512 at a 0.12 s horizon, 0.483 at 0.40 s and 0.383 at 1.20 s, against the cheap arm's 0.379. It closes the gap as the horizon grows and never overtakes, at 144, 480 and 1440 times the compute.
The shape is the lesson twice over. Below the body's maneuver time a slow path is an expensive fast path, and clearing that time is necessary rather than sufficient: on a near-linear plant a tuned linear law is already near optimal, so there is no dividend left to collect.
On the slack body at kp=0.30 it is worse on both counts at every horizon: 0.25 falls against 0.00, tracking 0.67 against 0.54. A plan built against a disturbance-free model cannot help a body falling for reasons the model does not contain.
The engineering conclusion is an ordering. Show your base scheme is the bottleneck before buying a second tier, because if it is not, no horizon will make the second tier pay.

05Retrieve & space

What sets the minimum useful horizon for a deliberative path?
Clearing the body's maneuver time is necessary for a dividend. Why is it not sufficient?
What should you establish about the base scheme before adding a deliberative tier?

06Mastery & project

Decide whether a body needs a deliberative tier at all, by establishing first whether its base scheme is the binding constraint.

Take a deployed dual-system policy with a published slow-path rate. Estimate the body's maneuver time constant and say whether the horizon clears it.