Correction count rises with relative ballistic endpoint error

What it is

In the submovement model, when an initial ballistic stroke ends far outside target tolerance, more corrective submovements are needed to converge. The important quantity is endpoint error relative to effective target width, not absolute pixel error alone. The same deviation needs more correction for a narrow than a broad target.

Why it happens

The fast stroke has endpoint variability. Visual feedback estimates residual error, and later small movements reduce it; where residual error is a larger proportion of tolerance, one correction cannot safely remove it, requiring more feedback cycles. Distance, device noise, gain, and target width all affect initial relative error.

This relationship has a more specific motor-control account behind it: the stochastic optimized submovement model treats the ballistic stroke as the outcome of a speed–accuracy trade-off — faster movement means more signal-dependent noise in the motor command, so endpoint variance rises with intended speed. Rather than aiming to land exactly on target, the system minimizes total movement time, tolerating a certain amount of ballistic-phase variance and leaving the remainder to be cleaned up by correction. Ballistic endpoint variability is therefore not a control failure; it is the deliberately chosen trade-off point, and the number of corrections that follow is a required consequence of that trade-off, not an incidental patch.

Studying it

Detect post-peak pauses, reversals, and small movements in trajectories; calculate relative distance from ballistic endpoint to hot zone, then model correction count, total time, and misses. Hold visual search stable while varying width and distance, and use effective rather than geometric width where needed.

A concrete pipeline: take the first peak in the velocity profile, then find the first point afterward where velocity drops near zero or reverses direction, and treat that as the end of the ballistic phase; divide its distance from the target centre by the target's half-width to get relative error; then count every subsequent local velocity peak that exceeds a set threshold (say, 5% of peak velocity) as one corrective submovement. Regressing relative error against correction count and total time tests the relationship directly, rather than relying on total reaction time alone.

Where it stops holding

Correction count is hard to identify from low-sampling or heavily smoothed logs; some users choose one slower initial movement rather than several corrections. Touch snapping, rejection filters, and automatic gain can hide physical error. The model is an average tendency, not an intent diagnosis for one trace.

Sampling rate is a hard constraint on whether this analysis works at all: corrective submovements typically last tens to a little over a hundred milliseconds, and logs sampled below roughly 100 Hz, or smoothed and predicted by OS-level pointer processing, will merge adjacent small corrections into one or drop them entirely — systematically undercounting corrections and weakening or erasing the relative-error relationship. Touch input has a floor mouse input does not: fingertip contact area imposes a few millimetres of physical error regardless of relative error, and that noise makes up a larger share of the budget on narrow targets, which is why this relationship tends to be weaker on touchscreens than with a mouse.

Related

• Same group: B1.16.1 Pointing usually comprises a fast ballistic stroke and slower corrections · B1.16.3 Proportional error reduction per correction yields a logarithmic time relation · B1.16.4 Enlarging a target primarily shortens correction, not the ballistic phase · B1.16.5 Without visual feedback, correction cannot occur and movement becomes ballistic
• Nearby: B1.01 Fitts's law · B1.05 Steering law
• Search terms: corrective submovement · endpoint error · effective width