C.29.3:5.1 - Carry a calculated count through a finite command interface

A robot’s supplied motion model uses rolling without slip, effective wheel radius 0.05 m, ten motor revolutions per wheel revolution and one thousand commanded increments per motor revolution. Positive motor increments produce forward travel. Successful execution completes the requested relative increment.

For a requested distance d, calculate and round:

N = nearest integer to ((d / (2π × 0.05 m)) × 10 × 1000)
modeled displacement = N / 10000 × 2π × 0.05 m

One metre gives 31,831 increments and a modeled displacement of approximately 1.0000003576 m. Rounding contributes at most half an increment, approximately 0.0000157080 m. This bound concerns command discretization under the supplied motion relation.

The interface accepts signed 16-bit values from -32,768 to 32,767. The one-metre input is representable. Two metres require 63,662 increments, which cannot be prepared as one positive value in that field.

Suppose the interface wraps modulo 65,536 and interprets the result as signed. Then:

63,662 − 65,536 = -1,874
modeled displacement ≈ -0.0588734463 m

The calculation of the desired count remains correct; input preparation changes the delivered count. An interface that rejects the value would produce a different failure, so the actual interface behavior matters.

One repair uses two completed commands of 31,831. Under the relative-addition rule, their total is 63,662 and their modeled displacement is approximately 2.0000007151 m. The preparation now supplies two representable values, and the execution argument uses addition across their completed displacements.

If a command means “reach this absolute count,” repeating 31,831 leaves the same target. For an absolute interface, reconstruct the target from the initial count and check that the resulting target is representable. B.5.MPC supplies the wider comparison when the motion model or the count’s physical meaning also changes.

A range restriction is another useful repair. The largest positive relative command corresponds to approximately 1.0294056648 m under this model. Whether that restriction is adequate depends on the requested motion. A deadline can make two commands unsuitable even when their displacements add.

The result is a command procedure with a stated input range and composition rule. For actual travel, determine whether the commands were completed and use a measurement or physical relation that accounts for consequential slip and effective radius.