# Complexity of Motion

I’ve done a ton of work on complexity of motion and gestures, some specifically applied to robotics, and a lot of it unpublished, which annoyed me, but I’m pleased that the paper I just got up and running (not quite final yet), Sorting, Information, and Recursion, includes an equation that is plainly the culmination of this work, and would allow you to measure the entropy of a sequence of motions, which can be found in Equations (1) and (2) of the paper.

Specifically, simply represent the velocity of a system over time as a sequence of velocity vectors $(v_1, \ldots, v_k)$, and if you can express the differences between all adjacent vectors $(v_i, v_{i+1})$ as either a real number or a real number vector, then you can use the equations in that paper to calculate an order dependent analog of entropy, that I discuss in some detail. It behaves exactly the way you’d want, which is if motions are highly volatile in sequence, you get a higher entropy, and if the velocity is constant, you get a zero entropy. I discuss this in more detail in the paper, and in particular, in Footnotes 5 and 6.

You can look at it three ways: (1) take the sequence $(v_1, \ldots, v_k)$ as it is observed, (2) sort it, which will minimize the entropy (see Corollary 3.1), or (3) apply another ordering that will maximize the entropy (see Footnote 8). These three measures tell you (1) what the real sequence entropy is, (2) what its theoretical minimum is, and (3) what its theoretical maximum is. This could be useful where you don’t have total control of the sequence itself, and instead can only set the individual velocities, giving you an objective criteria that would allow you to compare the complexities of two sets of motions. The lower the entropy, the smoother the motion should look to an observer, which is important not just for robotics, but also all modes of transportation, where people naturally feel frightened by sudden acceleration.