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Characterizing and addressing dynamic singularities in the time-optimal path parameterization algorithm

The algorithm for finding the time-optimal parameterization of a given path subject to dynamics constraints developed mostly in the 80's and 90's plays a central role in a number of important robotic theories and applications. A critical issue in its implementation is associated with the so-called dynamic singularities, i.e. the points where the maximum velocity curve is continuous but undifferentiable and where the minimum and maximum accelerations are not naturally defined. Since such singularities arise in most real-world problem instances, characterizing and addressing them appropriately is of particular interest. Yet, from original articles to reference textbooks, this has not yet been done completely correctly. The contribution of the present article is two-fold. First, we derive a complete characterization of dynamic singularities. In particular, we show that not all zero-inertia points are dynamically singular. Second, we suggest how to appropriately address these singularities. In particular, we derive the analytic expressions of the correct optimal backward and forward accelerations from such points.