pysähtymismatka

Pysähtymismatka is a term used in some Finnish discussions of dynamical systems, robotics and control theory to describe the distance a system travels from a given state until it reaches a stopping state under a specified control policy. It is not a widely standardized term in Finnish technical literature, but it appears in niche texts and discussions that focus on the spatial aspect of convergence rather than purely temporal measures.

Etymology and scope: The word combines pysähtyä (to stop) and matka (distance or journey), yielding a concept that emphasizes how far the system moves before halting. It is often used in contexts where the trajectory length, rather than time to stop, is of interest—for example when evaluating efficiency of a controller or the smoothness of a motion plan.

Definition and formulation: Consider a discrete-time or continuous-time dynamical system with state x in a metric space. Under a given control policy, the system halts when its state reaches a predefined stopping set or satisfies a convergence criterion within a tolerance. The pysähtymismatka M is the accumulated spatial progress along the trajectory until halt, commonly defined as M = sum over steps of the norm of state transitions, or equivalently the length of the trajectory up to the stopping time. Depending on the application, the stopping criterion may involve position, velocity, or other features of the state.

Applications and relation to other concepts: Pysähtymismatka is used to assess path length in robotics, autonomous navigation, and optimization trajectories, offering a spatial counterpart to settling time or convergence time. It relates to, but is distinct from, braking distance, stopping distance, and traditional halting or convergence metrics.

See also: settling time, convergence, stopping distance, halting problem.