  Understanding the global dynamics of a robot controller, such as identifying
attractors and their regions of attraction (RoA), is important for safe
deployment and synthesizing more effective hybrid controllers. This paper
proposes a topological framework to analyze the global dynamics of robot
controllers, even data-driven ones, in an effective and explainable way. It
builds a combinatorial representation representing the underlying system's
state space and non-linear dynamics, which is summarized in a directed acyclic
graph, the Morse graph. The approach only probes the dynamics locally by
forward propagating short trajectories over a state-space discretization, which
needs to be a Lipschitz-continuous function. The framework is evaluated given
either numerical or data-driven controllers for classical robotic benchmarks.
It is compared against established analytical and recent machine learning
alternatives for estimating the RoAs of such controllers. It is shown to
outperform them in accuracy and efficiency. It also provides deeper insights as
it describes the global dynamics up to the discretization's resolution. This
allows to use the Morse graph to identify how to synthesize controllers to form
improved hybrid solutions or how to identify the physical limitations of a
robotic system.
