Persistent homology of coarse-grained state-space networks.

This work is dedicated to the topological analysis of complex transitional networks for dynamic state detection. Transitional networks are formed from time series data and they leverage graph theory tools to reveal information about the underlying dynamic system. However, traditional tools can fail to summarize the complex topology present in such graphs. In this work, we leverage persistent homology from topological data analysis to study the structure of these networks. We contrast dynamic state detection from time series using a coarse-grained state-space network (CGSSN) and topological data analysis (TDA) to two state of the art approaches: ordinal partition networks (OPNs) combined with TDA and the standard application of persistent homology to the time-delay embedding of the signal. We show that the CGSSN captures rich information about the dynamic state of the underlying dynamical system as evidenced by a significant improvement in dynamic state detection and noise robustness in comparison to OPNs. We also show that because the computational time of CGSSN is not linearly dependent on the signal's length, it is more computationally efficient than applying TDA to the time-delay embedding of the time series.

Low-cost double pendulum for high-quality data collection with open-source video tracking and analysis.

The double pendulum is a system that manifests fascinating non-linear behavior. This made it a popular tool in academic settings for illustrating the intricate response of a seemingly simple physical apparatus, or to validate tools for studying nonlinear phenomena. In addition, the double pendulum is also widely used in several modeling applications including robotics and human locomotion analysis. However, surprisingly, there is a lack of a thoroughly documented hardware that enables designing, building, and reliably tracking and collecting data from a double pendulum. This paper provides comprehensive documentation of a research quality bench top double pendulum. The contributions of our work include (1) providing detailed CAD drawings, part lists, and assembly instructions for building a low friction double pendulum. (2) A new tracking algorithm written in Python for tracking the position of both links of the double pendulum. This algorithm measures the angles of the links by examining each frame, and computes uncertainties in the measured angles by following several trackers on each link. Additionally, our tracking algorithm bypasses the data transmission difficulties caused by instrumenting the bottom link with physical sensors. (3) A derivation of the equations of motion of the actual physical system. (4) A description of the process (with provided Python code) for extracting the model parameters-e.g., damping-with error bounds from physical measurements.