The experimental multi-arm pendulum on a cart: A benchmark system for chaos, learning, and control.

The single, double, and triple pendulum has served as an illustrative experimental benchmark system for scientists to study dynamical behavior for more than four centuries. The pendulum system exhibits a wide range of interesting behaviors, from simple harmonic motion in the single pendulum to chaotic dynamics in multi-arm pendulums. Under forcing, even the single pendulum may exhibit chaos, providing a simple example of a damped-driven system. All multi-armed pendulums are characterized by the existence of index-one saddle points, which mediate the transport of trajectories in the system, providing a simple mechanical analog of various complex transport phenomena, from biolocomotion to transport within the solar system. Further, pendulum systems have long been used to design and test both linear and nonlinear control strategies, with the addition of more arms making the problem more challenging. In this work, we provide extensive designs for the construction and operation of a high-performance, multi-link pendulum on a cart system. Although many experimental setups have been built to study the behavior of pendulum systems, such an extensive documentation on the design, construction, and operation is missing from the literature. The resulting experimental system is highly flexible, enabling a wide range of benchmark problems in dynamical systems modeling, system identification and learning, and control. To promote reproducible research, we have made our entire system open-source, including 3D CAD drawings, basic tutorial code, and data. Moreover, we discuss the possibility of extending our system capability to be operated remotely, enabling researchers all around the world to use it, thus increasing access.

SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics.

Accurately modelling the nonlinear dynamics of a system from measurement data is a challenging yet vital topic. The sparse identification of nonlinear dynamics (SINDy) algorithm is one approach to discover dynamical systems models from data. Although extensions have been developed to identify implicit dynamics, or dynamics described by rational functions, these extensions are extremely sensitive to noise. In this work, we develop SINDy-PI (parallel, implicit), a robust variant of the SINDy algorithm to identify implicit dynamics and rational nonlinearities. The SINDy-PI framework includes multiple optimization algorithms and a principled approach to model selection. We demonstrate the ability of this algorithm to learn implicit ordinary and partial differential equations and conservation laws from limited and noisy data. In particular, we show that the proposed approach is several orders of magnitude more noise robust than previous approaches, and may be used to identify a class of ODE and PDE dynamics that were previously unattainable with SINDy, including for the double pendulum dynamics and simplified model for the Belousov-Zhabotinsky (BZ) reaction.