Collaborative Research: Ergodic Trajectories in Discrete Mechanics

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The objective of this research project is to develop efficient and robust numerical methods for searching and exploring an environment using a mechanical system, while ensuring that the average temporal coverage reproduces a prescribed spatial distribution. This is achieved by developing methods for controlling mechanical systems so that they are ergodic with respect to the given spatial distribution, and combining them with geometric structure-preserving numerical integrators which have good backward error properties, and preserve geometric invariants like the symplectic structure, energy, and momentum. Furthermore, these methods preserve the nonlinear structure of the configuration manifold, such as the Lie group or homogeneous space structure. The key technical goals include: (i) the development and analysis of structured integrators to accurately predict ergodic properties of a given system; (ii) the development of simulation-based optimization of system parameters, and controls to maximize efficiency of ergodic search; (iii) the generalization of these techniques to Lie groups and homogeneous spaces, enabling ergodic search for a rich class of robotic systems; (iv) experimental validation on realistic systems.These techniques will directly be applicable to a broad range of real-world industrial applications, including joint space exploration for robotic systems, fault detection in manufacturing, and optimal search, coverage, and information extraction for autonomous sensor networks. This industrial outreach will be facilitated by the release of public-domain software that will lower the barrier to adapting geometric numerical integration techniques in a variety of applications. Furthermore, we will engage in public outreach activities by teaming up with the Museum of Science and Industry in Chicago to develop an interactive exhibit demonstrating search algorithms for mechanical systems. These concrete applications serve to inspire high-school students (in particular, underrepresented minorities and women) to pursue STEM degrees, and this in turn will help to secure the long-term economic innovation and competitiveness of American industry.