When do Lyapunov Subcenter Manifolds become Eigenmanifolds?

Multi-body mechanical systems have rich internal dynamics, which can be exploited to formulate efficient control targets. For periodic regulation tasks in robotics applications, this motivated the extension of the theory on nonlinear normal modes to Riemannian manifolds, and led to the definition of Eigenmanifolds. This definition is geometric, which is advantageous for generality within robotics but also obscures the connection of Eigenmanifolds to a large body of results from the literature on nonlinear dynamics. We bridge this gap, showing that Eigenmanifolds are instances of Lyapunov subcenter manifolds (LSMs), and that their stronger geometric properties with respect to LSMs follow from a time-symmetry of conservative mechanical systems. This directly leads to local existence and uniqueness results for Eigenmanifolds. Furthermore, we show that an additional spatial symmetry provides Eigenmanifolds with yet stronger properties of Rosenberg manifolds, which can be favorable for control applications, and we present a sufficient condition for their existence and uniqueness. These theoretical results are numerically confirmed on two mechanical systems with a non-constant inertia tensor: a double pendulum and a 5-link pendulum.