Properties of Lyapunov Subcenter Manifolds in Conservative Mechanical Systems

Abstract

Multi-body mechanical systems have rich internal dynamics, whose solutions can be exploited as efficient control targets. Yet, solutions non-trivially depend on system parameters, obscuring feasible properties for use as target trajectories. For periodic regulation tasks in robotics applications, we investigate properties of nonlinear normal modes (NNMs) collected in Lyapunov subcenter manifolds (LSMs) of conservative mechanical systems. Using a time-symmetry of conservative mechanical systems, we show that mild non-resonance conditions guarantee LSMs to be Eigenmanifolds, in which NNMs are guaranteed to oscillate between two points of zero velocity. We also prove the existence of a unique generator, which is a connected, 1D manifold that collects these points of zero velocity for a given Eigenmanifold. Furthermore, we show that an additional spatial symmetry provides LSMs with yet stronger properties of Rosenberg manifolds. Here all brake trajectories pass through a unique equilibrium configuration, which can be favorable for control applications. These theoretical results are numerically confirmed on two mechanical systems: a double pendulum and a 5-link pendulum.

Figures (9)

• Figure 1: Summary of the article: for conservative mechanical systems with configuration $x \in \mathcal{Q}$, momentum $P\in T^*_x \mathcal{Q}$, and projecting all data to $\mathcal{Q}$ for ease of visualization. Lyapunov subcenter manifolds (panels a and d) are families of general periodic oscillations $(x(t),P(t))$ springing from an equilibrium $(\bar{x},0)$. We prove conditions for LSMs to have stronger properties: Theorems \ref{['thm:LSM_mechanics_geometric']} and \ref{['thm:Eigenmanifold_Condition']} make it highly common for LSMs to become weak Eigenmanifolds (panels b and e) that collect periodic brake trajectories (oscillating between brake points $(x,0)$), and Eigenmanifolds that collect periodic brake trajectories whose configuration trajectory does not self-intersect (called geometric eigenmodes). Theorem \ref{['Theorem:1']} shows that yet stronger conditions turn LSMs into (weak) Rosenberg manifolds (panels c and f), where all modal configurations pass through $\bar{x}$. In both (weak) Eigenmanifolds and (weak) Rosenberg manifolds, brake points are collected on a connected 1D submanifold that we call the generator.
• Figure 2: Example of a weak geometric eigenmode in panel (a) and a $\varphi$-equivariant weak geometric Rosenberg mode in panel (b). Extending their non-weak counterparts, these allow for self-intersections in configuration space.
• Figure 3: Examples with $\dim(\mathcal{Q}) = 2$: the system of two masses in panel (a) has a Euclidean configuration space, $\mathbb{R}^2$ with a constant inertia tensor. The double pendulum in panel (b) has a non-Euclidean configuration space, $T^2$ with a non-zero curvature, where the associated inertia tensor is non-constant in any choice of coordinates.
• Figure 4: Equilibrium configurations of the three tested potentials: $V_{s1}$ with nonlinear terms from gravity and $V_{s2}$ with only quadratic spring terms are symmetric under $(S_2,\tau_2)$, in the appropriate coordinates. $V_a$ is obtained from $V_{s1}$ by shifting the equilibrium of the spring to $\pi/2$ such that $V_a(\bar{q} + q) \neq V_a(\bar{q} - q)$ for the minimum $\bar{q}$ of $V_a$, making $V_a$ not symmetric.
• Figure 5: Results of numerical continuation for different example systems generated by combining different potential functions and inertia tensors. The solid lines show computed generators of the system. The pair of blue and orange solid lines show the two generators associated to the first Eigenmanifold and the pair of purple and red the two associated to the second Eigenmanifold. The dashed blue line shows an example modal oscillation of the system for one energy. The conditions stated by Theorem \ref{['Theorem:1']} are satisfied for the cases (a,b) and (h,i).