Continuation of Periodic Orbits in Conservative Hybrid Dynamical Systems and its Application to Mechanical Systems with Impulsive Dynamics

TL;DR

The paper advances the understanding of periodic orbits in conservative hybrid dynamical systems by introducing a hybrid first integral and proving that periodic orbits are non-isolated within a Poincaré-map framework. It extends normal conservative orbit concepts to the hybrid setting and shows how embedding the system in a dissipative one-parameter family preserves the solution space, enabling robust numerical continuation. A key methodological contribution is the development of a time-based continuation approach that avoids grazing-related issues inherent to state-based formulations, while still enabling the construction of initial periodic orbits from equilibria or analytic grazing solutions. The results are demonstrated on four mechanical systems with impulsive dynamics (bouncing ball, rocking block, bouncing rod, and SLIP), highlighting the practical utility for designing and analyzing conservative gaits and locomotion strategies in legged mechanisms.

Abstract

In autonomous differential equations where a single first integral is present, periodic orbits are well-known to belong to one-parameter families, parameterized by the first integral's values. This paper shows that this characteristic extends to a broader class of conservative hybrid dynamical systems (cHDSs). We study periodic orbits of a cHDS, introducing the concept of a hybrid first integral to characterize conservation in these systems. Additionally, our work presents a methodology that utilizes numerical continuation methods to generate these periodic orbits, building upon the concept of normal periodic orbits introduced by Sepulchre and MacKay (1997). We specifically compare state-based and time-based implementations of an cHDS as an important application detail in generating periodic orbits. Furthermore, we showcase the continuation process using exemplary conservative mechanical systems with impulsive dynamics.

Figures (10)

• Figure 1: Graphical illustration of a recurrent hybrid trajectory of $\Sigma$ that starts and ends in the same phase $\mathcal{X}_1$. While all components of a HDS $\Sigma =\left(\mathcal{X},\mathcal{F},\mathcal{E},\mathcal{D}\right)$ are recurrent, e.g., $\boldsymbol{f}_{m+1}=\boldsymbol{f}_{1}$ and $\boldsymbol{\varphi}_{m+1}=\boldsymbol{\varphi}_{1}$, the transition states $\bar{\boldsymbol{x}}_{i}$ are trajectory specific. For this reason, the states $\bar{\boldsymbol{x}}_{m+1}$ and $\bar{\boldsymbol{x}}_{1}$ are different points in general.
• Figure 2: Graphical illustration of a recurrent hybrid trajectory in phase $\mathcal{X}_1$ that starts and ends on the Poincaré section $\mathcal{A}$.
• Figure 3: (a) illustrates a predictor-corrector step as described by \ref{['eq:predictor']} and \ref{['eq:corrector']}. (b) illustrates how a predictor-corrector step jumps over a simple bifurcation (SB) point $\boldsymbol{u}_\text{SB}$ and the tangent vectors $\boldsymbol{\tau}$ flip orientations. Both illustrations abbreviate tangent vectors by $\boldsymbol{\tau}^{i}:=\boldsymbol{\tau}\circ\mathrm{d}\boldsymbol{r}(\boldsymbol{u}^i)$.
• Figure 4: Shown are four conservative mechanical systems with impulsive dynamics due to lossless collisions with the ground: (a) bouncing ball, (b) rocking block, (c) bouncing rod and (d) one-legged hopper. The system's configuration variables are shown in blue.
• Figure 5: Key frames illustrating a normal conservative orbit of the bouncing ball and the rocking block. In the rocking block example, the system's symmetric motion is exploited, allowing us to compute only its pivoting motion about the left corner.

Theorems & Definitions (37)

• Definition : Recurrent Hybrid Trajectory
• Remark 1
• Definition : Recurrent Hybrid Flow
• Remark 2
• Remark 3
• Remark 4
• Definition : Conservative Hybrid Dynamical System
• Lemma 2.1
• proof
• Remark 5