Energy-conserving intermittent-contact motion in complex models

TL;DR

This work analyzes collisionless, energy-conserving intermittent-contact motion in $N$-DOF linear systems with ground contacts, extending prior $N=2$ analyses to the harmonic regime by linking collisionless solutions to the normal-mode spectrum with frequencies $\omega_i=\sqrt{\lambda_i}$. It derives an existence condition requiring at least one oscillatory mode in the most constrained phase and reduces the search for collisionless trajectories to solving two impact-time equations that depend only on the spectra $\lambda^p$. The authors illustrate the approach by solving a planar $N=3$ biped with an armed torso, demonstrating feasibility in higher-dimensional models and detailing asymptotic behavior in the critical region via determinant-based conditions and contour plots. This framework has potential to guide the design of energy-conserving locomotion and to inform extensions to multi-dimensional impacts and frictional effects in more complex systems.

Abstract

Some mechanical systems, that are modeled to have inelastic collisions, nonetheless possess energy-conserving intermittent-contact solutions, known as collisionless solutions. Such a solution, representing a persistent hopping or walking across a level ground, may be important for understanding animal locomotion or for designing efficient walking machines. So far, collisionless motion has been analytically studied in simple two degrees of freedom (DOF) systems, or in a system that decouples into 2-DOF subsystems in the harmonic approximation. In this paper we extend the consideration to a N-DOF system, recovering the known solutions as a special N = 2 case of the general formulation. We show that in the harmonic approximation the collisionless solution is determined by the spectrum of the system. We formulate a solution existence condition, which requires the presence of at least one oscillating normal mode in the most constrained phase of the motion. An application of the developed general framework is illustrated by finding a collisionless solution for a rocking motion of a biped with an armed standing torso.

Figures (6)

• Figure 1: Schematic depiction of the models considered in Section \ref{['n=2case']}, with the squares and circles representing masses. a) Hopping and b) juggling. c) Extended rimless wheel. Only three spikes are displayed, with the rest shown as dots. d) Coronal bipedal rocking. In c) and d) the white squares are rigidly affixed together, forming a single part. Each model is shown in both $P^p$ configurations, with $P$ configuration shown on the right in a-b) and on the left in c-d).
• Figure 2: The zero contour of $\phi(o_N,o'_{N-1})$ from Eq.(\ref{['phioo']}), $N = 4$. The spectrum is random and near critical with $|\lambda_{i}| = \mathcal{O}(1)$ and $\lambda'_{N-1} = 0.01$. The curves that run (predominantly) top-to-bottom and left-to-right represent the solutions of $\det{B_{(N)}} = 0$ and $\det{B_{(N+1)}} = 0$ respectively. Their intersections are the solutions of the impact equations. The cross symbols depict the large-$\tau$ solution derived in \ref{['critical']}, see Eq.(\ref{['largetauo']}).
• Figure 3: $N = 3$ biped with a standing armed torso. The legs are rigidly affixed together at an angle $2\theta$. Angles $x = [x_1; x_2; x_3]$ are measured relative to the static equilibrium configuration, shown on the right.
• Figure 4: The zero contour of $\phi(o_N,o'_{N-1})$ for $N = 3$ rocking motion. Physically realizable collisionless solutions are represented by the bottom row of curve intersections. We focus on the solution marked by a red circle.
• Figure 5: Complete collisionless solution, in terms of $x(t)$ (solid), $\dot{x}(t)$ (dashed) and $\ddot{x}(t)$ (dotted) plotted in the units of $\theta$, for the rocking motion of $N = 3$ armed biped, shown from $P$ (at $t = 0$) to $P'$ (at $t = \tau+\tau'$) separated by the impact (vertical gray line at $t = \tau$). Different colors are used for different components: red for $x_1$, green for $x_2$ and blue for $x_3$.