Bifurcations in Latch-Mediated Spring Actuation (LaMSA) Systems

TL;DR

The paper addresses the fundamental question of what makes LaMSA systems impulsive by analyzing a constrained-lagrangian contact-latch model. It shows that energy mediation arises from moving saddle fixed points in the latched mode, and that varying the latch force $F_L$ induces a saddle-node bifurcation that eliminates latched equilibria and triggers impulsive takeoff. The authors derive necessary and sufficient saddle-node conditions, formulate a nominal fixed-point equation with implicit differentiation to track $p^*$ as a function of $F_L$, and validate the predictions through numerical simulations, highlighting how design parameters control impulse. This framework provides a rigorous mathematical basis for designing and controlling bio-inspired LaMSA mechanisms and sets the stage for extending the analysis to other latch types and applications.

Abstract

In nature, different species of smaller animals produce ultra-fast movements to aid in their locomotion or protect themselves against predators. These ultra-fast impulsive motions are possible, as often times, there exist a small latch in the organism that could hold the potential energy of the system, and once released, generate an impulsive motion. These types of systems are classified as Latch Mediated Spring Actuated (LaMSA) systems, a multi-dimensional, multi-mode hybrid system that switches between a latched and an unlatched state. The LaMSA mechanism has been studied extensively in the field of biology and is observed in a wide range of animal species, such as the mantis shrimp, grasshoppers, and trap-jaw ants. In recent years, research has been done in mathematically modeling the LaMSA behavior with physical implementations of the mechanism. A significant focus is given to mimicking the physiological behavior of the species and following an end-to-end trajectory of impulsive motion. This paper introduces a foundational analysis of the theoretical dynamics of the contact latch-based LaMSA mechanism. The authors answer the question on what makes these small-scale systems impulsive, with a focus on the intrinsic properties of the system using bifurcations. Necessary and sufficient conditions are derived for the existence of the saddle fixed points. The authors propose a mathematical explanation for mediating the latch when a saddle node exists, and the impulsive behavior after the bifurcation happens.

Figures (4)

• Figure 1: The contact latch-based LaMSA model (a) preloaded, (b) in latched mode, (c) unlatched: takeoff.
• Figure 2: Fixed points in the Contact latch LaMSA system in Latched (Red) and Unlatched (Blue) mode. The stationary saddle is the green circle, the moving fixed point in Latched mode is the black square, and the fixed point in the Unlatched mode is the pink circle.
• Figure 3: Trajectory of the nominal equation with varying initial conditions $(p^*,F_L^*)$. The yellow region satisfies the saddle conditions, and the purple region does not satisfy the saddle conditions.
• Figure 4: Quiver plot of the nominal equation with the trajectory of saddle points $(p^*,F_L^*)$.

Theorems & Definitions (15)

• Definition 1: Latched Mode
• Definition 2: Unlatched Mode
• Proposition 3.1
• proof
• Corollary 3.1.1
• proof
• Proposition 3.2
• proof
• Lemma 3.3
• proof