Dynamic single-input control of multi-state multi-transition soft robotic actuator

TL;DR

This work addresses the challenge of controlling multi-DOF soft robotic actuators with a single input by introducing a chain of $N$ interconnected bi-stable elements whose dynamic response and pre-designed variability enable any desired sequence of state transitions. The authors develop a nonlinear dynamical model with rate-dependent damping and a tri-linear approximation, derive stability criteria, and show that the system can realize up to $2^{N}(1+N/2)$ states, including stable or unstable intermediate spinodal states. They validate the framework through experiments with 2- and 4-element actuators, demonstrating various transition trajectories and coupling to a 3-link mechanism that achieves traveling-wave-like motions with a single control input. The findings demonstrate a scalable, energy-efficient approach to high-dexterity actuation in soft robotics, avoiding external fields and enabling vast trajectory possibilities $(N!)^2$ for complete cycles.

Abstract

Soft robotics is an attractive and rapidly emerging field, in which actuation is coupled with the elastic response of the robot's structure to achieve complex deformation patterns. A crucial challenge is the need for multiple control inputs, which adds significant complication to the system. We propose a novel concept of single-input control of an actuator composed of interconnected bi-stable elements. Dynamic response of the actuator and pre-designed differences between the elements are exploited to facilitate any desired multi-state transition, using a single dynamic input. We show formulation and analysis of the control system's dynamics and pre-design of its multiple equilibrium states, as well as their stability. Then we fabricate and demonstrate experimentally on single-input control of two- and four-element actuators, where the latter can achieve transitions between up to 48 desired states. Our work paves the way for next-generation soft robotic actuators with minimal actuation and maximal dexterity.

Figures (6)

• Figure 1: (a) Schematic illustration of a multi-stable chain, comprised from bi-stable elements that are connected in series. The endpoints of each element are also connected to dampers. (b) The force-displacement relation $F(\varepsilon)$ of a bi-stable element is non-monotonic, with two states of positive stiffness, states '0' and '1', separated by a branch of negative stiffness termed the spinodal region or state 's'. The tri-linear approximation of this non-monotonic relation, denoted by the dashed straight-line segments, may offer useful analytical insights. (c) Pre-designed variability between the bi-stable elements which makes each element weaker or stronger compared to its neighbors, as in Eq. \ref{['eq:variability']}. (d)-(f) Examples of several state transition diagrams for a multi-state actuator with $N=3$ bi-stable elements, which offers 8 (or $2^N$) possible binary states. Transitions between these states form a trajectory. (d) State-of-the-art single-input actuators with ordered variability or dissipative damping as in ben2020single enable reaching any state, but the sequence of quasi-static transitions from initial to target state is limited to a specific order, which results in long, often impractical, sequences. Moreover, a complete cycle, from (000) to (111) and back to (000), is possible only through a single trajectory, thus termed single-input single-cycle actuators. (e) Single-input multiple-transition actuator. Our concept enables following any desired complete-cycle trajectory with a single input. The number of complete-cycle trajectories is 36 for $N=3$ (generally $(N!)^2$). Transitions occur by snap-through of an element from one stable phase to the other via an intermediate 's' phase of unstable equilibrium (spinodal), thus providing rapid transition. (f) High-resolution single-input multiple-transition actuator, where intermediate states having one element at 's' phase are deliberately designed to be stable, enabling enhanced trajectory resolution with $20$ stable states for $N=3$ (generally $2^{N-1}(N+2)$).
• Figure 2: A multi-stable actuator with two bi-stable elements, $N=2$. (a) Photo of our experimental setup. (b) Each bi-stable element is a 3-D printed double curved-beam structure. Variability between the force-displacement relations of the two bi-stable elements was introduced by means of differences in the geometry of the curved beams. Also, removal of the "safety pin" activates the serially-connected element acting as a linear spring, which changes the equivalent stiffness of phase 's' and therefore alters the stability of intermediate states. (c) The element with unlocked linear spring, clamped to Instron device for conducing force-displacement measurements. (d) Measurements of the bi-stable force-displacement profile for each of the two elements, with and without the safety pin, are shown for several extension-contraction cycles, demonstrating high repeatability and small hysteresis.
• Figure 3: Experimental results of the two-element actuator. (a)-(c) Measured elongations of the elements $\varepsilon_i$ versus time $t$, for input rates $v$={24, 26, 29}mm/s, respectively. (d)-(f) Measurements of displacements plotted as dots in the plane of $(\varepsilon_1,\varepsilon_2)$, for input rates $v$={24, 26, 29}mm/s, respectively. The three different trajectories demonstrate the ability to follow each of the sequences of state transitions by merely controlling the rate of the input (extension/contraction) rate $v$. For reference, the equilibrium curves (solid and dashed black curves for stable or unstable equilibrium, respectively) and the predictions of the numerical simulations of our theoretical dynamic model (orange and green solid curves with arrows) are illustrated as well. (g)-(h) Experiment results with unstable intermediate states (without the safety pin). Note the abrupt transitions (snap-through) between binary states, denoted by double-arrows, in contrast to the smooth transition through intermediate states observed in (d)-(e)-(f). (i) State transition diagram of a two-element actuator with stable intermediate states.
• Figure 4: A 3-link mechanism controlled by a single input using the proposed actuator, allowing for any desired sequence of strokes. (a) A transitions diagram between binary states of the three-link mechanism. (b) Plots of measured the mechanism's two joint angles $\theta_i$ versus time $t$ for low input rate $|v|=20$mm/s, leading to state sequence $\{(00)\to (01)\to (11) \to (10) \to (00)\}$. (c) Plots of measured the mechanism's two joint angles $\theta_i$ versus time $t$ for low input rate $|v|=28$mm/s, leading to state sequence $\{(00)\to (10)\to (11) \to (01) \to (00)\}$; Snapshot pictures from the movie of the mechanism's motion (d) for low input rate, (e) for high input rate. Below each snapshot sequence, we show illustrations of the bi-stable states of the actuator's two elements.
• Figure 5: A multi-stable actuator with four bi-stable elements, $N=4$: (a) Diagram with all possible binary states and transitions (not including intermediate 's' states). Each colored sequence of arrows indicates a different order of the snapping sequence, obtained experimentally. (b) The measured force-displacement profiles of the four elements, which were designed with ordered variability. (c) Plots of four time responses, each associated with a different experiment and a different snap-thorough sequence (see colored arrows in (a) corresponding to the colored circles in (c.1) through (c.4) ), showing the elongation, $\varepsilon_{i}(t)$, of each of the bi-stable elements in a complete extension from state (0000) to state (1111). different choice of time-varying input rate of extension $v(t)$, shown in the bottom graphs. Thus, each experiment results in a different trajectory with a chosen order of elements' snapping. (d) Snapshots from the experiment for which the order of snapping elements is $\{ 3\rightarrow 2 \rightarrow 4 \rightarrow 1\}$. That is, the sequence of binary state transitions is (0000)$\rightarrow$(0010)$\rightarrow$(0110)$\rightarrow$(0111)$\rightarrow$(1111). The small and large arrows in the pictures mark the closed and snapped-open elements, respectively.