Spiking control systems for soft robotics: a rhythmic case study in a soft robotic crawler

Abstract

Inspired by spiking neural feedback, we propose a spiking controller for efficient locomotion in a soft robotic crawler. Its bistability, akin to neural fast positive feedback, combined with a sensorimotor slow negative feedback loop, generates rhythmic spiking. The closed-loop system is robust through the quantized actuation, and negative feedback ensures efficient locomotion with minimal external tuning. Using bifurcation analysis, we characterize how the sensorimotor gain-coupling body and controller dynamics-governs the emergence of qualitatively distinct dynamical regimes, including resting and crawling behaviors associated with peristaltic waves. Dimensional analysis formalizes a separation of mechanical and electrical timescales, and Geometric Singular Perturbation theory explains the geometry of the relaxation oscillations leading to endogenous crawling. Within this singularly perturbed framework, we further formulate and analytically solve an optimization problem, proving that locomotion speed is maximized when mechanical resonance is achieved via a matching of neuromechanical scales. Given the importance and ubiquity of rhythms and waves in soft-bodied locomotion, we envision that spiking control systems could be utilized in a variety of soft-robotic morphologies and modular distributed architectures, yielding significant robustness, adaptability, and energetic gains across scales.

Figures (5)

• Figure 1: (A) Schematic of the body of the two-segmented soft crawler analyzed in this work. The index 1 (resp., 2) refers to the crawler's tail (resp., head). The inset represents the nonlinear anisotropic friction model as given in \ref{['eq:sigma_def']}. (B) The excitable crawler: information flow in the closed-loop system between the crawler's body and the neuromorphic excitable controller proposed in §\ref{['subsec:excitable_controller']}. The excitable controller has the realization of a flip-flop electrical circuit. (C) The static hysteresis I/O map relating strain and voltage utilized in §\ref{['sec:setting_the_pace']} is hardware-realizable with a Schmitt trigger---a comparator with hysteresis. $\beta$ denotes the (dimensionless) switching strain threshold and $\mathsf{M}$ the (dimensionless) output voltage magnitude.
• Figure 2: Behavior of the excitable crawler in a neighborhood of the symmetric fixed point $\boldsymbol{\mathsf{x}}_+$ projected into the $\mathsf{V}$-$\mathsf{s}$ plane for two different values of the sensorimotor gain $\pi_\mathsf{s}$, right before and after the Hopf bifurcation at $\pi_\mathsf{s}^\mathsf{H}$. $\boldsymbol{\mathsf{x}}_+$ is located close to the fold of the slow manifold. (A) Critical manifold $\mathcal{S}_0$, with the attracting (repelling) branches in green (red). White (purple) circles correspond to fixed points (folds). The trajectory in yellow corresponds to global relaxation oscillations at $\pi_\mathsf{s} \gtrsim \pi_\mathsf{s}^\mathsf{H}$. Panels (B) and (C) zoom into the region highlighted in blue in panel (A), for two different values of $\pi_\mathsf{s}$. The respective initial condition is represented by a black cross. (B)Resting regime. Convergence to one of the stable symmetric fixed points when $\pi_\mathsf{s} \lesssim \pi_\mathsf{s}^\mathsf{H}$. (C)Crawling regime. The top panel shows the transient trajectory of the excitable crawler at $\pi_\mathsf{s} \gtrsim \pi_\mathsf{s}^\mathsf{H}$ when initialized close to the unstable symmetric fixed point $\boldsymbol{\mathsf{x}}_+$, before the system eventually locks into global relaxation oscillations, as shown in panel (A)---yellow trajectory. (D) Convergence $\mathsf{V}_+ \to \mathsf{V}_+^\mathsf{F}$ as $\varepsilon \to 0^+$, as per Proposition \ref{['prop:canards']}.
• Figure 3: Relaxation oscillations in the excitable crawler. Parameter values: $\zeta = 4.7, \pi_{\mathsf{f}} = 2.5, \pi_{\mathsf{V}} = 0.5 , \pi_{\mathsf{\epsilon}} = 4.7 \cdot 10^3, \mathsf{n_f} = 1.5, \pi_{\mathsf{c}} = 10^4, \pi_{\mathsf{l}} = \pi_{\mathsf{s}} = 2 \cdot 10^4$, and initial condition $\mathsf{V}(0) = 2, \mathsf{s}(0) = \mathsf{v_s}(0) = \mathsf{v_{com}}(0) = 0$. All panels display dimensionless variables. (A) Limit cycle in the state variables $\mathsf{s}, \mathsf{v_s}, \mathsf{V}$ in red. Single (double) arrows on the limit cycle correspond to trajectories of the slow (fast) dynamics. The critical manifold \ref{['eq:critical_manifold']} is represented in gray. The initial condition is in black. (B) Schematic of the strain-driven voltage switching. Circled numbers indicate the temporal ordering of the snapshots. The value of the voltage is indicated by the dot in each panel. Purple (red) dots correspond to the value of the voltage right before (after) the switch between attracting branches of the manifold. (C) Time evolution of different state entries of the excitable crawler: (C.1) Displacements of head and tail, as indicated; (C.2) Relaxation oscillations in the voltage. The fast (slow) dynamics correspond to segments going from purple (red) to red (purple) dots; (C.3) Strain in the soft crawler's body.
• Figure 4: (A) Block diagram of the singularly-perturbed excitable crawler dynamics under Assumption \ref{['ass:nonlinearities']}, splitting linear and nonlinear blocks. The voltage dynamics \ref{['eq:V_dynamics']} are replaced by a static I/O map---a bi-level hysteretic relay, as shown in the red block. The sigmoidal friction model \ref{['eq:dimensionless_friction']} is approximated by an asymmetric relay with parameter $\Delta$, given in \ref{['eq:friction_relay']}, shown in the blue block. $p$ denotes the Laplace variable and the $p$-blocks denote time differentiation. (B) Bi-level hysteretic relay relating strain and voltage: (1) sinusoidal strain input $\mathsf{s}$ of amplitude $\mathsf{S}$; (2) bi-level hysteretic relay characteristic, defined by the parameters $\mathsf{M}$ and $\beta$; (3) voltage output of the relay for the sinusoidal strain input provided in panel (1)---in black; its fundamental harmonic is in orange. (C) Asymmetric relay relating local speed and frictional force: (1) speed input $\mathsf{u}'$, with a non-zero average $\bar{\mathsf{v}}$ and harmonic profile, with a phase $\phi$ with respect to the strain; (2) asymmetric relay $\sigma_{\Delta}$ relating speed and frictional force; (3) output of the asymmetric relay---in black; its harmonic approximation is in orange and the dotted blue line represents the speed input, showing that input and output are in phase.
• Figure 5: Describing function analysis and optimization. Parameter values: $\zeta = 2, \pi_\mathsf{V} = 0.5, \mathsf{M} = 2, \pi_\mathsf{f} = 2.5, \mathsf{n_f} = 1.5$. (A)$\mathsf{g}$ as defined in \ref{['eq:g_def']} as a function of $\mathsf{S}$ for the feasible values of $\mathsf{Z} \in \{0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9; 1 \}$, as indicated. Vertical axis is in logarithmic scale. Per Lemma \ref{['lemma:nonempty_set']}, each curve has a unique intersection with the abscissa---respectively highlighted by a circle. The optimal $\mathsf{S}$ as given in \ref{['eq:maximizer']} which corresponds to $\mathsf{Z}^* = 1$ is highlighted by a star. For the parameter values selected in this example, $\mathsf{S}^* \approx 0.51$. (B) Quantities of interest as a function of $\mathsf{Z}$, as approximated by the describing function and harmonic balance analysis: (B.1) the (dimensionless) frequency of the state oscillations, $\omega$. The optimal $\omega^* = 1$ corresponds to crawling at resonance; (B.2) amplitude $\mathsf{S}$ of the strain in the soft crawler's body; (B.3) (purple) relative phase between the fundamental harmonics of voltage $\mathsf{V}$ and the strain rate $\mathsf{s}'$. The fundamental harmonic approximations of these two signals align (i.e., $\phi_\mathsf{Z} = 0$) at the optimum $\mathsf{Z}^* = 1$; (green) average speed of the excitable crawler's center of mass, $\bar{\mathsf{v}}_{\mathsf{com}}$; (yellow) average muscle power $\bar{\mathcal{P}}$, maximized at $\mathsf{Z}^*=1$. (C) Instantaneous muscle power $\mathcal{P}$ as a function of time for different values of $\mathsf{Z}$ as highlighted in the legend. $\mathcal{P}$ oscillates at $2 \omega$. $\mathcal{P}$ is a non-negative signal only at optimality ($\mathsf{Z}^*=1$, purple curve). (D) Under periodic motion the actuator force versus strain trajectories form a closed curve. The work over a cycle done by the actuator force on the crawler's segments---for different values of $\mathsf{Z}:= \beta/\mathsf{S}$, as indicated---is the area enclosed clockwise by the respective curve. Dotted lines represent the respective values of $\beta$. The actuator's force work is maximized at $\mathsf{Z}^* = 1$. As such matching condition, the amplitude of the strain is maximal and the closed curve tangent to the vertical at $\mathsf{S} = \beta$, indicating maximal actuator work per cycle.

Theorems & Definitions (22)

• Definition II.1: Equivariant Mapping, from GolubitskyM.andStewart2002
• Definition II.2: $\mathbb{G}$-equivariant mapping, from GolubitskyM.andStewart2002
• Definition II.3: Directional derivatives of a vector field
• Theorem II.1: Pitchfork bifurcation, from Golubitsky1985
• Theorem II.2: Hopf bifurcation theorem, Theorem 3.4.2 in Guckenheimer1983
• Remark 1
• Proposition IV.1: Bistability
• proof
• Theorem IV.2: Paired Hopf bifurcation at $\boldsymbol{\mathsf{x}}_{\pm}$
• proof