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

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.

💡 Research Summary

The paper presents a rigorous control‑theoretic framework for generating rhythmic locomotion in a soft robotic crawler using a spiking controller inspired by neuronal excitability. The controller is modeled after the FitzHugh‑Nagumo (FHN) equations, featuring a fast positive feedback loop that creates bistability and a slow negative feedback loop that couples the controller output to a proprioceptive sensor measuring body strain. This combination yields self‑sustained spikes that, when fed back to the soft body, produce peristaltic waves and forward motion.

The authors first describe the mechanical model of a two‑segment soft crawler. Each segment is represented by a mass‑spring‑damper system with anisotropic friction, allowing forward propagation of a wave when one segment contracts while the other expands. The sensor‑motor gain (k_{sm}) is introduced as the primary bifurcation parameter linking the controller to the body.

A bifurcation analysis is carried out with respect to (k_{sm}). For low gain the closed‑loop system possesses a single stable equilibrium corresponding to a resting state. As the gain crosses a critical value, a pair of complex‑conjugate eigenvalues of the linearized system cross the imaginary axis, giving rise to a Hopf bifurcation. The resulting small‑amplitude limit cycle corresponds to the onset of rhythmic spiking and peristaltic motion. Because the overall system is symmetric under exchange of the two segments, a pitchfork bifurcation also occurs, generating two symmetric crawling modes (left‑to‑right and right‑to‑left). The authors provide formal proofs of these bifurcations (deferred to appendices) and illustrate the transition with numerical simulations.

Next, the paper exploits a clear separation of timescales between the fast electrical dynamics of the FHN controller and the slower mechanical dynamics of the soft body. By nondimensionalizing the equations, a small parameter (\epsilon = \tau_{elec}/\tau_{mech}) is identified. Geometric Singular Perturbation Theory (GSPT) is then applied: in the (\epsilon \to 0) limit the fast subsystem exhibits rapid jumps (spikes) while the slow subsystem evolves on a critical manifold. The concatenation of fast jumps and slow drifts produces relaxation oscillations, which manifest physically as traveling peristaltic waves along the crawler. This geometric picture explains why the system can generate robust, self‑sustained locomotion without external timing signals.

The final contribution is an optimization of locomotion speed using describing‑function analysis. The nonlinear spiking element is approximated by its fundamental harmonic, allowing the closed‑loop transfer function to be expressed in linear terms with an effective gain and phase shift. By imposing the Nyquist condition for sustained oscillations, the authors derive an explicit relationship between the controller’s time constant, the mechanical natural frequency, and the sensor‑motor gain. The optimization problem—maximizing the average center‑of‑mass velocity—reduces to matching the electrical oscillation frequency to the mechanical resonance frequency of the crawler. Analytically solving this yields a closed‑form expression for the optimal gain and time constant. At this resonance point, the phase of the actuator force aligns with the strain rate of the body, ensuring that instantaneous power input is non‑negative throughout a cycle and that average power consumption is minimized while speed is maximized. This result mirrors observations in biological locomotion where animals often operate near mechanical resonance for energetic efficiency.

Overall, the paper makes four key contributions: (1) introduction of an excitable spiking controller with bistability and slow negative feedback for soft robots; (2) systematic bifurcation analysis identifying Hopf and pitchfork transitions that delineate resting and crawling regimes; (3) application of singular perturbation theory to reveal the geometric structure of relaxation oscillations underlying peristaltic motion; and (4) a closed‑form optimality condition showing that maximal crawling speed is achieved when neuromechanical time scales are matched (mechanical resonance). The authors argue that this framework can be extended to multi‑segment, multi‑input‑multi‑output soft robots, and they suggest future work on distributed spiking controllers, adaptive gain tuning, and application to other soft morphologies such as soft arms or fins. The combination of rigorous mathematical analysis with biologically inspired control promises robust, adaptable, and energy‑efficient locomotion for the next generation of soft robotic systems.