Serpenoid Gait Function: Theory & Application

• Serpenoid gait function is defined as a traveling sine wave that prescribes joint angles along a snake’s body to generate propulsion in high-drag, low-inertia environments.
• It employs fixed amplitude and phase lag with dynamically adjusted frequency, serving as a foundation for both articulated and soft robotic locomotion.
• Recent techniques using reinforcement learning and evolutionary algorithms have achieved a 38% reduction in power consumption and a 7.5% velocity improvement over 10 m traverses.

The serpenoid gait function is the canonical mathematical model of undulatory body deformation underpinning both biological and robotic snake locomotion. Defined originally to model the traversal of traveling waves of curvature along an articulated or continuous backbone, the serpenoid function prescribes body shapes that efficiently generate propulsive force in high-drag, low-inertia environments. Its foundational role spans geometric analysis of animal motion, optimization-based control of articulated and soft robotic snakes, and the design of scalable, adaptable reactive controllers for cluttered environments.

1. Mathematical Characterization of the Serpenoid Gait

The serpenoid gait specifies the temporal and spatial evolution of joint angles (or local tangents) along the snake's body as a traveling wave, mathematically formulated as a phase-shifted sinusoid. The classical Hirose serpenoid formula is:

$$\theta_i(t) = A\,\sin\left(\omega t - (i-1)\phi\right)$$

where

• $\theta_i(t)$: angular position of joint $i$ at time $t$,
• $A$: joint amplitude (maximum angular deflection, typically 0.4–0.6 radians in practical robots (Liu et al., 2021, Sinha et al., 15 Nov 2025)),
• $\omega$: temporal angular frequency (radians/s),
• $\phi$: constant spatial phase lag between joints,
• $i$: joint index along the chain.

In recent control frameworks, especially for robot platforms, amplitude $A$ and phase lag $\phi$ are fixed, while frequency $\omega$ and, in some cases, an offset angle (e.g., heading steering parameter $\phi_0$ for the head) may be varied dynamically for locomotion modulation and steering (Sinha et al., 15 Nov 2025). In soft robot models, the serpenoid centerline is generalized as an integral-defined planar curve:

$$x(s)=\int_0^s \cos(a \cos(b\sigma) + c\sigma)\, d\sigma$$

$$y(s)=\int_0^s \sin(a \cos(b\sigma) + c\sigma)\, d\sigma$$

with typical parameters $a=-\pi/4$, $b=\pi/4$, $c=0$ and arc-length parameter $s$ (Arachchige et al., 2020).

This framework supports both spatially discretized models (articulated joints) and spatially continuous models (soft-bending backbones).

2. Dynamical Principles and Geometric Analysis

The serpenoid gait encapsulates a template for generating efficient axial body waves that maximize net displacement in rate-independent (highly damped) environments. Recent work has formalized the geometry of these gaits through principal-component or modal decomposition:

$\theta(s,t) = a_1(t)\phi_1(s) + a_2(t)\phi_2(s)$

with shape space trajectories traced by $(a_1(t), a_2(t))$ (Rieser et al., 2019).

Locomotion performance (forward speed, turning efficacy) is predicted by integrating a “local connection” derived from drag-based resistive force theory:

$$\xi = A(\alpha)\dot\alpha, \quad \Delta g \approx \oint_\Gamma A(\alpha)\, d\alpha \approx \iint_S (\nabla \times A)(a_1,a_2)\,dS$$

An optimal serpenoid gait corresponds to a near-circular loop in modal shape space that maximizes net geometric phase—measured as the surface integral of the curvature of the local connection (Rieser et al., 2019). Across undulatory organisms from C. elegans to desert snakes, this optimal loop radius is selected to maximize performance within both muscle power and environmental drag constraints.

3. Robotic Implementation and Parameterization Strategies

In articulated robotic snakes, the serpenoid gait is implemented directly as a sequence of joint commands varying sinusoidally in time and lagging in phase along the body. For example, the joint policies in energy-efficient control frameworks set

$$\theta_i(t) = A\,\sin\left(\omega_i t - (i-1)\phi\right)$$

with $\omega_i$ either learned or adaptively set per segment (Liu et al., 2021, Sinha et al., 15 Nov 2025).

Parameterization typically involves:

• Fixing amplitude $A$ and phase lag $\phi$ (e.g., $A=0.6$ rad, $\phi=\pi/6$ rad (Sinha et al., 15 Nov 2025)).
• Dynamically varying frequency $\omega$ to control stride and speed, and introducing a head offset $\phi_0$ for steering.
• Mapping sine-generated target positions to specific motor commands via analytical kinematic or numerical optimization chains, especially for soft continuum segments (Arachchige et al., 2020).

NEAT-based evolutionary control replaces hand-tuned or policy-learned $(\omega, \phi_0)$ with direct mappings from high-dimensional body- and environment-state (e.g., LiDAR scan, head pose) to these two control parameters, supporting real-time reactive obstacle avoidance (Sinha et al., 15 Nov 2025).

4. Optimization and Learning in Gait Control

Parameter optimization in the serpenoid framework is central to energy, velocity, and adaptability performance:

• Deep reinforcement learning (PPO, TRPO) permits the serpenoid frequency schedule $\omega(t)$ to be learned online, under reward structures penalizing backward motion and favoring energy efficiency (Liu et al., 2021). Fixed-amplitude and phase, variable-frequency serpenoid gaits outperform hand-tuned references, reducing mean power by 38% and increasing velocity by 7.5% over 10 m traverses.
• NEAT trains feedforward networks to map sensor and state data to serpenoid frequency and steering offset parameters, demonstrating superior collision avoidance in densely-obstructed simulated environments compared to Coach-Based RL and deep learning direct controllers, with a minimal memory footprint (~19 KB) and rapid convergence (Sinha et al., 15 Nov 2025).
• In continuum-body soft snakes, least-squares optimization matches sampled serpenoid backbone curves to actuator-space trajectories under regularization, translating analytically-specified gaits into pneumatic actuation schedules (Arachchige et al., 2020).

5. Biomechanical and Physical Insights

Serpenoid gaits reproduce a traveling sine-wave of body curvature that is well-matched to the low-inertial, high-drag context of biological and robotic snakes. Key properties include:

• Scale invariance: optimal modal loop radii match predictions across more than four orders of magnitude in organism body length, from nematodes to sidewinder rattlesnakes (Rieser et al., 2019).
• Net propulsion efficacy is maximized near the serpenoid geometric optimum, subject to muscle or actuator power limitations.
• Turning and non-linear maneuvers are achieved by phase offsetting the modal loop center or adding time-varying head bias (Rieser et al., 2019, Sinha et al., 15 Nov 2025).
• Pure serpenoid waveforms alone may fail to generate net forward motion on substrates with isotropic friction (e.g., carpet), but achieve locomotion if frictional anisotropy is introduced or if rolling/sidewinding variants are used (Arachchige et al., 2020).

6. Applications and Adaptations in Robotic Locomotion

The universality and flexibility of the serpenoid gait function make it foundational for:

• Snake robots employing articulated joints (with direct sine-based angle commands) for search, rescue, and confined space traversal (Liu et al., 2021).
• Soft robotic snakes with continuous bending, using sampled serpenoid curves mapped to local actuator lengths and then to input pressure via direct calibration (Arachchige et al., 2020).
• Adaptive, sensor-rich control systems employing evolutionary algorithms (NEAT) or deep RL to map environmental observations to stride and steering parameters, enabling real-time obstacle-avoidance and path-following in cluttered domains (Sinha et al., 15 Nov 2025).

Implementation Framework	Core Parameterization	Optimization Method
Articulated link robots	$A$, $\omega$, $\phi$, $\phi_0$	RL/PPO, NEAT evolution
Soft continuum robots	Backbone curve $(a,b,c)$, actuator extensions	Least-squares, calibration
Biomechanical analysis	Modal amplitudes $a_1, a_2$, shape basis	Geometric phase maximization

Further extension of the serpenoid framework includes evolving additional gait parameters for increased agility, multi-objective control trading off energy, speed, and collision rate, and deployment to hardware with real sensory noise and actuation lag (Sinha et al., 15 Nov 2025).

7. Limitations and Prospects

While the serpenoid gait function provides an efficient, parsimonious generative model for undulatory motion, limitations are evident in isotropic environments and under severe actuator or power constraints. NEAT and RL-based controllers improve adaptability but may converge to suboptimal solutions if hyperparameters are not well-tuned. Real-world deployments must additionally address sensor noise, frictional variability, and latency. A plausible implication is that coupling serpenoid templates with dynamic environmental estimation and multi-modal gait switching (e.g., rolling, sidewinding) will further extend robustness and versatility in future snake robot designs.

References: (Arachchige et al., 2020, Rieser et al., 2019, Liu et al., 2021, Sinha et al., 15 Nov 2025)