Embodied intelligence solves the centipede's dilemma

TL;DR

A dynamical model of centipede locomotion is presented that integrates leg-ground interactions, passive body mechanics, and active lateral musculature, and finds that centipedes actively modulate body mechanics to achieve rapid, efficient locomotion, highlighting how complex control can emerge from embodied physical properties rather than solely from neural computation.

Abstract

Although commonly associated with limbless animals like snakes and fish, multi-legged organisms like centipedes also utilize undulatory locomotion. Whether these undulations are actively reinforced or resisted by the axial musculature remains an open question. We present a dynamical model of centipede locomotion that integrates leg-ground interactions, passive body mechanics, and active lateral musculature. By varying stepping rate, actuation, and body stiffness, we examine how locomotor strategies affect speed and an effective energetic efficiency. Coordination emerges only when body stiffness is tuned to stepping frequency: overly flexible bodies lose synchrony, while overly rigid ones move slowly and inefficiently. This leads to the prediction that centipedes utilize speed dependent active stiffness to maintain this coordination. Our results suggest that lateral muscles also have a speed dependent function, revealed by optimizing speed and an effective cost, that resists a phase lag between leg touchdowns and body curvature. Together, we find that centipedes actively modulate body mechanics to achieve rapid, efficient locomotion, highlighting how complex control can emerge from embodied physical properties rather than solely from neural computation.

Figures (13)

• Figure 1: Minimal model of centipede locomotion with legged propulsion and body undulation. (a) Schematic of a running centipede showcasing the stepping wave pattern. Legs are colored according to their touchdown time $t_c / \tau_a$, where $\tau_a$ is the actuation cycle period. Red dots indicate legs currently in ground contact, which remain in contact for duration $\tau_s$. The schematic represents a 21-segment centipede with $\tau_a/\tau_s = 12$, corresponding to twelve segments spanning between consecutive ground contacts along one side. Scale bar: 2 cm, based on Scolopendra herosAnderson1995. (b) The model's planar geometry. Body segments are modeled as rigid rods. The rod's center is node $n_i$ and has Cartesian position $\textbf{r}_{n,i}$. Segments can rotate in the plane, encoded by the counter-clockwise angle $\theta_i$ relative to horizontal. Two length scales define the geometry: the segment length $\ell_s$ and leg length $\ell_L$. Each body segment has two attached legs, and we denote ground contact's Cartesian position by $\textbf{r}_{c,i}$. (c) Leg-ground interaction. During ground contact (duration $\tau_s$), the attached segment pivots in a circular arc about the contact point due to no-slip constraints (Eq. \ref{['eq:no-slip']}) and active leg torque $T_L$ and force $F_L$ (Eq. \ref{['eq:leg_torque']}). (d) Body bending. Passive bending between segments is modeled by a flexural elasticity (spring constant $k$) and a viscous damping (coefficient $\eta$), governed by the inter-segment angle $\sigma_i = \theta_{i+1} - \theta_{i}$ (Eqs. \ref{['eq:flexure']} and \ref{['eq:damp']}). Active bending results from lateral flexor muscles, modeled as a sinusoidal bending moment $M[x,t]$ with amplitude $T_B$ that propagates along the body (Eq. \ref{['eq:bend-moment']}). The phase relationship $\phi$ between muscle activation and leg stepping determines whether body undulation assists or resists forward locomotion.
• Figure 2: Coordinated locomotion requires a sufficiently high contact-elastic ratio $\tau_s/\tau_k$ and a sufficiently low contact-actuation ratio $\tau_s/\tau_T$. Model solutions obtained by integrating Eqs. \ref{['eq:nonsmooth']}, \ref{['eqn:motion']} with initial conditions $\mathbf{q}, \dot{\mathbf{q}} = 0$. Parameter values fixed throughout are found in Table \ref{['tb:params']}, and the varied parameters used here are $\lambda/\ell_s = 11, \tau_s/\tau_k \in [0.2,1.6], \tau_s/\tau_{T_L} \in [0.1,0.7], \tau_{T_L}/\tau_{T_B} = 1.36, \phi = 4\pi/5.$ (a) Phase diagram of the body's center of mass speed $(\ell_s/\tau_a)$ versus contact-elastic ratio $(\tau_s/\tau_k)$ and contact-actuation ratio $(\tau_s/\tau_T)$. Three distinct regimes emerge: uncoordinated motion (UNC.), coordinated locomotion (C.), and stasis. The dashed line marks the coordination boundary, determined by the transition from multi-periodic to single-periodic segment oscillations (Fig. S3). In the stasis regime, speed approaches zero as segment flexion increases to the point where leg pivots no longer align to give forward propulsion. (b) Comparison of uncoordinated ($\square$) and coordinated ($\triangle$) locomotion showing body configurations and temporal dynamics. Time series show segment angles $\theta_i$ for posterior ($i=1$), middle ($i=11$), and anterior ($i=21$) segments over five actuation cycles. Coordinated locomotion exhibits regular periodic oscillations, while uncoordinated motion shows irregular behavior. The posterior segment consistently exhibits the largest amplitude oscillations. (c) Maximum posterior segment flexion $(\mathrm{max}_t[\theta_1[t]])$ across the same parameter space as panel (a). The sharp increase in posterior flexion at the coordination boundary indicates that loss of coordination is driven by excessive bending of the posterior segment.
• Figure 3: Running speeds require longer stepping wavelengths and proportionally stiffer bodies, with optimal stiffness scaling as $\tau_k \sim \tau_s$. Parameter values fixed throughout are found in Table \ref{['tb:params']}, and the varied parameters used here are $\lambda/\ell_s \in [5,13], \tau_s/\tau_k \in [0.3,1], \tau_s/\tau_{T_L} = 0.3, \tau_s/\tau_{T_B} = 0.3, \phi = \pi/2.$ (a) Speed landscape as a function of actuation wavelength $\lambda/\ell_s = \tau_a/\tau_s$ and contact-elastic ratio $\tau_s/\tau_k$. The black curve traces the maximum achievable speed $s_{\text{max}}$ for each wavelength, revealing an optimal contact-elastic ratio that varies with gait. (b) Optimal relationship between elasticity and actuation wavelength: the value of $\tau_a/\tau_k$ that maximizes speed for each wavelength. The dashed line ($y = x$) represents the condition $\tau_s = \tau_k$, showing that optimal speed occurs when the contact timescale $\tau_s$ approximately matches the body's elastic response time $\tau_k$. (c) Optimal speed scaling suggests active stiffness that increases with speed. Top panel shows experimental results using Scolopendra heros data from Anderson et al. Anderson1995, which demonstrate a decreasing actuation period $\tau_a$ as speed increases. Bottom panel shows the stiffness multiplier required at each speed to achieve the optimal $\tau_a/\tau_k$ relationship predicted in panel (b) given the measured $\tau_a$ from the top panel. This analysis predicts that centipedes must increase body stiffness by up to 7-fold to achieve their highest experimental speeds. (d) Inspired by Gray and Lismann Gray1950, we propose an experimental test using pendulum force measurements to directly quantify effective inter-segment stiffness in order to test our model's prediction (see Discussion).
• Figure 4: Pareto-optimal locomotion strategies reveal speed-dependent transitions from leg dominated to bending dominated locomotion. (a) Speed-cost trade-offs across parameter space $\{\lambda/\ell_s, \tau_s/\tau_k, \tau_s/\tau_{T_L}, \tau_s/\tau_{T_B}, \phi\}$ with fixed $\eta = 1.25 \tau_a / \tau_k$. The parameter space is adaptively sampled using a genetic algorithm (SI). (b) Net bending work $W_{\text{bend}}$ over a steady state cycle along the Pareto frontier exhibits speed-dependent sign changes. For $s_{\text{com}} < 6$, to the left of the dashed green line, the work is small in magnitude and can be resistive or assistive. For $6 < s_{\text{com}} < 9$, between the green and purple dashed lines, bending work is negative and as such in aggregate resists the body's shape change. For top speeds $s_{\text{com}} > 9$, to the right of the purple line, bending work is positive and in net assists the body's shape change. (c) By ablating active bending, i.e. setting $\tau_s/\tau_{T_B}=0$, for parameter values along the frontier, we find a decrease in the center of mass speed and lateral undulatory amplitude. As such, although active bending does negative work for $s_{\text{com}} > 6$, it does not resist undulations but rather increases them. (d) The normalized impulse energy loss over a cycle lateral to the body's motion, $\Delta K_{\perp}/W^+$, increases in magnitude until $s_{\text{com}} \approx 6$ to oppose $\approx 60\%$ of the actuator's positive work $W^+$. As active bending begins resisting undulations, this proportion decreases at higher speeds.
• Figure S1: Acceleration to steady state is approximately self-similar under scaling, with at most 30 actuation cycles required for acceleration to reach zero. We simulate locomotion for bending actuation only, legged actuation only, and both. Plotting each solution's speed over time, we find that speed over time is self-similar for the different propulsion strategies. This is visible in the main plot, within which each curve is plotted such that the curves all obtain the same maximum and reach 95% of that maximum at the same time. The inset displays the unscaled curves. Parameters used: $\lambda/\ell_s = 8, \tau_s/\tau_k = 1.5, \tau_s/\tau_{T_L} = 0.25, \tau_s/\tau_{T_B} = 0.55, \phi = \pi/2, \zeta = 1.25.$