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We develop a simple hexapedal model for the dynamics of insect locomotion in the horizontal plane. Each leg is a linear spring endowed with two inputs, controlling force-free length and "hip" position, in a stereotypical feedforward pattern. These represent, in a simplified manner, the effects of neurally activated muscles in the animal and are determined from measured foot force and kinematic body data for cockroaches. We solve the three-degree-of-freedom Newtonian equations for coupled translation-yawing motions in response to the inputs and determine branches of periodic gaits over the animal's typical speed range. We demonstrate a close quantitative match to experiments and find both stable and unstable motions, depending upon input protocols. Our hexapedal model highlights the importance of stability in evaluating effective locomotor performance and in particular suggests that sprawled-posture runners with large lateral and opposing leg forces can be stable in the horizontal plane over a range of speeds, with minimal sensory feedback from the environment. Fore-aft force patterns characteristic of upright-posture runners can cause instability in the model. We find that stability can constrain fundamental gait parameters: our model is stable only when stride length and frequency match the patterns measured in the animal. Stability is not compromised by large joint moments during running because ground reaction forces tend to align along the leg and be directed toward the center of mass. Legs radiating in all directions and capable of generating large moments may allow very rapid turning and extraordinary maneuvers. Our results further weaken the hypothesis that polypedal, sprawled-posture locomotion with large lateral and opposing leg forces is less effective than upright posture running with fewer legs.

New models and theories of legged locomotion are needed to better explain and predict the robustly stable legged locomotion of animals and some bio-inspired robots. In this paper we observe that a hip-torque and leg-damping mechanism is fundamental to many legged robots and some animals and determine its affect on locomotion dynamics. We discuss why this hip-torque-and-leg-damping mechanism is not so easily understood. We investigate how hip-torque and leg-damping affect the stability and robustness of locomotion using a mathematical model: First, we extend the canonical spring-loaded-inverted-pendulum model to include constant hip torque and leg damping proportional to leg length speed. Then, we calculate the stability and robustness of locomotion as a function of increasing levels of torque and damping, starting from zero-the energy conserving and marginally stable special case-to high levels of torque and damping. We find that the stabilizing effects of hip-torque and leg-damping occur in the context of the piecewise-continuous dynamics of legged locomotion, and so linear intuition does not apply. We discover that adding hip torque and leg damping changes the stability of legged locomotion in an unexpected way. When a small amount of torque and damping are added, legged locomotion is initially destabilized. As more torque and damping are added, legged locomotion turns stable and becomes increasingly more stable and more robust the more torque and damping are added. Also, stable locomotion becomes more probable over the biologically-relevant region of the parameter space, indicating greater prediction and explanatory capabilities of the model. These results provide a more clear understanding of the hip-torque-and-leg-damping mechanism of legged locomotion, and extend existing theory of legged locomotion towards a greater understanding of robustly stable locomotion.

SUMMARY In nature, cockroaches run rapidly over complex terrain such as leaf litter. These substrates are rarely rigid, and are frequently very compliant. Whether and how compliant surfaces change the dynamics of rapid insect locomotion has not been investigated to date largely due to experimental limitations. We tested the hypothesis that a running insect can maintain average forward speed over an extremely soft elastic surface (10 N m−1) equal to 2/3 of its virtual leg stiffness (15 N m−1). Cockroaches Blaberus discoidalis were able to maintain forward speed (mean ± s.e.m., 37.2±0.6 cm s−1 rigid surface versus 38.0±0.7 cm s−1 elastic surface; repeated-measures ANOVA, P=0.45). Step frequency was unchanged (24.5±0.6 steps s−1 rigid surface versus 24.7±0.4 steps s−1 elastic surface; P=0.54). To uncover the mechanism, we measured the animal's centre of mass (COM) dynamics using a novel accelerometer backpack, attached very near the COM. Vertical acceleration of the COM on the elastic surface had a smaller peak-to-peak amplitude (11.50±0.33 m s−2, rigid versus 7.7±0.14 m s−2, elastic; P=0.04). The observed change in COM acceleration over an elastic surface required no change in effective stiffness when duty factor and ground stiffness were taken into account. Lowering of the COM towards the elastic surface caused the swing legs to land earlier, increasing the period of double support. A feedforward control model was consistent with the experimental results and provided one plausible, simple explanation of the mechanism.

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