Dynamics and stability of insect locomotion: a hexapedal model for horizontal plane motions.

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.