Dynamics Of Inverted Pendulum Research Articles

Biomechanical and energetic determinants of the walk–trot transition in horses

We studied nine adult horses spanning an eightfold range in body mass (M(b)) (90-720 kg) and a twofold range in leg length (L) (0.7-1.4 m). We measured the horses' walk-trot transition speeds using step-wise speed increments as they locomoted on a motorized treadmill. We then measured their rates of oxygen consumption over a wide range of walking and trotting speeds. We interpreted the transition speed results using a simple inverted-pendulum model of walking in which gravity provides the centripetal force necessary to keep the leg in contact with the ground. By studying a large size range of horses, we were naturally able to vary the absolute walking speed that would produce the same ratio of centripetal to gravitational forces. This ratio, (M(b)v2/L)/(M(b)g), reduces to the dimensionless Froude number (v2/gL), where v is forward speed, L is leg length and g is gravitational acceleration. We found that the absolute walk-trot transition speed increased with size from 1.6 to 2.3 m s(-1), but it occurred at nearly the same Froude number (0.35). In addition, horses spontaneously switched between gaits in a narrow range of speeds that corresponded to the metabolically optimal transition speed. These results support the hypotheses that the walk-trot transition is triggered by inverted-pendulum dynamics and occurs at the speed that maximizes metabolic economy.

Dynamics and stability state-space of a controlled inverted pendulum

A study is conducted into the dynamics of an inverted pendulum under the regulation of a linear-feedback control system. Analysis of the resulting dynamic equations indicates the presence of stable regions within the four-dimensional state space of the system, which are confirmed through simulation. System trajectories are examined in this state space, and complex, apparently chaotic behaviour is observed outside the stable regions of the state space. An algorithm is developed to map the boundaries of the stable state-space hyper-volume; this algorithm is broadly applicable to the study of other systems. Results are presented, generated by executing the algorithm on the inverted pendulum dynamics. These comprise a series of three-dimensional plots, outlining the four-dimen-sional stability volume. The boundaries of this volume are seen to exhibit a jagged, possibly fractal, morphology.