Passive Dynamic Locomotion

In this project, we systematically investigated passive dynamic gaits that emerge from the natural mechanical dynamics of a bipedal legged system. To this end, we developed an energetically conservative, yet complete dynamical model of a biped. We achieved this by extending the established Spring-Loaded Inverted Pendulum (SLIP) model to include two legs and by adding a foot mass and a hip spring to enable passive swing leg dynamics. By letting the foot mass and hip stiffness go to zero while keeping their ratio (and thus the leg swing frequency) constant, I prevented energy losses at touchdown. Through a targeted continuation of periodic motions, I showed that a range of different bipedal gaits emerged in this model from a simple bouncing-in-place motion with different discrete footfall patterns. Among others, these passive dynamic gaits included walking, running, hopping, skipping, and galloping.

The different gaits arose along with one-dimensional manifolds of solutions. These manifolds bifurcated into different branches with distinctly different types of motions. That is, the gaits were obtained as different oscillatory motions (or nonlinear modes) of a single mechanical system with a single set of parameters. As this biped model has neither actuation nor control, it supports the hypothesis that different gaits are primarily a manifestation of the underlying natural mechanical dynamics of a legged system. The occurrence and prevalence of certain gaits in nature are thus possibly the consequence of animals exploiting passivity based gaits in order to move in an energetically economical fashion.

It is also notable, that despite the vast differences in morphology, the gaits of bipedal and quadrupedal animals share some important similarities.  Heglund (1982) investigated the dynamic similarity between walking in bipeds and quadrupeds and hypothesized that they utilize the same mechanism similar to an inverted pendulum in which kinetic energy is exchanged for potential (gravitational) energy and vice versa. This implies that fluctuations in kinetic and potential energy happen out of phase. These energy-based observations can be extended to other gaits: in bipedal running or hopping and in quadrupedal trotting, fluctuations of potential and kinetic energy happen in phase and both are exchanged for elastic energy.  However, this analysis breaks down for asymmetrical gaits of quadrupeds.

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Due to the lack of the additional pair of legs, a biped cannot move in a fashion that is dynamically similar to a galloping quadruped. In this project, we also explore the dynamic similarity between bipedal gaits and asymmetrical quadrupedal gaits by using simplistic passive models. These models are built on an extensive body of previous work that investigates the passive dynamics of legged locomotion.  In the present work, we employ our two models to reveal potential dynamic relationships between bipedal gaits on the one side and quadrupedal asymmetrical gaits on the other. By letting the inertia of the torso in the quadrupedal model vary from zero to infinitely large, we explicitly connect the two models and link all bounding gaits of the quadrupedal model to the two-legged gaits of the bipedal model.

Symmetry manifests itself in legged locomotion in a variety of ways. No matter where a legged system begins to move periodically, the torso and limbs coordinate with each other’s movements in a similar manner. Also, in many gaits observed in nature, the legs on both sides of the torso move in exactly the same way, sometimes they are just half a period out of phase. Furthermore, when some animals move forward and backward, their movements are strikingly similar as if the time had been reversed. This work aims to generalize these phenomena and propose formal definitions of symmetries in legged locomotion using group theory terminology. Symmetries in some common quadrupedal gaits such as pronking, bounding, half-bounding, and galloping have been discussed. Moreover, a spring-mass model has been used to demonstrate how breaking symmetries can alter gaits in a legged system. Studying the symmetries may provide insight into which gaits may be suitable for a particular robotic design, or may enable roboticists to design more agile and efficient robot controllers by using certain gaits.

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