Universal Dynamics of Damped-Driven Systems: The Logistic Map as a Normal Form for Energy Balance

Damped-driven systems are ubiquitous in engineering and science. Despite the diversity of physical processes observed in a broad range of applications, the underlying instabilities observed in practice have a universal characterization which is determined by the overall gain and loss curves of a given system. The universal behavior of damped-driven systems can be understood from a geometrical description of the energy balance with a minimal number of assumptions. The assumptions on the energy dynamics are as follows: the energy increases monotonically as a function of increasing gain, and the losses become increasingly larger with increasing energy, i.e. there are many routes for dissipation in the system for large input energy. The intersection of the gain and loss curves define an energy balanced solution. By constructing an iterative map between the loss and gain curves, the dynamics can be shown to be homeomorphic to the logistic map, which exhibits a period doubling cascade to chaos. Indeed, the loss and gain curves allow for a geometrical description of the dynamics through a simple Verhulst diagram (cobweb plot). Thus irrespective of the physics and its complexities, this simple geometrical description dictates the universal set of logistic map instabilities that arise in complex damped-driven systems. More broadly, damped-driven systems are a class of non-equilibrium pattern forming systems which have a canonical set of instabilities that are manifest in practice.Comment: 26 pages, 31 figure

Stabilizing Highly Dynamic Locomotion in Planar Bipedal Robots with Dimension Reducing Control.

In the field of robotic locomotion, the method of hybrid zero dynamics (HZD) proposed by Westervelt, Grizzle, and Koditschek provided a new solution to the canonical problem of stabilizing walking in planar bipeds. Original walking experiments on the French biped RABBIT were very successful, with gaits that were robust to external disturbances and to parameter mismatch. Initial running experiments on RABBIT were cut short before a stable gait could be achieved, but helped to identify performance limiting aspects of both the physical hardware of RABBIT and the method of hybrid zero dynamics. To improve upon RABBIT, a new robot called MABEL was designed and constructed in collaboration between the University of Michigan and Carnegie Mellon University. In light of experiments on RABBIT and in preparation for experiments on MABEL, this thesis provides a theoretical foundation that extends the method of hybrid zero dynamics to address walking in a class of robots with series compliance. Extensive new design tools address two main performance limiting aspects of previous HZD controllers: the dependence on non-Lipschitz finite time convergence and the lack of a constructive procedure for achieving impact invariance when outputs have relative degree greater than two. An analytically rigorous set of solutions - an arbitrarily smooth stabilizing controller and a constructive parameter update scheme - is derived using the method of Poincare sections. Additional contributions of this thesis include the development of sample-based virtual constraints, analysis of walking on a slope, and identification of dynamic singularities that can arise from poorly chosen virtual constraints.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58477/1/morrisbj_1.pd

The ecomorphology of facultative bipedality in Lepidosauria: implications for the evolution of reptilian bipedality

Bipedality is a distinctive locomotor characteristic of some of the most noteworthy animals of all time, including dinosaurs and humans. However, the evolution of a bipedal locomotor mode is poorly understood in reptiles. It has been repeatedly hypothesised that a facultative locomotor mode, where an animal moves both bipedally and quadrupedally under different conditions, forms an intermediate stage in the evolution of obligate bipedality. I demonstrate that the evidence supporting this hypothesis is lacking, recovering facultative bipedality as an intermediate stage only once in multiple independent evolutions of bipedality, under two different topologies. In order to better understand facultative bipedality and the associated anatomies, I performed multiple studies into the ecomorphology and evolution of this behaviour in a modern clade: Lepidosauria. Linear morphometric studies accounting for variation in body size indicate that forelimb segment lengths across locomotor modes do not differ for lepidosaurs of the same size, but that distal hindlimbs segments differ greatly, contrasting with historical tropes. Using 3D landmark-based geometric morphometrics, I demonstrate that arboreal and facultatively bipedal species share many characteristics in the bony elements of the pelvis, including a straight-to-concave iliac blade and large ischial base. These shared anatomies are functionally qualified based on anatomical studies of lepidosaur pelvic girdle myology, and indicate a similarity in mechanical demands of both arboreality and facultative bipedality. Finally, I tested for the correlated evolution of facultative bipedality with substrate preference in a derived clade of squamates: Episquamata. Findings suggest that there is no correlation between substrate and a facultative locomotor mode, instead indicating that facultative bipedality is an exaptation of anatomies associated with vertically diverse environments. This echoes the evolution facultative bipedality in hominin, macropods and rodents, and is distinct from any current hypotheses concerning the evolution of archosaurian bipedality