Probabilistic modeling, Lie groups, and design: Applications in biomolecular modeling and advances in self-reconfigurable modular robots

The fields of structural biology and robotics may seem to be very different and in many ways are. However, a number of mathematical tools can be used to describe phenomena that arise in both. For example, if we consider a set of DNA fragments that have equilibrated onto a planar surface and a series of trajectories observed for a robot subjected to noisy inputs, the end-to-end distributions look very similar. Thus, the probability density functions used to describe both may be taken to be of similar form. This same scenario arises for many systems that evolve on the motion groups of rotations and rigid-body transformations. This dissertation primarily addresses uncertainty and, in particular, uncertainty that arises for things in motion.;These issues are explored in a variety of ways including: (1) improving methods for updating probabilistic estimates of systems evolving on motion groups; (2) developing methods to describe and compare models of various biological macromolecules, namely nucleic acids and proteins; (3) observing and designing solutions to reduce uncertainty with respect to a new modular robotic system; and (4) investigating methods to compensate for uncertainty when encoding rotary motion. Rotations and rigid-body transformations have been used and studied from a number of perspectives. The first part of this work deals with probability density functions taken over motion groups. Approximations are presented for fusing two distributions with a form similar to a typical multivariate Gaussian, but whose argument is a motion group element. Similar ideas and distributions are then applied to four coarse-grained models of double-helical DNA and RNA. These models and the motion group framework allow for reconciliation that has not been provided previously. The first part concludes with new metrics for comparing protein conformations when the conformations are not deterministic but rather represented as probabilistic ensembles.;An overview of an independently mobile self-reconfigurable modular robotics system is also presented. The wheels of each module serve as docking surfaces; this presents a new nonholonomic path-planning problem. We provide a solution to this problem along with a design for a low-cost variant of the system. Experiments were conducted to demonstrate the uncertainty inherent in this prototype and design decisions were made that help to reduce this uncertainty. Finally, a new rotary encoding scheme is demonstrated that reduces the effect of noise introduced when sampling through an imperfect communication channel.;Uncertainty arises in many real-world applications. This work serves to provide methods for better quantifying this uncertainty and coping with it through design. It also demonstrates that a better understanding of the probabilistic nature of many systems may allow for more information to be obtained about those systems.