| This dissertation is mainly concerned with the kinematic and probabilistic study of chain-like systems, especially robotic and bio-macromolecular systems such as DNA, RNA, and protein molecules. As a major tool in this study, Lie-group-theoretic methods are extensively used both in deterministic and probabilistic scenarios. First, we address how to find inverse kinematic solutions of 6-D.O.F. short segments of a biopolymer structure. We propose new methods for this study. One of our methods is based on inverse kinematic solution techniques which have been developed for "general" 6R serial robotic manipulators. The other is a Jacobian-based method which can be applied to both the general 6R manipulators and biopolymers. We show that these two methods can be combined to obtain the inverse kinematic solution more efficiently and accurately. Secondly, we present a method of determining the equilibrium conformations of chiral semi-flexible polymers such as double stranded DNA molecules with end constraints. We model this kind of polymer as a continuous thin elastic rod, which can be considered to be either inextensible or extensible. We also develop a new inverse kinematics procedure to obtain solutions which meet given end constraints. In this work, we utilize variational calculus on Lie groups. The proposed method can incorporate any type of elastic thin rod model of semi-flexible polymers with quadratic elastic potential energy. The third part is devoted to the study of conformational statistics of classical polymer and polypeptide chain models. This work employs probabilistic approaches on Lie groups. Specifically, we utilize the generalized convolution concept and the Fourier transform for the rigid-body motion group to obtain the probability density function of the end-to-end distance. We also apply this technique to the workspace density generation of redundant robotic manipulators, which is useful in their inverse kinematics and path planning. The kinematics of continuous chains is also applied to another interesting field, the kinematic modeling of flexible needle steering. First, two basic kinematic models previously developed are presented, in which Lie-group-theoretic description is employed. Then we present a new kinematic model, the "integral" kinematic model, which introduces a small deformation along the needle trajectory. We compare the predictions from this model with the experimental results. This theoretical model can be useful in minimally invasive surgical execution and planning. This work enhances the medical applications of robotic systems. |
