M.S. Thesis Presentation by John Alexiou
Monday, March 15, 1999

(Dr. Harvey Lipkin, advisor)

"Projective Articulated Dynamics"

Abstract

In this thesis, it is proposed to solve for the accelerations and internal forces of multiple connected rigid bodies using recursive formulations and screw theory. Screw theory allows for compact notation of the equations of motion and kinematics. The concept of a single articulated inertia modeling the behavior of an entire subchain is used to simplify the equations such that a recursive solution is possible for systems with no kinematic loops. In planar cases, the graphical interpretation of the equations introduces the connection between projective geometry and dynamics with screw theory. Projective geometry uses subspace decompositions and projections to extract useful information from problems. Similarly in dynamics, to find the reaction forces on all the joints, the splitting of all the internal forces into active and reactive subspaces is required. The splitting of body accelerations into active and reactive parts is also needed in order to understand how each solution affects the next recursion. Together the two decompositions form a symmetric and dualistic set of projections that is used for both single rigid bodies and multiple articulated rigid body chains. These projections are applied to the equations of motion yielding a new set of recursive equations with fewer steps. Each projected articulated equation maintains the two separate parts of active and reactive components. Examining the special cases presented when either of the two parts is zero for either the forces or accelerations indicates the physical interpretation of the parts. A similar non-recursive formulation is introduced to solve systems with kinematic loops. Several planar and spatial examples are included to illustrate solutions to projections, open and closed loop accelerations, and articulated inertias. A MATLAB toolbox is developed, described, and used in some of the examples implementing screw theory and projective articulated dynamics. Suggestions on further development are made that enhance the usability and understanding of multibody dynamics with screw theory.