AbstractThis dissertation deals with the problem of task centered design of mechanisms and robotic systems via novel concept of variational kinematic geometry. The results can be extended to other field such as biomechanics, structure chemistry and protein kinematics, as well as micro- and nano- systems, in which kinematics playsan important roles. Since the early days of Industrial Revolution, machine theorists and kinematicians have sought to develop a theory to analyze and synthesize mechanisms so that engineers could approach the problem in a rational way. This dissertation introduces a task-centered approach to mechanism design using a constraint basedparadigm, and conducts a comparative study on the kinematic and the geometric constraints of the motion.Kinematic-constraint based synthesis approach, derived from the classical viewpoint that a kinematic mechanism is a collection of kinematic links connected with kinematic pairs (or joints), deals with the determination of mechanism types (type synthesis) and/or their link dimensions (dimensional synthesis). Hence, this approachis referred to as two-step-based mechanism-centric design paradigm.This dissertation advocates a geometric-constraint based approach. Following this approach, a designer would focus on the analysis of point (or line) trajectories associated with the motion, the goal of which is to obtain a trajectory that can be constructed as a geometric condition or constraint that best describes the motion.Typically this is done in a geometric constraint identification and acquisition process, i.e., by comparing various trajectories of a specified motion with known constraints from a library of mechanically realizable constraints. The resulting feasibleconstraints can be used directly for the simultaneous type and dimensional synthesis of a physical device such as mechanical linkage that generates the specified motion task. This effectively reduces the problem of mechanism synthesis to that of constraint identification and acquisition and thus bridges the gap betweentype and dimensional synthesis. Furthermore, as a mechanism is defined by a combination of geometric constraints, the geometric-constraint based approach reduces the complexity in type synthesis significantly. This approach to motion modeling has similarity to constraint based shape modeling in Variational Geometry and istherefore referred to as Variational Kinematic Geometry.