| This thesis presents a detailed kinematic analysis of three degree-of-freedom planar parallel manipulator platforms possessing topological symmetry, called general planar Stewart-Gough platforms (PSGP). A specific super-set of topologically asymmetric platforms and one with actuated holonomic higher pairs are included in the analysis.; After PSGP are described and classified, the remainder of the first portion is devoted to the review of the geometric and mathematical tools used in the analysis.; A single univariate polynomial is derived which yields the solutions to the forward kinematics problem of every PSGP platform. Kinematic mapping is used to represent distinct displacements of the platform as discrete points in a three-dimensional projective image space. Separate motions of each leg map to skew one-sheet hyper-boloids, or hyperbolic paraboloids, depending on the kinematic architecture of the leg. After two elimination steps the three quadric surfaces are reduced to a sixth order univariate. The roots of this polynomial reveal all solutions to the forward kinematics problem. The procedure leads to a robust algorithm which can be applied to the abovementioned super-set.; The inverse kinematics problem of these platforms is solved, in closed form, using the same kinematic mapping. The procedure can be applied to any three-legged planar platform with lower pairs, regardless of symmetry.; A workspace analysis and simple criteria for the determination of the existence of a dextrous workspace are presented. Finally, a geometric singularity and self-motion detection method, which does not employ Jacobian matrices, is discussed. |
