## LSU Historical Dissertations and Theses

1993

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mechanical Engineering

William P. Mounfield, Jr

#### Abstract

Optimal design of a six degrees-of-freedom, fully parallel manipulator, called a Stewart platform is investigated. In order to optimize the mechanism, new performance measures are introduced since use of the previous methods suffer from lack of physical meaning due to dimensional inhomogeneity. To overcome the dimensional inhomogeneity problem, an Euclidean norm definition of each output space with homogeneous dimension is used to find input-output norm relation. As a result, four sets of eigenvalues are obtained which characterize translational and rotational velocity, force and torque, and position and orientation accuracy. From the four sets of eigenvalues, four determinant measures are defined, which represent the magnitude of the input-output transformation and four condition number measures are defined which are indices of uniform transformation. The invariant property of the new measures is investigated under the scaling operation. By the simplification of the design problem, the explicit equations of performance measures are derived which provide the valuable tools to analyze the parametric space of the design variables. Using the explicit formulation, singular configurations can be identified at home positions of the manipulator. It is shown that parameters satisfying the isotropic condition form the surfaces of the simple geometric entities, called the cones and cylindroids in cylindrical coordinates. These geometric entities provide the insight to figure out the behavior of the performance measures in the parametric space. Using the geometric entities, three optimum solutions are found: one for force capacity and position accuracy and one for the torque capacity and rotation accuracy and one for both aspects. It is shown that there are two isotropic surfaces corresponding to each given condition number not equal to one and in all the region bounded by the two surfaces the condition numbers are less than the given condition number. Using these facts, a minimax problem is solved for condition number measures. It is shown that the achievable minimum condition number is obtained when the geometric average of the upper and lower limit of the operating height is on the isotropic surface. The result is used to determine the adequate operating range.

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