Mechanical Engineering ETDs

Publication Date



Conventional numerically controlled machine tools use straight line approximations for machining contoured surfaces or outlines. This thesis uses modern control theory to develop an optimal tracking system which will follow any surface defined by a second degree polynomial without resort to straight line approximation. To develop the optimal tracking system, the mathematical expressions required to define the dynamics of a two axis machine tool following a parabolic path are formulated. Next, the third order differential equation representing the slide, driven by a hydraulic motor which is controlled by an electrohydraulic servovalve, is changed into a first order matrix differential equation, called the state equation. This state equation is combined with the performance index, chosen for the system, to give a linear optimal control law. Before this optimal control law can be applied, a matrix Riccati type algebra equation must be solved for the optimal feedback matrix. A time response computer program (GTRESP) was used to compare the optimal tracking system, using the optimal control law, with a conventional straight line following system. For this comparison, data covering 2.6 seconds of travel from the apex of the parabolic path was used. With a travel rate of one inch per minute along the path, the tool would travel .043 inches in 2.6 seconds. Even in this short distance, the superiority of the optimal tracking system over the conventional system was very apparent. Errors in the conventional system were 4.5 times greater than in the optimal tracking system, and the optimal tracking system provided a much smoother path.

Degree Name

Mechanical Engineering

Level of Degree


Department Name

Mechanical Engineering

First Committee Member (Chair)

Glenn Frank Cochrane, Jr.

Second Committee Member

William Ernest Baker

Third Committee Member

Daniel Paul Peterson

Fourth Committee Member

Frederick Dsuin Ju

Document Type