Computational Geometry for Optimal Workpiece Orientation

Workpiece orientation is formulated as an optimal design problem based on a discrete approximation of design surface geometry, the kinematic capabilities of the process machine tool, and processing cost. The primary process application addressed is three-and four-axis numerically controlled (NC) milling, although the techniques presented may be applied to machines with more general articulation. Recent developments in applied spherical geometry are employed to formulate a constrained problem, and furthermore, a nonlinear optimization problem. For three-axis milling applications, a weight is assigned to each surface normal of the discrete model corresponding to the actual area it represents. Workpiece/machine orientation is optimized such that the angle between the weighted normals and the milling tool axis is minimized. This formulation is augmented, for four-axis milling, to incorporate limitations of the rotational degree of freedom, into the optimization formulation. The influence of tool geometry is also discussed and incorporated within constrained orientation algorithm.