On the design and analysis of freeform motions using subdivision schemes.

摘要

This dissertation aims to develop new approaches for motion design and analysis suitable for CAD-CAM integration. Therefore, not only shape but also kinematic information should be conveyed in an efficient manner that can be easily implemented in CAD-CAM systems and eventually interpreted by CNC manufacturing equipment.;Dual quaternions, quaternions, and planar quaternions are used to represent spatial, spherical, and planar displacements, respectively. In this way, displacements of a rigid body in Cartesian space are mapped into points in the image space of displacements, transforming the kinematic problem of motion interpolation into a geometric problem where the powerful techniques for curve and surface interpolation from Computer Aided Geometric Design (CAGD) can be readily applied.;Motivated by this consideration, initially in this dissertation a set of orientations with angular velocity constraints are interpolated by means of quaternion biarcs. The resulting quaternion curve represents a piecewise spherical line-symmetric rational motion with C1 continuity which is used for tool path generation in 5-axis machining. Next, the four point interpolatory subdivision scheme for curve generation is adapted to the interpolation of a given set of positions of a cylindrical tool represented by dual quaternions. It is shown that the resulting discrete model of the tool path lends itself naturally to an algorithm for computing the characteristic curve belonging to the boundary surface of the swept volume at each of the discrete positions as well as the contribution from the top and bottom planes, and circular edges of the cylinder. Then, the dual tensor-product extension of such subdivision scheme is also used for two-parameter motion generation and swept surface computation.;A preliminary approach for mechanism simulation using subdivision schemes is also attempted by analyzing the motion of the coupler link of a planar 4R closed kinematic chain from the viewpoint of constrained motion interpolation. In the early stage of this refinement process, each new in-between position must be made to satisfy the 4R kinematic constraints exactly to ensure the correct motion. When there is sufficient number of coupler positions, one can use the unconstrained four-point interpolatory scheme to generate the in-between positions to allow for fast animation of the coupler motion.;The results obtained have applications in CNC tool path generation, robot path planning, and computer animation.