Construction of G~2 rounded corners with Pythagorean-hodograph curves

摘要

The problem of designing smoothly rounded right-angle corners with Pythagorean-hodograph (PH) curves is addressed. A G~1 corner can be uniquely specified as a single PH cubic segment, closely approximating a circular arc. Similarly, a G~2 corner can be uniquely constructed with a single PH quintic segment having a unimodal curvature distribution. To obtain G~2 corners incorporating shape freedoms that permit a fine tuning of the curvature profile, PH curves of degree 7 are required. It is shown that degree 7 PH curves define a one-parameter family of G~2 corners, facilitating precise control over the extremum of the unimodal curvature distribution, within a certain range of the parameter. As an alternative, a G~2 corner construction based upon splicing together two PH quintic segments is proposed, that provides two free parameters for shape adjustment. The smooth corner shapes constructed through these schemes can exploit the computational advantages of PH curves, including exact computation of arc length, rational offset curves, and realtime interpolator algorithms for motion control in manufacturing, robotics, inspection, and similar applications.