Two moving coordinate frames for sweeping along a 3D trajectory
Computer Aided Geometric Design
Computing frames along a trajectory
Computer Aided Geometric Design
Intersection approach to multi-point machining of sculptured surfaces
Computer Aided Geometric Design
Performance analysis of CNC interpolators for time-dependent feedrates along PH curves
Computer Aided Geometric Design
Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves
Computer Aided Geometric Design
Computation of rotation minimizing frames
ACM Transactions on Graphics (TOG)
Nonexistence of rational rotation-minimizing frames on cubic curves
Computer Aided Geometric Design
A three-dimensional configuration-space method for 5-axis tessellated surface machining
International Journal of Computer Integrated Manufacturing
Advances in Computational Mathematics
Inverse kinematics for optimal tool orientation control in 5-axis CNC machining
Computer Aided Geometric Design
| Hi-index | 0.00 |
When a ball-end milling tool cuts a given path on a smooth surface, it is desirable to maintain a fixed angle @j between the tool axis a and the local surface normal n at each point, to ensure a constant speed of the tool cutting edge against the surface. This means that the tool axis a must lie on a cone of angle @j about the surface normal n at each point, but its azimuthal position on this cone remains indeterminate. To resolve this indeterminacy, while minimizing actuation of the rotary axes that orient the workpiece relative to the tool, the component of a in the surface tangent plane is specified through the parallel transport of a given initial state along the path. This amounts to the integration of coupled first-order differential equations that involve the Christoffel symbols for the given surface. Alternatively, the tangent plane component of the tool axis a is shown to be rotation-minimizing with respect to the surface normal n, and its orientation relative to the Darboux frame along the tool path can be determined by integrating the geodesic curvature along that path. The method is illustrated by closed-form solutions for simple analytic surfaces, and numerical integration using an object-oriented software implementation for free-form surfaces. The real-time implementation of such rotation-minimizing 5-axis tool motions for free-form surfaces is well within the scope of modern CNC systems.