| On a 3-axis CNC milling machine, the cutting tool and work piece move relative to one another along the three Cartesian axes in accordance with the specified tool path r(xi) and feedrate v. The motion is controlled by a computer and actuated by DC motors. This study concerns specification of the feedrate v(xi) providing minimum-time traversal of a curved path, provided axes constraints deriving from actuator physical limitations. The study develops and implements generic feedrate computation and machine interpolator algorithms for three axis constraint cases: (1) constant acceleration limits, (2) constant acceleration and speed limits, and (3) constant and speed-dependent acceleration limits. The solution employs bang-bang control, demanding maximum or minimum acceleration of any one axis throughout traversal. In each case, cutting forces, friction, and other effects are neglected, thus only inertial effects are considered. Implementation examples are provided for several representative curves in each axis constraints case.; The study applies, and extends, algorithms developed in the robotics field in the 1980s concerning minimum-time traversal of an n-link robotic manipulator end-effector along a specified path, provided symmetric joint torque limits. The unique application to Cartesian machines admits closed-form optimal feedrate expressions, with several novel outcomes: feedrates are expressed as a piecewise-analytic functions of the curve parameter; feedrate computation shifts from numerical integration to linear programming and polynomial root solving; an error is highlighted in the established feedrate computation algorithm, prompting a correction and summary; and the feedrate furnishes interpolator algorithms for real-time machine control. The study applies the corrected robotics algorithm to the first axis constraint case, then extends the algorithm to consider the second and third cases. |
