Oscillator-based path planning for machining of a pocket shape in workpiece

1. A method for determining a path of a tool machining a pocket shape in a workpiece, wherein the pocket shape is a closed curve in an x-y plane, comprising:

determining positions and velocities of a motion of an oscillator having an energy defined by a function that is positive within a boundary of the pocket shape and equals zero on the boundary of the pocket shape;

determining coordinates of the path of the tool based on corresponding pairs of position and velocity values of the motion, wherein a position value of each pair represents an x coordinate of the path of the tool in the x-y plane, and a velocity value of each pair represents a y coordinate of the path of the tool in the x-y plane;

determining an equation describing the motion of the oscillator, wherein the equation includes a damping term;

solving the equation with respect to time to determine a set of position and velocity values of the motion of the oscillator for a corresponding set of time instants; and

determining coordinates of the path of the tool at each time instants based on corresponding position and velocity values of the motion;

updating the damping term;

updating the path of the tool by resolving the equation with the updated damping term;

comparing a metric of a performance of the machining with a threshold; and

repeating the updating the damping term and the path of the tool based on a result of the comparing, wherein steps of the method are performed by a processor.

2. The method of claim 1 , wherein the positions and the velocities of the motion of the oscillator are determined as a function of time, further comprising:

determining the coordinates as the function of time based on times of the corresponding pairs of position and velocity values.

3. The method of claim 1 , further comprising:

selecting an initial engagement point between the tool and a workpiece as a starting point of the motion of the oscillator.

4. The method of claim 1 , further comprising:

determining a metric of a performance of the machining with a threshold; and

repeating the updating of the damping term and the path of the tool if the metric does not satisfy a constraint.

5. The method of claim 4 , wherein the constraint is one or combination of a degree of a curvature, a length of the path, an angle of engagement of the tool, an amount of a removed volume, and a cutting force of the tool.

6. The method of claim 1 , further comprising:

determining an Rvachev (R)-function as the function of the energy of the oscillator;

determining an equation describing the motion of the oscillator using the R-function, wherein the equation includes a damping term;

solving the equation with different values of the damping term to produce a set of paths of the tool; and

selecting the path of the tool from the set corresponding to an optimal metric of performance.

7. The method of claim 6 , further comprising:

simulating the machining by the motion of the tool according to each path from the set of paths to produce a set of simulations; and

determining a metric of performance of each simulation from the set of simulations to produce a set of metrics; and

selecting an optimal metric of performance from the set of metrics according to a cost function.

8. The method of claim 7 , wherein the metric of performance includes an engagement surface of the tool moving along the path or a volume removed by the motion of the tool.

9. The method of claim 8 , wherein the metric of performance includes the engagement surface, comprising:

arranging a set of points on at least a part of a surface of the tool;

determining a distance between each point in the set of points and a surface of the workpiece modified by the motion; and

determining the engagement surface based on a subset of the points having the distance less than a threshold.

10. The method of claim 8 , wherein the metric of performance includes the volume removed by the motion of the tool, comprising:

generating a swept volume for a segment of the path of the tool, the segment defines the motion of the tool from a previous position to a next position of the tool during the simulation;

casting rays through the swept volume to represent a motion of the tool in relation to the workpiece;

determining a portion of volume removed from the workpiece by the swept volume based on intersections of at least some rays with the workpiece; and

repeating the generating, the casting and the determining the portion of the volume steps for all segments of the path to produce the volume removed by the motion of the tool.

11. A system for determining for determining a path of a tool machining a pocket shape given by a closed curve on an x-y plane, comprising:

a processor for determining positions and velocities of a motion of an oscillator having an energy defined by a function that is positive within a boundary of the pocket shape and equals zero on the boundary of the pocket shape and for determining coordinates of the path of the tool based on corresponding pairs of position and velocity values of the motion, wherein a position value of each pair represents an x coordinate of the path of the tool on the x-y plane, and a velocity value of each pair represents a y coordinate of the path of the tool on the x-y plane,

wherein the processor determines an equation describing the motion of the oscillator, wherein the equation includes a damping term, and the processor solves the equation with respect to time to determine a set of position and velocity values of the motion of the oscillator for a corresponding set of time instants, determines coordinates of the path of the tool at each time instants based on corresponding position and velocity values of the motion, updates the damping term, updates the path of the tool by resolving the equation with the updated damping term, compares a metric of a performance of the machining with a threshold, and repeats the updating the damping term and the path of the tool based on a result of the comparing.

12. The system of claim 11 , further comprising:

a simulation system for simulating the machining of a workpiece according to the path to determine a metric of performance of the path.

13. A non-transitory computer readable recording medium for determining a path of a tool machining a pocket shape in a workpiece, having stored thereon, instructions that when executed by a processor, cause the processor comprising steps of:

determining positions and velocities of a motion of an oscillator having an energy defined by a function that is positive within a boundary of the pocket shape and equals zero on the boundary of the pocket shape, wherein the pocket shape is a closed curve in an x-y plane;

determining coordinates of the path of the tool based on corresponding pairs of position and velocity values of the motion, wherein a position value of each pair represents an x coordinate of the path of the tool in the x-y plane, and a velocity value of each pair represents a y coordinate of the path of the tool in the x-y plane;

determining an equation describing the motion of the oscillator, wherein the equation includes a damping term;

solving the equation with respect to time to determine a set of position and velocity values of the motion of the oscillator for a corresponding set of time instants; and

determining coordinates of the path of the tool at each time instants based on corresponding position and velocity values of the motion;

updating the damping term;

updating the path of the tool by resolving the equation with the updated damping term;

comparing a metric of a performance of the machining with a threshold; and

repeating the updating the damping term and the path of the tool based on a result of the comparing.