S-Curve Algorithm of Acceleration/Deceleration With Smoothly-Limited Jerk in High-Speed Equipment Control Tasks

Abstract

The needs of increase in accuracy, quality and shape complexity of parts manufactured in high-speed equipment is a trend of industry. Problems of ensuring the accuracy at high velocities of machine organs motions are formulated. Machine organs movements accuracy is determined by smoothness of their motions and depends on the smoothness of the trajectory and the law of velocity changing along the trajectory. Algorithm of velocity changing along trajectory using S-curve acceleration-deceleration with smoothly-limited jerk is proposed. Experimental verification of the accuracy CNC machine organs motions at their control with using proposed S-curve algorithm showed a decrease in positioning errors.

Spline Curves Formation Given Extreme Derivatives

“…Figure 8 depicts visual rendering of the result of the calculation. It demonstrates the sought three segments and their subsequent connection with the order of smoothness 4 C . The values of curvature and torsion at boundary points j P , 1,..., 4 j = of the connected segments are listed in Table 1.…”

“…Simultaneously, these equations unambiguously define the unknown coefficients of the polynomials of the connected segments. A general cubic spline curve consisting of ( 1) j − segments, where j is the number of points of the curve, is formed through a total number of N conditions, where N is obviously defined through formula 4( 1) N j = − . This number is also the total number of the unknown coefficients of all polynomials of the third degree describing the connected segments.…”

“…In order to construct a closed Hermite spline curve of i segments subsequently connected in the given j knot points with smoothness 4 C , it is required to fulfill the following boundary conditions at connection points: 1 1 ( 0)…”

“…One of the main problems in development of high-speed cutting NC systems is achieving seamless cutting tool motion. The models of NC systems developed for this purpose ensure smoothness of acceleration a(t) and continuity of jerk defined by the third vector derivative [3][4][5]: j = da dt = d 3 r dt 3 . The fourth vector derivative is called snap dj dt = d 4 r dt 4 .…”

“…ED and SC (4) ED formation in the form of B-spline curves interpolating discrete arrays of points in the plane [20]. The paper considers vector derivatives of orders one to three as given extreme derivatives.…”