Theory of Equations. By J. V. Uspensky. Pp. vii, 353. 27s. 1948. (McGraw-Hill)

Real time control of five-axis machine tools requires smooth generation of feed, acceleration and jerk in CNC systems without violating the physical limits of the drives. This paper presents a feed scheduling algorithm for CNC systems to minimize the machining time for five-axis contour machining of sculptured surfaces. The variation of the feed along the five-axis tool-path is expressed in a cubic B-spline form. The velocity, acceleration and jerk limits of the five axes are considered in finding the most optimal feed along the tool-path in order to ensure smooth and linear operation of the servo drives with minimal tracking error. The time optimal feed motion is obtained by iteratively modulating the feed control points of the B-spline to maximize the feed along the tool-path without violating the programmed feed and the drives’ physical limits. Long tool-paths are handled efficiently by applying a moving window technique. The improvement in the productivity and linear operation of the five drives is demonstrated with five-axis simulations and experiments on a CNC machine tool.

http://research.engr.oregonstate.edu/mpcl/resources/Feed-optimization-for-ﬁve-axis-CNC-machine-tools-with-drive-constraints.pdf

Feed optimization for five-axis CNC machine tools with drive constraints

Straight lines, or G01 blocks, are the most widespread representation form for the tool path in CNC machining. At the junctions between consecutive segments, the tangency and curvature discontinuities may lead to feedrate fluctuation and acceleration oscillation, which would deteriorate the machining efficiency and quality. To solve this problem, a real-time path-smoothing method is proposed, which adopts a curvature-continuous B-spline with five control points to blend the adjacent straight lines. The advantage of the transition scheme is that, G2 continuity, analytical calculation of the curvature extrema, approximation error control and real-time performance are considered simultaneously. Then, a bidirectional scanning algorithm for jerk limited S-shape feedrate profile is proposed to evaluate the feedrate constraints. On this basis, a real-time look-ahead scheme, which comprises of path-smoothing, bidirectional scanning and feedrate scheduling, is developed to acquire a feedrate profile with smooth acceleration. Also, an arc-length based interpolation algorithm for mixed linear and parametric segments is proposed to overcome the difficulty of crossing different segments. With these schemes, the smoothness of both tool path and feedrate is guaranteed. Simulation and experiments on an X–Y–Z platform are conducted. The results demonstrate the feasibility and efficiency of the present algorithms.

A real-time look-ahead interpolation methodology with curvature-continuous B-spline transition scheme for CNC machining of short line segments

Competitive pressure requires manufacturers to simultaneously address increasingly stringent constraints on both productivity and quality. From the perspective of numerically controlled (NC) machine tools, this means higher machining performance in terms of speed and accuracy. Conventional approaches to programming NC operations involve selecting a constant feedrate for a given operation to produce acceptable performance (operation time and contouring accuracy). In this paper, we examine the possibility of scheduling or varying the feedrate by taking into consideration the geometry of the contour that the machine is expected to follow and the physical capabilities of the machine (i.e., its maximum velocity, acceleration and jerk constraints). Previous work by the authors has addressed the efficient, off-line computation of time-optimal trajectories with constraints on velocity and acceleration. This paper introduces additional constraints on the permissible jerk (rate of change of acceleration) on the machine's axis. From a practical perspective, excessive jerk leads to excitation of vibrations in components in the machine assembly, accelerated wear in the transmission and bearing elements, noisy operations and large contouring errors at discontinuities (such as corners) in the machining path. The introduction of jerk into the feedrate scheduling problem makes generating computationally efficient solutions while simultaneously guaranteeing optimality a challenging problem. This paper approaches this problem as an extension of our previous bi-directional scan algorithm [23] , [29] . A new acceleration-continuation procedure is added to the feedrate optimization algorithm to address jerk constraints and remove discontinuities in the acceleration profile. The algorithm maintains computational efficiency and supports the incorporation of a variety of state-dependent (such as position, velocity, acceleration and jerk) constraints. By carefully organizing the local search and acceleration continuity enforcing steps, a globally optimal solution is achieved. Singularities, or critical points, and critical curves on the trajectory, which are difficult to deal within optimal control approaches, are treated in a natural way in this algorithm. Several application examples and tests are performed to verify the effectiveness of this approach for high-speed contouring.

Feed-rate optimization with jerk constraints for generating minimum-time trajectories

A key role of the CNC is to perform the feedrate interpolation which consists in generating the setpoints sent to each axis of a machine tool based on a NC program In high speed machining, the feedrate is limited by the velocity, acceleration and jerk of each axis of the machine tool The algorithm presented in this paper aims to obtain an optimized feedrate profile which makes best use of the kinematical characteristics of the machine This minimum time feedrate profile is computed by intersecting all the constraints due to the drives in an iterative algorithm It is worth noting that both tangential jerk and axis jerk are taken into consideration The proposed VPOp (Velocity Profile Optimization) method is universal and can be applied to any articulated mechanical structure as it is demonstrated in the examples Moreover the algorithm has been implemented for various formats: linear interpolation (G1) and NURBS interpolation in 3- and 5-axes The effectiveness of the algorithm is demonstrated thanks to a comparison with an industrial CNC and can be freely tested using the VPOp software which is available on the Internet http://webservlurpaens-cachanfr/geo3d/premium/vpop 

/pdf/feedrate-interpolation-with-axis-jerk-constraints-on-5-axis-52ejjhzc01.pdf

Feedrate interpolation with axis jerk constraints on 5-axis NURBS and G1 tool path

The problem of specifying the feedrate variation along a curved path, that yields minimum traversal time for a 3-axis CNC machine subject to constraints on the feasible acceleration along each axis, is addressed. In general, this time-optimal feedrate incurs bang-bang control, i.e., maximum acceleration/deceleration is demanded of at least one axis throughout the motion. For a path defined by a polynomial parametric curve r(xi), we show that the (square of the) time-optimal feedrate can be determined as a piecewise-rational function of the curve parameter xi, with break-points corresponding to the roots of certain polynomial equations. Furthermore, this type of feedrate function is amenable to a real-time interpolator algorithm that drives the machine directly from the analytic curve description, eliminating the need for linear/circular G code approximations. The theoretical and computational aspects of such time-optimal feedrate functions are presented, together with experimental results from their implementation on a 3-axis mill driven by an open-architecture software controller. (C) 2004 Elsevier Ltd. All rights reserved.

/pdf/algorithms-for-time-optimal-control-of-cnc-machines-along-54seda7375.pdf

Algorithms for time-optimal control of CNC machines along curved tool paths

Algorithms are developed to compute the feedrate variation along a curved path, that ensures minimum traversal time for a 3-axis CNC machine subject to both fixed and speed-dependent axis acceleration bounds arising from the output-torque characteristics of the axis drive motors. For a path specified by a polynomial parametric curve, the time-optimal feedrate is determined as a piecewise-analytic function of the curve parameter, with segments that correspond to saturation of the acceleration along one axis under constant or speed-dependent limits. Break points between the feedrate segments may be computed by numerical root-solving methods. For segments that correspond to fixed acceleration bounds, the (squared) optimal feedrate is rational in the curve parameter. For speed-dependent acceleration bounds, the optimal feedrate admits a closed-form expression in terms of a novel transcendental function whose values may be efficiently computed, for use in real-time control, by a special algorithm. The optimal feedrate admits a real-time interpolator algorithm, that can drive the machine directly from the analytic path description. Experimental results from an implementation of the time-optimal feedrate on a 3-axis CNC mill driven by an open-architecture software controller are presented. The algorithm is a significant improvement over that proposed in [Farouki RT, Rajan VT. Algorithms for polynomials in Bernstein form. Comput Aided Geometric Des 1988;5:1-26], since the addition of motor voltage constraints precludes the possibility of arbitrarily high speeds along linear or near-linear path segments.

Time-optimal traversal of curved paths by Cartesian CNC machines under both constant and speed-dependent axis acceleration bounds

Minimum-time traversal of curved paths by Cartesian CNC machines, subject to prescribed bounds on the magnitude of acceleration along each axis, usually involves a “bang-bang” control strategy in which the acceleration bound is realized by one or another of the machine axes at each instant during the motion. For a path specified by a polynomial parametric curve and prescribed acceleration bounds, the time-optimal feedrate may be expressed in terms of a C0 piecewise-rational function of the curve parameter. This function entails sudden changes in either the identity of the limiting axis, or the sign of acceleration on a single limiting axis, incurring demands for instantaneous changes of motor torque that may not be physically realizable. A scheme is proposed herein to generate smoothed C1 (slightly sub-optimal) feedrate functions, that incur only finite rates of change of motor torque and remain consistent with the axis acceleration bounds. An implementation on a 3-axis CNC mill driven by an open-architecture software controller is used to illustrate this scheme.

Smoothing of time-optimal feedrates for cartesian CNC machines

The authors consider the problem of specifying the feedrate variation along a curved path, that yields the minimum traversal time for a multi-axis CNC machine subject to given bounds on the feasible velocity and acceleration along each axis. The torque-speed characteristics of the axis drive motors are first discussed, and are used to determine constraints on the velocity and acceleration for each axis. For a path specified by a polynomial parametric curve r(ξ), it is shown that (the square of) the time-optimal feedrate can be determined as a piecewise-rational function of the curve parameter ξ, with break-points that correspond to the roots of certain polynomials. In general, each feedrate segment is characterized by the saturation of either velocity or acceleration on one machine axis at each instant throughout the motion. The feedrate function admits a real-time interpolator algorithm that can drive the machine from the analytic curve description, eliminating the need for piecewise-linear/circular G code approximations. The theoretical and computational aspects of such time-optimal feedrates are presented, along with experimental results from implementation on a three-axis CNC mill driven by an open-architecture software controller. Compared to prior time-optimal feedrate algorithms (based on acceleration bounds only), the new algorithms give physically valid feedrates at high speeds, where the motor voltage ratings become a limiting factor.

Time-optimal feedrates along curved paths for Cartesian CNC machines with prescribed bounds on axis velocities and accelerations

Erratum to “Time-optimal traversal of curved paths by Cartesian CNC machines under both constant and speed-dependent axis acceleration bounds”: [Robot. Comput. Integr. Manuf. 24(1) (2008) 16–31]

Sebastian D. Timar

Papers

Algorithms for time-optimal control of CNC machines along curved tool paths

Time-optimal traversal of curved paths by Cartesian CNC machines under both constant and speed-dependent axis acceleration bounds

Smoothing of time-optimal feedrates for cartesian CNC machines

Time-optimal feedrates along curved paths for Cartesian CNC machines with prescribed bounds on axis velocities and accelerations

Erratum to “Time-optimal traversal of curved paths by Cartesian CNC machines under both constant and speed-dependent axis acceleration bounds”: [Robot. Comput. Integr. Manuf. 24(1) (2008) 16–31]