Optimal tool orientation control for 5-axis CNC milling with ball-end cutters

doi:10.1016/j.cagd.2012.11.003

Computer Aided Geometric Design

Volume 30, Issue 2, February 2013, Pages 226-239

https://doi.org/10.1016/j.cagd.2012.11.003 Get rights and content

Abstract

When a ball-end milling tool cuts a given path on a smooth surface, it is desirable to maintain a fixed angle ψ 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 ψ 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.

Highlights

► Inclination of ball-end tool maintains constant cutting speed. ► Azimuthal position of tool axis about surface normal determined by parallel transport in surface tangent plane. ► Tool axis orientation is also rotation-minimizing with respect to the surface normal. ► Closed-form solutions possible for simple analytic surfaces. ► Numerical integration scheme for free-form (NURBS) surfaces.

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Tool orientation optimization with interference-free condition and good machining performance is one of the key issues in the five-axis machining processes. The existing tool orientation optimization algorithms usually reduce the contour errors indirectly through reducing each order derivative of the trajectory. In this article, a tool orientation optimization algorithm, which achieves the direct constraining of the tool tip and tool orientation contour errors while avoiding the interference between the tool and workpiece, is proposed for the first time. Two prediction models, i.e. an analytical model and a numerical model, are developed to predict the contour errors according to the tool orientation splines. Through considering multiple factors such as the real feedrate, the dynamics of the servo control system, the friction disturbance torque, the velocity, acceleration and jerk of the servo axes, the developed prediction models realize the precise prediction of contour errors. Through transforming the contour errors constraints and interference-free constraints into penalty terms, the constrained optimization problem is transformed to an unconstrained problem. Experiments validate that through optimizing the tool orientations, the maximum tool tip and tool orientation contour errors can be reduced by 52.2% and 75.6%.
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Citation Excerpt :
It have been demonstrated that the simplification of the GFDs can facilitate indeed the solution of the constrained optimization of tool orientation, but there is still room for improving, e.g. simpler linear description for irregular GFDs to be discussed in detail in this paper. In addition, although it have been noticed that tool orientations can change the cutter-workpiece engagement, therefore affecting the cutting performance [11–14,28,29], most of works still focuses separately on eliminating the machining interferences [5,15], improving the kinematics performance [20–22,25–27] or cutting performance [11,14,28,29]. To best of our knowledge, no work takes into account the requirement of no machining interferences, kinematics capacities of rotary axes and cutting performance at the same time in adjusting tool orientations.
For 5-axis ball-end machining, it is desired to maintain the expected cutting performance of tool orientation when adjusting tool orientation for improving the motions of rotary axes of 5-axis machine. For this purpose, a cutting performance maintained (CPM) method is proposed to adjust the tool orientations, the objective of which is to minimize the sum of the absolute deviations between the initial and adjusted coordinates of the rotary axes while improving the kinematics performance of the rotary axes and ensuring no machining interferences. In order to speed up the solving of the optimization objective, the analytical linear representations for the drive limits of rotary axes and especially irregular geometry feasible domains (GFDs) of tool orientations are first discussed in detail. The nonlinearity of the objective function is then eliminated by introducing two new auxiliary variables for further simplifying the computation of optimal tool orientation. After rewriting the drive limits and GFD constraints with the two auxiliary variables, the linear objective function can be efficiently solved by the simple linear programming method. The tool orientations adjusted by the proposed CPM method can not only improve the interference-free motions of the rotary axes, but also can maintain the expected cutting performance. Finally, the computer simulation and real milling were conducted to validate the proposed method.
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This paper presents a novel jerk minimization algorithm in the context of multi-axis flank CNC machining. The toolpath of the milling axis in a flank milling process, a ruled surface, is reparameterized by a B-spline function, whose control points and knot vector are unknowns in an optimization-based framework. The total jerk of the tool’s motion is minimized, implying the tool is moving as smooth as possible, without changing the geometry of the given toolpath. Our initialization stage stems from measuring the ruling distance metric (RDM) of the ruled surface. We show on several examples that this initialization reliably finds close initial guesses of jerk-minimizers and is also computationally efficient. The applicability of the presented approach is illustrated by some practical case studies.
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When adopting 5-axis machine to mill the parts, it is desired to avoid the drastic change of tool orientation for improving the kinematics performance of 5-axis machining while ensuring no machining interferences. For this purpose, a kinematics performance oriented smoothing method is proposed to plan the tool orientations, which is focused specifically on minimizing the angular accelerations imposed on the rotary axes of 5-axis machine. In this method, with several specified representative tool orientations (RTOs), two B-spline curves, which represent the displacements of the rotary axes, are used to join smoothly the RTOs together and then to determine the tool orientations at other areas. The solutions for the two B-spline curves are achieved by solving a least-square objective function which minimizes the angular accelerations of the rotary axes. To restricting simultaneously the interpolated tool orientations in the geometric feasible domains (GFDs) of tool motion, a simple alternate strategy of first smoothing the tool orientation and then checking the machining interference is developed so that tool orientation planning and its geometric constraints are decoupled and the complicated constraint optimization process of tool orientation can be greatly simplified. Since the proposed method works in the machine coordinate system (MCS), it can not only ensure the smooth motions of the rotary axes without the machining interferences, but also can generate directly the rotary axis orders. Finally, the proposed method is validated by the experiments.

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^☆: This paper has been recommended for acceptance by B. Juettler.

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Optimal tool orientation control for 5-axis CNC milling with ball-end cutters☆

Abstract

Highlights

Comput. Aided Design

Comput. Aided Geom. Design

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Graph. Models

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Comput. Aided Design

Comput. Aided Geom. Design

Comput. Aided Geom. Design

Int. J. Mach. Tools Manuf.

Comput. Aided Design

Comput. Aided Geom. Design

Comput. Aided Design

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Int. J. Mach. Tools Manuf.

Comput. Aided Geom. Design

Comput. Aided Geom. Design