Elsevier

Computer-Aided Design

Volume 45, Issue 10, October 2013, Pages 1222-1237
Computer-Aided Design

Global obstacle avoidance and minimum workpiece setups in five-axis machining

https://doi.org/10.1016/j.cad.2013.05.007Get rights and content

Highlights

  • Classification of different types of obstacles encountered in five-axis NC machining.

  • Efficient numerical algorithms on how to represent dynamic obstacles.

  • Efficient computing of feasible regions considering all types of obstacles.

  • A heuristic optimization algorithm for finding the best workpiece setup.

  • A thorough analysis of both static and dynamic obstacles.

Abstract

The state-of-the-art tool path computation algorithms for five-axis machining consider only the workpiece and clamping device for collision avoidance. However, in a real five-axis machining process, there are many other types of obstacles beyond workpiece and clamps that the tool assembly must avoid, e.g., sensors and other intrusive devices. In such cases, the only solution at present is by means of computer simulation of the machining process after the tool path has been computed. If collision is found, it requires re-computing the tool path and/or changing the setup of the workpiece. This process is then re-iterated until all the collisions are resolved. As a result, the process is time consuming and requires excessive human intervention. In this paper, we present rigorous analyses of the obstacles in five-axis machining and propose efficient numerical algorithms for calculating and representing them. Using our results, the obstacle-free tool orientations can be determined completely at the tool path planning stage, rather than relying on the simulation afterward. In addition, as a direct application of our mathematical modeling, we present a heuristic-based solution to the optimal workpiece setups problem: finding a minimum number of workpiece setups for an arbitrary sculpture part surface so that it can be machined completely on a given five-axis machine without colliding with the obstacles. We use orthogonal table–table five-axis machines as an example and work out a numerical experiment using the proposed solution.

Introduction

Five-axis machining enjoys tremendous advantages over traditional three-axis machining, including larger range of tool accessibility, faster material removal rate, better surface finishing quality, and so on. It is especially suitable for machining parts with high precision sculptured surfaces such as aircraft structure parts, turbine blades, propellers and impellers, molds and dies, etc. In five-axis machining, at the tool path planning stage, when determining the tool orientation for the tool to contact any particular point on the part surface, the minimum yet also the most critical condition is that the tool must avoid the obstacles during the entire machining process.

Before any further discussion, it is imperative to give a clear classification of the obstacles encountered during a five-axis machining operation, as follows.

Type  0 Obstacle (Obs0): The obstacle is part of the workpiece, including the part surface itself or in-process geometry model of the workpiece.

Type  1 Obstacle (ObsI): The obstacle is attached to or fixed on the machine table (e.g., clamps and fixtures). Once the workpiece is set up on the machine table, an ObsI can be considered the same as an Obs0.

Type  2 Obstacle (ObsII): The obstacle is attached to or fixed on the machine’s base (e.g., sensors of the drive system).

Obs0 is static with respect to the part surface, since it is defined on the workpiece, and it is the most common obstacle type. ObsI and ObsII are those defined on the table and on the machine’s base, respectively. They are usually the clamps, the online measurement devices and some ancillary pieces of equipment used in the machining process. ObsI and ObsII do exist in the machining process. However, they are both ignored in almost all the existing tool path generation algorithms. Specifically, current five-axis tool path generation algorithms bear the following two major caveats:

  • (1)

    When deciding the obstacle-free orientations of the tool, only static obstacles (i.e., Obs0) are considered, leaving both ObsI and ObsII unattended.

  • (2)

    The relative orientation and position of the part surface (i.e., the setup of the workpiece) on the machine table are usually disregarded.

The fact that only Obs0 is considered when deciding the tool orientations causes two serious problems. First, since ObsI is not considered, after the workpiece is mounted and clamped on the machine table, some originally collision-free tool axes at some contact points may now collide with the ObsI. The second, which occurs during the machining process and is perhaps more troublesome, is that the tool assembly, according to an originally assigned tool axis, will either collide with the ObsII or the required machine table rotation angles are out of the working ranges of the rotary axes. At present, the only solution to both these two is by means of computer simulation of the machining process after the tool path has been computed. If collision or out-of-range is found, it requires re-computing the tool axes and/or changing the setup of the workpiece. This process is then re-iterated until all the collisions are resolved. As a result, the process is time-consuming and requires excessive human intervention.

Since both ObsI and ObsII vary (with respect to the part surface) depending on how the workpiece is set on the machine table, the workpiece setup problem itself is integrally related to the general global obstacle avoidance problem. Different workpiece setups on the machine table bring different tool accessibility and multiple setups may be required to cover the whole part. Generally speaking, when machining a part, fewer workpiece setups means less auxiliary time and less errors resulting from the remounting and recalibrating operations. Therefore, finding the minimum number of setups for a given workpiece (ideally only a single one) has great significance in five-axis machining.

Motivated by the above two problems and also the related workpiece setup problem, this paper has set forth two objectives: (1) to establish a rigorous classification of the algebraic structures of obstacles of all the three types and then propose efficient numerical algorithms for calculating and representing them; and (2) as a direct application of (1), to devise an optimization algorithm that will find a minimum number of workpiece setups for machining a given sculpture surface, attending to all the three types of obstacles.

While past works pertaining to the first objective are scarce, the minimum workpiece setup problem has drawn a great deal of attention over the last two decades. Past works primarily have been focusing on the calculation of static Visibility Maps (V-map) and finding the relationship between the V-maps, the (set of) Tool Access Directions (TAD), and feasible tool orientations in tool path generation, to be reviewed next.

Previous works in workpiece setup focus on different objectives. For the dynamic error reduction problem, Anotaipaiboon et al.  [1] minimized the kinematics errors by choosing an appropriate workpiece setup, and the optimization process is carried out by the least square method. While considering the design specification of the part, Zhang et al.  [2] used the tolerance analysis chart to get the optimal workpiece setup from the perspective of tolerance control. Contini and Tolio  [3] investigated and analyzed the machining requirement before generating a workpiece setup plan in the machining process. For the problem of accessibility and minimal number of workpiece setups, Radzevich and Goodman  [4] utilized the Gaussian map (G-map) of the part surface and the cutting surface of the cutter, taking both the number of setups and the optimal cutting condition into consideration, to determine the optimal workpiece orientation on the table; Gupta et al.  [5] used the concept of geometric duality and proposed a greedy heuristic method to find the optimal workpiece orientation which maximizes the number of (polygonal) faces machined in a single setup; Tang et al.  [6], [7], [8] conducted a series of works in finding the minimal number of workpiece setups for four-axis machining. However, all these works in accessibility and the minimum workpiece setup problem ignored the obstacles on the machine table and machine’s base; in other words, they can only treat static obstacles. There are also works done in workpiece setup for other purposes. Lee et al.  [9] proposed a manufacturability evaluation method and built a software module for the workpiece setup analysis for three, four and five axis machines; Kang and Suh  [10] gave a brute-force based method for finding a workpiece setup on the AB type and CA type five-axis machines that would minimize the maximum rotation angles of AB, or AC axis; Hu and Tang  [11] incorporated the workpiece setup into the kinematics chain of a five-axis machine, and improved the dynamics of machine’s rotary axes by finding the optimal workpiece setup; Cai et al.  [12] proposed an adaptive method for the setup planning in the machining of prismatic parts.

TAD is the set of feasible tool orientations provided by the machine in the workpiece coordinate system (WCS), which is decided by the travel ranges of the machine’s rotary axes and can be represented by a great arc or spherical area on the Gaussian sphere (G-sphere) in four-axis or five-axis machining, respectively. In the existing works, Tang et al.  [6], [7], [8] divided the TAD of four-axis machining, i.e. a great arc, into three types according to the travel range of the rotary axis; Woo  [13] classified the TAD with different degrees of freedom (DOFs) into a great arc or spherical area on the G-sphere; Kang and Suh  [10] classified the TAD of five-axis machining into two types of CA and AB according to the configuration of the machine’s two rotary axes, and further work by Cai et al.  [12] tested the effectiveness of such a division of TAD in a five-axis machine.

For each point on the part surface, the V-map is the set of collision-free orientations of the tool on the G-sphere defined in the WCS. The notion of the V-map was introduced by Chen et al.  [14] and Woo  [13], deriving from the definition of the G-map. In the existing research, the V-map was usually calculated as part of the work in the planning of the collision-free tool path  [15], [16], [17], [18], [19]. Wang and Tang  [20] proposed a fast iterative method of calculating the V-map. Based on the continuity property of the V-map on smooth surfaces, they calculated the V-map of one point by taking its neighboring points as the references and proposed a boundary tracking algorithm to calculate the V-map for the entire surface. Elber proposed an algorithm to calculate the V-map of external obstacles with the restriction of tool orientation along the surface normal  [21], and later, he proposed another global solution to extend the method for the case of arbitrary tool orientations  [22]. Lacalle  [23] analyzed the issues in the V-map, and pointed out that the interference between the tool and fixture on the machine table and base was still an unsolved problem in the CAM system. Again, in all these works, only static obstacles are considered.

Augmenting V-maps, some other types of constraints, which are equally important in tool path generation, have been added to further restrain or guide the tool orientation. Gong and Wang  [24] took the ball-end machined surface as a constraint for bounding the tool orientation in flank milling. Castagnetti et al.  [25] introduced the concept of Domain of Admissible Orientation (DAO) to restrain the allowable tool orientations at each CC point, which is defined according to the functional constraints specified by the machining requirements, and the tool orientation smoothing is carried out in the DAO. Hu and Tang  [11] defined the Domain of Geometrical Constraint (DGC) for the tool orientation based on the variation of the effective cutting radius in the iso-parametric tool path, and tool orientation smoothing for the dynamics of machine’s rotary axis was then carried out within the DGC. Considering the local geometry constraints of the surface,  [26], [27], [28], [29] restricted the tool orientation in the local gouging-free region and carried out optimizations for different purposes. Lasemi et al.  [30] presented a comprehensive summary of the tool path generation algorithms in five-axis machining, together with a variety of constraints for the tool orientation.

This paper is organized as follows. In Section  2, rigorous mathematical models of the minimum workpiece setup problem with Obs0 only and with all the three–Obs0, ObsI and ObsII–are established, respectively. A brief description about how to compute a V-map is given in Section  3. In Sections  4 , 5 , two important algebraic structures called Forbidden Map Type 1 (FI-map) and Forbidden Map Type 2 (FII-map) are introduced, which cater to ObsI and ObsII, respectively. In Section  6, we present our heuristic optimization algorithm for the minimum workpiece setups problem. Finally, in Section  7, we present a preliminary computer simulation experiment, followed by the summary.

Section snippets

Preliminary

Throughout this paper, without loss of generality, we exclusively focus on orthogonal table–table five-axis machines—namely, the machine with two rotary axes A and C on the table, and three linear axes (XYZ) of the tool, as shown in Fig. 1.

The workpiece setup on the table can be defined by a 6-tuple vector: setup=(a,b,c,φ,θ,ψ) where: setupI=(a,b,c) is the first part of setup, which represents the position of the origin of the WCS, e.g., ow in Fig. 1, in the Table Coordinate System (TCS); while

V-map calculation

For the part surface as well as any obstacles or fixtures that are static with respect to the part (together they form the workpiece), they are first sampled into triangles with n nodes, which will be the sample points for checking the tool accessibility. For example, shown in Fig. 6(a) is the upper part of a car seat, and in Fig. 6(b) is the corresponding triangulation with the nodes as the sample points. Since these nodes are uniformly sampled and the surface is smooth enough, we can use the

FI-map calculation

To simplify the computation of the FI-map, we approximate the shape of an ObsI (and all other types of obstacles as well) with its convex hull. This will guarantee safety with the sacrifice of slightly smaller visibility region, which is worthwhile for the greater benefit of performance and ease. For any obstacle of concave shape, it can be decomposed into multiple convex components. Hereafter, we use a box as an example of an ObsI but the algorithm can be easily extended to cover any other

Forbidden condition for ObsII

ObsII is defined in the MCS and fixed on the machine’s base. Akin to ObsI, a tetrahedron H2 with vertices {b1,b2,b3,b4} is used to represent ObsII, as shown in Fig. 1. Similar to that of ObsI, the first step for calculating the FII-map is to transform ObsII from the MCS to the LCS of the sample node, e.g., point p in Fig. 1. From Eq. (7), it is obvious that the kinematic chain between ObsII and the LCS, in addition to the setup, also includes the A and the C axis.

To characterize the FII-map, we

Minimum workpiece setups

We have accomplished our primary task. To summarize, given a mesh part surface S (the workpiece), the specific AC type five-axis machine (Fig. 1), the Type 1 obstacle H1 and Type 2 obstacle H2 represented as convex polyhedrons, and an arbitrary 6-tuple setup parameters (a,b,c,φ,θ,ψ), for any vertex p on S, all the tool orientations for the tool to contact it without any obstacle-collision are uniquely represented by its global visibility map Vg-map(p), which is a continuous region in its

Experimental results and discussion

To ratify the results obtained in this paper, a prototype computer program is implemented using MATLab on an averagely configured PC and a preliminary experiment in simulation is conducted to test the presented heuristic optimization algorithm for the minimum workpiece setups problem. Refer to Fig. 17 for the setting of the experiment. There are four ObsI (yellow boxes) defined in the TCS and two ObsII (red tetrahedrons) defined in the MCS. (Note that physically ObsI should never collide with

Summary

Traditionally in five-axis machining, when determining the tool orientations that are accessible to a point on a part surface, only static obstacles (i.e., only the part surface itself) are considered. In the presence of non-static obstacles such as fixtures and clamps on the machine table (which are not static with respect to the part surface depending on how it is set up on the machine table) and devices fixed in the space (which are dynamic even with respect to the rotation of the machine

Acknowledgment

This work is partially supported by the Hong Kong Government RGC General Research Fund619212.

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