The rapid development of modern industry has raised higher requirements for the performance and appearance of products, leading to the widespread application of free-form surfaces in aerospace, shipbuilding, automotive, energy, defense, and other industries^[1].

The so-called free-form surfaces are those surfaces that cannot be composed of primary analytic surfaces, but are free to change in complex ways.

The machining of free-form surfaces, which generally have complex shapes, high accuracy requirements, and are difficult to accurately represent mathematically, has always been a challenging problem in production.

It not only prompted the generation of CNC machining technology, but also has been the primary research and application object of CNC machining and technology.

For the CNC machining of free-form surfaces, the traditional method is mainly accomplished by using a ball-end cutter on a three-coordinate CNC machine tool.

With the development of computer technology and CNC technology, it is necessary to overcome the shortcomings of low machining efficiency and poor machining quality of the ball tool. Therefore, the multi-coordinate machining theory and machining methods for free-form surfaces are proposed.

The term “multi-coordinate machining” refers to machining performed on CNC machine tools with more than three coordinate linkages.

The tools used include ball-end cutters, ring cutters, end mills, cylindrical and conical flat-bottomed bar milling cutters, ring bar milling cutters, and a variety of targeted specialized tools.

Depending on the cutting edges used in the machining process, they can be categorized into multi-coordinate end milling and multi-coordinate side milling.

Ball, ring, and end milling cutters are mainly used in multi-coordinate end milling. In contrast, cylindrical or conical flat-bottomed bar milling cutters, ring bar milling cutters, and spherical bar milling cutters are used primarily in multi-coordinate side milling.

In end milling, spherical cutters are suitable for 3-5 coordinate machining, while the other cutter types are mainly used in 5-coordinate machining.

In this paper, we review the recent research progress on free-form five-coordinate end milling, side milling, and collision interference analysis.

Based on analyzing and summarizing the advantages and disadvantages of various methods, we highlight the shortcomings of the current research and outline the direction for developing tool trajectory planning technology for free-form five-coordinate machining.

5-coordinate ball-end machining of free-form surfaces

The five-coordinate machine tool consists of three translational axes and two rotary axes. According to the configuration of the different axes of motion, the five-coordinate machine tool has various structural types, as illustrated in Figure 1 for the table rotary and tool swing types of five-coordinate machine tools.

From a mathematical point of view, CNC machining and CNC programming theory are essentially the application of curved surface geometry in the machinery manufacturing industry ^[2], so the generation of free-form multi-coordinate machining trajectories is mainly based on the geometric engagement between the tool surface and the surface being machined.

Due to the normal-vector adaptation of the ball-end tool, the change in the tool axis attitude does not affect the geometrical meshing between the tool and the machined surface, and programming and avoidance of local interference are relatively simple. Therefore, it has a wide range of applications in traditional multicoordinate machining.

This method is mainly used for milling narrow channels and surfaces that are not monotonous in the z-coordinate direction.

The calculation of the tool center follows the same approach as in three-axis machining.

However, the key difference is that in five-axis machining, the tool axis can be controlled according to the critical line of the machined surface or the restraining surface.

This control ensures that the tool axis does not interfere with or collide with the surface being machined or the restraining surface.

Although this method can process some surfaces that require multiple clamping on a three-coordinate machine, its machining efficiency and accuracy are still constrained by the shape of the tool.

Additionally, the cutting speed of the ball-end tool is uneven, with zero cutting speed at the tip of the tool, which not only affects the surface quality but also increases tool wear.

Although the ball tool machining is not an ideal multi-coordinate machining method, it still has a wide range of applications in engineering practice due to the simplicity of programming.

In the study of ball end milling, SURESH et al ^[4] first introduced the concept of equal residual height machining systematically into the three-axis ball cutter machining of free-form surfaces, and according to the first-order Taylor expansion equation of the surface, derived the relationship between the tool step, residual height, and the curvature of the surface, which improves the machining efficiency.

LIN et al. ^[5] developed this method, utilizing the second-order Taylor expansion of surfaces to derive the analytical forms of the expressions for travel step and travel step length in both Cartesian and parametric coordinate systems for three-axis ball tool machining of planar, convex, and concave surfaces.

SARMA et al. ^[6] introduced the concept of a tool swept section (i.e., cutting profile) in the equal residual height machining method for ball-end cutters, thereby developing the equal residual height machining method for ball-end cutters into a multi-coordinate machining approach.

After comparing it with the equidistant method, it is found that the tool trajectory with equal residual height is the shortest, and machining accuracy can be easily controlled.

Five-coordinate end milling of free-form surfaces

As modern production demands for the quality and efficiency of surface machining continue to increase, traditional ball tool machining is far from meeting the needs of the cutting process.

Therefore, in free-form surface machining, there is an urgent need to utilize other types of cutting tools to supplement the deficiencies of the ball cutter in the cutting process, thereby participating in 4-5-axis multi-coordinate CNC machining.

The complexity of the geometrical engagement between the tool and the machined surface, as well as the unpredictability of the tool axis oscillation concerning the workpiece, leads to interferences between the tool and the machined surface and its constraint surfaces in the multicoordinate machining of non-ball-end tools.

To date, no mature algorithm has been developed for generating interference-free tool position trajectories for multi-coordinate non-spherical tools on free-form surfaces.

The five-coordinate end milling of flat-bottomed cutters has been a research hotspot in five-coordinate machining in recent years [7], which is widely used in the machining of large-scale free-form surfaces, such as turbine blades, impellers, and propellers. A large number of experts have researched this area.

This is achieved by adjusting the heel angle and side deflection angle, which enables the flat-bottom cutter to align more closely with the surface being machined, resulting in higher cutting efficiency.

Vickers et al ^[8] through the flat-bottom cutter and ball cutter in the same cutting conditions of the comparison, found that the flat-bottom cutter end milling efficiency can reach 24 times the ball cutter; at the same time, the flat-bottom cutter using the outer circle cutting, there is no cutting speed of zero point, high quality of the processed surface.

Five-coordinate machining of the flat-bottomed tool was first used in the Sturz method ^[9], to avoid the center of the bottom surface of the tool in contact with the machining surface and the cutting edge of the cutting edge and the machining surface interference, the tool axis should be tilted by an angle in the plane of the pendulum in the direction of feed and machining the surface normal vector, known as the angle of the heel.

CHOI et al. ^[10] proposed a method to optimize the calculation of tool position for five-coordinate end milling of a flat-bottomed cutter by adjusting the heel angle and side deflection angle, thereby minimizing the residual height of two adjacent trajectories.

Marciniak ^[11] studied the machinable domain and concluded that the machining bandwidth is largest when the direction of travel of the tool is along the line of minimum principal curvature of the surface being machined. The maximum machining bandwidth is mainly dependent on the principal curvature at the tangent point.

To avoid the instability of parametric surface calculations and the locality of geometric properties at the contact point, LI et al. ^[12] proposed a tool path generation method for five-axis machining of freeform surfaces using flat-end cutters, based on triangular mesh surfaces.

Local interference detection is performed by checking for intersections between the tool bottom surface and the triangular mesh elements of the machined surface within the influence region.

To correct local interference, two methods are employed: adjusting the tool orientation and retracting the tool.

In the determination of row spacing, LI simplified the calculation of a flat-bottomed cutter to a ball-end cutter with the same effective cutting radius, which is defined as the radius of curvature of the projection of the cutting edge on the plane perpendicular to the direction of travel at the point of tangent.

LEE et al ^[13-17] defined the projection of the cutting edge on the plane perpendicular to the direction of travel as the effective cutting profile, and Fig. 2 gives the commonly used method of determining the cutter axis vector and the corresponding shape of the effective cutting profile in the machining of flat-bottomed tools.

Through an in-depth study of the effective cutting contour, a method for processing free-form surfaces using a five-coordinate flat-bottomed cutter with an equal residual height approach is developed. A systematic research effort is conducted on local interference in machining and the optimization of tool axis vector selection.

In the five-coordinate end milling machining of a flat-bottom tool, the machining row spacing is the primary factor affecting machining efficiency, which primarily depends on the tool’s heel angle λ. In contrast, the dependence on the side deflection angle ω is relatively small.

LEE concluded that in five-coordinate machining with flat-bottomed tools, a slight heel angle corresponds to a large cutting radius, which contributes to improved cutting efficiency.

For this reason, LO ^[18] and others developed an adaptive heel angle determination method based on the equal residual height method to minimize the heel angle without interference, thereby achieving high machining efficiency.

Fig. 2 Machining diagram of a flat-bottom cutter

An effective cutting profile remains the primary method for studying the end milling machining of a flat-bottomed cutter, which simplifies the calculation of tool trajectory generation from a three-dimensional problem to a two-dimensional one.

However, this simplification also has obvious shortcomings, mainly in that the effective cutting contour can only study the tangential relationship between the tool and the surface being machined at the current tangential point, and cannot account for the influence of the tool on the current point at other tangential points.

For this reason, SARMA (19I) introduced the concept of localized cutting contour and compared it with the traditional cutting contour.

The local cutting contour is defined as the line of intersection between the surface and the normal plane perpendicular to the direction of travel when the tool passes through the tangent point during the cutting process.

Comparative analysis reveals that the local cutting contour is more accurate than the conventional cutting contour in determining the machining bandwidth, particularly when the tool oscillates more vigorously or when the surface to be machined has greater curvature.

However, since the difference in effective cutting radius between the local cutting contour and the conventional cutting contour is not very large, there is no significant difference between the two when performing local interference checks.

When the surface to be machined is smooth and the tool oscillation is not very violent, the traditional cutting profile calculation method is not a good choice, because the local cutting profile needs to be calculated numerically iteratively, which is much more computationally intensive than the traditional cutting profile.

RAO et al. ^[20] conducted an in-depth study on the avoidance of local interference in flat-bottomed cutter machining.

They focused on the swept surface generated by the cutting edge of the tool as the object of study. In this context, they derived the curvature of the swept surface in any direction.

Additionally, they provided an accurate calculation of the effective cutting radius in machining flat-bottomed cutters.

They argued that a sufficient condition to avoid local interference in flat-bottomed cutter end milling is as follows: in the direction perpendicular to the tool’s travel, the effective cutting curvature of the tool must be greater than the normal curvature of the surface to be machined.

The effective cutting curvature of the tool in the direction perpendicular to the travel direction is greater than the normal curvature of the machined surface.

STEPHEN et al ^[21] decomposed the tool trajectory problem of optimizing multi-axis CNC machining of complex surfaces into three sub-problems: local, regional, and global trajectory generation problems.

For the first two problems, an expression is provided for the feature line that measures the consistency between the machined surface and the tool surface at a given tangent point.

The properties of this feature line are then analyzed using the Dupin indicatrix.

Based on this analysis, the integral form solution for the tool orientation is achieved. This ensures the optimal tool position and tool attitude.

At the same time, the optimal instantaneous feed direction at the tangent point is also determined.

In terms of tool selection, JENSEN et al. ^[22] proposed a method based on the principle of curvature matching.

The goal of this method is to minimize machining errors and maximize material removal rates.

To achieve this, the method calculates the effective cutting radius of the tool that best matches the machining profile. This is done through a second-order Taylor expansion of both the tool surface and the machined surface in a plane perpendicular to the direction of travel.

By doing so, the method enables optimal tool selection.

In Prof. BEDI’s group at the University of Waterloo, RAO et al. ^[23-25] and WARKENTIN et al. ^[26,27] proposed the spindle method and multi-point method, respectively, for five-coordinate end milling of free-form surfaces.

The spindle method is an improvement of the Sturz method.

To minimize the residual height, the tool inclination should be selected in such a way that the minimum principal curvature of the tool at the tangent point is equal to the maximum principal curvature of the surface to be machined. Therefore, the change of tool inclination during machining is an automatic adjustment process.

It has been demonstrated in practice that incorporating surface curvature information into tool trajectory planning offers significant advantages over fixed inclination machining, resulting in a small and uniformly distributed residual height of the machining trajectory.

The multi-point method means that during the cutting process, the flat-bottomed tool or ring tool maintains the maximum number of contact points with the surface being machined, thereby achieving the maximum material removal speed.

The efficiency of the multipoint method is slightly improved compared to the spindle method; however, its implementation is complicated, and the selection of numerous parameters requires further study ^[28].

GRAY et al. ^[28] showed that the tool trajectory length for three-axis machining was 247% longer than that for five-axis machining using the spindle method, under the same conditions, through the machining of hydroformed molds.

Unlike a flat-bottomed cutter, the cutting edge of a toroidal cutter is a smooth surface, and the tool remains tangent to the machined surface at the tangent point.

Therefore, it is possible to overcut the surface in all directions of the cutting plane, unlike flat-bottomed cutters, where it is only necessary to ensure that the tool does not overcut the surface perpendicular to the cutting direction, i.e., no overcutting occurs.

To address this, GLAESER et al. ^[29] introduced the concept of forward Dupin indicatrix. Based on this, they conducted interference detection and tool selection for three-axis machining using tools with strictly convex cutting edges.

YOON et al ^[30] developed it to the five-coordinate circular cutter end milling machining of free-form surfaces, and gave a sufficient condition for no overcutting and an optimization method for tool axis ansatz selection through the second-order Taylor expansion of the surface of cutting edge of the circular cutter and the surface to be machined at the tangent contact point.

In China, Prof. Yuan Zhejun and Dr. Liu Xiongwei of the Harbin Institute of Technology have conducted a comprehensive and systematic theoretical analysis of the Sturz method based on the second-order approximation of the surface, thereby enhancing the method’s perfection ^[31,32].

Prof. Zhengcheng Duan of Huazhong University of Science and Technology proposed a point-involving method for generating overcut-free tool position trajectories for free-form aspherical knives ^[33,34].

The main principle of the point-contact method is to construct a tangent plane to the surface at a given step point. On this plane, a space is then searched where a non-spherical tool can be accommodated without interfering with the machined surface.

If such a space cannot be found, the tangent plane is rotated around the contact point until an aircraft is found where the non-spherical tool can both fit within the space and avoid interference with the machined surface.

To give full play to the potential of five-coordinate CNC machining, Prof. Xiaochun Wang ^[35-37] of Xi’an Jiaotong University proposed the close curvature method for five-coordinate CNC machining.

The close curvature method utilizes a concave disk milling cutter as the tool for machining free-form surfaces.

In this method, the tool axis is allowed to oscillate relative to the workpiece according to a specific law during each stroke.

As a result, in the typical cross-section perpendicular to the feed direction, the envelope formed by the cutter tip trajectory shares the same first to third-order derivatives as the typical cross-section of the theoretical surface.

This approach significantly reduces the number of actual strokes while maintaining the same level of machining accuracy.

Professor Chen Wuyi of Beijing University of Aeronautics and Astronautics proposed a macro-geometry-based tool position algorithm, known as the principle of wide-area curvature coincidence, to address the shortcomings of local differential geometry algorithms, which are unable to avoid theoretical errors and incomplete optimization ^[38].

Five-coordinate side milling of free-form surfaces

Five-coordinate side milling involves using the side edge of the rotary tool to cut the surface of the part. This method is primarily used for machining free-form surfaces with narrow cavities or the presence of flat parameter directions, such as the free-form impeller parts shown in Fig. 3.

Currently, research on five-coordinate side milling primarily focuses on machining straight-grained surfaces ^[39,40]. Under a given allowable error, the straight surface can realize one-turn molding, high efficiency, and good machining surface quality.

Side milling of free-form surfaces is less studied, and in general, free-form surfaces cannot be formed in one cut.

ELBER et al ^[41] gave a bar milling cutter side milling method for convex and hyperbolic surfaces by utilizing the technique of approximating free-form surfaces with segmented rectilinear surfaces under a given accuracy.

LIU Xiongwei ^[42] derived the fitting relationship between the tool and the surface to be machined when using a cylindrical bar milling cutter for side milling.

Based on this, he developed a single-point bias method for side milling of convex free-form surfaces and rectilinear surfaces using a bar milling cutter.

Additionally, he proposed a double-point bias method and a multi-point bias method, specifically designed for rectilinear surfaces.

WANG et al. ^[43] proposed a method for side milling of free-form surfaces using cylindrical bar milling cutters or tapered cutters, tailored to the surface’s microstructure. They provided a calculation method for the machining bandwidth.

For convex surfaces, the direction of the tool axis is aligned as closely as possible with the principal direction corresponding to the absolute minimum principal curvature of the surface. This alignment enables the maximum machining bandwidth to be achieved.

For concave surfaces, they are categorized into two types: non-hyperbolic surfaces and hyperbolic surfaces. An initial tool position is assigned to each of these two cases.

The distance from the surface to the tool axis is then used to determine whether the cutter interferes with the surface. If interference is detected, the tool axis must be adjusted accordingly.

Both bar milling cutters and conical cutters, in the processing of concave surfaces, due to the existence of interference, can not realize the true meaning of side milling.

Zhang Dinghua et al. from Northwestern Polytechnical University studied several key aspects of multi-axis machining. These included calculating tool position, determining tool step length, and correction methods for machining errors and interference in four-axis and five-axis machining.

They proposed a method that combines critical constraints with distance monitoring. This approach helps determine the optimal direction of the cutter axis and effectively eliminates collision interference ^[44,45].

Xiongwei Liu ^[31] of Harbin Institute of Technology proposed a tool position calculation method for five-coordinate side milling machining.

Cai Yonglin et al. ^[46-48] from Xi’an Jiaotong University proposed a drum taper cutter for free-form side milling. They used this cutter, based on the equal residual height method, to study the five-coordinate side milling machining and collision interference analysis of a free-form impeller.

Research status of collision interference analysis

The two rotational degrees of freedom in five-coordinate machining, on the one hand, allow the tool to achieve arbitrary spatial orientation and position, and on the other hand, make the tool’s movement more complex and unpredictable, thereby increasing the possibility of collision interference between the tool and the workpiece.

In five-coordinate machining, the complexity of tool movement caused by local interference prevents the practical application of five-coordinate machining from fully realizing its advantages ^[15].

Therefore, the interference processing ability is a key indicator to measure the technical level of a CNC machining programming system ^[49].

Tool interference is typically categorized into two types: local interference and global interference.

Local interference primarily refers to the tool cutting into the part of the machined surface that should not be removed due to an inappropriate tool size or position. In contrast, global interference refers to the collision between the tool (including the toolholder) or fixture and the workpiece or fixture.

In these two types of interference, although local interference leads to machining errors exceeding the permissible value or the scrapping of the part, collision interference is often more severe, potentially resulting in damage to the tool, fixture, machine structure, or even personal injury ^[50].

Many scholars have conducted in-depth research on local interference, with many approaches based on differential geometry and curvature analysis.

Techniques for detecting and correcting local interference in five-axis machining have been successfully applied.

However, these methods do not yield good results when dealing with collision interference. This is because collision interference is a global problem — it can occur at any time and in any location, rather than being confined to the neighborhood of the tangential contact point.

For collision interference, finding an optimal position where the tool does not collide with any surface is an iterative process of checking and correcting, which is computationally intensive.

Sometimes, the optimal position without collision may not exist by itself. Therefore, the collision is much more complicated and challenging to deal with than local interference.

Most existing commercial CAD/CAM software is limited to collision checking without an automatic correction function ^[51].

Therefore, in the field of multi-coordinate CNC machining, there are relatively few published research results on collision interference.

Collision checking between geometric models has been extensively studied across various fields, including robot path planning, computer graphics, computer games, and virtual reality.

In recent years, the development of robot motion planning, computer games, and virtual reality has significantly promoted research on collision interference checking.

There are several main approaches to collision interference research in the existing literature.

• (1) distance-based collision interference detection methods ^[47,52]

The principle of the distance-based collision interference detection method is to calculate whether the distance from the points on the machined surfaces and constrained surfaces to the tool axis is greater than the tool radius. If the distance is smaller than the tool radius, it is considered that a collision exists.

The main advantage of this method is that the theory is simple and easy to understand.

And when there is interference can directly determine the maximum in the depth of the interference, to facilitate the correction of interference: the disadvantage is that the calculation speed depends on the number of sampling points, when the sampling point is dense, the results are relatively reliable, but the calculation speed is low: when the sampling point is sparse, the computational speed is fast, but the calculation results are not reliable.

• (2) Convex packet method ^[53,54].

The convex packet method mainly utilizes the convex packet nature of Bezier, B-spline, and NURBS curved surfaces to achieve the detection of collisions.

In the first stage of the quick check, using the convex envelope of the control vertex, check whether the current tool position and the convex envelope of the surface there are in interference, if not, it means that the current tool position will not interfere with the surface:

If the quick check fails, that is, if there is interference between the tool and the convex envelope, it will be transferred to the detailed detection of uniform sampling points in the given inspection area to calculate the distance from these points to the tool axis and determine whether any interference occurs.

• (3) Vector method

Mathematically speaking, the detection of collision interference is essentially the intersection detection between the tool and the part being machined.

Due to the complexity of the workpiece’s shape, directly implementing the tool with surface intersection between the parts of the calculation is almost impossible.

For this reason, many scholars have converted this surface-to-surface intersection calculation into a line-to-surface intersection detection, which significantly reduces the calculation burden.

Literature ^[55] uses a series of parallel vectors evenly distributed along the z-axis to approximate the workpiece, and the interference calculation between the tool and the workpiece is approximated by the calculation between the tool vectors, which greatly simplifies the calculation.

Tan Guangyu et al. ^[56] used multiple normal vectors perpendicular to the tool surface. They determined whether a collision occurred by checking the intersections between these vectors and the machined surface.

Meanwhile, Chen Hanjun et al. ^[57] from Nanjing University of Aeronautics and Astronautics (NUAA) adopted a different approach. They used 12 straight line segments, uniformly distributed along the circumferential direction of the tool, as checking lines.

These checking lines were then used in combination with the constrained surfaces to perform either an open-angle or closed-angle collision check. This method allowed them to determine whether a crash would occur during the machining process.

The main disadvantage of the vector method is that computational accuracy depends on the density of the selected vectors, which leads to memory problems and a decrease in computational speed when high accuracy is required.

• (4) Mapping method

The mapping method primarily involves transforming a three-dimensional space, with the complex shape of the part and the tool’s movement, into a two-dimensional plane or sphere for collision-free interference and tool orientation selection. There are mainly the C-space method and the Gaussian mapping method.

The C-space method ^[58-61] was first introduced for robot path planning, and its main idea is to map the relationship between moving objects and obstacles onto a two-dimensional plane, thereby simplifying the complex three-dimensional motion into a distribution of points in the two-dimensional plane.

The main disadvantage of this method is that mapping obstacles and complex tools in 2D is challenging in practical machining.

WOO ^[62] of the University of Michigan introduced the concept of visual cone through the Gaussian mapping of points on the unit sphere, and utilized this method to study the workpiece clamping angle problem in four- and five-coordinate machining ^[63].

In CNC machining, collision-free tool position data, also known as reachability, refers to the condition where the tool arrives at a given point without colliding with surrounding constrained surfaces, as well as the surface being machined itself.

From this point of view, ELBER et al ^[64,65] of the Israel Institute of Technology investigated the reachability of five-axis machining using a fading algorithm for curved surfaces.

He first studied the machining method of the tool axis along the normal vector direction in five-coordinate end milling of convex surfaces and simplified the five-coordinate problem to a three-coordinate problem through accessibility mapping.

Then, the machining method of free-form surface tool axis along an arbitrary direction is investigated, and the process of surface subdivision obtains the boundary between the reachable and unreachable regions of the surface.

VAFAEESEFA et al. ^[66] investigated feasible tool positions for machining free-form surfaces in five coordinates using a toroidal cutter, considering global accessibility and the shape of the tool, as well as the local shape of the surface being machined and other obstacles. The practicability of this method needs to be further verified.

WANG ^[67] of General Electric transformed the tool radius to the obstacle surface by constructing a bias surface. Then it determined the feasible space of the radius-free tool by rotating the corresponding Euler angles of the coordinate system to find a workable tool position without collision interference.

Jimenez et al. ^[68] investigated the collision between two translating polyhedra using Gaussian mapping and visual mapping.

If the tool radius is not considered, machinability is consistent with visibility; therefore, the algorithm presented in this paper can be applied to the study of machinability in multicoordinate machining.

If it is reachable, it must be visible.” Based on this assumption, BALASUBRAMANIAM et al. ^[69,70] from MIT transformed earlier research on reachability into research on visibility.

They utilized the surface hiding algorithm from computer graphics and combined it with graphics card acceleration. This approach helped reduce computational complexity and improve processing speed.

The method involved computing the visibility of a given point by projecting it onto a Gaussian sphere in several discrete directions.

The visible cone of the point on the Gaussian sphere is formed, and then the cone is shrunk to obtain the center of the contour of the cone as the direction of the tool axis.

Using this method, BALASUBRAMANIAM et al. investigated the selection of tool position without collision interference in the roughing and finishing of five coordinates.

However, visibility is not necessarily achievable because the tool radius is not taken into account in the visibility calculation, and visibility only provides an initial potentially interference-free tool position for which specific collision interference detection has yet to be performed.

This method is equivalent to obtaining an initial, possibly non-interfering tool position through the calculation of the visibility cone; however, whether or not there is a real collision requires a specific collision interference analysis, which makes the method less practical.

Moreover, the tool position determined according to this method is not necessarily the optimal tool position for machining, especially when the feasible space of the tool is more open.

• (5) Enveloping volume method ^[71-73]

The enveloping volume method is widely used in computer graphics, robotics, computer games, and virtual reality. Especially in recent years, the development of virtual reality technology has made the fast collision detection technology based on the enveloping volume method increasingly emphasized.

The enveloping volume method is an approximate simplification of complex three-dimensional objects into simple geometric shapes, facilitating collision detection calculations between objects. It achieves this by constructing a hierarchical tree structure that gradually approximates the actual geometry of the object, allowing for the near-complete determination of the geometric characteristics of the collision area.

The encompassing volume method has three main advantages.

• (i) Fast determination of whether a collision has occurred.
• (ii) Fast determination of the collision area.
• (iii) Only the collision region needs to be processed subsequently ^[74].

As shown in Fig. 4, the commonly used types of enclosing boxes are enclosing spheres, axial enclosing boxes, fixed-direction convex envelopes, and directed enclosing boxes.

Although the intersection test between oriented wraparound boxes is more costly, it has the best tightness, can exponentially reduce the number of wraparound boxes and the number of basic geometric units involved in the intersection test, and its overall performance is better than several other options ^[75].

In addition, when the geometric object undergoes a rotational motion, it is sufficient to perform the same rotation on the base of the oriented enveloping box, without the need to reconstruct the enveloping box ^[76].

Therefore, it is a suitable choice to adopt the directed-surround box as a means of detecting collision interference in multicoordinate machining.

In recent years, the oriented bracket box has gradually gained applications in collision interference checking for multi-coordinate machining.

Balasubramanian and Ho et al. ^[77,78] of the Massachusetts Institute of Technology (MIT) employed a hierarchical directed box tree to detect collision interference in their realization of a five-coordinate virtual machining system.

DING et al. ^[50] from the National University of Singapore studied collision interference detection in five-axis machining. They used the method of directed enveloping box and octree for this purpose. In their approach, octree representation was applied to the machined surfaces, while a hierarchical directed enveloping box tree was used for the tools.

In the enveloping volume method, the application of the enveloping box serves as a quick way to rule out regions where collision between the two objects is not possible. At the same time, it helps determine the areas where a collision is likely to occur.

The collision detection of the triangular cells in the box must be performed accurately when a collision occurs.

A complete collision analysis should not only include the detection of collision interference, but should also be able to realize the corresponding correction of the existing collision interference. The principle of correction is to achieve the purpose of interference elimination through the slightest change of tool orientation and position.

Conclusion

After reviewing the recent research progress on tool trajectory planning technology for free-form five-coordinate CNC machining, the deficiencies and development directions are summarized as follows.

• Proven algorithms for generating interference-free tool position trajectories for multi-coordinate non-spherical tools on free-form surfaces have not yet been summarized.

Due to the complex shape of the free-form surface, the tool geometry is diverse, and the relative motion between the two is complex. Therefore, most current research on free-form surface multi-coordinate non-spherical tool machining for specific parts focuses on targeted research, and the algorithm cannot meet the generality of the requirements.

At the same time, most existing tool path planning methods for non-ball-end cutters are based on two-dimensional approaches — specifically, analyzing the geometric engagement between the tool and the freeform surface within a normal cross-section perpendicular to the tool path direction.

However, this method cannot account for the contact conditions between the tool and the machined surface in other directions, nor can it consider the influence of different contact points on the current contact position.

As a result, the two-dimensional analysis based on regular cross-sections cannot accurately represent the relative motion between the tool and the machined surface.

In terms of collision and interference analysis, the existing algorithms are complex to implement, and their robustness cannot meet the requirements of engineering practice, so in practical applications, they still rely on empirical and simulation means to verify the toolpath, and cannot realize the accurate detection and correction of collision and interference.

• The planning of the tool axis vector does not account for the kinematics and dynamics of the CNC machine tool.

In current research on multi-axis CNC machining of freeform surfaces, tool axis vector planning typically relies on determining the tilt and lead angles within a local coordinate system. This approach only ensures good geometric contact between the tool and the machined surface.

However, it does not take into account the kinematic and dynamic characteristics of the machine tool. As a result, the generated NC program may not align with the machine’s kinematic and dynamic capabilities, potentially causing the machine’s motion axes to exceed allowable limits in terms of speed and acceleration, especially during high-speed machining.

This issue has even affected the practical application of five-axis machining in engineering, where five-axis machines are often used as four-axis or four-and-a-half-axis machines, leading to a significant waste of equipment capabilities.

Therefore, during the tool axis vector planning stage, the machine’s motion characteristics should be fully considered.

The planned tool axis vectors should not only ensure good contact between the tool and the machined surface but also comply with the machine’s kinematic and dynamic properties, enabling the five-axis machine to operate at high speed and high efficiency.

• The microscopic tangential state cannot represent the actual tangential state

At present, most of the overcutting analysis and row distance calculation of non-ball head tools are based on the differential geometric properties of the tool and the machined surface at the cutting point, such as curvature, normal vector, etc.

However, meeting the no-overcut condition at this point does not guarantee that the other positions of the cutting edge will not overcut the machined surface, especially in cases of drastic changes in surface curvature.

Therefore, a wider range of non-spherical cutting tools and machined surfaces is examined in the tangential state of the research, providing a macroscopic understanding of the tool’s actual movement process under non-interference conditions. Tools and machined surfaces are matched according to the optimal matching principle.

• The research should consider the actual needs of the engineering field

Although a significant amount of theoretical research has been conducted on the free surface of the five-coordinate non-spherical tool in recent years, the number of projects that have achieved practical applications remains limited in practice.

In multi-coordinate machining, the most commonly used method in factories remains ball tool machining, particularly in the final finishing process. In non-ball tool machining, the Sturz method is still widely used.

Therefore, for the five-coordinate machining of free-form surfaces, on the one hand, the depth and breadth of theoretical research should be strengthened.

At the same time, it should be combined with the actual characteristics of the products in factories, to do a good job of transforming the theoretical results to the actual production of engineering.