Method for machining a blank by means of a tool

The present invention relates to a method for machining a blank (10) by means of a tool (12) for producing a finished part, wherein the tool (12) is moved during the machining on a guide path (14) comprising at least three successive path segments (16, 18, 20; 16-1, 18-1, 20-1; 16-2, 18-2, 20-2; 18′) in the form of two machining segments (16, 20; 16-1, 20-1; 16-2, 20-2) and one connecting segment (18; 18-1; 18-2; 18′), which connects the two machining segments (16, 20; 16-1, 20-1; 16-2, 20-2) to one another, and wherein the connecting segment (18; 18-1; 18-2; 18′) of the path segments (16, 18, 20; 16-1, 18-1, 20-1; 16-2, 18-2, 20-2; 18′), which connecting segment connects the two machining segments (16, 20; 16-1, 20-1; 16-2, 20-2), is determined in terms of its shape by the forward feed (F1) of the tool (12) at the end (24) of the first machining segment (16) and by the forward feed (F2) of the tool (12) at the start (30) of the second machining segment (20).

CROSS REFERENCE TO RELATED APPLICATIONS

This application is the U.S. national stage of International Application No. PCT/EP2014/002985, filed on Nov. 7, 2014, and claims the priority thereof. The international application claims the priority of German Application No. DE 102013112232.9 filed on Nov. 7, 2013; all applications are incorporated by reference herein in their entirety.

BACKGROUND

The invention relates to a method for machining a blank by means of a tool for producing a finished part.

To machine a blank by means of a tool for producing a finished part, the tool is moved on predefined guide paths and paths, respectively, which may be calculated using software. Each single guide path comprises an arbitrary number of successive path segments and path sections, respectively.

Regarding the path segments, there are machining segments and connecting segments. Along a machining segment, the tool engages the material of the blank to perform material removal. A connecting segment leads from the end point of a machining segment to the start point of a subsequent machining segment. Along a connecting segment, usually the tool does not engage the material of the blank. This can be achieved by retracting the tool to some predefined clearance distance above the blank or the workpiece. In that case, usually a higher forward feed is used, e.g. the predefined “rapid” forward feed. However, along a connecting segment, the tool may as well engage the material of the blank and remove material thereof. The connecting movement then takes place directly within the machining area between two machining segments following to each other, e.g. when using a line-by-line machining pattern.

Generally, it is advantageous to perform connecting movements as efficient as possible in order to reduce total machining time. This is even more important for machines used in the production mode of a series production. However, reducing machine load and respecting the acceleration characteristics of the machine is just as important as efficiency. This is especially relevant in the context of high-speed machining, where extremely high forward feeds and strong accelerations are common. To clarify the technological background, in the following some essential technical aspects of machine tools will be explained in more detail.

Machine tools used for milling differ in manifold ways in terms of structure and way of operation. First of all, the number of controllable linear and rotational axes may be different. Three axis machines are able to control three linear axes (X, Y and Z) but no rotational axes (A/B and C). Accordingly, the tool is always oriented orthogonal to the machine table, in other words parallel to the Z axis of the Cartesian machine coordinate system. Four axis machines are additionally able to control one rotational axis (A/B) in order to tilt the tool relative to the Z axis of the machine coordinate system. Five axis machines are most capable in that also the second rotational axis (C) can be operated as well, preferably in such a way that a simultaneous control of all these five available axes is enabled. Such simultaneous control of all axes provides maximal freedom in guiding the tool. However, toolpaths to be calculated for the purpose five axis machines have to contain not only positional data for each machining point but also information with respect to angular values to control both the rotational axes.

A further distinctive feature of five axis machines is the kinematic behavior, i.e., the technical implementation of the control of rotational axes. For each of these axes, a given angle can be adjusted either in swiveling the tool head or, alternatively, in (inversely) swiveling the machine table. Depending on the machine manufacturer, there are many different possibilities: head-head kinematics (both rotational axes located in the tool head), table-table kinematics (both rotational axes located in the machine table) and numerous kinds of mixed kinematics. In this context, in particular the so-called dynamics of a machine is of importance. The dynamic properties determine the acceleration characteristics of the linear and rotational machine axes. There are great differences between the machines depending on the machine kinematics and drive technology as used. However, within a machine, i.e. between the individual axes, the dynamics may vary. For example, often the Z linear axis will have different acceleration properties than the X and Y linear axes. This is called anisotropic axis acceleration.

Machines for high-speed processing (high-speed cutting) represent yet another special category because of the specific requirements. They are characterized by extremely high spindle speeds and strong dynamics, i.e. very high forward feeds and strong accelerations of axes. This still further increases machine load effects resulting from unsuitable tool guidance strategies.

In known methods for machining a blank110by means of a tool112for producing a finished part according to the preamble of claim1, as for example described in DE 696 24 093 T2 and/or shown inFIGS. 7ato7c,the tool112is moved on a guide path114comprising at least three successive or subsequent path segments116,118,120in form of two machining segments116,120and one connecting segment118which connects the two machining segments116,120with each other. For example, the tool112may be a milling tool, a drilling tool or a laser tool.

FIGS. 7ato 7cshow different application examples for the machining of holes122,122′.

In its simplest form, the connecting segment118has a linear shape, corresponding toFIG. 7a. The connecting movement of the tool112includes vertical retraction to some clearance distance above the blank110. First, using an increased forward feed, the tool112is lifted vertically from an end point124of the first machining segment116, which coincides with the start point of the connecting segment118, to the first target point126at the defined clearance distance. Next, a linear horizontal connecting movement at this distance or height leads to the second target point128. Finally, the tool112is lead and, respectively, moved down vertically from the second target point128to the start point130of the second machining segment120, which coincides with the end point130of the connecting segment118.

As shown inFIG. 7b,the connecting segment118again has linear shape. However, to avoid a collision between the blank110and the tool112in case of an obstacle during connection and the connecting movement, respectively, the clearance distance is enlarged.

Similar situations frequently arise in 5-axis machining, if the tool112is moved from one side132of the blank110to be machined to the next side134to be machined. In this case the critical portion of the blank has to be bypassed while considering the defined clearance distance, in the simplest case following a sequence of lines passing along a target point136, such that the resulting connecting movement of the tool112has a polygonal shape as shown inFIG. 7c.

FIGS. 7dand 7eshow further application examples relating to the machining of surfaces138of the blank110.

According toFIG. 7d,the connecting movement of the tool112in form of for example a milling tool may be carried out for surface machining of surfaces138between the two machining segments116,120via the connecting segment118.

Furthermore, as shown inFIG. 7efor instance, in the context of surface machining, connecting movements of a tool112in form of a milling tool may also occur within a local machining area140, where machining segments are connected directly, i.e., without vertical retraction and thus possibly by engaging the material. Exemplary simple linear connecting movements are provided on the connecting segments118,118-1,118-2etc. between the machining segments116,120,116-1,120-1,116-2,120-2,116-3etc. of a line-by-line zigzag machining pattern with alternating advance direction.

With all of these methods the abrupt direction changes at the start and the end of the horizontal linear connecting movements, e.g. after the vertical retraction and before the vertical touch down, respectively, and at the target points126,128,136of the afore described linear and polygonal, respectively, shaped connecting segments118, have been proved as most disadvantageous. Accordingly, such abrupt direction changes are unfavorable because they lead to high machine load, especially at high forward feeds.

For its improvement, in particular for a more machine gentle movement of the tool112, other known methods therefore propose using curved connecting segments118, as for example illustrated inFIGS. 8ato 8cfor drilling process and inFIGS. 8dand 8efor surface machining.

In this context, in the embodiments ofFIGS. 8ato8e,which correspond to these ones ofFIGS. 7ato7e,curves are used that enable a smooth, i.e. tangent-continuous, course or run of the connecting movements of the tool112. The transition between the connecting segment118and the foregoing and following machining segment116,120, respectively, thereby is (also often) tangent-continuous.

As shown inFIG. 8c,curved connecting segments118may be also used for a multilateral machining of the blank110from one side132to the other side134, if the portion to be circumscribed or bypassed has to be taken into account when calculating the connecting segment118and curve, respectively.

Curved connecting segments118may also be applied within local machining areas140, as shown inFIG. 8e.

However, in practice all these methods have proven to be disadvantageous as well. The reason is that the calculation of the connecting segments118or connecting curves is based on geometric aspects only. The shape of a connecting segment or a connecting curve exclusively depends on the positions of the points to be connected and possibly on the local tangents adjacent thereto. A calculation of the connecting segments118or connecting curves does not respect any non-geometric aspects. In particular, forward feed relationships are not considered. This is very disadvantageous with respect to a machine load reducing and efficient process, also considering the acceleration characteristics of the machine. Additionally, in general the calculated connecting segments118or connecting curves have a much greater length than it is actually required, considering the given forward feed values. Moreover, such methods also do not consider the individual dynamic properties of a machine and a possibly anisotropic axis acceleration profile of a machine.

SUMMARY

The present invention relates to a method for machining a blank (10) by means of a tool (12) for producing a finished part, wherein the tool (12) is moved during the machining on a guide path (14) comprising at least three successive path segments (16,18,20;16-1,18-1,20-1;16-2,18-2,20-2;18′) in the form of two machining segments (16,20;16-1,20-1;16-2,20-2) and one connecting segment (18;18-1;18-2;18′), which connects the two machining segments (16,20;16-1,20-1;16-2,20-2) to one another, and wherein the connecting segment (18;18-1;18-2;18′) of the path segments (16,18,20;16-1,18-1,20-1;16-2,18-2,20-2;18′), which connecting segment connects the two machining segments (16,20;16-1,20-1;16-2,20-2), is determined in terms of its shape by the forward feed (F1) of the tool (12) at the end (24) of the first machining segment (16) and by the forward feed (F2) of the tool (12) at the start (30) of the second machining segment (20).

DETAILED DESCRIPTION

It is therefore an object of the present invention to provide a method for machining a blank by means of a tool for producing a finished part, which allows to avoid the above mentioned disadvantages, is able to use machine tools with less wear and increase efficiency in order to reduce machining time compared to known methods, and allows consideration of machine-specific characteristics and operational parameters of a machine, thus leading to a significant reduction of operating and manufacturing costs in total.

This object is achieved in a surprisingly simple manner by the features of claim1.

By the embodiment of the method according to the invention for machining a blank by means of a tool for producing a finished part, wherein the tool is moved during the machining on a guide path comprising at least three successive path segments in the form of two machining segments and one connecting segment, which connects the two machining segments to one another, and wherein the connecting segment of the path segments, which connects the two machining segments with one another, is determined in terms of its shape by the forward feed of the tool at the end or end point of the first machining segment and by the forward feed of the tool at the start or start point of the second machining segment, wherein the connecting segment of the path segments, which connects the two machining segments with one another, is deformed towards the higher forward feed at the end or at the start of the two machining segments, the given forward feed values of two successive machining segments are considered in constructing the connecting segment connecting the two machining segments with one another. Moreover, in the method according to the invention, specific dynamic properties of the used machine can be taken into account, such that the calculated shape of the connecting segment is optimal with regard to the individual acceleration profile, thus further contributing to achieve a significant load reduced and efficient machining process. By the way, that the connecting segment of the path segments connecting the two machining segments to one another is deformed towards the higher forward feed at the end or at the start of the two machining segments, with respect to a very machine gentle movement of the tool and tool movement that further reduces machine load, respectively, this results in a steep or steeper course and shape, respectively, of the connecting segment having a lower curvature, where the forward feed is high, and a flat or more flat course and shape, respectively, where the forward feed is lower.

In other words, a significant gentle and efficient use of a machine drivable with tools is reached. This is mostly due to the fact that the forward feed-dependent calculation of the connecting segment complies with the behavior of such machines. Moreover, the method ensures significant reduction of machining time compared to known used methods. Accordingly, the connection segments are individually adapted or matched to the given forward feeds in shape, in course and, not least, resulting therefrom also with respect to segment length. Furthermore, the method according to the invention considers machine-specific characteristics and operational parameters of a machine, for example individual dynamic properties of a machine, anisotropic axis acceleration profiles of a machine etc. Finally, the method according to the invention also helps to reduce total operational and manufacturing costs significantly. Additionally, reducing wear and strain on the machine importantly increases its service life.

Further particularly advantageous features of the method according to the invention are described in claims2to12.

Of particular interest are the measures of claim2, according to which the connecting segment of the path segments connecting the two machining segments to one another is deformed according to a ratio of the forward feed of the tool at the end of the first machining segment to the forward feed of the tool at the start of the second machining segment. This ratio defines an amount by which the connecting curve is deformed or warped towards to the first segment. In other words, the magnitude of such a deformation can be defined by the ratio between the forward feed values F1and F2to one another.

According to claim3, it is within the scope of the invention that the connecting segment of the path segments connecting the two machining segments to one another is deformed towards the first machining segment, if a ratio of forward feed of the tool is larger than 1, and is deformed towards the second machining segment, if a ratio of the forward feed of the tool is smaller than 1.

Furthermore, of particular advantage are the measures of claim4, according to which the height or length of the connecting segment of the path segments is determined by the amount of the forward feed of the tool at the end of the first machining segment and/or at the start of the second machining segment.

Preferably the connecting segment of the path segments is adapted according to the features of claim5to a corresponding anisotropic acceleration profile of a tool machine holding the tool.

Furthermore, it is within the framework of the invention, to move the tool according to claim6during the machining on the guide path with the first machining segment and the second machining segment in engagement with the material of the blank.

Additionally, it is within the scope of the invention, to move the tool according to claim7during the machining on the guide path with the connecting segment between the first machining segment and the second machining segment without engagement or, alternatively, with engagement with the material of the blank.

In a very beneficial way, according to the features of claim8the tool is moved on the guide path in the area of the connecting segment of the path segments, while the tool orientation is interpolated evenly (linearly) and/or in a forward feed-dependent manner.

Furthermore, it is within the scope of the invention to move the tool according to claim9on the guide path having at least two machining segments having alternating advance directions.

Of particular advantage are the measures of claim10, according to which the tool is moved on a guide path that is embodied in the form of a tangent-continuous or curvature-continuous curve.

Furthermore, the tool according to claim11is moved on the guide path in a collision-free manner.

Finally, it is envisaged according to the invention that the tool according to claim12is a milling tool, a drilling tool or a laser tool.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following description of various embodiments of a method according to the invention for machining a blank10by means of a tool12for producing a finished part (not shown), matching equal components are each given identical reference numbers. The type of the machining process, which may be a drilling process and/or a surface machining process, does not affect the method according to the invention. Without restricting the invention, the tool12may be a milling tool, a drilling tool or a laser tool.

FIG. 1aschematically shows a first embodiment of a method according to the invention, wherein the tool12is moved on a guide path14during a drilling process. The guide path14comprises at least three successive or subsequent or following path segments16,18,20in the form of two machining segments16,20and a connecting segment18connecting the two machining segments16,20with one another.

The tool12in the form of a drilling tool or a milling tool is fed or pulled out of the first machining segment16, i.e., out of the first hole22, with a forward feed F1and is guided directly along the connecting segment18to the second machining segment20, i.e. to the second hole22′, to continue machining with a forward feed F2.

As additionally shown inFIG. 1a,the tool12thereby is moved on the guide path14from the first machining segment16to the connecting segment18, passing the end or the end point24, respectively, of the first machining segment16, which coincides with the start or start point, respectively, of the connecting segment18. Then, the tool12is moved from the connecting segment18to the second machining segment20, passing the start or start point30, respectively, of the second machining segment20, which coincides with the end or end point, respectively, of the connecting segment18.

Corresponding toFIG. 1a,on the connecting segment18a retraction of the tool14to a certain clearance distance occurs. Thereby, the connecting segment18of the path segments16,18,20connecting the two machining segments16,20to one another, particularly its shape and course, respectively, is determined by the forward feed F1of the tool12at the end24of the first machining segment16and the forward feed F2of the tool12at the start30of the second machining segment20.

Preferably, in this context, the connecting segment18of the path segments16,18,20, which connects the two machining segments16,20with one another, is deformed towards or in direction to the higher of the forward feed values F1, F2at the end24or at the start30of the two machining segments16,20.

Without providing any further detail, the forward feed F1of the tool12at the end24of the first machining segment16and the forward feed F2of the tool12at the start30of the second machining segment20are predefined. Thus, it is determined which one of the forward feeds F1, F2at the end24of the first machining segment16or at the start30of the second machining segment20is higher.

Such a deformation of the connecting segment18may be also preferably determined by the ratio F1/F2of the forward feeds. Accordingly, the connecting segment18of the path segments16,18,20is deformed towards the first machining segment16, in case of a ratio F1/F2being larger than 1, and is deformed towards the second machining segment20, in case of a ratio F1/F2being smaller than 1.

In each of the embodiments presented inFIGS. 1ato6b,the forward feed F1is chosen higher than the forward feed F2, which is indicated by the different lengths of the arrows F1and F2. According to the invention, therefrom results a deformation of the connecting segment18each towards or in direction to, respectively, the first machining segment16. The amount and the magnitude, respectively, of such a deformation may preferably be determined, for example, based on a ratio of the forward feed values F1and F2to each other. Mathematically, such a deformation may be described by shifting the control points of a spline curve. Tangent-continuity at the transition between the first machining segment16and the connecting segment18at end point24and start point, respectively, and between the connecting segment18and the second machining segment20at the end point and start point30, respectively, is not affected by the deformation.

The embodiment of the present invention ofFIG. 1b,which also represents a drilling process, differs from that one shown inFIG. 1ain that there is an obstacle between the machining segments16,20or two holes22,22′, respectively. In order to avoid collisions, it is envisaged according to the embodiment of the method according to the invention to increase the clearance distance, leading to a much steeper course of the path.

With the embodiment of the method according to the invention shown inFIG. 1crepresenting a5-axis drilling process a changeover of the tool12, e.g. in the form of a drilling tool or a milling tool, is provided from one side32of the blank10to another side34on different sides machining. As apparent from the surfaces of the two sides32,34, the two holes22,22′ are essentially orthogonal to each other. Although not shown in detail, it is easily conceivable that the holes22,22′ are inclined to each other within the same plane. Analogous toFIG. 8c,the portion to be bypassed is taken into account in the calculation of the connecting segment18or the curve, respectively.

The embodiment of the method according to the invention shown inFIG. 1ddiffers from these ones ofFIGS. 1ato 1cin a surface machining of surfaces38of the blank10. In this case, the connecting movement of the tool12, which is for example designed as a milling tool or a drilling tool, may lead out of or in a machining operation of surfaces38.

In the embodiment shown inFIG. 1e,the method according to the invention is also applied to surface machining by means of a tool12, preferably in the form of a milling tool. Thereby, connecting movements are directly performed within a local machining area40without vertical retractions (and consequently where required by engaging the material), for example, between the machining and connecting segments16,18,20of a line-by-line zigzag machining pattern path with alternating advance directions. In practice, both shown, in amount differing forward feeds F1and F2of the machining segments16,20result from the fact that depending on the advance direction, either conventional or climb milling is performed, and with conventional milling a lower forward feed F2is often required to reduce wear of the tool12.

In the following, the process is correspondingly repeated. The second machining segment20thus becomes the first machining segment16-1and is connected via another, second connecting segment18-1to a subsequent machining segment20-1, which quasi represents the second machining segment20-1. The second machining segment20-1then becomes the first machining segment16-2and is connected via another, third connecting segment18-2to an again subsequent machining segment20-2, which quasi represents the second machining segment20-2. The second machining segment20-2then becomes the first machining segment16-3, and so on.

The existing forward feed F1at the end24of the first machining segment16and/or the existing forward feed F2at the start30of the second machining segment20is/are also determined for the height or length of the connecting segment18. In the examples of embodiments ofFIGS. 2aand2b,forward feeds F1and F2are only half as large as these ones ofFIGS. 1aand1d.Accordingly, the resulting connecting segment18is only half as high. The technological reason is that the smaller forward feed F1at the end24of the first machining segment16allows a stronger curvature and therefore a stronger change of direction, and a flatter shape of the connecting segment18.

InFIG. 3is shown another embodiment of the method according to the invention, which is used for a drilling process, as an alternative to these ones ofFIGS. 1ato1e.Accordingly, the method according to the invention is not restricted to the two-dimensional case, wherein the two machining segments16,20and—consequently, also the connecting segment18—are in the same plane. Instead, the method according to the invention is also applicable to the three-dimensional case, wherein the two machining segments16,20and the connecting segment18may be arbitrarily located in the space.FIG. 3shows such an example of such an embodiment by means of two blanks10having two holes22,22′ on the sides32,34, the axes of which are skewed. The resulting connecting segment18between the two machining segments16,20extends across the three-dimensional space.

As an example, yet another embodiment of the method according to the invention is shown inFIGS. 4aand4b,again subjected to surface machining by means of a tool12, e.g. a milling tool. Connecting movements of the tool12are performed within two (or more) successive local machining areas40,40′ on one side or, as here, on several different sides32,34of a blank10or a workpiece. The connecting segments18,18-1,18-2etc. within the respective local machining areas40,40′ correspond to those ones in the example of embodiment shown inFIG. 1e.The forward feed F1is always chosen to be larger than the forward feed F2.

The change or movement of the tool12from the end point24of a local machining area40on the side32to the start point30of the other local machining area40′ on the other side34of the blank10is carried out by the connecting segment18′. As shown inFIG. 4a,its shape is determined by the forward feed F2at the end point24of the last machining segment20-2of the one local machining area40on the first side32and the forward feed F1at the start point30of the first machining segment16of the other local machining area40′ on the next side34. According toFIGS. 4aand4b,the forward feed F2is because of the surface machining in the local area40smaller than the forward feed F1.

In order to consider specific dynamic properties of a machine tool, the method according to the invention further allows the possibility to consider an anisotropic axis acceleration profile in the calculation of the connecting segments(s)18. A machine axis having strong acceleration capability enables more quickly changes of the forward feed and of the direction, respectively, of the tool12in the direction of this axis. In the method according to the invention, this can be exploited by integrating the acceleration capabilities of the axes into the guide path calculation process in order to optimize the shape of the connecting segments with regard to the machine.FIG. 5ashows at first the typical case of a uniform (isotropic) axis acceleration profile and a connecting segment18calculated in accordance with this configuration using the method of the invention, comparable to that one shown inFIG. 1a.

In contrast,FIG. 5bshows an anisotropic axis acceleration profile, wherein the maximal acceleration az in z-axis direction is much smaller than the maximal acceleration ax in x-axis direction (this description is easily applied to other configurations as well, particularly those ones in that also the y-axis is involved). The connecting segment18is now additionally deformed dependent on the maximal acceleration values. This deformation directly follows the forward feed-dependent curve calculation or deformation, but—without being shown in detail—may also be integrated into the forward feed-dependent calculation or deformation process.

In yet another embodiment of the method according to the invention, concerning a drilling process according toFIGS. 6aand 6band/or a surface machining process (not shown), the progression of the tool orientation and the adjustment of the rotational axes, respectively, respectively related to the respective connecting movement of the tool12can also be optimized.

Generally, as shown inFIG. 6a,the orientation of the tool12is strictly defined only at the end or end point24of the first machining segment16and at the start or start point30of the second machining segment20. Consequently, the orientation of the tool12may be interpolated in an arbitrary way along the connecting segment18between the start and end points24,30. Known methods propose to interpolate often simply evenly, using a constant angular step per length unit, wherein the dotted lines indicating the progression of the respective axis42of the tool12.

However, since different large forward feeds exist at the end or end point24of the first machining segment16and at the start or start point30of the second machining segment20, the resulting angular step per time unit may vary along the connecting segment18when using such methods. Obviously, this results in uneven motion of the tool12. For this reason, with the method according to the invention the forward feeds are additionally incorporated. For example, the angular step may be calculated inversely proportional to the local forward feed. As illustrated inFIG. 6b,a high forward feed only results in a small angular change per length unit and a lower forward feed results in a larger angular change per length unit.

Preferably, the tool12is moved on the guide path14in a collision-free manner.

Finally, it is envisaged according to the method of the invention that the tool12is moved on a guide path14that is embodied in the form of a tangent- or curvature-continuous curve.

The invention is not limited to the embodiments of the method according to the invention according toFIGS. 1ato6b.Thus, it is possible to arbitrarily combine the embodiments of the method according to the invention with each other. Furthermore, the invention is totally independent from the type of the machining process, that being in particular a drilling process or a surface machining process. Without restricting the invention, the tool12may be for example a milling tool, a drilling tool or a laser. Finally, with the method according to the invention, the connecting segment18may as well, with the same effect, be determined in its shape depending on the forward feed F1of the tool12at the start or start point of the connecting segment18, which coincides with the end point24of the first machining segment16, and the forward feed F2of the tool12at the end or end point of the connecting segment18, which coincides with the start point30of the second machining segment20, and consequently be deformed towards the higher forward feed F1, F2of the connecting segment18at its start or its end.