Method, device, and program for converting coordinates from a Cartesian coordinate system to a Freinet coordinate system

Abstract

To enable a coordinate conversion from an orthogonal coordinate system to a Frenet coordinate system for identifying coordinates of a point in the vicinity of a spiral, taking a configuration of a machine component having a spiral shape into consideration.SOLUTION: A coordinate conversion method performs a coordinate conversion from coordinates of an orthogonal coordinate system to parameters indicative of coordinates of a Frenet coordinate system as to a point in the vicinity of a spiral. The coordinate conversion method has: a first conversion step of performing a coordinate conversion of said coordinates of the orthogonal coordinate system of the point into the parameters indicative of the coordinates of the Frenet coordinate system; a second conversion step of performing the coordinate conversion of the parameter indicative of the coordinates of the Frenet coordinate system obtained in the first conversion step to coordinates of the orthogonal coordinate system; a setting step of setting provisional coordinates in the orthogonal coordinate system on the basis of a differential between said coordinates of the orthogonal coordinate system of the point and the coordinates obtained in the second conversion step, of the orthogonal coordinate system; and a third conversion step of performing the coordinate conversion of the provisional coordinates to the parameter indicative of the coordinates of the Frenet coordinate system, and thereby, calculating each parameter indicative of said coordinates in the Frenet coordinate system.SELECTED DRAWING: Figure 10

Description

[Technical field]

[0001] The present invention relates to a method, an apparatus, and a program for transforming coordinates from a Cartesian coordinate system to a Freinet coordinate system. [Background technology]

[0002] Conventionally, mechanical components such as ball screws have a spiral-shaped portion (for example, a groove in which a rolling element can roll). Various methods have been disclosed for appropriately analyzing the operating state and shape of each portion of such a mechanical component having a spiral shape.

[0003] For example, Patent Document 1 discloses a method for extracting data on the root portion of a helical shape in a threaded portion having the helical shape, and analyzing the helical shape. [Prior art documents] [Patent documents]

[0004] [Patent Document 1] Patent No. 3971968 Summary of the Invention [Problem to be solved by the invention]

[0005] For example, in a mechanical part having a helical shape such as a ball screw, in order to calculate the contact state with the internal rolling elements, it is necessary to properly identify the position of the rolling elements in the helical shape. In this case, in order to calculate the contact between the screw shaft or nut at an arbitrary position and the rolling elements, it is necessary to search for the position of the rolling elements in the vicinity of the screw shaft's helix. In particular, when performing a mechanism analysis, all of the components of the mechanical part, such as the screw shaft and rolling elements, can move to any position, so the number of repetitions of the calculation process is increased in order to identify the position each time. Furthermore, high accuracy is required in such calculations for neighborhood search.

[0006] However, such a calculation is an inverse problem, and since it is a solution to a nonlinear simultaneous equation with three unknowns, the solution cannot be found algebraically.On the other hand, when we look at the configuration of a mechanical part having a helical shape, it has the characteristic that the range of movement of the rolling elements is limited to a certain range near the helix.

[0007] On the other hand, ball screws are used in the drive system of the conveyor table of machine tools, for example, and the feed accuracy of the ball screw is one of the most important factors that directly affects the processing quality. In order to calculate or predict the feed accuracy, it is essential to calculate the mechanical behavior such as the load on the rolling elements and friction. In addition, strict contact calculation is essential for calculating the mechanical behavior, and contact calculation requires consideration of machining errors such as lead error and roundness error. In the calculation of the approach amount between the ball and the spiral groove in the conventional Cartesian coordinate system, it is assumed that the nut and the screw shaft are the same spiral, so the effect of the minute twist between the actual nut and the screw shaft is ignored, and there was a problem that it was not possible to calculate appropriately when the lead between the two components, the nut and the screw shaft, is not the same (for example, when a machining error occurs in the actual nut or screw shaft). Therefore, it is possible to use a coordinate system that can consider the effect of the minute twist between the actual nut and the screw shaft. The Freinet coordinate system is known as one such coordinate system. However, it is not necessarily easy to perform accurate coordinate conversion between the Cartesian coordinate system and the Freinet coordinate system.

[0008] In view of the above problems, an object of the present invention is to provide a conversion method that takes into account the configuration of a mechanical part having a spiral shape, such as a ball screw, and enables highly accurate conversion from a Cartesian coordinate system to a Frenet coordinate system in order to identify the coordinates of a point near a spiral. [Means for solving the problem]

[0009] In order to solve the above problems, the present invention has the following configuration: A coordinate transformation method for transforming coordinates of a point near a spiral from coordinates (x, y, z) in a Cartesian coordinate system to parameters (θ, η, ζ) indicating coordinates in a Frenet coordinate system, comprising the steps of: A first transformation step of transforming coordinates of the point in a Cartesian coordinate system into parameters indicating coordinates in a Freinet coordinate system; a second conversion step of converting the parameters indicating the coordinates of the Freinet coordinate system obtained in the first conversion step into coordinates of a Cartesian coordinate system; a setting step of setting a temporary coordinate in the orthogonal coordinate system based on a difference between the coordinate of the point in the orthogonal coordinate system and the coordinate of the point in the orthogonal coordinate system obtained in the second conversion step; a third transformation step of transforming the provisional coordinates into parameters representing coordinates of a Frenet coordinate system to calculate parameters representing coordinates of the point in the Frenet coordinate system; has.

[0010] Another aspect of the present invention has the following configuration: A coordinate conversion device that converts coordinates (x, y, z) of a Cartesian coordinate system into parameters (θ, η, ζ) indicating coordinates of a Freinet coordinate system for a point near a spiral, comprising: A first conversion means for converting coordinates of the point in an orthogonal coordinate system into parameters indicating coordinates in a Freinet coordinate system; a second conversion means for converting the parameters indicating the coordinates of the Freinet coordinate system obtained by the first conversion means into coordinates of a Cartesian coordinate system; a setting means for setting a temporary coordinate in the orthogonal coordinate system based on a difference between the coordinate of the point in the orthogonal coordinate system and the coordinate of the point in the orthogonal coordinate system obtained by the second conversion means; a third conversion means for converting the provisional coordinates into parameters indicating coordinates of the Frenet coordinate system, thereby calculating parameters indicating the coordinates of the point in the Frenet coordinate system; has.

[0011] Another aspect of the present invention has the following configuration. A program for executing a coordinate transformation method for transforming a point near a spiral from coordinates (x, y, z) in a Cartesian coordinate system to parameters (θ, η, ζ) indicating coordinates in a Frenet coordinate system, The coordinate transformation method includes: A coordinate transformation method for transforming a point near a spiral from coordinates (x, y, z) in a Cartesian coordinate system to parameters (θ, η, ζ) indicating coordinates in a Frenet coordinate system, comprising: A first transformation step of transforming coordinates of the point in a Cartesian coordinate system into parameters indicating coordinates in a Freinet coordinate system; a second conversion step of converting the parameters indicating the coordinates of the Freinet coordinate system obtained in the first conversion step into coordinates of a Cartesian coordinate system; a setting step of setting a temporary coordinate in the orthogonal coordinate system based on a difference between the coordinate of the point in the orthogonal coordinate system and the coordinate of the point in the orthogonal coordinate system obtained in the second conversion step; a third transformation step of transforming the provisional coordinates into parameters representing coordinates of a Frenet coordinate system to calculate parameters representing coordinates of the point in the Frenet coordinate system; has. Effect of the Invention

[0012] According to the present invention, it is possible to provide a conversion method that enables highly accurate conversion from a Cartesian coordinate system to a Freinet coordinate system for the coordinates of points near a spiral. [Brief description of the drawings]

[0013] [Figure 1] FIG. 1 is a schematic diagram showing an example of an apparatus configuration according to an embodiment of the present invention. [Diagram 2] 1 is a schematic diagram showing an example of a mechanical part to which the coordinate transformation method according to the present invention can be applied; [Diagram 3] FIG. 2 is a diagram for explaining each element of a spiral. [Figure 4] FIG. 2 is a diagram showing an x-axis that coincides with the direction of progression of a spiral and a unit vector at an arbitrary point q according to an embodiment of the present invention. [Diagram 5] FIG. 2 is a diagram for explaining coordinate transformation according to an embodiment of the present invention. [Figure 6] FIG. 2 is a diagram for explaining a coordinate plane according to an embodiment of the present invention. [Figure 7] FIG. 2 is a diagram for explaining movement of an xyz coordinate system according to an embodiment of the present invention. [Figure 8] 4 is a flowchart of an overall process according to an embodiment of the present invention. [Figure 9] 11 is a flowchart of a process for calculating a minute spiral angle and coordinates according to an embodiment of the present invention. [Figure 10] FIG. 2 is a diagram for explaining a relationship when conversion is performed between coordinate systems according to an embodiment of the present invention. [Figure 11] 4 is a flowchart of a repeat process according to an embodiment of the present invention. [Figure 12] FIG. 11 is a diagram showing an example of a calculation result according to the embodiment of the present invention. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0014] Hereinafter, the embodiment for carrying out the present invention will be described with reference to the drawings. Note that the embodiment described below is one embodiment for explaining the present invention, and is not intended to be interpreted as limiting the present invention, and all configurations described in each embodiment are not necessarily essential configurations for solving the problems of the present invention. In addition, in each drawing, the same components are given the same reference numbers to indicate their correspondence.

[0015] <First embodiment> A first embodiment of the present invention will be described below.

[0016] [Device configuration] 1 is a schematic diagram showing an example of the overall configuration of a coordinate conversion device capable of executing the coordinate conversion method according to the present embodiment. The coordinate conversion device 1 according to the present embodiment may be, for example, an information processing device such as a PC (Personal Computer), and the configuration is not particularly limited.

[0017] The coordinate conversion device 1 includes a central processing unit (CPU) 10, a read only memory (ROM) 11, a random access memory (RAM) 12, a hard disk drive (HDD) 13, an input device 14, a display device 15, and a communication device 16. The CPU 10 is a part that controls the entire coordinate conversion device 1, and may realize various functions by, for example, reading and executing a program stored in the HDD 13. The ROM 11 is a non-volatile storage area. The RAM 12 is a volatile storage area, and is used as a temporary data storage location. The HDD 13 is a non-volatile storage area, and various programs and data are stored and managed. The input device 14 is a part that accepts input from the outside, and is composed of, for example, a mouse and a keyboard. The display device 15 is a part for displaying various information, and is, for example, a liquid crystal display. A touch panel display in which the input device 14 and the display device 15 are integrated may be used. The communication device 16 is a part for communicating with an external device (not shown) via a network (not shown). The communication here may be wired or wireless, and there are no particular limitations on the communication standard.

[0018] The coordinate conversion method according to this embodiment can be used, for example, to simulate the operation of a mechanical part having a helical structure, such as a ball screw. FIG. 2 is a diagram showing an example of the configuration of a ball screw 20 to which the coordinate conversion method according to this embodiment can be applied. The ball screw 20 includes a nut 21, a screw shaft 22, and a plurality of rolling elements 23. A helical groove 22a is formed on the outer circumferential surface of the screw shaft 22, with which the rolling elements 23 contact and roll. The helical groove 22a draws a three-dimensional curve in the shape of a helical line (normal helix). FIG. 2 also shows the x-axis, y-axis, and z-axis in a Cartesian coordinate system. In the following description, the axes of each coordinate system correspond to each other.

[0019] FIG. 3 is a diagram for explaining each element of the spiral used in this embodiment. Here, in the xyz coordinate system, a helical string winding is shown in a counterclockwise direction around the x-axis. l indicates the pitch of the spiral. r indicates the radius of the spiral. θ indicates the spiral angle.

[0020] Coordinate System Two coordinate systems according to this embodiment will be described. The first coordinate system is an orthogonal coordinate system in which the helical axis is used as a reference and the x-axis and the helical axis coincide with each other. In the orthogonal coordinate system, the three axes are the x-axis, the y-axis, and the z-axis, and this is also called the xyz coordinate system. The second coordinate system is a coordinate system shown in the Freinet-Serret frame with an arbitrary point on the helix as a reference (hereinafter, this is called the Freinet coordinate system). In the Freinet coordinate system, the three axes are the ξ-axis, the η-axis, and the ζ-axis, and this is also called the ξηζ coordinate system. In this embodiment, the conversion of coordinates between these two coordinate systems is called coordinate conversion.

[0021] Based on the elements shown in Figure 3, if the pitch of the spiral is l, the starting angle of the spiral is α, the radius of the spiral is r, and the parameter corresponding to the spiral angle is θ, then in the xyz coordinate system, any point q on the spiral is defined by the following equation (1).

[0022]

number

[0023] l: pitch of the helix α: The starting angle of the spiral r: radius of the spiral θ: Parameter (helix angle)

[0024] As shown in equation (1), as the parameter θ increases, the spiral moves in the positive x-direction. The starting angle α in equation (1) corresponds to the phase angle at x=0 in the xz plane when the spiral is viewed along the y-direction.

[0025] On the other hand, in the Freinet coordinate system, if the unit vectors in the directions of the ξ, η, and ζ axes are t, n, and b, respectively, they are defined as the following equations (2), (3), and (4), respectively. Note that each variable is the same as in equation (1).

[0026]

number

[0027]

number

[0028]

number

[0029] FIG. 4 is a diagram showing the x-axis coinciding with the direction of progression of the spiral according to this embodiment and the unit vector at an arbitrary point q. In FIG. 4, the spiral is shown so that the x-axis of the xyz coordinate system and the spiral axis coincide with each other. An arbitrary point q is also shown on the spiral. The direction of the ξ-axis corresponds to the direction of progression of the spiral, as shown in FIG. 4 and the above formula (2), and t is the tangent vector at point q. The direction of the η-axis corresponds to the center direction of the spiral, as shown in FIG. 4 and the above formula (3), and n is the normal vector at point q. And, as shown in FIG. 4 and the above formula (4), the vector b(θ) corresponding to the direction of the ζ-axis is the binormal vector at point q, and is the cross product of the tangent vector t(θ) and the normal vector n(θ).

[0030] Here, an arbitrary point on the periphery of the spiral in three-dimensional space is x0, and its coordinates in the xyz coordinate system are (x0, y0, z0). To associate this with the ξηζ coordinate system, it is expressed as in the following formula (5) using parameters θ0, ζ0, and η0. In this embodiment, the conversion process based on formula (5) is defined as coordinate conversion between the xyx coordinate system and the ξηζ coordinate system.

[0031]

number

[0032] Figure 5 shows the η-ζ plane perpendicular to the direction of progression of the spiral (the direction of the ξ axis) with point q on the spiral as the reference point. Point x0 is located on this η-ζ plane. From the relationship shown in Figure 5, the coordinate of point x0 in the ζ axis direction and the coordinate of point x0 in the ξηζ coordinate system are defined. In other words, the η-ζ plane has point q on the spiral as its origin, and the coordinate of x0 is defined as the distance from each of the ζ axis and the η axis. Furthermore, the position of the η-ζ plane can be specified by the parameter θ, which corresponds to the spiral angle.

[0033] According to formulas (1) to (5), multiple solutions may exist depending on the value of x0. Therefore, when multiple solutions exist, the solution with the smallest ζ0 and η0 is treated as a single solution. In the case of a ball screw, which is an example of an application target of the coordinate transformation method according to this embodiment, as shown in FIG. 2, the rolling element 23 is located sufficiently close to the spiral defined by the spiral groove 22a. When such a mechanical component is assumed, the possibility of multiple solutions existing is extremely low. In this embodiment, the calculation is performed using an approximation assuming that the rolling element 23 is located sufficiently close to the spiral. By using the approximation, the solution can be limited to one.

[0034] [Coordinate transformation using approximation] Coordinate transformation using approximation according to this embodiment will be described. First, consider an arbitrary point x in the xyz coordinate system. Point x is located at a position different from the helix, but is located sufficiently close to the helix. For example, in the example of FIG. 2, the arbitrary point x may be the center position of the rolling element 23, or may be a point on the surface of the rolling element 23 that is closest to the helical groove 22a. In the ball screw 20, depending on the configuration, the rolling element 23 and the screw shaft 22 may contact each other, or a gap may be generated between them. When a gap is generated, the point on the surface of the rolling element 23 that is closest to the helical groove 22a may be assumed to be the arbitrary point x.

[0035] (When x=0, α=0) FIG. 6 is a diagram showing the coordinates of point x in the xyz coordinate system. FIG. 6 shows the yz plane where x=0, and the starting angle of the spiral is α=0. The coordinates (x, y, z) of point x in the xyz coordinate system at this time are defined as coordinates (0, y0, z0). As described above, the assumption that point x is located sufficiently close to the spiral can be replaced with the conditions that x0=0, y0≒r, and z0≒0 without losing generality. In other words, when the coordinate in the x-axis direction in the xyz coordinate system is 0, the coordinates of point x can be treated as being approximate to (0, r, 0), which is the coordinate on the spiral.

[0036] When the above formula (5) is applied to the yz plane shown in FIG. 6, the following formula (6) can be derived as the correspondence relationship between the xyz coordinate system and the ξηζ coordinate system with respect to the point x.

[0037]

number

[0038] At this time, since the rolling element 23 is located sufficiently close to the spiral defined by the spiral groove 22a, θ0<<1, and the following approximate value can be used in the above equation (6). That is, in this embodiment, in the Taylor expansion of the trigonometric function, approximation is performed using values ​​up to the second order, and this second-order approximate solution is defined algebraically for use.

[0039]

number

[0040]

number

[0041] From the above equations (6) to (8), the following equation (9) is obtained.

[0042]

number

[0043] From equation (9), the simultaneous equations shown in equation (10) below are derived.

[0044]

number

[0045] By solving equation (10), equation (11) is defined to perform coordinate transformation between the following two coordinate systems.

[0046]

number

[0047] (When x=x0, α=α0) The calculation for the case where x=0 and α=0 has been explained above. Next, the calculation for the case where it is expanded to the case where x=x0 and α=α0 will be explained. Figure 7(a) shows the coordinates of an arbitrary point x in the xyz coordinate system. The coordinates (x, y, z) of point x in the xyz coordinate system at this time are set to (x0, y0, z0). Also, the starting angle of the spiral is set to α=α0. In this case as well, the explanation will be given assuming that point x is located sufficiently close to the spiral.

[0048] In this case, the entire xyz coordinate system is rotated around the x-axis by the angle (spiral angle β) shown in the following equation (12). As above, α indicates the starting angle of the spiral, which is α0 in this example. Equation (12) indicates that the entire xyz coordinate system is rotated in the opposite direction to the progression of the spiral. For example, as shown in Figure 7(a), in the case of a spiral progressing counterclockwise, the entire xyz coordinate system is rotated clockwise.

[0049]

number

[0050] Next, the entire xyz coordinate system is moved by -x0 so that the x-axis value of point x in the xyz coordinate system becomes 0. As a result, the coordinates of point x have the relationship shown in FIG. 7(b), and the coordinates of point x in this case are shown as (0, y1, z1). This results in the same relationship as shown in FIG. 6. Then, in equations (10) and (11) described above in the case of x=0, α=0, a similar coordinate transformation can be performed by substituting (0, y1, z1) for (x0, y0, z0). Specifically, the following equation (13) is used.

[0051]

number

[0052] [Processing flow] 8 is a flowchart of the coordinate conversion process according to this embodiment. This process is executed by the coordinate conversion device 1, for example, by a control device (not shown) included in the coordinate conversion device 1 reading out a program for implementing the process according to this embodiment from a storage device (not shown) and executing the program.

[0053] In S801, the coordinate conversion device 1 sets the xyz coordinate system so that the helical axis of the target helix coincides with the x-axis of the xyz coordinate system. For example, in the case of Fig. 2, the helical axis corresponds to the central axis of the screw shaft 22. At this time, the coordinates of the coordinate point x of interest in the xyz coordinate system are set to (x0, y0, z0) as shown in Fig. 7(a).

[0054] In S802, the coordinate conversion device 1 calculates a tentative helical angle β from the height of a coordinate point x of interest in the xyz coordinate system. The height of the coordinate point here corresponds to the value x0 of the coordinate point x in the x-axis direction. The tentative helical angle β can be determined based on the configuration of the helix (pitch l, starting point angle α) shown in FIG. 3 and the height of the coordinate point (=x0).

[0055] In S803, the coordinate conversion device 1 rotates the entire xyz coordinate system in the direction opposite to the progression direction of the spiral by the provisional spiral angle β. That is, the xyz coordinate system set in S801 is rotated by the value shown in the above formula (12).

[0056] In S804, the coordinate conversion device 1 translates the entire xyz coordinate system by x0, which is the value in the x-axis direction of the coordinate point x of interest. Through the processing up to this point, the coordinates of the coordinate point x of interest in the xyz coordinate system can be converted to (0, y1, z1) as shown in FIG.

[0057] In S805, the coordinate conversion device 1 calculates the minute spiral angle and the coordinates. The processing of this step will be described in detail with reference to FIG.

[0058] In S806, the coordinate transformation device 1 adds the infinitesimal helical angle derived in the process of S805 and the provisional helical angle β to calculate the value of θ among the parameters (θ, η, ζ) indicating the coordinates of the coordinate point x in the ξηζ coordinate system.

[0059] In S807, the coordinate conversion device 1 outputs the parameters (θ, η, ζ) indicating the coordinates of the coordinate point x in the ξηζ coordinate system calculated in S805 and S806. The output method here is not particularly limited, and the parameters may be output in association with the coordinates of the coordinate point x in the xyz coordinate system, or conversion conditions for other coordinates may be output together. Then, this processing flow ends.

[0060] (Micro-spiral angle and coordinate calculation processing) FIG. 9 is a flowchart showing details of the process in step S805 of FIG.

[0061] In S901, the coordinate conversion device 1 sets approximate values ​​for minute quantities involved in conversion processing. In this embodiment, as shown in the above formulas (7) and (8), the approximate values ​​are set using quadratic approximate values ​​in which third-order or higher terms in Taylor expansion of trigonometric functions are set to 0.

[0062] In S902, the coordinate conversion device 1 calculates the infinitesimal helical angle θ0 by using the approximation value of the Taylor expansion set in S901. The infinitesimal helical angle θ0 here corresponds to the difference between θ of the parameters (θ, η, ζ) indicating the coordinates of the coordinate point x in the ξηζ coordinate system and the provisional helical angle β.

[0063] In S903, the coordinate conversion device 1 calculates the displacement in the slight twisting direction. The twisting direction here corresponds to the ζ-axis direction. Therefore, the displacement in the slight twisting direction corresponds to the value of ζ of the parameters (θ, η, ζ) that indicate the coordinate of the coordinate point x in the ξηζ coordinate system.

[0064] In S904, the coordinate conversion device 1 calculates the displacement in the slight bending direction. The bending direction here corresponds to the η-axis direction. Therefore, the displacement in the slight bending direction corresponds to the value of η of the parameter (θ, η, ζ) indicating the coordinate of the coordinate point x in the ξηζ coordinate system. Then, the process proceeds to S806 in FIG. 8.

[0065] In the flowchart shown in FIG. 9, the slight twist direction is calculated before the slight bending direction is calculated, but the order is not limited to this, and the calculations may be performed in the reverse order.

[0066] In the above calculation method, an example is shown in which a quadratic approximation value is used for the value of the trigonometric function, but the present invention is not limited to this. In addition, for third or higher order terms included in the calculation process, an approximation value may be used depending on the configuration of the mechanical part to which the coordinate conversion method according to the present embodiment is applied.

[0067] (Repeat processing) In this embodiment, the accuracy of the coordinate transformation is further improved by repeatedly applying the above coordinate transformation method. Hereinafter, the repeated processing according to this embodiment will be described with reference to the drawings.

[0068] FIG. 10 is a diagram for explaining the relationship between the conversion between the ξηζ coordinate system and the xyz coordinate system. Here, the explanation will be given focusing on θ in the ξηζ coordinate system and x in the xyz coordinate system. In FIG. 10, x is assumed to be the coordinate value in the x-axis direction of a point to be converted from the xyz coordinate system to the ξηζ coordinate system. By applying the conversion process using the approximation described above in FIG. 8 and FIG. 9 to this x, a parameter θ0 is obtained as the conversion result (FIG. 10(1)).

[0069] Next, consider transforming from the ξηζ coordinate system to the xyz coordinate system using the parameter θ0. The transformation method at this time may be a known method and is not particularly limited. There are various methods for obtaining each parameter of the xyz coordinate system from each parameter of the ξηζ coordinate system, but it is generally known that transformation with high accuracy is possible. As a result of transforming from the ξηζ coordinate system to the xyz coordinate system, it is assumed that a parameter x0 in the xyz coordinate system is obtained ((2) of FIG. 10). In the transformation process described in FIG. 8 and FIG. 9 above, the parameter θ0 is calculated using approximation, so there may be a difference between the original parameter x in the xyz coordinate system and the parameter x0 obtained by retransformation. The difference here is d1. In other words, due to the characteristics of the transformation process, the correspondence between x0 and θ0 is more accurate than the correspondence between x and θ0.

[0070] Therefore, in this embodiment, in order to more accurately calculate the parameter θ corresponding to the parameter x, a tentative parameter x is calculated based on the difference d1. t Here, we set x0 and x as shown in Figure 10. t So that the midpoint of is x, t (3) in FIG. 10). That is, taking into consideration the conversion error that occurs when using approximation, the parameters (coordinates) for the coordinate conversion in the xyz coordinate system are adjusted.

[0071] And this x tBy applying the conversion process described with reference to FIGS. 8 and 9 above, parameter θ1 is obtained as a conversion result (FIG. 10(4)). When using this parameter θ1 to perform a conversion from the ξηζ coordinate system to the xyz coordinate system using a known method as shown in FIG. 10(2), it is assumed that parameter x1 in the xyz coordinate system is obtained. At this time, a difference d2 may occur between the original parameter x in the xyz coordinate system and the parameter x1 obtained by the above process. However, as shown in FIG. 10, (0≦)d2 < d1, and it becomes possible to obtain a parameter θ (that is, θ1) corresponding to the original parameter x with higher accuracy.

[0072] That is, in the present embodiment, by using the relational expressions shown below, a more accurate coordinate conversion is performed. Note that θ1 shown in Equation (14) corresponds to the variables shown in FIG. 10, and η1 and ζ1 are parameters calculated by the conversion process together with θ1.

[0073]

Equation

[0074] FIG. 11 is a flowchart of the process described with reference to FIG. 10.

[0075] In S1101, the coordinate conversion device 1 performs a conversion process to the Frenet coordinate system by approximation. In this step, the process described with reference to FIGS. 8 and 9 is performed. Note that this step corresponds to FIG. 10(1), and each parameter of the ξηζ coordinate system is calculated from the original parameter (coordinate) of the xyz coordinate system.

[0076] In S1102, the coordinate conversion device 1 performs a conversion process by calculating each parameter of the xyz coordinate system using each parameter of the ξηζ coordinate system calculated in S1101. This step corresponds to FIG. 10(2), and each parameter of the xyz coordinate system is calculated. As described above, the conversion method here may use a known method and is not particularly limited.

[0077] In S1103, the coordinate conversion device 1 sets temporary parameters (coordinates) in the xyz coordinate system for reconversion based on the difference between the parameters (coordinates) of the original point in the xyz coordinate system used in S1101 and the parameters in the xyz coordinate system obtained in the coordinate conversion process in S1102. This process corresponds to (3) in Fig. 10.

[0078] In S1104, the coordinate conversion device 1 performs a conversion process to the Frenet coordinate system by approximation using the parameters set in S1103. In this step, the process described with reference to Figs. 8 and 9 is performed again. This step corresponds to (4) in Fig. 10, and each parameter of the ξηζ coordinate system is calculated from the temporary parameters of the xyz coordinate system. Then, each parameter of the ξηζ coordinate system obtained by this process is output as a conversion result corresponding to the original parameters of the xyz coordinate system used in S1101. Then, this processing flow ends.

[0079] [Conversion example] 12 is a diagram for explaining the accuracy of the transformation result using the coordinate transformation method according to the present embodiment. Here, the subject is a ball screw, and calculations were performed under the following conditions. Spiral pitch (l): 600 [mm] Spiral radius (r): 800 [mm] Helix angle (θ): 1000[°] Distance between the spiral and the coordinate point: 1μm~100mm

[0080] In FIG. 12, the horizontal axis indicates (distance from the helix of the coordinate point) / (helical radius), and the vertical axis indicates (coordinate conversion error) / (helical radius). The upper graph in FIG. 12 shows the accuracy of the results obtained by performing the conversion process shown in FIG. 8 and FIG. 9 once. The lower graph in FIG. 12 shows the accuracy of the results obtained by performing the conversion process in two stages as shown in FIG. 11.

[0081] As shown in Fig. 12, the error and distance were normalized by the spiral radius, and the error was 10 -11 ~10 -3In particular, the closer the coordinate point is to the spiral, the more accurate the coordinate conversion can be. The two-stage conversion process makes it possible to further improve the accuracy, and the conversion rate is 10 -17 ~10 -5 Coordinate transformation could be performed within the range.

[0082] In particular, the 10 -3 In the vicinity, a deviation as small as the discretization error in the case of 64-bit double-precision floating point is realized. This is the calculation result of a deviation on the order of ~10 μm for a standard ball screw with a spiral radius on the order of ~10 cm. Since the clearance of a typical ball screw is created to be ~10 μm, the method according to this embodiment achieves sufficient accuracy in normal numerical simulations.

[0083] As described above, in this embodiment, an approximation is used to reduce the processing load, taking into consideration the configuration of a mechanical part having a helical shape, such as a ball screw. Furthermore, a two-stage coordinate transformation is performed. As a result, in this embodiment, a highly accurate transformation from a Cartesian coordinate system to a Frenet coordinate system is possible to identify the coordinates of a point near the helix.

[0084] <Other embodiments>

[0085] In addition, the present invention can also be realized by supplying a program or application for realizing the functions of one or more of the above-mentioned embodiments to a system or device via a network or a storage medium, etc., and having one or more processors in a computer of the system or device read and execute the program.

[0086] Moreover, it may be realized by a circuit that realizes one or more functions (for example, an application specific integrated circuit (ASIC) or a field programmable gate array (FPGA)).

[0087] As such, the present invention is not limited to the above-described embodiments, and the present invention also contemplates mutual combinations of the various components of the embodiments, as well as modifications and applications by those skilled in the art based on the descriptions in the specification and well-known technologies, and these are included in the scope of protection sought.

[0088] As described above, the present specification discloses the following: (1) A coordinate transformation method for transforming a point near a spiral from coordinates (x, y, z) in a Cartesian coordinate system to parameters (θ, η, ζ) indicating coordinates in a Freinet coordinate system, comprising the steps of: A first transformation step of transforming coordinates of the point in a Cartesian coordinate system into parameters indicating coordinates in a Freinet coordinate system; a second conversion step of converting the parameters indicating the coordinates of the Freinet coordinate system obtained in the first conversion step into coordinates of a Cartesian coordinate system; a setting step of setting a temporary coordinate in the orthogonal coordinate system based on a difference between the coordinate of the point in the orthogonal coordinate system and the coordinate of the point in the orthogonal coordinate system obtained in the second conversion step; a third transformation step of transforming the provisional coordinates into parameters representing coordinates of a Frenet coordinate system to calculate parameters representing coordinates of the point in the Frenet coordinate system; A coordinate transformation method comprising the steps of: According to this configuration, it is possible to provide a conversion method that enables highly accurate conversion from a Cartesian coordinate system to a Freinet coordinate system for the coordinates of points near the spiral.

[0089] (2) The coordinate transformation method described in (1), wherein in the setting step, the provisional coordinates are set so that the x-axis value of the point in a Cartesian coordinate system is the midpoint between the x-axis value of the coordinate obtained in the second transformation step and the x-axis value of the provisional coordinates. According to this configuration, it is possible to perform highly accurate coordinate transformation by taking into account errors in coordinate transformation using approximation.

[0090] (3) The coordinate transformation method according to (1) or (2), wherein the first transformation step and the third transformation step perform transformation processing using the same transformation method. According to this configuration, the configuration for coordinate transformation from the Cartesian coordinate system to the Frenet coordinate system can be standardized, and by using the same method, the characteristics of the error caused by approximation can be appropriately reflected.

[0091] (4) In the first conversion step and the third conversion step, An orthogonal coordinate system is set so that the coordinate value with respect to the x-axis of the orthogonal coordinate system corresponding to the helical axis of the spiral is 0; When the coordinates of the point of interest in the set Cartesian coordinate system are (0, y0, z0) and the parameters indicating the coordinates of the point of interest in the Freinet coordinate system are (θ0, η0, ζ0), a coordinate transformation is performed based on the following formula (1):

[0092]

number

[0093] The coordinate transformation method according to (3), characterized in that in the formula (1), parameters (θ0, η0, ζ0) indicating the coordinates of the point of interest in the Freinet coordinate system are derived by solving a simultaneous equation obtained by using an approximation value in which the third or higher order terms are set to zero when the cubic function is expanded into a Taylor series. According to this configuration, it is possible to take into account the configuration of a mechanical part having a spiral shape, such as a ball screw, and to perform coordinate transformation of the coordinates of points near the spiral with a relatively low processing load.

[0094] (5) A coordinate transformation method according to any one of (1) to (4), characterized in that the Freinet coordinate system is defined as a point on the spiral as the origin, the direction of progression of the spiral as the ξ axis, the direction of the center of the spiral as the η axis, and a direction perpendicular to each of the ξ axis and the η axis as the ζ axis. This configuration enables coordinate transformation from a Cartesian coordinate system to a Frenet coordinate system that corresponds to spiral points.

[0095] (6) A coordinate conversion device for converting coordinates (x, y, z) of a Cartesian coordinate system into parameters (θ, η, ζ) indicating coordinates of a point near a spiral, comprising: A first conversion means for converting coordinates of the point in an orthogonal coordinate system into parameters indicating coordinates in a Freinet coordinate system; a second conversion means for converting the parameters indicating the coordinates of the Freinet coordinate system obtained by the first conversion means into coordinates of a Cartesian coordinate system; a setting means for setting a temporary coordinate in the orthogonal coordinate system based on a difference between the coordinate of the point in the orthogonal coordinate system and the coordinate of the point in the orthogonal coordinate system obtained by the second conversion means; a third conversion means for converting the provisional coordinates into parameters indicating coordinates of the Frenet coordinate system, thereby calculating parameters indicating the coordinates of the point in the Frenet coordinate system; A coordinate conversion device comprising: According to this configuration, it is possible to provide a conversion method that enables highly accurate conversion from a Cartesian coordinate system to a Freinet coordinate system for the coordinates of points near the spiral.

[0096] (7) To the computer: A program for executing a coordinate transformation method for transforming a point near a spiral from coordinates (x, y, z) in a Cartesian coordinate system to parameters (θ, η, ζ) indicating coordinates in a Frenet coordinate system, The coordinate transformation method includes: A coordinate transformation method for transforming a point near a spiral from coordinates (x, y, z) in a Cartesian coordinate system to parameters (θ, η, ζ) indicating coordinates in a Frenet coordinate system, comprising: A first transformation step of transforming coordinates of the point in a Cartesian coordinate system into parameters indicating coordinates in a Freinet coordinate system; a second conversion step of converting the parameters indicating the coordinates of the Freinet coordinate system obtained in the first conversion step into coordinates of a Cartesian coordinate system; a setting step of setting a temporary coordinate in the orthogonal coordinate system based on a difference between the coordinate of the point in the orthogonal coordinate system and the coordinate of the point in the orthogonal coordinate system obtained in the second conversion step; a third transformation step of transforming the provisional coordinates into parameters representing coordinates of a Frenet coordinate system to calculate parameters representing coordinates of the point in the Frenet coordinate system; A program comprising: According to this configuration, it is possible to provide a conversion method that enables highly accurate conversion from a Cartesian coordinate system to a Freinet coordinate system for the coordinates of points near the spiral. [Explanation of symbols]

[0097] 1... Coordinate conversion device 10…CPU(Central Processing Unit) 11...ROM (Read Only Memory) 12…RAM(Random Access Memory) 13...HDD (Hard Disk Drive) 14...Input device 15...Display device 16...Communication equipment 20...Ball screw 21...Nut 22...Screw shaft 22a...Spiral groove 23...Rolling element

Claims

1. A coordinate transformation method for transforming a point near a spiral from coordinates (x, y, z) in a Cartesian coordinate system to parameters (θ, η, ζ) indicating coordinates in a Frenet coordinate system, comprising the steps of: a first transformation step of transforming coordinates of the point in a Cartesian coordinate system into parameters indicating coordinates in a Freinet coordinate system; a second conversion step of converting the parameters indicating the coordinates of the Freinet coordinate system obtained in the first conversion step into coordinates of an orthogonal coordinate system; a setting step of setting a temporary coordinate in the orthogonal coordinate system based on a difference between the coordinate of the point in the orthogonal coordinate system and the coordinate of the point in the orthogonal coordinate system obtained in the second conversion step; a third transformation step of transforming the provisional coordinates into parameters representing coordinates of a Frenet coordinate system to calculate parameters representing coordinates of the point in the Frenet coordinate system; A coordinate transformation method comprising the steps of:

2. 2. The coordinate transformation method according to claim 1, wherein in the setting step, the provisional coordinates are set so that the x-axis value of the point is the midpoint between the x-axis value of the coordinates obtained in the second transformation step and the x-axis value of the provisional coordinates in a Cartesian coordinate system.

3. 3. The coordinate transformation method according to claim 1, wherein the first transformation step and the third transformation step perform transformation processing using the same transformation method.

4. In the first conversion step and the third conversion step, An orthogonal coordinate system is set so that a coordinate value with respect to an x-axis of the orthogonal coordinate system corresponding to the helical axis of the spiral is 0; The coordinates of the point of interest in the set orthogonal coordinate system are (0, y ０ , z ０ ), and a parameter (θ ０ , η ０ , ζ ０ ), coordinate conversion is performed based on the following equation (1): [0010] In the above formula (1), a simultaneous equation is obtained by using an approximation value in which the third or higher order terms are set to 0 when the cubic function is expanded into Taylor series, and a parameter (θ ０ , η ０ , ζ ０ 4. The method for transforming coordinates according to claim 3, further comprising the step of: deriving

5. The coordinate transformation method according to any one of claims 1 to 4, characterized in that the Freinet coordinate system is a coordinate system defined in such a way that a point on the spiral is the origin, the direction of progression of the spiral is the ξ axis, the direction of the center of the spiral is the η axis, and a direction perpendicular to each of the ξ axis and the η axis is the ζ axis.

6. A coordinate transformation device that transforms coordinates (x, y, z) of a Cartesian coordinate system into parameters (θ, η, ζ) indicating coordinates of a Freinet coordinate system for a point near a spiral, a first conversion means for converting coordinates of the point in an orthogonal coordinate system into parameters indicating coordinates in a Freinet coordinate system; a second conversion means for converting the parameters indicating the coordinates of the Freinet coordinate system obtained by the first conversion means into coordinates of a Cartesian coordinate system; a setting means for setting a temporary coordinate in the orthogonal coordinate system based on a difference between the coordinate of the point in the orthogonal coordinate system and the coordinate of the point in the orthogonal coordinate system obtained by the second conversion means; a third conversion means for converting the provisional coordinates into parameters indicating coordinates of a Frenet coordinate system, thereby calculating parameters indicating coordinates of the point in the Frenet coordinate system; A coordinate transformation device comprising:

7. On the computer, A program for executing a coordinate transformation method for transforming a point near a spiral from coordinates (x, y, z) in a Cartesian coordinate system to parameters (θ, η, ζ) indicating coordinates in a Frenet coordinate system, the program comprising: The coordinate transformation method includes: A coordinate transformation method for transforming a point near a spiral from coordinates (x, y, z) in a Cartesian coordinate system to parameters (θ, η, ζ) indicating coordinates in a Frenet coordinate system, comprising the steps of: a first transformation step of transforming coordinates of the point in a Cartesian coordinate system into parameters indicating coordinates in a Freinet coordinate system; a second conversion step of converting the parameters indicating the coordinates of the Freinet coordinate system obtained in the first conversion step into coordinates of an orthogonal coordinate system; a setting step of setting a temporary coordinate in the orthogonal coordinate system based on a difference between the coordinate of the point in the orthogonal coordinate system and the coordinate of the point in the orthogonal coordinate system obtained in the second conversion step; a third transformation step of transforming the provisional coordinates into parameters representing coordinates of a Frenet coordinate system to calculate parameters representing coordinates of the point in the Frenet coordinate system; A program comprising:

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