JP7484750B2 - Method, device, and program for converting coordinates from a Cartesian coordinate system to a Freinet coordinate system - Google Patents

Description

This invention relates to a method, a device, and a program for converting coordinates from a Cartesian coordinate system to a Freinet coordinate system.

Conventionally, mechanical components such as ball screws have spiral-shaped parts (e.g., grooves in which rolling elements can roll). Various methods have been disclosed for appropriately analyzing the operating state and shape of each part of such spiral-shaped mechanical components.

For example, Patent Document 1 discloses a method for extracting data on the root portion of a helical thread and analyzing the helical shape.

Japanese Patent No. 3971968

For example, in calculating the contact state of the rolling elements inside a mechanical part with a helical shape, such as a ball screw, 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 at an arbitrary position and the rolling elements, it is necessary to search for the position of the rolling elements near the helix of the screw shaft. In particular, when performing mechanical analysis, all of the components of the mechanical part, such as the rolling elements, can move to any position, so the calculation process is repeated many times to identify the position each time. Furthermore, high accuracy is required in such calculations for neighborhood search.

However, this type of calculation is an inverse problem, and since it involves the solution of 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 with a spiral shape, it has the characteristic that the range of movement of the rolling elements is limited to a certain range near the spiral.

In view of the above problems, the present invention aims to provide a method of conversion that takes into account the configuration of a mechanical part having a helical shape, such as a ball screw, and enables high-speed conversion from a Cartesian coordinate system to a Frenet coordinate system in order to identify the coordinates of a point near a helix.

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 setting step of setting the orthogonal coordinate system so that a coordinate value of the point with respect to an x-axis of the orthogonal coordinate system corresponding to a helical axis of the spiral is 0;
a conversion step of performing a coordinate conversion based on the following formula (1) when the coordinates of the point in the Cartesian coordinate system set in the setting step are (0, y ₀ , z ₀ ) and the parameters indicating the coordinates of the point in the Frenet coordinate system are (θ ₀ , η ₀ , ζ ₀ )

having
In the conversion process, the coordinates (θ 0 , η 0 , ζ 0 ) of the point in the Frenet coordinate system _are derived by solving a simultaneous equation obtained by using an approximation in which the third-order or higher terms in the Taylor expansion of the cubic function in equation ( ₁ ) are set to _zero .

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 point near a spiral in a Freinet coordinate system,
a setting means for setting the orthogonal coordinate system so that a coordinate value of the point with respect to an x-axis of the orthogonal coordinate system corresponding to a helical axis of the spiral is 0;
a conversion means for performing coordinate conversion based on the following formula (1) when the coordinates of the point in the Cartesian coordinate system set by the setting means are (0, y ₀ , z ₀ ) and the parameters indicating the coordinates of the point in the Frenet coordinate system are (θ ₀ , η ₀ , ζ ₀ );

having
The conversion means derives the coordinates (θ 0 , η 0 , ζ 0 ) of the point in the Frenet coordinate system by solving simultaneous equations obtained by using approximate values in which third-order or higher terms _are set to zero when a cubic function is expanded in Taylor series in equation ( _{1 )} .

Another aspect of the present invention has the following configuration. That is, a computer includes:
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 setting step of setting the orthogonal coordinate system so that a coordinate value of the point with respect to an x-axis of the orthogonal coordinate system corresponding to a helical axis of the spiral is 0;
a conversion step of performing a coordinate conversion based on the following formula (1) when the coordinates of the point in the Cartesian coordinate system set in the setting step are (0, y ₀ , z ₀ ) and the parameters indicating the coordinates of the point in the Frenet coordinate system are (θ ₀ , η ₀ , ζ ₀ )

having
In the conversion process, parameters (θ 0 , η 0 , ζ 0 ) indicating the coordinates of the point in the Frenet coordinate system _are derived by solving simultaneous equations obtained by using approximate values in which third-order and higher terms are set to zero when the cubic function in formula ( ₁ ) is expanded into _a Taylor series.

The present invention makes it possible to provide a conversion method that enables high-speed conversion from a Cartesian coordinate system to a Freinet coordinate system for the coordinates of points near a spiral.

FIG. 1 is a schematic diagram showing an example of an apparatus configuration according to an embodiment of the present invention. 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; FIG. 1 is a diagram for explaining each element of a spiral. FIG. 2 is a diagram for explaining a coordinate system according to an embodiment of the present invention. FIG. 2 is a diagram for explaining coordinate transformation according to an embodiment of the present invention. FIG. 2 is a diagram for explaining a coordinate plane according to an embodiment of the present invention. FIG. 2 is a diagram for explaining movement of an xyz coordinate system according to an embodiment of the present invention. 4 is a flowchart of an overall process according to an embodiment of the present invention. 11 is a flowchart of a process for calculating a minute spiral angle and coordinates according to an embodiment of the present invention. FIG. 11 is a diagram showing an example of a calculation result according to the embodiment of the present invention.

The following describes the embodiment of the present invention 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 not all of the configurations described in each embodiment are 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.

First Embodiment
A first embodiment of the present invention will be described below.

[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.

The coordinate conversion device 1 includes a CPU (Central Processing Unit) 10, a ROM (Read Only Memory) 11, a RAM (Random Access Memory) 12, a HDD (Hard Disk Drive) 13, an input device 14, a display device 15, and a communication device 16. The CPU 10 is responsible for controlling 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, a keyboard, etc. The display device 15 is a section for displaying various types of 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 also be used. The communication device 16 is a section 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.

The coordinate transformation 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. Figure 2 is a diagram showing an example of the configuration of a ball screw 20 to which the coordinate transformation 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 come into contact and roll. The helical groove 22a describes a three-dimensional curve in the shape of a helical line (normal helix). Figure 2 also shows the x-axis, y-axis, and z-axis in a Cartesian coordinate system. In the following explanation, the axes of each coordinate system correspond to each other.

Figure 3 is a diagram for explaining each element of the spiral used in this embodiment. Here, in the xyz coordinate system, the 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.

[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 based on an arbitrary point of the helix (hereinafter, referred to as 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.

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).

l: helix pitch α: helix starting angle r: helix radius θ: parameter (helix angle)

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.

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

Figure 4 is a diagram for explaining the relationship between the two coordinate systems according to this embodiment. In Figure 4, a spiral is shown so that the x-axis of the xyz coordinate system coincides with the spiral axis. An arbitrary point q is also shown on the spiral. The direction of the ξ axis corresponds to the direction of the spiral as shown in Figure 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 Figure 4 and the above formula (3), and n is the normal vector at point q. And, as shown in Figure 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).

Here, an arbitrary point on the periphery of the spiral in three-dimensional space is designated as _x0 , and its coordinates in the xyz coordinate system are designated as ( _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.

Fig. 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 a reference. Point _x0 is located on this η-ζ plane. From the relationship shown in Fig. 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 corresponding to the helix angle.

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 helix defined by the helical 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 helix. By using an approximation, the solution can be limited to one.

[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 come into contact with 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.

(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 α of the spiral is 0. The coordinates (x, y, z) of point x in the xyz coordinate system at this time are defined as coordinates (0, y ₀ , z ₀ ). As described above, the assumption that point x is located sufficiently close to the spiral does not lose generality even if it is replaced with the conditions that x ₀ =0, y ₀ ≒ r, and z ₀ ≒ 0. In other words, in the xyz coordinate system, when the coordinate in the x-axis direction is 0, the coordinates of point x can be treated as being approximate to (0, r, 0), which is the coordinate on the spiral.

When the above formula (5) is applied to the y-z plane shown in Figure 5, the following formula (6) can be derived as the correspondence between the xyz coordinate system and the ξηζ coordinate system for point x.

At this time, since the rolling element 23 is located sufficiently close to the spiral defined by the spiral groove 22a, θ ₀ << 1, and the following approximate value can be used in the above equation (6). In other words, in this embodiment, in the Taylor expansion of trigonometric functions, approximations are made with values up to the second order, and this second-order approximate solution is defined algebraically and used.

The following equation (9) can be obtained from the above equations (6) to (8).

From equation (9), the simultaneous equations shown in equation (10) can be derived.

By solving equation (10), we define equation (11) to perform coordinate transformation between the following two coordinate systems.

By substituting the coordinate values of the xyz coordinate system into equation (11) defined by solving equation (10), the parameters θ ₀ , η ₀ , and ζ ₀ corresponding to the coordinates in the Frenet coordinate system can be derived.

(When x = x ₀ , α = α ₀ )
The calculation when x=0 and α=0 has been described above. Next, the calculation when expanded when x= _x0 and α= _α0 will be described. Fig. 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.

In this case, the entire xyz coordinate system is rotated around the x axis by an angle (spiral angle β) shown in the following formula (12). As above, α indicates the starting angle of the spiral, and in this example, α is _0. Formula (12) indicates that the entire xyz coordinate system is rotated in the direction opposite 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.

Next, the entire xyz coordinate system is moved by _-x0 so that the value of the x-axis of point x in the xyz coordinate system becomes 0. As a result, the coordinates of point x have the relationship shown in FIG. 7B, 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 the formulas (10) and (11) described above in the case of x= ₀ , α= ₀ , by substituting (x0, _y0 , z0) with (0, _y1 , _z1 ), it becomes possible to perform a similar coordinate transformation. Specifically, the following formula (13) is used.

[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.

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).

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 1, starting point angle α) shown in FIG. 3 and the height of the coordinate point (= _x0 ).

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

In S804, the coordinate conversion device 1 translates the entire xyz coordinate system by the value of 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.

In S805, the coordinate conversion device 1 calculates the minute spiral angle and coordinates. The processing of this step will be explained in detail using FIG. 8.

In S806, the coordinate conversion device 1 adds the infinitesimal spiral angle derived in the process of S805 and the provisional spiral angle β to calculate the value of θ, one of the parameters (θ, η, ζ) that indicate the coordinates of the coordinate point x in the ξηζ coordinate system.

In S807, the coordinate conversion device 1 outputs parameters (θ, η, ζ) indicating the coordinates of 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 coordinate point x in the xyz coordinate system, or conversion conditions for other coordinates may be output together. Then, this processing flow ends.

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

In S901, the coordinate conversion device 1 sets approximate values for minute quantities involved in the conversion process. In this embodiment, the approximate values are set using quadratic approximation values that set third-order and higher terms to zero when performing Taylor expansion on trigonometric functions, as shown in the above formulas (7) and (8).

In S902, the coordinate conversion device 1 calculates the infinitesimal helical angle θ ₀ by using the approximation value of the Taylor expansion set in S901. The infinitesimal helical angle θ ₀ here corresponds to the difference between θ of the parameters (θ, η, ζ) indicating the coordinate of the coordinate point x in the ξηζ coordinate system and the provisional helical angle β.

At 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 parameter (θ, η, ζ) that indicates the coordinate of the coordinate point x in the ξηζ coordinate system.

At 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 in the parameter (θ, η, ζ) that indicates the coordinate of the coordinate point x in the ξηζ coordinate system. Then, the process proceeds to S806 in FIG. 8.

In this embodiment, the micro-twist direction is calculated before the micro-bend direction is calculated, but this order is not limited to this and the calculations may be performed in the reverse order.

In addition, in the above calculation method, an example was shown in which a quadratic approximation value was used for the value of the trigonometric function, but this 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 transformation method according to this embodiment is applied.

[Conversion example]
10 is a diagram for explaining the accuracy of the transformation result using the coordinate transformation method according to the present embodiment. Here, the target is a ball screw, and calculations are 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: 1 μm to 100 mm

In Fig. 10, the horizontal axis indicates (distance from the spiral to the coordinate point)/(spiral radius), and the vertical axis indicates (coordinate conversion error)/(spiral radius). As shown in Fig. 10, as a result of normalizing the error and distance by the spiral radius, it was possible to perform coordinate conversion with high accuracy in the error range of 10 ^-11 to 10 ^-3 . In particular, the closer the coordinate point is to the spiral, the more accurate the coordinate conversion can be.

As described above, in this embodiment, an approximation is used to reduce the processing load, taking into account the configuration of a mechanical part having a spiral shape, such as a ball screw. This enables high-speed conversion from a Cartesian coordinate system to a Frenet coordinate system to identify the coordinates of a point near the spiral.

<Other embodiments>

The present invention can also be realized by providing a program or application for implementing the functions of one or more of the above-described embodiments to a system or device via a network or storage medium, and having one or more processors in the computer of the system or device read and execute the program.

It may also be realized by a circuit that realizes one or more functions (for example, an ASIC (Application Specific Integrated Circuit) or an FPGA (Field Programmable Gate Array)).

As such, the present invention is not limited to the above-described embodiments, and the invention also contemplates 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 the protection sought.

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 coordinates (θ, η, ζ) in a Frenet coordinate system, comprising the steps of:
a setting step of setting the orthogonal coordinate system so that a coordinate value of the point with respect to an x-axis of the orthogonal coordinate system corresponding to a helical axis of the spiral is 0;
a conversion step of performing coordinate conversion based on the following formula (1) when the coordinates of the point in the Cartesian coordinate system set in the setting step are (0, y ₀ , z ₀ ) and the coordinates of the point in the Frenet coordinate system are (θ ₀ , η ₀ , ζ ₀ )

having
a transforming step of transforming the coordinates (θ 0 , η 0 , ζ 0 ) of the point in the Frenet coordinate system _by solving a simultaneous equation obtained by using an approximation obtained by setting the third or higher order terms to zero when a cubic function is expanded in _a Taylor series in the formula ( ₁ ).
This arrangement allows for a fast transformation from Cartesian to Frenet coordinates to identify the coordinates of points near the helix.

(2) In the setting step,
setting the orthogonal coordinate system so that the x-axis coincides with the helix axis of the helix;
Calculating a first helical angle based on a coordinate value of the point relative to the x-axis of the Cartesian coordinate system;
Rotating the orthogonal coordinate system in a direction opposite to a progression direction of the spiral based on the first spiral angle;
translating the orthogonal coordinate system based on the value of the x-axis coordinate of the point;
The coordinate transformation method according to (1), wherein the orthogonal coordinate system is set by:
According to this configuration, the orthogonal coordinate system can be adjusted and reset to a position where coordinate transformation is easy, thereby making it possible to reduce the load of calculation processing.

(3) The coordinate transformation method according to (2), wherein the first helical angle is calculated based on a coordinate value of the point with respect to the x-axis of the Cartesian coordinate system, a start angle of the spiral, and a pitch of the spiral.
According to this configuration, it is possible to set a provisional helical angle when performing coordinate transformation in accordance with the configuration of the helix, making it possible to reduce the load of calculation processing.

(4) The coordinate transformation method according to (2) or ( ₃₎ , characterized in that in the transformation step, the value of θ before rotating the orthogonal coordinate system in the setting step is calculated by adding up the first helical angle and the θ 0.
According to this configuration, even if the orthogonal coordinate system is adjusted, the coordinate conversion can be performed appropriately.

(5) The coordinate transformation method according to any one of (1) to (4), wherein the Freinet coordinate system is a coordinate system defined with a point on the spiral as the origin, the direction of progression of the spiral as the ξ axis direction, the η axis direction of the spiral, and the ζ axis direction perpendicular to the ξ axis and the η axis.
This configuration enables coordinate transformation from a Cartesian coordinate system to a Frenet coordinate system that corresponds to spiral points.
(6) A coordinate transformation device for transforming a point near a spiral from coordinates (x, y, z) in a Cartesian coordinate system to coordinates (θ, η, ζ) in a Freinet coordinate system, comprising:
a setting means for setting the orthogonal coordinate system so that a coordinate value of the point with respect to an x-axis of the orthogonal coordinate system corresponding to a helical axis of the spiral is 0;
a conversion means for performing coordinate conversion based on the following formula (1) when the coordinates of the point in the Cartesian coordinate system set by the setting means are (0, y ₀ , z ₀ ) and the coordinates of the point in the Frenet coordinate system are (θ ₀ , η ₀ , ζ ₀ );

having
The coordinate transformation device is characterized in that the transformation means derives the coordinates (θ 0 , η 0 , ζ 0 ) of the point in the Frenet coordinate system by solving simultaneous equations obtained by using an approximation value in which third-order _or higher terms in a Taylor expansion of _{a cubic function in equation (1 )} are set to zero.
This arrangement allows for a fast transformation from Cartesian to Frenet coordinates to identify the coordinates of points near the helix.

(7) to a 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 coordinates (θ, η, ζ) in a Frenet coordinate system, comprising:
The coordinate transformation method includes:
a setting step of setting the orthogonal coordinate system so that a coordinate value of the point with respect to an x-axis of the orthogonal coordinate system corresponding to a helical axis of the spiral is 0;
a conversion step of performing coordinate conversion based on the following formula (1) when the coordinates of the point in the Cartesian coordinate system set in the setting step are (0, y ₀ , z ₀ ) and the coordinates of the point in the Frenet coordinate system are (θ ₀ , η ₀ , ζ ₀ )

having
the program comprising: a transforming step for deriving coordinates (θ 0 , η 0 , ζ 0 ) of the point in the Frenet coordinate system _by solving a simultaneous equation obtained by using an approximation obtained by setting the _third or higher order terms to zero when a cubic function is expanded in Taylor expansion of the formula ( ₁ ).
This arrangement allows for a fast transformation from Cartesian to Frenet coordinates to identify the coordinates of points near the helix.

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 device 20... Ball screw 21... Nut 22... Screw shaft 22a... Spiral groove 23... Rolling element

Claims (7)

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 setting step of setting the orthogonal coordinate system so that a coordinate value of the point with respect to an x-axis of the orthogonal coordinate system corresponding to a helical axis of the spiral is 0;
a conversion step of performing a coordinate conversion based on the following formula (1) when the coordinates of the point in the Cartesian coordinate system set in the setting step are (0, y ₀ , z ₀ ) and the parameters indicating the coordinates of the point in the Frenet coordinate system are (θ ₀ , η ₀ , ζ ₀ );

having
a transforming step of deriving parameters (θ 0 , η 0 , ζ 0 ) indicating the coordinates of the point in the Frenet coordinate system by solving a simultaneous equation obtained by using an approximation in which third-order _or higher terms in a Taylor expansion of _a cubic function in equation ( ₁ ) are set to zero. In the setting step,
setting the orthogonal coordinate system so that the x-axis coincides with the helix axis of the helix;
Calculating a first helical angle based on a coordinate value of the point relative to the x-axis of the Cartesian coordinate system;
Rotating the orthogonal coordinate system in a direction opposite to a progression direction of the spiral based on the first spiral angle;
translating the orthogonal coordinate system based on the value of the x-axis coordinate of the point;
2. The coordinate transformation method according to claim 1, wherein the orthogonal coordinate system is set by: The coordinate conversion method according to claim 2, characterized in that the first spiral angle is calculated based on the coordinate value of the point with respect to the x-axis of the Cartesian coordinate system, the start angle of the spiral, and the pitch of the spiral. 4. The coordinate transformation method according to claim 2, wherein in the transformation step, the value of θ before rotating the orthogonal coordinate system in the setting step is calculated by adding up the first helical angle and θ ₀ . 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 with a point on the spiral as the origin, the direction of the spiral as the ξ axis direction, the direction of the center of the spiral as the η axis, and the direction perpendicular to the ξ axis and the η axis as the ζ axis. 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 setting means for setting the orthogonal coordinate system so that a coordinate value of the point with respect to an x-axis of the orthogonal coordinate system corresponding to a helical axis of the spiral is 0;
a conversion means for performing coordinate conversion based on the following formula (1) when the coordinates of the point in the Cartesian coordinate system set by the setting means are (0, y ₀ , z ₀ ) and the parameters indicating the coordinates of the point in the Frenet coordinate system are (θ ₀ , η ₀ , ζ ₀ );

having
The conversion means derives parameters (θ 0 , η 0 , ζ 0 ) indicating the coordinates of the point in the Frenet coordinate system _by solving simultaneous equations obtained by using approximate values in which third-order or higher terms are set to zero when a cubic function _{is expanded into a Taylor series in equation (1 )} . 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 setting step of setting the orthogonal coordinate system so that a coordinate value of the point with respect to an x-axis of the orthogonal coordinate system corresponding to a helical axis of the spiral is 0;
a conversion step of performing a coordinate conversion based on the following formula (1) when the coordinates of the point in the Cartesian coordinate system set in the setting step are (0, y ₀ , z ₀ ) and the parameters indicating the coordinates of the point in the Frenet coordinate system are (θ ₀ , η ₀ , ζ ₀ )

having
the program, in the conversion step, deriving parameters (θ 0 , η 0 , ζ 0 ) indicating the coordinates of the point in the Frenet coordinate system _by solving simultaneous equations obtained by using approximate values in which third-order or higher terms are set to ₀ when a cubic function is expanded in Taylor expansion of formula ( ₁ ).

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