WO1993017392A2 - Process for making tracking devices and traffic lanes for vehicles and process for the construction of objects - Google Patents

Abstract

The objects of the invention are a process for making tracking devices and traffic lanes for vehicles and a process for the construction of objects using a CAD system, and a CNC control process. According to the invention, a tracking approximation is made using a process working with directional differences. This leads to more accurate tracking.

Description

 METHOD FOR THE CONSTRUCTION OF TRACKING DEVICES AND

TRAFFIC BELTS FOR VEHICLES AND METHOD FOR

CONSTRUCTION OF OBJECTS

The invention relates to a method for the construction of lane guidance devices and traffic bands for vehicles, further to a method for the construction of objects using a CAD system and for CNC control.

To plan and stake out the axis or route of traffic bands such as roads, railroad tracks, etc., the individual plans are carried out on the basis of graphic preliminary designs. The compulsory points through which the route is to be taken must be taken into account.

It is known to use straight lines (with zero curvature) and circular sections for the initial concepts for such routing work.

EBSAT2BLATT te (with constant curvature) as well as additional independent elements or as transition arches clotoids (with linearly changing curvature). Such transitional arches are intended to prevent or at least to minimize any jerky changes in the centrifugal force between the individual route sections or elements. If the curvature transitions between the individual route elements are not sufficiently smooth, such jerky jolts and unstable driving operation and an impairment of the driving experience as well as disadvantageous stresses, for example, of the rails, which occur at the relevant points as a result of the changing inertia forces are subject to increased wear.

To avoid such bumps, flat circular arches are often used in the route. Nevertheless, the transitions are often not sufficiently smooth and the driving speed must be reduced as a result of excessive lateral acceleration.

Routing designs are currently being carried out with interactive computer support, the initial concepts usually being created as described above. The optimization of the designs in accordance with the geometric and technical requirements is essentially empirical, in some cases. even through manual measures on site. Therefore, the experience of the respective processor is crucial in the progress of the routing work.

The computational elaboration of route layouts currently consists in the fact that polynomial approximations with the compulsory points ^{• are used} as interfaces (cf., for example, M. Ingwersen and T. Rickert, "New Routing Technology for Local Railways", ZfV 2, 1990, pages 60 to 68; H. Schmidt, "Transition arc calculation by numerical integration with a quadrature formula by Gauss", VR 53/8, 1991, Pages 361 to 372). For example, Ingwersen and Rickert use a tangent polygon. For the road axis calculation from clotoids, for example Fresnel integrals for the x, y coordinates of the clotoid single points with respect to the long tangent are developed. These are asymptotic calculations. The Fresnel integrals cannot be solved in closed form and are therefore numerically calculated as series developments with the sequence broken off by the third or fourth term. Accordingly, in order to determine the route that is most favorable in terms of driving dynamics, a considerable amount of work and computing effort must be carried out with a large number of compulsory points. It is conventionally assumed that the approximation errors add up in the case of a combination of a plurality of calculated route elements.

H. Heckmann proposed in "An algorithm for route calculation on EDP systems", AVN 1/90 pages 19 to 26, 1990 a route algorithm in which station points on straight lines as corner points of an equilateral polygon with constant Direction. Station points on circles are counted as corner points of an equilateral polygon with a constant change of direction. Station points on transition arcs are counted as corner points of an equilateral polygon with a constant second change in direction. This results in an algorithm which, controlled by lawfully changing directional differences, continuously generates the coordinates of station points which are equidistant from one another on the route axis. This is done for straight lines, transition arcs and circles according to the same working principle, ie the coordinates are generated according to a uniform law of education. The equally spaced axis points can therefore be placed on mutually tangent sequences of straight lines, transition arcs and circles. This algorithm can be used to move along the route axis and directly, at equal distances, the Land- or Gauß-Krüger coordinates or other right-angled Generate coordinates of the axis points. The point spacing AN, the constant step size of the directional difference algorithm, can be 20m, 10m, 5m, 2m or less.

As stated in the above-mentioned article, the maximum deviations of the route determined in this way lie from an ideal route for increments of e.g. AN = 5 m at a few cm in the area intended for the practical use of the transition arch. In any case, by reducing the step size, an arbitrary approximation to works of art can be achieved which contain the results of Taylor series calculations with seven-digit coordinate values (Kasper, Schürba, Lorenz, "The Clotoids as a Trassie¬ rungselement", Ferd. Dümmlers Verlag, Bonn, 1968).

The invention has for its object to provide a method for precise route determination, which enables easy production of high-speed routes with a minimal maximum curvature. Such a route results in minimal lateral acceleration and the lowest possible wear.

This object is achieved in a method having the features of claim 1. Advantageous variants and applications of the method according to the invention are the subject of the dependent and further claims.

In the method according to the invention for the construction of lane guidance devices and traffic bands for vehicles, the course of the route is thus spatially determined by starting and ending points. The course of the route is determined by means of a processing device using a directional difference method in such a way that the route curvature and curvature change do not exceed predetermined values for stable driving behavior of the vehicles. According to the invention, the course of the route in the terrain is determined by Intermediate points successively determined. The route intermediate points are connected by route sections. Certain distances between the route points and the first, second and third directional difference are determined in this method in such a way that they contain the predetermined compulsory points.

In the routing method according to the invention, routes are thus laid through the constraint points. Each has a point where its curvature is greatest. This maximum curvature is the characteristic value for the quality of the route, i.e. the smaller the maural curvature, the better the route quality. The route with the smallest maximum curvature is therefore desirable. An equidistant sequence of points, which this route can represent, is achieved by the microsine polygon, which passes through all the constraint points and whose maximum change in direction is the smallest.

The geometry of the route axis through the corner points of the equilateral tendon polygon is determined by the side length AN = Δ. --ι ^d i ^e direction difference DRI = τ, and the change of the direction difference D2RI = - T described.

The alignment method according to the invention is characterized by an extremely high level of accuracy. It is e.g. possible to trace a circular path with a circumference of 400 km with an accuracy of approx. 5 μm at the start / end point of the trace.

A comparison of the method with tables of Fresnel's integrals showed that the comparison calculation over equidistant points with a constant second difference in direction according to the invention in all cases brought up to the greatest arc lengths and to the last digit the values which are in the published tables of the Fresnel integrals. The simplicity of the route determination according to the invention is based on the fact that the route element points are determined by the hypotensuses of right-angled triangles (cf. FIG. 3), the points being arranged at constant intervals according to a preferred embodiment. The breakpoints of the specific tendon polygon lie on the trajectory. The selected small side lengths of the micro-tendon polygon enable a smooth curve connection. Since the constraint points are also the mentioned break points, the centrifugal forces acting on the vehicle run smoothly. That is, the centrifugal forces that are effective are minimal. In the case of a railway line, this leads to minimal track stress.

The method according to the invention makes use of an interpolation between the constraint points, the curvature and change in curvature between the constraint points being controlled by law. In the case of conventionally used polynomial approximations, on the other hand, there may be function-related deviations (waviness, breaking out) between the support points and the curvature can become very large in some places. This would in turn lead to undesired bumps and stresses in the driving dynamics.

According to a variant of the method according to the invention, the direction and curvature of the route or surface are determined in a constraint point by the circle which passes through the constraint point and its two constraint point neighbors. The circle through the constraint point and its two constraint point neighbors is the line of smallest, maximum curvature that passes through these three constraint points, ie the line with the smallest, maximum steering wheel stop and thus the greatest driving stability and minimal wear. The function of the method according to the invention is explained further below using the digitization of the curvature of the determined route by means of directional differences.

On a finished, continuously curved, smooth route axis, the measuring tape can be used to mark points that always have the same spatial distance AN of, for example, 5 m. These spatial equidistance points could easily be used for the usual point-by-point setting out of the route axis. They are connected to one another by an equilateral chord polygon with the side length AN. Their mutual position in the plane is determined by the directional angle of the chords of the same length and the change in DRI of this directional angle from equidistance point to equidistance point. This direction remains unchanged on straight line sections (cf. FIG. 3). On circular route sections, the direction changes after each piece of tendon AN by the amount DRI (cf. FIG. 4). AN and DRI use the tendon formula AN = 2 * R * sin (DRI / 2) to determine the curvature and the radius R of the arc at this point. The DRI, which is the interim result, shows the curvature of the route when applied to the station on a suitable scale. Like the radius R, DRI can be positive and negative. On transition arches of the simplest kind, the change in direction increases by D2RI after each chord length, whereby according to [H. Heckmann, loc. Cit.] D2RI = (AN) ² / A ^{2 determines} the parameter A of the transition curve (cf. FIG. 5). The second change in direction D2RI can be positive and negative.

With these directional angles and their differences, the route in the straight and circular area can be determined mathematically precisely and with surprisingly high calculation accuracy. In the transition arch area, the driving dynamics-sanctioned clotoid is achieved with at least the accuracy that practice wants, as more than seven-digit plates with which could be compared, do not exist. The equidistant point sequence generated with the two-stage recursion is therefore at least as accurate as the clotoid used in practice. The consideration directed at the application can stop at this point.

If you have 3rd, 4th, ... n. If differences in direction are introduced, that is to say the number of stages of the recursion is increased accordingly, transition arches of a higher order are reached, in which the bogies of the railway carriages begin to rotate smoothly. As a result, the tracks wear less. In the FORTRAN program, this is an additional line in the curvature cascade.

A further check of the routing method according to the invention was carried out using the Cornu spiral.

« 0,5642. Sie gilt als besonders elegante Darstellung der "sehr ausgiebig un¬ tersuchten" Fresnelschen Integrale [Born, M. : Principles of Optics, Pergamon Press Oxford, U.K., 1989; Hecht, E.: Op- tics, Addison-Wesley, Reading, MA 1990]In the Cornu spiral, which is important in Fresnel's diffraction theory, the curvature is proportional to the arc length measured from the winding point. The Cornu spiral is a toilet with a special parameter

«0.5642. It is considered a particularly elegant representation of the "very extensively examined" Fresnel integrals [Born, M.: Principles of Optics, Pergamon Press Oxford, UK, 1989; Hecht, E .: Optics, Addison-Wesley, Reading, MA 1990]

L y (L) = sin (2 i *) l

ΓL x (L) = J cos (- l ² ) dl

0 2

Integration variable 1, upper integration limit L. For this purpose, a unitary clotoid and Cornu spiral with differences in direction were drawn. The FORTRAN source program required for this is 14 lines long. It generates the control commands required for the plotter in Hewlett-Packard Graphics Language (HP-GL):

OPEN (2, FILE = 'C0M1:') IY = 0 IX = 0

AN = 100.0 RIANBO = 0.0 DRBO = 0.0

WRITE (2, *) 'IN;SP1;PA0.0;'100 IY = IY + AN * SIN (RIANBO) IX = IX + AN * COS (RIANBO) WRITE (2, *)' PD ¹ , IY, ^• , ', IX, ^• ; ^» DRBO = DRBO + 0.00015625 RIANBO = RIANBO + DRBO GO TO 100 END

During the production run, this program sends the following commands in HP-GL via the serial interface COM1 to the HP 7586 DIN AO plotter:

IN; SP1; PA0.0;

As illustrated in FIG. 6, a spiral of aesthetic evenness is created, which reveals the performance of the directional difference method.

FIG. 7 illustrates the flexibility of applications with predefined directional differences of higher order using the example of a hyperclotoid for the third directional difference. The method according to the invention is particularly suitable for the production of high-speed lines (high-speed and trams), including rails, switches and other track-guiding means of rail-bound means of transport, the axis and surface coordinates of which are calculated by means of the method of program-controlled coupled directional differences. This is especially true when third and higher constant, but also function-dependent directional differences are used. Further applications are the production of road axes, the boundary boundaries of roads and other track-guided traffic bands, insofar as the directional difference method is used in the design and manufacture. There are special advantages in narrow buildings and many constraints.

There are still further possibilities of the method according to the invention if the fourth and / or higher directional difference are additionally specified, the alternative of a functionally determined directional change difference also being obtained. The specification of a constant fourth difference already leads to a multitude of new application possibilities.

Further possible variations of the method according to the invention include that the nth difference in the direction is constant. In this way, straight lines, circles, clotoids, hyperclotoids, etc. can be formed as route sections. On the other hand, the directional differences can also be varied depending on the function. The distances between the points can also be varied to determine the route. The specification of values or parameters can be changed from section to section.

The invention also creates a method for constructing objects using a CAD system using the directional difference method been. In this, the surface and / or shape of an object to be produced is spatially defined using a directional difference method, with certain distances between the intermediate surface points and the first, second to nth directional difference being defined.

This results in surprising shaping possibilities due to the high accuracy of the approximation. Any curved line and surface can be realized with the highest accuracy. Non-uniform curvatures, such as result from interpolation with cubic splines (cf. FIG. 1 from G. Pomaska, "Computer Graphics ...", page 108, Vogel-Verlag, Wuerzburg, 1984) are eliminated.

The construction method according to the invention is advantageously used in the production of flat and spatial industrial products such as body parts e.g. of motor vehicles and airplanes, the surfaces and boundaries of which consist of curved flat and spatial curves and surfaces, the lines and surfaces of which were calculated using the directional difference method during design or manufacture. The method proves to be particularly expedient if the directional difference method is used to control automatic manufacturing machines.

The directional difference method during the design and the production step by step calculates x, y coordinates or polar coordinates of the lines to be produced in almost any density and almost any small point spacing and used to control the production process. The inclusion of the third coordinate z enables the expansion to the description and production of spatial structures. A very advantageous application of the invention consists in the use of the method according to the invention for CNC controls. As a result of the high accuracy that is made possible, new objects with a more complicated shape can be manufactured or the production costs can be reduced due to the lower computing effort and machine wear.

The method according to the invention can be used very advantageously for the computer-controlled production of prostheses.

The invention is further illustrated in the accompanying drawing and the description below with an appended table. The drawing shows:

1 is an illustration of an Archimedean spiral appoximized by spline interpolation,

2 shows a schematic illustration of the definition of route sections for a route according to the invention,

3 shows an example for the determination of a route element which represents a straight line by means of the direction difference method,

4 shows an example for the determination of a route on the basis of a circular arc by means of the direction difference method,

5 shows an example of the combination of a straight line, a transition arc and a circle by means of the direction difference method,

6 shows an example of a Cornu spiral developed by means of the directional difference method,

Fig. 7 shows a ^'realization of a Hyperklotoide,

8 is an illustration of a further variant of the alignment according to the invention and

9 shows the circles through the compulsory triplet Z _j __ ₁ , Z ^ _r ^z i + l- 2, 8 and 9 are explained below.

2 illustrates the determination of the route according to the invention. The location, direction and curvature in the starting point A and end point E as well as the so-called compulsory points Z ₂ , Z ₃ ,. , , , through which the route is to be led, whereby the starting and ending points are at the same time the compulsory points Z- ^ and Z _n . The dash-dotted polygon through the constraint points already shows the changes in direction that the route has to go through. With the aid of the difference in direction method, route elements are then determined whose maximum curvature is as small as possible in order to reduce the undesired changes in lateral acceleration and thus wear etc. to a minimum. The route element sought between two compulsory points is shown with a solid line. The route element is again divided into sections for the route construction, each with two intermediate route points A ₁ , A ₂ ; Connect A ₂ , A ₃ , ... and form a second, finer micros tendon polygon. The side length of this micro-tendon polygon is, for example, I and allows the route to be set out directly in the terrain.

Fig. 8 illustrates the routing according to the invention in the event that Z ^ _ ₁ , Z ^, Z ^ ₊₁ ; Z ^, ^ ₊₁ , Z ^ ₊₂ each have an arc with center M _lf M ₂ and radius r ^, ^ ₊₁ . The tangent T-T ₂ in the central constraint Z _j _, Z ^ ₊₁ then supplies the direction of the route in this constraint point. The curve element defined by the two constraint points Z, Z ₂ with the same slope as the tangents - _j _, T ₂ and with a minimal change in curvature is the route element sought. As the corresponding solid line in FIG. 8 shows, in this case the transition arcs between the two circular sections are a clotoid. The route element is finally in the micro-polygon already mentioned divided with the chord length required for the construction. This is not shown in Fig. 8.

In Fig. 9, the circles through the compulsory triplet Z ^ _ ^ -, Z _j _, _j _ _{+1 are} shown, which cut out linear areas in which the best route lies, and for which Z ^ indicate direction and curvature in the central compulsory point .

3 to 7, no further explanation and description follows below. Rather, in order to avoid repetition, reference is made to the explanations in the introduction to the description.

The manufacturing method according to the invention is further illustrated with the aid of the source program for a route algorithm reproduced below, which has as its object the determination of a constant distance sequence with a constant nth directional difference.

As a FORTRAN 77 source program, the route algorithm has the following form:

DOUBLE PRECISION RHO, STAT, Y, X, AN DOUBLE PRECISION RIGO, DRIGO, D2RIGO, RIBO RHO = (200.D0) / (4.D0 * DATAN (l.D0)) READ (*, 10) STAT, Y _Λ X, AN, RIGO, DRIGO, D2RIGO

10 FORMAT (7F)

WRITE (*, 11) STAT, Y, X, AN, RIG0, DRIGO, D2RIGO

11 FORMAT (1X, 7F)

100 RIBO = RIGO / RHO

Y = Y + AN * DSIN (RIBO)

X = X + AN * DCOS (RIBO) C

STAT = STAT + AN C D3RIGO = ... C D2RIGO = ...

DRIGO = DRIGO + D2RIGO

RIGO = RIGO + DRIGO C

WRITE (*, 20) STAT, Y, X, RIGO, DRIGO, D2RIGO 20 F0RMAT (1X, 6F)

OFF = 0.0 READ (*, 12) FROM 12 FORMAT (F16.8) C <ENTER KEY> or 9.99 <ENTER>

IF (AUS) 100,100,1 C Change of D2RIGO and DRIGO 1 READ (*, *) D2RIG0 GO TO 100 END

Save, translate, link. Call of the production run. Enter the initial values in the selected FORMAT.

<ENTER> or 9.99 <ENTER> new D2RIGO and <ENTER> DRIGO are changed with D2RIGO in two or more steps. In a circle you make D2RIGO zero. Termination by switching off, warm start or <Ctrl + C>.

This algorithm generates an equidistant sequence of points with a constant nth difference in the direction according to the n-stage recursion listed below. The directions RIGO in Gon and RIBO in radians are denoted by T (r = RIBO)

^y i + _l ^{= y} i + AN * sin τ _i ^τ i = ^r il ⁺ A 'i, i-ι

AX, il = Δ 'il, i-2 ⁺ Δ ² etc. x ^ ₊₁ = X _j _ + AN * cos T ^

with the initial values y _Q , x _Q , AN, τ _Q , τ _Q ,. ^{2 r} *

Z _m ^{2 τ =} 0 means n = 2 the (simplest) transition arches. Δ ² T - 0 and Δ. ^{r =} ° means n = l constant change in direction, ie points on the circle. T = 0 means τ _n = r. and thus unchangeable direction, i.e. straight line. In the following, considerations regarding the numerical stability of the method according to the invention are made using circular routes.

After it had been shown that the route algorithm via directional differences provides exactly the same numerical values as the clotoid over series developments, the numerical stability of the interaction of so many different steps became interesting. Since circles return in themselves, the numerical stability of the route can be checked particularly well over long distances.

The equilateral polygon in the circular route is a polygon with many very short sides. However, this polyline is free of centering errors and therefore behaves differently than a polygon in the field.

An overview of the test calculations shows: If all variables and constants work with DOUBLE PRECISION, the algorithm in particular π and with 16 significant digits are made available, the circles calculated with directional differences still return after over 100 km and over 100,000 Difference steps back with precision mechanical precision. This repeats an experience that was surprising when recalculating the tables of the Fresnel integrals: The algorithm has great numerical stability.

The numerical stability is the result of the interaction of the computing algorithm and the computing system. An IBM Personal System / 2 model 80X21 with 80387 math co-processor and IBM FORTRAN / 2 compiler was used. This compiler processes all calculations via the 80387 co-processor in the "temporary real" format, regardless of the data type of the argument or the result [IBM 1987 D-13.14]. The 8 registers of the 80387 for "temporary real" have 80 bits, 64 of are mantissa, 1 is sign, 15 are exponent. 64 bit means 19 decimal places [Sargent, M., Shoemaker, R.: The IBM Personal Computer from the inside out, page 162, Addison-Wesley 1988]. With this accuracy, two values A and B are formed for the argument X over the partial tangent FPTAN (X), so that TAN (X) = A / B. C = SQRT (A * A + B * B), SIN (X) = A / C and COS (X) = B / C occurs in the registers of the 80387 with 19 significant decimal places. For DOUBLE PRECISION, these values are rounded to 16 decimal places (8 bytes hex).

Each sine and cosine value thus has an error 5 with -0.5 * 10 ^"16 <5 <+ 0.5 * 10 ~ ¹⁶ = e. With the algorithm from A to the neighboring point N with DY = AN * sinr and DX = AN * cosτ, each of the two coordinate differences has a maximum error of AN * e and the relative point position of A and N has an error of AN * e * - / 2.

was für AN im ungünstigsten Fall r*e* 2*2 ergibt. Die Relation für die Nachbargenauigkeit beider Berechnungswege verhält sich wie (AN*e* 2 ) / (r*e*2 *j2 ) = AN/2r. Bei r=60km und AN=lm ist das ein Verhältnis von 1/120 000. Der mit dem Richtungsalgorithmus über seinen Rand gerechnete Kreis hat die besseren Nachbarschaftseigenschaf- ten. Er ist runder. However, if one calculates A and N from the center of the circle, with y = r * sinτ and x = r * cosτ, the gap for A and N is maximal

which results in r * e * 2 * 2 for AN in the worst case. The relation for the neighboring accuracy of both calculation methods behaves like (AN * e * 2) / (r * e * 2 * j2) = AN / 2r. At r = 60km and AN = lm this is a ratio of 1/120,000. The circle calculated with its directional algorithm over its edge has the better neighborhood properties. It is rounder.

Claims

Expectations 1. Procedure for the construction of lane guidance devices and traffic bands for vehicles running through location, direction and curvature in the start and end point (A, E) as well as by predetermined constraint points (Z- _j , Z ₂ , ...), is so determined at the mit¬ means of a processing device of the route using a direction difference method is that the path curvature and change in curvature not exceed vorge added values for stable driving of the vehicles, of the route in the terrain by Trassenzwischenpuήkte (a, a ^, a ₂ , ...) is successively determined and the route intermediate points by route sections (A, A- _j _; A _lr A ₂ ; ...) are connected, whereby certain distances between the intermediate path points and the first, second and third directional difference are determined in such a way that they meet the specified compulsory points (Z- ^, Z ₂ , ...) contain. 2. Method for the construction of objects using a CAD system, characterized in that the surface and / or shape of an object to be manufactured is spatially defined using a directional difference method by means of intermediate surface points, with certain distances between the Surface intermediate points and the first, second and third directional difference are determined. 3. The method according to claim 1 or 2, characterized ge ¬ indicates that the route sections (A, A- _j _ A _lf A _{2 ;.} ••) or sections between the surface intermediate points are straight pieces, the sections to each other result in a polyline. 4. The method according to any one of claims 1 to 3, characterized in that the route or upper Intermediate surface points for the determination of the route or surface course are arranged at equidistant intervals. 5. The method according to any one of claims 1 to 4, characterized in that the fourth and / or higher directional difference are additionally specified. 6. The method according to any one of claims 1 to 5, characterized in that the nth difference in the direction is constant. 7. The method according to any one of claims 1 to 6, characterized in that the distances between the route or surface intermediate points are varied. 8. The method according to any one of claims 1, 3 to 7, characterized g e k e n n z e i c h n e t that the intermediate path points are used directly for point-by-point stakeout of the route axis. 9. The method according to any one of claims 1 to 8, characterized in that the directional differences are varied, in particular depending on the function. 10. The method according to any one of claims 1 to 9, characterized in that the direction of the route or the surface in a constraint point (Z ^) is given by the direction of the tangent of a circle through the constraint point (Z ^) and its both neighboring Zwangs¬ points ( _j ^, Z ^ ₊₁ ) runs. 11. The method according to any one of claims 1 to 10, characterized in that the route sections (A, A- ^; A. _j _, A ₂ ; ...) as interconnected micro-tendon polygons through the constraint points Z ^ - , Z ₂ , ..., Z _n ) whose maximum change of direction is a minimum. 12. Use of the method according to any one of claims 1, 3 to 11 for the construction of rail routes from the Hochgeschwindig¬ keitstrasse to the tram route. 13. Application of the method according to one of claims 2 to 10 to CNC controls. 14. Use of the method according to one of claims 2 to 10 for the production of body parts, workpieces, engine parts and industrial design. 15. Use of the method according to one of claims 2 to 10 for the manufacture of prostheses.