The national standard for crane rails from the Eurocode 3 standards [1] requires the monitoring of:

The span between rails;

The actual span can deviate from the projected one by a maximum of 10 mm. The maximum elevation difference must not exceed the value determined by equation. Δh_max= s/600.

The crane rail is measured using a classical polar surveying method for detail points. Characteristic points are determined indirectly by measuring the position of the “L” platform. According to the required precision and the principles of the method, the proposed approach may be used if two conditions are met:

A total station providing adequate basic measurement precision of at least 1 mm must be used.

The homogeneity of the measurement precision is maintained by measuring all points from a single instrument station. The instrument must be set in a stable position which enables the visibility of all desired detail points of the rail.

According to the required measurement precision derived from standard [1] we define the requirements that the instrument should meet. Given the dimensions of the crane rail, the required accuracy of the point determination can be computed.

Cartesian coordinates of a single detail point are calculated from the measured polar coordinates (s—horizontal direction, z—zenith angle. d—slope distance): (1) [ x y z ] = [ sin s ⋅ sin z ⋅ d cos s ⋅ sin z ⋅ d cos z ⋅ d ] .

Using error propagation law, the precision of coordinate determination can be calculated according to the measurements precisions: (2) ∑ xyz = J ⋅ ∑ szd ⋅ J T .

Matrix J represents Jacobian and contains derivatives of Equations (1) with respect to each measurement. If J is an invertible matrix, the procedure can be inverted. Therefore, the required measurement precisions can be determined according to the desired coordinates precisions: (3) ∑ szd = J − 1 ⋅ ∑ xyz ⋅ ( J T ) − 1 .

We are mainly interested in differences between coordinates. The rail span is the difference between the x coordinates and the elevation difference equals to the z coordinate differences (see Section 3.2) of two points, point 1 and point 2. Coordinate differences between two points of each profile can be written as: (4) [ Δ x Δ y Δ z ] = [ x 2 y 2 z 2 ] − [ x 1 y 1 z 1 ] = [ sin s 2 ⋅ sin z 2 ⋅ d 2 − sin s 1 ⋅ sin z 1 ⋅ d 1 cos s 2 ⋅ sin z 2 ⋅ d 2 − cos s 1 ⋅ sin z 1 ⋅ d 1 cos z 2 ⋅ d 2 − cos z 1 ⋅ d 1 ] .

When we try to use the inversed error propagation law in Equations (4), we face a problem. The Jacobian is neither square nor an invertible matrix. System (4) is therefore expanded with three additional equations for the averages of all three coordinates. Equations complement the system in a way that all the equations are independent and J becomes invertible again.

According to the desired coordinate difference precisions, we calculate the required precisions of measurements. They are represented in Table 1.

Simulation results (Table 1) clearly show that in order to achieve millimeter precision of coordinate difference we need the instrument to assure an angular precision of 1″ and distance precision below one millimeter. The dimensions of the treated crane rail are relatively small. In practice crane rails of much larger dimensions are encountered. The required measurement precisions are adequately higher. The proposed approach can therefore be performed only if we can use a total station of the highest precision possible.

A special platform (Figure 1) is used for the signalization of detail points. It consists of a holder and two precise prisms force-centered on it. The holder is made of an L-shaped 2 cm thick iron profile. It is 6.5 cm wide and its arms are 7 and 14 cm long. Adapters for precise prisms are attached at both outer surfaces. At the edge a circular level is fixed to the holder to ensure the horizontality of the platform.

The position of the characteristical point, representing the upper inner edge of the rail, can be derived from the measured positions of both precise prisms. Accurate dimensions of the platform need to be known, i.e., offset of both prisms from the edge of the rail.

We used the Leica Geosystems TS30 total station, with the technical characteristics given in Table 2. Two precise Leica Geosystems GPH1P reflectors were fixed on the “L” platform.

The instrument was stabilized with a tripod on a stationary crane. The platform was set on the rail every 1.4 m and put in a horizontal position using the circular level. The prisms were pointed towards the instrument. Figure 2 shows the stabilization of the instrument and the measurement point signalization on the rail.

The positions of both prisms were measured twice using the ATR function, which provides homogenous sighting precision for each measuring point on the rail. The platform was placed on each rail 38 times. For each stabilization of the platform each prism was measured twice. As a result we got 304 measured points with the polar coordinates [s, z, d].

The results of the polar surveying method (Figure 3) are the horizontal directions s_ijkl, zenith angles z_ijkl and slope distances d_ijkl, where i denotes the left or the right rail, j a consecutive point on the rail or profile, k the upper or side prism on the platform, and l the consecutive measurement in each setting of the platform.

For the presentation of the positions of characteristical points on each rail we chose a rectangular Cartesian coordinate system with the following properties:

the coordinate system unit is the meter,

The computation of point coordinates in the chosen coordinate system can be achieved using the following steps:

The transformation of points from polar to Cartesian coordinates in the local coordinate system of the instrument can be performed using Equations (1). Since each prism was measured twice, we can immediately check the deviations of repeated measurements and compute the arithmetic mean.

Azimuth angle for each set is computed from slope coefficient a of the line using the following Equation: (13) ν = arctan 1 a ± 180 ° .

The value of orientational direction o, used for the rotation of the coordinate system, was computed as the average of all four azimuth angles.

The coordinates of the measured points in the chosen local coordinate system were computed in the same way as in the first step. The only difference was that orientational directions were subtracted from all horizontal directions and that the coordinate system origin was translated 10 meters to the left (the x coordinates increased by 10 m) and 5 meters down (the z coordinates increased by 5 m): (14) [ x y z ] = [ sin ( s − o ) ⋅ sin z ⋅ d + 10 cos ( s − o ) ⋅ sin z ⋅ d cos z ⋅ d + 5 ] .

Twofold measurements of each prism are averaged. Measurement control has already been performed during the computation of the coordinates in the local coordinate system. The results are Cartesian coordinates of all the measured prisms in the desired coordinate system.

According to Equation (12) the coordinates of the centres of the two prisms are computed for each position of the platform. Each prism is always measured twice, so we have two positions of each point t₁=[x_t1, y_t1, z_t1] and t₁=[x_t2, y_t2, z_t2]. The positional difference between them is computed as: (15) δ t = t 2 − t 1 = [ δ x , δ y , δ z ] .

According to differences δt for all measured points, we compute parameters, which describe the measurement precision—the average value of differences, δ̄ standard deviation σ_δt and the maximum absolute difference max ‖δ‖The listed values are represented in Table 3.

Given the coordinates of the prism centres and the constants of the platform, the coordinates of the characteristic points for each position of the platform are computed using Equations (5) and (6). The position of the characteristic point for every single measurement is determined in two ways. According to the differences between different solutions the precision of the x and z coordinates can be analyzed. The precision parameters—the average value of differences δ̄, standard deviation Ơ_δ and the maximum absolute difference max |δ| are represented in Table 4.

We assume that the differences between the computed values of the coordinates of characteristical point from both prisms on the platform are small enough. The definite values of coordinates are computed from Equation (11), which eliminates the influence of platform non-horizontality.

The precision estimation of the characteristical points follows from the precision of the measured prism centres (Table 4) and from the precision of parameters dx₁, dx₂, dz₁, and dz₂. Based on the error propagation law (propagation of variances and covariances) using Equation (11) the precision of the coordinates of characteristical point can be estimated. The non-horizontality error is ignored: (16) [ σ x K T σ y K T σ z K T ] = [ σ x 2 + σ d x 2 2 1 2 σ y 2 σ z 2 + σ d z 1 2 ] = [ 0.55 mm 0.31 mm 0.51 mm ] .

It can be noted that the measurement precision is very high. The standard deviations of all three coordinates are far below 1 mm. These values are comparable with the precision of orthogonal measuring method and the method of geometrical levelling.

All characteristic points of rails were divided into two groups, each representing its own rail. For each rail we have 38 points. Although the position of points (y coordinates) of the left and the right rail do not coincide exactly, we assume that the i-th points on the left and on the right rail together represent their own profile.