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# Coordinate Transformation

## Background

GPS determines the three-dimensional positions of surveyed points based on the

WGS84 datum. These coordinates are either presented as geocentric Cartesian coordinates (X,Y,Z) values or geodetic coordinates (latitude, longitude, ellipsoidal height).

There are circumstances where it would be desirable to have positions represented in a different reference frame or format, i.e. based on a different datum or projected onto a plane (grid coordinates).

The ZXW-Receivers provide the following on-board tools to transform WGS84 coordinates into various formats and reference frames:

1.

Datum-to-Datum Transformation

Using this feature, WGS84 coordinates can be transformed into coordinates based on another datum.

2.

Datum-to-Grid Conversion

With this tool, a grid system can be defined to convert geodetic coordinates into grid coordinates.

3.

Elevation Modeling

Using an on-board geoid model, ellipsoidal heights can be transformed into orthometric heights using this capability.

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Table 7.1 provides an overview of user coordinate transformation functions for your

Table 7.1. User Coordinate Transformation Functionalities

Transformation

Datum to Datum

Datum to Grid

Elevation Modeling

Description

3D (7-parameter) datum transformation between two Cartesian XYZ systems associated with the WGS84 datum and local datum defined by the user.

Data projected from a geodetic system, associated with WGS-84 or a user-defined datum and a specified grid system.

Map Projections Supported

• Mercator (EMER)

• Transverse Mercator (TM83)

• Oblique Mercator (OM83)

• Sterographic (Polar and Oblique) (STER)

• Lambert Conformal Conic (2 standard parallels) (LC83)

Special Map Projections Specific to NAD27

• Transverse Mercator 27 (TM27 and TMA7)

• Oblique Mercator 27 (OM83)

• Lambert Conformal Conic 27 (LC27)

Interpolation of geoidal undulations

The remainder of this chapter describes in more detail the coordinate transformation features of your receiver.

## Datum to Datum

The receiver normally computes and outputs positions in the WGS-84 coordinate reference frame. However, it is possible to output positions in NMEA messages in a number of different pre-defined datums, as well as in a user defined datum.

To set the receiver to output positions in a different datum, use the \$PASHS,DTM command. Once set to a different datum, then all position outputs in NMEA messages such as GGA and GLL and the position displayed on the LED screen are referenced

to the chosen datum. For a list of Datums, refer to Appendix A, Reference Datums &

Ellipsoids.

If the list of datums does not include a datum of interest to you, you can define a datum and load it on the receiver, using the \$PASHS,UDD command along with the

\$PASHS,DTM command. Prior to using these commands, define the required parameters including the length of the semi-major axis and amount of flattening in the reference ellipsoid and the translation, rotation, and scale between the user defined system and WGS-84.

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The rotation and scale parameters are only available in version ZC00 or later.

The generic formula used to translate and rotate from coordinate system 1 to coordinate system 2 is as follows: x y z

2

=

∆x

∆y

∆z

+

(

1 + m

×10

– 6

)

1

ε rz

ε rz

ε ry

1

ε rx

ε ry

ε rx

1 x y z

1 where e rx

= e x expressed in radians, similarly for e ry

and e rz

.

Example: Define local datum as the WGS-72 datum

Send the following commands to the receiver:

\$PASHS,UDD, 0,6378135.0, 298.26,0,0,4.5,0,0,-0.554,0.23

\$PASHS,DTM,UDD

This implements the transformations listed in Table 7.2 and below.

Table 7.2. Ellipsoid Parameters for WGS-72 and WGS-84

Datum

WGS-72

WGS-84

Reference

Ellipsoid

WGS-72

WGS-84

a[m] 1/f

6378135.0

6378137.0

298.26

298.257223563

x y z

WGS84

∆x=∆y=0

ε x

=

ε y

=0

∆z= 4.5 meters m= 0.23 x 10

-6

ε z

= –2.686

x 10

-6 radians = –0.”554 in the following equation:

=

0

0

4,5

+

(

1 + 0,23

×10

– 6

)

1

2,686

×10

– 6

0

– 2,686

×10 0

1

0

0

1 x y z

1WGS72

After issuing the \$PASHS,DTM,UDD command, the receiver internally transforms positions from the reference datum (WGS-84) to the user-defined datum. In standard text books, however, the datum transformations are given from local datums to WGS-

84. To simplify entering the transformation parameters, the translation, rotation, and scale parameters are defined from the local datum to WGS-84.

Coordinate Transformation 95

Figure 7.1 illustrates the change in the coordinate systems.

Figure 7.1. Rotation and Translation Between Coordinate Systems

After transforming the datum, the receiver computes geodetic coordinates in the defined system. All coordinates output by the receiver are in this new system.

Do not forget to issue the \$PASHS,DTM,UDD command after defining the transformation parameters with the \$PASHS,UDD command. Otherwise, the newly entered parameters are not used.

## Datum to Grid

Use this transformation to generate coordinates in an <x,y> rectangular system, based on the user’s location and mapping requirements or local standard. You can select any projection along with any base datum for output.

Convert geodetic coordinates into grid coordinates by defining a grid system utilizing one of the supported projection types (Figures 7.2 - 7.6).

CAUTION

Although almost any projection or combination of datums and projections is mathematically possible, some are inappropriate with respect to the project scope or geographic area.

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To set the receiver to supply grid coordinates:

1.

Select the projection type that best fits your needs.

2.

Define the grid system, using this projection type, with the \$PASHS,UDG command. This command defines the grid system to be used.

3.

Enable the grid system with the \$PASHS,GRD,UDG command. The receiver computes grid coordinates in the system defined.

4.

To access the grid coordinates, use either the \$PASHQ,GDC command to query for one output of the current coordinates, or use the

\$PASHS,NME,GDC command to set the receiver to continuously output the current coordinates.

There is one exception when configuring the receiver to compute and output grid coordinates. If you are interested in computing and outputting WGS-84 based UTM coordinates, there is no need to define the grid system in the receiver. The parameters for WGS84 UTM are pre-set in the receiver. To use them, set the receiver to output grid coordinates using either the \$PASHQ,UTM command to query for one output of the current coordinates, or the \$PASHS,NME,UTM command to set the receiver to continuously output the current coordinates.

Check the GDC message for the currently assigned datum.

Coordinate Transformation 97

## Projection Types

The following graphics represent the different types of projections available for the receiver.

Figure 7.2. Mercator

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Figure 7.3. Transverse Mercator

Coordinate Transformation

Figure 7.4. Oblique Mercator

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Figure 7.5. Stereographic

Figure 7.6. Lambert Conformal Conic

## Elevation Modeling

In addition to computing and outputting geodetic and cartesian coordinates in different systems, the receiver can compute and output elevations in different systems.

By default, the receiver computes and outputs ellipsoidal heights. In some messages, the geoid separation is included, computed from the internal global model, relative to

WGS-84.

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To set the receiver to compute and output orthometric heights, use the

\$PASHS,HGT,GEO command. After setting this command, the receiver outputs orthometric heights using the internal global geoid model. Be aware that the internal geoid model used in this calculation is very coarse. Orthometric heights derived from this model could be in error by a meter or more.

If separation is included in the message, it is calculated by adding the difference between WGS-84 and a user- or pre-defined datum to the WGS-84-based geoid separation. An exception to this is the

GGA message which ONLY outputs WGS-84 based geoid heights and separation, as per NMEA specifications.

Coordinate Transformation 101

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### Key Features

• Real-time Kinematic (RTK) Positioning
• Differential Correction
• Data Logging
• High-Precision Positioning
• Multiple Data Output Options
• User-Friendly Interface
• Flexible Configuration Options

### Related manuals

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The ZXW Sensor requires a 12-24 VDC power supply.
###### What is the accuracy of the RTK solution?
The accuracy of the RTK solution depends on various factors, including the quality of the base station, the signal strength, and the environment. In ideal conditions, sub-centimeter accuracy can be achieved.
###### How do I set up a differential base station?
To set up a differential base station, you need to configure the receiver in base station mode and connect it to a radio or modem for data transmission.
###### What is the difference between RTK and Fast RTK?
RTK (Real-Time Kinematic) provides centimeter-level accuracy, while Fast RTK offers faster positioning updates but with slightly lower accuracy.