Radio and optical alignment method based on GPS

Graphical abstract

the transmitter. At first, the z -axis is aligned in the direction of the weight force parallel to the position vectorR TX as shown in Fig. 2 and the x -axis is aligned with a collimation laser (L 1 ) in the direction of the reference station located nearby the transmitter. The direction of the tracking laser L 1 is the same as the maximum gain direction of the antenna or the direction of the transmission laser used for the radio and optical communication respectively.
The procedure of the coarse alignment is based on undergoing a rotational transformation around the y-axis, the local reference frame translates to x 0 , y 0 and z 0 where x 0 aims at the position of the reference station on the Earth surface. The plane where x 0 and y 0 lie is called p 0 and is not coplanar with plane p identified by the positionsR TX ,R RX and R Rf . Through a rotation f around the axis x 0 the plane p and p 0 becomes coplanar. The z 0 0 axis of the new reference frame x 00 , y 0 0 and z 00 is orthogonal with the plane p 00 , that is parallel to p as shown in Fig. 3. Another rotation of an angle u around z 00 ensures the alignment of the transmitter and the receiver as shown in Fig. 4.

Three points GPS alignment
To emphasis this method of coarse alignment, a fixed orthogonal reference frame X, Y and Z which rotates in accordance to the rotation of the earth is chosen to identify the position vectorR TX ,R RX and R Rf . The origin of the reference frame is located on the center of the Earth where the X -axis has the direction of the Greenwich meridian, the Z -axis crossed through the geographical north pole and the Y -axis points in the direction 90 east with respect to the X -axis. This is shown in Fig. 5. The Earth is not a perfect sphere and, to locate the position of the receiver, transmitter and the reference station, a standard WGS-84 was employed. This standard considers the Earth as an ellipse  with the semi-major axis of a = 6378137 m and the semi-minor axis of b = 6356752.3142 m [7]. The altitude from the GPS is given from the ellipsoid of reference. The eccentricity e is expressed as: And the curvature of the first vertical is: where l is the latitude. The Cartesian coordinates of the position vector of the reference station are given by Here, h Rf represents the altitude from the ellipsoid to the Earth's surface of the reference station. The latitude and longitude are indicated by l Rf and m Rf respectively. The position vector of the transmitter R TX have the coordinates: Fig. 4. The last step involves the alignment of the transmitter to the receiver: After the previous rotation, the laser will aim in the direction of the reference station. By the rotation around z 00 of an angle u, the transmitter is aligned with the receiver. and the coordinates of the receiver are: The subscripts RX, TX and Rf represent the position of the receiver, transmitter and the reference station respectively. The vectorR TX is the position vector of the transmitter with respect to the chosen reference frame.
The angle u; denoting the rotation of the transmitter in the direction of the receiver, is calculated by the inverse of the following inner product: The direction of rotation (clockwise or anticlockwise) is calculated by the component Z of the cross productṼ 3 ¼Ṽ TXRf ÂṼ TXRX ; whereṼ TXRf ¼R Rf ÀR TX andṼ TXRX ¼R RX ÀR TX , as shown in Fig. 5. If the transmitter is located on the northern hemisphere and the component ofṼ 3 along Z is positive the direction of rotation is clockwise. If the transmitter is located on the southern hemisphere and the component ofṼ 3 along the Z -axis is negative, the direction of rotation is anticlockwise. Initially the direction of the laser L 1 is orthogonal to the vectorR TX and subsequently is tilted in order to be directed to the reference station. The local reference frame, y and z change to x 0 , y 0 and z 0 . The axis x 0 has the same direction ofṼ TXRf as shown in Fig. 5. For the transmitter, to be directed towards the receiver, it needs to be rotated around the vectorṼ TXRf once the angle u is calculated. By the cross product, the vectorṼ 1 ¼R TX ÂR Rf is orthogonal to the plane p 000 (see Fig. 5). Similarly, by the cross product the vectorṼ 2 ¼Ṽ TXRf ÂṼ TXRX is orthogonal to the plane. The angle ' 0 between the vectorsṼ 1 andṼ 2 is calculated by the rules of the inner product. The plane p 000 is aligned with the plane given by x and z (see Fig. 2). The plane p 0 is coplanar with p when the vectorsṼ 1

Validation of the algorithm
The algorithm was tested by using the Google TM Maps software in order to measure the angle u and compare the results calculated by using SciLab [8]. In the absence of infrastructure such as an internet connection or the presence of a strong magnetic field deviation the proposed method will adequately be able to coarse align a transmitter and receiver. As a proof of principle, an experiment was performed for a short distance in a soccer field, which is discussed in the next section. For long distances using Google TM Maps the locations for the transmitter, receiver and the reference station were determined. The coordinates were converted in decimal degrees and by Eqs.
(3)-(5) such that the components of the vectors were calculated. The simulation was done twice on the northern hemisphere and twice on the southern hemisphere. For the initial test the graphical software of Google TM Maps was used to measure the angle u G between the direction of the transmitter-reference and transmitter-receiver stations. The angle between the vectors transmitterreference station and transmitter-receiver u M was calculated using the double precision. Table 1 shows the algorithm for four testing conditions, and the error e uM ¼ u G À u M j j between the measured and the calculated angle was performed.

Validation under real environmental conditions
In the absence of infrastructure, as mentioned previously, in remote areas the proposed method will be appropriate for the alignment of the transmitter and receiver. To validate this process a field test was implemented using a classical GPS, a laser lasing at 532 nm and a tripod with a protractor for civil constructions. The laser was fixed on top of the protractor and using a GPS the position of the transmitter TX was measured. A second GPS position was taken at the reference station R Rf and at the position of the receiver RX. This experiment shows the feasibility of a coarse alignment using the GPS. The experiment was performed on the soccer field of the University of KwaZulu-Natal, Westville Campus (Durban, South Africa), over a distance of 18.91 m between the transmitter and the reference station. The distance of the link between transmitter and receiver was 75 m for TEST 1 and 93 m for TEST 2. Once the coordinates were acquired, the angle u between the line of sight of the transmitter to reference station and the transmitter to receiver station was calculated by the proposed algorithm. The position of the reference station and the transmitter was fixed, while the receiver was relocated in order to test the algorithm for two different positions of the receiver. The alignment process was divided into four steps: Phase 1: Alignment of the transmitter R TX with the reference station R Rf Phase 2: Alignment of the laser with the reference station located on the goal post, 18.91 m away from the transmitter.
Phase 3: Propagation of the laser in the calculated direction u, where the receiver RX1 was located.
Phase 4: Repetition of phase 3 for an alternative position of the receiver position RX2.
The results of the two measurements are shown in the Table 2 and graphically in Fig. 6 which is a further illustration of the accuracy of the system and the algorithm. Table 1 Verification of the algorithm in single precision using the Google TM maps software for four runs. The angles l X , m X , u X and f X are given in Degrees. The field D represents the direction of rotation between the direction of the transmitterreference and transmitter-receiver stations. The time, t(ms) is the calculation time, necessary for the microcontroller to calculate the angle u and w.
TEST (1) TEST (2) TEST (3) TEST ( Table 2 Experimental verification of the algorithm in single precision at two positions of the receiver station along the soccer field.