Given the gps position: longitude, latitude and altitude of a target, our objective was to always point to that target regardless of our position and orientation.

Luc Jaulin (my thesis advisor) and I developed this project as a scientific collaboration between the ENSIETA and DETI SA. The applicability of such system includes missile launching and guidance, video tracking, ultra-high speed wireless communication links, among others.

The following video is a demonstration of the prototype presented by my advisor Luc Jaulin (since I don't like speaking to cameras and my advisor likes showing off )

The problem to be solved is known by the name of Inverse Problem. What we need is to determine the Euler angles pitch and yaw given a vector u that points to our target (roll wasn’t used since our mechanical arm only had two freedom degrees, i.e. vertical and horizontal movements). This vector is represented in a global coordinate system with X pointing to the magnetic north and Z pointing to the sky.

This vector is found using the target GPS coordinates as well as ours. This vector could also be calculated by other means, like for example, some radar data from the target plus our GPS coordinates and orientation.

The inertial measurement unit (IMU) provides orientation data with respect to a global coordinate system (not necessarily the same with X pointing to the north). This orientation data allows, by constructing a rotation matrix R12, the representation of any vector in the IMU’s local coordinate system R2 on its global coordinate system R1. Then these coordinates can be represented in R0 (if needed, since the IMU's global coordinate system could match R0) thanks to the rotation matrix R01.

The pan and tilt also has its own local coordinate system, which can freely rotate along the Y and Z axis. One last rotation matrix R23 will then allows passing from the pan and tilt coordinate system R3 to the IMU’s local coordinate system R2.

Our problem now seems pretty clear. Given u, we need to find R23 Euler’s angles so that the X axis of the pan and tilt equals u (supposing that whatever we have on the mechanical arm, like a missile, camera or antenna is pointing in the direction of the X axis of the pan and tilt).

I developed a little animated program on Scilab in order to validate the angle expressions. You can download it here: Pan_and_Tits.sce

The following video is a demo while using the program:

It can be appreciated that the mechanical arm that lies in a platform with arbitrary orientation is able to follow the moving line representing the vector that we must point at.