Vision-based Control of an Industrial Robot Arm

As we have have seen above, to know where a given point in the real world will be projected on the camera sensor is almost straightforward, using simplified models; nonetheless, the inverse problem is more difficult, and, by using a single camera, to solve it is actually not even possible; the reason for that can be intuitively explained considering that all the points lying on the line connecting a point on the sensor with the focus will be projected on that point of the sensor; i.e., to know which point in the three-dimensional world generated a given point on the sensor is impossible; this can be explained more formally looking at the relationships 3.3: if we want to project a 3D coordinate on the sensor, then these relationships represent a system of two equations in the unknowns u and v, but if we know u and v and we want to calculate x, y and z, then we have three unknowns and only two equations, so the system does not have a unique solution. To overcome this problem is possible by using multiple cameras; in such a system the problem described above is called stereoscopic problem.
In the simplest case of only two cameras, the system we need to solve becomes a set of four equations in three unknowns; this means that generally the system does not have any solution; however an algorithm (like the least squares method) that finds the solution that best fits this system of equations can be used. [...] The system is quite complex, so it will not be shown here, but the interested reader can find it in the material listed in the bibliography (i.e., Siciliano et al.).
Another approach to solve this problem is to use a feedforward neural network trained for this purpose. [...]