Where to place the responders

The geometric arrangement of responders determines the “dilution of precision” (DOP, or noise gain), that one can expect in various parts of the volume of interest.
On the left is the annulus within which the initiator position is constrained when a single, noisy distance measurement is available. In the middle is the situation when two measurements are available from responders that are more or less at right angles in directions as seen from the initiator. Plausible solutions in this favorable case are confined to a small area. On the right is the less fortunate situation where the directions to the responders are similar, and not much new information is provided by the second measurement. Correspondngly, the likely position of the initiator is not as well confined.
When close to one of the responders, the geometry becomes more intricate, and, counter-intuitively, the solution may be less well determined.

It is generally not a good idea to have all the responders close together, since then the distance measurements will be correlated and somewhat redundant. The effect of errors typically not isotropic, but is stronger in some directions than others (as, for example, in the case of GPS, where the vertical DOP is considerably larger than the horizontal DOP, as a result of the fact that the “visible” satellites are not distributed evenly over a sphere of possible directions). In some cases ellipses of constant error may be quite elongated, meaning that while the position may be well determined in some directions, it is not in others. Finding the “best” layout of responders in a given 3-D volume is an open research problem.

In the above 2-D examples, the green spots are the positions of responders (APs), while the red dots are potential positions for the initiator (smartphone, STA). The constant error ellipses show how position may be poorly localized in some direction yet well constrained in a direction at right angles. In placing the responders, the aim is to make the constant error ellipses small and round in most of the work space. Symmetrical layouts for the responders seem to work well, as shown on the left, while somewhat suprising results may be achieved with asymmetrical layouts, as shown on the right. Note also that the method is useable even outside the convex hull of the responders — up to a point.

For a 3-D, cubic volume of interest (or a rectangular brick shaped volume with not-too-different side lengths), placing four responders at the vertices of a tetrahedron embedded in the cube has appealing properties (these points are at the four “even” vertices of the cube — see left figure below). With six responders, the vertices of an octahedron have good properties (these six points are at the face centers of a cube — see right figure below). Both of these configurations avoid placing any subset of (more than three) responders in a plane.

Adding a responder somewhere in the middle of the volume also improves overall position accuracy.

Placing responders at regular intervals along a line, while providing simplicity of installation, is not a good idea if position accuracy is of importance.

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Berthold K.P. Horn, bkph@ai.mit.edu