Linear-grammetry and calibration by simultaneous multilateration using only edge distance estimates through two-way ranging

1. A computer implemented method for relatively locating a plurality of objects in a space having a plurality of dimensions, using one-dimensional distance estimates between selected pairs of objects, said method comprising:

by the computer:

identifying a set of objects (clique) for which a distance estimate between pairs of individual objects in the set is known;

relatively localizing the objects within the clique by their respective coordinates;

unifying the coordinate systems among the clique;

wherein the unifying coordinate system between two cliques comprises finding two cliques that have a common clique of size equal to the dimension of the space plus one;

wherein unifying the coordinate system between cliques performed successively by initializing every clique to be a union clique and generate a set of all union cliques;

unifying the coordinate system between two union cliques to generate a combined union clique with a unified coordinate system;

replacing the two union cliques with their combined union clique in the set of all union cliques; and

outputting an indicia of the locations of the objects.

2. The method of claim 1 wherein the cliques are maximal cliques wherein no more objects can be added to the clique to generate a new clique.

3. The method of claim 1 , wherein the set of objects comprises of a set of anchors and the relative localization is first performed on the set of anchor points to form a union cliques between the anchor points.

4. The method of claim 3 , wherein at least one anchor is placed in particular locations in a planning phase.

5. The method of claim 3 , wherein the anchors are placed such that an intersection of at least two subset of anchor points that form a clique with respect to the distance measurements contain at least n+1 anchors where n is the dimension of the space.

6. The method of claim 3 , wherein an auxiliary anchor is dynamically added to the set of anchors.

7. The method of claim 6 , wherein the auxiliary anchor satisfies connectivity of a graph comprising the anchors as nodes and edges where measurement is available between two anchors.

8. The method of claim 6 , wherein the auxiliary anchor is added such that the size of a clique is increased.

9. The method of claim 6 , further comprising adding at least an auxiliary anchor only during calibration and removing that added anchor after calibration.

10. The method of claim 1 , further comprising filtering measurements of relative localization of objects in a clique and unification of relatively localized cliques.

11. The method of claim 1 , further comprising scheduling measurement between a pair of objects.

12. The method of claim 1 , wherein the unifying the size of the clique comprises choosing a clique that includes at least n+1 objects such that they are not in a same hyperplane having a dimension one less than that of its ambient space of dimension n.

13. The method of claim 12 , wherein a chosen clique is one that contains at least 3 objects that are not in a same line in 2-dimensional space or one that contains at least 4 objects that are not in a same plane in 3-dimensional space.

14. The method of claim 1 , further comprising placing objects at corners of a polytope such that a measurement associated with the polytope includes an area of a face of a polytope, or volume of a polytope.

15. The method of claim 1 , further comprising relatively localizing a subset of objects using rank deficient matrix approximation wherein the rank is less than or equal to the dimension of the space.

16. The method of claim 15 , further comprising identifying a maximal clique and relatively localizing objects in the maximal clique, from the remaining objects finding an object with maximal number of distance measurement to the localized objects and relatively localize this object in the same coordinate system, and continue the process of relatively localizing the remaining unlocalized objects one by one.

17. The method of claim 1 , wherein the unification between two union cliques further comprises selecting two union cliques with maximal number of distance measurements between the objects from the two union cliques.

18. The method of claim 1 , further comprising unifying the coordinates by selecting the union cliques with different orders and finally selecting the one with lowest mean squared error.