1 Cramér-Rao Bound for Hybrid Peer-to-Peer Positioning Henk Wymeersch 1 , Federico Penna 2 , Mauricio Caceres 2 1 Chalmers University of Technology - Dept. of Signals and Systems, Gothenburg, Sweden 2 Politecnico di Torino - Dept. of Electronics, Turin, Italy Email: henkw@chalmers.se, {federico.penna, mauricio.caceresduran}@polito.it I. I NTRODUCTION Novel cooperative positioning methods have been proposed to operate in GPS-challenged environments. However, such cooperative schemes can be also used in combination with GPS, so as to improve positioning accuracy in cases where GPS measurements are available (i) intermittently; or (ii) from a limited number of satellites; or (iii) are strongly affected by noise or multi-path. “Hybrid cooperative positioning” schemes can thus be designed to fuse information from peers and from GPS satellites. This contribution provides a theoretical characterization of achievable performance of hybrid cooper- ative positioning, by expressing the Cramér-Rao lower bound (CRLB) for the aforementioned scenario. Our results extend [?], by including the unknown clock bias, and [?], by taking into account satellites in addition to terrestrial devices. II. PROBLEM FORMULATION Given a heterogeneous network (Fig. ??), comprising satel- lites with known clock bias and known position, anchor nodes with known position but unknown clock bias, and agents with unknown clock bias and unknown position. Let M be the set of agents, S the set of satellites, A the set of anchors; denote by S m the set of satellites that agent m can see, by A m the set of anchors that agent m can communicate with, and by M m the set of peers it can communicate with. The position of a satellite s ∈S , an anchor a ∈A, and an agent m ∈M, are indicated respectively by x s , x a , x m . Our focus will be on 2-dimensional positioning, from which the extension to 3-dimensional case is straightforward. The variable b m represents the clock bias of agent m, expressed in distance units. Two types of measurements are available to agent m: r n→m is the measured distance between agent m and node n ∈ A m ∪M m , with r n→m = kx n - x m k + v n→m , where v n→m is measurement noise; ρ s→m is a pseudorange measurement between node m and satellite s ∈S m , ρ s→m = kx s - x m k + b m + v s→m , where v s→m is measurement noise.We assume that: (i) all measurement noise is zero-mean Gaussian; (ii) for peer-to-peer measurements, the link variance is symmetric: σ 2 n→m = σ 2 m→n . Our goal is to compute the CRLB of the deterministic unknown [X, b], where X = {x m∈M } and b = {b m∈M }, as The authors thank the European Space Agency (ESA), in particular Jaron Samson, and Prof. Roberto Garello for their support to this work in the framework of the “Peer to Peer Positioning” project. 1 2 3 4 5 6 Sat. 2 Sat. 1 Sat. 3 Sat. 4 Sat. 5 Sat. 6 Sat. 7 Figure 1. Example network topology. Agents’ positions (in m): 1: [-5 1]; 2: [0 3]; 3: 5 4]; 4: [3 -2]; 5: [-3 -4]; 6: [0 0]. Satellites’ positions (in m): 1: [-50 -10]; 2: [-50 20]; 3: [-20 50; 4: [12 50; 5: [50 0]; 6: [10 -50]; 7: [-10 -50]. Measurement noise: σs→m =3m ∀m ∈M,s ∈Sm; σn→m =0.10m ∀m ∈M,n ∈Mm. a function of the (range and pseudorange) measurement noise variances σ 2 a→m , σ 2 n→m , σ 2 s→m , and of the network geometry. III. FISHER I NFORMATION MATRIX The CRLB of any unbiased estimator of [X, b] is obtained by inverting the corresponding Fisher information matrix (FIM). Let F the FIM for our hybrid scenario. We will first compute the FIM under a non-cooperative setting, and then extend this result to the cooperative case. A. Non-cooperative Case We focus on a single agent, say m. Then the log-likelihood function of its measurements with respect to anchors and satellites is log p ( {r a→m } a∈Am , {ρ s→m } s∈Sm |x m ,b m ) = X a∈Am log p (r a→m |x m )+ X s∈Sm log p (ρ s→m |x m ,b m ) . =Λ m (x m ,b m ) . The Fisher information matrix is given by F m = -E { H m (Λ m (x m ,b m ))} , where the expectation is with respect to the measurements, and H m (·) is the Hessian operator containing the second-order