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ERROR SCALING LAWS FOR LINEAR OPTIMAL ESTIMATION FROM RELATIVE MEASUREMENTS
, 2009
"... We study the problem of estimating vectorvalued variables from noisy “relative” measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the diffe ..."
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Cited by 8 (1 self)
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We study the problem of estimating vectorvalued variables from noisy “relative” measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the difference between two variables. We take an arbitrary variable as the reference and consider the optimal (minimum variance) linear unbiased estimate of the remaining variables. We investigate how the error in the optimal linear unbiased estimate of a node variable grows with the distance of the node to the reference node. We establish a classification of graphs, namely, dense or sparse in R d, 1 ≤ d ≤ 3, that determines how the linear unbiased optimal estimation error of a node grows with its distance from the reference node. In particular, if a graph is dense in 1,2, or 3D, then a node variable’s estimation error is upper bounded by a linear, logarithmic, or bounded function of distance from the reference, respectively. Corresponding lower bounds are obtained if the graph is sparse in 1, 2 and 3D. Our results also show that naive measures of graph density, such as node degree, are inadequate predictors of the estimation error. Being true for the optimal linear unbiased estimate, these scaling laws determine algorithmindependent limits on the estimation accuracy achievable in large graphs.
ON THE EFFECT OF ASYMMETRIC COMMUNICATION ON DISTRIBUTED TIME SYNCHRONIZATION
"... Abstract — Several distributed algorithms have been recently proposed to estimate clock offsets and skews in a network of processors from a set of noisy measurements of the difference between clock offsets and of the ratios of clock skews. These algorithms are designed to converge to the optimal, i. ..."
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Cited by 5 (3 self)
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Abstract — Several distributed algorithms have been recently proposed to estimate clock offsets and skews in a network of processors from a set of noisy measurements of the difference between clock offsets and of the ratios of clock skews. These algorithms are designed to converge to the optimal, i.e., the best linear unbiased, estimates even in the presence of node and link failures. However, they require symmetric communication between nodes for convergence. We examine the case when communication is asymmetric, i.e., when a node can receive information from another node but not vice versa. We first show that in the presence of asymmetric communication links, these algorithms converge to an unbiased but suboptimal estimate. In fact, we show that with a distributed algorithm that is constrained to use only local information, it is generally impossible to converge to the optimal estimate when communication is asymmetric. We characterize the resulting estimate that these algorithms converge to in the presence of asymmetry, and node and link failures, and its error covariance. I.
From angular manifolds to the integer lattice: Guaranteed orientation estimation with application to pose graph optimization,” supplementary material, http://www.lucacarlone.com/index
, 2013
"... Abstract—Pose graph optimization from relative measurements is challenging because of the angular component of the poses: the variables live on a manifold product with nontrivial topology, and the likelihood function is nonconvex and has many local minima. Due to these issues, iterative solvers are ..."
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Abstract—Pose graph optimization from relative measurements is challenging because of the angular component of the poses: the variables live on a manifold product with nontrivial topology, and the likelihood function is nonconvex and has many local minima. Due to these issues, iterative solvers are not robust to large amounts of noise. This paper describes a global estimation method, called MOLE2D, for the estimation of the nodes ’ orientation in a pose graph. We demonstrate that the original nonlinear optimization problem on the manifold product is equivalent to an unconstrained quadratic optimization problem on the integer lattice. Exploiting this insight, we show that, in general, the maximum likelihood estimate alone cannot be considered a reliable estimator. Therefore, MOLE2D returns a set of point estimates, for which we can derive precise probabilistic guarantees. Experiments show that the method is able to tolerate extreme amounts of noise, far above all noise levels of sensors used in applications. Using MOLE2D’s output to bootstrap the initial guess of iterative pose graph optimization methods improves their robustness and makes them avoid local minima even for high levels of noise. Index Terms—pose graph optimization, SLAM, mobile robots, orientation estimation, SO(2) manifold, multihypothesis estimation, integer quadratic programming I.
ERROR SCALING LAWS FOR OPTIMAL ESTIMATION FROM RELATIVE MEASUREMENTS
"... We study the problem of estimating vectorvalued variables from noisy “relative” measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the differ ..."
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Cited by 3 (2 self)
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We study the problem of estimating vectorvalued variables from noisy “relative” measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the difference between two variables. We take an arbitrary variable as the reference and consider the optimal (best linear unbiased) estimate of the remaining variables. We investigate how the error in the optimal estimate of a node variable grows with the distance of the node to the reference node. We establish a classification of graphs, namely, dense or sparse in R d, 1≤d≤3, that determines how the optimal estimation error of a node grows with its distance from the reference node. In particular, if a graph is dense in 1,2, or 3D, then a node variable’s estimation error is upper bounded by a linear, logarithmic, or bounded function of distance from the reference, respectively. Corresponding lower bounds are obtained if the graph is sparse in 1, 2 and 3D. Our results also show that naive measures of graph density, such as node degree, are inadequate predictors of the estimation error. Being true for the optimal estimate, these scaling laws determine algorithmindependent limits on the estimation accuracy achievable in large graphs.
RESISTANCEBASED PERFORMANCE ANALYSIS OF THE CONSENSUS ALGORITHM OVER GEOMETRIC GRAPHS∗
"... Abstract. The performance of the linear consensus algorithm is studied by using a Linear Quadratic (LQ) cost. The objective is to understand how the communication topology influences this algorithm. This is achieved by exploiting an analogy between Markov Chains and electrical resistive networks. In ..."
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Abstract. The performance of the linear consensus algorithm is studied by using a Linear Quadratic (LQ) cost. The objective is to understand how the communication topology influences this algorithm. This is achieved by exploiting an analogy between Markov Chains and electrical resistive networks. Indeed, this permits to uncover the relation between the LQ performance cost and the average effective resistance of a suitable electrical network and, moreover, to show that, if the communication graph fulfils some local properties, then its behavior can be approximated by that of a grid, which is a graph whose associated LQ cost is wellknown. Key words. Multiagent systems, consensus algorithm, distributed averaging, largescale graphs AMS subject classifications. 68R10, 90B10, 94C15, 90B18, 05C50 1. Introduction. The
1 An asynchronous consensusbased algorithm for estimation from noisy relative measurements
"... Abstract—In this work we address the problem of optimal estimating the position of each agent in a network from relative noisy vectorial distances with its neighbors. Although the problem can be cast as a standard leastsquares problem, the main challenge is to devise scalable algorithms that allow ..."
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Abstract—In this work we address the problem of optimal estimating the position of each agent in a network from relative noisy vectorial distances with its neighbors. Although the problem can be cast as a standard leastsquares problem, the main challenge is to devise scalable algorithms that allow each agent to estimate its own position by means of only local communication and bounded complexity, independently of the network size and topology. We propose a consensusbased algorithm with the use of local memory variables which allows asynchronous implementation, has guaranteed exponential convergence to the optimal solution under mild deterministic and randomised communication protocols, and requires minimal packet transmission. In the randomized scenario we then study the rate of convergence in expectation of the estimation error and we argue that it can be used to obtain upper and lower bound for the rate of converge in mean square. In particular, we show that for regular graphs the convergence rate in expectation is reduced by a factor N, which is the number of nodes, which is the same asymptotic degradation of memoryless asynchronous consensus algorithms. Additionally, we show that the asynchronous implementation is also robust to delays and communication failures. We finally complement the analytical results with some numerical simulations comparing the proposed strategy with other algorithms which have been recently proposed in the literature. Index Terms—Wireless sensor networks, distributed localization algorithms, consensus algorithms I.
MultiRobot Localization via GPS and Relative Measurements in the Presence of Asynchronous and Lossy Communication
"... Abstract — This work addresses the problem of distributed multiagent localization in presence of heterogeneous measurements and wireless communication. The proposed algorithm integrates low precision global sensors, like GPS and compasses, with more precise relative position (i.e., range plus bear ..."
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Abstract — This work addresses the problem of distributed multiagent localization in presence of heterogeneous measurements and wireless communication. The proposed algorithm integrates low precision global sensors, like GPS and compasses, with more precise relative position (i.e., range plus bearing) sensors. Global sensors are used to reconstruct the absolute position and orientation, while relative sensors are used to retrieve the shape of the formation. A fast distributed and asynchronous linear leastsquares algorithm is proposed to solve an approximated version of the nonlinear Maximum Likelihood problem. The algorithm is provably shown to be robust to communication losses and random delays. The use of ACKless broadcastbased communication protocols ensures an efficient and easy implementation in real world scenarios. If the relative measurement errors are sufficiently small, we show that the algorithm attains a solution which is very close to the maximum likelihood solution. The theoretical findings and the algorithm performances are extensively tested by means of MonteCarlo simulations. I.
1 DISTRIBUTED CUT DETECTION IN SENSOR NETWORKS
"... Abstract — We propose a distributed algorithm to detect “cuts” in sensor networks, i.e., the failure of a set of nodes that separates the networks into two or more components. The algorithm consists of a simple iterative scheme in which every node updates a scalar state by communicating with its nea ..."
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Abstract — We propose a distributed algorithm to detect “cuts” in sensor networks, i.e., the failure of a set of nodes that separates the networks into two or more components. The algorithm consists of a simple iterative scheme in which every node updates a scalar state by communicating with its nearest neighbors. In the absence of cuts, the states converge to values that are equal to potentials in a fictitious electrical network. When a set of nodes gets separated from a special node, that we call a “source node”, their states converge to 0 because “current is extracted” from the component but none is injected. On the other hand, the state of a nodes connected to the source node converges to a different value. These trends are used by every node to detect if a cut has occurred, and if so, whether it is disconnected from the source or not. Although the algorithm is iterative and involves only local communication, its convergence rate is quite fast and is independent of the size of the network. I.
Distributed Localization from Relative Noisy Measurements: a Gradient Based Approach
"... Abstract — In this work we address the problem of optimal estimating the position of each agent in a network from relative noisy vectorial distances with its neighbors. Although the problem can be cast as a standard leastsquares problem, the main challenge is to devise scalable algorithms that allo ..."
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Abstract — In this work we address the problem of optimal estimating the position of each agent in a network from relative noisy vectorial distances with its neighbors. Although the problem can be cast as a standard leastsquares problem, the main challenge is to devise scalable algorithms that allow each agent to estimate its own position by means of only local communication and bounded complexity, independently of the network size and topology. We propose a gradient based algorithm that is guaranteed to have exponentially convergence rate to the optimal centralized leastsquare solution. Moreover we show the effectiveness also in presence of delays. We finally provide numerical results to support our work. I.