Showing papers by "Brian D. O. Anderson published in 2013"

Analysis of Noisy Bearing-Only Network Localization

Abstract: Graph theory has been used to characterize the solvability of the sensor network localization problem with ideal (i.e., precisely known) bearing-only measurements between certain pairs of sensors and a limited amount of information about the position of certain nodes, i.e., anchors. In practice, however, bearing measurements will never be exact, and the equations whose solutions deliver sensor positions in the noiseless case may no longer have a solution. This technical brief argues that if the same conditions for localizability that exist in the noiseless case are satisfied and the bearing measurement errors are small enough (as will be formalized later in the technical brief), then the network will be approximately localizable, i.e., sensor position estimates can be found which are near the correct values. In particular, a bound on the position errors is found in terms of a bound on the bearing errors. Later, this bound is used to propose a method to select anchors to minimize the effect of noisy bearing measurements on the localization solution.

Doppler Shift Target Localization

Abstract: The problem of Doppler-based target position and velocity estimation using a sensor network is outlined. The minimum number of Doppler-shift measurements at distinct generic sensor positions in order to have a finite number of solutions, and later, a unique solution for the unknown target position and velocity are stated analytically. Furthermore we study the same problem, where not only Doppler-shift measurements are collected, but also other types of measurements are available, e.g,. bearing or distance to the target from each of the sensors. Later we study the Cramer-Rao inequality associated with the Doppler-shift measurements to a target in a sensor network, and we use the Cramer-Rao bound to illustrate some results on optimal placements of the sensors when the goal is to estimate the velocity of the target. Some simulation results are presented at the end.

Adaptive range-measurement-based target pursuit

Abstract: SUMMARY This paper presents an adaptive scheme for localization of a target from distance measurements and motion control of a mobile agent to pursue this target. The localization and motion control task of interest is approached within a parameter-identifier-based adaptive control framework, where the localization is formulated as a parameter identification problem and the motion control is achieved using an adaptive controller based on the produced location estimates of the target. First, a robust adaptive law is designed to generate location estimate of the target using distance measurements. Then, following the standard certainty equivalence approach, a motion control law is developed considering substitution of the estimate generated by the localization algorithm for the unknown location of the target. Noting that there is some incompatibility between the persistence of excitation requirements of the localization algorithm and the target pursuit goal of the motion control law, the base motion control law is (re)designed to eliminate the effects of this incompatibility. The novelty of this paper is in this motion control design eliminating the persistence of excitation incompatibility. Stability and convergence analysis for the overall adaptive control scheme is presented. The results are valid in both two and three dimensions of motion space. The applications of the adaptive scheme include rescue localization, surveillance of signal sources, and formation acquisition of autonomous multi-robot/vehicle systems. Copyright © 2012 John Wiley & Sons, Ltd.

Connectivity of Large Wireless Networks Under A General Connection Model

Abstract: This paper studies networks where all nodes are distributed on a unit square A=Δ [- [1/2], [1/2]]2 following a Poisson distribution with known density ρ and a pair of nodes separated by an Euclidean distance x are directly connected with probability grρ(x)=Δg(x/rρ), independent of the event that any other pair of nodes are directly connected. Here, g:[0,∞)→ [0,1] satisfies the conditions of rotational invariance, nonincreasing monotonicity, integral boundedness, and g(x)=o(1/(x2log2x)) ; further, rρ=√{(logρ+b)/(Cρ)} where C=∫ℜ2g(||x||)dx and b is a constant. Denote the aforementioned network by G(Xρ,grρ,A). We show that as ρ→ ∞, 1) the distribution of the number of isolated nodes in G(Xρ,grρ,A) converges to a Poisson distribution with mean e-b ; 2) asymptotically almost surely (a.a.s.) there is no component in G(Xρ,grρ,A) of fixed and finite order k >; 1; c) a.a.s. the number of components with an unbounded order is one. Therefore, as ρ→ ∞, the network a.a.s. contains a unique unbounded component and isolated nodes only; a sufficient and necessary condition for G(Xρ,grρ,A) to be a.a.s. connected is that there is no isolated node in the network, which occurs when b→ ∞ as ρ→ ∞. These results expand recent results obtained for connectivity of random geometric graphs from the unit disk model and the fewer results from the log-normal model to the more general and more practical random connection model.

Equivariant Morse theory and formation control

Abstract: In this paper we study the critical points of potential functions for distance-based formation shape of a finite number of point agents in Euclidean space ℝd with d ≤ 3. The analysis of critical formations proceeds using equivariant Morse theory for equivariant Morse functions on manifolds of configuration spaces. We establish lower bounds for the number of critical formations. For d = 2 these bounds agree with the bounds announced in [3], while for d = 3 we obtain new bounds. We also propose a control law of the form of a decentralized gradient flow that evolves on a configuration manifold for agents in ℝd such that collisions among the agents do not occur. By computing the equivariant cohomology of the configurations spaces we establish new lower bounds for the number of critical collision-free formations in the configuration space. Our work parallels earlier research in geometric mechanics by Pacella [19] and McCord [18] on enumerating central configurations for the N-body problem.

Stabilization of stiff formations with a mix of direction and distance constraints

Abstract: Heterogenous formation shape control with a mix of inter-agent distance and bearing constraints involves the design of distributed control laws that ensure the formation moves such that these inter-agent constraints are achieved and maintained. This paper looks at the design of a distributed control scheme to solve the mixed constraint formation control problem with an arbitrary number of agents. A gradient control law is proposed based on the mathematical notion of a stiff formation structure and a corresponding stiff constraint matrix (which has origins in graph theory). This work provides an interesting and novel contrast to much of the existing work in formation control where distance-only or bearing-only constraints are typically maintained. A stability analysis is sketched and a number of other technical results are given.

Understanding Error Propagation in Multihop Sensor Network Localization

Abstract: In multihop localization procedures where not every node at unknown positions (i.e., sensors) can directly measure distances to nodes at known positions (i.e., anchors), sensor localization errors normally propagate (i.e., increase) as sensors progressively more distant from anchors are localized. To understand error propagation, we consider a primitive localization scenario: Nodes are deployed within a disk according to a homogeneous Poisson point process, the nodes around the disk center are anchors, the other nodes are sensors, and sensors are localized from the disk center to the outside in a hop-by-hop manner. Supposing noisy distance measurements between adjacent nodes, we analyze the quantitative relationship among sensor localization errors, minimal hop counts from sensors to anchors, the sensor density, and the noise level of distance measurements. This relationship clearly reflects the properties of error propagation and is greatly helpful to the design and deployment of large-scale sensor networks. Finally, a simulation analysis based on actual localization procedures and the Cramer-Rao lower bound confirms our results.

Non-robustness of gradient control for 3-D undirected formations with distance mismatch

Abstract: Gradient control laws can be used for effectively achieving undirected formation shape, by assuming that interagent distances between a certain set of joint agent pairs can be accurately specified and measured. This paper examines the formation behavior in a 3-D space context in the case that the neighboring agent pairs have slightly differing views or estimates about the desired interagent distances they are tasked to maintain. It is shown, by using a tetrahedron formation example, that the final formation shape will be slightly distorted as compared to the desired one. Further, in general each agent's motion will be a combination of rotation and translation. Specifically, a helical movement can be observed in the presence of distance mismatch.

Critical Density for Connectivity in 2D and 3D Wireless Multi-Hop Networks

Abstract: In this paper we investigate the critical node density required to ensure that an arbitrary node in a large-scale wireless multi-hop network is connected (via multi-hop path) to infinitely many other nodes with a positive probability. Specifically we consider a wireless multi-hop network where nodes are distributed in \mathbb{R}^d (d = 2,3) following a homogeneous Poisson point process. The establishment of a direct connection between any two nodes is independent of connections between other pairs of nodes and its probability satisfies some intuitively reasonable conditions, viz. rotational and translational invariance, non-increasing monotonicity, and integral boundedness. Under the above random connection model we first obtain analytically the upper and lower bounds for the critical density. Then we compare the new bounds with other existing bounds in the literature under the unit disk model and the log-normal model which are special cases of the random connection model. The comparison shows that our bounds are either close to or tighter than the known ones. To the best of our knowledge, this is the first result for the random connection model in both 2D and 3D networks. The result is of practical use for designing large-scale wireless multi-hop networks such as wireless sensor networks.

On distributed cluster consensus for multiple double-integrator agents

Abstract: This paper investigates the consensus for multiple interacting clusters of double-integrator agents under two different frameworks, viz, the framework that all agents share the same position and velocity interaction topology and the framework that the position and velocity topologies are modeled by totally independent graphs. Different cluster consensus algorithms are designed and analyzed accordingly. A consistent structural result is shown for both frameworks that cluster consensus can be reached if the interaction topologies satisfy some connectivity assumptions and further, compared to the interactions among different clusters, the interactions within each cluster are sufficiently strong. Some lower bounds for such strengths are specified as well.

Systematic Bias Correction in Source Localization

Abstract: A novel analytical approach is proposed to approximate and correct the bias in localization problems in n-dimensional space (n = 2 or 3) with N (N >= n) independently usable measurements (such as distance, bearing, time difference of arrival (TDOA), etc.). Here, N is often but not always the same as the number of sensors. This new method mixes Taylor series and Jacobian matrices to determine the bias and leads in the case when N = n to an easily calculated analytical bias expression; however, when N is greater than n, the nature of the calculation is more complicated in that a further step is required. The proposed novel method is generic, which means that it can be applied to different types of measurements. To illustrate this approach we analyze the proposed method in three situations. Monte Carlo simulation results verify that, when the underlying geometry is a good geometry (which allows the location of the target to be obtained with acceptable mean square error (MSE)), the proposed approach can correct the bias effectively in space of dimension 2 or 3 with an arbitrary number of independent usable measurements. In addition the proposed method is applicable irrespective of the type of measurement (range, bearing, TDOA, etc.).

Multiagent self-localization using bearing only measurements

Abstract: This paper proposes a two stage approach to solving a simple network localization problem arising in the control of multi-vehicle formation shapes using bearing-only measurements. While it is impossible for one agent to localize, in its own coordinate basis, a second agent undergoing arbitrary plane motion using bearing-only measurements, this paper shows how to use a combination of a Fourier Transform and an overdetermined linear system of equations to allow two agents undergoing plane circular motion to localize each other. It is postulated that each agent only knows the parameters fully describing its own motion and must determine enough parameters of the other agent to localize it. A Fourier Transform of the measured bearing is used by each agent to obtain an approximate magnitude of the other agent's angular velocity and a two-dimensional search grid is used in an overdetermined linear equation system to solve the localization problem. The paper investigates the effect of noise in bearing measurements on the accuracy of the proposed method, offering some potential methods of decreasing the effect of noise.

Translational velocity consensus using distance-only measurements

Abstract: This paper proposes a strategy to achieve translational velocity consensus in a multi-agent formation using distance-only measurements. Since with agents executing arbitrary motions, distance-only measurement cannot provide enough information for velocity consensus, we postulate that agents engage in a combination of circular motion and linear motion. When energy saving is the first priority, the linear motion component should dominate. On the other hand, when measurement accuracy is the first priority, the circular motion must be more prominent.

On the Zero Properties of Tall Linear Systems with Single-rate and Multirate Outputs

Abstract: The zero properties of tall discrete-time multirate linear systems are studied in this paper. In the literature, zero properties of multirate linear systems are defined as those of their corresponding blocked systems, which are time-invariant systems whose behavior is broadly equivalent to that of the generating multirate system. In this paper, we review some recent scattered results of the authors and their colleagues dealing with the zero properties of the blocked systems associated with multirate systems. First, we show that tall linear time-invariant unblocked systems are zero-free when the parameter matrices A,B,C,D assume generic values. Then it is argued that tall blocked systems obtained from blocking of tall linear timeinvariant systems with generic parameter matrices A,B,C,D, are also zero-free. Finally, we illustrate that tall blocked systems associated with multirate systems generically have no finite nonzero zeros.

Opportunistic broadcast in mobile ad-hoc networks subject to channel randomness

Abstract: Broadcast in mobile ad-hoc networks is a challenging and resource demanding task, due to the effects of dynamic network topology and channel randomness. In this paper, we consider 2D wireless ad-hoc networks where nodes are randomly distributed and move following a random direction mobility model. A piece of information is broadcast from an arbitrary node. Based on an in-depth analysis into the popular Susceptible-Infectious-Recovered (SIR) epidemic routing algorithm for mobile ad-hoc networks, an energy and spectrum efficient broadcast scheme is proposed, which is able to adapt to fast-changing network topology and channel randomness. Analytical results are provided to characterize the performance of the proposed scheme, including the fraction of nodes that can receive the information and the delay of information propagation. The accuracy of analytical results is verified using simulations.

Certifying non-existence of undesired locally stable equilibria in formation shape control problems

Abstract: A fundamental control problem for autonomous vehicle formations is formation shape control, in which the agents must maintain a prescribed formation shape using only information measured or communicated from neighboring agents. While a large and growing literature has recently emerged on distance-based formation shape control, global stability properties remain a significant open problem. Even in four-agent formations, the basic question of whether or not there can exist locally stable incorrect equilibrium shapes remains open. This paper shows how this question can be answered for any size formation in principle using semidefinite programming techniques for semialgebraic problems, involving solutions sets of polynomial equations, inequations, and inequalities.

Mixed frequency structured AR model identification

Abstract: This paper is concerned with identifiability of an underlying high frequency multivariate stable singular AR system from mixed frequency observations. Such problems arise for instance in economics when some variables are observed monthly whereas others are observed quarterly. In particular, this paper studies stable singular AR systems where the covariance matrix associated with the vector obtained by stacking observation vector, yt, and its lags from the first lag to the p-th one (p is the order of the AR system), is also singular. To deal with this, it is assumed that the column degrees of the associated polynomial matrix are known. We consider first that there are given nonzero unequal column degrees and we show generic identifiability of the system and noise parameters. Then we extend the results to allow zero column degrees corresponding to fast components. In this case, we first show generic identifiability of the subsystem of the components with nonzero column degree. Then we show how to obtain those components of the parameter matrices of the components corresponding to zero column degree by regression.

Distributed estimation and control for preserving formation rigidity for mobile robot teams

Abstract: Inspired by the concept of network algebraic connectivity, we adopt an extended notion named rigidity preservation index to characterize the rigidity property for a formation framework. A gradient based controller is proposed to ensure the rigidity preservation of multi-robot networks in an unknown environment, while the rigidity metric can be maximized over time during robots' motions. In order to implement the controller in a distributed manner, a distributed inverse power iteration algorithm is developed which allows each robot to estimate the global rigidity index information. Simulation results are provided to demonstrate the effectiveness of the estimation and control scheme.

Local average consensus in distributed measurement of spatial-temporal varying parameters: 1D case

Abstract: We study a new variant of consensus problems, termed `local average consensus', in networks of agents. We consider the task of using sensor networks to perform distributed measurement of a parameter which has both spatial (in this paper 1D) and temporal variations. Our idea is to maintain potentially useful local information regarding spatial variation, as contrasted with reaching a single, global consensus, as well as to mitigate the effect of measurement errors. We employ two schemes for computation of local average consensus: exponential weighting and uniform finite window. In both schemes, we design local average consensus algorithms to address first the case where the measured parameter has spatial variation but is constant in time, and then the case where the measured parameter has both spatial and temporal variations. Our designed algorithms are distributed, in that information is exchanged only among neighbors. Moreover, we analyze spatial frequency response and noise propagation associated to the algorithms. The tradeoffs of using local consensus, as compared to standard global consensus, include higher memory requirement and degraded noise performance.

On the Zero-freeness of Tall Multirate Linear Systems

Abstract: In this paper, tall discrete-time linear systems with multirate outputs are studied. In particular, we focus on their zeros. In systems and control literature zeros of multirate systems are defined as those of their corresponding time-invariant blocked systems. Hence, the zeros of tall blocked systems resulting from blocking of linear systems with multirate outputs are mainly explored in this work. We specifically investigate zeros of tall blocked systems formed by blocking tall multirate linear systems with generic parameter matrices. It is demonstrated that tall blocked systems generically have no finite nonzero zeros; however, they may have zeros at the origin or at infinity depending on the choice of blocking delay and the input, state and output dimensions.

Zeros of networks of linear multi-agent systems: Time-invariant interconnections

Abstract: This paper investigates the zero properties of networks of linear multi-agent control systems, where the coupling parameters between the agents are assumed to be constant. We characterize the zeros both for heterogeneous and homogeneous networks. Moreover, for homogenous networks with time-invariant interconnection dynamics and SISO agents, we illustrate how zeros of each individual agent and zeros of interconnection dynamics contribute to the zero properties of the whole network. We also investigate the effects of blocking on the zeros.

A new splitting-merging paradigm for distributed localization in wireless sensor networks

Abstract: This paper proposes a new merging (stitching) scheme for distributed localization of wireless sensor networks. In splitting-merging localization techniques, the network is first split into small sub-networks and each sub-network self-localizes itself, possibly in its own rather than a global coordinate basis. Then by using a stitching strategy they are merged back to compute the position of the nodes in a global coordinate basis. Unlike the existing techniques in which the stitchable sub-networks must have at least 3 nodes in common for the stitching to succeed, the proposed method can systematically address all possible scenarios (even when the two sub-networks are disjoint) while keeping the computational complexity fairly low. The scheme puts the idea of four-bar linkage mechanism and bilateration together in tackling the problem. Therefore, theoretically the proposed method can localize a broader class of networks which are localizable by any splitting-stitching technique. Simulation comparisons with trilateration and wheel graphs show a considerably higher percentage of the localized nodes in this technique. This technique in conjunction with some existing distributed splitting technique, provides a total distributed localization algorithm.

Certifying non-existence of undesired locally stable equilibria in formation shape control problems

Abstract: A fundamental control problem for autonomous vehicle formations is formation shape control, in which the agents must maintain a prescribed formation shape using only information measured or communicated from neighboring agents. While a large and growing literature has recently emerged on distance-based formation shape control, global stability properties remain a significant open problem. Even in four-agent formations, the basic question of whether or not there can exist locally stable incorrect equilibrium shapes remains open. This paper shows how this question can be answered for any size formation in principle using semidefinite programming techniques for semialgebraic problems, involving solutions sets of polynomial equations, inequations, and inequalities.

Local Average Consensus in Distributed Measurement of Spatial-Temporal Varying Parameters: 1D Case

Abstract: We study a new variant of consensus problems, termed `local average consensus', in networks of agents. We consider the task of using sensor networks to perform distributed measurement of a parameter which has both spatial (in this paper 1D) and temporal variations. Our idea is to maintain potentially useful local information regarding spatial variation, as contrasted with reaching a single, global consensus, as well as to mitigate the effect of measurement errors. We employ two schemes for computation of local average consensus: exponential weighting and uniform finite window. In both schemes, we design local average consensus algorithms to address first the case where the measured parameter has spatial variation but is constant in time, and then the case where the measured parameter has both spatial and temporal variations. Our designed algorithms are distributed, in that information is exchanged only among neighbors. Moreover, we analyze both spatial and temporal frequency responses and noise propagation associated with the algorithms. The tradeoffs of using local consensus, as compared to standard global consensus, include higher memory requirement and degraded noise performance. Arbitrary updating weights and random spacing between sensors are analyzed in the proposed algorithms.