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Finite-Time Attitude Synchronization with Distributed Discontinuous Protocols ; ; et al in IEEE Transactions on Automatic Control (in press) The finite-time attitude synchronization problem is considered in this paper, where the rotation of each rigid body is expressed using the axis-angle representation. Two discontinuous and distributed ... [more ▼] The finite-time attitude synchronization problem is considered in this paper, where the rotation of each rigid body is expressed using the axis-angle representation. Two discontinuous and distributed controllers using the vectorized signum function are proposed, which guarantee almost global and local convergence, respectively. Filippov solutions and non-smooth analysis techniques are adopted to handle the discontinuities. Sufficient conditions are provided to guarantee finite-time convergence and boundedness of the solutions. Simulation examples are provided to verify the performances of the control protocols designed in this paper. [less ▲] Detailed reference viewed: 81 (1 UL)Dynamic controllers for column synchronization of rotation matrices: a QR-factorization approach Thunberg, Johan ; Markdahl, Johan ; Goncalves, Jorge in Automatica (2018), 93 In the multi-agent systems setting, this paper addresses continuous-time distributed synchronization of columns of rotation matrices. More precisely, k specific columns shall be synchronized and only the ... [more ▼] In the multi-agent systems setting, this paper addresses continuous-time distributed synchronization of columns of rotation matrices. More precisely, k specific columns shall be synchronized and only the corresponding k columns of the relative rotations between the agents are assumed to be available for the control design. When one specific column is considered, the problem is equivalent to synchronization on the (d-1)-dimensional unit sphere and when all the columns are considered, the problem is equivalent to synchronization on SO(d). We design dynamic control laws for these synchronization problems. The control laws are based on the introduction of auxiliary variables in combination with a QR-factorization approach. The benefit of this QR-factorization approach is that we can decouple the dynamics for the $k$ columns from the remaining d-k ones. Under the control scheme, the closed loop system achieves almost global convergence to synchronization for quasi-strong interaction graph topologies. [less ▲] Detailed reference viewed: 128 (1 UL)Almost Global Consensus on the n-Sphere Markdahl, Johan ; Thunberg, Johan ; Goncalves, Jorge in IEEE Transactions on Automatic Control (2018), 63(6), 1664-1675 This paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multi-agent systems that evolve in continuous time on the n-dimensional unit ... [more ▼] This paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multi-agent systems that evolve in continuous time on the n-dimensional unit sphere or n-sphere. For any connected, undirected graph and all n 2 N\{1}, each protocol in said class is shown to yield almost global consensus. The feedback laws are negative gradients of Lyapunov functions and one instance generates the canonical intrinsic gradient descent protocol. This convergence result sheds new light on the general problem of consensus on Riemannian manifolds; the n-sphere for n 2 N\{1} differs from the circle and SO(3) where the corresponding protocols fail to generate almost global consensus. Moreover, we derive a novel consensus protocol on SO(3) by combining two almost globally convergent protocols on the n-sphere for n in {1, 2}. Theoretical and simulation results suggest that the combined protocol yields almost global consensus on SO(3). [less ▲] Detailed reference viewed: 102 (6 UL)A lifting method for analyzing distributed synchronization on the unit sphere Thunberg, Johan ; Markdahl, Johan ; et al in Automatica (2018) This paper introduces a new lifting method for analyzing convergence of continuous-time distributed synchronization/consensus systems on the unit sphere. Points on the d-dimensional unit sphere are lifted ... [more ▼] This paper introduces a new lifting method for analyzing convergence of continuous-time distributed synchronization/consensus systems on the unit sphere. Points on the d-dimensional unit sphere are lifted to the (d+1)-dimensional Euclidean space. The consensus protocol on the unit sphere is the classical one, where agents move toward weighted averages of their neighbors in their respective tangent planes. Only local and relative state information is used. The directed interaction graph topologies are allowed to switch as a function of time. The dynamics of the lifted variables are governed by a nonlinear consensus protocol for which the weights contain ratios of the norms of state variables. We generalize previous convergence results for hemispheres. For a large class of consensus protocols defined for switching uniformly quasi-strongly connected time-varying graphs, we show that the consensus manifold is uniformly asymptotically stable relative to closed balls contained in a hemisphere. Compared to earlier projection based approaches used in this context such as the gnomonic projection, which is defined for hemispheres only, the lifting method applies globally. With that, the hope is that this method can be useful for future investigations on global convergence. [less ▲] Detailed reference viewed: 69 (0 UL)Experimental design trade-offs for gene regulatory network inference: an in silico study of the yeast Saccharomyces cerevisiae cell cycle Markdahl, Johan ; Colombo, Nicolo ; Thunberg, Johan et al in Proceedings of the 56th IEEE Conference on Decision and Control (2017, December) Time-series of high throughput gene sequencing data intended for gene regulatory network (GRN) inference are often short due to the high costs of sampling cell systems. Moreover, experimentalists lack a ... [more ▼] Time-series of high throughput gene sequencing data intended for gene regulatory network (GRN) inference are often short due to the high costs of sampling cell systems. Moreover, experimentalists lack a set of quantitative guidelines that prescribe the minimal number of samples required to infer a reliable GRN model. We study the temporal resolution of data vs.quality of GRN inference in order to ultimately overcome this deficit. The evolution of a Markovian jump process model for the Ras/cAMP/PKA pathway of proteins and metabolites in the G1 phase of the Saccharomyces cerevisiae cell cycle is sampled at a number of different rates. For each time-series we infer a linear regression model of the GRN using the LASSO method. The inferred network topology is evaluated in terms of the area under the precision-recall curve (AUPR). By plotting the AUPR against the number of samples, we show that the trade-off has a, roughly speaking, sigmoid shape. An optimal number of samples corresponds to values on the ridge of the sigmoid. [less ▲] Detailed reference viewed: 110 (10 UL)On definition and inference of nonlinear Boolean dynamic networks Yue, Zuogong ; Thunberg, Johan ; et al in On definition and inference of nonlinear Boolean dynamic networks (2017, December) Network reconstruction has become particularly important in systems biology, and is now expected to deliver information on causality. Systems in nature are inherently nonlinear. However, for nonlinear ... [more ▼] Network reconstruction has become particularly important in systems biology, and is now expected to deliver information on causality. Systems in nature are inherently nonlinear. However, for nonlinear dynamical systems with hidden states, how to give a useful definition of dynamic networks is still an open question. This paper presents a useful definition of Boolean dynamic networks for a large class of nonlinear systems. Moreover, a robust inference method is provided. The well-known Millar-10 model in systems biology is used as a numerical example, which provides the ground truth of causal networks for key mRNAs involved in eukaryotic circadian clocks. In addition, as second contribution of this paper, we suggest definitions of linear network identifiability, which helps to unify the available work on network identifiability. [less ▲] Detailed reference viewed: 133 (5 UL)Distributed synchronization of euclidean transformations with guaranteed convergence Thunberg, Johan ; Goncalves, Jorge ; in 56th IEEE Conference on Decision and Control (2017, December) This paper addresses synchronization of Euclidean transformations over graphs. Synchronization in this context, unlike rendezvous or consensus, means that composite transformations over loops in the graph ... [more ▼] This paper addresses synchronization of Euclidean transformations over graphs. Synchronization in this context, unlike rendezvous or consensus, means that composite transformations over loops in the graph are equal to the identity. Given a set of non-synchronized transformations, the problem at hand is to find a set of synchronized transformations approximating well the non-synchronized transformations. This is formulated as a nonlinear least-squares optimization problem. We present a distributed synchronization algorithm that converges to the optimal solution to an approximation of the optimization problem. This approximation stems from a spectral relaxation of the rotational part on the one hand and from a separation between the rotations and the translations on the other. The method can be used to distributively improve the measurements obtained in sensor networks such as networks of cameras where pairwise relative transformations are measured. The convergence of the method is verified in numerical simulations. [less ▲] Detailed reference viewed: 75 (1 UL)Linear Dynamic Network Reconstruction from Heterogeneous Datasets Yue, Zuogong ; Thunberg, Johan ; et al in Linear Dynamic Network Reconstruction from Heterogeneous Datasets (2017, July) Detailed reference viewed: 80 (1 UL)Distributed methods for synchronization of orthogonal matrices over graphs Thunberg, Johan ; Bernard, Florian ; Goncalves, Jorge in Automatica (2017), 80 This paper addresses the problem of synchronizing orthogonal matrices over directed graphs. For synchronized transformations (or matrices), composite transformations over loops equal the identity. We ... [more ▼] This paper addresses the problem of synchronizing orthogonal matrices over directed graphs. For synchronized transformations (or matrices), composite transformations over loops equal the identity. We formulate the synchronization problem as a least-squares optimization problem with nonlinear constraints. The synchronization problem appears as one of the key components in applications ranging from 3D-localization to image registration. The main contributions of this work can be summarized as the introduction of two novel algorithms; one for symmetric graphs and one for graphs that are possibly asymmetric. Under general conditions, the former has guaranteed convergence to the solution of a spectral relaxation to the synchronization problem. The latter is stable for small step sizes when the graph is quasi-strongly connected. The proposed methods are verified in numerical simulations. [less ▲] Detailed reference viewed: 100 (3 UL)Shape-aware surface reconstruction from sparse 3D point-clouds Bernard, Florian ; Salamanca Mino, Luis ; Thunberg, Johan et al in Medical Image Analysis (2017), 38 The reconstruction of an object’s shape or surface from a set of 3D points plays an important role in medical image analysis, e.g. in anatomy reconstruction from tomographic measurements or in the process ... [more ▼] The reconstruction of an object’s shape or surface from a set of 3D points plays an important role in medical image analysis, e.g. in anatomy reconstruction from tomographic measurements or in the process of aligning intra-operative navigation and preoperative planning data. In such scenarios, one usually has to deal with sparse data, which significantly aggravates the problem of reconstruction. However, medical applications often provide contextual information about the 3D point data that allow to incorporate prior knowledge about the shape that is to be reconstructed. To this end, we propose the use of a statistical shape model (SSM) as a prior for surface reconstruction. The SSM is represented by a point distribution model (PDM), which is associated with a surface mesh. Using the shape distribution that is modelled by the PDM, we formulate the problem of surface reconstruction from a probabilistic perspective based on a Gaussian Mixture Model (GMM). In order to do so, the given points are interpreted as samples of the GMM. By using mixture components with anisotropic covariances that are “oriented” according to the surface normals at the PDM points, a surface-based fitting is accomplished. Estimating the parameters of the GMM in a maximum a posteriori manner yields the reconstruction of the surface from the given data points. We compare our method to the extensively used Iterative Closest Points method on several different anatomical datasets/SSMs (brain, femur, tibia, hip, liver) and demonstrate superior accuracy and robustness on sparse data. [less ▲] Detailed reference viewed: 156 (20 UL)Integration of sparse electrophysiological measurements with preoperative MRI using 3D surface estimation in deep brain stimulation surgery Husch, Andreas ; ; Thunberg, Johan et al in Webster, Robert; Fei, Baowei (Eds.) Medical Imaging 2017: Image-Guided Procedures, Robotic Interventions, and Modeling (2017, February 14) Intraoperative microelectrode recordings (MER) have been used for several decades to guide neurosurgeons during the implantation of Deep Brain Stimulation (DBS) electrodes, especially when targeting the ... [more ▼] Intraoperative microelectrode recordings (MER) have been used for several decades to guide neurosurgeons during the implantation of Deep Brain Stimulation (DBS) electrodes, especially when targeting the subthalamic nucleus (STN) to suppress the symptoms of Parkinson’s Disease. The standard approach is to use an array of up to five MER electrodes in a fixed configuration. Interpretation of the recorded signals yields a spatiallyvery sparse set of information about the morphology of the respective brain structures in the targeted area. However, no aid is currently available for surgeons to intraoperatively integrate this information with other data available on the patient’s individual morphology (e.g. MR imaging data used for surgical planning). This integration might allow surgeons to better determine the most probable position of the electrodes within the target structure during surgery. This paper suggests a method for reconstructing a surface patch from the sparse MER dataset utilizing additional a-priori knowledge about the geometrical configuration of the measurement electrodes. The conventional representation of MER measurements as intervals of target region/non-target region is therefore transformed into an equivalent boundary set representation, allowing efficient point-based calculations. Subsequently, the problem is to integrate the resulting patch with a preoperative model of the target structure, which can be formulated as registration problem minimizing a distance measure between the two surfaces. When restricting this registration procedure to translations, which is reasonable given certain geometric considerations, the problem can be solved globally by employing an exhaustive search with arbitrary precision in polynomial time. The proposed method is demonstrated using bilateral STN/Substantia Nigra segmentation data from preoperative MRIs of 17 Patients with simulated MER electrode placement. When using simulated data of heavily perturbed electrodes and subsequent MER measuremen [less ▲] Detailed reference viewed: 118 (10 UL)Local Lyapunov Functions for Consensus in Switching Nonlinear Systems Thunberg, Johan ; Goncalves, Jorge ; in IEEE Transactions on Automatic Control (2017) This note presents two theorems on asymptotic state consensus of continuous time nonlinear multi-agent systems. The agents reside in Rm and have switching interconnection topologies. Both the first ... [more ▼] This note presents two theorems on asymptotic state consensus of continuous time nonlinear multi-agent systems. The agents reside in Rm and have switching interconnection topologies. Both the first theorem, formulated in terms of the states of individual agents, and the second theorem, formulated in terms of the pairwise states for pairs of agents, can be interpreted as variants of Lyapunov’s second method. The two theorems complement each other; the second provides stronger convergence results under weaker graph topology assumptions, whereas the first often can be applied in a wider context in terms of the structure of the right-hand sides of the systems. The second theorem also sheds some new light on well-known results for consensus of nonlinear systems where the right-hand sides of the agents’ dynamics are convex combinations of directions to neighboring agents. For such systems, instead of proving consensus by using the theory of contracting convex sets, a local quadratic Lyapunov function can be used. [less ▲] Detailed reference viewed: 153 (4 UL)A Combinatorial Solution to Non-Rigid 3D Shape-to-Image Matching Bernard, Florian ; ; Thunberg, Johan et al in The proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2017) We propose a combinatorial solution for the problem of non-rigidly matching a 3D shape to 3D image data. To this end, we model the shape as a triangular mesh and allow each triangle of this mesh to be ... [more ▼] We propose a combinatorial solution for the problem of non-rigidly matching a 3D shape to 3D image data. To this end, we model the shape as a triangular mesh and allow each triangle of this mesh to be rigidly transformed to achieve a suitable matching to the image. By penalising the distance and the relative rotation between neighbouring triangles our matching compromises between the image and the shape information. In this paper, we resolve two major challenges: Firstly, we address the resulting large and NP-hard combinatorial problem with a suitable graph-theoretic approach. Secondly, we propose an efficient discretisation of the unbounded 6-dimensional Lie group SE(3). To our knowledge this is the first combinatorial formulation for non-rigid 3D shape-to-image matching. In contrast to existing local (gradient descent) optimisation methods, we obtain solutions that do not require a good initialisation and that are within a bound of the optimal solution. We evaluate the proposed combinatorial method on the two problems of non-rigid 3D shape-to-shape and non-rigid 3D shape-to-image registration and demonstrate that it provides promising results. [less ▲] Detailed reference viewed: 95 (6 UL)Fast Correspondences for Statistical Shape Models of Brain Structures Bernard, Florian ; Vlassis, Nikos ; et al in SPIE Medical Imaging (2016, March) Detailed reference viewed: 183 (17 UL)Consensus and Formation Control on SE(3) for Switching Topologies Thunberg, Johan ; Goncalves, Jorge ; in Automatica (2016), 66 This paper addresses the consensus problem and the formation problem on SE(3) in multi-agent systems with directed and switching interconnection topologies. Several control laws are introduced for the ... [more ▼] This paper addresses the consensus problem and the formation problem on SE(3) in multi-agent systems with directed and switching interconnection topologies. Several control laws are introduced for the consensus problem. By a simple transformation, it is shown that the proposed control laws can be used for the formation problem. The design is first conducted on the kinematic level, where the velocities are the control laws. Then, for rigid bodies in space, the design is conducted on the dynamic level, where the torques and the forces are the control laws. On the kinematic level, first two control laws are introduced that explicitly use Euclidean transformations, then separate control laws are defined for the rotations and the translations. In the special case of purely rotational motion, the consensus problem is referred to as consensus on SO(3) or attitude synchronization. In this problem, for a broad class of local representations or parameterizations of SO(3), including the Axis–Angle Representation, the Rodrigues Parameters and the Modified Rodrigues Parameters, two types of control laws are presented that look structurally the same for any choice of local representation. For these two control laws we provide conditions on the initial rotations and the connectivity of the graph such that the system reaches consensus on SO(3). Among the contributions of this paper, there are conditions for when exponential rate of convergence occurs. A theorem is provided showing that for any choice of local representation for the rotations, there is a change of coordinates such that the transformed system has a well known structure. [less ▲] Detailed reference viewed: 103 (12 UL)Global Optimality Bounds for ICA Algorithms Colombo, Nicolo ; Thunberg, Johan ; Goncalves, Jorge in 22nd International Symposium on Mathematical Theory of Networks and Systems (2016) Independent Component Analysis is a popular statistical method for separating a multivariate signal into additive components. It has been shown that the signal separation problem can be reduced to the ... [more ▼] Independent Component Analysis is a popular statistical method for separating a multivariate signal into additive components. It has been shown that the signal separation problem can be reduced to the joint diagonalization of the matrix slices of some higher-order cumulants of the signal. In this approach, the unknown mixing matrix can be computed directly from the obtained joint diagonalizer. Various iterative algorithms for solving the non-convex joint diagonalization problem exist, but they usually lack global optimality guarantees. In this paper, we introduce a procedure for computing an optimality gap for local optimal solutions. The optimality gap is then used to obtain an empirical error bound for the estimated mixing matrix. Finally, a class of simultaneous matrix decomposition problems that admit such relaxation procedure is identified. [less ▲] Detailed reference viewed: 165 (9 UL)Optimal output consensus for linear systems: a topology free approach Thunberg, Johan ; in Automatica (2016), 68 In this paper, for any homogeneous system of agents with linear continuous time dynamics, we formulate an optimal control problem. In this problem a convex cost functional of the control signals of the ... [more ▼] In this paper, for any homogeneous system of agents with linear continuous time dynamics, we formulate an optimal control problem. In this problem a convex cost functional of the control signals of the agents shall be minimized, while the outputs of the agents shall coincide at some given finite time. This is an instance of the rendezvous or finite time consensus problem. We solve this problem without any constraints on the communication topology and provide a solution as an explicit feedback control law for the case when the dynamics of the agents is output controllable. It turns out that the communication graph topology induced by the solution is complete. Based on this solution for the finite time consensus problem, we provide a solution to the case of infinite time horizon. Furthermore, we investigate under what circumstances it is possible to express the controller as a feedback control law of the output instead of the states. [less ▲] Detailed reference viewed: 110 (5 UL)An Iterative Projection method for Synchronization of Invertible Matrices Over Graphs Thunberg, Johan ; Colombo, Nicolo ; Yue, Zuogong et al in 22nd International Symposium on Mathematical Theory of Networks and Systems (2016) This paper addresses synchronization of invertible matrices over graphs. The matrices represent pairwise transformations between n euclidean coordinate systems. Synchronization means that composite ... [more ▼] This paper addresses synchronization of invertible matrices over graphs. The matrices represent pairwise transformations between n euclidean coordinate systems. Synchronization means that composite transformations over loops are equal to the identity. Given a set of measured matrices that are not synchronized, the synchronization problem amounts to fining new synchronized matrices close to the former. Under the assumption that the measurement noise is zero mean Gaussian with known covariance, we introduce an iterative method based on linear subspace projection. The method is free of step size determination and tuning and numerical simulations show significant improvement of the solution compared to a recently proposed direct method as well as the Gauss-Newton method. [less ▲] Detailed reference viewed: 128 (10 UL)Inverse Problems for Matrix Exponential in System Identification: System Aliasing Yue, Zuogong ; Thunberg, Johan ; Goncalves, Jorge in 22nd International Symposium on Mathematical Theory of Networks and Systems (2016) This note addresses identification of the A-matrix in continuous time linear dynamical systems on state-space form. If this matrix is partially known or known to have a sparse structure, such knowledge ... [more ▼] This note addresses identification of the A-matrix in continuous time linear dynamical systems on state-space form. If this matrix is partially known or known to have a sparse structure, such knowledge can be used to simplify the identification. We begin by introducing some general conditions for solvability of the inverse problems for matrix exponential. Next, we introduce “system aliasing” as an issue in the identification of slow sampled systems. Such aliasing give rise to nonunique matrix logarithms. As we show, by imposing additional conditions on and prior knowledge about the A-matrix, the issue of system aliasing can, at least partially, be overcome. Under conditions on the sparsity and the norm of the A-matrix, it is identifiable up to a finite equivalence class. [less ▲] Detailed reference viewed: 130 (15 UL)Linear Shape Deformation Models with Local Support using Graph-based Structured Matrix Factorisation Bernard, Florian ; ; Hertel, Frank et al in Linear Shape Deformation Models with Local Support using Graph-based Structured Matrix Factorisation (2016) Representing 3D shape deformations by linear models in high-dimensional space has many applications in computer vision and medical imaging, such as shape-based interpolation or segmentation. Commonly ... [more ▼] Representing 3D shape deformations by linear models in high-dimensional space has many applications in computer vision and medical imaging, such as shape-based interpolation or segmentation. Commonly, using Principal Components Analysis a low-dimensional (affine) subspace of the high-dimensional shape space is determined. However, the resulting factors (the most dominant eigenvectors of the covariance matrix) have global support, i.e. changing the coefficient of a single factor deforms the entire shape. In this paper, a method to obtain deformation factors with local support is presented. The benefits of such models include better flexibility and interpretability as well as the possibility of interactively deforming shapes locally. For that, based on a well-grounded theoretical motivation, we formulate a matrix factorisation problem employing sparsity and graph-based regularisation terms. We demonstrate that for brain shapes our method outperforms the state of the art in local support models with respect to generalisation ability and sparse shape reconstruction, whereas for human body shapes our method gives more realistic deformations. [less ▲] Detailed reference viewed: 124 (14 UL) |
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