State Estimation for Systems on Lie Groups with Nonideal Measurements

This thesis considers the state estimation problem for invariant systems on Lie groups with inputs in its associated Lie algebra and outputs in homogeneous spaces of the Lie group. A particular focus of this thesis is the development of state estimation methodologies for systems with nonideal measurements, especially systems with additive input measurement bias, output measurement delay, and sampled outputs. The main contribution of the thesis is to effectively employ the symmetries of the system dynamics and to benefit from the Lie group structure of the underlying state space in order to design robust state estimators that are computationally simple and are ideal for embedded applications in robotic systems. We address the input measurement bias problem by proposing a novel nonlinear observer to adaptively eliminate the input measurement bias. Despite the nonlinear and non-autonomous nature of the resulting error dynamics and the complexity of the underlying state space, the proposed observer exhibits asymptotic/exponential convergence of the state and bias estimation errors to zero. To tackle the output measurement delay problem, we propose novel dynamic predictors used in an observer-predictor arrangement. The observer provides estimates of the delayed state using the delayed output measurements and the predictor takes those estimates, compensates for the delay, and provides predictions of the current state. Separately, we propose output predictors employed in a predictor-observer arrangement to address the problem of sampled output measurements. The output predictors take the sampled measurements and provide continuous predictions of the current outputs. Feeding the predicted outputs into the observer yields estimates of the current state. Both methods rely on the invariance of the underlying system dynamics to recursively provide predictions with low computation requirements. We demonstrate applications of the theory with examples of attitude, velocity, and position estimation on SO(3) and SE(3). A key contribution of this thesis is the development of C++ libraries in an embedded implementation as well as experimental verification of the developed theory with real flight tests using model UAVs

Practical implementation of nonlinear time series methods: The TISEAN package

Nonlinear time series analysis is becoming a more and more reliable tool for the study of complicated dynamics from measurements. The concept of low-dimensional chaos has proven to be fruitful in the understanding of many complex phenomena despite the fact that very few natural systems have actually been found to be low dimensional deterministic in the sense of the theory. In order to evaluate the long term usefulness of the nonlinear time series approach as inspired by chaos theory, it will be important that the corresponding methods become more widely accessible. This paper, while not a proper review on nonlinear time series analysis, tries to make a contribution to this process by describing the actual implementation of the algorithms, and their proper usage. Most of the methods require the choice of certain parameters for each specific time series application. We will try to give guidance in this respect. The scope and selection of topics in this article, as well as the implementational choices that have been made, correspond to the contents of the software package TISEAN which is publicly available from http://www.mpipks-dresden.mpg.de/~tisean . In fact, this paper can be seen as an extended manual for the TISEAN programs. It fills the gap between the technical documentation and the existing literature, providing the necessary entry points for a more thorough study of the theoretical background.Comment: 27 pages, 21 figures, downloadable software at http://www.mpipks-dresden.mpg.de/~tisea

A mathematical model for breath gas analysis of volatile organic compounds with special emphasis on acetone

Recommended standardized procedures for determining exhaled lower respiratory nitric oxide and nasal nitric oxide have been developed by task forces of the European Respiratory Society and the American Thoracic Society. These recommendations have paved the way for the measurement of nitric oxide to become a diagnostic tool for specific clinical applications. It would be desirable to develop similar guidelines for the sampling of other trace gases in exhaled breath, especially volatile organic compounds (VOCs) which reflect ongoing metabolism. The concentrations of water-soluble, blood-borne substances in exhaled breath are influenced by: (i) breathing patterns affecting gas exchange in the conducting airways; (ii) the concentrations in the tracheo-bronchial lining fluid; (iii) the alveolar and systemic concentrations of the compound. The classical Farhi equation takes only the alveolar concentrations into account. Real-time measurements of acetone in end-tidal breath under an ergometer challenge show characteristics which cannot be explained within the Farhi setting. Here we develop a compartment model that reliably captures these profiles and is capable of relating breath to the systemic concentrations of acetone. By comparison with experimental data it is inferred that the major part of variability in breath acetone concentrations (e.g., in response to moderate exercise or altered breathing patterns) can be attributed to airway gas exchange, with minimal changes of the underlying blood and tissue concentrations. Moreover, it is deduced that measured end-tidal breath concentrations of acetone determined during resting conditions and free breathing will be rather poor indicators for endogenous levels. Particularly, the current formulation includes the classical Farhi and the Scheid series inhomogeneity model as special limiting cases.Comment: 38 page

Sub-Nyquist Sampling: Bridging Theory and Practice

Sampling theory encompasses all aspects related to the conversion of continuous-time signals to discrete streams of numbers. The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. In modern applications, an increasingly number of functions is being pushed forward to sophisticated software algorithms, leaving only those delicate finely-tuned tasks for the circuit level. In this paper, we review sampling strategies which target reduction of the ADC rate below Nyquist. Our survey covers classic works from the early 50's of the previous century through recent publications from the past several years. The prime focus is bridging theory and practice, that is to pinpoint the potential of sub-Nyquist strategies to emerge from the math to the hardware. In that spirit, we integrate contemporary theoretical viewpoints, which study signal modeling in a union of subspaces, together with a taste of practical aspects, namely how the avant-garde modalities boil down to concrete signal processing systems. Our hope is that this presentation style will attract the interest of both researchers and engineers in the hope of promoting the sub-Nyquist premise into practical applications, and encouraging further research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin

On-Manifold Preintegration for Real-Time Visual-Inertial Odometry

Current approaches for visual-inertial odometry (VIO) are able to attain highly accurate state estimation via nonlinear optimization. However, real-time optimization quickly becomes infeasible as the trajectory grows over time, this problem is further emphasized by the fact that inertial measurements come at high rate, hence leading to fast growth of the number of variables in the optimization. In this paper, we address this issue by preintegrating inertial measurements between selected keyframes into single relative motion constraints. Our first contribution is a \emph{preintegration theory} that properly addresses the manifold structure of the rotation group. We formally discuss the generative measurement model as well as the nature of the rotation noise and derive the expression for the \emph{maximum a posteriori} state estimator. Our theoretical development enables the computation of all necessary Jacobians for the optimization and a-posteriori bias correction in analytic form. The second contribution is to show that the preintegrated IMU model can be seamlessly integrated into a visual-inertial pipeline under the unifying framework of factor graphs. This enables the application of incremental-smoothing algorithms and the use of a \emph{structureless} model for visual measurements, which avoids optimizing over the 3D points, further accelerating the computation. We perform an extensive evaluation of our monocular \VIO pipeline on real and simulated datasets. The results confirm that our modelling effort leads to accurate state estimation in real-time, outperforming state-of-the-art approaches.Comment: 20 pages, 24 figures, accepted for publication in IEEE Transactions on Robotics (TRO) 201

A Comprehensive Introduction of Visual-Inertial Navigation

In this article, a tutorial introduction to visual-inertial navigation(VIN) is presented. Visual and inertial perception are two complementary sensing modalities. Cameras and inertial measurement units (IMU) are the corresponding sensors for these two modalities. The low cost and light weight of camera-IMU sensor combinations make them ubiquitous in robotic navigation. Visual-inertial Navigation is a state estimation problem, that estimates the ego-motion and local environment of the sensor platform. This paper presents visual-inertial navigation in the classical state estimation framework, first illustrating the estimation problem in terms of state variables and system models, including related quantities representations (Parameterizations), IMU dynamic and camera measurement models, and corresponding general probabilistic graphical models (Factor Graph). Secondly, we investigate the existing model-based estimation methodologies, these involve filter-based and optimization-based frameworks and related on-manifold operations. We also discuss the calibration of some relevant parameters, also initialization of state of interest in optimization-based frameworks. Then the evaluation and improvement of VIN in terms of accuracy, efficiency, and robustness are discussed. Finally, we briefly mention the recent development of learning-based methods that may become alternatives to traditional model-based methods.Comment: 35 pages, 10 figure

Control Theory: A Mathematical Perspective on Cyber-Physical Systems

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently the control field is facing new challenges motivated by application domains that involve networks of systems. Examples are interacting robots, networks of autonomous cars or the smart grid. In order to address the new challenges posed by these application disciplines, the special focus of this workshop has been on the currently very active field of Cyber-Physical Systems, which forms the underlying basis for many network control applications. A series of lectures in this workshop was devoted to give an overview on current theoretical developments in Cyber-Physical Systems, emphasizing in particular the mathematical aspects of the field. Special focus was on the dynamics and control of networks of systems, distributed optimization and formation control, fundamentals of nonlinear interconnected systems, as well as open problems in control

Multi-Photon Multi-Channel Interferometry for Quantum Information Processing

This thesis reports advances in the theory of design, characterization and simulation of multi-photon multi-channel interferometers. I advance the design of interferometers through an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. This procedure effects an arbitrary $n_{s}n_{p}\times n_{s}n_{p}$ unitary matrix on the state of light in $n_{s}$ spatial and $n_{p}$ internal modes. I devise an accurate and precise procedure for characterizing any multi-port linear optical interferometer using one- and two-photon interference. Accuracy is achieved by estimating and correcting systematic errors that arise due to spatiotemporal and polarization mode mismatch. Enhanced accuracy and precision are attained by fitting experimental coincidence data to a curve simulated using measured source spectra. The efficacy of our characterization procedure is verified by numerical simulations. I develop group-theoretic methods for the analysis and simulation of linear interferometers. I devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU$(n)$ basis states, which reduce the canonical subgroup chain, for arbitrary $n$. The boson realizations are employed to construct $\mathcal{D}$-functions, which are the matrix elements of arbitrary irreducible representations, of SU$(n)$ in the canonical basis. I show that immanants of principal submatrices of a unitary matrix $T$ are a sum of the diagonal $\mathcal{D}(\Omega)$-functions of group element $\Omega$ over $t$ determined by the choice of submatrix and over the irrep $(\lambda)$ determined by the immanant under consideration. The algorithm for $\mathrm{SU}(n)$ $\mathcal{D}$-function computation and the results connecting these functions with immanants open the possibility of group-theoretic analysis and simulation of linear optics.Comment: PhD thesis submitted and defended successfully at the University of Calgary. This thesis is based on articles arXiv:1403.3469, arXiv:1507.06274, arXiv:1508.00283, arXiv:1508.06259 and arXiv:1511.01851 with co-authors. 145 pages, 31 figures, 11 algorithms and 4 tables. Comments are welcom

A Sub-optimal Algorithm to Synthesize Control Laws for a Network of Dynamic Agents

We study the synthesis problem of an LQR controller when the matrix describing the control law is constrained to lie in a particular vector space. Our motivation is the use of such control laws to stabilize networks of autonomous agents in a decentralized fashion; with the information flow being dictated by the constraints of a pre-specified topology. In this paper, we consider the finite-horizon version of the problem and provide both a computationally intensive optimal solution and a sub-optimal solution that is computationally more tractable. Then we apply the technique to the decentralized vehicle formation control problem and show that the loss in performance due to the use of the sub-optimal solution is not huge; however the topology can have a large effect on performance