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Submersible vehicles may be regarded as complex systems because of their complex interaction with the surrounding fluid. This paper presents a mathematical model of a submersible vehicle formulated in a coordinate-free manner through the language of Lie groups and Lie algebras. The d'Alembert virtual-work principle was applied in conjunction with the minimal-action principle for a rigid body in order to incorporate into the mathematical model external influences such as fluid-current-induced deflection and control inputs. Such a method from mathematical physics can also take into consideration how a vehicle interacts with the fluid it is immersed in under the form of added (or virtual) mass. The resulting equations of motion were given over the Lie group of three-dimensional rotations as (non-pure) Euler-Poincaré relations. A numerical simulation technique based on Lie-group integrators was also briefly recalled and deployed to simulate the behavior of such mathematical model of an existing, academic-design-type submersible vehicle.

The aim of this research paper is to propose a framework to model, simulate and control the motion of a small spacecraft in the proximity of a space station. In particular, rendezvous in the presence of physical obstacles is tackled by a virtual potential theory within a modern manifold calculus setting and simulated numerically. The roto-translational motion of a spacecraft as well as the control fields are entirely formulated through a coordinate-free Lie group-type formalism. Likewise, the proposed control strategies are expressed in a coordinate-free setting through structured control fields. Several numerical simulations guide the reader through an evaluation of the most convenient control strategy among those devised in the present work.

The present paper describes, in a theoretical fashion, a variational approach to formulate fourth-order dynamical systems on differentiable manifolds on the basis of the Hamilton–d’Alembert principle of analytic mechanics. The discussed approach relies on the introduction of a Lagrangian function that depends on the kinetic energy and the covariant acceleration energy, as well as a potential energy function that accounts for conservative forces. In addition, the present paper introduces the notion of Rayleigh differential form to account for non-conservative forces. The corresponding fourth-order equation of motion is derived, and an interpretation of the obtained terms is provided from a system and control theoretic viewpoint. A specific form of the Rayleigh differential form is introduced, which yields non-conservative forcing terms assimilable to linear friction and jerk-type friction. The general theoretical discussion is complemented by a brief excursus about the numerical simulation of the introduced differential model.

The purpose of the present research endeavor is to propose a novel control strategy for a DC-DC electrical converter realized as a switched circuit. The present endeavor is based on an early work by Leonard and Krishnaprasad where a prototypical DC-DC converter was modeled as a state space dynamical system and controlled by an open-loop strategy based on Lie group theory. In this work, we shall rather introduce a closed-loop control strategy based on maximum convergence rate, study some features of the novel method, and illustrate its behavior by numerical simulations. A numerical comparison of the two methods complements the paper. The numerical comparison shows how the proposed feedback control method outperforms the static one in terms of convergence rate and resiliency against parameters mismatch.

The present paper treats a black-box estimation of the three independent parameters of a reciprocal lossy two-port network whose terminals are supposed to be accessible to an impedance measurement device. The discussed estimation method is based on the availability of a number of data pairs made of external load admittances paired to equivalent external admittances affected by measurement errors. The proposed method is framed as a squared-estimation-error minimization problem that leads to a system of three nonlinear equations in the three unknown parameters. A key observation is, however, that a core subsystem of two equations may be turned exactly to a linear form and hence may be solved in closed form. The purely real-valued case is treated first since it serves to clarify the optimization problem at hand and the structure of its solution. In the purely real-valued case, a statistical analysis is carried out as well, which affords the evaluation of the effects of the measurement errors. The results of the statistical analysis afford quantifying the dependence of the estimation errors from the number of samples and from the variance of the measurement errors. Subsequently, the full complex-valued case is treated. Results of numerical simulations complement and illustrate the theoretical findings. The obtained numerical results confirm the statistical analysis and that the proposed external identification method is effective.

The problem addressed in the present paper is the design of a controller based on an evolutionary neural network for autonomous flight in quadrotor systems. The controller’s objective is to govern the quadcopter in such a way that it reaches a specific position, bearing on attitude limitations during flight and upon reaching a target. Given the complex nature of quadcopters, an appropriate neural network architecture and a training algorithm were designed to guide a quadcopter toward a target. The designed controller was implemented as a single multi-layer perceptron. On the basis of the quadcopter’s current state, the developed neurocontroller produces the correct rotor speed values, optimized in terms of both attitude-limitation compliance and speed. The neural network training was completed using a custom evolutionary algorithm whose design put particular emphasis on the cost function’s definition. The developed neurocontroller was tested in simulation to drive a quadcopter to autonomously follow a complex path. The obtained simulated results show that the neurocontroller manages to effortlessly follow several types of paths with adequate precision while maintaining low travel times.

The present paper deals with the external identification of a reciprocal, special passive, 2n-port network under measurement uncertainties. In the present context, the multiport model is represented by an admittance matrix and the condition that the network is ‘reciprocal special passive’ refers to the assumption that the real part of the admittance matrix is symmetric and positive-definite. The key point is to reformulate the identification problem as a matrix optimization program over the matrix manifold S+(2n)×S(2n). The optimization problem requires a least-squares criterion function designed to cope with over-determinacy due to the incoherent data pairs whose cardinality exceeds the problem’s number of degrees of freedom. The present paper also proposes a numerical solution to such an optimization problem based on the Riemannian-gradient steepest descent method. The numerical results show that the proposed method is effective as long as reasonable measurement error levels and problem sizes are being dealt with.

The present tutorial paper constitutes the second of a series of tutorials on manifold calculus with applications in system theory and control. The aim of the present tutorial, in particular, is to explain and illustrate some key concepts in manifold calculus such as covariant derivation and manifold curvature. Such key concepts are then applied to the formulation, to the control, and to the analysis of non-linear dynamical systems whose state-space are smooth (Riemannian) manifolds. The main flow of exposition is enriched by a number of examples whose aim is to clarify the notation used and the main theoretical findings through practical calculations.

The aim of the present research work is to design a control law for a quadcopter drone based on the Virtual Attractive-Repulsive Potentials (VARP) theory. VARP theory, originally designed to enable path following by a small wheeled robot, will be tailored to control a quadcopter drone, hence allowing such device to learn flight planning. The proposed strategy combines an instance of VARP method to control a drone’s attitude (SO(3)-VARP) and an instance of VARP method to control a drone’s spatial location (R3-VARP). The resulting control strategy will be referred to as double-VARP method, which aims at making a drone follow a predefined path in space. Since the model of the drone as well as the devised control theory are formulated on a Lie group, their simulation on a computing platform is performed through a numerical analysis method specifically designed for these kinds of numerical simulations. A numerical simulation analysis is used to assess the salient features of the proposed regulation theory. In particular, resilience against shock-type disturbances are assessed numerically.

The present paper is concerned with an instance of automatic control for autonomous vehicles based on the theory of virtual attractive-repulsive potentials (VARP). The first part of this paper presents a review of the VARP control theory as developed specifically by B. Nguyen, Y.-L. Chuang, D. Tung, C. Hsieh, Z. Jin, L. Shi, D. Marthaler, A. Bertozzi and R. Murray, in the paper ‘Virtual attractive-repulsive potentials for cooperative control of second order dynamic vehicles on the Caltech MVWT’, which appeared in the Proceedings of the 2005 American Control Conference, (Portland, OR, USA) held in June 2005 (pp. 1084–1089). The aim of the first part of the present paper is to recall the mathematical and logical steps that lead to controlling an autonomous robot by a VARP-based control theory. The concepts recalled in the first part of the present paper, with special reference to the physical interpretation of the terms in the developed control field, serve as the starting point to develop a more convoluted control theory for (second-order) dynamical systems whose state spaces are (possibly high-dimensional) curved manifolds. The second part of this paper is, in fact, devoted to extending the classical VARP control theory to regulate dynamical systems whose state spaces possess the mathematical structure of smooth manifolds through manifold calculus. Manifold-type state spaces present a high degree of symmetry, due to mutual non-linear constraints between single physical variables. A comprehensive set of numerical experiments complements the review of the VARP theory and the theoretical developments towards its extension to smooth manifolds.

The present paper proposes a novel kernel adaptive filtering algorithm, where each Gaussian kernel is parameterized by a center vector and a symmetric positive definite (SPD) precision matrix, which is regarded as a generalization of scalar width parameter. In fact, different from conventional kernel adaptive systems, the proposed filter is structured as a superposition of non-isotropic Gaussian kernels, whose non-isotropy makes the filter more flexible. The adaptation algorithm will search for optimal parameters in a wider parameter space. This generalization brings the need of special treatment of parameters that have a geometric structure. In fact, the main contribution of this paper is to establish update rules for precision matrices on the Lie group of SPD matrices in order to ensure their symmetry and positive-definiteness. The parameters of this filter are adapted on the basis of a least-squares criterion to minimize the filtering error, together with an ℓ1-type regularization criterion to avoid overfitting and to prevent the increase of dimensionality of the dictionary. Experimental results confirm the validity of the proposed method.

The aim of the present tutorial paper is to recall notions from manifold calculus and to illustrate how these tools prove useful in describing system-theoretic properties. Special emphasis is put on embedded manifold calculus (which is coordinate-free and relies on the embedding of a manifold into a larger ambient space). In addition, we also consider the control of non-linear systems whose states belong to curved manifolds. As a case study, synchronization of non-linear systems by feedback control on smooth manifolds (including Lie groups) is surveyed. Special emphasis is also put on numerical methods to simulate non-linear control systems on curved manifolds. The present tutorial is meant to cover a portion of the mentioned topics, such as first-order systems, but it does not cover topics such as covariant derivation and second-order dynamical systems, which will be covered in a subsequent tutorial paper.

Helicopters are extraordinarily complex mechanisms. Such complexity makes it difficult to model, simulate and pilot a helicopter. The present paper proposes a mathematical model of a fantail helicopter type based on Lie-group theory. The present paper first recalls the Lagrange–d’Alembert–Pontryagin principle to describe the dynamics of a multi-part object, and subsequently applies such principle to describe the motion of a helicopter in space. A good part of the paper is devoted to the numerical simulation of the motion of a helicopter, which was obtained through a dedicated numerical method. Numerical simulation was based on a series of values for the many parameters involved in the mathematical model carefully inferred from the available technical literature.

The aim of the present paper is to improve an existing blind image deblurring algorithm, based on an independent component learning paradigm, by manifold calculus. The original technique is based on an independent component analysis algorithm applied to a set of pseudo-images obtained by Gabor-filtering a blurred image and is based on an adapt-and-project paradigm. A comparison between the original technique and the improved method shows that independent component learning on the unit hypersphere by a Riemannian-gradient algorithm outperforms the adapt-and-project strategy. A comprehensive set of numerical tests evidenced the strengths and weaknesses of the discussed deblurring technique.

The present paper deals with nonlinear, non-monotonic data regression. This paper introduces an efficient algorithm to perform data transformation from non-monotonic to monotonic to be paired with a statistical bivariate regression method. The proposed algorithm is applied to a number of synthetic and real-world non-monotonic data sets to test its effectiveness. The proposed novel non-isotonic regression algorithm is also applied to a collection of data about strontium isotope stratigraphy and compared to a LOWESS regression tool.

The present paper recalls a formulation of non-conservative system dynamics through the Lagrange−d’Alembert principle expressed through a generalized Euler−Poincaré form of the system equation on a Lie group. The paper illustrates applications of the generalized Euler−Poincaré equations on the rotation groups to a gyrostat satellite and a quadcopter drone. The numerical solution of the dynamical equations on the rotation groups is tackled via a generalized forward Euler method and an explicit Runge−Kutta integration method tailored to Lie groups.

The present article studies the problem of computing empirical means on pseudo-orthogonal groups. To design numerical algorithms to compute empirical means, the pseudo-orthogonal group is endowed with a pseudo-Riemannian metric that affords the computation of the exponential map in closed forms. The distance between two pseudo-orthogonal matrices, which is an essential ingredient, is computed by both the Frobenius norm and the geodesic distance. The empirical-mean computation problem is solved via a pseudo-Riemannian-gradient-stepping algorithm. Several numerical tests are conducted to illustrate the numerical behavior of the devised algorithm.

The lurch index has recently been introduced in applied kinematics as an integral descriptor of the fluency of the motion of a rigid body in space. It may be defined in different versions, according to the component of motion under investigation. In the present paper, we analyze a rotational lurch index, which describes the fluency of the spin component of motion and whose value depends, through involved relations, on the dynamics of three canonical descriptors of the orientation of a rigid body in space. The aim of the present paper is to offer a closed-form expression of the instantaneous component of the rotational lurch, which leads to the namesake index upon integration and normalization. The closed form of the index is, then, used to evaluate its practical calculation, based on numerical approximations on a number of data sets.

Statistical bivariate numerical modeling is a method to infer an empirical relationship between unpaired sets of data based on statistical distributions matching. In the present paper, a novel efficient numerical algorithm is proposed to perform bivariate numerical modeling. The algorithm is then applied to correlate glomerular filtration rate to serum creatinine concentration. Glomerular filtration rate is adopted in clinical nephrology as an indicator of kidney function and is relevant for assessing progression of renal disease. As direct measurement of glomerular filtration rate is highly impractical, there is considerable interest in developing numerical algorithms to estimate glomerular filtration rate from parameters which are easier to obtain, such as demographic and `bedside’ assays data.

This paper presents an improved version of a statistical trivariate modeling algorithm introduced in a short Letter by the first author. This paper recalls the fundamental concepts behind the proposed algorithm, evidences its criticalities and illustrates a number of improvements which lead to a functioning modeling algorithm. The present paper also illustrates the features of the improved statistical modeling algorithm through a comprehensive set of numerical experiments performed on four synthetic and five natural datasets. The obtained results confirm that the proposed algorithm is able to model the considered synthetic and the natural datasets faithfully.

The present contribution focuses on the estimation of the Cartesian kinematic jerk of the hips’ orientation during a full three-dimensional movement in the context of enabling eHealth applications of advanced mathematical signal analysis. The kinematic jerk index is estimated on the basis of gyroscopic signals acquired offline through a smartphone. A specific free mobile application is used to acquire the gyroscopic signals and to transmit them to a personal computer through a wireless network. The personal computer elaborates the acquired data and returns the kinematic jerk index associated with a motor task. A comparison of the kinematic jerk index value on a number of data sets confirms that such index can be used to evaluate the fluency of hips orientation during motion. The present research confirms that the proposed gyroscopic data acquisition/processing setup constitutes an inexpensive and portable solution to motion fluency analysis. The proposed data-acquisition and data-processing setup may serve as a supporting eHealth technology in clinical bio-mechanics as well as in sports science.

The chaos-based optimization algorithm (COA) is a method to optimize possibly nonlinear complex functions of several variables by chaos search. The main innovation behind the chaos-based optimization algorithm is to generate chaotic trajectories by means of nonlinear, discrete-time dynamical systems to explore the search space while looking for the global minimum of a complex criterion function. The aim of the present research is to investigate the numerical properties of the COA, both on complex optimization test-functions from the literature and on a real-world problem, to contribute to the understanding of its global-search features. In addition, the present research suggests a refinement of the original COA algorithm to improve its optimization performances. In particular, the real-world optimization problem tackled within the paper is the estimation of six electro-mechanical parameters of a model of a direct-current (DC) electrical motor. A large number of test results prove that the algorithm achieves an excellent numerical precision at a little expense in the computational complexity, which appears as extremely limited, compared to the complexity of other benchmark optimization algorithms, namely, the genetic algorithm and the simulated annealing algorithm.

Capsules were prepared from chitosan (QTS)-poly(vinyl alcohol) (PVA) blend by saline coacervation and then by formalization. A adsorbent based on chitosan, insoluble on acid solution, was obtained. The morphology, average diameters of QTS/PVA capsules and their pores were studied by using scanning electron microscopy. The entrapment-adsorption of dimethylglioxime and ethylenediaminetetracetate by the capsules were studied. The removal of the ion nickel (II) and copper (II), was more effective than by using unloaded capsules.

Algorithms for independent component analysis (ICA) based on information-theoretic criteria optimization over differential manifolds have been devised over the last few years. The principles informing their design lead to various classes of learning rules, including the fixed-point and the geodesic-based ones. Such learning algorithms mainly differ by the way in which single learning steps are effected in the neural system’s parameter space, i.e. by the action that a connection variable is moved by in the parameter space toward the optimal connection pattern. In the present paper, we introduce a new class of learning algorithms by recalling from the literature on differential geometry the concept of mapping onto manifolds, which provides a general way of acting upon a neural system’s connection variable in order to optimize the learning criteria. The numerical behavior of the introduced learning algorithms is illustrated and compared with experiments carried out on mixtures of statistically-independent signals.

The aim of this paper is to examine the performance of an independent component analysis algorithm based on neural networks applied to the solution of an electrical engineering problem related to non-destructive evaluation of conductive objects. The proposed application is assessed through computer experiments carried out on real-world data, which prove the usefulness of this non-destructive evaluation technique.

Abstract This review examines the role of defective carbon‐based electrodes in sodium‐ion and vanadium flow batteries. Methods for introducing defects into carbon structures are explored and their effectiveness in improving electrode performance is demonstrated. In sodium‐based systems, research focuses primarily on various precursor materials and heteroatom doping to optimise hard carbon electrodes. Defect engineering increases interlayer spacing, porosity, and changes the surface chemistry, which improves sodium intercalation and reversible capacities. Heteroatom functionalisation and surface modification affect solid electrolyte interface formation and coulombic efficiencies. For flow batteries, post‐fabrication electrode enhancement methods produce defects to improve electrode kinetics, although these methods often introduce oxygen functional groups as well, making isolation of defect effects difficult. Continued research efforts are key to developing carbon‐based electrodes that can meet the unique challenges of future battery systems.