Trajectories In Space Research Articles

From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part I. Stationary Hamiltonians

It is known that the Frenet–Serret apparatus of a space curve in three-dimensional Euclidean space determines the local geometry of curves. In particular, the Frenet–Serret apparatus specifies important geometric invariants, including the curvature and the torsion of a curve. It is also acknowledged in quantum information science that low complexity and high efficiency are essential features to achieve when cleverly manipulating quantum states that encode quantum information about a physical system. In this paper, we propose a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by dynamically evolving state vectors. Specifically, we propose a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced by a parallel-transported pure quantum state evolving unitarily under a stationary Hamiltonian specifying the Schrödinger equation. Our proposed constant curvature coefficient is given by the magnitude squared of the covariant derivative of the tangent vector [Formula: see text] to the state vector [Formula: see text] and represents a useful measure of the bending of the quantum curve. Our proposed constant torsion coefficient, instead, is defined in terms of the magnitude squared of the projection of the covariant derivative of the tangent vector [Formula: see text], orthogonal to both [Formula: see text] and [Formula: see text]. The torsion coefficient provides a convenient measure of the twisting of the quantum curve. Remarkably, we show that our proposed curvature and torsion coefficients coincide with those existing in the literature, although introduced in a completely different manner. Interestingly, not only we establish that zero curvature corresponds to unit geodesic efficiency during the quantum transportation in projective Hilbert space, but we also find that the concepts of curvature and torsion help enlighten the statistical structure of quantum theory. Indeed, while the former concept can be essentially defined in terms of the concept of kurtosis, the positivity of the latter can be regarded as a restatement of the well-known Pearson inequality that involves both the concepts of kurtosis and skewness in mathematical statistics. Finally, not only do we present illustrative examples with nonzero curvature for single-qubit time-independent Hamiltonian evolutions for which it is impossible to generate torsion, but we also discuss physical applications extended to two-qubit stationary Hamiltonians that generate curves with both nonzero curvature and nonvanishing torsion traced by quantum states with different degrees of entanglement, ranging from separable states to maximally entangled Bell states. In the Appendix C, we examine the different curvature and torsion characteristics of the three qubit [Formula: see text] and [Formula: see text] states under evolution by a quantum Heisenberg Hamiltonian.

The Impact of Second-Order Analytical Derivatives on Regularized Direct Multiple Shooting Method for Impulsive Spacecraft Trajectory Optimization

While the first-order derivatives of an objective function and constraints have often been analytically provided in the gradient-based optimization algorithms, the joint use of the second-order derivatives can further improve the computational efficiency and robustness. This paper implements the first- and second-order analytical derivatives in the direct multiple shooting-based, regularized method of minimizing the fuel expenditure for impulsive space trajectories. The high-order dynamical information expresses the Hessian matrix of the Lagrange function in the nonlinear programming problem. The result is an efficient tool for robustly computing fuel-optimal, multi-impulse trajectories in the regularized framework of removing singularities associated with null thrust impulses from the derivatives of the objective function. The computational performance is compared for the cases of optimizing impulsive transfers between cislunar periodic orbits within the regularized framework implementing the analytical derivatives in different ways over various initial guess trajectories and computational conditions. The results indicate the superiority of adopting both the first- and second-order analytical derivatives in terms of the efficiency and robustness.

Quantum Dynamics of Cavity–Bose–Einstein Condensates in a Gravitational Field

We theoretically studied the quantum dynamics of a cavity–Bose–Einstein condensate (BEC) system in a gravitational field, which is composed of a Fabry–Pérot cavity and a BEC. We also show how to deterministically generate the transient macroscopic quantum superposition states (MQSSs) of the cavity by the use of optomechanical coupling between the cavity field and the BEC. The quantum dynamics of the cavity–BEC system specifically include phase space trajectory dynamics, system excitation number dynamics, quantum entanglement dynamics, and quantum coherence dynamics. We found that the system performs increasingly complex trajectories for larger values of the Newtonian gravity parameter. Moreover, the number of phonon excitations of the system can be increased by coupling the cavity–BEC system to Newtonian gravity, which is analogous to an external direct current drive. The scattering of atoms inside the BEC affects the periodicity of the quantum dynamics of the system. We demonstrate a curious complementarity relation between the quantum entanglement and quantum coherence of cavity–BEC systems and found that the complementarity property can be sustained to some extent, despite being in the presence of the cavity decay. This phenomenon also goes some way to show that quantum entanglement and quantum coherence can be referred to together as quantum resources.

On the dynamics of spring-pendulum system: an overview of configuration space and phase space

The dynamics of the spring-pendulum system with two degrees of freedom were studied. The motion of this system is restricted to be in a vertical plane so that the chosen generalized coordinates are the increased length of the spring and the swing angle of pendulum . Hamiltonian of the system is obtained from the Legendre transformation of Lagrangian. Hamilton’s equation yields four differential equations that represent the dynamic of the system. The obtained results were visualized in configuration space and phase space trajectories. It is shown that generally the greater the initial swing angle, the more complex pattern will occur followed by the appearance of chaotic phenomena.

Geography and philosophy – space, time and being: the symbolism of the ideal man in the religious being from Athanasio of Alexandria in Vita Antoni

The philosophy of Western monasticism was born on the fringes of the official Church, especially when the Church had just become imperial. The search for a lived space referenced in the symbolism of the desert and for an ideal Christianity was also influenced by the hagiography of St. Anthony, in a broad space-time relationship. Athanasius of Alexandria, the Coptic bishop, when faced with the embryonic counterfeits of the friction between Church and State, projects in Vita Antoni an ideal presupposition of being Christian as humanity. This anthropological symbolism will crystallize in a trajectory of lived space and time, directly influencing Western civilization from an idealism that develops as a materiality and phenomenon in the deserts: the monasteries and the fuga mundi lifestyle. In the late period of the history of religions, a "father of the church" emerges who boldly describes the nuances of the pious character of the one who will one day be considered the father of monks, St. Anthony. This proposal analyzes the phenomenon of fuga mundi and its articulation with the locality of the monastery, from the perspective of Athanasius of Alexandria, with reference to the biography of Saint Anthony, observing its practical manifestations and how this enunciation occurred in the geohistory of Western Christianity, as well as identifying how this symbolic perspective influenced the Christian religious tradition. The methodology used in this work was applied research, using as a method the systematic analysis of the work of Athanasius of Alexandria and the hagiography present on St. Anthony, considering the evidence of socio-anthropological and symbolic aspects and the scientific convergence of geography, philosophy and the science of religion as an analytical proposal that unites categories such as space, time and being.

Optimal trajectory planning of robot energy consumption based on improved sparrow search algorithm

In order to reduce the energy consumption of the welding robot and ensure the cooperative movement of the robot joints, a trajectory planning method with optimal energy consumption based on improved sparrow search algorithm is proposed. Firstly, the trajectory planning model with optimal energy consumption is established based on the joint torque and angular velocity of the robot. To make the velocity, acceleration and jerk of each joint of the robot be bounded and continuous, the joint space trajectory is constructed with seventh degree B-spline curve. The total energy consumption of the robot is calculated by combining kinematic and dynamic parameters. On the basis of improved sparrow search algorithm, the time series corresponding to the optimal energy consumption is solved by using elite reverse learning, non-dominated sorting and Gaussian-Cauchy variation strategy, and then the optimal continuous motion trajectory of energy consumption is planned. The simulation results show that the proposed method can not only achieve continuous smooth control objective, but also effectively reduce energy consumption.

Geometrical Structure of Bifurcations during Spatial Decision-Making

Animals must constantly make decisions on the move, such as when choosing among multiple options, or “targets,” in space. Recent evidence suggests that this results from a recursive feedback between the (vectorial) neural representation of the targets and the resulting motion defined by this consensus, which then changes the egocentric neural representation of the options, and so on. Here we employ a simple model of this process both to explore how its dynamics accounts for the experimentally observed abruptly branching trajectories exhibited by animals during spatial decision-making, and to provide new insights into spatiotemporal computation. Essential neural dynamics, notably local excitation and long-range inhibition, are captured in our model via spin-system dynamics, with groups of Ising-spins representing neural “activity bumps” corresponding to target directions (as in a neural ring-attractor network, for example). Analysis, employing a novel “mean-field trajectory” approach, reveals the nature of the spontaneous symmetry breaking bifurcations in the model that result in literal bifurcations in trajectory space—and how it results in new geometric principles for spatiotemporal decision-making. We find that all bifurcation points, beyond the very first, fall on a small number of “bifurcation curves.” It is the spatial organization of these curves that is shown to be key to determining the shape of the trajectories, such as self-similar or space filling, exhibited during decision-making, irrespective of the trajectory's starting point. Furthermore, we find that a non-Euclidean (neural) representation of space (effectively an elliptic geometry) considerably reduces the number of bifurcation points in many geometrical configurations (including from an infinite number to only three), preventing endless indecision and promoting effective spatial decision-making. This suggests that a non-Euclidean neural representation of space may be expected to have evolved across species in order to facilitate spatial decision-making. Published by the American Physical Society 2024

Enhanced trajectory tracking for quadrotors: disturbance observer state feedback control.

This study investigates the dynamics and trajectory tracking of quadcopters by utilizing the Disturbance Observer-based Control (DOBC) algorithm. The quadcopter's dynamic model, which consists of six degrees of freedom, includes both disturbances and uncertainties in the model. The DOBC approach incorporates the disturbance model into the system by introducing it at the input variables. It then compensates for disturbances to achieve accurate tracking of different reference trajectories. The simulated trajectories span a range of motion, varying from simple straight paths to complex spiral paths. In order to verify and evaluate the efficacy of the suggested control technique, simulations are performed using MATLAB. The simulations conclusively show that the disturbance observer-based method effectively achieves the tracking of specified reference trajectories in three-dimensional space. The study highlights the effectiveness of the DOBC algorithm in reducing the effects of disturbances and uncertainties, thereby improving the quadcopter's capacity to accurately track various trajectories.

Characterization of Gauss–Markov stochastic sequences for mission analysis

In real scenarios, the spacecraft deviates from the intended paths owing to uncertainties in dynamics, navigation, and command actuation. Accurately quantifying these uncertainties is crucial for assessing the observability, collision risks, and mission viability. This issue is further magnified for CubeSats because they have limited control authority and thus require accurate dispersion estimates to avoid rejecting viable trajectories or selecting unviable ones. Trajectory uncertainties arise from random variables (e.g., measurement errors and drag coefficients) and processes (e.g., solar radiation pressure and low-thrust acceleration). Although random variables generally present minimal computational complexity, handling stochastic processes can be challenging because of their noisy dynamics. Nonetheless, accurately modeling these processes is essential, as they significantly influence the uncertain propagation of space trajectories, and an inadequate representation can result in either underestimation or overestimation of the stochastic characteristics associated with a given trajectory. This study addresses the gap in characterizing process uncertainties, represented as Gauss–Markov processes in mission analysis, by presenting models, evaluating derived quantities, and providing results on the impact of spacecraft trajectories. This study emphasizes the importance of accurately modeling random processes to properly characterize stochastic spacecraft paths.

Simulating human mobility with a trajectory generation framework based on diffusion model

Most mobility modeling methods are designed to solve specific tasks, leading to questions regarding their deficiency in generalizability. Inspired by the bloom of foundation models, we proposed a Trajectory Generation framework based on the Diffusion Model (TrajGDM) to capture the universal mobility pattern in a trajectory dataset by learning the trajectory generation process. The process is modeled as a step-by-step uncertainty-reducing process, in which a deep learning network with a novel training method is proposed to learn from the process. We compared the proposed trajectory generation method with six baselines on two public trajectory datasets. The results showed that the similarity between the generated and real trajectory movements measured by the Jensen-Shannon Divergence improved significantly on both datasets. Moreover, we applied zero-shot inferences on two basic trajectory tasks: trajectory prediction and trajectory reconstruction. The accuracy improved by a maximum of 25.6% on two tasks. The universal mobility pattern that is suitable for solving multiple trajectory tasks is verified, inferring the strong generalizability of our model. Finally, the study provides insights into artificial intelligence’s understanding of human mobility by exploring the way the model maps the trajectory in the latent space into reality.

A cyclic perspective on transient gust encounters through the lens of persistent homology

Large-amplitude gust encounters exhibit a range of separated flow phenomena, making them difficult to characterize using the traditional tools of aerodynamics. In this work, we propose a dynamical systems approach to gust encounters, viewing the flow as a cycle (or a closed trajectory) in state space. We posit that the topology of this cycle, or its shape and structure, provides a compact description of the flow, and can be used to identify coordinates in which the dynamics evolve in a simple, intuitive way. To demonstrate this idea, we consider flowfield measurements of a transverse gust encounter. For each case in the dataset, we characterize the full-state dynamics of the flow using persistent homology, a tool that identifies holes in point cloud data, and transform the dynamics to a reduced-order space using a nonlinear autoencoder. Critically, we constrain the autoencoder such that it preserves topologically relevant features of the original dynamics, or those features identified by persistent homology. Using this approach, we are able to transform six separate gust encounters to a three-dimensional latent space, in which each gust encounter reduces to a simple circle, and from which the original flow can be reconstructed. This result shows that topology can guide the creation of low-dimensional state representations for strong transverse gust encounters, a crucial step towards the modelling and control of aerofoil–gust interactions.

3D Space Trajectories and beyond: Abstract Art Creation with 3D Printing

3D Space Trajectories and beyond: Abstract Art Creation with 3D Printing

High-level aftereffects reveal the role of statistical features in visual shape encoding

Visual shape perception is central to many everyday tasks, from object recognition to grasping and handling tools.1,2,3,4,5,6,7,8,9,10 Yet how shape is encoded in the visual system remains poorly understood. Here, we probed shape representations using visual aftereffects-perceptual distortions that occur following extended exposure to a stimulus.11,12,13,14,15,16,17 Such effects are thought to be caused by adaptation in neural populations that encode both simple, low-level stimulus characteristics17,18,19,20 and more abstract, high-level object features.21,22,23 To tease these two contributions apart, we used machine-learning methods to synthesize novel shapes in a multidimensional shape space, derived from a large database of natural shapes.24 Stimuli were carefully selected such that low-level and high-level adaptation models made distinct predictions about the shapes that observers would perceive following adaptation. We found that adaptation along vector trajectories in the high-level shape space predicted shape aftereffects better than simple low-level processes. Our findings reveal the central role of high-level statistical features in the visual representation of shape. The findings also hint that human vision is attuned to the distribution of shapes experienced in the natural environment.

Decomposition of Equations of Nonlinear Affine Control Systems and its Application to the Synthesis of Regulators

The article deals with the decomposition of nonlinear differential equations based on the group-theoretic approach. At the beginning, the decomposition of differential equations of linear systems using a transition matrix of state is presented, and then, based on the theory of continuous groups (Lie groups), the process of decomposition of differential equations of nonlinear systems is shown. The decomposition approach is based on the isomorphism theorem of the space of vector fields and Lie derivatives, which allows us to consider vector fields as differential operators of smooth functions. A formula is derived about the adjoin representation of a Lie group in its Lie algebra, which actually determines the finding of a vector field that characterizes the interaction of two or more vector fields. The Lie algebra of derivatives makes it possible to determine the infinitesimal action of the Lie group, i.e. the linearization of this action is carried out (transformation of the points of the trajectory space of the original system in a small neighborhood). Decomposition allows, as in the linear case, to separate the finding of an action (only locally) of a group of transformations from the transformed points themselves. For linear systems, this separation is global. It is also shown that the decomposition of linear equations is a particular case of the decomposition of nonlinear equations. An algorithm of the method of model predictive control with Gramian weighting using this decomposition is presented. A practical example of decomposition and application of the model predictive control for stabilization of a nonstationary nonlinear system is considered.

Orthogonal Mixed-Effects Modeling for High-Dimensional Longitudinal Data: An Unsupervised Learning Approach.

The linear mixed-effects model is commonly utilized to interpret longitudinal data, characterizing both the global longitudinal trajectory across all observations and longitudinal trajectories within individuals. However, characterizing these trajectories in high-dimensional longitudinal data presents a challenge. To address this, our study proposes a novel approach, Unsupervised Orthogonal Mixed-Effects Trajectory Modeling (UOMETM), that leverages unsupervised learning to generate latent representations of both global and individual trajectories. We design an autoencoder with a latent space where an orthogonal constraint is imposed to separate the space of global trajectories from individual trajectories. We also devise a cross-reconstruction loss to ensure consistency of global trajectories and enhance the orthogonality between representation spaces. To evaluate UOMETM, we conducted simulation experiments on images to verify that every component functions as intended. Furthermore, we evaluated its performance and robustness using longitudinal brain cortical thickness from two Alzheimer's disease (AD) datasets. Comparative analyses with state-of-the-art methods revealed UOMETM's superiority in identifying global and individual longitudinal patterns, achieving a lower reconstruction error, superior orthogonality, and higher accuracy in AD classification and conversion forecasting. Remarkably, we found that the space of global trajectories did not significantly contribute to AD classification compared to the space of individual trajectories, emphasizing their clear separation. Moreover, our model exhibited satisfactory generalization and robustness across different datasets. The study shows the outstanding performance and potential clinical use of UOMETM in the context of longitudinal data analysis.

Computation of the Largest Lyapunov Exponent Using SPICE-Like Programs

This paper presents a novel method for calculating the positive largest Lyapunov exponent (LLE) using only SPICE-like programs. The LLE is a measure of the rate of divergence of nearby trajectories in phase space and is an important indicator of chaotic behavior. The proposed method calculates the LLE directly from systems represented in schematic circuits, inspired by interval extensions that exploit divergent trajectories to calculate the lower-bound error. The exponent is obtained from the slope of the line derived from the lower-bound error, using a differential amplifier to quantify it, along with a straightforward linear fit to the logarithm of the divergence of two circuits’ trajectories. This approach is unique as it requires only user skills in SPICE-like programs, unlike most other approaches, providing a simple, efficient, and accurate way to analyze and characterize chaotic dynamics. Results demonstrate that the suggested method successfully estimated the LLE for two well-known systems.

Движение гиростата под действием следящего момента силы

The gyrostat moves around a fixed point under the forces influence in pseudo-Euclidean space with signature (3,1). The resulting moment of the external forces system and the gyrostatic moment are specified relative to the coordinate orthoreper, invariably associated with the gyrostat carrier, and are constant. Three types of gyrostat motion are given, in each of which the resulting force vector-moment is collinear to of the main axis of the gyrostat inertia, and the gyrostatic moment is orthogonal to this axis. The vector manifold of the gyrostat angular velocities and the field of its phase trajectories in the state space are found with the parametric integration method using. Integration of the gyrostat motion equations is carried out in quadratures and elliptic functions of time.

GraphMM: Graph-Based Vehicular Map Matching by Leveraging Trajectory and Road Correlations

Map matching of sparse vehicle trajectories is a fundamental problem in location-based services, such as traffic flow analysis and vehicle routing. Existing literature mainly relies on sequence-to-sequence (Seq2Seq) models to capture the intra-trajectory correlation of an input trajectory and to sequentially predict the matched road segments. Due to the limited expressive capability of sequential models, these methods fall short of extracting inter-trajectory and trajectory-road correlations as well as correlation between road segments. We present <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GraphMM</b> , a graph-based approach that explicitly utilizes all aforementioned correlations. Our model exploits the graph nature of map matching and incorporates graph neural networks and conditional models to leverage both road and trajectory graph topology, while manages to align road segments and trajectories in latent space. We formally analyze the expressive power of our model in capturing various correlations and propose efficient algorithms for model training and inference. In particular, our optimization techniques dramatically reduce the computational complexity, making our model feasible on datasets with thousands of road segments. Extensive experiments show that our model significantly enhances prediction accuracy, while improving training and inference efficiency by up to an order of magnitude over both the industrial implementation of the hidden Markov model and state-of-the-art Seq2Seq-based methods.

An Upper Limb Exoskeleton Motion Generation Algorithm Based on Separating Shoulder and Arm Motion.

Many rehabilitation exoskeletons have been used in the field of stroke rehabilitation. Generating human-like motion is necessary for exoskeletons to help patients perform activities of daily living (ADL) while maintaining interaction quality and ergonomics. However, most of the current motion generation algorithms utilize inverse kinematics (IK) to solve the final configuration before generation, and do not consider the movement of shoulder girdle. Separately considering the shoulder girdle motion and arm motion, this paper proposes an algorithm integrated IK to generate human-like motion. The arm moves towards the target with a bell-shaped velocity in the absence of the final configuration, and the shoulder girdle maintain natural passive motion. Moreover, the generated motion can be mapped to the configuration space of exoskeletons. Compared with the experimental data collected using a motion capture system, the values of RMSE and HPDI of the generated wrist trajectory in the task space are within 0.2 and 0.17, respectively, while those of RMSE in the joint space are within 15 deg, which demonstrates the human-like nature of the generated motion.

Spatiotemporal dynamics of subarctic intertidal macrobenthos: going their own ways

Abstract Intertidal macrobenthos at the small Chernaya Bight (the White Sea) was surveyed six times during 1993–2018 in order to study spatiotemporal variability. Distributions of sediments and macrophytes were highly variable in both space and time, as were most macrofaunal community attributes. Biomass slightly increased with time, while no long-term trends were found in total abundance, diversity, or functional structure. All community attributes were patchily distributed across the beach, and their patterns were not spatially autocorrelated and poorly associated with sediment properties, but changed considerably from year to year. Temporal changes in the community composition were considerable but less substantial compared with the spatial variations. The overall dynamics of species structure did not show any regular trend-like pattern but formed quasicyclic trajectories in ordination space, with nondirectional, spatially noncorrelated fluctuations around some relatively stable state. Comparison with two other neighbouring intertidal sites, studied annually in 1987–2017, showed that macrofauna at every site had similar average biomasses and common dominant species; however, the communities maintained their specificity in structure and exhibited distinct types of dynamics. In particular, the communities demonstrated different long-term trends in total biomass and diversity and followed their own paths in dynamics, appearing as differently oriented interannual trajectories. Nine most abundant species revealed no significant among-site correlations in abundance, and only two bivalve species showed good intersite agreement in dynamics of biomass. We suggest that local benthic communities are largely influenced by site-specific environmental conditions, resulting in independent and even opposite patterns of dynamics in neighbouring localities.