Stable Heteroclinic Channel-Based Movement Primitives: Tuning Trajectories Using Saddle Parameters

Movement primitives are a well-established approach to robot motion planning, as they offer a modular basis for robot motion planning. Dynamic movement primitives (DMPs) are a popular control framework based on nonlinear differential equations which, when scaled in time, produce a smooth kinematic movement plan. In this letter, we introduce an adjusted movement primitive framework which replaces the stable equilibria of DMPs with saddle points. These saddle points are linked together in stable heteroclinic channels (SHCs) which are in turn part of dynamical systems called stable heteroclinic networks. SHC-based movement primitives (SMPs) form a framework where the weights of the kernel functions have spatial significance in the task space. In this letter, we show that SMPs and DMPs perform comparably according to a kinematic-based cost function. We also show that SMP kernel weights follow a given trajectory when plotted in the task space. The plotted kernel weights provide an intuitive tool for managing the controller.

Simulated Control of an Aquatic Serpentine Robot with Stable Heteroclinic Channels

The increasing number and variety of spacecraft that are expected to operate within cislunar space and other multi-body gravitational environments throughout the solar system necessitates the continued development of strategies for rapid trajectory design and design space exploration. In the field of robotics, similar needs have been addressed using motion primitives that capture the fundamental building blocks of motion and are used to rapidly construct complex paths. Inspired by this concept, this paper leverages motion primitives to construct a framework for rapid and informed spacecraft trajectory design in a multi-body gravitational system. First, motion primitives of fundamental solutions, e.g., selected periodic orbits and their stable and unstable manifolds, are generated via clustering to form a discrete summary of segments of the phase space. Graphs of motion primitives are then constructed and searched to produce primitive sequences that form candidate initial guesses for transfers of distinct geometries. Continuous transfers are computed from each initial guess using multi-objective constrained optimization and collocation. This approach is demonstrated by constructing an array of geometrically distinct transfers between libration point orbits in the Earth-Moon circular restricted three-body problem with impulsive maneuvers.

Linear stability analysis of helium-filled cavities in SiC

Traitement de la tuberculose active : étude comparative de 2 schémas thérapeutiques utilisant les drogues séparées ou l’association de drogues fixes

Collaborative robotics is one of the fastest developing but at the same time one of the most challenging fields in robotics. The reason for the latter is that a good robot collaborator should precisely understand the intentions of the human coworker. In this paper, human-robot collaboration is addressed by the application of a dynamical phase state system guided by stable heteroclinic channel networks to the motion of a humanoid robot. With this approach, a person can control the underlying dynamical system by applying forces to the grippers of the humanoid robot and this way guide the robot. The robot motions presented in this paper are lifting up the forearms and squatting, but the dynamical model can be further extended to incorporate an arbitrary amount of motions. The presented approach, therefore, provides a suitable way for the realization of efficient human-robot collaboration and should be further explored and developed.

Dirac Bracket for Pedestrians

Relaxation of few-body quantum systems can strongly depend on the initial state when the system's semiclassical phase space is mixed, i.e., regions of chaotic motion coexist with regular islands. In recent years, there has been much effort to understand the process of thermalization in strongly interacting quantum systems that often lack an obvious semiclassical limit. Time-dependent variational principle (TDVP) allows to systematically derive an effective classical (nonlinear) dynamical system by projecting unitary many-body dynamics onto a manifold of weakly-entangled variational states. We demonstrate that such dynamical systems generally possess mixed phase space. When TDVP errors are small, the mixed phase space leaves a footprint on the exact dynamics of the quantum model. For example, when the system is initialized in a state belonging to a stable periodic orbit or the surrounding regular region, it exhibits persistent many-body quantum revivals. As a proof of principle, we identify new types of "quantum many-body scars", i.e., initial states that lead to long-time oscillations in a model of interacting Rydberg atoms in one and two dimensions. Intriguingly, the initial states that give rise to most robust revivals are typically entangled states. On the other hand, even when TDVP errors are large, as in the thermalizing tilted-field Ising model, initializing the system in a regular region of phase space leads to slowdown of thermalization. Our work establishes TDVP as a method for identifying interacting quantum systems with anomalous dynamics in arbitrary dimensions. Moreover, the mixed-phase space classical variational equations allow to find slowly-thermalizing initial conditions in interacting models. Our results shed light on a link between classical and quantum chaos, pointing towards possible extensions of classical Kolmogorov-Arnold-Moser theorem to quantum systems.

The Weyl representation in classical and quantum mechanics

<p>Weather and climate are well known exemplars of chaotic systems exhibiting extreme sensitivity to initial conditions. Initial condition errors are subject to exponential growth on average, but the rate and the characteristic of such growth is highly state dependent. In an ideal setting where the degree of predictability of the system is known in real-time, it may be possible and beneficial to take adaptive measures. For instance a local decrease of predictability may be counteracted by increasing the time- or space-resolution of the model computation or the ensemble size in the context of ensemble-based data assimilation or probabilistic forecasting.</p><p>Local Lyapunov exponents (LLEs) describe growth rates along a finite-time section of a system trajectory. This makes the LLEs the ideal quantities to measure the local degree of predictability, yet a main bottleneck for their real-time use in  operational scenarios is the huge computational cost. Calculating LLEs involves computing a long trajectory of the system, propagating perturbations with the tangent linear model, and repeatedly orthogonalising them. We investigate if machine learning (ML) methods can estimate the LLEs based only on information from the system’s solution, thus avoiding the need to evolve perturbations via the tangent linear model. We test the ability of four algorithms (regression tree, multilayer perceptron, convolutional neural network and long short-term memory network) to perform this task in two prototypical low dimensional chaotic dynamical systems. Our results suggest that the accuracy of the ML predictions is highly dependent upon the nature of the distribution of the LLE values in phase space: large prediction errors occur in regions of the attractor where the LLE values are highly non-smooth.  In line with classical dynamical systems studies, the neutral LLE is more difficult to predict. We show that a comparatively simple regression tree can achieve performance that is similar to sophisticated neural networks, and that the success of ML strategies for exploiting the temporal structure of data depends on the system dynamics.</p>

Most dynamical systems in real-life applications are usually nonlinear and nonstationary, such as biomechanical systems, neuroscience, healthcare, economics, and engineering. Nowadays, the availability of data makes it possible to investigate and discover the hidden dynamics of target systems. However, numerous concerns arise in modeling and analysis of such systems, including complex system structure, data-inherent uncertainty in observations, poor interpretability, etc. To this end, we will investigate dynamical behaviors by proposing data-driven machine learning and dynamical system techniques. Specifically, we present various approaches for extracting characteristics or patterns that have good interpretability in various complex dynamical system applications.In the first study, we consider a pattern recognition problem in biomechanical system using multivariate human kinematic data and propose data-driven analytical methods to model human movement. We first perform recurrence quantification analysis on the phase space of the kinematic system to visualize and evaluate the recurrence behavior of the folded phase space trajectory. However, this method could miss out the relationship between variables. We therefore introduce a new modeling approach based on the self-expressive assumption to discover the interactions between variables and characterize the patterns of human motion. Our model measures correlations between variables by underlying linear subspaces, which can be further extended to nonlinear subspaces. Based on these correlations, our model improves the classification accuracy by 20% - 30% over other standard state-of-art correlation measurements and clustering methods. In the second study, we consider a specific neuroscience problem, seizure detection, to recognize abnormal brain states by learning from highly nonlinear electroencephalography. Based on dynamical systems theory, Koopman operator theory, and dynamic mode decomposition, we project the complex brain dynamics into an infinite-dimensional linear space, where the intrinsic linear dynamic shares the same dynamical behavior as the original nonlinear dynamical system. We propose a new system measurement to unify the projections in different Koopman spaces. Our measurement indicates the stability of the dynamical system, which represents the system state trajectory with respect to the corresponding attractor. We demonstrate the effectiveness of our approach with a seizure onset detection task. In this case study, our model gives high accuracy with a low false-positive rate. In the last study, we consider dynamical systems with multiple modalities. We design a Koopman-based deep learning network to identify and simplify the intrinsic nonlinear dynamics from different sources/observations of a multi-modal dynamical system. Since these different modalities describe the same system, they jointly share the same intrinsic dynamics. We develop an autoencoder framework with custom loss functions to find proper project functions/operators and to obtain the intrinsic Koopman coordinates. Our framework can reconstruct the original system states based on the joint dynamics. Once the intrinsic linear dynamics is learned, future system states can be predicted by the evolution in the Koopman space. This approach allows us to uncover and linearize the hidden nonlinear dynamics, which enables linear dynamical tools, for example, system control, spectrum analysis, to analyze complex dynamical systems.--Author's abstract

In physics, experiments ultimately inform us about what constitutes a good theoretical model of any physical concept: physical space should be no exception. The best picture of physical space in Newtonian physics is given by the configuration space of a free particle (or the center of mass of a closed system of particles). This configuration space (as well as phase space) can be constructed as a representation space for the relativity symmetry. From the corresponding quantum symmetry, we illustrate the construction of a quantum configuration space, similar to that of quantum phase space, and recover the classical picture as an approximation through a contraction of the (relativity) symmetry and its representations. The quantum Hilbert space reduces into a sum of one-dimensional representations for the observable algebra, with the only admissible states given by coherent states and position eigenstates for the phase and configuration space pictures, respectively. This analysis, founded firmly on known physics, provides a quantum picture of physical space beyond that of a finite-dimensional manifold and provides a crucial first link for any theoretical model of quantum space-time at levels beyond simple quantum mechanics. It also suggests looking at quantum physics from a different perspective.

As remarked the Introduction, dynamical systems are the systems whose motion is pre-determined by a set of rules (an algorithm). For dynamical systems we can introduce the notion of state determined by a set of quantities called dynamical variables. The space of dynamical variables is said to be the phase space. It follows from the definition of dynamical system that its state at each instant t must be uniquely determined by its state at some earlier instant t 0. Obviously all real systems are not dynamical, on two counts: for one thing, these systems are always acted upon by uncontrolled forces (fluctuations); for another, the initial state of these systems cannot be determined with pinpoint accuracy. By and large these circumstances are not essential, and the state of the real system at each instant t can be uniquely predicted, starting from its initial state. This situation occurs if the system motion is stable with respect to small perturbations. Otherwise the circumstances indicated above play a crucial role in the system’s behaviour, and determination of the system’s state starting from its initial state becomes impossible. This means that the system’s behaviour is unpredictable, a random behaviour. If this unpredictability is preserved when the intensity of perturbations approaches zero and the accuracy of designation of initial conditions tends to infinity, then such behaviour of dynamical system has been identified by Neimark [Neimark, 1987] as stochastic.

Resurgence theory implies that the non-perturbative (NP) and perturbative (P) data in a QFT are quantitatively related, and that detailed information about non-perturbative saddle point field configurations of path integrals can be extracted from perturbation theory. Traditionally, only stable NP saddle points are considered in QFT, and homotopy group considerations are used to classify them. However, in many QFTs the relevant homotopy groups are trivial, and even when they are non-trivial they leave many NP saddle points undetected. Resurgence provides a refined classification of NP-saddles, going beyond conventional topological considerations. To demonstrate some of these ideas, we study the SU(N) principal chiral model (PCM), a two dimensional asymptotically free matrix field theory which has no instantons, because the relevant homotopy group is trivial. Adiabatic continuity is used to reach a weakly coupled regime where NP effects are calculable. We then use resurgence theory to uncover the existence and role of novel ‘fracton’ saddle points, which turn out to be the fractionalized constituents of previously observed unstable ‘uniton’ saddle points. The fractons play a crucial role in the physics of the PCM, and are responsible for the dynamically generated mass gap of the theory. Moreover, we show that the fracton-anti-fracton events are the weak coupling realization of ’t Hooft’s renormalons, and argue that the renormalon ambiguities are systematically cancelled in the semi-classical expansion. Our results motivate the conjecture that the semi-classical expansion of the path integral can be geometrized as a sum over Lefschetz thimbles.

Microswimmers are encountered in a wide variety of biophysical settings. When interacting with flow fields, they show interesting dynamical features such as hydrodynamic trapping, clustering, and preferential orientation. One important step towards the understanding of such features is to clarify the interplay of hydrodynamic flows with microswimmer motility and shape. Here, we study the dynamics of ellipsoidal microswimmers in a two-dimensional axisymmetric vortex flow. Despite this simple setting, we find surprisingly rich dynamics, which can be comprehensively characterized in the framework of dynamical systems theory. By classifying the fixed-point structure of the underlying phase space as a function of motility and microswimmer shape, we uncover the topology of the phase space and determine the conditions under which microswimmers are trapped in the vortex. For spherical microswimmers, we identify Hamiltonian dynamics, which are broken for microswimmers of a different shape. We find that prolate ellipsoidal microswimmers tend to align parallel to the velocity field, while oblate microswimmers tend to remain perpendicular to it. Additionally, we find that rotational noise allows microswimmers to escape the vortex with an enhanced escape rate close to the system’s saddle point. Our results clarify the role of shape and motility on the occurrence of preferential concentration and clustering and provide a starting point to understand the dynamics in more complex flows.