Double-Ring Polyhedral Linkages

The overall goal of photonics research is to understand and control light in new and richer ways to facilitate new and richer applications. Many major developments to this end have relied on nonlinear optical techniques, such as lasing, mode-locking, and parametric downconversion, to enable applications based on the interactions of coherent light with matter. These processes often involve nonlinear interactions between photonic and material degrees of freedom spanning multiple spatiotemporal scales. While great progress has been made with relatively simple optimizations, such as maximizing single-mode coherence or peak intensity alone, the ultimate achievement of coherent light engineering is complete, multidimensional control of light–light and light–matter interactions through tailored construction of complex optical fields and systems that exploit all of light’s degrees of freedom. This capability is now within sight, due to advances in telecommunications, computing, algorithms, and modeling. Control of highly multimode optical fields and processes also facilitates quantitative and qualitative advances in optical imaging, sensing, communication, and information processing since these applications directly depend on our ability to detect, encode, and manipulate information in as many optical degrees of freedom as possible. Today, these applications are increasingly being enhanced or enabled by both multimode engineering and nonlinearity. Here, we provide a brief overview of multimode nonlinear photonics, focusing primarily on spatiotemporal nonlinear wave propagation and, in particular, on promising future directions and routes to applications. We conclude with an overview of emerging processes and methodologies that will enable complex, coherent nonlinear photonic devices with many degrees of freedom.

This paper presents a new approach to motion planning for a manipulator with many degrees of freedom, which is based on defining several types of reactive behavior, such as, moving toward a target, avoiding obstacles, folding up links, following edges, and so forth. Motion planning with many degrees of freedom suffers mainly from kinematical redundancy, i.e., it is hard to determine which links and how many links should be moved so that the end-effector can reach a specific target point from a start point in complex environments. Using reactive behavior with simple features, complex motion planning can be qualitatively formulated with ease. In order to formulate reactive behavior quantitatively, a manipulator with many degrees of freedom is first decomposed into several two-link sub-manipulators, and motion planning of these two-link manipulators is performed by an algorithm for few degrees of freedom. This method can be easily implemented, and in combination with sensors it is suitable for robots with many degrees of freedom that operate in uncertain environments. Some examples are reported to demonstrate the effectiveness of the proposed method.

Reinforcement learning has recently been receiving much attention as a learning method for not only toy problems but also complicated systems such as robot systems. It does not need priori knowledge and has higher capability of reactive and adaptive behaviors. However, increasing of action-state space makes it difficult to accomplish the learning process. In most of the previous works, the application of the learning is restricted to simple tasks with a small action-state space. Considering this point, we present a new reinforcement learning algorithm: Q-learning with dynamic structuring of exploration space based on genetic algorithm. The algorithm is applicable to systems with high dimensional action and interior state spaces, for example, a robot with many redundant degrees of freedom. To demonstrate the effectiveness of the proposed algorithm simulations of locomotion patterns for a 12-leged robot were carried out. As the result, an effective behavior was obtained by using our proposed algorithm.

A rotobreather in a lattice of coupled particles with rotational degrees of freedom is a dynamical regime when one particle rotates and the other particles vibrate with amplitudes exponentially decaying with distance from the rotating particle. In addition to rotational, particles can have internal degrees of freedom. Here, the influence of internal degrees of freedom on the possible rotational frequencies of rotobreathers is discussed for two models: a chain of coupled elastic rotators and a molecular crystal of carbon nanotubes (CNTs). It is shown that additional degrees of freedom create forbidden frequency bands for rotobreathers. In the case of a CNT crystal, due to resonances with many internal degrees of freedom, the frequency spectrum of a rotobreather has a complex structure resembling a Cantor set.

In [Gup90, GG92], we have presented a sequential framework that allows us to develop planners for manipulator arms with many degrees of freedom. The essence of this framework is to exploit the serial structure of manipulator arms and decompose the n- dimensional problem of planning collision-free motions for an n-link manipulator into a sequence of smaller m-dimensional sub-problems, each of which corresponds to planning the motion of a sub-group of m-1 links. In this paper, we present extensive experimental results within our sequential framework for a variety of manipulators with up to eight degree of freedom manipulators. Two main goals of these simulations are (1) to show the effectiveness of the sequential approach with the backtracking mechanism, and (2) to quantify the improvement of the backtracking mechanism and the trade-off between number of backtrackings and the execution time of the planner. Our experiments show that the sequential framework with the backtracking mechanism is quite efficient for manipulator arms with many degrees of freedom. For a given maximum backtracking level, the run time and memory requirements vary roughly linearly with the degrees of freedom. The planner succeeds for 91% of the examples in our simulations with a maximum backtracking level of 2. Typical run times for a six degree of freedom manipulator in quite cluttered environments are of the order of tens of minutes. Our sequential thus provides a framework within which practical motion planners for many degree of freedom manipulators can be developed.

In this thesis, we present a dynamical systems approach to adaptive controllers for locomotion control. The approach is based on a rigorous mathematical framework using nonlinear dynamical systems and is inspired by theories of self-organization. Nonlinear dynamical systems such as coupled oscillators are an interesting approach for the on-line generation of trajectories for robots with many degrees of freedom (e.g. legged locomotion). However, designing a nonlinear dynamical system to satisfy a given specification and goal is not an easy task, and, hitherto no methodology exists to approach this problem in a unified way. Nature presents us with satisfactory solutions for the coordination of many degrees of freedom. One central feature observed in biological subjects is the ability of the neural systems to exploit natural dynamics of the body to achieve efficient locomotion. In order to be able to exploit the body properties, adaptive mechanisms must be at work. Recent work has pointed out the importance of the mechanical system for efficient locomotion. Even more interestingly, such well suited mechanical systems do not need complicated control. Yet, in robotics, in most approaches, adaptive mechanisms are either missing or they are not based on a rigorous framework, i.e. they are based on heuristics and ad-hoc approaches. Over the last three decades there has been enormous progress in describing movement coordination with the help of Synergetic approaches. This has led to the formulation of a theoretical framework: the theory of dynamic patterns. This framework is mathematically rigorous and at the same time fully operational. However, it does not provide any guidelines for synthetic approaches as needed for the engineering of robots with many degrees of freedom, nor does it directly help to explain adaptive systems. We will show how we can extend the theoretical framework to build adaptive systems. For this purpose, we propose the use of multi-scale dynamical systems. The basic idea behind multi-scale dynamical systems is that a given dynamical system gets extended by additional slow dynamics of its parameters, i.e. some of the parameters become state variables. The advantages of the framework of multi-scale dynamical systems for adaptive controllers are 1) fully dynamic description, 2) no separation of learning algorithm and learning substrate, 3) no separation of learning trials or time windows, 4) mathematically rigorous, 5) low dimensional systems. However, in order to fully exploit the framework important questions have to be solved. Most importantly, methodologies for designing the feedback loops have to be found and important theoretical questions about stability and convergence properties of the devised systems have to be answered. In order to tackle this challenge, we first introduce an engineering view on designing nonlinear dynamical systems and especially oscillators. We will highlight the important differences and freedom that this engineering view introduces as opposed to a modeling one. We then apply this approach by first proposing a very simple adaptive toy-system, consisting of a dynamical system coupled to a spring-mass system. Due to its spring-mass dynamics, this system contains clear natural dynamics in the form of resonant frequencies. We propose a prototype adaptive multi-scale system, the adaptive frequency oscillator, which is able to adapt its intrinsic frequency to the resonant frequency of the body dynamics. After a small sidetrack to show that we can use adaptive frequency oscillators also for other applications than for adaptive controllers, namely for frequency analysis, we then come back to further investigation of the adaptive controller. We apply the same controller concept to a simple spring-mass hopper system. The spring-mass system consists of a body with two legs attached by rotational joints. The legs contain spring-damper elements. Finally, we present results of the implementation of the controller on a real robot, the experimental robot PUPPY II. This robot is a under-actuated robot with spring dynamics in the knee joints. It will be shown, that due to the appropriate simplification and concentration on relevant features in the toy-system the controller concepts works without a fundamental change on all systems from the toy system up to the real robot.

We present quantum mechanical calculations of the absorption line shape of an electron ‘‘solvated’’ in several sodalites. Photon absorption by the electron modifies the forces acting on the nuclei, setting the counterions in motion. This nuclear motion causes broadening and gives vibrational structure to the absorption spectrum of the electron. The major effort in the computation of the absorption spectrum is directed toward the evaluation of an overlap integral that evolves in time because of nuclear motion. The systems considered here have a very large number of nuclear degrees of freedom, and this makes a brute-force quantum mechanical calculation of the overlap impossible. Good results can be obtained with a method that exploits the fact that in a system with many degrees of freedom the overlap integral decays rapidly to zero, and can therefore be evaluated accurately and efficiently by short-time methods. The short-time method that seems most advantageous is the Gaussian wave packet (GWP) procedure proposed some time ago by Heller. This simplifies the nuclear dynamics and also substantially diminishes the number of electron energy calculations needed for determining the forces acting on the nuclei. When the GWP method is used, the electronic wave function is calculated only for a small number of nuclear configurations along the classical trajectory on which the center of the nuclear wave packet evolves. The present calculation is the first use of this method to compute the absorption spectrum of a complex system. We study the absorption line shape for an electron solvated in a dry sodalite, and in chloro-, bromo-, and iodo-sodalite. We find that the homogeneous linewidth due to the nuclear motion is narrower than that observed experimentally. This implies that the measured linewidth is due to inhomogeneous broadening. For the dry sodalite the main inhomogeneity is the disorder in the position of the counterions, and for halo-sodalites, the presence of defects introduced during synthesis. Our results imply that a careful synthesis can improve the contrast in displays based on the cathodochromic effects in zeolites.

Reinforcement learning is very interesting for robot learning. However, there are some significant problems in applying conventional reinforcement learning algorithms to the robot with many degrees of freedom, because the size of exploration space increase exponentially with increase of degrees of freedom, and it makes it impossible to accomplish learning process. On the other hand, animals and humans can learn and accomplish various tasks using many redundant degrees of freedom of the body in spite of the exploration space is very huge. In this paper, we consider how to solve the state explosion problem in applying the reinforcement learning to the redundant robot and propose new framework of reinforcement learning, which is inspired by the body image of animals, by summarizing our previous works of reinforcement learning. To demonstrate the effectiveness of proposed method, simulations and experiments have been carried out and as a result effective behaviors have been obtained.

Recently, the robot with many links and degrees of freedoms is actively produced. Generally, the robot that has many degrees of freedoms owns many redundant joints. So we make a suggestion for using redundant joints effectively. The paper proposes using redundant joint as a counter weight. We make a trajectory in momentum space, and we use Fourier Basis Algorism as the method for an optimization. This trajectory generating method has an effect on reducing a driving energy for throwing motion.

We argue that if the UV cutoff of an effective field theory with many low energy degrees of freedom is of the order, or below, the scale of the stretched horizon in a black hole background, which in turn is significantly lower than the Planck scale, the black hole radiance rate may not be enhanced by the emission of all the light IR modes. Instead, there may be additional suppressions hidden in the UV completion of the field theory, which really control which light modes can be emitted by the black hole. It could turn out that many degrees of freedom cannot be efficiently emitted by the black hole, and so the radiance rate may be much smaller than its estimate based on the counting of the light IR degrees of freedom. If we apply this argument to the Randall-Sundrum II (RS2) brane world, it implies that the emission rates of the low energy conformal field theory modes will be dramatically suppressed: its UV completion is given by the bulk gravity on ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$, and the only bulk modes which could be emitted by a black hole are the 4-dimensional (4D) $s$ waves of bulk modes with small 5-dimensional momentum, or equivalently, small 4D masses. Further, their emission is suppressed by bulk warping, which lowers the radiation rate much below the IR estimate, yielding a radiation flux $\ensuremath{\sim}({T}_{\mathrm{BH}}L{)}^{2}{\mathcal{L}}_{\mathrm{Hawking}}\ensuremath{\sim}({T}_{\mathrm{BH}}/{M}_{\mathrm{Pl}}{)}^{2}N{\mathcal{L}}_{\mathrm{Hawking}}$, where ${\mathcal{L}}_{\mathrm{Hawking}}$ is the Hawking radiation rate of a single light species. This follows directly from low conformal field theory cutoff $\ensuremath{\mu}\ensuremath{\sim}{L}^{\ensuremath{-}1}\ensuremath{\ll}{M}_{\mathrm{Pl}}$, a large number of modes $N\ensuremath{\gg}1$ and the fact that 4D gravity in RS2 is induced, ${M}_{\mathrm{Pl}}^{2}\ensuremath{\simeq}N{\ensuremath{\mu}}^{2}$.

Applications of nonlinear optimization problems with many degrees of freedom have become more common in the process industries, especially in the area of process operations. However, most widely used nonlinear programming (NLP) solvers are designed for the efficient solution of problems with few degrees of freedom. Here we consider a new NLP algorithm, IPOPT, designed for many degrees of freedom and many potentially active constraint sets. The IPOPT algorithm follows a primal-dual interior point approach, and its robustness, improved convergence, and computational speed compared to those of other popular NLP algorithms will be analyzed. To demonstrate its effectiveness on process applications, we consider large gasoline blending and data reconciliation problems, both of which contain nonlinear mass balance constraints and process properties. Results on this computational comparison show significant benefits from the IPOPT algorithm.

While statistical mechanics describe the equilibrium state of systems with many degrees of freedom, and dynamical systems explain the irregular evolution of systems with few degrees of freedom, new tools are needed to study the evolution of systems with many degrees of freedom. This book presents the basic aspects of chaotic systems, with emphasis on systems composed by huge numbers of particles. Firstly, the basic concepts of chaotic dynamics are introduced, moving on to explore the role of ergodicity and chaos for the validity of statistical laws, and ending with problems characterized by the presence of more than one significant scale. Also discussed is the relevance of many degrees of freedom, coarse graining procedure, and instability mechanisms in justifying a statistical description of macroscopic bodies. Introducing the tools to characterize the non asymptotic behaviors of chaotic systems, this text will interest researchers and graduate students in statistical mechanics and chaos.

To analyze trajectories for systems of many degrees of freedom, we propose a new method called wavelet local principal component analysis (WlPCA) combining the wavelet transformation and local PCA in time. Our method enables us to reduce the dimensionality of time series both in degrees of freedom and frequency so that characteristic features of oscillatory behavior can be captured. To test the new method, we apply WlPCA to a non-autonomous model of multiple degrees of freedom, the Froeschlé maps of $N=2$ and $N=4$, which correspond to autonomous systems of three and five degrees of freedom, respectively. The eigenvalues and eigenvectors obtained by WlPCA reveal those times when frequency variation exhibits switching between relatively stationary features. Moreover, further analyses indicate which degrees of freedom and frequencies are involved in the switching. We confirm that the switching corresponds to the onset of transport in phase space. These findings imply that, even for systems of larger degrees of freedom, barriers can exist in phase space that block transport for a finite time, thereby dividing the phase space into multiple quasi-stationary regions. Thus, our method is promising for understanding dynamics in systems of many degrees of freedom, such as vibrational-energy redistribution in molecules.

Signal processing for LED lighting systems : illumination rendering and sensing

A stochastic technique is described for planning collision-free paths of robots with many degrees of freedom (DOFs). The algorithm incrementally builds a graph connecting the local minima of a potential function defined in the robot's configuration space and concurrently searches the graph until a goal configuration is attained. A local minimum is connected to another one by executing a random motion that escapes the well of the first minimum, succeeded by a gradient motion that follows the negated gradient of the potential function. All the motions are executed in a grid shown through the robot's configuration space. The random motions are implemented as random walks which are known to converge toward Brownian motions when the steps of the walks tend toward zero. The local minima graph is searched using a depth-first strategy with random backtracking. In the technique, the planner does not explicitly represent the local-minima graph. The path-planning algorithm has been fully implemented and has run successfully on a variety of problems involving robots with many degrees of freedom. >