Periodic motion planning and control for underactuated mechanical systems

We consider the problem of periodic motion planning and of designing stabilising feedback control laws for such motions in underactuated mechanical systems. A novel periodic motion planning method is proposed. Each state is parametrised by a truncated Fourier series. Then we use numerical optimisation to search for the parameters of the trigonometric polynomial exploiting the measure of discrepancy in satisfying the passive dynamics equations as a performance index. Thus an almost feasible periodic motion is found. Then a linear controller is designed and stability analysis is given to verify that solutions of the closed-loop system stay inside a tube around the planned approximately feasible periodic trajectory. Experimental results for a double rotary pendulum are shown, while numerical simulations are given for models of a spacecraft with liquid sloshing and of a chain of mass spring system.     

Optimal control on Lie groups with applications to attitude control

In this paper we solve a class of optimal control problems on Lie groups in the sense that we derive differential equations which the optimal controls must satisfy. These results are applied to the attitude control of a spacecraft modeled as a rigid body. Specifically, we derive control laws (both in open-loop and closed-loop form) to maneuver the spacecraft between two given rotational states in finite time. The laws are such that a cost functional measuring the over-all angular velocity during the spacecraft’s motion is minimized. They do not require recourse to numerical methods and hence can be easily implemented in an on-board attitude control system. After dealing with a three-axis controlled spacecraft we also discuss the case that only torques about two principal axes of an axisymmetric spacecraft can be exerted.     

Almost Periodic Solution for Shunting Inhibitory Cellular Neural Networks with Time-varying Delays and Variable Coefficients

In this paper, a class of two-dimensional shunting inhibitory cellular neural networks with time-varying delays and variable coefficients as following system

Slow periodic motions in variable structure systems

Singularly perturbed relay control systems (SPRCS) with stable periodic motion in reduced systems are studied here. The slow motions integral manifold of such systems consists of parts that correspond to different values of control and the solutions of SPRCS contain the jumps from one part of the slow manifold to the other. The theorems about existence and stability of the slow periodic solutions are proved. An algorithm of asymptotic representation for this periodic solutions using a boundary layer method is suggested.     

Manipulator-actuated Adaptive Integrated Translational and Rotational Stabilization for Spacecraft in Proximity Operations with Control Constraint

This paper tackles the problem of integrated translation and rotation stabilization of the spacecraft in proximity operations by proposing a novel manipulator actuation strategy. To do so, by theoretically integrating the attitude/position motion of the spacecraft and the joint motion of the manipulator, a coupled translational and rotational kinematics of the spacecraft with a single space manipulator mounted is formulated, where system unknown parameters and residual system momentum are taken into account and analyzed. Taking the joint motion as the control input, a projection-based adaptive control scheme is then developed such that the translation and rotation of the spacecraft can be robustly stabilized with the manipulator-based actuation. The closed-loop asymptotic stability is guaranteed within Lyapunov framework. Meanwhile, considering the constrained joint motion of the manipulator, the resulting control constraint issue is handled by developing an optimization based bound analysis method, which also facilitates the determination of control parameters. Two scenario numerical simulations demonstrate the effect of the designed control scheme.     

Overdamped nonlinear systems

Overdamped solutions toddot{z} + KF(z, dot{z}) = 0, z(0) = z_{0}, dot{z}(0) = dot{z}_{0}, are defined to be those that are asymptotically stable to the origin and which are zero at most once for0 < t < x. The main result is a lower bound on the positive constantKto ensure that solutions to the equation are overdamped.     

Online Search with Time-Varying Price Bounds

Online search is a basic online problem. The fact that its optimal deterministic/randomized solutions are given by simple formulas (however with difficult analysis) makes the problem attractive as a target to which other practical online problems can be transformed to find optimal solutions. However, since the upper/lower bounds of prices in available models are constant, natural online problems in which these bounds vary with time do not fit in the available models.We present two new models where the bounds of prices are not constant but vary with time in certain ways. The first model, where the upper and lower bounds of (logarithmic) prices have decay speed, arises from a problem in concurrent data structures, namely to maximize the (appropriately defined) freshness of data in concurrent objects. For this model we present an optimal deterministic algorithm with competitive ratio \(\sqrt{D}\), where D is the known duration of the game, and a nearly-optimal randomized algorithm with competitive ratio \(\frac{\ln D}{1+\ln2-\frac{2}{D}}\). We also prove that the lower bound of competitive ratios of randomized algorithms is asymptotically \(\frac{\ln D}{4}\).The second model is inspired by the fact that some applications do not utilize the decay speed of the lower bound of prices in the first model. In the second model, only the upper bound decreases arbitrarily with time and the lower bound is constant. Clearly, the lower bound of competitive ratios proved for the first model holds also against the stronger adversary in the second model. For the second model, we present an optimal randomized algorithm. Our numerical experiments on the freshness problem show that this new algorithm achieves much better/smaller competitive ratios than previous algorithms do, for instance 2.25 versus 3.77 for D=128.     

无结构内阻尼磁性刚体航天器姿态动力学稳定性分析

在地球万有引力场和磁场中，考虑大气阻力的条件下，讨论无结构内阻尼的磁性刚体航天器在圆轨道上出现的混沌问题．根据动量矩定理建立动力学模型，利用Melnikov方法证明发生混沌的可能性，并利用数值仿真分析了系统的动力学行为．理论结果与数值仿真结果相一致．     

Lower and Upper Bounds on FIFO Buffer Management in QoS Switches

We consider the management of FIFO buffers for network switches providing differentiated services. In each time step, an arbitrary number of packets arrive and only one packet can be sent. The buffer can store a limited number of packets and, due to the FIFO property, the sequence of sent packets has to be a subsequence of the arriving packets. The differentiated service model is abstracted by attributing each packet with a value according to its service level. A buffer management strategy can drop packets, and the goal is to maximize the sum of the values of sent packets. For only two different packet values, we introduce the account strategy and prove that this strategy achieves an optimal competitive ratio of if the buffer size tends to infinity and an optimal competitive ratio of for arbitrary buffer sizes. For general packet values, the simple preemptive greedy strategy (PG) is studied. We show that PG achieves a competitive ratio of which is the best known upper bound on the competitive ratio of this problem. In addition, we give a lower bound of on the competitive ratio of PG which improves the previously known lower bound. As a consequence, the competitive ratio of PG cannot be further improved significantly. Supported by the DFG grant WE 2842/1. A preliminary version of this paper appeared in Proceedings of the 14th Annual European Symposium on Algorithms (ESA), 2006.     

Spin-axis stabilization of symmetric spacecraft with two control torques

It is a well-known fact that a symmetric spacecraft with two control torques supplied by gas jet actuators is not controllable, if the two control torques are along axes that span the two-dimensional plane orthogonal to the axis of symmetry. However, feedback control laws can be derived for a restricted problem corresponding to attitude stabilization about the symmetry axis. In this configuration, the final state of the system is a uniform revolute motion about the symmetry axis. The purpose of this paper is to present a new methodology for constructing feedback control laws for this problem, based on a new formulation for the attitude kinematics.     

Some orbital test problems

Problems with periodic solutions are convenient as test problems for differential equation software because of the ease with which the accuracy of computed results can be assessed. Even the motion of a single planet around a heavy sun is useful as a test problem because orbits of varying eccentricity make varying demands on numerical software. The orbits discussed here are based on this same simple problem but with the essential difference that the distance from the planet to the sun is based on the norm ‖ . ‖_∞ rather than the usual Euclidean norm ‖ . ‖₂. Specifically, we explore orbits based on each of the differential equation systems $$X = \nabla \left( {\frac{1}{{\left\| X \right\|}}} \right)$$ and $$X = - \frac{1}{{\left\| X \right\|^3 }}.$$ A feature of both these systems, when the ‖ . ‖_∞ norm is used, is the occurrence of discontinuities in the higher derivatives of the solution. This is why they have a potential value as difficult test problems. With this application in mind, some periodic solutions are identified. For an arbitrary choice of norm, the second of the two differential equation systems considered in this paper is shown to possess periodic orbits.     

Statistical analysis of inherent ambiguities in recovering 3-Dmotion from a noisy flow field

The inherent ambiguities in recovering 3-D motion information from a single optical flow field are studied using a statistical model. The ambiguities are quantified using the Cramer-Rao lower bound. As a special case, the performance bound for the motion of 3-D rigid planar surfaces is studied in detail. The dependence of the bound on factors such as the underlying motion, surface position, surface orientation, field of view, and density of available pixels are derived as closed-form expressions. A subset of the results support S. Adiv's (1989) analysis of the inherent ambiguities of motion parameters. For the general motion of an arbitrary surface. It is shown that the aperture problem in computing the optical flow restricts the nontrivial information about the 3-D motion to a sparse set of pixels at which both components of the flow velocity are observable. Computer simulations are used to study the dependence of the inherent ambiguities on the underlying motion, the field of view, and the number of feature points for the motion in front of a nonplanar environment     

Slow periodic motions with internal sliding modes in variable structure systems

Singularly perturbed relay systems (SPRS) in which the reduced systems have the stable periodic motions with internal sliding modes are studied. The slow motion integral manifold of such systems consists of the parts which correspond to the different values of relay control and the solutions may contain the jumps from one part of the slow manifold to another. For such systems a theorem about existence and stability of the periodic solutions is proved. An algorithm of asymptotic representation for this periodic solutions using boundary layer method is presented. It is proved that in the neighbourhood of the break away point the asymptotic representation starts with the first order boundary layer function.     

Response time analysis of digraph real-time tasks scheduled with static priority: generalization,approximation, and improvement

Graph-based task models have been studied to better model and analyze the schedulability of real-time systems. Among them, the digraph task model, with its powerful expressiveness to describe the behavior of a large class of real-time tasks, receives a wide range of interests recently. However, the exact schedulability analysis of digraph tasks on a uni-processor with preemptive static-priority scheduling has been shown to be coNP-hard. Approximate analyses based on the request and interference bound functions (\(\textit{rbf}\) and \(\textit{ibf}\)) have been proposed to improve the analysis efficiency. In this work, we summarize the existing results on these analysis techniques, and seek to further improve their generality, complexity, and accuracy. Specifically, we develop analysis techniques for tasks with arbitrary deadlines. We prove the periodicity of interference bound function such that it can be expressed as a finite aperiodic part and an infinite periodic part, which makes the asymptotic complexity of its calculation independent from the length of the time interval. Moreover, we develop a linear upper bound on \(\textit{ibf}\) that is tighter than that of \(\textit{rbf}\), to derive a better response time bound.     

Two-Parameter Discrete Systems over a Residue Ring and Their Properties

A class of two-parameter discrete systems defined on the ring of class of residues of integers modulo m is studied. All solutions are shown to be periodic, stability conditions (equality of solutions to zero, beginning from a certain instant) and a controllability condition are formulated. Controllability is shown to guarantee stabilizability.     

Spin-axis stabilisation of underactuated rigid spacecraft under sinusoidal disturbance

Spin-axis stabilisation of spacecraft is a problem of partial stabilisation for non-linear dynamical systems. In this article the analysis of spin-axis stabilisation of underactuated rigid spacecraft in the presence of sinusoidal disturbances is presented. By using the Euler–Poisson form to describe the equations of motion and assuming the disturbances in three axes are decoupled with known frequencies, the paper first studies the problem of the underactuated rigid axisymmetric spacecraft by applying the internal modal principle to eliminate the sinusoidal disturbance. Then the paper turns to the more complicated asymmetric spacecraft, where the boundedness of the angular velocity for the underactuated axis is analysed in detail. The paper also proves the global asymptotic stability of the closed-loop systems for both axisymmetric spacecraft and asymmetric spacecraft by combining the Lyapunov direct method with the LaSalle's theorem. The simulation results show that the proposed control law is effective in the presence of sinusoidal disturbance.     

Optimal space station detumbling by internal mass motion

This investigation deals with the use of a movable mass control system to stabilize a tumbling asymmetric spacecraft about the maximum inertia axis. A first-order gradient optimization technique is used to minimize angular velocity components along the intermediate and minimum inertia axes, thus, permitting a wide range of initial guesses for mass position history. Motion of the control mass is along a linear track fixed in the vehicle. The control variable is taken as mass acceleration with respect to body coordinates. Motion is limited to defined quantities and a penalty function is used to insure a given range of positions. Numerical solutions of the optimization equations verify that minimum time detumbling is achieved with the largest permissible movable mass, length of linear track, and positions of the mass on the two coordinates perpendicular to the linear motion. The optimal method permits detumbling in about one-fourth the time when compared to a force control law formulation available in the literature.     

The space complexity of unbounded timestamps

The timestamp problem captures a fundamental aspect of asynchronous distributed computing. It allows processes to label events throughout the system with timestamps that provide information about the real-time ordering of those events. We consider the space complexity of wait-free implementations of timestamps from shared read-write registers in a system of n processes. We prove an lower bound on the number of registers required. If the timestamps are elements of a nowhere dense set, for example the integers, we prove a stronger, and tight, lower bound of n. However, if timestamps are not from a nowhere dense set, this bound can be beaten: we give an implementation that uses n − 1 (single-writer) registers. We also consider the special case of anonymous implementations, where processes are programmed identically and do not have unique identifiers. In contrast to the general case, we prove anonymous timestamp implementations require n registers. We also give an implementation to prove that this lower bound is tight. This is the first anonymous timestamp implementation that uses a finite number of registers.     

On Compensating Long Actuator Delays in Nonlinear Control

We are interested in finite-escape open-loop unstable plants that are globally stabilizable in the absence of actuator delay but require controller redesign in the presence of delay. The simplest such plant is $ {mathdot Z} (t)= Z(t)^{2} + U(t-D)$, where $D$ is actuator delay of arbitrary length. For this system we present a control law that compensates the delay and achieves feedback linearization (of the entire ODE+delay infinite-dimensional cascade). However, even though exponential stability is achieved, the result is not global because the plant can have a finite escape with an initial condition $Z(0)geq 1/D$ before the feedback control “reaches” it at $t=D$ . We prove a stability result whose region of attraction is essentially $Z(0)≪ 1/D$ and for which we derive an asymptotic stability bound in terms of the system norm $Z(t)^{2} + int _{t-D}^{t} U(theta )^{2} dtheta $.      

Exponential Stability on Stochastic Neural Networks With Discrete Interval and Distributed Delays

This brief addresses the stability analysis problem for stochastic neural networks (SNNs) with discrete interval and distributed time-varying delays. The interval time-varying delay is assumed to satisfy $0≪d_{1}leq d(t) leq d_{2}$ and is described as $d(t)= d_{1}+h(t)$ with $0leq h(t) leq d_{2}-d_{1}$. Based on the idea of partitioning the lower bound $d_{1}$, new delay-dependent stability criteria are presented by constructing a novel Lyapunov–Krasovskii functional, which can guarantee the new stability conditions to be less conservative than those in the literature. The obtained results are formulated in the form of linear matrix inequalities (LMIs). Numerical examples are provided to illustrate the effectiveness and less conservatism of the developed results.