On the robust adaptive control of a contour-following robotic system

To demonstrate the force sensing and control, a possible model of a contour-following system is represented by a fourth-order linear continuous-time time-invariant system, in which stiffness k_t, robot natural frequency, ω_n, linear accommodation gain k_x, and angular accommodation gain k_φ are all constants obtained by measurement and experiment. This model works well for following a short contour. To satisfactorily follow a longer contour, k_t, ω_n, k_x and k_φ can be treated as unknown constants or time-varying variables. When they are considered as unknown constants, a robust model reference adaptive controller can be used to achieve both stability and tracking without having to know or find the true values of those constants, given bounded input or output disturbances and stable unmodeled dynamics. If ω_n is assumed to be a given constant but K_t, k_x, and k_φ are assumed to be unknown variables, then one has a linear time-varying plant and other types of model reference adaptive controllers have to be used to achieve the same purpose. In this paper, the schemes from various robust model reference adaptive control design will be studied and comparison and suggestions will also be made based on the simulation results for the contour-following robotic system mentioned above.     

A multi-tree routing scheme using acyclic orientations

We propose a mathematical model for fault-tolerant routing based on acyclic orientations, or acorns, of the underlying network G=(V,E). The acorn routing model applies routing tables that store the set of parent pointers associated with each out-neighborhood defined by the acorn. Unlike the standard single-parent sink-tree model, which is vulnerable to faults, the acorn model affords a full representation of the entire network and is able to dynamically route around faults. This fault tolerance is achieved when using the acorn model as a multi-tree generator for gathering data at a destination node, as well as an independent tree generator for global point-to-point communication. A fundamental fault-tolerant measure of the model is the capacity of an acorn, i.e., the largest integer k such that each vertex outside the neighborhood N(v) of the destination v has at least k parent pointers. A capacity-k acorn A to destination v is k-vertex fault-tolerant to v. More strongly, we show A supports a k independent sink-tree generator, i.e., the parent pointers of each vertex w V−N(v) can be partitioned into k nonempty classes labeled 1,2,…,k such that any set of sink trees T₁,T₂,…,T_k are pairwise independent, where tree T_i is a sink tree generated by parent pointers labeled i together with the parent pointers into v. We present an linear time optimization algorithm for finding an acorn A of maximum capacity in graphs, based upon a minimax theorem. We also present efficient algorithms that label the parent pointers of capacity-k acorn A, yielding a k-independent sink tree generating scheme.     

Experimental numerical studies on a supercomputer of natural convection in an enclosure with localized heating

We present particle simulations of natural convection of a symmetrical, nonlinear, three-dimensional cavity flow problem. Qualitative studies are made in an enclosure with localized heating. The assumption is that particles interact locally by means of a compensating Lennard-Jones type force F, whose magnitude is given by −G/r^p + H/r^q.

In this formula, the parameters G, H, p, q depend upon the nature of the interacting particles and r is the distance between two particles. We also consider the system to be under the influence of gravity. Assuming that there are n particles, the equations relating position, velocity and acceleration at time t_k = kΔt, K = 0, 1, 2, …, are solved simultaneously using the “leap-frog” formulas. The basic formulas relating force and acceleration are Newton's dynamical equations F_i,k = m_ia_i,k, I = 1, 2, 3, …, n, where m_i is the mass of the ith particle.

Extensive and varied computations on a CRAY X - MP/24 are described and discussed, and comparisons are made with the results of others.     

Assessment of turbulence-transport models including non-linear rng eddy-viscosity formulation and second-moment closure for flow over a backward-facing step

A computational study is performed in which the predictive capabilities of a range of eddy-viscosity and second-moment-closure models are examined by reference to a separated flow behind a backward-facing step in an expanding channel. The models include three second-moment-closure variants, all being of the ‘Launder-Reece-Rodi’ type, two RNG k—ε forms, one combining the RNG approach with a non-linear eddy-viscosity formulation, and a low-Re k—ε model. The study demonstrates that to achieve a solution similar to that returned by second-moment closure, the RNG formulation needs to be implanted into a non-linear eddy-viscosity framework; neither returns, on its own, the correct behaviour, not even for mean-flow features. Moreover, relatively minor variations within second-moment closure—specifically, such relating to wall-induced effects on turbulence isotropisation and to stress diffusion—can significantly alter the overall performance of the closure. All models specifically designed to return realistic solutions for normal stresses seriously over-estimate anisotropy.     

On the number of occurrences of all short factors in almost all words

We previously proved that almost all words of length n over a finite alphabet A with m letters contain as factors all words of length k(n) over A as n→∞, provided limsup_n→∞ k(n)/log n<1/log m.

In this note it is shown that if this condition holds, then the number of occurrences of any word of length k(n) as a factor into almost all words of length n is at least s(n), where lim_n→∞ log s(n)/log n=0. In particular, this number of occurrences is bounded below by C log n as n→∞, for any absolute constant C>0.     

Distribution of black nodes at various levels in a linear quadtree

The distribution of black leaf nodes at each level of a linear quadtree is of significant interest in the context of estimation of time and space complexities of linear quadtree based algorithms. The maximum number of black nodes of a given level that can be fitted in a square grid of size 2ⁿ × 2ⁿ can readily be estimated from the ratio of areas. We show that the actual value of the maximum number of nodes of a level is much less than the maximum obtained from the ratio of the areas. This is due to the fact that the number of nodes possible at a level k, 0≤k≤n − 1, should consider the sum of areas occupied by the actual number of nodes present at levels k + 1, k + 2, …, n − 1.     

Cardinal interpolation with differences of tempered functions

In this paper, we investigate the existence and uniqueness of cardinal interpolants associated with functions arising from the k^th order iterated discrete Laplacian ^k applied to certain radial basis functions. In particular, we concentrate on determining, for a given radial function Φ, which functions ^kΦ give rise to cardinal interpolation operators which are both bounded and invertible ℓ₂ (Z³). In addition to solving the cardinal interpolation problem (CIP) associated with such functions ^kΦ, our approach provides a unified framework and simpler proofs for the CIP associated with polyharmonic splines and Hardy multiquadrics.     

Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues

In many calculations, spectral discretization in space is coupled with a standard ordinary differential equation formula in time. To analyze the stability of such a combination, one would like simply to test whether the eigenvalues of the spatial discretization operator (appropriately scaled by the time step k) lie in the stability region for the o.d.e. formula, but it is well known that this kind of analysis is in general invalid. In the present paper we rehabilitate the use of stability regions by proving that a discrete linear multistep ‘method of lines’ approximation to a partial differential equation is Lax-stable, within a small algebraic factor, if and only if all of the -pseudo-eigenvalues of the spatial discretization operator lie within O() of the stability region as → 0. An -pseudo-eigenvalue of a matrix A is any number that is an eigenvalue of some matrix A + E with E ; our arguments make use of resolvents and are closely related to the Kreiss matrix theorem. As an application of our general result, we show that an explicit N-point Chebyshev collocation approximation of u_t = −xu_x on [−1, 1] is Lax-stable if and only if the time step satisfies k = O(N⁻²), although eigenvalue analysis would suggest a much weaker restriction of the form k CN⁻¹.     

New upper bounds on feedback vertex numbers in butterflies

Butterflies are undirected graphs of bounded degree. They are widely used as interconnection networks. In this paper we study the feedback vertex set problem for butterflies. We show that the feedback vertex set found by Luccio's algorithm [Inform. Process. Lett. 66 (1998) 59–64] for the k-dimensional butterfly B_k is of size . Besides, we propose an algorithm to find a feedback vertex set of size either or for B_k depending on whether k is even or odd.     

A new way of counting n

In this paper, we shall give a combinatorial proof of the following equation:

,

where m and n are positive integers, m ≤ n, and k₁, k₂, …, k_n-1 are nonnegative integers.     

Research on auto-reasoning process planning using a knowledge based semantic net

A process-planning model (PP model) is proposed to convert the geometric features into manufacture machining operations and sequence the machining operations of the part in a feasible and effective order. The process-planning model (PP model) construct a feature framework that makes a mapping from geometric features into machining operations. A semantic net named the Precedence-Relations-Net is established to reflect the precedence relationships among the machining operations. The vectors and the matrixes are employed to construct a mathematical sequencing model. A part is decomposed into several basic geometrical units, namely, U₁, U₂, … , U_N. For each unit U_i, two vectors, named F_i and P_i, represent the features and machining operations of U_i. Finally, a matrix named PP is used to memorize the process plan, and a matrix – PO (performing objects) – represents the object of machining operations.     

Subspace selection algorithms to be used with the nonlinear projection methods in solving systems of nonlinear equations

The nonlinear projection methods are minimization procedures for solving systems of nonlinear equations. They permit reevaluation of n_k, 1 ≤ n_k ≤ n, components of the approximate solution vector at each iteration step where n is the dimension of the system. At iteration step k, the reduction in the norm of the residue vector depends upon the n_k components which are reevaluated. These n_k components are obtained by solving a linear system.

We present two algorithms for determining the components to be modified at each iteration of the nonlinear projection method and compare the use of these algorithms to Newton's method. The computational examples demonstrate that Newton's method, which reevaluates all components of the approximate solution vector at each iteration, can be accelerated by using the projection techniques.     

Separating k-separated eNCE graph languages

An eNCE graph grammar is k-separated (k1) if the distance between any two nonterminal nodes in any of its sentential forms is at least k. Let SEP_k denote the class of graph languages generated by k-separated grammars. Then, SEP₁ (SEP₂) is the class of eNCE (boundary eNCE) graph languages, and so SEP₂SEP₁. Recently, Engelfriet (1991) showed that SEP₃SEP₂ and conjectured that, in fact, SEP_k+1SEP_k for each k 1. We prove this conjecture affirmatively.     

Minimum cost source location problem with vertex-connectivity requirements in digraphs

Given a digraph (or an undirected graph) G=(V,E) with a set V of vertices v with nonnegative real costs w(v), and a set E of edges and a positive integer k, we deal with the problem of finding a minimum cost subset SV such that, for each vertex vV−S, there are k vertex-disjoint paths from S to v. In this paper, we show that the problem can be solved by a greedy algorithm in time in a digraph (or in time in an undirected graph), where n=|V| and m=|E|. Based on this, given a digraph and two integers k and ℓ, we also give a polynomial time algorithm for finding a minimum cost subset SV such that for each vertex vV−S, there are k vertex-disjoint paths from S to v as well as ℓ vertex-disjoint paths from v to S.     

Blended Hermite interpolants

In (Röschel, l997) B-spline technique was used for blending of Lagrange interpolants. In this paper we generalize this idea replacing Lagrange by Hermite interpolants. The generated subspline b(t) interpolates the Hermite input data consisting of parameter values t_i and corresponding derivatives a_i,j, j=0,…,_i−1, and is called blended Hermite interpolant (BHI). It has local control, is connected in affinely invariant way with the input and consists of integral (polynomial) segments of degree 2·k−1, where k−1max{_i}−1 denotes the degree of the B-spline basis functions used for the blending. This method automatically generates one of the possible interpolating subsplines of class C^k−1 with the advantage that no additional input data is necessary.     

A learning scheme for a fuzzy k-NN rule

The performance of a fuzzy k-NN rule depends on the number k and a fuzzy membership-array W[l,m_R], where l and m_R denote the number of classes and the number of elements in the reference set X_R respectively. The proposed learning procedure consists in iterative finding such k and W which minimize the error rate estimate by the leaving ‘leaving one out‘ method.     

Piecewise hierarchical p-version axisymmetric solid element for laminated composites

This paper presents a piecewise hierarchical p-version finite element formulation for laminated composites axisymmetric solids for linear static analysis. The element formulation incorporates higher order deformation theories and is in total agreement with the physics of deformation in laminated composites. The element geometry is defined by eight nodes located on the boundaries of the element. The lamina thicknesses are used to create a nine-node p-version configuration for each lamina of the element. The displacement approximation for the element is piecewise hierarchical and is developed by first establishing a hierarchical displacement approximation for the nine-node configuration of each lamina of the laminate and then imposing interlamina continuity condition of displacements at the interfaces between laminas. The hierarchical approximation functions and the corresponding nodal variables for each lamina are derived from the Lagrange family of interpolation functions and can be of arbitrary polynomial order p_c and ^kp_η in the ε and ^kη directions for a typical lamina k. The formulation ensures C⁰ continuity, i.e., continuity of displacement across interelement as well as interlamina boundaries.

The element properties are constructed by assembling individual lamina properties which are derived using the principle of virtual work and the hierarchical displacement approximation for the laminas. Transformation matrices, formed based on interlamina continuity conditions, are used to transform each lamina's degrees of freedom into the degrees of freedom for the laminate. Thus, each individual lamina stiffness matrix and equivalent load vector are transformed and then summed to establish the laminate stiffness matrix and equivalent load vector. There is no restriction on either the number of laminas or their lay-up pattern. Each lamina can be generally orthotropic and the material directions and the layer thickness may vary from point to point within each lamina.

Numerical examples are presented to demonstrate the effectiveness, modeling convenience, accuracy, and overall superiority of the present formulation for laminated composite axisymmetric solids and shells.     

A small gain theorem for linear stochastic systems

A well-known result in linear control theory is the so-called “small gain” theorem stating that if given two plants with transfer matrix functions T₁ and T₂ in H^∞ such that T₁ < γ and T₂ < 1/γ, when coupling T₂ to T₁ such that u₂ = y₁ and u₁ = y₂ one obtains an internally stable closed system. The aim of the present paper is to describe a corresponding result for stochastic systems with state-dependent white noise.     

McClellan transformation and the construction of biorthogonal wavelet bases of LR

Bidimensional wavelet bases are constructed by means of McClellan's transformation applied to a pair of one-dimensional biorthogonal wavelet filters. It is shown that under some conditions on the transfer function F(ω₁,ω₂) associated to the McClellan transformation and on the dilation matrix D, it is possible to construct symmetric compactly supported biorthogonal wavelet bases of L²(R²). Finally, the construction method is illustrated by means of numerical examples.     

Multiple sequence comparison — a peptide matching approach

We present in this paper a peptide matching approach to the multiple comparison of a set of protein sequences. This approach consists in looking for all the words that are common to q of these sequences, where q is a parameter.

The comparison between words is done by using as reference an object called a model. In the case of proteins, a model is a product of subsets of the alphabet Σ of the amino acids. These subsets belong to a cover of Σ, that is, their union covers all of Σ. A word is said to be an instance of a model if it belongs to the model.

A further flexibility is introduced in the comparison by allowing for up to e errors in the comparison between a word and a model. These errors may concern gaps or substitutions not allowed by the cover. A word is said to be this time an occurrence of a model if the Levenshtein distance between it and an instance of the model is inferior or equal to e. This corresponds to what we call a Set-Levenshtein distance between the occurrences and the model itself. Two words are said to be similar if there is at least one model of which both are occurrences. In the special case where e = 0, the occurrences of a model are simply its instances. If a model M has occurrences in at least q of the sequences of the set, M is said to occur in the set.

The algorithm presented here is an efficient and exact way of looking for all the models, of a fixed length k or of the greatest possible length k_max, that occur in a set of sequences. It is linear in the total length n of the sequences and proportional to (e + 2)(2e+ 1)k^e+1p^e+1 g^k where k n is a small value in all practical situations, p is the number of sets in the cover and g is related to the latter's degree of nontransitivity.

Models are closely related to what is called a consensus in the biocomputing area, and covers are a good way of representing complex relationships between the amino acids.