Showing papers on "Piecewise published in 1983"
22 Jun 1983
TL;DR: In this paper, the authors present a methodology of feedback control to achieve accurate tracking for a class of nonlinear time-varying systems in the presence of disturbances and parameter variations.
Abstract: A methodology is presented of feedback control to achieve accurate tracking for a class of nonlinear time-varying systems in the presence of disturbances and parameter variations. The methodology uses in its idealized form piecewise continuous feedback control laws, resulting in the state trajectory `sliding' along a discontinuity surface in the state space. The idealized form of the methodology results in perfect tracking of the required signals; however certain non-idealities associated with its implementation cause the trajectory to 'chatter' along the sliding surface resulting in the generation of an undesirable high-frequency component which may excite high-frequency unmodelled dynamics of the control systems. To rectify this situation, it is shown how continuous control laws which approximate the discontinuous control law may be used to obtain disturbance and parameter variation insensitive tracking. At the same time, the continuous control laws decrease the extent of unwanted high-frequency signals.
1,636 citations
TL;DR: In this paper, the problem of joint trajectory planning for industrial robots is divided into two parts: optimum path planning for off-line processing followed by on-line path tracking, where the path planning is done at the joint level and the path tracking can be achieved by adopting the existing approach.
Abstract: Because of physical constraints, the optimum control of industrial robots is a difficult problem. An alternative approach is to divide the problem into two parts: optimum path planning for off-line processing followed by on-line path tracking. The path tracking can be achieved by adopting the existing approach. The path planning is done at the joint level. Cubic spline functions are used for constructing joint trajectories for industrial robots. The motion of the robot is specified by a sequence of Cartesian knots, i.e., positions and orientations of the hand. For an N -joint robot, these Cartesian knots are transformed into N sets of joint displacements, with one set for each joint. Piecewise cubic polynomials are used to fit the sequence of joint displacements for each of the N joints. Because of the use of the cubic spline function idea, there are only n - 2 equations to be solved for each joint, where n is the number of selected knots. The problem is proved to be uniquely solvable. An algorithm is developed to schedule the time intervals between each pair of adjacent knots such that the total traveling time is minimized subject to the physical constraints on joint velocities, accelerations, and jerks. Fortran programs have been written to implement: 1) the procedure for constructing the cubic polynomial joint trajectories; and 2) the algorithm for minimizing the traveling time. Results are illustrated by means of a numerical example.
474 citations
TL;DR: In this article, the authors studied functional differential equations of advanced type with piecewise constant argument deviations and showed that they are closely related to impulse, loaded and especially to difference equations, and have the structure of continuous dynamical systems within intervals of unit length.
Abstract: Functional differential equations of advanced type with piecewise constant argument deviations are studied. They are closely related to impulse, loaded and, especially, to difference equations, and have the structure of continuous dynamical systems within intervals of unit length.
195 citations
TL;DR: A new effective method for sequential adaptive segmentation is proposed, which is based on parallel application of two sequential parameter estimation procedures, which demonstrates the good detection properties of the algorithm and in particular an excellent ability to allocate the segment boundaries even within a sequence of short segments.
189 citations
PARC1
TL;DR: An algorithm is developed that takes a set of sample points, plus optional endpoint and tangent vector specifications, and iteratively derives a single parametric cubic polynomial that lies close to the data points as defined by an error metric based on least-squares.
Abstract: Parametric piecewise-cubic functions are used throughout the computer graphics industry to represent curved shapes. For many applications, it would be useful to be able to reliably derive this representation from a closely spaced set of points that approximate the desired curve, such as the input from a digitizing tablet or a scanner. This paper presents a solution to the problem of automatically generating efficient piecewise parametric cubic polynomial approximations to shapes from sampled data. We have developed an algorithm that takes a set of sample points, plus optional endpoint and tangent vector specifications, and iteratively derives a single parametric cubic polynomial that lies close to the data points as defined by an error metric based on least-squares. Combining this algorithm with dynamic programming techniques to determine the knot placement gives good results over a range of shapes and applications.
188 citations
Book•
01 Jun 1983
TL;DR: In this article, the authors proposed piecewise constant orthogonal basis functions (PCF) for linear and non-linear linear systems, and the optimal control of linear lag-free and time-lag systems.
Abstract: I Piecewise constant orthogonal basis functions.- II Operations on square integrable functions in terms of PCBF spectra.- III Analysis of lumped continuous linear systems.- IV Analysis of time delay systems.- V Solution of functional differential equations.- VI Analysis of non-linear and time-varying systems.- VII Optimal control of linear lag-free systems.- VIII Optimal control of time-lag systems.- IX Solution of partial differential equations (PDE) [W55].- X Identification of continuous lumped parameter systems.- XI Parameter identification in distributed systems.
188 citations
TL;DR: In this paper, an approximation of the Hamilton-Jacobi-Bellman equation connected with the infinite horizon optimal control problem with discount is proposed, and the approximate solutions are shown to converge uniformly to the viscosity solution, in the sense of Crandall-Lions, of the original problem.
Abstract: An approximation of the Hamilton-Jacobi-Bellman equation connected with the infinite horizon optimal control problem with discount is proposed. The approximate solutions are shown to converge uniformly to the viscosity solution, in the sense of Crandall-Lions, of the original problem. Moreover, the approximate solutions are interpreted as value functions of some discrete time control problem. This allows to construct by dynamic programming a minimizing sequence of piecewise constant controls.
178 citations
TL;DR: It is shown in this paper that non-conforming finite elements on the triangle using second-degree polynomials can be easily built and used and that this element exhibits a very peculiar regularity property.
Abstract: It is shown in this paper that non-conforming finite elements on the triangle using second-degree polynomials can be easily built and used. Indeed they appear as an ‘enriched’ version of the standard piecewise quadratic six-node element. This work is divided into two parts. In the first we present the basic properties of the element, namely how it can be built and basic error estimates. We also show that this element exhibits a very peculiar regularity property. In the second part we apply our element to the approximation of viscous incompressible flows and more generally to the approximation of incompressible materials.
174 citations
01 Jul 1983
TL;DR: Using a restricted form of quintic Hermite interpolation, it is possible to allow distinct bias and tension parameters at each joint without destroying geometric continuity, which provides a new means of obtaining local control of bias and pressure in piecewise polynomial curves and surfaces.
Abstract: The Beta-spline introduced recently by Barsky is a generalization of the uniform cubic B-spline: parametric discontinuities are introduced in such a way as to preserve continuity of the unit tangent and curvature vectors at joints (geometric continuity) while providing bias and tension parameters, independent of the position of control vertices, by which the shape of a curve or surface can be manipulated. Using a restricted form of quintic Hermite interpolation, it is possible to allow distinct bias and tension parameters at each joint without destroying geometric continuity. This provides a new means of obtaining local control of bias and tension in piecewise polynomial curves and surfaces.
168 citations
TL;DR: In this paper, a new multiplicative efficiency formulation is developed wherein the efficiency values are invariant under changes in the units of measurement of outputs and inputs, and it is shown that the associated Data Envelopment Analysis (DEA) implies that optimal envelopments are of piecewise Cobb-Douglas type.
147 citations
TL;DR: In this article, the authors showed that piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or = RHo continuous will not approximate certain smooth functions at all unless RHo is kept below (r-3)/2.
Abstract: : One of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh. This report shows this to be untrue in the multivariate setting. In particular, it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant. In an extreme case, piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or = RHo continuous will fail to approximate certain smooth functions at all (as the grid goes to zero) unless RHo is kept below (r-3)/2. During the analysis of approximation on a certain regular triangular grid, a novel kind of bivariate B-spline is introduced. This B-spline, in contrast to the established multivariate B-spline derived from a simplex, can be made to have all its breaklines in a given regular grid. This makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation, and its properties will be explored further.
TL;DR: In this article, the collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous, and a satisfactory sense of point evaluation is given for elements of the integral equation.
Abstract: The collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous. Difficulties with the usual function space setting of $L_\infty (D)$ are discussed, and a satisfactory sense of point evaluation is given for elements of $L_\infty (D)$. Other approaches which are discussed include (i) reformulation as a degenerate kernel method, (ii) the prolongation-restriction framework of Noble, (iii) other function space settings, and (iv) reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation.
TL;DR: This paper presents a new generation of parametric piecewise-cubic functions that are used throughout the computer graphics industry to represent curved shapes and some examples show how these functions can be modified for curved topographies.
Abstract: Parametric piecewise-cubic functions are used throughout the computer graphics industry to represent curved shapes. For many applications, it would be useful to be able to reliably derive this repr...
TL;DR: In this paper, the divergence operator acting on continuous piecewise polynomials on a fixed triangulation was characterized and proved to have a maximal right-inverse whose norm grows at most algebraically with the degree of the piecewise permutation.
Abstract: In the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation; more specifically, we characterize the range and prove the existence of a maximal right-inverse whose norm grows at most algebraically with the degree of the piecewise polynomials.
In the last part of this paper we apply these results to thep-version of the Finite Element Method for a nearly incompressible material with homogeneous Dirichlet boundary conditions. We show that thep-version maintains optimal convergence rates in the limit as the Poisson ratio approaches 1/2. This fact eliminates the need for any "reduced integration" such as customarily used in connection with the more standardh-version of the Finite Element Method.
TL;DR: In this article, a biharmonic boundary integral equation (BBIE) method is used to solve a two dimensional contained viscous flow problem and analytic expressions are used for the piecewise integration of all the kernel functions rather than the more time-consuming method of Gaussian quadrature.
TL;DR: In this paper, the determinants of Toeplitz and Wiener-Hopf operators with piecewise continuous symbols were obtained for the case of Wα(σ) and Wα (σ) is the WienerHopf operator defined on L2(0, α) with piece-wise continuous symbol σ having a finite number of discontinuities at ξr.
TL;DR: This paper describes a method for connecting an M OSFET 2-D device simulator to a circuit simulator via a 3-D table look-up MOSFET model via a proposed monotonic piecewise cubic interpolation technique.
Abstract: This paper describes a method for connecting an MOSFET 2-D device simulator to a circuit simulator via a 3-D table look-up MOSFET model. The computational cost of the device simulator is drastically reduced by a proposed monotonic piecewise cubic interpolation technique. With this technique, the device simulator needs to calculate only 100 ~ 200 points to make up an accurate 3-D table look-up MOSFET model. The computational time necessary for the interpolation is only about one third of the time for calculating one current point by the device simulator.
TL;DR: The algorithm to create knots is a modification of de Boor's knot placement scheme and has been realized in the CADCAM system SYRKO, a Daimler-Benz development for car body design and manufacturing.
Abstract: For piecewise polynomial representation of curves, an algorithm to create knots is presented. The aim is to minimize the interpolation error for a given number of knots or, conversely, the number of knots needed to interpolate within a tolerance. The method used is a modification of de Boor's knot placement scheme. The algorithm described in this paper has been realized in the CADCAM system SYRKO, a Daimler-Benz development for car body design and manufacturing.
01 Jan 1983
TL;DR: Piecewise Smooth Homotopies for finding Zeros of Entire Functions as discussed by the authors, where solving for Stationary Points by LCPs is Mixing Newton Iterates.
Abstract: Piecewise Smooth Homotopies -- Global Convergence Rates of Piecewise-Linear Continuation Methods: A Probabilistic Approach -- Relationships between Deflation and Global Methods in the Problem of Approximating Additional Zeros of a System of Nonlinear Equations -- Smooth Homotopies for Finding Zeros of Entire Functions -- Where Solving for Stationary Points by LCPs Is Mixing Newton Iterates -- On the Equivalence of the Linear Complementarity Problem and a System of Piecewise Linear Equations: Part II -- Relations between PL Maps, Complementary Cones, and Degree in Linear Complementarity Problems -- A Note on Stepsize Control for Numerical Curve Following -- On a Class of Linear Complementarity Problems of Variable Degree -- Linear Complementarity and the Degree of Mappings -- Sub- and Supersolutions for Nonlinear Operators: Problems of Monotone Type -- An Efficient Procedure for Traversing Large Pieces in Fixed Point Algorithms -- The Application of Fixed Point Methods to Economics -- On a Theory of Cost for Equation Solving -- Algorithms for the Linear Complementarity Problem Which Allow an Arbitrary Starting Point -- Engineering Applications of the Chow-Yorke Algorithm -- Availability of Computer Codes for Piecewise-Linear and Differentiable Homotopy Methods.
TL;DR: The suitability of B-splines as a basis for piecewise polynomial solution representation for solving differential equations is challenged in this article, where two alternative local solution representations are considered in the context of collocating ordinary differential equations: "Hermite-type" and "Lmonomial" representations.
Abstract: The suitability of B-splines as a basis for piecewise polynomial solution representation for solving differential equations is challenged. Two alternative local solution representations are considered in the context of collocating ordinary differential equations: “Hermite-type” and “Lmonomial”. Both are much easier and shorter to implement and somewhat more efficient than B-splines.A new condition number estimate for the B-splines and Hermite-type representations is presented. One choice of the Hermite-type representation is experimentally determined to produce roundoff errors at most as large as those for B-splines. The monomial representation is shown to have a much smaller condition number than the other ones, and correspondingly produces smaller roundoff errors, especially for extremely nonuniform meshes. The operation counts for the two local representations considered are about the same, the Hermite-type representation being slightly cheaper. It is concluded that both representations are preferable,...
01 Apr 1983
TL;DR: It is shown that the space of bivariate C1 piecewise cubic functions on a hexagonal mesh of size h approximates to certain smooth functions only to within O(h3) even though it contains a local partition of every cubic polynomial.
Abstract: : It is shown that the space of bivariate C1 piecewise cubic functions on a hexagonal mesh of size h approximates to certain smooth functions only to within O(h3) even though it contains a local partition of every cubic polynomial. One measures the approximation power of a family S of piecewise polynomial approximating functions on some partition in terms of the meshsize h of that partition. Typically, the error of approximation goes to zero like hr as the meshsize goes to zero, with r depending on the smoothness of the function being approximated. There is a maximal r typical for the space S used, and faster convergence rates are possible only for very special functions. Naturally, one would like this optimal rate or approximation order hr to be as fast as possible, i.e., would like the maximal r to be as large as possible. In any case, it is an important practical question to ascertain, for a given approximating space S, what its optimal approximation order is.
DOI•
01 Nov 1983TL;DR: A parameter-embedding approach to the optimal control of linear time-delay systems, and a simple computational algorithm via Walsh functions that employs the concept of Walsh operational matrices for delay and advance.
Abstract: The paper presents a parameter-embedding approach to the optimal control of linear time-delay systems, and develops a simple computational algorithm via Walsh functions. The algorithm employs the concept of Walsh operational matrices for delay and advance. These operational matrices provide us with a means of numerically integrating differential equations with delayed and advanced arguments; a major task in the process of computing optimal controls for time-delay systems. An attractive feature of the present method is its ultimate simplicity and the resulting piecewise constant controls are convenient in practical implementation.
01 Sep 1983
TL;DR: In this paper, the eigenfunctions are given in terms of Bessel functions and the coefficients of integration as well as eigenvalues are determined accurately such that the boundary conditions are satisfied.
Abstract: The solution of the three-dimensional linear hydrodynamic equations which describe wind-driven flow in a homogeneous sea are solved using the eigenfunction method. The eddy viscosity is taken to vary piecewise linearly in the vertical over an arbitrary number of layers. Using this formulation the eigenfunctions are given in terms of Bessel functions. The coefficients of integration as well as the eigenvalues are determined accurately such that the boundary conditions are satisfied. Values of the eigenfunctions at any depth can then be determined very fast and to a high degree of accuracy. Current profiles at any position can hence be computed accurately. The expansion of the horizontal component of current converges very fast at all depths.
TL;DR: In this article, an approximation theorem of stochastic differential equations driven by semimartingales is proved, based on approximation of semimARTingales by a sequence of processes with piecewise monotonic sample functions.
TL;DR: In this paper, maximum norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable, using the continuous in time Galerkin method based on piecewise polynomial approximating functions on a quasi-uniform mesh.
Abstract: Maximum-norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable, using the continuous in time Galerkin method based on piecewise polynomial approximating functions on a quasi-uniform mesh.
TL;DR: The first part of this paper is concerned with global characterizations of both the multivariate B-spline and the multiivariate truncated power function as smooth piecewise polynomials.
Abstract: The first part of this paper is concerned with global characterizations of both the multivariate B-spline and the multivariate truncated power function as smooth piecewise polynomials. In the second part of the paper we establish combinatorial criteria for the linear independence of multivariate B-splines corresponding to certain configurations of knot sets.
TL;DR: In this paper, a logical extension of the piecewise optimal linearization procedure leads to the Gaussian decoupling scheme, where no iteration is required, which is equivalent to solving a few coupled equations.
Abstract: We show that a logical extension of the piecewise optimal linearization procedure leads to the Gaussian decoupling scheme, where no iteration is required. The scheme is equivalent to solving a few coupled equations. The method is applied to models which represent (a) a single steady state, (b) passage from an initial unstable state to a final preferred stable state by virtue of a finite displacement from the unstable state, and (c) a bivariate case of passage from an unstable state to a final stable state. The results are shown to be in very good agreement with the Monte Carlo calculations carried out for these cases. The method should be of much value in multidimensional cases.