Self-Tuning Manipulator Control in Cartesian Base Coordinate System

1985 ◽  
Vol 107 (4) ◽  
pp. 316-323 ◽  
Author(s):  
A. J. Koivo
Keyword(s):  
Coordinate System ◽  
Digital Simulation ◽  
External Input ◽  
Input Vector ◽  
Unknown Parameters ◽  
Output Vector ◽  
Self Tuning ◽  
On Line ◽  
Cartesian Coordinate ◽  
Manipulator Control

A discrete-time stochastic model for the motion of a robotic manipulator system is introduced. The input vector consists of the voltages to the joint motors, and the output vector has the velocities (positions) of the gripper expressed in the world (xyz) coordinate system as the components. The model is a linear multivariate autoregressive model with external input. The unknown parameters of the model can be calculated recursively on-line by the least squares algorithm. An adaptive self-tuning type controller is then designed by minimizing the expected value of a quadratic criterion. This performance index penalizes the deviations of the actual path of the gripper from the desired values expressed in the Cartesian coordinate system and the use of the energy associated with the input vector. Digital simulation results using the parameter estimation and the control algorithms are presented, and discussed.

Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

scholarly journals Vector Arithmetic in the Triangular Grid

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 373
Author(s):  
Khaled Abuhmaidan ◽  
Monther Aldwairi ◽  
Benedek Nagy
Keyword(s):  
Coordinate System ◽  
Triangular Grid ◽  
Vector Addition ◽  
Plane Space ◽  
Physical Simulations ◽  
Rectangular Grids ◽  
Cartesian Coordinate ◽  
Zero Sum ◽  
Coordinate Geometry ◽  
Point Lattice

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.

On the Flexural-Torsional Behavior of a Straight Elastic Beam Subject to Terminal Moments

1993 ◽  
Vol 60 (2) ◽  
pp. 498-505 ◽  
Author(s):  
Z. Tan ◽  
J. A. Witz
Keyword(s):  
Coordinate System ◽  
Cross Section ◽  
Equilibrium Equation ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Elastic Beam ◽  
Cartesian Coordinate System ◽  
Kinematic Constraints ◽  
Torsional Behavior ◽  
Cartesian Coordinate

This paper discusses the large-displacement flexural-torsional behavior of a straight elastic beam with uniform circular cross-section subject to arbitrary terminal bending and twisting moments. The beam is assumed to be free from any kinematic constraints at both ends. The equilibrium equation is solved analytically with the full expression for curvature to obtain the deformed configuration in a three-dimensional Cartesian coordinate system. The results show the influence of the terminal moments on the beam’s deflected configuration.

Natural Frequencies of Parallelogrammic Plates Using Characteristic Orthogonal Polynomials in Rayleigh-Ritz Method

1992 ◽  
Author(s):  
L. T. Lee ◽  
W. F. Pon
Keyword(s):  
Boundary Conditions ◽  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Energy Functional ◽  
Comprehensive Treatment ◽  
Energy Functionals ◽  
Simply Supported ◽  
Cartesian Coordinate

Abstract Natural frequencies of parallelogrammic plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generalted by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. The strain energy functional and kinetic energy functionals are transformed from Cartesian coordinate system to a skew coordinate system. The natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods with all four edges clamped boundary conditions and greet agreements are found between them. The natural frequencies for parallelogrammic plates with other boundary conditions, such as four edges simply supported, clamped-free and simply supported-free, are also obtained. This method is considered as a better and accurate comprehensive treatment for this type of problems.

scholarly journals A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate

2018 ◽  
Vol 1 (4) ◽  
Author(s):  
Debabrata Datta ◽  
T K Pal
Keyword(s):  
Diffusion Equation ◽  
Coordinate System ◽  
Lattice Boltzmann ◽  
Spherical Coordinate System ◽  
Cartesian Coordinate System ◽  
Spherical Coordinate ◽  
Cartesian Coordinate ◽  
Lattice Boltzmann Models ◽  
Boltzmann Method ◽  
Good Agreement

Lattice Boltzmann models for diffusion equation are generally in Cartesian coordinate system. Very few researchers have attempted to solve diffusion equation in spherical coordinate system. In the lattice Boltzmann based diffusion model in spherical coordinate system extra term, which is due to variation of surface area along radial direction, is modeled as source term. In this study diffusion equation in spherical coordinate system is first converted to diffusion equation which is similar to that in Cartesian coordinate system by using proper variable. The diffusion equation is then solved using standard lattice Boltzmann method. The results obtained for the new variable are again converted to the actual variable. The numerical scheme is verified by comparing the results of the simulation study with analytical solution. A good agreement between the two results is established.

A Simple Algorithm for Self-Tuning Control of Low-Order Models

1988 ◽  
Vol 202 (4) ◽  
pp. 301-303
Author(s):  
M A Magdy ◽  
J Katupitiya
Keyword(s):  
Simple Algorithm ◽  
Estimation Algorithm ◽  
Recursive Least Squares ◽  
Discrete Time System ◽  
Time System ◽  
Simplified Approach ◽  
Estimated Parameters ◽  
Self Tuning ◽  
On Line ◽  
Low Order

A simplified approach to the design of self-tuning controllers or low-order systems is presented. The parameters of the continuous time system rather than the discrete time system are identified on-line using a recursive least-squares estimation algorithm. The estimated parameters are then used to calculate the controller parameters so that the system is forced to have a pre-specified closed-loop system performance. In some applications, adopting this approach reduces the number of parameters to be estimated. Further, the controller parameters are obtained using closed-form equations, thus avoiding the on-line solution of polynomial equations. An example is included.

Non-Linear Control of a Neutralisation Process

1971 ◽  
Vol 4 (9) ◽  
pp. T151-T157 ◽  
Author(s):  
P D Roberts
Keyword(s):  
Simulation Study ◽  
Digital Simulation ◽  
Linear Control ◽  
Attractive Alternative ◽  
Error Signal ◽  
Linear Controller ◽  
Linear Characteristic ◽  
Non Linear ◽  
On Line ◽  
Extra Parameter

The paper describes a digital simulation study of the application of a non-linear controller to the regulation of a single stage neutralisation process. In the controller, the proportional gain increases with amplitude of controller error signal. The performance of the non-linear controller is compared with that of a conventional linear controller and with the performance obtained by employing a linear controller with a linearisation network designed to compensate for the non-linear characteristic of the neutralisation curve. Although the performance of the non-linear controller is inferior to that obtained by employing a perfect linearisation network, its performance is still considerably superior to that obtained by using a conventional linear controller when operating at a symmetrical point on the neutralisation curve. In contrast to the linearisation network technique, the non-linear controller contains only one extra parameter and can be readily tuned on-line without prior knowledge of the neutralisation curve. Hence, it can be considered as an attractive alternative for the control of neutralisation processes.

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