Topic

# Transformation matrix

About: Transformation matrix is a research topic. Over the lifetime, 3132 publications have been published within this topic receiving 36427 citations. The topic is also known as: matrix of a linear operator & matrix of linear operator.

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TL;DR: New hierarchical solid modeling operations are developed, which simulate twisting, bending, tapering, or similar transformations of geometric objects, and the chief result is that the normal vector of an arbitrarily deformed smooth surface can be calculated directly from the surfacenormal vector of the undeformed surface and a transformation matrix.

Abstract: New hierarchical solid modeling operations are developed, which simulate twisting, bending, tapering, or similar transformations of geometric objects The chief result is that the normal vector of an arbitrarily deformed smooth surface can be calculated directly from the surface normal vector of the undeformed surface and a transformation matrix Deformations are easily combined in a hierarchical structure, creating complex objects from simpler ones The position vectors and normal vectors in the simpler objects are used to calculate the position and normal vectors in the more complex forms; each level in the deformation hierarchy requires an additional matrix multiply for the normal vector calculation Deformations are important and highly intuitive operations which ease the control and rendering of large families of three-dimensional geometric shapes

823 citations

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19 Mar 2008

TL;DR: In this article, Mittag-Leffler functions and fractional calculus are used for estimating density and order statistics in time series and wavelet analysis, respectively, in the context of matrix arguments.

Abstract: Basic Ideas of Special Functions and Statistical Distributions.- Mittag-Leffler Functions and Fractional Calculus.- An Introduction to q-Series.- Ramanujan's Theories of Theta and Elliptic Functions.- Lie Group and Special Functions.- Applications to Stochastic Process and Time Series.- Applications to Density Estimation.- Applications to Order Statistics.- Applications to Astrophysics Problems.- An Introduction to Wavelet Analysis.- Jacobians of Matrix Transformations.- Special Functions of Matrix Argument.

388 citations

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01 Jan 1932

TL;DR: In this paper, the unitary equivalence of self-adjoint transformations is proved for linear transformations in Hilbert space, and the operational calculus is used to prove the equivalence.

Abstract: Hilbert space and its realizations Transformations in Hilbert space Examples of linear transformations Resolvents, spectra, reducibility Self-adjoint transformations The operational calculus The unitary equivalence of self-adjoint transformations General types of linear transformations Applications Index.

386 citations

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TL;DR: A common mathematical framework is developed to solve for the hand-eye calibration problem using either of the two formulations and the nonlinear optimization method, which solves for rotation and translation simultaneously, seems to be the most robust one with respect to noise and measurement errors.

Abstract: Whenever a sensor is mounted on a robot hand, it is important to know the relationship between the sensor and the hand. The problem of determining this relationship is referred to as the hand-eye calibration problem. Hand-eye calibration is impor tant in at least two types of tasks: (1) map sensor centered measurements into the robot workspace frame and (2) tasks allowing the robot to precisely move the sensor. In the past some solutions were proposed, particularly in the case of the sensor being a television camera. With almost no exception, all existing solutions attempt to solve a homogeneous matrix equation of the form AX = X B. This article has the following main contributions. First we show that there are two possible formulations of the hand-eye calibration problem. One formu lation is the classic one just mentioned. A second formulation takes the form of the following homogeneous matrix equation: MY = M'YB. The advantage of the latter formulation is that the extrinsic and intrinsic parameters of the camera need not be made explicit. Indeed, this formulation directly uses the 3 x4 perspective matrices ( M and M' ) associated with two positions of the camera with respect to the calibration frame. Moreover, this formulation together with the classic one covers a wider range of camera-based sensors to be calibrated with respect to the robot hand: single scan-line cameras, stereo heads, range finders, etc. Second, we develop a common mathematical framework to solve for the hand-eye calibration problem using either of the two formulations. We represent rotation by a unit quaternion and present two methods: (1) a closed-form solution for solving for rotation using unit quaternions and then solving for translation and (2) a nonlinear technique for simultane ously solving for rotation and translation. Third, we perform a stability analysis both for our two methods and for the lin ear method developed by Tsai and Lenz (1989). This analysis allows the comparison of the three methods. In light of this comparison, the nonlinear optimization method, which solves for rotation and translation simultaneously, seems to be the most robust one with respect to noise and measurement errors.

382 citations

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TL;DR: Closed-form equations of motion are presented for planar lightweight robot arms with multiple flexible links based on standard frame transformation matrices describing both rigid rotation and flexible displacement, under small deflection assumption.

Abstract: Closed-form equations of motion are presented for planar lightweight robot arms with multiple flexible links. The kinematic model is based on standard frame transformation matrices describing both rigid rotation and flexible displacement, under small deflection assumption. The Lagrangian approach is used to derive the dynamic model of the structure. Links are modeled as Euler-Bernoulli beams with proper clamped-mass boundary conditions. The assumed modes method is adopted in order to obtain a finite-dimensional model. Explicit equations of motion are detailed for two-link case assuming two modes of vibration for each link. The associated eigenvalue problem is discussed in relation with the problem of time-varying mass boundary conditions for the first link. The model is cast in a compact form that is linear with respect to a suitable set of constant parameters. Extensive simulation results that validate the theoretical derivation are included. >

359 citations