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

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.

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.

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.

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. >