We show that the zero-point errors in the proper motions are ≤1 mas yr−1 for R>17, and are no larger than ∼10 mas yr−1 for R<17 mas yr−1. Monthly Notices of the Royal Astronomical Society Oxford University Press Introduction in astrometry, vectors are extensively used to describe the geometrical relationships among celestial bodies, for example between the observer and the observed object. Practical calculations using computer software are today mainly carried out with the help of vector and matrix algebra, rather than the trigonometry formulae typically found in older textbooks. It turns out that this often provides a better insight into the problem, and hence reduces the risk of errors in the derived algorithms, in addition to being advantageous in terms of computational speed and accuracy. This chapter provides a brief introduction to the use of vectors and matrices in astrometry. It broadly uses the notational conventions from C. Murray's Vectorial Astrometry (1983), which seem to provide a particularly clear and consistent framework for theoretical work as well as practical calculations. By way of illustration, some useful transformations are explained in detail, while references to the general literature are provided for other applications. Only vectors in three-dimensional Euclidean space are considered. What are vectors? In this section we define classical vectors, unit vectors, matrices and present some important formulae for manipulating them. Vectors and matrices Classically, a vector is defined as a physical entity having both magnitude (length) and direction, as opposed to a scalar that only has magnitude. Vectors can be visualized as arrows that exist in space quite independently of any coordinate system. The usual vector operations – addition, subtraction, multiplication by a scalar, scalar (dot) product, and vector (cross) product – have simple geometrical interpretations that are independent of the coordinate system. In this paper, we present a novel differential geometric characterization of two- and three-degree-of-freedom rigid body kinematics, using a metric defined on dual vectors. The instantaneous angular and linear velocities of a rigid body are expressed as a dual velocity vector, and dual inner product is defined on this dual vector, resulting in a positive semi-definite and symmetric dual matrix. We show that the maximum and minimum magnitude of the dual velocity vector, for a unit speed motion, can be obtained as eigenvalues of this dual matrix. Furthermore, we show that the tip of the dual velocity vector lies on a dual ellipse for a two-degree-of-freedom motion and on a dual ellipsoid for a three-degree-of-freedom motion. In this manner, the velocity distribution of a rigid body can be studied algebraically in terms of the eigenvalues of a dual matrix or geometrically with the dual ellipse and ellipsoid. The second-order properties of the two- and three-degree-of-freedom motions of a rigid body are also obtained from the derivatives of the elements of the dual matrix. This results in a definition of the geodesic motion of a rigid body. The theoretical results are illustrated with the help of a spatial 2R and a parallel three-degree-of-freedom manipulator.*Communicated by S. ![]() In this paper a new algebraic criterion is proposed for existence of the robust controller of the linear system, with is formed using the output vector. The criterion obtained does not only checks the problem for a possibility of solving, but synthesis design the desired controller with respect to the given phase constraints. ![]() This criterion is base on the properties of input - value matrices to. Have a maximum real eigenvalue with the appropriate eig does not only check the problem for a possibility of solving, but synthesis design the desired controller with respect to the given phase constraints.
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