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Introduction
Published in Sing-Tze Bow, Pattern Recognition and Image Preprocessing, 2002
Each spectrum component value can be considered as a variable in n-dimensional space, known as pattern space, where each spectrum component is assigned to a dimension. Each pattern then appears as a point in the pattern space. It is a vector composed of n component values in the n-dimensional coordinates. A pattern x can then be represented as x=|x1x2⋮xn|
Decentralized force and motion control of multiple cooperative manipulators
Published in Automatika, 2021
Hamid AzizZadeh, Mohammad Bagher Menhaj, Heidar Ali Talebi
Representing every EE in object frame means , there is due to constrains on relative displacement of EE about the object. Kinematic constraints could be shown as wherein denotes rotation matrix from frame to frame . For representation of orientation, we use unit quaternion , where is the real part and is the vector part of it. The represents cross product. As shown above, angular velocity of the object and end-effectors are equal, lead to and also angular acceleration of them show another constraint .
Re-LSTM: A long short-term memory network text similarity algorithm based on weighted word embedding
Published in Connection Science, 2022
Weidong Zhao, Xiaotong Liu, Jun Jing, Rongchang Xi
Each feature word in the text corresponds to a point on the vector space following word2vec model training. N-dimensional vectors can be used to represent points in a vector space. As in the formula (9). Among them, represents a single n-dimensional word vector, represents the feature word, and represents the corresponding dimension. The set of word vectors is , and the feature word weight set is .
Attitude estimation of connected drones based on extended Kalman filter under real outdoor environments
Published in Advanced Robotics, 2022
Kento Fukuda, Shin Kawai, Hajime Nobuhara
The quaternion expresses the attitude of the Body coordinate system from the World coordinate system. The advantage of using the quaternion is that no singularity point occurs when using Euler angles [33]. Generally, attitude control is impossible in singularity points. The quaternion expresses the attitude by using a four-dimensional vector . The quaternion, means θ rotation about the direction vector , where , and are the unit direction vector, which is the axis of rotation of the quaternion. All the quaternions treated in this study are in the unit of size , where represents the Euclidean norm of the vector. The rotation matrix is given as, Here, is a unit matrix of , and is a representation matrix of the vector product satisfying in the vector product. For example, the expression matrix of the vector product of is equal to Furthermore, the product of the quaternions is defined as At this time, is given. In a quaternion, the relationship between the true value , the estimated value , and the residual is If the difference between and is significantly small, the rotation angle of the residual is so small that Equation (1) yields . Therefore, if we truncate the small quantities of the order two or more from Equation (2), an approximation like can be held for the rotation matrix of the residuals.