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Triad Method
The Triad method is one of the earliest and simplest solutions to the spacecraft attitude determination problem. Given the knowledge of two vectors in the reference and body coordinates of a satellite, the Triad algorithm obtains the direction cosine matrix relating to both frames. Harold Black played a key role in the development of the guidance, navigation, and control of the U.S. Navy's Transit satellite system at Johns Hopkins Applied Physics Laboratories. Triad represented the state of practice in spacecraft attitude determination before the advent of Wahba's problem and its several optimal solutions. Covariance analysis for Black's solution was subsequently provided by Markley. Summary We consider the linearly independent reference vectors \vec_ and \vec_2 . Let \vec_1, \vec_2 be the corresponding measured directions of the reference unit vectors as resolved in a body fixed frame of reference. Following that, they are then related by the equations, for i = 1,2 , wher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wahba's Problem
In applied mathematics, Wahba's problem, first posed by Grace Wahba in 1965, seeks to find a rotation matrix (special orthogonal matrix) between two coordinate systems from a set of (weighted) vector observations. Solutions to Wahba's problem are often used in satellite attitude determination utilising sensors such as magnetometers and multi-antenna GPS receivers. The cost function that Wahba's problem seeks to minimise is as follows: : J(\mathbf) = \frac \sum_^ a_k\, \mathbf_k - \mathbf \mathbf_k \, ^2 for N\geq2 where \mathbf_k is the ''k''-th 3-vector measurement in the reference frame, \mathbf_k is the corresponding ''k''-th 3-vector measurement in the body frame and \mathbf is a 3 by 3 rotation matrix between the coordinate frames. a_k is an optional set of weights for each observation. A number of solutions to the problem have appeared in literature, notably Davenport's q-method, QUEST and methods based on the singular value decomposition (SVD). Several methods for solvi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity matrix. This leads to the equivalent characterization: a matrix is orthogonal if its transpose is equal to its inverse: Q^\mathrm=Q^, where is the inverse of . An orthogonal matrix is necessarily invertible (with inverse ), unitary (), where is the Hermitian adjoint (conjugate transpose) of , and therefore normal () over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation. The set of orthogonal matrices, under multiplication, forms the group , known as the orthogonal gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormality
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis. Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be ''perpendicular'' if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Attitude Dynamics And Control
Attitude control is the process of controlling the orientation of an aerospace vehicle with respect to an inertial frame of reference or another entity such as the celestial sphere, certain fields, and nearby objects, etc. Controlling vehicle attitude requires sensors to measure vehicle orientation, actuators to apply the torques needed to orient the vehicle to a desired attitude, and algorithms to command the actuators based on (1) sensor measurements of the current attitude and (2) specification of a desired attitude. The integrated field that studies the combination of sensors, actuators and algorithms is called guidance, navigation and control (GNC). Aircraft attitude control An aircraft's attitude is stabilized in three directions: '' yaw'', nose left or right about an axis running up and down; ''pitch'', nose up or down about an axis running from wing to wing; and ''roll'', rotation about an axis running from nose to tail. Elevators (moving flaps on the horizontal tai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation (geometry)
In geometry, the orientation, angular position, attitude, bearing, or direction of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies. More specifically, it refers to the imaginary rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement. It may be necessary to add an imaginary translation, called the object's location (or position, or linear position). The location and orientation together fully describe how the object is placed in space. The above-mentioned imaginary rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, and its location does not change when it rotates. Euler's rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis. This gives one common way of representing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spacecraft Attitude Control
Spacecraft attitude control is the process of controlling the orientation of a spacecraft (vehicle/satellite) with respect to an inertial frame of reference or another entity such as the celestial sphere, certain fields, and nearby objects, etc. Controlling vehicle attitude requires sensors to measure vehicle orientation, actuators to apply the torques needed to orient the vehicle to a desired attitude, and algorithms to command the actuators based on (1) sensor measurements of the current attitude and (2) specification of a desired attitude. The integrated field that studies the combination of sensors, actuators and algorithms is called guidance, navigation and control (GNC). Overview A spacecraft's attitude must typically be stabilized and controlled for a variety of reasons. It is often needed so that the spacecraft high-gain antenna may be accurately pointed to Earth for communications, so that onboard experiments may accomplish precise pointing for accurate collectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
