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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] |
Applied Mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grace Wahba
Grace Goldsmith Wahba (born August 3, 1934) is an American statistician and now-retired I. J. Schoenberg-Hilldale Professor of Statistics at the University of Wisconsin–Madison. She is a pioneer in methods for smoothing noisy data. Best known for the development of generalized cross-validation and "Wahba's problem", she has developed methods with applications in demographic studies, machine learning, DNA microarrays, risk modeling, medical imaging, and climate prediction. Biography Grace Wahba had an interest in science from an early age, when she was in Junior High she was given a chemistry set. At this time she also interested in becoming an engineer. Wahba studied at Cornell University for her undergraduate degree. When she was there women were severely restricted in their privileges, for example she was required to live in a dorm and had a curfew. She received her bachelor's degree from Cornell University in 1956 and a master's degree from the University of Maryland, Col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Matrix
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end rotates points in the plane counterclockwise through an angle with respect to the positive axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates , it should be written as a column vector, and multiplied by the matrix : : R\mathbf = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end \begin x \\ y \end = \begin x\cos\theta-y\sin\theta \\ x\sin\theta+y\cos\theta \end. If and are the endpoint coordinates of a vector, where is cosine and is sine, then the above equations become the trigonometric summation angle formulae. Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. One w ... [...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] |
Satellite
A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisotope thermoelectric generators (RTGs). Most satellites also have a method of communication to ground stations, called Transponder (satellite communications), transponders. Many satellites use a Satellite bus, standardized bus to save cost and work, the most popular of which is small CubeSats. Similar satellites can work together as a group, forming Satellite constellation, constellations. Because of the high launch cost to space, satellites are designed to be as lightweight and robust as possible. Most communication satellites are radio Broadcast relay station, relay stations in orbit and carry dozens of transponders, each with a bandwidth of tens of megahertz. Satellites are placed from the surface to orbit by launch vehicles, high enough to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flight Dynamics (aircraft)
Flight dynamics is the science of air vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of gravity (cg), known as ''pitch'', ''roll'' and ''yaw''. These are collectively known as aircraft attitude, often principally relative to the atmospheric frame in normal flight, but also relative to terrain during takeoff or landing, or when operating at low elevation. The concept of attitude is not specific to fixed-wing aircraft, but also extends to rotary aircraft such as helicopters, and dirigibles, where the flight dynamics involved in establishing and controlling attitude are entirely different. Control systems adjust the orientation of a vehicle about its cg. A control system includes control surfaces which, when deflected, generate a moment (or couple from ailerons) about the cg which rotates the aircraft in pitch, roll, and yaw. For example, a pitching mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetometer
A magnetometer is a device that measures magnetic field or magnetic dipole moment. Different types of magnetometers measure the direction, strength, or relative change of a magnetic field at a particular location. A compass is one such device, one that measures the direction of an ambient magnetic field, in this case, the Earth's magnetic field. Other magnetometers measure the magnetic dipole moment of a magnetic material such as a ferromagnet, for example by recording the effect of this magnetic dipole on the induced current in a coil. The first magnetometer capable of measuring the absolute magnetic intensity at a point in space was invented by Carl Friedrich Gauss in 1833 and notable developments in the 19th century included the Hall effect, which is still widely used. Magnetometers are widely used for measuring the Earth's magnetic field, in geophysical surveys, to detect magnetic anomalies of various types, and to determine the dipole moment of magnetic materials. In an air ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Positioning System
The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite systems (GNSS) that provides geolocation and time information to a GPS receiver anywhere on or near the Earth where there is an unobstructed line of sight to four or more GPS satellites. It does not require the user to transmit any data, and operates independently of any telephonic or Internet reception, though these technologies can enhance the usefulness of the GPS positioning information. It provides critical positioning capabilities to military, civil, and commercial users around the world. Although the United States government created, controls and maintains the GPS system, it is freely accessible to anyone with a GPS receiver. The GPS project was started by the U.S. Department of Defense in 1973. The first prototype spacecraft was lau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion Estimator Algorithm
The quaternion estimator algorithm (QUEST) is an algorithm designed to solve Wahba's problem, that consists of finding a rotation matrix between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a quadratic form, using the Cayley–Hamilton theorem and the Newton–Raphson method to efficiently solve the eigenvalue problem and construct a numerically stable representation of the solution. The algorithm was introduced by Malcolm D. Shuster in 1981, while working at Computer Sciences Corporation.Shuster and Oh (1981) While being in principle less robust than other methods such as Davenport's q method or singular value decomposition, the algorithm is significantly faster and reliable in practical applications,Markley and Mortari (2000) and it is used for attitude determination problem in fields such as robotics and avionics.Xiaoping et al. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Value Decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an \ m \times n\ complex matrix is a factorization of the form \ \mathbf = \mathbf\ , where is an \ m \times m\ complex unitary matrix, \ \mathbf\ is an \ m \times n\ rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \ \mathbf\ is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted \ \mathbf^\mathsf\ . The diagonal entries \ \sigma_i = \Sigma_\ of \ \mathbf\ are uniquely determined by and are known as the singular values of . The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Procrustes Problem
The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A and B and asked to find an orthogonal matrix \Omega which most closely maps A to B. Specifically, :R = \arg\min_\Omega\, \Omega A-B\, _F \quad\mathrm\quad \Omega^T \Omega=I, where \, \cdot\, _F denotes the Frobenius norm. This is a special case of Wahba's problem (with identical weights; instead of considering two matrices, in Wahba's problem the columns of the matrices are considered as individual vectors). Another difference is, that Wahba's problem tries to find a proper rotation matrix, instead of just an orthogonal one. The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed by either stretching their limbs or cutting them off. Solution This problem was originally solved by Peter Schönemann in a 1964 thesis, and shortly after appeared in the journal Psychometrika. This problem is equivalent to find ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
