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Invariant Extended Kalman Filter
The invariant extended Kalman filter (IEKF) (not to be confused with the iterated extended Kalman filter) was first introduced as a version of the extended Kalman filter (EKF) for nonlinear systems possessing symmetries (or ''invariances''), then generalized and recast as an adaptation to Lie groups of the linear Kalman filtering theory.Barrau, A., & Bonnabel, S. (2016). The invariant extended Kalman filter as a stable observer. IEEE Transactions on Automatic Control, 62(4), 1797-1812. Instead of using a linear correction term based on a linear output error, the IEKF uses a geometrically adapted correction term based on an invariant output error; in the same way the gain matrix is not updated from a linear state error, but from an invariant state error. The main benefit is that the gain and covariance equations have reduced dependence on the estimated value of the state. In some cases they converge to constant values on a much bigger set of trajectories than is the case for the EKF, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Kalman Filter
In estimation theory, the extended Kalman filter (EKF) is the nonlinear version of the Kalman filter which linearizes about an estimate of the current mean and covariance. In the case of well defined transition models, the EKF has been considered the ''de facto'' standard in the theory of nonlinear state estimation, navigation systems and GPS. History The papers establishing the mathematical foundations of Kalman type filters were published between 1959 and 1961. The Kalman filter is the optimal linear estimator for ''linear'' system models with additive independent white noise in both the transition and the measurement systems. Unfortunately, in engineering, most systems are ''nonlinear'', so attempts were made to apply this filtering method to nonlinear systems; most of this work was done at NASA Ames. The EKF adapted techniques from calculus, namely multivariate Taylor series expansions, to linearize a model about a working point. If the system model (as described below) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
White Gaussian Noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, including physics, acoustical engineering, telecommunications, and statistical forecasting. White noise refers to a statistical model for signals and signal sources, rather than to any specific signal. White noise draws its name from white light, although light that appears white generally does not have a flat power spectral density over the visible band. In discrete time, white noise is a discrete signal whose samples are regarded as a sequence of serially uncorrelated random variables with zero mean and finite variance; a single realization of white noise is a random shock. Depending on the context, one may also require that the samples be independent and have identical probability distribution (in other words independent a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simultaneous Localization And Mapping
Simultaneous localization and mapping (SLAM) is the computational problem of constructing or updating a map of an unknown environment while simultaneously keeping track of an agent's location within it. While this initially appears to be a chicken-and-egg problem, there are several algorithms known for solving it in, at least approximately, tractable time for certain environments. Popular approximate solution methods include the particle filter, extended Kalman filter, covariance intersection, and GraphSLAM. SLAM algorithms are based on concepts in computational geometry and computer vision, and are used in robot navigation, robotic mapping and odometry for virtual reality or augmented reality. SLAM algorithms are tailored to the available resources and are not aimed at perfection but at operational compliance. Published approaches are employed in self-driving cars, unmanned aerial vehicles, autonomous underwater vehicles, planetary rovers, newer domestic robots and even insid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inertial Navigation
An inertial navigation system (INS) is a navigation device that uses motion sensors ( accelerometers), rotation sensors ( gyroscopes) and a computer to continuously calculate by dead reckoning the position, the orientation, and the velocity (direction and speed of movement) of a moving object without the need for external references. Often the inertial sensors are supplemented by a barometric altimeter and sometimes by magnetic sensors ( magnetometers) and/or speed measuring devices. INSs are used on mobile robots and on vehicles such as ships, aircraft, submarines, guided missiles, and spacecraft. Other terms used to refer to inertial navigation systems or closely related devices include inertial guidance system, inertial instrument, inertial measurement unit (IMU) and many other variations. Older INS systems generally used an inertial platform as their mounting point to the vehicle and the terms are sometimes considered synonymous. Overview Inertial navigation is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Attitude And Heading Reference Systems
An attitude and heading reference system (AHRS) consists of sensors on three axes that provide attitude information for aircraft, including roll, pitch, and yaw. These are sometimes referred to as MARG (Magnetic, Angular Rate, and Gravity) sensors and consist of either solid-state or microelectromechanical systems (MEMS) gyroscopes, accelerometers and magnetometers. They are designed to replace traditional mechanical gyroscopic flight instruments. The main difference between an Inertial measurement unit (IMU) and an AHRS is the addition of an on-board processing system in an AHRS, which provides attitude and heading information. This is in contrast to an IMU, which delivers sensor data to an additional device that computes attitude and heading. With sensor fusion, drift from the gyroscopes integration is compensated for by reference vectors, namely gravity, and the Earth's magnetic field. This results in a drift-free orientation, making an AHRS a more cost effective solution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry-preserving Filter
In mathematics, Symmetry-preserving observers,S. Bonnabel, Ph. Martin and E. Salaün, "Invariant Extended Kalman Filter: theory and application to a velocity-aided attitude estimation problem", 48th IEEE Conference on Decision and Control, pp. 1297-1304, 2009. also known as invariant filters, are estimation techniques whose structure and design take advantage of the natural symmetries (or invariances) of the considered nonlinear model. As such, the main benefit is an expected much larger domain of convergence than standard filtering methods, e.g. Extended Kalman Filter (EKF) or Unscented Kalman Filter (UKF). Motivation Most physical systems possess natural symmetries (or invariance), i.e. there exist transformations (e.g. rotations, translations, scalings) that leave the system unchanged. From mathematical and engineering viewpoints, it makes sense that a filter well-designed for the system being considered should preserve the same invariance properties. Definition Consider ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transformation Group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional structure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
