Definition
The manipulability ellipsoid is defined as the set [1]
where q is the joint configuration of the robot and J is the robot Jacobian relating the end-effector velocity with the joint rates.
The manipulability ellipsoid is a concept in robotics that represents the manipulability of a robotic system in a graphical form. Here, the manipulability of a robot refers to its ability to alter the position of the end effector based on the joint configuration. A higher manipulability measure signifies a broader range of potential movements in that specific configuration. When the robot is in a singular configuration the manipulability measure diminishes to zero.
The manipulability ellipsoid is defined as the set [1]
where q is the joint configuration of the robot and J is the robot Jacobian relating the end-effector velocity with the joint rates.
A geometric interpretation of the manipulability ellipsoid is that it includes all possible end-effector velocities normalized for a unit input at a given robot configuration. The axis of the ellipsoid can be computed by using the singular value decomposition of the robot Jacobian. [1] [2]
An industrial robot is a robot system used for manufacturing. Industrial robots are automated, programmable and capable of movement on three or more axes.
In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system. Notice that this is a notion of "unrestricted" configuration space, i.e. in which different point particles may occupy the same position. In mathematics, in particular in topology, a notion of "restricted" configuration space is mostly used, in which the diagonals, representing "colliding" particles, are removed.
In computer animation and robotics, inverse kinematics is the mathematical process of calculating the variable joint parameters needed to place the end of a kinematic chain, such as a robot manipulator or animation character's skeleton, in a given position and orientation relative to the start of the chain. Given joint parameters, the position and orientation of the chain's end, e.g. the hand of the character or robot, can typically be calculated directly using multiple applications of trigonometric formulas, a process known as forward kinematics. However, the reverse operation is, in general, much more challenging.
In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized.
In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named after the mathematician Kurt Friedrichs. This extension is particularly useful in situations where an operator may fail to be essentially self-adjoint or whose essential self-adjointness is difficult to show.
In robotics, robot kinematics applies geometry to the study of the movement of multi-degree of freedom kinematic chains that form the structure of robotic systems. The emphasis on geometry means that the links of the robot are modeled as rigid bodies and its joints are assumed to provide pure rotation or translation.
Screw theory is the algebraic calculation of pairs of vectors, such as angular and linear velocity, or forces and moments, that arise in the kinematics and dynamics of rigid bodies.
In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution needs to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.
In engineering, a mechanical singularity is a position or configuration of a mechanism or a machine where the subsequent behaviour cannot be predicted, or the forces or other physical quantities involved become infinite or nondeterministic.
Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the worst of cases. ES is an alternative to value at risk that is more sensitive to the shape of the tail of the loss distribution.
Serial manipulators are the most common industrial robots and they are designed as a series of links connected by motor-actuated joints that extend from a base to an end-effector. Often they have an anthropomorphic arm structure described as having a "shoulder", an "elbow", and a "wrist".
A parallel manipulator is a mechanical system that uses several computer-controlled serial chains to support a single platform, or end-effector. Perhaps, the best known parallel manipulator is formed from six linear actuators that support a movable base for devices such as flight simulators. This device is called a Stewart platform or the Gough-Stewart platform in recognition of the engineers who first designed and used them.
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper. Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates. In some special cases solutions can be expressed in terms of polynomials called Lamé polynomials.
In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to
In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.
In applied mathematical analysis, shearlets are a multiscale framework which allows efficient encoding of anisotropic features in multivariate problem classes. Originally, shearlets were introduced in 2006 for the analysis and sparse approximation of functions . They are a natural extension of wavelets, to accommodate the fact that multivariate functions are typically governed by anisotropic features such as edges in images, since wavelets, as isotropic objects, are not capable of capturing such phenomena.
The product of exponentials (POE) method is a robotics convention for mapping the links of a spatial kinematic chain. It is an alternative to Denavit–Hartenberg parameterization. While the latter method uses the minimal number of parameters to represent joint motions, the former method has a number of advantages: uniform treatment of prismatic and revolute joints, definition of only two reference frames, and an easy geometric interpretation from the use of screw axes for each joint.
Paden–Kahan subproblems are a set of solved geometric problems which occur frequently in inverse kinematics of common robotic manipulators. Although the set of problems is not exhaustive, it may be used to simplify inverse kinematic analysis for many industrial robots. Beyond the three classical subproblems several others have been proposed.
In mathematics, the Fuchs relation is a relation between the starting exponents of formal series solutions of certain linear differential equations, so called Fuchsian equations. It is named after Lazarus Immanuel Fuchs.
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