Configuration space (physics) on arvau

Wikipedia / Configuration space (physics)

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.

Examples

A particle in 3D space

The position of a single particle moving in ordinary Euclidean 3-space, and therefore its configuration space for a point in configuration space; this is the convention in both the Hamiltonian formulation of classical mechanics, and in Lagrangian mechanics refers to velocities.

A particle might be constrained to move on a specific manifold is the sphere, i.e..

For n. In general, however, one is interested in the case where the particles interact: for example, they are specific locations in some assembly of gears, pulleys, rolling balls, etc., but the subspace (submanifold) of allowable positions that the points can take.

Rigid body in 3D space

, and is said to have six degrees of freedom.

in that configuration space is described using generalized coordinates; thus, three of the coordinates might describe the location of the center of mass of the rigid body, while three more might be the Euler angles describing its orientation. There is no canonical choice of coordinates; one could also choose some tip or endpoint of the rigid body, instead of its center of mass; one might choose to use quaternions instead of Euler angles, and so on. However, the parameterization does not change the mechanical characteristics of the system; all of the different parameterizations ultimately describe the same (six-dimensional) manifold, the same set of possible positions and orientations.

Some parameterizations are easier to work with than others, and many important statements can be made by working in a coordinate-free fashion. Examples of coordinate-free statements are that the tangent space, while the cotangent space corresponds to momenta. (Velocities and momenta can be connected; for the most general, abstract case, this is done with the rather abstract notion of the tautological one-form.)

Robotic arm

-rigid-body configuration space.

Note, however, that in robotics, the term configuration space can also refer to a further-reduced subset: the set of reachable positions by a robot's end-effector. This definition, however, leads to complexities described by the holonomy: that is, there may be several different ways of arranging a robot arm to obtain a particular end-effector location, and it is even possible to have the robot arm move while keeping the end effector stationary. Thus, a complete description of the arm, suitable for use in kinematics, requires the specification of all of the joint positions and angles, and not just some of them.

The joint parameters of the robot are used as generalized coordinates to define configurations. The set of joint parameter values is called the joint space. A robot's forward and inverse kinematics equations define maps between configurations and end-effector positions, or between joint space and configuration space. Robot motion planning uses this mapping to find a path in joint space that provides an achievable route in the configuration space of the end-effector.

Formal definition

In classical mechanics, the configuration of a system refers to the position of all constituent point particles of the system.

Phase space

. The set of positions and momenta of a mechanical system forms the cotangent bundle. This larger manifold is called the phase space of the system.

Quantum state space

In quantum mechanics, configuration space can be used (see for example the Mott problem), but the classical mechanics extension to phase space cannot. Instead, a rather different set of formalisms and notation are used in the analogous concept called quantum state space. The analog of a "point particle, the complex projective line, also known as the Bloch sphere. It is complex, because a quantum-mechanical wave function, thus making it projective.