Configuration space

"C-space" redirects here. For the art gallery, see C-Space, Beijing.

In classical mechanics, the configuration space of a physical system is the space of generalized coordinates that define its configurations, possibly subject to external constraints. The configuration space of a typical system has the structure of a manifold; for this reason it is also called the configuration manifold.

Configuration spaces in physics

The configuration space of a single particle moving in ordinary Euclidean 3-space is just R³. For n particles the configuration space is R³ⁿ, or possibly the subspace where no two positions are equal. More generally, one can regard the configuration space of n particles moving in a manifold M as the function space Mⁿ.

To take account of both position and momenta one moves to the cotangent bundle of the configuration manifold. This larger manifold is called the phase space of the system. In short, a configuration space is typically "half" of (see Lagrangian distribution) a phase space that is constructed from a function space.

In quantum mechanics one formulation emphasises 'histories' as configurations.

Robotics

In Robotics, configuration space often refers to the set of positions reachable by a robot's end-effector. The set of joint parameter values is called the joint space. The forward kinematics equations of the robot map its joint space to the configuration space of the end-effector. Robot motion planning seeks a path in the configuration space of the end-effector and then determines the joint trajectory in joint space using the robots inverse kinematics solution.

Configuration spaces in mathematics

The configuration space of 2 not necessarily distinct points on the circle is the orbifold T² / S₂, which is the Möbius strip.

In mathematics a configuration space refers to a broad family of constructions closely related to the state space notion in physics. The most common notion of configuration space in mathematics C_nX is the set of n-element subsets of a topological space X. This set is given a topology by considering it as the quotient C_nX = F_nX / Σ_n where $F_n X = \{(x_1,\cdots,x_n) \in X^n : x_i \neq x_j \forall \ i \neq j \}$ where Σ_n is the symmetric group acting by permuting the coordinates of F_nX. Typically, C_nX is called the configuration space of n unordered points in X and F_nX is called the configuration space of n ordered or coloured points in X; the space of n ordered not necessarily distinct points is simply Xⁿ.

If the original space is a manifold, the configuration space of distinct, unordered points is also a manifold, while the configuration space of not necessarily distinct unordered points is instead an orbifold.

Configuration spaces are related to braid theory, where the braid group is considered as the fundamental group of the space $C_n \Bbb R^2$.

A configuration space is a type of classifying space or (fine) moduli space. In particular, there is a universal bundle $\pi\colon E_n\to C_n$ which is a subbundle of the trivial bundle $C_n\times X^n\to C_n$, and which has the property that the fiber over each point $p\in C_n$ is the n element subset of X_n classified by p.

The homotopy type of configuration spaces is not homotopy invariant – for example, that the spaces $F_n \Bbb R^m$ are not homotopic for any two distinct values of m. For instance, $F_n\Bbb R$ is not connected, $F_n\Bbb R^2$ is a K(π,1), and $F_n \Bbb R^m$ is simply connected for $m \geq 3$.

It used to be an open question whether there were examples of compact manifolds which were homotopic but had non-homotopic configuration spaces: such an example was found only in 2005 by Longini and Salvatore. Their example are two three-dimensional lens spaces, and the configuration spaces of at least two points in them. That these configuration spaces are not homotopic was detected by Massey products in their respective universal covers.

See also

• Feature space (topic in pattern recognition)
• Parameter space
• Phase space
• State space (physics)

External links

• Intuitive Explanation of Classical Configuration Spaces.
• Interactive Visualization of the C-space for a Robot Arm with Two Rotational Links from UC Berkeley.