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.
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.
The position of a single particle moving in ordinary Euclidean 3-space is defined by the vector
A particle might be constrained to move on a specific manifold.
For example, if the particle is attached to a rigid linkage, free to swing about the origin, it is effectively constrained to lie on a sphere.
For n disconnected, non-interacting point particles, the configuration space is
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.
The set of coordinates that define the position of a reference point and the orientation of a coordinate frame attached to a rigid body in three-dimensional space form its configuration space, often denoted
represents the coordinates of the origin of the frame attached to the body, and
A configuration of the rigid body is defined by six parameters, three from
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.
Examples of coordinate-free statements are that the tangent space
For a robotic arm consisting of numerous rigid linkages, the configuration space consists of the location of each linkage (taken to be a rigid body, as in the section above), subject to the constraints of how the linkages are attached to each other, and their allowed range of motion.
except that all of the various attachments and constraints mean that not every point in this space is reachable.
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.
[1] 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.
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.
[2] The configuration space is insufficient to completely describe a mechanical system: it fails to take into account velocities.
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.
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 complex projective line, also known as the Bloch sphere.