Configuration space (physics)

In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector 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.

Particle

The position of a single particle moving in ordinary Euclidean 3-space is defined by the vector r=(x, y, z), and therefore its configuration space is R³. If the particle is constrained to lie on a sphere, then its configuration space is the subset of coordinates in R³ that define points on the sphere S².

For n particles the configuration space is R³ⁿ, or possibly the subspace where no two positions are equal.

An important problem in physics considers the set of all trajectories of a particle between two points, which is a configuration space that is also known as a function space M. In quantum mechanics one formulation uses histories, or trajectories, as configurations.

Rigid body

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 $\mathbb {R} ^{3}\times \mathrm {SO} (3)$ where $\mathbb {R} ^{3}$ represents the coordinates of the origin of the frame attached to the body, and $\mathrm {SO} (3)$ represents the rotation matrices that define the orientation of this frame relative to a ground frame. A configuration of the rigid body is defined by six parameters, three from $\mathbb {R} ^{3}$ and three from $\mathrm {SO} (3)$ , and is said to have six degrees of freedom.

In robotics, configuration space refers to the set of reachable positions by a robot's end-effector, considered as a rigid body in three-dimensional space.^[1] Thus positions of the end-effector of a robot can be identified with the group of spatial rigid transformations, often denoted SE(3).

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.

Phase space

In mechanics, the configuration of a system consists of the positions had by all components subject to kinematical constraints.^[2] The set of velocities available to a system defines a plane tangent to the configuration manifold of the system. Momentum vectors are linear functionals of the tangent plane, known as cotangent vectors. Thus, the set of positions and momenta of a mechanical system forms the cotangent bundle of the configuration manifold.

This larger manifold is called the phase space of the system. The joint space and configuration space can be viewed as a bijection on the phase space of a mechanical system.

References

↑ John J. Craig, Introduction to Robotics: Mechanics and Control, 3rd Ed. Prentice-Hall, 2004
↑ Sussman, Gerald (2001). Structure and interpretation of classical mechanics. Cambridge, Massachusetts: MIT Press. ISBN 0262194554.

External links

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[1] John J. Craig, Introduction to Robotics: Mechanics and Control, 3rd Ed. Prentice-Hall, 2004

[2] Sussman, Gerald (2001). Structure and interpretation of classical mechanics. Cambridge, Massachusetts: MIT Press. ISBN 0262194554.