CSpace

C-space, short for configuration space, is a conceptual space used in robotics and motion planning to describe all possible configurations of a mechanical system. Each point in C-space corresponds to a unique arrangement of the robot, typically specified by its degrees of freedom, such as joint angles, link lengths, and the pose of the base. The dimension of C-space equals the number of independent parameters required to define the configuration.

Obstacles in the real world map to obstacles in C-space, known as C-space obstacles. The subset of configurations that do not result in a collision with the environment is called C-free, while configurations that cause a collision form C-obst. By analyzing C-free, planners can determine a collision-free path from a start configuration to a goal configuration.

Path planning in robotics commonly operates in C-space. A feasible path in C-space corresponds to a continuous trajectory of the robot in the real world. Standard algorithms include sampling-based planners such as probabilistic roadmaps (PRM) and rapidly-exploring random trees (RRT), which explore C-free without requiring a complete map of C-space. In low-dimensional problems, exact methods like cell decomposition or visibility graphs may be used, but high-dimensional systems rely on approximations.

Examples: a planar robotic arm with two rotary joints has a two-dimensional C-space defined by the two joint angles. A mobile robot moving in the plane with position and orientation has a three-dimensional C-space (x, y, theta).

Challenges include the curse of dimensionality as DOF increases, and the potentially complex shape of C-space obstacles. C-space theory provides a unifying framework for reasoning about feasibility and collision avoidance across diverse robotic platforms.

See also: configuration space, motion planning, PRM, RRT.