configurationspace

Configuration space, or C-space, is a representation used in robotics and motion planning that encodes every possible configuration of a robot as a point in a multidimensional space. The configuration vector q collects the values of all independent joints and links, such as joint angles and translations. The dimension of C-space equals the robot’s degrees of freedom (DOF). For planar robots with revolute joints, C is typically a product of circles; for prismatic joints, Euclidean coordinates; for rigid-body pose in 3D, components may belong to SE(3) and SO(3).

In C-space, workspace obstacles map to C-space obstacles (C-obstacles); free space C_free consists of configurations that do not cause collision. Planning a collision-free motion becomes a path planning problem in C-space from a start configuration to a goal configuration, staying within C_free. Collision checks test whether the configuration yields a collision when the robot is placed in the corresponding pose.

Construction and properties: The C-space is the product of the individual joint spaces, subject to joint limits and kinematic constraints. Nonholonomic constraints, closed chains, and inequality bounds create manifolds with boundaries. Time can be added as an extra dimension for dynamics, yielding a time-augmented C-space. Dimensionality grows with DOF, making exact computation difficult and motivating sampling-based planners.

Applications and limitations: C-space methods underpin many motion-planning algorithms such as PRM and RRT, which sample configurations and connect feasible ones to build routes. The concept is powerful but leads to high-dimensional, nonconvex spaces in practice, requiring approximations, heuristics, and online planning for complex robots.