Configuration space is a powerful concept in robotics, representing all possible robot states as points in an abstract space. It simplifies motion planning by reducing complex 3D geometry to a lower-dimensional space, where the robot becomes a single point navigating among obstacles.

C-space obstacles map physical obstacles into regions where the robot collides. This representation enables efficient path planning algorithms. Free space, the complement of C-obstacles, defines areas for collision-free movement. Visualization techniques help understand these abstract spaces for different robot types.

Understanding Configuration Space

Concept of configuration space

• Configuration space (C-space) encompasses all possible robot configurations representing robot's state using parameters
• C-space dimensions equal robot's degrees of freedom (DOF) simplifying motion planning by representing robot as a point
• C-space reduces complex 3D geometry to lower-dimensional space enabling efficient path planning algorithms
• Types include joint space (joint angles as parameters) and task space (end-effector position and orientation)

Obstacle representation in configuration space

• C-space obstacles (C-obstacles) map physical obstacles into configuration space regions where robot collides
• Representation methods include geometric shapes (polygons, polyhedra), occupancy grids, and implicit functions
• Free space comprises regions for collision-free robot movement complementing C-obstacles
• Boundary representation defines interface between free space and C-obstacles
• Visualization techniques: 2D plots (planar robots), 3D surfaces (3 DOF spatial robots), higher-dimensional methods (complex robots)

Advanced Concepts in Configuration Space

Robot geometry and configuration space

• Robot shape and size influence C-obstacle boundaries while articulated robots create complex C-spaces
• Higher DOF increases C-space dimensionality leading to curse of dimensionality in high-DOF systems
• Workspace (physical operating space) relates to C-space (abstract configuration representation)
• Redundant robots allow multiple configurations for same end-effector pose creating self-motion manifolds
• Singularities appear as lower-dimensional subspaces where robot loses DOF

Transformation of real-world obstacles

• Minkowski sum expands obstacles by robot's geometry for translational robots
• Swept volume approach traces robot's motion through workspace for rotational and articulated robots
• Analytical methods derive explicit C-obstacle boundary equations for simple geometries and low DOF
• Sampling-based techniques (Monte Carlo) approximate C-obstacles in high-dimensional spaces
• Distance computation calculates minimum robot-obstacle distance for potential field methods
• Hierarchical representations use bounding volumes to simplify complex geometries enabling efficient collision checking