Configuration Space (C-Space)

The dimensionality of a C-Space is determined by the number of degrees of freedom (DOF) of the robot. Each independent parameter needed to specify the robot's configuration becomes a dimension in this abstract space.

• A free-floating rigid body in 3D space has 6 DOF (3 for position, 3 for orientation), resulting in a 6D C-Space.
• A typical industrial robotic arm with 6 revolute joints has a 6D C-Space.
• A simple mobile robot moving on a flat plane has a 3D C-Space (2D position + 1D orientation).

High dimensionality is the primary source of computational complexity in motion planning, often referred to as the curse of dimensionality.

In C-Space, physical obstacles in the workspace (the robot's physical environment) are mapped to corresponding C-obstacles. A configuration is in collision if the robot, placed in that pose, intersects a physical obstacle.

• The C-obstacle for a given workspace obstacle is the set of all configurations where collision occurs.
• The free space (C_free) is the set of all collision-free configurations.
• The forbidden space is the union of all C-obstacles.

This transformation turns the complex problem of checking the collision of a multi-link robot against irregular shapes into a simpler problem of checking if a single point lies within a pre-computed forbidden region.

The topology of a C-space describes its global shape and connectivity, which is often not a simple Euclidean space (R^n).

• For a joint that rotates 360°, its configuration is an angle θ ∈ S¹ (a circle).
• The C-space of a 2D mobile robot's orientation is S¹.
• The C-space of a 3D rigid body's orientation is SO(3), the Special Orthogonal group of 3x3 rotation matrices, which has a spherical topology.

Therefore, a robot's full C-space is often a Cartesian product of simpler spaces: e.g., R² × S¹ for a 2D mobile robot, or (S¹)⁶ for a 6-joint arm. Planning algorithms must respect this underlying topology to avoid discontinuities.

A metric (distance function) is defined on the C-space to measure how "far apart" two configurations are. This is crucial for search, optimization, and defining what constitutes a "small" motion.

• For translational DOF (in R^n), Euclidean distance is typically used.
• For rotational DOF (in S¹ or SO(3)), geodesic distance (shortest angular path) is used.
• A composite metric for a mixed C-space must weight different units meaningfully (e.g., meters vs. radians).

Common metrics include weighted Euclidean distance or the L2 norm on parameter vectors. The choice of metric directly affects the performance and path quality of planning algorithms like RRT or PRM.

A fundamental question in motion planning is whether a continuous, collision-free path exists between two configurations in C_free. This is a question of connectivity.

• Two configurations are path-connected if there exists a continuous curve between them lying entirely within C_free.
• The planning problem is to find such a path.
• C_free can be disconnected by C-obstacles, making some goals unreachable from certain starts.

Algorithms like sampling-based planners (e.g., PRM, RRT) explicitly probe and map the connectivity of C_free by building graphs or trees of sampled configurations.

C-space explicitly encodes the robot's kinematic constraints. A configuration is feasible only if it satisfies all constraints.

• Joint limits create hard boundaries in C-space (e.g., θ_min ≤ θ ≤ θ_max).
• Self-collision constraints forbid configurations where the robot's own links intersect, creating internal C-obstacles.
• Dynamic constraints (e.g., velocity, acceleration limits) are not part of the basic C-space but define a state space (which includes derivatives of configuration).

Motion planning must find paths within the feasible subset of C-space, which is the intersection of C_free with all other constraint manifolds.

The set of all possible states that a dynamical system, such as a robot, can occupy. While Configuration Space (C-Space) defines the space of possible positions, State Space is a superset that also includes dynamic variables like velocity, acceleration, and internal system modes. For a robot arm, the C-Space is defined by its joint angles; the State Space includes those angles and their angular velocities.

• Key Distinction: C-Space is geometric; State Space is dynamic.
• Planning Implication: Motion planning in State Space must account for inertia and actuator limits, making it more complex but more physically accurate.

The algorithmic process of computing a sequence of valid configurations or states for a robot to move from a start to a goal while avoiding obstacles and respecting constraints. Configuration Space (C-Space) is the abstract domain where this planning occurs; obstacles in the physical world become forbidden regions in C-Space.

• Core Algorithms: Includes sampling-based planners like RRT and PRM, and search-based planners like A*.
• Hierarchical Relationship: Often follows task decomposition, translating high-level goals into geometric motion problems within C-Space.

A class of motion planning algorithms designed for high-dimensional Configuration Spaces. Instead of modeling obstacle geometry explicitly, these algorithms probe the C-Space with random or quasi-random samples to build a graph or tree of collision-free configurations.

• Primary Methods: Rapidly-exploring Random Tree (RRT) and Probabilistic Roadmap (PRM).
• Advantage: Efficiently handles complex, non-convex C-Spaces where analytical solutions are intractable.
• Limitation: Provides probabilistic completeness (finds a solution if one exists, given enough time) rather than guaranteed completeness.

The process of refining a geometric path from a motion planner into a time-parameterized trajectory that minimizes a cost function (e.g., energy, time, jerk) while satisfying dynamic constraints. It operates on the output of C-Space planning.

• Input: A collision-free path in C-Space.
• Output: A smooth trajectory specifying positions, velocities, and accelerations over time.
• Methods: Includes direct collocation, shooting methods, and is often implemented online as Model Predictive Control (MPC) for dynamic environments.

The computational process of determining the joint angles (a point in Configuration Space) required to position a robot's end-effector at a desired location and orientation in Cartesian (task) space. It is the inverse of Forward Kinematics.

• Central Challenge: For redundant manipulators, multiple C-Space points can map to the same end-effector pose, leading to an infinite set of solutions.
• Role in Planning: IK solvers are used to convert high-level task goals (e.g., "grasp the cup") into specific target configurations for motion planning.

The computational geometry process of determining whether two or more objects, or a robot and its environment, intersect or are in contact. It is a fundamental and computationally expensive subroutine for validating points and paths in Configuration Space.

• C-Space Mapping: Each collision check in the physical world corresponds to checking if a sampled point lies within a C-Space obstacle region.
• Efficiency Techniques: Use bounding volume hierarchies (BVHs), spatial hashing, and signed distance fields to accelerate millions of checks required for sampling-based planning.