Robotics Configuration Space Quiz

World Components

• The world is divided into occupied and free space, where robots can't and "might" be able to go, respectively.
• Configuration space (C) represents all possible positions and orientations (configurations) of a robot within the world.

Configuration Space (C)

• Notation:
  • A: robot
  • W: Euclidean space where A moves (e.g., $R^2$ or $R^3$)
  • B1,...Bm: obstacles in W
• Frames:
  • Fw: fixed world frame
  • FA: robot frame attached to the robot
• C-space for a point robot:
  • 2D: $R^2$
  • 3D: $R^3$

Free Space (Cfree)

• Cfree is the subset of C where the robot is free of obstacles.
• Cfree = C \UCB₁ = {q∈ C: A(q)NUCB₁ = $}
• C-obstacle represents the space in C that is occupied by an obstacle.

Motion Planning Problem

• Input:
  • Robot and environment geometry
  • Start and goal configurations
• Output:
  • Path from start to goal, or indication that no path exists.

Motion Planning Methods

• Roadmap approaches: simplify C by constructing a network of paths, reducing search complexity.
• Cell decomposition: divide C into cells, allowing exploration.
• Potential Fields: create artificial forces to guide the robot towards the goal and away from obstacles.
• Bug algorithms: limited-knowledge planning using simple behaviors.

Visibility Graphs

• Construct a graph connecting vertices that are "visible" to each other, forming paths along obstacle perimeters.
• Uses Dijkstra's algorithm for graph search (O(N^2) complexity, where N is the number of vertices).

Visibility Graph Drawbacks

• May not find optimal solutions in higher dimensions.
• Paths may be "semi-free," meaning they may be in contact with obstacles.

Trapezoidal Decomposition

• Exact cell decomposition method dividing free space into trapezoids and triangles.
• Creates a connectivity graph to represent cell adjacency.
• NP-complete to find the minimum number of convex cells.

Potential Fields

• Principle:
  • Attractive potential: pull towards the goal.
  • Repulsive potential: push away from obstacles.
  • The negative gradient of the total potential acts as a guiding force.
• Pros:
  • Real-time path generation.
  • Merges planning and control.
  • Smooth paths.
• Cons:
  • Susceptible to local minima.
  • Commonly used for local path planning.

Bug Algorithms

• Insect-inspired path planning with limited knowledge (local sensing, obstacle detection).
• Uses simple behaviors:
  • Moving towards goal.
  • Circumnavigating obstacles.

Summary

• Configuration space is fundamental for robot motion planning.
• Various motion planning methods exist, each with its strengths and weaknesses.
• Visibility graphs, cell decomposition, potential fields, and bug algorithms are key approaches.

References

• "Computational Principles of Mobile Robots" by G.Dudek, M.Jenkin, MIT Press, 2000 (Chapter 5)
• "Robot Motion Planning" by J.C.Latombe, Kluwer Academic Publishers, 1991.
• S.Kambhampati, L.Davis, "Multiresolution Path Planning for Mobile Robots", IEEE Journal of Robotrics and Automation, Vol.RA-2, No.3, 1986, pp.135-145.
• O.Khatib, "Real-Time Obstacle Avoidance for Manipulators and Mobile Robots", Int.Journal of Robotics Research, 5(1), pp.90-98, 1986.
• P.Khosla, R.Volpe, "Superquadratic Artificial Potentials for Obstacle Avoidance and Approach" Proc.

Test your understanding of configuration space in robotics, including concepts of occupied and free space, C-space, and motion planning problems. This quiz challenges your knowledge on how robots navigate through different environments while avoiding obstacles.