Key Concepts in Motion Planning Algorithms to Know for Computational Geometry

Motion planning algorithms are essential for navigating complex environments. They leverage computational geometry to find efficient paths, whether in robotics or game design. Key methods include Dijkstra's, A*, RRT, and PRM, each with unique strengths for different scenarios.

1. Dijkstra's Algorithm

   • Finds the shortest path from a starting node to all other nodes in a weighted graph.
   • Utilizes a priority queue to explore the least costly paths first.
   • Guarantees the optimal solution but can be slow for large graphs due to its O(V^2) complexity.

2. A Search Algorithm*

   • Combines the benefits of Dijkstra's algorithm and a heuristic to improve efficiency.
   • Uses a cost function f(n) = g(n) + h(n), where g(n) is the cost from the start node and h(n) is the heuristic estimate to the goal.
   • Particularly effective in pathfinding on grids and maps, balancing exploration and exploitation.

3. Rapidly-exploring Random Tree (RRT)

   • A sampling-based algorithm that incrementally builds a tree by randomly sampling the space.
   • Efficiently explores high-dimensional spaces, making it suitable for complex environments.
   • Can handle dynamic obstacles and is often used in robotics for real-time motion planning.

4. Probabilistic Roadmap (PRM)

   • Constructs a roadmap of feasible paths by randomly sampling the configuration space and connecting nearby samples.
   • Works well in high-dimensional spaces and is particularly useful for multi-query scenarios.
   • The roadmap can be reused for different start and goal configurations, enhancing efficiency.

5. Potential Field Methods

   • Models the environment as a field of attractive and repulsive forces to guide the motion of the agent.
   • The goal exerts an attractive force, while obstacles exert repulsive forces, creating a gradient for navigation.
   • Simple to implement but can suffer from local minima, where the agent gets stuck.

6. Sampling-Based Methods

   • Involves generating random samples in the configuration space to explore possible paths.
   • Includes techniques like RRT and PRM, which are effective in high-dimensional and complex environments.
   • Provides a probabilistic guarantee of finding a solution as the number of samples increases.

7. Cell Decomposition

   • Divides the environment into simpler, manageable cells (either convex or non-convex).
   • Each cell is analyzed for feasibility, allowing for easier path planning within each cell.
   • Can be used in both static and dynamic environments, but may require complex preprocessing.

8. Visibility Graph

   • Constructs a graph where nodes represent vertices of obstacles and edges represent direct lines of sight.
   • Efficient for environments with polygonal obstacles, allowing for quick pathfinding.
   • The complexity increases with the number of obstacles, making it less suitable for highly cluttered spaces.

9. Voronoi Diagram-based Planning

   • Utilizes the Voronoi diagram to create a path that maximizes distance from obstacles.
   • Provides a natural way to navigate through free space while avoiding collisions.
   • Effective for environments with complex shapes, but can be computationally intensive to construct.

10. Configuration Space (C-space) Representation

    • Represents all possible positions and orientations of a robot as a multi-dimensional space.
    • Essential for understanding the robot's movement and collision avoidance in a given environment.
    • Allows for the application of various motion planning algorithms by transforming the problem into a geometric one.