6.1 Graph-based path planning

Graph-based path planning is a crucial technique for autonomous robots to navigate complex environments. It involves representing spaces as interconnected nodes and edges, allowing robots to find optimal routes while avoiding obstacles.

This approach offers flexibility in modeling various environments, from indoor spaces to outdoor terrains. By applying search algorithms to these graph representations, robots can efficiently plan paths, adapting to changes and uncertainties in real-time.

Graph representations of environments

• Graph representations provide a structured way to model and analyze environments for autonomous robot navigation
• Graphs consist of nodes representing locations and edges representing connections or paths between nodes
• Different graph representations offer trade-offs in terms of resolution, computational complexity, and suitability for various environments

Occupancy grid maps

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• Discretize the environment into a grid of cells, each representing a small area
• Each cell is marked as either occupied (obstacle) or free (navigable space)
• Provide a high-resolution representation of the environment
• Suitable for indoor environments with structured layouts and static obstacles
• Examples:
  • 2D occupancy grid map of a room with furniture as obstacles
  • 3D occupancy grid map of a multi-story building

Topological maps

• Represent the environment as a graph of nodes and edges
• Nodes correspond to distinct places or landmarks in the environment
• Edges represent connections or paths between nodes
• Provide a simplified and compact representation of the environment
• Suitable for large-scale environments with well-defined paths and landmarks
• Examples:
  • Topological map of a city with intersections as nodes and roads as edges
  • Topological map of a factory floor with workstations as nodes and pathways as edges

Voronoi diagrams

• Partition the environment into regions based on the proximity to obstacles
• Voronoi edges represent the points equidistant from two or more obstacles
• Provide a roadmap for robot navigation that maximizes clearance from obstacles
• Suitable for environments with sparse obstacles and wide open spaces
• Examples:
  • Voronoi diagram of a warehouse with shelves as obstacles
  • Voronoi diagram of an outdoor terrain with trees and buildings as obstacles

Visibility graphs

• Represent the environment as a graph of vertices and edges
• Vertices correspond to the corners of obstacles
• Edges connect vertices that are mutually visible (line of sight)
• Provide a graph representation that captures the visibility relationships between obstacles
• Suitable for environments with polygonal obstacles and clear visibility
• Examples:
  • Visibility graph of a maze-like environment with walls as obstacles
  • Visibility graph of a cluttered indoor environment with furniture as obstacles

Graph search algorithms

• Graph search algorithms are used to find optimal paths or solutions in graph-based representations
• Different algorithms have varying properties, such as optimality guarantees, computational complexity, and ability to handle dynamic environments
• The choice of algorithm depends on the specific requirements and constraints of the autonomous robot application

Dijkstra's algorithm

• Finds the shortest path from a single source node to all other nodes in a weighted graph
• Explores the graph in a breadth-first manner, expanding nodes in order of increasing cost
• Guarantees the optimal solution for graphs with non-negative edge weights
• Has a time complexity of O(E + V log V) using a priority queue, where E is the number of edges and V is the number of vertices
• Suitable for static environments with known edge weights

A* search algorithm

• Finds the shortest path from a start node to a goal node using a heuristic function
• Combines the actual cost from the start node (g-cost) with an estimated cost to the goal node (h-cost) to guide the search
• Explores the graph in a best-first manner, expanding nodes with the lowest total cost (f-cost = g-cost + h-cost)
• Guarantees the optimal solution if the heuristic function is admissible (never overestimates the cost to the goal)
• Has a time complexity of O(E + V log V) using a priority queue, where E is the number of edges and V is the number of vertices
• Suitable for environments with a known goal location and an informative heuristic function

D* search algorithm

• Finds the shortest path in dynamic environments where edge weights can change during the search
• Maintains a priority queue of nodes to be expanded and updates their costs when changes occur
• Efficiently reuses previous search information to adapt to environmental changes
• Guarantees the optimal solution in dynamic environments
• Has a time complexity similar to A* search, with additional overhead for updating costs
• Suitable for autonomous robots operating in dynamic environments with changing obstacles or costs

Bidirectional search

• Simultaneously searches from both the start node and the goal node until the two search frontiers meet
• Reduces the search space by exploring the graph from both directions
• Can be combined with other search algorithms, such as Dijkstra's algorithm or A* search
• Requires a mechanism to detect when the two search frontiers meet and to reconstruct the complete path
• Suitable for environments where the start and goal nodes are known and the graph is relatively symmetric

Heuristics for graph search

• Heuristics are estimates of the cost or distance from a node to the goal node in a graph search algorithm
• Well-designed heuristics can significantly improve the efficiency of search algorithms by guiding the search towards promising directions
• Admissible heuristics never overestimate the actual cost to the goal, ensuring optimal solutions
• Consistent heuristics satisfy the triangle inequality, enabling efficient search algorithms like A*

Manhattan distance

• Calculates the sum of the absolute differences in the x and y coordinates between two nodes
• Assumes a grid-based environment where movement is allowed only in horizontal and vertical directions
• Admissible and consistent heuristic for grid-based environments
• Example: In a 2D grid, the Manhattan distance between (3, 4) and (7, 2) is |3 - 7| + |4 - 2| = 6

Euclidean distance

• Calculates the straight-line distance between two nodes using the Pythagorean theorem
• Assumes a continuous environment where movement is allowed in any direction
• Admissible heuristic for environments with no obstacles, but may overestimate the actual cost in the presence of obstacles
• Example: In a 2D plane, the Euclidean distance between (3, 4) and (7, 2) is $\sqrt{(3 - 7)^2 + (4 - 2)^2} \approx 4.47$

Diagonal distance

• Calculates the distance between two nodes considering diagonal movement in addition to horizontal and vertical movement
• Assumes a grid-based environment where diagonal movement is allowed
• More accurate than Manhattan distance for environments with diagonal paths
• Example: In a 2D grid with diagonal movement, the diagonal distance between (3, 4) and (7, 2) is max(|3 - 7|, |4 - 2|) = 4

Admissible vs consistent heuristics

• Admissible heuristics never overestimate the actual cost from a node to the goal
• Consistent heuristics satisfy the triangle inequality: h(n) ≤ c(n, n') + h(n'), where n and n' are neighboring nodes and c(n, n') is the cost between them
• Consistent heuristics are always admissible, but admissible heuristics may not be consistent
• A* search guarantees optimal solutions with admissible heuristics and expands fewer nodes with consistent heuristics

Graph pruning techniques

• Graph pruning techniques aim to reduce the size and complexity of graph representations while preserving important information for robot navigation
• Pruning helps to improve the efficiency of graph search algorithms and reduces memory requirements
• Different pruning techniques offer trade-offs between graph size, resolution, and computational complexity

Hierarchical graphs

• Construct a multi-level graph representation with different levels of abstraction
• Higher levels represent coarser abstractions of the environment, while lower levels provide more detailed information
• Allows for efficient search at higher levels to quickly identify promising regions, followed by detailed search at lower levels
• Suitable for large-scale environments with hierarchical structures
• Example: A hierarchical graph of a multi-story building, with floors as higher-level nodes and rooms as lower-level nodes

Adaptive resolution grids

• Dynamically adjust the resolution of the occupancy grid based on the robot's location and surroundings
• Use higher resolution in areas with dense obstacles or critical regions, and lower resolution in open spaces
• Reduces the overall size of the graph representation while maintaining necessary details
• Suitable for environments with varying obstacle densities and robot navigation requirements
• Example: An adaptive resolution grid for a warehouse, with high resolution near shelves and low resolution in open corridors

Quadtrees and octrees

• Recursively subdivide the environment into quadrants (2D) or octants (3D) based on the presence of obstacles
• Each node in the tree represents a region of the environment, with leaves representing the smallest subdivisions
• Allows for efficient compression of large environments by merging homogeneous regions
• Suitable for environments with non-uniform obstacle distributions and multi-resolution requirements
• Example: A quadtree representation of a large outdoor environment, with dense subdivisions near buildings and sparse subdivisions in open areas

Real-time graph-based planning

• Real-time graph-based planning focuses on generating and updating paths in dynamic environments with time constraints
• Autonomous robots often operate in changing environments and need to adapt their plans based on new information
• Real-time planning algorithms balance the trade-off between plan quality and computational efficiency

Anytime planning algorithms

• Generate an initial suboptimal solution quickly and iteratively improve it as more time becomes available
• Provide a valid plan at any point during the planning process, allowing for interruption and execution
• Suitable for situations where a suboptimal plan is acceptable, and the robot needs to react quickly to changes
• Example: Anytime Repairing A* (ARA*) algorithm, which starts with a suboptimal solution and improves it over time

Incremental search methods

• Reuse information from previous search iterations to efficiently update the plan when the environment changes
• Maintain a graph structure and update the costs and paths as new information becomes available
• Avoid recomputing the entire plan from scratch, saving computational resources
• Suitable for environments with localized changes and incremental updates
• Example: Incremental A* algorithm, which updates the search tree based on changes in the environment

Replanning strategies

• Detect when the current plan becomes invalid or suboptimal due to environmental changes
• Trigger a replanning process to generate a new plan based on the updated information
• Decide when and how often to replan based on the frequency and magnitude of changes
• Suitable for environments with frequent or significant changes that require substantial plan adjustments
• Example: Adaptive replanning using D* Lite algorithm, which efficiently updates the plan when the robot encounters new obstacles

Applications of graph-based planning

• Graph-based planning techniques have diverse applications in various domains of autonomous robotics
• The choice of graph representation and search algorithm depends on the specific requirements and constraints of each application
• Real-world applications often involve integrating graph-based planning with other components, such as perception, control, and decision-making

Indoor robot navigation

• Navigate robots in structured indoor environments, such as homes, offices, or hospitals
• Use occupancy grid maps or topological maps to represent the environment
• Plan paths to avoid obstacles, reach target locations, and optimize criteria like shortest distance or energy efficiency
• Example: A domestic service robot using an occupancy grid map and A* search to navigate between rooms and perform tasks

Outdoor vehicle routing

• Plan routes for autonomous vehicles in outdoor environments, such as roads, highways, or off-road terrains
• Use topological maps or visibility graphs to represent the environment
• Consider factors like road network, traffic conditions, and vehicle constraints when planning paths
• Example: An autonomous delivery vehicle using a topological map and Dijkstra's algorithm to plan efficient routes in a city

Multi-robot coordination

• Coordinate the actions and paths of multiple robots in a shared environment
• Use graph-based representations to model the environment and the interactions between robots
• Plan paths that avoid collisions, optimize resource utilization, and achieve common goals
• Example: A team of warehouse robots using a centralized graph-based planner to coordinate their movements and optimize item retrieval

Autonomous warehouse systems

• Optimize the operations of autonomous robots in warehouse environments for tasks like inventory management and order fulfillment
• Use adaptive resolution grids or hierarchical graphs to represent the warehouse layout and shelf locations
• Plan efficient paths for robots to navigate between storage areas, picking stations, and shipping zones
• Example: An autonomous warehouse system using quadtrees and anytime planning algorithms to adapt to changing inventory levels and order demands

Limitations and challenges

• Graph-based planning techniques face several limitations and challenges that need to be addressed for effective autonomous robot navigation
• These challenges arise from the complexity of real-world environments, the need for real-time performance, and the integration with other components of the robotic system

High-dimensional state spaces

• Many autonomous robot applications involve high-dimensional state spaces, such as joint configurations of manipulators or combined position and orientation of mobile robots
• Graph-based representations and search algorithms become computationally expensive in high-dimensional spaces
• Dimensionality reduction techniques, such as sampling-based methods or motion primitives, can help alleviate this challenge
• Example: Planning the motion of a robotic arm with multiple degrees of freedom using a combination of graph-based planning and sampling-based techniques

Dynamic and uncertain environments

• Autonomous robots often operate in dynamic environments with moving obstacles, changing terrain, or uncertain sensor information
• Graph-based planning techniques need to adapt to these changes and handle uncertainty in the environment representation and robot localization
• Probabilistic graph representations, such as belief roadmaps or dynamic Bayesian networks, can model uncertainty and support decision-making under uncertainty
• Example: An autonomous underwater vehicle navigating in a changing ocean environment with currents and limited sensor visibility

Computational complexity

• Graph-based planning algorithms can be computationally expensive, especially for large-scale environments and high-dimensional state spaces
• Real-time performance requirements limit the available computational resources and the complexity of the planning algorithms
• Parallel and distributed computing techniques, as well as hardware acceleration, can help improve the computational efficiency of graph-based planning
• Example: Implementing A* search on a GPU to accelerate the planning process for a large-scale autonomous warehouse system

Integration with other planning methods

• Graph-based planning is often used in combination with other planning techniques, such as sampling-based methods, optimization-based approaches, or machine learning-based planners
• Integrating multiple planning paradigms requires careful design and coordination to leverage their strengths and compensate for their limitations
• Hybrid planning architectures can combine graph-based planning for high-level decision-making with other techniques for low-level motion planning and control
• Example: Integrating a graph-based planner for global path planning with a local sampling-based planner for obstacle avoidance in an autonomous mobile robot