Path Planning Algorithms for Robots: A Beginner's Guide

Path planning is a fundamental challenge in robotics, essential for enabling autonomous navigation in a variety of applications, from mobile robots to drones. In this beginner’s guide, we will demystify the core path planning algorithms, delve into their applications, and provide practical tools and code examples to help you get started. Whether you are a hobbyist or a professional developer, this overview will equip you with the knowledge you need to implement effective path planning solutions.

Path Planning Basics: Definitions and Taxonomy

Before diving into the algorithms, let’s clarify some key concepts and terminology.

Key Terms

• Configuration Space (C-space): Represents all possible robot configurations. For a 2D point robot, C-space = (x, y). For a 6-DoF arm, C-space = joint angles vector. Obstacles translate to forbidden regions in C-space where collision checking occurs.
• State vs Configuration: A configuration denotes a static pose (e.g., joint angles), while a state includes dynamics like velocity and acceleration (crucial for kinodynamic planning).
• Feasibility: A configuration or trajectory is feasible if it does not collide with obstacles and meets kinematic or dynamic constraints.

Planner Taxonomy

• Global vs Local:
  • Global planners operate on a static map to compute a path from start to goal (e.g., A*, PRM).
  • Local planners react to real-time sensor data and dynamic obstacles (e.g., dynamic window approach).
• Guarantees:
  • Complete: Guarantees finding a path if one exists; grid-based algorithms are typically complete.
  • Probabilistically Complete: Sampling planners like RRT and PRM become reliably successful as sampling increases.
  • Optimal: Returns the best path by some metric (e.g., shortest path). Some planners like A* are optimal, while others like RRT* are asymptotically optimal.

Graph-Based Algorithms: Dijkstra’s and A*

Graph-based planners discretize space into nodes and edges, applying graph search methodologies.

Grid and Graph Representations

• Occupancy Grid: Divides the workspace into uniform cells, categorizing them as free, occupied, or unknown.
• Waypoint Graphs: Consist of hand-designed or sampled waypoints connected by collision-free paths, beneficial for roadmap methods.

Dijkstra’s Algorithm

• Purpose: Finds the shortest path from a source node to all other nodes in a weighted graph.
• Complexity: O(E + V log V) using a binary heap/priority queue.
• Pros: Optimal on the graph, straightforward to implement.
• Cons: Uniform exploration can be inefficient for large grids.

A* Algorithm

A* enhances Dijkstra by incorporating a heuristic that estimates the remaining cost to the goal.

• Functionality: f(n) = g(n) + h(n) where g(n) is the cost from the start to n, and h(n) is the heuristic estimate to the goal.
• If h is admissible (never overestimates), A* remains optimal on the graph.
• Typical Heuristics: Euclidean distance, Manhattan distance (for 4-neighbors), octile distance (for 8-neighbor grids).

Practical Tips

• Heuristic Quality: A closer heuristic reduces node expansions; choose an admissible yet informative heuristic.
• Tie-Breaking: Prefer nodes with larger g(n) to favor longer expansions that may reach the goal faster.
• Grid Resolution Trade-Off: Higher resolution enables more accurate paths but also increases node count and computation costs.
• Variants: Consider using weighted A* for speed with some loss of optimality or Jump Point Search (JPS) for uniform grids.

A Simple Python A* Implementation on a 2D Occupancy Grid

import heapq

def astar(grid, start, goal):
    rows, cols = len(grid), len(grid[0])
    def neighbors(r,c):
        for dr,dc in ((1,0),(-1,0),(0,1),(0,-1)):
            rr, cc = r+dr, c+dc
            if 0<=rr<rows and 0<=cc<cols and grid[rr][cc]==0:
                yield (rr,cc)
    def h(a,b):
        return abs(a[0]-b[0]) + abs(a[1]-b[1])  # Manhattan

    open_heap = [(h(start,goal), 0, start, None)]
    came_from = {}
    gscore = {start: 0}

    while open_heap:
        f, g, current, parent = heapq.heappop(open_heap)
        if current in came_from:
            continue
        came_from[current] = parent
        if current == goal:
            path = []
            n = current
            while n:
                path.append(n)
                n = came_from[n]
            return path[::-1]
        for nb in neighbors(*current):
            ng = g + 1
            if ng < gscore.get(nb, float('inf')):
                gscore[nb] = ng
                heapq.heappush(open_heap, (ng + h(nb, goal), ng, nb, current))
    return None

This example is simplified for clarity; production code should handle diagonals and account for the robot’s footprint.

Sampling-Based Algorithms: PRM and RRT Families

Why Sampling?

Grid-based methods struggle with high dimensions; sampling-based methods avoid exhaustive search, making them ideal for high-DoF robots like manipulators.

Probabilistic Roadmap (PRM)

• Multi-query Method: Builds a roadmap by sampling collision-free configurations, connecting nearby samples with feasible paths.
• Query Phase: Links start and goal to the roadmap, executing a graph search (e.g., A*) on it. Ideal for numerous queries in static environments.

Rapidly-Exploring Random Trees (RRT)

• Single-query Method: Grows a tree from the start by sampling random points and extending the closest tree node.
• Effective for quickly finding paths in high-dimensional spaces while managing kinodynamic constraints.

Optimal Variants: RRT* and PRM*

• Introduce rewiring to optimize connections for lower costs, maintaining asymptotic optimality.
• More computationally intensive due to rewiring and neighborhood searches.

Practical Points and Limitations

• Collision Checking: Optimizing collision checking is critical as it often dominates runtime; leverage spatial indexing to speed up checks.
• Sampling Strategies: Use goal bias and informed sampling to enhance convergence speed.
• Post-Smoothing: Sampling can lead to jagged paths; apply trajectory optimization for smoother results.

The OMPL (Open Motion Planning Library) is an excellent resource for implementing sampling-based planners; find it at OMPL.

Example OMPL Usage (Python-like Pseudocode):

from ompl import base as ob, geometric as og

# Define state space (2D)
space = ob.RealVectorStateSpace(2)
space.setBounds(ob.RealVectorBounds(2, 0, 10))

si = ob.SpaceInformation(space)
start = ob.State(space)
start()[0], start()[1] = 1, 1
goal = ob.State(space)
goal()[0], goal()[1] = 9, 9

pdef = ob.ProblemDefinition(si)
pdef.setStartAndGoalStates(start, goal)

planner = og.RRTstar(si)
planner.setProblemDefinition(pdef)
planner.setup()
planner.solve(1.0)  # seconds

path = pdef.getSolutionPath()
print(path)

Consult OMPL documentation for implementation details on state validity functions.

Grid-Based & Incremental/Dynamic Replanning: D*, Wavefront, and Variants

Wavefront and BFS

• Wavefront propagates distance fields on occupancy grids, offering a simple method for uniform-cost grids.

D*, D*-Lite, and Dynamic Replanning

• These methods efficiently update existing paths when the map changes, beneficial for environments with dynamic obstacles.

When to Use Dynamic Replanners

• Opt for dynamic replanners in unknown or changing environments; otherwise, static planners like A* or PRM are sufficient.

Potential Fields and Optimization-Based Approaches

Potential Field Methods

• Treat the robot as a point influenced by fictitious forces, with goals attracting and obstacles repelling; it follows the negative gradient.
• Pros: Simple and computationally cheap.
• Cons: Susceptible to local minima traps.

Common Fixes

• Introduce random perturbations or combine methods with global planners (e.g., A* for waypoint navigation).
• Use harmonic potential fields to avoid local minima in simple environments.

Trajectory Optimization and MPC

• Frame planning as an optimization problem, minimizing costs while observing constraints. Popular options include CHOMP and STOMP.
• MPC (Model Predictive Control): Produces dynamically-feasible control actions; favored in robotics for high-speed vehicles and drones.

When to Prefer Optimization

• Opt for optimization when smooth, feasible trajectories are required, and a solid initial guess is available.

Practical Considerations and Robot Constraints

Kinematics vs. Dynamics

• Holonomic Robots (e.g., omnidirectional) can move freely; planning is relatively straightforward.
• Non-holonomic Robots (e.g., differential-drive) have movement constraints that require specialized planning.

Obstacle Representation and Collision Checking

• Using simple bounding shapes expedites collision checks and is often sufficient for effective navigation.

Sensor Integration, Mapping, and Localization

• Integrate SLAM and localization for timely updates on obstacle information. A common architecture involves global planners computing paths which local planners execute while adapting to new information.

Real-Time Constraints and Compute

• To accommodate limited CPU resources, employ a multi-rate architecture; run global planning at lower frequencies and local planners at higher frequencies.

Safety and Deployment

• Conduct initial tests in simulation for reliable evaluation before hardware implementation. Include safety measures such as emergency stops and watchdog timers.

Tools, Libraries, and Implementation Tips

ROS / ROS2, Navigation2, and MoveIt

• ROS2 Navigation (Navigation2): A complete navigation stack for mobile robotics. Documentation available at ROS Navigation.
• MoveIt: Integrates arm planning and execution. Visit MoveIt for more details.
• For ROS2 newcomers, consider our beginner guide to ROS2 for integration advice.

OMPL

• OMPL provides various sampling-based planners and benchmarking tools at OMPL.

Visualization and Debugging

• Use RViz for visualizing grids, planned trajectories, and sensor readings. Assure robust visualization of planner processes.

Starter Example: Run a ROS2 Navigation Demo in Simulation

# Launch simulation and navigation demo
os2 launch nav2_bringup navigation_launch.py use_sim_time:=true
# Visualize and send a goal using RViz
ros2 run rviz2 rviz2

Where to Start Coding

• Begin with implementing A* in a 2D occupancy grid, simulating a point robot’s path following.
• Explore OMPL examples to gauge behavior with different planners, utilizing Python bindings for versatility.

Evaluation, Common Pitfalls, and Next Steps

Metrics to Evaluate Planners

• Path length
• Smoothness (such as curvature changes)
• Clearance to the nearest obstacle
• Time and memory usage statistics
• Success rate on randomized queries

Common Mistakes

• Overlooking robot dynamics can lead to infeasible geometric paths.
• Ineffective heuristics or coarse grids may result inefficiencies or collisions.
• Improper collision checks may create unsafe paths.

Testing Strategies

• Initiate testing in simulation with varied scenarios.
• Implement hardware-in-the-loop for practical integration testing while simulating sensor feedback.
• Develop unit and integration tests for critical components.

Next Steps for Learning

• Implement A*, RRT, and RRT* on simple problems for practical experience.
• Use OMPL for planner exchanges and benchmarking under consistent checks.
• Transition into trajectory optimization topics like CHOMP and STOMP as skills advance.

Planner Comparison (Trade-offs)

Conclusion

There is no one-size-fits-all planner in robotics. The choice of algorithm depends on dimensionality, the need for optimality, static vs. dynamic environments, and available compute resources. A good starting point is to implement A* on a 2D grid, followed by experimenting with RRT/RRT* via OMPL, and eventually exploring optimization-based planners for producing smooth, dynamic trajectories. Utilize ROS2/Navigation2 and MoveIt to integrate planning effectively into your robot platform, ensuring to test extensively in simulation before hardware deployment.

References and Further Reading

• Steven M. LaValle, Planning Algorithms — Planning Algorithms
• OMPL — Open Motion Planning Library documentation: OMPL
• ROS Navigation (Navigation2) docs: ROS Navigation
• MoveIt official site: MoveIt

Suggested Next Tutorials

• Beginner tutorial: Implement A* on a 2D occupancy grid (step-by-step with code)
• RRT and RRT* explained with interactive demos
• How to use OMPL with ROS2 and MoveIt — a hands-on guide
• Collision checking strategies and efficient geometry representations for robotics

Relevant Internal Links

• ROS2 beginner guide: ROS2 Beginner Guide
• Camera sensors and perception: Camera Sensors and Perception
• Linux security/AppArmor hardening: Linux Security/AppArmor Guide

Happy planning! Start with basic implementations in simulation, iterating upon them for continuous improvement.