Modified 3D A* Algorithm

• The paper introduces an enhanced 3D A* algorithm by integrating octree spatial representation with a novel height-based penalty to favor ground-level traversal.
• It demonstrates over 90% reductions in memory usage and computation time while maintaining path optimality within a 1% difference compared to traditional methods.
• The approach supports real-time and embedded applications for ground vehicles and legged robots navigating obstacle-rich, complex three-dimensional terrains.

A modified 3D A* algorithm is a computational strategy for optimal path planning in three-dimensional environments, enhanced to account for application-specific constraints such as non-holonomic mobility, obstacle avoidance, multiobjective trade-offs, smoothness, real-time resource limits, and complex topographies. These algorithms depart from the classical A* formulation by integrating additional cost functions, search heuristics, data structures, and environmental representations—most notably octree decompositions and distance fields—that yield more efficient, feasible, and context-aware paths in robotic and autonomous systems.

1. Algorithmic Modifications and Cost Function Enhancements

Numerous modifications to the 3D A* algorithm address challenges unique to spatial navigation, particularly for ground vehicles and legged robots. The central architectural advance involves the extension of A* to operate over an octree spatial structure, where the environment is hierarchically partitioned to compress representation and reduce node counts (Ham et al., 5 Sep 2025). The cost function is adapted by introducing a height-based penalty term: $f(n) = g(n) + h(n) + \alpha r(n)$ where:

• $g(n)$ is the accumulated cost to node $n$,
• $h(n)$ is the Manhattan distance heuristic to the goal,
• $r(n)$ is the vertical distance from node $n$ to the nearest traversable surface,
• $\alpha$ is a weight parameter.

This alteration ensures the planner prioritizes ground-level traversal and properly accounts for traversable obstacles, penalizing paths through nodes with excessive elevation that are unsuitable for ground-based agents.

2. Octree Spatial Representation and Compression

The adoption of the octree structure is fundamental for achieving memory- and computation-efficient 3D navigation. Octrees hierarchically compress large free-space regions and represent them with higher-level nodes, while denser obstacle regions retain fine granularity. Key algorithms provided include coordinate-driven leaf searches to identify which octree leaf a given coordinate resides within, and robust neighbor identification across leaves of varying resolutions. Such spatial partitioning drastically limits the number of nodes that must be generated, expanded, and evaluated during search, resulting in significant performance improvements.

Approach or Scenario	Memory Usage (kB)	Computation Time (ms)
Traditional 3D A*	8192	927.7 / 7495
Octree-A* (modified)	367.4 / 608.7	68.4 / 661

These reductions, measured over two benchmark scenarios, correspond to >90% decreases in both metrics, with path optimality maintained ($<1\%$ difference in length).

3. Height-Based Penalty and Traversable Obstacles

The height-based penalty $r(n)$ is defined as the Euclidean vertical distance from the node center to the ground or to the nearest traversable obstacle surface. Nodes proximate to the ground or stable surfaces receive minimal penalties and are favored in path generation. Conversely, nodes that occupy volume above impassable regions or far from legitimate surfaces are heavily penalized, effectively constraining the planner to realistic locomotion regimes. The planner also allows certain obstacles to be leveraged (e.g., for traversal if permitted), enhancing adaptability in real-world terrestrial environments.

4. Benchmark Performance and Optimality

Quantitative results substantiate the efficiency and reliability of the modified 3D A* algorithm. For Scenario 1, Octree-A* incurred only a 0.41% path length increase relative to standard A*, and in Scenario 2, a 0.80% decrease, demonstrating preservation of optimality. Memory and computation times dropped by over 90% in both test cases. These improvements are directly attributable to octree compression and the tailored cost function that constrains exploration to ground-adjacent pathways.

Metric	Scenario 1	Scenario 2
Path length change (%)	+0.41	-0.80
Memory reduction (%)	>95	>92
Computation time reduction (%)	>92	>91

5. Real-Time and Embedded Applications

The architecture supports efficient, real-time path planning in complex 3D domains, making it especially applicable in computationally constrained settings. Embedded systems for unmanned ground vehicles (UGVs), search and rescue mobile robots, or quadrupeds benefit from reduced memory footprints and rapid computation, enabling robust operation in pipe networks, collapsed buildings, or cluttered urban environments where terrain and obstacle distribution vary widely.

6. Algorithmic Generality and Extensibility

While the described modifications specifically remedy ground vehicle path planning in voxelized 3D space (Ham et al., 5 Sep 2025), the underlying design extends to other platforms and operational domains. The use of hierarchical spatial representation (octrees), penalty-based cost function stratification, and dynamic neighbor identification suggests broad applicability for aerial robots, underwater vehicles, and multi-agent systems requiring ground-aware or context-sensitive navigation.

A plausible implication is that further integration of non-holonomic constraints, motion primitives, or environment-specific objectives (energy, solar exposure, etc.) could yield even greater improvements in feasibility and operational robustness for autonomous systems in high-dimensional, structured spaces.

7. Practical Considerations and Future Directions

The combination of octree spatial compression and height-penalized cost minimization provides a concrete framework for efficient 3D spatial navigation. Central practical considerations include:

• Tuning the penalty weight $\alpha$ to match vehicle capabilities,
• Maintaining up-to-date octree representations under dynamic environmental changes,
• Ensuring accurate calculation of traversable surfaces, particularly for legged robots that may exploit a broader class of obstacle types.

Future research may address integration with multiobjective search (e.g., Pareto-based frameworks), on-the-fly octree refinement in response to sensor input, and adaptation to environments with complex topologies or varying surface properties.

In conclusion, the modified 3D A* algorithm employing a ground-aware cost function and octree-driven spatial representation achieves real-time, memory-efficient path planning for ground vehicles and similar agents. The approach maintains path optimality while allowing for context-sensitive exploitation of the environment, with performance gains substantiated by empirical benchmarks. It constitutes a significant methodology for practical autonomous navigation in three-dimensional, obstacle-rich domains (Ham et al., 5 Sep 2025).