An Overview and Comparison of Traditional Motion Planning Based on Rapidly Exploring Random Trees
Abstract
:1. Introduction
2. Background
2.1. System Dynamics
2.2. Linear Quadratic Regulator
2.3. Control Barrier Functions
2.4. Control Lyapunov Functions
3. The RRT-Based Motion Planning
3.1. The RRT Algorithm and Its Variations
- Sample: The procedure provides independent identically distributed samples from .
- Steer: Given two states , the Steer procedure returns a state that lies between and while maintaining , for a prespecified constant . It is typically defined by
- Nearest node: Given a tree with a set of nodes and a set of edges , the procedure provides a node in that is closest to , i.e.,
- Collision test: Given two points , the procedure returns True if and only if the line segment between and lies entirely within , i.e.,
Algorithm 1: Body of RRT and RRT* algorithms |
Algorithm 2: ExtendRRT |
3.2. The RRT* Algorithm and Its Variations
- Near nodes: Given a tree with a set of nodes and a set of edges , the procedure utilizes a pre-defined distance to identify a set of near nodes in that are close to . This is typically defined as the set of all nodes within a closed ball of radius centered at , i.e.,
- Choose Parent: The procedure tries to find collision-free paths between a selected node and all its near nodes, and selects the near node that yields the lowest cost of as the parent of .
- Rewire: The procedure attempts to connect a selected node with each node in its neighborhood . If the trajectory from to the near node results in a lower cost for , then becomes the new parent of .
Algorithm 3: ExtendRRT* |
4. The RRT-Based Motion Planning with Dynamics
4.1. The LQR-RRT* Algorithm
- LQRNearest: The procedure identifies a node in that is closest to according to the cost function, i.e.,
- Near nodes: the procedure employs the cost function to find a set of near nodes in that are close to , i.e.,
- Steer: Given two states, , the LQR controller generates a sequence of optimal controls that steer a state trajectory from to .
4.2. The CBF-RRT Algorithm and Its Variations
- Safe steer: Given a state , a time horizon , and a control reference , the procedure steers the state to an exploratory new state . Concretely, it generates a sequence of control inputs by solving a sequence of CBF-QPs (Equation (4)) that steers a collision-free trajectory to at time .
Algorithm 4: CBF-RRT |
- LQR-CBF-Steer: Given two states , the LQR controller generates a sequence of optimal controls that steer a state trajectory from to . The CBF constraints are checked to ensure that the generated trajectory is collision-free. If none of the CBF constraints are violated, then is added to the tree. Otherwise, the trajectory is truncated so that the end state is safe, and the end state is then added to the tree.
5. Experimental Results
5.1. Path Planning
5.2. Double Integrator Model
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Chu, Y.; Chen, Q.; Yan, X. An Overview and Comparison of Traditional Motion Planning Based on Rapidly Exploring Random Trees. Sensors 2025, 25, 2067. https://doi.org/10.3390/s25072067
Chu Y, Chen Q, Yan X. An Overview and Comparison of Traditional Motion Planning Based on Rapidly Exploring Random Trees. Sensors. 2025; 25(7):2067. https://doi.org/10.3390/s25072067
Chicago/Turabian StyleChu, Yang, Quanlin Chen, and Xuefeng Yan. 2025. "An Overview and Comparison of Traditional Motion Planning Based on Rapidly Exploring Random Trees" Sensors 25, no. 7: 2067. https://doi.org/10.3390/s25072067
APA StyleChu, Y., Chen, Q., & Yan, X. (2025). An Overview and Comparison of Traditional Motion Planning Based on Rapidly Exploring Random Trees. Sensors, 25(7), 2067. https://doi.org/10.3390/s25072067