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Potential Functions based Sampling Heuristic For Optimal Path Planning

Ahmed Hussain Qureshi, Yasar Ayaz

arXiv:1704.00264v1cs.ROcs.AI

TL;DR

RRT* can converge slowly to optimal paths, requiring many iterations and substantial memory and time. This paper proposes P-RRT*, which integrates APF-guided sampling into RRT*, and reports faster convergence with fewer iterations across varied environments and non-holonomic constraints.

  • Problem

    RRT*’s slow convergence to optimal path solutions requires many iterations, increasing memory and time demands for motion planning.

  • Method

    P-RRT* incorporates Artificial Potential Fields into RRT* to guide random samples toward the goal for further path optimization.

  • Results

    P-RRT* is reported to converge faster than RRT*, use less memory, retain asymptotic optimality, and avoid local-minima problems.

  • Takeaways & Limitations

    The authors identify P-RRT* as a potentially efficient solution for real-time motion-planning problems because it determines optimal paths quickly.

  • Takeaways & Limitations

    Weak δ-clearance cases require a very small dobs value, and the evaluation also includes a non-holonomic differential-drive setting.

Abstract

from arXiv · show

Rapidly-exploring Random Tree Star(RRT*) is a recently proposed extension of Rapidly-exploring Random Tree (RRT) algorithm that provides a collision-free, asymptotically optimal path regardless of obstacle's geometry in a given environment. However, one of the limitations in the RRT* algorithm is slow convergence to optimal path solution. As a result, it consumes high memory as well as time due to a large number of iterations utilised in achieving optimal path solution. To overcome these limitations, we propose the Potential Function Based-RRT* (P-RRT*) that incorporates the Artificial Potential Field Algorithm in RRT*. The proposed algorithm allows a considerable decrease in the number of iterations and thus leads to more efficient memory utilization and an accelerated convergence rate. In order to illustrate the usefulness of the proposed algorithm in terms of space execution and convergence rate, this paper presents rigorous simulation based comparisons between the proposed techniques and RRT* under different environmental conditions. Moreover, both algorithms are also tested and compared under non-holonomic differential constraints.

1 Introduction

RRT* achieves asymptotic optimality but converges slowly because it relies on extensive exploration, increasing iteration, memory, and time demands. P-RRT* incorporates APF-guided sampling to accelerate convergence while retaining the underlying optimization framework.

  • RRT* continues sampling after finding an initial RRT path to optimize it and ensure asymptotic optimality.
  • RRT*’s pure exploration causes slow convergence and high memory requirements because optimal solutions require many iterations.Fast convergence is important for online motion-planning problems.
  • P-RRT* incorporates Artificial Potential Fields into RRT* to guide samples directionally toward the goal.Directionalized samples reduce the iterations and execution time needed to reach an optimal path.
  • Guiding random samples with APF is presented as a novel approach for improving and theoretically characterizing convergence in asymptotically optimal sampling-based algorithms.
  • P-RRT* is evaluated under different scenarios, including local-minima environments, and is reported as more efficient than RRT* in almost all cases.

2 Problem Definition

The paper formalizes feasible, optimal, and fast path-planning problems in an obstacle-free state space. These problems require reaching a goal, minimizing Euclidean path cost, and determining the optimum in as little time as possible.

  • The formal setting defines X as a d-dimensional state space, with obstacle space Xobs, free space Xfree, initial state xinit, and goal region Xgoal.
  • Feasible path planning seeks a path from xinit to Xgoal that remains entirely within Xfree.
  • Optimal path planning seeks a feasible trajectory with minimum cost c*, where cost is based on Euclidean path length.
  • Fast path planning requires determining the optimal path solution in the least possible time.

3 Related Work

The paper presents RRT* and APF as the foundations of P-RRT*. RRT* samples and rewires a tree for path optimization, while APF uses attractive and repulsive potentials to guide motion toward goals and away from obstacles.

  • 3.1 RRT*: RRT* extends RRT*’s random-sampling framework with nearby-node selection, parent selection, tree insertion, and rewiring.
  • 3.1 RRT*: RandomSample draws independent, uniformly distributed configurations from the obstacle-free region Xfree.
  • 3.1 RRT*: NearbyNodes selects tree vertices within a ball around a configuration, while NearestNode returns the closest tree vertex by Euclidean distance.
  • 3.1 RRT*: GetTuple sorts candidate states by path cost, ExtendTo creates straight paths between states, and CollisionFree checks whether a path lies in Xfree.
  • 3.2 Artificial Potential Fields: APF assigns attractive potential to the goal and repulsive potential to obstacles, producing a force equal to the negated potential gradient.
  • 3.2 Artificial Potential Fields: The attractive potential is quadratic outside a goal-centered radius and conical inside it, promoting rapid approach and slower motion near the goal.
  • 3.2 Artificial Potential Fields: Repulsive potential becomes zero when the robot is farther than d* from the nearest obstacle, allowing faster movement toward the goal.
  • 3.2 Artificial Potential Fields: Gradient descent iterates using the global force until the robot reaches a configuration with zero potential energy.

4 P-RRT*

P-RRT* extends RRT* with Artificial Potential Fields to guide random samples toward the goal while preserving obstacle-geometry independence. Its randomized gradient-descent procedure directs samples incrementally using attractive potential, obstacle proximity, and an exploration–exploitation parameter.

  • 4 P-RRT*: P-RRT* adds Artificial Potential Fields to RRT* and uses them to guide random samples toward the goal for further optimization.The guided sample is treated as the algorithm’s random sample after potential-based augmentation.
  • 4 P-RRT*: Randomized Gradient Descent moves each independently sampled x_rand incrementally toward decreasing potential using a small step size λ, producing x_prand.Unlike iterative gradient descent from a previous state, each RGD iteration seeds a random sample independently.
  • 4 P-RRT*: Quadratic attractive potential increases with distance from the goal, directing farther samples more strongly toward the goal region.The proposed method uses quadratic rather than conical attractive potential because random samples are not subject to goal-region overshooting.
  • 4 P-RRT*: The nearest-obstacle procedure computes only the distance from x_rand to the closest obstacle point, so P-RRT* does not require obstacle geometry information.This procedure replaces the robot configuration variable with the random sample when computing the nearest-obstacle distance.
  • 4 P-RRT*: RGD terminates when obstacle distance reaches d*_obs or after at most k iterations, with d*_obs kept small to permit motion near obstacles.The parameter k balances exploitation and exploration: larger values increase exploitation, while smaller values increase exploration.

5 Implementation using non-holonomic wheeled mobile robot (WMR) Poineer 3-DX

The paper evaluates P-RRT* and RRT* under non-holonomic differential-drive constraints using a Pioneer 3-DX robot model. Candidate controls are applied to nearby nodes to estimate feasible future states while extending toward sampled configurations.

  • 5 Implementation using non-holonomic wheeled mobile robot (WMR) Poineer 3-DX: P-RRT* and RRT* are implemented for a non-holonomic differential-drive Pioneer 3-DX robot.The robot orientation θ is included in the non-holonomic constraint formulation.
  • 5 Implementation using non-holonomic wheeled mobile robot (WMR) Poineer 3-DX: For P-RRT*, a collision-free random configuration is directed toward the goal before nearby-node expansion and control evaluation.RRT* samples are not described as goal-directed in this implementation passage.
  • 5 Implementation using non-holonomic wheeled mobile robot (WMR) Poineer 3-DX: Each nearby node is treated as the current robot state, and allowed control inputs estimate future states through the robot’s kinodynamic model.These estimated states are used while extending toward the random sample.

6 Analysis

The analysis establishes that P-RRT* retains probabilistic completeness and asymptotic optimality while using goal-directed sampling to improve convergence toward feasible and optimal paths. Its added sampling procedure has the same asymptotic computational complexity as RRT*, but performance depends on clearance-related assumptions.

  • 6.1 Probabilistic Completeness: P-RRT* retains probabilistic completeness because its samples are connected to the tree and directed toward the goal, making feasible-path discovery probability approach one.The argument parallels RRT* while adding goal-directed sampling.
  • 6.1 Probabilistic Completeness: P-RRT* fails to find a feasible path exponentially faster than RRT* when a feasible path and an attraction sequence exist.This comparison is stated for the limit as iterations approach infinity.
  • 6.2 Asymptotic Optimality: P-RRT* is asymptotically optimal under the same assumptions and γ condition as RRT*: d ≥2 and γ > γ∗.The proof relies on the same asymptotic sampling and rewiring reasoning used for RRT*.
  • 6.3 Fast convergence to optimal path solution: A very small dobs is needed in weak δ-clearance regions because larger obstacle-clearance thresholds can prevent samples from reaching those regions.The stated boundary concerns robustly optimal paths whose nodes may lie on obstacle boundaries.
  • 6.3 Fast convergence to optimal path solution: 60 samples let P-RRT* compute an optimal solution, compared with 2500 samples for RRT*, because its Voronoi bias guides sampling toward the goal.The analysis attributes the difference to goal-directed rather than uniform Voronoi bias.
  • 6.4 Computational Complexity: P-RRT* and RRT* have the same asymptotic computational complexity because RGD(xinit) adds only a constant number of steps independent of tree size.The proposed method changes sampling direction but leaves the remaining RRT* procedures unchanged.

7 Experimental Results

Experiments compare P-RRT* with RRT* across cluttered, complex, and other environments using repeated runs and shared experimental settings. The reported results show faster convergence, fewer iterations, and lower time or memory use for P-RRT* in the supplied cases.

  • Experimental setup: 50 runs per environment compared both algorithms using identical experimental parameters and configuration-space sizes.The evaluation recorded maximum, minimum, and average iterations and running time for reaching an optimal path solution.
  • 2D cluttered environment: P-RRT* found an initial path in 200 iterations and 0.045s, versus RRT* in 16,063 iterations and 0.57s, in a 2D cluttered environment.For the optimal trajectory, P-RRT* used 3,000 iterations and 0.65s, while RRT* found no optimal solution within 5 million iterations.
  • Complex environments: P-RRT* found optimal paths with fewer iterations and less time than RRT* in the complex maze environments.In one reported maze, RRT* still had not returned an optimal path after 5 million iterations.
  • Convergence: P-RRT* converged significantly faster than RRT* after initial feasible-path computation.The convergence rate is calculated from the point at which the initial path is found.
  • Memory and running time: P-RRT* consumed less memory than RRT* across twenty environments because it used fewer iterations.A separate fixed-cost comparison reports that RRT* required more time to converge.
  • Exploration and exploitation: A lower k biases P-RRT* toward exploration, whereas a higher k produces more exploitation; balancing both is important across environments.The supplied results also report that RRT* required barely 1.6 times the time of P-RRT* to determine a viable path.

8 Conclusions and Future work

The paper positions P-RRT* as a response to RRT*’s slow convergence and memory inefficiency while retaining its asymptotic-optimality framework. Its conclusions report faster convergence, no inherited local-minima limitation, and applicability to real-time motion planning, alongside tests in several environments.

  • Motivation: RRT* provides asymptotic optimality but is reported as memory inefficient and slow to converge.These limitations motivate the proposed P-RRT* algorithm.
  • Contribution: P-RRT* incorporates the Artificial Potential Field Algorithm into RRT* to address these limitations.The paper reports experimental and analytical evaluation of the resulting method.
  • Theoretical properties: P-RRT* retains the same asymptotic computational complexity as RRT* and inherits its asymptotic optimality.The paper also states that P-RRT* does not suffer from the local-minima problem.
  • Empirical conclusions: P-RRT* is reported to converge faster to an optimal path solution than RRT*.The paper presents cost-versus-running-time and running-time-ratio comparisons, alongside tests in 3D environments with barriers, narrow passages, and complex mazes.
  • Practical implication: The authors state that faster convergence and rapid optimal-path determination make P-RRT* an efficient solution for real-time motion-planning problems.The supplied figures also include non-holonomic robot demonstrations and local-minima comparisons.
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