[Paper Review] Stochastic Functional Gradient Path Planning in Occupancy Maps

Gilad Francis, Lionel Ott+1 more|arXiv (Cornell University)|May 17, 2017

Robotic Path Planning Algorithms19 references3 citations

TL;DR

This paper proposes a stochastic functional gradient path planner using approximate kernel-based path representation to enable efficient, scalable trajectory optimization in occupancy maps. By leveraging stochastic gradient descent with fixed-linear computational cost, the method achieves faster convergence and superior path smoothness compared to non-parametric Gaussian process planners and sampling-based methods like RRT*, while maintaining safety and efficiency in both simulation and real-world data.

ABSTRACT

Planning safe paths is a major building block in robot autonomy. It has been an active field of research for several decades, with a plethora of planning methods. Planners can be generally categorised as either trajectory optimisers or sampling-based planners. The latter is the predominant planning paradigm for occupancy maps. Trajectory optimisation entails major algorithmic changes to tackle contextual information gaps caused by incomplete sensor coverage of the map. However, the benefits are substantial, as trajectory optimisers can reason on the trade-off between path safety and efficiency. In this work, we improve our previous work on stochastic functional gradient planners. We introduce a novel expressive path representation based on kernel approximation, that allows cost effective model updates based on stochastic samples. The main drawback of the previous stochastic functional gradient planner was the cubic cost, stemming from its non-parametric path representation. Our novel approximate kernel based model, on the other hand, has a fixed linear cost that depends solely on the number of features used to represent the path. We show that the stochasticity of the samples is crucial for the planner and present comparisons to other state-of-the-art planning methods in both simulation and with real occupancy data. The experiments demonstrate the advantages of the stochastic approximate kernel method for path planning in occupancy maps.

Motivation & Objective

• To address the scalability limitations of non-parametric functional gradient planners in occupancy map path planning.
• To develop a computationally efficient path representation that supports fast, stochastic updates without fixed sampling resolution.
• To enable trajectory optimization in occupancy maps despite sensor coverage gaps and incomplete data.
• To improve path quality by minimizing obstacle occupancy while maintaining smooth, safe trajectories.
• To demonstrate the superiority of stochastic sampling and kernel approximation over deterministic or non-parametric approaches.

Proposed method

• The method uses kernel approximation to represent paths as a weighted sum of basis functions, enabling a fixed-linear computational cost independent of data size.
• Stochastic Gradient Descent (SGD) is employed to iteratively update the path model using random samples across the entire path domain.
• The path model is updated using stochastic samples drawn from the full trajectory space, avoiding commitment to a predefined sampling resolution.
• The objective function balances path length and obstacle avoidance, with cost computed via occupancy map probabilities.
• The approach generalizes the Gaussian Process Motion Planner by replacing non-parametric inference with an approximate kernel representation.
• Convergence is guaranteed under standard SGD assumptions, ensuring stable optimization despite stochasticity.

Experimental results

Research questions

• RQ1 Can a kernel-based path representation reduce computational complexity in functional gradient path planning while preserving expressiveness?
• RQ2 How does stochastic sampling across the full path domain improve optimization compared to fixed-resolution sampling?
• RQ3 What is the trade-off between convergence speed and computational cost in stochastic functional gradient planners for occupancy maps?
• RQ4 Can the proposed method achieve safer and smoother paths than RRT* and GP-based planners in environments with sensor gaps?
• RQ5 Does the approximate kernel model maintain performance with increasing numbers of observations, unlike non-parametric alternatives?

Key findings

• The proposed approximate kernel method achieves a maximum occupancy of 0.46±0.01, significantly safer than RRT* (0.50±0.01), indicating better obstacle avoidance.
• The path length is 20.48±0.07 meters, slightly shorter than RRT* (20.17±0.14 m), demonstrating improved efficiency.
• The method requires 3,530±160 samples on average, which is less than RRT* (12,180±650) and more than the GP planner (1,080±250), indicating a balance between convergence speed and computational cost.
• Despite requiring more iterations than the GP planner, the proposed method has lower runtime due to its fixed-linear complexity, unlike the cubic cost of the GP-based approach.
• The stochastic planner produces smooth, mid-line paths between obstacles, while RRT* paths are jagged and dangerously close to walls.
• Convergence is stable across repeated runs, with consistent performance in both simulation and real occupancy map data.