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A Semi-Lagrangian Approach for Time and Energy Path Planning Optimization in Static Flow Fields

Víctor C. da S. Campos, Armando A. Neto, Douglas G. Macharet

TL;DR

This work tackles time- and energy-optimal path planning for autonomous agents in static flow fields with obstacles by formulating a multi-objective optimal control problem. It introduces a Harmonic Transformation $\bar{v}(\mathbf{x}) = v(\mathbf{x})/(1+v(\mathbf{x}))$ to map value functions into $[0,1)$ and derives a transformed DP/HJB solved with a semi-Lagrangian scheme on unstructured grids. Two Pareto-front computation approaches are proposed: a deterministic Concurrent Policy Iteration (CPI) and a Multi-objective Evolutionary Policy Iteration (MEPI) using NSGA-II, both exploiting the Harmonic Transformation. Five numerical examples, including obstacle-laden and time-varying flow scenarios, demonstrate convergence, interpretable Pareto fronts, and favorable comparison to RRT$^*$ in energy-time tradeoffs, validating the method's potential for real-time, robust closed-loop planning in realistic environments.

Abstract

Efficient path planning for autonomous mobile robots is a critical problem across numerous domains, where optimizing both time and energy consumption is paramount. This paper introduces a novel methodology that considers the dynamic influence of an environmental flow field and considers geometric constraints, including obstacles and forbidden zones, enriching the complexity of the planning problem. We formulate it as a multi-objective optimal control problem, propose a novel transformation called Harmonic Transformation, and apply a semi-Lagrangian scheme to solve it. The set of Pareto efficient solutions is obtained considering two distinct approaches: a deterministic method and an evolutionary-based one, both of which are designed to make use of the proposed Harmonic Transformation. Through an extensive analysis of these approaches, we demonstrate their efficacy in finding optimized paths.

A Semi-Lagrangian Approach for Time and Energy Path Planning Optimization in Static Flow Fields

TL;DR

This work tackles time- and energy-optimal path planning for autonomous agents in static flow fields with obstacles by formulating a multi-objective optimal control problem. It introduces a Harmonic Transformation to map value functions into and derives a transformed DP/HJB solved with a semi-Lagrangian scheme on unstructured grids. Two Pareto-front computation approaches are proposed: a deterministic Concurrent Policy Iteration (CPI) and a Multi-objective Evolutionary Policy Iteration (MEPI) using NSGA-II, both exploiting the Harmonic Transformation. Five numerical examples, including obstacle-laden and time-varying flow scenarios, demonstrate convergence, interpretable Pareto fronts, and favorable comparison to RRT in energy-time tradeoffs, validating the method's potential for real-time, robust closed-loop planning in realistic environments.

Abstract

Efficient path planning for autonomous mobile robots is a critical problem across numerous domains, where optimizing both time and energy consumption is paramount. This paper introduces a novel methodology that considers the dynamic influence of an environmental flow field and considers geometric constraints, including obstacles and forbidden zones, enriching the complexity of the planning problem. We formulate it as a multi-objective optimal control problem, propose a novel transformation called Harmonic Transformation, and apply a semi-Lagrangian scheme to solve it. The set of Pareto efficient solutions is obtained considering two distinct approaches: a deterministic method and an evolutionary-based one, both of which are designed to make use of the proposed Harmonic Transformation. Through an extensive analysis of these approaches, we demonstrate their efficacy in finding optimized paths.
Paper Structure (20 sections, 1 theorem, 75 equations, 13 figures, 2 algorithms)

This paper contains 20 sections, 1 theorem, 75 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Problem formulation
  4. Harmonic Transformation and Dynamic Programming
  5. Numerical approximation
  6. Energy and time value functions
  7. Proposed approaches
  8. Concurrent Policy Iteration
  9. Genetic Algorithm and Evolutionary Policy Iteration
  10. Numerical simulations
  11. Example 1 - comparing Harmonic and Kruzkov transformations
  12. Example 2 - simple linear drift dynamics
  13. Example 3 - scenario with obstacles
  14. Comparison with RRT$^\star$
  15. Example 4 - marine navigation problem
  16. ...and 5 more sections

Key Result

Theorem 1

Consider the optimal control problem represented by the equation eq:HJB_harmonic. As long as $\mathbf{f}$ and $\ell$ are Lipschitz, and $\ell$ is positive definite, with regards to the states, there exists a unique viscosity solution to this equation, representing the transformed value function of t

Figures (13)

  • Figure 1: Time to reach the origin employing the Harmonic transformation and the Kruzkov transformation in Example 1. The obstacles are presented in black whereas the level curves of the time to reach the origin value functions are presented as the colored curves. The left plot represents the solution found by the Harmonic transformation while the right plot represents the solution found by the Kruzkov transformation.
  • Figure 2: Time to reach the origin employing the Harmonic transformation and the Kruzkov transformation in Example 1. The left plot represents the solution found by the Harmonic transformation while the right plot represents the solution found by the Kruzkov transformation.
  • Figure 3: Result running the Concurrent Policy Iteration for Example 2. The left plot represents the solution set found, while the right plot shows how each solution corresponds to a different path in state space, for initial point $(-0.9, -0.9, 0.9)$. The red arrows represent the flow vector field, and the solutions are color-coded to match each trajectory in state space with a point on the objective function space, with yellow being the fastest trajectory and red the trajectory that spends the least amount of energy.
  • Figure 4: Result running the Multi-objective Evolutionary Policy Iteration for Example 2. The left plot represents the final population found while the right plot shows how each solution corresponds to a different path in state space, for initial point $(-0.9, -0.9, 0.9)$. The red arrows represent the flow vector field, and the solutions are color-coded to match each trajectory in state space with a point on the objective function space, with green being the fastest trajectory and blue the trajectory that spends the least amount of energy.
  • Figure 5: Result found when running Concurrent Policy Iteration for Example 3. The left plot represents the solutions found while the right plot shows how each solution corresponds to a different path in state space, for initial point $(0, 0.9)$. The blue arrows represent the flow vector field, the black region represent an obstacle, and the solutions are color-coded to match each trajectory in state space with a point on the objective function space, with yellow being the fastest trajectory and red the trajectory that spends the least amount of energy.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3