Continuous Path Planning

Optimal Control in Spatially Varying Environments

Introduction

Continuous path planning studies path-planning problems on a continuous domain $\Omega \subseteq \mathbb{R}^n$, where environmental conditions $e(x,t)$ are defined at every position and the motion of a vehicle is governed by given dynamics. For a state $x(t) \in \Omega$ and an action $u(t) \in \mathcal{U}$, a typical formulation is

\[\dot{x}(t)=f\bigl(x(t),u(t),e(x(t),t)\bigr), \qquad u^\star=\underset{u}{\operatorname{argmin}} \left[\Phi(x(T))+\int_0^T L(x(t),u(t),e(x(t),t))\,dt\right].\]

Instead of finding only a single path from one initial position, the objective is to determine an optimal action $\pi^\star(x,t)$ at every position in the domain. The resulting feedback policy $\pi^\star:\Omega\times[0,T]\rightarrow\mathcal{U}$ can guide the vehicle from different initial states while satisfying its dynamic and environmental constraints.

The objective $J(u)$ may represent travel time, energy consumption, distance, or risk. Typical applications include marine routing under ocean currents, aircraft routing under wind fields, and motion through heterogeneous media in which the attainable speed varies by location or direction.

Prerequisites

  • Optimal Control (Undergraduate Level)
  • Optimal Design (Undergraduate Level)
  • Numerical Analysis
  • Differential Equations
  • Linear Algebra
  • Calculus

References

Curriculum

No. Title Source View
1 Dynamic Programming and Discrete LQR
Info
  • Bellman의 최적성 원리와 cost-to-go를 설명한다.
  • Bellman recursion과 상태·제어 격자화를 구성한다.
  • Discrete LQR의 Riccati recursion을 유도한다.
 MIT 16.323, Lec. 3
2 HJB Equation and Continuous LQR
Info
  • 연속시간 DP에서 HJB 방정식을 유도한다.
  • Continuous LQR의 DRE와 CARE를 유도한다.
  • LQR gain, 정상상태, weighting 선택을 분석한다.
 MIT 16.323, Lec. 4