[Paper Review] Approximate Time-Optimal Trajectories for Damped Double Integrator in 2D Obstacle Environments under Bounded Inputs

Vishnu S. Chipade, Dimitra Panagou|arXiv (Cornell University)|Jul 10, 2020

Reinforcement Learning in Robotics4 citations

TL;DR

This paper presents a method for computing near time-optimal trajectories for agents in 2D environments with polygonal obstacles under damped double integrator dynamics and bounded acceleration. It introduces the C¹-Tangent graph to ensure C¹-smooth paths and derives analytical velocity profiles that minimize travel time while respecting acceleration bounds, achieving near-optimality within a small ε of the theoretical optimum.

ABSTRACT

This article provides extensions to existing path-velocity decomposition based time optimal trajectory planning algorithm \\cite{kant1986toward} to scenarios in which agents move in 2D obstacle environment under double integrator dynamics with drag term (damped double integrator). Particularly, we extend the idea of a tangent graph \\cite{liu1992path} to $\\calC^1$-Tangent graph to find continuously differentiable ($\\calC^1$) shortest path between any two points. $\\calC^1$-Tangent graph has a continuously differentiable ($\\calC^1$) path between any two nodes. We also provide analytical expressions for a near time-optimal velocity profile for an agent moving on these shortest paths under the damped double integrator with bounded acceleration.

Motivation & Objective

• To enable time-optimal trajectory planning for agents in 2D obstacle environments with damped double integrator dynamics.
• To ensure C¹-smooth (continuously differentiable) paths between start and goal points despite obstacle constraints.
• To develop analytical, near time-optimal velocity profiles that respect bounded acceleration inputs.
• To extend the ellipse filter to convex polygonal obstacles for efficient path search on the C¹-Tangent graph.
• To provide a computationally efficient alternative to numerical integration-based velocity profile computation.

Proposed method

• Introduces the C¹-Tangent graph by constructing C¹-continuous, convex boundaries around polygonal obstacles and computing common tangents between them.
• Develops a novel quadratic function to analytically compute common tangents between two C¹-continuous, closed, convex curves approximating obstacles.
• Applies a modified ellipse filter to reduce the search space for shortest path computation on the C¹-Tangent graph in environments with convex obstacles.
• Derives analytical expressions for C⁰-continuous, near time-optimal velocity profiles along shortest paths, ensuring acceleration bounds are respected.
• Uses piecewise constant acceleration profiles on straight and circular segments, with closed-form solutions for terminal speeds based on path length and dynamics constraints.
• Solves a constrained optimization problem to maximize intermediate speeds under acceleration and velocity bounds, minimizing total travel time.

Experimental results

Research questions

• RQ1 How can C¹-smooth trajectories be generated in 2D obstacle fields under damped double integrator dynamics?
• RQ2 What is an efficient and analytically tractable method to compute common tangents between C¹-approximated obstacle boundaries?
• RQ3 How can the search space for shortest path computation be reduced in generic convex obstacle environments?
• RQ4 What analytical velocity profile achieves near time-optimality under bounded acceleration and damped double integrator dynamics?
• RQ5 Can a closed-form solution for near time-optimal velocity profiles be derived without iterative numerical integration?

Key findings

• The C¹-Tangent graph guarantees the existence of a C¹-smooth path between any two nodes in the graph, enabling motion planning under second-order dynamics.
• The proposed quadratic function enables exact analytical computation of common tangents between C¹-approximated obstacle boundaries.
• The extended ellipse filter reduces the search space for shortest path computation on the C¹-Tangent graph in convex polygonal obstacle environments.
• Near time-optimal velocity profiles are derived with analytical expressions, ensuring travel time τ satisfies τ* ≤ τ ≤ (1+ε)τ* for small ε > 0.
• The velocity profile computation avoids costly numerical integration, significantly reducing computational time compared to prior methods.
• The method achieves maximum feasible speeds along the path while respecting acceleration bounds, with terminal speeds determined via a two-pass algorithm.