A convex reformulation for speed planning of a vehicle under the travel time and energy consumption objectives

TL;DR

This paper presents a convex reformulation for vehicle speed planning that optimizes travel time and energy consumption, providing an efficient solution method with proven theoretical guarantees.

Contribution

It introduces a convex relaxation of a non-convex speed planning problem and develops a dynamic programming approach with polynomial time complexity.

Findings

Convex relaxation is exact for the speed planning problem.

Feasibility-based bound-tightening effectively characterizes the feasible region.

The proposed method efficiently computes approximate solutions with $O(n^2)$ complexity.

Abstract

In this paper we address the speed planning problem for a vehicle along a predefined path. A weighted sum of two conflicting objectives, energy consumption and travel time, is minimized. After deriving a non-convex mathematical model of the problem, we prove that the feasible region of this problem is a lattice. Moreover, we introduce a feasibility-based bound-tightening technique which allows us to derive the minimum and maximum element of the lattice, or establish that the feasible region is empty. We prove the exactness of a convex relaxation of the non-convex problem, obtained by replacing all constraints with the lower and upper bounds for the variables corresponding to the minimum and maximum elements of the lattice, respectively. After proving some properties of optimal solutions of the convex relaxation, we exploit them to develop a dynamic programming approach returning an approximate solution to the convex relaxation, and with time complexity $O(n^2)$, where $n$ is the number of points into which the continuous path is discretized.