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

TL;DR

The paper tackles speed planning along a fixed path with travel-time and energy consumption as competing objectives, modeling the problem in a discretized form using variables $w_i$ that represent half the squared speed. It develops a feasibility-based bound-tightening procedure to obtain lattice-structured bounds $y$ and $z$, proving an exact convex relaxation exists when replacing non-convex constraints with these bounds and introducing reference speeds $w^+$ and $w^-$. Leveraging the lattice structure, a dynamic programming approach with $O(n^2)$ complexity yields an approximate solution to the convex relaxation, with theoretical guarantees that the DP solution is close to the true optimum as the discretization step $h$ vanishes. Computational experiments show the DP method achieves substantial speedups (orders of magnitude faster) compared to general convex solvers while maintaining small relative optimality gaps, suggesting viability for real-time multi-objective speed planning.

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.

Figures (8)

• Figure 2: Computational times of the DP and the commercial solver.
• Figure 3: Maximum relative objective function value differences.
• Figure 4: Maximum absolute speed differences.
• Figure 5: Computational times of the DP for different values of $n$.
• Figure 6: Example paths with circles and crosses denoting starting and ending points, respectively.

Theorems & Definitions (40)

• Remark 1.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Proposition 3.1
• proof
• Remark 3.2