A Computationally Efficient and Human Implementable Minimum-lap-time Control Policy for Energy-limited Race Cars

TL;DR

This paper first formulate the energy-constrained minimum-lap-time control problem via Pontryagin's Minimum Principle and derive the optimal policy and costate dynamics using Karush-Kuhn-Tucker (KKT) optimality conditions, and shows that the optimal control policy follows a bang-bang structure that is easily implementable by a human driver.

Abstract

This paper presents a provably optimal, real-time capable energy management policy for race cars that provides simple human-driver-implementable control cues. Specifically, we first formulate the energy-constrained minimum-lap-time control problem via Pontryagin's Minimum Principle (PMP) and derive the optimal policy and costate dynamics using Karush-Kuhn-Tucker (KKT) optimality conditions. We show that the optimal control policy follows a bang-bang structure that is easily implementable by a human driver, eliminating the need for potentially dangerous active throttle pedal overwrites or distracting signals. Moreover, the analytical formulation of the optimal system dynamics allows us to recast the problem as a sequence of boundary-value problems, which can be efficiently solved using root-finding methods. Our results show that our proposed approach can compute the same globally optimal control strategies of existing numerical methods based on direct optimal control, whilst drastically reducing computation time from the order of seconds to milliseconds.

Figures (5)

• Figure 1: Electric endurance racing car (InMotion) InMotion, typically severely energy-limited. Optimal coasting and regenerative braking points along the circuit are highlighted in blue and gold respectively. The dark and light regions respectively indicate slow and high speeds.
• Figure 2: Optimal bang-bang control policy as a function of the costate ratio.
• Figure 3: Optimal trajectories for a single straight, between two costate jumping points at the apexes of subsequent corners, found by integrating the optimal solution dynamics. Two discrete signals suffice to instruct the driver to manage energy, minimizing cognitive load and distraction. The result of a direct solving method (NLP) on the problem is shown in a lighter color, matching and verifying the derivation of the costate dynamics.
• Figure 4: Velocity and costate ratio trajectories over a full lap of the Zolder circuit for three different energy budgets. Lap-time-optimal coasting and regenerative braking phases are triggered as the costate ratio passes the thresholds.
• Figure 5: Lap-time gains over energy budget for the different approaches. An adaptive costate trajectory is very advantageous, and its benefits are bigger when energy limits are tighter. For fixed costate trajectories and severe energy limitations, a regenerative braking phase may be disadvantageous due to the car slowing down too quickly and early in some corners.