Path Tracking for Automated Driving: A Tutorial on Control System Formulations and Ongoing Research

Abstract

In the United States, the DARPA (Defense Advanced Research Projects Agency) Grand Challenges [Thrun et al. (J. Field Robot. 23(9):661–692, 2006); Urmson et al. (J. Field Robot. 25(8):425–466, 2008); Urmson et al. (J. Field Robot. 23(8):467–508, 2006); Campbell (Steering Control of an Autonomous Ground Vehicle with Application to the DARPA Urban Challenge. Massachusetts Institute of Technology, 2007)] demonstrated that autonomous driving can be achieved through vision and sensor systems capable of detecting and interpreting the vehicle operating environment, rather than through autonomous driving options relying on the infrastructure (e.g., through magnets installed on the road surface to indicate the lanes), or vehicle-to-vehicle or vehicle-to-infrastructure communication systems. The latter options are very useful to further enhance the performance, safety, and energy efficiency, but are not strictly required. In a typical automated driving system, a reference path and a reference speed profile are defined based on the sensed environment. At a lower level of the control system hierarchy, a path tracking controller is responsible for calculating the steering angle for achieving the reference trajectory, while a speed controller determines the wheel torque demand for tracking the reference speed. Speed control implementations are already quite common in production vehicles equipped with cruise control and adaptive cruise control systems. Hence, the core element of novelty for autonomous driving in the area of vehicle control is represented by the steering control function for path tracking. Different steering-based path tracking algorithms, ranging from geometrical methods to model-predictive controllers, are presented and discussed in this contribution, together with the expected future research and vehicle implementation directions in the field.

Appendix: Definitions of Invariant Sets, Minkowski Sum ⊕, and Pontryagin Difference ⊖

The following definitions are provided:

• (a) Reachable set for systems with external inputs. Consider a system ξ _k + 1 = f(ξ _k, u _k) + w _k, with ξ _k ∈ Ξ, ${u}_k\mathcal{\in}\mathcal{U}$, ${w}_k\mathcal{\in}\mathcal{W}$. The one-step robust reachable set from a given set of states $\mathcal{S}$ is $\mathrm{Reac}{\mathrm{h}}_{\text{f}}\left(\mathcal{S},\mathcal{W}\right)\triangleq \big\{\xi \in {\mathbb{R}}^n|\linebreak {\xi}_0\mathcal{\in}\mathcal{S}, \mathcal{\exists}u\in \mathcal{U},\mathcal{\exists}w\in \mathcal{W}:\xi =f\left({\xi}_0,u,w\right)\big\}$;

• (b) Robust positively invariant set. A set $\mathcal{Z}\mathcal{\subseteq}\Xi$ is said to be a robust positively invariant set for the autonomous system ξ _k + 1 = f _a(ξ _k) + w _k, with ξ _k ∈ Ξ and ${w}_k\mathcal{\in}\mathcal{W}$, if ${\xi}_0\mathcal{\in}\mathcal{Z}\mathcal{\Rightarrow}{\xi}_k\mathcal{\in}\mathcal{Z}$, $\forall {w}_k\mathcal{\in}\mathcal{W},\mathcal{\forall}k\ge \in {\mathbb{N}}^{+}$

• (c) Minimal robust positively invariant set. The set ${\mathcal{Z}}_{\mathit{\infty}}\subseteq \Xi$ is the minimal robust positively invariant set for the defined autonomous system, if ${\mathcal{Z}}_{\mathit{\infty}}$is a robust positively invariant set and ${\mathcal{Z}}_{\mathit{\infty}}$ is contained in every closed robust positively invariant set in Ξ (see [89] for the details)

• (d) Minkowski sum. The Minkowski sum of two polytopes, ℘ and $\mathfrak{H}$, is the polytope $\mathfrak{\wp}\mathfrak{\bigoplus}\mathfrak{H}:= \left\{x+h\in {\mathbb{R}}^n|x\in \mathit{\wp},h\in \mathfrak{H}\right\}$;

• (e) Pontryagin difference. The Pontryagin difference of two polytopes, ℘ and $\mathfrak{H}$, is the polytope $\mathit{\wp}\ominus \mathfrak{H}:= \left\{x\in {\mathbb{R}}^n|x+h\in \mathit{\wp},\forall h\in \mathfrak{H}\right\}$.