Keywords

1 Introduction

The vehicle control community has long looked at human racing for inspiration in designing autonomous control for limit maneuvering to improve vehicle safety. One concept that emerges from this analysis is the notion of a racing line, which defines a minimum-time trajectory through the track. Researchers have leveraged this concept in formulating controllers for extreme maneuvering, largely due to its amenability with path-planning and tracking paradigms popular in control systems [2]. And indeed, recent work has confirmed the efficacy of this approach, with Kegelman et al. demonstrating that a racing line-based vehicle controller could beat a skilled amateur racecar driver on a closed course [5].

While these results present a major achievement, the authors importantly noted that their controller could not beat a professional human driver. And more recently, comparisons to human data by Hermansdorfer et al. reached a similar conclusion [4]. This difference between automated control and professional drivers indicates that state-of-the-art controllers may not be extracting all of the vehicle’s dynamic potential; this observed performance gap has thus continued to motivate research.

In order to address this difference, additional literature has focused on studying the human racing style, instead of purely the performance, in more detail. In a related study, Kegelman et al. instrumented human-driven racecars at the Laguna Seca Raceway and observed that human drivers, in actuality, do not track a static line when driving [6]. Instead, they appear to anchor their racing line on certain parts of the track, such as apexes, but drive more freely between. This statistical finding is consistent with some engineering handbooks on racing strategy (e.g. Smith [8]) and broaches whether a path planning and tracking paradigm is the most effective way to extract peak performance from a vehicle.

While this empirical compliance of the racing line has been observed and quantified, it has not yet, to the best of the authors’ knowledge, been rationalized from an optimization point of view. In this paper, we assess the importance of the racing line by delineating the set of near-optimal solutions from a standard minimum-time trajectory optimization. We find, perhaps intuitively, that the basin of near-optimality is wide, indicating that there is a family of vehicle trajectories that are all competitive. This suggests that what might separate the best autonomous controllers from professional drivers is not a difference in ability to track a predefined trajectory, but rather an ability to constantly operate the vehicle at its limits in a meaningful way. These results have implications for more general vehicle control that demands extreme maneuvering capability, motivating the potential of alternate control paradigms that can better extract the most performance out of the vehicle.

The rest of this paper is structured as follows. First, we briefly review the phenomenon observed in Kegelman et al.; namely, human drivers’ apparent consistency in lap time but variance in trajectory. We then set out to understand this phenomenon better using the language of trajectory optimization. In Sect. 3, we build a minimum-time trajectory optimization to find a racing line; then, in Sect. 4, we present a technique to compute the maneuvering time sensitivity to lateral variations. In Sect. 5, we analyze the results of this method when applied to the Laguna Seca Raceway, which shows consistency with the work presented in [6]. Lastly, we offer concluding thoughts in Sect. 6.

2 Observations from Human Driving

As mentioned previously, observations from human racing behavior indicate that humans do not place a uniform weight on tracking a predefined racing line. Kegelman et al. quantified this by instrumenting a racecar and collecting data from skilled human drivers [6]. As depicted in Fig. 1, the drivers exhibited significant path deviation lap-over-lap, on the order of one meter. This deviation was correlated to the track geometry, however, appearing tightest near the apexes and widest on the straight sectors. Remarkably, despite the variance in driven trajectories, the recorded mean lap times across drivers were extremely consistent—within 0.5%.

Fig. 1.
figure 1

Left: A visualization of the mean absolute dispersion in lateral position from the skilled human driver’s trajectories around the Laguna Seca Raceway studied in [6]. Right: The lap time sensitivity analysis proposed in this paper identifies a similar trend in lateral position compliance, suggesting that there is a tube of solutions that exhibits varying degrees of dispersion at different points along the track.

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3 Minimum-Time Trajectory Optimization

To better comprehend this observation, we evaluate the compliance of a racing line from an optimization point of view. The first step in this process is to solve for a minimum-time line. In this section, we accomplish this using standard trajectory optimization techniques [9]. To simplify the analysis, we model the vehicle as a friction-limited point mass using parameters derived from the Grand Sport Corvette instrumented in [6].

The point mass’s state is characterized by its longitudinal velocity \(v_x\), lateral velocity \(v_y\), track centerline progress distance s, and lateral position to the centerline e. We assume direct force control over the point mass’s motion in the longitudinal and lateral directions (\(F_x\) and \(F_y\)). To facilitate straightforward slew rate constraints on the force inputs, we define the control vector as the force derivatives and the actual force inputs are appended to the state vector. The resulting state vector is thus defined as \(\boldsymbol{x} {:}{=}(v_x, v_y, s, e, F_x, F_y)\) with the control vector as \(\boldsymbol{u} {:}{=}(\dot{F}_x, \dot{F}_y)\). The state evolution is given by

$$\begin{aligned} &\dot{v}_x=({F_x - F_{\text {d},x}})/{m} + r v_y \qquad \dot{s}={v_x}/({1-\kappa e}) \qquad \dot{F}_x = \dot{F}_x \\ &\dot{v}_y=({F_y - F_{\text {d},y}})/{m} - r v_x \qquad \dot{e}=v_y \qquad \qquad \qquad \dot{F}_y = \dot{F}_y, \end{aligned}$$

where m is the mass and \(\kappa \) is the local centerline curvature. The drag terms \(F_{\text {d},\{x, y\}}\) model the effects of aerodynamic resistance, and we assume a simplified model of drag which is quadratic in velocity.

In this derivation, we have added a notion of directionality to the point mass. This better facilitates encoding the difference between lateral force constraints (due to the friction circle) and longitudinal force constraints (due to the engine power limit and rear-wheel drive) in the trajectory optimization. The point mass’s xy-frame is here assumed to align with the local centerline’s se-frame; hence, the rotation rates r of these frames is given by \(\kappa \dot{s}\).

The racing line optimization is now defined as

figure a

where \(i = 0, \ldots , N-1\) and \(j = 0, \ldots , N\). The trajectory is discretized into \((N+1)\) stages with the sole objective to minimize the total maneuvering time. We approximate the system’s dynamics with a trapezoidal integration function \(h_\text {dyn}\) that uses a spatial reformulation of the dynamics as described in [9]. The final constraint imposes a drive traction limit on the \(F_x\) input, which incorporates the limits of the vehicle’s rear-wheel drive and the effects of steady-state load transfer. This maximum traction force \(F_{x,\text {max}}^\text {RWD}\) is defined as

$$\begin{aligned} F_{x,\text {max}}^\text {RWD} &{:}{=}\mu \left( F_{z,\text {r,static}} + \varDelta F_{z, \text {long}}\right) = \mu \left( \frac{a}{a+b} m g + \frac{F_x h_\text {com}}{a+b}\right) . \end{aligned}$$

4 Computing the Sensitivity of the Racing Line

Given a baseline racing line, we now investigate its lap time sensitivity to lateral perturbations. In principle, the solution gradients from Opt. 1 could be assembled to obtain a sensitivity at each position along the racing line. While these gradients generally provide useful information, in the context of a dynamical system, it is not immediately clear how to extract the absolute spread of trajectories from them. An impulse change in the lateral position e at one spatial index is nonphysical, as the vehicle’s dynamics require time to develop and transition.

To address this, we explore a related method that is more dynamically consistent. The core idea is to construct optimization-based subproblems around the baseline solution, whose objectives are to delineate the boundary of the \(\epsilon \)-suboptimal level set. Skaf and Boyd [7] provide an excellent discussion on this concept for the general optimization setting; below, we formulate an extension of this approach to the racing problem.

The main strategy is to convexify the original optimization problem around a solution and then batch solve for dispersion-maximizing subtrajectories subject to a time suboptimality constraint. For a given index-of-maximization k, the resulting subproblem is as below:

figure b

The constraints are nearly identical to those in Opt. 1 apart from the added time suboptimality limit (2d). This subproblem relies on the linearized dynamics \(h_\text {dyn,lin}\) about a solution from Opt. 1 using perturbational states \(\boldsymbol{\delta x}\) and inputs \(\boldsymbol{\delta u}\). The constraints (2c) are convexified versions of constraints (1c)-(1g) using these perturbational quantities. The resulting trajectories are hence locally dynamically feasible and better reflect the system’s dynamics than pointwise sensitivities.

5 Results for Laguna Seca and Human Data Comparison

After solving for a baseline racing line at Laguna Seca, we now solve the dispersion-maximizing subproblems at \(n_\text {sp}\) evenly spaced indices to produce a bundle of \(\epsilon \)-suboptimal trajectories. The optimization problems are constructed and solved with the CasADi and CVXPY packages [1, 3]. Table 1 lists further parameters.

Fig. 2.
figure 2

Lateral path deviation plotted against the local curvature. Our analysis, right, reveals a similar inverse relationship as observed in the human data from [6], left.

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Figure 1 indicates that the lateral spread of suboptimal trajectories is not uniform across the whole track, appearing tightest in the corners and widest on the straights. This observation is consistent with the conclusion in [6] that driven trajectories “anchor” at the apexes and flow more freely in between. Figure 2 plots the lateral deviance against the path curvature showing that the basin of suboptimal solutions is narrowest at high curvature. These data suggest that the variability displayed by human drivers is not tracking error on an ideal line but instead may correspond to the selection of a nearly optimal solution most suited to the track and car conditions at that moment.

6 Conclusion

While the racing line’s compliance has been described qualitatively in handbooks on racing and quantified with analysis of instrumented race cars, in this paper we aimed to validate this phenomenon using mathematical optimization. Our results are consistent with prior observations and suggest that there is a tube of solutions that achieves the minimum-time objective. Automated vehicles tracking a fixed line have no guarantee that their closed-loop trajectory, inclusive of corrections, falls into this tube. This analysis motivates investigating control schemes that move beyond path-planning and tracking to consider optimality of the closed-loop trajectories.

Table 1. Vehicle and optimization parameters
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