Keywords

1 Introduction

There has been great technical progress in automated driving. In terms of user acceptance, the question of optimum functional behavior is increasingly being raised, with the focus largely on occupant comfort [1, 2].

Using the example of stop-and-go situations, it is shown that the entire 3D parameter space [3] (driver, driven vehicle, driving environment) should be taken into ac-count when designing automated driving behavior. Therefore, a follow-up driving controller is presented that can be optimized both in terms of occupant comfort and traffic efficiency in stop-and-go situations. Different variants of the controller are analyzed objectively and a recommendation for an optimal functional behavior is derived.

2 Stop-and-Go Follow-Up Control

The following vehicle controller is designed on the basis of findings from 20 h of real driving data in stop-and-go traffic and relates to the longitudinal control of an ego vehicle behind an object vehicle in front.

2.1 Follow-Up Concept

The basic idea behind the following distance control for automated vehicles in stop-and-go situations presented here is, to soften the conventional adaptive cruise control (ACC) follow-up concept with the aim of maintaining a constant time gap in favor of larger absolute distances, depending on the driving situation. These are used in the further course of the journey to compensate higher object dynamics, by allowing the ego vehicle to approach a more decelerating object vehicle with less deceleration instead of also decelerating sharply immediately.

With further increasing distances or large relative velocity of the object vehicle, an increase in ego dynamics ensures that the absolute object distance does not exceed selectable limits, which benefits overall traffic efficiency [4].

2.2 Follow-Up Control Using MPC

The tracking controller is based on an MPC acceleration controller, that was parameterized very defensively with the help of auxiliary conditions that mainly concern a reduction of the longitudinal acceleration of the ego vehicle. The parameterization of the dynamic MPC model is based on [5] and is completely unchanged for different function variants.

With the relative speed Δvx between object and ego vehicle and the distance error Δd, which describes the deviation between the current and desired distance based on safety distance and time gap (Eqs. 1, 2), the driving state under consideration is obtained according to Eq. (3).

$$ \Delta v_{x} = { }v_{object} - { }v_{ego} $$
(1)
$$ \Delta d = d - { }t_{gap} \,{* }\,v_{ego} + { }d_{0} $$
(2)
$$ \vec{x} = { }\left[ {\begin{array}{*{20}c} {\Delta d} \\ {\Delta v_{x} } \\ {\begin{array}{*{20}c} {a_{ego} } \\ {v_{ego} } \\ \end{array} } \\ \end{array} } \right] $$
(3)

Since the desired acceleration of the follow-up controller aego,MPC does not affect the vehicle directly, the longitudinal dynamics are approximated by a first-order delay with the proportional factor KL and the time constant TL, so that the state equation is summarized according to Eqs. (4) and (5). For use in real-time simulation, the continuous-time state equation is discretized using Euler forward method [6].

$$ \dot{\vec{x}}\left( t \right) = \left[ {\begin{array}{*{20}c} {\Delta \dot{d}\left( t \right)} \\ {\Delta \dot{v}_{x} \left( t \right)} \\ {\dot{a}_{ego} \left( t \right)} \\ {\dot{v}_{ego} \left( t \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 1 \\ \end{array} } & {\begin{array}{*{20}c} { - t_{gap} } & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ {{\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/ {\vphantom {{ - 1} {T_{L} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{L} }$}}} \\ 1 \\ \end{array} } & {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \right] \left[ {\begin{array}{*{20}c} {\Delta d\left( t \right)} \\ {\Delta v_{x } \left( t \right)} \\ {\begin{array}{*{20}c} {a_{ego} \left( t \right)} \\ {v_{ego} \left( t \right)} \\ \end{array} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ {\begin{array}{*{20}c} {{\raise0.7ex\hbox{${K_{L} }$} \!\mathord{\left/ {\vphantom {{K_{L} } {T_{L} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{L} }$}}} \\ 0 \\ \end{array} } \\ \end{array} } \right] a_{ego, MPC} \left( t \right) + \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \right] a_{object} \left( t \right) $$
(4)
$$ \vec{y}\left( t \right) = I_{4}\, {* }\,\vec{x}\left( t \right) $$
(5)

2.3 Target Velocity Depending on Driving Status

The different driving behavior of various function variants is realized exclusively by the target speed setting; the MPC controller and the associated auxiliary conditions and factor weightings remain constant.

In contrast to conventional ACC, the target velocity is not selected by the driver but according to the driving situation (Fig. 1). The evaluation of 20 h of real stop-and-go situations on German highways, including the ego driving behavior as well as target object data, has shown that the average speed is less than 3.24 m/s in 75% of cases and less than 11.58 m/s in 90% of cases. On this basis, the maps shown in Fig. 1 were generated.

Fig. 1.
figure 1

Defensive (v1) and dynamic (v2) map for selecting target speed

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The target velocity for the MPC longitudinal controller is selected depending on the distance and relative speed to the object vehicle in front. Figure 1 shows the limit curves between the target velocity ranges. These are mathematically formulated in such a way that different characteristic maps and thus different ego behavior can be achieved by adjusting five parameters. In principle, an increase in the ego dynamics requires a shift of the limit curves in the direction of smaller abscissa and/or ordinate values.

3 Validation

The longitudinal controller presented here is validated on the basis of several objective criteria for occupant comfort and traffic efficiency. The simulations are carried out on the basis of real measured object vehicle speeds in stop-and-go traffic.

Figure 2 shows an example of the simulated driving behavior for a defensive and a dynamic parameterization of the target speed map. It becomes clear that there is a trade-off between high comfort (low ego dynamics, v1) and high traffic efficiency (good following behavior, low object distances, v2).

Fig. 2.
figure 2

Defensive (v1) and dynamic (v2) configuration of the longitudinal controller

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3.1 Evaluation Criteria

Evaluation criteria are defined below for further objective analyses of the longitudinal controller. These are based on the longitudinal dynamics of the ego vehicle and the following behavior in relation to the object vehicle in front. This addresses occupant comfort and energy consumption of the ego vehicle on the one hand and traffic efficiency on the other, adding aspects from all areas of the 3D parameter space [3].

Based on the relationship between the longitudinal acceleration and the energy consumption of the ego vehicle [7], the parameter AENERGY is calculated as a measure of energy efficiency according to Eq. (6):

$$ A_{ENERGY} = { }\frac{{\smallint |a_{ego} |{ }dt}}{{t_{SZENEARIO} }} $$
(6)

According to an internal evaluation of real driving data in the low speed range up to 60 km/h, the probability of incoming vehicles in the front area increases significantly if the distance to the object vehicle in front exceeds 45 m. An incoming vehicle in the front area leads to a deceleration of the ego vehicle and thus possibly to reduced driving comfort and reduced traffic efficiency [4, 8]. The integral of critical distances DCUT IN is therefore calculated according to Eq. 7:

$$ D_{{CUT{ }IN}} = { }\frac{{\smallint \left( {d \ge 45{ }\,m} \right){ }dt}}{{t_{SZENEARIO} }} $$
(7)

For the objective analysis of driving comfort, critical [9, 10] accelerations and decelerations are integrated in ACOMFORT (8) and the overall jerk of the ego vehicle in JCOMFORT (9):

$$ A_{COMFORT} = { }\frac{{\smallint \left( {a_{ego} \ge 0,\,14\frac{m}{{s^{2} }}} \right)dt + { }\smallint \left( {a_{ego} \le - 0,\,25\frac{m}{{s^{2} }}} \right)dt}}{{t_{SZENEARIO} }} $$
(8)
$$ J_{COMFORT} = { }\frac{{\smallint |j_{ego} |{ }dt}}{{t_{SZENEARIO} }} $$
(9)

3.2 Evaluation of Variants

For the objective evaluation of the functional variants presented, a representative scenario is derived from the 3D real driving data in the form of an acceleration curve of the object vehicle in front, which was calculated from the relative velocity from the real driving data. The ego behavior in this scenario was then simulated.

For better comparability, the calculated parameters are normalized according to Eq. (10). By subtracting the normalized parameter value from 1, the normalized parameter xSCALED is maximized for results to be evaluated as positive.

$$ x_{SCALED} = { }1 - { }\frac{{x - { }x_{min} }}{{x_{max} - { }x_{min} }} $$
(10)

The results are shown in Fig. 3. The subjective trade-off between high comfort and high traffic efficiency from Fig. 2 is also objectively significant. Three functional variants are compared with each other and with human driving behavior from the above-mentioned database. The human behavior represents a comparison option and is not necessarily representative of all drivers since it is a random sample. In addition to the parameters above, the driving behavior is evaluated with consideration of the maximum object distance (dMAX) and the absolute number of stops (NSTOPS).

Fig. 3.
figure 3

Relative objective comparison of different function parameterizations

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The human driving behavior shown has the worst energy efficiency and the lowest driving comfort. With regard to the number of stops and the object distance, the evaluation is best compared to the automated driving function.

Compared to the comfort-oriented variant of the automated driving function (v1), the dynamic variant (v2) shows a 21.2% reduction in jerk-related comfort and a 46.5% reduction in acceleration-related comfort. In contrast, the evaluation of the object distances is better with the dynamic variant. The maximum distance is 31.6% less and the time share of critical distances over 45 m is 61% less. The energy efficiency of the dynamic variant is 52.2% lower than that of the comfort-oriented driving strategy. Finally, it can also be observed that the dynamic driving strategy results in a higher number of standstills of the ego vehicle.

Overall, the ability of humans to only have to stop completely rarely even with small object distances is obvious. This is achieved by overall higher accelerations, which in turn result in poor energy efficiency. The automated driving strategy presented shows a trade-off between ego acceleration (comfort) and object distances (traffic efficiency). This can be adjusted with the help of the target speed maps presented, so that an (individually) good compromise is achieved. Furthermore, higher energy efficiency can be realized.

4 Conclusion

This paper presents the concept for a follow-up control system for automated vehicles in stop-and-go situations. The driving strategy is based on a defensively parameterized MPC acceleration controller. The desired speed, depending on the distance and relative velocity to a leading object vehicle, is used to parametrize different function variants. It is shown that objectively significant differences in the following behavior for automated vehicles in stop-and-go situations can be achieved with this strategy.

However, it must be taken into account that the available database of 20 h is not yet large enough to be considered fully representative, so that this publication focuses on the basic methodology as well as the controlling concept and further work in terms of parametrization is necessary.

In the further work on the controller shown, also the user acceptance for the function variants presented is recorded by subjective evaluations within a subject study.