Keywords

1 Introduction

Autonomous driving systems or advanced driver assistance systems in urban environment presents a significant challenge, particularly when navigating narrow roads shared with other traffic participants, including vulnerable road users (VRUs). Human drivers, when navigating these scenarios, maintain a safe longitudinal and lateral distance from objects to prepare for unexpected crossing. At the same time, drivers select an appropriate speed for overtaking objects. In particular, we focused on the scenario when a bicycle suddenly drifts out or crosses the ego vehicle’s driving lane while the ego vehicle is overtaking. According to a near-miss incident database HHDB, the near-miss incidents involving overtaking bicycles have reached 624 cases out of 3005 of entire bicycles’ near-miss incidents that are recorded in the database [1].

As legislative measures for overtaking bicycle, the Japanese Cabinet decided to impose on car drivers a duty to make an effort to overtake the bicycles at a safe speed when the lateral gap is insufficient [2]. In the United States, 3-foot passing laws were widely employed across states [3]. However, neither the safe speed nor the sufficient lateral gap has been quantitatively defined by the Japanese Cabinet. Also, the research investigated the effect of 3-foot law suggested that the effect to the driver behavior by the law was limited [4]. In addition, much research has been conducted on autonomous driving for overtaking bicycles [5, 6]. This research required huge amounts of driving data of the real drivers. Moreover, motion prediction and path planning with MPC requires more computational resources. Furthermore, the stochastic method seems to reasonably defines the motion of the object, however, we must note that most high-risk scenarios occur due to behaviors that happen in low probabilities. Therefore, it may be difficult to avoid the collision like the case that unusually happens with the stochastic method.

Therefore, this research aims to define, both theoretically and geometrically, the risk of sudden crossing in risk predictive scenario especially when overtaking bicycle in the same driving lane. Also, it proposes a method to minimize this risk to avoid collisions within the context of Autonomous Driving Systems and Advanced Driver Assistance Systems in urban driving. Previous research conducted by our team has revealed the safe speed and safe lateral gap to manage potential risks from blind object, such as blind intersections and parked vehicles [7]. Building on this research, this paper aims to geometrically determine and calculate the safe speed and lateral overtaking distance to safely overtake the ongoing bicycles, with a rationally lower computational cost.

2 Definitions of Safe Lateral Gap and Safe Speed

2.1 Conditions for Overtaking Bicycle

In this study, a following collision avoidance scenario by an autonomous vehicle was considered. A bicycle is driving straight on the same lane of the autonomous vehicle and the vehicle is catching up to the bicycle. When an onboard sensor on the autonomous vehicle detects a target bicycle, the system assumes that the object virtually moves into the driving trajectory. In generating the trajectory and speed profiles, the system considers three conditions to handle the scenario, as shown in Fig. 1. To prevent collisions, the system is designed to ensure side-face collision avoidance by maintaining sufficient lateral time gap, as in condition (a), or to decelerate to a safe speed to ensure frontal collision avoidance, as in condition (b). As another option, when the lateral gap is not enough to ensure both frontal and side-face collision, the system needs to judge to follow the object as condition (c).

Fig. 1.
figure 1

Three conditions to handle the preceding slower bicycle.

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As previously explained, we considered three conditions related to the width of the lateral drivable space. The virtual target object is presumed to start crossing at a certain time while maintaining the speed vobj. The timing t1 is derived from the geometrical relationships as depicted in the figures. Subsequently, the virtual target object sways out at an angle ψvir. The crossing angle is determined from the statistical measurements in a previous study [8]. The following subsections introduces the definitions of safe speed and safe lateral gap for each condition.

2.2 Condition (a): Avoiding Through Steering Maneuvers

When the road has enough width, the system aims to avoid the virtual crossing object by providing a safe-sufficient time-to-conflict of the virtual object tvir. We have defined the lateral gap distance when the ego vehicle can completely avoid the collision as the safe lateral gap Dsafe (see Fig. 2). When the vehicle can maneuver to keep the safe lateral gap, it avoids entering the virtual crossing area and traverses the area at the current speed or higher as in condition (a). The safe lateral gap Dsafe is calculated using the following equation:

$$ D_{safe} \left( t \right) = v_{obj} \left( t \right)t_{vir} \sin \left( {\psi_{vir} } \right) + C $$
(1)
Fig. 2.
figure 2

Schematic definitions of safe speed, lateral gap, and parameters at condition (a).

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Under condition (a), the vehicle needs to complete the overtaking, therefore, the collision avoidance is guaranteed only when the vehicle is driving faster than the current velocity vego. In this scenario, the target speed vtar and target position (Xtar, Ytar) for the vehicle control are as:

$$ \left\{ {\begin{array}{*{20}l} {v_{tar} \left( t \right) = v_{ego} \left( t \right)} \hfill \\ {X_{tar} \left( t \right) = X_{ego} \left( t \right) + D_{ego1} \left( t \right)} \hfill \\ {Y_{tar} \left( t \right) = Y_{obj} \left( t \right) - D_{safe} \left( t \right) - {{W_{ego} } \mathord{\left/ {\vphantom {{W_{ego} } 2}} \right. \kern-0pt} 2}} \hfill \\ \end{array} } \right. $$
(2)

2.3 Condition (b): Avoiding Through Deceleration and Steering Maneuvers

Conversely, as in the scenario (b), when the situation presents insufficient drivable space, the system maintains the maximum possible lateral gap Dmax without deviating from the road boundary. Then the vehicle decelerates to the safe speed vsafe that can avoid the collision with the virtual crossing object by performing emergency braking (see Fig. 3). The formula for calculating the safe speed vsafe is as follows:

$$ v_{safe} \left( t \right) = - a_{EB} \left( {t_{2} \left( t \right) - \tau_{EB} } \right) + v_{obj} \left( t \right)\cos \psi_{vir} $$
(3)

where aEB is the estimated acceleration of emergency braking and τEB is the latency before emergency braking activation. The time t2 can be calculated using the following equation:

$$ t_{2} \left( t \right) = {{\left( {D_{max} \left( t \right) - C} \right)} \mathord{\left/ {\vphantom {{\left( {D_{max} \left( t \right) - C} \right)} {\left( {v_{obj} \left( t \right)\sin \psi_{vir} } \right)}}} \right. \kern-0pt} {\left( {v_{obj} \left( t \right)\sin \psi_{vir} } \right)}} $$
(4)
Fig. 3.
figure 3

Schematic definitions of safe speed, lateral gap, and parameters at condition (b).

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By decelerating the vehicle speed to the safe speed vsafe and taking the safe lateral gap Dsafe = Dmax at the collision estimated point, the system can guarantee the frontal collision avoidance by performing emergency braking. In this scenario, the target speed vtar and target position (Xtar, Ytar) are as:

$$ \left\{ {\begin{array}{*{20}l} {v_{tar} \left( t \right) = v_{safe} \left( t \right)} \hfill \\ {X_{tar} \left( t \right) = X_{ego} \left( t \right) + D_{ego1} \left( t \right)} \hfill \\ {Y_{tar} \left( t \right) = Y_{obj} \left( t \right) - D_{safe} \left( t \right) - {{W_{ego} } \mathord{\left/ {\vphantom {{W_{ego} } 2}} \right. \kern-0pt} 2}} \hfill \\ \end{array} } \right. $$
(5)

2.4 Condition (c): Following Object

When the time t2 is shorter than the latency τEB, the emergency braking may not be able to stop the vehicle if the ongoing object starts crossing in front of the vehicle. In this scenario, regarding t2 = τEB, the target speed vtar and target position (Xtar, Ytar) are as:

$$ \left\{ {\begin{array}{*{20}l} {v_{tar} \left( t \right) = v_{obj} \left( t \right)} \hfill \\ {X_{tar} \left( t \right) = X_{ego} \left( t \right) + D_{ego1} \left( t \right)} \hfill \\ {Y_{tar} \left( t \right) = Y_{ego} \left( t \right)} \hfill \\ \end{array} } \right. $$
(6)

2.5 Summary

The proposed algorithm can determine the target speed, positions, and longitudinal acceleration simply from the onboard sensor information and several assumptions of the virtual motion of the bicycle. This simple calculation method enables real-time calculations in practical environment. Therefore, the target speed and positions will be iteratively updated until the loss of detection of the target object.

The target points can be used as control waypoints, and the path generation methods that follow the waypoint enable automated driving [7]. Moreover, the proposed algorithm allows easy combination with other waypoint generation methods and can extend the scenario to more complex and practical scenes [7].

3 Simulations and Results

From the definitions of safe speed and lateral gap, and considering the environmental factors, the safe speed and lateral gap are determined from the road width and speed of the ongoing target object. Simulations were conducted to demonstrate the calculation of safe speed and safe lateral gap. In the simulations, parameters for the algorithm were set as follows: ψvir = 30 [deg], tvir = 1.5 [s], aEB =  − 5.0 [m/s2], and the lateral position of target object was set to 1.0 m from the left road boundary.

Figure 4 schematically explains the features of how the safe speed and safe lateral gap are influenced by the object speed and road width. The figure represents three areas corresponding to the conditions (a), (b), and (c). In the area under of condition (a), the safe speed cannot be defined, so the safe speed was stated as 0 km/h, and for the area of condition (c), it is not possible to define the safe lateral gap, therefore, the safe lateral gap is stated as 0 m.

Fig. 4.
figure 4

Characteristics of safe speed and safe lateral gap regarding road width and object speed.

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Fig. 5.
figure 5

Path of the autonomous driving vehicle and target object in the simulation.

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As an example case of the simulation, we set the target object speed vobj to 12.0 km/h and the road width to 6.0 m. From the definitions, the scenario is classified as condition (b), and the safe speed and safe lateral speed were calculated as vsafe = 20.2 [km/h] and Dsafe = 2.9 [m]. Therefore, the target speed and target positions are defined from Eq. (4). Figure 5 shows the example simulation of the overtaking scenario. In the simulation, the vehicle model simply decelerates to the safe speed before reaching the target point and reaccelerates after overtaking. For the lateral control, the Triclothoidal path generation was used to maintain the safe lateral gap [9]. From the figure, the method overtook the target with a natural path within the road.

4 Conclusions

This paper proposed a method for autonomous driving system to safely overtake the bicycles. The method can theoretically and geometrically determine the safe speed and safe lateral gap with a low-calculation cost by anticipating the virtual sudden crossing. The method classifies the situation into three conditions related to the drivable space of the road and bicycle speed. The method simply provides the target position and speed that can guarantee collision avoidance with the assumed sudden crossing. Simulations revealed that the method could generate the natural overtaking speed and lateral gap. The advantage of the proposed method is that the method can designate the safe target as a point and speed, therefore, it is compatible with waypoint-based path generation methods. As a future extension of the research, the compatibility of the method and test on the real automated vehicle will be discussed.