Vision Zero: on a Provable Method for Eliminating Roadway Accidents without Compromising Traffic Throughput

12/09/2018 ∙ by Shai Shalev-Shwartz, et al. ∙ 0

We propose an economical, viable, approach to eliminate almost all car accidents. Our method relies on a mathematical model of safety and can be applied to all modern cars at a mild cost.

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1 Introduction

In 1997 the Swedish Parliament introduced a “Vision Zero” policy that requires reducing fatalities and serious injuries to zero by 2020. One approach to reduce the number of serious car accidents, which has been advocated by the “Vision Zero” initiative, is to enlarge the tolerance to human mistakes by combining regulative and infrastructure changes. For example, installing speed bumps in urban areas, which reduces the common speed from 50 kph to 30 kph, may make the difference between a mild injury and a fatality when a car hits a pedestrian. Another example is not allowing a green light for two routes at the same time (like “turn right on red” scenarios). The disadvantage of this approach is that it compromises the throughput of the road system — for example, reducing the speed limit from 50 kph to 30 kph increases traveling time by 66%.

Another approach to reduce the number of car accidents is to rely on Advanced Driving Assistant Systems (ADAS). For example, a Forward Collision Warning (FCW) system alerts the driver when the car is dangerously close to a frontal car and an Automatic Emergency Braking (AEB) system applies a strong autonomous braking at the last moment in case an accident is likely to happen. A recent study of the Insurance Institute for Highway Safety (IIHS) shows that vehicles equipped with FCW and AEB systems have a 64% fewer front-to-rear crashes with injuries 

[1]. The advantage of the ADAS approach is that it does not affect the throughput of the road system.

The main goal of this paper is to propose a roadmap for reaching an ADAS system that can substantially reduce fatalities and serious injuries, at a reasonable cost, while sustaining the usefulness and throughput of the road system. We emphasize that the industry as a whole keeps enhancing ADAS systems and comes up with novel new approaches (a notable example is Toyota’s Guardian [3]). Our goal is to put on the table a complete proposal with a clearly stated goal, which is fully accessible to the public and to regulatory bodies, and which comes with some formal mathematical guarantees.

Our starting point is exiting AEB systems. As mentioned previously, AEB systems already eliminate roughly 64% of front-to-rear crashes. While this is an impressive achievement, we believe that the main reason some crashes are not eliminated is because AEB is an “emergency” system rather than a “preventive” system. That is, AEB systems are designed to react to an imminent collision (hence the term “emergency” in the acronym) rather than to prevent a dangerous situation to occur in the first place. Indeed, the AEB test specifications by regulatory bodies (like ENCAP [2]) are geared towards avoiding or mitigating a collision that is expected if no action is taken. To balance safety and comfort, normally the brake activation is applied at a very short time-to-collission (TTC) in order to reduce the chance of a false detection causing an unexpected brake activation111Car makers adopt a variety of ”braking profiles”, but as shown in [5] all braking profiles begin braking at TTCs not higher than 2s.. As a result, AEB avoids collisions at relatively low speeds and somewhat mitigates collisions at higher speeds but even those are much lower than highway driving speeds. Electronic Stability Control (ESC), on the other hand, is designed to reduce skidding, operates in a “preventive” fashion by limiting the maneuverability of the vehicle. Most drivers do not feel constrained and mostly are not even aware that ESC was activated while those who seek full unencumbered control of the vehicle can always deactivate ESC if they so desire.

The first step in our roadmap is therefore a “preventive” collision avoidance system in which a mild brake activation is applied before the imminent danger occurs while keeping the comfort of driving intact. We wish to do so while providing formal guarantees of safety in the sense of proving that if all cars comply with the proposed system then collisions rates will drop considerably below current AEB rates. Our approach embraces the “prevention” rational of the “Vision Zero” initiative — unlike AEB systems, that are activated at the last moment, we would like to keep vehicles from entering dangerous situations in the first place. A notable advantage of this “preventive” approach over AEB is that unlike AEB systems that prevent an accident by braking very strongly (1.5g in some systems) and as a result might cause someone else to hit us from behind, the preventive approach will eliminate the danger, of being hit from behind, in advance by applying a mild brake. At the same time, we embrace the “keep the normal flow of traffic” rational approach of ADAS systems in general, and AEB in particular, as we refrain from reducing speed without distinction. Instead, we intervene in the driver’s decisions only when he/she is not “careful” according to the specific scenario.

At the heart of our approach is the Responsibility Sensitive Safety (RSS) model described in [4]. RSS formalizes an interpretation of reasonable human common sense. RSS is a rigorous mathematical model formalizing an interpretation of a set of principles of reason, and has been designed to achieve several goals: first, the interpretation of how humans interpret caution. Second, the interpretation should lead to a desired result (we call “Utopia”) that, if all agents comply with RSS parameters, accidents, resulting from decision making processes, could be eliminated. Third, the interpretation should be useful, meaning it will enable agile driving rather than enforcing an overly-defensive driving which inevitably would block traffic and in turn limit the scalability of system deployment. Finally, the interpretation should be efficiently verifiable in the sense that we can rigorously prove that a car will always obey the interpretation of human caution without the need to role out future actions of all agents involved.

Originally, RSS has been designed as a safety seal for the decision making process of self driving cars. However, we show in this paper how a variant of RSS can be leveraged to ADAS, to enhance the safety of human drivers as well while providing formal mathematical guarantees for safety when all agents comply with RSS. In a nutshell, RSS introduces a parameter-based methodology for defining safe distances which when breached constitute a “dangerous” situation. Safe distance was originally couched in a specific ”braking profile” that fits a robotic control. In this paper we extend the definitions of RSS to include any braking profile and in particular we propose a jerk-bounded profile that allows for relatively large TTC yet supports smooth braking. The benefit of couching a braking profile within the RSS framework is that all the formal guarantees associated with RSS apply as well. Thus, rather than proposing

some heuristic

of a braking profile, RSS enables a preventive braking system with formal guarantees. Furthermore, we argue that the proposed RSS-for-ADAS system may be similar to ESC in the sense where most drivers would have no problem with mild interventions of the car as a safety mechanism, and the few drivers which would like full unabated control for competitive driving will be able to shut off the system and be exposed to a higher level of accident risk.

The rest of the sections describe the roadmap for an ADAS system that enforces compliance with RSS rules. We envision two stages of deployment where at first we propose merely an enhancement of existing AEB systems in the sense of handling front-to-rear crashes only using existing front-facing sensing (camera or camera+radar). Our proposal for stage 1, can lead to a significantly higher elimination rate of front-to-rear crashes, comparing to existing AEB systems. The main improvement of stage 1 is due to the preventive approach, which will allow technology providers to balance the false positive / false negative tradeoff in a better way. By considering the sensing capabilities of existing systems, we estimate that the elimination rate of front-to-rear accidents will be roughly

. For the second stage we envision a full surround camera sensing fused with a crowd sourced map, to enable the full implementation of RSS. The advantage of the full implementation is that RSS comes with the elegant mathematical guarantee, stating that if all players fully comply with RSS rules, then accidents resulting from driving decision making process would become rare, thus achieving Vision Zero 222As we explain later, accidents can still occur if some agents do not comply with RSS rules due to hardware failure, sensing error, software bugs or some “act of god”. But, it is reasonable to estimate that such failure will happen in less than of the dangerous situations, and under this assumption, at least of the accidents would be eliminated. for all possible crashes. The cost of installing a camera surround system, in high volume, would be around times the cost of existing AEB systems which is negligible compared to the cost of accidents to society.

2 Stage I: preventing front-to-rear crashes by Automatic Preventive Braking (APB)

The goal of stage I is to prevent front-to-rear crashes as an enhancement of existing AEB systems. As of 2018, AEB systems are activated when the Time-To-Collission (TTC) to the front vehicle/pedestrian is very small. Typically, a number between to seconds. The time to contact is the distance between the cars divided by their relative speed. It follows that if the front and rear vehicles are driving at the same speed, the TTC is infinity. To illustrate why this approach is problematic, consider an extreme case, in which the rear vehicle is driving slightly faster than the front vehicle. Say, the rear vehicle is driving at ( kmh) while the front vehicle is driving at . If the distance between them is , then the TTC is . Taking an AEB system which is being activated at TTC of seconds, we obtain that the system will not be activated as long as the distance is larger than . It follows that driving behind a car at a speed of kmh and at a distance of cm is considered “safe” by an AEB system. This does not feel right.

The source of the aforementioned problem in AEB systems is that they do not consider a “what would happen if” type of reasoning. Indeed, if the front car would suddenly brake, the AEB system would need a time to respond to the change and it will most likely be too late. The common sense of a safe human driver is to be a little bit paranoid — a good driver keeps a safe distance from a frontal car so as to be ready for the unexpected. Of course, being over protective is also not good, as it leads to an extremely defensive driving. The secret sauce is to be ready for the unexpected, yet reasonable, events, while ignoring completely un-reasonable events.

RSS [4] is a mathematical, interpretable, model, formalizing the “common sense” arguments above. In particular, RSS formalizes

  • What is a dangerous situation ?

  • What is the proper response to a dangerous situation ?

  • What are the reasonable assumption a good driver can make on the future behavior of other road agents ?

  • What does it mean to be cautious ?

For a detailed exposition of RSS we refer the reader to [4]. We do not intend to repeat all the definitions in this paper, but only to introduce a generalization of RSS that will be useful for human drivers.

We first consider the simplest case of a front car, , driving in front of a rear car, , where both cars driving at the same direction along a straight road333see [4] on homomorphism from general to straight roads., without performing any lateral maneuvers444to support lateral maneuvers see Sec.3.. RSS defines the notion of a dangerous situation by relying on the following definition:

Definition 1 (RSS Safe Distance)

A longitudinal distance between a car that drives behind another car , where both cars are driving at the same direction, is safe w.r.t. a response time if for any braking of at most , performed by , if will accelerate by at most during the response time, and from there on will brake by at least until a full stop then it won’t collide with .

Relying on the above definition, RSS states that:

  • A situation is dangerous if the distance is not safe.

  • The proper response to a dangerous situation for the rear car is to brake (after a response time) by at least until either reaching a full stop or gaining again a safe distance

Crucially, RSS’s proper response does not depend on the underlying driving policy. In fact, it can be embedded on top of any driving policy. Our main observation is that we can embed RSS on top of a human driving policy. Basically, whenever a human driver brings the car to a non-safe distance, the RSS-ADAS system will apply braking so as to bring the car back to a safe distance.

The problem with this naive implementation is that applying a strong brake without any warning might be dangerous by itself.555

This is one of the main reason why existing AEB systems are tuned to have an extremely low probability of false positives. The price for extremely low false positive rate is a much higher false negative rate (meaning, the system fails to detect a car which is at a non-safe distance).

To tackle this problem, we first propose a generalization of RSS (Section 2.1), and then we specify a particular member of this RSS family that enables to intervene in advance but in a smooth manner without inconveniencing the human driver (Section 2.2). Another advantage of this approach is that it mitigates the danger of false positives.

2.1 Generalized RSS

In the original definition of RSS, the rear car is assumed to accelerate during the response time and then to brake until reaching a full stop. This is an example of a braking profile. More generally:

Definition 2 (Braking profile)

A braking profile, , is a mapping from initial kinematic state of the car (mainly, initial velocity, , and acceleration, ) to a pair , s.t. is the future velocity of the car and is the first time in which .

Some examples for braking profiles are given below:

  • The braking profile applied by the front car in the original definition of RSS is defined by and .

  • The braking profile applied by the rear car in the original definition of RSS is defined by , and

We now define a generalization of RSS’s safe distance.

Definition 3 (Generalized RSS Safe Distance w.r.t. Braking Profiles )

A longitudinal distance between a car that drives behind another car , where both cars are driving at the same direction, is safe w.r.t. braking profiles if in case the front car applies braking profile and the rear car applies braking profile , then the cars will reach a full stop without colliding.

The proper response is defined as follows:

Definition 4 (Proper response)

Suppose the first time that the distance between and is non-safe is , and w.l.o.g. set . Then, the proper response for the front car is to have its velocity at least , where is the velocity defined by , and the proper response for the rear car is to have its velocity at most , where is the velocity defined by .

It is easy to verify that if both cars apply proper response then there will be no accident.

2.2 Jerk-bounded Braking Profile

In this section we describe the jerk-and-acceleration-bounded braking profile, denoted . The idea is that we start decreasing our acceleration linearly (with slope ), until reaching a max brake parameter (denoted ), and then we continue to brake with a constant deceleration until reaching zero velocity. In the following we provide closed form formulas for calculating this braking profile.

Braking with a Constant Jerk

Suppose we start braking at jerk of . Suppose also that we can immediately arrive to zero acceleration (leave the throttle), hence below we assume . Then, the dynamics of the car is as follows:

Braking profile

Based on these equations, braking distance will be defined as follows. Let be the first time in which either or , that is,

The time for reaching full brake is

And the speed is as follows:

Braking distance

Finally, the car will move the following distance until reaching a full stop:

2.3 Safe distance w.r.t. and

Suppose that the front car brakes with braking profile (braking with a constant deceleration of ), and the rear car relies on braking profile . Then, the safe distance formula is

2.4 The Automatic Preventive Braking (APB) System

As soon as the distance becomes non-safe (according to the jerk-based formula), the APB system will start to brake with a jerk of until one of the following happens: (1) the car stops (2) the distance becomes safe, or (3) the driver turns off the RSS system (by a button or some other mechanism).

An inductive proof, similar to the one given in [4], can show that if we start with a safe distance from a front vehicle, and apply the proper response of the APB System, then under the assumption that the front vehicle will not brake stronger than , we will never hit the front vehicle from behind.

2.5 A better balance between false positives and false negatives

It is worth noting that the same hardware that implements AEB nowadays can be adjusted to implement APB. Since AEB systems brake abruptly at the last seconds before a crash, an activation of the AEB system is scary and dangerous (someone might hit us from behind), and hence AEB systems are tuned to have an extremely small rate of false positives (a false positive event is when the system detects that we are going to crash into a “ghost” vehicle, which is not really there). The price for an extremely small rate of false positives is a higher rate of false negatives (a false negative event is when the system fails to detect a car, even though we are going to crash into it). Furthermore, since false positive events are much more dangerous when driving at a high speed, AEB systems are only activated when driving at a low speed.

In contrast, because the braking profile of APB is jerk bounded, the danger of false positives is much milder (the driver will not be jolted, and a car from behind will also not be surprised because of the bound on the jerk). Therefore, we can tune the system to have much less false negatives than an AEB system, and we can also activate the APB system when driving at a high speed.

3 Stage II: Beyond Front-Rear Accident Prevention

In stage II of the system, we require a surround (camera-based due to the required resolution) sensing system and a crowd sourced High Definition (HD) map. The surround system enables to prevent a car from performing a reckless cut-in and to merge in junctions unsafely. The major contribution of the map is that the system has full knowledge on where to expect potential dangers: it knows in advance where there should be lanes that might merge to our lane, who has the priority, where to expect traffic lights, where to expect occluded pedestrians, etc. Planning in advance enables the system to adjust speed mildly in advance and not to be surprised. In addition, a map enables to know the geometry of the road even without explicitly detecting lane marks. Finally, RSS is a formalization of reasonable common sense in such scenarios. With the use of a crowd sourced map, we can gain access to road conditions and the actual data on the behavior of road users at every road, thus helping regulators to define the appropriate rules for every road (for example, to specify “school zone” areas in an automatic way).

Unlike geo-fenced autonomous driving applications limited to particular areas, an ADAS system would require the support of an HD-map in all driving locations. An HD map that can offer coverage of the entire globe and is perpetually maintained fresh and updated by means of crowd sourced data. With a surround sensing and a map, we can implement the full stack of RSS. The details can be readily inferred following [4] and the exposition of Stage-I above.

4 Utopia is (almost) Possible — Toward Reaching Vision Zero

The premise of RSS is that “utopia is possible”, in the sense that if all agents fully comply with RSS’s proper response,666By “following proper response” we mean that the agents’ actual decelerations match the requirements of proper response. then accidents resulting from wrongful driving decisions would become rare. And so, geographic regions enforcing RSS on all vehicles (by regulation), will benefit from a substantially high level of road safety. It should be noted that accidents can still occur as a result of conditions outside the driving decision making process, such as accidents caused by vehicles that do not apply proper response due to a hardware failure (e.g. malfunctioning braking system) or due to perception mistakes (e.g., a system does not detect a car at a non-safe distance in front of us). However, suppose that the probability of such a failure is one in every 100 dangerous situations. While this level of accuracy is insufficient for autonomous driving (an accident once in every 100 dangerous situation is a very bad result), it can be shown that if this failure rate is maintained in ADAS systems, where the driver is still active and responsible, we can obtain an elimination rate of roughly 99% of car accidents. This is a huge step toward fulfilling the “Vision Zero” initiative.

References