Motion constraints for self-driving vehicles during safe stop maneuvers

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DEGREE PROJECT IN ENGINEERING

PHYSICS SA114X, FIRST CYCLE, 15 CREDITS STOCKHOLM, SWEDEN 2017

Motion constraints for self-driving vehicles during safe stop maneuvers

A proprietary method and a modification of an existing method for finding speed limitations JONATHAN BLIXT

SARA JOON

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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www.kth.se

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Acknowledgments

Both authors are equally involved in producing this report. Jonathan has been the main producer of the discrete method and Sara has been the main implementer of the method in MATLAB. Both authors have contributed to each section of the report.

We would like to thank associate professor Mats Jonasson at Volvo Cars for all his guidance, encouragement and help.

Stockholm, 19 May 2017 Jonathan Blixt and Sara Joon

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Abstract

At any given moment, self-driving vehicles need to be able to perform a safe stop maneuver in order to come to a complete stop in case of an emergency. It is important to find the maximum allowed entry speed in order to follow the given emergency path and stop in time without losing grip or rolling over. A new method is developed for this purpose and compared to an existing method developed by Joseph Funke and Christian Gerdes.

It is found that the new method is more general but does not always converge to a solution for bad guesses and extreme paths. Also, the existing method cannot determine the initial speed for all paths. The increased generality lies in optimizing the emergency path given by a safe zone and considering different friction coefficients in lateral and longitudinal directions. Plots are presented visualizing the maximum speed´s dependence of various parameters for a specific path. The software CarMaker by IPG Automotive is used to validate the results for the developed method. The simulations done show that the method works well for paths in two dimensions but limits the initial speed more than necessary in three dimensions. Both methods find the accelerations needed at every point which may be translated into control signals as an additional use. Methods of knowing the friction coefficients in advance are also discussed.

Sammanfattning

Självkörande fordon ska när som helst kunna utföra en säker stoppmanövrering för att komma till ett stanna helt vid en nödsituation. Det är viktigt att hitta den maximala tillåtna ingångshastigheten för att följa den angivna nödbanan och stanna i tid utan att förlora greppet eller välta. En ny metod utvecklas för detta ändamål och jämförs med en befintlig metod som utvecklats av Joseph Funke och J. Christian Gerdes.

Det är konstaterat att den nya metoden är mer allmän men konvergerar inte alltid till en lösning för dåliga gissningar och extrema vägar. Den befintliga metoden kan inte heller bestämma ingångshastigheten för alla banor. Den ökade generaliteten ligger i att optimera nödbanan inom en säker zon och tar hänsyn till olika friktionskoefficienter i sidled och färdriktning. Plottar presenteras för att visualisera maxhastighetens beroende av olika parametrar för en viss bana. Programvaran CarMaker av IPG Automotive används för att validera resultaten från den utvecklade metoden. Simuleringarna visar på att metoden fungerar väl för banor i två dimensioner men begränsar hastigheten mer än nödvändigt i tre dimensioner. Båda metoderna hittar de accelerationer som krävs vid varje punkt vilket kan översättas till styrsignaler, som en extra användning. Metoder för att känna till friktionskoefficienterna i förväg diskuteras också.

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The demand for transport is increasing each day. While drivers’ salaries are steadily growing technologies like cameras, ultra-sonic sensors and artificial intelligence are becoming increasingly more advanced while the prices are dropping. Meanwhile, traffic accidents have become one of the main causes of injury and death and each year more than a million people lose their lives in traffic accidents globally [1]. Several vehicle companies are today racing against each other to produce cars that are capable of driving safely on their own, at least to some degree, to cope with this situation and increase their sale [2]. Assisted driving or autopilot are examples of names used today to describe semi- autonomous driving capabilities. These systems are rapidly becoming more advanced and commonly spread as many car manufacturers claim that they will be able to produce fully autonomous vehicles within the next few years[3].

However, there are strict requirements that must be satisfied by these vehicles such as keeping the number of fatal accidents close to zero and being able to handle countless of different traffic, road or weather conditions. To prevent accidents from occurring, autonomous vehicles must have an emergency path to follow if it gets into a situation it cannot handle. The emergency path is the calculated optimal path to follow in order to safely come to a complete stop as soon as possible. It is of interest to know the maximum speed allowed when entering this given path. If the speed exceeds the restriction there is a risk that the vehicle will lose its grip, reduce the speed insufficiently or even rollover.

There are several factors that affect the braking distance such as the slope and curvature of the road. The road may be wet and slippery, or dry with good grip, which also affects the available braking force.

1.1 Problem formulation

A vehicle that is traveling at a certain speed may detect an obstacle or encounter some sort of malfunction and need to stop as soon as possible. There is a risk that this speed exceeds the maximum speed that enables a complete stop within the safe zone which in turn may cause an accident. To avoid this, it is of interest to predetermine the maximum speed allowed when following a specific path so that the vehicle is always able to come to a complete stop in time.

1.2 Aim

The purpose of this work is to determine the maximum speed a vehicle may have and still be able to stop at a given point. The vehicle is to follow a given path known as the emergency path. This path is to be determined by using several parameters and a known safe zone as input. Friction coefficients of the tires against the road, slope and road bank are the variables that will determine an upper limit of the speed for each path. Given a safe zone, and these parameters at every point, it is desired to develop a method that describes the maneuver in terms of accelerations and finds a mathematical solution of the maximum initial speed.

1.3 Restrictions

The work done has been restricted to the study of a particle. Therefor the dynamics due to the physical extent of a rigid body are neglected. These dynamic characteristics include the moment of inertia, force distribution individually on the wheels and aerodynamic effects. Some effects or characteristics play greater parts than others.

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The risk of roll-over is not considered in the model. To ensure that this does not occur one must look at the height of the center of mass which means that the vehicle can no longer be viewed as a particle.

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2. Background

2.1 Autonomous vehicles

Fully autonomous vehicles are, by definition, fully capable of driving without any human input. This means that if any emergency occurs the system needs to take action independently and safely come to a complete stop. This system needs to be failsafe even if there are technical problems, where the most sever one would be the main Electronic Control Unit shutting down or, equivalently, losing all inputs given by the sensors.

All autonomous vehicles need to meet the safety standard ISO 26262 which says that maximum one serious accident may occur in 10⁹ operational hours. This is equivalent to 114 000 years in operation.

2.2 Motion constraints of vehicles

One may think of many different motion constraints to be applied to vehicles but only the following were examined in this work, as they play the greatest roll. During normal driving conditions, it is reasonable to put further and stricter constraints on the powertrain and steering in order to lessen the wear and tear on the vehicle and allow for margins. But during emergencies lives could be at stake, hence it is of importance to make use of every margin and wear and tear loses their importance in comparison.

The friction coefficient

To calculate the braking distance the friction coefficient between the road and the tires needs to be known as it limits the maximum braking force. By the same logic, when the maximum initial speed is to be determined given a braking distance, the friction coefficient is needed. To ensure that the vehicle may perform a safe stop maneuver at any given moment during its journey the friction coefficient is desired for all positions along the entire road.

To gain information about the friction coefficient it can either be measured by sensors in the vehicle or by receiving values in advance via the internet. These values may be based on temperature and humidity measurements etc. from weather stations. The trend

“Internet of Things” imply that the possibility of vehicles being connected to the internet in the future is large. “Internet of Things” means that objects such as refrigerators, televisions and vehicles collect information from the surroundings and further communicates to other objects via the internet. [4]

From weather updates the friction coefficient can be predicted as an estimation but may not be accurate enough to fulfill the ISO 26262 standard [5]. Therefore, it would be preferable to measure the friction coefficient from built in sensors because this yields a more accurate value. It may not be realistic for all vehicles in the future to have such measuring equipment, but if only a portion of the vehicles have these types of sensors they can share the information with all other connected vehicles. The advantage of receiving the friction coefficient from other vehicles lies in knowing the value arbitrarily far ahead allowing vehicles to reduce the speed in advance if the coefficient is lower ahead. In Sweden, a project, called Road Status Information [6], is currently in progress where hundreds of cars measure how slippery the roads are. The collected data is combined with weather forecasts to form a “friction map” that is continuously updated.

The friction coefficient is noted by 𝜇 and is often different in the longitudinal direction than in the lateral direction, though the difference may be small. It is more important to

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have a larger braking force than turning force for safety reasons which leads to a larger friction coefficient between tires and road in the longitudinal direction.

Safe zone

It is of importance to have a safety margin to other vehicles, objects and delimitations such as the lane divider line and roadside ditch. The extent of the margins depends on the situation. The distance from children running on the sidewalk must be greater than from a lamp-post since humans are unpredictable and vulnerable. There are countless of factors that need to be considered, such as the velocity and behavior of other drivers and it is assumed that the safe zone is provided with all necessary considerations.

Rollover avoidance

Having enough friction force between the wheels and road to follow a path without skidding does not necessarily mean that it can be followed by the vehicle. This is due to the possibility of rolling over. There are many ways of preventing rollovers such as reducing the speed or shifting the frictional force between the wheels. An Electronic Stability Program (ESP) can detect when there is a risk of rollover and control the traction and breaking on the wheels individually to stabilize the vehicle. [7] An active suspension system can also be a part of the ESP. This allows for a lowering of the center of gravity as well as shifting it inward the curve to counteract the outward tilt. There are several systems that intend to alert the driver when there is a significant risk of rollover, such as Lane Departure Warning, but these are of no interest in vehicles that handle all situation independently, e.g. self-driving vehicles.

System delays

It is impossible to momentarily change the steering angle and the braking force from a given value to another since it would require an infinitely great force. The maximum rate of change of the steering angle and braking force depends on vehicle under consideration.

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3. Vehicle modelling

3.1 Friction ellipse

The maximum force a vehicle can use to brake is partly dependent on the properties of the tires and road. All tires together can brake with a force that is equal to the total friction force generated when driving straight. The vehicle can also turn with a force that corresponds to the maximum total friction force in the lateral direction when not braking.

But combining braking and steering resolves in a reduction of either forces upper limit.

The sum of the force in the direction of motion and the lateral force are bound by the following inequality:

𝐹_𝑥²

𝜇_𝑥²𝐹_𝑧²+ 𝐹_𝑦²

𝜇_𝑦²𝐹_𝑧² ≤ 1 (1)

where 𝐹_𝑥, 𝐹_𝑦 and 𝐹_𝑧 denote the force parallel to the motion, lateral force and the normal force respectively [8]. 𝜇_𝑥 and 𝜇_𝑦 denote the friction coefficients in the direction parallel and perpendicular to the motion. Introducing the angle of slope of the road and the road bank yields two additional parameters 𝛼_𝑥 respectively 𝛼_𝑦. The slope is defined to be positive when going uphill and the road bank positive when the vehicle is tilting to the right.

For simplicity, the vehicle is considered as a particle rather than a rigid body neglecting effects from physical size such as moment of inertia, see Figure 1, as mentioned in Section 1.3.

Figure 1. An illustration of the forces acting on the particle representing the vehicle. Either 𝑖, 𝑗 = 𝑥, 𝑦 or vice versa.

This means that the vehicle is either seen from the side traveling in the x-direction or from the front of the vehicle, i.e. when 𝑖 = 𝑦. 𝒈̂ is the unit vector in the direction of gravity.

From Figure 1 the force in 𝑥- and 𝑦-direction can be determined as

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𝑚𝑎_𝑥= 𝐹_𝑥− 𝑚𝑔 cos 𝛼_𝑦sin 𝛼_𝑥 𝑚𝑎_𝑦 = 𝐹_𝑦− 𝑚𝑔 cos 𝛼_𝑥sin 𝛼_𝑦

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where 𝑚 is the mass of the vehicle and 𝑎_𝑥 and 𝑎_𝑦 are the accelerations in 𝑥- and 𝑦- direction respectively. If the aerodynamic force 𝐹_{𝑎𝑒𝑟𝑜} and the force due to rolling resistance 𝑅 were included in equation (2) it would look as follows [9]

𝑚𝑎_𝑥= 𝐹_𝑥− 𝑚𝑔 cos 𝛼_𝑦sin 𝛼_𝑥− 𝐹_{𝑎𝑒𝑟𝑜}− 𝑅 𝑚𝑎_𝑦 = 𝐹_𝑦− 𝑚𝑔 cos 𝛼_𝑥sin 𝛼_𝑦

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however, these forces are neglected in the models. The normal force is

𝐹_𝑧 = 𝑚𝑔 cos 𝛼_𝑥cos 𝛼_𝑦 (4)

omitting 𝑎_𝑧 since it cannot be directly controlled as 𝑎_𝑥 (gas or brake) or 𝑎_𝑦 (steering), instead it depends on the speed and change of slope that is assume to be reasonably small.

These forces in equation (1) yield

(𝑚𝑎_𝑥+ 𝑚𝑔 cos 𝛼_𝑦sin 𝛼_𝑥)²

𝜇_𝑥² +(𝑚𝑎_𝑦+ 𝑚𝑔 cos 𝛼_𝑥sin 𝛼_𝑦)² 𝜇_𝑦²

≤ (𝑚𝑔 cos 𝛼_𝑥cos 𝑎_𝑦)².

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Dividing by 𝑚 yields the friction ellipse (𝑎_𝑥+ 𝑔 cos 𝛼_𝑦sin 𝛼_𝑥)²

𝜇_𝑥² +(𝑎_𝑦+ 𝑔 cos 𝛼_𝑥sin 𝛼_𝑦)²

𝜇_𝑦² ≤ (𝑔 cos 𝛼_𝑥cos 𝑎_𝑦)². (6)

3.2 Methods of describing a path

There are many ways to describe a path. For example, a series of straight segments and arcs can describe a path. Also, polynomials can be used, as can many other functions. For the work done, the methods of interest are discrete paths and paths described as a series of clothoids and straight segment. A body fixed coordinate system is used in both methods.

Body fixed coordinate system

To describe the motion of a car mathematically a body fixed coordinate system is used.

This system is denoted by 𝑥𝑦 where 𝑥 is in the direction of motion of the body and 𝑦 is the lateral direction following the right-hand rule as illustrated in Figure 2. The inertial, or global, coordinate system is denoted by capital letters 𝑋𝑌. The orientation of the body realtive to the global system can be described by the yaw, or heading, angle 𝜓 between the axes 𝑥 and 𝑋.

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Figure 2. An illustration of the global coordinate system 𝑋𝑌 and the body fixed coordinate sytem 𝑥𝑦 with the heading angle 𝜓.

Discrete path

The discrete path may be given by a series of positions with associated headings described by (𝑋, 𝑌, 𝜓) in a global coordinate system. However, if the heading, or yaw angle, 𝜓 is not given one may calculate the velocity vector at each point to keep track of the yaw.

For the method described in Section 4.2, this type of path description is used (𝑋, 𝑌).

The discrete path can also be described by a radius at every point along the path, except the first and last point. To describe the radius at a point (𝑋_𝑛, 𝑌_𝑛), the previous point (𝑋_𝑛−1, 𝑌_𝑛−1) and succeeding point (𝑋_𝑛+1, 𝑌_𝑛+1) is needed to fit a circle. The radius of the fitted circle is the radius of the path at the point, (𝑋_𝑛, 𝑌_𝑛). By solving the following system of equations, the radius 𝑅_𝑛 can be determined.

{

(𝑋_𝑛− 𝑋_𝑐)²+ (𝑌_𝑛− 𝑌_𝑐)² = 𝑅_𝑛² (𝑋_𝑛+1− 𝑋_𝑐)²+ (𝑌_𝑛+1− 𝑌_𝑐)² = 𝑅_𝑛² (𝑋_𝑛−1− 𝑋_𝑐)²+ (𝑌_𝑛−1− 𝑌_𝑐)² = 𝑅_𝑛²

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where (𝑋_𝑐, 𝑌_𝑐) are the coordinates of the center of the circle. The curvature 𝑘(𝑋_𝑛, 𝑌_𝑛) is the inverse of the radius.

Reference path derived from safe zone

A discrete path with an allowed deviation 𝑑_𝑛^𝐴 assigned to every point (𝑋_𝑛^𝑟𝑒𝑓, 𝑌_𝑛^𝑟𝑒𝑓) can be used as a reference to an area with arbitrary accuracy by choosing the total number of points 𝑁 high enough. This means that the safe zone can be represented by discrete points and associated deviations, as shown in Figure 3. It is only a manner of fitting circles that touch the safe zone boundary and calculating their center point and radius. The reference points need not be equidistance.

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Figure 3. An example of a safe zone where the black dots represents the discrete path as points (𝑋, 𝑌). 𝑑𝑛𝐴 is the allowed devience at point (𝑋𝑛, 𝑌_𝑛). The red lines indicate the safe zone boundary.

Continues path of clothoids and straight segments

A series of clothoids and straight segments can describe a continuous path. The clothoid- part is described by an entry clothoid where the curvature 𝑘 increases linearly, an arc where the curvature is constant and an exit clothoid where the curvature decreases linearly, as shown in Figure 5. The curvature is the inverse of the radius 𝑅,

𝑘 = 1

𝑅 (8)

and is dependent on the successive distance 𝑠 along the path,

𝑘 = 𝑘(𝑠). (9)

The curvature of a general elementary path can be described by:

𝑘(𝑠) =

{

𝜎𝑠 𝑖𝑓 𝑠 ∈ [0, 𝐿1 − 𝜆 2 ] 𝜎𝐿 1 − 𝜆

2 𝑖𝑓 𝑠 ∈ (𝐿1 − 𝜆

2 , 𝐿1 + 𝜆 2 ) 𝜎(𝐿 − 𝑠) 𝑖𝑓 𝑠 ∈ [𝐿1 + 𝜆

2 , 𝐿]

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where 𝜎 is the rate of curvature change, 𝜆 is the ratio between the arc length and the total length of the clothoid segment 𝐿. These parameters are shown in Figures 4 and 5. Also the exiting angle 𝜃 and distance between starting point 𝑞₁ and ending point 𝑞₂ of a clothoid 𝑑 are illustrated in Figure 5. Derivations to obtain 𝜎 and 𝐿 from the position of the endpoints of the segments and a given 𝜆 can be found in Joseph Funke and J. Christian Gerdes article [10].

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When two elementary paths are combined, a bi-elementary path is formed. For an example of a bi-elementary path, see Figure 4.

Figure 4. An example of a path consisting of a straight segment and two clothoids, i.e. a bi-elementary path. 𝛾 is the lane change ratio 𝑌/𝑋, 𝜓𝑙𝑎𝑛𝑒 is the angle between the direction of the path at the end point relative to the 𝑋-axis.

Image courtesy of J. Funke and J. C. Gerdes. [10]

Figure 5.The figure to the left illustrate the relationship between the curvature 𝑘 and the distance along the path, 𝑠 for a clothoid. The figure to right illustrates the form of a clothoid. Image courtesy of J. Funke and J. C. Gerdes. [10]

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4. Method

The method consists of developing a new method that satisfies the aim of the work, studying a similar existing model and implementing both models in MATLAB. The two methods are described in Sections 4.1 and 4.2. It is desired to calculate the maximum initial speed, or entry speed of the path, under the given constraints that reduces the speed to zero at the end of the path. However, the opposite problem of starting at zero velocity and accelerating backwards along the path and maximizing the final speed is a lot easier to implement. The problem becomes less complex since methods solving the problem are iterative and need a starting value. It is more sensible to set the initial value to zero rather than guessing the maximum speed and hope for the final speed to be zero.

4.1 Clothoid method

Joseph Funke and J. Christian Gerdes [10] have developed a method for determining the maximum speed at the end of a followed path, which in their case was a lane change trajectory. The path consists of clothoids and/or straights as described in Section 3.2 under Continuous path of clothoids and straight segments. Since a lane change trajectory is similar to an emergency path this method is applicable to the problem of emergency stopping. In this method accelerations in 𝑥 and 𝑦 directions correspond to those parallel and perpendicular to the direction of motion as shown in Figure 2. Their steps are as follows except for a modification of the friction ellipse, that has been added to take into account the slope and road bank:

1. All calculations are done at successive distances 𝑠_𝑛 along the trajectory, where 𝑛 indicates the point number.

2. The accelerations are restricted by the friction-ellipse determined in Section 3.1, equation (6).

3. The lateral acceleration 𝑎_𝑦 is determined by the acceleration required to stay on track and is approximately equal to the centripetal acceleration

𝑎_𝑦(𝑠_𝑛) ≈(𝑈_𝑥(𝑠_𝑛))²

𝑅(𝑠_𝑛) = (𝑈_𝑥(𝑠_𝑛))²⋅ 𝑘(𝑠_𝑛)

where 𝑈_𝑥(𝑠_𝑛) is the longitudinal speed and 𝑘(𝑠_𝑛) is the curvature at point 𝑛 along the path.

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4. With the lateral acceleration inserted in equation (6), the longitudinal acceleration follows

𝑎_𝑥(𝑠_𝑛) = −𝑔 cos 𝛼_𝑦sin 𝛼_𝑥

+ √(𝑔 cos 𝛼_𝑥cos 𝛼_𝑦)²− (𝑈_𝑥²⋅ 𝑘 + 𝑔 cos 𝛼_𝑥sin 𝛼_𝑦

𝜇_𝑦 )

2

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where 𝑈_𝑥, 𝑘, 𝜇_𝑥, 𝜇_𝑦, 𝛼_𝑥 and 𝛼_𝑦 all depend on 𝑠_𝑛. The equality sign here indicates that the longitudinal acceleration is the maximum allowed and therefore will generate the maximum possible speed at the end of the path. Equation (12) is not the same as in J. Funke and J. C. Gerdes paper since the friction-ellipse used there

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does not depend on road banks and slopes. Besides equations (12) and (15) below, this method is the same as in J. Funke and J. C. Gerdes paper [10].

5. The time between two successive distances, 𝑠_𝑛+1 and 𝑠_𝑛, is given by

𝑡(𝑠_𝑛) =

−𝑈_𝑥(𝑠_𝑛) + √(𝑈_𝑥(𝑠_𝑛))²+ 2𝑎_𝑥(𝑠_𝑛)(𝑠_𝑛+1− 𝑠_𝑛) 𝑎_𝑥(𝑠_𝑛)

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6. The longitudinal speed for the next successive distance, 𝑠_𝑛+1, is given by 𝑈_𝑥(𝑠_𝑛+1) = 𝑈_𝑥(𝑠_𝑛) + 𝑎_𝑥(𝑠_𝑛) ⋅ 𝑡(𝑠_𝑛)

= √(𝑈_𝑥(𝑠_𝑛))²+ 2𝑎_𝑥(𝑠_𝑛)(𝑠_𝑛+1− 𝑠_𝑛)

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7. This integration continues either until

(𝑔 cos 𝛼_𝑥cos 𝛼_𝑦)²− (𝑈_𝑥²⋅ 𝑘 + 𝑔 cos 𝛼_𝑥sin 𝛼_𝑦

𝜇_𝑦 )

2

< 0 (15) for then the longitudinal acceleration in equation (12) becomes imaginary or until the end of the path is reached. Here as well 𝑈_𝑥, 𝑘, 𝜇_𝑥, 𝜇_𝑦, 𝛼_𝑥 and 𝛼_𝑦 are dependent on the successive distance.

Funke and Gerdes take into account that the entry and exit angle may differ and also the lane change ratio which described the ratio between 𝑌 and 𝑋 marked in Figure 5. These factors have not been considered in this work.

4.2 Discrete method constrained by a safe zone

This method has been developed by us for the specific problem of finding the maximum final speed of a particle moving along a given discrete path under given motion constraints. The method also computes the optimal path to follow from the reference points with allowed deviations based on a safe zone. The reference path is given by points in a global (inert), two-dimensional coordinate system (𝑋_{𝑛,𝑟𝑒𝑓}, 𝑌_{𝑛,𝑟𝑒𝑓}) and the conditions are as follows:

 The particle must move from one point to the following with a maximum lateral deviation of length 𝑑_𝑛^𝐴 at point 𝑛 in order to stay in the safe zone.

 The maximum accelerations in lateral and parallel direction, relative to the direction of travel, are limited by the friction between the tire and road which is given by the friction ellipse in Section 3.1, equation (6).

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Figure 6. An illustration of the reference points (black) and associated allowed deviations 𝑑𝑛𝐴 derived from the safe zone. The green line represents an example of a possible vehicle trajectory. The velocity 𝑣̅∥𝑛is the component of the total velocity 𝑣̅𝑛 parallel to the vector pointing towards the next point and the velocity 𝑣̅⊥𝑛 is the component in the

corresponding perpendicular direction.

All parameters are shown in Figure 6. The steps are as follows:

1. Calculate the angle 𝜃_𝑛 between the direction of 𝑣̅_𝑛 and the new desired direction of travel by

𝑣̅_𝑛∙ 𝑠̅ = 𝑣_𝑛 _𝑛𝑠_𝑛cos 𝜃_𝑛 → 𝜃_𝑛 = 𝑎𝑟𝑐𝑐𝑜𝑠 [^𝑣̅^𝑛^∙(𝑋^𝑛+1

𝑟𝑒𝑓−𝑋𝑛,𝑌_𝑛+1^𝑟𝑒𝑓−𝑌𝑛)

𝑣𝑛‖𝑋_𝑛+1^𝑟𝑒𝑓−𝑋𝑛,𝑌_𝑛+1^𝑟𝑒𝑓−𝑌𝑛‖] . (16) 2. Use 𝜃_𝑛 to calculate

𝑣_𝑛ǁ = 𝑣_𝑛cos 𝜃_𝑛 𝑣_𝑛⊥ = { 𝑣_𝑛sin 𝜃_𝑛

−𝑣_𝑛sin 𝜃_𝑛

(17)

If the lateral speed is in the positive 𝑦-direction, body fix, 𝑣_𝑛⊥ is positive and if the lateral speed is in the negative 𝑦-direction 𝑣_𝑛⊥ is negative.

3. Find the duration 𝑡_𝑛 of the travel along 𝑠̅ as a function of 𝑎_𝑛 _∥ from the distance equation

𝑠_𝑛 = 𝑣_∥𝑛𝑡_𝑛 +1

2𝑎_∥𝑛𝑡_𝑛² (18)

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13 𝑡_𝑛(𝑎_∥𝑛) =

{

√(𝑣_∥𝑛 𝑎_∥𝑛)

2

+2𝑠_𝑛 𝑎_∥𝑛 −𝑣_∥𝑛

𝑎_∥𝑛, 𝑎_∥𝑛 ≠ 0 𝑠_𝑛

𝑣_∥𝑛, 𝑎_∥𝑛 = 0

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4. Write the lateral deviation 𝑑_𝑛+1 condition as a restriction of the accelerations

𝑑_𝑛+1 = |𝑣_⊥𝑛𝑡_𝑛+1

2𝑎_⊥𝑛𝑡_𝑛²| 𝑎𝑛𝑑 𝑑_𝑛+1 ≤ 𝑑_𝑛+1^𝐴 (20) where 𝑑_𝑛+1^𝐴 is the allowed maximum deviance at point 𝑛 + 1, and can vary from point to point.

5. When the accelerations, governed by the conditions, are selected one may calculate where the particle ends up, and what the velocity will be. The 𝑋̅_𝑛 vector is described in the global coordinate system (𝑋,𝑌) and so is the velocity 𝑣̅_𝑛.

𝑋̅_𝑛 = [𝑋_𝑛 𝑌_𝑛]

(21) 𝑋̅_𝑛+1 = 𝑋̅_𝑛+ (𝑣_⊥𝑛𝑡_𝑛+1

2𝑎_⊥𝑛𝑡_𝑛²) 𝑅 ∙ 𝑠̂_𝑛+ (𝑣_∥𝑛𝑡_𝑛+1

2𝑎_∥𝑛𝑡_𝑛²) 𝑠̂_𝑛 (22) 𝑣̅_𝑛+1= 𝑣̅_𝑛 + 𝑎_∥𝑛𝑡_𝑛𝑠̂_𝑛+ 𝑎_⊥𝑛𝑡_𝑛(𝑅 ∙ 𝑠̂_𝑛)

(23) Here

𝑅 = [0 −1 1 0 ]

(24) is the rotation matrix. This rotation is done so that the vector 𝑅 ⋅ 𝑠̂_𝑛 is pointing in positive 𝑦-direction, i.e. in the body fix coordinate system, as shown in Figure 2.

6. The objective of maximizing the final speed can now be expressed as minimizing

−‖𝑣̅_𝑒𝑛𝑑‖ where

𝑣̅_𝑒𝑛𝑑 = ∑ 𝑎_∥𝑛𝑡_𝑛𝑠̂_𝑛+ 𝑎_⊥𝑛𝑡_𝑛𝑅 ⋅ 𝑠̂_𝑛

𝑒𝑛𝑑

𝑛=0

(25)

After implementing this method and testing it for a reference path, the results were verified in a vehicle simulation program called CarMaker by IPG Automotive [11], see Section 5.4.

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5. Results

The methods described in Section 4 are general and in this section, the results of implementing the methods are shown for examples of paths described in respective subsection.

The results from the method described in Section 3.2 implemented on a path of distance 50 m is shown in Figure 7, where the maximum speed of entry is plotted for different values of friction coefficient 𝜇 and slope 𝛼_𝑥. Figures 8 and 9 are easier to survey and show Figure 7 from two different angles. Here the friction coefficients in longitudinal and lateral directions are assumed to be the same, i.e.

𝜇_𝑥= 𝜇_𝑦 = 𝜇 (26)

For this path the slope and friction is assumed to be the same at all points. The path that is followed is described by the curvature as

𝑘(𝑠_𝑛) =

{

𝑠_𝑛⋅ 0.001 10 ⋅ 0.001 (25 − 𝑠_𝑛) ⋅ 0.001

−(𝑠_𝑛− 25) ⋅ 0.001

−10 ⋅ 0.001

−(50 − 𝑠_𝑛) ⋅ 0.001

𝑓𝑜𝑟 0 ≤ 𝑠_𝑛 < 10 𝑓𝑜𝑟 10 ≤ 𝑠_𝑛 < 15 𝑓𝑜𝑟 15 ≤ 𝑠_𝑛 < 25 𝑓𝑜𝑟 25 ≤ 𝑠_𝑛 < 35 𝑓𝑜𝑟 35 ≤ 𝑠_𝑛 < 40 𝑓𝑜𝑟 40 ≤ 𝑠_𝑛 ≤ 50

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This path has a continuous curvature and has the unit [𝑚⁻¹]. The maximum turning radius 𝑅_𝑚𝑎𝑥 is

𝑅_𝑚𝑎𝑥 = 1

10 ∗ 0.001= 100 𝑚 (28)

The method implementation in MATLAB can be found in appendix A.

Figure 7. A view over the maximum speed as a function of the friction coefficient and the slope in percentage. The maximum entry speed is expressed in km/h. Negative slope means that the path is downhill and positive, uphill.

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Figure 8. Another angle of view of Figure 7. The maximum entry speed is shown as a function of the slope of the road.

Figure 9. Another angle of view of Figure 7. The maximum entry speed is shown as a function of the friction coefficient between tires and road. The thick line in the middle corresponds to no slope. For lines above this line

correspond to increasing slope and the lines below to decreasing slope.

5.2 Discrete method constrained by a safe zone

The method described in Section 4.3 is implemented with a path that is described in table 1 by 𝑋-coordinates, 𝑋_𝑟𝑒𝑓 and 𝑌-coordinates, 𝑌_𝑟𝑒𝑓. The method is implemented in MATLAB using the function fmincon, which minimizes a cost function with respect to some parameters while satisfying the constraints described by equations (6) and (20). The cost function is described by equation (25). For the MATLAB code, see appendix B. The velocity at all points are multiplied by −1 to obtain the velocity as if the particle was performing a braking, rather than speeding up since the method implemented in MATLAB only works on speeding up because an initial speed is needed.

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𝑋_𝑟𝑒𝑓 [𝑚] 𝑌_𝑟𝑒𝑓 [𝑚] 𝛼_𝑥 [%] 𝛼_𝑦 [%]

0.0 0 0 0

0.0 5 3.49 −1.75

0.3 10 3.49 −3.49

1.0 15 0 −1.75

2.5 20 −6.97 0

5.0 25 −12.16 1.75

6.5 30 −17.28 3.49

7.0 35 −12.16 1.75

7.2 40 −6.97 0

7.2 50 0 0

Table 1. The coordinates and angles (slope and road bank) of the path used to demonstrate the method in Section 4.2.

Figure 10. A plot of the path a particle takes with velocities illustrated by the black vectors at each point. The velocity is scaled. The reference path is marked with blue stars and the calculated path is marked with black rings. The red

dashed lines represent the boundaries of the lane.

The optimized path is shown in Figure 10 together with the velocities at each point. The friction coefficients are both 1 in longitudinal and lateral direction. The total length of the path is approximately 51 𝑚 and the maximum possible speed 𝑣_𝑚𝑎𝑥 obtained for this path is approximately

𝑣_𝑚𝑎𝑥 ≈ 99 𝑘𝑚/ℎ (29)

If the path were equally long, but straight and horizontal in both longitudinal and lateral direction, the maximum speed would be approximately 114 𝑘𝑚/ℎ. This is derived from

{𝑠 = 𝑣⁰𝑡 + 𝑎𝑡 2

2

𝑣 = 𝑣₀+ 𝑎𝑡

(30)

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17

where 𝑠 = 51 𝑚, 𝑣₀ = 0 𝑚/𝑠, 𝑎 = 9.81 𝑚/𝑠² and 𝑣 is to be determined.

Comparison between small and large safety zone

The path from table 1 is used here as well with the exception that the road bank at all points are zero, i.e.

𝛼_𝑦 = 0 (31)

for all points. Now we allow the maximum deviation from all the reference points to be 0.15 𝑚 and 1.00 𝑚. The illustration of the optimal path is shown in Figure 11.

Figure 11. An illustration of the optimized paths, black rings, with different allowed deviances. The reference path is shown with blue stars. The path in the upper graph has an allowed deviance of 0.15 𝑚 and the path in the lower

graph has an allowed deviance of 1.00 𝑚.

The maximum initial speed differs depending on how large the deviance is. For the two paths described above, and illustrated in Figure 11, the speeds were

𝑣_0.15 ≈ 92 𝑘𝑚/ℎ (32)

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18 and

𝑣_1.00 ≈ 109 𝑘𝑚/ℎ. (33)

Here 𝑣_𝑑 denotes the maximum initial speed with deviance 𝑑 measured in meters.

5.3 Straight path

Both methods are implemented on a straight path of length 50 𝑚, with no slope nor road bank and both friction coefficients are 1. The speed denoted by 𝑣_{𝑐𝑙𝑜𝑡ℎ𝑜𝑖𝑑} represents the speed calculated using the clothoid method described in Section 4.1 and 𝑣_{𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒} is the speed calculated using the discrete method described in Section 4.2. The results were

{𝑣_{𝑐𝑙𝑜𝑡ℎ𝑜𝑖𝑑} = 112.7553102962337 𝑘𝑚/ℎ 𝑣_{𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒}= 112.7553100545006 𝑘𝑚/ℎ

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The difference between the speeds obtained by the two methods is

𝑣_{𝑐𝑙𝑜𝑡ℎ𝑜𝑖𝑑} − 𝑣_{𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒}≈ 2.41733 ⋅ 10⁻⁷ 𝑘𝑚/ℎ. (35) The maximum speed obtained from the standard equations in equation (30) is

𝑣 = 112.7553102962339 𝑘𝑚/ℎ (36)

with 𝑠 = 50, 𝑣₀ = 0 𝑚/𝑠 and 𝑎 = 9.81 𝑚/𝑠². The deviation of the speed calculated from the discrete method from this speed is approximately the same as in equation (35).

5.4 Validation in CarMaker by IPG Automotive

Two different paths without road bank were tested in the software CarMaker by IPG Automotive to validate that the vehicle can follow the given reference path and stop at the end of the safe zone with the calculated initial speed and without rolling over. The software is implemented to minimize the braking distance and the deviation from the center of the road. Since the braking distance can be reduced by allowing greater deviations there must be a trade-off. One of the simulated paths did not have any slope on the road and the car successfully stopped at the end of the path. A freeze frame of this simulation is shown in Figure 12.

Figure 12. A freeze frame of the simulation on a flat road. The vehicle enters the path with the speed approximately 90 𝑘𝑚/ℎ and stops at the end of the path. The simulation was performed by associate professor Mats Jonasson at

Volvo Cars in the program CarMaker by IPG Automotive.

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The other path had different slopes at all points. In Figure 13, it is shown that the vehicle stops before the end of the path. This is also visualized in Figure 14 where the red line representing the simulated path is shorter than the reference path.

Figure 13. A freeze frame of the simulation of a road with different slopes where the vehicle has come to a stop. The vehicle starts with speed 106 𝑘𝑚/ℎ and stops before the path ends. Also, this simulation is performed by associate

professor Mats Jonasson in the program CarMaker by IPG Automotive.

Figure 14. An illustration of the different paths when the road has slope. The blue line represents the reference path, the yellow line the path calculated by the discrete model and the red line the path obtained from the simulation. The

𝑦-axis starts on 500 𝑚 because the vehicle needs to accelerate up to the initial speed.

A comparison of the accelerations in longitudinal direction is shown in Figure 15 and the accelerations in lateral direction in Figure 16. The path used in the discrete model is divided in 10 steps and therefore 10 accelerations are obtained, i.e. one at each distance and time. The simulated accelerations have values for hundreds of different times because the software program continuously regulates the accelerations. This also leads to a more continuous and smooth steering and braking.

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Figure 15. An illustration of the accelerations in longitudinal direction versus the time. The red curve represents the accelerations calculated by the discrete model and the blue curve is the accelerations from the simulation.

Figure 16. An illustration of the accelerations in lateral direction versus the time. The red curve represents the accelerations calculated by the discrete model and the blue curve is the accelerations from the simulation.

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6. Discussion

6.1 Common limitations of the vehicle modeling in both methods

An important influence on the initial speed is the friction coefficient and an overvalued friction coefficient can lead to the vehicle driving off the road, i.e. if the calculations were made considering a road with better grip than the actual road or if the tires are more worn out than expected. Therefore, the vehicle should update its unique friction characteristics occasionally. The friction ellipse, although more general than a friction circle, is a simplification that could affect the actual maximum initial speed. It is of importance to validate the results in real world conditions.

A model can never include all characteristics of an actual safe stop maneuver since the world is infinitely complex. The road is not perfectly flat which may cause a wheel to lose grip in a pothole or a wind gust may affect the motion of the vehicle. The major characteristics are considered in the models, but many notable characteristics have also been omitted due to the complexity it would evoke. It would be desirable to take into account the moment of inertia and that the vehicle is not actually a particle but a rigid body with 4 or more wheels with specific forces on each individual wheel. The former consideration may not be necessary for future vehicles though if they were to be fitted with spherical wheels. Vehicles with spherical wheels would be very well simulated using these methods since they would not need to turn (change yaw orientation), having no effect of the moment of inertia.

Throughout the report, forces due to rolling resistance and air resistance have been omitted. If these forces were to be included equation (3) would be used to described the friction ellipse rather than equation (2). Although these forces do effect the maximum entry speed, it would be in a way that increases the maximum possible speed. This is because forces due to resistance are always pointing in the opposite direction of the motion of the vehicle, i.e. help decelerate the vehicle.

The problem studied is independent of the mass of the vehicle since both the inertia and friction forces are linearly dependent of the mass, hence cancelling the influence of mass.

If we were to consider the aerodynamical effects that are only dependent of the geometry of the vehicle, not the mass, there would be a dependence of the mass included in the problem since it would not be cancelled out.

The risk of rollover is not accounted for in either model. However, the validation using CarMaker shows that this is not a problem for the derived example paths. Since the method from Section 4.2 optimizes the path by reducing the lateral accelerations this risk of rollover is significantly reduced. Future driverless vehicles are to have satisfying safety margins to other object so that safe stop maneuvers should not be too extreme, or sharp, hopefully. Furthermore, driverless vehicles will most likely be electric and electric vehicles have the advantage of an extremely low center of gravity thanks to the heavy battery under the floor. Anyhow, if there is a risk of rollover one may introduce an additional restriction on the lateral acceleration to ensure that no flip-over may occur.

This is easily implemented by writing the allowed interval |𝑎_𝑦| ≤ 𝑎_𝑦^𝑚𝑎𝑥 as an additional condition for fmincon.

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The delays in the steering and braking are not considered in the methods. Since it is impossible to change the steering angle and braking force from one value to another, without assuming all intermediate values, it is not realistic to let the accelerations vary from maximum to minimum between two adjacent points. This implies that a model with continuously varying angle and force would be needed to simulate the vehicles motion correctly. By only allowing clothoid paths, i.e. only continuously varying curvature, the accelerations automatically vary continuously. But in addition, the maximum rate of change of the steering angle and braking force is needed as they are both limited. It could be that the tires can generate the friction needed to follow a clothoid while the maximum steering rate might not be sufficient to cope with the rapidly changing curvature. It is possible to add this constraint to either model to account for this delay.

The lack of ability to find an optimal path is acknowledged by the authors Joseph Funke and J. Christian Gerdes:

“The clothoid method is restricted to elementary and by elementary paths, which is a subset of clothoid based paths. This means that we cannot find, for example, the fastest route as is desired in racing.”

Also, this model cannot handle a turn greater than 90° as stated in their report:

“𝐷(𝛼, 𝜆) can be negative, resulting in a negative (and thus infeasible) length, but this does not occur for forward facing points.”

6.3 Discrete method

The method of implementing the method from Section 4.2 is to use fmincon in MATLAB which tries different values for the lateral and longitudinal accelerations at all points and finds the maximum speed for a combination of accelerations. Therefore, the method does not always converge to a solution for all paths, which is a disadvantage. If the method converges depends on how near the initial guess is to the solution. This is not a problem for the clothoid method described in Section 4.1 since it is an iterative method.

Another application of the discrete method may be in racing. The safe zone in this context would be the track and the method would not only find the optimal path in a curve, but also the maximum entrance speed. In the process of accelerating out from the curve the method could be used backward to optimize the path and longitudinal acceleration.

It is worth mentioning that the time required to maximize the initial speed depends on how many points the path is divided into. The more points, the longer time the calculations take but the path and accelerations will become more accurate and continuous.

If the method from Section 4.2 is not appropriate to use, the method from Section 4.1 may be used with the discrete path described by radiuses at all points. From Section 3.2 Discrete path, it is known that a discrete path can be described by a radius at each point and there, it is described how to obtain these radiuses. The first point does not have a previous point and hence the radius at this point cannot be calculated. Although the radius is undefined, the speed at the first point is zero and thus the lateral acceleration 𝑎_𝑦,

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equation (11), becomes zero. The last point also has an undefined radius since there does not exist a point after the last. An appropriate assumption would be that the vehicle continues along the same arc as the previous point and therefore has the same radius as that point which means that

𝑅_𝑁= 𝑅_𝑁−1 (30)

where 𝑁 is the last point along the path. The curvature will not be continuous, but if the path is gentle the differences in steering may not be to extreme. Also, if the number of points tend to infinity the curvature will tend to continuity, i.e. more points make the path gentler.

6.4 Comparison of the two methods

From Section 5.3 the speeds obtained from the two methods implemented on a straight path differ little. The difference was, equation (35), 1.114 ⋅ 10⁻⁶ 𝑘𝑚/ℎ which is not noticeable and may affect the braking distance by no more than a millimeter.

6.5 Validation in CarMaker in IPG Automotive

From Section 5.4 the results from the simulation of the first path, i.e. the path without slopes, the discrete model obtains the correct initial speed in order to stop at the end of the path. For the second path, with slopes, the vehicle stopped several meters before the end of the path, which can be seen from Figure 14. This is because in the method and the implementation, the path is considered as two dimensional with slopes corresponding to each point. In the method, the slope only plays roll in the friction ellipse whilst it should also be considered in the all equations leading to the final speed, i.e. for example, calculating the distance between points and angle 𝜃_𝑛 in equation (16).

The vehicle stops earlier in the simulation which may be because the distance it has travelled in three dimensions is the same as the distance calculated in the discrete method in two dimensions but it has not moved as far along the two-dimensional path because it has moved up and down also. So therefore, to achieve a realistic method to calculate initial speed of a path that is in three dimensions, a three-dimensional method is needed.

The method developed by us is correct for two dimensional paths which is validated by the simulation done in Section 5.4 for the first path but not for three-dimensional paths.

6.6 Processing sensor signals

Delays due to information transfer time and time needed for calculations have been omitted deliberately. This is because the system knows the path and accelerations needed in advance since this information is continuously updated preventively while the vehicle is driving. As stated before, an autonomous vehicle is required to be able to handle a situation of total loss of input which in turn demands a system capable of functioning solely based on previous inputs.

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7. Sustainability

This work concerns a small fraction of the area of scientific research developing technology for automated driving. Though the methods described could be used in other contexts, the report is written with the intention of studying automation of vehicles, hence it is sensible to mention what impact autonomous vehicle could have on society and the environment.

The ethical aspects of introducing autonomous vehicles in the society mainly concerns safety and employment. In 2014, there were almost 1.25 billion road vehicles in use around the world with 330 million of these being used as commercial vehicles [12]. All drivers of these commercial vehicles could be losing their jobs in the process of introducing driverless vehicles. There are countless of other professions that depend on today’s structure of human drivers such as traffic wardens, gas station attendants, rental car agencies.

It is worth pointing out that a structural change of society often results in reemployment rather than unemployment since the demand of workers in other areas is usually raised as a result. Now, there are more than 33 companies developing technology for autonomous driving employing thousands of engineers. Several companies, such as Uber and Tesla, are planning to provide a mobility service, similar to that of taxi companies, but using their own fleet of self-driving vehicles.

Each year 1.3 million people worldwide lose their life due to traffic accident. It is among the 10 most common causes of death killing approximately the same number of people as tuberculosis or diarrheal diseases [13]. It is also estimated that up to 50 million people suffer injuries [14]. The annual cost of roadway crashes in the US was estimated at $871 billion in 2010 including both economic loss and societal harm [15]. The purely economic cost alone is equivalent to $900 for each American citizen per year. Since research show that 94% of car crashes are caused by human choice or error the great majority could have been avoided using self-driving technology [16]. It would be unethical not to use this kind of technology to save lives, prevent injuries and lessen the strain on society.

The environmental benefits of automating the worlds vehicle fleet mainly composes of shared mobility, optimal driving behavior and easier transition to electric vehicles. Most vehicles today are rarely used or mainly used for short distances causing cars to be parked 95% of the time [17]. Apart from that, cars usually have 5 seats of which only one is used most of the time and the average American spend 42 hours a year in traffic jams [18]. The average American car only uses (5% − ^{42 ℎ}

365∗24 ℎ)¹

3= 1.5% of its potential transportation capacity if we assume that a third of the seats are occupied on average. This means that if we could effectively share our vehicles, one American vehicle could be share by up to a 67 people which becomes more reasonable considering that no one would need to be occupying a seat just to drive someone else where they need to go. One may argue that the transportation demand is higher in rush hours, but if we were to have flexible vehicles that could transport both goods and people they could adapt the number of goods being transported keeping the total transportation demand constant. Having vehicles of different sizes and information regarding every commuter´s needs and budgets could dramatically increase the efficiency of cars and busses transporting a lot more people while taking shorter routes. Eco-driving is easily programed into a self-driving car and connected cars would enable optimization of the total fuel consumption of the fleet.

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Finally, the biggest disadvantages of electric cars, the range and duration of charging, would no longer be an issue. You would never need to wait while your car is charging since they would go charging themselves when needed. The range limitations today are mainly caused by the cost of batteries, but since electricity is a lot cheaper than gas and the maintenance costs are minimal the total cost is already today lower than internal combustion engine (ICE) vehicles after a certain number of miles. In a shared fleet the purchase price becomes less important since it is shared among many. Hence, there would be an economical benefit of using electric vehicles in favor of ICE vehicles in a shared mobility network.

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8. Conclusion and future work

From the results obtained by the clothoid method, the maximum initial speed increases when the friction coefficient is high or when the road slope is high and positive. This is intuitive because good grip increases the friction force and gravity helps brake the vehicle when driving up a hill. The relationship between the initial speed and slope or friction coefficient is not linear. The maximum speed decreases quicker for negative slopes than for positive. The difference between speeds for two friction coefficients that differ by 0.1 is approximately 10 𝑘𝑚/ℎ for the studied path of 50 m. It is therefore very important to have information about the path ahead to be able to stop in time. The magnitude of the slope has a greater effect on the maximum speed for low friction coefficients than high.

We fulfilled to aim to produce a method that calculates initial speed but we would have preferred for it to works well for three dimensional paths in addition to working for two dimensional paths. The calculated initial speeds for two dimensional paths are almost identical to those determined by the clothoid method. The method can calculate the speed fast, using fmincon in MATLAB, but there are convergence problems if the starting guess for the accelerations at all points is far from the solution or if the path is too extreme for the model to handle. With too extreme, it is meant that the path has, for example, a too sharp curve when the speed is relatively high. Other optimization programs may be used to obtain faster calculations or that converges in more situations. The clothoid method, also, cannot be used to calculate the initial speed for certain extreme paths.

For future work, it is important to expand the method to a more general three-dimensional model and do more validation simulations. Without the simulations to validate the model it is not useful. The simulations can be made either in a simulation program such as CarMaker by IPG Automotive or in a control robot in a real vehicle. It is also important to add the most influential properties of vehicles into the model to make it more applicable. Yaw angle and force distribution are major influences. Also, more general friction ellipse can be used considering rolling resistance on the wheels and the force due to aerodynamics although these effects are not as large as the previous mentioned. The model developed by us is a foundation that can be adjusted to different types of vehicles, such as cars or trucks, by considering the properties and constraints that defines that type of vehicle.

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[3] P. A. Eisenstein ‘The Race for Autonomous Vehicles Picks Up Speed’, NBC News, [website], 9 April 2017, http://www.nbcnews.com/business/autos/race-autonomous- vehicles-picks-speed-n743076 (accessed 28 April 2017).

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Motion constraints for self-driving vehicles during safe stop maneuvers