Optimally Fuel Ecient Speed Adaptation

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Optimally Fuel Efficient Speed Adaptation

ASSAD AL ALAM

Masters’ Degree Project

Stockholm, Sweden March 2008

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Optimally Fuel Efficient Speed Adaptation

ASSAD AL ALAM

Master’s Thesis at Automatic Control Supervisor: Per Sahlholm Examiner: Karl H. Johansson

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iii

Abstract

An optimal velocity trajectory for a heavy duty vehicle, obtained with the aid of modern GPS and digital map devices, depends on several variables. Curvature speed limitations, road grade, and posted road speed are common constraints imposed by the road travelled. This thesis presents a method for modelling and analysing a switching controller through the use of the former mentioned constraints. A non-linear model for the heavy duty vehicle is derived, enabling suitable control methods to be applied. Pontryagin’s Principal and LQR are discussed to get a profound understanding of how the controller should be de-signed. It is discovered that a switching controller based on optimal control and engineering experience is most favourable for the problem at hand. The con-troller is designed to address the main objectives set in this paper of minimising fuel consumption, travelling time, and brake wear.

Gauss-Newtons’s algorithm for non-linear equations is used to estimate curve radii. Other input parameters are presumed to be available. GPS data error is discussed to perform a sensitivity analysis. An electronic horizon is pro-duced on three road segments, entailed with data of the future road topology. Finally the switching controller is applied to the road segments. Experimental results show that the controller produces a velocity trajectory, which reduces fuel consumption by 5-15% and brake wear by 15-35%, while the travelling time is only increased by 1-2%.

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Acknowledgements

The work described in this master’s thesis has been conducted at the System Pre-development Department, REP, at Scania CV AB in Södertälje, Sweden. It was supervised by the Automatic Control Systems department, at the Royal Instititute of Technology (KTH), in Stockholm. I would first and foremost like to thank my supervisor Per Sahlholm at Scania, for all his help and guidance during this Master’s Thesis. His input have been invaluable and inspiring. A deep gratitude is extended to Jon Andersson for his significant input to this project. Along with Per Sahlholm and Jon Andersson, other department members at REP namely Joseph Ah-King, Rickard Lyberger, Håkan Gustavsson, Daniel Thuresson, Erik Persson and the se-nior manager Nils-Gunnar Vågstedt, have been most helpful and supportive. I would therefore like to extend my gratitude towards them as well. Finally I would like to thank my supervisor Karl H. Johansson at KTH for his guidance, creative input, time and support.

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Introduction

Fascinating, inspiring and at times petrifying could be said about the development of the technology today. Day by day new breakthroughs are made, new inventions are pursued, and new intriguing applications of old technology and methodology are found.

A relatively new technology called the Global Positioning System (GPS) has been the interest of many fields. The GPS is becoming increasingly relevant in road-navigation systems. Traffic is becoming intense and more complex throughout the world, making it increasingly difficult for the driver to focus on all the relevant aspects of driving. A lot of research is therefore being conducted regarding the use, reliability and accuracy of the information a GPS can provide. Future navigation systems may not only guide the vehicle along the best route, but also direct it in a cost-efficient manner.

By enriching the information to the driver and the vehicle with e.g. future curvature information and traffic signs could most probably improve the driver behaviour significantly. Hence implementing control systems as a next step by using the GPS information and advanced digital maps as input could not only reduce costs, but also increase safety and improve on environmental aspects. Emission regulations are becoming more stringent making the preceding field of research a top priority in the modern vehicle industry.

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Chapter 2

Background

2.1 Look-Ahead

Look-Ahead could be described as a process of acquiring ’Preview Information’. It is a concept for providing control strategies with information about future dis-turbances and inputs. The information can be properties such as road topology, curvature, and speed regulations, which are important input signals when designing a speed controller. Look-Ahead provides information based on GPS and digital maps data, which is then processed intelligently. The information is subsequently conveyed to the driver or the vehicle, thus enhancing the perception of future dis-turbances and enabling suitable future actions by extending the drivers information input-horizon. It is mainly utilized in Advanced Driver Assistance Systems (ADAS).

2.1.1 Premise

It is assumed that the driver selects a drive mission, i.e. a given route. Information regarding the topology, curvature and legal speed limitations is acquired through the Look-Ahead system. Accounting for the fact that the system is a heavy truck, it is possible to derive a mathematical model for the system and use the available electronic information horizon to calculate a proper and possibly optimal velocity trajectory.

2.1.2 Limitations

If the information provided by the GPS is accurate to a certain extent, proper esti-mations such as curvature, grade, speed-limits, etc. can be made with a justifiable reliability. However, physical limitations such as the time required to receive signals from the satellite, signal distortion, delays in the actuator data processing subsys-tems, amongst others make room for significant possibility of errors. The loss of reliable information could lead to dangerous situations such as the heavy vehicle not reducing the speed in time.

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4 CHAPTER 2. BACKGROUND

Therefore this thesis is set to be a primary study, resulting in simulations and not physical implementation on an actual vehicle. The simulations and results are limited to smaller winding roads. Solely road topology constraints are focused upon, i.e surrounding traffic and its behaviour is not accounted for in the objectives. The effects of other vehicles in the traffic are discussed briefly, but an adaptive methodology is nonetheless considered to be out of the scope of this thesis. Road grade and posted road speeds are presumed to be given. Conducting research and designing methods for measuring such data inputs would require more time than what is allowed for this thesis.

2.1.3 Implementation

To implement a designed controller, the road topography information is obtained by the combination of an on-board database with altitude information and a global positioning unit (GPS). It is assumed that road information is available and the current route is predicted or supplied by the driver as stated in 2.1.1. By creating a model from existing parameters based upon the heavy vehicle’s characteristics, a prediction of vehicle motion and energy consumption as a function of control signals and known disturbances can be made. Road topology information, desired constraints on comfortable lateral acceleration, and maximum deceleration limits, are presumed to be available control inputs. The road grade is assumed available through various mathematical tools. Finally the conventional cruise controller is fed with set points from an optimisation algorithm.

from a GPS unit Global position received

feeding the vehicle with set speeds Laptop calculating optimal trajectories and

Road slope stored in on−board database

Truck receiving set speeds and reporting current velocity, gear and estimated mass

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2.2. RELATED WORK 5

A GPS unit connected to a laptop receives the global position as shown in Figure 2.1. The position is then matched to the road database to obtain slope information about the road ahead. The computer subsequently calculates a most favourable speed trajectory for a desired electronic horizon ahead through an appropriate method.

2.2 Related Work

Using GPS positioning and digital maps have been proposed in various works. The problem of how to drive a heavy truck over various road topographies such that the fuel consumption is minimised has been addressed by Fröberg et. al., 2006 [4], amongst others. An optimisation problem is formulated, which has the advantage of enabling explicit analytical solutions. Another presented approach is developing a predictive cruise controller for a heavy truck, using dynamic programming to numerically solve the optimal control problem. Thereby, several strategies are found that might be appropriate for various types of road segments. The results presented in the article show that there is a potential to reduce fuel consumption by utilising information about the topology ahead.

However, fuel consumption is not the only beneficial factor. Research has been conducted to create advanced driver speed assistance in curves. One article pre-sented by Aguilera et. al., 2007 [7]; discusses driver interaction modes, i.e. ways in which a driver assistance may interact with the driver. The method uses a vehicle infrastructure driver speed profile, which gathers information from the electronic horizon, the driver behaviour, and the vehicle dynamics. A risk function is con-structed to determine what action is suitable to convey to the driver. The risk function is based on the maximum possible speed in a curve, i.e. a curve speed warning system (CSWS) of sorts.

Another article presented by the Mitsubishi Motors Corporation in Japan [5], investigates how cornering can create discomfort and, under extreme circumstances, result in loss of vehicle control. It is believed that conventional implementations of the active safety approach, e.g. antilock braking and traction-control systems can be improved by complementing such control systems with strategies that perform vehicle control prior to entering the turn. Therefore, an investigation is conducted in the article involving empirical studies regarding location of brake activation before various curves and mathematically based maximum speed limits based on the curve radius and the preferred lateral acceleration.

The research presented above is however not based on deriving an optimal con-trol strategy for speed reductions to posted speed limits and curvatures.

2.3 Thesis Outline & Objective

A lot of research within the subject matter have already been done. A model pre-dictive control strategy has previously been designed [10]. However, the constraint of the speed limits inflicted from the legal road speed limitation and curvature were

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6 CHAPTER 2. BACKGROUND

never accounted for. A curve speed assistant prototype with similar goals but a fo-cus on hardware implementation has previously been built at Scania. Information regarding topology were taken into account when designing a controller which gath-ered information about curvature and road speed limitations through a software tool known as Advanced Driver Assistance System Research Platform (ADASRP). The controller was based upon the truck model and inversely calculated a fuel opti-mal velocity trajectory. However, the physical limitations discussed in 2.1.2 caused certain problems which the controller could not handle. In addition the road slope data was used as an input to obtain a reasonable fuel cut-off point for slowing down, but it was not sufficient in certain situations. The assumption that the heavy vehi-cle always had the possibility to reduce its speed in time through coasting was not always valid. In the situation of facing a long and steep downhill, the system would reach its lower speed set point, rendering a non-existing solution. To obtain the desired speed at the foot of the hill, the truck had to reduce its speed significantly. This resulted in an unacceptably early control input and an unreasonable vehicle road behaviour. Thus it was discovered that in some cases using the brakes can be justified, despite the effect on fuel economy [12].

Hence, new relevant parameters such as driver comfort, time optimality, road dynamics and driver behaviour must be highlighted. Also, the effects of the physical limitations have to be accounted for. Most notably were the delays in the processing and actuation parts, which consequently have to be addressed. Thus the objective of the thesis will be to account for and evaluate these newly emphasised parameters. Previous experience have also proven that it is very important to imply a failsafe system of sorts in the case of estimation error. It has to be noted however that these precautions will be applied according to the complexity of the case. Too complex situations will not be dealt with in great depth due to the time constraint issued upon this research. The main focus will lie on designing a controller which allows for travelling an assigned route in a comfortable, fuel-, time-, and brake-efficient manner.

The thesis is hence structured as follows: in chapter three a description of the vehicle modelling is presented, which serves as a premise for all the forthcoming calculations. Thereafter a study of optimal control approaches are undertaken. However, the analytical control inputs derived from the optimal control strategies are found to increase in complexity as additional constraints are implemented within the model. Further the optimal control presented in this case is based on a linear model and cannot account for the non-linear behaviour that arises primarily from the road grade and gear shifting. The varying road grade and gear shifting have a significant impact on the vehicle behaviour and cannot be neglected. Therefore a new non-linear switching controller is designed based upon optimal control theory and engineering experience, which is discussed in chapter five. Chapter six presents results and sensitivity analysis based on the controller’s performance. Finally the limitations and strengths of the controller are discussed in chapters seven and eight, along with the future possibilities of the presented topic.

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Chapter 3

Vehicle Modelling

The main parts of a heavy duty vehicle (HDV) consist of engine, clutch, transmission shafts and wheels. The combination of all these parts creates the driveline or also known as the powertrain. The powertrain is a fundamental part when evaluating the dynamics of the vehicle. It can be modelled in various ways depending on the purpose and use. The main interest of this study is to create a discrete model of the powertrain, based upon the simple model depicted in Figure 3.1, which will be used as the base for the controller design.

3.1 Powertrain

General powertrain modelling can be found in Vehicular Systems [3]. The main parts of interest within this study of the powertrain are depicted in Figure 3.1.

Figure 3.1. A basic model of the powertrain. The engine utilized in this model is a diesel engine.

Engine: The engine produces a torque through combustion of diesel mixed with a surplus of air in a very high pressurised chamber. The highly explosive combustion

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8 CHAPTER 3. VEHICLE MODELLING

drives the crank shafts, which in turn are connected to the clutch by a shaft, causing a desired torque. The output torque from the engine is characterised by the torque resulting from the combustion, the internal friction from the chamber walls, and the external torque from the clutch. Thus if the inertia is obtained, Newton’s second law gives:

Je˙ωe= Me− Mc (3.1)

where Me(ωe, δ) is the output engine torque obtained empirically through an

engine map, which depends on the angular speed ωe and the engine fuelling δ.

Clutch: The clutch involves two frictional discs, which are pressed together and connects the flywheel of the engine with the transmission’s input shaft. Such clutches are commonly found in vehicles equipped with manual transmission. The connection between the transmission and the clutch is considered to be stiff, i.e.:

Mt= Mc (3.2)

ωt= ωc (3.3)

where Mt denotes the torque output and ωt denotes the output angular speed

from the transmission.

Transmission: The transmission is the connection between the clutch and the propeller shaft. It consists of a set of cogwheels (gears) which are connected such that the output torque is transformed depending on which gear is engaged. It is modeled in this case as a conversion ratio it, which varies according to the specific

gearbox transmission characteristics. The transmission in this case is modelled as an optimal procedure for gear changing during coasting, i.e. allowing the HDV to only operate under the ideal range of operation for most of the gearboxes at Scania CV AB. The ideal range is empirically determined to be between 1100 and 1400 RPM. Thus the 12 geared transmission box is mapped with respect to velocity through (3.4). RP M = 60 2π(itif v rw ) (3.4)

The mapping is illustrated in Figure 3.2.

Another characteristic of the gear box is the efficiency ηt. The inertia of the

trans-mission is neglected and the gear shifts are assumed to be instantaneous, i.e. an immediate change of conversion ratio and efficiency, hence

Mp = itηtMt (3.5)

ωp = itωe (3.6)

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3.1. POWERTRAIN 9

Figure 3.2. A mapping diagram between gears, RPM, and velocity. i = 1, . . . , 12 denotes the active gear. As the RPM decreases during coasting and reaches RP Mn=

1100, the gear is instantaneously shifted to a lower gear, increasing it to RP Mn+1=

1400. The current velocity is unaffected since the gear shift is considered to be instantaneous, i.e. vn= vn+1.

Propeller Shaft: The propeller shaft connects the transmission to the final drive. No friction is assumed and the connection is considered to be stiff.

Mp = Mf (3.7)

ωp = ωf (3.8)

Final Drive: Like the transmission, the final drive is characterised by a con-version ratio if, and an efficiency ηf. The value for the ratio and the efficiency

depends on the final drive design. Neglecting inertia the following relation could be made by the input and output.

Md= ifηfMf (3.9)

ωd= ifωf (3.10)

Drive Shafts: The drive shafts connects the final drive to the wheels. In this simplified model it is assumed that the wheel speed is the same for both wheels. It should be kept in mind that the wheel speed differs when the vehicle enters a curve. However, it is negligible compared to other simplifications within the model. The connection between the wheels and the drive shafts is considered to be stiff and can therefore be modelled as:

Mw = Md (3.11)

ωw = ωd (3.12)

Wheels: The connection between the road and the wheels is modeled by as-suming no slip, i.e.:

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10 CHAPTER 3. VEHICLE MODELLING Jw˙ωw = Mw− Mb− rwFw (3.13) v = rwωw= rwωe itif (3.14) The braking torque Mb is hard to measure, often difficult to model, and is most

often zero. Therefore, it is neglected in this model.

3.2 Longitudinal Forces

The external, i.e. longitudinal forces on the heavy vehicle is modelled according to Figure 3.3. Froll Fgravity Fairdrag Fbrake Fengine α

Figure 3.3. The longitudinal forces inflicted upon a heavy vehicle in motion.

Thus applying the generalized Newton’s second law gives the state-equation:

mt˙v = Fengine− Fbrake− Fairdrag− Froll− Fgravity (3.15)

where Fbrake is considered to be zero due to the explanation given in 3.1 and α

denotes the road grade. During coasting the engine exerts a brake force as opposed to a driving force when the accelerator is applied. The total accelerated mass is:

mt= Jw rw2 + m + it 2_i f2ηtηfJe rw2 (3.16) The aerodynamic force is given by:

Fairdrag =

1

2cwAaρav

2 _(3.17)

where cw denotes the airdrag coefficient, Aa denotes the maximum cross-sectional

area of the vehicle and ρadenotes the air density. The rolling resistance is given by:

Froll = crmg cos(α) (3.18)

cr denotes the corresponding coefficient, g denotes the gravitational constant,

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3.2. LONGITUDINAL FORCES 11

Fgravity = mg sin(α) (3.19)

All of the constants are obtained empirically. Combining equations (3.1)-(3.17) a final mathematical vehicle model could be derived as:

˙v =dv

dt =

1

mt(Fengine

− Fairdrag(v) − Froll(α) − Fgravity(α)) =

= 1

Jw

rw2 + m +

it2if2ηtηfJe

rw2

(Fengine− Fairdrag(v) − Froll(α) − Fgravity(α) =

= rw 2 Jw+ mrw2+ it2if2ηtηfJe i_ti_fη_tη_f rw Me(ωe, δ)− − 1 2cwAaρav 2_{− c} rmg cos(α) − mg sin(α) (3.20)

Note that (3.20) is a nonlinear time varying state-space equation. The data from the GPS and the digital maps is spatially sampled rather than with respect to time and must therefore be transformed using the chain rule:

dv dt = dv ds ds dt = v dv ds ⇒ dv ds = 1 vmt

(Fengine− Fairdrag(v) − Froll(α) − Fgravity(α))

(3.21) where v > 0 is evident due to physical properties. Hence using a first order Euler approximation results in a discrete spatially sampled model with sampling distance ∆s, the difference equation is deduced as:

vk= vk−1+ ∆s∆vk−1 (3.22) where ∆vk−1 = c1 M_ek−1 v_k−1 − c2vk−1− c3 1 v_k−1 − c4 sin(αk−1) v_k−1 (3.23)

which is the discretized version of equation (3.21). c1, c2, c3, c4 is clarified in

(3.24). They denote the vehicle parameters, which are a function of the gear.

c1 = rwitifηtηf Jw+ mrw2+ i2ti2fηtηfJe c2 = 1 2rw2cwAaρa Jw+ mrw2+ i2ti2fηtηfJe c3 = cr rw2mg Jw+ mrw2+ i2ti2fηtηfJe c4 = rw2mg Jw+ mrw2+ i2ti2fηtηfJe (3.24)

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12 CHAPTER 3. VEHICLE MODELLING

3.3 Summary

By modelling the driveline as presented in 3.1, an expression for the output force from the engine can be derived. Inserting that expression into Newton’s second law, after deriving expressions for the longitudinal forces acting upon the HDV, a non-linear discretized equation is derived. Equation (3.22) presents a non-linear dis-cretized equation, which allows for the estimation of a velocity trajectory throughout a given stretch. Thus, changes in the varying road grade and gear shifts can be ac-counted for, which partly characterises the non-linearity in the equation. Thereby a continuous (3.21) and a discrete (3.22) model is derived, which will serve as a base for the mathematical procedures in the forthcoming chapters. Equation (3.22) will prove to be the most useful equation throughout this thesis.

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Chapter 4

Optimal Control

Optimal control is one of the most useful systematic methods for control design. Often control problems become very complex and hard to solve using pure intuition. Optimal control presents a systematic method and logical reasoning to solve complex control problems. A control problem could have many solutions. However, it is often desirable to find the best solution according to certain criterion.

To obtain a better understanding of how an optimal controller is to be designed a mathematical study is undertaken. Two possible optimisation strategies for op-timal control are implemented to give a more profound perspective of the opop-timal solution. Constraints are forced upon the solution space to obtain optimal results with respect to fuel, time, driver comfort, and brake application. To solve this opti-mal control problem, it needs to be stated in the standard form shown in equation (4.1). Standard form: min u:[0,tf]→U Z tf 0 L(x(t), u(t))dt + φ(x(tf)) subject to: ˙x(t) = f (x(t), u(t)), x(0) = x0 (4.1) Remarks

• U ⊂ R set of admissible control

• Infinite dimensional optimisation problem: Optimisation of functions u : [0, tf] → U

• Constraints on x from the dynamics. • Final time tf fixed

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14 CHAPTER 4. OPTIMAL CONTROL

4.1 Pontryagin’s Principal

In optimal control, the problem at hand could be solved by using the Pontryagin minimum principle (PMP). By investigating local properties and setting empirical constraints, necessary conditions for optimality can be obtained and an optimal trajectory can be calculated. One advantage amongst many of using PMP is that it can be used even though dynamic programming fails due to lack of smoothness of the optimal cost function. However, it should be noted that it only gives a necessary condition for optimality.

The current problem could hence be stated in (4.2), using the same notation as in [6]: J = min Z (u(t)T Ru(t))dt ˙x = Ax + Bu, u = Feng x(tf) = vf, x(0) = v0, ˙x(t) ≥ −kdecc x ∈ Xn, u ∈ Un

X = {x : vmin< v < vmax} , U = {u : Feng,min < Feng < Feng,max}

(4.2)

where the state vector x is chosen to be the vehicle velocity and the input-control signal is the produced force on the vehicle, i.e. momentum, from the vehicle engine. The produced force from the engine is allowed to be both positive and negative, indicating whether work is being done on or by the engine. In other words, the control signal could also be denoted as u = [Feng Feng,brake]T, i.e. having

the characteristic of a driving and braking force.

The objective is to find the smallest possible input energy, which will make the vehicle decelerate to the desired final velocity within a desired distance. This ap-proach requires a fixed start point and a fixed final point. The calculated non-linear state equation (3.20) needs to be linearized to apply the PMP. It is evident that the equation (3.20) is rather linear within small segments. The optimal vehicle behaviour for a speed decrease from 90km/h to 70km/h is of vast interest for the problem at hand, since it is very common event on the Swedish highways. There-fore, investigating a linear equation within the segment 70-90 km/h should give satisfactory results. Linearizing the equation yields:

dv(t)

ds(t) =

1

vmt

(Fengine− Fairdrag(v) − Froll(α) − Fgravity(α)) ≈

≈ 1 mt (−c1M0 v02 − c2+ c3 v02 )∆x + c1 mtv0 ∆u = = 1 mt (−c1M0 v02 − c2+ c3 v02 )˜x + c1 mtv0 ˜ u = A˜x + B ˜u (4.3)

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4.1. PONTRYAGIN’S PRINCIPAL 15 where ˜ x = x − x0 = x − v0, ˜u = u − u0= u − M0 v0 = 80km/h, α0 = 0 M0 = c2v02+ c3cr+ c3sin(α0) c1

Thus the optimization problem can be stated accordingly:

J∗_{= min} Z (˜u(s) + u0)2dt subject to: ˙˜x(s) = A˜x(s) + B˜u(s) x(0) = x0, x(sf) = xf (4.4)

The input is minimized in (4.4) and not the deviation from the linearization point. There is no restriction on setting the starting point to s0= 0, since the system allows

for arbitrary time translation of its solutions. Hence, if u(s) ∈ U transfers the initial state x(0) = xi to the final state x(sf) = xf, then ˜u = u(s − si) transfers ˜x(si) = xi

to ˜x(si+ sf) = xf and the trajectories are related accordingly as ˜x(s) = x(s − si).

(4.4) is now in the standard form. The optimal input can thus be calculated analytically by first stating the hamiltonian:

H(x, u, ˜λ) = λ0(˜u + u0)2+ λT(A˜x + B ˜u) (4.5)

The adjoint equation is hence:

˙λ(s) = −dH

dxλ(s) = −A

T_{λ(s) ⇒ λ(s) = e}−As_λ(0) _(4.6)

Choosing λ0 = 1, due to the fact that the system is controllable, the calculation

could be furthered accordingly:

arg min ˜ u+u0 H(x, u, ˜λ) = arg min ˜ u+u0 λ0(˜u + u0)2+ λT(A˜x + B(˜u + u0) − Bu0) ⇒ ˜u + u0 = −1 2B T_{λ(s) = −}1 2B T_e−As_λ(0) ⇒ ˙˜x = A˜x + B ˜u = A˜x − 1 2B( 1 2B T_e−As_{λ(0) + u} 0) (4.7)

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16 CHAPTER 4. OPTIMAL CONTROL x(sf) = eAsfx(0) − Z sf 0 e A(sf−τ )_B(1 2B T_{λ(τ ) + u} 0)dτ = . . . = = eAsf_{x(0) −}1 2 Z sf 0 eA(sf−τ )_BBT_eAT(sf−τ )_{dτ e}ATsf_{λ(0) −} Z sf 0 eA(sf−τ )_{dτ Bu} 0 = = eAsf_{x(0) −} 1 2W (sf, 0)e AT_s f_{λ(0) − G(s} f, 0)Bu0

It is evident that λ(0) can be solved for, thus the optimal input is: ˜

uopt,P M P = −BTeA

T_(s

f−s)_W−1_(s

f, 0)(eAsfx(0) − G(sf, 0)Bu0− x(sf)) − u0 (4.8)

Implementing ˜uopt,P M P and simulating the state equation gives a trajectory

shown in Figure 4.1. The distance for which no input is required, i.e. the vehicle simply decelerates to the final desired velocity due to the existing frictional forces with no fuelling, is determined to be sd = 1094m. The results for simulating the

system for a deceleration stretch longer than sd are shown in Figure 4.1.

0 200 400 600 800 1000 1200 1400 1600 60 80 100 120 140 u opt Distance [m] Torque [Nm] 0 200 400 600 800 1000 1200 1400 1600 70 75 80 85 90 x trajectory Distance [m] Velocity [km/h] 0 500 1000 1500 2000 2500 3000 3500 100 200 300 400 500 u opt Distance [m] Torque [Nm] 0 500 1000 1500 2000 2500 3000 3500 70 80 90 100 x trajectory Distance [m] Velocity [km/h]

Figure 4.1. Plot of the velocity trajectory and the actual optimal input from the engine for a longer fixed deceleration stretch than sd. The figure to the left is

simu-lated for a fixed final deceleration strecth of sf = 1500 and the figure on the right for

sf = 3000

It can be observed that a deceleration distance longer than sd, mandates a positive

declining control input, i.e. the engine has to work to maintain the speed initially since the deceleration stretch is too long. It becomes evident by studying the right figure in Figure 4.1, that if a much longer deceleration stretch is mandated, the optimal solution is to keep the top speed as long as possible and successively de-celerating the velocity as late as possible. Intuitively it is understood that the fuel consumption is minimised by slowing down to the final desired speed without using the brakes. Also, by maintaining the maximum allowed speed as long as possible the travelling time is minimised.

The results for simulating the system for a deceleration stretch shorter than sd

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4.2. LQR 17 0 100 200 300 400 500 600 −500 −450 −400 −350 u opt Distance [m] Torque [Nm] 0 100 200 300 400 500 600 60 70 80 90 x trajectory Distance [m] Velocity [km/h]

Figure 4.2. Plot of the velocity trajectory and the actual optimal input from the engine for a shorter fixed deceleration stretch than sd, i.e. for sf = 500

The trajectories depicted in Figure 4.2 illustrates the case when the deceleration distance is shorter than sd. As expected, the optimal control input is a negative

declining force input, i.e. a relatively large "braking" force is initially required to slow down to the final speed during the short stretch.

4.2 LQR

The optimal controller calculated through the PMP disclosed a deeper insight and revealed key characteristics of the optimal trajectory. However, the cost function in (4.4) only constrains the input by minimising it. Consequently all the objec-tives presented in 2.3 are not fulfilled. Mandating additional constraints in (4.4) increases the complexity of the optimal solution in the PMP, rendering an analytical expression difficult.

Thus the discrete linear quadratic regulator (LQR) follows as an alternative approach. Hence, the non-linear discrete model (3.22) is linearized in (4.9) through a first order Taylor approximation. It is conducted around a desired equilibrium point. As explained in 3.2, each gear exerts a braking force of different order of magnitude when coasting. This is denoted as a mode in the discrete linear model (4.9).

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18 CHAPTER 4. OPTIMAL CONTROL Am = 1 + df dx|x=xm∆s = 1 + ∆vm∆s Bm = c1 vm ∆s where ∆vm = −c1,m_v₀M2 m − c2+ _vc3 02(cr+ sin(α0)), and Mm = c2v02+c3(cr+sin(α0)) c1 . ˜

x = x − xm denotes the linearized state variable, and ˜u = M − Mm denotes the

relative engine torque. Subsequently the linearized discrete state space model is: ˜

xk+1= Amx˜k+ B ˜uk (4.9)

Having a linear discrete difference equation, the costfunction can be stated and modified to address additional constraints. By changing the equilibrium to xm =

70km/h, a constraint could be put on the final state. A constraint could also be put on maximum allowed deceleration, which tends to the driver comfort requirement. By adding these new constraints along with the constraint on the input force from the engine, the issues of minimizing fuel consumption, driver comfort, and brake efficiency are addressed. The mathematical representation of the cost function is given by (4.10). min sf−1 X s=0 ( ˙˜xTQ ˙˜x + ˜uTR˜u) + ˜x(sf)TQfx(s˜ f) (4.10) subject to: x(0) = x0 ˜ xk+1= Amx˜k+ B ˜uk

The problem could be solved in several ways. A Dynamic Programming (DP) solution is applied, since it gives an efficient, recursive method to solve LQR prob-lems [11]. Accordingly a value function is stated in (4.11).

Vs(z) = min u(s),...,u(sf−1) sf−1 X τ=s ( ˙˜xT(τ )Q ˙˜x(τ ) + ˜uT(τ )R˜u(τ )) + ˜x(sf)TQfx(s˜ f) (4.11)

subject to ˜x(s) = z, and ˜x(τ + 1) = Amx(τ ) + B ˜˜ u(τ )

Vs(z) gives the min-cost-to-go starting from state z at point s, hence V0(x0) is

the minimum LQR cost. Thus the dynamic programming principle could be stated as:

Vs(z) = min_w (zTQz + wTRw + ˙zTQ1˙z + Vt+1(Az + Bw)) (4.12)

where w = ˜u + u0. Solving the Hamilton-Jacobi equation, a Riccati recursion is

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4.2. LQR 19

P_s−1= Q + AT_P_˜

sA − ATP˜sB(R + BTP˜sB)−1BTP˜sA) (4.13)

By solving the recursive Riccati equation, the optimal input solution is given in (4.14)

uopt,LQR(s) = −(R + BTPs+1B)−1BTPs+1Ax(s) (4.14)

For a more detailed explanation of how the optimal solution was derived from (4.12)-(4.14), the reader is kindly referred to Appendix A. Implementing ˜uopt,LQR

and simulating the state equation gives a trajectory shown in the figures below. The main focus lies on what new information the simulation with additional constraints will prevail. Therefore, similar cases presented in 4.1 are studied for comparison. Figure 4.3 shows the simulation for when given deceleration stretch is longer than stretch for which the vehicle simply decelerates to the final desired velocity due to the existing frictional forces.

0 500 1000 1500 70 75 80 85 90 x lqr Distance [m] Velocity [km/h] 0 500 1000 1500 80 100 120 140 u lqr Distance [m] Torque [Nm]

Figure 4.3. Plot of the velocity trajectory and the actual optimal input from the engine for a longer fixed deceleration stretch than sd, i.e. for sf = 1500

It can be seen in Figure 4.3 that the optimal input is positive and slightly increased, i.e. roughly kept constant in the beginning. The velocity decrease trajectory shifts from a non-linear to a linear deceleration as the final velocity is reached. Figure 4.4 shows the optimal trajectories for a fixed deceleration stretch shorter than sd.

It is noticed in Figure 4.4 that the input is constantly negative. Due to the fact that the deceleration stretch is too short, a braking force must be applied. The neg-ative force is increased in a manner that makes the optimal velocity trajectory seem rather linear. A constraint is put on the deceleration, i.e. driver comfort, in the

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20 CHAPTER 4. OPTIMAL CONTROL 0 100 200 300 400 500 70 75 80 85 90 x lqr Distance [m] Velocity [km/h] 0 100 200 300 400 500 −400 −350 −300 −250 u lqr Distance [m] Torque [Nm]

Figure 4.4. Plot of the velocity trajectory and the actual optimal input from the engine for a shorter fixed deceleration stretch than sd, i.e. for sf = 500

cost-function (4.10). Consequently the optimal input reveals an increasing deceler-ating force, i.e. it is favourable to slowly increase the braking force as deceleration is commenced.

4.3 Optimal Solution

Both the continuous linearized model in 4.1 and the discrete linearized model in 4.2 shows an optimal behaviour of the control input. The optimal input is based on a linearized model which utilizes no dynamic information regarding the topology of the road. Variations in the road grade have significant effect on the control input. It is however concluded that implementing a road grade dependence in the controller is out of the scope of this thesis. Also additional necessary conditions are very hard to implement in the cost-function to give analytical solutions.

An essential factor concerning optimality in the problem at hand is a well de-fined time-constraint. Given a long deceleration stretch, the optimal control input according to the LQR will allow the speed to slowly decrease on the given stretch for fuel-efficiency, while the PMP would suggest maintaining the current speed. Main-taining the maximum legal speed a little longer and then decelerating "comfortably" would result in a higher average speed, hence decreasing the total travelling time, which is a significant factor in some cases. Such behaviour is also more accepted by experienced drivers.

Another known crucial factor is driver comfort. The vehicle cannot be allowed to decelerate too fast. Studies have shown that a maximum deceleration of 0.5

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4.3. OPTIMAL SOLUTION 21

m/s2 _{is considered to be comfortable. This issue puts another constraint on the}

controller, which was hard to implement in the PMP but possible in the LQR. Ac-knowledging the fact that there are many constraints on the optimal input signal, it can be understood that a favourable solution does not lie in one single optimal control strategy, but in the combination of several strategies. Therefore, a switch-ing control algorithm based upon the constraints should give a more advantageous control behaviour.

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Chapter 5

Switching Controller

As discovered in the previous chapter, an optimal controller addressing all the objec-tives discussed in 2.3 is very difficult to implement analytically. The road topology must be taken into consideration when designing an optimal controller. Therefore, grade, road speed, and curvature must be utilised as input parameters to produce an optimal velocity trajectory through a controller as depicted in Figure 5.1.

Figure 5.1. Block diagram for the controller design

The non-linear vehicle model (3.20) presents a method to calculate the velocity with the grade as an input parameter. Hence, an optimal speed selection algorithm can be designed with the aid of an electronic horizon.

5.1 Road Speed

5.1.1 Coasting

The point of deceleration explicitly due to frictional forces can be calculated by modifying (3.22). The modified formula is stated in (5.1).

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24 CHAPTER 5. SWITCHING CONTROLLER

vk= vk−1+ ∆s(−∆vk−1) (5.1)

Thus information regarding the topology, i.e. the grade up to the point of interest, varying vehicle parameters, and the desired speed at a given location can be used to calculate a fuel cut-off point. Wasting energy through unnecessary braking is thereby avoided and the fuel consumption is minimised. The method utilises the input of a future decrease in posted road speed and calculates the appropriate speed at every time-step inversely from that point until the current maximum speed is reached. 200 400 600 800 1000 1200 1400 1600 1800 2000 65 70 75 80 85 90 95 Distance [m] Velocity [km/h] v inv v Road 200 400 600 800 1000 1200 1400 1600 1800 2000 −10 −5 0 Distance [m] Altitude [m]

Figure 5.2. Speed adaption for a road speed decrease from 90 km/h to 70 km/h

Figure 5.2 shows a simple case for the applied method, where a legal road speed decrease from 90 km/h to 70 km/h occurs. It clearly shows that cutting off the fuel at approximately 900 meters, causes the HDV to decelerate by coasting down to the desired velocity. Fuel is saved and braking is reduced during coasting, creating a cost efficient and safe solution.

However, when faced with a steep downhill the magnitude of the gravitational force will increase with the increased magnitude of the grade. If it is increased to a certain extent, it will prohibit the vehicle from deceleration. The consequence might be several different scenarios. The speed has to be decreased to an unacceptable level before the downhill in order to obtain the legal speed. In such a scenario, the speed decreases far below the final desired velocity resulting in an uncomfortable driver experience. This is also unacceptable road behaviour according to the traffic behind the heavy vehicle. Another scenario is when the grade of the hill produces a gravitational force upon the HDV, which is equal to the frictional forces. The vehicle would therefore keep a constant speed and never decelerate. A third scenario is when no solution exists at all. It could occur if the vehicle is travelling on a road with a

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5.1. ROAD SPEED 25

speed limit of 90 km/h and there is a decrease in the limit at the bottom of a hill to 70 km/h. In this case the steep grade of the road causes the vehicle to accelerate past the legal speed limit which is an unacceptable solution. Hence,merely using (5.1) as a control method might have either no solution or unacceptable solutions such as an undesirably long deceleration stretch. Owing to the fact that the inverse method cannot be implemented all the time, an alternative method must be switched to. The alternative methods have to address and optimise three major areas of concern, which is depicted in Figure 5.3.

Figure 5.3. Three decision regions when facing a steep downhill

When the inverse method is not obtainable, the new method must determine: (I) When to initiate the fuel cut-off point: Maximum fuel reduction is obtained

by implementing the fuel cut-off point as soon as possible.

(II) How long is coasting possible: It is most favourable to coast as long as possible without breaking the legal speed limit.

(III) If braking is deemed to be necessary, when is it most advantageous to initiate braking: Braking instigates wasting energy and should thus be minimised.

5.1.2 Fuel Cut-Off Point

Maximum fuel reduction is obtained by implementing the fuel cut-off point as soon

as possible. The fuel consumption will be minimized by allowing the vehicle to

coast as long as possible. However, as presented in 5.1.1, it might imply that the vehicle decelerates during a long stretch. If the road grade produces a gravitational force which is slightly less than the opposing frictional forces, the inverse method might suggest a cut-off point as far as 3 km from the speed change. Reducing

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the speed over a 3 km long stretch will certainly agitate the held up traffic behind the truck, induce driver discomfort, and increase the total travelling time. The controller behaviour will therefore most probably not be perceived as warranted by the driver. Hence the driver will disengage the controller, making the controller obsolete. On the other hand a short fuel cut-off point will not minimise the fuel consumption.

Driver experience shows that it is very difficult for a driver to predict the length of the deceleration stretch if the road is a road grade present. However, the experi-enced driver estimates the deceleration stretch of a vehicle on a level ground rather easily and intuitively. Hence, calculating the distance it takes for a heavy vehicle to decelerate on a level ground could be used as a reference fuel cut-off point. Thus, the vehicle will not be allowed to apply any decelerating control input before the calculated cut-off point. Modifying equation (3.15) by removing the gravitational force contribution and setting the roll force to a constant, the distance of interest could easily be calculated and is referred to as SMax. Therefore the vehicle will not

be allowed to commence decelerating, i.e. cutting off the fuel until it reaches SMax.

This preserves the use of the inverse velocity estimation as an optimal solution while driving in an incline, since SMax will clearly be longer.

5.1.3 Maximum Roll-Off

It is most favourable to coast as long as it does not break the legal Speed limit. Setting

the fuel cut-off point to SMax, will enable the vehicle to start coasting from that

point and onwards. Consequently a forward velocity estimation method is applied through equation (3.22), making use of the topology information and allowing the possibility of a fuel cut-off point. However, due to the topology, a few previously mentioned problems might arise, which are depicted in Figure 5.4.

Figure 5.4. Three common problems that arises within the forward method.

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5.1. ROAD SPEED 27

forward simulation results in a speed decrease followed by a larger speed increase due to the topology. In such a scenario the driver might experience discomfort due to the sudden decrease followed by an increase in speed. It is counterintuitive to suddenly have a drop in speed only to regain it moments after. Hence, SMax

must be redefined from that point which will avoid such fluctuations in the desired velocity. 5.4 b) shows the scenario when the topology demands that the speed be decreased below the final desired velocity. This is a similar situation to which the inverse controller failed. A proper action would be to shorten SMax to that point

and restarting the forward simulation from the new point until the situation does not arise. Hence, the number of iterations would be minimised while maintaining a cost-effective method. 5.4 c) simply shows the consequence of implementing SMax.

Naturally the velocity at the final point where the mandated speed reduction occurs will be overshot due to the short deceleration stretch. Therefore, the controller will have to switch to a favourable deceleration method by an intelligent application of the several braking systems.

5.1.4 Most Favourable Braking

Braking instigates wasting energy and should thus be minimised. It is not within the

objectives of this paper to assess how the various vehicle brake systems should be applied, but rather what the effects upon the driver comfort will be. Applying the brakes will lead to loss of the produced energy from the fuel. Therefore, they should be applied as late as possible. However, there is a trade-off between the time when brakes are applied and the magnitude of the decelerating force required to bring the vehicle to the desired velocity. If the brakes are applied at a relatively late stage, the driver will experience great discomfort due to the large deceleration. Experts in the field state that a maximum deceleration of 0.5m/s2 _{is acceptable, without the}

driver feeling any discomfort. Braking procedure with high performance regarding driver discomfort and fuel-cost can easily be calculated. The resulting speed change from a certain deceleration can be determined from (5.2).

adec = dv dt = dv ds ds dt = v dv ds ⇒ dv ds = adec v ⇒ vk= vk-1+ ∆s(−∆vk) = vk-1+ ∆s(− adec vk-1) (5.2)

It has to be noted that situations may arise when a larger deceleration is induced by the topology. In such cases the controller must allow the vehicle to decelerate according to what the road grade mandates. If a larger deceleration through coast-ing e.g. due to a steep incline before the point of the speed reduction arises, the controller should switch to coasting. Such a behaviour is deemed to be intuitive by the driver.

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5.1.5 Implementation

The controller switches between several methods which are optimal with respect to various objectives mentioned in 2.3. Within every method several unique problems that may arise are solved. By not allowing the vehicle to decelerate until it reaches

Smax, the traffic behind the vehicle will not get upset, since the vehicle deceleration

is kept relatively short. However, by not reducing the speed in time, the vehicle will overshoot the desired end velocity. Therefore the brakes have to be applied in order to maintain the legal speed limit. Consequently the vehicle will have a fuel cut-off point, but will have to apply the brakes eventually. Such vehicle behaviour is acceptable according to the surrounding traffic dynamics and it also increases the average speed, i.e. reduces the travelling time. The vehicle coasts as long as possible, minimising the fuel consumption. The overall procedure is demonstrated in Figure 5.5

Figure 5.5. An overview the Switching Controller methodology

If an electronic horizon is available, the controller will be able to produce an optimal trajectory with respect to time, fuel consumption, driver comfort, and braking. Figure 5.6 shows the optimal velocity trajectory for when the vehicle faces a road speed decrease at the bottom of the hill.

The hill is too steep for the inverse method to be applied. The velocity vinv

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5.2. CURVATURE 29 1100 1200 1300 1400 1500 1600 60 65 70 75 80 85 90 95 100 Distance [m] Velocity [km/h] Actual Velocity_v

brake vmax vforward vinv 1100 1200 1300 1400 1500 1600 −8 −6 −4 −2 0 2 4 Distance [m] Altitude

Figure 5.6. Optimal velocity trajectory for a steep downhill. The red (dotted) line depicts the posted road speed. The green (dotted) line depicts the velocity trajectory from inverse method. A blue (dash-dotted) line depicts the favourable braking strategy. The purple crosses depicts the forward method trajectory. A (solid) black line illustrates the resulting total velocity strategy produced by the switching controller.

The controller switches to the forward method after calculating the optimal point of control action, i.e. Smax. The optimal velocity is therefore indicated with vforward

(purple crosses) in Figure 5.6 Evidently the shortened deceleration stretch compels the vehicle to overshoot the final desired velocity. Thus, the optimal braking velocity trajectory (blue dashed line) is switched to in the end. The estimated optimal velocity is finally given by the black line.

5.2 Curvature

The grade and road speed reductions are not the only significant inputs to the switching controller shown in Figure 5.5. The curvature dictates a maximum speed. There are several ways to determine a maximum allowed speed through road curves. In this study, it is determined by setting a speed that is mandated by a comfortable centripetal force. Curve speed warning systems (CSWS) is a vast and current field of research [8]. The centripetal force along with the banking of the road is taken into consideration within many of those studies. There are scientific methods based on models of various complexity that determine what exact velocity is required for the HDV to tilt.

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5.2.1 Curve-Radius Estimation

The issue that arises in determining a maximum velocity through a curve is whether the radius of the curve can be obtained. There is a database table within ADASRP, which has a measured radius attribute for certain roads within Europe. However, the applicability of the data is questionable. The radius can vary significantly depending on the path chosen through the curve, i.e. middle or the side. It is known that a driver usually cuts through the curve to maintain a higher velocity throughout the curve. Therefore the geographically mapped road curvature is not the always actual travelled path. A method of estimating the road curvature can thereby be established.

The radius is estimated from a GPS-trace, which contains the actual positioning of the vehicle. Mind that there is an inaccuracy with GPS-positioning, but the error is handled through averaging that seems to be adequate. By estimating a best curve fit on five data points, a circle can be estimated. Gauss-Newton’s method is modified and applied in order to find the desired curve-radius in all GPS points. The modification is carried out in (5.3).

(x − xc)2+ (y − yc)2= R2 ⇒x2+ y2 _{= R}2_{+ 2x} cx + 2ycy − (x2c+ y2c) ⇒ " x2_{+ y}2 .. . # = " 1 x y .. . ... ... #    R2_{− (x} c2+ yc2) 2xc 2yc    (5.3)

The least square method is used to serve as a guess for the value of R. Due to the non-linear nature of this problem, the error of the estimate can further be reduced by the use of iteration. Gauss-Newton’s algorithm [2] is as follows:

• Write the non-linear equation on the standardform f (z) = 0.

• Determine the analytic expression for the elements in the Jacobian J(z) • Use reasonable values as starting points (in this case the values obtained in

the least square method above).

• (*) Calculate f and J with the measured zi-values, i.e. the five points obtained

in the GPS-map data.

• Solve the linear equation Jδz = −f • Update: z = z + δz

• Repeat (*) until desired accuracy is obtained.

Hence, the circle radius is presented as a minimization problem. The error of the curve fit is reduced iteratively by find using a least square estimation as an initial

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5.2. CURVATURE 31

solution to the minimization. Applying the method on data measured from a known test road at Scania, the result depicted in Figure 5.7 is obtained. The radii of various lengths are depicted as (red dashed-dotted) lines with the magnitude corresponding to the length of the radius.

−400 −200 0 200 400 600 800 1000 1200 1400 1600 −400 −200 0 200 400 600 800 1000 1200 1400 1600

Figure 5.7. Radii plotted on a known 10.3 km long test-road.

A straight road could be perceived as having an infinite radius. Therefore, the vehicle faces a rather long radius entering a curve.

900 950 1000 1050 1100 1150 1200 1250 1300 150 200 250 300 350 400 450 500 X−coordinate Y−coordinate

Figure 5.8. Radii plotted for a S-curve of the test-road.

The radii decreases as the vehicle proceeds through the curve and increases as it exists as shown in Figure 5.8. Slightly bending curves does not have any impact on

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the maximum speed limit constraints. Therefore, only radii of less than 700 meters are taken into consideration. Hence, the true curvature experienced by the vehicle is obtained through measuring the actual path travelled by the vehicle. Data is also collected from a test-run around an island of known radius. By comparing the calculated radius with the measured radius, the error is thereby estimated to be

±0.5m, which is an acceptable tolerance.

5.2.2 Maximum Curve Speed

For the purpose of this study a maximum speed is derived simply by determining a maximum centripetal force exerted on the driver. A threshold of 0.15g − 0.2g is determined in this case [12]. Thus a maximum velocity when travelling through a curve can be calculated according to equation 5.4.

0.2 = v 2_C g − E ⇒ vmax= r 0.2g C (5.4)

where g is the gravitational constant, C = 1

R is the curvature, and R is the

radius of the road. The road crossfall E is neglected due to lack of data. Hence an additional maximum speed limit is set on the road. The constraints set from the curvature is illustrated for a single curve in Figure 5.9.

2400 2500 2600 2700 2800 2900 3000 3100 3200 20 40 60 80 Distance [m] Velocity [km/h] 700 800 900 1000 1100 1200 1300 1400 1500 1600 −200 0 200 400 X−coordinate Y−coordinate

Figure 5.9. Radii plotted for a segment of the known 10.3 km long test-road.

Figure 5.9 shows the maximum comfortable velocity calculated by (5.4), for the curve segment depicted in the lower part of the figure. Unlike the mandated posted road speed, which instantaneously changes the maximum comfortable speed for entering a curve is smooth. Clearly, the speed only has to be reduced to ap-proximately 60km/h when entering the first curve. However, entering the narrower curve subsequently, a much lower speed is necessary not to feel discomfort. Hence,

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5.2. CURVATURE 33

applying (5.4) on the test-road depicted in Figure 5.7, the combined constraint of the mandated legal speed and the maximum curve velocity is depicted in Figure 5.10. 0 2000 4000 6000 8000 10000 12000 10 20 30 40 50 60 70 80 Distance [m] Velocity [km/h] vcurve vlegal

Figure 5.10. The maximum speed allowed on a given road inflicted by legal con-traints and curvature. A (solid) blue line depicts the speed mandated by the curva-ture. The (dotted) red line illustrates the posted road speed.

Figure 5.10 clearly shows that the original speed limit shown by the dashed red line, is highly altered by the constraints mandated by the road curvature. The new smooth velocity constraint implemented on the controller induces new problems upon the strategy of the switching controller’s forward method.

5.2.3 Method Strategy

The aim is to reach the maximum allowed velocity set by the shortest radius, i.e. the min-point, of every encountered curve on the given electronic horizon in an efficient and strategically sound manner. Naturally the switching controller will apply the same procedure as suggested in 5.1.5. First and foremost the inverse method is chosen if the deceleration stretch is shorter than Smax and the maximum

speed constraint set by the legal road speed in addition to the curvature constraint is not exceeded. However, if the afore mentioned criteria is not fulfilled, the forward method is switched to.

Approaching a curve, the objective is to be able to decrease the speed of the vehicle in time in an energy efficient manner so that it does not overshoot the maximum velocity set by the shortest radius.

Figure 5.11 shows an attempt to reach the desired final velocity through the inverse method. However, the maximum allowed velocity is set by the curvature is trans-gressed, creating an unacceptable velocity trajectory. The bound set by the legal road velocity and the curvature must never be exceeded. Therefore the minimum of

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34 CHAPTER 5. SWITCHING CONTROLLER 2600 2650 2700 2750 2800 2850 20 30 40 50 60 Distance [m] Velocity [km/h] vmax vinv 2600 2650 2700 2750 2800 2850 50 55 60 65 70 75 80 85 90 95 Distance [m] Altitude

Figure 5.11. Plot of the maximum speed allowed along with the inverse method velocity trajectory. The (dotted) red line illustrates the maximum allowed speed mandated by the curvature and posted road speed. The (dashed-dotted) green line depicts the velocity trajectory produced by the inverse method.

the velocity trajectory and the velocity constraint should always be chosen. On the other hand resorting to that option might create an undesirable fast decelerating trajectory as depicted in Figure 5.12

2550 2600 2650 2700 2750 2800 2850 25 30 35 40 45 50 55 60 Distance [m] Velocity [km/h] vbrake vmax vinv 2550 2600 2650 2700 2750 2800 2850 50 55 60 65 70 75 80 85 90 95 Distance [m] Altitude

Figure 5.12. Plot of the maximum speed allowed and the inverse method velocity trajectory, along with the comfortable deceleration (braking) trajectory. The (dotted) red line depicts the curvature and posted road speed. A (dashed-dotted) green line illustrates the velocity trajectory calculated by the inverse method. The (dashed-dotted) blue line illustrates a favourable braking trajectory.

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5.2. CURVATURE 35

line, dash-dotted line) as well as the inverse method trajectory (green, dash-dotted line) and the maximum allowed velocity (red, dotted line). The maximum allowed velocity is lower than the inverse method trajectory and should therefore be cho-sen at the 2770 meter marker in Figure 5.12. However, that would imply that the requested deceleration between the 2770 m and the 2790 m markers for the up-per limit would clearly be greater than the comfortable deceleration. Thus if the maximum allowed velocity is lower than the inverse method trajectory and it decel-erates uncomfortably fast, a suitable strategy is applied. The final desired velocity is reached by applying the inverse method to a point one discretisation step ahead and then keeping a constant velocity during the last part. Thereby the vehicle will still coast as long as possible and not exceed any of the constraints. It implies a safe, comfortable and economic vehicle behavior. Note that if the grade of the road induces a faster deceleration than the calculated comfortable deceleration speed, it is still considered to be intuitive, and therefore allowed.

In many cases a mandated speed reduction set by the curvature from e.g. 70 km/h to 20 km/h will result in an unacceptably long deceleration stretch Smax

presented in 5.1.2. An upper limit of 700 meters is therefore set on the maximum allowed deceleration stretch.

Hence, many large speed reductions will result in switching to the forward method. In that case the inverse method will no longer apply and can therefore not be shifted to resolve the problem of too fast deceleration due to the curvature con-straints. Thus, an appropriate braking strategy is formed. By calculating the slope of the mandated curvature speed, a new point of interest can be determined. The point where the slope of the upper limit exceeds the comfortable braking trajectory will thereby serve as a new point of reference to where a new comfortable braking trajectory will be calculated. The final desired speed will therefore be reached in advance and subsequently held until the original point of interest is reached. If the inverse method could not be applied in the case depicted in Figure 5.12 due to the topology, the intersection point at the 2770 meter marker would serve as a new reference point to which the braking velocity trajectory would be shifted.

As seen earlier in Figure 5.4 b), there might arise a situation where an action would be to decrease the velocity below the final velocity. The action adapted in that situation was to reduce Smaxto that point and continue the velocity trajectory

from that point onward. However, when the maximum velocity profile changes due to the dynamics entering a curve, the action of reducing Smax to that point is no

longer an effective choice in certain situations as illustrated in Figure 5.13.

The top plot in Figure 5.13 clearly shows that if the forward simulation starts just before 7000 m, the estimated velocity trajectory of the forward method (purple crosses) will drop below the final speed at approximately 7050 m. Hence, the forward method will shift Smax to that point and starts over, which is not the best solution.

A proper action in this case would be to shorten Smax an arbitrary fixed distance

and then starting the velocity trajectory from that point onward. Doing so will allow the forward method to eventually follow the more efficient trajectory (dotted-green line). That trajectory is considered to be a better solution, since the fuel

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Figure 5.13. Plot of the trajectory resulting from the forward method. The purple (crosses) depicts the velocity trajectory calculated by the forward method in each sampling point. A red (dotted) line illustrates the mandated maximum speed. The (dotted) green line illustrates a more fuel efficient alternative. The blue (dashed-dotted) line depicts the comfortable braking trajectory. The (solid) black line illus-trates the final produced velocity trajectory.

cut-off point arises earlier and the increase of speed is smaller from the last minima, i.e. both the fuel consumption and the brake wear is lowered.

It should also be noted that the forward method trajectory might drop below the final velocity on the flat part of the comfortable braking trajectory due to the road grade, as can be seen after the 7400 m marker in the top plot of Figure 5.13. If such situations arises, Smax should obviously not be shortened and the higher velocity

of the brake-trajectory should be the natural choice. Such a situation might also arise if the upper limit is chosen as a part of the optimal trajectory and not deemed to create an unacceptable deceleration. An incline in the road might then create a larger deceleration, which would in turn create a trajectory from that forward method that drops beneath the final desired velocity. Smax should not be reduced

in such a situation. Instead, the braking trajectory should be maintained.

Figure 5.14 shows the importance of separating the three cases presented in 5.1.1. In this case the forward method is applied, since the inverse method is not applicable. The switching controller is always calculating suitable strategies to undertake in each discritization step. Before the velocity trajectory drops beneath the tolerance set on the final desired velocity at the 9110 meter marker, the road grade creates a higher deceleration than the comfortable braking trajectory, making the coasting trajectory an optimal choice for the braking strategy. Evidently the controller switches to the braking method and consequently makes an unacceptable instantaneous increase in velocity of approximately 30 km/h. Such behaviour is obviously not tolerated since

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5.2. CURVATURE 37

Figure 5.14. Plot of the trajectory resulting from a controller misunderstanding. The purple (crosses) depicts the velocity trajectory calculated by the forward method in each sampling point. A red (dotted) line illustrates the mandated maximum speed. The blue (dashed-dotted) line depicts the comfortable braking trajectory. The (solid) black line illustrates the final produced velocity trajectory.

it creates a physically impossible velocity request. A misunderstanding of which method, i.e. inverse, forward, or braking, has priority, can lead to an unacceptable velocity output. Therefore, the current active method strategies are must be -isolated and prioritised from other methods.

Hence, the controller first have to make a choice of whether to choose the inverse method velocity profile or the forward method depending on the predicted decel-eration distance for every legal speed reduction and every maximum allowed curve velocity set by the minimum radius within that curve. It then has to decide how far the vehicle can coast before activating proper braking action. If the road grade dic-tates that the vehicle must lower its speed below the final desired velocity along the calculated velocity trajectory, the deceleration stretch must be reduced and a new optimal trajectory must be calculated from a new reference point. The switching controller constantly compares suitable strategies and chooses the most appropriate one. If the topology compels the controller to switch to the braking method, several new strategies discussed above are investigated. However, it is imperative that the method strategies are separated as discussed above.

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