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Optimization Tool for Gear Shift Strategy Control Design DAVID HULTMAN Master of Science Thesis Stockholm, Sweden 2014 Optimization Tool for Gear Shift Strategy Control Design DAVID HULTMAN Master’s Thesis in Systems Engineering (30 ECTS credits) Master Programme in Aerospace Engineering (120 credits) Royal Institute of Technology year 2014 Supervisor at Volvo construction equipment was Björn Brattberg Supervisor at KTH was Per Engvist Examiner was Per Engvist TRITA-MAT-E 2014:63 ISRN-KTH/MAT/E--14/63--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kthse/sci Abstract Modern construction vehicles are required to perform very well in tough conditions. It is therefore of great importance that they have well developed gear shift strategies to manage with these requirements. These strategies are updated often and they are normally based on empirical knowledge and extensive testing. It is of interest to know if

software tools can be used to provide theoretical analysis to further modernize and improve the gear shift strategy design process. This thesis investigates this possibility by creating an optimization tool that can provide various optimized simulations. These simulations finds the theoretical optimal velocities and gear shifts for driving specific distances. It is done by creating a vehicle model that is used to form an optimal control problem. This optimal control problem uses a dynamical programming algorithm as a solver and the results are optimized simulations. The optimization tool has the possibility to investigate ways to minimize travel time, fuel consumption and durability losses. It is concluded that there are ways to drive that can reduce the fuel consumption and durability losses by a large amount while not affecting the travel time very much. ii Sammanfattning Moderna byggfordon har höga krav på att prestera väl i tuffa miljöer. Det är därför viktigt

att de har väl utvecklade växlingsstrategier så att dessa krav kan uppfyllas. Strategierna är uppdaterade ofta och bygger normalt på empirisk kunskap samt noggranna tester. Det är utav intresse att undersöka om mjukvaruverktyg kan användas för att göra teoretiska analyser som kan modernisera och förbättra utvecklingsprocessen utav växlingsstrategier. Det här arbetet undersöker den här möjligheten genom att skapa ett optimeringsverktyg som kan bidra med diverse optimerade simuleringar. Dessa simuleringar görs genom att hitta de teoretiskt bästa hastighetsprofilerna samt växlingar för att köra specifika sträckor. Detta möjliggörs genom skapandet utav en fordonsmodell som sedan används till att formulera ett optimeringsproblem. Optimeringsproblemet löses genom en dynamisk programmeringsalgoritm. Det resulterande optimeringsverktyget kan undersöka sätt att minimera körtid, bränsleförbrukning samt hållbarhetsförluster. Det

visas att det finns sätt att köra som kan minska bränsleförbrukningen och hållbarhetsförlusterna mycket medans körtiden bara ökas lite. iii Acknowledgements This work has been performed at Volvo Construction Equipment in Eskilstuna. I have gotten extensive support from several individuals here and I have made some great friends. I am very grateful for all the help I have received and the friendly environment that exists here. Firstly I would like to thank my supervisor at Volvo, Björn Brattberg for always providing the assistance I have needed. I would like to thank Anders Fröberg for the invaluable tips and feedback and for providing me with the opportunity to come here. Others that have helped me and deserve a thanks are Anton Hugo, David Berggren and Mats Åkerblom. A thanks goes out to the entire department at Controls Driveline and everyone else located at the floor for making me feel so welcome. I am also very grateful from the help I have gotten from

KTH through the work of my supervisor there, Per Enqvist. I would also like to thank my sisters Marta and Natalia for their support, and lastly, Magdalena Sierant for brightening my days. David Hultman Eskilstuna, October 2014 iv Contents 1 Introduction 1.1 2 Outline of the thesis . 2 The Construction Equipment 3 4 2.1 Articulated Haulers . 4 2.2 Wheel Loaders . 5 2.3 Motor Graders . 7 3 Modeling 3.1 3.2 8 Powertrain Modeling . 8 3.11 The Engine . 8 3.12 The Torque Converter . 10 3.13 The Gearbox . 13 3.14 Gear Shifts . 13 3.15 Total Powertrain Ratio and Inertia . 14 3.16 The Wheels . 14 3.17 Resisting Forces . 14 3.18 Longitudinal motion .

15 Criterion Modeling . 16 3.21 Travel Time . 16 3.22 Fuel Consumption . 16 3.23 Durability Losses . 18 3.231 18 Fatigue of rolling bearings . v 3.232 Clutch Wear . 19 3.233 Total Durability Losses . 19 4 The Optimal Control Problem 4.1 4.2 20 Problem Formulation . 20 4.11 Designing and choosing the weight and scale parameters . 21 Dynamical Programming . 22 4.21 The Inverse Method . 23 4.22 Computational Complexity . 24 4.23 State Space Representation . 25 4.24 Limiting the Search Space . 26 5 Simulations and Analysis 5.1 5.2 5.3 27 Model verification . 27 5.11 Gear Shift Model Analysis .

28 Optimized Acceleration From Standstill . 30 5.21 Minimizing Travel Time . 31 5.22 Minimizing Fuel Consumption and Durability Losses . 32 Pareto Fronts of Criteria Trade-off’s . 34 5.31 Travel Time versus Fuel Consumption . 34 5.32 Varying All Criteria Weights . 36 6 Conclusions 37 6.1 Summary . 37 6.2 Discussion . 38 6.3 Future Work . 38 7 References 40 1 Chapter 1 Introduction Creating intelligent gear shift schedules is of great importance to the design of any vehicle using an automatic gearbox. It is a key part in generating desired and efficient performance, and much effort is put into finding the best gear shifting strategies These strategies highly affect important properties like fuel consumption and travel time, but also some less noticeable ones, like gearbox

lifetime. The conventional way to implement such strategy schedules into the automatic gearboxes is to do it through the use of shift maps. These maps consist of a schedule of points which tells the transmission when to shift up or down based mainly on the velocity and power output of the vehicle. The traditional way to create these is to do it based on empirical knowledge and extensive testing [1]. Passenger cars often have 3 different gear shifting maps, corresponding to the balanced, eco and sport modes. Off-road construction vehicles which drive in a more complex way in unpredictable terrain can have up to as many as 10 different maps. Research is being done in several fields to find ways to improve and optimize the gear shifting strategies. It is often focused towards passenger cars or long-range trucks with focus on reducing fuel consumption. Previous work includes [2], where a Dynamical Programming (DP) algorithm is developed to numerically solve optimization problems on fuel

consumption in trucks. A similar algorithm is used in [1] to create a gear shift schedule for a passenger car that results in lowering its fuel consumption by up to 11.2% In [3], the concept is further developed to take into consideration 2 not only fuel consumption, but also driveability. This thesis will look into the possibility of using a DP algorithm as help in the design process of creating gear shifting maps for off-road construction vehicles. A software tool will be developed that can find the theoretically optimal ways to drive the vehicles over specific distances and terrain based on certain criteria. The criteria will cover not only fuel consumption and travel time, but also durability losses of the gearbox and other components. Extending the lifetime of different vehicle parts is something that is of high interest for both manufacturers and users, but is often disregarded when designing gearshift maps. In this work, a model for durability losses is implemented This

gives the possibility to analyse the impact that different drive styles have on vehicle lifetime. The vehicles covered in the scope of this thesis are articulated haulers, wheel loaders and motor graders. 1.1 Outline of the thesis The chapters of this report are organized as following: Chapter 2 introduces the three different construction vehicles which are covered in the scope of this work and describes their areas of usage. It is followed by a chapter where a generalized vehicle model is created. This model is made to be able to simulate all three different vehicle types described previously. The way in which the different optimization criteria are modeled is also presented. In Chapter 4, an optimal control problem is formed and the dynamical programming algorithm that is used as a solver is described. This control problem uses the model created in the precedent chapter and is able to simulate various different ways to drive the construction vehicles in an optimal way. A

description of the numerical complexity is also given. In Chapter 5 various example simulations are presented and some analysis is done. A model verification is made to show how well the model of this work compares to previous test data. This is followed by the last chapter which cover conclusions and recommendations for future work. 3 Chapter 2 The Construction Equipment The optimization tool which is created in this thesis is applicable to 3 different construction equipment vehicles: haulers, loaders and graders. Their work tasks and objectives differ from each other and this chapter will give a description of each of the vehicle types. 2.1 Articulated Haulers A hauler is similar to a dump truck, but is designed to be driven off-road over rough terrain. Its objective is to transport material from one point to another and its most common loads include gravel, stone and soil. The vehicle consists out of two parts, the front tractor including the driver cab and the back trailer

with the dump body. The two parts are steered relative to each other by articulation, providing good path-tracking which is desired in off-road driving. 4 Figure 2.1: A Volvo A40F Articulated Hauler 2.2 Wheel Loaders The objective of a wheel loader is mainly to load material. See Figure 23 for an illustrated scheme of one of the most common drive cycles: the short loading cycle. The wheel loader can be equipped with a wide range of attachments including buckets, forks, grapples and more. The most common material it loads include gravel, stone and soil, but it can also be used for timber and much more. A wheel loader often loads material into a hauler which then transports it. 5 Figure 2.2: A Volvo L250H Wheel Loader Figure 2.3: A short loading cycle, picture from [4] 6 Figure 2.4: A Volvo G940B Grader 2.3 Motor Graders A grader is a type of vehicle which is used mainly for creating gravel roads, preparing the ground before creating an asphalt road or, in some

cases, for snow ploughing and similar activities. Some variations in equipment exist, but most commonly it has one large blade which it uses to grade the road. 7 Chapter 3 Modeling In this chapter, a mathematical model of the necessary parts needed to derive the longitudinal motion of the vehicles is created. This is equal to modeling a powertrain and the external forces acting on it. The resulting equation of the longitudinal motion is what will be used to simulate the driving of the vehicles and it is created with a level of detail which will suit the complexity of the optimization problem. The modeling process will be similar to that described in [5] with addition to a torque converter. The torque converter equations will be created like in [6] and [7] together with experimental data taken from Volvo CE. 3.1 Powertrain Modeling A Powertrain is a combination of all the parts that are used to generate power and transfer it to the ground. It consists of the engine, shafts,

torque converter, gearbox, axles and the wheels An illustration of the layout of a powertrain is show in Figure 3.1 3.11 The Engine A combustion process inside the engine consumes fuel and generates torque to an outgoing shaft. This generated torque is dependent on the current angular rotation 8 Figure 3.1: A Powertrain speed of the engine, ωe , and the amount of fuel combusted, uf . It is assumed that at any given point, any desired torque level can be reached within the limitations of the engine by adding the related amount of fuel. Transient behaviour is disregarded This allows the outgoing engine torque to be described as Te = f (ωe , uf ) (3.1) A typical map of the maximum torque available as a function of engine speed is shown in Figure 3.2 9 Figure 3.2: An example of engine performance 3.12 The Torque Converter A torque converter (TC) is a type of fluid coupling with the ability to multiply torque. It is positioned between the engine and the gearbox and it is

used in vehicles with automatic transmissions as a replacement of the traditional clutch used for manual transmissions. Where such a clutch normally has friction plates to connect the shaft going out from the engine to the shaft going in to the gearbox, a TC instead connects the two through a hydraulic fluid. It consists of a container of oil where both shafts meet and transfer torque through the fluid. The engine shaft going into the TC is connected to an impeller which makes the oil rotate. The shaft going out of the TC, towards the gearbox, is connected to a turbine that gains a rotation because of the 10 movement of the oil. This construction allows the two shafts to have different rotation speeds and therefore prevents engine stalls at low vehicle speeds. It removes the need of disconnecting the engine and the transmission every time the vehicle stops. The speed difference is defined as Φ= ωT C ωe (3.2) where ωT C is the rotation speed of the shaft going out of the TC.

Inside the TC, there is a stator which ensures that the flow of the returning fluid from the turbine is directed into the same direction as the impeller is rotating. This ensures that the returning fluid does not slow down the impeller and cause energy losses. Another effect of this is that at a large difference in rotation speeds of the two shafts, the transferred torque is multiplied. As described in [6], the speed difference Φ plays a big role in how much torque is multiplied. The converter output torque TT C is determined by the relation TT C = ξ(Φ) · Te (3.3) where ξ is a function that must be experimentally calculated for each converter and will be modeled using data from Volvo CE. A typical graph of how much torque is being multiplied as a function of Φ is shown in Figure 3.3 11 Figure 3.3: An example of torque converter performance The TC has a built-in lockup clutch system which gives the possibility of locking the two rotating shafts together into a fixed one

and transfer the torque directly instead of through the fluid. This is because in some cases this gives a better efficiency In general the TC is most useful at low vehicle speeds and in situations where frequent stops and starts occur. At higher vehicle speeds the two rotating shafts reach almost the same speed and then it is often preferred to lock them together to remove slippage and additional losses caused by the TC. Lockup is engaged at different situations in different vehicles. Some engage it directly after start and some never do it The lockup will be modeled as a simple switch which can change between leading 12 the power through the TC or through a locked shaft.   T =T if Lockup is engaged TC e  TT C = ξ(Φ) · Te if Lockup is disengaged 3.13 The Gearbox The next component in the powertrain is the gearbox. It consists of a number of gears of different sizes and enables a discrete choice of the ratio between ingoing and outgoing angular velocity of the shafts

connected to it. This allows the speed conversion between the engine shaft and the wheel axles to be controlled, making it possible to provide more accurately desired torque values to the wheels at a wide range of vehicle velocities. The gearbox is modeled as a discrete choice of the gear ratio igear (g), where g is the current gear. The corresponding torque and angular rotation speed equations are given by TGB = igear (g) · TT C ωGB = ωT C igear (g) (3.4) (3.5) where TGB and ωGB are the output torque and angular rotation speed from the gearbox. 3.14 Gear Shifts The process of changing the gear ratio is done during a gearshift. This is traditionally done by disconnecting the gearbox axle at a mechanical clutch and then connecting it again when the gear ratio has been changed. Such a disconnection creates an undesired drop in the transferred torque and in modern automatic gearboxes a so called powershift technique is used instead. This way of shifting is done in a way in which

the current gear is being disconnected at the same time as the new one is connected. This is a more efficient technique than to completely disconnect the axles while shifting and produces a smaller reduction of the transferred torque. In this work, the shift process will be modeled as a one second time period during 13 which the transferred torque is reduced by 50%. At the end of the one second period, the gear ratio is changed. The value of 50% was chosen after comparing different ones in a gear box model test, in which the one of 50% was shown to give closest results to that of existing test data. The results of this test is shown in section 511 3.15 Total Powertrain Ratio and Inertia Besides the gearbox and the TC, some other parts in the powertrain also have a ratio of the transferred torque and angular rotation speed. These are fixed and are modeled simply as a single total ratio between the gearbox and the wheels. It will here be denoted iaxle . The corresponding equations

become Tw = iaxle · TGB ωGB iaxle where Tw and ωw are the wheel torque and angular velocity. ωw = (3.6) (3.7) The inertias of the powertrain are modeled into two parameters. One is the engine inertia Je and one is a lumped inertia of the other components combined, denoted Jw . Jw is the sum of the inertias of the wheels, the shafts and the gearbox components. 3.16 The Wheels Slippage is disregarded and the wheels are modeled with a constant effective radius rw . The velocity of the vehicle then becomes v = ωw · rw 3.17 (3.8) Resisting Forces The external forces acting on the vehicles in the longitudinal direction are air resistance, Fa , rolling resistance, Fr , and gravity, Fg [8]. Air resistance is estimated as 1 Fa (v 2 ) = cw Av 2 2 14 (3.9) Where cw is the air drag coefficient of the vehicle and A is the cross section area of the vehicle front. Rolling resistance comes from the fact that the rubber wheels of a vehicle are elastic and get slightly deformed at

the contact surface with the road. This creates some resistance which can be modeled as Fr = cr mg cos(α) (3.10) where cr is the rolling resistance coefficient, m the vehicle mass, g the gravitational acceleration and α the road slope. The resisting force that comes from the gravitational forces is Fg = mg sin(α) (3.11) The expression for the total resisting forces, Fres , becomes Fres = Fa + Fr + Fg 3.18 (3.12) Longitudinal motion The motion of the vehicles is derived from Newton’s second law. The acceleration is described by m dv = Fw − Fres dt (3.13) where Fw is the driving force on the wheels and Fres = Fa + Fr + Fg is the sum of the resisting forces. The dynamics of the wheels are Jw ω˙w = Tw − Tb − rw Fw (3.14) where Tb is the brake torque. The dynamics of the engine are Je ω˙e = Te − TT C 15 (3.15) By putting together equations 3.3 - 314 an expression for the acceleration can be determined to be rw dv (itot (g)ξ(Φ)Te (ωe , uf ) − Tb (ub )

− rw (Fres )) = 2 dt Jw + mrw + itot (g)2 Je (3.16) where itot (g) = igear (g) · iaxle and Tb (ub ) is the brake torque depending in the brake control signal ub . With lockup engaged it becomes dv rw = (itot (g)Te (ωe , uf ) − Tb (ub ) − rw (Fres )) 2 dt Jw + mrw + itot (g)2 Je 3.2 (3.17) Criterion Modeling The optimal control problem which is formed in Chapter 4 is based on minimizing the following 3 criteria: travel time, fuel consumption and durability losses. These are modeled in this section. The control problem uses a discretization in step length of the distance travelled and all the criteria costs are calculated at each step. The total number of steps is denoted as N and the step index as k. 3.21 Travel Time The time it takes to drive a certain step distance ds is calculated as t(k) = ds v(k) (3.18) where v(k) is the vehicle velocity at that step. The travel time for the entire distance is the sum of the time for each step ttot = N X t(k) (3.19) k=1

3.22 Fuel Consumption The amount of fuel required to drive the vehicles is calculated from a fuel map which is built up from test data from Volvo CE. At each step, the fuel consumption Mf (k), 16 Figure 3.4: An example of a fuel consumption map is extracted from the map based on engine rotation speed and the generated torque. It is measured in liters consumed for step k. Mf (k) = f (ωw , Te ) (3.20) The amount of fuel used for the entire distance is summed up as Mf,tot = N X k=1 A typical fuel map is shown in figure 3.4 17 Mf (k) (3.21) 3.23 Durability Losses The method used to estimate the durability loss in each simulation assumes that there are only two causes of losses: fatigue of rotating parts, and a discrete loss each time a clutch is opened and closed during a gear shift. All other failure modes are disregarded. Engine durability loss is also disregarded and the modelling will approximate the rest of the driveline losses to only occur in the gearbox and the

rolling bearings at its ingoing and outgoing shafts. The losses will be measured in the estimated price in SEK that the wear has costed. 3.231 Fatigue of rolling bearings The estimation of durability losses of the two rolling bearings are based on the SKF formula for rolling bearing life [12], which states that their lifetime L is inversely proportional to their stress as 1 ∝ ωT 10/3 (3.22) L To be able to compare different simulations to eachother, the durability loss of each rolling bearing will be measured as a duty value D for each step as D(k) = t(k)ωT 10/3 (3.23) This gives an estimate of how much damage the rolling bearing has taken at each step. The concept is further developed to estimate the price cost of the wear. It is done by taking the price of a gearbox and dividing it by the average amount of duty value that the rolling bearings reach in their lifetime. This associates a unit of duty value to a price κ in SEK . duty It is then possible to estimate the price

of the wear of each bearing at every step as Pbearing (k) = κ · D(k) (3.24) For the two different bearings, the prices become 10/3 Pbearing,in (k) = κ · t(k)ωT C TT C 18 (3.25) 10/3 Pbearing,out (k) = κ · t(k)ωGB TGB 3.232 (3.26) Clutch Wear Each time a clutch is opened and closed during a gear shift, some durability loss occurs. The magnitude of the loss is related to the amount of torque that is being transferred at the moment and also to the rotation speed difference of the two parts. A large torque and a large speed difference give a large loss and vice versa. The clutch wear will here be modelled as a fixed price for making a gear shift. The price will be set to an estimated average calculated as the price of a clutch divided by the average amount of gear shifts in a clutch lifetime, denoted Pgearshif t . 3.233 Total Durability Losses The total durability loss at a step k is calculated as the price of the bearing wear plus the gear shift price times a help

variable gs(k) that tells if a gear shift has occurred or not. where P (k) = Pbearing,in (k) + Pbearing,out (k) + Pgearshif t · gs(k) (3.27)   1 if a gear shift has occured at step k gs(k) =  0 otherwise (3.28) The total price of the wear caused during the entire distance is summed up as Ptot = N X k=1 19 P (k) (3.29) Chapter 4 The Optimal Control Problem This chapter covers the formulation of the optimal control problem, which is the core of the optimization tool that is being created. The objective function which is the target to minimize and optimize is presented, and the dynamical programming algorithm used as a solver is analysed. The problem complexity is discussed and methods used to lower the computation time are explained. 4.1 Problem Formulation The objective of the optimization tool is to be able to run calculations that will result in data of the optimal way to drive over a given distance. Optimal refers to the ’best’ way to drive, and the

definition of that will in this work be decided by the 3 critera mentioned earlier: travel time, fuel consumption and durability losses. Having 3 criteria to minimize leads to a multiobjective optimization problem and to formulate the objective function, the so called scalarization technique and a weighting method will be used. The concept of this method is to summarize the values of each criteria cost function into one scalar. Since it can be desirable to prioritize the various criteria differently, each of them is multiplied by a weight and scale factor qi . The objective function cost C is then calculated as C = q1 ttot + q2 Mf,tot + q3 Ptot 20 (4.1) where ttot , Mf,tot and Ptot are calculated as in Equations (3.19), (321) and (329) The optimal control problem is to minimize this cost while obeying the limitations of the vehicle dynamics modeled in Equation (3.16) The problem can be formulated as minimize q1 ttot + q2 Mf,tot + q3 Ptot subject to dv dt = rw 2 2 +i Jw +mrw

tot (g) Je (itot (g)Te (ωe , uf ) − Tb (ub ) − rw (Fres )) Te ≤ Te,max (4.2) ωe ≤ ωe,max Tb ≤ Tb,max g = {1, ., gmax } 4.11 Designing and choosing the weight and scale parameters The weight and scale parameters qi can be designed in various different ways and opinions on which is the best way differ. This is discussed in [9] and there, several approaches are presented. It is clear that the parameters should be scaled to get an easier overview of the problem and to create unit-less costs that makes more sense to summarize. A suggestion is made that each criteria should be scaled with its ideal value. This would put the ranges of the costs to between 0 and 1, but is only applicable to problems with non-zero ideal cost values, which is not the case here. Instead, the approach used in this problem will be to scale each criteria with their average values. These average values are calculated based on driving a step with cruising speed of the particular vehicle. qi = wi

qi,scale (4.3) Here, wi are weighting parameters which decide how much each of the criteria will be prioritized. They will be normalized to give a sum of 1 X wi = 1 (4.4) The interpretation of wi is that it is the relative importance of the criteria it is associated with. While qi,scale are fixed in the optimization tool, wi are allowed to be 21 changed. By changing these weights, the priority of the criteria functions are changed and the optimization will result in a different solution of the ’best’ way to drive. This opens up the possibility to analyse many different drive styles. 4.2 Dynamical Programming The algorithm which will be used to solve the optimal control problem (4.2) is dynamical programming It breaks down the problem into subproblems and is suitable for finding the optimal solution of a complex computation. The algorithm follows the principle of optimality which states that for any initial state and decision, each following decision must obey an optimal

policy with regard to the initial state. [10] For dynamical programming to be applicable to (4.2), the equations and simulations will first be further discretized and re-written into a deterministic multi-stage decision problem. This is done by dividing the driving distance into N steps with fixed length, (the step index will be denoted k), and also quantizing the allowed vehicle velocities and engine rotation speeds into a state matrix X(k) together with the already quantified gear choices. The goal now becomes to find a series of decisions that takes the inital state x0 through each step k to the end state xN via an optimal path. The method of the algorithm is to go backwards from the last state while calculating the cost to take each step in every possible way. The running-cost for taking the step k is denoted ξk (x, u) and is based on the objective function cost (4.1) The costs of each passed step are then added together and the lowest cost from the current point, denoted Jk (x),

is stored. This creates subproblems at each grid point in the state matrices and the lowest cost of driving through the entire distance, which will be J0 (x0 ), is found when the algorithm has gone through all steps and reaches the initial state. The standard dynamical programming algorithm as stated as in [11] is 1. Let JN (xN ) = 0 2. Let k = N − 1 22 3. Let Jk (x) = minu∈Uk {ξk (x, u) + Jk+1 (C(x, u))} x ∈ Xk 4. Repeat (3) for k = N − 2, N − 3, , 0 5. The optimal cost is J0 and the sought control vector is the optimal control from the initial state to the final state Here, u is the control signal between the states and Uk is the set of allowed, dizcretized controls. With this standard approach it is not certain that the discrete controls will make the states transition exactly into other grid points in the state matrices, creating the need for interpolations to be made. This interpolation need and the fact that the number of calculations increases exponentially with

the size of Uk has made an alternative method more preferable. In this work, the so called inverse method which is presented in [2] is used. 4.21 The Inverse Method In this method, instead of going through all possible controls in Uk and interpolating the results, only the required control to exactly transition between each grid point in Xk to each point Xk+1 are calculated. If the running cost to make the transition between xi ∈ Xk and xj ∈ Xk+1 , which is denoted ξki,j , is found to be not feasible, the cost will be set to infinity. By using this inverse method, the algorithm becomes 1. Let JN (x) = 0 2. Let k = N − 1 (i,j) 3. Let Jk (xi ) = minxj ∈Xk+1 {ξk + Jk+1 (xj )} xi ∈ Xk 4. Repeat (3) for k = N − 2, N − 3, , 0 5. The optimal cost is J0 and the sought control vector is the optimal control from the initial state to the final state 23 in Figure 4.1 an illustration of the algorithm is made where the discretized driving distance is shown with the state

matrices at each step. Figure 4.1: An illustration of the algorithm 4.22 Computational Complexity The total number of calculations that is performed using the dynamical programming with the inverse method will be 2 Q = N KX (4.5) where KX is the number of gridpoints inside the state matrix X. This is the result of that at each of the N steps, the trajectory from each state in Xi to each state in Xj is computed. Therefore the number of calculations increase quadratically with the number of gridpoints in the state matrix, and linearly with the number of steps. 24 4.23 State Space Representation 2 states, (vehicle velocity and gear choice), are required to do the optimizations while driving with the torque converter lockup engaged, and 3 states (vehicle velocity, gear choice and engine rotation speed) are required if the torque converter lockup is disengaged. (Generally, only the computations of the wheel loaders require the use of 3 states, the hauler and grader computations

will most often only require 2 states). Finding a suitable number of gridpoints when quantifying the state space is of high importance. Because the algorithm only finds trajectories from point-to-point, it is important to have a dense enough space with gridpoints as close to eachother as possible to get realistic results. The drawback of having many points is that the computation time increases quadratically with the number of points, therefore picking a number that provides good results while not taking unnecessarily long time to compute is important. The number of gears is already quantized. For most of the gearboxes this is 9 and will be representing one dimension in the state matrix. Quantizing the velocity and engine rotation speed is more complex and a priori knowledge of the drive pattern currently being analyzed is very helpful. If for example a hauler driving on a flat road is being analyzed, the velocity range will be large and probably reach values between 0 and 50 km/h.

Therefore a large number of points is needed to get a dense space and somewhere around 200 steps in the velocity dimension has been tested and found suitable. Because of the locked up torque converter, only two states are needed and the final amount of gridpoints in the state matrix in this example is 9 × 200. If a wheel loader is being analyzed, 3 states are required. A priori knowledge will say that the velocity probably does not reach more than 25 km/h, making a suitable quantification of the velocity and engine rotation speed dimensions to have 100 steps each. This makes the amount of gridpoints in this example to be chosen to 9 × 100 × 100 25 4.24 Limiting the Search Space The calculation time can easily become quite long, especially in the cases where 3 states are being used. Therefore some further methods to limit the computational complexity can be used. One way that will be used is to limit the so called search space. The search space is the amount of points in the

to-go state matrix Xk+1 that trajectories are being calculated to. Normally this covers all of the state matrix at that step, but a priori knowledge tells that many of these are not feasible and can be skipped beforehand. One of the largest limitations to make is to only compute trajectories to the reachable velocity points. This is done by checking the possible accelerations dv ds for the current state xi ∈ Xk and see which states xj ∈ Xk+1 that the maximum and minimum acceleration can reach, then limit the calculations to cover only gridpoints in between these values. A visualization for this is shown in Figure 42 where only the points between the maximum and minimum velocity change dv are allowed. Figure 4.2: Search space limitation 26 Chapter 5 Simulations and Analysis This chapter will present various results of analysis made about the optimization tool as well showing example simulations. A model verification is made where the vehicle model in the optimization tool

is compared with existing test data to see how well it simulates a real vehicle, different example optimizations are presented on various ways to drive a hauler and an analysis is made about what happens if the weights of the different optimization criteria are varied. Pareto fronts are used to visualize the results of the optimizations with varied weights. (A pareto front is a collection of optimal points found from making many simulations with varying criteria weights.) 5.1 Model verification To check that the vehicle model that is created in section 3.1 and used in the optimal control problem behaves in a realistic way, a comparison is made with a Volvo CE in-house Simulink model which is known to behave similarly to a real vehicle. The case that is tested is to accelerate a hauler from standstill for 200m on a flat road with full throttle. The simulations are all made with an unloaded hauler with a mass of 30 tonnes. The optimal control model is in the comparison limited to make

the same gear choices as the simulink model and it is set to prioritize only travel time. This means that the weight associated to travel time is set to 1 while the other weights 27 are set to 0. The result is shown in figure 51 It is seen that the two models have very similar results, with the only exception being that the simulink model has a slightly bigger loss of vehicle speed at the gearshift between gear 7 and 8. This is due to complicated behavior of the mechanical components in the planetary gearbox that occurs in real life but which is not modelled in the optimal control model. Figure 5.1: A comparison between the optimal control model and a Volvo CE in-house simulink simulation 5.11 Gear Shift Model Analysis As described in section 3.14 the gearshift is modeled as a 1 second time period during which the transferred torque is reduced by 50%. To find this value, some different gear shift models are tested in the same driving case as in section 5.1 where an unloaded

hauler is accelerated on flat ground. 3 different shift models is tested with the transferred torque being 50%, 75% and 100%. The results are shown in Figure 52 28 where it can be seen that the simulation with the model with 50% transferred torque comes closest to the existing test data and therefore best represents a real vehicle. Figure 5.2: Gearshift model analysis A 2D-pareto front is also created for each of the different gear shift models, where the two critera, fuel consumption and travel time, are compared. There, the weighting parameter for the travel time in the optimization, w1 , is varied from 1 to 0 and the weighting parameter for the fuel consumption, w2 is varied from 0 to 1. The weighting parameter for durability losses ,w3 , is kept at 0. This means that the cases that are tested are first with full prioritization of travel time, then this is gradually shifted towards fuel consumption and in the last case, the fuel consumption has full prioritization. A point is

plotted with the value of the optimal time taken and fuel used for each of the simulations. This creates a front of points with the optimal values for different weights. They are connected by a line for better visualization and this is repeated for each of the three different gear shift models, resulting in three different 29 fronts. They are simulated in a case of full acceleration from standstill for 60 meters and the results are shown in figure 5.3 Figure 5.3: Pareto fronts for different gear shifting models What can be concluded from this is that the different gear shift models tested give simulation results that vary by up to about 9% (for the point where only travel time is prioritized). This means that it is important to choose a model that is very similar to the test data. The model that best does this is the one with 50% transferred torque and is the one that is chosen. 5.2 Optimized Acceleration From Standstill In this section, some further tests are made on the

same case as previous, where a a hauler accelerates from standstill. The best ways to drive and to shift gears are investigated based on different weights on the criteria. The optimal control model and its dynamical programming optimization is from now on allowed to freely start 30 at any gear and make any gear shifts possible. It is not limited to make the same gear shifts as the Volvo CE Simulink model. 5.21 Minimizing Travel Time To find the optimized way to accelerate based on only minimizing the travel time, the weights are kept at 1 for the travel time and 0 for fuel consumption and durability losses. With the model now being free to take any gear shifts, the theoretically fastest way to accelerate based on this model is found. In Figure 5.4 the results are shown, plotted against the Simulink model and the limited optimal control model from the previous section. Figure 5.4: Acceleration from standstill What can be concluded from this simulation is that the optimal way

to shift in this specific situation, according to the optimization algorithm, is to start at the fourth 31 gear and then go to sixth, eighth and finally ninth gear. This gives the simulation with the fastest acceleration and it shows that for an empty hauler as used in this case, taking fewer gear shift steps and skipping more gears gives a better result. Skipping gears in this way is often the best case for any vehicle that has a powerful engine and is unloaded. The same case was also tested for a fully loaded hauler that in total has a mass of 60 tonnes which is twice as much as an empty one. The results of this simulation is shown in 5.22 Minimizing Fuel Consumption and Durability Losses In this section, two simulations are made to investigate the optimal ways to drive when also taking into consideration either fuel consumption or durability losses. The case is still to accelerate a hauler from standstill and the results show the differences in how to drive to save fuel

compared to saving durability. While making tests on these other criteria, it is of interest to not only find the best gear shifts, but to also investigate the engine torque and engine RPM. When minimizing travel time it is quite straight forward that the optimal way to drive is to always use the maximum available power and engine torque, but when considering the other criteria it is not obvious how much power should be used. To investigate this, the optimal engine torque and engine RPM are also plotted in these tests. In Figure 5.5 the two new simulations are plotted together with the previous one which prioritizes only travel time. The two new ones consist of one with the weights being 0.85 for travel time and 015 for fuel consumption, and one with the weights being 0.85 for travel time and 015 for durability losses In Table 51 the values of the total time, fuel consumption and durability losses for the three different simulations are listed. 32 Figure 5.5: Acceleration from

standstill The results show that the optimal engine torque and optimal engine RPM differ a lot between the two simulations. It is seen that to save fuel, the engine should be driven with low RPM and high torque. To save durability, the engine should be driven with high RPM and low torque. The reason why it is good to drive in such a way to save durability is because the durability losses of the rolling bearings are high when a high torque is used and therefore a low torque is preferred. This is the opposite strategy compared to saving fuel where instead high torques are preferred due to the engines being more fuel efficient in those situations. 33 Criteria weights Time Taken [s] Fuel Cons. [l] Dur. Loss [SEK] 100% Time 20.22 0.3816 0.7126 85% Time 15% Fuel Cons. 20.62 0.3083 0.2051 85% Time 15% Dur. Loss 21.00 0.3134 0.1514 Table 5.1: Results from 3 simulations with different criteria weights 5.3 Pareto Fronts of Criteria Trade-off ’s One of the major

objectives of the optimization tool is to be able to compare how different ways of prioritizing the different optimization criteria results in different optimal ways of driving. In this section, further analysis of this is made by creating Pareto fronts which will show how the different criteria are correlated and what tradeoff’s can be made between them. 5.31 Travel Time versus Fuel Consumption By only looking at travel time and fuel consumption, it is possible to get a good idea of how these two affect each other. By prioritizing only travel time, one can find what can be described as the performance driving mode which gives the fastest way to drive. By giving increased prioritization towards the fuel consumption it is possible to find various ways of balanced driving modes and also more eco-focused modes. To do a more detailed investigation of what happens when these prioritization weights are varied, one can look at a Pareto front like the ones in figure 5.3 As described in

Section 5.11, the same simulation is run with different criteria weights The results are then plotted and by fitting a line through the optimal points, a pareto front is created. in Figure 56 a pareto front of travel time versus fuel consumption for the final vehicle model is shown. The caste tested is to accelerate a hauler from standstill for 60 meters. 34 Figure 5.6: Pareto front of travel time versus fuel consumption What can be gained from this front is a better understanding of how the trade-off of one criteria affects the other. It also gives some information which can be of value when designing gear shift maps. It is for example seen that in the area of maximum speed, which is the point in the top left end of the front, the slope of the front is very steep. This means that by just changing the drive style a little bit, a lot of fuel can be saved with a small trade-off in travel time. By comparing some points in that region of the front, it is found that a reduction in

fuel consumption by about 20% can be made while only lowering the travel time with about 2%. It is of course of high importance that one keeps in mind what exactly has been simulated. The case tested in Figure 56 is acceleration from standstill for 60 meters The good trade-off that can be achieved by moving along the steep area of the front might not be the same in other driving conditions. It is also needed to make a further analysis of what causes the good trade-off. It could for example be to drive slower, or it might be that the engine RPM and torque is adjusted to follow a path in the fuel consumption map that is more fuel economical. 35 5.32 Varying All Criteria Weights In this section, the same case as in Section 5.31 is investigated, but this time all 3 criteria are active. Many combinations of the the 3 weights are used to create a three-dimensional Pareto front which will show what trade-off’s can be made when taking all criteria into consideration . The front is show

in Figure 57 Figure 5.7: Pareto front for all 3 critera A front like this is more complicated, but will still give information in what directions good trade-off’s can be found. Again, the most interesting areas will be the ones with steepest slopes. There, a lot can be gained in one criteria while only losing a little in the others. 36 Chapter 6 Conclusions In this chapter the work is summarized and the most important conclusions are presented and discussed. A suggestion on what future work can be done as a continuation is given. 6.1 Summary This thesis work started with an introduction of different types of construction vehicles and the standard way of implementing gear shift strategies were presented. This was followed by an in-depth chapter of how a general vehicle model was created by mainly modeling a powertrain and its components. This powertrain model was used to find expressions for the vehicle dynamics in a longitudinal direction which were then implemented in a

optimal control problem. The optimal control problem uses a dynamical programming algorithm as a solver to find the optimal way to drive a distance that the user inputs. The optimal way is found by calculating the optimal velocity and gear shifts based on the criteria of travel time, fuel consumption and durability losses. These criteria can be weighted as the user wants. The work also include some example simulations and analysis where driving situations with varying criteria weights are investigated. 37 6.2 Discussion The goal of the thesis was to create an optimization tool which is capable of making theoretical analysis that can help when designing gear shift strategies. This has been accomplished and the optimal way to drive in many different situations can be found by using this optimization tool. One of the most interesting things that the tool can be used for is to vary the criteria weights to find the best trade-off’s. This can be useful when creating shift maps

because the vehicle can be designed to drives in ways that are particularly beneficial. For example, it was seen in the example simulations that accelerating in a way that took 2% longer time than the fastest way saved about 20% fuel. Similarly, about 70% durability could be saved by accelerating in a way that also only took about 2% longer time. Creating vehicles that drive in these ways could be of interest The optimization tool opens up for various interesting other analysis to be made. It must however always be remembered that the result of the optimal way to drive by the tool only holds for that specific situation that has been used as an input. To create a shift strategy based on this tool, it is needed to analyse many different situations. One must also keep in mind that the durability model is a based on big generalizations and only take into consideration a few components. To get a more reliable estimation of the durability losses, the model for that needs to get further

developed. 6.3 Future Work To further develop the optimization tool, the main thing that will be suggested to work on is the model for durability losses. Durability is a complex concept and many layers of details can be added to the current model that will improve it. More components can be added to the model and a further study can be made about which other causes exist which create durability losses on the vehicles. Something else that can be added is to create a model for using the hydraulic work functions of the vehicles, for example to use and move the arm and bucket of a 38 wheel loader or to tilt the trailer of a hauler. This would open up the possibility to investigate more complex driving situations and work cycles. 39 Chapter 7 References [1] D.V Ngo et al Improvement of Fuel Economy in Power-Shift Automated Manual Transmission through Shift Strategy Optimization - An Experimental Study IEEE VPPC Conference, Lille, France, 2010. [2] Erik Hellström. Look-ahead

control of heavy trucks utilizing road topography Licentiate thesis, Linköping University, 2007. LiU-TEK-LIC-2007:28, Thesis No. 1319 [3] Ngo Dac Viet. Gear Shift Strategies for Automotive Transmissions PhD thesis, Eindhoven Technical University, 2012. [4] Reno Filla. Optimizing the trajectory of a wheel loader working in short loading cycles The 13th Scandinavian International Conference on Fluid Power, SICFP2013, Linköping, Sweden, 2013. [5] Uwe Kiencke and Lars Nielsen. Automotive Control Systems, For Engine, Driveline, and Vehicle Springer Verlag, 2nd edition, 2005 [6] Lino Guzzella and Antonio Sciarretta. Vehicle Propulsion Systems Introduction to Modeling and Optimization Springer Verlag, 2nd edition, 2007 [7] Dag Myrhman et al. Terrängmaskinen 1 SkogForsk, 2nd edition, 1993 40 [8] Karl Popp and Werner Schiehlen. Ground Vehicle Dynamics Springer Verlag, 2nd edition, 1993. [9] Kaisa M. Miettinen Nonlinear Multiobjective Optimization Springer Science+Business Media, LLC,

1998 [10] Richard Bellman. Dynamic Programming Dover Publications, 1957 [11] Mengxi Wu and Gustav Norman. Optimal driving strategies for minimizing fuel consumption and travelling time. Master thesis, Kungliga Tekniska Högskolan, 2013. [12] Ioannides E., Bergling G and Gabelli A An analytical formulation for the life of rolling bearings. Acta Polytechnica Scandinavica, ME 137, Espoo, 1999 41 TRITA-MAT-E 2014:63 ISRN-KTH/MAT/E14/63-SE www.kthse