# Modeling Hydrostatic Transmission in Forest Vehicle

## Full text

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### in Forest Vehicle

Department of Electrical Engineering, Link¨opings Universitet Erik Carlsson

LITH - ISY - EX - - 06/3801 - - SE

Examensarbete: 20 p Level: D

Supervisors: Anton Shiriaev,

Deparment of Applied Physics and Electronics Ume˚a Universitet

Johan Sj¨oberg,

Control & Communication,

Department of Electrical Engineering, Link¨opings Universitet

Examiner: Svante Gunnarsson, Control & Communication,

Department of Electrical Engineering, Link¨opings Universitet

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Institutionen f¨or systemteknik 581 83 LINK ¨OPING SWEDEN May 2006 x x http://www.diva-portal.org/liu/undergraduate/index.xsql?lang=en LITH - ISY - EX - - 06/3801 - - SE

Modeling Hydrostatic Transmission in Forest Vehicle

Erik Carlsson

Hydrostatic transmission is used in many applications where high torque at low speed is demanded. For this project a forest vehicle is at focus. Komatsu Forest would like to have a model for the pressure in the hose between the hydraulic pump and the hydraulic motor. Pressure peaks can arise when the vehicle changes speed or hit a bump in the road, but if a good model is achieved some control action can be developed to reduce the pressure peaks.

For simulation purposes a model has been developed in Matlab-Simulink. The aim has been to get the simulated values to agree as well as possible with the measured values of the pressure and also for the rotations of the pump and the motor.

The greatest challenge has been due to the fact that the pressure is a sum of two flows, if one of these simulated flows is too big the pressure will tend to plus or minus infinity. Therefore it is necessary to develop models for the rotations of the pump and the motor that stabilize the simulated pressure.

Different kinds of models and methods have been tested to achieve the present model. Physical modeling together with a black box model are used. The black box model is used to estimate the torque from the diesel engine. The probable torque from the ground has been calculated. With this setup the simulated and measured values for the pressure agrees well, but the fit for the rotations are not as good.

Hydrostatic transmission, Forest Vehicle, Model, Pressure, Simulation Nyckelord Keyword Sammanfattning Abstract F¨orfattare Author Titel Title

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## Abstract

Hydrostatic transmission is used in many applications where high torque at low speed is demanded. For this project a forest vehicle is at focus. Komatsu Forest would like to have a model for the pressure in the hose between the hydraulic pump and the hydraulic motor. Pressure peaks can arise when the vehicle changes speed or hit a bump in the road, but if a good model is achieved some control action can be developed to reduce the pressure peaks.

For simulation purposes a model has been developed in Matlab-Simulink. The aim has been to get the simulated values to agree as well as possible with the measured values of the pressure and also for the rotations of the pump and the motor.

The greatest challenge has been due to the fact that the pressure is a sum of two flows, if one of these simulated flows is too big the pressure will tend to plus or minus infinity. Therefore it is necessary to develop models for the rotations of the pump and the motor that stabilize the simulated pressure.

Different kinds of models and methods have been tested to achieve the present model. Physical modeling together with a black box model are used. The black box model is used to estimate the torque from the diesel engine. The probable torque from the ground has been calculated. With this setup the simulated and measured values for the pressure agrees well, but the fit for the rotations are not as good.

Keywords: Hydrostatic transmission, Forest Vehicle, Model, Pressure, Simu-lation

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## Acknowledgements

I would like to thank several people that have supported me during the work on my thesis:

My examiner Svante Gunnarsson for valuable opinions on the report, my super-visor at Ume˚a University Anton Shiriaev for letting me be a part of this project and my supervisor at Link¨oping University Johan Sj¨oberg who has helped me very much with both knowledge and encouragement during the creation of this report.

Komatsu Forest for putting forward this very interesting project and a special thanks to G¨oran Blomberg and Joakim Johansson for the practical help during our experimentation with the forwarder.

Rebecka Domeij B¨ackryd for her help with LATEX.

My coworkers at the institution of Applied Physics and Electronics, Pedro, Anders, Ian, Leonid and Uwe for making my spare time here in Ume˚a more interesting with squash and lunches.

I would also like to thank my family for their support and especially my girlfriend Maria for her love and invaluable help.

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## Nomenclature

Symbols and abbreviations are described here. SI units are used throughout the report.

### Symbols

p charge pressure in the hose [Pa] preturn pressure in the return hose [Pa]

∆p difference between charge and return pressure [Pa] V volume of the hose [m3]

β bulks modulus [Pa] C leakage coefficient [Nm5/s] TL torque from the wheels [Nm]

Td torque produced by the diesel engine [Nm] Tp torque from the pump to the engine [Nm] Tm torque from the motor to the wheels [Nm] ωe angular velocity of the engine [rad/s] Qv flow through the safety valve [m3/s] Ql leakage out from the hose [m3/s]

ωp angular velocity [rad/s] ωm angular velocity [rad/s] Jp moment of inertia [Nms2] Jm moment of inertia [Nms2] Jd m.o.i. of the engine [Nms2] Jω m.o.i. of the wheels [Nms2]

Jpd Jp+ Jd Jwm Jm+ Jω

Qp flow from the pump [m3/s] Qm flow to the motor [m3/s] Dp displacement [m3/rad] Dm displacement [m3/rad] ip current to the pump [A] im current to the motor [A] Bp friction coefficient [Nms] Bm friction coefficient [Nms] ηvp volumetric efficiency [-] ηvm volumetric efficiency [-] ηtp torque efficiency [-] ηtm torque efficiency [-] Tf p friction torque [Nm] Tf m friction torque [Nm] µp friction coefficient [Nms] µm friction coefficient [Nms] Kp friction coefficient [Nm] Kp friction coefficient [Nm]

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### Abbreviations

MS5050 Multi-System 5050

CVT Continuously Variable Transmission SITB System Identification toolbox

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## Contents

1 Introduction 1 1.1 Background . . . 1 1.2 Purpose . . . 1 1.3 Method . . . 2 1.4 Delimitations . . . 2

1.5 Outline of the Report . . . 2

2 System overview 3 2.1 Forest vehicle . . . 3

2.2 Transmission line . . . 4

2.3 Previous work, modeling . . . 5

2.4 Previous work, identification . . . 7

3 Physical model of the hydrostatic transmission 9 3.1 Introduction model . . . 9 3.2 Diesel engine . . . 10 3.3 Hydraulic Pump . . . 12 3.4 Hose . . . 13 3.5 Hydraulic Motor . . . 13 3.6 Traction model . . . 14 3.7 Complete model . . . 15 4 Identification 19 4.1 Introduction . . . 19 4.2 Linear identification . . . 20

4.2.1 Linear black box modeling . . . 20

4.3 Non-Linear identification . . . 22

4.3.1 Non-linear black box modeling . . . 22

4.3.2 Neural Networks . . . 23

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xiv Contents 5 Data collection 25 5.1 Multi-System 5050 . . . 25 5.1.1 Sensors . . . 25 5.2 Test day . . . 27 5.2.1 Sensor locations . . . 27 5.2.2 Performed tests . . . 27

5.2.3 Modifications of the data . . . 29

6 Model modifications, Simulations and Results 31 6.1 Modifications . . . 31

6.1.1 Swashplate angle . . . 31

6.1.2 Flows from valve and leakage . . . 32

6.1.3 Return pressure . . . 32

6.1.4 Efficiencies . . . 33

6.2 Simulations for the pressure . . . 34

6.3 Black box models . . . 35

6.3.1 Pump . . . 36

6.3.2 Motor . . . 39

6.4 Grey box models . . . 41

6.4.1 Pump . . . 41

6.4.2 Motor . . . 44

6.5 Complete model . . . 46

7 Discussion and Conclusions 51 7.1 Discussion . . . 51 7.2 Conclusions . . . 52 8 Future work 53 Bibliography 55 A Data sheet 59 B Simulink models 61 C .m-files 65 C.1 filtrera.m . . . 65

C.2 umu fetch matrix.m . . . 66

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## Introduction

In this thesis, a model of the hydrostatic transmission in a forest vehicle will be developed. The introduction chapter presents the background, purpose and delim-itations, the method used to solve the problem and an outline of the report.

### Background

Komatsu Forest develops and manufactures forest vehicles. The production in Ume˚a started 1961 in a small family-owned company. Since then there have been some different owners. 2004 Komatsu Ltd bought the company and Komatsu Forest was founded. The development and production have changed course from Slash-bundler ”Skruven” to today’s high technology forest machines. The development and production for Komatsu is in Ume˚a and Wisconsin. In Ume˚a, the production mainly consists of wheel-based machines and harvester heads. Some of the research projects in Ume˚a have been placed at Ume˚a University where extra time and knowledge can be found. This thesis is a part of one of these projects. 

A forest vehicle needs much more torque in comparison to for example a road truck. Because of that a hydrostatic transmission is used. Komatsu wants a model of this transmission. One of the motivations for the detailed modeling comes from the fact that shifting speed usually results in a peak of pressure that has an undesirable effect on the system and should be avoided. Such rise (drop) in pressure cannot be explained by simple static models and dynamical modeling is therefore needed. If a model that matches the real values in a good way is obtained then hopefully a controller that significantly reduces this pressure peak can be developed.

### Purpose

The main purpose of this thesis will be to build a model for the hydrostatic transmission in a forest vehicle using available measurement signals. If this

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2 Chapter 1. Introduction

model agrees well with reality and there still is time left, then the next task will be to control the input signals to reduce the pressure peak.

### Method

To achieve the objectives of this thesis a mathematical description of the sys-tem has been developed. For simulation purposes this model is implemented in Matlab-Simulink. Experiments were needed to estimate unknown parame-ters. These experiments have been done together with Komatsu Forest on one of their test vehicles. Adjustments of the model and estimation of the parameters have been made to obtain a better model.

### Delimitations

When developing the model six measurement signals were used. One of them was the rotation for the pump, which in opposite to the others also is available online. This signal could be used when simulating the model, but that approach has not been tested in this thesis.

A model for the pressure of the return flow is not offered, but instead measured values are used.

Another restriction made in this thesis is to only consider rotation of the hy-draulic pump and motor in one direction. This has simplified the equations describing the system and made it easier to understand.

The report has been written to suit students from an engineering program with some experience of modeling systems.

### Outline of the Report

The main topics dealt with are presented in the chapters below.

Chapter 2: A forest vehicle and the transmission line are shortly described. Chapter 3: The theoretical model for the hydrostatic transmission is

pre-sented.

Chapter 4: Some identification methods are explained.

Chapter 5: The experiments and the data collection are described. Chapter 6: Simulations are performed and the results are given. Chapter 7: Discussions and conclusions of the work are presented. Chapter 8: Describes possible ideas for future work.

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## System overview

In this chapter, an overview of both a forest vehicle and its transmission will be given. The intention of this chapter is to give the reader a better understanding of how a hydrostatic transmission works and why it is used in a forest vehicle.

### Forest vehicle

In Sweden two types of forest vehicles are used, harvesters and forwarders. They are usually wheel based. The harvester is used to take trees down, remove twigs and cut the trees in logs of suitable length. The task of the forwarder is to move the logs from the harvester area to the road where a truck will come and pick them up. The focus of this thesis will be to build a model for a forwarder although the difference to other vehicles is small so an extension to these vehicles will be possible.

Figure 2.1: The forwarder used in this project: Valmet 840.2

The forwarder considered in this project has model number 840.2, see Figure 2.1. The weight is almost 14 tons and the payload 11 tons. The source of power is a 6.6 liter diesel engine, developing 170 hp. The forwarder is using energy in

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4 Chapter 2. System overview

two ways. One way is to move the crane and the other is to move the vehicle forward. Usually these two cases do not occur at the same time, but if this is the case there will be a problem with power supply for both of them. That problem is outside the scope of this thesis. 

### Transmission line

As mentioned above it is the diesel engine that delivers the power for the ma-chine. The diesel engine is connected to a hydraulic pump via a drive shaft. Inside the pump there is a swashplate that determines whether the vehicle is moving forward, backwards or is stationary. The angle of the swashplate also controls how much oil that flows out from the pump. A short description how this works will be offered below. To the swashplate a number of pistons are attached. An angle of the swashplate will make the pistons go back and for-ward with the rotation. An increase of the angle makes the stroke of the pistons longer which results in larger flow of oil of the pump. Figure 2.2 shows how it works in a simplified way.

Figure 2.2: Simplified diagram of the transformation from swashplate angle to an high pressure flow into the hose.

The flow out of the pump is a product of the rotation of the pump and the volume/displacement of the pump, where the displacement is determined by the swashplate angle. The flow created by the pump passes through a security valve that ensures that the pressure is not too high. After that it is connected to the hydraulic motor. A hydraulic motor is basically the same thing as a hydraulic pump, but it works in the opposite way. Consequently the hydraulic motor is converting the flow back into rotation of the wheels. The motor as well as the pump has a swashplate. These two swashplates are used to achieve the demanded speed from the driver. If really high speed is desired then the swashplate angle as well as the volume of the motor decreases, and the flow passes through a smaller volume and the rotation increases. To transform the rotation of the motor to lower speeds the motor shaft is connected to a gearbox with two gears. From the gearbox there is a connection to the wheels.

The flow from the hydraulic motor is connected to the input of the pump, which means that it is a closed loop system. A possible problem would then be that there is a lack in the oil supply to the pump. To ensure that oil always is available there is an extra pump called charge pump connected to the same

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2.3. Previous work, modeling 5

shaft as the first one. The charge pump guarantees that the pressure on the inlet side of the pump is around 30 bar. To avoid damage a safety valve is placed on the high-pressure side, and it opens when the pressure becomes too high. A simple schematic overview of the hydrostatic transmission is given in Figure 2.3.

Figure 2.3: Schematic overview of the hydrostatic transmission.

A hydrostatic transmission has advantages compared to a normal mechanical transmission line, and some of them will now be presented. Hydrostatic trans-mission is generally used in low-speed and high-torque applications. The rea-son is the continuously variable transmission, CVT, which makes it possible to achieve desired torque. A CVT is a little bit less efficient than a mechanical transmission but the possibility to drive at an optimal combination of torque and speed makes the diesel engine to work in an more efficient range, and there-fore the whole vehicle becomes more efficient [2, 3]. The choice of torque and speed can be reached with high accuracy . Another motive is that a hydraulic motor produces up to ten times more power compared to an electrical motor with the same dimensions .

### Previous work, modeling

The final goal for this project is to control the pressure in the hose between the hydraulic- pump and motor. To do this, a model of the hydrostatic transmission is needed. Development of models for the hydrostatic transmission started in the late 1940’s. Huge progress was made by Merritt in the 1960’s . His work has then later been updated by Manring . Both Merritt and Manring focused on theoretical issues. However, I will mostly describe what has been done for physical machines. More specifically, the discussion will be focused at the work done by Prasetiawan et al.  at the University of Illinois and Lennevi et al.  at Link¨oping University. For more details see [8, 9] and references therein. The work done at Illinois [10, 11, 12] and Link¨oping [13, 14] has resulted in many theses and articles.

In Illinois, a laboratory setup with an Earthmoving Vehicle Powertrain Simu-lator (EVPS) has been used. The EVPS has an induction motor as its prime mover for the system and three hydraulic motors. A variable displacement pump then provides flow for the three motors. The hydraulic motors have fixed

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6 Chapter 2. System overview

displacement and are connected to valves that determine which one to use. A load simulator is connected to each hydraulic motor.

The setup used at Link¨oping university is very similar to the one used in Illi-nois. The largest difference is that the group in Link¨oping has used a variable displacement motor instead of three fixed displacement motors. An induction motor controlled by a servo-valve has been used to simulate a diesel engine and a load simulator is connected to the hydraulic motor. Figure 2.4 shows the setup used in both Illinois and Link¨oping.

Figure 2.4: The setup used by the groups in Illinois and Link¨oping.

To the author’s knowledge no detailed modeling using data from a machine, driven in a realistic situation, has been done. Nevala et al.  have developed an antislip control for a forest vehicle. Fuzzy control was used to minimize the slip and therefore no model of the hydrostatic transmission was needed. A Matlab-Simulink package with a model of a hydrostatic transmission has been developed by Jedrzykiewicz et al. . This model has many similarities with the models developed in Illinois and Link¨oping. These three models are the basis for the mathematical model presented in Chapter 3. In that chapter, a simplified model for the transmission given by Egeland and Gravdahl  will also be presented.

A phenomenon described in the literature is the influence of the fluid properties on the efficiency of the hydrostatic transmission. This has been investigated by Dahl´en . Another issue that needs to be considered if a more detailed model is desired is the propagation of pressure. This has briefly been described by Egeland and Gravdahl  and more carefully by Weddfelt . Weddfelt has also approached the problem with pressure ripple. However, I have had no immediate use of the information found in these references.

Of the references presented in this section the references from Illinois and Link¨oping are most closely related to this project, but there are some major differences in the basic conditions. First, the diesel engine in this project is not very well known. This would not have been a problem if it was possible to measure the torque from the engine, but unfortunately this is not the case. The setups in both Illinois and Link¨oping use induction motors for which the control signal is the delivered torque. Moreover, a torque sensor is used in Link¨oping. Another important difference is also due to sensor signals. The other setups have measurement signals for the swashplate angles of the pump and the mo-tor. That is not possible to get from the forest vehicle used in this project. It is

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2.4. Previous work, identification 7

however still possible to measure the control signal to the hydraulic pump and the motor which controls the swashplate angels. The problem is to determine the dynamics of the swashplate. In this thesis, the swashplate dynamics will be neglected and hopefully this assumption will not affect the model too much. If the effect would be large it would most likely only cause a small delay in the pressure.

The last bigger difference is that the Illinois and Link¨oping approaches use load simulators. Load simulators produce a stable torque which is rather easy to adjust to a desired level. The Link¨oping group does also have a torque sensor connected. In our case, a forest vehicle without a torque sensor is connected. The load torque when driving will be quite unpredictable.

It may seem like the difference between this project and those projects above is very big. Therefore, it is necessary to stress that the main part of the trans-mission line is very similar for all three setups. The hydrostatic transtrans-mission consisting of a hydraulic pump and a hydraulic motor is basically the same. Therefore the models will approximately be the same.

### Previous work, identification

A model of the hydrostatic transmission will contain a lot of parameters. The values for some of them can be found in different documents from manufactures, and some are more or less known from experiments etc. However, some param-eters are unknown and need to be estimated.

An electro-hydraulic servo system (EHSS) as well as a hydrostatic transmission contains nonlinear hydraulic dynamics. Therefore, it is reasonable to see what has been presented in the literature on identification of an EHSS. An EHSS consists of a hydraulic valve, a hydraulic cylinder and a mass. The valve receives a control signal determining which side of the cylinder that should receive the oil flow. The pressure in the receiving side rises and the mass starts to move. See figure 2.5.

Figure 2.5: Overview of the electro-hydraulic servo system.

To model the EHSS, Reuter  has used bilinear canonical forms. Recursive prediction error methods are then used to identify the parameters in the bilinear

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8 Chapter 2. System overview

canonical forms. This concept can cause problems with convergence if the initial values are not ’good enough’. To avoid this difficulty Jelali and Schwarz  have used a modified recursive instrumental variables algorithm. Linear inte-gral filters were used to handle derivatives of measurement. The final model for the EHSS is represented in observer canonical form. Another way to approach the problem with identification of a nonlinear system is to use neural networks. This has been done by Anyi et al. . During the training of the network backpropagation was used.

A problem that can occur when identifying parameters in a model is hysteresis effects. This has been approached by Park and Lee  when modeling a single-rod cylinder. To avoid these effects they used a modified signal compression method to estimate different dynamics during expansion and retraction. In this project a hydrostatic transmission should be modeled and identified. As described in Section 2.3 this problem has previously been approached by Prase-tiawan in Illinois and he has used frequency and time responses to identify the system. Instead of estimating the parameters in the developed model, transfer functions were shaped using system identification toolbox (SITB) in Matlab. For example, transfer functions from the reference signal for the swashplate an-gle to the pressure and to the speed of the motor were derived. The model that had been created before provided information about which order the transfer functions should have. An observation made by Prasetiawan was that the model before improvements had good agreement between simulated and measured val-ues for the speed of the motor but the simulated and measured valval-ues for the pressure did not have the same accuracy. This shows that it is important to build and identify a model that is adapted to the goal of the project. In , Lennevi and Palmgren have created a controller for the speed of the motor. Therefore, a good model for the speed of the motor was needed but the pressure simulations were less important. In our case, the goal is to control the pressure in the hose.

In , Cidr´as and Carrillo describe their model of a hydrostatic transmission. To estimate the unknown parameters a method based on the Melder-Mead sim-plex algorithm was used. Cidr´as and Carrillo evaluated if the rotation of the motor in the hydrostatic transmission could be kept constant. The reason to build such a controller was that an electrical generator should be attached to the hydrostatic transmission. Another, not so successful approach was tested by Luigi del Re . He developed a controller based on a model derived by using black box identification of the hydrostatic transmission.

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## hydrostatic transmission

In this chapter a model describing the hydrostatic transmission will be presented. The model will consist of smaller blocks where each block is a model for a part of the system. This will make it easy to exchange parts of the model, for example, the pump without changing the other elements. It will also give a fine overview of the system.

### Introduction model

In  Egeland and Gravdahl present a simple model of the hydrostatic trans-mission. To get an introduction to the system the model is described in detail below by (3.1) – (3.3). The model has three variables that describe the rotation of the pump ωp, the rotation of the motor ωm, and the pressure in the hose between the pump and the motor p respectively. Figure 3.1 shows an overview of the system. The equation for the pump is

Jpw˙p= Td− Bpωp− Dpp, (3.1) where Td is the torque from the diesel engine, Jpis the moment of inertia of the pump, Bp is a friction coefficient for the pump and Dp is the displacement of the pump. The displacement depends linearly of the swashplate angle. Equa-tion (3.1) shows that the pump starts to rotate when a torque from the diesel engine is present. The hydraulic motor is described by

Jmw˙m= −Bmωm+ Dmp − TL, (3.2) where TL is the load torque from the wheels, Jmis the moment of inertia of the motor, Bm is the friction coefficient of the motor and Dm is the displacement of the motor. The two equations above demonstrate the similarity between the pump and the motor. The pressure in the hose between the pump and the motor is described by

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10 Chapter 3. Physical model of the hydrostatic transmission

V

βp = −D˙ mωm+ Dpωp− Cp, (3.3) where V is the volume of the pump, β is the bulk modulus and C is a leakage coefficient. This equation illustrates that the pressure goes up when the flow from the pump is greater than the flow into the motor plus the leakage.

Figure 3.1: Egeland and Gravdahls model.

Now when the basics are presented a deeper exploration of the different parts of the system will be offered.

### Diesel engine

The engine was not included in the introduction model described in Section 3.1. When trying to include the diesel engine in the model it is important to consider that the engine and the pump are connected by a shaft. This means that they have the same rotation speed. Prasetiawan et.al. in  approach the modeling problem by assuming that the rotation produced by the engine is affected by the torque from the pump, see Figure 3.2.

Figure 3.2: Connection between the diesel engine and the pump.

In Figure 3.2 the control signal to the engine is left out. In a private car the control signal is represented by the gas pedal that affects the throttle which controls the air flow into the motor. As a response the control-box for the engine injects more or less fuel to keep the air/fuel-mixture constant. This is not the case with Komatsu’s forest machines. Instead of controlling airflow the gas pedal is input to the control-box that affects the swashplate angles of the pump and the motor. Komatsu’s control-box also sends a reference to the engines control-box which rotation speed is desired. The control-box of the engine then controls air and fuel input to match the desired rotation speed. See Figure 3.3 for an illustration. The result of the control system described is that a drop in speed will make the engine work harder to regain the reference speed given by Komatsu’s control-box.

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3.2. Diesel engine 11

Figure 3.3: Overview of the different control signals in the transmission that affect the final speed of the vehicle.

To describe the rotation of the engine and the pump an equation similar to (3.1) is used,

(Jp+ Jd) ˙we= Td− Tf p− Tp, (3.4) where Jd is the moment of inertia of the diesel engine, ωeis the angular veloc-ity of the engine, Td is the torque produced by the engine, Tf p is the friction affecting the engine and the pump and Tp is the torque from the pump to the engine. The relation ωe= ωp holds because of the connection between the two units.

Tf pis the joint friction of the engine and the pump, and a reasonable assumption is to approximate it with a coulomb and a viscous friction. Then Tf p can be expressed as

Tf p= µpωe+ Kpsign(ωe), (3.5) where µpand Kpneed to be estimated from measurements. The term sign(ωe) delivers the correct sign depending on the rotation of the diesel engine, but the engine will only rotate in one direction, and therefore this term can be removed.

It will not be necessary to take special care and model the charge pump since the estimate of Jd and Tf p will include the charge pump without extra effort.

One way to get an accurate value for Td is to create a good model, that requires a lot of information about the engine. Another option is to measure the motor’s internal signals. Unfortunately, neither of these options are available. This means that some kind of black/grey-box model needs to be developed for Td, see Chapter 6. Until that problem is solved, Tdwill be treated as an input signal to the system.

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12 Chapter 3. Physical model of the hydrostatic transmission

### Hydraulic Pump

The rotation of the pump is generated by the diesel engine, but more modeling for the hydraulic pump is necessary. The torque Tprepresents the torque which the diesel engine senses from the pump and it will be a function of displacement and difference between charge- and return-pressure. Sauer Danfoss, the devel-oper of the hydrostatic system, provides a formula for Tp in the documentation for the pump 

Tp= Dp∆p

ηtp

, (3.6)

where ηtpis the mechanical efficiency for the pump. This is basically the same as the term Dpp in (3.1). The big difference is the efficiency ηtpthat makes the formula more accurate. An intuitive argumentation for (3.6) is that, when the pressure rises in the hose it is harder to rotate the pump. This means that Tp goes up and the rotation initially goes down. It also happens when the displace-ment of the pump Dp increases, then the pump should provide more flow and therefore decreases in speed. The initial drop in speed will be counteracted by the diesel engine that increases the torque Td in order to try to keep the speed constant.

Equation 3.7 that describes the flow out of the pump is offered by Sauer Dan-foss in  and in almost the same shape by Jedrzykiewicz et al in . The difference between the two equations is mainly notational. The equation looks like

Qp= Dpωpηvp, (3.7)

where Qp is the flow out from the hydraulic pump and ηvp is the volumetric efficiency of the pump.

One big advantage with the Sauer Danfoss representation of the pump is the existence of the efficiencies in a datasheet, see Appendix A. That will give extra information about the behavior of the pump.

Modeling the pump contains a problem which has not been addressed so far. The displacement Dpis linearly dependent on the swashplate angle but to control the angle of the swashplate a current is used. The relationship between the current and the angle is approximately known but small errors in the estimate of Dp can result in large errors in the pressure. Also the displacement of the motor Dm is controlled by a current. More about the relation between the currents and the angles can be found in Section 6.1.1.

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3.4. Hose 13

### Hose

The flows of oil into and out from the hose will determine its pressure. There are four flows to consider. The pressure in the hose is determined by the flow from the pump Qp, the flow into the motor Qm, the leakage Ql, and the flow through the valve Qv. See Figure 3.4 for a sketch of the flows.

Figure 3.4: The signals to and from the hose.

Changes in pressure propagate in the hose with the speed of sound1. Therefore, it is realistic to assume the same pressure in the 1.2 meter long hose. Expansion of the hose is another phenomenon that will not be encountered for in this thesis. High pressures will affect the rubber hose but hopefully not enough to make significant changes in the volume of the hose. With these two assumptions made, the physics behind pressure changes can be described by

p = β V

Z t

0

(Qp− Qm− Qv− Ql) dτ , (3.8)

where V is the volume of the hose, β is the bulk modulus and p is the pressure in the hose. This is basically the same equation as (3.3) with some small modi-fications. As explained in Section 6.1.2, Qv and Ql can be set to zero.

It could be important to stress one property of (3.8). If the pressure in the hose is constant it means that the flow out of the hose is equal to the flow into the hose. This is of course true in the opposite way, if the flows are equal the pressure will be constant.

### Hydraulic Motor

The equations describing the hydraulic pump can almost without changes be used for the motor as well. The flow Qmlooks like

Qm= ωmDm

ηvm

, (3.9)

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14 Chapter 3. Physical model of the hydrostatic transmission

where ηvmis the volumetric efficiency of the motor. The reason to have ηvmin the denominator instead of the numerator is that the flow is the input to the motor contrary to the pump where it is the output. An analogous discussion can be used for ηtm in the equation that express Tm

Tm= Dm∆pηtm, (3.10)

where ηtm is the mechanical efficiency for the motor and Tmis the torque that the motor delivers to the traction model. An overview of the signals into and out from the motor can be found in Figure 3.5.

Figure 3.5: Connections with the hydraulic motor.

### Traction model

A torque from the motor is delivered to the traction model, and that torque should be used to calculate a rotation for both the wheels and the motor. The torque TL stands for the torque from the wheels and Jω is moment of inertia for the wheels. An equation for the angular velocity ωmcan be written

(Jm+ Jω) ˙wm= Tm− Tf m− TL, (3.11) where Tf mdescribes the friction and could be assumed to look like

Tf m= µmωm+ Kmsign(ωm), (3.12) e.g. a Coulomb friction and a viscous friction. The coefficients µm and Km need to be estimated. The hydraulic motor together with the wheels can rotate in both directions, and therefore the term sign(ωm) is needed.

The problem left to handle is then TL. Unfortunately TLcauses a lot of trouble. It depends on several parameters that possibly could be found. It also depends on how slippery the ground is, whether it is a slope or not and if there are holes or stumps. Therefore, it is necessary to treat TL as an immeasurable input to the system or possibly use a blackbox/greybox model, see Chapter 6.

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3.7. Complete model 15

### Complete model

To sum up, the full set of equations that constitute the model are presented be-low. Some modifications will be presented later because Tdand TLare unknown. The pump related equations are

(Jp+ Jd) ˙wp= Td− Tf p− Tp, Tf p= µpωp+ Kp, Tp= Dp∆p ηtp . (3.13)

The motor related equations are

(Jm+ Jω) ˙wm= Tm− Tf m− TL, Tm= Dm∆pηtm,

Tf m= µmωm+ Kmsign(ωm). (3.14) The equations describing the pressure are

p = β V Z t 0 (Qp− Qm) dτ , ∆p = p − preturn, Qp= Dpωpηvp, Qm= ωpDm ηvm . (3.15)

How signals and sub-models are connected for the full model are shown in Figure 3.6.

Figure 3.6: Input signals to the system together with the signals that connect different subsystems.

No model has been developed for the return pressure. Instead, the return pres-sure is seen as a input to the system. The torques Tdand TLwill at this time be

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16 Chapter 3. Physical model of the hydrostatic transmission

treated as disturbances. The two last signals into the system are Dp and Dm. These are the two controllable signals primary used to avoid pressure peaks. The equations presented above represent the model for the hydrostatic trans-mission. This model can be rewritten in a better way. Introducing two new variables Jpd = Jp+ Jd and Jmw = Jm+ Jω together with substitutions of Td, Tf p, Tm, Tf m, Qp, Qmand p gives ˙ wp= Td Jpd −µpωp Jpd −Kp Jpd −Dp∆p Jpdηtp , ˙ wm= − TL Jmw −µmωm Jmw −Kmsign(ωm) Jmw +Dm∆pηtm Jmw , ˙ ∆p =βDpωpηvp V − βωmDm V ηvm − ˙preturn. (3.16)

The goal is to rewrite (3.16) in state space form to more clearly show the struc-tures of the model. It will also make it easier to apply identification methods. Before a state space form can be given some dependencies need to be clarified. The controllable input signals to the model (3.16) are the displacement of the pump and the displacement of the motor. These displacements are determined by electrical signals. The functions from current to displacement are approxi-mately known, see Section 6.1.1, and will therefore not render more unknown parameters. Still Dz, where z is either p or m, will be written Dz(iz) in the following state space representation to keep in mind that the displacements are functions of the currents. In this thesis the dynamics of the swashplates have been neglected, otherwise a model from iz to Dz would be necessary. Hope-fully this assumption is reasonable when no measurements for the angles of the swashplates are available.

As mentioned before the values for the efficiencies are given in a data sheet provided by Sauer Danfoss. These values are probably not totally reliable and therefore it can be of interest to see them as unknown parameters. It could happen that the efficiencies are dependent on velocities, pressures and displace-ments. Therefore, the following notation will be used,

ηtp= ηtp(ωp, ∆p, Dp), ηvp= ηvp(ωp, ∆p, Dp), ηtm= ηtm(ωm, ∆p, Dm), ηvm= ηvw(ωm, ∆p, Dm).

The pressure in the return hose is not modeled but it is measured in the tests, and therefore it will be used as input signal to the system. The following vectors are now introduced

x =   wp wm ∆p  , u =   Dp(ip) Dm(im) ˙ preturn  , d =  Td TL  , θ = µp µm Kp Km Jpd Jmw β θηtp θηvp θηtm θηvm T ,

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3.7. Complete model 17

where x represents the states, u are the input signals, d are the disturbances and θ contains the unknown parameters. Because ηtp is not constant it is pa-rameterized using θηtm which can be a vector.

The model can now be expressed as

˙ x = f (x; θ) + g(x; θ)u + ϕ(θ)d, f (x; θ) =    −µpx1 Jpd − Kp Jpd −µmx2 Jmw − Kmsign(x2) Jmw 0   , g(x; θ) =     − x3 Jpdηtp(x,u;θηtp) 0 0 0 x3ηtm(x,u;θηtm) Jmw 0 βx1ηvp(x,u;θηvp) V − βx2 V ηvm(x,u;θηvm) −1     , ϕ(θ) =   1 Jpd − 1 Jmw 0  . (3.17)

To simplify this representation some assumptions has been done. The two torque efficiencies are rather constant when the speed of the vehcile is above 0.2 km/h. Therefore, they will be approximated by a constant. The two volumetric effi-ciencies are probably most dependent of the speed of the pump and the speed of the motor, respectively. Because of that the pressures and displacements will be neglected in the function. A delimitation made in this thesis is that the rotation of the wheels will only be considered in one direction, and therefore sign(x2) can be removed. As a result of the discussion above new θ, f (x; θ) and g(x; θ) can be formed as θ = µp µm Kp Km Jpd Jmw β ηtp θηvp ηtm θηvm T , f (x; θ) =    −µpx1 Jpd − Kp Jpd −µmx2 Jmw − Km Jmw 0   , g(x; θ) =    − x3 Jpdηtp 0 0 0 x3ηtm Jmw 0 βx1ηvp(x1;θηvp) V − βx2 V ηvm(x2;θηvm) −1   , (3.18)

As we can see there are a lot of unknown parameters in θ. Estimation of these are necessary, and therefore an introduction to identification will be given in the next chapter.

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## Identification

In this chapter an introduction to linear and non-linear identification will be given.

### Introduction

A lot of research has been done on identification of systems and parameters. An introduction to the aspects of identification is given by Ljung and Glad . Further information can be found in  by Ljung and for non-linear identifi-cation in  by Sj¨oberg. The information presented in this chapter is found in this literature.

Different kinds of models can be developed for a system. These can be catego-rized into three different groups.

- White Box models - Grey Box models - Black Box models

A white box model is used when all the information about the system structure is available. This is almost never the case when it comes to modeling in the real world. Even with good knowledge about the system there will be unknown parameters that need to be estimated in some way. This kind of model is called grey box model. Another type of model that also falls under the category of grey box models is when some physical information about the system is avail-able and that knowledge is used to estimate a model of black box nature. A black box model is used when no information and knowledge about the system exist.

Which kind of model that should be used depends on the knowledge of the system, how much time one has and what the model should be used for. Maybe it is enough to use a black box identified model and then there is no need to

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20 Chapter 4. Identification

spend more time to determine the physical relations of the system. If one has some physical insight it should probably be used when developing the model. Then a more realistic model can be achieved and fewer parameters need to be estimated.

### Linear identification

If the developed model for the system is linear but have some unknown param-eters it could be written as

˙

x(t) = A(θ)x(t) + B(θ)u(t) + d(t)

y(t) = C(θ)x(t) + D(θ)u(t) + h(t), (4.1) where u(t) stands for the inputs to the system, y(t) for the outputs, x(t) are the states of the system, d(t) and h(t) are disturbances and θ is a vector containing the unknown parameters. The task is to estimate these parameters through dif-ferent experiments. How this should be done depends on the system and which measurements that can be collected.

If less knowledge about the system is available, then black box modeling could be used to approach the problem, this will be presented below.

### Linear black box modeling

When using an identification program to estimate a black box model, for ex-ample SITB , the model structure needs to be chosen. Some of the possible structures will be presented below and an example will be given for one of them. For the rest of this chapter discrete time models with q as the shift operator will be used, where q works as

qy(t) = y(t + 1), q−1y(t) = y(t − 1). (4.2) A general model can be described by

A(q)y(t) =B(q) F (q)u(t) +

C(q)

D(q)e(t), (4.3)

where y(t) is the output, u(t) is the input and e(t) is a white noise disturbance. As can be seen in (4.3) this general model will result in some special cases. If an OE (output error) model is desired, B and F are used and A, C and D is set to one. Other examples are: the ARX model (C=F=D=1), the ARMAX model (F=D=1) and the BJ (Box-Jenkins) model (A=1).

Which model structure that should be chosen depends a lot on the behavior of the disturbances on the system. The question is how the disturbances enter to

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4.2. Linear identification 21

the system. If for example a measurement sensor is affected by a white noise disturbance a good choice of model would be an OE model. This is because an OE model has dynamics for the input signal u(t) while the disturbance e(t) directly affects y(t). An ARX model can be a good alternative when the distur-bance probably enters the system in the same way as the control signal. Then both u(t) and e(t) will have the same pole dynamics. A problem can then be that A(q) needs to describe the properties of disturbance as well. An ARMAX model can be used if some extra flexibility for the estimate of the disturbance is desired. In an BJ model the dynamics for u(t) and e(t) to y(t) are separate. Therefore, a BJ model can be good to use when the disturbances enter late in the process.

A deeper investigation of how to estimate the parameters in an ARX model will now be performed. The polynomials A and B look like

A(q) = 1 + a1q−1+ a2q−2+ · · · + amq−m

B(q) = b1q−1+ b2q−2+ · · · + bnq−n (4.4) where a1to amand b1to bn are unknown parameters and will therefore be our θ, which can be written as

θ = [a1 a2 · · · am b1 b2 · · · bn]T. (4.5) Using (4.3) and (4.4) the model can be written as a difference equation

y(t) + a1y(t − 1) + · · · + amy(t − m) =

b1u(t − 1) + b2u(t − 2) + · · · + bnu(t − n) + e(t) (4.6) or as

y(t) = −a1y(t − 1) − · · · − amy(t − m)+

b1u(t − 1) + b2u(t − 2) + · · · + bnu(t − n) + e(t). (4.7) Introducing ϕ(t) makes it possible to write (4.7) as

y(t) = ϕT(t)θ + e(t) (4.8)

with

ϕ(t) = [−y(t − 1) · · · − y(t − m) u(t − 1) · · · u(t − n)]T. (4.9) Because e(t) directly affects y(t) and e(t) is unpredictable white noise the best prediction ˆy(t|θ) for y(t) is

ˆ

y(t|θ) = ϕT(t)θ. (4.10)

From (4.8) it is possible to get an estimate for θ using linear regression. Terms in ϕ(t) are called regressors and ϕ(t) is the regression vector.

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22 Chapter 4. Identification

The estimate ˆθ will converge to the true value of θ when the number of samples approaches infinity. This is true if the system could be completely modeled by an ARX model. Another restriction is that the dynamics of the system are excited by the input signal. When estimating a black box this is a really important issue. If the identification should have a chance to be successful it is necessary that the system dynamics is available in the output signal. To ensure that large variations of the input signal should be performed.

### Non-Linear identification

The goal when it comes to non-linear models is the same as for linear model, that means the prediction should be as close to the real values as possible. To achieve that a cost function is formed

Vn(θ) = 1 n n X k=1 ky(tk) − ˆy(tk|θ)k2. (4.11)

This cost function will be large when our prediction for y(t) is bad. It is now possible to optimize θ using Vn(θ) as a measurement of how good ˆy(tk|θ) is. The minimization of Vn(θ) by improving θ could be performed using

ˆ

θ(i+1)= ˆθ(i)− ν(i)[V00 n(ˆθ

(i))]−1V0

n(ˆθ

(i)), (4.12)

where ν is the step length, Vn0(θ) is the derivative of Vn(θ) with respect to θ and Vn00(θ) is the second derivative. Calculation of (4.12) is not trivial but the actual problem is to define ˆy(t|θ). It could be done using either a physical model or a black box approach (for example neural networks).

### Non-linear black box modeling

The prediction ˆy(t|θ) is now a function of θ and ϕ(t), (4.10) will therefore look like

ˆ

y(t|θ) = g(ϕ(t), θ). (4.13)

Also ϕ(t) can be a function

ϕ(t) = ϕ(y(t − 1), · · · , y(t − m), u(t), u(t − 1), · · · , u(t − n). (4.14) As before, different choices for the regression vector ϕ(t) will result in different kinds of models, for example NARX (Non-linear ARX), NOE and NARMAX. Another choice that needs to be considered is how the nonlinear mapping func-tion g(ϕ(t), θ) should look like. To illustrate this problem and give an introduc-tion to a nowadays popular method, neural networks will be explained.

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4.3. Non-Linear identification 23

### Neural Networks

The problem that will be dealt with is how g(ϕ, θ) should take us from regressor space to output space. A reasonable assumption is that it will work well to use parameterized functions to describe g(ϕ, θ)

g(ϕ, θ) = n X k=1 αkgk(ϕ), (4.15) θ = [α1 · · · αn]T,

where α1to αnare parameters in the expansion and gkare called basis functions. A good choice is to use the same function for all gk but to use two parameters β and γ to make each gk individual. Equation (4.15) can then be written as

g(ϕ, θ) = n X k=1 αkκ(βk(ϕ − γk)), (4.16) θ = [α1 · · · αn β1 · · · βn γ1 · · · γn]T, The parameter γ locates the function κ and β is a scale parameter.

A short example of a function approximation using (4.16) will now be given. Assume that ϕ is a scalar and κ is chosen to be a unit pulse, then our prediction ˆ

y will be approximated with a piece-wise constant function. The parameters α will determine the level of each step, γ the position and 1/β the length. Figure 4.1 shows an example.

1 1.5 2 2.5 3 3.5 4 0.8 1 1.2 1.4 1.6

Figure 4.1: Approximation of a function using a neural network.

If a smoother function is desirable κ can be set to be for example a Gaussian bell,

κ(x) = √1 2πe

−x2/2

. (4.17)

In this scalar example it is relatively easy to realize that the approximation can be arbitrary good, but this is also true for higher dimensions. This is the main reason why neural networks are so popular.

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## Data collection

In this chapter the experiments and the method to collect the data will be described. There are some important issues to consider before tests can be performed. Which signals should be measured, which signals are even possible to measure, if a signal is not measurable can it be estimated from other signals and in that case which tests are good for achieving unknown parameters and verifying the model?

### Multi-System 5050

Development of the model and estimation of the parameters assume measure-ments from a real forest vehicle. These measurements need to be collected somehow. In this thesis Komatsu’s measurement tool, Multi-System 5050 , has been used, but in the future DSpace will probably be used. MS5050 is a product from the German company Hydrotechnik.

Multi-System 5050 is a hand held computer with a graphical interface, see Fig-ure 5.1. It is possible to connect up to six measFig-urement signals. Channel one to channel four should have analogue input. Channel five and six should have pulses as input, which will be converted to the corresponding frequency of the pulses by MS5050. If all six inputs are used with the frequency 1000 Hz (e.g. storing one value every ms) the memory can store values for approximately 2 minutes. A USB-interface is available in order to get the stored values from MS5050 into a PC. The data files are then converted to excel files which are easily accessible from Matlab. 

### Sensors

Komatsu has sensors for measuring rotation speed, pressure, and electrical sig-nals. The rotation sensor uses an infrared signal to measure the rotation. The signal is sent out from the sensor, then it bounces on a reflex attached to the rotating part, and is then detected by the sensor. It is possible to attach more reflexes on equal distances on the rotating part to pick up changes in the rota-tion quicker. The sensor is placed on a magnet that makes it possible to put

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26 Chapter 5. Data collection

Figure 5.1: Multi-System 5050, the measurement tool used in this project.

the sensor close to the rotating part.

The forest machine has several positions where it is possible to connect pressure sensors. The cap is unscrewed and replaced by the sensor, see Figure 5.2. The sensor delivers an analogue current signal to the MS5050 unit.

Figure 5.2: One of the possible positions where a pressure sensor can be placed. In this case it is on the hydraulic motor.

The interface of the electrical sensor has many similarities with the pressure sensor. The electrical sensor is also analogue and has contacts on the vehicle where it can be connected.

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5.2. Test day 27

### Test day

The test area was located in J¨amteb¨ole, forty kilometers northwest of Ume˚a.

### Sensor locations

As mentioned in Section 5.1 there are two frequency inputs on a MS5050-unit, which were used for rotation sensors. One sensor measured the rotation of the diesel engine, although that one is connected to the hydraulic pump, which means they have the same speed. The other rotation sensor measured the speed of the drive shaft. The rotation ratio between the hydraulic motor and the drive shaft is 5.17:1 when the low gear is used and 1.67:1 for the high gear. These are all the interesting rotations in the forest vehicle, so there would not have been any use for more rotation sensors.

Two electrical sensors were used. They were placed to measure the signals to the pump and the motor that command which angle the swashplates should have.

The two last positions were used for pressure sensors. One should of course be placed to measure the charge pressure to the motor. The other was placed after the motor to measure the pressure of the return flow. This setup has one big advantage; the pressure drop over the motor is available. Some other pressures could also have been of interest, for example to see if a valve is open or closed. If there had been more time available for testing, the other interesting pressures could have been measured by replacing the pressure sensor for the return flow.

### Performed tests

Test 1-7: Repetitive tests at three different speeds were performed, three for the middle speed and two for the other speeds. The reason was to check if the same signals were obtained when the test was repeated. One further motive for these tests was that it hopefully would be easier to achieve the efficiencies for the hydraulic transmission. One example of a test is shown in Figure 5.4.

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28 Chapter 5. Data collection

Figure 5.3: The track at which the tests where performed.

Test 8-9: In test 8 the goal was to perform velocity steps up and in test 9 steps down. These tests were made to get dynamic information. Previously described steps were not easily performed, but hopefully they are good enough.

Test 10-12: These tests were done on the second gear and at the same speeds as tests 1-7. Under ideal circumstances this would give the relationship for the friction in the motor under different rotation speeds.

10 15 20 15 20 25 30 35 Time [s]

Return pressure [Bar]

10 15 20 0 50 100 150 Time [s]

Charge pressure [Bar]

10 15 20 0.4 0.6 0.8 1 Time [s]

Current to the pump [mA]

10 15 20 0.2 0.4 0.6 0.8 1 Time [s]

Current to the motor [mA]

10 15 20 1340 1360 1380 1400 1420 Time [s]

Rotation speed pump [rpm]

10 15 20 1000 1050 1100 1150 1200 Time [s]

Rotation speed pump [rpm]

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5.2. Test day 29

### Modifications of the data

In some of the measuring points the rotation sensor on the drive shaft missed the reflex resulting in that the rotation value was too low. In these cases the incorrect value has been replaced with the value of the previous sample. This will not affect the data because the sampling frequency 10 Hz is used and the rotation of the motor is pretty constant most of the time.

Because of the rough road, some of the data can be hard to get good estimates from. However, it can be interesting to see what happens in the signals when for example a stump is hit.

As can be seen in Figure 5.4, the pressures and currents are measured with high frequency, 1000 Hz. The rotations are measured with 10 Hz. The current signals look pretty messy, and the reason for that is that they are pulse width modulated. To make the signals smoother and more useful they have to be filtered. Different Butterworth filters have been used for this purpose. Figure 5.5 shows the same data as in Figure 5.4 but filtered.

10 15 20 25 25.5 26 26.5 27 Time [s]

Return pressure [Bar]

10 15 20

50 100 150

Time [s]

Charge pressure [Bar]

10 15 20 0.6 0.62 0.64 0.66 Time [s]

Current to the pump [mA]

10 15 20 0.41 0.42 0.43 0.44 0.45 Time [s]

Current to the motor [mA]

10 15 20 1340 1360 1380 1400 1420 Time [s]

Rotation speed pump [rpm]

10 15 20 1000 1050 1100 1150 1200 Time [s]

Rotation speed pump [rpm]

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## Simulations and Results

The essence of the work done in this thesis will be presented in this chapter. First some adjustments of the mathematical model in Chapter 3 will be done, and after that different ways to simulate the model will be explained.

### Modifications

This section will add some extra information about the forest vehicle that affects the model. It will also explain some previous statements.

### Swashplate angle

The amount of oil flowing into and out from the hydraulic pump and motor is determined by the displacement. The displacement is a linearly function of the swashplate angle. As seen in Section 5.2.1 neither the displacements nor the swashplate angles are measured. Instead the electrical control signals to the pump and the motor are measured. Because of this, the relationship between the current and the displacement is needed. The swashplate angle does not only depend on the current but also on the pressure difference over the pump and the rotation of the pump. For the experiments in this thesis the hydrostatic transmission was a new prototype with a new pump. This new setup was pretty tolerant to changes in the rotation speed making it possible to neglect the in-fluence of the pump rotation on the swashplate angle.

A hydraulic override exists in the system. It exists to avoid damage on the transmission in case the pressure goes up. When the pressure is over 300 bar the override decreases the angle of the pump. In test run 1-9 the pressure is al-most always under 300 bar but in test run 10-12 it is sometimes over. Since this hydraulic override is not included in the model, the time periods with several pressure peaks well over 300 bar have been excluded. Changes in displacement

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32 Chapter 6. Model modifications, Simulations and Results

due to changes in pressure will therefore be disregarded.

Now to the displacement’s function of the currents. The discussion is based on the experience available at Komatsu. The pump starts to move the angle of the swashplate from zero when the current exceeds 300 mA. The function is linear to the pump’s maximum displacement at 147 cm3 which is reached at 800 mA. Contrary to the pump the motor starts with maximum displacement, 160 cm3, when the current is less then 330 mA. Minimum displacement for the motor, 68 cm3, is achieved at 560 mA. However, this function is not linear. The displacement’s function of the currents are shown in Figure 6.1.

0.2 0.4 0.6 0.8 1

0 50 100 150

Current to pump [A]

Displacement in pump [cm3] 0.2 0.3 0.4 0.5 0.6 0.7 60 80 100 120 140 160

Current to motor [A]

Displacement in motor [cm3]

Figure 6.1: Illustration of the functions from currents to displacements for the pump respectively the motor. Note that the two have contrary reactions to a rise in current.

### Flows from valve and leakage

As written in Section 3.4 the flow through the security valve Qvand the leakage Qlcan be set to zero. The leakage Qlis really small, and definitely negligible in comparison to the normal flow through the hose. In contrast to Ql, Qv can be large. The security valve is there to make sure that the hydrostatic transmission does not break if something happens. The valve opens at 415 bar and therefore Qv can be set to zero because the high pressure starts to affect the swashplate angles before the security valve opens. Therefore that data have already been excluded as explained in Section 6.1.1.

### Return pressure

A model for the charge pressure (3.8) was built in Chapter 3, but we have no dynamic model for the pressure in the return hose. One reason for this is that the charge pump is difficult to model and would result in more unknown parameters since much information about the charge pump and the valves are missing. The most important reason to skip a model for the return pressure is however in this thesis that the pressure is quite constant. The pressure will have very small variations over time and will not be noticed in comparison to the forward pressure. Still, it happens that the return pressure makes a peak,

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6.1. Modifications 33

which probably is when the forwarder hit a stump in the track. For this reason the measurements of the return pressure will be used directly as input instead.

### Efficiencies

The model developed in Chapter 3 uses efficiencies. The efficiencies are pro-vided by Sauer Danfoss in a data sheet that is attached in Appendix A. These efficiencies are given for the old system, although it is reasonable to assume that the different systems have quite similar structure. Even so it must be kept in mind that this new pump has a bigger volume then the old one.

There are two kinds of efficiencies, volumetric efficiencies and torque efficiencies. The parameters ηtp and ηtm represent the torque efficiency for the pump and for the motor respectively. From the data sheet in Appendix A it is seen that the torque efficiencies have fairly constant levels over the working range of the pump and the motor and will therefore be set to constants

ηtp= 0.92, ηtm= 0.97.

The volumetric efficiencies render larger problems. The variations of the effi-ciencies are larger and the effieffi-ciencies given for the pump and the motor are not perfectly covering all driving cases. For the pump the efficiency is given as a function of displacement, but just for the rotation speed 2100 rpm. Most of the time the pump rotation is in the range of 1100 - 1600 rpm. For the motor it is the opposite situation, then the displacement is fixed and the function of the rotation speed is given. This is true except for really high speeds, because then the displacement decreases at the same time as the rotation increases. To get around this problem the rotation dependency of the motor will be used to adjust the volumetric efficiency for the pump. This concept is used the other way around as well.

To implement this, two dimensional lookup-tables are used. The tables for the pump and the motor have the same structure and only the values differ. Three levels for the displacement are used: minimum, maximum and an appropriate level in between. The rotation has five levels. The lowest level is zero rotation and the highest is higher than any measured rotation. That will ensure that no values are outside the range of the table. The other three levels are set to cover most of the driving cases.

The efficiency values for the highest rotation level are set equal to the efficiency values for the second highest rotation level. This is also done for the efficiency values for the lowest and second lowest rotation levels. The reason is to make it easier to optimize the efficiency values later on. The discussion above has resulted in two Tables, 6.1 and 6.2.

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34 Chapter 6. Model modifications, Simulations and Results

Pump, Rotation speed [rpm] 0 1100 1350 1600 2500 Displacement, 0 cm3 0.56 0.56 0.58 0.59 0.59 Displacement, 85 cm3 0.84 0.84 0.86 0.87 0.87 Displacement, 147 cm3 0.90 0.90 0.92 0.93 0.93 Table 6.1: Volumetric efficiency for the pump, ηvp, given in a table.

Motor, Rotation speed [rpm] 0 600 1200 2000 4000 Displacement, 68 cm3 0.77 0.77 0.88 0.97 0.97 Displacement, 105 cm3 0.81 0.81 0.91 0.97 0.97 Displacement, 160 cm3 0.85 0.85 0.94 0.97 0.97 Table 6.2: Volumetric efficiency for the motor, ηvm, given in a table.

### Simulations for the pressure

A simulation model for the charge pressure in the hose was the main goal of this project. Measurements of rotations and currents that can be transformed to displacements of the pump and the motor are available. Using this information it should be possible to simulate the pressure using

Qp= Dpωpηvp, Qm= ωpDm ηvm , p = β V Z t 0 (Qp− Qm) dτ ,

where the volume of the hose V can be calculated as

V = LD 2

4 π ≈ 600cm

3. (6.1)

The parameter L stands for the length of the hose and D is the diameter of the hose. All parameters are known with more or less accuracy except for β. The bulk modulus β depends on which oil that is used. The range of the bulk mod-ulus is 107-1010 Bar. The flows are not affected by β, only the rate of change in pressure.

An estimate of β should make it possible to simulate the pressure for the sys-tem. Unfortunately it does not. The simulated pressure can be in the same range as the measured pressure in one or a couple of the test runs, but for the

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6.3. Black box models 35

rest of the test runs the simulated pressure goes to either plus or minus infinity. The problem is that the simulated flow out from the pump is not equal to the simulated flow into the motor when the measured pressure is constant and the measurement values for the rotations are used for the calculations.

One idea that could improve the simulation results would be to use some iden-tification method from Chapter 4 to estimate better values for the parameters. The two efficiency parameters ηvp and ηvm are the most uncertain ones, but also the functions from currents to displacements are uncertain. There are two problems when the efficiencies should be estimated, as seen in (3.17). The ef-ficiencies probably depend on many variables and are non-linear. Some linear models have been tested but with negative result. Non-linear identification is necessary to get a better estimate for the efficiencies. Due to time limits non-linear identification has not been tested in this thesis, instead the efficiencies from Section 6.1.4 have been used.

Even if the parameters would be improved the simulated pressure will not be ac-curate for all test runs. This statement may seem strange, but when considering that the pressure is an integral over time of two flows it is quite obvious why it will not work. These flows are only dependent on parameters and measurements and there is no feedback involved in this part of the system. Consequently, if a stationary error is present, for example the simulated input flow is larger than it is supposed to be, then the pressure will go to infinity. Therefore, this approach is unrealistic since perfect models are not possible to build.

To continue it is necessary to realize that the problem is that there is no feed-back from the simulated pressure. One idea to solve the problem is to add the models for the pump and the motor. If the pressure in the hose goes up the resistance for the pump will increase and it will therefore rotate slower. This in turn will result in a decreased flow as well as a lower pressure. On the motor side the same thing will happen, the higher pressure will rotate the motor faster and the flow out of the hose increases, with lower pressure as a result.

To summarize, it can be realized that if only a model for the hose is used, it will make the simulations unbounded. Hopefully models on the sides of the hose will create that necessary feedback to make the simulated pressure stable.

### Black box models

The advantage with a black box model is that no knowledge about the system is necessary. Instead the computer has the freedom to adjust a model to fit measurement data. Still there are choices that need to be considered; what kind of model should be used, what the model order should be and which data the model should be based on. In this thesis the graphical interface for system identification toolbox in Matlab  has been used to create the black box models. SITB uses signal processing to estimate the black box models with the

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