Modelling of Test Bench for Road Load Simulation

Master of Science Thesis in Electrical EngineeringDepartment of Electrical Engineering, Linköping University, 2017

Modelling of Test Bench forRoad Load Simulation

Dennis Åberg Skender

Master of Science Thesis in Electrical Engineering

Modelling of Test Bench for Road Load Simulation

Dennis Åberg Skender

LiTH-ISY-EX--17/5092--SE

Supervisor: Angela Fontanisy, Linköpings universitet

Kristin BäckströmToyota Material Handling

Kristian OhlssonToyota Material Handling

Examiner: Svante Gunnarssonisy, Linköpings universitet

Division of Automatic ControlDepartment of Electrical Engineering

Linköping UniversitySE-581 83 Linköping, Sweden

Copyright © 2017 Dennis Åberg Skender

Sammanfattning

Lagertruckar drivs ofta av batterier. Batterikapaciteten kan testas genom att an-vända en testbänk där trucken körs i en körcykel. För att simulera samma has-tighetskurva på testbänken som trucken erhåller vid riktig körning på väg måstetestbänken kunna simulera vägmotstånd.

I detta examensarbete modelleras en testbänk i Simulink genom grey-box model-lering och valideras med uppmätt data. Även en hastighetsregulator utvecklasoch implementeras i modellen för att simulera vägmotstånd. Simuleringar medmodellen och hastighetsregulatorn visar på hög träffsäkerhet jämfört med upp-mätt data. Resultaten visar dock på att den monterade momentsensorn inte äroptimalt belägen och att växeloljans temperatur är av intresse att mäta för attkunna modellera friktionen som en funktion av temperaturen.

iii

Abstract

Warehouse forklifts are often powered by batteries. By using a test bench wherethe forklift can be driven in a certain driving-cycle, the battery capacity can betested. To obtain the same speed curve on the test bench as a forklift obtainswhen driving on a real road, the test bench must be able to simulate road load.

In this master’s thesis, a test bench is modelled in Simulink using grey-box mod-elling and validated with measured data. Also, a speed regulator is developedand implemented in the test bench model to simulate road load. Simulations withthe model and the speed regulator show high accuracy when compared againstmeasured data. However, the results show that the pre-attached torque sensoris not optimally located, and that the gear oil temperature is of interest to mea-sure to be able to model the friction torque as a function of the temperature.

v

Acknowledgments

This thesis has been a roller coaster with both ups and downs. During this ride,I have received a lot of help from my supervisors at Toyota Material Handling,Kristian Ohlsson and Kristin Bäckström, together with the rest of the staff atToyota Material Handling who I would like to thank for all the support and help.I would also like to thank Svante Gunnarsson and Angela Fontan at LinköpingUniversity for reading, correcting and discussing the thesis over and over againuntil it made any sense, thank you.

Secondly, I would like to thank my mother, father and brother for all the sup-port and love they have given me during these five years at Linköping and forremaining me that there is a life beside school.

Lastly, I would like to thank all the friends and staff that I have met during mystudies and my girlfriend Lina. Without you, I would have never been able to getthrough this education.

Norrköping, October 2017Dennis Åberg Skender

vii

Contents

Notation xi

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.2 Road Load Simulation . . . . . . . . . . . . . . . . . . . . . 4

1.5 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 System Description 72.1 Test Bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Pre-equipped Controllers . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Drum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Modelling of Test Bench 113.1 Pre-equipped Controllers . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Drum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Temperature Dependency . . . . . . . . . . . . . . . . . . . . . . . 193.6 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 213.7 Model Evaluation and Validation . . . . . . . . . . . . . . . . . . . 23

3.7.1 Model Validation with Data from Program 1 . . . . . . . . 233.7.2 Model Validation with Data from Program 2 and Program 3 27

3.8 Validation using Normalised Root-Mean-Square Deviation and Sumof Relative Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Road Load Simulation 35

ix

x Contents

4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Speed Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Measurement Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4 Estimation of Forklift and Road Load Parameters . . . . . . . . . . 424.5 Evaluation of Road Load Simulation . . . . . . . . . . . . . . . . . 44

5 Conclusions 475.1 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . 475.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

A Appendix 51

Bibliography 57

Notation

Nomenclatures

Nomenclature Meaning

a(t) Accelerationα Backlash gapb Time delayc Shaft damping constantCrr Rolling resistance constantCrr,v Rolling resistance constant dependent on velocityγ Inclination angleδg Friction constante(t) Electromotive forcefg Friction constantfg,c Friction constantFd(t) Drum forceFw(t) Forklift forceg Gravitation constantG Shear Modulus

ηf orklif t Forklift efficiencyηg Gear efficiencyθbase Base angular velocityθd(t) Drum angleθd(t) Drum angular velocityθd(t) Drum angular accelerationθg (t) Gear angleθg (t) Gear angular velocityθg (t) Gear angular accelerationθgd(t) Shaft angle between gear and drumθm(t) Motor angleθm(t) Motor angular velocityθm(t) Motor angular acceleration

xi

xii Notation

Nomenclatures


θmg (t) Shaft angle between motor and gearθmg (t) Shaft angular velocity between motor and gearθw(t) Forklift angular velocityia(t) Armature currentif (t) Field currentJd Drum moment of inertia

Jf orklif t Forklift moment of inertiaJg Gear moment of inertiaJgd Shaft polar inertia between gear and drumJmg Shaft polar inertia between motor and gearJm Motor moment of inertiaka Armature constantkm Motor constant

kstif f ,gd Shaft stiffness constant between gear and drumkstif f ,mg Shaft stiffness constant between motor and gearK1 Magnetic flux constantK2 Magnetic flux constantKm Magnetic flux constantlgd Shaft length between gear and drumlmg Shaft length between motor and gearLa Armature inductanceLf Field inductancem Forklift mass

ξf orklif t Forklift gear ratioξg Gear ratiord Drum radiusrgd Shaft radius between gear and drumrmg Shaft radius between motor and gearrw Forklift wheel radiusRa Armature resistanceRf Field resistanceT (t) Electromagnetic torqueTd(t) Drum torqueTf (t) Friction torque

Tf orklif t(t) Forklift motor torqueTg (t) Gear torqueTgd(t) Shaft torque between gear and drumTL(t) Load torqueTm(t) Motor torqueTmeas(t) Measured torque

Notation xiii

Nomenclatures


Tmg (t) Shaft torque between motor and gearTRL(t) Road load torqueTw(t) Forklift wheel torqueua(t) Armature voltageuf (t) Field voltagev(t) VelocityΦ(t) Magnetic flux

Abbrevations

Abbrevation Meaning

DC Direct CurrentNRMSD Normalised Root-Mean-Square Deviation

PI Proportional-IntegralPLC Programmable Logic Controller

RMSD Root-Mean-Square DeviationSEDM Separately Excited DC Motor

1Introduction

This master’s thesis has been carried out at Toyota Material Handling Manufactur-ing Sweden AB in Mjölby and the division of Automatic Control, the departementof Electrical Engineering at Linköping University. This chapter will explain themain subject of the thesis together with related research within the area.

1.1 Background

Toyota Material Handling is one of the largest forklift manufacturers in the world.They produce outdoor forklifts, warehouse forklifts and hand pallet trucks. InMjölby the company focuses on battery powered warehouse trucks, hand pallettrucks and specially designed trucks. The Test and Verification department is re-sponsible of testing and verification of subsystems and complete vehicles. Biggerfocus on energy-efficient products requires new effective and repeatable analysisand measurement methods of power consumption. Warehouse forklifts are oftenpowered by batteries, which means that minimisation of power consumption isof big interest. By minimising the power consumption, battery charging can bedone less frequently. This benefits companies since the forklifts’ effective work-ing time increases, which means more profit for each bought or rent forklift. Italso benefits the climate since lower power consumption and less frequent charg-ing imply longer battery life span i.e. less battery consumption and hence lessemissions and pollution from battery manufacturing.

One way to test the power consumption is to drive the forklift in a special driving-cycle until the battery is empty. This can be done by having an operator drivingthe forklift. However, this is expensive and time consuming. Also, equal repeata-bility of the driving-tests is very difficult. Instead a test bench that simulates road

1

2 1 Introduction

load can be used.

The test bench the thesis will focus on contains a pre-installed controller con-nected to a separately excited DC motor which through a shaft is connected to agear. This gear is then connected to a drum via a shaft which the forklift’s drivingwheel is spinning on. At present, the test bench is controlled with a PLC by feed-ing the DC motor with different speeds and then map the outgoing signals fromthe motor and drum. This works well for measurements where different types ofspeeds are tested separately. However, to be able to test dynamical processes inthe test bench and simulate rolling resistance, a good model and controller forthe test bench must be developed.

1.2 Problem Formulation

The goal of this master thesis is to create a complete model of the test bench.The second step is to develop a regulator for the test bench to make it possibleto reproduce dynamic processes and simulate rolling resistance. The aim is tocontrol the torque from the DC motor to the drum when a forklift mounted onthe drum is accelerating so that the given torque from the drum to the forkliftsimulates the rolling resistance a road gives and hence the correct speed curve onthe forklift can be obtained. The model will be implemented and controlled inSimulink and MATLAB.

A schematic image of the different conditions that the test bench should be able tosimulate is shown in the left part of Figure 1.1, while the right part of Figure 1.1shows the forklift mounted on the test bench drum.

Figure 1.1: Left figure shows the different conditions that the test benchshould be able to simulate including steady state, acceleration, decelerationand inclination. Right figure shows the forklift mounted on the test benchdrum where these conditions should be simulated.

The following sub-goals in the thesis are:

• Create a model of the complete test bench. This includes the pre-equippedcontrollers, the separately excited DC motor, the gear, the shafts and thedrum. Validate the model with measured data.

• Implement data of delivered torque from a forklift measured during driv-ing on road and design and evaluate a regulator that can control the DC

1.3 Delimitations 3

motor so that the correct speed curve on the drum is obtained when theforklift acts as an input signal to the system.

1.3 Delimitations

There are a few different delimitations that exist and are considered in the thesis.

• The interval for the input voltage to the motor is [-400,400] V while theinterval for the armature current in the motor is [-72,72] A. The speed in-terval for the DC motor is ± 4830 rpm while the speed interval for the pre-equipped control system is 0 − 4830 rpm. The test bench is therefore mod-elled to work in the speed interval ± 4830 rpm but validated in the interval0 − 4830 rpm.

• The friction in the gear is dependent on the gear oil’s viscosity and hencetemperature, which cannot be measured. Therefore, the developed Simulinkmodel will be evaluated using validation data at a certain approximate geartemperature. Hence, the Simulink model will not necessarily work equallywell at other gear temperatures.

• The DC drive connected to the DC motor uses a 3-phase thyristor bridgecontrolled by a phase-controlled rectifier to feed the DC motor’s armaturevoltage. This 3-phase thyristor will be neglected and therefore not mod-elled.

• There are limited measured data from driving the forklift on road, i.e., thereare limited possibilities for validation of the model and the simulation ofroad load. Hence, the simulation of road load will be validated for steadystate, acceleration and deceleration phase only.

Another limit is time. Since this thesis is made during a limited number of weeks,simplifications and approximations in the models will be considered.

1.4 Related Work

A number of other theses, scientific papers and books have developed differentmodels and control strategies within this area. By reading these and take knowl-edge of their results and information, a well chosen selection of models and strate-gies can be made.

1.4.1 Modelling

In a separately excited DC motor (SEDM), the armature and field circuits arefed from different voltage sources and can thus be controlled separately. In apermanent magnet DC motor, only the armature circuit can be controlled. In [7],a linear model of a SEDM is explained.

For combined armature and field controlled DC motors, the nonlinear transient

4 1 Introduction

behaviour and the magnetic core saturation should be taken into account to cor-rectly model the nonlinear characteristics. Hence, a more advanced mathemati-cal model of a SEDM is developed in [1] where the magnetic flux is modelled asa function of the field current based on experimental data.

Mathematical models for gears, shafts and different types of rotating systems aredeveloped in [8]. The book also contains approaches to filter data, parameterestimation methods and validation methods.

The most common types of static and dynamic friction models are derived andstudied in [14].

Models of backlash and gearplay in gears are derived, simulated and evaluatedin [13].

PI-controllers are explained and modelled in [3].

1.4.2 Road Load Simulation

The basic structures of how to simulate inertia and road load using a chassis dy-namometer are described in [2]. The paper derives a control strategy for bothspeed regulating dynamometer control and torque regulating dynamometer con-trol.

Two different ways to control a driveline test bench are analyzed in [5]. Thefirst method is to control the resistive torque values from a predetermined speedprofile or drive cycle. The other method is to control the desired speed by usingtorque feedback from the system. The first method works well if speed profiledata is known before testing of the system. However, if the aim is to use thetest bench without using a predetermined speed profile, the second method is ofmore interest to investigate.

In [6] an inertia and road load simulation test bench with speed control is devel-oped and explained using feedback from measured speed and torque.

1.5 Method

First, the most basic physical models that could describe the system were im-plemented in Simulink and compared against measured data. The models werethen replaced or modified to better describe the real system. During these tests,parameter estimation was also done. The complete model was then validatedusing different running scripts on the test bench.

When a sufficiently accurate model was obtained, estimation of road load param-eters using measured data was made. The control system for regulation of speedcould then be developed and validated using measured data.

1.6 Thesis Outline 5

1.6 Thesis Outline

The thesis is divided into four chapters. Chapter 2 contains a short summariseddescription of the main parts included in the test bench.

Chapter 3 contains the modelling of the test bench where used physical modelsare described and evaluated using validation data. The dependency of tempera-ture and measurement noise is also discussed and analysed.

The control strategy for speed control is described and implemented in Chapter 4.This chapter also includes parameter estimation and validation of the developedcontrol system.

Finally, Chapter 5 contains discussion and conclusions of the obtained resultstogether with examples of future work.

2System Description

A brief description of the complete system and its main parts is presented in thissection.

2.1 Test Bench

The test bench consists of three main parts and two pre-equipped controllers. ADC motor with separate excitation fed by two PI-controllers, a gear and a drum.A picture of the test bench is shown in Figure 2.1 and a simple block diagram isshown in Figure 2.2. By applying a reference angular velocity to the controllerand thereby feeding the DC motor with a voltage, a torque can be obtained fromthe motor which through a shaft is transmitted to a gearbox. The gearbox in-creases the input torque to the drum by decreasing the output speed on the shaftconnected to the drum. There is also a torque sensor mounted on the shaft be-tween the motor and the gear.

On the drum there is a measuring wheel that measures the number of revolutions.This number can be scaled to the drum’s angular velocity.

7

8 2 System Description

Figure 2.1: Test bench including DC motor (1), torque sensor (2), gear (3)and drum (4).

Separately Excited

DC Motor

ua

θm

Gear

Pre-equipped Controllers ia

θref

Drum

Shaft

Shaft θd θg

Td

θmg

Tmg Tg

Tgd

θgd θmg

Tg

.

Figure 2.2: Principal outline of the complete system including pre-equippedcontrollers, electric motor, gear, drum and shafts.

2.2 Pre-equipped Controllers 9

2.2 Pre-equipped Controllers

The DC motor is pre-equipped with a PLC connected to a DC driver that usesclosed loop control of armature current ia(t) and motor angular velocity θm(t).The control system has a cascade control structure where the input is the speedreference and the output is the voltage fed to the DC motor. The controllers alsosaturate the reference angular velocity, the controlled armature current and thefed armature voltage.

2.3 Motor

A DC motor is separately excited if the field winding is physically and electricallyseparate from the armature winding, see Figure 2.3 and [10]. For a DC motor, themaximum speed is determined by the maximum supplied voltage to the motor.To be able to reach speeds higher than the base speed without increasing the sup-plied voltage, the magnetic flux produced by the armature and field current isweakened in a SEDM by decreasing the field current. This is called field weaken-ing and means that the produced torque will also be weakened at higher speeds,see Figure 2.4. Under the motor’s base speed the given torque will be constantwhile over the given base speed the torque will be reduced. The power will there-fore be linear under base speed and constant over base speed.

uf Lf

Rf

e

Ra

La

ua

+

+

-

-

Tm , θmField Armature

ia

if

Figure 2.3: Electric circuit of a separately excited DC motor. uf (t), Rf , if (t)and Lf is the field voltage, resistance, current and inductance respectivelywhile ua(t), Ra, ia(t) and La is the armature voltage, resistance, current andinductance respectively. Tm(t) is the produced motor torque, θm(t) is themotor angle and e(t) is the electromotive force.

10 2 System Description

Torque

Speed

Constant Torque Region Constant Power Region

Base Speed

Figure 2.4: The plot shows the torque and speed characteristics for a SEDMat field weakening.

2.4 Gear

Gears are used to increase or decrease the ratio between the input speed and theoutput speed. The gear in the system is a helical-bevel 3-stage gearbox with agear ratio of 1:62.60, see [17]. This means that 62.6 turns on the input shaft equalone turn on the output shaft. However, the friction in the gear is temperaturedependent since the gear oil’s viscosity depends on the temperature. The gearwill therefore be modelled using data from experiments where the test bench hasbeen warmed up for a certain time. The gear also has backlash which will betaken into consideration.

2.5 Drum

The drum is the part where the forklift’s wheel is spinning on during road loadsimulation. It has a diameter of 1.29 meters and is made of heavy steel. The drumis affected by the torque given from the shaft between the gear and the drum. Thefriction in the bearings between the shaft and drum is assumed to be small andtherefore neglected.

3Modelling of Test Bench

The complexity that modelling often implies can be reduced by dividing the com-plete system into subsystems. The reference angular velocity to the DC motor isknown and the motor’s angular velocity and armature current from the DC motorcan be measured. Also, the torque in the shaft connected between the motor andthe gear can be measured with the mounted torque sensor. Moreover, the drumangular velocity can be measured. However, the torque given from the gear tothe drum cannot be measured.

3.1 Pre-equipped Controllers

The test bench is pre-equipped with a cascade controller including two PI-controllersthat regulate the speed and the current given by the motor, as seen in Figure 3.1.The controllers are defined by equation (3.1), where the speed controller param-eters are Kp,speed = Kp and Ti ,speed = Ti while the current controller parametersare Kp,current = Kp and Ti ,current = Kp/Ti . These parameters are all known andgiven by the control system. e(t) indicates the error.

u(t) = Kp ·(e(t) +

1Ti

t∫t0

e(τ)dτ)

(3.1)

Inserting the parameters into the general equation (3.1) yields equation (3.2) and(3.3).

11

12 3 Modelling of Test Bench

iref (t) = Kp,speed ·(θerror (t) +

1Ti ,speed

t∫t0

θerror (τ)dτ)

(3.2)

ua(t) = Kp,current ·(ierror (t) +

1Ti ,current

t∫t0

ierror (τ)dτ)

(3.3)

uaPI-Controller PI-Controller

ierrorirefθerrorθref

iaθm

∑ ∑

. .

.

Figure 3.1: Block representation of the controllers connected to the DC mo-tor that regulates the amount of armature voltage ua(t) to the motor to reachthe reference angular velocity θref (t) controlled by a PLC.

The speed reference θref (t) is saturated at 105% of the maximum motor speedθm,max, the current reference iref (t) is saturated at ±ia,max and the voltage refer-ence which is the armature voltage ua(t) is saturated at ±ua,max. Hence, the ref-erence signals for the controllers can be written as equation (3.4), (3.5) and (3.6).The Simulink model of the controllers can be seen in Figure A.1 in Appendix A.

ua(t) =

Kp,current ·

(ierror (t) + 1

Ti ,current

∫ tt0ierror (τ)dτ

), |ua| ≤ ua,max

ua,max, ua > ua,max−ua,max, ua < −ua,max

(3.4)

iref (t) =

Kp,speed ·

(θerror (t) + 1

Ti ,speed

∫ tt0θerror (τ)dτ

)|iref | ≤ ia,max

ia,max iref > ia,max−ia,max iref < −ia,max

(3.5)

θref (t) =

θref (t) |θref | ≤ θm,maxθm,max θref > θm,max−θm,max θref < −θm,max

(3.6)

3.2 Motor 13

The signal from the PLC to the controllers is also delayed for a certain time. Thisdelay is modelled as equation (3.7) where θref ,P LC(t) is the signal sent from thePLC and b is the time delay estimated from measured data.

θref (t) = θref ,P LC(t − b) (3.7)

3.2 Motor

A separately excited DC motor with constant field current and variable armaturecurrent can basically be modelled as an ordinary DC motor, see [15]. Kirchhoff’svoltage law on the armature circuit in Figure 2.3 gives equation (3.8).

ua(t) − e(t) = Ra · ia(t) + La ·dia(t)dt

(3.8)

A schematic image of the motor is seen in Figure 3.2. The motor armature hasa moment of inertia Jm. As seen in Figure 2.2, the motor is affected by the loadtorque TL(t) which is the torque from the shaft connected to the gear Tmg (t). Themotor friction is assumed to be small and therefore neglected. Hence, the com-plete torque equation for the motor is given by equation (3.9) where Tm(t) is themotor torque and T (t) is the electromagnetic torque.

θm T

Jm

Figure 3.2: Schematic image of the motor where T (t) is the electromagnetictorque, Jm is the motor moment of inertia and θm(t) is the motor angle.

T (t) = Tm(t) + TL(t) (3.9)

Inserting the motor angular velocity θm(t) and the moment of inertia Jm in equa-tion (3.9) gives equation (3.10).

T (t) = Jm ·dθm(t)dt

+ Tmg (t) (3.10)

The electromagnetic torque can be calculated by multiplying the magnetic fluxΦ(t), the armature current ia(t) and the armature constant ka while the back elec-tromotive force is calculated by multiplying the angular velocity θm(t), the mag-


netic flux Φ(t) and the motor constant km. However, since the numerical valuefor the armature constant ka is equal to the motor constant km in SI units, theycan be written as the same constant k. Hence, the electromagnetic torque T (t) isgiven by equation (3.11) and the back electromotive force e(t) by equation (3.12).

T (t) = k ·Φ(t) · ia(t) (3.11)

e(t) = k ·Φ(t) · θm(t) (3.12)

The magnetic flux is constant for speeds under base speed θbase. Over base speedit decreases since the field current is weakened. The magnetic flux is therefore afunction of the field current and is modelled as equation (3.13) where C1 and C2are constants, see [1].

k ·Φ(t) =

Km, θm(t) ≤ θbaseC1 · if (t)C2+if (t) , θm(t) > θbase

(3.13)

Kirchhoff’s voltage law for the field circuit gives equation (3.14).

uf (t) − Rf · if (t) = Lf ·dif (t)

dt(3.14)

However, since the field voltage uf (t) is fed by an independent supply input, theresistance Rf can vary and the inductance Lf in the field is unknown, the mod-elling of the field current is difficult. Instead, since the field current is constantunder base speed and weakened over base speed, it can be modelled as a func-tion of the motor speed. By measuring the field current if (t) at different angularvelocities θm(t), the relation between the field current and the motor angular ve-locity can be calculated. According to [9] and as seen in Figure 2.4, there exists aninversely proportional relation between the torque, and hence the field current,and the angular velocity for speeds over base speed. Measurements of field cur-rent for different speeds prove that this is the case, see Figure 3.3 which showsthe relation between measured field current and motor angular velocity.

3.2 Motor 15

2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

Angular Velocity m

(RPM)

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Fie

ld C

urr

en

t i f (

A)

Figure 3.3: The plot shows the relation between field current and angularvelocity for the SEDM at field weakening.

The relation between the field current and the motor’s angular velocity is there-fore modelled as equation (3.15) where C3 is a constant. By inserting equation(3.15) into equation (3.13) and rearranging the constants and variables, the mag-netic flux Φ(t) results to be dependent on the angular velocity θm(t) as shown inequation (3.16) where K1 and K2 are constants.

if (t) =C3

θm(t)(3.15)

k ·Φ(t) =K1

K2 + θm(t)(3.16)

Since the test bench should be able to rotate in both directions, equation (3.16) isrewritten as equation (3.17).

k ·Φ(t) =K1

K2 + |θm(t)|(3.17)

Moreover, the inequalities in equation (3.13) must be taken into consideration.Hence, the magnetic flux is modelled as equation (3.18), where k ·Φ(t) is a con-stant value until k ·Φ(t) is lower than Km.

k ·Φ(t) =

Km, θm(t) ≤ θbaseK1

K2+|θm(t)| , θm(t) > θbase(3.18)


In conclusion, the complete model of a separately excited DC motor with variablemagnetic flux and variable armature current without dependency on field currentcan be written as equation (3.19). The Simulink model of the motor can be seenin Figure A.2 in Appendix A.

dia(t)dt

=1La

· ua(t) −RaLa

· ia(t) −k ·Φ(t)La

· θm(t)

dθm(t)dt

=k ·Φ(t)Jm

· ia(t) −1Jm

· Tmg (t)

k ·Φ(t) =

Km, θm(t) ≤ θbaseK1

K2+|θm(t)| , θm(t) > θbase

(3.19)

3.3 Gear

The gear is connected to the DC motor through a shaft. It is modelled usingEuler’s second law as shown in equation (3.20) where Jg is the gear’s moment ofinertia, Tg (t) is the torque from the shaft connecting the motor to the gear, Tgd(t)is the load torque from the shaft connecting the gear to the drum, Tf (t) is thefriction and ξg is the gear ratio from the gear. The gear torque Tg (t) is definedin equation (3.21) while the gear angle θg (t) is defined in equation (3.22). Aschematic image of the gear is seen in Figure 3.4.

Jg · θg (t) = Tg (t) − Tgd(t) − Tf (θg (t)) (3.20)

Tg (t) = ξg · Tmg (t) (3.21)

θg (t) =θmg (t)

ξg(3.22)

Since the shaft between the motor and gear is made out of heavy steel and has alarge diameter, the damping constant is large and therefore neglected. Hence, theshaft torque Tmg (t) is modelled as a torsion spring according to equation (3.23)where kstif f ,mg is the shaft stiffness constant, θm(t) is the motor angle, θmg (t) isthe shaft angle connected to the gear’s input and θg (t) is the gear output angle.

Tmg (t) = kstif f ,mg · (θm(t) − θmg (t)) = kstif f ,mg · (θm(t) − ξg · θg (t)) (3.23)

The stiffness constant kstif f ,mg is calculated according to equation (3.24) whereG is the shear modulus of steel, lmg is the length of the shaft and Jmg is the polarmoment of inertia of the shaft given by equation (3.25) where rmg is the shaftradius, [4].

3.3 Gear 17

θg

θm θmg

Jg

g

kstiff,mg

Tf

Tmg Tg

Figure 3.4: Schematic image of the gear where kstif f ,mg is the shaft stiffnessconstant, θm(t) is the motor angle, Jg is the gear moment of inertia, ξg is thegear ratio, Tf (t) is the friction torque, θmg (t) is the shaft angle connected tothe gear’s input and θg (t) is the angle on the gear’s output.

kstif f ,mg =G · Jmglmg

(3.24)

Jmg =π2

· rmg4 (3.25)

The shaft torsion is measured by a torque sensor mounted on the shaft. Sincethe torsion emerges due to load on the opposite side of the shaft, the measuredtorque Tmeas(t) is defined as equation (3.26).

Tmeas(t) = −Tmg (t) = kstif f ,mg · (ξg · θg (t) − θm(t)) (3.26)

The friction in the gear Tf (t) is modelled as a nonlinear viscous friction togetherwith a nonlinear Coulomb friction, i.e., as equation (3.27) where fg,c is the Coulo-mb friction constant, fg is the viscous friction constant and δg is an exponent con-stant to make the viscous friction nonlinear, [14]. The constants will be estimatedusing measured data. The factor sgn(θg (t)) is included to ensure that the frictiontorque Tf (t) has the correct sign with respect to the gear angular velocity θg (t).

Tf (t) = (fg,c + fg · |θg (t)|δg ) · sgn(θg (t)) (3.27)

Hence, the complete model of the gear and shaft is described by equation (3.28).The Simulink model of the gear can be seen in Figure A.3 in Appendix A.


Jg · θg (t) = ξg · kstif f ,mg · (θm(t) − ξg · θg (t)) − Tgd(t)−

(fg,c + fg · |θg (t)|δg ) · sgn(θg (t))(3.28)

3.4 Drum

The shaft between the gear and drum is affected by backlash in the gear. It ismodelled using a modified dead-zone model, [13]. The drum torque Td(t) isdescribed by equation (3.29) where kstif f ,gd is the stiffness of the shaft betweenthe gear and the drum and c is the damping constant in the shaft. ∆θ(t) is definedby equation (3.30) where θg (t) and θd(t) are the angles for the gear’s output anddrum respectively. The backlash equation Dα is defined in equation (3.31) where2α is the total backlash gap. A schematic image of the drum is seen in Figure 3.5.

Td(t) = Tgd(t) = kstif f ,gd ·Dα(∆θ(t) +

ckstif f ,gd

·d∆θ(t)dt

)(3.29)

∆θ(t) = θg (t) − θgd(t) = θg (t) − θd(t) (3.30)

Dα(x) =

x − α, x > α

0, |x| ≤ αx + α, x < −α

(3.31)

θgd kstiff,gd

Jd

θd θg Dα

Tgd Td

Figure 3.5: Schematic image of the drum where kstif f ,gd is the shaft stiffnessconstant, θg (t) is the gear angle, θgd(t) is the shaft angle, θd(t) is the drumangle, Jd is the drum moment of inertia and Dα is the backlash.

The stiffness constant is calculated according to equation (3.32) where lgd is thelength of the shaft and Jgd is the polar moment of inertia of the shaft given byequation (3.33) where rgd is the shaft radius.

kstif f ,gd =G · Jgdlgd

(3.32)

3.5 Temperature Dependency 19

Jgd =π2

· rgd4 (3.33)

Using Euler’s second law, the total torque on the drum can be defined as equation(3.34) where θd(t) is the drum’s angular acceleration and Jd is the drum’s momentof inertia. The Simulink model of the drum can be seen in Figure A.4 in AppendixA.

Jd · θd(t) = Td(t) = kstif f ,gd ·Dα(∆θ(t) +

ckstif f ,gd

·d∆θ(t)dt

)(3.34)

3.5 Temperature Dependency

The gearbox in the system includes three gearwheels installed in a shell made ofcast iron [17]. To reduce the friction between the gearwheels, the gearbox is filledwith gear oil. As the gear oil temperature rises, the viscosity decreases and hencethe friction in the gear decreases. Since the transmitted torque from the motor tothe drum is dependent of the friction in the gear Tf (t), the gear oil temperatureis of interest to measure. However, since there is no available device that canmeasure the gear oil temperature, the temperature is measured outside the gearbox using attached thermocouples on each side of the gear, see Figure 3.6. Hence,by comparing the armature current from the motor ia(t), the measured torqueTmeas(t) and the temperature on the gearbox’s shell under a certain time, it ispossible to make an approximate model of how long it takes for the gear to reacha certain temperature.

Figure 3.6: Thermocouples connected on each side of the gearbox’s shell.Thermocouple 1 in left figure and Thermocouple 2 in right figure.


Figure 3.7 shows how the armature current ia(t) decreases as the temperatureincreases. However, after roughtly 55 minutes the current stabilises to a con-stant value even though the temperature continues to increase. To understandwhy this happens, equation (3.19) and (3.20) describing the motor and the gearmust be analysed. Since the friction term in equation (3.20) decreases as the gearoil’s temperature increases, the armature current ia(t) must be reduced by thePI-controller to obtain constant motor angular velocity θm(t). Hence, the frictiontorque in the gear Tf (t) obtain its lowest value after 55 minutes with a constantθref (t) set to 4025 rpm.

The data used to validate the system and estimate the parameters will be col-lected after pre-running the test bench so that the temperature on the thermo-couples reaches a temperature of approximately 32 C. The friction is there con-sidered low enough.

0 10 20 30 40 50 60

Time (Minutes)

-10

0

10

20

30

40

50

60

70

80

Te

mp

era

ture

(°

C)

an

d C

urr

en

t (A

)

Long time test

Thermcouple 1 (°C)

Thermcouple 2 (°C)

Armature Current ia (A)

Figure 3.7: The plot shows the temperature from thermocouple 1 and 2 andthe armature current produced by the motor during testing with a referenceangular velocity θref (t) set to 4025 rpm.

3.6 Parameter Estimation 21

3.6 Parameter Estimation

The model contains a number of known and unknown constants. By using out-put data from experiments and comparing with the physical model developed inSimulink, unknown parameters can be estimated.

The parameter estimation is done in Simulink using the Parameter EstimationToolbox. The toolbox uses the function lsqnonlin which is a nonlinear least-squaresolver. It calculates the residual ri in equation (3.35) for each data point i wherem is the number of data points, yi is the measured output from the real systemand f (xi , β) is the simulated output from the developed model dependent on thevariables xi that are used in the physical equations used to describe the system,and the parameter vector containing n parameters β = (β1 β2 ... βn)T which areof interest to estimate, [11]. The initial values of the parameters are arbitrarilyguessed.

ri = yi − f (xi , β), i = 1, 2, ... m (3.35)

The solver finds the parameter vector β that makes the model function fit thegiven data in a optimal way in the least squares sense, i.e., minimising the sum ofsquares S in equation (3.36). The minimum value of S is reached when the finalchange in S relative to its initial value is less than a default value of the functiontolerance.

S =m∑n=1

r2i (3.36)

Table 3.1 shows the value for each parameter used in the model together withinformation about the type which indicates if the parameter is given, estimatedor calculated.


Parameter Value Unit TypeRa 0.424 Ω GivenLa 0.004 H GivenJm 0.128 kgm2 GivenKm 1.8506 (Nm/A) (V/(rad/s)) GivenK1 366.83 - EstimatedK2 8.0644 - EstimatedJg 0.02242 kgm2 Estimatedξg 62.6 - GivenG 79.3 · 109 Pa Given

kstif f ,mg 8.1096 · 104 N/m Calculatedlmg 0.6 m Givenrmg 0.025 m Given

kstif f ,gd 1.3493 · 106 N/m Calculatedlgd 0.577 m Givenrgd 0.05 m Givenfg,c 35.989 Nm Estimatedfg 345.38 Nms Estimatedδg 0.4418 - Estimatedc 1.2032 · 105 Ns/m EstimatedJd 312.69 kgm2 Estimatedrd 0.645 m Givenα 0.15073 rad Estimated

Kp,speed 15 - GivenTi ,speed 1 - Given

Kp,current 30 - GivenTD,current 3 - Given

b 0.063947 s EstimatedTable 3.1: Parameter values used in model.

3.7 Model Evaluation and Validation 23

3.7 Model Evaluation and Validation

The parameter estimation and the evaluation of the simulated data comparedwith the measured data are done using 100 Hz sampled measurements of the ar-mature current ia(t), motor speed θm(t), drum speed θd(t) and measured torqueTmeas(t). The data is then zero-phase filtered using the command filtfilt in MAT-LAB. The three PLC scripts tested on the test bench are named Program 1, Pro-gram 2 and Program 3, see Figure 3.8. Data set from Program 1 is used for theparameter estimation while data set from Program 2 and Program 3 are used forthe validation. Finally, the model is also validated by calculating the normalisedroot-mean-square deviation and the sum of the relative error between measuredand simulated data.

θm

t t t

Program 1 Program 2 Program 3

.θm.

θm.

Figure 3.8: The plots show the motor angular velocity θm(t) for Program 1,Program 2 and Program 3.

3.7.1 Model Validation with Data from Program 1

Figure 3.9 shows the comparison between the measured and simulated motor an-gular velocity θm(t) during acceleration, constant and deceleration phase. It ispossible to notice that the formulated model approximates the measured systemin a good way even if, as shown in Figure 3.9, there are discrepancies at the be-ginning of each phase. Furthermore, the model does not simulate the overshootthat can be observed in the measured signal.

The comparison between the measured and simulated drum angular velocityθd(t) shows almost the same result as the comparison of the motor angular ve-locity. However, studying the beginning of the deceleration phase, which startsafter around 14 seconds in the test, a decrease of the velocity occurs before itcorrectly decelerates as a ramp. This jump in the system is probably due to thebacklash in the gear which the used backlash model fails to model exactly. Thesame behaviour also occurs in the acceleration stage. Even though the model hassome problems simulating these jumps in velocity, it shows high accuracy.


0 2 4 6 8 10 12 14 16 18

Time (seconds)

-50

0

50

100

150

200

250

300

350

400

Angula

r V

elo

city (

rad/s

)

Measured and simulated motor angular velocity

Measured

Simulated

Reference signal

Figure 3.9: The plot shows a comparison between measured and simulatedmotor angular velocity θm(t) when the model, estimated using data fromProgram 1, is simulated using data from Program 1.

0 2 4 6 8 10 12 14 16 18

Time (seconds)

-1

0

1

2

3

4

5

6

7

Angula

r V

elo

city (

rad/s

)

Measured and simulated drum angular velocity

Measured

Simulated

Figure 3.10: The plot shows a comparison between measured and simulateddrum angular velocity θd(t) when the model, estimated using data from Pro-gram 1, is simulated using data from Program 1.


Figure 3.11 shows the comparison between the simulated and measured arma-ture current ia(t). The model manages to describe the behaviour of the measuredcurrent in almost all the stages. However, it does have an overshoot in the begin-ning of deceleration.

0 2 4 6 8 10 12 14 16 18

Time (seconds)

-40

-30

-20

-10

0

10

20

30

40

50

Curr

ent (A

)

Measured and simulated armature current

Measured

Simulated

Figure 3.11: The plot shows a comparison between measured and simulatedarmature current ia(t) when the model, estimated using data from Program1, is simulated using data from Program 1.

Figure 3.12 representing the measured torque Tmeas(t), shows that the modelmanages to imitate the shape of the measured torque during all the stages. Fig-ure 3.12 also shows a lot of spikes in the real system during the acceleration anddeceleration phase. These are probably the results of backlash in the gear whichthe dead-zone model fails to describe. However, the model manages to simulatethe first and largest spike.

As seen directly after the spikes, the friction torque model manages to describethe nonlinear behaviour of the friction in the system. The torque during constantvelocity is also simulated correctly together with the size of the spikes whichis simulated almost correctly by the model. The spikes are however difficult tomodel exactly since their size depend on the starting position of the gearwheelsin the gearbox. The difference in starting position means that the backlash gapnot necessarily will be equal every time the test bench is started. This differencecan be shown in Figure 3.13 where two different tests are made. The first testshown in stage 1 is measured with the drum rotated so that the backlash gap is asminimal as possible, while stage 2 is measured with the drum rotated so that thebacklash gap is as maximal as possible. The figure clearly shows that the size of


the spike depends on the test bench’s starting position while deceleration is notdependent of this.

0 2 4 6 8 10 12 14 16 18

Time (seconds)

-50

-40

-30

-20

-10

0

10

20

30

Torq

ue (

Nm

)Measured and simulated torque

Measured

Simulated

Figure 3.12: The plot shows a comparison between measured and simulatedtorque Tmeas(t) when the model, estimated using data from Program 1, issimulated using data from Program 1.

0 5 10 15 20 25

Time (seconds)

-0.005

0

0.005

0.01

0.015

0.02

0.025

Angula

r V

elo

city (

rad/s

)

Drum angular velocity during starts with different size of backlash gap

0 5 10 15 20 25

Time (seconds)

-2

-1.5

-1

-0.5

0

0.5

Torq

ue (

Nm

)

Measured torque during starts with different size of backlash gap

Figure 3.13: The plots show starts of test bench from standing still to run-ning with different backlash gap where the drum has been rotated beforestart so that backlash gap is minimal or maximal. Stage 1 between 0-6 sec-onds is with minimal backlash gap in start and stage 2 between 17-22 sec-onds is with maximal backlash gap in start. The drum angular velocity θd(t)is in the left figure and the measured torque Tmeas(t) is in the right figure.


3.7.2 Model Validation with Data from Program 2 and Program 3

The model is validated with two different PLC scripts Program 2 and Program 3to see how it performs. As a first stage, the model will be compared with the realsystem by doing a graphical comparison between simulated and measured data.Thereafter, a more precise analysis will be done where the model fit and accuracylevel will be calculated.

The motor angular velocity for Program 2 is shown in Figure 3.14. As shown,the model manages to imitate the motor angular velocity almost perfectly. Theonly problem arises during the first change of velocity where the real system hasan overshoot. Moreover, the plot for the drum angular velocity shows high accu-racy from the model compared with the real system even though there are someovershoots, see Figure 3.15. However, it is possible to observe some discrepan-cies during the first acceleration phase, in particular in the interval 0-1 seconds.This behaviour reminds of the case of the measured data for the motor angularvelocity θm(t) used during parameter estimation.

0 5 10 15 20 25

Time (seconds)

0

50

100

150

200

250

300

350

400

Angula

r V

elo

city (

rad/s

)


Measured

Simulated

Reference signal


The simulated motor current shown in Figure 3.16 is also considered sufficientlygood where the only considerable problem is during the first acceleration ramp inthe interval 0-3 seconds where the real system has higher current than the mod-elled current. This difference probably occurs due to inaccurate friction torquein the model.


0 5 10 15 20 25

Time (seconds)

0

1

2

3

4

5

6

7

Angula

r V

elo

city (

rad/s

)


Measured

Simulated


0 5 10 15 20 25

Time (seconds)

0

5

10

15

20

25

30

35

Curr

ent (A

)


Measured

Simulated

Figure 3.16: The plot shows a comparison between measured and simulatedarmature current ia(t) when the model, estimated using data from Program1, is simulated using data from Program 2.


The comparison between the measured and simulated torque in Figure 3.17 showsthat the model manages to simulate the measured torque sufficiently well, in par-ticular since the simulated first spike has the same size as the measured torque.

0 5 10 15 20 25

Time (seconds)

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Torq

ue (

Nm

)Measured and simulated torque

Measured

Simulated

Figure 3.17: The plot shows a comparison between measured and simulatedtorque Tmeas(t) when the model, estimated using data from Program 1, issimulated using data from Program 2.

The motor angular velocity for Program 3 that is used for validation is shown inFigure 3.18. The model manages to simulate all velocity changes and accelera-tion phases with high accuracy except the overshoots occured at velocity change.The same analysis can be done for the drum angular velocity seen in Figure 3.19where the overshoots remain in the simulation together with the velocity jumpthat occurs during the acceleration and deceleration phases.


0 5 10 15 20 25 30 35 40 45 50

Time (seconds)

-50

0

50

100

150

200

250

300

350

400

Angula

r V

elo

city (

rad/s

)


Measured

Simulated

Reference signal


0 5 10 15 20 25 30 35 40 45 50

Time (seconds)

-1

0

1

2

3

4

5

6

Angula

r V

elo

city (

rad/s

)


Measured

Simulated



As seen in Figure 3.20, the model manages to reproduce the armature current inmost phases where some spikes are reproduced correctly while other are incor-rect. For example the model fails to describe the measured current at around 27seconds where a large spike rises in the simulated armature current. Also, thesimulated armature current during constant velocity fails to describe the mea-sured current in the interval 4-13 seconds and 30-40 seconds. The reason forthis behaviour can be due to the fact that the temperature, and hence the fric-tion torque in the gearbox, is not exactly equal to the one measured during thecollection of data used for parameter estimation.

0 5 10 15 20 25 30 35 40 45 50

Time (seconds)

-30

-20

-10

0

10

20

30

40

Curr

ent (A

), A

ngula

r velo

city (

rad/s

)


Measured, (Ampere)

Simulated, (Ampere)

Drum angular velocity, (rad/s)

Figure 3.20: The plot shows a comparison between measured and simulatedarmature current ia(t) when the model, estimated using data from Program1, is simulated using data from Program 3. Drum angular velocity θd(t) isplotted to simplify identification of armature current characteristics duringdifferent velocity phases.

Figure 3.21 shows that the model manages to reproduce the torque of the real sys-tem better than the current. However, it still has some problem with some of thespikes’ size in the same way as the armature current shown in Figure 3.20. Thiscan be due to the dead-zone model which fails to model the backlash correctly insome cases.


0 5 10 15 20 25 30 35 40 45 50

Time (seconds)

-40

-30

-20

-10

0

10

20

Torq

ue (

Nm

), A

ngula

r velo

city (

rad/s

)

Measured and simulated torque

Measured, (Nm)

Simulated, (Nm)

Drum angular velocity, (rad/s)

Figure 3.21: The plot shows a comparison between measured and simulatedtorque Tmeas(t) when the model, estimated using data from Program 1, issimulated using data from Program 3. Drum angular velocity θd(t) is plottedto simplify identification of measured torque characteristics during differentvelocity phases.

3.8 Validation using Normalised Root-Mean-SquareDeviation and Sum of Relative Errors

Even though graphical comparison is a useful tool to validate model performance,there are other methods to analyse and validate performance. One way is tocalculate the normalised root-mean-square deviation NRMSD of the model de-fined in equation (3.37), where RMSD is the root-mean-square deviation definedin equation (3.38). yt are the predicted values and yt are the measured values.The normalised root-mean-square deviation is usually expressed as a percentagewhere a low value indicates less residual variance which implies a good model,[16]. Equation (3.38) can be compared with equation (3.35) where the criterion inequation (3.38) is a value of how small the sum of square of ri is for the estimatedparameters.

NRMSD =RMSD

ymax − ymin(3.37)

RMSD =

√√1n

n∑t=1

(yt − yt)2 (3.38)

3.8 Validation using Normalised Root-Mean-Square Deviation and Sum of RelativeErrors 33

The total accuracy and hence the model fit is measured using equation (3.39)where V al1 is the model fit in percent.

V al1 = 100 · (1 − NRMSD) (3.39)

Studying Table 3.2, the V al1 value is high for the angular velocity on both the mo-tor θm(t) and drum θd(t) which confirms the conclusions derived from the graph-ical analyses. The values for the armature current ia(t) and measured torqueTmeas(t) are however not as high. Nevertheless, the values of the V al1s for bothcurrent and torque are high which indicate high accuracy and low residual vari-ance in the model.

θm(t) θd(t) ia(t) Tmeas(t)Program V al1 (%) V al1 (%) V al1 (%) V al1 (%)

1 99.37 99.07 94.44 94.662 99.45 99.02 91.70 93.873 99.36 98.79 94.48 95.05

Table 3.2: Values of V al1 for measured data where Program 1 is used formodel estimation and validation and Program 2 and Program 3 are used forvalidation.

Another method to validate model performance is to calculate the sum of the rel-ative errors, where the absolute value of the difference for each data sample be-tween the predicted data yt and measured data yt is calculated and then dividedwith the absolute value of the measured data, [12]. This gives the total error inthe model and is calculated with equation (3.40). By subtracting the error valuefrom 1 and multiply with 100 according to equation (3.41), the total accuracyand model fit can be defined as a percentage.

error =n∑t=1

|yt − yt ||yt |

(3.40)

V al2 = 100 · (1 − error) (3.41)

Table 3.3 also shows high values on V al2 and hence high accuracy for the angularvelocity on both the motor θm(t) and drum θd(t). Compared with the previoustable, the values for the armature current ia(t) and measured torque Tmeas(t) arelower than the V al1 values, which indicate more residual variance. Nevertheless,the V al2 values show high accuracy.


θm(t) θd(t) ia(t) Tmeas(t)Program V al2 (%) V al2 (%) V al2 (%) V al2 (%)

1 99.55 99.36 88.55 85.962 99.57 99.48 90.47 90.893 99.48 99.17 82.39 86.86

Table 3.3: Values of V al2 for measured data where Program 1 is used formodel estimation and validation and Program 2 and Program 3 are used forvalidation.

4Road Load Simulation

This chapter covers the development of the speed regulator that is used to simu-late road load.

4.1 Problem Formulation

The aim for the road load simulation is to control the torque delivered from theDC motor to the drum so that the speed profile a forklift has on a real road canbe emulated on the test bench, [2].

A basic schematic drawing of the forklift driving wheel acting on a real road canbe seen in Figure 4.1. γ is the inclination angle on the road while the total torqueacting on the wheel is the torque given from the motor Tf orklif t(t) minus the roadload torque TRL(t) given from the road which include rolling resistance, air-dragand gravitation. These torques give the wheel a certain angular velocity θw(t).

Forklift wheel on

road

Tw θw

-TRL

.

TRLTw

ɣ

Figure 4.1: Block representation and schematic image of torques acting onforklift wheel on road.

35

36 4 Road Load Simulation

Figure 4.2 shows the same forklift wheel, this time mounted on the test benchdrum. The same torque from the forklift motor is acting on the forklift drivingwheel. However, in this case the road load torque TRL(t) is not present. Instead,the drum torque Td(t) will act on the forklift wheel. The goal is to mimic theroad load torque TRL(t) using the drum torque Td(t) to receive the same angularvelocity on the forklift wheel θw(t) as the case showed in Figure 4.1.

Forklift wheel on

drum

Tw θw

Td

. TdTw

Figure 4.2: Block representation and schematic image of torques acting onforklift wheel on test bench drum.

To fulfil this task, feedback from the torque sensor attached between the motorand gear and feedback from the sensor measuring the angular velocity attachedon the drum are used, see Figure 4.3. No slip between the wheel and drum isassumed.

Tmeas

Speed Regulator

θref

.θd

.

Figure 4.3: Block representation of speed regulator including input signalsand output signal.

Hence, the complete test bench including the forklift mounted on the drum andthe speed regulator implemented can be seen in Figure 4.4.

4.2 Speed Regulation

The basic principle for how to regulate the speed for road load simulation is de-scribed in [2]. A controller computes a road load value dependent on the mea-sured drum angular velocity and hence forklift velocity. The controller subtractsthe road load from the measured torque. It then integrates the difference and di-vides the result with the selected emulated inertia to obtain the reference angularvelocity

4.2 Speed Regulation 37

Separately Excited

DC Motor

ua

θm

Gear

Pre-equipped Controllers Ia

θref

Drum

Shaft

Shaft θd θg

Td

θmg

Tmg Tg

Tgd

θgd

Tf

θmg

Tg

θw Tw

Forklift

Speed Regulator

Tmeas

θd

. .

Figure 4.4: Block diagram of test bench including speed regulator and fork-lift.

The force from the forklift acting on the drum Fw(t) can be described as equation(4.1) where FRL(t) is the road load equation dependent on the gravitation g andthe forklift velocity v(t), m is the forklift mass and a(t) is the forklift acceleration,see [2].

Fw(t) = FRL(t) + m · a(t) (4.1)

Rearranging equation (4.1) and multiplying it with the drum radius rd , yieldsequation (4.2), where θd(t) is the drum angular velocity, TRL(t) is the road loadtorque and Tw(t) is the torque given from the forklift which is multiplied withthe factor rd

rwto correctly scale the forklift torque to the drum.

θd(t) =1

rd2 ·m

t∫0

( rdrw

· Tw(τ) − TRL(τ))dτ (4.2)

Since the reference angular velocity θref (t) is a reference signal to the motor angu-lar velocity θm(t) and not to the drum angular velocity θd(t), the relation betweenthese must be derived. An approximate relation between θref (t) and θd(t) can bedefined as equation (4.3) where ξg is the gear ratio.


θref (t) = ξg · θd(t) (4.3)

Using equation (4.3), the reference angular velocity θref (t) can be defined as equa-tion (4.4).

θref (t) =ξg

rd2 ·m

t∫0

( rdrw

· Tw(τ) − TRL(τ))dτ (4.4)

Since the torque sensor is located between the motor and gear and not on theforklift, the scaled forklift torque rd

rw· Tw(t) must be defined as a function of the

measured torque Tmeas(t).

Figure 4.5 shows a schematic description of the torques acting on the drum whichcan be defined as equation (4.5). No slip between the wheel and drum is assumed,i.e. θw(t) ∝ θd(t).

θd ,Tdrd

Drum

Forklift wheel

rw

θw , Tw

Fw , Fd

Figure 4.5: Representation of forklift wheel mounted on drum.

Jd · θd(t) = Td(t) +rdrw

· Tw(t) (4.5)

By rearranging equation (4.5) and inserting equation (4.6), i.e. equation (3.20)and (3.21) describing the total torques obtained on the gear defined in section3.3, equation (4.7) can be derived where Tmg (t) is the torque to the gear from the

4.2 Speed Regulation 39

shaft connected to the motor, ξg is the gear ratio, and Tf (t) is the friction torquein the gear.

Jg · θg (t) = ξg · Tmg (t) − Tgd(t) − Tf (t) (4.6)

rdrw

· Tw(t) = Jg · θg (t) + Jd · θd(t) + Tf (t) − ξg · Tmg (t) (4.7)

The measured torque Tmeas(t) is the negative shaft torque −Tmg (t). Hence, thetorque delivered from the forklift can be described as a function of the measuredtorque. However, equation (4.7) includes the gear angular velocity θg (t) whichcannot be measured. It also includes the friction torque Tf (t) that also dependson θg (t). Since the gear angular velocity can not be measured, the gear angularvelocity θg (t) will be assumed equal to the drum angular velocity θd(t) for thefriction torque implemented in the control system Tf control(t), see equation (4.8).

Tf control(t) =(fg,c + fg · |θd(t)|δg

)· sgn(θd(t)) (4.8)

Since the gear angular velocity is not measured, also the gear angular accelera-tion θg (t) included in the term describing the total torques obtained on the gearJg · θg (t) in equation (4.7) will be assumed equal to θd(t).

By inserting equation (4.8), the approximation of the gear angular accelerationand the measured torque Tmeas(t) into equation (4.7), equation (4.9) can be de-rived.

rdrw

· Tw(t) = (Jg + Jd) · θd(t) + Tf control(t) + Tmeas(t) · ξg (4.9)

The road load torque TRL(t) is derived from the road load force FRL(t), which canbe described as equation (4.10) where Fv(t) includes forces dependent on velocityand Fa(t) includes forces dependent on acceleration.

FRL(t) = Fa(t) + Fv(t) (4.10)

It is possible to neglect air-drag forces due to the fact that the forklift operatesat low speeds. Fv is defined in equation (4.11) where Crr is the rolling resistancecoefficient, Crr,v is the rolling resistance coefficient dependent on velocity, γ isthe inclination angle on the road and sgn(θd(t)) is included to make sure that theapplied torque is zero at θd(t) = 0 and has the correct sign, dependent on thedirection of the drum angular velocity.


Fv =(Crr · cos(γ) + sin(γ)

)·m · g · sgn(θd(t)) + Crr,v ·m · g · θd(t) (4.11)

By multiplying equation (4.10) with the drum radius rd , the torque emulationequation TRL(t) can be written as equation (4.12), which can be seen as a combi-nation of Coulomb and viscous friction.

TRL(t) = rd ·((Crr · cos(γ) + sin(γ)

)·m · g · sgn(θd(t)) + Crr,v ·m · g · θd(t)

)(4.12)

Since there is a limited amount of measured data from driving the forklift onroad, there are limited possibilities for validation of the model and the simulationof road load. The inclination angle γ will therefore be zero during modelling andsimulation, i.e., γ = 0. Hence, the simulation of road load will not be validatedfor road conditions with inclination. It will only be validated for steady state,acceleration and deceleration phase. The Simulink model of the speed regulatorcan be seen in Figure A.5 in Appendix A.

4.3 Measurement Data

The speed regulator is evaluated using torque data Tf orklif t(t) and velocity dataθw(t) received from Toyota Material Handling AB using a Toyota BT Reflex RRE160 forklift with a 930 Ah battery. The velocity data is collected with a samplefrequency of 1000 Hz using a measuring wheel that measures the number ofrevolutions on the forklift wheel which is then scaled to the forklift’s velocityv(t). The torque is measured with a sample frequency of 500 Hz using a load cellattached between the forklift motor and gear. This measured torque Tf orklif t(t) ismultiplied with the gear ratio ξf orklif t and gear efficiency ηf orklif t on the forkliftto receive the correct applied torque on the wheels, see equation (4.13).

Tw(t) = ξf orklif t · ηf orklif t · Tf orklif t(t) (4.13)

The load cell used for measuring the torque was calibrated using a torque wrench,with the driving wheel lifted up in the air, in the interval 20-90 Nm with 10 Nmstep size. As seen in Figure 4.6, the load cell has some deviation on all measuredsteps. This deviation will be taken into consideration as a disturbance during theevaluation and the validation of the road load simulation.

The data is measured during a specific driving-cycle on a straight road withoutinclination where the forklift accelerates to its maximum velocity in one directionbefore it brakes until it stops. It then accelerates to its maximal velocity in theopposite direction before it brakes until it stops. This test is done once withoutload and once with 1600 kg extra load on the forklift. Because of the deviation

4.3 Measurement Data 41

in the measured torque data, the torque value differs for each time the forkliftstands still. Hence, only a part of the driving-cycle is used for parameter estima-tion and evaluation of the road load simulation. Also, since no inclination existedduring the test, the model will be simulated and validated only for steady state,acceleration and deceleration.

Figure 4.6: The plot shows the calibration steps for the torque sensor re-ceived from Toyota Material Handling AB. It shows that the torque wrenchis not optimally calibrated where deviation exists for all torque sizes.


4.4 Estimation of Forklift and Road Load Parameters

Estimation of parameters is done considering the road load torque acting on theforklift wheel, i.e., equation (4.10) multiplied with the forklift wheel radius rw,which corresponds to equation (4.14).

The parameters Crr and Crr,v in equation (4.14) and the efficiency ηf orklif t in theforklift’s gearbox are estimated using equation (4.15) where the forklift’s momentof inertia Jf orklif t is given by equation (4.16).

TRL(t) = rw ·Crr ·m · g · sgn(θd(t)) + rw ·Crr,v ·m · g · θd(t) (4.14)

v(t) =rw

Jf orklif t

t∫0

(ξf orklif t · ηf orklif t · Tf orklif t(τ) − TRL(τ)

)dτ (4.15)

Jf orklif t = m · rw2 (4.16)

The parameter estimation is done in Simulink using the Parameter EstimationToolbox and uses data from the same forklift without load and with 1600 kg load.Since the road load equation (4.14) depends on the velocity, the measured veloc-ity and the measured torque are used as input signals to the model described byequation (4.15). The parameters are then estimated to find the highest model fitbetween the simulated and measured velocity v(t). However, because of devia-tion in the load cell measuring the torque and since the efficiency in the forkliftgear box ηf orklif t not necessarily is equal for both tests, the parameters are esti-mated independently for each data set. The Simulink model used for estimationcan be seen in Figure A.6 in Appendix A.

Figure 4.7 shows the measured velocity compared with the simulated velocity forthe forklift without load while Figure 4.8 shows the comparison between mea-sured and simulated velocity with 1600 kg load. As seen in the figures the modelmanages to simulate the real velocity almost perfectly for both data sets with dif-ferent loads. This is also shown in Table 4.1 where the V al1 and V al2 valuesare high for both simulations. Moreover, in Table 4.1, the estimated parametersare very similar for both tests except the forklift gear efficiency ηf orklif t , with analmost 3 percentage difference between the two data sets.

Load (kg) Crr Crr,v ηf orklif t v(t), V al1 (%) v(t), V al2 (%)0 0.011721 9.3882e-06 0.88763 99.28 99.20

1600 0.011447 4.3296e-06 0.91695 99.46 99.32Table 4.1: Estimated parameters and values on V al1 and V al2 for tests usingdifferent loads on forklift.

4.4 Estimation of Forklift and Road Load Parameters 43

0 5 10 15 20 25 30

Time (seconds)

-2

0

2

4

6

8

10

12

14

16

Velo

city (

km

/h)

Velocity v for forklift without load

Measured

Simulated

Figure 4.7: The plot shows a comparison between measured and simulatedvelocity with estimated parameters for forklift without load using equation(4.15).

0 5 10 15 20 25 30

Time (seconds)

-2

0

2

4

6

8

10

12

14

16

Velo

city (

km

/h)

Velocity v for forklift with 1600 kg load

Measured

Simulated

Figure 4.8: The plot shows a comparison between measured and simulatedvelocity with estimated parameters for forklift with 1600 kg load using equa-tion (4.15).


4.5 Evaluation of Road Load Simulation

The simulated drum angular velocity θd(t) is multiplied with the drum radius rdto get the velocity for the drum and hence forklift, see equation (4.17).

v(t) = rd · θd(t) (4.17)

The simulated velocity and the measured velocity can be studied in Figure 4.9and Figure 4.10. Both plots show a spike during both acceleration and decelera-tion. These spikes probably arise due to the backlash in the gear. The measuringwheel that measures the drum angular velocity is affected by the backlash whilethe torque sensor between the motor and gear and the sensor measuring the mo-tor angular velocity are not affected by it. Hence, the reference angular velocityfails to adjust the spikes occurred due to backlash.

Except for that, the simulation manages to emulate the speed curve for the dataset with no load sufficiently well. The maximum constant speed is higher thanthe measured speed while the deceleration between 23-25 seconds is done slowerthan the real system. One reason for these problems can be the approximaterelation between the drum and the gear angular velocity in the friction torqueterm used in the simulation system and the approximate relation between themotor and drum angular velocity.

0 5 10 15 20 25 30

Time (seconds)

-2

0

2

4

6

8

10

12

14

16

Ve

locity (

km

/h)

Velocity v for forklift without load

Measured

Simulated

Figure 4.9: The plot shows a comparison between measured and simulatedvelocity in developed test bench model with speed regulator implementedand estimated parameters for forklift without load.

4.5 Evaluation of Road Load Simulation 45

Figure 4.10 shows the velocity for the forklift with a 1600 kg load. Here themaximal velocity is lower compared with the measured velocity. The same spikesduring acceleration and deceleration also appear, probably due to the backlashgap. The deceleration in the time interval 27-30 seconds is however much betterin this case with 1600 kg load, than in the previous case, forklift with no load.

0 5 10 15 20 25 30

Time (seconds)

-2

0

2

4

6

8

10

12

14

16

Ve

locity (

km

/h)

Velocity v for forklift with 1600 kg load

Measured

Simulated

Figure 4.10: The plot shows a comparison between measured and simulatedvelocity in developed test bench model with speed regulator implementedand estimated parameters for forklift with 1600 kg load.

Table 4.2 shows the values for V al1 and V al2 for the tests with different loadson the forklift. These values are high for both data sets which shows that thesimulation emulates the measured velocity with high accuracy and low residualvariance.

Load (kg) v(t), V al1 (%) v(t), V al2 (%)0 97.05 96.50

1600 97.93 98.09Table 4.2: Estimated parameters and values on V al1 and V al2 for tests usingdifferent loads on forklift.

In conclusion, the spikes in the beginning of the acceleration and decelerationphase, which probably depend on the backlash, are a disturbance. However, themodel manages to stabilise the velocity sufficiently fast so that steady state, accel-eration and deceleration phases are simulated with high accuracy both with loadand without load.

5Conclusions

In this chapter, conclusions and results are discussed together with future workand improvements.

5.1 Discussion and Conclusions

In this thesis, a test bench has been modelled using grey-box modelling wherephysical equations have been used to describe the system. Unknown parametersin the equations have been estimated, and the complete model of the test benchhas been validated in Simulink by comparing the results from the model simula-tion with measurements from the real system. Also, a speed regulator has beendeveloped and implemented in the Simulink test bench model to simulate roadload.

The developed model of the test bench shows high model fit compared with thereal system. Since it is not possible to measure the gear oil temperature, mod-elling of the friction torque as a function of temperature is not possible. Hence,the model’s validation interval is limited if high model fit is wished to be main-tained.

The road load simulations using the developed speed regulator also shows highmodel fit. However, since the torque sensor is not attached between the gear andthe drum, the backlash in the gear will have a negative impact on the simulation.Nevertheless, the speed regulator manages to simulate the correct speed curvewith high accuracy.

In conclusion, the tasks in the problem formulation are completed with respectto the delimitations that are made in the thesis.

47

48 5 Conclusions

5.2 Future Work

There are some interesting parts in the thesis that can be investigated more thor-oughly. One is to implement the developed speed regulator in the real systemto simulate road load with a truck on the test bench. Another is to implementa measuring device in the test bench that can measure the gear oil temperature,and by using this, develop a friction torque model as a function of the tempera-ture. Also, a control system that minimises the impact from the backlash can beof interest to investigate.

Appendix

AAppendix

Figure A.1: Simulink model of pre-equipped control system.

51

52 A Appendix

Figure A.2: Simulink model of DC motor.

53

Figure A.3: Simulink model of gear.

54 A Appendix

Figure A.4: Simulink model of drum.

55

Figure A.5: Simulink model of speed regulator.

56 A Appendix

Figure A.6: Simulink model used for estimation of forklift and road loadparameters.

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Modelling of Test Bench for Road Load Simulation

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